Текст
                    Handbook of Finsler Geometry
Volume 1
Edited by
P. L. Antonelli
Department of Mathematical Sciences,
University of Alberta, Edmonton, Canada
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON I LONDON


A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 1-4020-1555-0 (Vol. 1) ISBN 1-4020-1556-9 (VoL 2) ISBN 1-4020-1557-7 (Set) Published by Kluwer Academic Publishers, P.O. Box 17.3300 AA Dordrecht. The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers. 101 Philip Drive, Norwcll. MA 02061. U.SA. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322.3300 AH Dordrecht, The Netherlands. Printed on acidfree paper All Rights Reserved © 2003 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.
TABLE OF CONTENTS Preface xv Part 1 Complex Finsler Geometry 3 Tadaski Aikou 1 Kähler Fibrations 9 1.1 Fibrations 9 1.2 Local Treatments 10 1.3 Bott Connections 13 1.4 Kähler Fibration 18 2 Complex Finsler Bundles 23 2.1 Vector Bundles Over Complex Projective Space 23 2.2 Complex Finsler Metrics 27 2.3 Bott Connections of Finsler Vector Bundles 35 2.4 Negativity of Vector Bundles 39 2.5 Special Finsler Vector Bundles 48 3 Kobayashi Metrics . 59 3.1 Poincaré Metrics 59 3.2 Kobayashi Metric 62 3.3 Bounded Domains 65 3.4 Holomorphic Sectional Curvature and Schwarz Lemma . . 67 Part 2 KCC Theory of a System of Second Order Differential Equations 83 P.L. Antonelli and L Bucataru 1 The Geometry of the Tangent Bundle 91 1.1 The Tangent Bundle 91 1.2 The Vertical Subbundle 93 1.3 The Almost Tangent Structure 94 1.4 Vertical and Complete Lifts 94 1.5 Homogeneity 95 2 Nonlinear Connections 97 2.1 Horizontal Distributions and Horizontal Lifts 97 2.2 Characterizations of a Nonlinear Connection 99 2.3 Curvature and Torsion for a Nonlinear Connection . . 102 2.4 Autoparallel Curves and Symmetries for a Nonlinear Connection 103 2.5 Homogeneous Nonlinear Connection 107 v
vi Anastasiei and Antonelli 3 Finsler Connections on the Tangent Bundle 109 3.1 The Berwald Connection Ill 3.2 The h and i^Covariant Derivation of a Finsler Connection 112 3.3 The Torsion of a Finsler Connection 113 3.4 The Curvature of a Finsler Connection 114 3.5 Finsler Connections Induced by a Complete Parallelism 116 3.6 The Cartan Structure Equations of a Finsler Connection 118 3.7 Geodesics of a Finsler Connection 120 3.8 Homogeneous Berwald Connection 121 4 Second Order Differential Equations 123 4.1 Semispray or Second Order Differential Vector Field . . 123 4.2 Nonlinear Connections and Semisprays 125 4.3 The Berwald Connection of a Semispray 127 4.4 The Jacobi Equations of a Semispray 129 4.5 Symmetries for a Semispray 131 4.6 Geometric Invariants in KCC-Theory 132 5 Homogeneous Systems of Second Order Differential Equations 135 6 Time Dependent Systems of Second Order Differential Equations 139 6.1 Sprays and Nonlinear Connections on Jets 139 6.2 Variational Equations 144 6.3 The “Film-Space” Approach to Type (B) KCC-Theory . 147 7 The Classical Projective Geometry of Paths 151 7.1 Paths, Parametrized Paths 151 7.2 The Various Geometries of Paths - Finite Equations . . 152 7.3 The Various Geometries of Paths - Differential Equations 153 7.4 Afhne Connections 155 7.5 The Fundamental Projective Invariants 158 7.6 The Projective Parameter and the Normal Spray Connection 161 7.7 Projective Deviation 165 Part 3 Fundamentals of Finslerian Diffusion with Applications 177 P.L, Antonelli and T.J. Zastawniak 1 Finsler Spaces 187 1.1 The Tangent and Cotangent Bundle 187 1.2 Fiber Bundles 189 1.3 Frame Bundles and Linear Connections . 191 1.4 Tensor Fields 192 1.5 Linear Connections 194 1.6 Torsion and Curvature of a Linear Connection .... 196
Table of Contents vii 1.7 Parallelism 197 1.8 The Levi-Cività Connection on a Riemannian Manifold 197 1.9 Geodesics, Stability and the Orthonormal Frame Bundle 199. 1.10 Finsler Space and Metric 200 1.11 Finsler Tensor Fields 202 1.12 Nonlinear Connections 202 1.13 Affine Connections on the Finsler Bundle 204 1.14 Finsler Connections 206 1.15 Torsions and Curvatures of a Finsler Connection . . . 208 1.16 Metrical Finsler Connections. The Cartan Connection . 210 2 Introduction to Stochastic Calculus on Manifolds 213 2.1 Preliminaries 213 2.2 Ito’s Stochastic Integral 216 2.3 Ito’s Processes. Ito Formula 219 2.4 Stratonovich Integrals 221 2.5 Stochastic Differential Equations on Manifolds .... 221 3 Stochastic Development on Finsler Spaces 227 3.1 Riemannian Stochastic Development 227 3.2 Rolling Finsler Manifolds Along Smooth Curves and Diffusions 233 3.3 Finslerian Stochastic Development 242 3.4 Radial Behaviour 246 4 Volterra-Hamilton Systems of Finsler Type 249 4.1 Berwald Connections and Berwald Spaces 249 4.2 Volterra-Hamilton Systems and Ecology 253 4.3 Wagnerian Geometry and Volterra-Hamilton Systems . 254 4.4 Random Perturbations of Finslerian Volterra-Hamilton Systems 260 4.5 Random Perturbations of Riemannian Volterra-Hamilton Systems 262 4.6 Noise in Conformally Minkowski Systems 266 4.7 Canalization of Growth and Development with Noise , 267 4.8 Noisy Systems in Chemical Ecology and Epidemiology 271 4.9 Riemannian Nonlinear Filtering ...... 279 4.10 Conformal Signals and Geometry of Filters 285 4.11 Riemannian Filtering of Starfish Predation 289 5 Finslerian Diffusion and Curvature 295 5.1 Cartan’s Lemma in Berwald Spaces 296 5.2 Quadratic Dispersion 298 5.3 Finslerian Development and Curvature 299 5.4 Finslerian Filtering and Quadratic Dispersion 300 5.5 Entropy Production and Quadratic Dispersion .... 302 6 Diffusion on the Tangent and Indicatrix Bundles 319 6.1 Slit Tangent Bundle as Ri eman ni an Manifold 320 6.2 hv-Development as Riemannian Development with Drift 321 6.3 Indicatrized Finslerian Stochastic Development .... 323
vîii Anastasiei and Antonelli 6.4 Indicatrized /^-Development Viewed as Riemannian . » 327 Appendix A Diffusion and Laplacian on the Base Space . . . . 335 A.1 Finslerian Isotropic Transport Process 336 A.2 Central Limit Theorem 338 A. 3 Laplacian, Harmonic Forms and Hodge Decomposition 340 Appendix B Two-Dimensional Constant Berwald Spaces .... 343 B. l Berwald’s Famous Theorem 343 B.2 Standard Coordinate Representation 344 B.3 B2(l) with Constant G^k 345 B.4 Class B2(2) with Constant 347 B.5 B2(r,s) with Constant Gjk 348 Part 4 Symplectic Transformation of the Geometry of ¿-Duality 359 D. Hrimiuc and ÏÏ. Shimada 1 The Geometry of TM and T*M 363 1.1 Connections on TM 363 1.2 Semisprays and Connections 368 1.3 Linear Connections on TM 370 1.4 The Geometry of Cotangent Bundle 373 1.5 Linear Connections on T*M 376 1.6 Lagrange Manifolds 378 1.7 Hamilton Manifolds 381 2 Symplectic Transformations of the Differential Geometry ofT*M 3S5 2.1 Connection-Pairs on Cotangent Bundle 385 2.2 Special Linear Connections on T*M 390 2.3 The Homogeneous Case 395 2.4 /-Related Connection-Pairs 398 2.5 /-Related ^-Connections 403 2.6 The Geometry of a Homogeneous Contact Transformation 405 2.7 Examples 409 3 The Duality Between Lagrange and Hamilton Spaces .... 413 3.1 The Lagrange-Hamilton ¿-Duality 413 3.2 ¿-Dual Nonlinear Connections 417 3.3 ¿-Dual ¿-Connections 421 3.4 The Finsler-Cartan ¿-Duality 426 3.5 Berwald Connection for Cartan Spaces. Landsberg and Berwald Spaces. Locally Minkowski Spaces . . . 431 3.6 Applications of the ¿-Duality 435
Table of Contents ix Part 5 Holonomy Structures in Finsler Geometry 445 L. Kozma 1 Holonomy of Positively Homogeneous Connections . . . . . 453 1.1 Connections .of a Tangent Bundle . 453 1.2 Holonomy Group of a Positively Homogeneous Connection 454 1.3 Curvature and Holonomy Algebra of a Positively Homogeneous Connection 455 1.4 Homogeneous Holonomy of Finsler Manifolds .... 458 1.5 Metrizability of Positively Homogeneous Connections . 458 2 Holonomies of Finsler V- Connections 463 2.1 A Topological Group and Its Lie Algebra 463 2.2 V-Connections 464 2.3 The V-Holonomy Group and V-Holonomy Algebra . . 465 3 Holonomies of the Finsler Vector Bundle 469 3.1 Linear Connections of the Finsler Vector Bundle . . . 469 3.2 Osculation of Finsler Pair Connections 470 3.3 ht^Holonomy Groups of the Finsler Vector Bundle . . 472 3.4 The Mixed Holonomy Groups 473 4 Holonomies of Special Finsler Manifolds 477 4.1 Berwald Manifolds 477 4.2 Landsberg Manifolds 481 Part 6 On the Gauss-Bormet-Chern Theorem in Finsler Geometry 491 Brad Lackey 1 Topological Preliminary 497 2 The Method of Transgression 499 3 The Correction Term 503 4 Special Cases 505 4.1 Riemannian Geometry 505 4.2 The Chern Connection 505 4.3 A Special Family of Finsler Connections 506 Part 7 The Hodge Theory of Finsler-type Geometries 513 Brad Lackey 1 Elliptic Complexes 521 1.1 The Hodge-deRham Complex 521 1.2 Elliptic Complexes ■ 523 1.3 Elliptic Operators . 527 1.4 The Hodge Decomposition Theorem 531 2 The Weitzenbock Formula 533 2.1 Complete Positivity 534 2.2. Covariant Formalism 536 2.3 Existence and Uniqueness of a Connection 539 2.4 A Bochner Vanishing Theorem 541
X Anastasiei and Antonelli 3 Complete Positivity of the Symbol 543 3.1 The Geometric Ratio 543 3.2 Computing the Geometric Ratio 545 3-3 An Example 547 Part 8 Finsler Geometry in the 20th-Century 557 M. Matsumoto 1 Finsler Metrics 565 1.1 Extremals 565 1.2 Finsler Metric ; 569 1.3 Randers Metric 574 1.4 (a,/3)-Metric 581 1.5 1-Form Metric 587 1.6 m-th Root Metric 592 1.7 Birth of Finsler Geometry 595 2 Connections in Finsler Spaces 601 2.1 Frame Bundles 601 2.2 Linear Connections 607 2.3 Vectorial Frame Bundles 61S 2.4 The Theory of Pair Connections 628 2.5 Standard Finsler Connections 644 2.6 Special Finsler Connections 661 3 Important Finsler Spaces 677 3.1 Finsler Space of Dimension Two 677 3.2 Riemannian Space and Locally Minkowski Space . . . 709 3.3 Stretch Curvature and Landsberg Space 717 3.4 Berwald Space 723 3.5 Wagner Space 735 3.6 Scalar Curvature and Constant Curvature 741 3.7 Finsler Space of Dimension Three 753 3.8 Indicatrix and Homogeneous Extension 775 4 Conformal and Projective Change 783 4.1 Conformal Change 783 4.2 Conformally Flat Finsler Space 790 4.3 Conformal Change and Wagner Space 796 4.4 Projective Change 799 4.5 Douglas Space 814 4.6 Finsler Space with Rectilinear Extremals 827 5 Finsler Spaces with 1-Form Metric and with m-th Root Metric 839 5.1 Finsler Spaces with 1-Form Metric 839 5.2 Curvature of Two-Dimensional 1-Form Metric .... 847 5.3 Conformal Change of 1-Form Metric 854 5.4 Finsler Space with m-th Root Metric 858 5.5 Stronger Non-Riemannian Finsler Space 867 5.6 Two-Dimensional m-th Root Metrics 874 5.7 Berwald Spaces of Cubic and Quartic Metrics .... 879
Table of Contents xi 6 Finsler Spaces with (a, ^-Metrics SS9 6.1 Fundamental Tensor of Space with (a, /3)-Metric . . . 889 6.2 C-Tensors of (a,/3)-Metrics 894 6.3 Connections for (a, /3)-Metrics 901 6.4 Douglas Space with (a, £)-Metric 913 6.5 Two-Dimensional Space with (a/?)-Metric 916 6.6 Strongly Non-Riemaanian (o/3)-Metric 924 6.7 Conformal Change of (a, ^-Metric 928 6.8 Projective Change of (a/3)-Metric 936 6.9 Randers Spaces of Constant Curvature 946 Part 9 The Geometry of Lagrange Spaces 969 Radu Miron, Mihai Anastasiei and Ioan Bucataru 0 Introduction 973 1 The Geometry of the Tangent Bundle 977 1.1 The Manifold TM 977 1.2 Semisprays on the Manifold TM 984 1.3 Nonlinear Connections 987 1.4 TV-Linear Connections 995 1.5 Semisprays, Nonlinear Connections and TV-Linear Connections 1002 1.6 Parallelism. Structure Equations 1007 2 Lagrange Spaces 1013 2.1 The Notion of Lagrange Space 1013 2.2 Geometric Objects Induced on TM by a Lagrange Space 1017 2.3 Variational Problem and Euler-Lagrange Equations . . 1019 2.4 A Noether Theorem 1021 2.5 Canonical Semispray. Nonlinear Connection 1023 2.6 Geodesics in a Finsler Space 1025 2.7 Hamilton-Jacobi Equations 1028 2.8 The Almost Kâhlerian Model of a Lagrange Space Ln . 1030 2.9 Metrical TV-Linear Connections 1033 2.10 Almost Finslerian Lagrange Spaces 1038 2.11 Geometry of ç>Lagrangians 1042 2.12 Gravitational and Electromagnetic Fields 1045 2.13 Einstein Equations of Lagrange Spaces 1047 3 Subspaces in Lagrange Spaces 1053 3.1 Subspaces L in a Lagrange Space Ln 1053 3.2 Induced Nonlinear Connection 1056 3.3 The Gauss-Weingarten Formulae 1060 3.4 The Gauss-Codazzi Equations 1061 3.5 Totally Geodesic Subspaces 1062 3.6 Lagrange Subspace of Codimension One 1064 3.7 Subspaces in Finsler Spaces 1067
XU Anastasiei and Antonelli 4 Generalized Lagrange Spaces 1073 4.1 The Notion of Generalized Lagrange Space 1074 4.2 Metrical ^-Connection in a GL-Space 1077 4.3 GL-Metrics Determining Nonlinear Connections . . . 10S0 4.4 GL-Metrics Provided by Deformations of Finsler Metrics 10S5 4.5 Almost Hermitian Model of a Generalized Lagrange Space 1091 5 Rheonomic Lagrange Geometry 1097 5.1 Semisprays on the Manifold TM x R 1097 5.2 Nonlinear Connections on E = TM x R 1099 5.3 Variational Problem 1101 5.4 Rheonomic Lagrange Spaces 1103 5.5 Canonical Nonlinear Connection 1104 5.6 An Almost Contact Structure on E 1105 5.7 TV-Linear Connection 1107 5.8 Parallelism. Structure Equations for TV-Linear Connections 1108 5.9 Metrical TV-Linear Connection of a Rheonomic Lagrange Space 1111 5.10 Rheonomic Finsler Spaces 1112 5.11 Examples of Time Dependent Lagrangians 1114 Part 10 Symbolic Finsler Geometry 1125 S.F. Rutz and R. Portugal 1 Computer Algebra for Finsler Geometry 1129 1.1 Introduction 1129 1.2 Computer Algebra 1130 1.3 Manipulation of Indices via Group Theory 1144 1.4 FINSLER Package 1150 Part 11 A Setting for Spray and Finsler Geometry 1183 J6zsef Szilasi 0 Introduction 1187 1 The Background: Vector Bundles and Differential Operators 1191 A Manifolds . 1191 B Vector Bundles 1195 C Sections of Vector Bundles 1204 D Tangent Bundle and Tensor Fields 1208 E Differential Forms • 1218 F Covariant Derivatives 1226 2 Calculus of Vector-Valued Forms and Forms Along the Tangent Bundle Projection 1237 A Vertical Bundle to a Vector Bundle 1237 B Nonlinear Connections in a Vector Bundle 1245
Table^of Contents xiii C Tensors Along the Tangent Bundle Projection. Lifts . . 1258 D The Theory of A. Frolicher and A. Nijenhuis ...... 1272 E The Theory of E. Martínez, J. F. Cariñena and W. Sarlet 1298 F Covariant Derivative Operators Along the Tangent Bundle Projection 1314 3 Applications to Second-Order Vector Fields and Finsler Metrics 1347 A Horizontal Maps Generated by Second-Order Vector Fields 1347 B Linearization of Second-Order Vector Fields 1362 C Second-Order Vector Fields Generated by Finsler Metrics 1369 D Covariant Derivative Operators on a Finsler Manifold . . 1383 Appendix 1399 A.l Basic Conventions 1399 A.2 Topology 1400 A.3 The Euclidean n-Space R” 1401 A.4 Smoothness 1402 A.5 Modules and Exact Sequences 1403 A.6 Algebras and Derivations 1408 A.7 Graded Algebras and Derivations 1409 A.8 Tensor Álgebras Over a Module 1411 A.9 The Exterior Algebra 1415 A.10 Categories and Functors . 1419
PREFACE It was over three years ago, at the annual meeting of the American Math¬ ematical Society in San Diego, California, that Dr. Paul Roos of Kluwer asked me to poll Finsler geometers around the world as to their interest in writing a HANDBOOK OF FINSLER GEOMETRY. The result of that query was a resounding affirmation, and here at long last, is the final result. You have in your hands,the most complete and authoritative exposition of state-of-the-art Finsler geometry that is possible, today. Each of the eleven parts is completely independent of the rest, and each has been written with mathematics and science students in mind. These articles are accessible! P.L. Antonelli Edmonton, Alberta, Canada June, 2003 xv
ACKNOWLEDGEMENTS The editors would like to express their sincere thanks to Vivian Spak, who typeset this book, and to Scott Berard, who kept our computers running. xvii
PART 1
Complex Finsler Geometry Tadashi Aikou
Contents 1 Kâhler Fibrations 9 1.1 Fibrations 9 1.2 Local Treatments 10 1.3 Bott Connections 13 1.3.1 Structure Tensors 14 1.3.2 Bott Connections 16 1.4 Kâhler Fibration 18 2 Complex Finsler Bundles 23 2.1 Vector Bundles Over Complex Projective Space 23 2.2 Complex Finsler Metrics 27 2.2.1 Complete Circular Domains and Minkowski Functionals . 27 2.2.2 Complex Finsler Metrics on Cr and Kahler Metrics on Pr_1 30 2.2.3 Complex Finsler Metric on Vector Bundles 31 2.3 Bott Connections of Finsler Vector Bundles 35 2.4 Negativity of Vector Bundles 39 2.4.1 Positive Line Bundles and Ample Line Bundles 40 2.4.2 Negative Vector Bundles . . 43 2.4.3 Vanishing Theorems 45 2.5 Special Finsler Vector Bundles 48 2.5.1 Finsler Vector Bundles Modeled on a Complex Minkowski Space 49 2.5.2 Flat Finsler Vector Bundles 50 2.5.3 Protectively Flat Finsler Vector Bundles 52 3 Kobayashi Metrics 59 3.1 Poincaré Metrics 59 3.2 Kobayashi Metric 62 3.3 Bounded Domains 65 3.4 Holomorphic Sectional Curvature and Schwarz Lemma 67 3.4.1 Generalized Schwarz Lemma 68 3.4.2 Holomorphic Sectional Curvature by Curvature Tensor . . 71 5
Preface In this short note, we shall discuss the geometry of Finsler vector bundle. The geometry of Finsler bundles are treated as the geometry of fibred mani¬ folds. In fact, in the geometry of Finsler manifolds, each tangent space TXM at x e M is considered as a Riemannian space TXM with a Riemannian metric Sijfay), where x is fixed. This Riemannian metric is given by the Hessian gij — ¿PF/dtfdyi of the fundamental function F — L2/2 of the Finsler met¬ ric. Since each tangent space TXM S has a natural flat structure, we can consider (szfixed) as a so-called Hessian manifold. The tan¬ gent bundle 7T : TM —► M may be considered as a fibred manifold with each fibre is a Hessian manifold such that Hessian structure depends smoothly on the base point x € M» In this interpretation, a Landsberg space may be con¬ sidered as the fibred manifold 7m with isometric fibre, i.e., any local horizontal mapping, which covers an arbitrary curve in the base space Af, is an isometry. A Landsberg space is said to be a Berwald space if the local horizontal mapping is defined by a linear connection on 7m- A Berwald space is said to be locally Minkowski if the linear connection is flat, and thus a Finsler manifold is locally Minkowski if and only if the fibred manifold 7m is locally trivial, i.e., it is locally a Riemannian product of the base space M and a fibre. In complex case, each tangent space TXM = C71 is considered as a Kahler manifold with a Kahler metric for the fundamental func¬ tion F of the convex Finsler metric on a complex manifold. Since any Kähler manifold is also a Hessian manifold, we also understand the geometry of com¬ plex Finsler manifold as a geometry of complex fibred manifold with each fibre is Kahler manifold such that Kahler structure depends smoothly on the base pint z € M, It is natural to generalize some special class of real Finsler metrics to complex case. The purpose of the present note is to discuss some special complex Finsler metrics.
Chapter 1 Kähler Fibrations In this chapter, for applications to Finsler geometry, we shall investigate the differential geometry of fibred complex manifold {Af, M, 7r} such that each fibre is a Kahler manifold. We call such a fibred complex manifold a Kahler fibra¬ tion. Especially, we are concerned with Bott connections on the relative tangent bundle TX/m> 1.1 Fibrations Let X and M be connected complex manifolds with dime Af = n+r > dime M — n. We assume that there exists a surjective holomorphic map % : Af —► M of maximal rank. Definition 1,1. We call % : Af —► M a fibration simply if the following condi¬ tions are satisfied. (1) 7T : X —> M is a differentiable fibre bundle, (2) every fibre 7r_;L(z) Xz (z € Af) is a connected complex submanifold of X of dime X£ = r. The complex manifold X is called the total space, M the base space and 7r the projection. We denote by Tx and Tm the holomorphic tangent bundles of X and M respectively. Since the differential dir : Tx —* is surjective, we have the fundamental sequence which is an exact sequence of vector bundles over X: O^Vx-^Tx^ir*TM-*O, (1.1) where Vx = ker{d7r : Tx —> is vertical subbundle of Tx- The exact sequence O - ® Vx -Û ®Tx-^ ® t-Tm -» O 9
10 Aikou induces a long exact sequence of cohomology groups >H°(X, ® Tat)) H°(X, ® %’T«)) H\X, 0(^(4 ® Vat)) —> • • • ■ The cohomology class k — 6*(Id) € (vr'iTw ® V*)) “ Ext1(«-*TM> Vx) is called the extension class of the sequence (1-1). This cohomology class is an. obstruction class to a global holomorphic splitting of the sequence (1.1). In fact, there exists a globally defined g e HQ (Af, <9(7r*Q^ ® Tx)) such that j"g = Id if and only if k = 5* o j*g = 0. We also denote by 0» the holomorphic tangential sheaf. Then we have the locally free extension 0 —► Gyx —> Ox -* —► 0 of it*Om by the relative tangential sheaf Ox/m and the exact sequence of direct image sheaves: > =3M -£-> R^Oxm —►' • • 7s i Tz J, (TM), H\xs,ex,) where rz is the restriction map. The connecting map p: Om —► Rfyv&x/M is the so-called Kodaira-Spencer map. The linear map p~ : (Tm)z —> Hl (X-, 3x<) is the Kodaira-Spencer map atztM. 1.2 Local Treatments Let 7T : X —► Af be a fibration. Since we treat the Kodaira-Spencer map p in local coordinate, we fix a point z G M and we work on a neighborhood U of z with local coordinate (s1, • • ■ >sn). Then we take a locally finite open covering {Ua} of Tr_1((7) so that an arbitrary point in Ua C tt“1 (C7) is given by (21, • • • , zn, , ¿^). On the intersection Ua A Us, the coordinate change is given by (1.2) for a family of holomorphic functions we Put the transition cocycles of Tx with respect to the natural bases {d/dza, are given by the GL(n + r, C)-valued holomorphic functions {j&ab} of the form
Complex Finsler Geometry 11 The vertical subbundle Vx is a holomorphic vector bundle over Af locally spanned by and the structure cocycle of Vx is given by {Fab}. These cocycles satisfy Aac = FabAbc + AabGbc (1.3) on Ua A Ub A Uc / 0- By this relation, the 1-cochain a = {«tab} € ® Vx) defined by satisfies aab + &bc + <tca = 0 whenever C7a A Ub n Uc / </>• Hence it defines a cohomology class « = [{^ab}] € H\X, ® Vx)) = Ext^Tw, VX), which is the extension class of the sequence (1.1). The sequence splits holo¬ morphically if and only if the extension class « is trivial (cf. [14]). If we have splittings : %*Tm|cJa —► on each Ua, the cocycle {hg —h^} represents the extension class k. Because of dirx ° (^b — ^a) “ 0, we have ha - ^a € her (cfrr^), and thus we may regard {Kb - ^a} as a O ® V^)- valued 1-cocycle. By easy computations, we see that Iib — ^A = ^ab- The exist¬ ence of a global holomorphic splitting is equivalent to w = 0, i.e., the existence of 0-cochain {Na G O ® V*)} satisfying ^ab = Na - Nb = ab- (1.4) Then hA + Na (= hB + defines an element g € ZT° (<< (9(7t*Q^ ® 2») and satisfies j*g — Id. For vv = £ v^z^d/dz* € r(U,^Af)? we define a holomorphic vector field o-abM on Ua A Vb by <tab{v) = € r(UA n US, ex/M). Since (1.3) implies oab(v) + ascfa) + ^caM = 0 on Ua A Ub A Uc / the collection {oab(^)} defines a cohomology class [<tab(v)] in H1(7r‘"1(U), f°r Vu € By the direct limit of this corresponding map au : H*(U, eM) 9 v -> [aXB(v)J € H\*-\U), ex/M\^m), we get the Kodaira-Spencer map a :== limacr: &m By restrict¬ ing a to each fibre Xx, we have Kodaira-Spencer map at z: <r*: (Tm); Wk-w, ex/M\.-^)) = H\xs, ex,). We denote by ® Vx) the sheaf of germs of smooth sections of the bundle Since ®Vat) is fine, the cohomology class «is trivial in C°°-categoiy. Hence there exists a 0-cochain .V » {TVa € 0 V^)} satisfying (1.4) on Ua A Ub / If « = 0, we can assume that the 0-cochain N is holomorphic. Since pab(v) = N&(v) — Na(v), we have
12 Aikou Proposition 1-1. If the extension class k is trivial, the Kodaira-Spencer map cr is trivial For a 0-cochain N = {Na 6 ® V*)} satisfying (1.4), the extension class k is given by [92V] in the Dolbeault cohomology 7r*Q^ ® Vx). We shall express the 0-cochain 2V = {Na} in the form (l.o) on 7r_1(Z7). Then we define a Tx,-valued 9-closed form <pa by where we put (1.6) Ri - J™'« ' (1.7) Then, from (1.4), we have Proposition 1.2. Let N = {TVa} be an arbitrary representative of the extension doss « of the sequence (1.1). The form represents az(d/dz°1') in Dolbeault cohomology Definition 1.2. A fibration tt : X —► M is said to be locally trivial if every point z € M has a neighborhood U C M such that %“x(iZ) is bi-holomorphic to U x X~. We suppose that 7r: X —* Ai is a locally trivial. By definition, each point z e M has a neighborhood U such that there exists a bi-holomorphism & : U x Xz —► 7T”1 (17) with the commutative diagram U*XZ I Pi J. it u u, where pi : U x Xs —> U is the projection to the first factor. We express S' as #(3,f) = (za,^(z,i)) for holomorphic functions on each Ua C 1 (CZ). Since ^a — £(b) on Ua H Ub / we have
Complex Finsler Geometry 13 Hence, if we define a O-cochain AT = {NA} by (1-9) then {JVa} defines a splitting h of (1.1). Since are holomorphic, h is holo¬ morphic, and so the extension class « is trivial. Proposition 1.3. If a fibration tt : X —► M is locally trivial, the sequence (1.1) splits holomorphically. 1.3 Bott Connections To investigate the geometry of fibrations, we need a partial connection on which is naturally defined from a connection of the sequence (1.1). A connection of the fibration % is equivalent to single out a n-dimensional tangent subspace called horizontal subspace at each point xeX, which is projected by dir on the tangent space at the base point z — 7r(z) G M. For any O-cochain N = {Na € 4 ® V*)} satisfying (1.4), the local morphism Jia = Id — Na defines a smooth splitting on each Ua- If the sheaf of splitting defines a globally defined smooth splitting h, we call h a connection of the fibrations tt. Definition 1.3. A connection h on a fibration % : X —> M is a smooth splitting of the sequence (1.1), i.e., h : it'Tm —► Tv is a smooth bundle morphism satisfying dir o h =: Id. A connection h on a fibration ir : X —► M is given by a O-cochain N = {Na} satisfying (1.4). Since ir : X —► M is a differentiable fibre bundle, we may assume that every point z € M has a small neighborhood U such that 7T”1(CZ) is diffeomorphic to U x Xz. We denote by $ : U x Xz ir^(JJ) the diffeomorphism, and we set = (z%l?x(2,s)) on eacb If we define a O-cochain N by the same equation (1.10), then N defines a connection h of the sequence (1.1). Proposition 1.4. Any fibration ir: X —> M admits a connection h. We suppose that a connection h is given by a O-cochain N. If we express N as in (1.5), the horizontal lift of a vector field on M is defined by (1-10)
14 Aikou For a connection h : 7T*7m —> Tx on the sequence (1.1), the horizontal vector bundle Hx is defined by Hx = h(ir*TM)> and it induces a smooth decomposition Tx^x&Hx- (1.11) By definition, Tix is a complex vector bundle over X locally spanned by {XQ} (a = 1, • * • ,m), where Xa — h (d/dza) is the horizontal lift of {d/dz*} (a = 1,— ,n) in (1.11). A connection h of the fibration tt is also defined by a splitting h of the exact sequence O —> Tr’n]^ —~ If we put V* — h (^vat) wbich is isomorphic to yAi, we get the dual splitting (1.12) The subbundle V* C is locally spanned by the forms {£’} (i = 1,• >• }r) defined by = h^) = de According to the decomposition (1.13), the differential operator d : Ox —► X(Q^) is decomposed as d = d° where <F : Ox X(V*) is the differential operator along Vx and dh : Ox —► -4(^*HA/) is the one along the distribution Hx* The operators d and d are also decomposed as d = dh+dv and S — dh+&u respectively. We denote by X& the complex conjugate For any function F on and ‘"•-Efp, "-E^ for the dual frame fields of {d/d^ in local holomorphic coordin¬ ates (z0,^) on X, 1.3.1 Structure Tensors The horizontal vector bundle ?ix C Tx is said to be integrable if it is stable under the Lie bracket. We shall investigate the integrability conditions of a connection h. We define the structure tensors JR^q and of Hx by respectively, where we put
Complex Finsler Geometry 15 Definition 1.4. A connection h of a fibration 7r is said to be flat if ft is holo¬ morphic and integrable. By this definition, we have Proposition 1.5. A connection h of a fibration it is flat if and only if — If we consider the following system of PDE gj = -N^) (1.14) with unknown ^’s, we have Proposition 1.6. A fibration tt is locally trivial if and only if ir has a fiat connection h. Proof: We suppose that tt has a flat connection h. By the previous propos¬ ition, h is holomorphic and = 0 is satisfied. Then the PDE (1.15) has holomorphic solutions {^} for every initial point (zo, <). Then ft# is an integ¬ rable holomorphic subbundle of T#, and it defines a holomorphic foliation whose leaves are complex submanifolds of X transversal to the fibres. If we define a holomorphic map # by #(z,£) ~ (Zj^^z,^)), the leaf through a point < € XZQ is given by i<(z,<), and it defines a local triviality & : U x XZQ vr”1^) for a sufficiently small U C M. The converse is trivial. Q.E.D We suppose that a connection h of the fibration n is flat. For the holomorphic solution S'* satisfying ^(zq,?) = C, we have Then we may consider the bi-holomorphic map # : (z“,£*) —> (za,C) = (za,^(z,$)) as a coordinate change for a sufficiently small U. Then we have Hence, if a fibration tt : X —► M admits a flat connection h if and only if there exists a coordinate system U = {Î7, (z*,£*)} on X such that With respect to such a covering U, we have IV* = 0 on each Ua € ZÏ, i.e., Na = 0. Then, from (1.4) we have a as = 0, i.e., on Ua ri U&.
16 Alkov 1.3.2 Bott Connections Suppose that a smooth splitting h is given on the sequence (1.1). We introduce the Bott connection on the vertical subbundle Vx- Let X and Y be sections of Hx and Vat respectively. Then we put JDxy=[X,y]v, (1.15) where ( • )v : Tx —* Vx is the natural projection. By this definition, we can easily show that DxY is O a-linear in X, and moreover it satisfies the Leibnitz rule Dx{fY) = fDxY + (XfjY for all smooth function f on X, Then we get a homomorphism D : >1(1^) —► X1^*) satisfying D (JY) = dhf 0 Y 4- fDY for all section Y € A(VAf) and all smooth function f on Af. Such a morphism D is called a partial connection m Vx- Definition 1.5. The partial connection D of the vertical subbundle Vx is called the Bott connection associated with the connection h. We denote by — (wj) the connection form of D. For any Y — ^Yi (d/dÇ*) we have The forms wj are given by the horizontal 1-forms wj = with the coefficients dN* = (1.16) By definition (1.7) and (1.14), by direct calculations, we have ^ + £ wj A ^' = £•8^“ A dz& + £ A dz& + £^jd«° A&. The form r=£^®(^+£^) is called the torsion form of D, From Proposition 1.5, we have Proposition 1.7. A connection h of a fibration rr is flat if and only the torsion form T of the Bott connection D associated with h vanishes identically. The curvature form QD = (f2j) of D is defined by DoD (d/d£f) — £ (£/££*) 0 and is given by the End(V^)-valued horizontal 2-form (1-17)
Complex Finsler Geometry 17- Definition 1.6. A (partial) connection D is said to be flat if D o D = 0. Then we have easily the following. Proposition 1.8. Let h be a connection of a fibration x. The Bott connection D associated with h is flat if and only if its curvature vanishes identically, and moreover h satisfies dhcdh = 0. We define a Vx 0 V£-valued horizontal (1, l)-fonn 3 — 0 $ 0 by and a Vat 0 Vj-valued horizontal (1,0)-form # = 0 ®d/dÇ by = (1-18) where is defined by (1.7). We shall represent the curvature form in terms of 3 and. Proposition 1.9. The curvature L2D of the complex Bott connection D is given by QD = e_#/\$. (1.19) Proof: Since = d(X&Nlf)ldzj + we have sty+52 A = | E ~^fdzat A dzfi- (k2°) Moreover, since X^a — d{X^N^)/d^ 4- we ^ave - E A - £ 0 A (1.21) Hence we have (1.20). Q.E.D. As an application of (1.20), we shall prove the following: Proposition 1.10. A connection h of a fibration tt is flat if and only if its Bott connection D is flat.
18 Aikou Proof: We suppose that h is a flat connection of a fibration 7r. By Proposition 1.5, the connection h satisfies R^ = R^ — Rla- = 0. Then, from = 0, (1.19) and (1.20), we get QD — 0. Conversely, we suppose that D is flat. By Proposition 1.7, we have — 0 and dh o dh = 0. Since this second condition is equivalent to we get $ — 0 from (1.20). Then, if we take a suitableHermitian metric (•, •) on the vertical subbundle V^, we have ||#||2 = £0 A & = 0, and so we have 0 — 0. This means that R^ — 0. By using this and the equation R*a& = 0, we get also dN^/dz^ =* 0. Consequently the coefficients N# are holomorphic, and so h is holomorphic. Q.E.D. From the results in this section, we get Theorem 1.1. Let h be a connection of a fibration 7r : A —► M. Then the following conditions are equivalent mutually. (1) h is fiat. (2) ‘ There exists a coordinate system {C7, (^a,C)} 072 X such that (3) The Bott connection D onV# associated with h is flat. (4) The torsion form T of D associated with h vanishes identically. 1.4 Kähler Fibration In this section, we shall consider the case where a fibration tt : X —► M is a family of Kahler manifolds {XÄ, IIZ} (z e M) parameterized by the base space M, here and in the sequel, we assume that the Kahler forms depend on z G M smoothly. Definition 1.7. A fibration % : X —» M is said to be a Kahler fibration if there exists a locally öä-exact real (1, l)-fonn IIx such that its restriction Hx\xx *= nx to each fibre X- = 7r_1(z) (z g M) is a Kähler form on Xz.
Complex Finsler Geometry 19 If 7T : X —► M is a Kahler fibration, then the real (1, l)-form fix induces a Hermitian metric (♦, ♦) on V*. The Bott connection D on associated with a connection h is said to be compatible with («, •) if it satisfies dh{A, B) = {DA, B) + {A, DB) (1.22) for A{VX). We denote by g^ the metric tensor of the Hermitian metric (♦, •) on Vx with respect to {0/5$*}. We also denote by (p^) the inverse matrix of (&j). Since the Bott connection D is (l,0)-type, the connection form cu of P is given by w*- * £^0^ “ E we Put = 2^%^“ 111 a local coordinate, the compatibility condition (1.23) is equivalent to Xag^ - = 0 which is written as - ZXÿ - c1-23) and thus we have = = (1-24) We shall show that the vertical subbundle Vx of an arbitrary Kahler fibration f : X —> M admits a Bott connection D which is compatible with respect to the metric induced on Vx- By definition, there exists a distinguished coordinate system {CT^on tt“1 (Ï7) and R-valued smooth functions G on each Ua such that wx = */-LddG on Ua and G|x, is the Kahler potential on Xz C\Ua> Each local function G is pluri-subharmonic on Ua A Xz for each z g M. In terms of distinguished coordinates (s*,^), the Kahler form IIz is given by the relative (1, l)-form where — d'G/d^dt?. We shall determine the Bott connection D which is compatible with this Hermitian structure (-,♦). For this purpose, we determine a connection h of tt. Since the coefficients I^a of D are given by (1.17), and moreover, since dg&/d£? = dg^/dg, the compatibility condition (1.24) can be written as Proposition 1.11. The local functions d2G ds^dç- (1.26) define a connection h of the fibration it.
20 Aikou Proof: Let {Ua} be a distinguished coordinate system on 7r^1(Cr). Then, in the proof, to distinguish the quantities on a neighborhood Ua* we use the subscripts A, B • * * in local computations. For the family of local functions {Ga} above, since Gb—Ga is pluri-harmonic on Ua^Ub^ there exists a holomorphic function <Pab on Ua A Ub </> satisfying Gb — Ga + <Pab + VaiL Then we have ^b = (dGA d& d^J- Differentiating by z“, we have ^b r^(&GA A dzad?s d^B \dz^d^A dz<* 9AlmJ ’ and if we put on Ua^Ubi we get the relation (14). Q.E.D. Remark 1.1. The coefficients defined by (1.27) and the components g$ are independent of the choice of potentials {<?} which represent the pseudo¬ metric Их- In fact, if we take another potentials {2/} adapted to the common open covering {Ka} of vr-1 (£7), then G — H are pluri-harmonic. Then, by d§- Poincare lemma, there exists a family of holomorphic functions {K} satisfying G ~ H = К + X, and so we have a2^ a2# a2^ а2я a^a^ “ э^а^’ a^a^r a^a^* Consequently the components g$ and functions are independent of the po¬ tentials of wx- If we denote by D the Bott connection associated with the connection h of (1.27), D is canonical in the following sense. Proposition 1.12. Let {% : A —> M, IIx} &e a Kahler fibration. The Bott con¬ nection D defined by the connection h of (1.27) is compatible with the Hermitian structure (•, •). Proof: The coefficients (1.27) satisfy the equation (1.24). Then, since the coefficients Tfa are given by T^a = dN^/d^ù we have (1.24). Hence D is compatible with the Hermitian structure (♦,-).
Complex Finsler Geometry 21 Q.E.D. We shall investigate the conditions for a Kahler fibration to be with isometric fibres, i.e., the parallel displacement, which covers an arbitrary smooth curve in the base space Af, is an isometry between the fibres. To this end, we shall compute the Lie derivative = 0, (1.27) where the notation Lx» denotes the Lie derivation with respect to the horizontal lift XH of a vector field X on the base M, and we use the notation g(Y instead of (K, Z}. Then, since the condition (1.28) is satisfied if and only if = 0 is satisfied. Hence the horizontal mapping is an isometry between the fibres if and only if TiS = 0. Proposition 1.13. The pseudo-Kahler manifold (X,J7) is a complex fibred manifold with isometric fibres if and only if = 0, i.e,f <pa — 0. As a special case of locally trivial fibrations, we shall investigate locally trivial Kahler fibrations. Definition 1.8. A Kahler fibration {% : X —► M, Л#} is said to be locally trivial if every point zq e M has a sufficiently small neighborhood Z7 such that each fibre (Х~,П~} (ztU) is holomorphically isometric to (X^, Л^). We suppose that a Kahler fibration : X —* M, Л#} is locally trivial. By definition, every point zq g M has a small neighborhood U С M and a bi-holomorphism : U x Xo тг_1(Л) which makes the diagram (1.8) commutative and induces a isometry between the fibres; We express the Kahler form ПЯо on X^ by TZso — Since $7 induces an isometry from (X~O,77ZO) to (Xz, Лг) for each z € Л, we can assume that the form Пх is expressed by Л# = y/^lddG for local functions G = (ЙН1)*^ on each тг“1 (U). The local real (1, l)-form y/^lddG defines a global (1, l)-form Л# and Пх\хл = y/^Id^dvG is a Kahler form on X-. Since 9$(2о,С) = Й =X2>0) and дФт dG , .дФ1 2-^ dtf dzad^m ~dza'
22 Aikou we have on Tr“1(i7). Hence the connection h defined by these fonctions {N^} & Ûat, and by Proposition 1.10, the Bott connection D is fiat. The converse is also true: Theorem 1.2. A Kahler fibration ir is locally trivial if and only if its metrical Bott connection D is flat. Proof: We suppose that D is fiat. Then, by Proposition 1.10, the connection h is also flat, and so by Proposition 1.6, the fibration 7r is locally trivial, i.e., there exist a bi-holomorphic map № tUx XZQ —> tt~ 1 (Ï7) for a sufficiently small neighborhood U of zq e M. In fact, for the local solution W* of d^/dz* = the map & is defined by ^(z.C) — (zaf^i(z1Ç)). Since {Xa} is the horizontal lift, we have = 0. Moreover, since X& tangents to the leaf defined by ÿ, we have № = Consequently, the holomorphic map ÿ defines a holomorphic isometry îF- : (XM for each z G U by Q.E.D.
Chapter 2 Complex Finsler Bundles La this chapter, we shall study the geometry of Finsler bundles as an application of the geometry of Kahler fibrations. The fundamental tool in this chapter is the Finsler connection, which is naturally defined as an extension of Bott connections. We also see that our connection is also a natural generalization of the so-called Rund connection of real Finsler geometry. 2.1 Vector Bundles Over Complex Projective Space Let Pr_1 the complex projective space of dimension r—1 (r > 2). Let^1,--- ,^) the homogeneous coordinate system on Pr_1. On Uj = {[£] 6 IF-1 | & / 0}, we define a function Kj : Uj —► R by On the intersection we have == Hence we have dd log Ki = ddlogÆj, that is, nFS = ^ÇÏaâiogKy is a global real (1, l)-form on IP7*“1. By definition, IIfs satisfies (HIfs = 0- We shall show that there exists a Hermitian metric on Pr“1 whose Kahler form is just the form IIfs- In fact, if we put & — ^/s1 on Uit we see that IIfs is given by The Schwartz’s inequality implies the positive-definiteness of IIfs, an<l thus Hfs defines a Kahler metric on P-1 which is called the Fubini-Study metric. 23
24 Aikou The components of the metric are given by Since H1 (P-1,©) = H~ (p-l,O) = 0, the exact sequence 0 Z —> O O* —> 0 implies the exact sequence 0 —> Jff1(P’_x,<9*) №(P_X,Z) —> 0. Then we have the isomorphism . of abelian groups. Thus a holomorphic line bundle over F”1 is determined by its Chem class. We shall list up some vector bundles over projective space P7""1 for later discussions. Example 2.1. (Tangent bundle Tpr-i) Let p : <Cr\{0} —► F”1 be the natural projection. We take an open covering U = {Uj} of F"1 defined by Uj — {(i1 : • • •: C) € F“11 & 7^ 0}. We use the homogeneous coordinate (i1, • • •, S’-) and set ? - g/g (i / j). Since we have (2.1) and the holomorphic tangent bundle Tpr-i of F‘ is spanned by with the relation (2.1). □ Example 2.2. (Hyperplane bundle) We use the notation in Example 2.1. Let F^1, • • • be a homogeneous polynomial of degree 1. The set V(F) = {£ € Cr\{0} [ F(f) = 0} is isomorphic to Cr\{0}. For the natural projection p : Cr\{0} — F"1, we put HQ = p(V(F)). The hyperplane HQ * F“2 C F"1 is defined by the equation Ri = ^.=0 on each Ui e U. Since on the intersection Utf\Uj, Ri/Rj = & /£? is non-vanishing holomorphic function, {ft} defines a divisor. The, line bundle determined by this divisor is called the hyperplane bundle over F“1. This line bundle is defined by the cocycle (2-2)
Complex Finsler Geometry 25 with the covering W. We denote this line bundle by H. All hyperplanes are linearly equivalent to each other as divisors so that H is well-defined (In fact, the line bundle H is defined by the cocycle defined by (2.2) which is inde¬ pendent of the polynomial F(£)). On each Ly, we put gj = |2 ||£||2. Because of gi = |2^ on UiQUj 0, {#} defines a n standard” Hermitian metric on H. The curvature form is given by O = dd log ||f [|~, and its Chem form is given by Thus ci(jy) = M is represented by the Kahler form 77fs of the Fubini-Study metric on F_1. A hyperplane 77q C F”1 defines a holomogy class in JHr2r_4(Pr"1,Z), and its Poincare dual of Hq is given by ci(77) 6 772(F"X, Z) (cf. [27]). Since cx(77) generates H2 (F-1,Z), every holomorphic line bundle S —> F”1 is a power of 77, i.e., S = H®171 for some m e %. For example, the canonical line bundle K^-i = Ar"1T*r_1 is given by □ Example 2.3. (Tautological line bundle) Let L be the disjoint union of lines in (T. For a line defined by vector £ € Cr\{0}, we define tt ; L —> F_1 by 7t(Zc) = p(£). In another way, L is defined by L = {([^], V) € F“1 x Cr |£g V}, i.e., for [£] = e F“1 the fibre %"1([i]) is given by the line 1$ C Cr- We show that 7T : L —> F“1 is a holomorphic line bundle. Since any point of L is represented uniquely in the form for (i1, “ • ,^r) € Cr\{0} and t € C, we have ?r“1(?7y) » {¿(f1, • * • ,£r) € Cr I i € C,& 0} on Uj, Since t(£\ < ,C) x (£! : ••• : f) e Cx where tj = is uniquely determined by the element of 7r 1(Uy). Then we define a homeomorphism <pj : ?r_1(U}) —* Uj x C by It is trivial that <pi is C-linear on fibres. On the intersection Uij = Ui n Z7y, if tft1, • • •, C) € Tr“1(l7^) then, since tj = if* and ti == we have This means that the coordinate change 1 is holomorphic, and thus %:£-* F"1 is a holomorphic line bundle, which is called the tautological line bundle
26 Aikou over Pr”1. The transition cocycle {¿(ij)} of L with respect to the covering {Uj} of P7*“1 is given by = | = (2-3) □ From (2.3) we have Proposition 2,1. The tautological line bundle L is the dual of the hyperplane bundle i.e., L = H*=H~\ (2.4) By this proposition, the fibre of H at [£] = lc 6 P* is given by l£ = Hom(Ze,C), the space of linear functionals on 1$. Let f = 3 ^near functional on Cr. For [£] = [i1, : • • * : Cl € F*1, the fibre 1$ of L over fc] is given by 1$ = {(if1, • • * , tCj; i € C} C Cr. Then, if we denote by the restriction of f to the line l^ we get a section erf of H by oy([f]) = f\i^ € Thus any element of Hom(Cr, C) determines a global holomorphic section of H. The converse is also true (cf. p. 86 of [70]). Proposition 2.2. H’°(Pr’1,O(jH’)) is naturally identified with Hom (Cr, C). The hyperplane bundle H has many global holomorphic sections, but the tautological line bundle L has no non-zero holomorphic section. In fact, if we suppose that L has a global section t, then, for every point [£] € Pr"1, r defines a point (t1 ([$]),♦♦ - ,Tr([£])) € Cr which lies on the line Z$. By projecting to the J-th component, we obtain a holomorphic function f$ : P1 —> C, that is» P ((£]) = ([£])• Since Pr“1 is compact, and so Pr_1 has no non-constant holomorphic function. Hence this function is constant. The functions f1, • * • , /** defined in this way are constant. The constant point defined by these functions should be the origin, since the point lying on all lines through the origin is the origin itself. Hence L has no non-zero global holomorphic sections, (see [27], [60]) Example 2.4. (Euler sequence) Let L be the tautological line bundle over the complex projective space Pr“1. We recall the Euler sequence (cf. [71]): 0 —> L —* — 0. (2.5) Because of H = £*, we have the exact sequence 0 —► Ijh-x -i. H®r X Tr-1 —+ 0,
Complex Finsler Geometry 27 where we put H®7' — H®- • (r-times). By Proposition 2.2, any holomorphic section of H is naturally identified with a linear functional on Cr. We consider a (1,0)-type vector field on Cr where a1, • * • , aT are linear functionals on Cn. Since p*(a(Ai)) — for all A € C, the definition p*(o-)([$]) = Mi)) is well-defined .The bundle morphism P : -+ Tpr~i is defined by PCa1,..- ,<Z) = d>(<7). (2.6) Since each local coordinate Ç7 (1 < j < r) is considered as a section of H, the morphism P is surjective. From (2.1) and the definition of P, if we denote the section • ■ • ,f) € Z7°(F-\0(Hr)} by £, we have P(£) = 0. The trivial line bundle lpr-i is spanned by the section 5. ’If we define a Hermitian metric on jff®7*, we get the smooth orthogonal decomposition: ^ = 7^-1 ©lpr-i. (2.7) Hence we have c (7pr-i) ♦ 1 = c (H®r\ where c ( - ) means the total Chern class. Then, 22 cs ffir-1) = (1 + oi(£r))r, and so we have ci (T?r-i) — rci (H). □ 2.2 Complex Finsler Metrics 2.2.1 Complete Circular Domains and Minkowski Func¬ tionals Let V be a complex vector space. A complex Finsler metric on V is a norm || • || satisfying the following conditions: (1) ||i|| > 0, and ||i|| = 0 if and only if £ = 0, (2) ||ДСП = )A| ||e|| for € C and ё V, (3) №1 is C~ on V\{0}. The pair (V, || • ||) is called a complex Minkowski space. The unit ball 7? = {i G C | ||i|| < 1} is called the indicatrix of || • ||. If we set /(£) = ||i(|2, then f satisfies the following conditions: 1* /(i) > 0, and /(i) — 0 if and only if £ = 0, 2. /(Ai) = |A|2/(i) for VA e C and vf € V, 3. f is C°° on V\{0}.
28 Aikou The function f is called a fundamental function of || • [|. A complex Finsler metric is said to be convex if its fundamental function f is strongly pluri¬ subharmonic outside of the origin. We fix a basis {$!>• • •> ,sr} of V and identify V with C7* with coordinate system (f1,5C)- Then the strong pseudo-convexity of f is equivalent to that its Levi form f = y/^ïddf is positive-definite, i.e., the complex Hessian (A?) defined by f is positive-definite. We also need the following definition. Definition 2.1. A domain T> in Cr satisfying the following conditions is called a complete circular domain. (1) If C € T> and A € C with |A| < 1, then A£ = (A?1, • • • ,AZT) (2) If £ € D and A € C with |A| < 1, then A< € D. In the sequel, we usually treat complete circular domains with smooth bound¬ aries. For such a bounded complete circular domain £>, its Minkowski functional m?> is defined by mo(e)“inf{| |tfiÉP,i>o},΀Cr. (2.9) If we set (2.10) it is trivial that fa satisfies A>(A£) = |A|2/p(^) for all £ € Cr and A € C. It is also true that $ G D if and only if A>(f) < 1, that is, the domain T> is the indicatrix of the corresponding fundamental function fa Moreover, if T> is strongly pseudo-convex, then ((A>)ij)>0, ((tog/p)#) > 0. (2.11) A Minkowski norm whose fundamental function f satisfies these condition is called a convex Finsler metric on Cr. There exists a one-to-one corresponding between the set of complete circular and strongly pseudo-convex domains with smooth boundaries and the set of convex Finsler metrics. The proof of the following proposition is given in [55]. Proposition 2.3. Let Pi and T>2 be two complete circular domains in Cr with smooth boundaries. Then. is biholomorphic to T>2 if and only if the Finsler metric fa ofT>i is related to fa ofT>2 by fa = fa°A for some A G GL(r, C).
Complex Finsler Geometry 29 By using this proposition, the following characterization of Hermitian inner product is obtained: Proposition 2.4. ([55]) Let L> be a complete circular domain with smooth boundary in Cr. The following statements are equivalent: (1) T> is bzholomorphic to the unit ball B — {£ € Cr | £ |C|2 < 1}.* I • I2 (2) the associated Finsler metric fa is of the form /p(i) = £ some A = (¿0 € GL^C), (3) /© is smooth at the origin. We shall give another characterization for Hermitian inner product. Let G = {A & GL(r,C); | = Hill for v£ G V} be the isometry group of || • ||. The continuity and the homogeneity of || • || imply that G is a compact Lie group(cf. [72], [70]), and so it is isomorphic to a closed subgroup of U{r) “ GL(r,C) 0 G(2r). Since ||^|| ||£|| = 1 for G S = &D and vp G G, G acts on the unit sphere S. The action is transitive if and only if (V, [| • |[) is an inner product space. Then we have. Proposition 2.5. Let (V, || • ||) be a complex Minkowski space. Then (V, ][ • ||) is a Hermitian inner product space if and only if the isometry group G is iso¬ morphic to the Unitary group U(y). Proof: Since G is compact, there exists a bi-invariant Haar measure dg. Then, for an arbitrary Hermitian inner product (-,■)? we define a G-invariant inner product < •, • > by <i,’7>= [ (¡faffing- Jg The indicatrix Do of < •,• > is the open unit ball centered at the origin with the isometric group Go — ^(r)- The group Go acts on the boundary dDo transitively. We can assume without loss of generality that PriPo because if it is necessary we multiply the inner product (♦, ♦) by a positive constant. Let ^o be a fixed point in D ADo- We suppose that G Z7(r). Then G also acts on ODq transitively. For an arbitrary point r/ G d'Do, there exists a g 6 G satisfying V = Then we have ||7/|| = ||^io|| “ ||fo || = 1* Hence p € from which we have =< >. Q.E.D.
30 Aikou 2.2.2 Complex Finsler Metrics on CT and Kahler Metrics on F_1 Let / be the fundamental function of a convex Finsler metric. We shall show that f induces a Kahler metric on the complex projective space F_1. We denote by p : C*\{0} -+ P7-“1 the natural projection. The tangent bundle Tpr-i is locally spanned by the vector fields {dp (д/d^)} with the relation (2.1). For the hyperplane bundle H Pr_1, w’e identify the fibre Яде over [fj G Pr_1 with the set of homogeneous functions of order 1 on p^1 ([£]). Since the given metric f on Cr is convex, we define a Hermitian metric (*, •) on by for sections X = (X1, • •« , Xr) and Y = (У\ , Уг) of Яфг. With respect to this Hermitian metric, the Euler sequence (2.5) implies the orthonormal decom- position (2.7). By the relation (2.1), the bundle lpr-i is the trivial line bundle locally spanned by 8 — (f1, ♦ - • , f7*) and, moreover we have (£, 8) = 1. Then any section X G Яфг is decomposed as X = (X, 8}8 4- X for X = P(X) G Tpr-i. Then it induces a Hermitian metric (*, on Tpr-i by - (x,£) = (^logi) (X, Y). For the homogeneous coordinate (f \ •, f7*) on Uj = {[f] G P7*-1 | f* / 0}, the local function gj (f) := log /(f)—log |f* |2 on Uj satisfies V^lddgi = y/^lddgj — yf=lddlog / on Ui П Uj. Hence the real (1, l)-form Прг-i = J^lddgi = v^ia^log/ (2.13) defines the Kahler metric (*, *)r-x* The functions {#} are called the Kahler potentials of (», Especially, if the function / is given by /(f) = (i.e., /(f) is the fundamental function of the flat metric ^d^^d^1 on C7*), the induced Kahler metric on F_1 is the Fubini-Study metric. In the sequel of this subsection, we shall show that the converse of this fact is also true. Proposition 2.6. A Kahler metric Прг-i on the projective space P7'"1 defines a convex Finsler metric on Cr which is unique up to a positive constant multiple. If we denote by S the sheaf of germs of pluri-harmonic functions on Pr~1, the exact sequence o— X.Jt) . —> —* я°(Г— *,<$)—► /МОГ“1.*) — 5 i II R СО implies Я°(РГ_1,5) = R. Непсе any pluri-harmonic function on P7'“1 is con¬ stant.
Complex Finsler Geometry 31 Proof of Proposition 2.6: We express locally ZTpr-i = y/—lddgj on Uj for a C00-function gj on Uj, Since gj — gL is pluri-harmonic, there exists a 1-cocycle Kij e Z\Ui 0 Uj, Op— i) satisfying gj -gi~ Kij + Kij on Ui n Uj 0 $>, Then {Kij} is a 1-cocycle on F“1, and since 2?1(Pr^1, Op—i) - 0, we may put Kij — (Kj - log^) - (Ki -logf*) for a 0-cochain {Kj} on F“1. Hence we have 9i - {Kj + AT) + log |f I2 = 9i - {Ki + Kl) + log If I2 If we put Wl)=exPto-(A<+^)} on Uj, we have |^|2/j(KI) = l$*I2/»([$])• Thus we have a function f(£) — ls5 l2/j(Kl) on Cr. It is clear thaty satisfies the condition (1) ~ (3). Moreover, because of >/-iddlog/ = ^/-I^log/; — y/^lddgj > 0 and V—lddf = (ddlog/ + ¿Hog/ A Slog/) , the function / defines a convex Finsler metric on Cr. We suppose that we get another Finsler metric / from another Kahler po¬ tential {gj}. Then, since y/^lddgj = V—lddgj, the function log / - log/ is pluri-harmonic function on F*“1. Hence it is a constant c. Consequently we have / = ecf, Q.E.D. 2.2.3 Complex Finsler Metric on Vector Bundles Let 7Fe : E —► M be a holomorphic vector bundle over a complex manifold. If rank(E) = 1, then any Finsler metric on E is reducible to a Hermitian metric, and so we assume rank(E) = r > 2 in the sequel. Definition 2.2. A complex Finsler metric on E is a smooth assignment to each fibre Ez = 7r^1(^) of a norm [| • ||x. We call (E, || • ||) a complex Finsler vector bundle. We define a function fz : Ez JR by fz(£) = ||£||2 on each fibre E- Cr. The function f- satisfies the following conditions: 1- /$(£) > 0 and /$(€) = 0 if and only if £ = 0, 2. /c(Ai) = |A|2A«)forvAeC, 3. f~ is smooth on E* = £!s\{0}.
32 Aikou The function F : E —► R defined by F(;s,£) = /-(£) is called the fundamental function of (£?, || • ||). Conversely, if a function F : E —► R satisfying these condition is given on E, then it defines a unique complex Finsler metric || * || on E. Thus, in the sequel, we sometimes identify a complex Finsler metric || • || with its fundamental function F, We shall fix a covering {Uy (ecj)} of E with an open covering U = {tZJ of Af and local holomorphic frame fields ey = (ei,• • ♦ , er) of E on each U € U. Then {Uy e&r} introduces a local trivialization tpy : tt^1 (U) -+ U x Cr by sending v-\U) 9 v = £ to (A ’ • • , sn, i1, • • • , C) € U x e: tt“1(CZ) CTxCr i % i Pi U Uy and it defines a canonical coordinate system {vr^1(C7), (s*,^)} on E, The projective bundle ^p(B) • P(F) -♦ M associated with E is defined by !?(£?) = E*/C*. The tautological line bundle • ¿(-E) is defined by L(E) = {(yy V) € ]?(£) x E | v G V}. We fix an open cover {U} of Ai. If we define an open covering {Uj} of P(F) by Uj = A / 0}, the transition cocycles {l(ift} of L(E) relative to {Uj} are given by [£]) = (2.14) Proposition 2.7. ([38]) Any Finsler metric onE is identified with a Hermitian metric on the tautological line bundle L(E\ Proof: For any Finsler metric F on Ey we define a positive function on Uj by F(zyÇ) = [£]). Then, from (2.14) it is easily verified that 9l(E) KI) = R(v) |20r(s)X*> KI) (2-15) for the transition cocycle {¿(¿;)} of Le, and thus the family defines a metric on Le- Conversely, any Hermitian metric on Le is defined by the family of positive functions satisfying (2.15), we can define a Finsler metric F on £ by Q.E.D.
Complex Finsler Geometry 33 Definition 2.3. A complex Finsler metric F is said to be convex if F is convex on each fibre Ez, i.e., the Levi form y/^lddf- is positive definite on Es, Remark 2.1. We say F is strongly convex if its real Hessian on each fibre E~ is positive-definite. The strong convexity implies the convexity. Instead of convexity, we sometimes assume the strong convexity'. In fact, in real Finsler geometry, we assume this strong convexity. An almost complex manifold (M, J) with a strongly convex Finsler metric F has been investigated by Ichijyo[30], and such a space (M, J, F) is called a Rizza manifold. □ It is easily shown that the definition of convexity is independent of the choice of local trivialization {U, } of E. If we define a Hermitian matrix (F^j) by - jgj. (2-16) F is convex if and only if (F$) is positive definite. Example 2.5. Let g be an arbitrary Hermitian inner product on E. With respect to an open cover {&/, {gu)} we put g$ = gle^ e>), the function F : F —> R defined by (2.17) defines a convex Finsler metric on E. We remark that this function F is smooth on the whole of the total space E. □ If a convex Finsler metric F is given on E, we can define a Hermitian metric {',•)$ on the vertical subbundle Vs by (2.18) By this definition, the metric F defines a Kahler metric IIs on each fibre Ez^Cr by nz = V—ldSand the bundle tt : E M is a Kahler fibration with the pseudo-Kahler metric ITs = y/^ldSF. Any convex Finsler metric F defines a pseudo-Kahler metric on P(E). To show this fact, we denote by H(E) = LIE)' the hyperplane bundle over P(F) defined by the transition cocycles KJ) ~ £i ~ (si) (2.19) The Euler sequence (2.5) is generalized to the sequence 0 —> L(E) —► *£(#)£ —► L(F) 0 > 0> (2.20),
34 Aikou and tensored by H(E) we have 0 1?(S) ® 7Tp^E -?-> Tp(e}/m —* 0, where Ip(^) is the trivial line bundle over P(2?) spanned by the Euler vector field i = (i1,1 jf) and the bundle morphism ? : H{E) ® Kp^E —> Vp(£) is defined as follows. Any section a of H(E) is defined by a function cr : Es —> C which is a linear functional on each fibre Es, and any section X of H E is naturally identified with a section X = of V& satisfying the homogeneity condition A£) — AX*(z,£) for all A G C*. Then we define P(X) = dp^X\z,£)^ for the natural projection p : E* —► P(S). If F is a convex Finsler metric on E, we can define a Hermitian metric (•, •) on H(E) ® by (X,Y}= r.,,1 f.y'-^rXiY3 = r^—r(x,Y\ (2.21) \ / F(z,£)^d£d& F(z,£)\ ’ /e ' ’ for sections X = (X1, • ■ •, Xr) and Y = (Y1, • • •, Yr) of £T(F) ® Ac- cording to the orthogonal decomposition H(E) ® ^p^E = lp(£) © the map P is also defined by P(X) = X- £ = X - 1 {x,£)s£. For any sections X and Y of Ifys), we take sections X and Y of H(E) such that P(X) = X and P(X) = Y. The induced Hermitian metric on Vp(£) is defined by i (x,Y)B - ± (2.22) which is a Kahler metric IIS on each fibre P(FS) ~ Pr"'1 of the fibration 7rp(^. Proposition 2.8. Let (EyF} be a convex Finsler vector bundle. Then the bundle 7Tp(£) : P(F) —* M is a Kahler fibration with the pseudo-Kähler metric ^p(jE) — \/"~ldd log F. We have obtained a Kahler fibration ^p(£) : P(2?) —► M with a pseudoKahler metric %/^lddlogF from an arbitrary convex Finsler metric F, and it induce a Hermitian metric as the restriction of to the bundle Vp^). Conversely, from an arbitrary pseudo-Kähler metric 27?(e) of the projective
Complex Finster Geometry 35 bundle !?(£) or Hermitian metric (•, •}₽(£) on Vp(Jg)> it induces a convex Finsler metric F on E, In fact, if we restrict to . any fibre IP(£7Z) = F*“1, we have a Kahler metric Hs on IP(Es). Then, by Proposition 2.6, H. determines a convex Finsler metric fz on Es Cr which is unique up to a positive constant multiple. Since this Finsler metric fs depends on the base point z € M smoothly, we have Proposition 2.9. If a pseudo-Kahler metric ITp(^) is given on the projective bundle ]?(£) associated with Ef then defines a convex Finsler metric F on E which is unique up the multiply of a positive function on M, 2.3 Bott Connections of Finsler Vector Bundles Let (£7, F) be a convex Finsler vector bundle. Then we have two Kahler fibra¬ tions. One is the fibration ke : E —> M with the pseudo Kahler metric IIe = V—lddF whose Kahler metric on each fibre Ez is defined by (2.18), and another one is the fibration 7rp(s) ' P(£7) —► M with the pseudo-Kähler metric 17p(S) = V'^lddlogF whose Kahler metric on each fibre P(^) — F is defined by (2.22), For local expression of complex Bott connection of these Kahler fibra¬ tion, it is convenient to treat the bundle (E, F) by considering (f1, • • • , $**) as the homogeneous coordinate of the fibre Let 7T : E -+ M be a holomorphic vector bundle over a complex manifold with rank(£7) = r. Setting X — Ei the total space of the bundle, we obtain a connection of the bundle tt : E —► M. Since each fibre of the fibration % is a complex vector space, we denote by 5 the sheaf of germs of functions on the total space E which are linear functionals along the fibres of 7r. Any connection of the sequence 0 —> VE —► TE 7T*TM 0 (2.23) is determined uniquely by the action of Ofe on the sheaf S. Let U be an open set in M with local coordinate (z1, • • • , zn), and let = (ei, - • ,er) be a local holomorphic frame field on U. Then the pair (t^ecz) induces a coordinate (z1, • • • , zn, , f7*) on 7r"1(CZ), where (z1, zn) is lifted from the base manifold M and (i1, • • ’ is the fibre coordinate. Then a germ f of S is written in the form / = S on 7r”1(^)» and the action on f is written as =E^w • e+E/iW • Since dfi € and € TQ;, there exists some functions {A^} on 71,-1 (tf) satisfying (2.24)
36 Aikou By this definition, the functions satisfy the homogeneity condition: Nl(z,À<)=ÀAX(^e) (2.25) for all À G C, and, in generally, is not linear in the variable If we take another open covering {(¡7, et/)} with the same open cover {¿7} of M, there exists a holomorphic function Au :U —> GL(r, C) such that ëu = eu Au* Let the coefficients of hs relative to the covering {(t7, e^)}, that is, Here we note that (z“,^) with k the coordinate on tt”1^) determined by (17, e^). Then, from (2.24), the relation (1.4) is written as «)=E 4(z)^(S, ô - E <2-26) Definition 2.4. A connection He of the bundle x : E —► M is called a non¬ linear connection of E. The functions are called the coefficients of h&. A non-linear connection h& is said to be linear if the coefficients are linear functionals along the fibres of ir. Since the coefficients N^(z,£) are homogeneous of degree one with respect to the variable £, hs is linear with respect to the variable £ if and only if are holomorphic with respect to i.e., — 0. We suppose that Ke is a linear connection of E. Let Xa be the horizontal lift of d/dz°\ By definition, there exists some local functions Fya(~) satisfying X^ = S ¿¿a (*)£*• By these functions the connection h : w*Tm -+Te is given by (2'27' From the relation (2.26), the local 1-form wj = 520a(z)^Q defines the con¬ nection V : A(£7) —► Ax(£?) of (l,0)-type. Thus, if Ke is linear, then there exists a connection wj = £TJa(z)dza on E such that and the Bott connection DE associated with As is given by the bull-back connection DE = 7r*V. Conversely, if a connection V : A(£) —* A1 (2?) is given by connec¬ tion forms wj — 2Fja(z)dza? then a connection hs • k*Tm —► 7b is defined by (2.27) which is linear. Consequently we have Proposition 2.10. Let h$ be a non-linear connection of a holomorphic vector bundle 7T: E —► M. Then, the following conditions are mutually equivalent. (1) hE is linear.
Complex Finsler Geometry 37 (2)^j = 0. (3) There exists a connection V on E such that the Bott connection DE as¬ sociated with kg is given by the pull-back DE = tt* V. Now we shall consider the Bott connection DE of the Kähler fibration tte : E —> M with IÏ& — y/^lddF. In this case, from (1.27) the connection h& of ke is defined by the coefficients (2.28) The derivation d^ = dfe + d$ associated with this non-linear connection is defined by dfe = £ Xa0dza for the horizontal lift Xa of d/dz*. The coefficients r%a of DE are defined by (1.17), and by the homogeneity of F, the coefficients and satisfy the relations (2.29) This condition is equivalent to £)*£ = 0. (2.30) Since {£,£) — F(z,£) and DE satisfies the metrical condition (1.23), we have (2.31) The following proposition is proved by direct calculations. Proposition 2.11. ([6], [9]) Let DE be the Bott connection of(E, F) associated with the non-linear connection Ke of (2.2S). Then we have (1) d% o = 0, i.e., R^p = 0, and the torsion form T of D is given by T* = £ R^dz* /\dzß + y^ R^dz* A A dz*. (2.32) (2) + w A u) = 0, i.e., R^aß — 0 and the curvature form is given by (2.33) The components of QD is given by the form = £ Rt^dz* A dz0, where the curvature tensor R*^ == — Xpr^* is expressed as (2.34)
38 Aikou by the identity (2.29), we have the relation (2-35) If the torsion form T of DB vanishes, then, by Proposition 2.10, there ex¬ ists a linear connection V such that DB — tt*V, and then (2.35) implies Since R*a& — 0, the connection V and so DB is flat. Conversely, if DB is flat, by Theorem 1.1 shows T — 0. Consequently we have Theorem 2.1. Let DB be the Bott connection associated with the non-linear connection He in (2.28). Then DE is flat if and only if its torsion form T vanishes identically» Next we shall consider the Bott connection of the vertical subbundle Vp(s) of the fibration 7Fp^) : P(E) —► M. For this purpose, we shall define the connection h?(E) of the fibration ttp(e)- We define a connection hp(^) of the fibration %?(£) by d$(B) = Y,dP ® d»“> (2.36) for the projection p : —► P(£), where the vector fields on T?(E) span locally the horizontal distribution Wp(s) = We shall define a partial connection of T^e) as follows. For any Ÿ € X(Tp(s)/M), there exists a section Y € A(H(E)®7fyB}E) such that P(Y) = dp(Y) = Ÿ. If we denote by (»)y the natural projection from TB to VB, we have \dp{Xa\Ÿ] = dp^.Y] and (dp[Xa, Y])y = P ([X*, Y]v). Then we shall define = P {DeY). (2.37) Because of (2.31), (2.32) and P (PSX) = PeX - 1 (DeX,£)b£, the following proposition is proved by direct calculations. Proposition 2.12. The partial connection satisfies - ("M»+ /ord/X,Ÿ€X(TP(B)). (2.3S)
Complex Finsler Geometry 39 The partial connection is just the Bott connection of the Kahler fibra¬ tion 7Tp(£) : P(E) —► M. The Bott connection DB is flat if and only if = 0 for certain local coordinate system. The flatness of is given as follows. Proposition 2-13. The Bott connection Dv^ of the Kahler fibration %?(£) : P(E) —* M is flat if and only if there exists a local coordinate system (z“^) such that = (2.39) for some local function <?a(z) on eachU^U. Proof: By Theorem 1.1, the Bott connection is flat if and only if ^p(£) is flat and this is equivalent to the condition «.-Es»*1- By the definition (2.36) and, since kerdp is spanned by £, this condition is equivalent to for some function satisfying the homogeneity a(z, A^) — 0tt(z, Ç) for A € Cx. Then, we get E « - E - E O’1) é (Sê) - « because of the homogeneity of aa, and from this we get which shows that a* is holomorphic with respect to the variable f. Since <ra is homogeneous of degree zero with respect to <ra depends only on the base point £ € M. Q.E.D. A convex Finsler metric whose non-linear connection is of the form (2.39) will be discussed in a later section. We shall show that, if N& is of the form (2.39), then there exists a local function <r(z) such that cra = dcr/dz01. 2.4 Negativity of Vector Bundles In this section, we shall discuss the negativity (or ampleness) of holomorphic vector bundles, and show a characterization of it by using complex Finsler geo¬ metry (see [21] and [38]).
40 Aikou 2.4,1 Positive Line Bundles and Ample Line Bundles Let L be a holomorphic line bundle with a Hermitian metric g. Let be an open covering of L with transition cocycle {guv}* If we put g(eu, ea) = gu(z) on each ¿Z, the local function gu is smooth and positive, and moreover it satisfies gv = gu\9uv\2 on U fl V. The Hermitian connection V of (L,^) is given by the local (1,0)-form cvv = # log 0a and its curvature is given by = dd log gu* The first Chern class ci(L) is represented by its Chem form ci(L,0) = ^~~^Ric(g) for its Ricci curvature Ric(p). 2tt Definition 2.5. A holomorphic line bundle L is said to be positive if its first Chern class Ci(L) is represented by a positive real (1, l)-form. By this definition, a holomorphic line bundle L is positive if and only if L admits a Hermitian metric g whose curvature « Sd log g is positive- definite. Then the form log g defines a Kahler metric on M. A complex manifold M is said to be a Hodge manifold if there exists a positive line bundle Example 2.6. Let H be the hyperplane bundle over a complex projective space F1. From Example 2.2, we have T 7T and thus H is positive. □ Let 7T : L —> M be a holomorphic line bundle over a compact complex manifold M. Since M is compact, dime #°(M, (P(L)) is finite. Let {so, • • *, sjv} be a set of linear independent sections of L of the complex vector space of global sections. The vector space spanned by these sections is called a linear system on M. If the vector space consists of all global sections of L, it is called a complete linear system on X. Then a rational map : M —► is defined by V|L| (2) = M«): • • •: sk(«)], (2.40) where we put ¥>(si) — /’) € U x C for a local trivialization : 7r“1(17) —. U xC. This rational map is defined on the open set in M which is the comple¬ mentary to the common zero-set of the sections (Q <i < N). It is verified that the rational map obtained from another basis {so? • • * is transformed by an automorphism of
Complex Finsler Geometry 41 Definition 2.6. A line bundle L over M is said to be very ample if the rational map ^i£| : M determined by its complete linear system |L| is an embed¬ ding. L is said to ample if there exists an integer m G Z such that L®m is very ample. Let L be a very ample line bundle over a compact complex manifold M, and {so, • • ‘ > swj a basis of (9(L)) which defines an embedding : M —> Pv. Under this imbedding, we can think of [so (s): • • •: sjv(s)] as a coordinate system on the embedded M in P7^. We define an open covering {Vj} of M by Vj = {z € M ] sj(z) 7^ 0}. With respect to this covering, the local trivialization : 7r”1(^) —> V> x C of L is given by <£,($$) = The transition cocycle [Ijk ' Vj A Vk —► C*} is given by The transition cocycle {hjk} for the hyperplane bundle H is given by the form (2.2) for the standard covering {llj} of Pn. Then, by definition, Vj = {z € M | Sj/s) / 0} — H M). Hence we get hjk ° V’l-DI = and thus we have L — Lemma 2.1, Let L be a very ample line bundle over a complex manifold. Then L is isomorphic to the pull-back bundle of the hyperplane bundle H over the target space PN ofy^. The following well-known theorem shows that any Hodge manifold M is algebraic, i.e., M is holomorphically embedded in a projective space P^. Theorem 2.2. (Kodaira’s embedding theorem) Let L be a holomorphic line bundle over a compact complex manifold. If L is positive, then it is ample, i.e., there exists an integer no such that for all m> no the map : M —► PA' is a holomorphic embedding. Conversely, if L is ample, then there exists a basis {so, * • * , s/v} of HQ(M, C?(L®m)) such that | : M -+ PN defined by (2.40) is an embedding. By Lemma 2.1, the line bundle L®™ is identified with (p*L0TTiiH. Thus there exists a Hermitian metric g on L®m such that ci(L^) = mci(L) = ^ïâaiogp(s) ,
42 Aikou where g(z) is defined by g = 53 |/*G5)|*. Since H is positive, the (1, l)-form y/^lSdlogg(z) is positive, and thus ci(£) = i m ^J-dd1ogg[z) is positive. Consequently we get Proposition 2.14. A holomorphic vector bundle L over a compact complex manifold M is positive if and only if L ample. Let E be a holomorphic vector bundle over a compact complex manifold M. The Serre duality and Dolbeault isomorphism = #g(M, t2p(E)) imply H<(M,np(E)) * 2P”9(M,Pn“p(E*))*. Putting p = q = 0, we have F°(M, (9(E)) * #n(M, Now we suppose that M is a compact Riemann surface. Since dime M — 1, we have HQ(M, O(E)) ~ KX(M, PX(E’))* = 22X(M, O(E* ® KM))* and dime0(E)) = dimcE^M, 0(E* ® Km)) for the canonical line bundle Km = of M. The genus of M is defined by the integer g := dime H1 (M, OM) = dime H* (M, O(KM)). The degree of a line bundle L is defined by degE = / ci(£) 6 2, If we apply Jm the well-known Riemann-Roch theorem O[L)) - dime EX(M, 0(2,)) = degL +1 - g to the case of £ = KM, we have dime Hx(Af, O(Km)) = dime K°(Mi PX(K^)) = dimc#°(M,0Af) = 1, since M is compact and thus Hq(M,Om) = C. Con¬ sequently we have deg Km = 2$ - 2, and the Euler characteristic x(M) is given by %(M)= [ ci(Tm) =-degKJtf = 2-2$. Jm Any compact Riemann surface M is determined completely by its genus $. For example, (1) if g = 0, then M is holomorphically isometric to the Riemannian sphere Px=CU{oo},
Complex Finsler Geometry 43 (2) if g = 1, then M is holomorphically isometric to a torus C/A, (3) if g > 1, then M is hyperbolic, that is, M admits a Kahler metric of negative curvature. In the case of g = 0, i.e., M =■ P1, then since ci(7\f) > 0, its tangent bundle is positive (or equivalently ample). If g > 1, then its tangent bundle Tm is negative since Ci(Tm) < 0. In the case of dime M > 2, Hartshone’s conjecture: • If the tangent bundle Tm is ample, then M is bi-holomorphic to the pro¬ jective space Pn was solved affirmatively by Mori. It is natural to investigate complex mani¬ folds whose tangent bundles are negative. Kobayashi ([38]) has investigated this problem by the method of complex Finsler geometry and shown the following: ♦ A holomorphic vector bundle : E M is negative if and only if E admits a convex Finsler metric with negative curvature. We shall discuss this problem in the next subsection. 2.4.2 Negative Vector Bundles The interest in complex Finsler geometry arises from the characterization of ample (or negative) vector bundles due to Kobayashi[38]. A holomorphic vector bundle 7r : E -+ M over a compact complex manifold M is said to be ample if its tautological line bundle L(E) —► P(E) is ample. A holomorphic vector bundle 7T : E —> M over a compact complex manifold M is said to be negative if its dual E* is ample. In this subsection, we shall investigate negative vector bundles by differential geometric method. Since the Chern class ci (£(£)) of L(E) is expressed by ddlogpj for an arbitrary Finsler metric F on E, L(E) is negative if and only if ^^aaiogF<o (2.4i) or equivalently this real (1, l)-form = x/^lddlogF defines a Kahler met¬ ric on the base manifold P(E). Kobayashi’s characterization is obtained by analyzing the positivity of the form Hp(£). To investigate (2.41), the following lemma is useful (cf. [9]). Lemma 2.2. The curvature 3d log F o/(L(F),F) is given by A (2.42) where we put f07' curvature tensor of the Bott connection DE.
44 Ajkou We note that the curvature tensor Rifag is given by = -XpXaF^ + ^F^X^F^XaFii. We define a Hermitian form 'F = £ A on P(£) by K]) := F&t) £ ^3^. (2.43) Then a convex Finsler metric is said to be negative curvature if # is negative, i.e., < 0 for all (Xa) / 0. From (2.22), the second term in (2.42) is negative definite, and thus we have Kobayashi’s theorem. Theorem 2.3. ([38]) A holomorphic vector bundle E is negative if and only if E admits a convex Finsler metric F with negative curvature For a negative vector bundle E over M, we shall construct a convex Finsler metric F with negative (cf. [8] and [79]). By definition the line bundle L(E) is negative, and so L(E)* is ample. Hence there exists a sufficiently large m € 2 such that L := ¿(E)*®171 is very ample. By the definition of very ampleness, we can take • , fN e if°(P(E), L) such that <fif : P(S) 3 (z, [$]) [£]):•■•: fN(z, [£])) e P" defines a holomorphic embedding ip/ : P(£) —► P77. Then, for the hyperplane bundle H —► PN, from Lemma 2.1, we have L S Since PAr admits the Fubini-Study metric, the first Chem form of H is given by on Va = {[T1: •• •: Tw] € P");?“ * 0}. On Uj := H 0} C P(l?), we put fb — {fj}, (6 = 0*• •, A*), where ff are holomorphic functions on Uj. Then a canonical Hermitian metric of v>jH is defined by a(z, [$]) . IWfl)l2 / on tpj1 (Va) D Uj< Since H is ample and <pj is holomorphic embedding, we have ^^^log^jH.aCs, KJ) > 0. The corresponding Hermitian metric gz on L is given by the functions (2.44)
Complex Finsler Geometry 45 on each Uj. Since L — the corresponding Hermitian metric on L(E) is given by the functions on Uj. Since {#£(£) j} satisfies (2.15) on Ui fl Uj fa we may define a complex Finsler metric F on E by t) := I2gL(EU = <2’45) Since F satisfies (2.41), we have Proposition 2.15. Let E be a negative vector bundle over a compact complex manifold M. For the holomorphic embedding tpf : P(£) —+ the function F defined by (2.46) is a convex Finsler metric with negative curvature S'. 2.4.3 Vanishing Theorems In this subsection, we shall investigate some vanishing theorems in complex Finsler geometry. Let (F, F) be a convex Finsler vector bundle. For the Bott connection DE of (F,F), we put _R(Z) = ^2 «« ® dza A dz13 for *Z € A(Ve)‘ Then we have Proposition 2.16. The Bott connection DE of (E,F) satisfies d^{Z. Z)& = (DEZ, DeZ}b ~ {R[Z\ Z}E (2.46) for all holomorphic section Z ofVE. Proof: This equation is obtained from (1.23) and (2.33). Q.E.D. Let (F, F) be a convex Finsler vector bundle over a compact Hermitian manifold (M, p), where g = 52 9^^° ® dz^. For the curvature tensor of D, we put := 52 and call it the mean curvature of (F,F). Then we shall define a Hermitian form K by K(Z, W) := 52 Kt^Wi (2.47) for *Z,W € A(Vj;). For any holomorphic section Q of E and the function /«) ® F(z. £(z)), we have the following Weitzenbock-type formula.
46 Aikou Proposition 2.17. Let (E,F) be a convex Finsler vector bundle over a Her¬ mitian manifold (M,g). For any holomorphic section Q of Ef the following identity holds: □/(2) = ||D^||2-K(C<), (2.48) where we put ll^r == Proof: Applying the formula (2.46) to the function f(z) = F(z,<(z)) = «(«), C(*)>S» 0£ = -<*(0>0B + (P£<, DbQp)e, (2.49) and by taking the ^-trace of the equation, we complete the proof. Q.EJD. For a holomorphic section £ of E, the equation (2.48) is written as = -F(z,C(Sy)^z, [<(*)]) + £^(^(s))P*0Pf0. If has at least one negative eigenvalue at every point of P(E), the complex Hessian ddf has a positive eigenvalue at every pint of P(E). Hence we have Kobayashi’s vanishing theorem. Theorem 2.4. ([38]) Let (E,F) be a convex Finsler vector bundle over a com¬ pact complex manifold. If has at least one negative eigenvalue at every point of P(£)> then there exists no non-zero holomorphic sections, that is. HQ(M,O(E)) = 0. Let C = C(s)$i be a non-vanishing holomorphic section of E over an open set U. We say that a holomorphic section C — SC(^)s* is parallel with respect to DB if it satisfies + = 0. By the formula (2.48) we get the following Bochner-type vanishing theorem for holomorphic sections: Theorem 2.5. ([6]) Let {E, F) be a convex Finsler vector bundle over a compact Hermitian manifold (M,p).
Complex Finsler Geometry 47 (1) If the mean curvature K is negative semi-definite on P(J5), then every holomorphic section £ of E is parallel with respect to DB, that is, DB£ — 0, and satisfies £) = 0. (2) If K is negative definite on then E admits no nonzero holomorphic section: H°(M,O(E)) = 0. Proof: Applying the maximum principle of E. Hopf (Theorem 1.10 of [39]) to the formula (2.4S) implies our assertions. Q.E.D. bi the sequel of this sub-section, we shall show a vanishing theorem for cohomology groups. Let (E, F) be a convex Finsler vector bundle over a compact Kahler manifold where Um is its Kahler form. We assume that & is semi-negative with rank > fc. Then the first Chern class Ci (£(£)) is semi¬ negative with rank > k + r — 1. Hence the bundle E is semi-negative of rank > fc(cf. [39], p. 83). Then, Theorem 6.17 of [39] may be generalized as follows: Theorem 2.6. ([8]) Let be a convex Finsler bundle o/rank(E) = r over a complex manifold M. . (1) The curvature & is semi-negative of rank >k if and only if the curvature SdlogF of the corresponding Hermitian metric in L(E) is semi-negative of rank > fc + r — 1. (2) If the curvature W is semi-negative of rank > k, then E is semi-negative of rank > k. (3) If the curvature & is semi-negative of rank > k, then Jf<7(M,Pi>(P)) = 0 for p + q<k — rt provided that M is compact Kahler. Proof: The first and second are trivial from the definition of semi-negativity. Hence we shall prove the third assertion. If we apply the Gigante’s vanishing theorem(cf. [39], p. 69) to the Hermitian line bundle (L(E),F), we get J7p(L(JE))) = 0 for P + Q < k + r — 2, where QP(L(E)) denotes the sheaf of E(E)-valued holo- morphic P-forms. On the other hand, we know the following isomorphism(cf. [39], p. 84): E«(M, PP(E)) ~ f2n-”(E')) « H"-’(P(E), i2”-₽((L(E))-)) ~ E’+r-1(lP(E), ^^(¿(E))),
48 Aikou where ~ and « denote the Sene duality and Le Potier isomorphism respectively. This implies the third assertion. Q.E.D. As a special case, if S' is negative definite, Theorem 2.3 implies that E is negative. Then Corollary 5.10 in [39] can be written as follows: Corollary 2-1. Let E be a holomorphic vector bundle of rank r over a compact complex manifold M of dimension n. If E admits a convex Finsler metric F with negative №. then QP(E)) = 0 for p + q <n — r. In the case where E is a holomorphic line bundle, any Finsler metric on E is a Hermitian metric, and so, Corollary 2.1 is a generalization of Nakano’s vanishing theorem. 2.5 Special Finsler Vector Bundles In this section, we shall investigate some special Finsler metrics on holomorphic vector bundles as natural generalizations of the case of real Finsler geometry. In real Finsler geometry, there exists three important notions of special Finsler spaces, i.e., Landsberg spaces* Berwald spaces (or modeled on a Minkowski space) and locally Minkowski spaces (or flat) ([16], [17], [51]). Let M be a smooth manifold with a Finsler metric. Then the total space of its tangent bundle is considered as a fibred manifold whose fibres are Hessian manifolds parameterized smoothly by the base point in M, where a Riemannian manifold is said to be Hessian if it is a flat manifold and the metric tensor is given by the Hessian of a function in an affine coordinate system. A convex Finsler metric defines a canonical horizontal subbundle of Tm, i.e., Cartan’s non-linear connection. A Finsler manifold is said to be Landsberg if Tm is a fibred manifold with isometric fibres, i.e., the parallel displacement along any smooth curve in the base space M is an isometry between the fibres (cf. [4]). A Landsberg space is said to be Berwald if the parallel displacement is linear. In this case, the Cartan’s non¬ linear connection is defined by the Riemannian connection of a Riemannian metric on M (cf. [66]). A Berwald space is said to be locally Minkowski if the non-linear connection is flat, i.e., the associated Riemannian metric is flat. In complex case, we have no idea for the notion corresponding to Landsberg spaces. As an analogy of real case, we suppose that the total space is a fibred manifold with isometric fibres. Then, we have — 0 (cf. Proposition 1.13), and the metric is modeled on a complex Minkowski space (cf. Proposition 2.10). Thus, in this section, we shall investigate two special cases, the one is the case where the metric is modeled on a complex Minkowski space and another one is the case where the metric is flat.
Complex Finsler Geometry 49 2.5.1 Finsler Vector Bundles Modeled on a Complex Minkowski Space Ichijyô[29] has introduced the notion of Finsler manifolds modeled on a Minkowski space. In this subsection, we shall generalize this notion to the case where the metric is a convex Finsler metric on a holomorphic vector bundle. Definition 2.7. A convex Finsler vector bundle (E, F) is said to be modeled on a complex Minkowski space if its the non-linear connection hs in (2.2S) is linear, equivalently, there exists a connection V on E such that Ds = it*V. We suppose that (E,F) is modeled on a complex Minkowski space. By Proposition 2.10, we have = 0, or equivalently, there exists a connection V on E such that DE — V. Then, by Proposition 1.13, the parallel displacement with respect to V is an isometry between the fibres, i.e., the Kahler metrics (Ex, lddF-), and thus it is natural that the parallel displacement preserves the Finsler metric F. We shall show that this connection V is the Hermitian connection of a Hermitian metric gF on E. To show this, we need a lemma. Let (E, F) be a convex Finsler vector bundle. For each z € M, the fibre E~ is a complex Minkowski space (Cr, /«)' with the norm ||s||2 = A(C)- We denote by G the compact Lie group of isometries of the norm: G = {g G GL(r, C); ||tf || = |]$||, € Ez}. (2.50) Let J be a complex structure on E, that is, J € End(E) satisfies J2 = —id. By the idea due to Szabd ([66]), we have Lemma 2.3. Suppose that E- admits a complex connection V on E which pre¬ serves the norm F invariant under the parallel displacement. Then there exists a Hermitian metric gp on E such that V is compatible with gF. Proof: Let H be the holonomy group of V with reference point z € M. Since V preserves the Finsler structure F invariant, H is a subgroup of G. Then we define an inner product •) on Ez by 'g for an arbitrary inner product («, •) on Ez and a bi-invariant Haar measure dg on G. Then we have [ (.g(JO>g(Jy))dg= [ (J(g$,J(gri))dg = i {gi,grf}dg = &ri)s. J G *G *G Thus (-,‘)x is a Hermitian inner product on Ez. By definition, (•,-)• is G- invariant and since H C <7, it is also H-invariant. Hence, by the help of parallel displacement with respect to V, we can extend the inner product (•, •)- to a
50 Aikou Hermitian metric gp on E. It is trivial V is compatible with respect to pF- Q.E.D. Suppose that (E, F) is modeled on a complex Minkowski space. Then, by Lemma 2.3, there exists a Hermitian metric g? on E such that DE = is compatible with pF, i.e., dgF^t ri) = 77) 4- 9f(£, V77) for G A(E). Since DB is (l,0)-type, V is also (l,0)-type and so the Hermitian connection of pF- Consequently we have a generalization of Szabd’s theorem ([66]) to complex case. Theorem 2.7. We assume that a convex Finsler vector bundle (F, F) is modeled on a complex Minkowski space. Then there exists a Hermitian metric qf on E such that V is the Hermitian connection of qf* Concerning to Finsler metrics modeled On a complex Minkowski space, the invariant group G defined by (2.50) is contained in the unitary group C7(r). From Proposition 2.5, we have Theorem 2.8. Suppose that a convex Finsler vector bundle (E,F) is modeled on a complex Minkowski space. Then, if the invariant group G is isomorphic to the unitary group U(r), then (E,F) is a Hermitian vector bundle. 2.5.2 Flat Finsler Vector Bundles In this section, we shall define the flatness of a Finsler metric, and characterize it by vanishing of the curvature 12s of complex Finsler connection DE of (B, F). Definition 2.8. A complex Finsler metric F is said to be flat if there exists an open cover {£/, (ea)} of E such that the fundamental function F relative to {¿/»(ey)} is independent of the base point z € M, i.e., F$ — F^(£). Such an open cover {ZY, (et/)} is said to be adapted. If F is fiat, the coefficients in (2.2S) vanish in an adapted coordinate system, and so the non-linear connection h& is fiat, and the Bott connection DE associated with h$ is flat. Then the Kahler fibration ve : E M with pseudo-Kahler metric y/^lddF is locally trivial.
Complex Finsler Geometry 51 Theorem 2-9, ([13]) A convex Finsler metric F is flat if and only if its complex Bott connection DB is flat. Proof: We suppose that DE is flat. Then» by Theorem 1.1, there exists a coordinate system (z*, £*) on E such that = 0 with respect to this coordinate system. In this case, from (2.28), we get (PF/dz^dtf — 0. Then we get dz* ’ and thus' F is independent of the base point z G M. To complete the proof, we shall show that such a coordinate system is adapted, i.e., such a coordinate system is obtained from a change of local frame field eu &u on each U. If we denote by (za, C) —► (za,l^(z,^)) the associated change of coordinate on C7, from (2.26), the corresponding transformation of the coefficients of h$ is given on each open set U. Since h$ is trivial, we may assume — 0, and so we have iVj = №i/dza. Since N& is homogeneous of degree 1 relative to the valuable £ and !₽* are holomorphic, we have i.e., there exists a holomorphic function Aj :U —> GL(r, C) such that = £ Ay (*)$*• The change of local frame field is given by ej = 53eMy(s). Q.E.D. A flat Finsler metric F on E is a special class of Finsler metric modeled on a complex Minkowski space. Since DB is given by the pull-back of the Hermitian connection of the associated Hermitian metric if V£ is flat, the associated Hermitian metric g? is also flat. Hence we have Corollary 2.2. A convex Finsler vector bundle (E, F) is flat if and only if it is modeled on a complex Minkowski space and, moreover its associated Hermitian metric gp is flat. Since any flat Hermitian metric is a flat Finsler metric, from Proposition 4.21 of [39], we have Theorem 2.10. ([6]) Let E be a holomorphic vector bundle ofrank(E) = r. The following conditions are equivalent. 1. 7r: E —> M is a locally trivial Kohler fibrations. 2. E admits a flat Hermitian metric.
52 Aikou 3. E is defined by a representation p: tfi(M) —► D’(r), i.e.} E MxpCr, where is the fundamental group of M and M is the universal covering of M. 2.5-3 Projectively Flat Finsler Vector Bundles Let P be the GL(r, C)-principal bundle associated to E. We denote by PGL(r, C) the projective linear group GL(r,C)/CIr} where Clr is the center of GL(r, C). We denote by p : GL(r, C) —> PGL(ry C) the natural projection and by p! : ^Z(r,C) —* pgl{r.C) the Lie algebra homomorphism. Any connection V on E with curvature Q induces a connection on the PGL(i\ C)-principal bundle P = P/QIT, and the curvature of the induced connection is given by //(/?). A vector bundle E is said to be projectively flat if P is provided with a fiat structure, and so E is projectively if and only if E admits a connection whose curvature Q is of the form Q = a 0 I for a complex 2-form a (cf. [39]). A Hermitian metric g on E is said to be projectively flat if its curvature form is also of this form. In [50], it is noticed that a Hermitian metric g on E is pro¬ jectively flat if and only if it is locally conformal-flat. In this section, we want to investigate an analogy of Finsler geometry to Hermitian geometry. Let (F, F) be a convex Finsler bundle. Then, by Proposition 2.8, the project¬ ive bundle 7Tp(j5) : P(F) —► M is a Kahler fibration, and the vertical subbundle admits a Hermitian metric (•, ’)p(jE) defined by (2.22) in which each fibre P(EZ) — F*“1 is a Kahler manifold with a Kahler metric = a2 log f (2.51) Corresponding to Definition 2.8, it is natural to define the projective flatness of F as follows. Definition 2.9. A complex Finsler metric F is said to be projectively flat if there exists an open cover {14, (6^)} of E in which the Hermitian matrix G# is a function of fibre point [£], i.e., G$ = <?#($). Since each fibre P(F~) P1"”1 is compact and any pluri-harmonic function on P(F^) is constant, by this definition and (2.51), F is projectively flat if and only if logF(z, ¿) - depends only on fibre point i.e., = F(f) (2.52) for some local function on each V € U> Hence the projective-flatness is equivalent to the local conformal-flatness (cf. [10]). This equivalence in Hermitian geometry is remarked in [50].
Complex Finsler Geometry 53 We also characterize the projective-flatness by the flatness of the projective Bott connection Dp(E\ We assume that F is projectively flat, i.e., F satisfies (2.52), Then, the definition (2.28) implies (2.53) i.e., TV = dcr{z) ® E which is of the form (2,39). Hence, by Proposition 2.13, the projective Bott connection is flat. Conversely we assume that is flat. By Proposition 2.13, the connection hg is defined by in (2.39). Therefore the Bott connection DE is given by and thus (F, F) is modeled on a complex Minkowski space. Then, by Theorem 2.7, there exists a Hermitian metric gp on E such that or — g^dgp. Hence we have crc^z) = -tr^d^), i.e., _ 1 â log det (pf) r dz* If we put cr(^) = 1 log det (<7r) on each U € W, the metric F = e~a^F is flat on {7, i.e., F is locally conformal-fiat. Therefore, corresponding to Theorem 2.9, we have proved the following. Theorem 2.11. A convex Finsler metric F is projectively flat if and only if its projective Bott connection is flat. Corresponding to Corollary 2.2, we have Proposition 2.18. A convex Finsler metric F on E is projectively flat if and only if (E, F) is modeled on a complex Minkowski space and its associated Her¬ mitian metric gp is projectively flat. By this proposition, if a convex Finsler vector bundle (E, F) is projectively fiat, then its associated Hermitian metric gp is also projectively flat, and so ac¬ cording to Proposition 2.8 in [39], the bundle E is projectively flat. Moreover P is defined by a representation p : tti(M) —> FiZ(r), where PU(f) = U(r) is the projective unitary group. This means that, if we consider the universal covering space M as a %i(M)-principal bundle M —► Af, the bundle P is defined by the representation p : iri(Af) —> PU(r). The flat structure of P is induced by the natural flat structure of M —► M. Conversely, any projectively flat Hermitian metric is also a projectively flat Finsler metric. Hence we have(cf. Proposition 4.22 of [39])
54 Aikou Theorem 2.12. The following conditions are equivalent: 1. : P(E) —> M is a locally trivial Kahler fibration. 2. E admits a projectively flat Hermitian metric. 3. The bundle P = P/Q*Ir is defined by a representation p : 7Ti(M) —► PU(r): P*MxfiPU(r). Example 2.7. (cf. [10]) Let M be a so-called Hopf manifold {Cn - 0}/Aa, where A a is the group generated by the holomorphic transformations (z1,— ,zn) (Az1,— ,Azn) on Cn - {0} for A e C, 0 < |A| < 1. Then M admits a standard Hermitian metric, say Boothby metric ds2 = -A y dz° ®dza = e-’osIM2 V dza ® da“. Hall2 This metric is locally conformal Kahler-flat (cf. [69]), and thus it is also pro¬ tectively flat. Its Hermitian connection is given by w = -5(log||z||2)®Z. (2.54) The norm function defined by the metric above is -Fb(s,f) = e~108 H'H* ||£||2. To obtain a projectively flat Finsler metric F, we shall modify Fo into the form F(z,i) = e-lo®»--«7(i) (2-55) for an arbitrary convex Finsler metric /(£) on C71. Since F is also invariant by the action of A^? it defines a convex Finsler metric on 7m. This Finsler metric is of the form (2.52), and thus F is also projectively flat. We shall compute the non¬ linear connection in (2.28) to make sure. Because of F$ = *“los^a/<$(£) and the definition (2.28), we obtain in the form (2.53): fji _ dloghl|2-fi 1 “ dza * ’ and (1.17) implies that the connection coefficients of DTm are given by h ■_ aiog||g||2H “ dza f Hence the connection form w of DTm is given by (2.54). The associated Her- mitian metric gp is given by the Boothby metric. □ Let L be an arbitrary holomorphic line bundle over M with a Hermitian metric pr. Since ]?(£ 0 E) = P(E) and L(L 0 F) = 0 L(£), we can define a complex Finsler metric on the product bundle L 0 E. Let {Luv}
Complex Finsler Geometry 55 the transition functions of L with respect to an open covering {U, ¿a}- For V£ = Etoto ® $Ui G A(L 0 E), we define its norm ||C||£0aB by IlClkos — IltolL IlCalls = ot/WF(i,Cy), where we put aa(^) — Pi Oto »to) and to = € A(U,E). Since ay = aa|Lav|2 and Qj =» Lav£EavJ$r, t^b“8 definition is well defined. Hence gi, • F — {ay • F} defines a Finsler metric on L 0 E, If {ay • F} defines a fiat Finsler metric on L ® E, then F is locally conformal-flat, i.e., projectively flat. . Now we shall consider the converse. We suppose that a holomorphic vector bundle E admits a projectively flat Finsler metric. We shall investigate what conditions imply that E 0 L admits a flat Finsler metric for a holomorphic line bundle L. A cohomology class c € №(M,1R) is said to be integral if c € J"E2(M,Z) for the induced map J* : #2(Af,Z) —► if2(M,R) obtained from the inclusion j : Z R. The following lemma is well-known (cf. Lemma 2.36 in [64]). Lemma 2.4. Let £ be a closed real (1, on a complex manifold M. If the de Rham class [£] G H2(M} R) is integral, then there exists a Hermitian line bundle L over M with the curvature QL such that For rank(E) — r of E, we say c is integral (mod r) if |c is integral. The first Chern class ci(E) is integral (mod r) if and only if there exists a line bundle L satisfying ci (L 0 E) = 0. Then we have Theorem 2.13. Let E be a holomorphic vector bundle with rank(E) = r (> 2). If L®E admits a flat Finsler metric for a holomorphic line bundle L, then E admits a projectively flat Finsler metric. Conversely, we suppose that E admits a projectively flat Finsler structure. Ifci(E') is integral (modr), then there exists a holomorphic line bundle L such that L®E admits a flat Finsler structure. Proof: We shall prove the second part of the theorem. We suppose that E admits a projectively flat Finsler structure F. Then, by Proposition 2.18, there exists a Hermitian metric g? on E such that the curvature is given by = 0 Ie for the Ricci curvature p = dd log det pF of gp. The first Chern class ci(E) is given by If ci(E) is integral (mod r), then — ^ci(E) is integral, and so by Lemma 2.4, there exists a Hermitian line bundle (L, g£) with curvature
56 Aikou such that Px, = (57,). The complex Finsler metric defined by the function p on the product bundle L ® E is fiat. In fact, since I2l®e = 1 ® P + Pl ® 7s = -p 0 Ie p 01# = 0, r r the curvature vanishes identically:Hence L® E admits a fiat Finsler met¬ ric. Q.E.D. Proposition 2.19. Suppose that a holomorphic vector bundle E over a compact Kahler manifold M admits a projectively flat Finsler metric. If ci (E) = 0, then E admits a flat Finsler metric. Proof: Since M is compact Kahler, the assumption ci(E) — 0 implies that there exists a smooth real function f(z) on M such that --p = ddf(z). For the trivial line bundle L = M x C with Hermitian metric g& = the product bundle L®E admits a flat Finsler metric FL®E « e^F. Since L is trivial, L®E~E. Hence e^F is a flat Finsler metric on E. Q.E.D. We shall investigate the condition for the projection 7tP(£;) : F(E) —* M is a Kahler submersion with respect to a Kahler metric Um on M under the assumption that the real (I, l)-form ZTp(s) is a Kahler metric on P(E). Definition 2.10. ([72]) A holomorphic submersion 7r : X —► M is said to be a Kahler submersion if X and M are Kahler manifolds, and % is a Rieman- nian submersion, that is, is an isometry at each point for the orthogonal complement Hx — (Va,)_L- Watson[73] proved that, if 7r : X —► M is a Kahler submersion with the horizontal subbundle ?ip(E) » then is integrable and totally geodesic. Now we suppose that a holomorphic vector bundle : E —> M admits a convex Finsler metric F which defines a Kahler metric ZTp(£) — V^ïââlogF on P(E). (By Theorem 2.3, if M is compact, this assumption is equivalent for E to be negative.) Then, from (2.42) dälogF = + A
Complex Finsler Geometry 57 defines a Kahler metric on P(£). The horizontal bundle 'Hp(e) = dp (7is) locally spanned by {dp(XQ)} is the orthogonal complement of Vp(£) with respect to i-e., ?fp(E) = (V?(.E)) • Moreover we suppose that F is projectively flat. Then, by Proposition 2.18, there exists an associated Hermitian metric gF on E which is also projectively flat. We denote by Rap its Ricci curvature: _ ^logdet^j?) “ dzadz& * Then the curvature R^a$ of DB is given by and the curvature is given by - p E^J Since Jp^ is negative-definite, gF is a Hermitian metric with negative Ricci curvature Rogiz). Then gF determines a Kahler metric g = (— on M, and 7Tp(jb) : (P(E),7Zp(£j) —> (M,p) is a Kahler submersion. Proposition 2.2Ö. Let F be a convex Finster metric which defines a Kahler metric Æp(£) = lddlogF on ]?(£?). If F is projectively flat, then the base space M is also a Kahler manifold and the projection kp(e) a Kahler submer¬ sion. Conversely we suppose that H^E) ~ -/^lââlogF is a Kahler metric on P(^) and the projection 7FP(£) is a Kahler submersion. Then &aß = —gaÿ(z) for a Kahler metric Um = g^dz* A dz& on Af. By Watson’s theorem, the horizontal subbundle Hp(E) is integrable, and thus R?a& is of the form R^ß = Then we have = IEFÄ^ = = A<& Hence we have A^p = and Rï^ — -gapc\ Then (2.34) implies - E^^F- If (E, F) is modeled on a complex Minkowski space, i.e., R^ = 0, then we have Rjctß
5S Aikou and the associated Hermitian metric gp is also projectively fiat. From Pro¬ position 2.IS, the Finsler metric F is projectively flat. Consequently we have Theorem 2.14. Let F be a convex Finsler metric such that (E, F) is modeled on a complex Minkowski space. Assume that Zrpçs) = V—TddlogF defines a Kahler metric on P(jE). Then the projection is a Kahler submersion if and only id F is projectively flat.
Chapter 3 Kobayashi Metrics In this section, we shall recall some properties of Kobayashi metric which is a typical example of complex Finsler metrics. It is, well-known that, for a strongly convex domain with a smooth boundary in Cn, its Kobayashi metric is a strongly convex Finsler metric (cf. [47]). 3.1 Poincaré Metrics First of all, we recall some facts on the Poincar6 metrics on the unit disc' A, upper half plane IE and the punctured disc A*(cf. [45] and [22]). Let M be a Riemann surface with a Kahler metric ds2 — 2g(z)dz ® dz. The holomorphic sectional curvature Hg at (z, f) € Tm is given by H3M = 1 d2 g(z) dzdz (3.1) Hence the sectional curvature of a Riemann surface is independent of the dir¬ ection £ and is just the Gaussian curvature Kg(z) of For the disc A(r) {£ e C | |C| < r} of radius r > 0 in C71 centered at the origin, the Poincaré metric pù(r) is defined by SA(r)’22(r2-Kl^®^ (3.2) Because of a2 ( > \ = 2r2 g ^2 p - KI2)2/ (r2 - KI2)2 ’ we have Hg± (z, f) = —4, that is, the holomorphic sectional curvature Hg± is negative constant. This shows that (A(r),pA(r)) is a hyperbolic manifold. 59
60 Aikou Remark 3.1. On the unit disk A = A(l), if we define a holomorphic map <p : A —► A(r) by ¥>(f) = then we have which shows that <p : (A, g&) —► (A(r),£A{r)) is a holomorphic isometry. □ For any two pints £ and 77 in A, we take a C'1-class curve 7 : (0.1) —> A such that 7(0) — £ and 7(1) =77. If we denote by ||*y(t) || the norm of tangent vector 7(t) with respect to g&, then its hyperbolic length La (7) is defined by ¿AW-/1 midt Jo (3.3) The hyperbolic distance ¿^(£>77) between Ç and 77 is defined by dù(Crç) = inf La (7), (3.4) where the infinimum is taken over for all C1 -class curves connecting £ and 77. For. a holomorphic map <p : A —> A, the following lemma is well-known. Proposition 3.1. (Schwarz-Pick Lemma) Let A be the unit disk with the Poincare metric g&. Then every holomorphic map <p : A —> A is metric¬ decreasing <p*g& < g±: 19/(01 < 1 i-WOI2" 1-KP (3.5) for all Ç € A, and thus <p is distance-decreasing <p*d& < d&: ^(^(Ç),^))<dA(Cî7) (3.6) for allCv e A. Let Aut(A) be the group of automorphisms of A: Aut(A) = |v(0 = Then we have Proposition 3.2. The Poincaré metric g^ on ïh is Aut(ty-invariant, that is} <p*g& — g& for all ip € Aiit(A), and thus we have d±(ip(C),<p(7i) — ¿aÎC7?)* Let 7(i) = r(t)^^ : (0,1) —► A be a C1-class curve connecting the origin 7(0) = 0 and 7(1) = a. Then, because of
Complex Finsler Geometry 61 we have ¿a(0, Ct) = | log and thus (3-7) l + l<*l The equality holds if & = 0 and r(t) > 0. In this case such a curve is a shortest curve between the origin and a, called a hyperbolic geodesic. If we take an automorphism <p e Aut(ti) such that y>(f) = 0 and <^(77) =* a, we have dA(C»7) = di(0, a) = I log|^> r = l-<7? (3.8) Then we can show that, for a sequence {2^} in A, the condition d&(zn,z) —> 0 (n —► 00) if and only if ► 0 (n —> 00), that is, d& and | • [ induce the same topology on A. Moreover it is shown that the metric space (A, ¿a) is complete. Since the holomorphic sectional curvature is negative constant —4, we have Proposition 3.3. The Poincaré disc (A, Pa) is a complete hyperbolic space. Let El = {z € C| Im(.s) > 0} the upper half plane of C. Let <p : El —► A be a bi-holomorphic mapping defined by z + a For the Poincar6 metric g± on A, we put g& — <p*g±. Then we have 5H - 1 l<fel2 4(Im(s))2 1 (l-l^)l2)2 The metric gu is also called the Poincar6 metric on EL The automorphism group >4itt(E[) of BI is given by Aut(H) = (a. b,c, d e R, ad - be = 1 | a PSL(2, R). 1 cz 4- d J Then, it is easily shown that g& is Aut (El)-invariant. Moreover (El, g^) is Cauchy complete and its holomorphic sectional curvature is also Bh = —4. Now let : BI —► C a holomorphic mapping defined by ^(z) = Since |^(z)| = |e~27ry| for z = x + (y > 0), we have 0 < |^(z)| < 1, and thus is a holomorphic mapping from El on to the punctured disc A* = A\{0}.
62 Aikou If we define an action of Z on H by z —> z+n (n G Z). Then we have JK/Z A*, that is, : El —> A* is a universal covering map. Then there exists a Hermitian metric 0a- on A* such that ip* g&s — g^, In fact, if we put w — e“2*v/zT~j then we have z = (log |w| + v^largw), and thus W) »~j£log|w|. 1 dw w ‘ dz = Hence, if we put ~ (log |w|2)2 I"“ ’ we have = <7h- This metric g&* is also called the Poincard metric on A*. We also see that (A*, <?£*) Is complete and its holomorphic sectional curvature is Has = —4. 3.2 Kobayashi Metric Let M be a complex manifold. A typical example of (pseudo) Finsler metric on its holomorphic tangent bundle Tm is given by Kobayashi metric. We recall the definition(cf. [22], [35], [41], [46]). We denote by A(r) C C the disk of radius r > 0 centered at the origin. Lemma 3.1. For any point p € M and any vector vP G T?tQM, there exists a holomorphic map f : A(r) -+ M satisfying /(0) » p and (3.9) for sufficiently small r > 0. Proof: Let p € U c M and denote by <p(z) — (z1, • • < , zm) a local coordinate on Z7 such that z*(p) = 0 (i ® 1, • • • tm). We denote *vp G by vp = X} sJ (d/d^)p f°r <* € C. Let : C -+ C”1 a holomorphic map defined by ^(A) = (A^1, • • •, A^™). If we take sufficiently small r > 0, we have ip (A(r)) C Then, if we define a holomorphic map f : A(r) —> M by /(f) := y?“1 o ^ta(r)(C)i we have /(0) = p and it satisfies (3.9). Q.E.D. The norm of the vector G T0A(r) at the origin £ = 0 is given by |[(d/^C)ollA(r) 555 r“1. We denote by Hom(A(r),M) thy space of holomorphic maps from A(r) to M. The Kobayashi metric is defined as follows.
Complex Finsler Geometry 63 Definition 3.1. The Kobayashi metric kw : Tm —> R of a complex manifold M is defined by kM (p, v) := inf | 3f € Hom (A(r), JW), /(0) = p, /. (0) = vp for v(p,t>p) G Tm, where the infinimum axe taken for all f 6 Hom(A(r),M) satisfying /(0) — p and /* (0) = vp. The Kobayashi metric kM is the maximum metric among the pseudo metrics which satisfy the decreasing principal (cf. [41]). In fact, if H is a pseudo-metric satisfying the decreasing principal, for the PoincarS metric pa(r) defined by (3.2) on A(r)S we have f*H < g^ry Then, since for v(p,v) G 7m there exists a holomorphic map f : A(r) —> M such that /(0) = p and A((®/^C))o = v, we have S(p,V)2 < gw ((Wo, (Wo) = Thus we have H < It is obvious that kM is absolutely homogeneous of degree 1, that is. for vAgC: fcM(p,A«) = |A|fcAf(p,v). But kM is not the Finsler metric in our sense. It is known that kM is upper semi-continuous, that is, for VX G Tm and ve > 0 there exists a neighborhood U of X such that kM(Y) < kM(X) + e for all Y G U (cf. [41], [46]). This metric is important notion on complex manifold, since the following Decreasing Principal holds. Theorem 3.1. ([41]) Let M and N be complex manifolds, and <p : N -* M a holomorphic map. Then k^ > tp*kM holds, that is, for all X G 7\ we have M*) > kM&*(XYL (3.10) 7/ 9?: TV —> M biholomorphic, then by = <p*kM> Proof: For any vp G T*'QN, we take a map f G Hom(A(r),7V) satisfying /(0) = p and (3.9). Then <p o f G Hom(A(r), M) satisfies (p o /(0) = y?(p) and (p ° /)* (0) = ^*(^p). Hence we have kM < i. r Since f is arbitrary, we have kw(yp) > kM(<p*(yP)). Hence we have (3.10). If <p is bi-holomorphic, we apply (3.2) to y?”1, we have kf/typ) = kN («;>))) < kM (V’.i®?)) > and thus we have = <p*kM*
64 Aikou Q.E.D. By this theorem kM is Aut(M)-invariant, and so depends only on the complex structure on M. In this sense, Kobayashi metric kM is an invariant metric on M. By the Decreasing Principal, the following is also obtained. Proposition 3.4. Let M and M be complex manifolds, and 7T : M —> M an un-ramified covering map, Then we have k^ — kM* For an arbitrary vector V on M, there exits a (1,0)-type vector X such that V = X + X = 2ReX. Then we define kM(V) := 2kM(X). Let c : (0,1) —> M be an arbitrary curve of C^-class on M. Then, the length L(c) with respect to kM is defined by Then for two point p, q € M, Kobayashi pseudo-distance d^(p, q) is defined by pseudo-distance also satisfies the decreasing principal (cf. [41]): <$(?,«) > (?(?)> ^(«)) for an arbitrary € Hol(N, M). Definition 3.2. A complex manifold M is said to be Kobayashi hyperbolic if the pseudo-distance is the distance in the strict sense. There exists a criterion for Kobayashi-hyperbolicy. Theorem 3.2. ([53]) A complex manifold M is Kobayashi-hyperbolic if and only if there exists a positive junction c(z) and a continuous Finsler structure F on Tm satisfying (3.H) for all e Tm-
Complex Piaster Geometry 3.3 Bounded Domains 65 In this section, we are concerned with completely circular domains in Cn. For a completely circular domain V c Cn, its Minkowski functional mp is defined by (2.9) and its fundamental function /p is defined by (2.10). If V = Cn, then mp = 0. Hence, in the following, we always assume that T> is bounded. Especially, if T> C Cn is bounded, then it will be proved that D is Kobayashi- hyperbolic. The following is well-known. Proposition 3.5. (cf. [35]) Let T> be a pseudo-convex and completely circular domain in Cn. Then we have m^X) — fcp(0,X) for € (3b)o, ie., the indicatrix of & at the origin coincides with the domain D. Proof: We suppose mp(X) / 0. Then a holomorphic map (p : A —► T> is defined by Since <¿>(0) — 0 and y>'(0) = X/m^X), we have M0,X) = inf {o > 0| y>(0) = 0, a^'(0) = X} = inf {^(X)} < m^X). Hence we have k&(X) < m©(X). Conversely, for ip e Hol(^T>) satisfying <^(0) = 0 and o<p'(0) = X (a > 0), if we define (p € ifo^AjC71) by ^(A) ® A£(A), we have y?'(0) = £(0). Since ^?(A) € we get mp (^>(A)) = lAjm© (<£(A)) < 1, and thus we have sup (mp o <p) = 1. A—»SA On the other hand, since T> is pseudo-convex, d2 (mp o <p) (Pm-D 0<pl f dip™ \ n dxox ~ dxldxm dx\dx )~ This shows that m© o <p is subharmonic, and thus from Maximum principal we havempo^ < 1. Consequently we have mp(X) = mp (ay>'(0)) = ccm-o (£(0)) < a. Then, by definition, ^(X) < inf {a > 0| p(0) = 0, c^'(0) = X} = fcp(0,X), that is, m©(X) < &d(0,X), Q.E.D. We suppose that the automorphism group Aut(Z>) of T> acts on V transit¬ ively, that is, the domain D is homogeneous. Then for an arbitrary z 6 T> there exists some e Aut(P) such that <p(z} = 0, and we have fe(z,X) = k^(z)^(z)X} = to(0,^'(^)X) = 77lz>(/(5)X) for an arbitrary X € TZT>.
66 Aikou Example 3.1. (cf. [35]) The unit ball Bn of Cn is given by Bn — {(z1, — , ;zn) € C1; ||z|| < 1}. The Kobayashi metric F^ of Bn is given by 11*11 -M2 2 l(s,*>l2 > where (•. ■) is the standard Hermitian inner product on Cn. Especially (0, X) = 11*11- □' For a pseudo-convex and complete circular domain 2? C C71, the following representation of Fg is given by Suzuki[65]: M0J)=W (3J2) where R(X) is the radial of the disc D n lx, here lx is the complex line CX in direction X through the origin. By using this formula, we shall show the following proposition. The proof of it is also due to [65]. Proposition 3.6. A pseudo-convex and complete circular domain 2? c Cn is Kobayashi-hyperbolic if and only if V is bounded. Proof: We assume that 2? is not bounded. We take a sequence {zn} in 2> such that lim ||zn|| = oo. Then, if we put Xn = zn/ ||zn ||, then we have M0,*„)=£^-y, since ||Xn || = 1. Since 2? is complete circular T> is defined by < 1. Then, by Proposition 6.4, we have ||3n|| /R(zrf) — Zcp(0,^n) = mvfzn) < 1. Because of R(Xn} = R(zn)t we have R(Xn) > ||zn||. Then we have M°’x») = K(k)<liZii->0(n->oo)- This shows that 2? is not hyperbolic. Hence, if V is hyperbolic, then 2? is bounded. Q.E.D. Here we shall recall some facts due to Lempert[47], Let T> be a strongly convex bounded domain with C^-class boundary (fc > 6). A holomorphic map f : A —> T> is said to be extremal with respect to X e TPD if it satisfies the following conditions:
Complex Finsler Geometry 67 (1) /(O)=p, (2) there exists a real number A > 0 such that /'(0) — XX, (3) for every holomorphic map g: A —> 2? such that $(0) = p and gf (0) = ¡¿X, the inequality p>< A holds. Then Lempert showed that for every X 6 TpP, there exists a unique extremal holomorphic map Jx : A —* which is C*“4-class with respect to X. Then the map (A £ C\{0}) is extremal with respect to XX, and thus we have Then the Kobayashi norm Zcp(p,AT) of X G Tp2?\{0} is given by fx(ty = v/fcz>(p,X),i.e., kofaX) - i = inf | ¿(0) = p, g'(0) -rX,ge Hol(/X,ty If 2? is a strongly convex domain with smooth boundary, then fcp(p, •): Tp2?\{0} - 1R is smooth and strongly convex. The indicatrix {X G TP7> | k&(p,X) < 1} C TPD = Cn is not bi-holomorphic to the unit ball with respect to the standard metric in Cn. 3.4 Holomorphic Sectional Curvature and Schwarz Lemma If the holomorphic sectional curvature Hs of a Hermitian manifold (M,g) is bounded above by a negative constant — k < 0, then for every (z, f) G Tm we have > y/k ||£||^, where ||f || is the Hermitian norm of £ relative to g. Hence, by Theorem 3.2, such a Hermitian manifold is Kobayashi-hyperbolic. Royden [57] defined the holomorphic sectional curvature Xf of a complex Finsler manifold (M, F) as a natural generalization of Hermitian case. In this section, we shall give a expression of Xf hi terms of the curvature tensor of the Bott connection DTm gtlE^Tm over (Af, P) (cf. [3]).
68 Aikou 3.4.1 Generalized Schwarz Lemma We suppose that a smooth Finsler metric F is given on a complex manifold. If a holomorphic map <p : A(r) —► M is given, we can define a conformal metric on the disk A(r) by ds2 = çj’,F(CX®dC, where we put /F(() = F(^(£),y?, (0)- The Gaussian curvature K#*? of ^*F is given by (3.1): (3.13) v _ 2 32logv"jF y*F ■ Then, we define the holomorphic sectional curvature of (M,F) at (z,£) € Tm as follows. Definition 3.3. The holomorphic sectional curvature /Cf(2,£) of F at (s,Ç) € Tm is defined by (3.14) £f(*}%) := sup{K^p(0)}, where supremum is taken all holomorphic map ip : A —► M satisfying y>(0) = z and ¥>W#C)o = In the following, we shall denote by F^ the square of Kobayashi metric that is, Fm = (&w)“. The following is the Schwarz Lemma for Finsler metrics: Lemma 3.2. Let F be a smooth Finsler structure o/Tm. If the holomorphic sectional curvature JCf is bounded by a negative constant —kf then we have 4F$ > kF Proof: (cf. [53]) The proof is given by the same as the Hermitian case. For the Poincae metric Pa(t) on A(r), we put _ 2(^)(C)P - KI2)2 ^^”r2/2(r2-|C|2)2 r2 The function is continuous on A(r) and /^(t) = 0 on dA(r). Hence there exists some Co € A(r) such that log^Co) = sup log(0- Since (^*F)(C) / C€A(r) 0, /¿r(C) is C°° near Hence, if we put C = x + y/^ly and Co = so + V—Uta, since « d2_ = 1 / a2 a2 \ d£<K 4 \&r2 + dy2 J and Mr(C) attains the maximum at Co, we have
Complex Fiosler Geometry 69 logMr(c) log^rxc) (j,2 _ ^2)2 implies (^1O6(^r)(<))0 ~ (^-ICbl2)2‘ On the other hand, from the assumption (3.15) 2 a2log(y-F)(<) <-fc we have *(<F)«o) < 2 (a^log^K))0 • Hence, from (3.15) and (3.16) we have (3.16) ^‘■F’)(Co)< (r2_|&|2)2> that is, Atr(fo) < 1/A. Consequently, for all r such that 0 < r < 1 and for all C e A(r), we have /¿r(C) < 1/&. For an arbitrary fixed £ e A(r), & (0) < (r2 ^|2)2 (3-17) implies < pA(r)- Hence kF(z, £) < 4/r2. By the definition of Icm* we have kF(z^< 4F$(z,C). Q.E.D. This lemma and Theorem 3.2 implies Theorem 3.3. ([38]) Let (M,F) be a complex Finsler manifold with the holo¬ morphic sectional curvature JCp which is bounded above by a negative constant —k. Then M is Kobayashi-hyperbolic. By this theorem, we have Theorem 3.4. If M is a complex manifold with negative Tm? then M is Kobayashi-hyperbolic.
70 Aikou Now we shall show a characterization of Kobayashi metric due to [24] and [541. Let F be a complex Finsler metric on M. A holomorphic map <p : A —*■ M is said to be F-extremal for (z, € Tm if it satisfies y>(0) = z and Proposition 3.7. Let (M, F) be a complex Finsler manifold. If its holomorphic sectional curvature JCp is bounded above by negative constant —4 and there exists a F-extremal for any (z, Ç) € Tm, then F coincides with the Kobayashi metric Fm* Proof: By assumption, for any (~,s) £ Tat, there exists a holomorphic map (p : A —> M such that y?(0) = z and y?*(0)F(z,i) = Then, by the definition of kjiif we have = inf {)A|; ¥?(0) = a,A<p*(0) = C} < |A| hence we have Ffâ < F On the other hand, by K>k < -4 and Lemma 3.2, we have F$ > F. Consequently we have F = F$. Q.È.D. Remark 3.2. The assumption of the existence for F-extremal can be replaced by the existence of complex F-geodesic in the sense of Vesentini. A holomorphic map <p : A —> M is said to be a complex F-geodesic if d&(a, b) = dw(^(a), ¥?(&)) for any a,b e A, where d& (resp. ¿m) is the distance defined by the Poincare metric on A (resp. the Finsler metric on M). In fact, by the assumption on JCf, we have F < F^, and djtf(p,g)<inf [ y/Ftftffidt < [ y/Ftfttydt < f fcikrfr'(*))<&• J q. J a I a The decreasing property for kM implies g& > <p*F^, and so J ^m(7 (t))dt < J — |r=t dt ¿a (ct, 5). Since d^a.b) == ¿Af(p,g) for a complex F-geodesic between p = p(d) and q = <?(&), hence we have Moreover, since > <p*F^ > ^>*F, we have g& « F. Consequently is an isometric along the interval (a,&), from which we have Fj$ < F. □
Complex Finsler Geometry 71 3-4.2 Holomorphic Sectional Curvature by Curvature Tensor The holomorphic sectional curvature £> is an important notion in Finsler geo¬ metry as shown in Theorem 3.3. We need a computational expression of The holomorphic sectional curvature of a Hermitian manifold is expressed by its curvature tensor. In the case of convex Finsler metric F at least of C3-class, we also have the explicit expression of JCp by its curvature (cf. [1]. [3], [76]). Let F be a convex Finsler metric on Tm with F^ = &F/d£d£?. The canonical splitting Nj in this case is given by Nj = E (3.18) This splitting is called Chem’s non-linear connection in [1], which is exactly the one defined by Rund[58] and Royden[57], The Bott connection DTm is given by the (1,0)-form wj = 2 withl For convenience in local computation, we shall introduce the normal coordin¬ ate system on complex Finsler manifold. A complex coordinate system is said to be normal at P if it satisfies 1. ^(P) = ^, 2. r;fc(P) = o. Proposition 3.8. For an arbitrary point Pq = (zo,£o) € Ты, we can always choose a complex coordinate system {tt“1(Z7), (5.$)} around Po satisfying ljfc(Po) + r4.(Po)=O (3.19) Proof: For a given complex coordinate system {17, (2)} on M, we define a new coordinate system {E7, (5)} around zq by P = (? - 4) - 5 E ^(W - 4)(*‘ - 4)- Then, we have (3,19) with respect to the new coordinate {17, (5)}. Q.E.D. A local coordinate (тг“'1(С7), (з4,О) around P € Tm said to be semi-normal at P if the Bott connection DTm satisfies (3.19) at P. By Proposition 3.8, in a complex Finsler manifold (M, F), for an arbitrary point P € Tm, there exists a semi-normal coordinate around P.
72 Aikou Remark 3.3. The Kahlerity of Hermitian manifolds will be generalized to complex Finsler manifold. We remark that, in [3], we used the term Finsler- Kahler manifold^ although in [1], it is called a strongly Finsler-Kahler manifold. We define a fundamental form S by the horizontal (1, l)-form By direct calculations, we see that dS = 0 if and only if F$ is a Kahler metric on M, A convex Finsler metric on M is said to be Finsler-Kahler if it satisfies dhE = 0. By direct calculations, we can show that a complex Finsler manifold (M.F) is Finsler-Kahler if and only if DTm is symmetric, that is, Tjk = Tjj. Since at each point P e 7m we can alway choose a semi-normal coordinate around P, we have an intrinsic characterization of Finsler-Kahler manifold: A complex Finsler manifold (M, F) is Finsler-Kahler if and only if around each point P 6 7m there exists a complex coordinate system which is normal at P. □ Now we shall investigate the holomorphic sectional curvature of Finsler man¬ ifolds. We define a function Hf by _ 2 V" f'V _ ^~Fi3 \ eicjckct ~ f(z, e)2 ¿A2-' dzi dz^ds1 P'*'' • From the last equality, we see that if is a Hermitian metric on 7m, Hf is just the holomorphic sectional curvature Hg of (M, g) (cf. [SO]). We prove the following theorem: Theorem 3.5. ([3]) The holomorphic sectional curvature lCp(z,£) € (7m) - coincides with the function HfM' Proof: Since JCp(z,X£) = £f(z,£) and Hp(z,Xty — we may assume F(s}£) = 1. In the following, we put <f>*F — E. Since we always choose a coordinate ( on A satisfying (dE/d^)^Q — (dE/d^\=Q = 0, the sectional curvature JCp(z,g) is defined by
Complex Finsler Geometry 73 By direct calculations, we have =E +E (¿o+E ^) - & where we put — (d2q?/âÇ2)çz=o. Form this equation, we have HF{z,^ = K^F{z^) + 21 £ (X + E • <3-2°) Hence Hp > K^f for all 92. If we take a semi-normal coordinate around the point (z,£) e Tm and 92 : A —► M as a complex line y>(C) = (z1 + Ci1, • • • ,zn + £$*) through the point, then the second term of (3.20) vanishes, and so the sup {Kp.f} attains to the maximum J?f(z, £). Consequently we have Hp — ICf* Q.E.D. Remark 3.4. By the Hermitian form № defined by (2.43), the holomorphic sectional curvature Kf is given by JCf = F(z,f)_1 s • Hence, by The¬ orem 3.3, if M is a compact complex manifold with negative Tm, then M is Kobayashi-hyperbolic. Let (M, F) be a complex Finsler manifold modeled on a complex Minkowski space (V,/). Then, by Theorem 2.7, there exists a Hermitian metric gp on M associated with F. In (V, /), the indicatrix T> = {£ € V; /($) < 1} is a bounded and strongly pseudo convex domain. And so there exists a unique Euclidean sphere S centered at the origin inscribed about the indicatrix T>, If necessary by multiplication by constant, we may assume that S is the indicatrix of gF- Let fM bo the function on Tm defined by fM = 2 for Since S is inscribed about 2? we have (3.21) Moreover, if the sectional curvature ICp is bounded above by a negative constant —k, Lemma 3.2 implies 4F$ > kfw- Consequently we have Proposition 3.9. ([5]) Let (M, F) be a complex Finsler manifold modeled on a complex Minkowski space. If the holomorphic sectional curvature JCp is bounded above by a negative constant —k, the we have the following inequality: 4F$ — kfM kF. (3.22)
74 Aikou As an application of Proposition 3.7, we have a characterization of unit ball: Theorem 3.6. ([8]) Let M be a simply, connected complete complex manifold with a convex Finsler metric F. Suppose that (M,F) satisfies the following conditions: 1. For each (z,£) € Tm, there exists an F-extremal. 2. Its holomorphic sectional curvature is negative constant —4. 3. (M, F) is modeled on a complex Minkowski space, and its associated hp is Kahler. Then (M,F) is holomorphically isometric to the unit ball with the standard metric in Cn. Proof: By the first and second condition, the given Finsler metric F coincides with Kobayashi metric F^. Then (3.22) implies F = fM = F^. Hence, by the third condition implies the manifold is a simply connected complete Kahler manifold with negative constant curvature —4. By the well-known uniformiza- tion theorem of Kahler manifolds(cf. [43]), (M, F) is holomorphically isometric to the unit ball with the standard metric in Cn. Q.E.D. By Lempert [47], if T) is a strongly convex domain in Cn with smooth bound¬ ary, then its Kobayashi metric F% is a complete and strongly convex Finsler metric. Since strong convexity implies convexity, the Kobayashi metric Fp is a Finsler metric in our sense. Moreover F$ satisfies the first and second assumption in the theorem above (see [1] and [24]). Hence we have Proposition 3.10. ([8]) Let D be a strongly convex domain in Cn with smooth boundary. The following statements are equivalent. 1. T> is biholomorphic to the unit ball in Cn. 2. (P, Fpf) is modeled on a complex Minkowski space, and its associated Her¬ mitian metric is Kahler.
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PART 2
KCC Theory of a System of Second Order Differential Equations P.L. Antonelli and I. Bucataru
Contents 1 The Geometry of the Tangent Bundle 91 1.1 The Tangent Bundle . 91 1.2 The Vertical Subbundle 93 1.3 The Almost Tangent Structure 94 1.4 Vertical and Complete Lifts 94 1.5 Homogeneity 95 2 Nonlinear Connections 97 2.1 Horizontal Distributions and Horizontal Lifts 97 2.2 Characterizations of a Nonlinear Connection 99 2.3 Curvature and Torsion for a Nonlinear Connection 102 2.4 Autoparallel Curves and Symmetries for a Nonlinear Connection 103 2.5 Homogeneous Nonlinear Connection 107 3 Finsler Connections on the Tangent Bundle 109 3.1 The Berwald Connection Ill 3.2 The h and v-Covariant Derivation of a Finsler Connection 112 3.3 The Torsion of a Finsler Connection 113 3.4 The Curvature of a Finsler Connection 114 3.5 Finsler Connections Induced by a Complete Parallelism 116 3.6 The Cartan Structure Equations of a Finsler Connection 118 3.7 Geodesics of a Finsler Connection 120 3.8 Homogeneous Berwald Connection 121 4 Second Order Differential Equations 123 . 4.1 Semispray or Second Order Differential Vector Field 123 4.2 Nonlinear Connections and Semisprays 125 4.3 The Berwald Connection of a Semispray 127 4.4 The Jacobi Equations of a Semispray 129 4.5 Symmetries for a Semispray 131 85
86 Antonelli and Bucataru 4.6 Geometric Invariants in KCC-Theory 132 5 Homogeneous Systems of Second Order Differential Equations 135 6 Time Dependent Systems of Second Order Differential Equations 139 6.1 Sprays and Nonlinear Connections on Jets . 139 6.2 Variational Equations 144 6.3 The “Film-Space” Approach to Type (B) KCC-Theory 147 7 The Classical Projective Geometry of Paths 151 7.1 Paths, Parametrized Paths 151 7.2 The Various Geometries of Paths - Finite Equations . 152 7.3 The Various Geometries of Paths - Differential Equations .... 153 7.4 Affine Connections 155 7.5 The Fundamental Projective Invariants 158 7.6 The Projective Parameter and the Normal Spray Connection . . 161 7.7 Projective Deviation 165
KCC Theory 87 To the reader: Here are two suggested approaches for reading this document. One (I) in¬ volves reading Chapter 7, last, while the other (H) has it first. The bridge between (I) and (II) is Chapter 5. Thus, approach (I) including Chapter 5 is completed by reading Chapter 7, last. Approach (II) is to read Chapter 7, first, then read Chapter 5 to get an idea of a global perspective. Approach (II) is possible, for the most part, with only a calculus background. It has been successfully used in teaching a one semester course at the senior undergraduate level.
Introduction The modem geometry of a system of second order differential equations (SODE) was initiated in the 1920’s by Synge [26], Knebelman [16], Douglas [14] and the geometric invariants of a SODE were obtained in the 1930’s by Kosambi [17], Cartan [10], and Chem [11]. In their papers they considered a system of second order differential equations (l* tri1 « dsf, -¿p + 2<7(t, $, —) = 0, * G {1,n} (1) where (i, xz) are the local coordinates on a real (n-Fl)-dimensional fibred man¬ ifold 7T : M —> jR. The main problem they studied was to find the geometric properties one can associate to the system (1) that are invariant under the following groups of transformations: { £* = x^x*) OT The problem is not completely solved and in the last decade, there have been a lot of papers on the subject [12], [18], [20], [25]. On the other hand, properties of (1) which are invariant under arbitrary (smooth) parameter transformation on all solutions simultaneously, t — (pfra), where a = {ai,..-¡¿m} are a set of parameters which suffice to distinguish the different solutions among them¬ selves, combined with arbitrary transformations, a = a(a), of these, are called projective properties of (1). However, in this case, (1) is required to have Gi independent oft and (positively) homogeneous of degree two in the dx/dt vari¬ ables. This is the setting for classical projective geometry of (1), pedagogically described in Chapter 7. But, it is possible to extend these and other classical results to systems occurring in engineering, biology and physics, etc. by use of Schooten’s film space technique. We leave as an open problem for the reader that of extending projective geometry to general SODE’s. The geometry of the system (1) under the action of the group of transforma¬ tions (A) is called the KCC theory of type (A), while the geometry of the system (1) under the group of transformations (B) is called the KCC theory of type (B). For a mechanical point of view the system (1) is called a dynamical system, the manifold M is the space of configurations. From this point of view the main 89
90 Antonelli and Bucataru problem is to find the symmetries of the system (1), that are the diffeomorphisms of the space of configurations M that map solutions of (1) into solutions of (1) and first integrals, i.e. functions which are constant on the solutions of (1). In KCC theory of type (A), the system (1) lies on the tangent bundle of a n-dimensional manifold Mn, while in KCC theory of type (B) the system (1) lies on the first jet bundle of a n + 1-dimensional fibred manifold tf : Mn+1 —► R.
Chapter 1 The Geometry of the Tangent Bundle The KCC theory of type (A) we shall present here appears as a subgeometry of the geometry of the tangent bundle. We start with a manifold M (the space of configurations of a dynamical system), then construct the tangent and cotan¬ gent bundle and study some natural geometric objects like: the almost tangent structure, the Liouville vector field, and the vertical distribution. 1.1 The Tangent Bundle Let M be a real, n-dimensional manifold and A = {(C7a, Mati} be an atlas of C°°-class on M. For every p 6 M, we denote by TPM and T£M the tangent and cotangent spaces at p to M, respectively. The disjoint union TM =; of tangent spaces to M is called the tangent bundle to M. Similarly, the disjoint union of cotangent spaces to M is called the cotangent bundle to M. For every local chart (£7, 0) on p € U C M we denote by the local coordinates induced by 0, that is 0(p) = (^(p)) 6 jRn. We shall denote this by 0 = (x*) or (CT,0 = (x*)). A local chart (CT, 0) at p € M induces an isomorphism h^p : TPM —► 7?n, such that (h^p o o^“1)(v), where (V, 0) it is another local chart at p e Af. In local coordinates, if 0 = (x£) and V» = (£*), then (h^p o We remark here that there is no canonical isomorphism between the tangent space TPM at a point p G M and and between two tangent spaces TPM and TqM at two points p and q to M. A connection provides one, as we shall see. Denote by 7r: TM —► M the canonical projection, that is if v e TPM then tt(v) = p, and we have 7r_1(p) = TPM, Vp 6 M. For a local chart (CT, 0) on M we define & : %_1(C7) C TM -► 0(17) x Rn by $(y) = (0(7r(v)),h^ff(v)(v)). It is easy to check that (7r"1(L7),$) is a local chart on TM and it is called the induced local chart. The set of all induced local charts determines a differentiable 91
92 Antonelli and Bucataru atlas of (7°°-class on TM. Consequently, we have that TM is a 2n-dimensional manifold of C°°-class. By abuse of language the triple (TM, 7r, M) is called the Tangent Bundle of the base manifold M, TM is the total space and % is the canonical submersion. Similarly, the cotangent space T”M carries a differentiable structure of C°°- class, the triple (T*M,tt*,M) is called the cotangent bundle, and the canonical submersion tt* : T*M M is defined by tt* (cu) = p if and only if w G T*M. Throughout this work the summation convention on upper and lower re¬ peated indices is assumed. Let (U, 0 = (x*)) be a local chart at p € M and h<t>,p : TPM —► the induced isomorphism. If is the natural basis of B?, we denote by Ip 388 h^p(ei). Then edited the natural basis of TPM. For a vector v G TPM, we denote by (y*) the local coordinates with respect to the natural basis, that is v = Ip* This is equivalent to h^p(v} = yiei. The local coordinates determined by an induced local chart (tt- 1 (U), $) on TM are then (xi,yi), because 4>(v) = (^(v^h^^v)) = (^(^(v)),^). So, an induced local chart on TM will be denoted by = (æ*,^)). With respect to the induced local coordinates, the canonical submersion has the expression tt : (x^y*) w (x*). The local coordinates on the cotangent space T*M are denoted by (xz,pi) and the canonical submersion ; T*M —> M has the local expression 7r* : (xSpi) >->• (x*). If (IZ, <£ = (x1)) and = (51)) are local charts around p G M, the local coordinates (x*) and (¿*) are related by 5* = x'fâ), with rank(^) — n. The corresponding change of coordinates on TM, induced by = (x\ y'f) and (7T-1(V), # = (£*,§*)) is given by: (1.1) We call (1.1) the change of coordinates formula on TM. As we have that the Jacobian of is always positive (it is equal to det(|—)2), so TM is an orientable manifold. The change of local coordinates formula on T*M (the corresponding formula of (1.1) on T*M) is: ®i = 2i(aS’), rank(fg-) = n, ' d&. J If u G TM, we denote by TUTM the tangent space at u to TM. This is a 2n-dimensional vector space and the natural basis induced by a local chart
KCC Theory 93 (Tr_1(C7)}$ = (x1,^)) atuis {-z-yIu, ■£"T|uh=rç? After a change of coordinates UX oyz (1.1) on TM. the natural basis changes as follows: d i _ r \ 9 \ t i 9 I a?1“ " l“ + a®’ (uW“’ d i _ dit. . d . ay*i“- dxi^dÿi^- (1-2) d d A vector Xu € TUTM has the form Xu = ^(u) + y*(u)w*H-u with ox1 ay1 respect to the natural basis. Under a change of coordinates (1.1) on TM, the coordinates of a vector Xu e TUTM change as follows: (1.3) A vector field X on M is a differentiable section of the tangent bundle, that is X : M —> TM, such that tt o X = Id m- Denote by x(M) the set of all vector fields over M and by ¿F(M) the set of all real differentiable functions on M. Then %(M) with the Lie bracket is a real Lie algebra of infinite dimension and a module over the ring .F(M). Similarly, we denote by x(TM) and 5(TM) the set of all vector fields on TM and the set of all real differentiable functions on TM, respectively.. We may remark here that the tangent space TTM carries two natural pro¬ jections. One is the natural projection r of the tangent bundle (TTM,r,TM) and the second one is the linear map tu induced by In local coordinates we have: r : (x,3/,X,y) e TTM i-> {x,y) € TM, and tt* : (x,y,X,Y) € TTM » {x,X} € TM. 1.2 The Vertical Subbundle Q From the last formula of (1.2) we can see that span a n-dimensional vector subspace VUTM of TUTM. We call it the vertical subspace and it determ¬ ines a n-dimensional, integrable distribution V : u € TM »-► VuTM C T^TM, which is called the vertical distribution. If we denote by VTM — Utt€TAf KTM, then VTM is a subbundle of the tangent bundle (TTM,r,TM) to TM. As 7r: TM —* M is a submersion it follows that : TUTM —► TV^M is an epi¬ morphism of linear spaces, for Vu € TM, where is the linear map induced by 7T at u 6 TM. The kernel of is exactly the vertical subspace, that is VuTM = Kerr^u, Vu € TM.
94 Antonelli and Bucataru We denote by xv(TM) the set of all vertical vector fields on TM. It is a real subalgebra of %(TM). Consider now T*TM the cotangent space of TM at it € TM and denote by {dz* 1^, dyiI«} the natural cobasis. In other words, cfy*!«} is the dual basis of ^7 luK ^l^ter a c^ianS€ local coordinates (1.1) on TM, the dual basis changes as follows: 1 (1-4) 1.3 The Almost Tangent Structure The almost tangent structure of the tangent bundle (or the vertical endomorph¬ ism) is defined as follows: and (1-5) Using (1.2) and (1.4) one can check that J is globally defined on TM. For the almost tangent structure J we have the properties: 1° J2 = 0; 2° Ker J= Im J=VTM. The Nijenhuis tensor field of the almost tangent structure J is given by: Nj(X, Y) = [JX, JY] - J[X, JY] - J[JX, Y],VX, Y € x(TM). (1.6) A direct calculation shows that Nj = 0, that is, the almost tangent structure J is integrable. 1.4 Vertical and Complete Lifts For every u € TM, one defines the linear map lVtU : T%(U)M —> TUTM as iv.vCXVC«))¿k«)) = I«- We can see that : TrwM -» KTM is a linear isomorphism. It is called the vertical lift of the tangent bundle. We may also think to the vertical lift lv as an 5'(M)-linear map between %(M) and x(^M). In this case lv is defined as follows: for every vector field X — € X(-M% (lvX)(u) = ¿v>ti(X%(tt)). The vertical lift of a vector field X € x(M) will be denoted also by Xv G x(TM).
KCC Theory 95 The complete lift Xc of a vector field X — X'-tt-t <= x(M) is defined as follows: X Xc = X*— J- A. dxi 1 dx^dtf (1.7) Consider f 6 .F(M) a function on the base manifold M. Denote by fv = f o tt and /c(x, y) — ^y1 the vertical and the complete lift, respectively. For the vertical and the complete lifts we have the following properties: 1° (JX)'' = fvX*, (fX)c = fvXc 4- fcXv, VX € x(M), / 6 ^(M); 2° J(XC) = Xv, [Xv,yv] - 0, [Xv,yc] = [X,y}v, [Xc,yc] = [X,y]c. A tensor field T of (r,s)-type on TM is said to be a Finsler tensor field (or a distinguished tensor field, or a d-tensor field for short) if under a change of local coordinates (1.1) on TM, its local components change as the local components of a (r,s)-type tensor field on the base manifold. 1.5 Homogeneity Denote by TM — TM \ {0} the tangent space with zero section removed. If A € (0, 4-oo), we define hx : TM —> TM by Aa(z, y) = (x, Xy) and we call hx the homothetic of ratio A. The set of all homothetics {hx, A € (0, +oo)} constitutes a one-parameter group. The vector field that has this group as a one-parameter group is called the LiouvUle vector field and in local coordinates it has the form C = A function f € F(TM) is said to be homogeneous of degree r if fohx — Xrf. There is a Euler type theorem for homogeneous functions, that uses the Lionville vector field €: a function f € ^(TM) is homogeneous of degree r if and only, if £cf — rft or equivalently, in local coordinates, ^y* = rf. Here, is the Lie derivative in direction C. We need also to define the homogeneity for a vector field X € So, a vector field X 6 x(TM) is homogeneous of degree r ifXohx — Ar”1(h<x)»X. A Euler type theorem says that X € x(TM} is homogeneous of degree r if and only if £qX = (i— 1)X. In local coordinates a vector field X — X* is homogeneous of degree r if and only if X* d As an example, we have that the Liouville vector field (D = yi^r7 is homo- vy* geneous of degree 1.
96 Antonelli and Bucataru More generally, a tensor field T of (l,s)-type is homogeneous of degree r if = (r — 1)T. As an example, we have that the almost tangent structure J is a (1,1)-type tensor field homogeneous of degree 0. In order to prove this we have to show that £^7= - J, that is equivalent to [C, JX] — J[(C, X] = —JX, VX € xlTM)) and we can prove this by taking X e If a function f € F(TM) is homogeneous of degree r, then there exist the functions synimetric with respect to such that /(z, y) = » • • y^, so f is a symmetric polynomial of degree r with respect to y. If we want to avoid this particular cases, we have to assume that the function f is of C°°-class on TM and continuous on the null section. A similar remark holds also for tensor fields. Next, if we oxe^referring to a homogeneous object this will be supposed to be of C°°-class on TM and continuous on the null section. If T is a (l,l)-type tensor field and X is a vector field then the Frolicker- Nijenhuis bracket of T and X is a (l,l)-type tensor field [T,X], defined as follows: [T,X](y) = [T(y),X]-T[y,X]. The Frolicker-Nijenhuis bracket of two (l,l)-type tensors K and L is a vector 2-form [X, IS] and is defined as follows: [A',L](X,y) = (K(X),L(y)] + [£(X),AT(y)] + (KoL)[X,y]+ (L o K) [X, y] - X[X, £(y)] - K[L(X), Y]-L[X, X(y)] - L[K(X), y]. In particular, K](X,Y') = [X(X), X(y)l + K*[X, y] - K[X, K(y)] - A[K(X),y], Njc ~ |[K, A] is called the Nijenhuis torsion of K. We already have seen that Wj = j[J, J] = 0 and [J, <CJ = J.
Chapter 2 Nonlinear Connections An important tool in the KCC-theory of type (A), as a subgeometry of the geometry of the tangent bundle, is the notion of nonlinear connection. The existence of a nonlinear connection on TM will allow us to extend some results and geometrical objects from the vertical subbundle VTM to the tangent bundle TTM. In this section we shall give some equivalent definitions for a nonlinear con¬ nection and study the main geometrical objects induced by it. As we intend to apply this theory to a dynamical system, we pay a special attention to the autoparallel curves of a nonlinear connection and their symmetries. The case of homogeneous nonlinear connection is also studied. 2.1 Horizontal Distributions and Horizontal Lifts Definition 2.1. A nonlinear connection on the tangent bundle TM is a sub¬ bundle (JHTM,rff,TM} of the tangent bundle ^TTM.r,TM} such that on fibers we have TUTM = HUTM © VUTM, Vtz € TM. (2.1) A nonlinear connection HTM on TM induces a distribution H : u € TM w HUTM C TUTM of constant rank n, provided M is connected. We call it the horizontal distribution. From (2.1) we can see that the horizontal distribution is supplementary to the vertical distribution. For a nonlinear connection HTM we denote by h and v the horizontal and the vertical projectors that correspond to (2.1), respectively. A vector field X E x(TAf) is called horizontal if h(X) X and vertical if v(X) = X. We denote by %h{TM} the ^r(TM)-module of horizontal vector fields. As : TUTM —> T„.(U)M is an epimorphism, from (2.1) -we can see that the restriction of to HUTM from HUTM to T^M is an isomorphism. We denote by l^u : T^yM —► HUTM the inverse map of the above mentioned 97
Antonelli and Bucataru isomorphism. We call lh.,u the horizontal lift induced by the given nonlinear connection. The horizontal lift lh can be thought also as a 7*(M)-linear map between x(M) and x(TM) and is defined as follow’s: if X = € x(M) define lh(X)(u) = The horizontal lift of a vector field X € x(M) will also be denoted by Xh € The horizontal ¿ft 1^ induced by a nonlinear connection H and the vertical lift lv are related by: Jointly (2.2) One can prove that if lh : x(M) —► x[TM) is a ^(M)-linear map such that (2.2) holds then H : u 6 TM wHUTM = is a nonlinear connection on W. X/5 X Denote by ^-r|u = We have that |u}i=T^ is a basis of HUTM^ Vu 6 TM and under a change of coordinates (1.1) on TM we have that: tix* dxi ’ (2.3) (2.4) The set of functions (Arj) are defined on domains of induced local charts and they are called the local coefficients of the nonlinear connection. Proposition 2.1. To give a nonlinear connection HTM on the tangent bundle TM it is equivalent to give a set of functions Nj on every domain of induced local chart such that on intersections of such domains, they are related by: (2.5) Proof: The “if” part is a consequence of (2.3) and the action of the group of coordinate transformations (1.1). For the “only if” part we suppose that on every domain of induced local chart we have a set of functions JVj such that on the intersection of any two domains the corresponding functions Nj and Nfc are related by (2.5). Then we & may define as in (2.4). It is a straight forward calculation to check that • fa? $ (2.3) is true and {—span a n-dimensional subspace HUTM of TUTM. As |U! ■a“d*} linearly independent, thenHUTM and VUTM satisfy (2.1). ox1 ay1
KCC Theory 99 Example 2.1. Let be the local coefficients of a symmetric linear con¬ nection on the base manifold M. Under a change of local coordinates on M we have that: A ’ dxl d& dxk dx?dxq d& dxk * If we denote by NJ(x,y) = 7jA(x)j/A and take into account the above law of transformation, we find that N*(x,y) satisfy (2.5) so. they are the local coeffi¬ cients of a nonlinear connection. 2.2 Characterizations of a Nonlinear Connection For a given nonlinear connection HTM^ we have a basis { |u} of TuTM adapted to the decomposition (2.1). We call it the Berwald basis of the nonlinear connection i HUTM = € TUTM, ^(Xtt) = 0}. The horizontal and the vertical projectors of the nonlinear connection can be expressed with respect to the Berwald basis as follows: v=-^®syi- <2'6)- From (1.5) we can see that the almost tangent structure J acts on the Berwald basis as follows: Then for Vu € TM, Ju : HUTM —► VUTM is an isomorphism. The inverse map of this isomorphism is denoted by 3V : VUTM —> HUTM. We may extend this structure 3 to the whole TUTM by taking 3U := 3U a vu> This is equivalent to We call the morphism 3 the adjoint structure. It has the properties: 1° 32 = 0, Im0 = Ker3 = HTM: 2° 3o J — h, J o3 = v and consequently Id = 3 o J + J o 3, Conversely, we have: Proposition 2.2. An FtJ'Myiinear morphism 3 : %(TM) —> x(TM) such that 32 = 0 and Id = 3°J+Jo3 determines a nonlinear connection HTM = Ker3.
100 Antonelli and Bucataru Proof. Let «(¿) - A>and »(¿) = + d^ A. »» + ■>'»(¿) - ® « >«" d^ + d A + Ai£, = «o C{ = d'f and Al = —Dl. From 0 = 02(^“?) we that &i = —^Aj. Denote by N? — A3if under a change of coordinates (1.1) on TM the set of functions N? obey the transformation rule (2.5) so they are the local coefficients of a d • d nonlinear connection HTM. We have also that — N? ■=—r) = 0 and Kdx* ’ d]p d 0 • d 0(-z-^) = -5-7 and the statement is proved. 'dy*' dx* ldtf Proposition 2.3. To give a nonlinear connection HTM on the tangent bundle TM it is equivalent to give for every u € TM a linear map Ku : TUTM -+ TK(U)M such that Ku o Ju = Proof: If we have a nonlinear connection HTM^ then we consider the structure 0 and we define Ku — tt*,« o 0U. As 0« o and tf*,u o we obtain JCu O «Txt = 7T*jxa’ Conversely, let Ku : TUTM -* T^M be a linear map such that Ku o Ju — 7r+)tl. As is an epimorphism then Ku is so, Vu € TM. If we denote by HUTM = KerKy, we have an «-dimensional distribution on TM. The vertical distribution VUTM = KerJu is «-dimensional, too and from Ku oju — 7r*?tx we have that HUTM A VUTM = 0 and then (2.1) it is satisfied. The map we used in the above proposition is called the connection map and appears for the first time in Dombrovsky’s paper, [13]. Now, let’s put all structures Ju, 0u, 7r*>u, Zv|W, Ku into a diagram: VUTM Ku il>tU K(u)M Next we present two other structures, the almost product structure and the almost complex structure whose existence is equivalent to the existence of a nonlinear connection. Proposition 2.4. To give a nonlinear connection HTM on the tangent bundle TM it is equivalent to give a T(fTM)-linear morphism IP : x(TM) -+ x(^7Vf)
KCC Theory 101 such that: JoF = J, IPo.7=-J. (2-7) Proof: If a nonlinear connection HTM is given, we define IP : x(TA7) xfT.«) » F(i i + r - r- A. Th„ M „ lnle. Conversely, let IP : x(TAf) —► x(TM) be a J7(TM)-linear morphism such that (2.7) is true. Then in the natural basis, the morphism IP has the form JP(tt7) = — 2AT/ and IP(-^r) = — It can be shown that under a ax* ax* dy3 ay* dy* change of induced local coordinates (1.1) on TM, the functions Nj satisfy the formula (2.5) so they are the local coefficients of a nonlinear connection HTM. The morphism IP defined in the “if” part of the above proof satisfies also IP2 = Id, and consequently it is called the. almost product structure of the nonlinear connection. It has the property that the distribution of eigenspaces corresponding to +1 is the horizontal distribution and the distribution of ei¬ genspaces corresponding to —1 is the vertical distribution. With respect to the Berwald basis of the nonlinear connection, the almost product structure JP has the expressions (2.S) Proposition 2.5. To give a nonlinear connection HTM on the tangent bundle TM it is equivalent to give a F(TM)-morphism IF : %(TM) —> xlTM), such that: 1^ = —Id, andIFoJ + JoIF= Id. (2.9) Proof: If we have a nonlinear connection HTM, we consider the adjoint struc¬ ture 0 and define F = 0—J. ThenF2 — 02-QoJ—JoO+J2 = -(h+v) = -Id. Also, we have that F o J-f- Jo F = 0o J + Jo 0 = h + v = Id. Conversely, consider a .7* (TM)-linear morphism F : x(TM) —> xC^M) that (2.9) are true. If we define 0 = J+F we have that 02 = 0 and 0o J+Jo0 = Id. According to Proposition 2.2, HTM — Ker0 is a nonlinear connection on TM. The structure F is called the almost complex structure of the nonlinear connection and it has the following expression with respect to the Berwald basis: Let T be a tensor field of (r, s)-type on TM, so T is a ^(TMJ-linear morph¬ ism T : AX(TM) x • • • x A1 (TM) x x(TM) x • ♦ • x x(TM) J*(TM). As for every 1-form cu € AX(TM) and every vector field X € xC^M) we have the decomposition into a horizontal and a vertical component cu = huj + vw and X — hX + vX, then T(7mji 4- wj,..., hur + vwr, hX^ + vXi,..., hXs + vX*) is a sum of 2r+4' terms, each of them being a Finsler tensor field on TM. Then we
102 Antonelli and Bucataru may define a Finsler tensor field of (r,s)-type as a tensor field that reduces to only one term from those 2r_H possible terms, that is T(cui,u>r, Xi,XkS) = T(£iw1,..-,£rwr,£1Xi,...,^Xtf), where si,...,€ {h,v}. 2.3 Curvature and Torsion for a Nonlinear Con¬ nection If HTM is a nonlinear connection, according to Frobenius theorem, we have that HTM is integrable if and only if %fe(TM) is a Lie subalgebra of x{TM). $ As {7-7} are generators for %\TM), we have that HTM is integrable if and 8 8 only if [^-r, ^—7] € We have that: (2.H) ±1=P‘± =6Nf_SNl w’Jar h Sxi Sxi ' So, a nonlinear connection is integrable if and only if its curvature tensor Rjif which is a Finsler tensor field of (l,2)-type, vanishes. The curvature tensor of a nonlinear connection is defined as (2.11)' JZ=-№ = -i[A,A], where h is the horizontal projector and № is the Nijenhuis tensor of h. We have that: R = R^dx* ftdx3 ® 3 dyk For a nonlinear connection on TM we call the weak torsion of the nonlinear connection, the vertical-valued 2-form: t(X,Y) = J[AX,AY] - v[hX, JY] - v[JX,hY\. (2.12) With respect to the Berwald basis we have that the weak torsion has the form * = ®® =: ® ® <2J2>' We have immediately that Jot — 0 and t(JX, K) = t(X, JY) = t(JrX, JY) — 0. A nonlinear connection is said to be symmetric if its weak torsion, t, vanishes, . + . dNj dNl that zs -¿—f- = -T— dy^ dyi The Nijenhuis tensors of the adjoint structure 0, the almost complex struc¬ ture, and the almost product structure are given by: Ne = Njf = ijk^k ® ® + R№yk ® Sy1 ® (2.13)
KCC Theory 103 Ntp = Qdx*®-^. (2.14) From (2.14) we can see that a nonlinear connection HTM is integrable if and only if the corresponding almost product structure TP is integrable, that is the Nijenhuis tensor, Njp, vanishes. From (2.13) we have that a symmetric nonlinear connection is integrable if and only if the almost complex structure IF is integrable, which is equivalent to the adjoint structure 0 being integrable. 2.4 Autoparallel Curves and Symmetries for a Nonlinear Connection Now let us consider the autoparallel curves of a nonlinear connection and their symmetries. First we have to define the dynamical covariant derivative induced by a nonlinear connection. Definition 2.2. The dynamical covariant derivative induced by a nonlinear connection HTM is V : xtM) —► VX = v(Xc), where v is the vertical projector and Xc is the complete lift of X. In local coordinates we have: if X — X*-^- g y(Af), then VX — VX*-^-, dx' h dy* where: (2.15) (2.16) 77ie dynamical covariant derivative has the properties: 1° V(X + Y) = VX + VY; V(/X) = feXv + f°VX-, rux-x^t th» X‘ - x>± + vx‘A. More generally, we may define the covariant derivative as V : ^(TAf) XV(TM), by Vtx^y)^) - VX*^, where: Q We can see that if X = ■^i(aj)'S77t that is X is the vertical lift of a vector oy on Af, then the formula (2.16) reduces to (2.15). Denote now S = = 3 ' ■ 0 US' * ~ We have that S is a global vector field on TM. With this notation, both formulae (2.15) and (2.16) can be written as follows: VX^SiX^ + NJX* (2.17)
104 Antonelli and Bucataru Definition 2.3. A smooth curve c : t e Z C IR h* c(t) = (x*(t)) G M is called an autoparallel curve of the nonlinear connection HTM if its natural lift to TM, C : t € I >-» c(i) = (**(*), € TM is a horizontal curve, which means that the tangent vector field to c(t) is horizontal. In local coordinates, a smooth curve c(i) = (^(t)) is an autoparallel curve if and only if: Using the dynamical covariant derivative, the invariant equivalent form of (2.18) V(^)=0. (2.18)' Consider now, c(t) = (x4(t)) a trajectory of (2.18), and let it vary into nearby ones, according to = (2.19) where s denotes a scalar parameter with small value |e|, and ^(¿) are compon¬ ents of a contravariant vector field along the curve c. If we substitute (2.19) into (2.18) and let € approach zero we get the so-called variational equations of (2.18): dt2 +<dyk dt ' dt dxi dtK (2.20) Theorem 2.1. For the variational equations (2.20), the equivalent invariant form (the Jacobi equations) are given by: + + %k^k = 0. (2.21) A vector field ^(¿) along a trajectory c(i) = (&*(*)) of (2.18) is called a Jacobi vector field if it satisfies (2.21). Proof: Along a trajectory of (2.18), we have that = S =: So, the equations (2.20) are equivalent to (2.22) SW+^k++=°- It is a straightforward calculation to check that (2.21) and (2.22) axe equivalent if we take into account that VC = 5(C)+Njtf and V2C = S2^) + + 2NfyS(£?) + . For every vector field on the base manifold X — X' -fg € x(M), we consider: aj(n,y,X) = k dX\Tk , 9XkKri , dN<vk , dNidXr dxidxkV dxkNj + dxi dakX + dyr dxk yk (2.23)
KCC Theory 105 The Lie derivative of NJ with respect to the complete lift Xc of a vector field X = X*-^ € x(M) 0331 shown to be + (2.23)' For every X e x(M) we have that aj(s,t/,X) = £x*NJ is a (l,l)-type Finsler tensor field on TM. The variational equations (2.20), or the equivalent invariant forms (2.21) or (2.22) can be written as follows, then: <>;(»(«), $.5«))^=- »■' (2«) Definition 2.4. A vector field X e x(M) is said to be a symmetry of a nonlinear connection HTM, if £X°Y = [Xc,Z] € VZ s As is a local basis for a nonlinear connection HTM on TM we have that a vector field X G x№) is a symmetry of HTM if and only if [Xc, ^-t] € xh(TM). Vi € [1, In local coordinates we have that d d v[Xc, = -a£(z,y,X)^j, where aJ(x,yyX) is defined by (2.23). Con¬ sequently we have that a vector field X € x( Af) is a symmetry of a nonlinear connection HTM if and only if, £x*NJ = 0. (2.25) Theorem 2.2. A vector field X € ) is a symmetry of a nonlinear connec¬ tion HTM if and only if £x*№ — 0, where IP is the almost product structure of the given nonlinear connection. Proof: We have to prove that X € x(Af) is a symmetry of HTM if and only if [Xc, IP(r)J - IP[XC, Y] = 0, VZ e x(TM), As £XeP(y) is ^(TM)-linear we need to check this only for Y = and Y — -—-t. dx1 dyl Fory = wehavethat IXMP(A)]-P[X‘, ¿1 = (Id -I?)([X‘, ¿]) = 2v([Xc, ]). As X is a symmetry of HTM> we have that [Xc, is horizontal, If y = A, [x. _ p[y, . _(Id + P)(|x., 2.n _ Conversely, consider £X<1P = 0. If Y € ^(TM), then [Xc,IP(y)] - P[Xc,y] =x 0. But, IP(y) = y, so, (Id - IP)[Xc,y] == 0, that is v[Xc,y] = 0 and the theorem is proved.
106 Antonelli and Bucataru Theorem 2.3. A vector field X € x(M) is a symmetry of a nonlinear connec¬ tion HTM if and only if £x'& — 0, 0 being the adjoint structure induced by HTM. Proof: We have that: Then we obtain [Xc, ?] = Cx.9 = £x.Nj-f^ ® W- From these we have that £x^ = 0 if and only if = £x«ATJ = 0, that is X is a symmetry of the given nonlinear connection. Theorem 2.4. A vector field X € x(№) is a symmetry of a nonlinear connec¬ tion HTM if and only if = 0, IF being the almost complex structure of HTM. Proof: As IF — 6- J and £x<= J = 0, VX € %(M), we have that £x°G = £x«IF- Theorem 2.5. Every symmetry of a nonlinear connection HTM is a Jacobi vector field along any autoparallel curve of HTM. Proof: A vector field X € x(M) is a symmetry of a nonlinear connection HTM if and only if afo^y/x) = £xc^> = 0* Consequently we have that aj(z, 2/, X)i/ — 0. The restriction of the last equation along an autoparallel curve of HTM gives us the equations (2.24) which is an equivalent form of the Jacobi equations (2.21). If X e x(Af) is a vector field on the base manifold and is its local one-parameter group of transformations, then the complete lift Xc has (&t) = ((&)*) as its local one parameter group of transformations. The vector field X is a symmetry of a nonlinear connection HTM if and only if c H^TM^u € TM. Consider c an autoparallel curve of the nonlinear connection HTM. Then the natural lift c of c to TM is a horizontal curve. This means that the tangent vector field is a horizontal vector and is horizontal too. As ($tkc(*)(;f) = and $t Qc(t) is the natural lift of <^oc(t) we have that the one-parameter group of transformations <f>t maps autoparallel curves into autoparallel curves.
KCC Theory 107 2.5 Homogeneous Nonlinear Connection If h\ : TM —► TM, h\(x,y) — (x,Xy) is the homothetic of ratio A, A e (0, oo), then : TUTM T^^TM is an isomorphism of linear spaces, Vu € TM. If for a nonlinear connection HTM, we have that faMTM) C Hhx{u)TM,Vu € TM, the nonlinear connection is said to be homogeneous. So, a nonlinear connection is homogeneous if and only if Nfa, Xy) = XNfa, j/), that is the local coefficients of the nonlinear connection are homogeneous functions of degree 1. Using the Euler type theorem for homogeneous functions, we have that a nonlinear connection is homogeneous if and only if^tyk = Nj. For homogeneous nonlinear connection we assume that the local coefficients N* are of C°°-class on TM and continuous on the null section. If the local coefficients NÎ- of a nonlinear connection are of C°°-class on the whole TM, then Nj(x,y) = yjk(z)yk. In this particular case, the connection is called linear. The functions 7jk(x) are the local coefficients of a linear connection on the base manifold M as we have seen in Example 1.1. Proposition 2.6. A nonlinear connection HTM is homogeneous if and only if — 0. where C = is the LiouvUle vector field and IF is the almost oy1 product structure. Proof: In local coordinates we have that = (ATJ — ® Thus, £clP = 0 if and only if ^¡kyk ~ Nj, that is, the nonlinear connection is homogeneous. The (l,l)-isflpe Finster tensor field CqffP is called the tension of the nonlinear connection HTM. Q For a vector field X = & homogeneous nonlinear con¬ nection HTM, we define: t . a2x£ ax* aw? di^dx?, Q& Qyk + 9xi + dN*3Xp d2N'p d2N* ax» 3 dyk dx$ dykfty? QypQyk Qr^a (2.26) The Lie derivative of the geometric object F%k(x, y) — with respect to the complete lift Xc of a vector field X =» € x(M) *s given by: rY pi + dX” , dXP F» dXi , ^Xi £x'F]k - X {F]k) + ■*₽* Qxj + Fti> Qxk Fjk dalp + orf fat ■ (2.26)'
108 Antonelli and Bucataru We have that for every vector field X G x(Af), 2/, X) is a (l,2)-type Finsler tensor field and y,X)yk — afaiyiX'). So, a vector field X G %(M) is a Jacobi vector field for HTM if and only if A vector field X G x(M) is a symmetry of HTM if and.only ifGjjt(.x>y*X)yk = afa,y,X)=0. The Jacobi equations for a symmetric, homogeneous nonlinear connection are now: V2^+^^ = 0. (2.21)'
Chapter 3 Finsler Connections on the Tangent Bundle For a manifold M there is no canonical isomorphism between two tangent spaces TPM and TqM at p, q € M. The existence of such isomorphism, which will be called a parallel transport, is equivalent to the existence of a linear connection on the manifold. If the tangent space TM is endowed with a nonlinear connection HTM, then at every point u e TM we have the decomposition TUTM = HUTM © VUTM. For two points u, v € TM we are interested to define a parallel transport between TUTM and TVTM that preserves the above decomposition. The linear connection that corresponds to such a parallel transport is called a Finsler connection (or a TV-linear connection) on TM. We shall see that every nonlinear connection HTM on TM determines a Finsler connection. , So, a Finsler connection on TM is a special linear connection D on TM that preserves by parallelism the horizontal and vertical distributions. We de¬ termine all components of curvature and torsion and give examples. The Cartan Structure Equations of a Finsler connection are discussed and their integrability conditions studied. • Throughout this section a nonlinear connection HTM with local coefficients NJ is fixed. Let h and v be the horizontal and vertical projectors induced by HTM, Consider also the almost product structure IP, the adjoint structure 6 and the almost tangent structure IF induced by the nonlinear connection HTM, Definition 3.1. A linear connection D (Koszul connection) on TM is called a d-connection if it preserves by parallelism the horizontal distribution, that is Dh = 0. Proposition 3.1. A linear connection D on TM is a d-connection if and only if one of the following conditions is true: 1° Dv = 0; 109
110 Antonelli and Bucataru 2° DJP = 0, Proof: As Id = A + v we have that Dh = 0 and Dv = 0 are equivalent. Also, from h = |(Id + IP) and v = |(Id - IP) we have that Dh = 0 and DIP = 0 are equivalent. As a d-connection preserves by parallelism the horizontal and the vertical distributions we have that: DXY = hDxhY + vDxvY,VX, Y € x(TM). Proposition 3.2. For a d-connection D on TM the following conditions are equivalent. 1° DJ = 0; 2* DO = 0; 3° DIF = 0. Proof: As IF =s 0 — Jy it is enough to prove that two of the above three conditions are equivalent. Suppose that DJ = 0 and let us prove that DO = 0. As Dx0Y — GDXY is horizontal, we have that Dx0Y — 0DxY = h(Dx0Y — 0DxY) = 0J(Dx0Y - 0DxY) = 0DXJGY - 0JGDxY = 0DxvY - 0DxY = 0vDxY - 0DXY = 0DxY - 0DxY = 0, and then DO = 0. Conversely, suppose that DO = 0. As D preserves the vertical distribution we have that DXJY — JDXY is a vertical vector field. Consequently, DXJY — JDXY = v(DxJY—JDxY) = J0(DxJY—JDxY) - JDX0JY-JOJDXY = JDxhY - JDXY = JhDxY - JDXY = JDXY - JDXY = 0. So, we have proved that DJ = 0. Definition 3.2. A d-connection is called a Finsler connection (or a ïV-Iinear connection) if one of the equivalent conditions of the Proposition 3.2. holds good. With respect to the Berwald basis a Finsler connection has the form: ~ 5xk ’ D~& dyi ~ Qyk ’ (3-1) From (3.1) formula we can see that a Finsler connection D transports by par¬ allelism horizontal vectors into horizontals and vertical vectors into verticals. Moreover, this parallelism acts on the same manner on horizontal and vertical vectors. The set of functions are called the local coefficients of a Finsler connection D, Sometimes we refer to a Finsler connection D by the set
KCC Theory 111 Dr = Under a change of coordinates (1.1) on TM, we have: pk _ pi &xP &X<1 ^xk &xP ^x<1 ij q%i m Qfcj Qx^dx^ dx* d& ’ r* - — r‘ dxt> dx<> ij dx‘ Md&d#' So, the horizontal coefficients Fkj of a Finsler connection D on TM, have the same rule of transformation as the local coefficients of a linear connection on the base manifold M. The vertical coefficients are the components of a (1,2)-type Finsler tensor field. 3.1 The Berwald Connection The next theorem will give us our first example of a Finsler connection on TM. Theorem 3.1. The map D : x(TM) x x(TM) —► %(TM), given by: DXY = v[hX, vY] + hY] + 0Y] + G{hX, JY] (3.2) is a Finsler connection on TM We call it the Berwald connection of the nonlinear connection HTM. Frooiz As all the operators involved in the right hand side of (3.2) are additive we have that D is additive too, with respect to both arguments. To prove that D/xY = fDxY, V/ 6 T’(TM), we have to use that vh — hv = Jv = Gh — 0. Now let us prove that DxfY ~ X(f)Y + /DxY. From (3.2) we have that DxfY = fDxY^(hXXf)v2(Y) + (vJf)(/)h2(Y) + (yX)(f)JG(Y) + (hX)(/)^J(Y). As v2 = v, h2 = h, JG = v, and GJ = h we have that DxfY = X(f)Y + fDxY. At this moment we have proved that D is a linear connection on TM. As v J = J and Gv = G we have that DxvY = v[hX, vY] + GY] = v{DxY)y that isDv = 0 and D preserves by parallelism the vertical distribution. Consequently, D is a d-connection on TM. Next, we have that DXGY = v[hX. vGY] + h[vX. hGY] + J[vX, 02Y] +G[hX, JGY] = + 0[hX,uY] = 0(D*Y), because vG = 0, hO = G, G2 — 0, and JG — v. So, DG = 0, and D is a Finsler connection. With respect to the Berwald basis, the Berwald connection has the expres- n li- fa? “ 1 dyi dyk ~ dyi 8xk ' r. If S <5 1 A Consequently, we have also _ d _dNt d d . dyi dyf dyk 3X1 Bv* dy3
112 Antonelli and Bucataru We have then that the lnnal eneflfinients nf the Rerwald ennnectinn are PK = In Example 2.1 we saw that if V is a linear connection on the base man¬ ifold Af, with local coefficients 7^(0;), then Nj(xyy) = 7jk(x)yk are the local coefficients of a nonlinear connection HTM on TAf. The Berwald connection that corresponds to this nonlinear connection HTM has the local coefficients 3.2 The h and v-Covariant Derivation of a Finsler Connection For a Finsler connection D on TM, there is associated a pair of operators: h- and v-covariant derivation in the algebra of Finsler tensor fields. For each X€xC™0,set: DhxY = DmcY, D^f = (ftX)(/), VT e X(TM), V/ € F(TM). (3.3) If we A1 (TM), define (Dj^)(Y) = (ftX)(w(K)) ~ "(DxY'l, *Y € x(™). (3.3)' So we may extend the action of the operator Dx to any Finsler tensor field by asking that preserves the type of Finsler tensor fields, is IR-linear, satisfies the Leibnitz rule with respect to tensor product and commutes with all contrac¬ tions. We shall keep the notation Dx for this operator on the algebra of Finsler tensor fields. We call it the operator of h-covariant derivation. In a similar way, for every vector field X € x(TM) set: DVXY = DvXY, Dxf = (vX)(f), W € x(TM), V/ € JF(TAf). (3.4) If or € A1 (TM), define = (^V)(u/(y)) - o^R), Vy € X(™). (3.3)' We extend the action of Dx to any Finsler tensor field in a similar way, as for Dx. We obtain an operator on the algebra of Finsler tensor fields on TAf, this will be denoted also by D* and will be called the v-covariant derivation. If T is a Finsler tensor field of (r, s)-type with local components (a, y)} then its h-covariant derivative is a (r, l)-type Finsler tensor field ¿>XT given by: № = ® • • • ® ® dr* ® • • - ® V«, where: (3.5)
KCC Theory 113 (3.5)' The v-covariant derivative of a Finsler tensor field T of (r, s)-type is a (r, s 4-1)- type Finsler tensor field D^T, given by: DVXT = where: (3.6) (3-6)' 3.3 The Torsion of a Finsler Connection For a Finsler connection D, consider the torsion T, defined as usual: T(X, Y) = DXY - DYX - [X, Y], VX} Y € (3.7) Breaking T down into horizontal and vertical parts gives the Theorem 3.2. The torsion of a Finsler connection D on TM is completely determined by the following Finsler tensor fields: hT(hX, hY) =* D^hY - D$hX - h[hX, hY], vT(hX,hY^ = -v[hX,hY], hT(JhX,vY) = -D^hX - h[hX,vY], vTfhX, vY) = D^vY -. v[fcX, vY], (h)h — torsion; (v)h — torsion; — torsion; (v)hv — torsion; (3.8) vT(yX> vY) = DxvY — DyvX - v[vX, vY], (v)v - torsion. Proof: Certainly, T(X, Y) = T(hX, hY)+T(hX, vY)+T(uX, hY)+T(yX,vY), ¥X,Y € Every vector field from the RHS of the previous equality has a horizontal and a vertical component. From these eight components two are zero because T is skew symmetric and one is zero because h[i>X, vY] = 0. As,. D preserves by parallelism the horizontal and the vertical distributions, the five components of torsion are given by formula (3.8). With respect to the Berwald basis, the five components of torsion are given
114 Antonelli and Bucataru by. A Finsler connection D is said to be symmetric if the (h)h-torsion and (v)v- torsion vanish, that is and = C^. (3.8)' 3.4 The Curvature of a Finsler Connection Next, we study the curvature of a Finsler connection 7?, and typically consider: R(X, Y)Z = DxDyZ - DyDxZ - D[x^ VX, Y, Z € (3.9) As D preserves by parallelism the horizontal and the vertical distributions, from (3.9) we have that the operator R(X,Y) carries horizontal vector fields into horizontal vector fields and vertical vector fields into verticals. Consequently, R(X> Y)Z =: hR(X> Y)hZ + vR(Xf Y)vZ, VX, Y,Ze X(TM). (3.9)' Noting that the operator R(X, K) is skew symmetric with respect to X and Y, a theorem follows: Theorem 3.3 The curvature of a Finsler connection D on the tangent space TM is completely determined by the following six Finsler tensor fields: R{hX, hY)hZ = DxDyhZ - D^D^hZ - D^^hZ. ' RthX, hY)vZ = DxDyvZ - D^DxvZ - PpucjiyjvZ, R(yX, hY)hZ » D^D^hZ - D$DvxhZ - D[vX.hY]hZ, R(yX,hY)vZ = DxDyvZ - D^LfyvZ - D[vXthY}vZ, R(vXs vY)hZ = DvxD^hZ - DyDxhZ - D^hZ, R(yX, vY)vZ « D^D^vZ - D^DvxvZ - D^x^vZ. 4 (3.10) As the almost tangent structure J is absolutely parallel with respect to the Finsler connection 7X i.e. DJ = 0, JR{X,Y)Z = R(X>Y)JZ, WCYZ €
KCC Theory 115 X(TM), Then the curvature tensor of a Finsler connection D has only three different components with respect to the Berwald basis. These are given by: \ $ R' $ - ' ^6xk,6^}6xh k dyk ’ 6x3} 6xh ' h jk 6xi ’ рЛЛ±-.с^ ± kdyk ’ дхз }6xh~' k jk 6xi * (3.11) These three components are those of the first, the third and the fifth Finsler tensors from (3.10). The other three Finsler tensors from (3.10) have the same local components Pffjk^ ^hjk* p/ 0 0 \ — p * {Sxk’ Sxi'dy11 ~ h jkdyi’ A. ' Qyk ’ Sri ' ~ h ik dyi ’ p/ & & \ & a i & (dyk,dri)dyh kikdyi' J (3.11)' So, a Finsler connection DT = (Nj,Fjk,Cjk) has only three local components Riijki Phjk< »“d i’fc* and these are siven D i _ A J °”hk i pm ni r^m rri , i тут. jk — faj~ + ^hj-^mk - “hk^mj + ^hm^jki dFi- pi "J ^y% i zM pm, ~ Qyk Qi ^^hk i rrnrra гтт rd &h, jk ^k “* Qyj ^hj'-'mk (ЗД2) Here, denotes the h-covariant derivative of the (l,2)-type tensor field Cjk. The Berwald connection BT = (JVJ, -^pO) induced by a nonlinear connec¬ tion has only two nonzero components of torsion: ± Л=t*-_L = - dN*y 5 6x* ’ 5x3 ' зг fak ' Qyj Qyi ' fak ,T(± 5A_L k 6xi' 6x3}~ 3* dyk k 6x* 6x3 } dyk h(h) — torsion; v(h) — torsion. (3.13) If the nonlinear connection HTM is symmetric, then the h(h)-torsion vanishes and the only nonzero component of torsion, for the Berwald connection D, is Rkj, the curvature of the nonlinear connection.
116 Antonelli and Bucataru The Berwald connection has only two nonzero components of curvature: (3.14) The four-index B-tensor was introduced in [14]. Now, consider a nonlinear connection HTM with curvature tensor and the induced Berwald connection Z?, with its curvature tensor R^ These two tensors are related by: If Xi{xyy) are the components of a Finsler vector field on TMb then from (3.10) we may derive the Ricci identities of X* with respect to a Finsler connection Д and although these may be written for every tensor field: yi yi ym-D i yi 'pm Y’tl P™ Jk - xfmc% - ► *U - jk - x*\ms%. (3.15) If we use the first set of Ricci identities (3.15)i for the Liouville vector field yf we have S'lyl* - J^li = As Vy — the nonlinear connection is symmetric and homogeneous then yfj =; 0 and consequently: RÂjkym = ^k- 3.5 Finsler Connections Induced by a Complete Parallelism So far we have spoken about linear connections on TM that preserves by paral¬ lelism the horizontal and vertical distributions. The parallelism induced by such a linear connection will preserve the direct sum (2.1). Consider now {#a}oc=T3^ a frame on TM. If this frame preserves by parallelism the horizontal and the vertical distributions, then with respect to the Berwald basis the frame has the form №} = {HXÀ.Kî^rh^-'Then Va(u)) is a basis of TUTM, adapted to the decomposition (2.1). We call it a nonholonomic frame, adapted to the vertical and the horizontal distributions. This means that we have also two nonsingular matrices (B^(u)) and (V^(îî)) such that Ha(u) =* and
KCC Theory 117 V^(u) = Denote by (J7|*(ti)), and (V<Q(u)) the inverses of these two matrices, that is: ITaH? = ¿j; VjV/ = <5}; V?V£ = (3.16) Next, consider only with nonholonomic frames for which V& — J (Ha), that is This condition means that the frame Ha commutes with the almost tangent structure J. As a Finsler connection D preserves by parallelism the horizontal and the vertical distributions, and the almost tangent structure J is absolutely parallel with respect to it, i.e. DJ = 0, then: (3.17) DvaV^ = C^aVy. The set of functions (Fp^C^) are called the nonholonomic coefficients of D with respect to the nonholonomic frame (Ha, Va). For a Finsler connection DV = and a nonholonomic frame #i(u), the nonholonomic coefficients of D, and Fja are given by: = = 1 ► (3*18) Theorem 3.4. There exists a unique Finsler connection D on TM such that the given frame is h- and v-covariant constant. For this Finsler connection D all components of curvature are zero. Proof: The nonholonomic horizontal frame is h-covariant constant if for all a € {1,n} we have » 0. This is equivalent to + F^H™ — 0. Solving for we have Similarly, the nonholonomic frame Ha is v-covariant constant if for all a G {1,n} we have H^\j = 0. This is equivalent to + C^jH^ = 0. Solving for If we use the Ricci identities (3.15) for H}*, ~ ^fc,^ = 0,and s£ kjHa = 0, Va e {1,n}. As is invertible, R^kj = kj = kj = 0. The Finsler connection defined in Theorem 3*4. is called the Crystallographic connection of the nonholonomic frame [6], [7] and this corresponds to the complete parallelism on TM induced by the field of frames (if«, Va).
11S Antonelli and Bucataru 3.6 The Cartan Structure Equations of a Finsler Connection 5 0 Now denote by {Xa}a_Y^j the vector fields of the Berwald basis {^r, induced by a nonlinear connection HTM and by the dual basis {dx’.Jy*}. For a Finsler connection D, the connection 1-fonns (w£) which cor¬ respond to this basis are defined as follows: w6a(X) = tF(DxXt), VX e It is a straightforward calculation to check that the connection 1-forms are given by wf = ( q J 0i ), where wj = Fjkdxk + Cjk8yk. For a vector field w = waxaex(TM), 3 DVW = (V(W°) + Wbuf (V))XO, that is 0“(DvW) = (V). Theorem 3.5. The Cartan’s first structure equations of a Finsler connection D are given by: —dxl'Aw}t = -&i, 1 -• r (3'19) W) -¿y/‘Aa>l = -et, ) where the 2-forms of torsions 0° = (0l, 0*) are defined by: 0a(X,y) = i»a(T(X,y)), and are given by : Q^^Tkdxi Adxk + Cjkdx? A<fyfc, (3.20) 0* = A dxk + P^dx’ A 8yk + A 6yk. Cartan’s second structure equations of a Finsler connection D are given by: du}~w^ A4 = -iij, (3.21) where Hie curvature 2-forms (flj) = q H*- ) ’ are fl?(X,y) = io(J?(X,y)Xd), and 12/ are given by : Qj = Adxh + P^&f A 8yh + Is^by” A 6yh. (3.22) Proof: Note that 0°(X,y) = 0a(7(X,y)) = 0a(.Dxy)-0o(r>yX)-0a([X,y]) = X(0«(y)) + ^(y)w£(X) - y(f*(X)) - eb(X)u>S (y) - 0°([X, yj) = dffa(X, Y) +
KCC Theory 119 (u?f A 0b)(X,Y). If we take 0® to be dx* and Syi) respectively, then we get the Cartan’s first structure equations (3.19). From Hg(X,y) = du;f(X,y) 4- (u£ Aa/J)(X,Z) we have Cartan’s second structure equations (3.21). The torsion two-forms 6^ contain the horizontal components of the torsion of the Finsler connection D. We shall call them the horizontal torsion two-forms of the Finsler connection D, For the Berwald connection D, the horizontal torsion two-form 0* vanishes if and only if the nonlinear connection is symmetric. The torsion two-forms 01 will be called the vertical torsion two-form. Proposition 3.3. If for a Finsler connection D on TM the curvature 2-forms Î2*- vanish, then there exists a nonholonomic frame (H^)such that the local coef¬ ficients of the connection D are given by: (3.23) Proof: If the curvature two-forms of D vanish, then the Cartan’s second struc¬ ture equations are: dwf + a;* A = 0. According to the general theory of linear connection, there exists a frame on the tangent space TM, whose components with respect to the Berwald basis are Hf{x,y), such that ¿Hf+ u£iff = 0. (3.24) The parallelism induced by the Finsler connection D is path independent and perfectly determined by the field of frames iff. As the Finsler connection pre¬ serves by parallelism the horizontal and the vertical distributions, then the / if j 0 \ frame iff has the form iff = I J I. We have also that the almost tangent structure J is absolutely parallel with respect to the Finsler connec¬ tion D. This will imply that the frame if and the almost tangent structure J commute. From this we have that H} = HÏ and the field of frames has the form iff = ( ¡^’ 2f2 ■ The connection one-form of the Finsler connection D is given by — iff ¿(if”1) J = — dH* (if“1) J. If we take into account the particular form of the connection one-form cvf and the field of frames iff, we have that = if/d(if“1)j = — dHl(if“1)*«. Consequently, the local coefficients of D are given by (3.23). The frame is said to be holonomie if there exist dd>a n functions <j>a on the base manifold M such that iff* = dz*-’ i.e. the one-form rf* — Hfdxi is exact.
120 Antonelli and Bucataru Proposition 3.4. A frame H^ is holonomic if and only if the horizontal torsion two-form O\, defined by (3.20) i of the Crystallographic connection induced by H\, both vanish. Proof: From (3.20)i we have that 0* = 0 if and only if Tjk — 0 and cjfc = o, where TL - FL -FJki and FL and CL are given by (3.23). But, CJk - 0 if and J J J QH? dH** only if H\ are functions of (x) only. So, TJk = 0 if and only if . This is equivalent to H? being a gradient of n functions </>* on the base manifold M. Theorem 3.6. Consider D a Finsler connection on TM with local coefficients (Fjk,CJk). The horizontal torsion two-forms 0* and the curvature two-forms £2} vanish if and only if there are load coordinates on the base manifold such that FJk = Cjk — 0, with respect to the induced coordinates on TM. Proof: If the curvature two-forms of the Finsler connection 7?, vanishes then from Proposition 3.3 there is a frame such that the local coefficients of the Finsler connection D are given by (3.23). From Proposition 3.4, the frame H? is holonomic, that is, there exist n functions such that Hf — Since, are coordinate functions on M, with respect to the induced coordinates on TM, the local coefficients of the Finsler connection D, vanish. Applying Theorem 3.6 to the Berwald connection induced by a symmetric nonlinear connection HTM, we have the following result: Proposition 3.5. Let HTM be a symmetric nonlinear connection with local coefficients Nj. There exists local coordinates (x*) on the base manifold M such that with respect to the induced coordinates on TM, the functions NJ are func¬ tions of (s) only, if and only if the Berwald connection has zero curvature. 3.7 Geodesics of a Finsler Connection Next we study the geodesics of a Finsler connection D. A smooth curve c:t € I I-+ c(i) = (m^t),^^)) € TM is a geodesic of I? if D±c = 0, where c(i) — i + 18 the tangent vector c- + c is a geodesic of D if and only if: . Erf , ri dxj 5^ dt2 jk dt dt ik dt dt ’ d , pi 6y^d^_ 5^ dt'' dt' ik dt dt dt dt (3.25) = 0.
KCC Theory 121 From these equations a horizontal curve is a geodesic of the Finsier connection D if and only if d2r'£ ~ dxj dap dfi + *jk dt ~dT dt = 0. We now consider smooth curves on M, c : t € I i-> c(i) = («*(<)) e M. We say that c is a geodesic of a Finsier connection D, if its natural lift to TM. c(t) — -T-) is a geodesic of D. (tt It follows that an autoparallel curve c(t) = (z’(i)) of a nonlinear connection (Pxi ► dxj dxk is a geodesic of a Finsier connection if and only if + Flk——— — 0. at- J at at If, for a Finsier connection D, the curvature two form Qi vanishes, then 6 J d there exists a nonholonomic frame (Ha — H^——^Va — Hi-^) such that % oxz oy1 its nonholonomic components vanish. If we denote by —rr — and at at fru<* fy? -- = the nonholonomic components of the tangent vector c(t), then at at the smooth curve c is a geodesic of D if and only if J (3.25/ If in addition, the torsion two-form © and the curvature two-form 2*, of a given Finsier connection P, both vanish, then around every point on the base manifold M there exists local coordinates (s’) such that the equations of geodesics are given by: (3.25)" 3.8 Homogeneous Berwald Connection Consider now a homogeneous nonlinear connection HTM, that is, the local coefficients Nfa^y) of HTM are homogeneous functions of degree one with respect to y. Recall here that for homogeneous functions we assume that the functions are of C°° class on TM and continuous on the null section of the tangent bundle TM. This is to avoid the particular case when a homogeneous function of degree r is a polynomial function of degree r in y. Let D be the Berwald connection associated to HTM. As Nj are homogeneous of degree one,
122 Antonelli and Bucataru dN{ -^±yk = N^ that is, F%kyk = Ni So the equations = 0, are equivalent. This means that the autoparallel curves of a homogeneous non¬ linear connection coincide with the geodesics of the induced Berwald connection, Q For this particular case, if X = X‘t-7 € x(M) then the (l,2)-type tensor field a^k{x,y^X) given in (2.21) is the Lie derivative of Fjk with respect to the complete lift Xc of X, that is: Consequently, a vector field X € %(Af) is a symmetry of the nonlinear connec¬ tion HTM if and only if
Chapter 4 Second Order Differential Equations The geometric theory of a system of second order ordinary differential equations (SODE) is named the KCC-theory after its initiators D.D.Kosambi, E.Cartan, and S.S.Chem [17], [10], [11]. It is well known [15] that a SODE determines a nonlinear connection HTM on TM and a Finsler connection D, namely the Berwald connection is induced by HTM. The Cartan structure equations of the Berwald connection D will give us the geometric invariants of the system. We also study the symmetries of a SODE, by specializing the results we have found for the symmetries of the induced nonlinear connection HTM. 4.1 Semispray or Second Order Differential Vec¬ tor Field We start with a n-dimensional manifold M, that is the configuration space of a dynamical system governed by a system of second order ordinary differential equations: •^r + 2i?(a;,— ) = 0. (4.1) It is more accurate to say that each system (4.1) is defined over a local chart on TM. So, we have a collection of systems (4.1) on local induced charts on TM, that are compatible on the intersection of induced local charts. This compatibility means that under a change (1.1) of local induced coordinates on TM, the (LHS) of (4.1) is a Finsler vector field on TM. This is equivalent to saying the functions ^(x, ^) transform according to: 123 (4.2)
124 Antonelli and Bucataru 0 0 Proposition 4.1. The vector field S = is globally defined on TM if and only if the functions G2(x,y), defined on domains of induced local charts, satisfy (4.2) under a change of local coordinates (1.1) on TM. Proof: It is a straight forward calculation to check that under a change of coordinates (1.1) on TM we have that if and only if the functions G* and <7* are related by (4.2). Definition 4.1. A vector field S G %(TM) is called a semispray, or a second order vector field if JS = 47. Proposition 4.2. A vector field S € xlfTM) is a semispray if and only if on every domain of local charts on TM, there are functions Gi such that S — The functions Gi (x,y) are called the local coefficients of the semispray. The functions G* are supposed to be of C7°°-class on TM and continuous on the null section. Proof: A vector field S = Ai(x9y)^-r + Bi(x,y)^ on TM is a semispray if ox* oy1 d d and only if JS — Ai(x, y)-£-? = 47 — yi^^ If we take B* = -2(7* we have that cfy uy* d Ô S is a semispray if and only if S = y'-x-? - ZG^x, j/)^-t. ox“1 uy Proposition 4.3. A vector field S G xfTM), that is, a section on the tangent bundle (TTM, r, TM) is a semispray if and only if S is a section of the bundle (TTM,n*,TM). Proof: Let S — A*(z,3/)^r - 2Gi(x,y)^ a vector field on TM, As x, : (¡r, y, X, Y) G TTM i-+ (x, X) € TM we have that S is a section of x, if and only if tt+ o S — IdrAf * that is (z\ A* (a;, j/)) = (xi, y*) and the proof is finished. Definition 4.2. A smooth curve c: t € I c(t) « («*($)) e M is said to be a path of a semispray 5 if its complete lift c: t G I *-> c(t) = (s*(t), ^) 6 TM is an integral curve of the vector field S. d S If S = “ 2<J*(z>^)^r then a smooth curve c on M is a path of S if and only if c is a trajectory of (4.1). From Proposition 4.1 we have seen that a collection of compatible systems (4.1) determine a semispray S with local coefficients (7*.
KCC Theory 125 4.2 Nonlinear Connections and Semisprays Theorem 4.1 [15]. If S is a semispray. then IP — —£sJ is an almost product structure on TM, that satisfies (2.7). Proof: We have to prove that the f(TM) morphism IP : x(TM) —> given by JP(X) = —(£gJ)(X) = — [S, JX] 4- J[S, X] satisfies (2.7). First we prove the formula: J[JX, S] = JX, VX € x(TM). (4.3) As the Nijenhuis tensor Nj of the tangent structure J, vanishes, we have; 0 — Nj(S,X) = [d7, JX] — J[d7.X] — J[S, JX]. But J is O-homogeneous, that is, [C, JX] - J[(D, X] = - JX. Consequently, J[JX, S] = JX. Now J1P(X) = -J[S, JX] = JX, VX G x(TM) so, .TIP = J. Also, IPJ(X) = J [S' JX] = - JX, VX € x(^M) and so IP J = - J. Then (2.7) formulae are true and according to Proposition 2.4 the almost product structure IP determine a nonlinear connection HTM on TM. of the induced nonlinear connection are Nj — We can check this directly. Let G? be the local coefficients of a semispray S, then under a change of local coordinates (1.1) on TM we have the formula (4.2). The functions Nj — satisfy the formula (2.5) and according to Proposition 2.1 they are the local coefficients of a nonlinear connection HTM on TM. According to the Theorem 4.1, a semispray determines a nonlinear connec- nonlinear connection is symmetric. Let HTM be the nonlinear connection induced by a semispray S. As we have seen in Section 2, a nonlinear connection HTM determines a horizontal distribution that is supplementary to the vertical distribution. This means that the direct sum holds good: TUTM = HUTM © VUTM. The horizontal and the vertical projectors that correspond to the above decom¬ position are given by: h(X) = l(X-[S,JX]-J[XIS]) v(X) = |(X + [S, JX\ + J[X,S]). (4.4) Theorem 4.2. If HTM is a nonlinear connection on TM with h the induced horizontal projector, then there exists a unique semispray S such that: (4.5)
126 Antonelli and Bucataru Proof: Let S' be an arbitrary semispray on TM and denote by S — hS'. Then S is a semispray on TM, too. Indeed as Jh = J and J S' — C. then JS — C. Furthermore, the semispray S does not depend on the semispray S'. That is, if S* is another semispray on TM, then hS' = hS*. This is true because if S' and S” are two semisprays on TM, then J (S' - S*) — 0 and their difference S' — Sn is vertical. Consequently, hS' — hS”. Now let’s prove that S = hS' = hS satisfies (4.5). FYom (4.3) if we take X = St then J[JS, S] = JS and because JS — G, we have that J[C, S] — G — 0, which is equivalent to J([(C, S] — S) — 0. Consequently, [C, S] — S is a vertical vectorjield, so ~ If S is a semispray on TM such that (4.5) holds true we have that S' = [<C, S] is a semispray and then S — h[(D, S] = hS' = S and the theorem is proved. In local coordinates the semispray induced by a nonlinear connection HTM with local coefficients Nfe.y) is given by: that is, the local coefficients of the induced semispray are 2Gi(xi y) = NUx. yjy3. . d d For a semispray S = y'-^ ~ 2<rconsider the induced nonlinear con- oxz ay1 nection HTM with local coefficients № = -r—r. We have the formula: 3 dy3 (4-6) The Finsler vector field Pfay) = 2Gi(x^y') - Njfayjy3 — 2G*(x,y) - is called the first invariant of the semispray. Definition 4.3. A semispray S is said to be a spray if the first invariant £* vanishes. dG* ■ We have that a semispray S is a spray if and only if2G*(xty) = -^y3 oy3 which means that the functions G{(xf y) are homogeneous of degree two. This is equivalent to say that S is a homogeneous vector field on TM. We may express this by saying that a semispray S is a spray if and only if C^S — [C, S] = 0. Proposition 4.4. a) If S. is a semispray and HTM the induced nonlinear connection, then S is a spray if and only if it coincides with the semispray induced by HT,M* b) Let HTM be a symmetric nonlinear connection on TM and S the induced semispray. The nonlinear connection induced by S coincides with the given nonlinear connection HTM if and only if this is homogeneous.
KCC Theory 127 Proof: a) Let G* be the local coefficients of the semispray S, Then the induced nonlinear connection HTM has the local coefficients JVJ = The .d'U3 . . semispray S( induced by HTM has the local coefficients 2G/£ = Nfa = 3^72?. We have that S — S* if and only if Gn = that is equivalent to OG*' 2Gi — 7-7^ and this means that S is a spray. dy3 b) Let HTM be a symmetric nonlinear connection with A’J the local coeffi- duced by HTM has the local coefficients — Njy?. Then the non- . 3G* linear connection induced by S has as local coefficients Nj = = means exactly that the nonlinear connection HTM is homogeneous. 4.3 The Berwald Connection of a Semispray From now on let S be a semispray with local coefficients G* and HTM the dG^ induced nonlinear connection with local coefficients M Consider the J ôy> Berwald connection D induced by HTM. This is a Finsler connection with the one-forms of the Berwald connection D are then given by As the nonlinear connection is symmetric, the h(h)-torsion (3.13) of the Ber¬ wald connection D vanishes and consequently the Berwald connection D is symmetric. The Berwald connection has only one component of torsion, the v(h)-torsion, which is also the curvature of the nonlinear connection: The horizontal two forms of torsion 6* of the Berwald connection vanish and the vertical two-forms of torsion of the Berwald connection are given by: Ada?.
128 Antonelli and Bucataru The two nonzero components of curvature for the Berwald connection D are: jç pii ' Di ■ h-j > rpm r^i tc*î . -K'h.jk— fak fap +'Thj*mk T)i _ h ~ dy*dyidyk' J (4.8) The curvature two forms of the Berwald connection are given by: Qj- = A dxh + D/^dar* A Syh. ¿* The Theorem 3.5 has the following particular form for the Berwald connection: Theorem 4.1. TAe Carton's first structure equations of the Berwald connection D are given by: —dxh A = 0, (4*9) Au^ = A dxk. The Cartan's second structure equations of the Berwald connection D are given by: dwj - A 4 = A dxh - DfuJaJ' A Sy11. (4.9)' The above Cartan’s structure equations of the Berwald connection D are useful to determine necessary and sufficient conditions in which the system of second order differential equations (4.1) is linearizable in velocities. Theorem 4.2. The Berwald connection of a semispray S has zero curvature (is fiat) if and only if about every point p Q M there are local coordinates (x*) in M such that with respect to the induced coordinates on TM, the local coefficients of the semispray S have the form: + (4.10) Proof: If there exist induced coordinates on TM such that the semispray S has the local coefficients 2Gi(x,y) = Afa)y? 4- B*(x) then the local coefficients ft-Qi of the Berwald connection D vanish, that is Fh = = 9« Hom (4.S) we cam see that the curvature components of D vanish so the Berwald connection is flat. Now let us assume that the curvature two forms QJ of the Berwald con¬ nection vanish. As the horizontal torsion two forms 0* are zero, according to the Theorem 3.6 there are induced coordinates on TM with respect to which the local coefficients of the Berwald connection vanish: Fjk = 0 and
KCC Theory 129 2C>(z, y) = A^x)^ + Bi(x). Let us consider now the system (4.1) we start this section with: According to the Theorem 4.2, this system is linearizable in velocities, that is it takes the form: . + + = (4.11) if and only if the curvature of the induced Berwald connection vanishes. The systems (4.11) are used to describe models in biology, such systems are known as Laird’s law in multidimensional growth, [4]. 4.4 The Jacobi Equations of a Semispray in Section 2. This covariant derivative was useful to determine an invariant form (2.IS)' for the autoparallel curves of a nonlinear connection. We studied also the variational equations of the autoparallel curves and we found an invariant form (2.21) using this covariant derivative. Now we apply all these considerations for the particular case when a semispray induces the nonlinear connection. This way we can get information about the system (4.1). If Xi(xf yj is a Finsler vector field on TM, we define its dynamical covariant derivative by: It can be easily proved that VX* is still a Finsler vector field. Then we can see the dynamical covariant derivative as a map V : xv(TM) —► x^(TM), defined by Or we can view the dynamical covariant derivative as a map V : —* Xv(TM)y defined by V(X*^r) — VX*-^. It can be seen that the first definition can be derived from the second one by composition with the vertical lift. Using the dynamical covariant derivative, the system of equations (4.1) takes the form: We may remark here that both sides of the equation (4.12) behave like a vector field, so they are Finsler vector fields. Let c(t) = (rr*(t)) be a trajectory of (4.1). If we perform a variation of this trajectory into nearby ones according to ^(t)=xi(t) + sC(t),
130 Antonelli and. Bucataru as we did for the autoparallel curves of a nonlinear connection, we get the variational equations: 4. dt2 ‘ dt = 0. (4.13) Theorem 4.3. The variational equations (4.13) have an equivalent invariant form (Jacobi equations)*. V2f + + E&& = 0. (4.13)' Here is the /¿-covariant derivative (with respect to the Berwald connection) of the first invariant A vector field (C(t)) along a path c(t) of the semispray S is called a Jacobi vector field if it satisfies (4.13). Proof: Denote by: i ^dy^ dyrdyl’ It can be proved that is a (l,l)*type Finsler tensor field. It has been intro¬ duced in [9], for the homogeneous case. This tensor field is called the second invariant of the given SODE in [17], [10], and [11], or the Jacobi endomorphism in [12]. It is easy to check that the equations (4.13) are equivalent to: V2f + B^ = o. (4.15) All we have to prove now is the following expression of the second invariant: ^ = ^+4. (4.16) Let X£(rc, y) be an arbitrary Finsler vector field, and consider the vector field X = Xi/? + 5(Xi)A dx* x dy* on TM. We have then: [S,X] = (V2Xi + Bj^')^7. If we consider the expression of S and X in the Berwald basis S = — £* and X = respectively, then the bracket [5, X] can be expressed as follows: [S,X] = {V2X‘ + {%kyk + If we compare the above two formulae and we take into account that X’(x, y) is an arbitrary Finsler vector field, then the second invariant BJ can be expressed as in (4.16).
131 KCC Theory 4.5 Symmetries for a Semispray Definition 4.4. 1° A Lie symmetry of the semispray S is a vector field X on the base manifold M such that [S, Xe] = 0, where Xe is the complete lift of X. 2° A dynamical symmetry of the semispray S' is a vector field X on TM such that [S,X] = 0. If X € is a Lie symmetry of S then Xe is a dynamical symmetry of S, As for X e x(M) we have that Xe — 2Xh -r [S, Xv], then X is a Lie symmetry of S if and oidy if 2[S, Xh] + [S, [S, Xv]] = 2£sXh + £S£SXV 0. (4.17) Theorem 4.4. 1° A vector field X = X'fay) + Yt(xiy)^r is a dynamical symmetry of S if and only if Y1 = VX\ and V2X< + B]X^ = 0. (4.1S) 2° If X = X*(x,y)-£? + Yi(xiy)^ is a dynamical symmetry of the semispray S and c(t) — (x’(i)) is a path of S, then the restriction of Xi(x,y) along c(t) = (x*(t), ^(i)) is a Jacobi vector field for S. Proof: 1° If we express the Lie bracket [S, X] using the Berwald basis, we have [5,X) = (VX4 - y4)^ + (VT4 + BjX^. (4.18)' So, X is a dynamical symmetry of S if and only if (4.18) is true. 2° If X’(x, y) axe the horizontal components of a dynamical symmetry X, then V2X* 4- (J&jkyk + SfyXi = 0. The restriction of this along the curve c gives us the equations (4.13)', and then X1 is a Jacobi vector field along c. The Jacobi equations (4.13)' are the invariant form of the variational equa¬ tions (4.13) using the dynamical covariant derivative. Also, in (4.18) we found the invariant equations of dynamical symmetries (or Lie symmetries) in terms of dynamical covariant derivative.
132 Antonelli and Bucataru Ô For a vector field X = £ x(M), we consider: .. _ d2Xi j * , ^dX* kd& a{x.y.X) -G dxj + 2dxjX +2dxkV dyl' The Lie derivative of 2& with respect to the complete lift Xc of a vector field X = № is defined as follows: /) X* X* « = X=(2Gi) - 2^ + For every X € we have that a^x^X) — Cx^G*) is a Finsler vector on TM. Proposition 4.3. a) A vector field X e x^M}. is a Lie symmetry for a semispray S if and only if: Lx<№(w)} = a^x.y.X) = 0. (4.19) b) A vector field $*(i) along a trajectory c(t) = (z’(i)) of (4.1) is a Jacobi vector field if and only if: £i«(2(?(x) ^)) = ¿(z, = 0. (4.19)' Proof: We have that for every X € x(M), [S,X«1 = ¿frv.X)-^ = £x«(2<?i)^. Then if we use the Theorem 4.4 for thé particular case of a complete lift of a vector field X € x(M) we get the statements a) and b) of the Proposition. 4.6 Geometric Invariants in KCC-Theory For a system (4.1) it is important to determine the geometric invariants under the group of transformations x* = ÿ(arJ), rank(|^-) = n. These geometric invariants where determined by Kosambi [17], Chem [11], and Cartan [10] using the equivalence method. We want to prove now, that for a given semispray, we can determine all these five geometric invariants using KCC-Theory. The first KCC-invariant is £* and it was defined in (4.6) as the vertical com¬ ponent of the semispray. The second invariant is B^, the Jacobi endomorphism, defined in (4.14) and it has been used to study the Jacobi equations and the symmetries of a semispray.
KCC Theory 133 The third, fourth and fifth invariants, as they were defined in KCC-papers, [17], [11], [10] are: .= ’ 3{dyk dyih * dy' ’ dF]k && dyL dy^dykdyi ' Bijft (4.20) The Tensor D^kl is called the Douglas tensor*, and we already saw in (4.8) that it is one of the nonzero components of the curvature of the Berwald connection. Theorem 4.5. 1° The curvature R%k of the nonlinear connection N (or the (v)h-torsion of the Berwald connection D) is the third invariant of the semispray S. 2° The Riemann-Christ off el curvature tensor R^kl of the Berwald connection D is the fourth invariant of the semispray S. Proof: We have to prove that Rjk = Bjk and = Bjki* First we prove that R%k and R%kl satisfy (4.20)2, that is R^k — From (2.11) we have ** 6xi fap 1 dy1 dy1' fap ' dy1' faP ' A® = F^> we haTO that $ = ~ &($■) + “ FtiFjp ~ Fiji- According to (4.16) we have for the second invariant Bj, the expression Bj = R^y* + So, = + F^y1 + ^|fc. Then, g - = 22^. + + £f.|fc -4|j. Using the Ricci identities (3.15) for the Berwald connection D and the first invariant & we have that ¿¿It - = D\jk8\ and - 5*1^ = As the Douglas tensor is symmetric we have that: - &|y|*. Consequently, we have, = ^1^ - £*1^ ~ Rflkjyl ~ Rfkj. Finally, we have that: = 3B}fc + (Bj^ + Rfa + Rj^)?/. Using the Bianchi identity for the Berwald connection D, we have that R]^ + R^ + = 0, so that Rfjk — j(-^ - and the theorem is proved. 1 Distinguished from the so-called Projective Douglas tensor.
Chapter 5 Homogeneous Systems of Second Order Differential Equations In this section we study systems of second order differential equations: (5.1) where the functions G are homogeneous of degree two with respect to As we have seen in Section 4, each system of (5.1) is defined on local charts on TM and these systems are compatible on the intersections of domains of local charts. The compatibility means that if <£*(£, y) and are defined on 7r“1(l7) and 7r_1(V) respectively, then on n V) we have: 2öi=£2ö,-ä5fe^fc- (5.2) From the above formula we can see that the required condition for the functions Gi to be homogeneous of degree two with respect to is chart invariance- So, if G are homogeneous of degree two, then G are also homogeneous of degree two. Using the Euler theorem we have that the functions G are homogeneous of degree two if and only if — 2G*. If the systems (5.1) are given on each domain of local chart with the com¬ patibility conditions (5.2) then we may consider the vector field: (5.3) Then S is a globally defined vector field on TM and it is called a spray. As G are homogeneous of degree two, are homogeneous of degree one, then S is a 135
136 Antonelli and Bucataru vector field homogeneous of degree .two. This is equivalent to say that = S, where — yi^ is the Liouville vector field. The spray S induces a nonlinear connection HTM on TM with the local coefficients Nj = and the horizontal and vertical projectors given by (4.4). This nonlinear connection is symmetric and homogeneous because the local coefficients N'j are homogeneous of degree one. This is equivalent, according dN"1 I to Euler’s theorem, to = Np We have seen in Section 2 that the ho¬ mogeneity of a nonlinear connection is equivalent with the homogeneity of the induced almost product structure IP. According to the Proposition 2.6, this is equivalent to £<pIP = 0. For a spray S, we consider the induced nonlinear connection HTM with g|r) the corresponding Berwald basis. FYom the homogeneity condition, the first invariant — 2(7* — — 0 so we have that the spray S is a horizontal vector field, that is: (5.4) Also the autoparallel curves of the nonlinear connection HTMy namely the solutions of the system of second order differential equations: (5.5) coincide with the paths of the given spray 5. Let D be the Berwald connection induced by HTM. Then the horizontal coefficients Fjk = are homogeneous of degree zero. This means that && k dG* dyidyk^ dyi Then the equations (3.25) for the geodesics of the Berwald connection are now: dV , drf dxk = 0. (5.6) Consequently, we have that the geodesics of the Berwald connection D are the same with the autoparallel curves of the nonlinear connection HTM and coincide with the solutions of (5.1), the paths of the given spray S. The systems of SODE (5.1), (5.5) or (5.6) are equivalent to: (5.7) Here V is the covariant derivative induced by the nonlinear connection HTM or by the Berwald connection D: VX* = S(A?) + N}X’ = (5.8)
KCC Theory 137. where is the /i-covariant derivation of X* with respect to the Berwald con¬ nection D. Proposition 5.1. For a spray S, the second, the third and the fourth invariant are related as follows: ■8} =^ikyk = ^ikymyk, K* =RÀjkym- Proof: The second formula (5.9) is a direct consequence of the Ricci identities (3.15) and the homogeneity of the nonlinear connection HTM. This homogen¬ eity appears here in the form: dN'- Proposition 5.2. a) The Jacobi equations of the system of SODE (5.7) have the form: V=r+Si,^-?=O. !5-10) b) A vector field X = e x(M) is a Lie symmetry for a spray S if and only if: V2X£ + R^ykXj = 0, (5.11) or in the equivalent form: Cx^) = W = 0. (5.11)'
Chapter 6 Time Dependent Systems of Second Order Differential Equations In this section we shall develop a geometrical theory for a system of second order differential equations: + (0.1) More precisely, we shall study the geometric properties of this system under the group of transformations: This theory is called the KCC-theory of type (B). 6.1 Sprays and Nonlinear Connections on Jets The system (6.1) lies on the first jet bundle of a n + 1-dimensional fibred man¬ ifold. It will be then expedient to develop our theory on a (n + l)-dimensional manifold Mn+1 which is a fiber bundle over R with t as coordinate on R and z1,..., xn as local coordinates on the n-dimensional fiber. This is a general¬ ization of Mn x R. Thus, given a bundle projection 7T : Mn+1 —► R, let ^1,0 ’ —► Mn+1 denote the induced bundle projection on the first jet bundle Recall, that a point in the total space may be regarded as a tangent vector of Mn+1 in the form dt+y'di, where yi are local coordinates on the fiber of The vertical distribution is given by Ker (n^o)*, the kernel of the differential, and is spanned by The natural basis on the tangent space Tp( Jxtt) at a point p in the total space of J1^ is 139
140 Antonelli and Bucataru Let us consider type (B) transformations & rank(f£)=n. Under change of coordinates (6.2) on Mn+1, the natural basis (^ |p, ^r|p) on TpMn+\ p = (t,®1’) G Mn+1 transforms according to £ = £ <?*l = A dt §t+ dt dxv d = dxi d dxi ~ dx* dx*' (6.3) The induced change of coordinates in Jx7r is given by t = t (6.4)- It follows that GL(n,R) x R*) is an affine bundle, [22]. For u = (t)xiJyt) G the natural basis transforms according to £ Ä£ + ^A + ^A dt dt dt dx* dt 0yi d _~dx? d dyi d dxi ~ dxi dxi dxi dyi d dxi d dyl “ dxidyi' (6.5) We remark that the vertical distribution Vu :=» Ker(^1,0)*,-a C is n- dimensional and integrable. The vertical endomorphism operator (or the almost tangent structure) is the linear map (6.6) Definition 6.1. By a semispray on M”+1 we shall mean a vector field S G %(J1^), the space of C°° sections of for which dt(S) 1 and J(S) = 0.
KCC Theory 141 A semispray S may also be regarded as a section of the affine bundle (J27r\7r2.i, In local coordinates S appears as where Gi are called the local coefficients of the semispray S. The local coefficients G* transform according to dx? 2 0x3 V3 2 dt * Definition 6.2. A nonlinear connection on J1 % is an (n 4- l)-dimensional distribution JT which is supplementary to the vertical distribution Tu © K, V u € J1^. (6.9) We remark that restriction of (tti.oK.-u to Hu is an isomorphism onto the (n 4- l)-dimensional vector space 0(ujAfn+1. In fact, the inverse linear map is called the horizontal lift by the nonlinear connection H and is denoted lh,u : GGrlioMMn+1 —> Hu, Also with a € {0,1,, n}, x° = t and any u € J1^, defines the Berwald basis of differential operators (6.10) Under type (B) transformations we see that S S 6 St &+ dt fâ* 6 _ dx3 ô Sxi dxi 5x$ ’ (6.11) Note that for type (B) transformation |£(a:, y) is a covariant vector. This is, of course, well known but was first pointed out by Berwald. We remark that the set of functions (A7,2Vo) are defined in all local charts and this is part of our definition of a C00 manifold. This set of pairs is called the set of local coefficients of the nonlinear connection H on if and only if (6.12)
142 Antonelli and Bucat&ru Proposition 6.1. 0 d ■ d a) Every semispray S — ~x~ 4- yl " 2G^ determines a nordinear connec- dt dxz oyz tion H with local coefficients A/'"** = N3 = 2^-^. (6.13) b) Every nonlinear connection H, with local coefficients (Nj, Nq) determines a semispray S with local coefficients 2Gi = Njyj+N^ (6.13/ Moreover, nection. then is the Berwald basis adapted to the nonlinear con- S = (6.14) Proof: a) Let G^ be the local coefficients of a semispray S. Under a coordinate change (6.4) Gi satisfy (6.8) so that 2 d& "dx? yi dt’ "dy? yi a-# t_ a2# ~ dy? dx> dxr d&dxi* dx^dt “ dyP dxi “dx? ’ because the last two terms sum to dxf^dxi^ dt 9xP' Hence, the first formula of (6.13) is verified. For the second formula of
KCC Theory 143 (6.13), consider the last equality from (6.4). This establishes the second formula. (b) One checks that under a change of coordinates (6.4) on the set of functions 2G* = + Nq constitutes a semispray S given by (6.7). This completes the proof. As we can see from (6.14) a semispray S is always horizontal with respect to the induced nonlinear connection H. This doesn’t happen for a time on dependent semispray S, where the vertical component of S is the first invariant which vanishes if and only if S is homogeneous.In KCC-theory of type (B) we cannot define the homogeneity and the place of the first invariant & is taken by the coefficients JVg. We remark here that the integral curves of a semispray S (i.e. the so-called paths) are jets of sections of t h or(t) = (t,s£(t), ^-) € J1^ for which the tangent vector field d(t) satisfies d(t) = S(a(i)). In local coordinates, we have (6.15) Moreover, a section t 7(t) = (¿,^(i)) is an auioparallel for the nonlinear connection (Nj, Nq) if and only if its first jet lies in the corresponding horizontal distribution N. In local coordinates, this means (6.15)' Theorem 6.1. Lei S be a semispray induced from a nonlinear connection N. The paths of S coincide with the autoparallels of N.
144 Antonelli and Bucataru Proof: According to the equations (6.13) and (6.13)' we have that (1.15) and (1.15)' are equivalent. In Section 5 we saw that the paths of a spray S are the same with the auto¬ parallel curves of the induced nonlinear connection, but this is a consequence of the homogeneity. Proposition 6.1. tells us that in KCC-theory of type (B) this always happens. Proposition 6.2. The horizontal distribution N is integrable if and only if locally, ■Rjfc = 0 and = 0, (6.16) where Sx* 8x1 Rlti (6.17) oi ^¿_ oi St ~ R’°- d 6 5 6 Proof: N is integrable if and only if the Lie brackets [—7,7-7] and ■=-?] <5 6 °X d 6 belong to the horizontal distribution. But, = and the result follows. We remark that under a coordinate change (6.4) Si _ ok dxP dx* v - Qxi gxk ryi _ Sm 1 Sr d®9 &xi ~R°ldxi 3xm + dt dxi dxT' so that (6.16) are coordinate invariant conditions in J1^. It also follows that the so-called “deviation tensor” or the second invariant of the system (6.1) is given by + (6.18) and it is a (1, l)-type Finsler tensor field. We may compare (4.16) and (6.18) to see the difference between the second invariant Bj in KCC-theory of type (A) and the second invariant in KCC-theory of type (B). 6.2 Variational Equations Let us first note that if f is a C00 real-valued function on Afn+1 then, t_d_L+j?L dt dt y dx*'
KCC Theory 145 while, if f is C°° on dt where a is a path of a semispray S. Secondly, the covariant derivative of a vector field can be expressed as the vertical part, relative to a nonlinear connection, of its complete lift. Let ns exploit this well-known fact as follows: If X = is a vector field on Mn+1 its complete lift and horizontal lift are given by d • , s d dP d On the other hand, if X = Vz(t, z) is a vector field on Mn+1 its complete lift and horizontal lift are given by vc \ d d x x" Under a change of coordinates (6.2), Since the covariant derivative of a vector field X on M is defined as: VX -^ = Xc-Xh = v(Xc) at we have in local coordinates: ™=^±=ff£+w+N<)± dt dt dy* [dt+ ^dyi' ¡¡x^t,^ dt dt 9yi 1 dt 3 ,dyi’ d if X = V’l(^aj)—Consequently, we have (6.19) (6.19)' (6.20) (6.20)! (6.20)2 vxi _ dxi vxj dt ~ dxi dt *
146 Antonelli and Bucataru where X* is either or so that the covariant derivative of a vector on Mn+1 is again a Finsler vector field under (6.2). Using this covariant derivative, the equations (6.15) and (6.15)’ are equivalent to (6.21) Now we are interested to obtain the variational equations of (6.21). Consider the path a(i) for a semispray 5, i.e. cr(t) = (tJxi(t)^ ^) € J1^, is a solution of (6.1) and a variation of the form t — t + T) and z*(t) = xl(t) + ^(t), (6.22) for |i7| small. d The variation vector field is X - -r- + ¿(t,^)-^-- E we evaluate the Lie Qt OXZ bracket ro vci _ Ä + 2^ + +2^? + ~Qyi dt +" dt ’dyv we can determine the variation of (6.21): ^+,^+Ä + Ä-0 dt* +" dt + ~dx^ dyi di (6.23). The reader may compare this with the variational equations (4.13) in Section 4. To obtain the normal form of (6.23) we must compute the Lie bracket [S', Xc] in the Berwald basis (¿, A). We have S = Xc = so that dt dtf [Ml -<f +IS(^> +N^ + + +s5"’>^ Consequently, we have Theorem 6.2. The variational equations (6.23) have the invariant form + (6.24) The reader can compare (6.24) with (4.15) from Section 4.
KCC Theory 147 6.3 The “Film-Space” Approach to Type (B) KCC-Theory In this section we generalize Schouten’s “film-space” concept to include homo¬ geneity of degree two for & and for type (B) transformations. We then derive variational equations and compare them to (6.24) above. Let us consider the system cPx* , i \ —) = 0 i 6 {1,2., n}, where (s’) are local coordinates on a C00 n-dimensional man¬ ifold, AT. Let (T(7? x AP), r, R x AP) be the tangent bundle of the (n+l)-dimensional manifold R x Mn. The local coordinates on T(R x AP) are (ra,/), a € {0,1,... ,n} where a;0 = t. Let T(R x Afn) denote T(Rx AP) with zero-section removed and consider G° = 0 and &&*, ya) = <№&(??, ^). (6.25) Then (6.26) 9 ar» 9 s~y d^~2G is a global vector field on T(R x Af). Since Gi are (positively) homogeneous of degree two in y, we call S the canonical spray. The vertical distribution, denoted Vu C TUT(R x Af), is spanned by { An important supplementary (n-Fl)- Oya dimensional distribution u € T(R x Af) satisfies TVT(R = and is called the horizontal distribution of S. Its Beruiald basis is given by where 6 dt °dyi d’i d d N?-— dxi 'df 6x* ^(3%y°) _ d& ~ dyi' (6.27)
148 Antonelli and Bucataru Remark 6.1: If we consider transformations of type (B) t = S* (t, re*), rank(^) = n, then the equation, y0 = 1, defines an immersed submanifold of T(R x M), which is locally difieomorphic to the first jet bundle of the trivial bundle projection % ; R x M —► R. Remark 6.2: If we restrict Nj, ATj, along yQ = 1, we get Nj» N] where N£ s £* in KCC-theory of type (A). We may remark here that 5* is not o Finsler vector field in type (B) theory. Define the curvature tensor of the nonlinear connection TV by ja 5xa Sxi Proposition 6.3. The nonlinear connection N is integrable if and only if 6 6 d Proof: Since ■=—1 = RL-x—r, the Lie bracket is horizontal, i.e. in N, if _ 6x3 6x<*1 3 dy' and only if R%a — 0. Consider the variation xa = xa of a path of the spray S, Its variational vector field is X = £°— + £*-5—. The variational equations for (6,1) are thus at oxa d? + 9t~+ dx^ +~dyi dt +~dy° dt ° ^ = 0 dt? U Using (6.28) (6.29) we rewrite (6.28) in the invariant form V2f° V2P -4- = 0 and—4- + Blf" = 0 *2 u dt- T "o'» where B*, = R^y0 = + ^ay°, or equivalently, (6.30) (6.31)
KCC Theory 149 Theorem 6.3. Along the immersed manifold = 1 in T(R x M), the variational equations (3.11) become, which are identical with (6.24) for the variation (6.22), and R-kj' Rjo ^o-
Chapter 7 The Classical Projective Geometry of Paths 7.1 Paths, Parametrized Paths We must distinguish between curve and parameterized curve. A curve is to be visualized as a thread, a parameterized curve as a thread together with a distribution of real numbers over the points of the thread. Let Sn be an open, connected simply connected subset of IRN provided with a coordinate system (x), then a parameterized curve is identified with a set of equations ** = /(*), (7.1) where f are C°° (or Cw), not all constants, i = 1,2,..., N. A curve, on the other hand, is represented by ¿-/w (7.2) where are fixed C°° (or C^), not all constants, and tp ranges over all noncon¬ stant C°° (or C*) functions : (a, 5) —► W C .n Dom(/‘) C IR1. Remark: In the notation of charts Sn is written as (U, h) where h : U —► IR77 is a homeomorphism and if (V,k) is another chart, hоk~l :VC\U—> ЖЛ is C°° (or Cw), if V П V / 0, U, V C Sjv* The collection of all such charts is called a C°°-structure on Sn, (or C^-structure on Sn). So a particular choice of y? is a particular parameterization of curve (7.2). We may change this parameterization by writing t-ip^a). (7.3) The resulting equation representing the same paths, each referred to a new parameter, r. 151
152 Antonelli and Bucataru 7.2 The Various Geometries of Paths — Finite Equations By a system of paths in Sx we mean any system of curves definable by equations ? = /(t,a) ¿ = of C°° (or Cw) curves with parameter t and a denotes a set of 2N - 2 (essential) parameters ai, ¿2»•••? a27V-2, which vary from curve to curve, and such that the following spray conditions hold: a) there is a unique curve in the system passing through any two given points in Sk, sufficiently close b) there is a unique curve through any point p € Sn, with dxi/dt\i^o, being (a direction at p) arbitrary. Imagine a system of paths (i.e. a local spray) in S^, We require in addition a reference system A with three components (Ai) coordinate (ar) (A2) a parameter t for each path (A3) a set of parameters a ~ {ai,..., a^-2} to distinguish between the paths. Once A has been selected, the system of paths is represented by the equation (finite equation) (7.4) according to conditions given above. For a given system of paths, the reference system A may be transformed as follows:’ (Ti) transformation of coordinates in Saz (non-singular Jacobian), (T2) simultaneous parameter transformation on all paths i = ^(r,a), (T3) By a transformation of parameters a: a = a(a). (7.5) Suppose (Ti) i = 1,2,3 converts (7.4) into =p\r,a).
KCC Theory 153 This is a representation of the same system of paths in a new reference system, B. • Remark: The condition (T3) does not effect the differential equations of the paths: w (’=$> ‘-1 N The main theorem of J. Douglas (192S) is that (*) with each Hl is p-homogeneous of degree two in p, is adequate to represent the most general system of paths with thé spray conditions. We shall prove this below. The Projective Geometry of a system of paths with spray conditions (i.e. a local spray) is the theory of properties of (7.4) which are invariant under an arbitrary transformation (T2) of the parameterization of the paths, combined with an arbitrary transformation (T3) of the u’s. The Affine Geometry of a local spray is the invariant theory of (7.4) under coordinate transformations (regularX (Ti) combined with parameter transform¬ ations t = ar + /3, (7.6) simultaneously, along all the paths of (7.4), where c^/3 are arbitrary functions (C°°) of the a’s. The relation (7.5) is called an affine parameter change. Generally, if G is a group of coordinate transformations G C GL(IR, n) and H is the additive group defined by (7.5). H is the so-called one dimensional affine group. The corresponding (G, H) invariant theory of a spray is exactly the classical affine connection theory. The case where H is reduced by a = 1, is the classical metric connection theory. We will show, in fact, there exists a (Finsler) metric function F(x, ^) which is constant along spray curves, in this metric theory. Remark: (GL(IR,n), (T2))-invariant theory is classical projective theory of sprays, with Weyl curvature tensors, etc. 7.3 The Various Geometries of Paths — Differ¬ ential Equations Case (A) Affine. Suppose we are given an affine space of paths (*) + a) (7.7)a This is a local spray, of course. Differentiating (7.7)a twice with respect to t we get Pi = = aX’ (a* + ft a) (7.8).
154 Antonelli and Bucataru and = a2_f(at + /?, a). (7.9)a Since there is a unique spray curve through each point in each direction, the 2n equations (7.7)a, (7.8)a are solvable uniquely for the 2n unknowns a, at + 0,a, as functions of x,p. Claim: Substitution of these values in (7.9)* will result in a system of the form - »'(»,»)• p.10). To prove this note that if in (7.8)O each pi is replaced by Xp\ the resulting equations can be written Pi + a) (7.sya and (7.7)O, (7.8)* are the same equations in a/X, ort + 0, a that (7.7)*, (7.8)* are in a, at + 0, a. Hence, at + /?, a have the same values as before, while a has X times the value it had before. Now because of the factor a2 in (7.9)* it follows that the effect of multiplying each p by X is to multiply <P‘xi/dt?‘ by X2, but this is the definition of homogeneity of 2nd degree, thus proving the claim. Conversely, suppose given any system of o.d.e. of the form (7.10)*. Since (7.10)* is invariant under t >-> at + /?, the integral curves are xi — /*(at + /?, a) - i.e. an affine system of paths. Case (B). Projective. The finite equations of a projective system of paths are *’ = /Wa),a] (7.7), where 99 is an arbitrary function of its arguments (C00). Now embed the para¬ meterization a) into an affine space of paths by setting S4 = r + 0,a), a) (7.8)6 (i.e. by writing at +0 in place of t). Differentiating (7.8), twice with respect to t, we obtain p* = a<p'f™ [<Xai + /3, a), a] (7.9)k = a2(^W)2/<<"’[<p(at + ,5,a),a] +a2sp(")/<<'’[9P(at + J0,a),a]. (7.10)b Now (7.8), and (7.9), give 2n equations in 2n unknowns a<p^\ <p(oct + £), a. We can solve these locally for expressions in terms of xi and p\ If we omit for the moment the second term on the right of (7.10),, we have the same elimination theory as in the affine case but with aif№\ a in place of a, at + 0, a. It therefore follows that the first term can be written in the form H^(x,p). Moreover, this form of H* is independent of the form of as a
KCC Theory 155 function of at+/3 and a, since , tp figure as whole symbols in the mechanics of elimination. The functions are determined by the arbitrarily chosen initial parameterization of the paths corresponding to <p(t, a) — t in (7.7)5. Now the second term can be written as i ^"Kat + M ¥?(')(at + /3, a) since we can solve for the 2n unknowns a, at + a in terms of xi and p* in (7.8)5, (7.9)5, we can substitute into (7.11)5 getting the expression piQiX'p)' (7.12)5 where G is independent of index i, but depends on the form of 92 Le. on the parameterization of the paths. In fact, G must be p-homogeneous of degree one. For, as before, the effect of multiplying each pi by A > 0 is to multiply a by A and leave at + /?, a fixed. Hence, the effect on G(a?,p), which is the 2nd factor of (7.11)5 is multiplication by A. We have shown that the projective system of paths has differential equation where £T(ir,p) is actually an arbitrary Junction of its kind, varying with the parameterization of the paths. Here is why: the two systems of o.d.e.s represent the same spray in two different parameterizations. If t, r denote the corresponding parameters on an arbitrary path, we have by (7.11)5 ag№ _ Pt/dr2 __ / dx\ <p№ dt/dr (i)\ ’ dr) so that dr -J H(x,dx) di=Ae.M (7.15)5 (7.16)» along a path 7 of the system of paths. This shows that r is determined along any path of the system 7 up to two arbitrary constants of integration, once K (x, ) is chosen. 7.4 Affine Connections A manifold S^n is affinely connected if, given any two points A and B and a C°° curve AB (parameterized), starting at A and ending at B, there is a linear map $ from tangent vectors at A to tangent vectors at B, i.e. (7.17)
156 Antonelli and Bucataru so that ^H^,AB) = A№,XB)+AW^) (7.18) for any real scalars A, A'. We say £b is obtained from ¿x by parallel translation along the curve AB s N = n B Figure A Let us write ^=Fi(®,^,i) i = l,...,n = .W (7.19) and see what affine connections arise from this infinitesimal definition. We choose a parametrized curve AB, expressed as f = xi(t) and note that substi¬ tution of this into (7.19) will yield a 1st order ode in as functions of t. These n equations can be solved uniquely with initial conditions t = to and C — Ci- This will supply a unique tangent vector to the curve AB at any intermediate point and we say is thereby parallel translated along AB. Furthermore, we require linearity of jF* in f. Thus, (7.19) must read Another natural requirement is that should be independent of the partic¬ ular parameter used for AB. If we replace t by y>(t) in (7.20) we get from Hence, (7.21) is independent of «=> G) (m, g) be ^-homogeneous of degree one, in . Note: <ff (t) is assumed positive for all t. We can now write (7.21) as (7.22) or, equivalently, (7.23)
KCC Theory 157 by homogeneity. Let us now define or Note that are of degree zero in so they are actually dependent on ratios ; no path parameters need by involved. By Euler’s theorem (7-25) We can further specialize $ by requiring to be symmetric in j, k i.e. (7.26) These are necessary and sufficient conditions for the existence, for each i 6 {1,..., n}, of a function ^)j suck that (1/ 2 '(2) (7.27) Note that this existence is local onlyt but can be pasted together to provide global description as in the early sections of this paper. Theorem (J. Douglas). The most general symmetric (spray) connection defined by odes (7.19) is df = i 9^^^, (7.28) 2 (2) where 1%==^^- (7-29) which are the parametrized curves whose tangent vectors dx/dt (velocity) are parallel under the connection (4.13): they are called the parametrized autopar¬ allels of the (spray) connection. From (7.25) taking g = we obtain from Euler's theorem dt2 (2) \ ’ dt ) (7.30) as the differential equations of the autoparallels. We reserve the term geodesics for the metric space of paths for which C* = 0. Geodesics are autoparallel curves of the connection (7.29).
158 Antonelli and Bucataru Remark A: In the case ^-7*^ = ^, ^ = 0, is not —7*^, unless the Levi-Civit& symbols are independent of in which case Tjfc == In general, (*) rfc = 7^++ -difyfa™. The law of transformation of the connection coefficients r(x, i) are dxa dxb ~ _ d&_ra _ d2^ dxi dxk dx* ik dx^dx* (7-31) or equivalently, (Z.32) Remark B: The Levi-Civita symbols do transform as an affine connection, iff gij is independent of as can be seen directly from (*) since is x- independent, iff is also. To sum up, we have shown that affine spray curves are autoparallels of the connection (7.2S) and (7.29). We shall call this connection the spray connection. It is linear and affine iff IDJ^ — 0. 7.5 The Fundamental Projective Invariants An arbitrary change of parameter on the paths is given by H^IT+piH. (2) (1) (7.33) Applying the operator | djdk to (7.33) we obtain FL = 1% + &Hk+rk&i+Pi H 3k (7-34) where Hj-.= \dhH and Hjk := dkHj (ir 2 ft(i) (i/k (7.35) with Euler’s Theorem and (7.34) we obtain r^=I^ + (n + l)Hfc. (7.36) Differentiation by p? yields ^ = ^ + («+1)^* (7.37)a
KCC Theory 159 where PH* ^jkt = Serjfc = djrk( = - dpjQ^dpf. is the so-called Douglas Tensor. Note that (7.37)à (7.38) Thus, we see that the quantities - r« - - ¡rniir:,- - p.39) depend only on the paths themselves, regardless of their parametrization. ^jk is called the fundamental protective invariant, m that every other projective invariant is expressible in terms of and its partial derivatives. We also call these quantities the projective connection after T.Y. Thomas. By contraction of (7.32) we get - dxb ^^(T^ + ftlogA) (7.40) where A is det (f|) and Exercise: ¿Mog A = > which is equivalent to d*A = db(d*xm). Cofactor of daxm in A. (To see this last, expand â&A for fixed b and write out righthand side for the case n — 2.) Using the expression (7.39) for 7rj& and combining (7.31), (7.40) and the law of transformation of H?ka we find = ft d2^ dxl de d& de dxi dxk ^ai> dxa dx^dxk dx? dxk dxk dxj where 6 = log A. Applying di to (7.41) we obtain a tensor (7.42)
160 Antonelli and Bucataru since the last three terms in (7.41) do not depend on x. This is the so-called Douglas projective tensor. Define: A system of paths is protectively quadratic if 3 a parametrization of the paths for which H1 are quadratic polynomials. We show this is equivalent to Claim: Differentiating (7.39) by pe we get - jp(^ p^, (7.43) where IP indicates the sum of three terms obtained by cyclic permutation of j, kyL Hence, if = 0 in some parametrization of the paths, then = 0. Conversely, suppose = 0. First, let fy act on (7.34) to get dtT^ = d^k + + d^Hjk +PUhHjk = irjkt + IP ’ (7.44) Now if we choose a particular H = - -Z— T^pk, (7.45) (1) n +1 k 7 we obtain from (7.44) = 0 = (7.46) (i.e. there exists a parametrization of the paths for which = 0). This proves the claim, since, Remark: The complete obstruction to the existence of coordinate and projective parameter change so that a given spray is quadratic is the vanishing of the Douglas projective tensor,
KCÇ Theory 161 Corollary. If a spray is (projectively) quadratic in some coordinate system it is in any coordinate system. Proof: (ftj-w = 0) Hjy — 0? is true in every coordinate system, if it holds in one, since it is a tensor. 7.6 The Projective Parameter and the Normal Spray Connection Given a local spray relative to coordinate (x) and parameter t : we make a projective change t —► tx, defined by (7.45): ^=-^ïr^fc- Thus, substitution into (7.44) results in the normal spray connection, T* - rj. - 4^ - where clearly, (7-47) (7.47)e (7.47)d r}fc = 7rjfc. (7.47)6 This normal spray connection T has for its coefficients precisely those of the projective connection. The corresponding parameter for this projectively trans¬ formed spray is given by at and this normal spray has the equation drxi _ * dxj dxk dt% ~ *ik dt? dt? ' Remark: If P' is projectively related to a given T then (7.34) holds by definition for some H. Hence (7.36) holds, i.e. (i) } = r;fc + (n + i)^. Suppose both T and P' have the normality condition satisfied, i.e. P“fc = = 0. Then = PkHk — 0 and (7.34) implies Tjk = Tj*. This proves there is only one connection projective to a given P such that normality holds. Also, dt — A-e const.
162 Antonelli and Bucataru so that the projective parameter tx of a normal connection t = tx is unique up to a linear transformation, tf = At + B, Remark: For a quadratic spray (Le. = 0) the values together with determine completely. But, generally 0, so this is not so. But, let us compute the effect of a regular coordinate transformation (x) —* (z) on the projective parameter. Combining (7.36) and (7.40) we find r« ak~ dxk (r26 + (n+l)Æ6 + ^logû) (7.48) soifrjj. = 0 = rjb. Then 1 n+ 1 so that Therefore, ¿Mog A afc(logA)ctafc n+i (7.49) (7.50) Conclusion. The projective parameter s = tx remains the same up to linear transformation under coordinates transformation A = constant and only those. Remark: (1) SL(IR, n) is a natural group of coordinate transformations for a geometry that makes s — tx fundamental. Theorem. Let F be a spray and consider the group (projective) of regular coordinate transformations: afê d-d* ca2z* + d with D = det d ci b1 ... bn~ /0 CA 7^ is a tensor relative to D and only these transformations. Remark: Suppose — 0 and make a ^-transformation to This neces¬ sarily vanishes and the spray curves of the normal connection are straight lines xi = azs + bz, i G {1.. .n} and in the new coordinate we have x* = a1 s 4- as well. But, s is not necessarily s (up to a linear transformation) because A=det(S)=-D* const-
KCC Theory 163 Proof; We see the types of transformation for which the coefficients of a pro¬ jective connection are components of a tensor. Obviously, we necessarily must have from (7.41) Px* ^dxa de dxa de dx'dxi ~ dx* + dx* dx? ’ Let us differentiate this expression by d* to obtain (7.51) Interchanging j and k we have 0 dididkxa = {dj^kxa) • did+ dkxa ■ {djdifi) 0 + M■ (djdkff). It follows that (*) dixa(dkdi9) + {dkdtx“} • djd = dkxa ■ {d^e) + ■ dke. From (7.51) multiplication by dkf) yields dke ■ did.jxa = • dke)+dimity ■ dke) substitution into the RHS of (*) gives RHS = dkX^dte) + dj^idiO ■ dk9) + dix0‘(dje)(dkff). From (7.51) we have didkXa = dkxa • did + 9iXa • dk6 so multiplying by dj9 yields drf ■ didkxa = • {9i9 ■ dj9) + dixa(dk9 ■ djff). Substituting into LHS (*) gives LHS = djSPidkdiff) + dkdix* ■ dj9 = djxa(dkdiff) + dkxa(di9 ■ drf) + dixa(dk9 • 9,6)
164 Antonelli and Bucataru noting the common term in LHS and RHS dixa(dj0 • dk0) we have LHS = RHS is equivalent to d^dkdiQ - dke • $0) = dkx^diO - d5e«dj). This equation must hold for each value of a = 1,2,... > n. There are no summa¬ tions over indices and dx^/dx^ is regular so this is equivalent to dkdiO - dke • diO = 0, (*)' Noting that = (3/)2 has y = —¿nx + c as a solution let us try e~e = akxk + d where d are constants« Substitution into (7.51) we end up with 5J[(aAx*+d)g]+o<g=0. Note: From (7.51) e~9 • didjx* = -djX* • die~9 - dix^e-9 since die“9 = —dtf • e“9. But, dj [(akxk + d) * 0^*] = dj [e~9 • dix*] — e~9 • djdixa + (dje~9) * dixa, from above, = —Qidjxa — <ijdiXa + (dje~9>) * dixa 11 = — ciidjxa so + ¿0 * d»z“] + Oidjx* = 0 from which we see that jl dxa (akxk + d) 4- aix01 = c? = constants and integrating we get _a= dj^x^ + d w Note: This last follows from , , ,-dy , , dx + d {ax + <i)-+dy = c^y=-^ I = elpW^ = e-T sfe <*» = |ao: + d|
KCC Theory 165 p(x) - a ax + d' y' + p(x)y = Q(x) f . v (az + d)c [y(ax + d)] = = c cm? 4- a y(ax + d) = ex + d _ cx + d ax + d* 7.7 Projective Deviation Given a local spray + 2Gl(x,x) =0 i — 1, • • * ti. The quantities ^(x, x) = Gi ~ GÇ(x, ¿)iS (7.52)» n 4" A where GJ *= ¿jGiJ remain invariant under projective changes G* —> G* where Gi = Gi + $(2?,±)ii (7.52)d and #(ar, x) denotes the common value — *r— • ('-52)‘ Of course, is p-homogeneous of degree one in x. We introduce the notation (7.53) It is easy to see that (7.54)» and that < = 0. (7.54)* Now considering £* 4-2G*(s,s) _ № 4- 2CP(xix) (-* — ~ ” dt ) to undergo the projective change (7.52) 6 we get s* 4- 2G* — 2$x* _ 4- 2G7 — 2$xi xi x$
166 Antonelli and Bucataru or & H- 2G* & + 2G> (7.54)c In particular, let us choose $ = — in (7.52);,. Then (7.54)c is written ... x x + 27Tl ^+2ttJ h\t) = — = r-— xl xi Introducing the canonical projective parameter p = A-J¿W^dt + B we can rewrite (7.54)d as dp2 (7.54) j (7.54) « (7.54) , cPxi ■ ( dx\ Since {dPp/dt2)/dp/dt = h(t). Now proceed with (7.54)j forming the variational equations in analogy to the Finslerian case to attain the Projective Jacobi field equations (or Projective Deviation Equations). (7.55)« where Bfc, x) — 2dj^ — drrf-xr + 27r}r7rr - (7..55)6 and (7.55)e is the equiprojective covariant derivative. Note that &rxr = 0 (7.55)«; follows from p-homogeneity of degree two: 2(drTTi)±r - da^xsxT + II. = 2(âr7f')ir — 2Ôa7rtis + 2ir^xT - 7rXær II II 27rj7T^ — 2^71^ = 0. Remark: By the equiprojective geometry of sprays we mean the totality of properties that remain unchanged under: 1) coordinate transformations with Jacobian equal to +1 2) parameter changes t* = i*(i) o)
KCC Theory 167 3) all regular transformation of 2n — 2 essential parameters by which the indi¬ vidual spray curves are identified. Thus, the transformation of coordinates has the effect 2T = 2%^ - (dr^1) xrx*. (7.56)« Note that differentiation of (7.56)a by x shows that 7r]w (= notation of previous chapter) the so-called projective Douglas tensor is a tensor. (This is even true with det(dtf/dx^) — +1 relaxed.) Equation (7.45) can now be written as = IDJfeg - + 3DJjr<% + IDJ^i’). (7.56)b Now* define the equiprojective curvature scalar B=^1 {2dr*r - (7.57) Claim: = -1=2^5. (7.58) To establish this relation take 9* of to get dk&j = - dr^kxr - + 2ir’-rJt9rr +2^k-^j~4^k- Contracting on k and i we have =2^< - dj&i8- + = -a.^+7rjx- But, - ^2 d;B = - i -(dM - = -[ar7rj - which establishes the clam. Now we define equiprojective torsion &jk (7.59)«
168 Antonelli and Bucata.ru and the equiprojective curvature tensor := dhB}k = - dt&ty (7.59)* Claim: bU=°- (7.59). Proof: Use (7.59)*, (7.59)a together with (7.58) and note didhBl - = ^ (- dkB) - a* (- diBj = o. Remark: The torsion and curvature equiprojective tensors are not tensors\ Why? Because (7.50) - the projective parameter p remains invariant (up to linear transformation) if and only if A = const. Thus, if we start with the (7.54)/ and proceed to obtain the deviation equation (7.55)O, it can only be invariant under A = const. (A — +1 is equiprojective SL(n, IR)) so and all others are at least tensors (i.e., equiprojective). It will happen that some of these equiprojective tensors are actually tensors. For example, is a tensor. There are others, as well. Let us now evaluate in terms of GF(x, m) and their derivatives. To this end we must substitute (7.52)a into (7.56)*. The result is using (7.54)a. 3 + TH ~ + nTi + TH “ 2d& ~ + + «71 Then, the equiprojective curvature scalar is seen to be (7.60) s-B + STi + (7.61). where E := IB? = -i-z- (2^(7 - arG?ir + 2GlrGF - GyG?)- n — 1 n — JL (7.61)* Now replace the factor of in RHS of (7.60) by its value from (7.61)a‘ Then differentiate (7.60) by dmi contract on i and m and notice that 1st degree p-homogeneity of the 2nd parenthesis in (7.60) yields “ 77x + nTT 3 (^ " W- (7.62)«
KCC Theory 169 Denote the LHS of (7.62) by Wj, Then wg = _ ©¿j - -L_ (e;t _ (7.62)* does not change under projective turns of the spray <7*. Brom (7.62)a and (7.58) we obtain = + (7-62)c which is also a projective invariant, and is 2nd degree homogeneous in ¿. We have the identities (i)wx = o, (2)w;-o, (3)ârwj = o (7.63) which follows from (7.55)a, (7.57) and (7.58), using the 2nd degree homogeneous of 5J and B in Proof of (1): B*rxr - BFrxr + \(drB)xrè = 0-5^4-1(2^ = 0. Proof of (2): BrT-Bn + \{drB)xT = (n-l)B-nB + B=0. À Proof of (3): 5*wg = dkB^ - + ^dkdjB}#- + contracting on i and k to obtain drwj = dr®- - i (drdjB)^ +1 fyB ■ n = ^b - %b+i dr&B)xr + J d^B Note: WJ = 0 for n = 2. Sketch Proof: Use relation (7.55)^ Bixr = 0 and 9TB^k = 0 = l^jk in (7.59)c.
170 Antonelli and Bucataru Remark: We give a different proof later on. Analogous to the formations of we form and Wijk-.= dhwjk. Note that = |(2w^' - dkwiih) = wi because dkWiÿ^-wi, This last follows from (7.62)c and (7.55)d using (7.64) a (7-64)ft (7.64) e (7.64) « (7.64) e Claim: (7.65) Proof: (=>) is clear from (7.64)a and (7.64)*. To prove (<=) note that (7.64)c and (7.64)a imply Wlhkxixh — so that Wj. => 0. Theorem (H. Weyl). = ^ + ¿1 № - W <7-66) where Bij 8^z and fydjB = ^z^Bij. Proof: Use (7.62)c, (7.64)a and (7.64)*. Note:: 8^ is not the same as In fact, Bij — Bji — ~BZij- = 0 by (7.59)c. In Riemannian geometry, we can write = + (7.67) using Riemannian curvature of the Levi-Civita connection. Corollary. For n-dimension Riemannian manifolds, n > 3, = 0 constant curvature. Proof: Constant curvature Rijkt = K(9jk9it - 9ji9ik) (if n > 3). Therefore, VXjki=K(Sieg:ik-6ikgjt) and A, * = K(ngik-gjk) = (n~l)Kgjk. Substitution leads to + Kfc - g^A = 0.
KCC Theory Conversely, if W = 0, then 71 “ 1 By the skew symmetric identity + Rjw = 0, we have Rjk9i£ ~ KjZptA: + ^ik9jl “ №i£9jk = 0. Transvection with gu yields nRjfc - Mt + ^ikSj - ^9jk = o n where K :— Now plug this in above to get R (**) _ J) (9jk9i£ ~~ 9j£9ik) > 3) Claim: must be constant [K = • Remark: (**) holds for all 2-manifolds. The claim is false for n = 2. Proof: It will suffice to show R Rjfc =» — gjkt n 3, implies R = const. The Bianchi Identity: 4” Rtj£j7l|fe 4“ ^¿J77lfc|Z = 0’ Transvection with gim yields II II II II Rj4|fc Now transvect with g^k to get = 0
172 Antonelli and Bucataru + =0 pm A£|?tx + II But, RJ* = 5 so ;RI'- and n / 2 => B — const. Remark: For the case n = 2, we know that ■R1212 = [511522 - (5i2)2]K and all others are zero’or ±Bi2i2 and K — is not constant in general. This also explains why Weyl’s Tensor is identically zero for n = 2 i.e. = 0 for Riemannian geometry. In the more general spray case & — 0 for n = 2 also and W% = 0.
Bibliography [1] Anastasiei, M. (1994) The geometry of time-dependent Lagrangians, Mathl. Comput. Modelling, 20 (4-5), 67-81. [2] Anastasiei, M. and Bucataru, I. (1997) Jacobi fields in generalized Lagrange spaces, Rev. Roumaine Math, Pures Appl., 42 (9-10), 689-695. [3] Antonelli, P.L., Auger, P. and Bradbury, R.H. (1998) Corals and starfish waves on the Great Barrier Reef: Analytical trophodynamics and 2-patch aggregation methods, Mathl, Comput. Modeling, 27 (4), 121-135. [4] Antonelli, P.L. and Bradbury, R.H. (1996) Volterra-Hamilton Models in the Ecology and Evolution of Colonial Organisms, World Scientific. [5] Antonelli, P.L. and Bucataru, I. (2001) Volterra-Hamilton production mod¬ els with discounting: general theory and worked examples, Nonlinear Ana¬ lysis: Real World Applications, 2, 337-356. [6] Antonelli, P.L. and Bucataru, I. (2001) On Holland’s frame for Randers space and its applications in physics, Steps in Differential Geometry, Kozma, L. (ed.) et al., Proceedings of the Colloquium on Differential Geo¬ metry, Debrecen, Hungary, July 25-30, 2000. Debrecen: Univ. Debrecen, Institute of Mathematics and Informatics, 39-54. [7] Antonelli, P.L. and Bucataru, I. (2001) Finsler connections in anholonomic geometry of a Kropina space, Nonlinear Studies, 8 (1), 171-184. [8] Antonelli, P.L., Ingarden, R.S. and Matsumoto, M. (1993) The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Press, Dordrecht-Boston-London. [9] Berwald, L. (1947) On systems of second order ODE’s whose integral curves are topologically equivalent to the system of straight lines, Annals of Math¬ ematics, 48 (22), 193-215. [10] Cartan, E. (1933) Observations sur le mémoir précédent, Math, Zeitschrift, 37, 619-622. [11] Chem, S.S. (1939) Sur la géométrie d’un système d’equations differentialies du second ordre, Bull. Sci. Math., 63, 206-212. 173
174 Antonelli and Bucataru [12] Crampin, M., Martinez, E. and Sarlet, W. (1996) Linear connections for systems of second-degree ordinary differential equations, Ann. Inst. Henri Poincare, 65 (2), 223-249. [13] Dombrowski,P. (1962) On the geometry of tangent bundle, J. Reine und Angewande Math., 210, 73-88. [14] Douglas, J. (1928) The general geometry of paths, Ann of Math., 29, 143- 169. [15] Grifone, J. (1972) Structure presque-tangente et connexions I, II, Ann. Inst. Henri Poincare, 22 (1), 287-334, 22, no. 3, 291-338. [16] Knebelman, M.S. (1929) Colineations and Motions in generalized spaces, Amer. J. Math., 51, 527-564. [17] Kosambi, D.D. (1933) Parallelism and path-space, Math.Zeitschrift, 37, 608-618. [18] Krupkova, O. (1997) The Geometry of Ordinary Variational Equations, Springer. [19] Kron, G. (1955) A physical interpretation of the Riemann-Christoffel curvature (the distribution of dumping and synchronizing torques in os¬ cillatory transmission system), Tensor, N.S., 4, 150-172. [20] Lackey, B. (1999) A model of trophodynamics, Nonlinear Analysis, 35 (1), 37-57. ’ [21] Matsumoto, M. (1986) Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press. [22] Miron, R. and Anastasiei, M. (1994) The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic Press, Dordrecht-Boston- London. [23] Miron, R. and Anastasiei, M. (1997) Vector Bundles and Lagrange Spaces with Applications to Relativity, Geometry Balkan Press. [24] Schouten, J.A. (1951) Tensor Analysis for Physicists, Clarendon Press, Oxford. [25] Shen, Z. (2001) Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht. [26] Synge, J.L. (1926) On the geometry of dynamics, Phil. Trans. Royal Soc., A 226, 31-106.
PART3
Fundamentals of Finslerian Diffusion with Applications P.L. Antonelli and T. J. Zastawniak
Contents 1 Finsler Spaces 187 1.1 The Tangent and Cotangent Bundle 187 1.2 Fiber Bundles 189 1.3 Frame Bundles and Linear Connections 191 1.4 Tensor Fields 192 1.5 Linear Connections 194 1.6 Torsion and Curvature of a Linear Connection 196 1.7 Parallelism 197 1.8 The Levi-Cività Connection on a Riemannian Manifold 197 1.9 Geodesics, Stability and the Orthonormal Frame Bundle 199 1.10 Finsler Space and Metric 200 1.11 Finsler Tensor Fields 202 1.12 Nonlinear Connections 202 1.13 Affine Connections on the Finsler Bundle 204 1.14 Finsler Connections 206 1.15 Torsions and Curvatures of a Finsler Connection 208 1.16 Metrical Finsler Connections. The Cartan Connection 210 2 Introduction to Stochastic Calculus on Manifolds 213 2.1 Preliminaries 213 2.2 Ito’s Stochastic Integral 216 2.3 Ito Processes. Itô Formula 219 2.4 Stratonovich Integrals 221 2.5 Stochastic Differential Equations on Manifolds 221 3 Stochastic Development on Finsler Spaces 227 3.1 Riemannian Stochastic Development 227 3.1.1 Deterministic Case 227 3.1.2 Stochastic Case 230 3.2 Rolling Finsler Manifolds Along Smooth Curves and Diffusions 233 3.2.1 Deterministic Case 233 3.2.2 Stochastic h-Rolling of Finsler Spaces 237 3.2.3 Stochastic hv-Rolling of Finsler Spaces 239 179
180 Antonelli and Zastawniak 3.3 Finslerian Stochastic Development 242 3.4 Radial Behaviour 246 4 Volterra-Hamilton Systems of Finsler Type 249 4.1 Berwald Connections and Berwald Spaces 249 4.2 Volterra-Hamilton Systems and Ecology 253 4.3 Wagnerian Geometry and Volterra-Hamilton Systems 254 4.4 Random Perturbations of Finslerian Volterra-Hamilton Systems . . . 260 4.5 Random Perturbations of Riemannian Volterra-Hamilton Systems 262 4.6 Noise in Conformally Minkowski Systems 266 4.7 Canalization of Growth and Development with Noise 267 4.8 Noisy Systems in Chemical Ecology and Epidemiology 271 4.9 Riemannian Nonlinear Filtering 279 4.10 Conformal Signals and Geometry of Filters 285 4.11 Riemannian Filtering of Starfish Predation 289 5 Finslerian Diffusion and Curvature 295 5.1 Cartan’s Lemma in Berwald Spaces 296 5.2 Quadratic Dispersion 298 5.3 Finslerian Development and Curvature 299 5.4 Finslerian Filtering and Quadratic Dispersion 300 5.5 Entropy Production and Quadratic Dispersion 302 6 Diffusion on the Tangent and Indicatrix Bundles 319 6.1 Slit Tangent Bundle as Riemannian Manifold 320 6.2 /^-Development as Riemannian Development with Drift 321 6.3 Indicatrized Finslerian Stochastic Development 323 6.4 Indicatrized h-u-Development Viewed as Riemannian 327 A.l Finslerian Isotropic Transport Process 336 A.2 Central Limit Theorem 338 A. 3 Laplacian, Harmonic Forms and Hodge Decomposition 340 B. l Berwald’s Famous Theorem 343 B.2 Standard Coordinate Representation 344 B.3 B2(l) with Constant 345 B.4 Class B2(2) with Constant G^k 347 B.5 B2(r,s) with Constant 348
Introduction The erratic motion of pollen grains and other tiny particles suspended in liquid is known as Brownian motion, after its discoverer, Robert Brown, a botanist who worked in 1828, in London. He turned over the problem of why this motion occurred to physicists who were investigating kinetic theory and thermodynam¬ ics, at a time when the existence of molecules had yet to be established. In 1900, Henri Poincare lectured on this topic to the 1900 International Congress of Physicists, in Paris [94]. At this time, Louis Bachelier, a thesis student of Poincare, made a monumental breakthrough with his Theory of Stock Market Fluctuations, which is still studied today, [36]. Norbert Wiener (1923), who was first to formulate a rigorous concept of the Brownian path, is most often cited by mathematicians as the father of the subject, while physicists will cite A. Einstein (1905) and M. Smoluchowski. Both considered Markov diffusions and realized that Brownian behaviour could be formulated in terms of parabolic 2ad order linear p.d.e.’s. Furthermore, from this perspective, the covariance of changes in position could be allowed to depend on the position itself, according to the invariant form of the diffusion introduced by Kolmogorov in 1937, [64]. Thus, any time-homogeneous Markov diffusion could be written in terms of the Laplacian, intrinsically given by the symbol (covariance) of the p.d.e., plus a drift vector. The theory was further advanced in 1949, when K. Yosida was able to study Brownian motion on a 3-sphere using semi-group methods, [99]. This linear operator approach was fundamental to the Russian school of E.B. Dynkin and led to many interesting theorems connecting geometry of the domain to properties of diffusions, [39]. Soon after in the 1940’s and 1950’s the subject of Stochastic Differential Equations was developed by K. Ito and his school and was used extensively in physics, engineering and finance, according to the method of “adding white noise,” or simply, the noise ansatz. In this period, Ito and McKean’s book on the Brownian path appeared, setting a high standard in the subject, [62]. Then came McKean’s book on stochastic (Ito) integrals [75], the books of N. Ikeda and S. Watenabe [59], and finally that of K.D. Elworthy [43]. They allowed for a systematic study of Stochastic Riemannian Geometry, Such subjects as Feynman-Kac solutions to PDE’s (Chapter 4) and the role of Riemann scalar curvature, 1Z, in the Onsager-Machlup Theory (Chapter 5) could then be used in applications in science and engineering, on problems such as filtering and large deviations [12], [92]. Through the 60’s, 70’s and 80’s Stochastic Riemannian 181
182 Antonelli and Zastawniak Geometry grew into a vast modem edifice because of the efforts of the ltd and Dynkin schools. For up to date reviews see the papers of Mark Pinsky [S3], [84]. By the late 19S0’s, Stochastic Riemannian Geometry had been applied to various subjects, for example, nonequilibrium thermodynamics, [49], [4S], and Stochastic Nelson Mechanics, [82]. Yet the work of some of the contributors of this book on coral reef ecology and chemical warfare in plants and marine invertebrates [19], [16], [17], [3] marked a return to the biological world from which Brownian motion had originally come. That reversion originated from Peter Antonelli’s reading a paper of N. Nagasawa (19S0) on spatial patterns in highly social and mobile monkey troops, [80], which prompted an interest in Nelson’s stochastic (conservative diffusion) approach to Quantum Mechanics and its application to invertebrate developmental models. Furthermore, it was through the study of social interactions in ecology that the importance of Finsler geometry was first realized, [8], [9], [7]. The breakthrough can be phrased in terms of VolterrarHamilton systems, [14], [6]. Letting (X*, N*) denote the natural phase space coordinates in a local chart of the tangent bundle TM, consider thé 2nd order system (not summed) -TihNiNk+rijN* + ei, where all coefficients (possibly) depend on the n3 functions are homogeneous of degree zero in the Nit1 and with smooth initial conditions Xj, TVg, io- For almost twenty years this system has played a major role in math¬ ematical theories of ecology, evolution and development in colonial invertebrates such as corals, starfish, bryozoans and other marine fauna, [7], [16], [17]. The coordinates X* are Volterra’s production variables, whose constant per capita rate of increase is fo, while the second part of the system is a description of how different species or subpopulations of a colonial organism (i.e. castes) №* > 0, grow (rj), interact (rjfc) and react (e*) to external influences. Generally, this scheme must entail competition, symbiosis or parasitism, while predator effects usually require additional equations coupled to 2 • The condition that rjftare functions of ratios of JV*, mentioned above, signals the presence of so-called social interactions, which are higher-order, density-dependent effects. Whereas classical theory would have the n* merely constants, the theory of density dependent social interactions, initiated by the great ecologist G.E. Hutchin¬ son in 1946, [56], [55], found experimental verification in subsequent work of Wilbur, Hairston and others, [96], [95], [50]. In 1991, work with marine biolo¬ gist Roger Bradbury indicated that Hutchinson’s theory must be recast using zero degree homogeneous interactions (i.e. the rjA.), in order to be consistent with this data. Unfortunately, the mathematical approach that Hutchinson had used in 1946 was intractable and his theory lay fallow for more than 40 years. Thus, 1991 marks the birth of a mathematically accessible theory of Hutchinso- nian social interactions, and also the realization that Finsler geometry describes cost-effective growth and physiology in socially interacting colonial organisms
Finslerian Diffusion 183 like siphonophores, ants and other social insects, as well as many species of (Acropora) corals. It also can be applied to the myxomatosis disease epidemic model (Chapter 4). In order to model environmental noise in the Volterra-Hamilton system ft is not enough to merely add white noise to the second part, or population equar tions. The fact can be simply illustrated with a coral colony. This grows mainly due to the sun’s energy which induces photosynthesis in each coral polyp’s en- dosymbiotic algae, allowing the whole colony to produce a calcium carbonate exoskeleton. The coral colony is very sensitive to sunlight, so that random fluctuations in cloud cover have a striking effect on a colony’s exoskeleton pro¬ duction (the X’s), causing erratic variations. The point of this example is that both portions of 2 must be augmented with noise. The question is how to do this. Somehow, the noise term added to the first part, or production equations, must reflect the interactions through which the fluctuations are propagated, for these interactions express the physiology which ultimately produces the cal¬ cium carbonate. Our answer to this is general enough to apply to all uses of the VolterrarHamilton model, and not only the coral example. The noise ansatz we use for the population equations, for the case of no social interactions (i.e. q* do not depend on ratios of population sizes AP/1W), is just the usual addition of white noise of constant variance. On the other hand, a point in production space (X’s) will be displaced by a distance proportional to the magnitude of the perturbation. But where does the concept “distance”, used here, derive from? The answer is that gTowth and physiology are generally cost- effective (or nearly so) so that it is natural to use an a priori given cost-functional whose associated Euler-Lagrange equations will provide the coefficients FJfc (but not generally the other terms) in the £ equations. In the case of the Finsler cost-functional needed to obtain cost-effective social interactions (the depend explicitly on ratios of AT*), the Finsler distance function is used in our noise ansatz in place of a Riemannian one called for in the absence of social interactions. Also, the noise added to the population equations of is Minkowskian, rather than Euclidean or white. Thus, our noise ansatz explains why all constructions take place on the tangent bundle TM or one of its associated bundles. This is the natural setting for Finsler geometry, as developed by R. Miron and his school, [78], [18]. It is here that we develop Finslerian Diffusion Theory (Chapter 3) to be applied in Chapter 4 to Volterra-Hamilton systems with noise for both the nonsocial (Riemannian) and social (Finslerian) types. There is in that chapter a detailed discussion of E. Nelson’s conservative diffusion theory, and filtering problems for Rieman¬ nian Volterra-Hamilton systems are discussed in detail, while the full power of Finslerian Diffusion Theory on TM is brought to bear on the problem of myxomatosis, the European Wild Rabbit disease. In Chapter 5 we prove an Onsager-Machlup formula for a class of Finsler spaces which arise in applications in the theory of evolution in biology, [29]. These are the so-called Berwald spaces with positive definite Finsler metric tensor. Such formulas in the Riemannian case have involved the Riemannian scalar curvature: for example, the term in the asymptotic sojourn time for¬
184 Antonelli and Zastawniak mula for a Riemannian diffusion, [91], or in the nonequilibrium statistical ther¬ modynamics [49], [48]. Also, Riemannian scalar curvature often enters non¬ linear filtering theory, through the Zakai equations. In fact, this is shown to be always true for 2-dimensional signals in Chapter 4. Our Onsager-Machlup formula for positive definite Berwald spaces (the best understood Finslerian spaces) applies to h-diffusions, only. We first formulate a stochastic version of kinetic energy, called quadratic dispersion, which in normal coordinates has the form where £<<{•} is the conditional expectation given (a^, $/<,), where (a;,^) is a Fins¬ lerian h-difiusion. Expanding in powers of h, one obtains -$> as the coefficient of the quadratic term. R is the Gauss-Berwald scalar curvature, which generalizes the familiar Riemannian scalar curvature invariant to Finsler spaces.. This result has not been generalized to more complicated Finsler spaces, largely because our proof, like all known proofs of Onsager-Machlup formulas, depends on the existence of normal coordinates and these exist only in Berwald spaces [38], [14]. Yet, this class of Finsler spaces is important in applications, for example, in coral reef ecology problems such as the crown-of-thoms starfish devastation of the Great Barrier Reef, [19], [16], [17], [3]. An example of how theOnsager- Machlop term enters a Finslerian filtering problem is given in Section 5.4. In Section 5.5 its role in entropy production is examined in detail. In Chapter 6 we answer the question of how the Finslerian Av-Brownian motion is related to the Laplace-Beltrami operator on TM, provided with the Riemannian metric defined by the Sasaki lift (i.e. diagonal lift) of the Finslerian metric tensor, ^»y(x,^)< For the Finslerian Laplacian expressed in Riemannian terms, one should always add to Laplace-Beltrami operators drift fields, coming from two torsion tensors intrinsic to any Finsler geometry, not both of which can be vanishing. The Finslerian Laplacian is, however, intrinsically defined by Finslerian concepts extended to a probabilistic setting. The Laplace-Beltrami operator on TM results only when gij is independent of y, which is exactly the case when the drifts both vanish (i.e. Riemannian). That chapter also develops the diffusion theory on the Indicatrfa Bundle of a Finsler manifold Af, in keeping with the profound insights of E. Cartan, who viewed the subject as one taking place on a sphere bundle associated with the tangent bundle, but with spheres replaced by unit length vectors relative to the given Finslerian metric function. The main result is again that Finslerian Diffusion on the Indicatrix Bundle has the Laplace-Beltrami operator plus non-zero torsion-induced drift fields as generator. The results of Chapter 6 are of importance for short-time asymptotics of Finslerian Diffusions because they reduce the problem to a Riemannian one, albeit with torsion-induced drifts, which is already well-understood. The main obstacle would be the (ordinary) line integrals of these drifts, which enter the asymptotic kernel, [79]. These would have to be numerically estimated is some cases. A. Milgram and P. Rosenbloom (1950) studied harmonic forms on a compact,
Finslerian Diffusion 185 oriented, Riemannian manifold without boundary, using heat kernel methods from p.d.e. theory, [77]. In the Appendix A, we make a start on the Milgram-Rosenbloom program, but for the more general case of Finslerian manifolds. The first requirement is to construct a Finslerian Laplacian, Дд^, acting on p-forms. This operator must live on M and not on TM, as do the generators of hv- and h-Brownian motions. The construction proceeds by first introducing a Finslerian version of a random walk and proving a Central Limit Theorem type result, which gives rise to our Laplacian Ддз« This generalizes Mark Pinsky’s theory of isotropic transport on a Riemannian manifold, [83]. Then, following arguments of D. Bao and B. Lackey, [31], we show that each real de Rham cohomology class of M has a representative form which is annihilated by Дди- For more information on Finslerian Hodge Theory, the reader may consult The Theory of Finslerian Laplacians and Applications, eds. P. Antonelli and B. Lackey, in Kluwer Aca¬ demic Press. Appendix В contains a complete proof of the classification of locally constant Berwald spaces of dimension two. Such spaces play a major role in applications throughout the text. Acknowledgements. Thanks to Paule Antonelli and Joseph Modayil for proof reading and to our friends Robert Elliott, Makoto Matsumoto, Robert Sey¬ mour, and Hideo Shimada who collaborated with us on various portions of the theory we here record. Financial support was provided in part by NSERC. The authors would also like to express their gratitude to Vivian Spak for her excellent typesetting.
Chapter 1 Finsler Spaces 1.1 The Tangent and Cotangent Bundle The Roman letter M will denote a smooth (i.e. C00) manifold of dimension n. It will be assumed Hausdorff, connected and separable in the sense of having a countable base for its topology. The smooth structure is a family ^(M) of coordinate charts {(Ua,ha')} which form an open covering of M, and for which all overlap maps hp o h-1 : n Up) JET, (1.1) are smooth (i.e. C°°), where each ha : Ua —► is a homeomorphism onto an open set of Euclidean rt-space Rn. It is also required that F is mammal with respect to the smoothness property of (1.1): if h^) has non-empty overlap with an element of 5*, then it is itself in 5*. A smooth (i.e. C°°) map f : M -> N, between smooth manifolds is a continuous map for which the collection of functions O f o h-1: ha(Ua) - kp{V0) (1.2) are smooth, when (Ua,ha) € ^(M) and (Vp,kp) € F(N)- Note that is open in and kp(Vp) is open in №. A tangent vector A to M at a point p in M is a map which assigns to each (lfa, ha) € 5*(M) for which p 6 an n-tuple A# of real numbers, such that if (Up, hp) is another chart containing p, then Ap = Ahpoh^^Aa, (1.3) where J(hp o h^1) denotes the Jacobian of the overlap map evaluated at ha(p). Here, Aa = (A*,..., A*) denotes the contravariant components of A relative to the chart (I7a, ha). The collection of tangent vectors at p is denoted by TPM and constitutes a real vector space of dimension n. It is called the tangent space at p. Its operations are induced from component-wise addition and multiplication by a real scalar. None of the concepts described above depend on particular choices of coordinates. 187
18S Antonelli and Zastawniak For any tangent vector A at p E M and choice of chart around p, one can uniquely write *■=* (¿),- 0-0 (Summation on repeated upper and lower indices.) The dual vector space is the so-called cotangent space at p and is denoted T*M. It has the dual basis du\ i = 1,..., n given by (¿,du^ = <5j (1.5) where (,) denotes evaluation of the linear functional dv? on the vector (d/du*) and Sj is the Kronecker delta or identity tensor. Elements of T*M may be uniquely written as B* = . (1.6) The Bf are called the covariant components of the cotangent vector Ba = ..., B%) relative to the chart (Ua, ha). The transformation from one chart to another is given by = (1.7) where Jj(h$ o h*1) is the (i, j)th entry in the Jacobian of (1.3) above. The collection of all tangent vectors to M is denoted TM = Up€Af ?pM while T*M = denotes the collection of all cotangent vectors to M. These are called, the tangent bundle and cotangent bundle, respectively. They are provided with a topology and smooth structure. First require the projection map ?r: TM —► M defined by tt(A)=p iff AeTpM to be continuous, while also requiring the map ha : %_1(EU -»Rn X Rn (1-8) given by (ha{ir{A)\A^ = ha(A) (1.8a) to be a C°° diffeomorphism. The requisite overlap maps are given by fyj o h~x(p, A) = (hp o /£*(?), J(hfi a h^1) • a) (1.9) where p — ha(p) and the Jacobian is taken at p. Since hpoh~l are C°° maps it follows that hp o h"1 are also smooth. Hence, the charts (tf”1^),^) generate a smooth structure on TM. It is easy to see that TM is Hausdorff, connected, separable and of dimension 2n and that the projection map 7T.is smooth. A similar construction can be carried out for the cotangent bundle, T*M, so that it too is a smooth 2n-manifold. Both these bundles are examples of fiber bundles which we now briefly discuss. The basic reference is Steenrod’s book, The Topology of Fibre Bundles.
Finslerian Diffusion 1S9 1.2 Fiber Bundles Recall that a topological transformation group G acting on a space F is, firstly, a topological group, meaning that G is a topological space for which the group operations (01,02) 91 • 92 and 91 0J"1 taking G x G —> <? and G —> G, respectively, are continuous. Secondly, G acts on F (from the left), if (pi • (02 * /)) = . g2) • / for all pi,p2 € G and f € F. Here, (p, /) w p • f is the action map G* F —> F. G is said to act freely, if I - f — f and p • f — f for some f € F implies p — I. On the other hand, if I • f = f and p • / — f for all f € F implies p — then G is said to act effectively. In this case, for any fixed p, the map y —> p • y is a homeomorphism whose inverse is p —> p_1 > y. Therefore, G is isomorphic to a group of homeomorphisms of F. Often G is a Lie group, meaning that G is a smooth manifold and that the two group operation maps are C°°, as well. A nice example of a Lie group is, CL(R,n), the group of n x n real nonsingular matrices provided with the topology it receives from being viewed as an open subset of It has two connected components, these being determined by the continuity of the real- valued determinant function defined on n). A fiber bundle is a 5-tuple (B, tt, B, F, <7) where E, B, F, G are topological spaces and tt : E —► B is a continuous map onto B with the following addi¬ tional requirements. There is an open covering of B, {VQ} so that there are homeomorphisms <f>a:VaxF-.ir-1(Vat) (1.10) for each Va and f)=P for all P € Va (1.10a) and f e F. Furthermore, there are continuous maps haP : Va fl V? -> G (1-11) defined by homeomorphisms of F by ha0(p) = fa^o^0,T>, (1.11a) with ¥>«,₽(/) = ¥>«(₽. /)> (1.11&) so that (1.11a) coincides with a unique element of G. The map of (1.11b) is a map of F into 7r_1(p). Thus, G is a topological transformation group acting (effectively) on F as a group of homeomorphisms. The space E is called the total space of the fiber bundle, B, the base space, G, the structural group and, F, the fiber space. The map 7r is called the projection map of the fiber bundle. The special covering {KJ is called the trivializing cover of the bundle. As stated above, the action of the structural group G on. the fiber is always effective.
190 Antonelli and Zastawniak One can show that = Z (1.12) holds on the triple overlap V# n V$ A Vy. The properties (1.10), (1.10a) and (1.12) characterize fiber bundles up to bundle equivalence. Two bundles with the same base, fiber and structural group are said to be bundle equivalent if their ^-functions are conjugate in G. That is to say, if for each pair (a, /3) there is a continuous map A : Va 0 Vp —► G such that M(p) = A 1(p) • ha0(p)\(p) for all p 6 Va n Of course, A can be defined only when Va and Vp have nonempty intersection. If E, B, F, G are smooth manifolds and 7r is a smooth map of E onto B then A and hap can be taken to be maps, as well. In this case, G acts as a group of diffeomorphisms of the fiber manifold, F. Note that dim E = dim B+ dim F and that % has maximal rank in the sense that its differential or Jacobian is a surjective linear map of TPE to T„^B for each p € E. In the special case where E is exactly B x F, the fiber bundle is said to be a product bundle or that it is trivial. This must always be the case if G — {I}, as can be seen directly from the definitions. However, it can be shown for B a Lie group, that the tangent bundle (a fiber bundle with GL(R, n) as structural group) is trivial. It follows that T*G is trivial, as well, because G is orientable. Here’s why. First, a manifold M is called orientable if its tangent bundle TM is bundle equivalent to the same bundle, but with structural group GL(^,n) replaced by SO(lR,n), the group of rotations. One says the structural group GL(№, n) has been reduced to SO(R, n). Second, if M is orientable, then TM and T*M are bundle equivalent. Last, observe that any Lie group is orientable because its tangent bundle is actually reducible to {/}, because it is a product bundle. An important class of fiber bundles is the so-called principle bundles, which have both fiber and group identical (F — G). Thus, G acts on itself effectively. In fact, this action is free. As an example let us consider (S3, tt, S2,51, S1), the so-called Hopf Bundle, where complex numbers of modulus one (i.e. S1) act by multiplication on the unit quaternions (i.e. S3), the projection map tt is just the quotient map of this action, which is without fixed points (i.e. free). The base space B = S2, the unit 2-sphere, is just the quotient space. The fiber space is S1 — F and the structural group G = S1 acts on F by left multiplication. On the other hand, where K denotes the non-orientable, 2- dimensional manifold known as the Klein bottle, is the total space of a bundle with base space, S1 and fiber space, S1, but whose structural group, G, can not be taken to be, S1. The reason being that, if it could, then the action of S1 on F = S1 C K would be free with orbit space B — S1 which is impossible because of non orientability of K. In fact, the group G can be reduced to 0(1), the orthogonal group of dimension one, i.e. Z%, and the bundle is not principle.
Finslerian Diffusion 191 In the next section we discuss the most important principle bundle in dif¬ ferential geometry, namely, the frame bundle of a smooth manifold. We follow A.I.M. rather closely, in this section. 1.3 Frame Bundles and Linear Connections A frame z at p € M is a basis for TPM, that is, a set {za} a = 1,..., n of n linearly independent tangent vectors at p. We let L denote the set of all frames on M, and let tt: L —► M denote the projection ?r(z) = p, p being the origin of the frame z. The set of all frames with origin p is just tf"1^), the fiber over p. If is a chart on M and d/dx\ i = 1,... ,n, the basis for p € Ua, then a frame z — evaluated at p is written z^d/dx^p. The subset {7T“1 (Ua), c L is a chart on the (n2 4-n)-dimensional manifold L. The fiber is an n2-dimensional submanifold with charts induced from those on L, i.e. (zj). Define the free right action of GL(№, n) = G(n) on L by /3 : L x <?(n) —> L with /?(«. 9) = zg = (x*, zig*) (1.14) and using the summation convention on repeated upper and lower indices. For a fixed g e G(n), 0(z,g) = 0g(z) — zg G L, maps the fiber ^(p) into itself. Therefore, M is the quotient space, L/G(n), of the free right action /3. The 5-tuple (L,7T,M,<?(n),(?(n)) is a principle fiber bundle known as the frame bundle on M. TZL is the tangent space to L at z. There is a subvector space of tangent vectors along the fiber containing z, LZ = {X G T~L|ci7r(X) = 0}. This is just the kernel of the differential of 7r, the frame bundle projection. It is called the vertical subspace of T-L. Now fix z G L and note that z/3 : G(n) —► 7r“'1(p) given by the free right action /3, has its differential, dzfi, mapping the tangent space, Te<7(n). onto L*. That is, the linear map d«/3 maps the tangent space of G(n) at the identity e onto the vertical subspace at z. The former is just the Lie algebra G(n) of G(n) and consists of all n x n real matrices (Af) = A. Then d-fi(A) := Z(A)-. From (3.1) we see that Z(A) = zlA^d/dzl), A=(Ag), z=(xi>4)€L, (1.15) since, dz/3(d/dg“)e = z* (d/dzj). This vertical vector field on L is called the fundamental vector field corresponding to A. What happens to Z(A) under the free right action? Consider the mapping h G C(n) ghg~^ G G(n) for fixed g G G(n). .This is an inner automorphism of G(n) and induces ad(p) : <?(n) —> G(n) of the Lie algebra, defined by ad(p)A = (pJAJ^“1)^) and is called the adjoint representation of g € G(n), Thus, ^/3(Z(A))=Z(ad(^1)A). (1.16)
192 Antonelli and Zastawniak Now, just as there are basic vector fields on L there are basic forms on L, Let’s briefly describe these. Define the map a ; x V -+ TPM (1.17) by a(z,v) = zv = z\y*(d/dx*) G TPM (1.17a) with z — v — v*ea, where V is a real vector space of dimension n with basis (ea), fixed once and for all. Now fix z € L and define the corresponding induced map from (3.4) za : v G V zv G TPM (1.18) and its inverse xOT1: Xi( A)₽ € TPM -+ z~xv e V. (1.18a) Here, ((2“1)*) is the inverse matrix of (zj). We define the basic \-form by 9 = zoTY o cfrr = {(z“1)* cfcrl}ea (1,19) from (1.18), (1.18a) and composition with the differential of the frame bundle projection, Note that since the vertical subspace Lvz is the kernel of drr, we must have, 9{X) = 0, for all vertical fields, i.e. X G Lt. Furthermore, since (Z9)a_1 = ff_1Ga_1)- we have (1.20) The relations (1.16) and (1.20), therefore settle the questions about action map /3 and its effects. 1.4 Tensor Fields Let V* denote the dual space of real vector space V of n dimensions. Thus, V* is the set of all linear maps V -+ R with the basis ea, a = 1,... ,n, dual to that of V, e&, i.e. such that, ea(e&) = Accordingly, e°(v) = va for vector v = vaea G V. Now we have the left action of G(n) on V defined by < : (p, v) G G(n) xV-+gv = (g£vb)ea G V (1.21) with g = (#£) and v = vaea.Thus, g acts on the basis (ea) by p(ea) ® (e^J). This action is effective but not generally free. Similarly to (1.21) we have : {g, v+) G G(n) x V* gv* G V* (1.22)
Finslerian Diffusion 193 given by pv*(v) = v*(p“xv) for all v € V' from (1.21) it follows that gv' = t>6(5_1)«ea, for v* = vaea. The mapping a defined in (1.17), (1.17a)) can be defined for V* by a’ : (z,v*) € ^_1(P) x V* zv* = e T*M (1.23) with z = (s’, zla), = vaea and 7JJM denoting the dual space of the tangent space TPM. Note that a* is characterized by zv^zv) — v*(v). (1.23a) Now we can define the tensors of type (r,$) as the set Vf of all multilinear mappings y x V* x • - x V* —► R provided with the basis (e^;;^r) 9 r defined by et-;.(ec,...,e‘i1...)=^...<5t.. (1.24) This cumbersome notation is usually replaced with XT = V y ® V* V* and s r 4::. = ea®---®e6®... The mapping £ and are also extended to (r, s)-tensors by (1.4a) (1-25) given by gvi(y,... ,«*,...) =w(p_1v p-1«’,...). (1.25a) Thus, for example, for g = (rf) and w = w^e^, € Vj. If (t?) are chart coordinates around point p € M, a basis for TPM is [d/du^p. while the dual basis for T*M is (du’)p according to (1.5). The tensor spaces (TPM)g have bases d/dtf 0 • • ♦ ® du? ®. By a tensor field T on M of type (r,s) we mean a ^-valued function, T, on the frame bundle L defined as follows: (1.26) with T.(y,Tp(zv,.. .,zv‘,...), (1.26a)
194 Antonelli and Zastawniak and 7r(z) = p, where, Tp is smooth as a function of u1 in a chart around p so that it sends u€ M —>TU G (TUM)J. If we let be components of T in coordinates («'*) then fs = Th^)e?;;;, W = ^:r(u)4 • • • (z-1),- • • • • (1-266) In fact, T-.9 = (s-1)^ • • • Tt:::ai ■ ■ ■ (i.26c) so that (1.25a) yields Toi3s=g-'T. (L27) We shall make good use of tensors from now on in our text. The first topic to consider is of profound importance. It is the theory of Linear Connections on Manifolds. 1.5 Linear Connections In a smooth manifold TV a smooth map D *.u € N —> K G TUN where Vu is a subspace of the tangent vector spaces TUN is called a distribution in N There is a special distribution called the vertical distribution Lv : z G L —► Lvz € TZL in the total space L of the frame bundle of M. It is identical to the kernel of the basic 1-form 0 and is spanned at each point z e L by the fundamental vectors Z(A). A distribution T : z G L —► G TSL in the total space L of the frame bundle is called a linear connection in L (or on M), if 1) TzL===r.©Z£ 2) d^(Ts)=r^ geG(n). The subspace Fz of TZL is called the horizontal subspace and X G F- are called horizontal vector fields. A vector field on M induces by scalar multiplication a smooth distribution of 1-dimensional subspaces, since it is a smooth map u G M —► Tu € (TUM)?. Likewise, a 1-form induces via scalar multiplication a 1-dimensional distribution on T*M, since it is a smooth map u G M —> G (TuM)q. The reader should note that on many occasions throughout this text, distributions, tensor fields, connections etc. will be only locally defined. By the lift with respect to F we mean a map ês : TPM T-, n(z) = p such that (1.28) Note that F is invariant under the free right action, according to (2) above, and is a complement of the vertical distribution. For a given linear connection T we can define a G(n)-valued differential 1- form, W, called the connection form of T, by 1) W(Z(A)) = A, A G G(n)
Finslerian Diffusion 195 2) = 0. Thus, (1.16) and the definition of W imply W o d/3g = ad(g~lyw. (1.29) There is therefore, a one-to-one correspondence between linear connections F and connection forms VZ Consider the canonical coordinates in the frame bundle, namely, (uS^) : W~ = (W^s)*) where WS := + *£%<&*), (1.30) where depend only on u\ These h? functions are the so-called connection coefficients of T. Now, (2) in the definition of W leads to (L31) where X = Xi{d/dui)p G TPM. Thus, to each choice of v € V, the standard fiber (Rn) of TM, we obtain a horizontal vector field, B(v), given by B(v)s = 4(zv) = ~ (1.32) with z = (ui, 3*), v = vaec. This B(v) is called the basic vector field corresponding to v G V, Thus, (1.28) gives (1.33) To sum up, a linear connection T yields a connection form, W, supplement¬ ary to the basic form 0, and the basic vector fields B(v) supplementary to the fundamental vector field Z(A), satisfying 1) W(£(A)) - A, WW(vY) = 0 2) ¿(2(A)) =0, 0(B(v)) = t;. (1.34) Now consider the V*-valued function T on L corresponding to a tensor field T of type (r,s) as in (1.26) - (1.26c). The fundamental vector field Z(A) acts on T so that Z(A)(T) = -A-T, (1.35) as follows from (1.26a) and (1.15). For instance, if (r,s) « (1,2) and w G V/, A ■ w is just A%w£. - u&Ag - vl^Ai and d(z~x)?/^ = In a similar manner we define covariant derivative of tensor field T of type (r, s) to be VT, where on the frame bundle L, V?(v) := B(v)(f). (1.36)
196 Antonelli and Zastawniak Thus, VT is a tensor of type (r, s + 1). For instance, for (r, s) = (1,1) we must have B(v2)(T(vi,v*)) =VT(vi,v2,v*)1 Vi,vs eV, v*<=V*. (1.37) Then from (1.32) and (1.26c) we obtain the components of the covariant deriv¬ ative VT as (VT)]k = dkTj + Trrrk - T^jk, (1.38) which is also expressed as Tj.fc in classical notation. 1.6 Torsion and Curvature of a Linear Connec¬ tion Recall that in the Lie algebra G(n) there is defined the Lie bracket, [Ai, As] := A1A2 — A2Ai. It is easy to see, then, that [Z^^Z^-ZQAiMd) (1-39) and [£(A), B(t>)] = B(A • v) (1.39a) where A * v is defined as in the example (1.35). Now consider the Lie bracket of B(vi) and B(vs) and note =B№,u2)) +Z(R(ux,^)) (1.40) where T(vi, f2) G V and K(ui, ^2) € C(n) which can be identified with a tensor of type (1,1). Since R o (33 = g~xR and T o = g^T from (1.16) and (1.34) we are lead to tensor fields R of (l,3)-type and T(«i,V2) of (1,2)-type called the V-curvature tensor and T-torsion tensor, respectively. Use of (1.32) give us for curvature (1) +r^rrA - 07*). where (j/k} denotes the terms obtained from those proceeding by interchanging indices j and k. Likewise, for the torsion we have (2) 3}fc = r’.fc-07fe). Note that both (1) and (2) are local expressions, in a fixed chart on M. Using the classical notation we can observe readily the Ried identity (for contravariant vector fields) Ai;ft;fc-(h/fe) = R)ZlfcAr-A’!r^. (1.41)
Finsleñan Diffusion 197 Similar expressions hold for other (r, s)-tensors. Using the Jacobi identity for the Lie bracket of vector fields, applied to basic vector fields, we get [ [Ufa), + (1,2,3) = 0, (1.42) where (1,2,3) denotes the terms obtained from the first one by cyclic per¬ mutation. Substitution from (1.40) into this leads to the Bianchi identities by splitting into horizontal and vertical parts, namely, T* k + ^Tjk - H§k + (i,j, k) = 0 (1.43) and RfcVik + »Mrlife + (i> J, V = 0- (1-44) 1.7 Parallelism If C : [0,1] —► £ is a smooth curve on the frame bundle, then 7T o (7 = <?, is a smooth curve on M which is said to be covered by C. If the tangent vectors of C are horizontal relative to a linear connection T, then C is called the horizontal lift of C. In a local coordinate chart (t?) on M and the canonical coordinates (tt\^l) in L, ?? is simply (t?(i)) whereas, C is (u*(t), z^tf) and (1.31) shows dtf/dt - X\ dzi/dt = (u(i)) X*. (1.45) A frame field along C on M is called parallel along C, if the smooth curve t —► z(i) is horizontal. A vector field v(t) along (7 is called parallel along C, if v(t) has constant components in a parallel frame field along C. It follows that (1.45) gives a parallel frame field, and if v(i) = vl{t} — z* (t)va, with constant then transvection with va gives dv'/dt 4- («(*)) duk/dt = 0, (1.46) as the equations for a parallel vector field. Generally, for a contravariant vector field A* along (7 in Af, the absolute differential relative to T is defined as DAi := dA? 4- A^k{u(t))duk, (1.47) in a local chart (i?). Infinitesimal parallel displacement of a vector field A re¬ quires DA1 s= 0. Likewise, the absolute differential of a covariant vector field, DAi :== dAi - (u(t)) duk. (1.48) 1.8 The Levi-Civita Connection on a Rieman- nian Manifold Suppose M is provided with a positive-definite tensor g of type (0,2), i.e. for X, Y vector fields g(X,Y)~g(Y,X), g(X,X)>0, (1-49)
198 Antonelli and Zastawniak with g(X, X) = 0, if and only if X — 0. In a local chart on M, (¿7, h) we can OTlte g(X, Y) = gtfdi, Jdj) = g^rf, (1.49«) where gij — gid^d/) and di — d/du\ This so-called Riemannian metric tensor, gij, enables one to define arc lengths along any smooth curve, ul = ux(t), by * - ft‘ <*, (1-50) where du/dt is the velocity vector field of the curve. One is also able to define angles between intersecting curves in M at the point p of intersection, as the angle 0 defined by, cos e =' g(Xp,Y>>) , (1.51) ■ yjgtmY^ where X? and Yp are the velocity fields of the two curves at intersection p. Another important idea is that the Riemannian metric tensor allows us to raise and lower indices of tensors. For instance, H&mjk£ — (1.52) define the purely covariant components of R and (1-53) defines the scalar curvature, R> and the Ricci curvature, R^. Of course, so that ($v) in a particular local chart is the matrix inverse of (p^) in that chart. But, the role of the metric tensor is even more profound than the above, because it can be used to construct a linear connection T which satisfies, via the associated T-covariant derivative, 9ij;k := dkg-ij — grfTik — 9ri^jk — 0- (1.54) This is the Ricci Condition. We say also that a connection T satisfying (1.54) is metrical. The fundamental theorem of Riemannian geometry is simply that there is exactly one linear metrical connection with T^k = 0, on a Riemannian Manifold, (M,g). A connection with zero.torsion is called symmetric. This unique symmetric metrical connection is called the Levi-Civit^ connection, after its founder. It has the local coefficients (i.e. T’s) ijk - 19U{pk9zi + &i9tk - di9ik)* (1.55)
Finsleriân Diffusion 199 More generally, if we relax the symmetry condition on the connection T one can show by direct calculation that T'ijk — 9jr^ik ~~ 'Yijk “F Aijfcj Aijk = q föijk (1.56) Tijk — 9jrTik) where 7yfc = Pjr7iA is given by (1.55). Thus, if the torsion tensor Tjk is specified a priori there is a unique linear connection T which is metrical, i.e. so that (1.54) holds. 1.9 Geodesics, Stability and the Orthonormal Frame Bundle It is often convenient to use a Riemannian metric on a given manifold M, as they always exist. Therefore, there are a lot of linear connections, too, by the results in Section 8 above. It is clear parallel displacement via the Levi-Civita connection does indeed preserve the length of vectors in a Riemannian manifold, because of its metricity. Similarly, the angle 0 between a pair of tangent vectors at p is preserved; as follows from (1.51). Therefore, since g-ij = 1 is preserved under parallel transport, the covariant derivative of dxz/ds = vanishes i.e. D(xz) — 0 or dßx* dx^ dxk The trajectories or solutions of (1.57) are called geodesics. In fact, (1.57) are Euler-Lagrange equations for the (regular) variational problem ds — 0, (1.58) with fixed endpoints and ds given by (1.50). The positive definiteness of g^ ensures that geodesics actually minimize length, locally. Let £*($; s) be a family of geodesics with initial conditions 2^(0; s) and ¿*(0; e) = C(0î e), t » 1,2,..., n. The #*($;£) are C°° relative to e. For |e| < 1 and each i, xi(s\ e) = a*(s) + sV^(s) + s2(* • • ) from which it follows .that 1% (*(r, e)) = rjfc (®(s)) + diTjkeVe(s) + ^(- • • ). Substituting into (1.59) and (1.60) into (1.57) we obtain ds2 Edtr3kv & ¿g -£ljk ¿g ¿g +- ( )• (1.59) (1.60) (1.61)
200 Antonelli and Zastawniak Dividing by s and letting e —> 0, we obtain the deviation equations or Jacobi field (V*) equations, Direct calculations now show (1.62) can be written P2V* ds2 (1.62) (1.62a) which is the invariant form of the deviation equations. Here, JRJW are just the coefficients given in (1) of Section 6. If 0 is a solution of (1.62), (1.62a) so that initially V*(sq) is orthogonal to the velocity field of geodesic 7, then this is true for all 5 > so. Stability of geodesics, in the sense of Liapunov, can be decided by JR, Rij, and JRjw. For example, if JR < 0, then geodesics are (locally) unstable while JR > 0 ensures stability, for the case n — 2. For higher dimensions one must require the so-called sectional curvatures, defined below, all be positive for stability. Just one of these being non-positive yields Liapunov instability. Let Xr := (d/duk) be n mutually orthogonal unit vectors at p € (U, h) on M. Thus, (1.63) and {Xi,..., Xn} is called an orthonormal frame at p. The sectional curvatures K are determined by the set of all pairs (X^X^), by K(r, s) = (1-64) Just as the frame bundle L over M has been defined above using arbitrary frames, one can restrict the frames to orthonormal frames defined by (1.63) for a Riemannian manifold (M, g). However, G£(R,n) would be replaced by O(R, n), the n-dimensional orthogonal group, in the construction. Alternatively, using the fact that O(JR,n) is a compact Lie subgroup, which is also a deformation retract of GL(JR,n), the bundle reduction theorem in Steenrod’s book (ibid.) shows that the orthonormal frame bundle, thus constructed, is bundle equivalent (in GL(R, n)) to L. We shall denote OM, for the orthonormal frame bundle over Af in the rest of this text. 1.10 Finsler Space and Metric Let M be an n-dimensional smooth, i.e. C°°, manifold.Then TM will denote the tangent bundle over M with projection ttt : TM -+ Af, so that 7Tr(y) = a if y € TMX, TMX being the tangent space at x G AT. Given any smooth curve [a,&] <r(t) G M with velocity a(t) G we define the length of cr by b 1(a) = [ L(ff(i),a(i)) dt,
Finslerian. Diffusion 201 where L(xt y) is a scalar function defined for any x € M and y eTMXi which can thus be regarded as a scalar field on TM, L : TM —► BL It is assumed that L(x, y) satisfies the conditions below. (LI) L(x, s/) is positively homogeneous of degree one in yf that is} L(x, kyj = kL(x, y) for any x € Mr y € TMXl and k> 0. We observe that condition (LI) ensures that I(X) is independent of the para¬ meterization of X, as long as the change of parameterization preserves the orientation of the curve. (L2) L(X) y) is smooth at every x € X and y € TMX such that y 0. Note that if L(x, y) is also differentiable at y = 0, then the homogeneity condi¬ tion (LI) implies that L(xy y) is linear in y. Next, let us consider a local chart (z*) : M D U -* on a coordinate neighbourhood U in M. For any x € U and y € TMX, we can write y = yidi> where di = d/dx\ Then (¡e\j0 : TM D —► R2n is a local chart on TM. We call the induced coordinates on TM. Since, as a consequence of (LI), L2(æ, y) is homogeneous of degree two in y, it can be written as L2(z, y) — gij(x, y)yzy^ in terms of the induced coordinates, gij(x,y) being homogeneous of degree zero in y. If gÿ(x,y) is assumed to be symmetric in £, J, then it is uniquely defined by this formula. By Euler’s theorem on homogeneous functions, it is seen that (1.65) It follows immediately that g can be defined globally as a function from the so-called slit tangent bundle TM, that is, TM with the zero section removed, to the bundle T§M of tensors of rank (0,2) over M such that gfay) € T§MX for any x € M and 0 / y € TMX. In addition to (LI) and (L2), we assume that (L3) gijtx.y) is a non-degenerate positive definite quadratic form for any x € M and y € TMX such that y / 0. The latter assumption is clearly a generalization of the positive definiteness condition for the metric tensor in Riemannian geometry. It proves crucial for the existence of a Brownian motion. Definition 1.1. A Finsler space is a finite-dimensional smooth manifoldM equipped with L : TM —► R such that conditions (LI), (L2), (L3) are satisfied. The scalar function L is then called the Finsler metric function and g defined by (1.65) is called the Finsler metric tensor.
202 Antonelli and Zastawniak 1.11 Finsler Tensor Fields We have seen above that the Finsler metric tensor g is a smooth function from the slit tangent bundle TM (that is, TM with the zero section removed) to the bundle 2jM of tensors of rank (0,2) on M such that p(x, y) € T§MX for any x € M and 0 y e TMX. Similarly, the Finsler metric function L is a smooth scalar function from TM to R. This observation gives rise to the following general definition. Definition 1.2. A Finsler tensor field A of rank (m,n) on TM is a smooth function from the slit bundle TM to the tensor bundle T^M such that A(m, y) € T£MX for any x G M and 0 / y G TMX, that is, the following diagram commutes: A TM T^M 7T1 \ / T2 M Here 7ri and tto are the projections of the corresponding bundles over M. The above definition should be compared with the standard definition of a tensor field A on M as a smooth function from M to T^M such that A(x) G T^Mx for any x GM, that is, a smooth section of the tensor bundle T^M: A M T^M ^2 By J^(TM) we shall denote the set of Finsler tensor fields of rank (m,n) on TM. We shall write simply F = Fq(TM) for Finsler vector fields.We shall denote by X£(M) the set of standard tensor fields of rank (m, n) on a manifold M and write simply = Aq (M) for standard vector fields on M. It follows that g : TM —► T$M defined by (1.65) is a Finsler tensor field in J$(TM). It is called the Finsler metric tensor. By condition (L3) of the previ¬ ous section,the matrix (¿fa) representing g in local coordinates is non-singular. Its inverse matrix (g^) determines another Finsler tensor field, which belongs to ^(TM). One can use (¿p) and (¿fa) to raise and lower the indices of any Finsler tensor field in the standard way. As another example, we consider the so-called supporting element 7? € a Finsler vector field defined by 7i(x,y) = y for any x € M and 0 y 6 TMX. 1.12 Nonlinear Connections Let (a;*) be local coordinates on M with the associated induced coordinates (x\ y*) on TM. We shall consider the tangent bundle TTM over TM refereed to as the double tangent bundle. TTMX<U will denote the tangent space to TM
Finslerian Diffusion 203 at (x, y). We shall write di — d/dxi meaning the canonical vector fields on M, and Tdi = d/dxi and Tdi = d/dyL for the induced vector fields on TM in local coordinates. Assigning di »-► one can readily obtain aglobally defined linear mapping v from the space 5* of Finsler vector fields on TM to the space X of standard vector fields on TM, defined locally by A'di *-> AfTdi, A1 being the local components of a Finsler vector field A G 7. The vectors Td^ i = span an n-dimensional subspace VTMx>y in TTMx,y for every (x,y) in the corresponding coordinate neighbourhood, the subspace being independent of the choice of local coordinates. The subspaces VTMx>y form the so-called vertical distribution VTM over TM. The linear mapping v : T —► X is called the vertical lift over TM. Talking of Tdiy let us also mention that, applying this operator to any Finsler field A of rank (m,n), we can get a new Finsler field of rank (m, n + 1) with components On the other hand, trying to use Tdi instead of Tdi as above, presents diffi¬ culties. The mapping di Tdi cannot be extended to a global linear mapping from F to The subspaces spanned by Tdi;i = 1..., n depend on the choice of local coordinates in M. And, lastly, Tdi applied to a Finsler tensor field does not, in general, produce a Finsler tensor field, not even if the given tensor field is a scalar Finsler field. The concept of a nonlinear connection as described below provides a solution to these problems. For any x € M and 0 / € TMX, VTMx,y is an n-dimensional subspace of TTMXt3i. We can write TTMXt2/ = HTMx>y © VTMx,y, where HTMx>y is an arbitrarily chosen n-dimensional subspace and © denotes the direct sum. The distribution HTM over TM formed by all the subspaces HTMx>y is called a horizontal distribution. We write TTM = HTM © VTM. Definition 1-3. By a nonlinear connection over TM we mean any smooth horizontal distribution HTM over TM. For any (x^y) in a local coordinate neighbourhood on TM, there is exactly one vector T bi in HTMXyV that projects onto Tdi. It can be written as TSi - Tdi - Nf (x, y)Tfy. We call N? (x, y) the nonlinear connection coefficients. The vectors Tbi span the horizontal space HTMx>y. We say that Tbi and Tdi form the adapted frame. We can now assign di !-► Tbi. This can be extended to a global linear mapping : T7 —> called the horizontal lift over TM. Corresponding to any nonlinear connection with coefficients Nj(x,y) is the notion of parallelism along a curve a : [a, b] 911-* x(t) € M. We say that a curve d : [a, &J 91 w (x(t), y(t)) € TM with 0 / y{t) e TMx(t) is a horizontal lift of a if a is tangent to the horizontal distribution HTM, that is, dd{t)/dt € HTM^ for all t € [a, &]. In local coordinates this can be written as dyi dt
204 Antonelli and Zastawniak 1.13 Affine Connections on the Finsler Bundle A modern approach to Finsler connections is to represent them as connections on the Finsler bundle FMy a principal bundle over TM whose total space consists of elements (x.j/,z), where x G M, 0 y G TMX, and z = (¿i,...,£n)Aare linear frames in TMX, with the projection 7rp : FM 9 (x, z) i-> (s, y) e TM. The structure group is GL(n), the group of non-singular linear transformations g : Rn —► Rn with composition as the group operation. The right action of GL(n) on FM is Rg(x,y,z) = (x,f/, zg), where (zg)j — zig^ with (#j) being the matrix of g G GL(n) in the standard basis of Rn. Then TM can be identified with the quotient space FM/GL{n). Given a local chart (x*) : M O U —► on M, we define a local chart (x*, y*, zj) : FM D jr^Sr^lT —► ]R2n+n* by writing y — y* di and Zj = zj d^ We call (x*,y\ zj) the induced coordinates on FM. In this way the total space FM can be equipped with the structure of a smooth manifold. The right action of the structure group on FM is then a smooth mapping. The Finsler bundle FM on which to introduce affine connections is used with a view to developing the concept of parallelism for Finsler vector fields. The slit bundle TM is the base space of FM because it is the domain of Finsler fields. The fibres are linear frames in the tangent spaces to M because TM is the range of Finsler vector fields. Once an affine connection on FM is specified, it determines linear frames parallel along a curve a in TM. Then Finsler vector fields parallel along a, and in fact Finsler tensor fields of any rank parallel along <r, are defined as those whose components are constant in any parallel frame along er. We consider the tangent bundle TFM over FM. Given local coordinates (x*) on M, we set Fdi = d/dx1^ Fdi — d/dy\ and F&1 = d/dz^ where (x\ y*, zj) are the induced coordinates on FM. The vectors Fd^ span an n2-dimensional subspace VFMU in TFMU for every u — (x, y, z) in the corresponding coordin¬ ate neighbourhood. The subspaces VFMU are independent of the choice of local coordinates and form the so-called vertical distribution VFM over FM. For any u G FM, we can choose a 2n-dimensional subspace HFMU such that TFMU = HFMU © VFMU. By a horizontal distribution HFM over FM we mean a distribution formed by such horizontal subspaces HFMU. We write TFM = HFM®VFM. Definition 1.4. By an affine connection over FM we mean any smooth hori¬ zontal distribution HFM over FM that is invariant under the right action of GL(n), i.e., TR9(HFM) = HFM for any g G GL(n), TRg : TFM —► TFM being the differential of Rg : FM —► FM. For any u — (x, y, z) in a local coordinate neighbourhood on FM, there is exactly one vector l(Tdi) in HFMU that projects onto Fdi and exactly one
Finslerian Diffusion 205 vector K^di) that projects onto Fd{. These vectors can be written as l(Tdi) = (1.66) l^di) = Fdi-C^x^dL (1.67) We call r%j(x,y) and C^(x,y) the coefficients of the affine connection. That they are independent of z follows from the invariance of HFM under the right action of GL(n). The vectors l(Tdi) and where i =• 1,.. ..n, span the horizontal space HFMU. On assigning Tdi >-> Z(Tft) and Tdi w l^dt), we can extend I to a global linear mapping I: X —> X called the horizontal lift of the affine connection HFM. An affine connection on FM with coefficients T^k(x,y) and Cik(x,y) gives rise to the notion of parallelism along a curve a : [a, 6] 9 t (x(t), y(t)) € TM with x(t) G M and 0 0 y{t) G TMx(ty A curve & : [a, &] 9 t h € FM ^h z(t) being a linear frame in TMX^ is said to be a horizontal lift of a if a is tangent to the horizontal distribution HFM, that is. daffi/dt G HFMzty for every t € [a, 5]. In local coordinates this can be written as pi /_ kA ,A Lkdyi We also say that the linear frame z(t) is parallel along a. Of a Finsler tensor field A G F^(TM) we say that it is parallel along a if the components of A(a(t)) with respect to a linear frame z(t) that is parallel along a are independent of t. For instance» a Finsler vector field A G F is a parallel along a if A(a(t)) — with o? = const. We define the so-called pull-back transformation along a by Ts4 : TMX^ 9 »-> G TMX^, a4 G R for any s,t G [a, 6], where z(t) is a linear frame parallel to a. The above definitions are independent of the choice of the linear frame z(t) parallel along a. A Finsler vector field A G F is a parallel along a if Ttf.tA(a(s)) = A(a(i)) for any s, t G [a, 6]. In local coordinates the condition for A G F to be parallel along <r(t) = (z(t),j/(t)) reads = -rjk(x,y}Ak(x,y)^. - qk(z,y)A\x,y)^. (1.68) An affine connection on FM determines the covariant derivative VzAeF of a Finsler vector field A G F relative to a vector field Z G X. This covariant derivative can be defined by taking any integral curve a of Z, that is, any curve a : [a, 6] —► TM such that da/dt = Z o a, and setting (VzX)(<r(t)) = Um (1-69) for any t 6 (a, &). This defines at every point (x. y) of TM, since there is always an integral curve of Z passing through (x,y). In local coordinates the
206 Antonelli and Zastawniak expression for VgA reads VZA = + rjfcA*) + Z& (^A* + Ak) ] di, (1.70) where Z — ZiTdi + Z^Tdi and A = Aidi. The definition of VzA can be extended in the standard way to Finsler tensor fields A e ^^(TM) of any rank (m,n); cf. (1.73) and (1.74) in the next section. It follows that the condition for a Finsler tensor field A € F£(TM) to be parallel along a smooth curve a on TM can be expressed as (V^A) o a = 0, where Z e X is any vector field on TM such that a is an integral curve of Z, 1.14 Finsler Connections Formula (1.69) cannot be applied directly to define the covariant derivative of a Finsier vector field A € F with respect to another Finsler vector field X E 5, for there is no integral curve a on TM associated directly to X.. One can, however, use a nonlinear connection over TM to lift X to a vector field in X, and then, given an affine connection over FM, use the covariant derivative (1.69) of A with respect to the lifted field. This leads to the following definition, which combines the connections of the preceding two sections. Definition 1.5. A Finsler connection on M is a pair consisting of a nonlinear connection HTM over i*M and an affine connection HFM over FM. Consider a Finsler connection on M whose nonlinear connection HTM has coefficients jV/(x, y) and whose affine connection HFM has coefficients (x, y) and C^(xyy) in local coordinates. Composing the horizontal and vertical lifts h and v of the nonlinear connection HTM with the horizontal lift I of HFMt we oh tain the linear mappings I * F —* X, which act as follows when applied to fy: loh(di) = l(Tii} = FSi-F^x,y)Z3lFdtk, loV(di) = l(rd^) = Fdi-C^x>y)z3lFdlk, (1.71) F6i = Fdi-NU^y)Fdj, where iy (», y} = ly(x, y) - N{ (x, y)Ckj (x, y). • We call N^FijiCij the coefficients of the Finsler connection. It is customary to refer to a Finsler connection simply by specifying the triple (Xf Since either h or v can be used to lift a Finsler vector field X € F to a vector field in X, there are two corresponding covariant derivatives of a Finsler vector field A € F with respect to X defined by VjrA = ^(X)A * ~ (1.72)
Finslerian Diffusion 207 These are called the horizontal and vertical covariant derivatives, respectively. The covariant derivative on the right-hand sides in (1.72) is defined by (1.69). It follows from (1.70) and (1.71) that VhxA = X^S^ + Fi^di, VJjA = Xi ^9^ + 0^) di in local coordinates. The definition of V^A and V^A can be extended in the standard way to Finsler tensor fields A G ^(TM) of arbitrary rank (m,n). Denoting the components of A in local coordinates by we set n (1-73) _ pl V=1 . 771 n (1.74) m=i Then, for any X G the components of V^A and VVXA are an^ 111 "tim7 respectively. Let o*: [a, 6] G t -+ (x(t), ?/(t)) G TM with x(t) G M and 0 / y G TMX^ be a curve in TM, To express the condition for a Finsler tensor field A G to be parallel along a in terms of the Finsler connection, we take a vector field Z G X for which a is an integral curve, that is, Z o a — da/dt^ and consider the Finsler vector fields X, Y G T uniquely determined by the relation Z = h{X} + v(K). It follows that where (1.75) (1-76) It also follows that VZA = V^-A + VyA. Then the condition (V^A) o j - 0 for A to be parallel along a can be written as (V^A + V£A)c(7=;0. (1.77) In particular, for a Finsler vector field A G T7, the latter condition expressed in local coordinates reads
208 Antonelli and Zastawniak which is equivalent to (1.68) via (1.71) and (1.76). As an example we consider the horizontal covariant derivative of the sup¬ porting element rj e 5*, a Finsler vector field defined in Sect. 1.11 by ??(a;, y) = y for any x e M and 0 / € TMX. There is a Finsler tensor field D € (TAf) called the deflection tensor such that = D(X) — D^X^di for any X € It follows that Dj(.x,y) = yi]j = F^k(xty)yk — N^(x,y) in local coordinates. A Finsler connection is said to be deflection-free if D = 0. From now on, when there is no risk of confusing vectors form different tan¬ gent bundles, we shall suppress the indices T and F attached to and d?. 1.15 Torsions and Curvatures of a Finsler Con¬ nection We consider a Finsler connection with coefficients NJ, Fjki Cjk and designate the horizontal and vertical covariant derivatives by j and | as in (1.73) and (1.74), respectively. For any Finsler vector field A G 7, one can easily verify the Ricci identities = A^^^A^T^-A^R^ (1.78) ¿likÄb = A^P^^A^C^-A^P^ (1.79) = A^S^-A^Sfi. (1.80) The five Finsler tensor fields TyRtCyPyS € with components T^ki R*jky Cjki Pjky S^k are called the torsions of the Finsler connection. We have T(XyY) = 7JkXJYkdi for any X,Y € 7, and similar formulae for B(X,Y), C(X, Y), P(X, Y), and S(X, Y).The components C^k of one of the torsion fields turn out to be the same as the connection coefficients Cjk. The components of the other four torsions are expressed in terms of the connection coefficients as follows: /pi _» Z?i ¿jk — ^jk^^kjt = wn-Sktf, sjk = cjk-c^ Pik .== The three Finsler tensor fields R,PyS 6 ^¿(TM) with components R^kly P^kli Sjkl are called the curvatures of the Finsler connection. We have R(A, Xy Y) = R%klAi XkYldi for any A,X,Y € P, and similar formulae for P(A,X, Y) and S(A, X, Y). The expressions for the curvature fields in terms of the connection
Finslerian Diffusion 209 coefficients read = d^SiCij + CLFt? + (1.81) sjkl = dkCij-ikCii +0^-0^. It is customary to use the same letters R, P, S to designate both the curvature fields and the corresponding three torsion fields. This abuse of notation is unlikely to lead to ambiguity, as the curvature and torsion fields have different rank. Below we present an index-free description of the torsion and curvature fields of a Finsler connection. For any Z,W € X such that Z = hX -I- vY and W = hU + vV with X.YU.Ve P, we set V^W = + vVhxV + vVyV e X. We call V the lifted covariant derivative. We define the lifted torsion and curvature tensors f G and R € Aj (TM) by f(Z,W) = VzW-VwZ-[Z,W], (1.82) R(Q,Z>W) « VzVwQ-VwVzQ-V[ZiW]Q (1.83) for any Q, Z, W G X. Please note that formulae (1.82) and (1.82) cannot be writ¬ ten directly for Finsler vector fields without farther elaboration of the concept of the Lie bracket [ •, • ] for such fields. The one-to-one correspondence between vector fields A G X and pairs of Finsler vector fields X, Y e 7 expressed by the relation A = hX + vY induces a one-to-one correspondence between tensor fields in X™(TM') and 2Tn+n-tuples of Finsler tensor fields in In the case of T G A^(TM) the corresponding eight Finsler fields in Pj(TAf) are determined by TQiX.hY) = hT(X,y) + vP(X,y), -T(hX,vY) = My,X) + vP(y,X), f y T(vX,hY) = hC(X,y) + vP(X,y), k } T(vXyvY) = v5(X,y) for any X,Y eP. It is seen that one of the eight Finsler fields on the right-hand side of (1.84) is equal to zero, and the remaining ones are expressed in terms of the Finsler torsion fields T, P, C.P^S G Pj (TM). In the case of the lifted curvature field R G A3 (TM), the corresponding 16 Finsler fields in ^(TM) are determined by R(hA,hX,hY) = hR^X.Y), R(vA,hX,hY) = vR^A.X.Y), —R(hAyhXyvY) = hP(A.Y,X), -R(vA,hXivY') = vP^^X), R(hA,vX,hY) = hP(A,X,y), RlvA.vX.hY) « vP^X^Y), R(hAivX,vY') = hS(AX,y), R^vA.vX.vY) = vS(A,X,Y) . (1.85)
210 Antonelli and Zastawniak for any A,X,Y G Eight of these 16 Finsler fields are seen to be zeros, while the remaining eight are expressed in terms of the Finsler curvature fields R,S,T € 5j(TM). Relations (1.84) and (1.85) can serve as definitions of the Finsler torsion and curvature fields appearing on the right-hand sides. 1.16 Metrical Finsler Connections. The Cartan Connection The theory of Finsler connections presented in the preceding sections is de¬ veloped with no reference to the metric of a Finsler manifold. Here we shall be concerned with Finsler connections compatible with the metric in the sense of the definition below. Definition 1.6. A metrical Finsler connection is one for which the Finsler metric tensor g is parallel along any smooth curve on TM. Given a smooth curve a on TM, we can choose two Finsler vector fields X,Y e 7 such that a is an integral curve for Z = hX + vY eX. According to (1.77), g is parallel along a if + Vyp) o a = 0. Since cr is arbitrary, it follows that V^5 = 0, V^p = 0 (1.86). for any X € F. Conditions (1.86) are necessary and sufficient for a Finsler connection to be metrical. It is said that a Finsler connection is h-metrical if = 0 and v-metrical if = 0 for any X € F. consider the submanifold OM of FM that consists of points u = (ar, j/, z) 6 FM such that x € M, 0 y € TM®, and z == (zi,..., Zn) is a linear frame in TM® orthonormal relative to p(x,j/). This submanifold can be equipped with the structure of a principal bundle over TM with projection ttq : OM 3 (x,i/,z) >-+ (ff,y) e TM and structure group O(ri) consisting of all orthogonal transformations g : Rn —► R” with composition as the group operation. The right action of O(n) on OM is Rg(x,y, z) = (s,3/,z0), where (zg)j — zig^, ($*♦) being the matrix of g G O(n) in the standard basis of Rn. Then TM can be identified with the quotient space 0M/0(n). We shall call OM the orthonormal Finsler bundle. Another equivalent condition for a Finsler connection to be metrical is that HFMU c TOMU for every u 6 OM, (1.87) where HFM is the corresponding horizontal distribution over FM and TOM denotes the tangent bundle to OM. To demonstrate the sufficiency of (1.86), we take a smooth curve c : [a, &] —> TM and consider a horizontal lift cr: [a, 6] -* FM of a, so that a(t) G FM^ and d&(t)/dt G HFM&W for all t G as described in Sect. 1.13. This determines a to within the initial condition a (a) 6 FMff^. We can choose a(a) G OK(a). Then, by (1.87), it follows that a(t) G OM for all t G [a, 6].
Finslerian Diffusion 211 Hence» for all t G [a,&], the components of p(a(t)) in the frame ¿r(t) are equal to i.e., constant. Thus, by definition, g is parallel along <t. which proves the sufficiency of (1.87). To verify the necessity of (1.87), we fix any u G OM and take an arbitrary vector field Z G <V. Let a : [a, 6] —► TM be an integral curve of Z for which u G and let 5- : [a, &] -* FM be the horizontal lift of a with a (a) *= u G OM. Since g is parallel along a, it follows that <r(t) G OM for all t G [a, 6]. Thus (¿2% = d&(a)/dt G TOMu-> where I: X —> X is the horizontal lift defined by (1.66) and (1.67). It foUows that (1.87) holds true, because every vector in HFMu is of the form (ZZ)^ for some Z G X. We note that condition (1.87) enables one to regard the affine part of a metrical Finsler connection as an affine connection over the orthonormal Finsier bundle OM, rather than over FM, the horizontal and vertical distributions HOM and VOM over OM being defined by HOMu = HFMU and VOMU = VFMU A TOMU for every u G OM. We then have TOM ® HOM © VOM. It is easily verified that HOM is invariant under the right action of O(n). We call the resulting connection the induced connection over OM. It will be referred to frequently when dealing with metric Finsler connections later on. It is also true that any affine (i.e. invariant under the right action of O(n)) connection HOM over OM can be extended to a metrical affine (i.e., satisfying (1.87) and invariant under the right action of GL(n)) connection HFM over FM. The reader may wish to check the details. The principal example of a metrical Finsler connection has been introduced by Cartan and called by him the Euclidean connection. Nowadays it is custom¬ ary to refer to it as the Cartan connection. It plays a similar role in Finsler geometry as the Levi-CivitA connection in Biemannian geometry. Below the Cartan connection is defined by a system of axioms due to Matsumoto. Definition 1.7. The Carton connection is the unique Finsler connection that (Cl) is h-metrical, V^g — 0 V X G (C2) is ^metrical, V^g = 0 V X G (C3) its torsion tensor T vanishes identically, T =. 0; (C4) its torsion tensor S vanishes identically, S = 0; (C5) its deflection tensor D vanishes identically, D — 0. We shall denote the coefficients of the Cartan connection by These coefficients can be computed from (C1)-(C5). Let us write (Cl) and (C2) as $k9ij = dk9ij — + &kj9n- (1.88) Since, by (C3) and (C4), F}& = F^ and — CjH, we can find in much the
212 Antonelli and Zastawniak same way as in the case of the Levi-Civita connection that , = ¿9* {SjSik-rSkgtj-Stg^, (1.89) C}* = -^gil (djgik + ¿fcSy — &i9jk) • (1.90) But = didjL2/2, and so (1.90) can be written simply as c*fc = \gildi3jk. (1.91) Since g^fay) is homogeneous of degree zero in it follows from (1.91) that Cj*y* = O (1.92) by Euler’s theorem on homogeneous functions. To determine Fjk and Nj, we write (C4) as NJ = FJfcyfc. Note that the formula (1.S9) for involves Nj via 6j = dj— NJ dj. We substitute this expression in (1.89), multiply by $/*, and use (1.92) to get NJ = Fjfe/ = 7Jfcyfc - CjfcNfj/, (1.93) where 7^ = 5/ &9ik + &k9ij “ di9jk} * (1-94) On multiplying (1.93) by and using (1.92) once more, we obtain = 7^3/^’ Thus, we finally find from (1.93) that nJ = 7j*yfc - cJ^j/y”1. (1.95) The coefficients NpF^CJ* are thus completely determined by (1.89), (1.91), and (1.95) in terms of the metric tensor g^ and its derivatives. This also proves the uniqueness of the Cartan connection.
Chapter 2 Introduction to Stochastic Calculus on Manifolds 2.1 Preliminaries The standard setting for stochastic calculus is a probability space (Q,P,P) equipped with a filtration (Pt)t>o- Here Q is a set, P is a a-field on Q, that is, a family of subsets of Q satisfying the conditions (Fl) G P, (F2) if A € P, then Q \ A G P, (F3) if Ax,A2,then U~xA, G P, and P : P —> R is a probability measure on P defined by the conditions (Pl) P(A) > 0 for every AtF, (P2) P(H) = 1, (P3) if Ai, Ao, ... G T are disjoint sets, then P(U2i — £Xi ^(^)- We shall always assume that the probability space is complete^ that is, if A C B G P and P(B) = 0, then A G P The sets A G P are called events and the elements cj G Q are called element¬ ary events. If a property holds for all cu G A\B, where A, B G P and P(B) = 0, it is said to hold P-almost surely (P-a.s.) on A, In particular, if A ~ Q, then we simply say that the property holds P-a.s. A random variable with values in a topological space M is a function f : Q —► M that is P-measurable, that is, such that ¿"X(B) e P for every B G B, where B denotes the cr-field of Borel-measurable subsets of M. Given a family of random variables ($)*€/ values in Af, we denote by a(^, i G I) the a-field 213
214 Antonelli and Zastawniak generated by the family. It is defined as the smallest cr-field that contains all the sets where i € I and B € B. It follows that c^i € /) C F. We shall write cr(£) for the cr-field generated by a single random variable We say that cr-fields 5i,.,. ,Fn c F are mutually independent if, for any A1eFu...,AneFn, P(Ai A Az A... O An) = P(Ai)P(A2) .. .P(A*). Random variables are said to be mutually independent if the ^-fields <r(^i), * • m are mutually independent. Sometimes it also is convenient to say that a random variable £ is independent of a a-field Q whenever the a-fields a(£) and Q are independent. We denote the expectation of a random variable £ : ft —> by provided the above integral exists. We say that f is integrable if E|£| < oo and square-integrable if B]£|2 < oo, where | • | is the standard Euclidean norm in Rd. Let Q be a cr-field on ft such that G c P and let £ : ft —► Rd be an integrable random variable. A ^-measurable random variable with values in is called the conditional expectation of£ given Q and denoted by E(f|S) if fA^dP = f E(i\S)dP for all AeS. The conditional expectation is unique to within equality P-a.s. and exists by virtue of the Radon-Nikodym theorem. It has the following properties: (El) £ w is a linear map, (E2) 2?(£(C|0)=m (E3) £(£ • 7}\S) = £ • E(r}\G) if $ is ^-measurable (where • denotes the scalar product in Rd), (E4) E(£\S) = P(<) if £ is independent of Q. (E5) .E(P(f |£)|7f) = £($|7f) if H is a a-algebra on ft contained in Q. A stochastic process with values in a topological space M is by definition a function £ : ft x [0, oo) M such that £( •, t) is a random variable for any t > 0. We shall often write £(*) instead of f , t). We can also consider stochastic processes for which the interval [0, oo) is replaced by some other subset of R, for example, a bounded interval or the set N of natural numbers. For any weft, the function [0, oo) 9 t f (w, t)- € M is called a sample path of By a filtration (^i)t>o we mean a family of a-fields, all of which are contained in P, such that c Ft if s < t. In addition, we will always assume that all the sets A € F with P(A) = 0 belong to Ft for every t > 0 and the filtration is right-continuous, which means that Ft = Fs for every t > 0.
Finsleri&n Diffusion 215 We say that a stochastic process £ : ft x [0, oo) —► M is measurable if the set {(w, i) : £(u>, t) € B} belongs to the product a-field F 0 5) for any B e 8. We say that a stochastic process £ is adapted to the filtration (Ft)t>o if C(t) is ^-measurable for every t > 0. Two stochastic processes £ and 7? are called stochastically equivalent if £(t) = 7](t) F-a.s. for all t e [0, oo), in which case the processes are also called modifications or versions of each other. We say that a stochastic process £ : fl x [0, oo) —> M has the Markov property if E (/«(t))|a(m « < *)) = E (/(C(t))|a(i(a))) P-a.s. (2.1) for any 0 < s < t and any bounded Borel-measurable function f : M —> R. Let £ : £1 x [0, oo) —► Rd be a stochastic process adapted to the filtration (Ft)t>o* It is called a martingale (to be precise, an (lFt)-martingale) if £(t) is integrable for any t > 0 and s(ew|p;)=^) p-a.s. for any 0 < s < t. A stopping time is a random variable r : Q -+ [0, oo] such that {r < ¿} 6 Ft for every t € [0, oo]. For a stopping time r, we put Fr — {A € F: A n {r < t} 6 Ft for every t € [0, oo]}. It is easily seen that FT C F is a a-field on Q.If ( : Q x [0, oo) —> M is a stochastic process, then : (u>,t) h* i(cu,t A r(w)) is said to be the process stopped by r, Here A denotes the minimum of two real numbers. Definition 2.1. A stochastic process w : Q x (0,oo) —► Rd adapted to the filtration (Ft)t>o is called a (standard) Brownian motion (or Wiener process) in Rd if the sample paths t w(i) are F-a.s. continuous, w(0) = 0 F-a.s., and the increment w(t)—w(s) is independent of Ftf for any 0 < s < t and has normal distribution such that for any Borel set A C Rd, | • | being the standard Euclidean norm in Since w(t) - w(s) is independent of F, it is independent of Fr for any 0 < r < $. It follows that w satisfies (2.1), that is, has the Markov property. Definition 2.1 also implies that the components w1,..., wd of w are mutually independent. Another important consequence is that S(w(i)|Z,) = w(s) (2.-3) E((wi(t)-wi(s))(w?'(t)-wJ’(s))|j;) = t-s (2.4) for any 0 < s < t. An alternative way of expressing (2.3) and (2.4) is to say that w’(t) and wi(t)uF(t) — are martingales for any t, j — 1,... ,d. By a
216 Antonelli and Zastawniak theorem of Lévy (see, for example, [37], Chap. 7 or [63], Chap. 3), conditions (2.3) and (2.4) characterize Brownian motion. More information on the notions mentioned in this section can be found, for example, in [37], [62], [75], [46], [45], [30], [53], [59], or [63]. These books, and a number of other books on stochastic processes and stochastic calculus, can also be consulted for the proofs of the existence of a Brownian motion in and its properties. 2.2 Ito’s Stochastic Integral Integrals of the form Ç(s) duz(s), where Ç(i) is a stochastic process and w(t) a Brownian motion, cannot be defined in the standard way as Lebesgue-Stieltjes integrals along each sample path t >-> w(t), since the paths of a Brownian motion are well known to be P-a.s. nowhere differentiable and to have infinite variation on any finite interval [0, t]. The difficulty has been resolved by ltd [60], [61], who gave an elegant definition of what is now known as Ito’s stochastic integral. Here we shah present the definition of the stochastic integral withe respect to a Brownian motion w(t) in Rd. The integrand £(t) will be a stochastic process of a certain class specified below with values in the set (Rd)* of linear transformations from to R. We shall write xy = xiy* for any x = (Xi) in (Rd)* and y = (2/) in Rd. By | • | we shall designate either the absolute value of a real number or the standard Euclidean norm in (Rd)*, depending on the context. In the latter case |æ|2 = Ô^XiXj for any x = (xi) in (Rd)*. Definition 2.2. The following spaces of processes are involved in the construc¬ tion of Ito’s stochastic integral: a) The space of simple processes f : Q x [0,oo) —► (Rd)*. A simple pro¬ cess is by definition a stochastic process adapted to the filtration bounded P-a.s., and such that there exists an increasing sequence of real numbers 0 = fe<ti <..♦<<*<.♦. with t* —► oo as n -+ oo such that £(i) = i (it) for any t G (ti-i, t»], (2.5) where i = 1,2,... . b) The space £2 of all measurable stochastic processes £ : Q x [0, oo) (R4)* adapted to the filtration (Pt)t>o such that for every t > 0, ||Î^,,t = £jr‘|<(S)|2<iS<oo. c) The space M2'c of continuous square-integrable martingales f : Q x [0, oo) —► R, that is, martingales with P-a.s. continuous sample paths such that for every t > 0, = W)|2 < 00-
Finslerian Diffusion 217 d) The space P2 of all measurable stochastic processes £ : Q x [0, co) —> (Rd)* adapted to the filtration such that for every t > 0, t j£(s)]2 ds < oo P-a.s. e) The space of continuous local martingales f : Q x [0, oo) —»R, which are defined as processes with P-a.s. continuous .sample paths such that there exists a sequence rn of stopping times such that P{7n < t} —► 0 as n oo for all t > 0, rn < Tn+i, and the stopped process £Tn (t) — ¿(i Arn) is a martingale for each n, The spaces £2 and M2,0 are equipped with the metrics p& (i >»?) = 522_n i1 A - n=l oo (2.6) 2 n (1A ||£ — yyll^c^) > n=l (2.7) In P2 and we define the metrics P?= (£>»?) ■I 1/2’ ds (2.S) 1 A Sup ¿<=M (2-9) Remark 2.1. The choice of simple processes as ones with left-continuous sample paths is not essential for the construction of Ito’s integral with respect to a Brownian motion. Right-continuous sample paths are also admissible. Remark 2.2. To be precise, processes <, rj € £2 should be identified if />¿2 (£, ij) = 0, and £2 should be regarded as a space of equivalence classes. Similar remarks apply to P2, jVt2’c, and Remark 2.3. A sequence Çn 6 P2 converges to f e P2 in the metric pps if and only if f* |Çn(s) - ¿(s) |2 ds —> 0 in probability for every t > 0. Remark 2.4. A sequence converges to in the metric if and only if sup^o^j |sn(-$) - £(s) | —► 0 in probability for every t > 0, in which case the sequence is said to converge in probability, uniformly on compact sets. Theorem 2.1. a) Co is a dense subspace of the metric space £? with metric b) The metric space M2'c with metric & complete»
21S Antonelli and Zastawniak c) £? is a dense subspace of the metric space P2 with metric p?z. d) with metric pM^« is a complete metric space. These results are now standard in stochastic calculus. For the proofs of a) and b) we refer, for example, to [59] or [63]. The proof of c) can be found in [45] or [70]. Assertion d) follows from Theorem 2.1,2 of [68]. We are now ready to construct the stochastic integral, first as a functional A42*c, and then to extend it to a functional I : P2 —> For a simple process f € £o defined by (2.5) we put (/¿Xi) = £?(ti) [w(t A ii+1) - w(t A ti)] (2.10) i=X) for any t > 0. Note that there are only finitely many non-zero terms in the sum for every t > 0. It is easily verified using (2.3) and (2.4) that for any £,ri e £q, and 0 < s < t, b((jo(*)|^) = (zow (2-n) E(|(/e)(t)- (Ii)(s)|2|^) = (2.12) It follows from (2.11) that If is a martingale for any f € £q. It is seen from (2.10) that it has P-a.s. continuous sample paths, and from (2.12) that it is square-integrable. Thus I : £o —► A42*c. Formula (2.10) implies that I is a linear mapping. By taking the expectation of both sides of (2.12), we find that = IKIU»,« for W < > 0, which yields pc^rj) = PM^sti for W f, Tj € £o. This means that I is a linear isometry from the dense subspace £o of ¿2 to the complete space M2>c. Therefore it has a unique extension to a linear isometry from £? to M2'c> which will be denoted by the same symbol I. Definition 2.3. The linear isometry I: £2 —► M2'c is called the Ito stochastic integral on £2. We shall often write jjf(s)dw(s) or just instead of A simple argument shows that formulae (2.11) and (2.12) are true for any i€£2. We need the following lemma to define the stochastic integral as a functional I :P2 —> Mby extending it from £2. Lemma 2.2. The functional I: P2 ‘D £? —* M2 C is uniformly continu¬ ous with respect to the metrics ppz and pM^ induced on £? and Ai2,c from P2 and respectively. Proof: The lemma is a simple consequence of the inequality sup |(ze)(s)|>4<^+p( f[i(s)|2ds> J (»€[0,4] J s UO J
Finslerian Diffusion 219 valid for every f G £2 and £,/2 > 0. The inequality is proved, for example, in [70], Chap. 4 and [45], Chap. 4. As a uniformly continuous functional from £2, which is a dense subspace of P2 in the ppa metric, to the complete metric space JVi^, I has a unique extension to a continuous functional from P2 to which we shall denote by the same symbol I. It is readily verified that this extended functional is also linear. Definition 2.4. The extended linear functional I : P2 —► is called the stochastic integral on P2. We shall often write /J f (s) dw(s) or J’j £ dw instead of (If)(t) for any f G P2. Note that when f G P2, the integral If will, in general, no longer satisfy (2.11) or (2.12). 2.3 Ito Processes. Ito Formula In stochastic calculus we frequently encounter processes f : ft x [0, oo) the form —> R of f(t)-f(O) — / a(s)dw(s)+ / b(s)ds, Jo Jo (2-13) where a: ft x [0, oo) —► (Rd)* and b : ft x [0, oo) —► R are measurable stochastic processes adapted to the filtration (Ji)*>o such that j |a($)|2ds < oo P-a.s. (2.14) [ ]&($)[ ds < oo P-a.s. Jo (2.15) for all t > 0. This means that a is a process of class P2 defined in Sect. 2.2. Similarly, we shall denote by Pi the class of all processes that satisfy the above conditions for b. Definition 2.5. We call a process f : ft x [0, oo) ~> R of the form (2.13), where a G P2 and b G P1, an Ito process. Note that every Ito process is measurable, adapted to the filtration (^i)t>o} and has F-a.s. continuous sample paths. For example, each of the components wi of a Brownian motion in Rd is an Ito process, and so is any deterministic (that is, independent of a; G ft) 'real-valued process with absolutely continuous paths. A multitude of other examples can be obtained with the aid of the Ito formula below (Theorem 2.3). It proves convenient to define the following stochastic integrals withe respect to Ito processes.
220 Antonelli and Zastawniak Definition 2.6. Let where i = 1,2,... be Itô processes such that f (t) - C(0) = a\s) dw(s) 4- [* i>£(s) ds with a* G P2 and 5* G P1, and let rj be a measurable process adapted to the filtration (^t)t>o with P-a.s. continuous sample paths. We put for any t > 0. Here {x, y) — ffixiyj is the ordinary Euclidean scalar product of x = (a^) and y — (^¿) in (Rd)*. The condition that r/ should have P-a.s. continuous paths can be relaxed. For the integrals on the right-hand sides of (2.16) and (2.17) to exist it suffices that 77a1 G P2, rjb1 G P1, and »/(a1, a2) G Pi. The following Ito rules follow directly from (2.16)-(2.18): a) / dwlduP = <FJi, b) [ dufdt — O, c) [ dtdt = O, (2.19) Jo Jo JO Here wl are the components of a Brownian motion in Rd. Theorem 2.3 (Ito formula). Let ..., : Q x [0,00) —► R 5e Zto processes and let f : Rn —> R be a function of class C2. We put £ = (£x,...,i*1). Then f (£) is an Ito process and /(£(<)) - /(?(0)) - jT $/«) + I f* (2-20) for any i > 0. The integrals on the right-hand side of (2.20) exist, since dif(£) and didjf(£) have P~a,s. continuous sample paths, dif and didjf being the partial derivatives of f. The proof of Ito’s formula can be found, for example, in [75], [46], [45], [59], [63], and almost any other book on stochastic calculus. It is customary to suppress the integral sign in equalities involving linear combinations of stochastic integrals on both sides. For example, since /(£(t)) — /(f(0)) = Jq d/(£)> the Ito formula (2.20) can be written as #($) = dif® (2.21) This convention, referred to as the stochastic differential notation, will often be used in the sequel. In this notation the Itô rules (2.19) will be written as a) dw'dw^ = Ö^dt, b) dvfdt = 0, c) dtdt = 0.
Finslerian Diffusion 221 2.4 Stratonovich Integrals The Ito formula (2.20) or (2.21) can be regarded as a stochastic analogue of the change-of-variables rule in ordinary calculus. The second term on the right¬ hand side of (2.20) or (2.21) is a purely stochastic feature known as the stochastic correction. Replacing the Ito integral by the Stratonovich integral defined below, one can dispense with the correction term (more precisely, absorb it into the integral). This is especially important when developing stochastic calculus on manifolds to ensure the correct transformation properties. Another property of Stratonovich integrals, which will also prove crucial to us, is their robust behaviour under approximation by processes with piecewise smooth paths. Definition 2.7. Let $ and ( be Ito processes. The Stratonovich integral is defined by ft ft i ft Jo = Jo ^+2j0 or, in the stochastic differential notation, The corollary below, which follows immediately from Theorem 2.3, states the Ito formula in terms of the Stratonovich integral. Corollary 2.4. If < = (i1,... ,$n), where are Ito processes, and f : Rn —> R is of class C3, then /(«(<)) - / (£(0)) = f dif® o df, (2.22) Jo or, equivalently, df(£) = di/(£) o d^. Note that more smoothness is required of f in (2.22) than in (2.20). This is necessary to ensure that £»/(£) is an Ito process. 2.5 Stochastic Differential Equations on Mani¬ folds We begin this section with the definition of a strong solution of a stochastic differential equation (SDE) in Rn. We confine ourselves to time-homogeneous equations. Then we proceed to discussing SDEs on manifolds, in which case the notion of a solution will be extended so as to include solutions that admit explosions. « Definition 2.8. Let w — (w1,..-,wd) be a Brownian motion in JRd and let ai: Rn —► (Rd)* and b*: Rn —> R be Borel-measurable functions for i = 1,..., n.
222 Antonelli and Zastawniak We say that £ = (S1,. *.,£*) : Q x [0,oc) —> Rrt, where are Ito processes, is a strong solution of the SDE df (t) = <?(£(*)) dw(t) + Fftty) dt (2.23) ifaWeP3, &*(<(•)) ePSaad C(i) - f (0) = f <?(£(«)) &»(«) + f b^(s)) ds P-: Jq Jo for any i = 1,..., n and t > 0. Theorem 2.5. Suppose that a* and b* in Definition 2.8 satisfy the Lipschitz and bounded growth conditions < K\x-y\, 1^)1 <*(l + |s|), (2.24) |^)-^)| < *|z-2/|, \b\x)\ <K(1 + |*|), (2.25) for every x<y € Rn and some K > 0, and let 17 : Q—>Rn be an Po-measurable random variable such that E|t7|2 < oo. Then there exists a unique (to within equality P-a.s.) strong solution f of the SDE (2.23) with initial condition $(0) — rj. Moreover, E|^(i)|2 < oo for every t > 0. The proof of this standard result can be found, for example, in [70], [45], [59], or [63]. Example 2.1. Let w be a one-dimensional Brownian motion and let a,b € R be constants. It is easily verified that the unique solution to the SDE d£(t) = a dw(t) — b£(t) dt with initial condition f(0) = x G R is the Omstein-Uhlenbeck velocity process £(t) = e“wa? + ae~bt [ e6ir dw(s). jo Example 2.2. The unique solution to the SDE d£(t) — £(t) dw(t) with initial condition ¿(0) == 1, w being a one-dimensional Brownian motion, is the so-called exponential martingale $(t) = exp[w(t) -t/2]. According to Definition 2.8, a strong solution to a SDE must be defined for all t > 0. This condition proves too stringent for SDEs on manifolds, in which case one frequently encounters solutions that ‘explode’ (that is, escape to infinity in finite time) with non-zero probability. Note that the standard global Lipschitz and bounded growth conditions (2.24) and (2.25), which ensure- the existence and uniqueness of a solution for all t > 0, do not readily generalize to manifolds. As solutions of a SDE on a manifold M we must therefore admit stochastic processes defined on a random interval [0,r), where r is a stopping time.
Finslerian Diffusion 223 Definition 2.9. Let w = (w1,.. .,wd) be a Brownian motion in JR* and let Ao, Ai,,.,, Ad GAf be vector fields on a finite-dimensional manifold M. A pro¬ cess £ : H x [0, r) —► M with r : Q [0, oo] a stopping time such that a) 0 < r P-a,s., b) £(t) : Q D {i < r} —► M is Ji-measurable for every t > 0, c) the sample paths [0,t) 9 t £(t) g M are P-a.s. continuous, is called a solution of the SDE = AotëW) dt + A^(t)) о dt?(t) if for every f G Cq°(M) and t > 0, /(«*))-жо))= f(Ao/)(i(S))^+ Лл/)«(а)) JO JO (2.26) о dwl(s) (2.27) P-a.s. on {t < t}, Oq°(M) being the set of smooth functions f : M —► JR with compact support. Remark 2.5. We shall often refer to the SDE (2.26) by writing it in local coordinates as (*) - 4(i(i)) dt + A&(t)) o where Aj are the components of the vector fields A,. The standard theorem on the existence and uniqueness of solutions of SDEs on manifolds reads Theorem 2.6. For any ^-measurable random variable 7}: О —> M there exists a solution $ x [0,т) M to the SDE (2.26) with initial condition ¿(0) = 77 such that if f : Q x [0,r) —► M is another solution to (2.26) with ¿(0) = £(0) P~a.s.f then r>r and C = f on [0,t) P-a.s, (2.28) The proof of Theorem 2.6 can be extracted from [59], [43], or [68]. Definition 2.10. A solution $ : П x [0, r) —> M to (2.26) that satisfies condition (2.28) is called a maximed solution and the corresponding stopping time r is called the explosion time of £. The theorem below is concerned with the behaviour of a maximal solution to a SDE near explosion time. It is a generalization of the well-known result for ordinary differential equations and, in fact, justifies the term ‘explosion time? Given a function f : [a, 6) —► M, we shall write Ит^ь/(¿) = 00 if for any compact set К с M there is s G [a, 5) such that /(t) G M \ К for all t G [s, 5).
224 Antonelli and Zastawniak Theorem 2.7. If £ : £7 x [0, r) —* M is a maximal solution to (2.26), then P — oo or lim ¿(t) = oo j- = 1. The proof of the above result follows, for example, from Theorem VII.6 of [43]. An important consequence of Theorem 2.7 is that for any maximal solution < : £7 x [0,r) —► M of (2.26) on a compact manifold M, r = oo P-a.s., that is, there are no explosions. The above definitions and theorems can readily be extended to the case of initial conditions at s > 0. Given any s > 0 and x G M, we denote by t >-* t) the maximal solution to the SDE (2.26) with initial condition £($, x; s) == x. Let r($, x) be the corresponding explosion time. „ Theorem 2.8. The system of maximal solutions £($, x; ♦) : Q x [s, r(s, a?)) —► M, where s > 0 and x G M, has a version (for which we retain the same symbols £(s, x; i) and t(s, x)) such that a) i(s, f (r, x; $); t) = £(r, x; i) P-a.s. on {t< r(r, x)} for any Q <r < s <t; b) M(s, t) = {x € M : t < r(s,x)} is an open subset ofM for every 0 < s <t and £(s, ♦; t) : M(s, t) —> M is a C^-diffeomorphism onto an open subset of M, P-a.s.; c) £(s, z;t) is P-a.s. continuous as a function of (s,x,t), and so are its dif¬ ferentials of any order withe respect to x. For the proof we refer the reader to Sect. VHI.2 of [43] and Sect. 4.8 of [68]. FYom now on we shall always use the version of £(s,x;t) described in The¬ orem 2.8 and refer to it as the stochastic flow on M associated to the SDE (2.26). An important consequence of Theorem 2.8 a) is that for any fixed r > 0 and a? € M, £(r, x\ ’) satisfies the Markov property (2.1) for any s and t such that r < s < t, P-a.s. on {t < r(r, a;)}. Finally, we introduce the notion of the differential generator of $(s, x\ t). On evaluating the Stratonovich integral in (2.27), we find that for any f G Cq°(M) / (£(s,a:;t)) -x = £ (^SijAiAjf + Al)f^^(s,x-,u)') du (2.29) + (A7) (€(«, x-,«)) dw(u) P-a,s. on {t < r(s,x)}< We extend f(£(s9x*,t)) to a process defined for all t > $ by setting it equal to zero if t > t(s,x). By Theorem 2.7, since f has compact support, this yields a process with P-a.s. continuous trajectories. The processes (Aif) (£(s,x;i)) and (^ijAiAjf + Acf) ($(s,z;t)) can be extended
Finslerian Diffusion 225 to any t > s in the same way We retain the same notation for the extended processes. Then (2.29) becomes an equality satisfied F-a.s. on the whole set ft. By taking the expectation of both sides of (2.29), we get Ef(£(stX',t))-x = E (i(s.x;tt)) du. It follows that (Df№) = Km x = Q&AiAif + Ao/) (x). The operator D = AiA$ +Ao is called the differential generator of £($, x; t).
Chapter 3 Stochastic Development on Finsler Spaces 3.1 Riemannian Stochastic Development In this section M will be a d-dimensional Riemannian manifold with metric tensor g(x)j x € M. The coefficients of the corresponding Levi-Civit& connection V will be denoted by 'ijk = + &k9jn — &n9jk) • (3*1) 3.1.1 Deterministic Case Let a; be an integral curve of a vector field X on M, that is, dx . . a’*» A vector field Y on M is said to be parallel along x if (Vxr)(x)=0. (3.2) Putting y = K(z); we can write (3.2) in local coordinates as The horizontal lift of a vector field X = X*di from M to TM is a vector field h(X) on TM defined by A(X)(®,v) = X*(x) (di - . 227
22S Antonelli and. Zastawniak An equivalent way of stating condition (3.2) is to say that is the horizontal lift of X in the sense that, = (3.4) For a smooth curve (xfy) on TM such that (3.4) holds we say that y is a horizontal lift of x to TM. Such a y is defined uniquely for each initial condition 3/(0.) € TMx(0). Equation (3.4) implies, in particular, that y(t) e TMX^ and in local coordinates. The mapping 9vhy(s,v;t) G TMX^ where y(s,v; ■) is the horizontal lift of x to TM such that y(s, s) — vy is called the h-parallel transport along x. The following proposition means that is a unitary mapping from TMX^ to TMX^. Proposition 3.1. Let x be a smooth curve on M. If y and y are horizontal lifts of x to TM, then dij^y^ = const. Proof: Differentiate gijfâÿÿ* with respect to t, use (3.3) to express and and apply the metricity condition of the Levi-Cività connection = dfcSij ~ dinYkj ~~ Snj^fik = 0 (3-6) (which follows from (3.1)) to show that (pv(®W) = o, which proves the claim. Let OM be the orthonormal frame bundle over M and X a vector field on M. The horizontal lift of X to OM is a vector field l(X) on OM defined by Z(X)(a:,z) = *’(*) (ft -^zfä) . Let x be a smooth curve on M and (¡r, z) a smooth curve on OM. We shall call z a horizontal lift of x to OM if (3.7)
Finslerian Diffusion 229 Here A is a vector field on M such that a; is an integral curve of A, In local coordinates equation (3.7) takes the form <¿4 dt (3.8) where zi, ..., zj. are the vectors forming the frame z. An equivalent condition is that each vector Zn is a horizontal lift of x to TM, that is, ^ = /l(A)(x,^n). (3.9) for n = 1,..., d. However, this equivalence will no longer be valid in the case of a Finsler manifold. The lift from M to OM makes it possible to lift any smooth curve x from M to any tensor bundle 7£M. Namely, for a smooth curve (a?,u) on T£M we say that u is a horizontal lift of x to T£M if for any horizontal lift z of x to OM the tensor u(t) has constant components in the frame z(t), This horizontal lift u depends on the initial condition u(0) € but not on the choice of z. Xt is clear that for TM ® T^M we obtain the same horizontal lift of x as that defined by (3.4). The parallel transport of tensors along x is defined by where v;«) is the horizontal lift of x to T^M such that u(s, v; $) — v. Let M and M be ¿-dimensional Riemannian manifolds. Suppose that they start rolling against one another without slipping, the point of contact tracing smooth curves x and x on M and M, respectively. We take horizontal lifts z and z of x and x to OM and OM. The condition that no slipping is involved means that the components of the velocity vector — in the frame z are the same as those of in 5. This can be expressed by the differential equation * dt * <ü ’ (3.10) where < and Ç are the dual frames to z and z. If the initial conditions z(0), z(0) and 5(0), ¿(0) are fixed, then this defines a one-to-one mapping between smooth curves x and x on the two manifolds, which is known as (Riemannian) rolling, A typical situation is when M = In this case z is constant and can be chosen so that z^ = 4 (the canonical basis). Then equations (3.10) and (3.S) reduce to (3.11) ¿4 dt (3.11a) For any initial conditions x(0), z(0) the system (3.11) defines a smooth curve (z, z) on OM called the (Riemannian) development of x. In particular, if x is a straight line in then x will be a geodesic on M.
230 Antonelli and Zastawniak 3.1.2 Stochastic Case Let M be a d-dimensional Riemannian manifold M. In the previous section we considered the horizontal lift of a smooth curve from M to TM. This will now be extended to the case -when the curve is replaced by a diffusion x on M. We shall approximate the diffusion by a stochastic process with piecewise smooth sample paths, take the deterministic [0,1] —► M, where V is a neighbourhood of the diagonal {(z, x): x 6 M} in M x M, such that (11) I fax fa = x for all t € [0,1], (12) Z(®,a/,0) = x and /(x,a/,i) = s', (13) d^Ifax'fa/dt™ = O fa™ fax')) for m = 1,2,3, where dfax') is the distance between x and x' measured along the shortest geodesic joining the two points. There is a multitude of interpolation rules to choose from. A natural one, which will be used in what follows, is the geodesic interpolation rule such that t h I fa x\ t) is the shortest geodesic on M starting from x at t = 0 and arriving at xl at t — 1, see [44]. In what follows x will be a diffusion on M satisfying by the SDE dx = Ao(z)di + Am fa o dwm, (3.12) where Ao?Ai,...,Ad are vector fields on M and w is a standard Brownian motion in Let 7r: 0 — ¿o < <i <tz < ••• be a division of the time interval [0, oo] with diameter M= sup [ti-ti-il. We define a piecewise geodesic approximation of x by for any t € ii] and ¿ = 1,2,... , The sample paths of xv are continuous piecewise geodesic curves on M. Each curve of this kind can be lifted to TM by taking the horizontal lift of each smooth segment and pasting the resulting pieces together into a continuous piecewise smooth curve yv on TM. Each smooth segment of y^ satisfies the differential equation The following result is well known, see [44].
Finslerian Diffusion 231 Proposition 3.2. As |tt| —► 0 the approximation (x%, yn) tends in probability uniformly with respect to t in any compact set to a diffusion (x,y) on TM satisfying the SDE dyl = Wj? ° dxk, (3.13) This SDE can be expressed in coordinate-free form as dy = h(AQ)(x, y)dt + у) о dw™. (3.14) If (x, y) is a diffusion on TM satisfying (3.13), then у is called a horizontal lift of x from M to TM. Proposition 3.1 can be extended to diffusions as follows. Proposition 3.3. Let x be a diffusion on M defined by (3.12). If у and у are horizontal lifts of x to TMf then 9^(х)угУ^ = const P-a.s. Proof: Compute the stochastic differential d (gij(x)yiyj>) with the aid of the Ito formula, using (3.13) to express dyi and dip and applying the metricity condition (3.6) of the Levi-Civit& connection to get d (,9ц(¿)yiy>') = (dkgi^y1^ о dxk + дцу> о dy1 + д^у* о dy? = {dkgij - ТадЛ» - УТ' ° dxk = 0. It follows that д^{х)угу^ is constant P-a.s. By Theorem 2.6 the horizontal lift у of x to TM exists for any initial con¬ dition 2/(0) G TMa.(o) and it is unique in the sense that if у is also a horizontal lift of s and P-a.s. 2/(0) = y(0), then P-a.s. y = y. Неге у and у are understood as maximal solutions of the SDE (3.14). Moreover, Proposition 3.3 implies that the length of у is constant P-a,s., so by Theorem 2.7 the explosion time of у must be the same as that of x. The mapping JXr,t :TMX(9) Э v h+ y(s,v;t) 6 ТМхщ, where ?/($, v; •) is the horizontal lift of x to TM such that y($, v; s) = v, is called the stochastic parallel transport along x. Proposition 3.3 implies that Пч<д is P-a.s. a unitary mapping from TMX^ to TMX^. A diffusion x on M can also be lifted to the orthonormal frame bundle OM in much the same way as a smooth function can. For a diffusion (x, z) on OM we say that z = (zi,..., z*) is a horizontal lift of x if to OM if each TW-valued diffusion Zm is a horizontal lift of x to TM, that is, dzn = A(Ao)(x, Zn)dt + h{A^(x, z^} о dw™
232 Antonelli and Zastawniak for each n = 1,..., d. An equivalent way of writing this condition is dz = Z(Ao)(a;, z)dt + l{Am){xy zfo du™ or, in local coordinates, ¿4 = -7>fc(®)4 ° <&*■ (3.15) These equations should be compared with (3.7), (3.S) and (3.9). Theorems 2.6, 2.7 and Proposition 3.3 ensure that z exists and is unique to within P-a.s. equality for any initial condition z(0) € OMX(0) that the explosion time of z is the same as that of re. We can also define stochastic versions of the horizontal lift of a diffusion to any tensor bundle T£M and the parallel transport of tensors along a diffusion. Namely, for a diffusion {x,u) on T^M we say that u is a horizontal lift of a diffusion x from M to T^M if for any horizontal lift z of x to OM the components of u{t) in the frame z{t) are F-a.s. constant. The stochastic parallel transport of tensors from T%M along x is defined by n»,t: W(>) => v I-» e TSMr(i), where u(s, v, •) is the horizontal lift of a: to T%M with initial condition u(s, v, s) = 27. Let M and M be d-dimensional Riemannian manifolds. The notion of Riemannian rolling can be extended to the case of ‘rolling’ the manifolds along diffusions x and x on M and M, respectively. Here we assume that x is given by the SDE (3.12) with vector fields Ao, Ai,..., Ad and x by a similar SDE with vector fields AotAi,..., Ad. We take z and z to be horizontal lifts of x and x to OM and OM, respectively. Then the no-slipping condition (3.10) can be generalized as follows: o = (j o d^, (3.16) where £ and Q are the dual frames to z and 5. If the initial conditions x(0), z(0) and ¿(0), 5(0) are fixed, then the system of SDEs (3.16) defines a one-to-one mapping between diffusions x and x on the two manifolds. This mapping is called the {Riemannian) stochastic rolling. It is independent of the choice of z and 5 as long as (3.16) is satisfied. In particular, in the most typical situation when M = and x — w is a Brownian motion in equations (3.16) and (3.15) reduce to dx^z^odw”, (3.17) ¿4 = -4fc(a:)44l 0 (3.17a) see (3.11). For any initial conditions x(0), z(0) the system of SDEs (3.17) defines a unique (to within equality F-a.s.) diffusion (x, z) on OM, which is called the {Riemannian) stochastic development
Finslerian Diffusion 233 It turns out that x has the Markov property (2.1) and the probability law of x depends only on s(0), but not on 2(0), see [59], [43] or [44]. From (3.17) we find that dxl = z\dwrl + ^dz'ridwn = z\dwn - ^kz&^dwmdwn = dx'dx3 = zinz^dwTLdwrri = g^dt, since dwmdwn = 5Tnndt and = gij. Taking any smooth function f from M to R and applying the Ito formula, we compute d/(x) = (dif)dxi+ ^(didjf')dxidx^ = (difrtdw* +(didjf - ^dkf) dt. It follows that the differential generator of x is D = 5A = Iff« (a&f - rf3.(x)dkf), where A the Laplace-Beltrami operator, A diffusion x on M with differential generator 5 A is called a Brownian motion on Af. 3.2 Rolling Finsler Maniiolus Along Smooth Curves and Diffusions From now on M will be a d-dimensional Finsler manifold with metric func¬ tion L(ic,y), metric tensor gij(x,y) and the Cartan connection with coefficients 3.2.1 Deterministic Case On a Finsler manifold the notion of parallelism along a curve splits into two alternative concepts, namely, /¿-parallelism and hv-parallelism together with the corresponding horizontal lifts, rolling and development. To begin with, we shall introduce these concepts in the deterministic setting. This will then be used to set up an approximation procedure, which gives rise to stochastic hr and hv-horizontal lifts and rolling along a diffusion. /¿-Rolling Let X be a Finsler vector field on M. (In particular, X may be an ordinary vector field on M.) We shall define the n-horizontol lift h(X) and the cor¬ responding n-horizontal lift of a smooth curve from M to TM. These will be
234 Antonelli and Zastawniak determined by the nonlinear connection Nj- Namely, h(X) is a vector field on TM defined by h(X)(x,y) = Х'(х,у) (ft -N*(x,y)^) . (3.18) Let x be an integral curve of a vector field X on M, that is = X, For a smooth curve (x, y) on TM we say that у is an n-horizontol lift of x to TM if ^ = h(X)(x,y), which in local coordinates takes the form or, equivalently, f = 0. (3.20) This defines у uniquely for any initial condition 3,(0) € TMX^. Proposition 3.4. If у is an n-horizontal lift of a smooth, curve x from M to TM, then L(x, y) — const. Proof: First we compute 5il? 6i (gjky’y*} = gik) yV + gjk (№} yk + ffifcSr’ (&£*) = (Sigik - gjnF^ - SfaJTy) yiyk = 0, since № « —N; = — (the Cartan connection is deflection-free) and tiigjia = gjnFfk + gknFij (the Cartan connection is h-metrical). But diL2 = 2L5iL, so 6iL = 0. Thus, by (3.20) we have (x, у) = йДх, у) + 9iL(x, ») ^- = 0, which proves the proposition, Then-parallel transporting x is defined by n£t:Э vm j/(s,v;t) €ТМлф9 where y(s, v; •) is the n-horizontal lift of з to TM -with initial condition у (s, v; s) = v.
Finslerian Diffusion 235 The h-horizontal lift Z(X) of a Finsler vector field X to OM is a vector field on the Finsler orthonormal frame bundle OM defined by l(X)(x,y,z) = Х\х,у) (8¡ -F^z.y)^). We say that a smooth curve z on OM is an h-horizontal lift of a smooth curve x on M if ■£=l(X)(x,y,z), where X is a vector field on M such that x is an integral curve of X and y is an n-horizontal lift of x to TM. In local coordinates this differential equation can be written as (3.21) where y satisfies (3.20). This defines z uniquely for any initial conditions z(0) € OMe(Q) and j/(0) € rMx(o)< The h-horizontal lift of any smooth curve x from M to T^M can be defined as follows. Let (x, u) be a smooth curve on T£M such that for any h-horizontal lift z of x to OM the components of u are constant in the frame z. Such a u is defined uniquely for any initial conditions u(0) G T£MX(O) and y(ty € TMx(Qy In particular, in TqM = TM the h-horizontal lift u satisfies the differential equation where y satisfies (3.20). Because the Cartan connection is deflection-free, if y is an n-horizontal lift of x, then it is also an hr-horizontal lift of x, but not the other way round. On a Riemannian manifold both the n-horizontal lift and the h-horizontal lift of x reduce to the differential equation (3.5), so they are the same. Let M and M be two_ d-dimensional Finsler manifolds. We take smooth curves x and x on M and M and their h-horizontal lifts z and z to OM and OM, respectively. Thus, z satisfies (3.21), where y is a solution of (3.19).^Similarly, z and y satisfy (3.21) and (3.19) with x,$/,Fjfc,Nj replaced by x,¿/,FJfc, Ñj. We shall say that the manifolds h-roll against one another without slipping if dx? _ -¿dx* dt “ Çj dt where Q and C are the dual frames to z and 2, just like in the Riemannian case, see (3.10). For initial conditions x(0),s/(0), z(0) and х(0),у(0), 2(0) this defines a one-to-one mapping between smooth curves x and x on M and M, which will be called the Finslerian h-rolling.
236 Antonelli and Zastawniak Av-Rolling The hv-horizontal lift l(Y) of a vector field Y = XiSi -5- Y*di on TM is a vector field on the Pins!^ orthonormal frame bundle OM defined by l(Y)(z,y,z) = X^yj^-F^y)^ (3.22) H-Y^y^-C^.y)^), see Sections 1.13 and 1.14. Let (z, y) be a smooth curve on TM and let (z, y, z) be a smooth curve on OM. We say that z is an hv-horizontol lift of (z, y) if ^ = I(Y)(®,y,4. where Y is a vector field on TM such that (z, y) is an integral curve of Y. In local coordinates this differential equation can be written as follows: ^-P;.(«).<-c-rt(«)<. (3.23) This defines z uniquely for any initial condition z(0) € OM^o^q)). Now the Av-horizontal lift of any smooth curve (z, y) from TM to T^M can be defined to be a smooth curve (z, u) on T^M such that for any Av-horizontal lift z of y to OM the components of u are constant in the frame z. This definition does not depend on the choice of z. In particular, if u is a Av-horizontal lift of (z, y) to TqM = TM, than it satisfies the differential equation if = - Cjfe(^^f • (3.24) When M is a Riemannian manifold, then y) = 0 and Fjfc(z,y) = ^k(x), so the Av-horizontal lift of (z, y) reduces to the horizontal lift of z as defined in Section 3.1.1. Proposition 3.5. Let (z, y) be a smooth curve on TM. Suppose that u and v are hv-horizontol lifts ofy to T^M = TM. Then 9ij (s, y)v№ — const. Proof: Differentiate gijfayfuW with respect to t, use (3.24) to express and and apply the A- and v-metricity conditions (1.88) of the Cartan connection to show that &9ijfay)v№ = 0, which proves the proposition.
Finslerian Diffusion 237 Let M and M be two d-dimensional Finsler manifolds. We take smooth curves (x, y) and (¿J/) on TM and TM, and let z and z be their ^horizontal lifts to OM and OM, respectively. This means that z satisfies equation (3/23) and z satisfies the same equation with x, y, Cj^ and Ô replaced by x, and 5. We shall say that the manifolds hv-roll against one another without slipping if dt -V dt’ where Ç and Ç are the dual frames to z and z. Given the initial conditions z(0),p(0),z(0) and z(0),ÿ(0),5(0) this defines a one-to-one mapping between smooth curves (a?, y) and (5, y) on TM and TM, which will be called the Fins- lerian hv-rolling. 3.2.2 Stochastic h-Rolling of Finsler Spaces In Section 3.2.1 we discussed the n-horizontal and h-horizontal lifts in the de¬ terministic setting. One of the goals of the present section is to extend these notions to the case of diffusions on a Finsler manifold. We shall proceed in a similar manner as in Section 3.1.2, that is, we shall approximate diffusions by processes with piecewise smooth sample paths and investigate the limit of the corresponding lifts. An interpolation rule on a Finsler manifold is defined by the same conditions (I1)-(I3) as on a Riemannian manifold in Section 3.1.2. We shall use the geodesic interpolation rule I on M such that t »-► I(æ, s', i) is the shortest geodesic with Z(æ,ÿ,0) — x and 7(z, æ', 1) =* x'. Proposition (7.13) in [44] shows in detail how to verify that conditions (II)—(13) are indeed satisfied in this case. We take x to be a diffusion on a Finsler manifold M such that dx = + An(z) o dwm, (3.25) where Aq, Aj, ..., A¿ are vector fields on M, and w is a standard Brownian motion in We define a piecewise geodesic approximation of x by M*) = I (®(t»-i),«(*»), for any t € [ti-i,and ¿ = 1,... ,n, where % : 0 — íq < ti < to < * * * is a division of [0, oo]. The sample paths of x* are continuous piecewise geodesic curves on M. Each curve of this kind can be lifted to TM by taking the n- horizontal .lift of each smooth segment and pasting the resulting pieces together into a continuous piecewise smooth curve on TM. Each smooth piece of ’yn satisfies the differential equation
238 Antonelli and Zastawniak The standard theorem on approximating solutions of Stratonovich SDE’s, see e.g. Theorem (7.24) in [44], implies the following result. Theorem 3.6. As |7r| —> 0 the approximation (x^.y^) tends in probability um- formly with respect tot in any compact set to a diffusion (¡r,y) onTM projecting onto x and satisfying the SDE dy1 = —NJ (a;, y) o dxk. (3.26) This SDE can be expressed in coordinate-free form as dy « h(Ao)(x, y)dt 4- h(Am)(x,y) o dwm, (3.27) where h is defined by (3*18). A diffusion (x^y) on TM such that y satisfies this SDE is called an n- horizontal lift of x from M to TM. For a diffusion (s, y) on TM we introduce the notation Syi = dy1 + Nj(x,y) O dxh. (3.28) Thus, for example, equation (3.26) can be written as (see (3.20)) 6yi = 0. (3.29) Proposition 3.7. If x is a diffusion on M and y is an n-horizontal lift of x to TMf then L(x, y) = const P-a.s. Proof: It was shown in Proposition 3.4 that biL(x,y) = 0. Thus, computing the stochastic differential of L(x, y), we obtain dL(x, y) = diL(x, y)odxi 4- diL(x, y) o dyi ® 6iL(x, y)odaf + diL(x, y) o fry1 — 0. by the ltd formula and (3.29). This proves that L(a;,y) = const P-a.s. By Theorem 2.6 the n-horizontal lift y of x to TM exists for any initial condition y(Q) 6 TMX(Q) and it is unique in the sense that if y is also a horizontal lift of x and P-a.s. j/(0) = J(0). then P-a.s. y = y. Here y and y are understood as maximal solutions of the SDE (3.27). Moreover, Proposition 3.7 implies that the length L(xyy] of y is constant P-a.s., so by Theorem 2.7 the explosion time of y must be the same as that of x. Let (x> y,z) be a diffusion on the Finsler orthonormal frame bundle OM. We say that z is an h-horizontal lift of x to OM if d4 = -Fjfc(^y)Z> (3.30) where y is an n-horizontal lift of x to TM. This SDE can be written in the coordinate-free form
Finslerian Diffusion 239 dz - l(Ao')(x,yiz')dt-\-l{Arn){xiy,z) odwTni where Ao,Ai,...,A„ are the vector fields defining x in (3.25). In this way z is defined uniquely for any initial conditions y(G) G TMX^ and z(0) € OM(x(0)13/(0))- The h-horizontal lift of a diffusion x from M to T^M can be defined as a diffusion (x, u) on T£M such that for any ¿-horizontal lift z of x to OM the components of u are constant in the frame z. Such au is defined uniquely for any initial conditions tt(O) G T^MX^ and y(0) € In particular, in TqM = TM the ¿-horizontal lift u satisfies the SDE <№ = -F}k(x,y)u3 o dx\ where y satisfies (3.26). As in the deterministic case, if y is an n-horizontal lift of x, then it is also an ¿-horizontal lift of x, but not the other way round. This is because the Cartan connection is deflection-free. On a Riemannian manifold the ^horizontal lift and the ¿-horizontal lift of x both reduce to the SDE (3.13), so they are the sama. Let M and M be two d-dimensional Finsler manifolds. We take diffusions x and x on M and M and their ¿-horizontal lifts z and z to OM and OM, respectively. Thus, z satisfies (3.30), where y is a solution of (3.26) .^Similarly, z and y satisfy (3.30) and (3.26) with x,7/,Fjfe,NJ replaced by x, J,FjA,Nj. We shall say that the manifolds h-roll against one another without slipping if (3.31) where C and £ are the dual frames to z and z. Given the initial conditions x(0),3/(0),z(0) and x(0),ÿ(0),z(0) this defines a one-to-one mapping between diffusions x and x on M and M, which will be called the Finslerian stochastic h-rolling. 3-2.3 Stochastic ¿^-Rolling of Finsler Spaces To generalize the ¿v-horizontal lift we shall approximate a diffusion (x, yj on TM (rather than on M, as in the previous section) by a process with continuous piecewise smooth sample paths. To this end we need an interpolation rule on TM rather than on M. Such an interpolation rule can be obtained by introducing the so-called diagonal lift metric G(x,y) - gij(xiy)dxl ® dx3 + gij(x>y)$yi ® Ôy3, Since the Finsler metric tensor g is assumed to^be non-degenerate and positive definite, this defines a Riemannian metric on TM with metric tensor G, Now we are in a position to consider the geodesic interpolation rule J on TM such that 1J((x,ÿ), (x'} s/),i) is a geodesic on the Riemannian manifold TM with the diagonal lift metric G for any x, xz G M and y 6 TMXi y' G That
240 Antonelli and Zastawniak this is indeed an interpolation rule follows directly from Proposition (7.13) in Let (or, y), where y G TMX1 be a diffusion on TM satisfying the SDE d(x, y) = Yq(x, y)dt + Ym(z, y) o dwm, (3.32) where io, Yx,... ¡Yj are vector fields on TM. Writing Ym = Am$i + Bmdi for m — 0,1,..., d. where the Am and Bm are Finsler vector fields on M, we can express (3.32) in local coordinates as &Y = A^(z.y)dt + A*n(xiy')odwm. 6y* = B^x^dt + B^x^odw™. We define a piecewise geodesic approximation of (xt y) by for any t G i, ii] and i = 1,..., n, and any division 7r:O = to<ii<*2<--- • The sample paths of (x*, y^) are continuous piecewise geodesic curves on TM. Now we can take the hv-horizontal lift of each sample path of (z^, y^ obtaining a piecewise smooth process (xVi y„, z^) on the Finsler orthonormal frame bundle OM. (To be precise» we lift each smooth segment of each piecewise smooth path and then paste the lifted segments together to obtain continuous piecewise smooth paths on OM.) Each smooth piece of the lifted process satisfies the differential equation From Theorem (7.24) in [44] we immediately obtain the following result. Theorem 3.8. As |tt| —> 0 the approximation tends in probability uniformly with respect to t in any compact set to a diffusion (x^ y, z) on OM satisfying the SDE dzn = ~Fjk (®> 3/)4 ° ^ - cjk(x> s/)4 ° svk- (3.33) This SDE can be written in coordinate-free form as dz — l(Y(f){x,y,z)dt 4- l(Ym)(xiyiz) o dwm, (3.34) where I is the hv-horizontal lift (3.22). We say that a diffusion z on OM satisfying (3.34) is an hv-horizontal lift of (x,y). Theorem 2.6 implies that z exists and is unique for any initial condition
Finslerian Diffusion 241 ¿(0) € OM(x(0),y(o))-Because the fibres of OM are compact, the explosion time of z cannot be less than that of (x, y). . In a similar way as in the previous sections we can define an hv-horizontal lift of a diffusion (x, y) from TM to any tensor bundle T^M. For a diffusion (x, u) on T£M we say that u is an hv-horizontal lift of (x, y) from Af to T^M if for any fry-horizontal lift z of (x, y) to OM the components of u in the frame z are P-a.s. constant. Such a diffusion u is defined uniquely (to within equality P-a.s.) for any initial condition iz(0) and is independent of the choice of z. I particular, if u is an hv-horizontal lift of (x, y) to TjM = TM, then it satisfies the SDE di? = -F}k (x, y)tf o dxk - C}k (x. y)uj o Syk. (3.35) The stochastic hv-parallel transport of tensors from T£M along (x, y) is defined by : St’H u(s,v;t) € T£Mx(t), where u(s,v; •) is the fry-horizontal lift of (x,y) to T^M with initial condition u(s,v;s) = v. Proposition 3.9. Let (x, y) be a diffusion on TM. Ifu and v are hv-horizontol lifts of (x,s/) to TM. then gijfayju'yi — const P-a.s. Proof: Applying the Ito formula and using (3.35) to express du* and dtp, we compute the stochastic differential d ($ij (a?, 2/)uV) ® o dxk + (9^)uV o 6yk +gijV^ o du* 4- gtju* o cfap = (<Wy - PtnF",- - Sn.i-F^)uV o da? +(.dk9ij - 9inCkj ~ 9njCki)uivi 0 ¿9* = °> which is equal to zero because of the fr- and v-metricity conditions (1.88) of the Cartan connection. It follows that P-a.s. gij(x^y)u*v^ = const, as required. Let M and M be two d-dimensional Finsler manifolds. We take diffusions (x. y) and (x, y) on TM and TM. Let z and z be their ^horizontal lifts to OM and OM, respectively. This means that z satisfies the SDE (3.33) and z satisfies the same equation with x,y,FJfc,Cjfc and 6 replaced by x}y,F^}Cjjj and 5. We shall say that the manifolds hv-roll against one another without slipping if Cj odx3 — QadxJ, (3.36) (3.37)
242 Antonelli and Zastawniak where £ and £ are the dual frames to z and 5. Given the initial conditions x(0),y(0),z(0) and ¿(0), ¿/(0),5(0) this defines a one-to-one mapping between diffusions (x, y) and (z, y) on TM and TM, which will be called the Finslerian stochastic hv-rolling. 3.3 Finslerian Stochastic Development We shall apply the concepts introduced in Section 3.2 to roll a Finsler manifold along a Brownian motion in a Euclidean space. Let M be a d-dimensional Finsler manifold and let Rd with the ordinary Euclidean metric play the role of another Finsler manifold. Then x can be identified with the tangent space TRd. We denote by (w, v) a Brownian motion in Rd x that is. w and v will be independent Brownian motions in JRd. According to the discussion in the previous section, M can either be h-rolled along w or Av-rolled along (w, v). This leads to two alternative approaches to Finslerian stochastic development. In the case in hand equations (3.31), (3.29), (3.30) of Section 3.2.2 reduce to dxl — z„ o dwn, <5^0, (3.38) (3.38a) (3.386) Definition 3.1. A diffusion (z, 3/, z) on the Finsler orthonormal bundle OM which is a maximal solution of the system of SDEs (3.38) is called a (Finslerian) h-stocha$tic development. On the other hand, equations (3.36), (3.37), (3.33) of Section 3.2.3 reduce to dz4 = zj, o dwn, fy* = z'n o dvn, = -V^y^z^dw”1 - Cik(x,y)^m °dvm. (3.39) (3.39a) (3.396) Definition 3.2. A diffusion (z. y, z) on the Finsler orthonormal bundle OM which is a maximal solution of the system of SDEs (3.39) is called a (Finslerian) hv-stochastic development For any initial conditions (z(0),y(0),^(0)) G OM both systems (3.3S) and (3.39) have unique (to within equality P-a.s.) maximal solutions (z,y,s). In general, the solutions are subject to explosions. The results of Section 3.4 will imply that, in fact, only the x part of (z, 2/, z) can be subject to explosions. In particular, this means that if M is a compact manifold, then (x,yy z) can never develop explosions and is therefore defined for all t G [0,00). Example 3.1 (Minkowski spaces). A Minkowski space M is by definition a finite-dimensional normed vector space whose norm L(y) is smooth for all y g M — M\ {0}. In this case the slit tangent bundle TM is trivial and can
Finslerian Diffusion 243 be identified with M x M. A Minkowski space is clearly a Finsler space with metric function MxM i-+ L(y) €. [0, oo). It is easily seen that the Cartan connection coefficients N j and FJÆ are identic¬ ally equal to zero, so ¿i = dt, while CJfc(y) = ^9ü{y)^9jk(y) and the Finsler metric tensor 9-ц(у) = depend only on y € TM = M, but not on x e M. Here we assume in addition that 9ij(y) is non-degenerate and positive definite. In this case the system (3.38) of SDEs for the Л-stochastic development takes the form dx* == dy* = 0, ¿4 = o, so y = const, and z = const P-a.s. and x is just a standard Euclidean Brownian motion on M. On the other hand, the system (3.39) of SDEs for the Tw-stochastic devel¬ opment reduces to dx* = z^odw71, dy1 = 4°^n, ¿4 = The last two equations of this system are independent of x and have the same form as the system of SDEs (3.17) for Riemannian stochastic development. Therefore y is a Brownian motion on M regarded as a Riemannian manifold with metric tensor py(y) and Levi-Cività connection Example 3.2 (Berwald spaces). A Berwald space M can be defined as a Finsler space such that = 0, where CJft are the vertical coefficients of the Cartan connection. As a consequence, there exists a coordinate system in which the horizontal coefficients of the Cartan connection are functions of x e M only, that is, they are independent of y € TM, see [73]. In particular, it follows that the nonlinear connection coefficients Nj are in fact linear functions of y, that is, Nj(s,j/) (х)^Л in this coordinate system. Then the SDEs (3.3S) for the Л-stochastic development take the form dx% = Zn odwn, W = 0,' (3.40) The first and third equations in this system are independent of y. By a theorem of Szabd [90], for any positive definite Berwald space M with compact smooth
244 Antonelli and Zastawniak indicatrix there exists a Riemannian metric gij on M such that the Fjfc coin¬ cide with the Levi-Civita connection coefficients 7^ in this Riemannian metric in the coordinate system in which the depend on x G M alone. Then the first and third equations in (3.40) are just the SDEs (3.17) for the stochastic development on M regarded as a Riemannian manifold with metric tensor while the second equation in (ЗЛО) means that у is the Riemannian horizontal lift of x. However, the initial conditions for (ЗЛО) corresponding to the Rieman¬ nian development and Finslerian h-development are different. Namely, in the Riemannian case the initial fame з(0) must be orthonormal with respect to the Riemannian metric while in the Finslerian case з(0) must be orthonormal with respect to the Finsler metric gij. The following results are concerned with the projection of an Л- or hv- development (s, y. z) from OM to TM. Theorem 3.10. If (x, y, z) is a Finslerian h-stochastic development^ then the probability law of(x,y) is independent of the initial frame z(0). Moreover (®,y) satisfies the Markov property and has the differential generator D =x Proof: Let (x,^,3) be a solution to the system of SDEs (3.3S) with initial conditions (®(0),^(0),3(0)) e OM. Take any ¿(0) € Then there exists h € O(Rd) such that ¿«(O) = ^3^(0). We put It is easily seen that (m, 2/, 5) is a solution to (3.38) with wn replaced by w™ such that w™ = and with initial conditions (s(0), j/(0),5(0)). But w is also a Brownian motion on so it has the same probability law as w. It follows that the probability law of (3, y) is independent on z(0). Because of this, since (z, y, z) has the Markov property as a solution of the system of SDEs (3.38), it follows (Xi y) also has the Markov property. To compute the generator of (m5 y) we express the stochastic differential dx1 in (3.38) and dx'dx^ as dx' = z'ndwn + ^dz„dwn = zildwn-^Fik^kdt, (kFdx? == = g^dt. Next we evaluate the stochastic differential of any smooth function f on TM
Finsleri&n Diffusion 245 with the aid of the Ito formula: d/(x,y) = (Sifydx* + - - V&kf)dt + (SMdwn. The differential generator | M is given by the expression multiplying dt above. Theorem 3.11. If (x,y, z) is a Finslerian hv-stochastic development, then the. probability law of(x,y) is independent of the initial frame s(0). Moreover, (x,y) satisfies the Markov property and has the differential generator . D = 5A'"’=l^(v?vi + = (W - ^sk) + y* (didj - c£A) • Proof: The proof is similar to that of Theorem 3.10. Let (or, y, z) be a solution to the system of SDEs (3.39) with initial conditions (x(0),y(0),«(0)) € OM. Take any 5(0) G Then there exists h e ¿(¿d) such that 5^(0) = h™£m(0). We put 5n = h™Zm- It is easily seen that (x,y,z) is a solution to (3.39) with initial conditions (x(0),3/(0),5(0)) and with wn,vn replaced by wn,vn, where wm = h™wn and vm — h™vn. But (w,v) is also a Brownian motion on x so it has the same probability law as (w,v). It follows that the probability law of (x,y) is independent on z(0). Because of this, since (s, y, z) has the Markov property as a solution of the system of SDEs (3.39), it follows (z, y) also has the Markov property. Next we evaluate the stochastic differentials dx* and 6y% in (3.39) and their products dxi = z^dwn + ^dz^dwn = z^dvT - ^k^dwmdwn - = z\dwn (3.41) Sy* ~ ¡sndv”' + X<izn^vn £ = z\dvn - ±F$kz?nz^dwmdvn - ^Cijkzjtz^.dvmdvn (3.41a) = zidvn-^k^dt dx'dx3 = Sy1 Sy3 — gl3dt (3.415) dx'Sy3 = Sy'dx? — 0. (3.41c)
246 Antonelli and Zastawniak We use these^ expressions to compute the stochastic differential of any smooth function f : TM —* R with the aid of the Ito formula: df(x,y) = {Sif'jdx1 + +(<V)iy’ + |(W)<W + • = ¿jW - ^Skf)dt+ +¿¡F&djf - c^dkf)dt+(ajXdv". The differential generator | Ahv is given by the expressions multiplying di above. Definition 3.3. A diffusion (x,y) on TM with generator jA^ (respectively, I Afev) is called a Finslerian h-Brownian motion (/w-Brownian motion). 3.4 Radial Behaviour By the radial behaviour of a diffusion (re, y) on TM we shall understand the behaviour of L(x,y), where L is the Finsler metric function. We shall study the radial behaviour of (re, y) for a stochastic ^-development or /^-development (x, j/, 2?). One of the purposes of this study is to ensure that the y part of the development cannot explode. Since the fibres of OM are compact, it is clear that z cannot develop explosions either. Because of this, it is only the x part of Finslerian development that may be subject to explosions. We observe that an explosion of the y part could a priori occur in two ways, namely, y could escape to infinity or to zero in the sense that L[x, y) —► oo or £(x, y) —► 0 as t / r, where r < oo. If this were the case, it would be impossible to extend the development beyond r. (In the latter case, when L(x,y) —> 0, this is because Finslerian objects, for example, gtj(x,y) are, in general, undefined for y — 0.) Therefore we have to ensure that y can neither escape to oo nor to 0 in finite time. Theorem 3.12. a) If (z, y,z) is a Finslerian h-stochastic development, then L(x, y) = const P-a.s. b) If (x,y,z) is a Finslerian hv-stochastic development, then L(x,y) is a Bessel process with index dirnM, that is, dL2 (ar, y) — 2L(z, y)du + (dim M )dt, where u is a standard Brownian motion in R. (On the Bessel process see [59], Example 8.3. J
Finslerian Diffusion 247 Proof: In the proof of Proposition 3.4 it was verified that SiL2 = 0. Moreover, diL- = di(gjkyiyk) = {digjk)yjyk+ ^^5^ (3.42) = 2Cÿfc3/V+2yi = 2yi by (1.92). Thus, by the Itô formula dL2 (s, y) = 6iL2(x. y) o dxi + diL2(z, y) o fry* = ftL2(z,S/)o<fyl = diL2{xt y)dyz + ^idjL^x^drfty* + ^dibjL^X'y^y'&y* = 2yiôyi + (x, yjdy^y^. (3.43) Here we use the fact that dx'&yi = 0 for both the h- and /^-development. a) Since fry* = 0 by (3.38a), it follows from (3.43) that dL2(x,y) = 0. This means that L2(x,y) = const P-a.s,, which proves assertion a) of the theorem. b) Because âyz « z^dvn - ^C^g^dt and = gijdt by (3.41a) and (3.41b), it follows from (3.43) that dL2(z, y) = (x, y)Syidyj - 2yiZ^dvn + gij (x> y)gij (x, y)dt = 2L(x, y)diL(x> y)z^dvn + (dim M) di, since ÿiCjfc — 0 and 2yi — 2LdiL. Putting = diL{x^ y)#n and defining u to be an R-valued such that du = Nndvn> we can write dL2(x, y) = 2L(x, y)du + (dimM) dt. We claim that u is a standard Brownian motion in R. If this is so, then the proof of b) is completed. To verify the claim we shall apply the Lévy theorem (see, for example, [37]). According to this theorem, it suffices to verify that u(t) and tr (t) -1 are mar¬ tingales. We observe that L2NrnNnômn « ViV^ztt™ = yiVjg^ = L2, which implies that NmNndmn = 1. It follows that u = f Nndvn is a square- integrable martingale. By the Itô formula du2 = 2uNndvn + NmNndmndt == 2uNndvn + dt, so tr — t is a martingale, which proves the claim.
248 Antonelli and Zastawniak Remark 3.1. a) An interesting consequence of Theorem 3.12 a) is that if (z, y, z) is a stochastic ^-development such that (z, y) starts from a point on the indicatrix bundle IM = {(z,y) € TM : L(x>y) = then (z>2/) will P-a.s. remain on IM. b) Theorem 3.12 b) means that if (z,j/,z) is a stochastic hv-development, then L(z, y) exhibits the same behaviour as the radial part (wn)2of a standard Brownian motion w in In particular, this means that P-a.s. L(z, y) can neither escape to oo nor to 0 in finite time.
Chapter 4 Volterra-Hamilton Systems of Finsler Type 4.1 Berwald Connections and Berwald Spaces In this chapter on Volterra-Hamilton theory emphasis will be on the class of Finsler connections known as Berwald connections, which arise locally from the geodesic equations of the Finsler metric function L in that the torsion and curvature tensors are intrinsically given by the equations of the geodesics them¬ selves. This connection is not generally metrical. Thus, in a local chart we start with the L-geodesic equations ^-+2G<(a:,y) = 0, i = l,2,...,n, (4.1) where G1 are positive homogeneous of degree two in y' = s being the arc-length and &{x^y) — ijk^y)y^yki with ijkM) = ^9a (dida + - di9jk} being the Christoffel symbols of the metric tensor g# for L. The Berwald con¬ nection is given by (^,^,^ = (^,^,0), where Gj = djG* and G$k = dkG^ = djd^G1. Consequently, 2Gi = Gfâ — G}kyW by Euler’s theorem on homogeneous functions. Note that the vertical ’connection is a tensor relative to admissible coordinate changes and that for the Berwald connection it is always vanishing. This connection has torsion tensors ^• = 0, Sjfc = O, ^k = B^k, 2^ = 0 (4.2) 249
250 Antonelli and Zastawnlak where B'ojk := with BU == -i- - (J/k). (4.3) The deflection tensor for the Berwald connection vanishes identically in any local chart: ■= t/Si = W + ^kyk = ~G} + = 0- (4.4) The double short bar indicates horizontal covariant differentiation in terms of the Berwald connection. The corresponding S and P curvatures are ^=0, <4-5) The Douglas tensor Dlh^k of the Berwald connection is actually the P curvature of this connection and its vanishing is a necessary and sufficient condition for & being a set of n quadratic forms in any coordinate system. This is a special circumstance for a Finsler space. Namely, such a space is referred to as being a Berwald space. Such spaces play an important role in VolterrarHamilton theory. The importance of the Douglas tensor is highlighted by Theorem 4.1. D}kl = 0 Cijk^ = 0 is independent of y. where the short bar is the Cartan horizontal covariant derivative and C^k = & the Cartan torsion tensor in pure covariant form [14]. The condition Cijk\i — 0 characterizing Berwald spaces means that Cijk is a parallel tensor in the Cartan sense. In such spaces = 0 for Berwald covariant differentiation [14]. Remark 4.1. Coordinates in which Berwald covariant differentiation reduces to ordinary differentiation exist if and only Djki — 0 [38]. These are called normal coordinates. There are spaces for which ~ 0 yet there are no normal coordinates. These must be Landsberg spaces which are not Berwald. A Finsler space is called locally Minkowski if there is a covering by local charts in each of which the metric function L is independent of x. Theorem 4.2. In a Berwald space Bjkl = 0 <=> = o the space is locally Minkowski [14]. The reader will readily verify Corollary 4.3. A Finsler space is locally Minkowski <=> = 0 and B^kl s 0 [14]. In the two-dimensional case we can compute Cijk in a simple form by us¬ ing the Berwald frame defined by I* := y'/Lfoy), giflzmf = 0, and
Finslerian Diffusion 251 gijm'm? — 1. This defines the frame to within the orientation of It- is easy to see that gijVF — 1. Furthermore, 9ij 25 lilj “F TfiiTHj (4.6) for the positive definite case [14]. Here k = gi$V and mi = gijm?. Then, using Cijklk — 0, from homogeneity we obtain LCijk “ 1772^771^771^, (4./) where I(x,^) is a scalar function homogeneous of degree zero in y called the principal scalar of the Finsler space. Obviously, Corollary 4.4. A two-dimensional (positive definite) Finsler space is Rieman- nian <==> I(x,y) = 0. For the Cartan covariant derivative it is clear that g^i = 0, l*\j = 0, = 0, and L\j = 0, which implies Corollary 4.5. Cijk\i = 0 <$=> Ip = 0 for the Cartan covariant derivative. Theorem 4.6 (Berwald). = LKrn'tymk — btmj). This defines the Gauss-Berwald curvature K(x,y) of a (positive definite) two-dimensional Finsler space. For a scalar field S(x,y) we obtain Berwald’s h-scalar derivatives (Sti,S,2) and v-scalar derivatives (Sti,S«2) by setting S|p = Sjli 4-Somi, S|i — S^li 4- (4.8) where Sjp = diS — G^djS and S']» = diS. Note that if S(xy y) is positive homogeneous of degree zero in y, then S;i = 0. Using this notation on S = I(x,y) in Theorem 4.6, we see that I,i,2- I2,x = -#I^ (4.9) From Corollary 4.5 we know that a two-dimensional positive definite Finsler space is Berwald iff I,i = 1,2 = 0. Therefore, ZCZo — 0, so that Theorem 4.7 (Berwald). A positive definite Finsler space of two dimensions is Berwald if and only if it is either locally Minkowski (If = 0) or I = const (K / 0). Furthermore, there are three classes of spaces: (A) I2 < 4, (B) I2 = 4, and (C) I2 > 4. The famous Berwald classification of the above cases is (A) L2 - (/?2 4- 72) (B) L2 = /32eW, (4.10) (C) where /3 and 7 are independent one-forms (that is, cross-sections of the cotan¬ gent bundle of the Finsler space) [14]. See Appendix B.
252 Antonelli and Zastawniak We note that (A) has interest in our biological examples below. Antonelli’s Berwald metric [4] in ecology is L = ^1)2 4. (y2)2, (4.H) where er is a smooth function of x and p is a constant called the perturbation parameter. In Sect. 3.1 of [4] cr(x) — atx1 with constant ai and ao, while in Sect. 3.6 of [4] ir(s) = OiX* + (x1)2 + (a2)2 with ft, fts, Æ being constants. (C) also have biological applications. Antonelli showed a relationship between I and p with Theorem 4.8. A two-dimensional (positive definite) Finsler space with metric junction (4.11) is of type (A) with I — 2p/ y/p? +1. Furthermore, the Berwald connection coefficients are constants for f4.11) iff <r(z) is linear in x. Finally, K = - (p2 +1) e-[2(’’2+1)''«+2>,“CTan(»,/»2)]/!k{r(a;), (4.12) where A is the two-dimensional Euclidean Laplacian. Let us prove (4.12). As in [14] (p. 128), the equation for the Landsberg angle 0 is Ldi9 = rm (4.13) can be integrated because Ldjrm = - {li - Irm) mj (4.14) follows from (4.7) and the properties of the Berwald frame (r,m*). F\irther- more, The Bianchi identities become quite simply [14] K-2 + IK + = 0. (4.15) In the case K 0 the Berwald space must therefore satisfy (by Theorem 4.7) K,2+IK = 0, (4.16) which upon use of (4.13) becomes d&K + IK = 0 (4.17) (see [88], p. 256, and [14]). Now, in the case in which arctan^/jr) is omitted the metric function (4.11) is Biemannian and it follows that ([2], p. 113) £(®) = - (p2 +1) Thus, K(z,y) = R(x)e~n. Since from (4.13) we find that 0 = y/pr +1 arctan setting the arbitrary constant of integration equal to zero, we finally obtain (4.12), as desired. (4.1S) (4-19) (4.20)
Finslerian Diffusion 253 4.2 Volterra-Hamilton Systems and Ecology Now we describe n-dimensional Volterra-Hamilton systems and give some ex¬ amples. The basic references here are [14] and [6]. Let us begin by writing (4.1) in terms of an arbitrary path parameter i: da? da* dt dt d2^4 _ z .. -^■+2Gljk(x,x) (4-21) (i — (i1,..., xn), x* = ^). The length s of the trajectory in the Finsler metric is called the production parameter and can be interpreted as the logarithm of total size. In the case of unlimited or open growth s = Be** solves drs ds dP“<it = 0, (4.22) allows (4.21) to be written as Volterra-HamÜton equations < N'. . £ (4-23) ^-=.-G'jk(x,N)N^>Nk + XNt. Here N* is interpreted as the number of individual production units of kind i and the Volterra variables xi(t) = Ni(r)dr + xi(Q') (4.24) Jo denote the total accumulation^ up to time t, of the ‘product’ of kind i. In general, we replace the first equation by the Volterra production equation — =k{^ z = l,...,n (4.25) (where the parentheses cancel the summation convention on repeated upper and lower indices), so that ki := is the per capita rate of increase of x4. Likewise, the last term of the second equation of (4.23) is generally written as where the 7] are constants. The second equation of (4.23) is the ecological equation. The case when the Berwald connection coefficients are constants plays an important role in the models of coral communities of n species, [14], [6]. Here the coefficients define ecological interactions between these species in a manner completely consistent with the original formulations of V. Volterra, the founder of modern mathematical ecology. The reader is invited to consult any standard text on ecology. We especially recommend G. E. Hutchinson’s famous book [56]. The case when the connection coefficients depend only on x4 naturally con¬ cerns the Finsler differential geometry of the associated Berwald space playing an important role in chemical ecology. which these systems model [6]. Especially
254 Antonelli and Zastawniak interesting is the problem of soft coral (Alcyonacea) encroachment on hard cor¬ als (Scleractinia) on reefs such as the Great Barrier Reef and Pandora Reef. It is the soft coral which has evolved chemicals with which it can kill the reef build¬ ing hard corals. The variables xl represent amounts of chemicals in this case and are allometrically related to soft coral biomass m (i.e. xi — logm + y%i > 0). This expresses the fact that such chemicals are uniformly distributed in soft coral tissues [6]. The same concepts have been used to model plant/herbivore interactions in the terrestrial environment. In this case the xi measure so-called defensive compounds which plants have evolved for protection by poisoning herbivores which try to devour them [6]. In Sect. 4.7 we discuss, stochastic perturbations of the Volterra-Hamilton system (4.23) with metric function (4.11). 4.3 Wagnerian Geometry and Volterra-Hamilton Systems A contravariant vector A? is T-parallel along a smooth curve C : x = x(t) on an n-dimensional manifold with a linear connection F(z) iff locally = (4.26) 'The connection can have non-zero torsion tensor := — and is defined by the classical connection transformation law d2x* dxidxk (4.27) under a local diffeomorphism from (x1,..., xn) to (s1,..., xn). Note that no reference to a metric tensor or function is necessary here. A vector B* is T-parallel along C to A* iff dßi 4.r> - dl°s*’ where B* — <X*)A* along C and <p(t) = etc dt. Let us inquire into the conditions under which parallelism is the same for two linear connections (possibly with torsion) T and T. It is a classical result that this can happen iff r|fc-rjfe + 2^ (4.29) where is some covariant vector field [40]. It is not possible for both connections to be torsion-free — 0, Tjk = 0) in this instance. Theorem 4-9 (Schouten). A necessary and sufficient condition that parallel¬ ism be the same for two connections and where the first is torsion-free, is that (4.30)
Finsleriân Diffusion 255 The connection T^k is said to be semi-svmmetrically related to Tjk in this case. Furthermore, the curvature tensors satisfy (4.31) If denotes the symmetrized connection, then sfy = 5 +n,.) = rjfc +1 (fy>k+4^) (4.32) The reader will note that = B^kl iff = di^ for some smooth scalar function ^(¿r). Now consider the tangent field dxi/dt along C. f-parallelism of these vectors along C means that d?xi drf dxk dxi di? di dt ~ $ dt (4.33) holds according to (4.23) and the fact that (4.34) can always be integrated to define a natural parameter s for T. Namely, (4.35) Choosing this parameter s for i, we obtain and solutions of these define curves C which are called autoparallels of f. Given a semi-symmetric change of T into f* given by a gradient the T-autoparallels with natural parameter s will map in one-to-one correspondence with those of f with natural parameter p given by ds The autoparallels of T can also be realized as solutions of (4.37) (4.38) which are just the autoparallels of the torsion-free connection ST. The trajectories of the F-connection are therefore preserved under a semi- symmetric change or, from another point of view, are reparametrized in a way very similar to a projective change of connection. We shall use the term semi- projective transformation in the semi-symmetric case. It is important to keep
256 Antonelli and Zast&wniak in mind the reparametrization perspective for these two concepts for they are fundamental to various models we shall study. The classical projective change of connection [40] is defined by rjfcW = rjfc(w) + Sfa(x) + «Ui (®) (4.39) with (4.40) as Clearly, (4.32) above defines such a change with i&(x) — Now let us consider a Finsler space with fundamental metric tensor g and Finsler connection (iVj, Vjk) that satisfies the following four axioms of Hashigu- chi;- (Hl) It is h- and ^-metrical, i.e. Svl* = 0 and py|*=0, (H2) The deflection tensor D vanishes identically, i.e. ^=^-2^=0, (H3) It is semi-symmetric, i.e. = Fjk~ *lj = for some specified covariant Finsler vector field ay (z, y), (H4) The vertical torsion tensor vanishes identically, i.e. ^=^-^ = 0. The connection is uniquely determined [51] and is called the Wagner connec¬ tion relative to a,(x,y). Note that Vfk « Cjk, the Cartan tensor. If ay = djff for some smooth scalar function and if F^k is independent of y, the space is called a Wagner space relative to a (sometimes, a-Wanner space). Fjk(x) is a linear connection. Note that any locally Minkowski space is a Wagner space relative to the zero function. More generally, a Berwald space is a Wagner space relative to the zero function. Theorem 4.10 (Hashiguchi). If (N^,F^k(x),Cjk) is the Wagner connection of a Wagner space with metric function L(x,y) and L = is a con¬ formally related metric. it is also a Wagner space equipped with connection withTjk = T^+i&a-S&a. Corollary 4.11. If L(x, y) is a locally Minkowski metric function, then any con¬ formally related metric L = e^L is a a- Wagner space with Wagner curvature zero.
Finslerian Diffusion 257 Proof: Any Wagner space arising from another by a conformal change of the metric function as above induces a semi-projective transformation of the geodesics via the gradient djo(x), Consequently, by (4.31). Since the Wagner connection for a locally Minkowski space can be expressed as (0,0. ), it follows that Rjkl =0 and the corollary is proved. Remark 4.2. The converse of Corollary 4.11 holds: A Wagner space with — dicr(x) for which the Wagner curvature vanishes is necessarily conform¬ ally locally Minkowski with L = e^L [51]. We now turn to examples of 2-dimensional Wagner spaces. The basic refer¬ ence is [72]. Theorem 4.12 (Wagner). Let I and 9 denote the principal scalar and Lands¬ berg angle of a 2-dimensional Finsler space and suppose that d$I 0. This space is a Wagner space iff del is a function of I. Furthermore, the vector field Oi is given by Remark 4.3. It is well known that I and 9 are conformal invariants [58]. For the 7nth-root metric function [23] ^y) = [(y1)’n + (y2r]1/m (4.42) we obtain m - 2 1 - g” 2y/m — 1 v/g™ (4.43) where z — y^/y1. Note that I may have the opposite sign if the Berwald frame vector m* has reversed orientation. However, I2 is always uniquely defined. Since the Landsberg angle is [23] (4.44) we see that m(m - 2) (l + gm)2 9 4(m -1) zm For m / 2 we obtain, therefore, (4.45) If we set m = 3, we obtain ö9/ = -|-3/2, (4.46)
258 Antonelli and Zastawniàk which was originally obtained by Wagner for any cubic metric function L(x,y}=[ai^}yW]l/3. (4.47) Of course it is only the case when m is even that yields positive definite metric tensors necessary for the stochastic theories developed herein. For general 2-dimensional Finsler spaces we have the following result [28]. Theorem 4.13 (Antonelli, Zastawniak). For the Finsler metric junction L = where L is locally Minkowskian, the Berwald- Gauss curvature is expressed as K — - (gFcij _j_ ImWaij - (del^n^rTijoiO-j^ , (4.48) where andfh* ~ e“a^9n*, (P,™1) being the Berwald frame for L and Oij(x) := didjtr(x). The reader may compute (4.12) from (4.4S). For L — with a(rr) =* o^x1 and L as in (4.42) we have [23] ~ _ m(m - 2) ftj - ai/aa)2 (zm +1)2 yxy2 . A “ 4(in-l)2 L2 ' 1 } If we start with a locally Minkowski metric L and perform a projective change of connection via 4>(x), we arrive at reparametrized straight lines + (4.»> where the new ‘tick’ of time or production measure is p = A + bJ e2^^dTdt along a trajectory 7. This serves as a model of heterochrony, time-sequencing change in growth and development for n-dimensional Gompertz growth d2xi dx* dt2 +*dt = 0, (4.51) called Laird's law [6]. First, we pass to the total production parameter s so that on 7 ds — L(x, dx) or L(x, dx/ds) = 1 and (4.51) becomes (4.52)
Finslerian Diffusion 259 Then we arrive at (4.50) and finally pass back to real time t to obtain the Volterra-Hamilton system dx* — = TV1 < dt. ’ (4.53) where A is some constant. A heterochronic change can be ‘reversible,’ as the above projective change procedure can be reversed, but usually such time-sequencing changes occur be¬ cause of specific environmental influences. Thus, we need to have some descrip¬ tion of these dynamic external influences. As a first attempt we could try to add to (4.50). But because we must require the heterochronic- ally transformed system to be derivable from a Lagrangian, in feet a Finslerian metric, we will set dp = L(x, dx) and try to determine X. Since L(x, dx/dp) = 1 along trajectories now, we observe that 0* = iFtj? — L2gtjdj<f> and that (4.50) is therefore equivalent to (4.54) with In other words., (4.54) are the geodesic equations of the Finsler metric e^L = L. Thus, a heterochronic transformation will be a passage from (4.51) to (4.55) It is known that the external environmental influences cause the solution of (4.51) to become curved. But we are not able to express this in the obvious way, i.e. by using normal coordinates for L = e^Lat an arbitrary point. However, if we use Wagnerian geometry, we can rewrite the geodesics of L as dx^ dxk dp dp (4.56) where ?L2 = = L2^ = L2?. Now the left-hand side can be written in terms of ST, the symmetrized Wagner connection, as dx? dx* dp dp (4.57)
260 Antonelli and Zastawniak At an arbitrary point we use normal coordinates x* for ST to obtain just = (4.58) for the geodesics through this point. Thus, (4.57) expresses a complete concept of heterochrony, one that embodies a semi-projective (time-sequencing) change of parameter along the Gompertz trajectories of (4.51) together with external environmental causative influence C1. The reader can verify that Cl is ortho¬ gonal (relative to gy or gij) to the solutions of (4.57). Notice that we have not claimed that is orthogonal to these geodesics. This is one reason why normal coordinates for the symmetric linear connection not S^ve correct description of how environmental influences cause the original straight lines to become curved upon heterochronic transformation. Remark 4.4. Normal coordinates exist for the Wagner connection of Wagner space in the sense that ST in (4.57) can be viewed as a torsion-free linear connection. Moreover, for the special case of the mth-root metric L defined by (4.42) with £ = e<r(x)L, <r(x) = aix\ ai,a2>0, (4.59) the xi being log biomasses, one can exhibit a consistent interpretation of these normal coordinates as multiples of true logarithms of biomasses. The reason is that the heterochronic transformation of the straight lines of the ¿-space to the geodesics of L as in (4.57) actually preserves a true straight line out of the origin of the ¿-space. This is because <7* = 0 has exactly one solution, the ratio yx/y2 = (ai/c^)1^17*”1)? so x1 = (ai/^)1^771“1^, and this is the image of x1 = in ¿-space, so x1 = ax1 and x2 = bar for some constants a, 5. This gives the consistency of interpretation. The geodesics of (4.59) have interesting ecological meaning because the in¬ teraction coefficients are not constant (n > 3) but do not depend on x\ Such spaces are called y-Berwald spaces [5]. 4.4 Random Perturbations of Finslerian Volterra-Hamilton Systems We consider a general Volterra-Hamilton system of Finsler type ' = i < <& V' (4-60) . using the total production parameter s. Here Ffa are the horizontal connection coefficients of a Finsler connection 0% that is h~ and v-metrical relative to a Finsler metric function ¿(x, yj with the corresponding metric tensor
Finslerian Diffusion 261 gij(x, y). As explained earlier in this chapter, the variables y* can be interpreted as population numbers and the xz as the (log-) biomasses. In the most general situation, random environmental and/or developmental perturbations can occur in both the production and population equations. This will be expressed by adding noise terms to the right-hand sides of both equations in (4.60). In our model we take two independent standard (Euclidean) Brownian motions t?(s) and w"(s) in perturbing the biomasses xx(s) and population numbers yz(s\ respectively. We assume that this external noise <?($), w7(s) is not affected by the behaviour of the Volterra-Hamilton system. However, the response dxi(s\dyi(s) to the perturbations will, in general, depend not only on the magnitude and direction of infinitesimal perturbations dvx(.$),dtox(s), but also on state sT(s), y*(s) of the system. But what are the responses as functions of d^x(s),dwx(s)? In the present section we address this question by establishing a general rule for noise addition in Volterra-Hamilton systems of Finsler type. The metric plays a key role in expressing the responses in terms of perturb¬ ations. Loosely speaking, the infinitesimal distance by which the state (z, y) of the Volterra- Hamilton will be displaced must be proportional to the magnitude of the perturbation dv^dw1 Since (z, y) € TM, a suitable metric on the tangent bundle TM is necessary to measure this distance. A natural choice is the diagonal lift G(x, y) of the Finsler metric tensor g(x. y) as described in Chap. 3. Thus, given the perturba¬ tions dvx, dwz with dtf dvi — dwidwj — tyds and dvzdwi — 0, the rule for noise addition can be expressed by the condition that the covariance matrix of x\ dyz must be equal to the metric of the reciprocal tensor to G(x>y) in the natural frame dj times dsf that is, dxtdx^ dx'dy* _ _ [* gij -gik№k _ dy'dx? dytdyi J " [ g* + gkiNlNl . ds (4.61) This can be achieved with the aid of hv-isometrical rolling (see Chap. 3). To begin with, we consider a solution z(s).3/(s) to the deterministic system (4.60). Then hv-rolling along this solution defines a smooth curve /¿(s),z/(s) in R2n given by the system of differential equations
266 Antonelli and Zastawniak where C* is given by (4.57). The second equation in (4.75) can clearly be under¬ stood as the geodesic equation (4.56) written in terms of Wagnerian geometry and perturbed by noise. 4.6 Noise in Conformally Minkowski Systems The metric function L of a conformally Minkowski space is defined by X(a?} y) = « g Rn, y e R, where = L(y) is homogeneous of degree one in y, i.e. it is the metric function of an n dimensional Minkowski space. In fact, it is enough to require L(a;,3/) to be locally Minkowski, i.e. L(xt y) — in .some system of coordin¬ ates z. Clearly, L(ar, y) defines the structure of a Finsler manifold on Rn called a conformally Minkowski space. The corresponding metric tensors Sij = ^didjL2 and gij = ^&idjL2 are related by = e2*(x)5yfe), in analogy to the conformally flat case (4.70), which makes it possible to con¬ struct random perturbations in much the same way as in the 'examples in Sect. 4.5. In particular, we refer to Example 4.2, which establishes a relationship between random perturbations and the Wagner connection. The Finsler space with metric L has the structure of a Wagner space with the natural Wagner connection , where (see Sect. 4.3) Fik^dk^ N1 = 9^, cjfc = (The vertical connection coefficients are written without a bar because they are conformal invariants.) The connection coefficients and Nj of the Minkowski metric L(y) are of course identically equal to zero. The geodesic equations can be written in terms of the Wagner connection as (see (4.56) < f da;* . ¿¿=y’ w pi ds C> (4.76) where and W dyi ,fridxj ds ds i ds (F - gi’gkmykymdj<l> - y^djt. (4-77)
Finslerian Diffusion 267 Because the fibres of the tangent bundle axe no longer Euclidean, but have the structure of a Riemannian manifold with metric function L{y), metric tensor gij(y) and connection coefficients Cjk(y), it is necessary to modify the system (4.75) of stochastic differential equations by introducing an auxiliary frame zj orthonormal under gy*. dxl = yzds + o dv*, < 6y* — C^ds + o dvF, (4.78) < dzj = -Ctklz‘jodyk. This system combines the idea of isometrical rolling of a fibre regarded as • a Riemannian manifold with metric tensor gij(y) and Levi-Civit4 coefficients Cjfc(2/) with a stochastic perturbation of the form put forward in Example 4.2. One can verify that dxldx^ = e“2^^ dwds = e~2t^g^ds = g^ds, dx'dyi = dx^dyi - N’k o da?) = -N^-^g^ds = -N^ds^ dy'dyi — (Sy* — Nko dxk)(Sy^ — o dxk) = e_2^(pv 4. g^ffyNfods = (g* +gtaNiNi)ds, which means that the rule of noise addition (4.61) is satisfied for the diagonal lift <5(a?, y) of the of the metric tensor g(xf y). The generator of a;(s), y(s) defined by (4.78) is easily seen to be G = ¿3* faj - fa - + y^i + C^di. (4.79) Remark 4.5.. An alternative way of introducing random perturbations in (4.76) is to use the h-v-isometrical rolling with respect to the Wagner connection {dx* = yzds + zj o dtf, Sy* — C^ds + zj o dw\ (4.80) dzj = o dxk - C^Zj o 6yk, Here the auxiliary frames zj are orthonormal in the metric gtj (rather than gij, as in (4.78). The Markov diffusion x^s^y^s) defined by (4.80) is equivalent to that defined by (4.78) in the sense that it has the same generator (4.79). 4.7 Canalization of Growth and Development with Noise Consider a system governed by the Finsler Lagrangian (4.11), i.e. L2(a;,y) = [(y1)2 + (y2)2] e2(J>3+i)<»(«)+2pMxtOa(91/v3)i (4.81)
268 Antonelli and Zastawniak where x = (s1, a?) and y — (p1,^2). The corresponding Euler-Lagrange equa¬ tions + (a2 - Qip) = 0, = 0 (fix1 da2 dx1 dx2 ds ds + 2(a2 + Ofip) dx1 dx2 ds ds (4.82) describe the growth of a two-coral community of sclaractinian corals [3], [lo], [13], [17]. According to the theorem on p. 209 of [14], the Finsler space with Lagrangian (4.81) is a Berwald space with locally constant coefficients. In this case there is a Riemannian metric = Sij exp [2(aj - azp)^ + 2(a2 + aip)®2], whose geodesic equations coincide with (4.82). We employ the above system to construct a model of growth and develop¬ ment in the presence of noise and demonstrate that it gives rise to the canal¬ ization of biomasses around their ‘targeted’ values, a phenomenon discovered experimentally by Medawar in his study of the growth of embryonic chicken heart tissue [6]. The system will be subject to constant environmental influences (temperat¬ ure, etc.) expressed mathematically by introducing constants on the right-hand sides of (4.82), which becomes d2®1 _ _ dx1 dx2 -y^r + 2ofs -y—y— ds- ds ds (fix2 ( dx1 dx2 ds2 + ds ds -r âi + &2 = -aôi, = -Mi2, (4.83) where &i = <21 — asP? «2 = Oi2 + aip, a > 0. These constants can be obtained from the potential V(x) = I exp (20«®*) by computing the gradient, ad? = V*V := g^djV. In addition, we standardize the external environmental influence by setting a2 + - (1 + P2)(<2i + <*2) - I« Our method of developmental noise and canalization rests on Nelson’s cel¬ ebrated conservative diffusion theory [81], [82]. Thus, we replace the-smooth trajectory x(s) in (4.83) by a Markov diffiision £($) with generator of the form
Finsleriân Diffusion 269 where A := (g^g^dj) with g — det^y] is the Laplace-Beltrami oper¬ ator, v is the noise intensity, and b — b'di is a vector field refereed to as the forward drift. We denote by p(s, x) the probability density of £(s) relative to the volume form dVg^^dx^dar. Let u := |Vlnp be the so-called osmotic velocity and v := b — u the current velocity. Following Guerra and Morato [47] and [82], we define The latter expression enables one to consider A as a functional of the drift field b. In place of solutions x(s) to the deterministic system (4.83) we consider diffusions C(s) that are critical pints for A under variations of the drift field b such that the initial distribution p(0, x) is fixed, which we write briefly as M(&) = 0. This criticality conditions implies the Nelson-Newton law (4.84) In addition, we have the continuity equation (4.85) from which one can see that v*p is the probability current We shall seek stationary solutions of (4.84), that is, those for which the probability current vip vanishes everywhere. The latter implies that p and u are both independent of s and (4.84) becomes Therefore, there is a constant A such that This can also be written as (4.86) where — y/p. Thus, the problem of finding stationary solutions of (4.S4) can be reduced to the eigenvalue problem for the operator — A + V. On applying the coordinate transformation r = exp (¿hx1 4- c^2) , 0 — — dix2,
270 Antonelli and Zastawniak the eigenvalue problem becomes where a - j.1 d Apoiur- ^0+ r Qr + d&2i which is just the eigenvalue problem for a harmonic oscillator in two dimensions. As is well known, this problem has an increasing sequence of eigenvalues An = VH*a(n-bl), n — 0,1,2,... . The eigenfunction corresponding to the smallest eigenvalue Ao is $0 = Coexp , where Co is a normalizing constant. For the second eigenvalue Ai there are two independent eigenfunctions ^10 — Cio^exp V>oi - Coiyexp BÆ4 (-^4 where x = rcosO and y = rsin0 are the Cartesian coordinates. For the third eigenvalue A2 there are three linearly independent eigenfunctions, and so on. The corresponding densities p = -02 provide stationary solutions of (4.84), see Figures 1-4. In general, a stationary solution p has nodal lines p = 0, which split the plane into disjoint domains. No sample path £($) of the corresponding Nelson diffusion can ever cross any of these lines (i.e. the event has probability zero). The diffusion £($) will therefore remain forever in one of these disjoint regions. Moreover, its is a basic result of Nelson’s theory that the sample paths Ç(s) con¬ verge to the deterministic solutions of the Euler-Lagrange equations (4.S3) with potential V, The confinement of £(s) in one of the regions bounded by nodal lines can be interpreted as a mathematical manifestation of the canalization of biomasses in the presence of developmental noise, in the sense of C.H. Wadding¬ ton [6].
Finslerian Diffusion 271 4.8 Noisy Systems in Chemical Ecology and Epi¬ demiology Our goal is to introduce noise into the Berwald space with metric (4.81) and into the geodesic dynamics of the Finsler metric L — • L, with L being the 7nth~root Minkowski metric (4.42) We will do this in a way which results in Markov Diffusions on TM, the slit tangent bundle over with M being diffeomorphic to R2. Here, we interpret the noisy perturbation as due to external environmental effect on the two subpopulations (n = 2). This is different in spirit than that of Section 4.7, where Nelson’s conservative diffusion was used to model developmental noise in the sense of C.H. Waddington. The geodesics of the two metrics, therefore, have an interpretation consistent with Volterra- Hamilton Theory as used in ecology [6]. For example, (4.81) has been used to model corals of the Great Barrier Reef on the famous problem of starfish devastation [3], while (4.42) has been used to model myxomatosis, the European wild rabbit disease [22]., [20], [27], in which the rabbit flea is the disease vector. In the absence of noise, the dynamics of either (4.81) or (4.42) is expressed on TM by dx* = yids, (4.87) W = 0, where бу* ® dy* + 2V^(o;,y)dsJ, TV* being the coefficients of the appropriate nonlinear connection used in each of the cases. For (4.81) we use the Cartan nonlinear connection while for (4.42) we use the Wagner nonlinear connection^ N} - dj&y** The way noise enters the deterministic equations must depend on the current state ir(s), y(s) of the two populations and on the geometry of the configuration space (production space) in the neighborhood of s($). In Finslerian spaces, this geometry depends on y(s), as well, but in a way that’s dictated by the Finsler metric functions. First observe that any geodesic of either metric can be obtained by /iv-rolling the Finsler space along a curve (7(s),t?(s)) in JR2 x R2, where 7(5) = as+b is a straight line and rfa) — c is a constant. Here, a, 6, c are points in R2 depending on the initial conditions in the equations for /iv-rolling. We now perturb (7(a),??(s)) by the noise (w(s),^($)) to obtain the process (7(s) + w(s), ??($) + v(s)) in R2 x R2, along which we ТмнгоИ the manifold using the same initial conditions. Thus, we obtain the process (s(s), $/($), z(s)) on OM, that solves the system of SDE’s. dx* = Zj о dw? + y*dsy бу* = + (4.8S) о dxk - о Зук, where (JVj, 7jk, Cjk) are the coefficients of the Cartan connection for the case of (4.81) and those of the Wagner connection for the case of (4.42). For the former
272 Antonelli and Zastawniak Figure 2. Density pu(x,y) and oodd Hues pufcy) -0- Figure 1: Density /tofoy) and nodal lines posfc, y) =*0. Figure 3: Density and nodal lines pufoy) = 0. 41 «2 Figure 4: Density y) and nodal lines pnfay) = 0.
Finslerian Diffusion 273 C* = 0, while for the Wagner case C* is given in (4.77). Note that Cjk = Cjk* because the Cartan torsion tensor is a conformal invariant so they can each be computed directly from the two Minkowski metrics associated with (4.81) and (4.42) The projected process (x(s),j/(s)) is a homogeneous Markov diffusion on TM with generator D = i5y(Mj - | - C$dk) + ädi + j6it (4.89) where g^ = e2<rgij is the metric tensor of L in either (4.81) or (4.42) Because yi(s) are interpreted as relative population sizes, the process (x(s),p(s)) must be restricted to the region where both ^-coordinates are pos¬ itive, i.e. the process must start at a point (zo, 2/o) with yj > 0 for i = 1,2, and will be considered up to the first moment r of hitting the set p1 jr = 0. Let Gaßtpc, 0 = 1,2,3,4) be the Riemannian metric tensor on TM obtained by the diagonal lift of gij (i>j — 1,2) and let g = det (p^), G = det (Gap) which means, p2 = G. Denote by p(s, x, y) the probability density of the process with generator (4.89) stopped at time r, to bejn a region A C TM O {y1 > 0, y2 > 0} relative to the measure VGdxdy in TM, i.e. p{(x(s),t/(s)) € A}= [ p{s,x,y)\/G{y)dady. J A The function p(s, x, y) satisfies the forward initial boundary value problem lim p(s, x,y) = pa (x, y) (4.90) „ =o> where the initial density po(x,y) is supported on TM n {p1 >0,^0 > 0} and D* is the adjoint of D relative to the measure VGdxdy. Let us now proceed to solve (4.90) in each of the two cases, via the Feynman- Kac formula. Let us consider (4.81) first. To this end, we compute D* to find 2?*/ = ©y - 2y^if - SgVviijf + g^C^f + Vf (4.91) where use of Theorem 4.13 for the curvature K yields the Feynman-Kac poten¬ tial, V = 2K + 21 — 4Imlj — 2I2 + 4ylai - | I2/L2. (4.92)
274 Antonelli and Zastawniak Here, when is the h-curvature tensor of the Cartan connection. In the special case we have here the main scalar = 2p/i/j? +T, is a constant function so the last term in the expression for K in (4.48) vanishes. We need now to introduce the auxiliary process (£(s), t?(s)) in TM which is an Av-Brownian motion with additional horizontal and vertical drifts fci = _2y*-4j%-, &T = ^C&. Thus, (i(s),r?(s)) satisfies the system of SDEs d£ = Zj o + &*(£, rj)ds Sr? = zjodvi + 'b*(£,rj)ds dzj = -F^rfizj od£k (4.93) (4.94) We claim that £(£($),?;($)) is a Bessel process with index n = 2. The proof is similar to the Bessel process result (Theorem 3.12 in Chapter 3), except that drifts & and 6l must be taken into account. But, yk&(x,y) — 0 so the proof of that result can be adapted to our case here. It follows that ??(s) cannot have explosions, nor can the orthonormal frame z(t). In order to rule out the possibility that £(s) might explode, we assume that cr(x) vanishes for [m| > B, where B is a large enough real number. Biologically, this amounts merely to a statement that the model is not applicable if biomass x exceed some a priori given maximum. With this support condition (4.94) reads df* — s‘- o dw? — 2rfds drf = z'j o dv^ + Q*(T})ds « <fyk where Since the right-hand sides of these SDEs are independent of £ and neither t?(s) nor z(s) has explosions, f ($) cannot have explosions either. We now denote by ¿^(s), ?7x>y(s) the solution of (4.94) with initial conditions f(0) = re, ??(0) = 3/, z(0) = z. Note, incidentally, that the initial condition for the frame z(s) can be omitted since (£($),??($)) is a Markov diffusion process on its own. In addition, we denote by r^y the first hitting time at the surface y^y1 = 0, by the process ^y{s). Then the solution to (4.90) can be written as the so-called Feynmah-Kac formula
Finslerian Diffusion 275 There are a number of conclusions about the qualitative behaviour of p(s, x, y) one can draw from this formula. For example, large positive curvature K will cause p to increase locally, for the short time asymptotics, while just the op¬ posite will happen in regions which have large negative curvature. In models of toxic soft corals competing for space against reef building hard corals [19], [6], a = aixi +1 i/fx1)2 - pz1^2 and p in (4.81) has the interpretation of a morpho¬ logical adaptation in the hard coral species, to fend off the poisonous soft corals’ encroachment. If p > 0 is not too large, then K is negative so that over the short-run the phase space transition density p(s, x, y) is relatively diminished. This parameter is soft-coral encroachment, [ibid]. This is consistent with observations made on Pandora Reef on the Great Barrier Reef of Australia, by Dr. P.W. Sammarco, [19], [6]. Now let us consider the rabbit/fiea model for myxomatosis using for the basic deterministic model the geodesics of metric L in (4.42) with a certain linear perturbation term to be described below. We must first, however, develop a notation suitable to this fairly complicated model. An introduction to the deterministic theory can be found in, [22], [27], [6]. Coevolution of the rabbit host and fiea parasite entails an uncertain environ¬ ment. Certainly, a portion of the parasite’s environment is the host population undergoing random fluctuations. These noisy effects can be added in an intrinsic manner directly from the deterministic dynamics, i.e. from the metric structure of the physiological space, spanned by x(t) = (xp(i)TxH(t)) which denote the (log) total caloric intake up to time t, for the parasite and host populations, prescribed by their ecological interactions. The metric (4.42) is now written and the deterministic equations are with i,j, k G Ep - -spyF, EH = -enyH • Here, r^A(ic,?/) are the Christoffel symbols for the Finsler metric (4.96) and sp and £# are metabolic rates (greater for the parasite). The system (4.97) can be written more explicitly as (4.98) Note that the linear terms will disappear under a transformation to a new time parameter if £p = sh, only. Thus, (4.98) axe not generally geodesics. In any
276 Antonelli and Zastawniak case» the metric remains biologically relevant, for it captures the ability of the host/parasite co-evolved system to respond to external perturbations from the environment. Namely, a state, which is represented by a point in the physiological space, M, will be displaced a distance (in terms of the metric) proportional to the magnitude of the perturbation. This is our noise ansatz. The physiological manifold M with metric L has the structure of a Wagner space with connection (ArJ, where F-k = 4>& and C*fc = j g^tgjk- We shall see that the Wagner connection, although deflection-free and h- and r-metrical relative to L, it has non-vanishing torsion The equations (4.98) can be written in terms of the Wagner connection as follows dx* — y*dt, W = giiTjkgimykymdt +^dt, (4.99) where. (4.100) and the perturbed equations have the form dx* = y*dt 4- du1 6yl = giiTj^lmykymdt + ^dt + dsi (4.101) where, (4.102) The noise terms du1 and ds* must transform like vectors and the Stratonovich circle notation must be used to ensure that the stochastic .differential equation (4.101) are covariant. According to our general noise ansatz, the perturbation terms du1 and ds* must be determined by the metric. Because the fibers of TM are non-Euclidean for m > 3 (each fiber is isometrically isomorphic to a Minkowski space with the mth root metric), the perturbations ds* added to the so-called population equations are not simple white noises. In order to find the expressions for du* and ds* in terms of L(z, y), we use hv-rolling of the Finsler space along a curve as introduced in Chapter 3, but this time it will be controlled by the Wagner connection. The result is (4.S8) with C* given by (4.77). Equivalently, we can write dx* = y*dt + Zj o duA, Sy* = g^T^g^y^dt + FFdt + a*, o dtf, dz] — -FkeSj o dxk - CiiZj o Hyk. (4.103)
Finslerian Diffusion 277 Noting that du1 = Zjo dw? = ZjdwJ -1“ ~ dZjdijji = z}dw> -ir^z^’dt = zjdwi - gM(x, y)T'kedt, we conclude that u^t) - u‘(0) - i i g:'k(x(s),y(s))7'jkds, (4.104) 70 is a martingale. The Markov process (4.103) can now be shown to have generator (4.89) or equivalently, 2> = Ig^SiSk-F^ + lg^dk-q^ (4.105) +3/^ + gijT^gem fy^di + tfdi. We can compute the adjoint operator to be 2>* = 2> + - 2E*di + V (4.106) where = -Sg^-W, Bi = + (4.107) & = and V = Qg^^i ~ ht^L2 + J2/2 L1 + 5j^ + 2s + (4.108) where yp = y**, yH = — yH, € — | (ep 4-eh) is the average metabolic rate, and £ — 1/sp — sr is the so-called efficiency of the system, [22], [6]. Here, 6 is the Landsberg angle (4.44) and I the main scalar (4.43). Once again we introduce an auxiliary process (X, Y) on TM defined by dX1 = ZjoiW + A’dt, 6Y = 2}od^ + Bidt-2Eidt, dZj = -F^od^-C^oiy*, (4.109)
278 Antonelli and Zastawniak where w and v are independent standard Brownian motions on R2 with respect to a fixed probability measure P. We denote by (Xx.y, Yx>y) the Markov diffusion on TM defined by (4.1Ô9) with initial conditions X(0) = x, Y“(0) — y where (æ, y) 6 TM+ (all coordinates of x, y are positive). Moreover, let Tx,y be the first hitting time for the boundary £(TAf+) by the sample paths of (X^iZc.y)- Then the solution p(t,xty) to (4.90) in the case at hand can be written as the Feynman-Kac formula p(t,x,y) = js{x{rœ,ï>t}po(^lj,(t),yx,î,(t)) exp [j V(XX1JZ(s),yx,!,(a!))ds]}, ° (4.110) where E is the expectation relative to P and \ is the indicator function of a set. We will now use (4.110) to obtain information about the myxomatosis epi¬ zootic. We demonstrate how to study the dependence on the average metabolic rate s and the efficiency £ of the system. We will find strong relationships between density dependence of the host/parasite system and Finsler geometric quantities. First, we extract the dependence of the Feynman-Kac potential V in (4.108) on e and by use of the Cameron-Martin-Girsanov Theorem. We set # = 2Çj£', duP^dw*, diï^dè-tfdt (4.111) where Çj is the frame dual to Zj. Then the auxiliary diffusion (Xx>2Z, YXtV) defined by (4.109) satisfies the following SDEs: dXi = Z^odw^ + A^dt SY* = Ziodvi+B'dt (4.112) dÿ = -F'k{Z^dXk-CltZjoJYk, where, by the Cameron-Martin-Girsanov Theorem, w, v are independent stand¬ ard Brownian motions on R2 with respect to P defined by dP = exp ( Sirfdwi - i dP. (4.113) The Feynman-Kac formula can now be written p(t,®,y) = ¿{x{r..v>t}Po(Xx.j,(i),yx.v(t))exp S^duP +| f MW + £ V(Xx>9(s), yx>9(«)>]}. (4.114) The dependence on s and £ is now only in the exponent in (4.114) because by (4.112) the probability law <£ [XXtinYx<y) relative to the new probability measure P is independent of s and From (4.111) we compute Si^duP -^Si^ijPdt = ig^odY1- 2gktEkBidt +2gklEkEtdi - ^7lEkdt (4.115)
Finslerian Diffusion 279 where = djA* + CjkAk is the vertical covariant derivative. Thus, substi¬ tution for and & from (4.107) and using the expression for V, it is possible to write the entire exponent above in the form * From this expression we can derive important consequences. Suppose ep » sjf. Then s is large and $ is small so the above exponent is dominated by ■2 / ^u(X,Y){eYk +Ÿkl^{éYt ^Y1/^ which is negative when Yp and YH are both positive. This means the transition density is degraded so that on average the process of myxomatosis is speeded up, relatively speaking. On the other hand, considering fixed values of sh and ep (or equivalently, of e and f) if yH is small but yp « C > 0, where C is some constant, then I2/2L~ and -Z^/2L“'both tend to +oo as y** —> 0 and yp —► C > 0. All the remaining terms can be shown to be bounded, so that these two Finsler terms involving the main scalar I will dominate the exponent. The result is increased probability density for the transition into the vicinity of the y11 = 0 axis, as one would expect in the myxomatosis epizootic when the rabbits have mostly died off. Increased transition density into a neighborhood of y** = 0, also means that the stochastic dynamics is slowed down, there. This slow down makes good biological sense for the end of the epizootic. The major point here is that it is due entirely to density-dependent (i.e. purely Finsler with m > 3) effects, for when m = 2,1 = Iq — 0. The above conclusions are consistent with the Lyapunov stability results for the deterministic model, when ep ~ ejy, as follows from (4.49). 4.9 Riemaimian Nonlinear Filtering The role of the scalar curvature, Tfc, in Graham’s formulation of path-integral theories in quantum mechanics and in nonequilibrium statistical thermodynam¬ ics is well known and important [4S], [49]. Recent work by Takahashi and Watanabe has made it possible for mathematicians to appreciate this, as well [91]. However, it was Huzurbazzar and Rao who first brought scalar curvature
280 Antonelli and Zastawniak into statistics via the maximum likelihood surface concept [57], [85]. This sec¬ tion may be viewed as an attempt to bridge a gap between these divergent statistical theories. Here, it is demonstrated that enters nonlinear filtering theory for the estimation of a signal process conditioned on information in the observations process. Although we do not dwell on this here, it is interesting to compare Fisher fs information matrix with the Ricci curvature tensor Rij, defined in Section 4.10. The interested reader is invited to consult [1], [57], [85] on these matters. The scalar curvature enters filtering theory via explicit formulas for the C00—densities of a large class of (adjoint) Zakai equations. This class is defined in terms of the type of signal processes we allow. They are homogeneous Markov diffusions whose noise terms are conformal, in the sense of differential geometry explained in Section 4.10. The conformal restriction does not apply to the ob¬ servations processes which are taken to be homogeneous diffusions, also. In fact, the conformal restriction is no restriction at all for two dimensioned Rieman- nian signals. We do require that signal and observational noise are statistically independent, however. Generally, it is proved here that positive 'll increases the value of C°°— densities for the signal estimates, while negative R does just the opposite. This will be seen to appear explicitly in the measure-valued solutions of the nonlinear Kushner equations, as well as in the C00—solutions of the adjoint Zakai equa- tions. Important for our results is the previous work of Kunita [68] on the hy¬ poellipticity problem for stochastic partial differential equations, together with that of Hormander [54]. Especially relevant to our approach is the backward stochastic calculus for Stratonovich integration of SDEs. We believe, however, that the present work is the first to bring scalar curvature into the theory of stochastic differential equations and nonlinear filtering theory. The Riemannian scalar curvature has significant application in mathematical biology [2], [25], [1], [10], [26]. Especially interesting is its interpretation in the growth and chemical ecological dynamics of sessile communities like forests and coral reefs [2], where it can be used to estimate community vigor. Large neg¬ ative R—values indicate rapid growth potential and quick (chemical) responses to predation or herbivory [11], [24]. Large positive 7?.—values indicate a relative metabolic passivity [2]. Results of the present study applied to dynamics of sessile organisms imply that signals from vigorous communities are more diffi¬ cult to estimate than for the less vigorous, all other things being equal. This seems intuitively correct. However, more interesting is that under normalized conditions this difficulty increases more than quadratically with the number of species in the community. In the last section we compute results for a much-studied model of starfish predation on an n-species community of corals [11], [15], [13], [16]. First a brief review of nonlinear filtering following [42], [67]. In Section 4.10, the necessary geometric definitions are given including the Levi-Cività connec¬ tion and the various curvature tensors leading to the Riemann scalar curvature R. This invariant is given explicitly for the class of Riemannian geometries known as locally conformally flat. All geometries, known to the authors, which
Finslerian Diffusion 281 arise in biological applications as above, are of this type. Thus, the coral/starfish example uses this expression for 1Z to obtain the (7°°—density for the related Za- kai equation, in Section 4.11. In Chapter 5 we shall consider Finslerian filtering problems. We suppose given, once and for all, a probability space (Q,P, P) and a complete, right-continuous, filtration {Pi} of sub a—fields of P, for t € [0,T]. All processes considered will be Pt—adapted. There are two distinguished classes of processes denoted generally {7G} and {Yi} and called signal and observation processes, respectively. Write yt = <?{Xn ' s <t} for the a-field generated by a given observation process Yt. Note that < Pt is usually proper inclusion. Denote by ^ = ^1^) (4.116) the least squares estimation of Xt conditioned on 34» and call Xt the mean filter of the process Xt rel 34- If Yt can be given in the form Yt= f'hsds + Lt. (4.117) Jq where Yt = (Y/...., Y™), and where Lt = (Lj,..., L™), is an m dimensional Wiener process (i.e., Brownian motion), then the process vt = Yt- ^h3ds (4.118) is a 34-“Brownian motion called the innovations process. This process will be bounded and Pt—measurable t € [0,T] and represents information in the observations yt concerning Xt. In general below we shall be interested in a class of homogeneous strong Markov processes of diffusion type Zt which are strong solutions of Ito type vector stochastic differential equations like dZt = k$ds + 6(Z3)dB^ (4.119) where 5(Z) is nonsingular d x d matrix on IRd, bounded away from zero as ||5(Z)||e > s > o, z e JRd (4.120) (Euclidean norm), and B# is a d—dimensional standard Brownian motion. Lipschitz conditions are also invoked. Namely, ||£(Zt) - 5(Z;)||C < K\\Z-Z'\\*t (4.121) for all t € [0,71] and K a constant, and where * denotes the supremum of norms over i. Zt will have initial conditions independent of Po- We also require ^jTlIM2*) <00.
282 Antonelli and Zastawniak (For the rest of Chapter 4 we suspend use of the summation convention on repeated upper and lower indices.) We use the symbol & for the n—dimensional signal process Xt governed by d^t=X0(t,^dt + Y,Xi°d7ft, (4.122) where 77J are n separate Brownian motions and o denotes Stratonovich integra¬ tion. We often will suppose the observation process Yt is m—dimensional and solves dy„ = h^ds + £ ° dB> (4.123) for some initial condition independent of Fq and Jh» and it will be supposed that sufficient smoothness and boundedness conditions are satisfied so that (4.122) and (4.123) define diffusions equivalent to (4.119), so that we may pass back and forth between Ito and Stratonovich theories, at will. Let Bs be a standard Brownian motion. The filtration 5* is the least complete a—field for which all random variables Bu — Bv : r < v < u < t are measurable. If f (r), r € [0,i] is a continuous stochastic process which is «7*—measurable for each r, then the Ito-Backwards integral is defined using partitions A with f evaluated at right, instead of left, endpoints, i.e. f{r)dBr = Hm Of course, the Backward Stratonovich integral is defined similarly, but with midpoint averaging, i.e. f(r) o dBr = lta V ± + /(**))(Btk+l - BtJ. ”U 1.—fl z We shall use Kunita’s method of solving forward sde and spde problems with backward integrals and Feynman-Kac formulas [67]. Letting irt(dy, w) denote the conditional distribution of the signal given the observation data we write 7Tt(dy,cd) = € ds/|3à)(<*>) for w e Q and JlR" As in [42], one shows 7rt(/) satisfies the Kushner equation
Finslerian Diffusion 283 where and - n A(s)f(x) - 5 £^(S)/(®) +X0(*j/(®) (4.125) is the infinitesimal generator of the diffusion signal process (4.122). Here we use Hormander notation so that Xj are C£'2— vector fields1 on IRn with *)(*)/(*) Further, D, = (Ol(s),...,Pm(s)), where n W(«)=£il%w/(«); (4.126) £ = 1,..., m is defined by dridBl-yi1. (4.127) Here absolute continuity of 7* is assumed relative Lebesgue measure on IRm [42]. If we define the at-process by at — exp [/0 i(y»)dYs') |lJ'1(^)’r’(A*)l|eds]- <4-128> Then, *(/>*(/)«* (4.129) solves the following Zakai equation (4.130) provided %*(/) solves the Kushner equation (4.124). The Zakai equation is pit/) = po(f)+£ +£ ["p^M^odY^, (4.130) *=1Jo where Mk(s) = Dk(s) + hk(s) and f € C£(IR”’). Also note that ’■•«-£$ (4-m) 1 Continuously differentiable in t and twice continuously differentiable in x and the first derivatives aro bounded.
284 Antonelli and Zastawniak so that 7Tt(/) may be recaptured from the measure solutions of the linear Zakai equation. Note that the operator 1 m I(4) = A(S)-i£M*2(3) ~ fc=l (4.132) is nontrivially elliptic [67]. If pt(dy) is a measure solution of (4.130) having a (7°°—density it must satisfy the adjoint Zakai equation dpt = L‘ (f)ptdi + MkPt ° k=l (4.133) where 1 n i=l (4.134) x5 = -Xo + è(è^)xy, j=l ¿=1 % = Ao+èè (È^xJ)2-È^*xo " J=1 iz=l ¿=1 (4.135) - n m n J=1 fct=i Z—l Mk = -Dk + hk, (4.136) (4.137) and, finally, (4.138) i=l The coefficients of the vector fields Xj and Dk are assumed bounded and con¬ tinuous, (71 in t, C4 in x, and C% in x. We are interested in computing (7°°—densities for the (forward) stochastic pde (4.130) which is in adjoint form. In order to do this we make use of the backward Stratonovich calculus [67]. We will need to solve the backward equation dt = -X0*(s,S)ds- J=i -¿Z>*(3,£)odB*, (4.139) A=1 with terminal condition *tj(x) “ x, The unique solution passing through (x, t)
Finslerian Diffusion 285 is then written in terms of Stratonovich backward integrals as = x + j xo (n &,t («)) dr + £ [ xi (’•i ?r.t («)) o drfr 5=1 J> + £ Dk(s,&,t)cdB*. (4.140) A?=l If we let z be a reaZ variatZe and write ia,i(x,2,w) = s-expi^r/" hj(r,fr.t(x)) odB* fc=iJa + f fto(n€,t(®))*-J, (4.141) where Kq and are given in (4.136) and (4.138), then the C00—density for the measure solution of the Zakai equation (4.130) must be pt(x,w) = B:[po(So.t(®,w)) •$t0,t(a?>l,w)]. (4.142) We will compute pt(®, w) for certain systems in the next section because it will be shown that C°°—densities exist for these special systems. 4.10 Conformal Signals and Geometry of Filters We consider signals of the form (4.122) for which Xo and X, = e^dj, j = 1, ...,n, are Cw-vector fields (i.e. real analytic) with bounded first and second partials. Also, we suppose drftdBi - 0 for the noise functionals of the signal and observations process K- We require Yt to be a homogeneous strong Markov process solving dya = ha(x)ds + Sk(.ya) o dB*. (4.143) Also, ha (®) = (hj (x),..., fe™(x)) are supposed Cu for all x G IRn. The operator for the conformal signal process Zt is A(s) = j £ e-*(ft(e-*ft)) + £^i- — 1=1 1=1 Such a linear differential operator is said to be hypoelliptic in an open set U C IRn+1 if every distributional function u in U is C°° in every open sub¬ set W C U in which A(s)u is C°° [54], p. 148. Thus, if A(s) is hypoelliptic with
286 Antonelli and Zastawniak distributional solution u of A(s)u = 0, it follows that u can be modified on a set of measure zero to be C00. Noting that Mk = hk because of (4.143), it follows that the operator L(s) of (4.132) is -elliptic, with scalar term if A(s) is -elliptic. Also, the adjoint operator, L*(s), has h*k = hk from (4.138) and Ao = fto - £ diXi + 1 e-2* ¿[2(^)2 - (4-144) i * j=l follows directly from (4.136) and the form of A(s) above. It is clear that L*(s) is elliptic, so by a theorem of Hormander the dimension of the Lie algebra generated by {Xq — d/dt. Xj,..., Xn} is maximal and equal to n +1 [54], ibid., [67], p. 150. Since Cw, implies C°°, the result of Kunita ensures the existence of a C00—density for the measure solution pt(/) of the Zakai equation (4.130). Recall that Dk 0, here. This density is given by (4.141) and (4.142). Direct substitution of hk for and (4.144) gives the density explicitly. We wish to describe the Zakai (7°°-density geometrically. Therefore, we digress into some background material on locally conformally flat Riemannian spaces^ which is basic to the type of signal processes we use in our main results. Let us rewrite the signal process (4.122) as =)dxi = £ 4^) o drf + xidt. fc—1 The corresponding infinitesimal generator for the Markov process solution & is the elliptic operator M i -1 * r,k (4.145) ■where n (4.146) and fc=l chm = £(^-4 ~ rJc (4.147) is the so-called Stratonovich termor Christoff el field [25], [1], [10],
Finslerian Diffusion 287 Both gij and its inverse g^ are second-order tensor fields on a Riemannian space (lRn, g^) whose arc-length element is given by ds2 - ^2^jdxz ® dx? tj (see [41] or [66]). A metric in the usual sense is obtained (1.50) by minimizing £ over all C°°— curves 7 joining xo = 7(^0) to xi — 7(^1). This definition can be shown independent of the particular parameter used. If xq and Xi are close enough in this metric then the minimizing curve 7 is a solution to the geodesic equations fx 4. V'ri n ds2 jk ds ' ds ’ it j, k = 1,..., n. Here, the Riemannian length of dx/ds, is unity, that is, d^/ds is a unit vector, and the solution starts at xo and ends at Xi. The’ 3-index symbol plays a fundamental role here, in that it defines Riemannian parallelism on the one hand, and the Stratonovich term, or Christoffel field, on the other. Given the Riemannian metric tensor g^, the Levi-Civitd connection is (1.55) rJfc = | 22 + dj9kt - dtgjk). (4.148) “ e-i Note that all vanish if g^ = the identity matrix, but that a smooth change of coordinates need not preserve this nullity because the T do not con¬ stitute a tensor. Nevertheless, the geodesic equations are invariant under the transformation law of the connection T [41], [66], The quantities g^ and and their first partial derivatives define the basic curvature notions of Rieman¬ nian geometry, and The most important of these is the (full) Riemannian curvature tensor (see Section 1.6) - ¿r^r-fc. (4.149) r=l r=l It is basic that Ufa = 0 for all i,j, k,£, if and only if g^ is transformable to the Euclidean metric tensor «5«. But, this is only a local statement It holds for S1 x JR1, the ordinary straight cylinder, for example. The next most important curvature is the so-called Ried curvature «=1 (4.150)
288 Antonelli and Zastawniak This tensor is powerful enough to distinguish (locally) between all 2- or 3- dimensional Riemannian spaces, as is [41], [66]. Finally, the Riemann scalar curvature, (4-151) j,k is good enough to distinguish all (2-dimensional) Riemann surfaces, locally. All such surfaces are locally conformally flat, in that it is possible to use the Cauchy- Riemann equations to prove the existence of so-called isothermal coordinates u1 and vr so that = e+2^3ij, « ¿(u1, w2) [66]. The Riemann scalar curvature is then 7£=-2e~2*(^ + ^). Note that there is always lurking behind Ryu,Rih, and a sign convention. It is simply a matter of how one arranges the minus signs in (4.149). In [41], the sign convention is opposite to that used here, for example. This carries over systematically to Rjk and R. In general, n-dimensional Riemannian spaces for which ga = e^Sij (4.152) holds in some coordinate patch are called locally conformally flat spaces. Their Levi-Civit& connections are then given (see [2]) as 1^=1^=^, igtj i^j (4.153) r*fc = 0, The Riemann scalar curvature is given by * = -(n- l)e-2*£>(^) + (n - 2)O,^)2) (4.154) j (see [41]). Returning to (4.144) and the conformal signal Xo, Xj — use of (4.154) allows us to write ha = ho - +1 {(n + 2)llgrad^ll2 + }, (4.155) Ugrad^H2 = ]Te_2*(a^)2 where (4.156)
Finslerian Diffusion 2S9 is the Riemannian norm of the Riemannian gradient of </>, grades £ff^0=e-2W). (4.157) It now follows that the function of (4.141) is 4 [(n + 2)||grad9<+ dr +52J °^r}> (4.158) 3 where solves (4.139) with £tit — xt and 0. It is clear that for conformal signals, regarding H as an independent variable, if His generally positive, $ has a relatively larger value, while if JR is negative generally, # has a relatively smaller value. This carries over to the C00—density of the Zakai equation. Furthermore, because Pt(f) = [ /{¿)pt(x)dx and JlR« solve the Kushner equation, this behaviour carries over to the estimation problem for any suitably smooth f, providing at = Pt(l) is independent of JR. The estimation of /(&) conditioned on observational data from Yt, is larger ifHis generally positive, and is smaller if H is generally negative, providing Pt(l) is independent ofH. Remark 4.6. Because any Riemannian metric on a 2-dimensional surface is locally conformally fiat, all two-dimensional signals of the form (4.122) can be considered to be conformal signals. This firmly establishes the role of H in the associated Zakai and Kushner equations for the nonlinear filtering problems of signal dimension two. 4.11 Riemannian Filtering of Starfish Predation We follow [26] in this section (but consult [11], [20], [22] also). Let IR”+ denote the subspace of !Rn defined by all x* > 0. Define the metric
290 Antonelli and Zastawniak where all > 0 are constants. Extend this metric to all of IRn by modification with C°°-bump functions as follows. Let C denote a unit size collar of the boundary of Define on on IRn-IR^+, where 0 btf) = 1 0 < xi < 1 for xi < 0 for xi > 1, and all i. Note dib^x*) > 0 for 0 < xi < 1 and all orders of derivatives are bounded. Thus, gij =• — e2t^6ij = g^ on IR"+ — and is the fiat metric outside IR"+. Clearly, then e~$, &(e~*), . djke~^ are all bounded in JR”. It is necessary to discuss the hypoellipticity of (4.159) on IRn in order to ensure the existence of a C00-density for the Zakai equation for the conformal signal (4.161) and observations process (4.163), below. The operator j4(s) is clearly elliptic on IR£+ and, by a well-known theorem, supports a C00—fundamental solution, (i.e., is hypoelliptic). A(.s) agrees with A(s) on I&++ — <0, defined in terms of g^ in the same way as ^4(s) is defined in terms of g^, But, X(s) is O'—elliptic. By Hormander’s theorem, this analyticity implies the Lie algebra generated by {Xq — d/dt, Xj_,... ,Xn} has maximal rank equal to n+1, on IR++ — C [54], p. 149, [67], p. 150. Outside R”+, A(s) has Euclidean symbol and again its associated Lie algebra has rank n + 1. We claim this is true in the collar C, as well. The argument applied to our special case is due to Hormander [54], p. 149. He shows that if A(s) is not of maximal rank at x e <P, then the Frobenius theorem can be used to show that X(s) is not hypoelliptic. But we know* that <A(s)u = 0 has a C00—solution which is in fact a density on IRn. Thus, A(s) has maximal rank everywhere. Now, using a theorem of Kunita [67], p. 153, the C°°—vector fields Xo, Xx,... Xni with Dk = 0, k = 1,... ,m, and the maximal rank Lie algebra condition implies the associated Zakai equation has a C°°—density for its measures solu¬ tion. The L(s) operator (4.132) for this problem is A($) — | £4X1 Mt (s) will be further described. Because we are interested in g^ only for biological reasons we may use g^ <t> instead of g^ </> in what follows. Consequently, it is the behaviour in IEt++ w*hich mainly concerns us in this example. The Levi-Civita connection is
Finslerian Diffusion 291 given by (4.153) to be Ta = Oi, rk = r^ = a>> rjy = -ait (4.160) i% = o, (see [2])- The signal process to be considered is defined initially only on 1R" + x IR£+ by dx* = N$dt + e~^drft = [ - r]k№Nk + XiNi- dt + drji, (4.161) where Xi > 0 and di > 0 are 2n fixed constants and F is defined on IR^. 2 Extension to all of EV* x IRrt is done by regarding the tangent bundle TIR" as identical to IR* x JR”, with the base space spanned by x1,... ,xn and the canonical fiber spanned by N1,.2Vn. We define a Riemannian metric tensor on TTRn as ds2 = e+2^ In this way TIRn becomes the product Riemannian manifold (JRn,gij) x (IRn, fy). Replacing <j> by in (4.161) yields the extended ¿¿—process, ¿¿. Note that ¿t is not a conformal signal, strictly speaking, rather it is an extended conformed signal. The scalar curvature 11 of the product metric is numerically identical with that for the p# factor alone because the canonical fiber ]Rn of TIRn is fiat EucEdeaa. Therefore, the curvature for ¿¿,7^, is 7*c = -(n-l)(n-2)e-«[f>)2]. 1=1 (4.162) Note that |7£c| | 0, as ||x||c T oo for x G IR*+, and that 72$ becomes more negative with increasing n. In fact, it does so at least quadraticcdly keeping constant as n varies. Finally, note that for n = 2, 7£$ = 0. Now the observations process to be considered here is (dYt ==) dFt = ^(M, Ft)Ftdt 4- ’ where > ^ = ■^ = -£ + 7^ + ^^ t (4.163) and £, 7, S'i are positive constants.3 We are supposing difdW = 0 for all i. Here, m = 1, Pi s 0, so Mi = hi = t/i. Therefore, | Mfc(s) = j and L(s) = A(s) - i ^2. is to be modified by multiplication with a C™ bump function so that it is zero for large enough jF—values, likewise for 7 in (4.163). is modified by multiplication with C°° bump function so it is zero for large enough ^-values; likewise for 7 in (4.163).
292 Antonelli and Zastawniak The adjoint Zakai equation is dpt - - i tfjptdt + (& • pjo dWt as follows from (4.132), (4.133), (4.137), (4.138). Note that integration may be taken to be of Ito type. The C00—density pt (x,tu) may now be constructed as in (4.142) using (4.133). Thus, for the extended conformal signal (4.161) we have Xo given by Xi = 2V\ i = l,...,n Xj+n = -£rjkN*Nk + XiNi-6iNiF, jjc and the divergence of Xq is given by Si=i = S?=i(^o+n/dN*), because we suppose a? and N* are independent variables. We leave it to the interested reader to compute this divergence in more detail from (4.153) and the definition of 0. The functional can be explicitly written as = z-expf [ (4.164) J* i=i +| ((n + 2) E *’2 W)2 + )] (n tr.i)dr f '^^r^dr+/ ^(r> &,*(«)) °dw’r}. In the region of biological interest, IRJ+ x JR^+, the positive quantity £7 e-25^^2 becomes e-2^ • £Xi(a»)2> an<^ IPven by (4.162). Also, & solves the backward Stratonovich SDE in Hn x IRn, dZ = -X0*(s,£)<fe - >1 as follows from (4.139). Xq is computed from (4.135) replacing n with 2n and defining Xj in the obvious way from the product metric “square root*. Note that, since the Tj/s in (4.161) are constants [see (4.160)], so that the second equation (4.161) is independent of x. It follows, in particular, that N and F are independent of the curvature 7^, which by (4.162) is a function of x only. Hence, the At—process in (4.163) is also independent of It now follows that ott = pt(l) [see (4.128), (4.129)] is independent of and so, as indicated in the discussion following (4.158), we can conclude that the Kushner measure ih(f) = Pt(/)/pe(l) (4.165) increases with increasing distance from the origin in IR++ x IR++, because becomes less negative with distance. Also, 7^ decreases at least guadratically
FinsleriâJi Diffusion 293 with other things being equal (see the discussion following (4.162)). Interpret¬ ation of this is straightforward. Namely, the “difficulty” in estimating the state (a:1, A7*)? i = 1,2,... ,n, of the community increases with the negativeness of 7^. Because — He is a measure of vigor in production [2], [26], the more vigorous communities are harder to estimate, regardless of the fixed observations process used to obtain information, all other things being equal. In the next chapter we develop a stochastic notion of vigor appropriate for a large class of Finslerian h—diffusions and use it in Finslerian filtering. The technicalities are based on quadratic dispersion.
Chapter 5 Finslerian Diffusion and Curvature As in the well-known Riemannian case, one would expect Finslerian diffusion to be closely related to the curvature of the manifold. In the present chapter we establish such a relationship for Finslerian h-development. A major difficulty is that all methods of relating curvature and diffusion involve normal coordinates, which, unfortunately do not exist for all Finsler manifolds. This is because the exponential map from TMX to M can, in general, develop a singularity at the origin. Because of this, we restrict ourselves to Berwald spaces, a class of Finsler spaces which can be characterized by the existence of normal coordinates at each point x eM [38]. The main tool used to relate diffusion and curvature on a Riemannian man¬ ifold is the Cartan lemma, which provides an expansion of the metric tensor gij in normal coordinates in terms of the curvature tensor Rjkl and its covari¬ ant derivatives. Because no analogue of the Cartan lemma for Finsler spaces has been known so far, our first task will be to extend the result to Finsler spacs of Berwald type. This is done in Section 5.1. Then, in Section 5.2 we define the quadratic dispersion of a diffusion and establish a stochastic version of the Taylor formula. In Section 5.3 the quadratic dispersion of Finslerian ¿-development will be related to the curvature through the Onsager-Machlup term where R is the Cartan scalar curvature. (Equal to the Berwald scalar curvature K in the case of Berwald spaces.) The results of the present chapter were first announced by the authors in [29]. In the case of ¿^-development the relationship with curvature can be studied on the basis of a result established in Chapter 6, which makes it possible to regard ¿‘v-development as a Riemannian development (with some additional drift) on TM equipped with a certain Riemannian metric, see Theorem 6.4. Because the curvature of this Riemannian metric can be expressed in terms of the Finslerian curvature tensors, we will be able to reduce the problem to the well-known case of diffusion on a Riemannian manifold. The details will be 295
296 Antonelli and Zastawniak presented in Chapter 6. In Section 5.4, we show quadratic dispersion enters a Finsler filtering problem. In the long Section 5.5 its relationship to entropy production is examined. 5.1 Cartan’s Lemma in Berwald Spaces Throughout this chapter we assume that M is a Berwald space with metric function L(x,y). For the definition and the main properties of Berwald spaces, see Section 4,1. We denote by expx the exponential map from a neighbourhood of 0 e TMX to a neighbourhood of x G M. Namely, for any x € M and y G TMX we consider the geodesic y; t\ starting from x with velocity y at t = 0, that is, - ¿v 1 ri dp +F* Xd^drf ) dt dt (5-1) with v(.x,y;0) =z, d dt n(x,y',t) = y, (5-2) t=0 and we put exps y = r){x,y,l), provided exists for t — 1. When M is a Berwald space, expx is a diffeomorphism from a neighbourhood of 0 G TMX to a neighbourhood of a: € M. The inverse map exp“1 form neighbourhood of x G M to a neighbourhood of 0 G TMX defines the normal coordinates at x, which will be denoted by NC® for brevity. For the definition of the Berwald connection coefficients and G^k appear¬ ing in the lemma below, see Section 4.1. Lemma 5.1. Let M be a Berwald space. Then for any x GM and any y G TMX (a) Nj(x,y) = Gj(x,y) = 0 in NCX, (b) F*fc(x,v) = Gfcw) = 0 in NCI( (c) Sigjkfay) = 0 in NCX, (d) Sidjf(z,y) = dj5if{x,y) in NC® for any smooth function f : TM —> R, (e) V) + y) + y) = 0 in NC,- Proof; In normal coordinates equation (5.1) becomes $£ = 0 inNCx (5.3)
Finslerian Diffusion 297 for any geodesic passing through x. For any y e TMX we can take a geodesic such that (5.2) holds. Putting t — 0 in (5.2) and (5.3), we find that (Tix, y) = F)fc(x, 2/)^^ = 0 in NCx for any y G TMX. Differentiating this with respect to y^ we immediately obtain (a) and (b), since Nj = Gj = djGi and Fjfc = G^k = djG^ = djdkGi in a Berwald space. Next, (c) follows from (b) because of the h-metricity condition ôiffjk = F^m* + F^jm, and (d) obviously follows form (a). To prove (e) we differentiate both sides of the geodesic equation (5.1) with respect to t to get d2??™ dr? dqk dt2 dt dt drtf dqk di2 dt = 0. But in normal coordinates (5.3) holds for any geodesic passing through x, so the above reduces to dqm drp dqk dt dt dt — 0 in NCX. Taking 7] to be a geodesic satisfying (5.2), we obtain SmFijk(.x,y)ymyiyk = 0 inNCs (5.4) for any y € TMX. Because, by (d) and (b), OnSmFjk (s, V) =* y) = 0 in NCX for all y € it follows that SmFjk is in fact independent of y. Thus, differentiating (5.4) three times with respect to y, we obtain (e). Now we are in a position to extend the Cartan lemma to Finsler spaces of Berwald type. Lemma 5,2. Let M be a Berwald space. Then for any x^M and y G TMX (a) g^g^ShSkgij = j-R in NCS, in NC«. Here R is the Cartan scalar curvature (equal to the Berwald scalar curvature K in the case of Berwald spaces.) and F£ = ^F’^. Proof: Contracting the indices in (e) of Lemma 5.1, we obtain 0x2^'+^ inNCx.
298 Antonelli and Zastawniak Moreover, R = K - ghkg^Khijk = g^F^-bF* in NCX. Adding both sides of these equalities, we obtain (a). Subtracting the second equality twice from the first one gives (b). Remark 5.1. The Cartan lemma on Riemannian manifolds involves fewer con¬ traction of indices, so the full curvature tensor features there. This is also possible in the case of Berwald spaces, but the proof is more involved and the above is just what is needed for our purposes. 5.2 Quadratic Dispersion The definition below is motivated by the notion of kinetic energy U = of a smooth trajectory x on a Riemannian manifold as used in classical mechanics. The kinetic energy U can also be defined by Z4W*)) = [expX(1t)a;(i+/i)]3 = h2U + o(h2). The latter definition can readily be extended to diffusions. Let M be a Berwald space and let (z, y) be a diffusion on TM with generator D. Definition 5.1. The quadratic dispersion of (®,y) is defined by Wh(a;(t), y(t)) = {[exp"^x(t + ft)] [exp^x(t + ft)]*}, for t,h > 0, where Et = E{* |«(t),y(t)} is the conditional expectation given In normal coordinates this expression takes the form %(a:(i),g(i)) = ^9ij(x(t\y(f))Ei{xi(t + h')xi + (5-5) Proposition 5.3. Let M be a Berwald space and let (x, y) be a diffusion on TM with generator D and initial conditions x(0) — xq, 2/(0) = yv. Then Uh{xQ,yo) = ^3v(®o,yo)£(a:i:c’)l o (5.6) 6 y°»W0 h2 | (®°> „,0 + in NCro- ’ V“W0
Finslerian Diffusion 299 Proof: For any smooth function f we have <t+h Etf(x(t + h),y(t + h')) = + / EtDf(x(s),y(s))ds t = y(t)) + hDf(x(t), y(i)) + o(A) Applying the same formula to the integrand above, we obtain the following stochastic version of the Taylor formula (note that D here is a second-order operator): - y(i)) + hDf(x(t), y(t)) + —DDf(x(f), y(t)) + o(A2). Now, for any fixed zq e M and ya € TMXQ we take /(x,y) = ®v mNCS(>. Then, by (5.5) and (5.7), we obtain (5.6). 5.3 Finslerian Development and Curvature In this section we establish a relationship between curvature and one kind of Finslerian development, namely, the h-development on a Finsler manifold of Berwald type. It is remarkable that this relationship involves the term ~ of Onsager-Machlup type, where R is the Cartan scalar curvature (equal to the Gauss-Berwald scalar curvature K in a Berwald space). Theorem 5.4. Let M be a Berwald space and let {x^y^z) be a stochastic h- development with initial condition (xo^yo^zo) € OM. Then (5.8) Proof: In what follows the arguments (zo, ya) will be omitted for brevity. By Theorem 3.10 the generator of (z, y) is so we can apply Proposition 5.3 with D First, we evaluate ДА(®^’) = gV -
300 Antonelli and Zastawniak The last two terms vanish in NCXo by Lemma 5.1 (b), so gijAft(a;V)I = gyffV = dimM in NCX0, which, by (5.6) gives the first term in (5.8). To find the second term we compute pvAfcAh(x^)|_o ^g™^^ -5^ -^g^SmSngn-S^ R 2R = R 3 + 3 3 in NCœo In this computation we discard all terms which vanish in the normal coordinates by virtue of Lemma 5.1 (b) and the fact that a;l|x=sx0 — 0 in NCXo, and then we apply Lemma 5.2 (a) and (b). By (5.6) it gives the second term in (5.8). We observe that even though (5.8) has been proved in normal coordinates, it is in fact independent of the choice of local coordinates. The interpretation of the coefficient of the h2-term in the quadratic disper¬ sion formula as the negative of vigor, —V, has its roots in [2] for the deterministic case and in [12], on filtering, and [21] on entropy-production. Its deep role in filtering any 2-dimensional signal is presented here in detail in the last part of Chapter 4. In the next section, we consider a 2-dimensional example of Finsler filtering and the role of quadratic dispersion. 5.4 Finslerian Filtering and Quadratic Disper¬ sion Consider a 2-dimensional Volterra-Hamilton system of Berwald type =\i)Ni-TijhWNiNk (5-9) where are the Berwald connection coefficients of the metric L2(æ, y) = [(y1)2 + (y2)2] exp [2(p2 + l)<r(m) + 2p arctan ^5] = Z2 • exp[2(p2 + 1)<t(o;)]. (5.10)
Finslerian Diffusion 301 The extended dynamical equations for our filtering problem are = AiN1 — 2r}2^№ — — T22(№)2 — iiFN1 = Aa^-srljW2-^^1)2-^^2)2-^^2 = ßF^+N^+^-sF. (5.11) Here Xit 6iy s are positive constants and 1$ = -didjB^k (5.12) Bhk =^L2-gkk-i/V with a* == dk<r. The Minkowski metric tensor is r. _ ^(,y1+py2')-y1i 4 t (y\)2 + (y2)2 J (5.13) P12(y) $21 (S') = [1 “ ^pTan-1 4 (y1)2 + (y2)21 The Berwald coefficients are explicitly, given as r« =^ = -4. (i ¿3), = Fji ~ &3 (* 7^ J), with 5i = dicr- (02<T)p} 52 = ^2<r + (dia)p, as in (4.82). We will now assume a is linear m Therefore, di<r — &i, i = 1,2 are constants, taken positive throughout this section. It follows that the vigor, V, vanishes because of Theorem (4.8). However, following the method of Section 4.7 on Stochastic Nelson Mechanics, we use the addition of noise ansatz dXf = N}dt + e*™ -difi & : (5.14) dN} = [-V^N* + A(i)JV* - S^N^dt + dr,i for the signal process and an observation process of the form (4.163) with n = 2 and ¿i = = Pi i.e. (dYt =) dFt = Ft)Ftdt + dWt (5.15) ht= V = -e + 7-ft + ßFt(Nt + N?)-
302 Antonelli and Zastawniak This is allowed only because the Finsler geodesic equations (4.S2) for the metric (5.10) are precisely those of the Riemannian metric lr = Ife1)2 + (p2)2] exp [251> (5.16) with a — ?i3;1+a2X2, £i — ai — c^p, ^2 — ao+^ïp* Naturally, the Riemannian scalar curvature vanishes for this metric, so the Riemannian and Finslerian quadratic dispersions agree. The filtering problem is solved by the expression (4.164) for the Zakai equa¬ tion (4.130) with n = 2 and ■£> IR^ = 0. Also, 7rt(/) given by (4.165). However, the h—diffusion for this filtering problem can be chosen Riemannian because the Berwald connection coefficients are constants. To see that this is true, suppose there exists a scalar function so that 01 — po^ 1+P2’’ 0's + po*i 1+ p2 ■ Integrability of (5.17) implies Ao- = didia + = 0, (5.18) so that a(®) must be harmonic. Therefore, the Riemannian metric, L 2 [(p1)2 + (p2)2]e2^*\ whose geodesics are + /(®) G/1)3 + - /(xXj/2)2 = o (5.19) 7&T - S&Ky1)2 + 2/WpV + ff(»)(v2)2 - o with 01 — po>2 1+P2 ’ S(x) = O~2 +PO~1 1+P2 /(®) = must have vanishing curvature. This follows directly from Theorem (4.S) with p — 0 and replacing ¿r(x). Therefore, if a(x) is taken to be a quadratic polynomial, for example, the above trick can not be used to solve the filtering problem. In the same way, filtering the myxomatosis model dynamics which involves the mth—root metric will not yield to this method, except for the m = 2 case, [20], [22]. Both these filtering problems, then, are open at the time of this writing. 5.5 Entropy Production and Quadratic Disper¬ sion Let M be an orientable n—dimensional C°° —manifold without boundary (e.g., lRn). Associated to a Riemannian geometry on Af, is the Riemannian volume form (or invariant measure), dp, defined by djjfx) — yfg (x)dx (5.20)
Finslerian Diffusion 303 where y/g (x) = (det (pii(z)))1^2 in local co-ordinates x = {**}, and where dx denotes the standard Lebesgue measure on !Rn. The function y/g (a;) does not define an invariant scalar function on M. In fact, its transformation law is J9 (£) = (5.21) where | J(s,2i)| is the determinant of the Jacobian matrix of the transformation x w x of local co-ordinates. However, if we are given a second Riemannian geometry g^ (defining the same orientation as gij), then it is clear from the universal relation (5.21) that the ratio (5.22) does define a C°° invariant scalar function on M, Furthermore, <•> is strictly positive everywhere. Let (tt, v)gf denote the £2—inner products defined by the invariant measures dp, dp' for the geometries gij and 5^, respectively (5.20). Here, u and v are suitable scalar functions on M which are square integrable with respect to both measures (e.g., C°° functions with compact support, or which decay to zero rapidly enough at infinity). If IL : Cr(M) -+ Cr~fc(M) is a linear operator on Cr-scalar functions on M (r > k > 0), denote by IL+, respectively £*, the Lo—adjoints of IL with respect to dp and dp'] i.e., (ILtz;v)5 = (u;IL+‘u)5 and (ILu;v)^ = (wIL*?;)^ for any pair u, v for which both inner products exist. We then have IL* (fai) — $L+ (u) (5.23) (5.24) for any u e Cr(M), where is given by (5.22). To establish (5.24), let v € <7°°(Af) have compact support. Then the inner products in (5.23) exist for any u € &(№). From (5.20) and (5.22), we have that dp = <j>dpf, whence, (5.25) Thus (v; IL* (¿u))^ = (IL v; 4ni}3> = \JUUU, tpUJgt = (W.IL+u).; = (v;^IL+^ by (5.23) by (5.25) by' (5.23) by (5.25). Since v is arbitrary, (5.24) now follows from the non-degeneracy of the inner product.
304 Antonelli and Zastawniak We apply (5.24) to the case L = © = © + V, where V is a continuous potential function on M, and © is a diffusion operator of the form, lD = i ù.g + h. (5.24a) Associated to such an operator are two forward diffusion equations (5.26) where it, v are scalar functions on M x (0, oo) which are C2 in the space variables and C1 in the time variable. The relation (5.24) allows us to establish a relation between solutions of the two equations (5.26). Thus, we have Proposition 5.5. Let v be a solution of the first equation (5.26) with initial distribution vq = limjio'Ut- Then u = is a solution of the second equation (5.26) with initial distribution uq = To use Proposition 5.5, we need to compute the adjoint ID+ in terms of invariants involving the geometry. To this end, we recall the definition of the Riemannian divergence of a contravariant vector field X; (5.27) in any local co-ordinate system. Now recall (5.24a) that D - | A5 4- H V, where A5 is the invariant Laplacian for the ^-geometry, and h is a contravariant vector field on M. We may regard h as a first order differential operator on (r > 1); h = h'di. We then have = -h- divph. (5.2S) To establish (5.2S), we have that, for suitable test functions u and v, {h(u); v)3 + {u',h(v) -I- v div3h)g = {h(u)v + uh(v) + uv div? h}dp &vg(uvh)dfj, using (5.27). Supposing u to have compact support, this integral is zero by the divergence theorem and (5.28) now follows. Recalling that is self-adjoint in the ^—geometry, (5.28) now shows that If ]D = i A^ + h + V, then ©+ = i As-h-divsh + V. (5.29) Note that (5.29) implies that © is self-adjoint if and only if h = 0. From (5.25), taking u to be the constant function 1, we obtain, D’(^) — • ©(1), and from (5.29), © +(1) = V — div5fc. Thus, we have shown
Finslerian Diffusion 305 Proposition 5.6. Let <l> be as in (5.22), and suppose that V — div^h = 0. Then ID *(0) = 0; i.e., <p is a C°°-stationary density for the forward diffusion operator ID * = ID* + V. Note in particular, that Proposition 5.6 holds if V = div^h — 0. As is well known, the requirement that h be divergence free means that h preserves . volumes in the -geometry. The more general statement of Proposition 5.6 can now be interpreted as stating that a sufficient condition for the existence of a stationary density is that any tendency of the (covariant) drift vector h to compress or expand volumes, should be compensated by the existence of a balancing gradient field. _ We resume the task of computing ID * in terms of invariants of the gij —geometry. From (5.28), we have = divs-fc + V. À (5.30) To compute div^/h, we note the formula div5(V>X) =s -0 divffX + X(^), (5.31) where X is a C1 contravariant vector field on M (regarded as a first order differential operator), and is a C1—scalar function. Using the fact that y/g = V/77 (5.22), (5.27), and (5.31), we find that (5.32) Now let (X; Y)g denote the Riemannian inner product for vector fields X and y, i.e., (X;y)5=P^y>, (5.33) and let |X|S = {X : X)J^2, denote the associate Riemannian norm. Finally, for a C1 -scalar function, let grad^ denote the Riemannian gradient of (5.34) We may write the second term of (5.31) as XW = (*; grad^),. From (5.30), we must compute AJ. We have A; = Ai7-2gradi7(£n^)+r5(0), (5.35) (5.36) where the vector field grad^n </>) is regarded as a first order differential oper¬ ator, and £g (</>) is the invariant scalar function (5.37)
306 Antonelli and Zastawniak To establish (5.36). recall that Ag(u) — divpgradÿ(u). Thus Д;(и) =^Д+(1и) (from (5.24)) = ^Д9(1«) since A^ is self-adjoint in the ^—geometry. Now, use of (5.31), (5.35) and the relation grad^(^) = grad^u+w grad^, shows that, for any C2 —scalar function A5^M = Ф&ди + 2(gradffi£; grad^w) + uA^v. Applying this with = l/ф, and noting that (l/ф) grad?0 = grad5(£n ф), we find that Д;(«) = Д9и - 2(gcaÂs(e^)-,giaÂau)a + {2|grad9(b $j2 - 1 &дф}и. Finally, we note, using (5.31) and (5.35), that дя(&1 ф) = 1 Ь.дф - [grads(£n ¿)2|j. (5.36) and (5.37) now follow. To summarize, from (5.30), (5.32), and (5.36), we have shown Proposition 5.7. Lei ID == | Aff + h. Then Ю- = 1д, - (h + grad,(Ai ф)) + {I£а(ф) + h(ÙL ф) - div9(ft)}, where ф is given by (5.22), £д(ф) by (5.37), and ID* is the adjoint of ID with respect to the metric gl^. Another useful expression for ID* is now easily derived from Proposition 5.7, namely ID* fa) = divj- {| ф gradfffa-fa) - uh}. (5.3S) To see this, express div^ in terms of divff using (5.32), then expand the right¬ hand side of (5.38) using (5.31) and (5.35), and compare the resulting expression with that given in Proposition 5.7. Finally, it is worth noting that nothing in the discussion of this section depends on any properties of the metric other than its volume from dp\ It therefore suffices just to postulate the existence of such a volume form (defining the same orientation as d/z), rather than a full Riemannian geometry. The essential point is that the forward diffusion equation has ID* as above, if the Chapman-Kolmogorov equations for the process {Xt} are defined with respect to the measure dp!. We return to the case M == IRn, and take to be the standard Euclidean metric. Thus, for a diffusion operator ID given by (5.24a), ID* refers to the formal Euclidean adjoint
Finslerian Diffusion 307 Given an initial distribution tto on lRn, we consider solutions u(x,t) to the forward diffusion equation (ID* — (d/£t))u = 0 with limeio^GM) — uq(x). (Here, as above, ID * = ID’ 4- V for some potential function V.) We suppose u(x, i) is C2 in x and C1 in t for t > 0. Under certain additional boundedness and smoothness conditions, an explicit expression for u(x,t) can be given in terms of a functional integral; e.gM [67], part I, Sect. 5. However, we will not make use of such expressions here, and we merely suppose that such solutions exist and are sufficiently well-behaved. Generally, we are interested in non-negative solutions which remain positive somewhere (and hence on some non-empty open set) throughout their evolution. Thus, we rule out solutions which decay to zero after a finite time. In order to normalize such a solution to form a probability density, we must find conditions under which Nt = i u(x,t)dx (5.39) is finite. Note that our assumptions on the positivity of u(x, t) imply that Nt > 0 for all t > 0. We shall consider the finiteness of (5.39) by making assumptions concerning the Riemannian geometry associated to the operator ID. To do this, it is convenient to use Proposition 5.5 to express the solution u{x, t) in the form u(x, t) = v(x, t)<j)(x), (5.40) where v(x,t) is a solution of (ID+ - (d/dtf)(v) = 0, with limtiovfot) = 0~1(e)wo(e). Here, = ^/g (x) is given by (5.22) with g^ the standard Euclidean metric (so that g^x) = 6% in standard co-ordinates). Since <j> is strictly positive everywhere, the assumptions concerning the positivity of u(x, t) are equivalent to similar assumptions for v(x,t). The reason for preferring (5.40) is that we shall assume that 0(e) has strong convergence properties. In particular, we want to assume that IRn has finite volume in the g±j geometry. But we shall require rather more than this. To state what is needed to justify the formal manipulations which follow, we first define two notions of domination for (Lebesgue measurable) functions on IRn, both of which imply that Such functions are integrable with respect to the measure dp>(x) = 0(e)dx, given suitable assumptions on the gtj metric. First, we shall say that a function f(x) is polynomially dominated (p.d.) if |/(®)[ <<7(l + |«|ft) ' (5.41) for some positive constant C, and non-negative integer k. The notion of p.d. is independent of the metric gtj, but we shall also need a (weaker) notion of domination which is geared to the specific metric we are considering. Thus, we shall say that f(x) is allowably dominated (a.d.) if |y(x)| < <7(1 + (5.42) with C and k as in (5.41) and 0 < a < 1. The idea is that |/(e)| should be allowed to grow very fast as |e| —► oo, but not quite as fast as 0_1(e).
308 Antonelli and Zastawniak In terms of these notions of domination, we shall make specific assumptions concerning the metric gij and the solutions t>(z,i). Thus, for the metric, we assume (i) (H) (in) (iv) Each is p.d. gll (x) and its first partial derivatives are a.d. £n <6 and its first and second order partial derivatives are p.d. (5.43) 0(e) < e"some positive constants K and a, and for all sufficiently large |a?|. Note that (iv) implies that 0(x) —► 0 as |x| f oo. Also, (i) implies that the gij(x) are fairly tame at infinity, while (ii) implies that g^(x) can be much wilder as |x| T oo (but not as wild as 0_1). From (iv), we have 0“"(e) > 1 for all positive a and sufficiently large |z|. Hence, (5.42) implies (5.41), so that a.d. is weaker than p.d. We collect together the technical properties we shall use, which follow from the assumptions (5.43). Lemma 5.8. (i) If f(x) is a.d. then f € L1(IRn;dju). (ii) If f(x) is p.d. and g(x) is a.d., then f(x)g(x) is a.d. (iii) If f(x) and its first partial derivatives are p.d., then ]gradff/| is a.d. (iv) If X is a vector field on IR" such that [X| is a.d., and f(x) is as in (iii), then X(J) is a.d. (v) If |X] and div X are a-d., then so is div5 X. (vi) If f(x) and its first and second order partial derivatives are p.d., then &gf is a.d. Proof: (i) follows from (5.42), (5.43)(iv) and the fact that for a, X(1 — a) > 0; and k a non-negative integer. (ii) is obvious. (iii) First observe that a vector field X has the property that |X| is a.d. if and only if each component X* is a.d. Now grad* f is a sum of a.d. functions by (5.43) (ii) and hence is a.d.
Finslerian Diffusion 309 (iv) |X(/)[ = |(X;grad f)\ < |X| • |grad f|, and this is the product of an a.d. function and a p.d function, so the result follows by Lemma 5.8(ii). (v) |div5 XJ = ¡div X + X(tn ¿)| < |div X] + |X(^n ¿)|, by (5.32). The first term is a.d. by hypothesis, while the second term is a.d. by (5.43) (iii) and Lemma 5.8(iv), (vi) A/ = div^grad/) = div grad/ + (grad5/)(^n <j>) by (5.32). The second term is a.d by (5.43) (iii) and Lemma 5.8(iii). For the first term, we have div grad/ = The result now follows from (5.43)(ii) and the fact that djf and didjf are p,d., and Lemma 5.8(ii). For the solution v^t) of (5.40) we shall assume that v(z,t) and its first and second order (space) partial derivatives are p.d. Note, in particular, that this implies that |grad5vt[ and Agvt are a.d. (by Lemma 5.8(iii) and (vi)). We shall also suppose that |fe[ and div h are a.d., and that V is measurable and a.d. Note that this implies that h(yt) and (V - div^h)vt are a.d. (by Lemma 5.8(ii) and (iv)). It now follows, from (5.29), that = dvt/dt is a.d., and hence is L* with respect to dp (Lemma 5.8(i)). It follows from (5.40) and the preceding discussion that 0 < Nt < oo. Thus, the probability density . p(xf t) = u(x, t)/Nt (5.44) exists. Note, however, that p(x, t) does not in general satisfy the forward dif¬ fusion equation (ID * — d/dt)p = 0. In fact, assuming that Nt is differentiable (which will be justified below), we have (5.45) The additional term on the right-hand side of (5.45) can be obtained by replacing V by the (time-dependent) potential V - (d/di)(^n iVt), and so it acts as an additional force acting to counterbalance any loss or gain of “total mass” in time. Note that, if total biomass is conserved in time; i.e., if Nt — No for all t > 0, then (5.45) reduces to the original diffusion equation, and almost all properties of u(x, t) are inherited by p(®, t). To see what is required for ^conservation of mass” recall that our assump¬ tions imply that dv(xyt)/dt « ID + (v(x, ¿)) is a.d. Hence, by Lemma 5.8(i), the differentiation under the integral sign is justified in the computation
310 Antonelli and Zastawniak Here {f)t denotes the expected value of the function /(2?) with respect to the measure v(x, t)dp,(x) = u(x, t)dx. Note that, our assumptions on V and h imply that both ID + (v(x,t)) and V(x)^(ir,t) are a.d., so that both right-hand integ¬ rals exist. Of course, (5.46) also shows that Nt is differentiable, and so justifies (5.45). We would now like to apply (5.3S), and conclude from the (Euclidean) diver¬ gence theorem, that the first integral on the right-hand side of (5.46) vanishes. Clearly, we must have fairly stringent boundary conditions “at infinity” for this to be justified. Assuming such conditions for the moment, (5.46) then integrates to give Nt = NQ + f\v}ads. (5.47) Jo This shows that the accumulation or loss in time is due entirely to the potential V (provided there are no sources or sinks “at infinity”). In particular, ifV = 0, then Nt — Nq for all t, and we have “conservation of total biomass.” Another notable case is when V — E, a non-zero constant, and we then have Nt — so that total biomass grows or decays exponentially. To see what conditions are required for (5.47) to hold, we apply the diver¬ gence theorem to (5.38) and use (5.40) to obtain f ID *(«(□;, t))dx = | Um /" (v; ¿grad9(vt))dÄÄ J]R,n * «Too JSr -Um [ (y;<puth)dAR, (5.48) •»Too J$R where Sr is the (Euclidean) sphere of radius jR, centre 0, in IRn, dA& the induced Euclidean measure on Sr, and v the outward pointing unit normal. Now note that, for any vector field X on Hn, we have \js (v;j>X)dAR\<V(S)Rn~ Tsup {|X(jR»)|^(Bv)}, vGS where S is the unit sphere in JRn, and V (S) its Euclidean volume. Thus, / div (^X)cte = 0 if ¿(z)|X|n~1|X(ir)| -► 0 as |X| T oo. (5.49) JlRn In particular, this is true if |X(x)| is a.d. (see (5.42) and (5.43)(iv)). From (5.48) and (5.49), we conclude that (5.46) holds if [v*7i| and [grad^vj are a.d. But this is implied by our assumptions on and h. Hence, with our stated assumptions, (5. 47) holds. □ We shall retain our previous assumptions, but now also suppose that V = 0, so that Nt = No for all t, and the probability density (5.44) is a stationary solution of the forward diffusion equation (5.45). Consider the (information)
Finslerian Diffusion 311 entropy St = - i in (p(z, t)}p(xrt)dx JJR,n = - [ &(p(a:,t))g(a:,i)<iM(a:), (5.50) JlRn where q(x,t) = v(z,t)/Nb> and p(pit) = 0(«)g(ir,t). It is usual to take (p £np)(x,t) = 0 if p(x,i) » 0 for some (re,t). However, if we allow pt to have finite zeros, certain additional complications arise in the arguments which follow. Although this can to some extent be overcome by additional assumptions, it seems best, to retain reasonable simplicity and avoid too many side issues, to assume that pt is strictly positive on IRn for all t > 0., Again, to justify the formal manipulations which follow, we shall need to assume not only , that qty but also that in qt is reasonably well behaved “at infinity”. Specifically, we must ensure that |ingt| does not grow “too fast” if qt —> 0 in some direction. The easiest way to ensure this is to suppose that in qt and its first and second order (space) partial derivatives are p.d. We also note the following useful fact 1/ /(x) is a.d., then so are f£n</> and f in|/|. (5.51) That f tn is a.d. follows from (5.43) (iii) and Lemma 5.8(ii). For f in we note first that (5.42) and (5.43) (iii), Lemma 5.8(ii) imply that (l/l^|/|)(x) <JD(l + Wi>-₽. But, from the fact that y in y > —1/e for any y > 0, we have that |/| in|/| is bounded below, and so (5.51) follows. The existence of the integral (5.50) follows from the estimate |<?t in p*| = 1^ in qt 4- qt </>\ < lit in it | + |it in <j>\, and the fact that all these functions are assumed p.d. (and hence a.d.). We now wish to compute dSt/dt. Differentiating (5.50) formally, we obtain f (s-52) To justify the differentiation under the integral sign, it is enough to show that each of the three terms in the integrand are a.d. Now, as noted previously, our assumptions imply that ID+(&) = dqt/dt is a.d. Hence, (dqt/dt) in is a.d. by (5.51), while (dqt/dt) tn qt is a.d. by Lemma 5.8(ii) and our assumption that in qt is p.d. We proceed with the formal analysis of (5.52), Noting that we have shown previously that Ld^d^L^ =°’
312 Antonelli and Zastawniak (5.52) reduces to = _^RTl = - J TD*(pt)£apt-dx. (5.53) To compute the latter integral, we first use (5.3S) to obtain the formula Thus, it follows from (5,49) that is (u grad^v — v grad^w) — twfc] wa.cL then (5.55) We wish to apply the above with u = ¿n pt = ¿n Qt + -fa 0, and v — qt, The vector field inside the { } in (5.54) then becomes Recalling that we are assuming that qtf £n qt and In </> are p.d. and that |h| is a.d., it follows from Lemma 5.8 (ii), that [(# £n qt + g*€n </>)h\ is a.d. Again, (5.43) (iii) together with Lemma 5.S(ii) and (iii) imply that qt |gradff(^n ^)| is a.d. Also, frgrad^ngt) = grad^gt is a,d. by Lemma 5.8(iii), while |gradp(gt).£n^| is a.d, by Lemma 5.8(iii) and (5.43)(iii). Finally, |grad5(gt)^n gt| is a.d. by the Lemma 5,8(iii) and the assumption that ¿n qt is p.d. We conclude that (5.56) is a.d., and we have therefore obtained the relation (5.55) with u = In pt and v = qt> Hence (5.53) reduces to - = -[ Pt^Pt)dX. at jjRn (5.57) Now note, from (5.24a) that D(inp) = A?(ûip)+À(£ip) = 5 (iA,(p)-| gradin p)|=} + ih(p) = H>(p) - 11 gradinp)ß. Thus, (5.57) reduces to (5.58) It remains to compute the first integral in (5.58). To this end, take v = u = pt ~ <j>qt in (5.54) to obtain PtlD *(1) — ID (pt) = — div [<l>qt{h+ gradin 0)}].
Finslerian Diffusion 313 Our assumptions that qt is p.d. and ]h| is a.d. imply that |gth| is a.d. Also, (5.43) (iii) and Lemma 5.8(iii) and (ii) imply that |gtgrad5(£n 0)| is a.d. Hence, we may apply (5.55) to conclude that (5.59) But, from Proposition 5.7, ID*(1) — | £$(<£) + h(€n <j>) — div5ft = | £</0) - divh (by (5.32)). Thus, combining (5.58) and (5.59), we have proved Theorem 5.9. Let the metric gij satisfy the assumptions (5.43), and let ID = j A5 +h for some C1 vector field h with |&| and div h a.d. Suppose that p(x, t) = $(z)q(x, t) is a strictly positive solution of the forward diffusion equation (ID * — (d/dtf)p = 0, and suppose that q(x, t), £n g(a?, i) and their first and second order (space) partial derivatives are p.d. Then the rate of change on entropy (5.50), is given by | <|grad9(&ipi)||)fl - where Mg(h) = |£p(^) — divh, and Cg^) given by (5.37). It is now straightforward to interpret Theorem 5.9. The expression Mg(h) splits into two parts, | £$(<£) which is an invariant of the diffusion geometry only, and div h, which is an invariant of the drift only. (In particular, if h preserves Euclidean volumes, then Mg(h) = | £^(0) is independent of h.) From the formula in Theorem 5.9 , we conclude that if Mg.(h) < 0 everywhere, dSt/dt is always positive so that entropy always increases with time. Further, the more negative, JAg{h\ the faster St increases. However, if Mg(h) > 0 everywhere, then it is possible for entropy to decrease with time (at the very least, it increases less rapidly than it would if the geometry and drift had no influence other than through the density pt itself). Further, the more positive Mg(h) is, the closer the system is to one for which the entropy will decrease over some time interval. Thus, in the positive case, we see that it is possible for the degree of uncertainty about the state of the system at time t (given some initial density po) to decrease with i, so that the system progresses to a state of greater order or canalization, all this, of course, is in the absence of external forces (V = 0). □ Now recall Example 4.1 or (4.152) that a metric on IRn is conformally flat if there is a smooth function -0 such that (5.60) in standard co-ordinates. We shall compute the invariant £5($) of (5.37) for •such a metric. We have ¿(z) = V5 (z) = (5.61)
314 Antonelli and Zastawniak and the Christofiel symbols are easily computed to be ri = ^ = -rv if i*j r:7. =rv=ö^ if = 0 otherwise. (5.62) We shall compute £g in terms of theRiemannian scalar curvature. 7Z. To this end, note that R = S% (5.63) where Ri, is the Ricci tensor (4.150), is given by = dkr§ - rrj^. + I*afc(& V5) - didi(£n Vg). (5.64) Use of (5.61), (5.62), (5.63), and (5.64) now yield R = -(n - l)e“2* ¿{2(^3^) + (n - 2)(0^)2}. (5.65) i=:l Using (5.61), and (5.62) we find that A,(& <f>) = ne~^ ¿(AM + (n - 2 W)2} i-1 and |grads£n <j>\2 = n2e~^ ^(ftv)2. (5.66) i=l Combining (5.65), (5.66) and using (5.37), we find an invariant closely related to quadratic dispersion (5.8) | {(n + 2) Igrad^l2 + ^}. (5.67) Thus, we see that, in view of Theorem 5.9 and the remarks following, the rate of production of entropy for a conformally flat metric, and hence the possibility for the production of order, is strongly influenced by the sign of Riemannian scalar curvature 1Z. Thus (on the surfaces Igrad^l^ — constant), £g(</>) is smaller in regions where 7Z is negative, and larger in regions where H is positive. To obtain a negative rate of change of entropy, we require £g(<j>) to be large and positive. Thus, for fixed drift vector h, the more positive 7Z is, the smaller the rate of entropy production, whereas, the more negative 1Z is, the greater the entropy production per unit time. It is interesting to compare the above remarks with results on filtering (4.15S), where the formula (5.67) occurs explicitly. There it is shown that, for a given signal process, the estimation problem becomes more difficult as 1Z becomes more negative. In particular, in relation to the chemical ecology of
Finslerian Diffusion 315 communities of sessile organisms, the negative of the quadratic dispersion, V, is a measure of community vigour, so that the more vigorous communities are harder to estimate (with a’ given observations process). Regarding entropy as a measure of the degree of uncertainty, we see that the amount of uncertainty inherent in a community increases faster for more vigorous communities, and so, from this point of view, it is not surprising that it becomes more difficult to estimate (i.e., less information is obtained from a given observation). Thus, the approach in Section 5.4 and that given here, reinforce each other in this context and show* the profound role of quadratic 'dispersion in these problems. To further justify the above discussion, we must consider when our metrics (5.60) satisfy the assumptions of Theorem 5.9, i.e., satisfy the conditions (5.43). Observe first that the conditions on 0 and <p (5.43(iii) and (iv)) imply that must satisfy -C|s|fc < ^(z) < -A'|z|<r as [z| T oo, (5.68) for some positive constants C, K, a and non-negative integer k. In particular, we must have gij{x) —► 0 as |x| f oo, and so the gij{x) are bounded Hence, (5.6S) implies that (5.43 (i)) is satisfied. Next, observe that dkffl) = — 2(dk^)gij- Thus, to conform with (5.43 (iii)) we must require that be p.d., and hence by Lemma 5.8(ii), £*(0V>) is a.d. if and only if is a.d. But (5.60) and (5.61) imply that = (5.69) and so g^ is a.d. provided n > 2. Thus, all hypotheses (5.43) are satisfied provided (5.68) holds and each dkfyfi is p.d., if n > 2. For n = 2, we cannot use Theorem 5.9 as it stands. The essential point about a.d. functions used is that such functions belong to ZA(IRn;dp). To retain this property, and obtain a formula for dSt/di like that in Theorem 5.9, we must impose very strict domination conditions on the solutions q(x,t) of the diffusion equation (ID + — d/dt)q = 0, to compensate for the lack of a good constraint on the g^. Thus, we can recover the results of Theorem 5.9, if we assume thatln qt is p.d. (as before) and that qt and its first and second order partial derivatives are dominated by C(1 + 1*1* W) (5.70) for some s > 0. The proof is essentially the same. The point is that the product of gv with a function dominated by (5.70) is a.d. Note that solutions q(x,t) dominated by (5.70) tend to zero as ]x| T oo, so that the result for n — 2 applies to a very much more restricted class of solutions than the result for n> 2. O Recall that, from Proposition 5.6, 0 is a stationary density for the diffusion operator ID * if V = i.e., ifID = |A£?+Zi+ div5h. For a conformally flat metric (5.60), this density is given by (5.61). We shall be particularly interested in the case in which (5.60) is a Gaussian metric; i.e., a metric for which ^(¿r) - i (x - p)u4(z - /¿)* + c (5.71)
316 Antonelli and Zastawniak for some m € Rn, constant c, and positive definite, symmetric matrix A == (%•) (here, * denotes transpose). Such metrics are of interest in connection with the chemical ecology. For a Gaussian metric, the normalized stationary probability density pÿt = ÿ/VÇ(IRn), is the given by the n-variate Gaussian distribution with mean /x and covariance matrix nA, ' (det A)1/2exp { - (x - ¿)A(x - //)•}. (5.72) We shall consider the entropy of (5.72) in terms of the ^temperature-like” para¬ meter T = (det A)"1/n, (5.73) i.e,, T is the reciprocal of the geometric mean of the eigenvalues of A. By analogy with classical thermodynamics, we shall refer to T as the absolute temperature of the Gaussian metric defined by (5.71). We shall show that, for fixed n, the entropy is a function ofT alone. Performing an orientation preserving orthogonal transformation of coordin¬ ates, if necessary, we may suppose that A = diag (Ai,..., An), and then, via (5.73), (5.72) reduces to the product (5.74) so, the entropy S is given by The integral is easily computed to be | V2%/nAi • yfa/nXj = (27rT/n)n/2, using (5.73). Thus, we obtain an expression for S as a function of T, (5.75) Note that S is an increasing function ofT. Thus, large temperatures means more disorder, while small temperatures mean more order.
Finslerian Diffusion 317 Also of interest is the variation of S with n for fixed T, Replacing n by a continuous positive variable u, we find that S(u) has a maximum value of T at u = 2ttT, and S(u) increases as u increases from 0 to 27rT, and decreases as u increases beyond 27rT. In fact, S(u) = 0 when u — ?7rTe, and becomes negative as u increases beyond this point. It follows that, if n > 27rT, the more species there are living in a community at a given temperature, the more orderly the system is. Furthermore, this effect is enhanced if the temperature decreases as more species are added, and is diluted otherwise. This type of result, with identical parities, was proved for the vigour V in the filtering sections. Remark 5.2. We note finally, that using the formulas (5.66), we may compute the invariant £ff(0) for the Gaussian metric (5.71). We find £,(«) = ne-W{2\(z - g)A|2 + TY (A)}, where Tr(A) = trace of A. It is clear then, that £ff(<£) depends on all the eigenvalues of A separately, and is not a simple function of T alone. Note also that >ne-^®>IY(A)>0 for all x, since A is positive definite. Thus, from the discussion after the proof of Theorem 5.9, we see that the rate of entropy production is strictly smaller for a Gaussian metric than for a metric with non-positive £g(0) but comparable density at some given time t
Chapter 6 Diffusion on the Tangent and Indicatrix Bundles Finslerian diffusion has been constructed in Chapter 3 by extending the notion of stochastic development from Riemannian manifolds to Finsler manifolds. In the present chapter we shall study Finslerian diffusion as a process on the slit tangent bundle TM and the indicatrix bundle IM of a Finsler manifold M. In Section 6.1 it is demonstrated that the Finslerian hu-Brownian motion in¬ troduced in Chapter 3 is equivalent to Riemannian Brownian motion with drift on the slit tangent bundle TM equipped with a suitable Riemannian metric, the so-called Sasaki lift or diagonal lift of the Finsler metric. In Section 6.3 an alternative approach to Finslerian diffusion will be developed with TM replaced by the indicatrix bundle IM, in line with the view that Finslerian objects de¬ pend in fact only on the direction of a tangent vector to M because of their homogeneity in y e TM. This leads to the definition of an indicatrized Finsler hv-Brownian motion. Similarly as in the case of 7w-Brownian motion on TM, in Section 6.4 it is demonstrated that the indicatrized Tiv-Brownian motion is equivalent to a Riemannian Brownian motion with drift on the indicatrix bundle IM equipped with the diagonal lift metric. The drift on TM and the drift on IM can both be expressed in terms of the torsion tensors P and C of the Cartan connection. Throughout this chapter we shall assume that M is an n-dimensional Finsler manifold with metric function £, positive definite metric tensor and the Cartan connection (NJ,FjA, Cjfc). Given a local coordinate system (a?) on M, we take the induced coordinates (a®, yt) on TM and consider the adapted frame on TM, where Here di — d/dxi and d^ == d/dy1. For ease of notation, in the present chapter we shall write in place of the symbol di used throughout the rest of this 319
320 Antonelli and Zastawniak book. The dual frame to will be denoted by 6y\ where 6yl = dy* + Nj (x, y)dx3. 6.1 Slit Tangent Bundle as Riemannian Mani¬ fold The goal of the present section is to demonstrate that Finslerian /tu-Brownian motion as defined in Chapter 3 by means of Finslerian stochastic development can also be regarded as Riemannian Brownian motion with a suitable drift on the slit tangent bundle TM equipped with the so-called diagonal lift metric, which turns TM into a Riemannian manifold. The drift can be expressed in terms of the torsion tensors Pik and Cjk of the Cartan connection. Definition 6.1. The diagonal lift metric on TM is defined by G(x,y) = gi,(x,y)dxi ® do?’ + (a;,y)Sy‘ ® 6y>, where g is the Finsler metric tensor. If g is positive definite, then so is G. Equipped with, the metric tensor G, the slit tangent bundle TM becomes a Riemannian manifold. We shall also need the following connection on TM. Definition 6.2. The connection V on TM such that = rfyji* + +. = r(i)üA + r^)0W = <%dw is called the horizontal lift of the Cartan connection to TM. The above formulae define the coefficients etc. of V in the adapted frame 6k* fyjt) on TM in terms of F^ and C-. Namely, pfc — pi*2) _ 1 1 (i)Ü) Proposition 6.1. V is a metrical connection with respect to the Riemannian metric G on TM, that is, VG = 0.
Finslerian Diffusion 321 Proof: The proposition follows immediately from the definitions of V and G, and the fact that the Cartan connection is h- and v—metrical. Even though V is a metrical connection, it differs from the Levi-Cività con¬ nection in that it has non-vanishing torsion, which is computed in the next proposition. Proposition 6-2. In the adapted frame d(i) the torsion tensor T of V can be expressed as follows: TWa) = T*6k + 4%) = Rikdw, = T^6k 4- = -c&k-P*dw, Wi) = T^,5k 4- = +c&k+p$dw, = TfawSk 4- T^dw = 0. where (6-1) are the coefficients of the three non-vanishing torsion tensors R, P, C of the Cartan connection in the adapted frame Proof: The formulae follow by direct computation from the formulae for the coefficients V in Definition 6.2 and the general definition T{X, y) = - VyX - [X, y] of the torsion tensor. 6.2 /^-Development as Riemannian Development with Drift According to Propositions 6.1 and 6.2 the horizontal lift V of the Cartan con¬ nection is a metrical connection with torsion on TM regarded as a Riemannian manifold with the diagonal lift metric G. The following theorem extends the results of Section 3.1.2 to the case when the Levi-Cività connection on a Riemannian manifold is replaced by a metrical connection FJ^ with torsion. Theorem 6.3. Let M be an n-dimensional Riemannian manifold with metric tensor gij and let F^ be the coefficients of a metrical connection with torsion tensor rrti ■ ■ Tit ini
322 Antonelli and Zastawniak Then for any diffusion (Xy Z) on the orthonormal frame bundle OM satisfying the system of SDEs dX{ = (6.2) dZ^ = X is a diffusion on M with generator n = - r^fc) = 1a+B, (6.3) where WWHfc) is the Laplace-Beltrami operator and B a vector field on M such that B^^T^di. (6.4) This is to say. X is a Riemannian Brownian motion on M with drift B given by (6.4). Note that Frame fields (Z^) are denoted as upper case Roman letters in this chapter. Proof: This result is well known. For the proof we refer, for example, to [59]. We are now in a position to present the main result of this section, accord¬ ing to which Finslerian Av-Brownian motion can be regarded as Riemannian Brownian motion with drift on TM. Theorem 6.4. A diffusion (X,Y)on TM is a Finslerian hv-Brownian motion if and only if it is a Riemannian Brownian motion with drift * = (s-5) on the slit tangent bundle TM regarded as a Riemannian manifold with the horizontal lift metric G of Definition 6.1. Here and Cjk are the coefficients (6.1) of the torsion tensors P and C of the Cartan connection in the adapted frame 6i, 9(f). Proof: It suffices to demonstrate that the generator D of Riemannian Brownian motion on TM with drift (6.5) is the same as that of the Finslerian Av-Brownian motion. To this end we apply Theorem 6.3 to the case of TM with the diagonal lift metric G and the horizontal lift V of the Cartan connection to TM. The drift B in Theorem 6.3 can be written as
Finslerian Diffusion 323 This follows immediately from Proposition 6.2. Now, expressing the generator D = + B in Theorem 6.3 in terms of the adapted frame 6i, 6^ and applying the formulae for the connection coefficients of V given in Definition 6.2, we find that D = - C^w). This is the generator of Finslerian /w-Brownian motion, see Theorem 3.11. 6.3 Indicatrized Finslerian Stochastic Develop¬ ment In the present section we adopt the point of view that Finslerian objects should depend only on the direction of a tangent vector y € TM, the dependence on the length of y being completely determined by the homogeneity conditions. In line with this point of view we reformulate the theory of Finslerian diffusion, using the indicatrix bundle IM with fibres IMX = {y GTMX : I(M) = 1J in place of the slit tangent bundle TM. While the system (3.39) of SDEs defines Finslerian hv-stochastic develop¬ ment in terms of ‘rolling’ along a Brownian motion (W, V) on the ‘tangent bundle’ Rn x over we shall introduce ‘rolling’ along Brownian motion (W, U) on the ‘sphere bundle’ R” x Sn“T over Rn, where S’1-1 = {¡r € F : = 1} is the unit sphere in Rra. Thus, we now assume that W is a standard Brownian motion on Rn and U is a Riemannian Brownian motion on the unit sphere S’1“1 C Rn such that U and W are independent. We consider the unit sphere 5n_1 c Rn as a Riemannian manifold with metric induced from Rn. By definition, a Riemannian Brownian motion U on S71"1 is an Sn"1-valued diffusion with generator D = |A, where A is the Laplace-Beltrami operator on Sn_1, which, as is well known, can be written as A = - x^didj - (n - where x* are the canonical Cartesian coordinates on Rn and di = d/dx1, Even though expressed in terms of the operators di on Rn, the above expression defines an operator on Sn_1, since, for any smooth functions /i,/s : —> R, if fx = f2 on S71’1, then A/i ~ A/i on S”"1. To construct a concrete specimen of Riemannian Brownian motion U on Sn~l, one can employ the system (3.17) of SDEs for Riemannian stochastic de¬ velopment. However, for our purposes, it proves more convenient to use another
324 Antonelli and Zastawniak representation of Brownian motion on S^1 due to Stroock [89] (see also [59], Section IIL2). Namely, let for any 0 / x € Rn and let V be a standard Brownian motion on Rn. Then the solution U to the SDE dJ7£=A}(i7)odVJ‘ (6.6) with initial condition 17(0) € S*1"1 is a Brownian motion on 5n_1. It is easily verified that dCT = -i(n-l)irdt, (6.7) dtTdZP = (S”'-UWjdt. Definition 6.3. Let (X, Y, Z) be a solution to the system dXi = SYi = Z]odUj (6.8) dZj = -Fii(X,r)2,jodXfc-Ci.i(X,Y)2,jo<5yfc of SDEs, where SY1 = dY1 + N}(X, Z) o dXj and where W is a standard Brownian motion on and U a Riemannian Brownian motion on the unit sphere 5n_1 C Rn such that U and W are in¬ dependent, with initial conditions X(0) = a?o € M, Z(0) = yo € IMXo, and £(0) — 2o, a frame in TMXo orthonormal with respect to the quadratic form 0(®Ch3fo) such that y*(0) « IP (0)£j(0). Then (X, Z, Z) is called the indicatrized Finslerian hv-development. As compared to Definition 3.2, we have used a Riemannian motion U on the unit sphere S”"1 in place of a Euclidean Browmian motion in Rn, and we introduced the initial condition Zi(0) — £P(0)Zj(0). As the theorem below demonstrates, the resulting process (X, y, Z) is restricted to a subbundle of the orthonormal Finsler bundle OTM. Definition 6-4. The orthonormal Finsler bundle OIM c OTM over IM C TM is, by definition, a subbundle of OTM induced by the projection 7r : OTM TM, that is, OIM = n~l(IM). Theorem 6.5. The indicatrized Finslerian hv-development (X, Y,Z) is a diffu¬ sion on the orthonormal Finsler bundle OIM over IM, and (X, Y) is a diffusion on IM with generator 0 = + i(i?y (6.9)
Finslerian Diffusion 325 Remark 6.1. Even though the generator D given by (6.9) is expressed in terms of the adapted frame on TM it can be regarded as an operator over IM in the sense that if /1, /2 : TM —> R are smooth functions such that /1 = /2 on IM, then D/i = D/o on IM, This property of D can be verified by direct computation, but it also follows from the proof of Theorem 6.5 as well as from assertion b) of Theorem 6.9 Lemma 6.6. Under the assumptions of Theorem 6.5. a) Yi = U’Z} P-a.s., b) = P-a.s., c) L(X,Y) = l P-a.s.. Proof: a) Given the solution (X, Y, Z) to (6.S) with the initial conditions de¬ scribed in Theorem 6.5, we consider the SDE dAi = —F|,z (X, Y)Al o dXk - (X, Y)Al o 6Yk + 2} o dUj. (6.10) Since Cjk^y)yk — 0 the Cartan connection is deflection-free, i.e., FJfc(a;, y)yk = NJ (x, y), it follows from the second equation in (6.7) that Ai — Yi is a solution to (6.10). By the Ito formula, d(tPZj) = IF o dZ] + Zj o dW. Substituting for dZj from the third equation in (6.8), we find that A* — IPZ* is also a solution to (6.10). But, by assumption, Y£(0) = CP(O)Zj(O), which, by uniqueness, means that Y* = IFZij, as required. b) The proof of assertion b) is exactly the same as that of (Prop. 3.9) above. c) By the Ito formula, dL2(X, Y) = 5iL2(X, Y) o dXi + 0(i)L2(X, Y) o JY\ Since the Cartan connection is deflection-free, it follows that <5«L2(m, y) = 0. We also have d^L2{x,y) — 2gij(x,y)y?. Thus, by the second equation in (6.7) and assertions a) and b) of the present lemma, dL2(X,Y) = 2^(X,Y)Yio6Y^ - 2gij[XiY)UkZikZjodUt = 26kiUkodUl ~ 2d(6kiUkUl) — 0, for 17 is a Brownian motion on the unit sphere, i.e., 5kiUkUl = 1. Because, by assumption, L(X(0), Y(0)) = 1, it follows that L(X, Y) = 1, as required. Proof:[Proof of Theorem 6.2.1]As a consequence of assertions b) and c) of Lemma 6.6, (X, Y,Z) is a diffusion on the orthonormal Finsler bundle 01M over IM with projection (X, Y) onto IM.
326 Antonelli and Zastawniak Let us now compute the stochastic differentials dX* and from (6.7): dX* - ZjdWi -^(X.YyZ^dV^dW^ 5Yi = Z^KF - \-Fikl{XiY)Z^ZlidWmdUi -^(X^Z^Z^dU^. But dWmdWi — S^dt by the Ito rules, and dUmdW2 = dW™dU3 — 0 because U and IV are independent. Without loss of generality, we can assume that U is the representation of Brownian motion on the unit sphere S^1 C Rn defined by (6.6), which means that w’e can use formulae (6.7) for dlP and dJJ^dW, It follows that dX* = Z^dW^ -^^(X^Z^Z^dt = ZidWi - ±gki(X, Y^X, Y)dt, SY* = Ztei-SuUWydV* 2> = Zj (Si - SuUWydV* -Ifn-l^dt —g^Y^X^dt, since Z^Z^S”* = 5W(X,y) by assertion b) of Lemma 6.6, Y* — U^Zj by assertion a) of Lemma 6.6, and = 0. It also follows that dX*dX> - ZiZ{dWkdWl = ZiZtS^dt = gij(X,Y)dt, SY'SY^ = ZikZi<ttJkdUl = Zj.Z{ (Skt - UkU‘ydt = (ffy(x,y)-yiy^<it1 dXW = ZiZ]dWkdUl = 0, STTdX’ = Zlzi^dW1 = 0,
Finslerian Diffusion 327 Thus, for any smooth function f : TM —► R, df(X, Y) = 6if(X, Y) o dXi + ¿to/(X, Y) o 5Yi $if(Xy Y)dXi + S(i)/(X, Y)^T + Y^dX'dXi + ^Sid^ftX, Y)dXi3Y’ + ¿d^ftXWrdXi + = Sif(X,Y)ZidWi + d{i)f(X,Y)Zi(& - SkiU’U^dV11 + t^(X, Y^S^X, Y) - F*.(x, Y)Skf(X,Y)) + |(?'(x,y) -y’y>)(5way)/(x,y) - c^{x,Y)d(k)f(x,Y)) -|(n-i)yfcaw/(x,y)]dt. As a consequence, (X, Y) is a diffusion with generator D given by the expression in square brackets multiplying dt above. Since, by assertion c) of Lemina 6.6, L(X, Y) = 1, that is, (X, Y) 6 ZAf, it follows from the above calculation that D is an operator on IM in the sense of Remark 6.1. Definition 6.5. The solution (X, Y, Z) to (6.S) with the initial conditions in Theorem 6.5 will be called an indicatrized Finslerian hv-stochastic development. Definition 6.6. A diffusion (X, Y) on IM with generator D given by (6.9) will be called an indicatrized Finslerian hv-Brownian motion. We have just proved that if (X, Y, Z) is an indicatrized ^stochastic devel¬ opment, then (X, Y) is an indicatrized hv-Brownian motion. In the next section we shall demonstrate that an indicatrized hv-Brownian motion can also be rep- resented as a Riemannian Brownian motion with drift on the indicatrix bundle IM equipped with the Riemannian metric induced from TM. 6.4 Indicatrized /^-Development Viewed as Rieman¬ nian As in the previous section, we assume that M is an n-dimensional Finsler man¬ ifold with metric function L, positive definite metric tensor and the Cartan connection (Nj, F^3 CJA). We shall use the standard summation convention on upper and lower indices supplemented with the rule that there is no summation with respect to n = dim M. We also assume that Latin indices t, J, Zc, Z, m run from 1 to n, while Greek indices x, A, /z run from 1 to n — 1. We consider the indicatrix bundle IM C TM as a Riemannian manifold equipped with the metric G induced by the diagonal lift metric G defined on TM by Definition 6.1. We shall prove the following theorem.
328 Antonelli and Zastawniak Theorem 6-7. A diffusion (X, Y) on IM is an indicatrized hv-Brownian mo¬ tion if and only if it is a Riemannian Brownian motion with drift B on the indicatrix bundle IM equipped with the Riemannian metric G.. The drift is ex¬ pressed as in terms of the torsion tensors and Cjk of the Cartan connection and the adapted frame Remark 6.2« The vector field B on TM defined in Theorem 6.7 is tangent to IM. which means that it can also be regarded as a vector field on IM. This can be verified by direct computation, but it also follows from the proof of assertion a) of Theorem 6.9 below. Theorem 6.7 will be proved at the end of the present section. Right now, let us introduce the notation and state some facts concerning the indicatrix bundle IM to be used in the remainder of this section. Let (xk) be local coordinates on M and let {xk,yk) be the induced coordin¬ ates on TM. The vector fields = 9w = a/ay\ sk = dk-Njk(x,y)d(J) on TM are defined as before. We also choose another system (xk,yk) of local coordinates on TM defined by $*(», y) = iFfay) = y*/L(x, y), y^x, y) = L(x, y), subject to the condition d(n)L(x, y) / 0. Since y / 0 and y)=yk/L for any k, reshuffling the indices if necessary, one can always ensure that the latter condition is satisfied in a neighbourhood of any given point in TM. Then, by the implicit function theorem, (xk$*) are well defined local coordinates in a neighbourhood of the given point. Also, the indicatrix bundle IM is given by the equation y^^y) = 1 and $\y*) are local coordinates on IM. Using the identities d(k)L{x,y) = y*/£(a;,y), dkLfay) - 'fok{x,y)/L(x>y), where = 7jfc(®»y)yil^> one finds that _ n - n 93? l’ &$ ' 9$ ’ 9x‘~ yn' dy* _ SML 9yn _ dyk _ dyx x ’ dy* yn' 9yn L' (6.12) We define the vector fields 9*. = d/9xk and 9(k) = on TM. From (6.12) it follows that dk = dk - (7o&/l/n)^(n), 5(X) = L5(X) - (Lyx/j/n)5(n), 3(n) = WIL)9($.
Finslerian Diffusion 329 Clearly, the vector fields dk and d^ are tangent to IM. Proposition 6.8. a) The vector fields 6k — dk - are tangent to IM\ b) The vector field â(n) = d/dy* is perpendicular to IM in the diagonal lift metric G. Proof: The proof depends on the condition = 7^(2;, y) satisfied by the nonlinear part of the Cartan connection. To prove assertion a), it suffices to express as a linear combination of dk and We get Sk = dk — N*0(0 = & - (Nè/L)3w + [(7g*/»n) - (N^/ïn) - NJ№(n) = &-(Nè/L)ôw. To prove assertion b), we shall demonstrate that is perpendicular to each of the vectors 6k, d^ tangent to IM, which clearly form a basis in TIM. But, G(M(n)) = G(6k, (?/L)^(i)) == 0, G{d^,d^) — G(Ld(rf — (i®x/yn)d(n)i ü/7£)d(i)) = y^ - ■ “ 3/x ~ y* — 0, as required. We set dxk = dxk and 6yk = (d^/dy^y1, where 6yl — dyl + N!m(xfy)dxm. Then 6yk is the dual frame to 6k, Moreover, 6y* is the dual frame to the frame 6k, d(x) on IM. Let G be the diagonal lift metric on TM. We set Then, by Proposition 6.8 b), the metric tensor G on TM and the induced metric tensor G on IM can be written, respectively, as G = gkidZk ® ¿S' + 9(x)(x)Sy>‘ ® Sy* + g(n^n')Syn ® ¿y", G = gkidxk Qdz1 + 9{>t)(X)Sy*c ® 5j/\ (6.13) (6.14)
330 Antonelli and Zastawniak From (6,13) and (6,14) it follows that the dual tensors G# and to G and G can be expressed as G* = sklSk ®6t+ g^d{x) ® dw = gW)d<n) ® 0(n), (6.15) G* = gkiSk ®5t + J(xW(x) 0 dw, where g*1 and ¡ft® axe, respectively, the inverse matrices to gki and 5(fe)(z}- By transforming the formulae for the horizontal lift connection V (Defini¬ tion 6.2) on TM into the frame and projecting the resulting expressions orthogonally onto TIM, we obtain jbhe following formulae for the induced con¬ nection V on IM (cf. [98]), where V is called the K-connection): = = öj,™ \>kdy>~ + &y>Fkl) KJ = = äFCfciötn = = d^dy1 dy™ dy^dy^ dy* dyx (6.16) Since V is a metrical connection on TM with respect to the metric tensor G, it follows that V is a metric connection on IM with respect to the induced metric tensor G. In general, the induced connection V has a non-vanishing torsion tensor T, which can be obtained by transformingthe formulae of Proposition 6.2 for the torsion tensor T of V into the frame 8k, and projecting the resulting
Finslerian Diffusion 331 expressions orthogonally onto TIM. This procedure yields f(«U0 = r^6m+T^d{lt} (6.17) ^(^(x),^)) — + r(xj(A>^W = 0 We are now in a position to consider the Riemannian stochastic development defined by (6.2) in the case of IM equipped with the Riemannian metric G and metric connection V. By Theorem 6.3, the projection (X, Y) of this Riemannian stochastic development from the orthonormal frame bundle 01M onto IM is a Riemannian Brownian motion with drift B on IM. Theorem 6.9. Let (X, K) be the above Riemannian Brownian motion with drift B on IM. Then a) B can be expressed by the formula in Theorem 6.7, b) (X, Z) is an indicatrized Finslerian Jw-Brownian motion. Proof: To prove assertion a), we compute the drift B using formula (6.4) from Theorem 6.3 in thejsase of the indicatrix bundle IM with metric tensor G and metric connection V. By (6.15) and (6.17), we have B = (6.18)
332 Antonelli and Zastawniak The last equality holds, since by1 dy* _ _ by* dy* by* dyj ~ dyk dyj dyn dyj (6.19) = gij -y^/L2 Because y) is homogeneous of degree one in y, it follows by the Euler theorem that ^^)N^(t,2/) = Nj^y), which implies that yly^m = №№ - FCJ = y^in ~ y^in = 0. We also have ymClmL(x,y) = 0. Since L(x,y) = 1 on IM, the formula for B therefore follows from (6.18). To prove assertion b), we compute the generator D of (X, Y) using formula (6.3) from Theorem 6.3 in the case of the indicatrix bundle IM with metric tensor G and metric connection V. Writing the first equality in (6.3) in terms of the frame 5*, by (6.15), (6.16), (6.19), (6.20), and the identity yiC^ix, y) = 0, we have D = - fs^n - rtfdw) + j$MW(W(A) - r£)Wim - f$waM) - - FB« + wt (Py* 3^ a 3j/fc dy1 dy* dym _,i a \ dy*dy* dyi dy?a(m) dy* dy‘ dy* dy? ^'kl0<‘mu = - F$Sm) + - Cß0(m)) To evaluate the last term in the above formula, we compute the following second- order partial derivative, the result being equal to zero, since L(xty) = J* is independent of ^A: o d2L2(x,y) nd fdyi \ yi) ~dy*dy*yi+“dyxdy*9i:i'
Finslerian Diffusion 333 Thus, dyi dtf &yx dyx9ij -(n-l). Since L{x, y) — 1 on ZM, it follows that D is given by formula (6.9), i.e., (X, V) is an indicatrized Finslerian Au-Brownian motion. Proof: [Proof of Theorem 6.2.1]If follows from Theorem 6.9 that the generator D of an indicatrized Av-Brownian motion given by (6.9) is equal to + B, where A is the Laplace-Beltrami operator on IM equipped with the Riemannian metric G and where B is the vector field in IM defined by (6.11). This proves Theorem 6.7.
Appendix A Diffusion and Laplacian on the Base Space Our aim in this appendix is to present a construction of the Laplace operator on the base space M of a Finsler manifold rather than on TM or IM, as in the main body of the book. The operator will be applied to define harmonic forms and to obtain a Hodge decomposition theorem for Finsler spaces. The vehicle we employ in our construction is diffusion theory, in particular, a result which can be understood as the Central Limit Theorem for geodesic random walks. This extends Pinsky’s results on isotropic random walks and their, limit on a Riemannian manifold [83], [84]. Diffusion theory makes it possible to resolve the well-known difficulties inherent in studying harmonic forms on a Finsler manifold. The Laplacian emerging from diffusion on the base space M provides a natural definition of harmonic forms as those that realize heat equilibria. We shall denote our Laplacian by Aaz to distinguish it from that proposed by Bao and Lackey in their ground-breaking paper [31]. We shall refer to the latter as the BL-Laplacian and denote it by A bl« It was constructed by consid¬ ering the indicatrix bundle IM as a Riemannian manifold with the diagonal lift metric G (for the definition, see Chapter 6), which gives rise to a Hodge star * on IM and thus an inner product of differential forms 77, 9 on IM. This can also be regarded as an inner product of differential forms tj, 9 on M, since any such form can be identified with a form on IM (depending only on x G M). Taking to be the exterior differential on M and the adjoint to ¿m in the sense that = {ri,dMe}' for any differential forms 77,9 on M, the BL-Laplacian Abl on M can be defined by Abl = + djwdjf • 335
336 Antonelli and Zastawniak Because an inner product has been introduced on the elliptic complex of differ¬ ential forms on M, a Hodge decomposition theorem follows by a general result about elliptic complexes in Wells [93], Chap, 4, Theorem 5.2. Is any of the two Laplacians Abl and Aaz more natural than the other? Or could perhaps yet another candidate on which to base Finslerian Hodge Theory be revealed? After all, as far as the Hodge decomposition theorem is concerned, it only matters that there should be a well-defined inner product on differential forms, which leaves plenty of room for choice. Our Laplacian Aaz emerging from diffusion theory appears to be the most natural one in that the resulting harmonic forms realize heat equilibria, that is, they are invariant under the heat flow generated by diffusion on M. This completes the first step of the Milgram-Rosenbloom program for Finsler Hodge theory. The construction of Aaz will proceed as follows. First we shall introduce a geodesic random walk on a Finsler manifold M extending Pinsky’s concept of an isotropic random walk on a Riemannian manifold [S3], [84]. Then a limit will be taken under suitable scaling as in the Central Limit Theorem, producing a diffusion generator on M, which is obviously a second-order operator. The quadratic form of this operator will give rise to an inner product of differential forms on M, leading eventually to the definition of our Laplacian Aaz m a similar way as for Abl above. Harmonic forms are then introduced and a Hodge decomposition obtained again by a general result about elliptic complexes in [93]. Throughout this appendix M will be a without boundary compact Finsler manifold with smooth metric function L : TM —> R positive definite metric tensor g : TM —► ToM. This ensures that for each x e M the indicatrix IMX is a smooth compact submanifold in TM. In addition, we shall assume that L homogeneous in y, that is, = |A|L(x,y), and not just positive homogeneous. A.l Finslerian Isotropic Transport Process To define this process we first introduce the geodesic flow on a Finsler manifold. This is a deterministic motion &(x,y) obtained by following the unit speed geodesic with initial position £o(z, — x € M and initial velocity ¿o(z, j/) =* y € IMX. It satisfies the geodesic equation =0, where are the Finslerian formal Christoffel symbols (1.55). If we put = №(*,»),&(*»»)) for any function f e then I^i-t = and u(t,x,y) = (2?/) (x,y)
Finslerian Diffusion 337 satisfies the p.d.e. dtu — Zu, where the first-order operator Z = y*di -'^k{x,y)yiyi.di = y*5i is known as the geodesic flow field or the canonical horizontal vector field on IM. It is the infinitesimal operator of the semigroup T?tt> 0. Next, for any x € M we consider the indicatrix IMX as a submanifold of IM with the diagonal lift metric G induced from IM. This means simply that gij(x,y) considered as a function of y with x fixed is taken as a metric tensor on TMX and then it is restricted to IMX c TMX. The indicatrix IM® equipped with this metric is a compact Riemannian manifold. We denote by wx the volume measure of IMX in this metric and assume that it is normalized so that / = 1. JIMK The canonical projection operator II: C°°(IM) —► C^tM} is then defined by (II/)O) = [ f(x,y)tox(y). JIM* Let us take a sequence of independent random variables ei, • • * with ex¬ ponential distribution > 0 = i > 0, on some probability space (Q, S, P). Then tn = ei 4 F en are the jump times of a Poisson process with unit rate. We also put to = 0 for convenience. The isotropic transport process can be defined by induction. For any given starting point x GM and starting velocity y 6 IMX we put xo = x and yo — y. Assuming that Xi 6 M and yi G IMX. have been constructed for i — 0,.,., n-1, we put «n = íe„(®n-l,í/n-l) and require that the conditional distribution of yn € ZMX. given ex,... , en and yi,...,yn-i should be uniform on IMXn under the normalized volume measure wXn, that is, ■E{/(sn>yn)|ei,. • - ,en,yi,... ,yn-i} = (n/)(®„) for any f G C°°(JMXn). Finally, we put Xt — ^t—tn (pm yn) and = Xt for t G [tn? ín+i)j n == 0,1,2,,.. . In this way we obtain a Markov process (Xt,Yt) on the indicatrix bundle IM defined for all t > 0, called the isotropic transport process. The sample paths of Xt áre continuous piecewise geodesic with jumps of the tangent vector Yt at the jump times tn of the Poisson process. The infinitesimal operator L of (Xt, Yi) can be computed by observing that - on {t < ei}, which has probability e“\ the process moves along the original
338 Antonelli and Zastawniak geodesic y), and on {t > ei}, which has probability 1—e *, a new direction Vi is chosen according to the measure wXl at time ti. For any f G C^fJM) we f(Xt,YJ = Шх,уШх,у)) = f(x,y) + t(Zf)(x,y) + o(t) on {t < ei} and f(Xt,Yt) = /(xljifl)+o(l) (Щ)(х^ = (П/)(®)+о(1) on {t > ei}. Using the fact that E{f(xi,yi)|ei} = (Hf)(xi), we obtain Ef(Xt,Yt) = [ f(Xt,Yt)dP+ [ f(Xt,Yt)dP = e_ + t(Zf)(x,y)] + [ №i,yi)dP4-o(t) = ^[f(x,y) + t(Zf)(x,y)}+ [ (Hf)(X1)dP + o(t) = e-t[/(x,?/) + t(Z/)(^j/)] + (1 - е-‘)(П/)(х) +o(t) = f(x,y)+t[(Zf)(x,y) + (П/)(®) - f(x,y)] + o(i). It follows that Ь/ = Шп1(/оТ^/) = Я/ + П/-/. Here Tt is the semigroup of (X£, Y£), (Ttf)(x,y) = Ef(Xt,Yt) for any f 6 A.2 Central Limit Theorem We shall scale the isotropic transport process in such a way that the mean time and distance travelled between consecutive jumps of direction will be a2 and e, respectively, where a > 0. In other words, the scaled process X® will be piecewise geodesic with speed 1/s and jumps of direction at the jump times of a Poisson process with rate 1/e2. The velocity part of the process will be defined by Yf == eXf to ensure that the joint process (Xf, Yf) lives on the unit-radius indicatrix bundle IM. The infinitesimal operator of the scaled process (Xf, Yf) is then found to be Г = (1/£)2 + (1/е2)(П-1). We claim that Xf tends weakly to a diffusion process on M as a 1 0. First we observe that П2 = П and 1ЮТ — 0. The latter equality depends on the fact that any linear function of у on IMX has integral zero with respect to the normalized volume measure wx. This will be so if the indicatrix IMX
Finslerian Diffusion 339 is symmetric about the origin in the tangent space TMX, which is the case if the Finsler metric function L is homogeneous in y (rather than just positive homogeneous). Using these two equalities, we obtain the identity 27(1 + sZ + e2Z2)n = n^2n + eZ3H. This implies the following lemma. Lemma A.L For any f G C^M) put fc = (1 + eZ + s2Z2)/ 6 Then = f and lim 27/' = TLZ2f. The operator HZ2 on the right-hand side is the generator of a diffusion Xt on M. The theory of convergence of semigroups, in particular, the limit theorem for semigroups in application to random evolution proved by Kurtz in [69] can now be applied to show that lime^7 = e‘nz7 eXO J for all f G which, in turn, implies that X* converges weakly to Xt in the sense that Um£{/(Xf)}-E{/(Xt)} for every f e C(M'). All technical assumptions leading to these results are trivially satisfied if M is compact. The diffusion generator IIZ2 can be expressed as (ILZ2/) (x) = [ yLy> (didj - (x, j/)5fc) f(x) dux (y) for any f G C°°(M). This limiting argument is based Pinsky’s approach to isotropic transport and diffusion on Riemannian manifolds, see [S3], [84], where further technical details can also be found. In particular, if M is a Riemannian manifold, then g*(x) = (dimM) [ y^d^^y) JIM* and A = (dimM)nZ2, the Laplace-Beltrami operator, so Xt/aim m is a Brownian motion on M. This has been extended above to the case of Finsler manifolds.
340 Antonelli and Zastawniak A.3 Laplacian, Harmonic Forms and Hodge De¬ composition The diffusion generator ILZ2 is a second-order differential operator with leading quadratic form Jimx which is clearly non-degenerate and positive definite. When M is Riemannian, it reduces to the metric tensor g^(x) = (dim M) H^(x). Similarly, we define = i ..y'+yb dwx(sz), Jim* which can be written briefly as [ //Ml1) . Jim* for multi-indices I and J. Because - [ (yTu/)2 dwx(y) > 0 (Al) Jim* for any p-form u / 0, it follows that HIJ(x) defines an inner product on the exterior algebra AgM, namely, (#) = [ m9jHIJjHdx, Jm where Vh= Vg^(.y), V9= J&rt.atj')- Jim* v It is easily seen that \/H is a tensor density of weight 1 and H1J is a contravari¬ ant tensor of rank 2p on M, so the product (t?|0) is well defined. It follows from (A.l)) that HIJ is invertible as a map from p-forms to p-forms. The inverse will be denoted by Hu = We denote by d the exterior differential on M and by 6 the codifferential defined to be the adjoint to d in the above inner product, (<M*) = W*) for any 7) e A^M and 0 € APM. Definition A.l. The AZ-Laplacian on the base manifold M is then defined by Aaz = Sd + d6.
Finslerian Diffusion 341 Proposition A.2. Let M be a without boundary compact Finsler manifold with smooth metric Junction L(x^y) homogeneous (rather than just positive homo¬ geneous) in y and such that the Finsler metric tensor metric tensor gij(xfy) is positive definite. Then &AZn = Q<^dr/ = &? = 0 (A.2) for any ri € №M: Proof. Indeed, if Aaz*? = 0, then 0 « (AAz7?W - {SdrM) + (¿fyW = (dr)\drfj + , so dr) — St) = 0. □ Definition A. 2. A form rj € APM satisfying either side of the equivalence (A. 22) will be called harmonic and the space of such forms will be denoted by №M. Finally, we are in a position to state the following Hodge decomposition theorem based on our Laplacian A az- Theorem A.3. Let M be a Finsler manifold satisfying the assumptions of Proposition A.2, with harmonic junctions defined by means of the AZ-Laplacian £±AZ’ Then (a) Each cohomology class HPM contains a unique harmonic representative. (b) is finite dimensional, its dimension being equal to the pth Betti number of M. (c) APM = © dAf^M © 5Ap+1M, the three spaces on the right-hand side being mutually orthogonal in L$. Proof. The proof is exactly the same as in Bao and Lackey’s paper [31]. It follows directly from the general result about elliptic complexes in Wells [93], Chap. 4, Theorem 5.2, or can be proved by the elegant and simple argument presented in [31]. Rather than repeating this word-by-word, we refer the reader to the sources. □
Appendix B Two-Dimensional Constant Berwald Spaces B.l Berwald’s Famous Theorem We consider an ^-dimensional Finsler space F" = (AT, L(x, j/)) : (x1) local coordinates and (jf) tangent vectors; therefore, F = L2/2, gi^didjF, (g^) = (gii)-1, 7jh = 9"{dk9rj + dj9rk — dTgik)/2, 2i? = 3ii{yr^rF') - djF} = ^ky>yk, Gj = Gjfc=^Gj, 2Gi = G‘*^fe, dx with the geodesics: + 2G*(x, —) = 0. Fn is called a Berwald space, if G^k depend on x — (zl) alone. We shall find all the Finsler spaces with constant Gjk from the set of Berwald spaces of dimension n — 2» Fn is called a Locally Minkowski space, if L depends on y — (yl) alone for some choice of x — (z*). Then of F71 vanish, hence a locally Minkowski space is, of course, a kind of Berwald space. We have the following celebrated theorem on two-dimensional Berwald spaces which are not locally Minkowski. All the two-dimensional Berwald spaces which are not locally Minkowski are divided into the following three classes according to the magnitude of the main scalar J: (1) B2(1):Z2<4, L = ^/(a1)2 + (a2)2 exp {J tan“1 (a1/«2)}, J = W4- P, 343
344 Antonelli and Zastawniak (2) B2(2) : I2 = 4, L = [a1] exp (ecr/a1),. f = //2-il, (3) B2(r,s) : I2 > 4, L= |(a1)r(a2)4'|, r + s = l, where aa(x,y) = a“(x)yx + o^{x)y2y a — 1,2, are independent 1-forms. This is the celebrated Theorem of Berwald, [74]. Theorem B.l. 2Gi(xiy) of the spaces belonging to B2(l), B2(2) and B~(r, s) are written in the form XTfay) = {rjfc(®) +Tijk(x,y)}yiyk, where we put rjk(x) = ti-dka?, (&l) = (a?r1, rjk(x) = rjA - rl,., T*jk = Putting d ~ det (a*) / 0 and A* = dsaf — djag, we have (1) *2(1) ■ /no T2 <00 = (A«6o)(^)/<i2(l + J2), ^-(A^Xa^y^l+J2), bi = a1 + Ja2, b^ — a2 — Ja1, (2) B2(2); { = (Aaca)(a$c0')/cft, = -(Aaca)(afcj3)/d2, d = a1 — ea2, cz = sa1, (3) B2(r,s): — —T^saza1 + raJa2)/rscP, = T(sa2ax + raja^/rsd2, T — rAx a2 + sA2«?. B-2 Standard Coordinate Representation Let l/a(x) and l/b(x) be integrating factors of a1 and a2, respectively. That is, we have the functions x1 (x) and x2(x) such that a1 (x, dx) = a} (x)dx1 + ai>(x)dx2 — a(x)dxx, a2(x,dx) = a2(x)dxx + c&(x}dx2 — &(x)dx2.
Finslerian Diffusion 345 The pair (s1,#2) may be regarded as a new local coordinate system, because the Jacobian dtx1, ®2)/^(a;1, x2) — d/ab does not vanish. Such a coordinate system is called a standard coordinate system of a two-dimensional Finsler space with 1-form metric. It is noted that for a standard (a;1,®2) coordinate system, (far1, kx2) with non-zero constant fa k is also standard. We shall write the expression of Gjk(x) of Finsler spaces belonging to the classes B2(l), B2(2) and B2(r, s) respectively, in a standard coordinate system: (®,y) s (s1,^2) and (i,y) = (j/1,?/2). (1) B2(l) : a6(l + J2)©2! = bax(l + J2) - <7a(aj, + Jbx), <z(l + J^)Gi2 = 05, + Jbx, a2(l + J2)G^ = 6(Jos-l>a), d2(i + J2)G?1 = -a(ns + J&a)1 b(l + J2)G212 = bx- Jay, + J2)G^2 = ods(l + J2) - Jb{Joy - bx), (2) J52(2) : The surviving (i.e. non-zero) G^k are <?n = Ox/<¡ + soy/b ~ b*/b, (^^-(a/bXoy/b-ebx/b), G?2 = eoy/b, (B.2) (3) B2(r, a) : The surviving are G11 = (ryl* + sBx)/r, A= log |a[, G22 — “i" B = log |6|. (B.3) B.3 B2(l) with Constant G^k We shall find the Finsler spaces belonging to B2(l) which have constant Gj-fc in a standard coordinate system. From the equations (B.l), we have generally a2G^ + b2Gl2 = a2Gj2 + b2Gh = 0. (B.4) Now, assume that all the G^k are constant. Then (B.4) leads us to the following two cases: (1°) G%2 or Gji / 0. Then c = b/a is constant. (2°) G?2 = = 0. Then G1^ = G|2 - 0.
346 Antonelli and Zastawniak First, we deal with the case (1°). Since (x,cy) may be regarded as the new standard coordinate system (x:y), we may take c = 1. Consequently, (B.l) reduces to <31 = GÎ2 = ~^2 = (*« - + *^2)> (B ^12“ ^22 = = Gfy + *7ax)/a(l + J2), which imply (Z®/d — ^11 “h *7^22» Cfy/& = ^22 *7^11* Thus ax/a = ai and Oy/a — 02 must be constant, hence we have a = exp (giîd+ a2y + Go) with another constant ao> Proposition B.2. A Finsler space which belongs to B2(l) and has constant Gjk in standard coordinates is such that L(x, VW y) = \/x2 + y2 exp {ais 4- a2y + ao + J ^^(x/ÿ)}, where ai, a2 and ao are constant and the dots denote d/ds. Also, G11 — C12 ” ~^*22 ~ (ai — *7^2)/(l + *72) ~ ci, ^22 = ^12 ~ “^11 = (a2 + *7ûl)/(l + <7“) = C2, and the geodesics are x + c\d? + 2c2xy — cry2 — 0, y - c2x2 + 2crxy + c2y2 = 0, or y" = <0/')2 + 1}(*2 ~ ciî/'), yf - dy/dx. Remark B.3. It is obvious that a$ may be reduced to zero by the homothetic transformation. Thus we obtain P.L. Antonelli's metric where oti = 0^/(1 + J2), [3], [4]. The case (2°) leads us only to locally Minkowski spaces. In fact, = (712 = 0 yield Oy — bx = 0, hence a = 0(2:) and b — b(y). Then = Oxja ==• Cl, (?22 = — &1, hence we have a = exp (ai® + ao) and & — exp (biy+bo) with constant ao and ba. If arbr 0, then we put ai5 == exp (aix + ao), brÿ = exp (fay + &0). If ai = 0, then we put x = (exp ao)x, then it is easy to show that £ does not contain either x or y.
Finslerian Diffusion 347 B.4 Class B2(2) with Constant G^k From (2) we can solve for aXi ay, bx and by as follows: de — gGJi + sbGrii) Oy ~ sbCf^i bx = b&G&a -r G?2), by = b((%2 + £bG212/a). Or, w*e put b — ca and obtain Ox/a — Gi! + £cG2i, a^/a = ecG^t c^G^-G^, cy/c = (^. Suppose that all the G$k be constant. Putting Cl = G*2 ^11» c2 — ^22, (B.6) (B.7) (B.S) two equations of (B.7) yield c = exp (ciz + C2p + co) with another constant cq. Similarly as in Section B.3. co may be taken co = 0. Hence c = exp (cis + C2j/). (B.9) Next, if we put A = log |a|, the remaining two equations of (B.7) are written as Ax = Gii + ¿v - «*>12, (B.10) which imply cyGl1 = CxG?3, that is, c2G?i=dGi2) (B.ll) which is a necessary condition for our assumption. Then (B.10) yields A = Gi1x + sGi1 J cdx +f(y)—sGlz J cdy + g(x). Therefore, we have (1°) ci / 0 : A = Gixa; + £Gh(c/ci) + co, where co is another constant. (2°) ci — cs = 0 : A = (Gfi + scG^x + ecG^y + co, where cq is a constant. On the other hand, we have c = exp co; this constant c may be taken as c = 1 because a = eA and b = ceÂ, as in Section B.3. Proposition B.3. A Finsler space which belongs to B2(2) and has constant G*jk in standard coordinates is such that
348 Antonelli and Zastawniak (1°) L = |i| exp (A + ecy/x), A = G^x + eG?, (c/ci), c = exp (d®4- dy), ci = G{2 - Gii 0, (B . C2 = G22, C1G12 = dGll, with the geodesics: Ci# 4- (coco — cj)#2 = 0, co — Giu Ciy 4- cq#(ci# 4- c2y) + czyfax 4- ci£) « 0, or yft 4- c2(s/)2 + (c? + coc2)y'/ci 4- co = 0. (2°) L = |#| exp (A + ey/x), A = (Gjx 4- ¿Ch)* 4- eG^y, (?12 = <^11 = Cl, G%2 = °> with the geodesics: x 4- cj#2 = 0, y 4- coir 4- 2ci#y = 0, or y" + cxyf + c2 = 0, c2 = Ch- B.5 B2(r, s) with Constant G^k Assume that G^k are constant in (B.3). Then we have rA 4- sB — rcix 4- sc2y 4- Co, Cl — <?i1} C2 = G^t CQ = COnst. Hence, we have |arb*| = exp (rcix 4- sc2y 4- co). Proposition B.4. A Finsler space which belongs to B2(r,s) and has constant G^k in a standard coordinate system is such that L = |#7y,| exp (rci® 4- sc2y + co), ci-Ch, c2 = G22, with the geodesics: ¿4-ci#2=0, y4-c2y2=0, or y" - -c2(^)2 + cis/'.
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PART 4
Symplectic Transformation of the Geometry of T*M; ^-Duality D. Hrimiuc and H. Shimada
Contents 1 The Geometry of TM and T*M 363 1.1 Connections on TM 363 1.2 Semisprays and Connections 36S 1.3 Linear Connections on TM 370 1.4 The Geometry of Cotangent Bundle 373 1.5 Linear Connections on T*M 376 1.6 Lagrange Manifolds 378 1.7 Hamilton Manifolds 381 2 Symplectic Transformations of the Differential Geometry of 385 2.1 Connection-Pairs on Cotangent Bundle ...... 385 2.2 Special Linear Connections on T*M 390 2.3 The Homogeneous Case 395 2.4 /-Related Connection-Pairs 398 2.5 /-Related 0 -Connections 403 2.6 The Geometry of a Homogeneous Contact Transformation .... 405 2.7 Examples 409 3 The Duality Between Lagrange and Hamilton Spaces 413 3.1 The Lagrange-Hamilton ¿-Duality 413 3.2 ¿-Dual Nonlinear Connections 417 3.3 ¿-Dual d-Connections 421 3.4 The Finsler-Cartan ¿-Duality 426 3.5 Berwald Connection for Cartan Spaces. Landsberg and Berwald Spaces. Locally Minkowski Spaces 431 3.6 Applications of the ¿-Duality 435 361
Chapter 1 The Geometry of TM and T'M Introduction The study of the geometry of Lagrange and Hamilton manifolds, by using the ideas from Finsler spaces, was initiated by IL Miron [34], [38] and then developed by many other authors. The basic concept used in these geometries is the nonlinear connection that generates simplifications of many geometrical objects and also yields other new geometrical structures. 1.1 Connections on TM This paragraph is a short description of the geometry of tangent bundle related to the concept of nonlinear connection. Let M be a C°°-differentiable manifold, n-dimensional and vr: TM —> M its tangent bundle. If (x*) is a local system of coordinates on a domain of chart (U, on M, the induced system of coordinates on 7r_1({7) in TM is (z\ ya). We put di := da := d/dya to denote a local frame of TM generated by (a:*,*/*). We may keep the notation for the standard frame of TXM, x — (x1). A change of coordinates on TM is given by 5i=5i(x1,...Ix")J = rank(g)=n. (1.1) (The range for all indices is {1,2,.,. ,n}.) Let dvr : TTM -+ TM be the differential of tt and VTM =s Ker(dr) the vertical bundle, which is a subbundle of (TTM, dvr, TM). Since 7r is a submersion 363
364 Hrimiuc and Shim&da we remark that the vertical bundle is in fact the bundle of tangent spaces tangent to the fibres induced by tf. A vector field X € X(TM) is locally written as and the vertical distribution u —> VUTM = Ker dvr (u) is locally spanned by hence it is integrable. Let x € -M and y G TXM. We can define a linear map TXM —> VyTM as follows: each u G TXM is mapped into the vertical vector uv, the tangent vector to the curve t —> y + tu, at t — 0. This map u € TXM. uv 6 VyTM is called the vertical lift. Now every vector field X G X(M) can be lifted to a vertical vector field Xv on TM such that X’(j/) = (X(ir(j/))1', y^TM. Locally if X = Xif^ then Xv = Xidi. (1.2) Now we can introduce a linear map J : X(TM) —> X(TM) defined as J(X) = O(X)f, X € X(TM). J is locally given by J(a) = ^ J(&)=0 or J = di ® da? (1.3) J is called the canonical almost tangent structure of TM. The following properties are immediate: J2 = 0, Ker J = Im J = VTM. ' (1.4) Let TM be the tangent space on M with the null cross-section removed. Definition 1.1.1. A nonlinear connection (shortly connection) on TM is an almost product structure .V on TM (i.e. N2 = id), smooth on TM, such that VTM= Ker(id + 2V). Remark 1.1.1. (i) If N is a connection on TM then HTM — Ker (id — N) is a subbundle over TM (or a distribution on TM) supplementary to the vertical bundle (or vertical distribution). Hence TTM = HTM ®VTM. (1-5)
Symplectic Transformation 365 HTM is called the horizontal subbundle (distribution) associated-to N. Each X € X(TM) can be written as x = xH+xv where Xa and Xv are sections in the horizontal and respectively vertical sub- bundle. • XH and Xv are called the horizontal and respectively the vertical com¬ ponent of X. • • If Xv = 0 then X is called a horizontal vector field, while if XH = 0 then X is called a vertical vector field. An important example of a vertical vector field is the Liouville vector field C. It is defined as following: for each y € C(y) = yv € VyTM. Locally we have 0 = ^ (1.6) (ii) A connection N on TM generates two morphisms (projectors) h,v : X(TM) X(TM) h(X)=XH, v(X)=Xv, XeX(TM). We have A=l(id+2\T), u = i(id-JV). (1.7) and Ker h = hn v = VTM, Im h = Ker v = HTM. (1.8) Proposition 1.1.1. An almost product structure N on TM smooth on TM is a connection if NJ — —J and JN = J. (1.9) Proof: Let N be an almost product structure on TM. Then for each X € X(TM) by using (1.4) and (1.8), we have . JN(X) = J(X - 2v(X)) = J(X) and NJ(X) = J(X) - 2«(J(X)) = J(X) - 2J(X) = -J(X). Conversely, let N be an almost product structure on TM such that NJ.= — J and JN = J. If N(X) = -X,X€ X(TM) then JN(X) = J(X) « -J(X).
366 Hrimiuc and Shimada Hence J(X) = 0 and therefore X is vertical. Now if X is vertical then, X € Im J that is X = JY, Y G *(TM). Hence NX = NJ(Y) = - J(Y) = -X. Remark 1.1.2. (i) The local expression of A’ in local coordinates (re*, y') on TM is N{di) = 2N&, (1.10) and the local vector fields Si := A(di) = 1 (di + N(di)) =&i- N{dj (1.11) generate a basis of HTM. • The frame at (x,y) generated by the local frame (<%,<%) and TV is called the adapted frame at (x, y). Note also that A^,) = h(5i) — 6^ v(5<) = 0. • The dual frame of the adapted frame dj) is (dz\ Sy') where Sy^dtf + Ntdx*. (1.12) If (1.1) is performed, the adapted frames and their duals changes under the rule: ^=SSyi- (L14) • The functions N? = N?(x,y) are called the local coefficients of the con¬ nection N. If a change of coordinates (1.1) is performed on TM then the coefficients of N are related as following: v dxi i dxi dx^xi (1-15) Notice that given a connection N on TM is equivalent of giving a set of n2 real functions (N?) =T,... ,n on every coordinate neighbourhood on TM such that on the intersection of two coordinate neighborhoods (1.15) holds. (ii) If HTM is the horizontal distribution generated by N then the restric¬ tion of ¿7T to the horizontal subbundle is an isomorphism of vector bundles
Symplectic Transformation 367 (HTM^p, TM) and (TM, %, M). Each X € X(M) can be lifted to a section Xh of (HTM, d7r, TM) called the horizontal lift of X, Locally if X = X1 £ then Xh = X^i. (iii) The local expression of A' in the standard frame can be written as = di ® dx* - 2N?dj todat-di® dyi (1.16) or in adapted frame N = 6i®dxi-di®dyi. (1.17) (iv) A connection N on TM defines a morphism F : X(TM) —> X(TM) as following: If X € X(TM), X = X* + Xv then F(h(X)) = - JX, F(JX) = hX. Since F2 — — id, F is a complex structure on TM associated to N. Locally F№) = -a, F(di)^6i or F — —di ®dxl + 6i® dy\ (1.18) Proposition 1.1.2. We have the brackets [6^]=^-, [di,dj]^0 (1.19) where = SjN? - 5iNj. (1.20) □ Let R(X,Y) = v[hX,hY], X,Y e X(TM). Locally R — R^jdk 0 dx1 0 R is called the curvature of and it is an obstruction to the integrability of HTM. HTM is integrable iff R^ — 0. Definition 1.1.2. A connection N on TM is called homogeneous of degree 1 if the Lie derivative of N with respect to the Liouville vector field C vanishes. We notice that LcN = 0 [C, JVX] - jV[C, X] = 0, VX € X(TM) or equivalently y^iNf = Nf. (1.21)
368 Hrimiuc and Shimada Hence A’ is homogeneous of degree 1 iff N!f are homogeneous functions of degree 1 in y. If Nl[x,y) = ykNlk(x) then N is just a linear connection on M. ’ A connection A’ define a covariant derivative of Y € X (M) with respect to X € #(M) given locally as following: p-22’ Let c : I —► M be a smooth curve on M. Y is said to be parallel along c if D±Y = 0. Locally this condition reads: where xi = xl(t) are the local equations of c. A curve c is said to be a path of A if C(t) is parallel along C(i), i.e., D^c = 0. This is locally equivalent to Notice: If N is a linear connection, (1.23) are the equations for geodesics of a linear connection on M. Any connection A = (AJ) on TM induces a linear connection D on the vertical bundle (VTM,7r,TM) with local coefficients where & = l,...,n and 1^=0 if k = n +1,...y2n. Then the covariant derivative with respect to this connection is Dg.di^dk, Ds.di = 0. The linear connection defined above is called the Berwald connection associated to N. More properties related to this connection can be found in [41], [43]. 1.2 Semisprays and Connections In this section we investigate the relationship between connection and semis- prays. For more details see [17], [18], [29], [41], [54]. Definition 1.2.1. A vector field £ € X(TM\ C°° on TM, is called a semispray on TM (or a second order differential equation) if = C. A semispray is locally given by where f7 — y) are C°° on TM. (2.1)
Symplectic Transformation 369 Remark 1.2.1. (i) Let be a semispray on TM. A curve c on M is called a path (or integral curve) of f if c(i) = i(c(t)) (2-2) or locally (ii) A semispray £ on TM is called a spray if the deviation of £ r-icci-e (2*3) vanishes and f is of class C1 on the zero section. If is a spray and is of class C2 on the zero section then it is called a quadratic spray. We notice that $ is a spray if the functions = ^(x, y) are 2-homogeneous and C1 on the zero section and it is a quadratic spray if are quadratic on y\ - □ If TV is a connection on TM and g a semispray then the vector field 4 = ft£' = l(id+^' (2.4) is a semispray on TM which does not depend on the selection of f £ given by (2.4) is called the associated semispray to N. Locally i = (2.5) If N is a linear connection then $ = (2.6) We remark that $ is a spray iff TV is homogeneous and it is a quadratic'spray iff N is linear. Now, to any semispray £ on TM we can associate a connection N defined as follows: N(X) = J = X] - [£, JX], X € X(TM) (2.7) It is easy to verify that N is an almost product structure which verifies (1.9) hence is indeed a connection on TM (see [29], [41]). If f is locally given by (2.1) then the local coefficients of N are: («» y) = - | (®, v)- (2.S)
370 Hrimiuc and Shimada If i is quadratic then N is a linear connection on M and its coefficients are = = Nir (2.9) If TV is the connection associated to £ then the associated semispray to N is Thus, the associated semispray associated to N is just £ iff £* = 0, that is N is homogeneous. On the other hand if £ is a spray and W is defined by (2.7) then This equation shows that any homogeneous connection and its associated spray have the same paths. 1.3 Linear Connections on TM In this section we consider a connection ¿V on TM previously given. The approach of the geometry of the tangent bundle TM endowed with a connection TV can be considerably simplified. Because of decomposition (1.5) we can introduce the algebra of a special tensor field called d-tensor field (or TV-tensor field). This algebra is locally generated by (1,over the ring 5(TM). For instance a vector field X e TV(TM) is locally written as X = + (3.1) and a one form w on TM is given locally as follows w = Widx1 + (3.2) The tangent vector to a curve c : / —► TM, c(t) = (ar(t), y(t)) is locally given by dx* "dt (3.3) Notice that if dt dt + N?(x(t),y(t)') = 0, tel (3.4) then c is called a horizontal curve. Let F be the almost complex structure on TM induced by N as in (1.18). Definition 1.3.1. Let V be a linear connection on TM. (i) We say that V is a N-connection (or a d-connectiori) if VjV « 0. (n) We say that V is a normal N-connection (or a normal d-connection) if VN = 0, VF = 0.
Symplectic Transformation 371 Remark 1.3.1. (i) A linear connection V on TM is a ^connection iff preserves by parallelism the vertical and the horizontal distribution generated by N on TM. (ii) A JV-connection V is normal iff V J = 0. □ A N-connection on TM induces two types of covariant derivatives. (i) an À-covariant derivative, given by Vxy := V^Y, X, y € X(TM) (3.5) (ii) a -v-covariant derivative, defined as follows V$Y :== VxvK X, Y € X(TM) where Xs and Xv are the horizontal and vertical projections of X. (3-6) A ¿-connection V on TM can be characterized by a quadruple of local coefficients (£jfc(*,y), L^x,y), C\a(z,s/), C*bc&, y)) given by Ôa$j = = O^bc^a» For a normal JV-connection, Lfk = and O4*, = S^C^. (3-7) (3.8) (3.9) For a change of coordinates given by (1.1), the coefficients obey the classical law of change for the coefficients of a linear connection while C^ja, Cabc are (1,2) ¿-tensor fields. For instance, if we have the ¿’tensor field K = K^6i ®da®dxi® 5j/6 its h-covariant derivative is (3.10) := = Kfyk6i ® da 0 dx3 0 Syb, where Tfia f jfia ri Tfla , ra rfic rt 7c Tria, Ajb\k — OkJijb + ¿'¿k^jb + ¿'ck^jb “ hjkJ^tb ~ ^bk-Kjc) and similar its v-covariant derivative: (3.U) *$|c = - Cli6K% - C^Kft. (3.12)
372 Hrimiuc and Shiznada The torsion tensor field T of V is defined as usual T(X,Y) = VXY - VyX - [X,y], X.Y e X(TM). In the adapted basis we put R{^kt ^j) = T1 jk$i "I" 'T<l'jkd(i r(^,^) = Pi^+P°A T(£C)$fr) = S^bc^i ~F S&bc&a We have T*jk ~ Ljk — ■L/Cj, Tajfc — R?jk> P*jb = jb9 S'bc s^c =0, (3.13) ft, =* R*^ R?jk - tkN? - 6jNS (3.14) is the torsion tensor field of N — (Nf). The curvature of a ^connection on TM, R(X, Y)Z = VxVyZ - VYVxZ - V[xr]2, X, y, Z e *(TM) is written in the adapted basis with only six non-vanishing components = P/wih P(<fre>£&)9& “ R^* kt^a P(^Cj &k)$j = Rj*kc$ii R(dci $k)db — Rbakc^a p(4, = s^i, R(dd, de)db = sbacdda. By a simple computation we get = II +WA)+^^k, (3.15) W,C) Rbajk = II + Сльс^к, (3.16) (5Л) Rj ко. — ^лЦк “ C*je|A + C*jt>pbkat (3*17) Rbakc = ^c^bk ~ C*bc\k + СамР^ka (3.1S) Sfbc = Ц + Скзь&кс, (3.19) (СЛ) Sb* cd = П {^‘bc + C'bcC'td, (3.20) (c,d) where TJ {• • *} indicates interchange of j and k for the terms in the brackets (M) and subtraction.
Symplectic Transformation 373 For normal JV-connection we get only three curvature tensors: RfSjk^Pj'ka, Sfab* Let us consider the metric tensor on TM G = gij(x,y)dxi ® dtf ^gab^v^y* 0 5yb. (3.21) The following result is well known: Theorem 1.3.1., If a nonlinear connection N = (Nf) is fixed, there exists only one normal N-connection DT — (L^k,Cabc) which satisfies the following properties: (i) 9ij\k = 0 — metrical) (ii) ^|c = 0 (v-metrical) (Hi) T^ = 0 (iv) 5^ = 0. The coefficients of this normal ^connection are given by £jk = q + fikgjh, ~ ^h9jk)i (3.22) — 2 d^^bddc + dcPbd ~ ddSbc)- (3.23) This connection will be called canonical A systematic presentation of the geometry of TM (from above point of view) is given in the monograph of Miron and Anastasiei [41]. Also, see [42], [43]. 1.4 The Geometry of Cotangent Bundle The geometry of cotangent bundle 7r* : T*M —► M can be developed similar as for the tangent bundle. However, the geometry of T*M cannot be obtained from that of TM by simple dualization because the geometrical structures of T*M, TT'M, and TM, TTM are different. If (x') is a local system of coordinates on a domain of chart (C7, <p) on M then the induced coordinates of a point u € tt*^1(L7) c T*M, 7r*(u) = x will be denoted by (rr*,Pi) (the range for all indices is {1,2, • - ,n}). If (f^tPi) are the local coordinates on T*M induced by another local chart, we have (4-1) We denote di := dx^ x — Pi =
374 Hrimiuc and Shimada According to (4.1) the local fields of frames (${,&) and (&,&) on T’M are related by: (4-2) (4-3) s*- For the dual frame (dz\dpi\ we have the transformation da? d3f = — dx3 02xk dx3 h dFdpi+pkd?d&dJdx • Let VT*M = Ker(d7r*) the vertical bundle, which is a subbundle of (TT*M, d7T\ TM). The vertical distribution u -+• Kerd7r*(u) is locally spanned by S’, hence is integrable. Definition 1.4.1. A nonlinear connection (or connection) on T*M is an almost product structure N* on smooth on such that VT*M = Ker(id + N*). Remark 1.4.1. (i) If N' is a connection on T*M then HT*M = Ker (id — JV*) is the horizontal subbundle associated to 2V* and (4-4) Each X 6 /¥(Z*M) can be written as X = XH + XV where XH and Xv are sections in the horizontal and respectively vertical sub¬ bundle. If XH = 0 then X is called a vertical vector field and if Xv — 0 it is called a horizontal vector field The vector field C* given locally by (4.5) is a vertical vector field called Lionville vector field . (ii) A connection TV* on T*M induces two projectors h\v* : X(T*M) X(T*M) h* (X) = XH, v* (X) = Xv. X e * (T*M)
Symplectic Transformation 375 such that ft‘ = i(id+lV*), v* = |(id-^) (4-6) and Kerb* = Imv* - VT*M; Imh* = Kero* = HT”M. (4-7) Locally, we have ^(ft) = di + 2^°, jV*(da) = -da (4-8) and the local vector fields 5- := hT {di) - i {di +’ (^)) = di + (4.9) generate a basis of HT*M~ • The frame ($■ ,0*) is called the adapted frame at fop). We also have v“W)=0. • The dual frame of the adapted frame (£*, 5°) is (dz\ $*pft) where rpa^dpa-Ni^dx*. (4.10) With respect to (4.1) the transformation formula for the coefficients = ^(a?,p) is dxh dxh dx1 dx3 Nhk(x,p) +ph d2xh d&dx3' (4.11) Conversely, if on the domain of each local chart of T*M there exist n2 differen¬ tiable functions Nij(x,p), i, j — 1, • • • ,n satisfying (4.9) on overlaps, then there exists a unique connection N* on T’M whose coefficients are JVy(x,p). Also, if (4.1) is performed then = dx* dx3 (4.12) • jV* can be locally expressed in the standard frame as - Ar’ = di ® dx' + 2Nijd3 0 dx' — d* 0 dpi (4-14) or in the adapted frame (4.15)
376 Hrimiuc and Shimada (iv) A connection TV* on T*M defines an almost symplectic structure w dx1/\6*Pi> (4.16) We remark that if =; Nji (i.e. TV* is symmetric) then or — dx1 A dpi = —d0 where 0 = pidxi is the canonical 1-form of T*Ttf and thus w is just the canonical symplectic form of T*M. (v) We have the brackets [¿?,#] = -#(№*)$*; = 0 (4.17) where (4.18) is the curvature of TV*. HT*M is integrable iff Rijk = 0* (vi) A connection TV* on T*M is called homogeneous if the Lie derivative of TV* with respect to the Liouville vector field C* vanishes. Locally this is equivalent to Pi^Njk = Njk. (4.19) 1.5 Linear Connections on T*M The algebra of d-tensor fields on T*M can be introduced in a similar way as for the tangent bundle, with respect to the horizontal and vertical distribution. This algebra is generated by (1,^3*) over the ring Jr(T*M). For instance the tangent vector to a curve c : I C IR —► T*TVf, c(i) == (®(t),p(t)), t € I is written as follows: dt \dt J \dtj dt 1 dt where S’Pi dpt t ,.^daP We say that c is horizontal if (~“)V = 0- Hence c is horizontal if and only if, locally ~ Nii= °- Suppose that we have fixed a connection TV* on T*M. (5.1) Definition 1.5.1. Let V be a linear connection on T*M.
Symplectic Transformation 377 (i) We say that V is a N*-connection (or a ¿-connection) if VTV* — 0. (“) We say that V_is a normal N*-connection (or a normal ¿-connection) if V7V* = 0 and Vu = 0. Similar as for tangent bundle we get an ft- and v-covariant derivative, asso¬ ciated to a TV*-connection. Also, we can characterize a TV*-connection by a quad¬ ruple of local coefficients DN* — ((ar, p), Hgk (x, p), Vy* (x. p), V*b (x, p)) where we have put: _ _ (5.2) = -ve^, For a normal jV*-connection we obtain: Hgk = S?8iH>k, V? = (5.3) For a ¿-tensor field K = ® 3d ® (fa? ® 8*pa we get the h-covariant derivative Aj6||* — ^k^jb “T H-ikAjb T MckKjb - ^jk^tb - ^bk^jc V>A) and the v-covariant derivative J$||c = + Vf'Kfi + V\CK^ - - VkjeKiS>. We also have the torsion tensors (similar to those in Section 1.3). T*jk = &jk - T^k = Rajk, T*,-“ = Vy. A? = Hbaj - d»Nja, S* = V* - Va* hCre AOJ-fc = - S*kNja and six kinds of curvature tensors (see Section 1.3) R^k = II {Wk+H&rtk} (W) = II ~ RcJkV^, (M) - vy n* + V^TV, >«*c = + vafc||A - K6<f?dfcc, S?“* = U {ddV^c + VkieVikd}, (d»c) (5.5) (5-6) (5.7) (5-8) (5-9) (5.10) (5-11) (5-12)
378 Hrimiuc and Shimada Sabcd = H + VacdVeae}. (5.13) (M) For a normal N*-connection we get Wjk = -T&w, Pj\a = -fyf, s1^ = -^d- (5.14) Let us consider the metric tensor on T*M G{x,p} = 9i^ (¡xtp)dx2 ^dx3 + Jph(xJ>)6*p* 0 6*pt. The following result can be proved [35], [43]. Theorem 1.5.1. If a connection N* — (Nia) is fixed on T*M, there exists only one normal N*-connection DN* = V^) having the following properties: (i) ^IIA: =0 (h—metrical) (ii) ^[1° = 0 (y-metrical) (iii) 7^ = 0 (iv) 3^ = 0. The coefficients of this d-connection are given by: Hik = | - ^gifc), (5.10) . V*1“ = -1 gad(aV= +d°3u- W). (5.16) This connection will be called canonical. 1.6 Lagrange Manifolds Some generalizations of Finsler geometry have been proposed in the last three decades by relaxing the definition of Finsler metric. The Lagrange manifolds introduced by J. Kern [27] and developed by R. Miron in [34], [39], [41] represent a first direction. In a Lagrange geometry the metric tensor is obtained by taking the Hessian with respect to the tangential coordinates of a smooth function L defined on the tangent bundle. The function is called a regular Lagrangian, provided the Hessian is nondegenerate, so no other conditions are envisaged. Many aspects of theory of Finsler manifolds apply equally well to Lagrange Manifolds. However a lot of problems may be totally different, especially those concerning the geometry of M. For instance, because of lack of the homogeneity condition, the length of a smooth curve on M, if defined as usual for Finsler manifolds will depend on the parametrization of thecurve. Let M be a smooth differentiable manifold and TM the slit tangent bundle i;e. the tangent space with the null cross-section removed. Definition 1.6.1. A regular Lagrangian on M is a continuous function L : TM -+ IR of class C°° on TM such that the matrix with the entries 3ij(x,y) = 6idjL(x.y) (6.1)
Symplectic Transfonnation 379 is nondegenerate on TM. A Lagrange manifold is a pair (M, L) where M is a smooth manifold and L is a regular Lagrangian. Examples. 1. Any Finsler space F* = (M,F(x,y)) is a Lagrange manifold. Here L(z,3/) s= ^F^x^y), positively homogeneous of degree two with respect to y and the matrix with the entries (6.1) is nondegenerate. 2. (M? L) with L=ir2(a;,ii) + 6i(x)S/i + c(x) (6.2) where F is the fundamental function of a Finsler space, bi = bi(x) are the components of a covector field and c — c(x) is a smooth function on M is a Lagrange manifold. □ For any smooth path 7 : [0,1] —► M the action integral may be considered I(7)= /1i(7(i),7(i))di- (6-3) Jo A geodesic of a Lagrange manifold (M, L) is an extremal curve of (6,3). The extremal curves with fixed endpoints are solutions of the Euler-Lagrange equations d /â£\ _ dL _ _ da? dt xdx1) dx* " ’ X ~~ dt (6.4) where (a?(t)) is a local, coordinate expression of 7. This system is equivalent to Gi{x,y) = gi’(ykdjdkL-djL), i = l,...,n. (6.6) Now, as in Finsler manifolds, we can derive from (6.5) a connection having the following local coefficients Nj = ^djGi(x,y). (6.7) This connection is called the canonical connection (nonlinear) of the Lagrange manifold (M, L) and will be considered in the next. On a Lagrange manifold we can consider the canonical 1-form = diLdx* (6.8)
380 Hrimiuc and Shimada which is globally defined on M. The exterior differential of is the canonical closed two form wl = d$L — didjLdxL A drr7 + didjLda? A dyi which is a symplectic structure on TM. Remark 1.6-1. Let E — yzdiL — L be the energy functional associated to L. The vector field $ solution of the equation ix<*>L ~ dE is a semispray (see [28]) locally e = y^i + fdi, è = - îfdidkL) where t? = gV(djL — ykdjdkL) = —G\ The local coefficients of connection (2.7) associated to £ are according to (2.8) Mj&y) = -^d£(x,y) = that is (6.7). Proposition 1.6.1. In adapted coordinates = 9ij^yz I\dxP. (6.9) Proof: From (6.8) we have = d$L — + 5i(âj£)dxx) A da?. Now using (6.7) we obtain ôidjL = $jdiL hence Wl = gijty* A da? 4* i - iy $£))<£? A da? 2 = 9ijtyl Adsb7. Proposition 1.6.2. For a Lagrange manifold the following properties hold: (i) 3hkRhij + gihl&jk + gjhRhki = 0. (Ü) fij9ik $i9jk = 9ih.dkNj (6.10) (6.11)
Symplectic Transformation 381 Proof: Since 6yl = dy1 + Njdrf by taking the exterior differential we obtain d($/) = I A <tofc + A fat. Now since 0 — d?u?L = d@L we get by using (6.9) ^k9ijd^k + dkgij&Uk) A fry1 A + gij (1 R}kfa? A fa* + dkNfaf A fa?) A fa? = 0. from which (i) and (ii) can be easily derived. If (M, L) is a Lagrange manifold, on TM a metric tensor can be defined as following: G(x, y) = gij(x, y)dx' 0 cfe* + g*b(xz y)6ya 0 6yb where gij(x.y) are given by (6.1). The vertical and horizontal distributions are orthogonal with respect to G. Moreover, there exists only one normal N-connection that verifies (i) - (iv) of Theorem 1.3.1. The coefficients of this connection are given by (3.22) and (3.23). 1.7 Hamilton Manifolds The geometry of cotangent bundle endowed with a Hamiltonian was investigated by R. Miron in [38], [40], [43] developing concepts and finding results which have similarities to those of a Lagrange space. In [23], [24], [43] D. Hrimiuc and H. Shimada derived the geometry of a Hamilton manifold from that of a Lagrange manifold using the Legendre duality. _ Let M be a smooth differentiable manifold and TM* the slit cotangent bundle. Definition 1.7.1. A regular Hamiltonian on M is a continuous function H : T*M —► R smooth on T*M such that the matrix with the entries (7.1) is everywhere nondegenerate on T*M. A Hamilton manifold is a pair (M, H) where M is a smooth manifold and J? is a regular Hamiltonian.
382 Hrimiuc and Shimada Examples. 1. Let H = 17V (x)pi₽i + V (z)pi + c(x) (7.2) where 7^(2;) are the components of a Riemannian metric on M, bz(x) are the components of a smooth vector field on M and C = c(z) is a smooth function on M. 2. A Cartan manifold is a pair (M, F) where F : T*M —> R is continuous, smooth on positively 1-homogeneous with respect to p and such that g^(xtp) « | d'&F“ is nondegenerate» These spaces were introduced by R. Miron [36], [37], [38]» A Cartan manifold is a Hamilton manifold with tf=|F2. □ Let (Af, H) be a Hamiltonian manifold. If 9 = dpi A dx* is the canonical symplectic structure of T*AT we can find an unique vector field X# € #(T*Af) such that iX/i9 = -dH. (7.3) The integral curves of Xu are the solutions of the Hamilton-Jacobi equations: d± = QH dpi=_dH dt dpi' dt dxi * Z< ' On a Hamilton manifold we can consider a connection (nonlinear) whose local coefficients are given by NV = | (dkgi;dkH- dkga&H) “ 1 (7.5) - (gik&djH+gjk&diH). This connection is called thecanoniccd connection of the Hamilton space (M, H). It was obtained by R. Miron [40] and it is the Legendre dual of (6.7). This connection will be obtained in Section 3.2 as a result of ¿-duality. Proposition 1.7.1. The canonical connection of the Hamilton space (Af, H) has the following properties (i) Rijk + Rjki H“ Rkij — 0 (7-6) (ii) Nij = Nji (symmetric connection)» (7.7) The above properties are the dual of (6.10) and (6.11) and will be justified in Section 3.2.
Symplectic Transformation 383 We also mention that the tensor field given by (7.1) generate a metric on G'(.x,p) = g^(x,y)dxiè>d^+g’ab{x,p)S‘pa^6,pb. (7.8) With respect to this metrical structure the distributions VT*M and HT'M are orthogonal. There is a unique normal ;V*-connection on T*M that verifies (i) - (iv) of Theorem 1.5.1. The coefficients of this connection are given by (5.15) and (5.16).
Chapter 2 Symplectic Transformations of the Differential Geometry of T*M It is well-known that symplectic transformations preserve the form of the Hamilton equations. However, the natural metric tensor (kinetic energy matrix) is not generally invariant nor is its associated differential geometry. In this chapter we address precisely the question of how the geometry of the cotangent bundle changes under symplectic transformation. As a special case, we also consider the homogeneous contact transformations. This chapter follows closely the subject as developed in [4], [5], [43]. 2.1 Connection-Pairs on Cotangent Bundle Let M be a n-dimensional C°°-differentiable manifold and тг* : T*M —> M the cotangent bundle. As we have seen (Section 1.4) a nonlinear connection on is a supplementary distribution HT'M of the vertical distribution VT*M = Ker d(%*) or an almost product structure N on T’M such that VT*M = Ker (id +2V). If f € Diff (T’M) and N is a connection on the push-forward of N by f generally fails to be a connection. Because of this, we will now define a new geometrical structure which nevertheless is an extension of the above definition for connections. Definition 2.1.1. A^connection-pair ф on T*M is an almost product structure on T*Af smooth on T'M such that Ker (id — ф) is supplementary to VT*Af. HT*M — Ker (id — ф) will be called the horizontal bundle and WT*M = Ker (id + ф) the oblique bundle. Remark 2.1.1. If ф is a connection-pair on T*M, then a unique connection N 3S5
386 Hrimiuc axid Shinxada can be associated to it, such that Ker (id — 2V) = Ker (id — 0), therefore <f> — N on Ker (id - 0). Conversely, if a connection N is given on T*Af, we can get a conn ection-pair on T*M by taking a complementary subbundle of Ker (id — N). Let 0 be a connection-pair on and N the associated connection. We will denote by h and v the projections induced by N : h=|(id+N), W=i(id-JV) (1-1) and by h', w those induced by <p h'= 5 (id +0)» w= 1 (id -$• (1-2) The local expression of N is given by N(di) = di + 2Ni^, N(&) = (1-3) and the local vector fields: % := h{di) = I (di 4- A^)) =di+ Ni^ (1-4) provide us with a frame for HT*M at (x,p). We also obtain: ^)=x(d?)=sr, m)=^- (1-5) On the other hand, <^(ai) = -ai+2irii;. (1.6) Indeed, from <!>(&) = we get + VkFk} + and now (1.6) follows easily. The local vector fields £ := w(^) = | (6»‘ - ^)) = & - ITJS*j (1-7) form a basis for WT*M at (x,p) and = —F. (1-8) Therefore, ($£,#) is a frame for 7T*M at (s,p), adapted to the connection¬ pair, The dual of this adapted frame is (¿>*z%£*pi) where Fx* = d'pi = dpi - NjiCW. (1-9)
Symplectic Transformation 387 (1.10) (1.11) Using the notation above we have the following local expression of <j> and its associated connection N : ^=5? 0<Tpi} TV = £* - & ®rPi. With respect to natural frame, <j> has the local form <№i) = $ - '2NijWk)dk + 2(M* - Ni^N^d*, = 2nifc& + (2TT’Njk - 31)0*. From (1.4) and (1.7) we get: Proposition 2.1.1. The adapted basis (£*,£*) and its dual (6*x\0*pi) trans¬ form under a change of coordinates on T*M as follows: 5$ = fyzV;, 8e = dix1'?, (1.12) ¿V' = S'p« = di^tfpi. . (1.13) Proposition 2.1.2. If a change of coordinates is performed on T*M> then the coefficients of the connection-pair </> obey the following rules of transformation Ni'j'(x\pf) - dx^xidj^Nij{x1p) +pkdi'dj'xk, (1.14) ir^(x',p') = dixi'djx’'lF(x,p'). (1.15) Remarks 2.1.2. (i) In spite of being an object on T*M, T& follows the same rule of transformation as a tensor of type (1,1) on M, therefore are the components of a d-tensor field. (ii) If M is paracompact, there exists a connection-pair on T*M if and only if, on the domain of each chart on T'M there exists 2n2-difierentiable functions Nij and IF5 satisfying (1.14) and (1.15) with respect to the transformation of coordinates on T"M. (iii) Explicit examples of connection-pair on T*M. Let 7 = (7^) a Rieman- nian metric on M and { ?,} the Christofiel symbols of g. We can define on every domain of a chart I. ’ • Na = { J?*, ii* = -.7, -. 1 (1.16) These are the local components of a connection pair [50]. More generally, if (M, H) is a Hamilton manifold, we can take Nij as the coefficients of the canonical nonlinear connection and = d'&H. Proposition 2.1.3. We have the brackets: №,5)]= Rij£TI.tk3*k+Rijk3kt (1-17)
38S Hrimiuc and Shiznada [$M;j = (¿‘IP* 4- ¡PRiorf* + #N¡¿1!*)% (1.18) + (&Nik + TPtRiik')6k, [6\&]= Rykrk + №T&NTk - H?TSiNrk)6k, (1.19) where Rijk= FiNjk-rjNik, (1.20) = ii-tfk _ £nj7= + (ITW„ - TPr5iNn>)U‘>k. (1.21) Let us put = *+*'. where [0, <j>] denotes the Nijenhuis bracket of </> and R, JR! are given by R(X, Y) = w[h% h'Y], R'(X, Y) = h'[wX, wY]. We call R the curvature and Rf the cocurvature of the connection-pair <j>. R and R! are obstructions to the integrability of HT*M and WT’M, respectively. Locally we have: R = R^S* 0 Fz* 0 R! = R*’k6*k 0 FPi 0 5*Pj. (1.22) HT*M and WT*M are integrable iScftis integrable, or equivalently, R — R! — Q. Definition 2.1.2. A connection-pair <f> on T*M is called symmetric if Nij = Nji and IT' = n>\ Let 0 — pidz1 be the canonical one form of T*M and w = d0 the canonical symplectic 2-fonn. The Definition 2.1.2 above is invariant because of Proposition 2.1.4. A connection-pair </> is symmetric if and only if = -cu. (1.23) Proof: w has the local expression oj = dpi Adz1. We obviously have From 5t) = -W(d;, - 0, ’ we get ir^nji + iri,tfr(№r-^) = 0i Nsr = Nrs. We obtain therefore, № « IF* and Nij === Nji. Corollary 2.1.1. The following statements are equivalent
Symplectic Transformation 389 (i) 0 is symmetric. (ii) WT*M and HT*M are Lagrangian (every subbundle is both isotropic and coisotropic with respect to w). Proof: If 0 is symmetric, using the proposition above we get = 0, = 0 and on account of dim WT*M = dim HT'M = j dïmTT*M it follows that HT*M and VT*M are Lagrangian. Now, if conversely HT* M and VT*M are Lagrangian, using again the pro¬ position above, we get that 0 is symmetric. Remark 2.1.3. A connection-pair <6 on T*M induces two almost symplectic forms, globally defined on : of == a dx\ of' = ô*pi A Î’a:4. (1-24) $ is symmetric iff u; .= of = a/'. Its associated connection N is symmetric (that is My = Nji, or equivalently N*oj = —a;) iff w = a/. Let C = pièf be the LiouvUle vector field, globally defined on T*M. We denote by T*M the slit cotangent bundle, that is, the cotangent bundle with zero section removed. Définition 2.1.3. A connection-pair on T*M is called homogeneous if the Lie derivative of <f> with respect to C vanishes, that is 2^0 = 0. (1.25) The following, characterize the property of homogeneity for connection-pairs in terms of homogeneity of its connectors. Proposition 2.1.5. A connection-pair $ is homogeneous iff N# and ITJ are 1-homogeneous, respectively, —1-homogeneous, with respect to p. Proof: From L^<f> = 0 and (1.5) we get But, @J>l] = (-Nkh+pidiNkk)dk and from (1.6) ¿(^) ==-£* +211^;. Therefore, (-Nkh + Pi&Nkh)#1 + (Nkh -Pid^n^ = 0
390 Hrimiuc and Shimada and thus Nij are 1-homogeneous. We also must have: 4- £*] = 0, but using the 1-homogeneity of Nij above, we get 5k] = (-n^ - 5k and thus that is, 11** is -1-homogeneous relative to p. 2.2 Special Linear Connections on T*M Let 0 be a fixed symmetrical connection-pair on T*M and TT'M = HT'M © WT*M, the splitting generated by it. HT*M is the horizontal bundle and WT*M is the oblique bundle. Every vector field X € X(T*M) has two components with respect to the above splitting X==Xh'+Xw (2.1) where Xh' = hf(X) is the horizontal component and Xw = w(X) is the oblique component of X. We can also introduce some special tensor fields, called ^-tensor fields as objects in the algebra spanned by {1, <5*, £*} over the ring of of smooth real valued functions on T*M. For instance K = K$St®5i® S'xk®8*ph (2.2) is a (2,2) ^-tensor field. For a change of coordinates given on T*M the com- ponents of a ^-tensor are transformed in exactly the same way as a tensor on M, in spite of Pi dependence, thus K is a ¿-tensor field. Definition 2.2.1. Let V be a linear connection of T*M and a connection¬ pair. We say that V is a ^-connection if V</> = 0 and Vcj — 0. (2.3) It can easily be proved that V0 = 0 is equivalent to Vw = 0 or VZ&' = 0. This definition extends to a general setting, the definition of so called Finsler connection for Cartan space. A ^-connection can be characterized locally by a pair of coefficients (Hjki Vjk) such that va;^ = ^5 VS'F = -HikV, v*.? = = v?k6*.
Syraplectic Transformation 391 Proposition 2.2.1. Under a change of coordinates on T*M the coefficients of a (¡¡-connection V change as follows: Hj,k, - diZ*'dj'xidk'ZkHjk + (2.5) V^tj = dix1'dk>xkVkj ^dkx%l5h{dk>xk), (2.6) Remark 2.2.1. (i) (2.6) is equivalent to v£j' = atx^d^d^v^+T^dh^dk'^did^'. (2.7) (ii) A ^-connection can be characterized by a couple of coefficients (Hjk> Vÿ*) which obey the transformation law of (2.5) and (2.6), if a change of co¬ ordinates on T*M, is performed. □ A ^-connection on T*M induces two types of covariant derivative: (a) the h-covariant derivative v£y := Vxv Y V X,y € X(T’M) (2.8) (b) the w-covariant derivative V^r := Vx-y V X y € X(T* M). (2.9) If K is the ^-tensor field of (2.2) , then the local expressions of its h -and w-covariant derivative have the following form: ® & ® ® S*Pi., = Kf-, ® # ® 5‘a/ ® FPif, where, xf |& - rkK$ + HikK%, + - HerkK%, (2.10) = 6kK$ + VjkK$ + Vf'“^ - VfkK%, - VpK%. (2.11) Let C = piU1 the Liouville vector field on T*M. Definition 2.2.2. A (^»-connection V on T*M is of Cartan type if = 0 and = id.
392 Hrimiuc and Shimada Proposition 2.2.2. A ¿-connection is of Carton type iff pi}j = 0 Hkpk -Nij = Q (2.12) = %*=> V^Pk + I&Nu = 0. (2.13) Remark 2.2.2. (i) If (2.12) is verified, we say that V is h-deflection free and if (2.13) is true then V will be called v-deflection free, (ii) When ITJ — 0, ÿ is just the connection N which arises as usual. We o will denote this JV'-'Connection by V. Locally, we have vs.d' = -H}k&>, = v?8;. 1 Theorem 2.2.1. Let ft be a connection-pair and N its associated connection, o Then a N-connection V induces a ¿-connection V on T*M, given by: VXY = w(Vxi™) + h'(yxYh'), X, Y 6 (2.15) The local connectors of V are the following: V?' =vV-ltâi&. (2.16) Proof: V from (2.14) is clearly a linear connection. Let us find the local form of this connection. We have v,/; = h'(yS;8ï) = Vj.55 = w(Vi(.<P) = w(Vj- (& - IP’£)) = w( - - W)J; - = h'(V^6k - tf*H^) = (VfS*k - T^H^k, = w(Vi4^) = w(V* (# - IP‘*^) - ITWi.(# - IP*i£)) ' = w((nisn’iÆjz + IT^XIF*) - IJ^vr - ^(IV*))^ +(-v*+ML)sm) »-(vË-n^HL)*”1.
Symplectic Transformation 393 Therefore, V is a ^-connection and also (2.16) are verified. □ Let V be a ^-connection and T(X, K) == - VyX - [X, y], (2.17) its torsion. Locally, with respect to the frame , ¿*), we have = T^+fyfci*, Ttf, Sj) = (2.1S) T(&,6i) -s*krk+sgsk, Where T* = H*j-H*i+Rijetfk-, Tijk = Rijk, Ff (¿¿II** + T^R^IL™ + I»kj =Hij-(&N.ik+TLieRjek')t <2"19) 5« = yii _ Vii _ - r&iiNrk). o Proposition 2.2.3. The torsion connectors of V and V are related as follows: =T^-Rjitntk, Tijk = fijk, O fci PF = P< -(5mik+'lPtRjemn.mk + ^Njentk), o. (2-20) P*kj = F*W-H^№, S% = Sÿ + (W№fc - K’t&Nek). However, V has an extra torsion tensor S^k — R'^ which does not occur o when IPJ = 0. It is clear that V is /i-deflection free iff V is h-deflection free. The following result gives the relations between v-deflection free tensors: o Proposition 2.2.4. Assume that V is h-deflection free. Then V is v-deflection o free iff V is v-deflection free» Proof: ttl* = s* <=*■ v?kPj ° = -nw№ VikPj - n«^ = -nw№ 4=> VikPj = 0 ■*"*P<|* = $*
394 Hrimiuc and Shimada (here J denote the v-covariant derivative induced by V). The curvature tensor of a ^-connection V, R{X, Y)Z = VyZ - Vy VXZ - V(x,y]Z has three essential components. We have: *(^W. = #jU& R{SktSk)g; ¡W =-R$kh6i, R(ik,6^ =-p;№, R(ih,6k)6i * * =-S?hS’, (2.21) where = II {5^ + H£H>mh} - Rhki(rfmjrjm + V"), (W - Ji(^) + H‘jkV? - H'akVfh - (i^nw + + 3^11^)^ - + TL^R^V^ = II {^(^*) + V^V?} - 8!»% - (UM^ - WNuM 3 (2.22) where U {• • •} indicates interchange of i and k for the terms in the brackets O.fc) and subtraction. Let us consider the diagonal lift metric tensor on T*M G = ga (x.p)^®4 ® <5*^ + gi3(x,p)S*pi ® 5'Pj, (2.23) where jy is a symmetric nondegenerate d-tensor field. Theorem 2.2.2. Let $ be a fused connection pair on T’M. Then there exists only one ¿-connection such that the following properties are verified: (i) 9ijfk =0, (ii) =0, (iii) T]k =0, (iv) Sf =0. The coefficients of this ¿-connection are the following: Hij = | (2.24)- Vik = " j 9im(Sj$mk + - WA) - i gim(gajB^k + gakB^ + gm"Bk3), (2.25)
Symplectic Transformation 395 where A*k ^.Rk^-, S'? := n.’er&Nri - n^Nri. (2.26) Proof: (i) is written in the following from: $k9ij = Htkdrnj + i^jkQiTn, (2.2/) and by using the same technique used to find the Christoffel symbols for Rieman- nian manifolds, and the first of (2.19), we get (2.24); similarly for (ii), but using o the last equation of (2.19). In particular, the N-connection V has the coeffi¬ cients hi = 5 g^i'iSmi + S]gim -¡M (2.28) hk = -~gim&gmk + dkgim-dmgik) (2.29) and verifies (i) - (iv); this connection is metrical with respect to G = gij(x9p)daf + g^(x,p)6*pi 0 5*pj. (2.30) a o Theorem 2.2.3. Let V be the G-metrical connection above, and V its- induced (^-connection (2.15). Then V is G-metrical Proof: We must show only that V is v-metrical. By virtue of (2.16) we have: Sy I* = 8kgij - Vfkgtj - V-kgu = &aij - TL^ga - (Vf* - T^H^gti - (Vjk - nk>Hj.)ga = d^ga - vikga - vfgu - n^g« - han ~ Hfa«) ok 0. = 5yl =0. o Remark 2.2.3. This ^-connection, induced by V, is the appropriate one for studying the geometry of T*M endowed with the metric tensor (2.23). Eisenhart [14] and also Yano-Davies [56] used a similar connection. 2.3 The Homogeneous Case We specialize, here, the results of previous sections in the particular case when ff*(x,p)= (3.1) and J? is a real smooth function, 2-homogeneous in p», and such that the tensor (gV(a?,p)) is everywhere nondegenerate on T*M. Spaces endowed with such kind
396 Hrimiuc and Shimada of functions are called Cartan. manifolds [37], [43]. Among nonlinear connections on these spaces the Cartan nonlinear connection is without doubt important. This connection has the following coefficients: (3-2) where we have put, as usual, 7,>- := l9kh(.digi>.j + d,gih - dhgij), 7y := 7kjPn, 7it> ••= TijPkP*, Pi := Note that N is deflection free, [35], [43]. However, the geometry of Cartan manifolds as given in [23], [40], is dra¬ matically changed under a diffeomorphism which is not fiber-preserving. In this case, the geometrical approach described in the previous section is the correct one to use. Theorem 2.3.1. Let be a homogeneous connection-pair on T*M, such that Pilft = 0. If V is the ^-connection given by (2.24), (2.25) and g^ are those of (3.1), then (i) V is h-deflection free iff Nij — P™ (RimtH?9Qsj ■+■ 9»i)' (3.3) (ii) If V is h-deflection free, then Vfkpk — 0. (Hi) 7/V is h-deflection free, then V is also v-deflection free. Proof: (i) By definition V is h-deflection free iff = HkjPk* From (3.1) we see that yijk = Qigjk (3.4) is completely symmetric and because g*k is 0-homogeneous we get Pi&g* = P&g* - P&g* = 0, and also jfrga = -p* gaging1™ = -Pidkgtmgjm = 0. (3.5) Taking into account that = 0 and using the above identities, from (2.24) transvecting by p*, we get: Nij = 7& “ - ipn*(^m£n^i + ÆiWn^).
Symplectite Transformation 397 Transvecting again by pf, we obtain N{q = 7^ and thus Nfj = 4-^wn^), that is, (3.3) holds true. To prove (ii) we need the following: Lemma. IfV is h-deflection free, then Rijtp* = 0. Proof: Because V is ¿-metrical and ¿-deflection free and H = g^PiPj we get 5^H — 0, that is, diH=-NijdjH. (3.6) Therefore, R,ktPe = (6tNM - = [(&№, - dkNa) + (N'i&Nu - N^N^H = ds(NkiOeH') - dkINted'H) - NM*H + N^dk&H + Nsi&tNu&H) - N/'i&yU&H) - NM^H + NkM^H - -d'dkH+dkd,H - Nkidt&H+Naidh&H - N^&dkH+ = 0. Let us now prove (ii). We have Pk^9mk = ¿’W**) - ypkd1^ = i - B?3N,k)gmk = | &dmH - | | 'K>aN3tdtdmH -gjm + Ii?),N3igtm = - i | iP3dm(NatdlH) ¿1 ¿1 Thus, pfci^m& -Pk5mgik = | - n"“^^)p£ + (IP’g™* - II”VZ)^, and by using the above lemma we get, Pk^g^-Pki"1^ = | (BT^Nrs - nro’W,)i>a + (IP'g™* - n"Vz)AU On the other hand, g™Bk/Pk =
398 Hrimiuc and Shimada and Pk9*V ~ | - TPrr*NrJp*. From the last four equalities, by again using (3.5) and pilf J = 0, we finally get, v?kpk = 0. (ill) From the last of Eq. (2.19) we get V?kPi = VkiPi + (lW*№i - nfcs S’Nsifa - -IL^N^ and thus V is ^-deflection. free. Remark 2.3.1. (i) The condition pilft = 0 was used by Yano-Davies [56] and Yano-Muto [57] and it holds when WT*M is the image of the vertical subbundle through a TV-regular homogeneous contact transformation (see the next section). In fact, in this case the Liouville vector field C = pi& belongs to WT*M. (ii) Eq. (3.3 ) shows how to select the connection TV such that (2.24), (2.25) hold and the connection is of Cartan type. 2.4 /-Related Connection-Pairs Let 0 be a connection-pair on T'M and WT*M —► T’M, HT*M -+ T*M, the oblique and horizontal bundles. We denote by W its associated connection (nonlinear). Definition 2.4.1. f 6 Diff (T*M) is called TV-regular if the restriction of (%/)♦ := d(7rf) to HT'M (tt/% : HT*M -> TM is a diffeomorphism. If f has the local expression f(x,p) « (z(z,p),p(z,p)), then it is TV-regular iff (tt/),(^), i 6 l,n are linearly independent, that is, the matrix with entries e* := = diXk + Nij&x* (4.1) has maximal rank. Theorem 2.4.1. Let </> be a connection-pair onT'M and f € Diff (T*M). The following statements are equivalent (i) <£ = is a connection-pair (ii) f is N-regular. Proof: <f> is clearly an almost product structure on T*M and it is a connection¬ pair iff Ker (id - is transversal to VT* M, or equivalently, tt* : Ker (id — 0) —► TM is an isomorphism. But this last condition is equivalent to (ii) because (Ker (id - 0)) = Ker (id - 0).
Symplectic Transformation 399 Definition 2.4.2. The connection-pair <j> given by (i) _above is called the pushr forward of (/> by /. The connection N associated to ÿ will also be called the push-forward of N byf. Also we will say that and are f-related. Theorem 2.4.2. The coefficients of two f-related connection-pairs </> and <f> are connected by the following equations: <$7^=^, (4.2) (&pk - = IP 6* - (4.3) Proof: From 0/» — f*<t> and (1.5) we get: </>/*(¿7) = that is A(<7) G HT*M := Ker (id — </>). On the other hand, = (<5‘^)(^-^) + Wpfc)> Therefore, AW) G HT*M dtfk = 5l^NZk, that is (4.2). Now, let us prove (4.3). We have ¿AW) = -/.(£) A(^) G WT'M := Ker (id +?). But, 7*(ji) = (^fc)% + №)èfc = ÿ&fâ - Nkt^ + H*^)) + + n*%) = (5*®* - $^iNiaTïsk + SipiT£,k'irk + (^pfc - S&N'kft. Thus /.(£) € WT'M iff <Fx* - + ^psîffc = 0. By using (4.2) we get (4.3). Corollary 2.4.1._jÇf №j^*) are adapted frames at (x,p) and (x.p) induced by ÿ and 0 respectively, then: (4.4)
400 Hrimiuc and Shimada (4.5) where (4.6) The regularity of 0j follows from (4.5) but is also a consequence of Proposition 2.4.1. (i) / is N-regulariff f"1 is ~N-regular. If f G Diff(T“M) is N-regular, then (u) (£?$*)$>') = <5?; and (Hi) (^pfc - &&Nhk)(Shpj - Proof: We have W?)) = ^a-1^) = (¿?®*)(&®%* and (i), (ii) follow. To prove (iii) we use the following equalities: r = tW) = (^ - = Generally, if N is a connection on T*M the push-forward of N is not a connection. Some consequences of Theorem 2.4.2 are the following. Proposition 2.4.2. Let N be a connection on T*M and f 6 Diff(T*M) N-regular. Then <t> = f^NfT1 is a connection-pair; the connectors of the con¬ nection N associated to <j> are given by (4.2) while has the following form: IT = (¡MN* - &pk)dka?. (4.7) Proposition 2.4.3. The push-forward of a connection-pair <f> by a N-regular diffeomorphism is a connection if and only if IP = faFfaj. (4.8) Corollary 2.4.2. The push-forward of a connection N by a N-regular diffeo¬ morphism is also a connection iff f is fiber preserving (that is. locally. f(x,p) — (x(x),p(x,p)j. Now we will study when the push-forward of symmetric connection-pair by f is also symmetric.
Symplectic Transformation 401 Theorem 2.4.3« Let ф be a symmetric connection pair on T*M and ф the push-forward о/фЬуа N-regular diffeomorphism f. The following statements are equivalent: (i) ф is symmetric (ii) HT*M and WT*M are f*и-Lagrangian. Proof: ф is symmetric if and only if HT*M and WT*M are both Lagrangian. Therefore, we must have: ^(2 ¿5) (¿i j ) = 0, = 0. By using (4.4) and (4.5) these conditions are equivalent to /М*Лф=0 and therefore HT*M and WT*M are isotropic and thus Lagrangian with respect to /*oz. The converse statement is immediate. Corollary 2.4.3. ф is a symmetric connection pair iff (4.9) 6кр^ = Ькр^. (4.10) Proof: We have: Гы = = (S^PiS*sk + S'pJ'p,') A (ffiTS-x* + Г&Грг). By using Theorem 4.3 we get the equalities above. Note: N is symmetric iff (4.9) is verified. Theorem 2.4.4. Let f be a N-regular symplectomorphism. Then ф is symmet¬ ric iff ф is symmetric. Proof: ф is symmetric iff HT*M and WT*M are /*o>-Lagrangian, that is w-Lagrangian which is equivalent to ф being symmetric. To summarize the results above we can state the following: Theorem 2.4.5. Let f 6 Diff (T’M), N-regular, such that (<?,£*) » Then each pair of the next statements implies the third: (i) f is a symplectomorphism (ii) ф is symmetric
402 Hrizmuc and Shimada (iii) is symmetric. Remark 2.4.1. (i) The condition /*o>(d'\<5J) = is equivalent to (4.U) and it is obviously verified when f is a symplectomorphism. (ii) If f is a TV-regular symplectomorphism then © WT'M) = A(2ÏT*M) © MWT'M). ’ Proposition 2.4.4. Let f be a N -regular symplectomorphism ofT*M. Then ~ ^4 = ^ (4.12) Proof: It follows from using (4.4) and (4.5). Note: Under the conditions of the above result and Proposition 2.4.1 we have: Q} = + N^x1 = iTpj - Nsj&x*, (4.13) and its reciprocal, 9} = - Nkjd^ = djx* + Nk&x*. (4.14) Proposition 2.4.5. If f € Diff (T*M) is a N-regular symplectomorphism and (S*xijrpi') is the dual offâ,#) then /.(i*®i) = ê}T^; f.(S*pi) = efffy. (4.15) Proof: L ~ /.(Fx4)^) = = S'x^S*} = and first equality (4.15) follows. A similar proof holds for the second. Let us now study the connection between curvature tensors Rf and Rf of N and TV. Proposition 2.4.6. Let <j> be a connection-pair and <j> its push-forward by a N-regular symplectomorphism. If R and R are the curvature tensors of N and N then: &jk = Rtm(4.16)
Symplectic Transformation 403 RiikT^ = ^(C) - + tiffl&RHjr*. (4.17) Proof: From (1,17) we get: Ç] = Rin&j"1 + Ri3kntkwTm. On the other hand, /♦MJ = [^,^i = - *$(0^ + ^iW”1+ and the relations above follow immediately. Corollary 2.4.4. If ST™ = 0 then Rii^ - riJ(C) - «7^(0- (4.18) 2.5 /-Related </> -Connections Let us now investigate the behaviour of geometrical objects described in Sec¬ tion 2.2 under symplectomorphisms. If is a connection-pair on T*Af and f : T*M —► T*M is a ^regular symplectomorphism, then the symmetry of the connection-pair <t> — MfT1 is preserved and also /.№*) . /.(*) = /.(<5*^) = x^, f.(6'pi) - e{s*pj, where = 3 and = We can construct a new geometry on T*M, generated by f, by pushing forward all geometrical objects described in Section 2.2, thereby extending to a more general setting, the results of [16], [50], [56]. For instance, if K is the tensor field, locally given by (2.2), then we can consider its push-forward: where (5.1)
404 Hrimiuc and Shimada In particular, the push-forward of G from (2.23) has the following local form: G = g^T ® T x3' + ® T pjt (5.2) where ^0/ = ^. (5.3) If V is a linear connection on T*M, we define its push-forward by f as follows: VYY:=fjyxY\ X = ? = A(n (5.4) V is clearly a linear connection on T*M. Proposition 2.5.1. (i) V is a ^connection iff V is a ^-connection. (ii) V is G-metrical iff V is G-metrical. Proof: (i) V0 = 0 v^(y) = 0(vxy)) = (Mf^)f.^xY) V<t> = 0. On the other hand, Vg/ = V/^ = A(Vw) because f is symplectomorphism and thus Vcj = 0 Vu> = 0. (ii) This follows from VG^A(VG). Proposition 2.5.2. The coefficients of V are related to those of V by the following relations: K = (5.5) V? = 0^,0^' + . (5.6) Similar theorems to those of Section 2.2 (Theorem 2,2.2 and 2.2.3) hold when V is replaced by V and G by 3.
Symplectic Transformation 405 2.6 The Geometry of a Homogeneous Contact Transformation In this section we will restrict our considerations to the slit tangent bundle T*M instead of T*M. _ Let 0 be the canonical one form of locally given by 0 = pi da?1. (6.1) Definition 2.6.1. A diffeomorphism f :T”M —> T*M is called a homogeneous contact transformation (h.c.t.) if 0 is invariant under /, that is pe = e. (6-2) Proposition 2.6.1. If f is a h.c.t. then f*ÇS) == C. Proof: We use the property of the Liouville vector field C — p>d\ as the only one such that i^d0 = 0 where V denotes the interior product of C and d0. We have i^WX) = <w(/.(c),x) = de(MC), /.(r1).^)) = (/•<#) (5, = d(f*e)d5, (f-z),x) = ae(c, (f~l),x) = i^e((f-^x) = e((f~1).x) = (f~1)’e(x) = e(x) for every X G X(T*M). Note: The set of h.c.t. is clearly a subgroup of the group of symplectomorphisms off*M. Corollary 2.6.1. If f fap) = (fc fap),9 fap)) is the local expression o/ah.c.t. then x = xfap) and p — pfap) are homogeneous of degree 0 and 1 with respect to p. Proof: /. (C) = c Pid&dk + P&PJ& = <=* pi&x*=o, pid^pk = Pt • Remark 2.6.1. (i) See also [14] for another proof of this result. (ii) A h.c.t. is a symplectomorphism, therefore we must have: ^¿a? * — d pi, dipk — dkPi, (6.3)
406 Hrimiuc and Shimada If x = x(z,p), p = p(x,p) are homogeneous of degree 0 and 1 with respect to p, Eq. (6.3) are also sufficient conditions for f(x,p) = (x,p) to be a h.c.t. In [14] it is proved that f is a h.c.t. then di^djPk - dj&diPk = o, d^&Pk-d^djp^fy (6.4) which in fact results from (6.3). (iii) If Jo € Diff (M) then the cotangent map induced by /o is a h.c.t. In fact, if x — x(x) is the local form of fo then, f(x, p) = (x (ar), p(x, p)), pk = Pi^kX*. In this case f is called an extended point transformation. It can easily proved that every fibre preserving map which is also a h.c.t. is an extended point transformation (see also [50]). (iv) The reason to use the word “contact” in the name of this transforma¬ tion is given by the property of preserving the tangency of some special submanifolds of T*M. (See [50].) Proposition 2.6.2. Let $ be a connection-pair, N its associate connection and f a TV-regular h.c.t. (i) </> is homogeneous <&=> <j> is homogeneous (ii) If V is a (^-connection ojy is the ^-connection defined by (5.4), then V is h(v)-deflection free iff V is h(y)-deflection free. Proof: Straightforward consequence of Definition 2.2.2 and Proposition 2.6.1 Let H : T*M —> JR be a 2-homogeneous regular Hamiltonian and H the push-forward of H by /, H = Hof~\ (6.5) H is also 2-homogeneous Hamiltonian, but the matrix with entries (6.6) c c may not be regular. Assume also that f is N-regular, where N is given by (3.2).
Symplectic Transformation, 407 Using the homogeneity property of f we get Pi = Pk&Pi = Pkt&Pi - d^N^) = efpk. Therefore, Pi-^iPk and Pfc = ^Pi. (6.7) Let be the metric tensor (5.2). The push-forward of G is given by (2.23), where = (6.8) We have = PiPj^Xfffc,l(®,p) = PfcPfc5fcft(x,p). Therefore, (6.9) Of course, we also have H= i^iPj, but gij may happen. In fact, the metric tensor induced by f is g* and not g^ in general. The tensor J*5 is 0-homogeneous with respect to p± and nondegenerate, but it may lack the property ^gV = îÿgV w’hich assures that following Section 2.5, the geometry, as in Section 2.3, can be derived from it. Therefore, it is from g^ that we can derive the geometry described in Sec¬ tion 2.2. Now let us find the relationship between g^ and g*. Proposition 2.6.3. (i) ikH = o f-1 (Ü) = r’ + - (M - ^№fc)^)pfc. Proofs (i) tkH = dsH&'x" + d’H&p,. But 5$H = 0 ==> d3H = —Nat&H = —NstpP and by using (3.2) we get, because of homogeneity of ffy, 9,H = -y°a0 = -Naipi. (6.10) Therefore, +p>tkp, = piffipi - (7y - £ = Pi (&Pi - nJFx*'} =
408 Hrimiuc and Shimada where we also have used (4.13). ' Therefore, we get (i). Now, 3&h = +s^dkHd^x^ + &H(dm0il&vm+amij^pro). Using Eq. (6.10) we obtain dmdkH - -dmNkipe ~ NktdmdeH and this equality transforms into (ii) after a straightforward calculation. Note: The relation (ii) above is just (3.19) combined with (3.20) of [15], if we start with a Riemannian metric g* ~ 7^(x). Remark 2.6.2. g* - - {de^ - tf^Nek)#^ = 0. By using (6.4) we see that this equality is equivalent to pedh&ie = Aid,lxk and p^dM-S^N^ Aides*, for some functions AJ.. But from these equalities we get 4 = «Wl - <FmdmNta + Nevj^p* (see also [15], (3.26)). Also note, gti =gV holds t -ftg^p^Q ^g^p, =pk = gkjPj- As a consequence of the discussion above and results of previous sections, we have the following summary: coo (a) If we start with a Cartan manifold (M,tf), we get the triple (2V,V, (7) o c o given by (3.2), (2.28), (2.29) and (2.30) V is a ^connection, G-metrical, o o . . of Cartan type and the torsion tensors Tfj, S± vanish. (b) TaJking a TV-regular h.c.t. we get a new triple (<£, V, <7). Here </> is a homo¬ geneous connection-pair those coefficients are given by (4.2) and (4.7); V is the ^connection of Theorem 2.2 (in (2.24) and (2.25) gij and are substituted by g^, J* and 6*, by J*, ?) and G is given by (5.2). This lin¬ ear connection is G-metrical and of Cartan type. Also, the torsion tensors and 5^ vanish. In fact, T*- and S% are contact transformation (as (5.1))ofT^aadSi<
Symplectic Transformation 409 We get a new function H as in (6,5) which may not be a regular Hamilto¬ nian, Also, S*^ = 0 and Hpp =23-1*7 = 2^ (6.11) where f denotes the ^covariant derivative with respect to V. (c) If — gt then Ji is a regular Hamiltonian and Theorem (3.1) is valid (baring all the coefficients). A simple consequence, for the deflection-free case, is : n = N iff (5iTOZn<X +^n^)r> = 0. (6.12) The relation (6.11) can be also written by using Proposition 2.4.6 and (5.3) in terms of similar objects derived from H. (Rimt and g9j are contact transformations of RimJ. and gsf). When (6.12) is verified, by virtue of Theorem 2.2.1, Proposition 2.2.2 c o o o and Theorem 2.2.3 we can pass to the triple when V' is a C_ Q .V-connection, G-metrical, hr and ^-deflection free, but generally fails to o. ’ have vanishing torsion tensor T^k, Sj. Therefore, it does not coincide with the Cartan linear connection for the Hamilton manifold (M,H) [23], [40]. If f is an extended point transformation, then n = 0, N = N and the push-forward of the geometry of Cartan manifold (M, H) is just the geometry of (M, H) so this geometry is invariant. 2.7 Examples We now construct a connection-pair on T*IR2 which is horizontally flat, but with complicated IT*. In fact, we construct a homogeneous contact transformation between (T*IR2, H), where = 0 and "Ny 0, and (T*IR2, if), where IT* / 0 and Tij = 0. Here, 1H » | (P2 4- P?) is the Euclidean Hamiltonian in T*IR2, spanned by (Q1, Q2,Pi,P2), and gij = fit Select, once and for all, a Finsler metric function A = ACg1,^2,?!,^) and the metric = e“2^*1 • (¿£d*A2) and set where, iX«1,«2, Q1. <Q2) = Ate1»«2)«!?1 + fetfrflQ2 = A is defined in terms of C°° functions /i, /2 and A, some constant. Noting that Pid<Ql — Pidqz = 0,
410 Hrimiuc and Shimada we have the possibility of constructing the desired contact transformation (Q \Pi) i-► (g\pi) and its inverse, locally. But, two side conditions will be necessary for this. Firstly, det' = det(|^)/0, must hold in some chart (Z7,h). Then has a unique solution in (Z7, A), <Ci = ^(pi.ps.«1,«2)- Of course, Pi-AAfcW), so that the transformation is determined by fa and djfa. It is also required that the transformation by N-regular. In this case, = 0, so this condition is merely Now, the push-forward of g^ — is required to be (by results of Section 8.6) g* = ^(JH o $ = I (5^ A2) • A where, = -jWl)2 + (/2)2]. We have supposed that g2) is known and defined in a chart (Ï7, h). Secondly, we must now select /¿(g1,#2) so that the N-regularity condition holds, locally. Set 1R « — (ps A) + (pi A) 92, to denote this linear operator and assume A isjndependent of g1, g2. Thus, X is a Minkowski metric function in the chart Note that d?1 « - l/det/IR(A) and Q2 = - l/det/IR(A). Proposition. Under the condition det/ 0 in N-regularity holds for A Minkowski <=> 9i(¿n det/) [(IRA) QR o - (IRA) • (H o %)(/2)] - 92(£n det/)[(R/2)(lRo91))(A) - (1RA)(IRo91)(A)L^ [(!Ro91)(A)] x[(IRo92)(A)]-[(1Ro91)(a)][(1Ro^)(A)] in (p,h). Corollary. Zn addition, assume fz — c * ._where c > 0 is a sufficiently small constant. Then N-regularity holds in (V,h) <$=> Hess (A) 0 in (V,h)
Symplectic Transformation 411 _ o o (Hessian determinant) and ft = y/e2* — (eg2)2 . Here. V C B C U (B is interior of closed 2-disk).' Proof: A short calculation shows that the condition of the proposition reduces to the non-zero Hessian condition. An easy continuityjargument shows that fi above is well-defined in some closed 2-disk in (i/,h). Merely note m < > i2) < M holds in any closed disk B C (U, h) and take the radius r = ^em so that c ♦q2 < in this. B (radius = r). Now choose a smaller chart V in the interior of B. This completes the proof. _Also note that by linear adjustment, we can always suppose that d> (center of B) == 0 in IR2. We can now state the Theorem. If <j> has a non-degenerate critical point x in (U, h) of IR2, then N-regularity holds in some neighborhood of x. Consequently, (T*IR2,H) is homogeneous contact equivalent to (T*IR2}H) where = kg^Pipj = le~2*(&$X2)piPj. Moreover, IT-7 is by (4.7) not zero generally and is completely determined by = 0, Nij = 0 and this transform¬ ation. Similar results are possible even if has no nondegenerate critical points. For example, if <j> = a#*, a» constants, the conclusion of the theorem above holds. It can be reformulated as Theorem. Any 2-dimensional constant Wagner space is the Legendre-dual of the homogeneous contact transformation of the fiat Cartan space (T*IR2, H) with non-trivial oblique distribution II. Similar reformulations can be made of the main theorem on ^regularity, as well, using the known result that Wagner spaces with vanishing Ti-curvature must have local metric functions of the form • A. These have been found to be of fundamental importance in the ecology and evolution of colonial marine invertebrates.
Chapter 3 The Duality Between Lagrange and Hamilton Spaces In this chapter we develop the concept of duality between Lagrange and Hamilton spaces (particularly between Finsler and Cartan spaces) and a new technique in the study of the geometry of these spaces. We will apply this tech¬ nique to study the geometry of Kropina spaces (especially the geometric objects derived from the Cartan Connection) via the geometry of Ränders spaces. These spaces are already used in many applications. 3.1 The Lagrange-Hamilton ^-Duality Let L be a regular Lagrangian on a domain D C TM and let H be a regular Hamiltonian on a domain P* C T’M. Hence, the matrices with entries and gab(x,y) := dadi>L(x,y) g^fap) d* & H(x,p) (i-i) (1-2) are everywhere nondegenerate on D and respectively P‘. If L € ^(P) is a differentiable map, we can consider the fiber derivative of Ly locally given by V(.^,y) = (xi,daL(x,y)>) '(1.3) which will be called the Legendre transformation. It is easily seen that L is a regular Lagrangian if and only if <p is a local diffeomorphism. 413
414 Hrimiuc and Shimada In the same manner if H € ^(D*) the fiber derivative is given locally by = (x\ ^H{x,p)) (1.4) which is a local diffeomorphism if and only if H is regular. Let us consider a regular Lagrangian L. Then p is a diffeomorphism between the open sets U CD and iZ* C T*M. We can define in this case the function: H: K(x,p) = pal/a _ (1 5) where y =» fea) is the solution of the equations pa = daL(z3 y). Also, if H is a regular Hamiltonian on M, $ is a diffeomorphism between same open sets CZ* c D* and U G TM and we can consider the function L:U —► jR, L(xt y) = paya - p), (1.6) where p = (pa) is the solution of the equations p* = ^tf(s,p). It is easily verified that H and L given by (1.5) and (1.6) are regular. The Hamiltonian given by (1.5) will be called the Legendre transformation of the Lagrangian L (also L given by (1.6) will be called the Legendre transfor¬ mation of the Hamiltonian H. Examples. 1. If L is zn-homogeneous, m / 1, regular Lagrangian, then locally, H(x,p) — (m — l)L(x,y), pa = daL(x,y). 2. If L(x, y) = ^Oij (x)ylyi + ¿¿Ji1 + c then its Legendre transformation is the Hamiltonian £■(«,₽) - ^aV(x)piPj - ttpi + d where b* := and d := btb* — c. In the following , we will restrict our attention to the diffeomorphisms <p : U —► IT and : tZ* —> CZ (1.7) (where is the Legendre transformation associated to the Hamiltonian given in (1.6)). We remark that U and CZ* are open sets in TM and respectively T*M and generally are not domains of charts. The following relations can be checked directly
Symplectic Transformation 415 930^ = 1^,, ^0^ = 1 [j (1.8) 9iH(xtp) - -diL(x,y); didaL(xty) = -di^’H(x,p)g*l>(x,p') (1.9) 5«*(®,!/)<?*6c(^p)=i: (1.10) where pa = daL(x,y), ya = 9aB'(x,p). Using the diffeomorphism tp (or we can pull-back or push-forward the geometric structures from U to C7* or from U* to U. A) if f 6 5*(U) we consider the pull-back of f by (or push-forward by p) r-.= fo^ = fo<p-\ reW). (1.11) Also, if f € F(U‘), we get /> € F(U) f° •- fo<p = fotp-1. (1.12) We have the following properties: (i) (A/ + pg)* = A/* + MT, w = /‘S’; VA,p € Si,Vf,g € F(U), (ii) (A/ + pg)0 = A/0 + pg°, \fg)° = f°g°; VA)/t e € ^(¡7‘), (iii) (/‘)° = /, (I?0)* = g, f e F(U),g e F(u*), (iv) (gab)‘— g*“*, (fta)’^. B) If X € X(U) the push-forward of X by <p (or pull-back by ^) is X* e *(17*) X‘ ¡=T<poXo<p-1=Ti>~1oXoi>. (1.13) (T<p is the tangent map of <p.) Also, if X e X(U*) we can consider the push-forward of X by t/> (or pull¬ back by <p), X° € *(17) X°-.= Ttl>oXotl>~l:=T<p-1oXo<p The following relations are easily checked: (i) (fX + gYY~FX-+g*Y-, Vf,g £?(?),VX,Y € X(U), (ii) (fX + gY)° = f°X° + g°Y°, Vf,g 6 F(U*),VX,Y € *(17’), r«n =ix*>y,i> vx>y e x^> w ix,y]0 = [X0,y°], VX,y G*(17*), (iv)' (X*)° = X, (K0)’ = Y, Xe X(U), Y € X(U'). (1.14)
416 Hrimiuc and Shimada C) If 0 € X* (U) the push-forward of 0 by y? (or pull-back by VO is 0* € X*(£7*) 0* — (Ttpy* o 0 o «p”1 = (2V1)* o 0 o (1-15) and if 0 € X* (£7*) we can consider 0° = (T^)* o 0 o -0"1 =: (TV1)’ o 0 o <p (1.16) where (Tip)* denotes the cotangent map of <p. We have similar properties as (i), (ii), (iv) above. D) Generally if K e 77(£7) is a tensor field on U we can define similar the push-forward of K by <p, K* G 77 (£7*) and, for K € 77 (C7*) we get X* € 77(£7). We have (X ® T)* =K*®T\ (K' ® T')° = X'° ® T'°. (1.17) Let V be a linear connection on U, We define a linear connection V* on £7* as follows: := (Vx»y°)*, X,Y e X(U*). (1.18) Also, if V is a linear connection on U* we get a linear connection V° on ¡7. V&K := (Vx-r-)°, X,YG X(U). (1.19) It is easily checked, using the above examples, that V* and V° are indeed linear connections on £7* and £7. For the torsion and curvature tensors of V* we have . T*(X, y) = [T(X°, y°)]*, VX, Y e X(£7’)> (1.20) R\X,Y)Z = [K(X0,y°)Z°r, VX,y,Z G X(£7*). (1.21) Generally, if K € 77(£7) and X* G 77(£7*) is its push-forward by then V*X* = (VX)*. (1.22) Definition 3.1,1. We will say that f and /* (or f and J0) X and X* (or X and X°), K and K' (or K and X°), V and V* (or V and V°) are dual by the Legendre transformation or are ¿-dual. In the next section we will look for geometric objects on £7 and £7* which are ¿-dual. These geometric objects will be obtained easily each one from the other.
417 Symplectic Transformation 3.2 £-Dual Nonlinear Connections Let N and N be two connections on the open sets U C TM and U* C and KZ7, HZ7* the induced horizontal subbundles. Definition 3,2.1. We say that N and TV are £-dudl if Ty(HU) = HU*. (2.1) Let (Nf) and (Ma) be the coefficients of N and N. Theorem 3.2.1. The following statements are equivalent: (i) N and N are £-dual; (ii) = -N*fT* - d&H or N^ = -Nfa + di9aL-, (iii) N = Tip o N o Tip-1 (i. e. N = <p*N). (iv) SitT = -Nf*; MM = <; (vi) Str = VfertU). Proof: N and N are £-dual«^=f-Tv>(fi’Z7) = HU*-^Tip(8i) e HMU', Vi € 1, n. We must have TiptSi) = = ai(d}- + N^). On the other hand, T<p(Si) = T<p(di - Nfda) = T<p(di) - NtT<p(da) = di + didbL^> - N?dttdbLdb = di + (didbL - Nfgab')db. Therefore, we get 4 = and N?a = -Ntgab + d^L (or, equivalent, N*‘ = —Nag'ba—didftH) and we have proved that (i) (ii). Now we have dtf” =* di^r + Nibfyf = d&H + Nit&d'H «= Nag'^ + d^H SiP°a = diP° - N?dbP° = didaL - NfdtdaL = -Nfa» + didaL Str = №/)* <=► {difr + (daf)-d&H+N^dhfY^H =* (difY-Nrtdkfr- Using these relations we get the proof. The ¿-dual of the nonlinear connection N will be denoted by Ar*. (Similarly, the ¿-dual of N will be denoted by TV0.)
418 Hnmiuc and Shbnada Corollary 3.2.1. If N = (Nf) and A* = (№») are two £-dual nonlinear connections we have (These properties are characteristic for two £-dual nonlinear connections, too J Proposition 3.2.1. The following equalities hold good: (d^y^dx1-, ¿v- = g^(àbf)-t dif = (dif)*+d&H&fy. Corollary 3.2.2. Let N = (Nf)^V* = (A^) be two £-dual nordinear connec¬ tions. The following assertions hold: (i) If X = X*6i + then X* = + g*bXa'db and (Xs)* = (%*)*, (Xy)* = (X’)y. (ii) If w = Wiada? A 5ya then — (wia)*p*abdx® A (iii) If K => ® da ® d& ® tyb then X* = (K^)*O*Wi< ®dc®dxi® 5*pd. Remark 3.2.1. We have There¬ fore the components of the £-dual of K in (x,p) are obtained from the com¬ ponents K'fb of K in (x,y), = daL{xyy\ unchanging the horizontal part and raising and lowering of indices for vertical part by using g^ Examples. 1. The £-dual of the metric tensor gab has the components g***. 2. If Ccte = then C‘*»° = ~&lg*bc. If = ¿gadddSbc then = —g^g'^. 3. If <5 is the Kronecker delta, with components 6^ then the components of its dual <5* are as follows: = = <Z**. Let c(t) — (x(i), y(t)), t G IC 2R be a differentiable curve on U. The tangent vector can be written as follows: c(t) = (2-2) 5ya We say that c is a horizontal curve if -7- = 0. at
Symplectic Transformation 419 A similar definition holds for differentiable curves on 17*. Proposition 3.2.2. (JV,JV) is a pair of £-dudl nonlinear connections if and only if the £-dual of every horizontal curve is also a horizontal curve. Proof: Let c(t) = (x(t),y(t)), c*(t) = (x(t),p(t)), tele JR two ¿-dual curves, therefore ya(f) — d^H^t^p^t)). We have: ^-0 «aS.H^ + ^a^ + i,.^_o <=► g*ba~ + (didaH + N^^-=o at at ^■ + ^dkdbH + g’abN^=0. Let suppose that N and .V are ¿-dual nonlinear connections. Using The- orem2.1, (ii), and the above relation we get = 0. Conversely, from -y- — 0, at at = 0 we obtain easily that N and N are ¿-dual. at Example. For a Lagrange manifold the geodesics are extremals of the action integral of L and coincide with the integral curves of the semispray XL = yidi-2Gada, (2.3) where Ga = ¿s“ - dbZ). (2.4) This semispray generates a notable nonlinear connection, called canonical, whose coefficients are given by N^dyGa (2.5) (see Section 1.6, Ch. 1). Using (ii) from Theorem 3.2.1, we get the coefficients Nia of its ¿-dual nonlinear connection: ^ = -diGiffba + &idkL and after a straightforward computation we obtain Ki = ^gljdkH - d^H) --(g^ff+g;kj*d)H). (2.6) We remark that is expressed here only using the Hamiltonian. This is the canonical nonlinear connection of the Hamilton manifold (M, H) obtained in [40]. We also remark that the canonical nonlinear connection (2.6) is symmetrical, that means Tij = Nij — ffji — 0. (2.7)
420 Hrimiuc and Shimada Taking the ¿-dual of (2.7) we get the “symmetry” condition for (N?) Nfai ~ N?9ij - djdiL - didjL . (2.S) ((2.8) can be also checked directly and thus (2.7) may be obtained as a con¬ sequence of (2.S)). Now, let us fix the nonlinear connection given by (2.5) and (2.6) on U and respectively ¿7*. The canonical two form u) = dxa A 6”pa is just the canonical symplectic form of T*M. The Hamilton vector field Xh can be obtained from the condition: ixHw = dH ix^dx* A 6*pi) = fr-Hdx* + dm^Pi. Consequently, Xh = (2.9) The integral curves of Xh are solutions of Hamilton-Jacobi equations ^ = 9% ^=-$2 (2-10) (equivalently with ~ ). ctt etc The ¿-dual of Xh is just X^ the Lagrange vector field from (2.3). In adapted frames (2.3) one reads XL^yi5i + ^Nf-2Ga)da and we remark that X& is horizontal iff Ga is 2-homogeneous. An integral curve of Xl verifies the Euler-Lagrange equations: i - * - "?<’«>• <2-n> which are ¿-dual of (2.10). The ¿-dual of the canonical one form, 0 = pidx1 is the canonical 1-form of the Lagrange manifold 0 = diLdrf (2.12) and the ¿-dual of w is the canonical 2-form of (M, L) = g^dx1 A fry*. Proposition 3.2.3. (i) If N and N are £-dual then (2.13)
Symplectic Transformation 421 (ii) = o, (Hi). (d-Nn,)0 = gae(Si9el> - gM9cN^. Proof: (i) [¿;, ii] = &,5fc]’ = (#,*&)* = Rajkd°. (ii) We use the symmetry of the tensor ¿kgjk in all indices. (iii) [d5, On the other hand, [<M]0 = ^da^] = -gbadaN& - ^(g^da and then we will get (iii). Proposition 3.2.4. Let N and N be two ¿-dual nonlinear connections. Then (i) Nv = Nji N^gid - Nfaj = djdiL - d^L, (ii) = d’Nji ^=> gih.dkNf - gjk9kN? = - Si9]k, (iii) g^N* - g^N* = - 6i9'k 9^ = 9^. Proof: (i) follows from Theorem 3.2.1, (ii), and (ii), (iii) are direct consequences of (iii) of Proposition 3.2.3. 3.3 £-Dual ¿-Connections Let (N, TV*) be a pair of ¿-dual nonlinear connections. Then ¿-dual of the almost product structure TV = da®$ya on C7 is AT* = 8>®dxi~da®6*pa. Let V be a linear connection on U and V* its ¿-dual on C/*, given by (1.18). Proposition 3.3.1. V is a d-connection if and only if V* is a d*-connection. Proof: We use V*N* (Vtf)*. Theorem 3.3.1. Let CT(N) = (L^,L^,C*jc, C%c) te a N-connection on U, and CT (TV*) = Va^) be a d-connection on U*. Then CT(N) and CT(N*} are ¿-dual if and only if the following relations hold: (i) (Hjk)°^Ljk (H) (H^^g^gv'-g^Lfo (iii) (iv) (V^)° = g^g^ld^gea) -gafCfed].
422 Hrimiuc and Shimada Remark 3.3.1. We see that is obtained very simple from Ljk and V'j0, as the ¿-dual of that is On the other hand, V* = -C*be-g*adddg*bc and therefore <=s- = —g^g’1” 4=* = ^9^. Corollary 3.3.1. (i) v**=VHw, V*y = Vv*. (ii) Let K* be a d*-tensor on U*f the C-dual of the d-tensor K on U, K* [* * and K* |c its h- and v-covariant derivative with respect toV\T:- K|fe, Tf K\e. Then K**k = T\ k*]c = t'*. A consequence of (1.20) and (1.21) is the following result: Proposition 3.3.2. Let V and V* be two C-dual d-connections, Then, the torsion and curvature tensors of^v* are C-dual of torsion and curvature tensors ofV, Remark 3.3.2. The proposition above states that the torsion and curvature tensors of V* can be obtained from those of V by lowering or raising vertical indices, using gat>. As we have seen (Theorem 3.3.1), the ¿-dual of a normal N-connection generally is not a normal iV*-connection. Proposition 3.3.3. Let CT(N) = (2^., C0^) be a normal N-connection on U and CT*(7V*) » Ve) its C-dual, Then RSk^a^g^-g^M, (3.1) (3.2) Conversely, if the coefficients ofV (the C-dual ofV) verify (3.1) and (3.2) then V is a normal N-connection. Proof: Let F* — —g^d* 0 dz* + g*^* ® ô*pa be the ¿-dual of thé almost complex structure F = —di ® dz* — fa 0 6y\
Symplectic Transformation 423 Then the N-connection V is normal if and only if Vr = 0 <=> V’F = 0 <=> (3.1) and (3.2). □ On the tangent bundle we have the metrical structure G = gtjdx1, ®dx? + g^ya ® &yb- (3.3) The ¿-dual of this metric tensor is <7* = g^dx* ®<&+ g'*bFpa 0 Fpb. (3.4) Therefore we are in the position to apply Theorem 1.5.1 and Theorem 3.3.1 and so we will get the canonical d-connection of the Lagrange and Hamilton manifolds (here restricted to U and [/*). Theorem 3.3.2. The £-dual of the canonical N-connection of a Lagrange ma¬ nifold is just the canonical N*-connection of its associated Hamilton manifold, (Only in this case = C*6®.) Proof: Using Theorem 1.3.1 and Theorem 3.3.1, (i) we get = yh^9hk + 5kgjh - 6hgjk), and using Theorem 3.2.1 we obtain +6’k9'jh - 5wjk). Also, making use of Theorem 3.3.1, (iv) and Theorem 1.5.1, we have Va* = + dV* - Pg'^ = = 0*<y and from (ii) and (ii) of Theorem 3.3.1 we obtain Theorem 3.3.3. Let V* be the £-dudl of a normal N-connection CT(N) = (LjA,Ca^c) on ¡7. Then we have: (») = Hik iffV* « h-metncal (g*j ¡fc = 0,p^> |* = 0); (ii) = V* iff V* is v-metrical ¡c = 0,^ |c = 0).
424 Hrimiuc and Shimada Proof: (i) The /¿-metrical condition for V* and (3.1) can be rewritten in the following forms: 9^k + = 6^ ( i* = 0) 9cb^k + 9lcPbk = $k9ab ( 9db I* = 0) 9ha^bk + 9bi^h.k = ^k9bh' If Hjk = from the last equality we get the first two and from the last condition and the first we get Hjk = Hjk. By a similar argument we can prove (ii). Consequently, we can conclude finally: Theorem 3.3.4. The class of normal N-connections which is preserved by ^-duality is only the class of normal metrical N-connections. Let u>£ be the canonical symplectic form of (M,L) given by (2.13). The following result is a consequence of the above theorems: Theorem 3.3.5. (i) If V is a normal N-connection onU c TM, then ~ 0 ■4=^ VC? = 0. (ii) If V is a normal N*-connection onU* C T*M then VF* = 0 VG* = 0. Proof: (i) Let V* be the ¿-dual of V. Then VG = 0 V*G* V* is normal <=> V*cu = 0 <=> Vljl = 0. (ii) We use a similar argument. □ Let us stand out some problems connected with the deflection tensor field. The h-deflection tensor field of a ¿-connection V, on the tangent bundle can be defined as follows: D : X(U) —► X(U), D(X) = where C is the Liouville vector field. Locally we have: C — y*da, D = Daida^dxi, Dai = y^L^-Nf. (3.5)
Symplectic Transformation 425 The deflection tensor field of a jV-connection V, on the cotangent bundle, P(X) = V*C where C is the Liouville vector field on T’Af, C — has the local form P — DaifF ® dx’, Dai — Pa||i — Nia “ f^aiPb- (3.6) Also we can consider the v-deflection tensor field d(x) = v%c, d=dabdb ® 6ya, dab = yfa = 5? + cfceayc (3.7) and its correspondent for cotangent bundle d(x) = v£c, d = &bdb ® s-Pa, &b=pbIIa = - vbMpe. (3.8) Using the ¿-duality we see that generally, the ¿-duals of D and d are different by D and respectively d. We have D*(X) = V^C* (3.9) where <7* is the £-dual of the Liouville vector field, C* = y>, yl-^g^h (3.10) and locally D* = ® dx*, D'ai = or %=y:î«=^:-^y;. (3.11) The ¿-dual of d is d* = 0 <5po where d-%=^*ae(dcc)-=2/0’|ö. (3.12) The following result holds: Proposition 3.3.4. C* — C if and only if L{x^y) = ip2(a:,y) +ti(sc) where F is 1-homogeneous and u is à scalar field. In this case D = D* and d =« d*. Proof: C* = (?■-<=>. g^yblK — pa ^=> yadadt>L — dbL <=> 9aL is 1-homoge- Remark 3.3.3. As we have seen from the last two sections there exists many geometrical objects (nonlinear connections, linear connections, metrical struc¬ tures and so on) which can be transfered by using ¿-duality from U to CZ* and also from CZ* to U. Now let suppose we have a regular Hamiltonian defined on a domain D* C T*M. The Legendre transformation : [Z* —> U is a diffeomorphism between some open subsets U*,U of P* and TM, Taking the Lagrangian L{x^y) = palP—H(¡r, p), pa* = tFHix^p) we can construct a Lagrange geometry restricted to U and then we pull-back by V’ the geometrical objects on U, to tZ*. These will depend only by Hamiltonian, therefore we will be able to extend them on the whole domain D’.
426 Hnmiuc and Shimada 3.4 The Finsler-Cartan ^-Duality In this section we will give an idea for the study of the geometry of a Cartan space using the ¿-duality and the geometry of its associated Finsler space. Let if be a 2 positively homogeneous Hamiltonian on a domain of T*M, : U* —► U the Legendre transformation and L^y^p^-H^p), y^ =diH(xtp) (4.1) its associate Lagrangian'. We remark, using the 2 positively homogeneous property of 7?, that L(x,y) = H(x,p). (4.2) Proposition 3.4.1. The Lagrangian given by (4.1) is a 2 positively homogenous Lagrangian. Proof. Let us put /(X.P) := y* = 9i(x,y) := p? = ¿Ofay). We know that /* is 1 positively homogeneous, then 9i(x, Ay) - 9i(xrXfi(x,p))=9i(tx,fj(x,Xp)) = Ap? = X9i(x,y) and thus gi is 1 positively homogeneous so, L is 2 positively homogeneous. Therefore, using the theory made in the previous sections we may carry some geometry of Finsler spaces on 2-homogeneous Hamilton manifolds. Remark 3.4.1. For 2 positively homogeneous Hamiltonian we have Pi=yi or Pi=Vi (4.3) PW == S^P* = 9ijVi = Vi (4.4) pi~g-Hp.=g*iiy>=y», (4.5) Among the nonlinear connections of a Finsler space one has the most interest. It is the Cartan nonlinear connection ^Wo-CWoo (4.6) where 7’^ = ^yi'“(diyW: + dk9jh - dh9,k), CPjk = ^g^dhSjk, ¥00 = 7jfcSr'y*> 7jo = ^k»*- Theorem 3.4.1. The £-dval of the Cartan nonlinear connection (4.6) is (4.7)
Symplectic Transformation 427 where we have put 7$ = ^9'kh(diS*hj + dj9;h - dh9^, 1$ - 7y%, 7*o° = 7*^^ and Vkij = Proof: We have (№j)° = -Nigkj + didjL and making use of (4.2) we get Mi MY + \9M,yryrda9ii - ^9lj (4.8) where A? := d^H. On the other hand, 2(7y»r)* — + (d?9ij)* ~~ (dj9ir)* (&i9jr + &r9ij “ fy9ir}~ So, we will have 2(7iir)*r = ~Pk^ + jT (Mr ~ ^№*9*)*} i'WTy')’ = -prfhF + y*sy^9lr ~ = -Pfc7^ +y’ay*rd9g"„. Then (4.S) becomes Mi =pfc7yfc. - |^7«W- +y*rMr) + \da9ijya{y*Td»9'„ + Akg'‘ba). But we can write: V*rMr = 9'TkPkdi9jr = ~Pkg*kr9jK9ltdi9’ht ' = -9^9'^) = ~9*jhdi^H = -Afaj and substituting it in the above equality we get (4.7). Among Finsler connections, the Cartan connection is without doubt very important. The following result is also well known [30]. Theorem 3.4.2. On a Finsler space there exists only one Finsler connection which verifies the following axioms (Matsumoto's axioms) Cl) 9ij\k = 0 (h-metrical); C^) g. j = 0 (v^metrical); v fc
428 Hrimiuc and Shimada C3) = —Ar£ + y^k = 0; Q) T^ - 0; C5) S^k = 0. The coefficients of this Finsler connection are: Fjk = ^9ih^j9hk + $k9jh - ¿h9jk) (4.9) & jk == ~^9^^h9jk (4.10) and the nonlinear connection is given by (4.6). The Finsler connection given by (4.9), (4.6), (4.10) is the Cartan connection of the Finsler space F* = (M, F). Using the results of the previous sections we can state the ¿-dual of The¬ orem 3.4.2 for the Cartan space C71 = (M, F). Theorem 3.4.3. On a Cartan space (M, F) there exists only one normal N- connection CT — (AT) = (#Jfc, VFk) which satisfies the following axioms: C£) s’yil* = 0; Q) ÿ^ll* = 0; Q) ¿u = -Nik +PiH^ = 0; <3)2^ = 0; C6Wfc = 0. It is the £-dual of the Cartan connection above. That is: Hjk = ¿g-^g'kk + w - rhg& (4-n) = (4.12) and the nonlinear connection is given by (4.7). Proof: The connection given by (4.11), (4.12) and (4.7), verifies CJ)-££). For this connection T^k — Sijk = S*ijk and Âa = IFik (Proposition 3.3.4). This connection is unique. Indeed, if there exists another one, taking the ¿-dual of it we will get two Finsler connections (restricted to an open set) which satisfy Matsumoto’s axioms. The AT*-connection of Theorem 3.4.3 is just the Cartan connection of the Cartan space C71 — (Af,F). We remark that conditions C^-CJ) are all the ¿-duals of Ci)-<7b). Consequently, all properties of the Cartan connection from Finsler spaces can be transfered on the Cartan spaces only by using the ¿-duality. Remark 3.4.2. When we look for a ¿-dual of a AT-tensor field we must pay attention to the vertical indices; in this section (and sometimes in the other sections) for the sake of simplicity we have omitted to use indices a, 6, c, d, e, f to stand out the vertical part.
Symplectic Transformation 429 Let c*(i) = (z(t),p(t)), t € I C JR a differentiable curve on Z>*. c* will be called h-path (with respect to Cartan connection CT*) if it is horizontal and drxi i idx dp\ dx^ dxk dt2 + ^k\dt' (<=^(t)c(t) = 0). Theorem 3.4.4. The ¿-duals ofh-paths ofCT^(N*') = (Hjki VPk) are h-paths Proof: It follows from Proposition 3.2.2 and Theorem 3.3.1, (i). Corollary 3.4.1. Let c*(t) = (x(t),p(t)) be an integral curve of the Hamilton vector field (2.9). then c“(t) is an h-path of CT*. Proof: c*(i) = (x(t),p(t)) is an integral curve of X& iff its dual c(i) — (æ(i), j/(t)) is an integral curve of and therefore an Æ-path so its C-dual c* will be also an h-path. □ The next results will give us an interesting field where the C-dual theory can be applied. As we know a Randers space is a Finsler space where the metric has the following form F(z,y)=a + /3 (4.13) (Randers metric) and a Kropina space is a Finsler space with the fundamental function F(x,y) = c?/0 (4.14) (Kropina metric) where a = \/aij{x)yiyi is a Riemannian metric and 0 — bitx'jy* is a differential 1-form. We can also consider Cartan spaces having the metric functions of the fol¬ lowing forms F(x,p) = yja^pipj + b*pi or (4.15) F(x,p) = (4.16) trpi and we will again call these spaces Randers and, respectively, Kropina spaces on the cotangent bundle T*M. Theorem 3.4.5. Let (M,F) be a Randers space and b = (a^d*^)1/2 the Riemannian length of bi. Then (i) If b2 = 1, the ¿-dual of (M,F) is a Kropina space on T*M with H(x,p) = (4.17)
430 Hrimiuc and Shhnada (ii) If b2 / 1, the C-duoI of (M, F) is a Handers space on T*M with H(x,p) = | ^/aVpiPj i&p^ , (4.18) wAere -aH = _2_ov + 5* = —^v- (in (4.18) ” corresponds tob2 < 1 and “+” corresponds tob2 > 1). Proof: We put a2 = y,#*, b' = aPbj, @ = biÿi, /3* = b'pi, p* = a^pj, a*2 = PiPi = aypipj. We have F-a + 0, p< = = (« + /?) (J+ fe) =-F’(§ + &i)- Contracting in (4.19) by pz and bz we get: a*2=-p(?+j8*)’^=f (ê+à2)’ (4-19) (4.20) Therefore» (4.21) JT*2 (i) If b2 = 1, from (4.21) we obtain /3* = — and using (4.20), we get a f,, a*2 atfpiPi rM> (n) If Ô2 / 1, from (4.20) and (4.21) we have: l«-2 = £ + /r;/3‘ = f + F(62-l) and by substitution - ia- - (»’ - 1)F - f ~ (f + ' . From this last relation we obtain (4.18). Theorem 3.4.6. The £-dual of a Kropina space is a Randers space on T*M with the Hamiltonian #(*,?) = | (^fa&PiPj ± av = = ±b\ 4 2 where (4.22)
Symplectic Transformation 431 (Here "+ * corresponds to /3 > 0 and to ¡3 < 0.) Proof: We use the same notations as in. the proof of Theorem 3.4.5. We have Pi = FdiF=^(2yi~Fbi). (4.23) Contracting by pi and then by b* we get: ■ a*2 = y (2F - IT), f = ^(2/3 - Fb2). (4.24) Using these relations, after a simple computation, we obtain (4.22). We must have b2 0 for regularity of a*. But, the regularity condition for the Kropina metric leads to b2 0. Remark 3.4.3« Using the Theorem 3.4.6, we can derive the geometrical pro¬ perties of Kropina spaces, very simply, from those of Handers spaces, by using ¿-duality. We will explain this more precisely in the next section. 3.5 Berwald Connection for Cartan Spaces. Landsberg and Berwald Spaces. Locally Minkowski Spaces Berwald connection BT — (dkNij,Nij,0) of Cartan space (here Nq are given by (4.6)) is not the ¿-dualof Berwald connection of its associated Finsler space like Cartan connection. There exist some important distinctions here, which are consequences of the nonexistence of a spray and thus, the nonlinear connection cannot be obtained as a partial derivative of a spray. Theorem 3.5.1. On Cartan space (M,F) there exists only one Finsler con¬ nection with the following properties: BJ) ^=o, b2‘) 5y=o, Bl) = 6Jg-k - Bl) Pak = Q, Bl) V?* = 0. Proof: It is easily checked that the connection BP = (dkNij,Nij,0) with Nij given by (4.6) satisfies all B{)—B£). Indeed, let us take the local diffeomorphism : U* - U and consider W = (N])> the ¿-dual of N = N = (Wj) is 'Cartan nonlinear connection of the associated Finsler space (M, F). We have F1Ji = ^F=№F)*=0. B4) and are obvious and BJ) is equivalent to = djNf (see (2.8)).
432 Hrimiuc and Shimada Let us prove the uniqueness of the Finsler connection which satisfies BJ) - BJ). If ST = (Hik,N^V/k) is another connection, then it must have the following form: Br=(dkNij)Nij,0). Taking the lift of this connection on ¿7*, we get a normal ¿-connection (d'Nij, dk Nij, 0, 0), and ((dfc JVij)0, dkify 0, gkh dh9ij\ is the ¿-dual of this connection, where (jV£) is the ¿-dual of (№y). This ¿-connection provides a Finsler connection of U Br==(^,j\y,o) which has the following properties: FK = 0, Pj-O, P^-0, cr^ = o. These conditions are sufficient to assure the uniqueness of Finsler connection of U. Now, we can easily prove the uniqueness. □ Let us put ©A :=aw^. Berwald connection for Cartan spaces has the following, generally nonvan¬ ishing curvature tensors: - II G^j + G^Gfk} and a torsion tensor Ajfe = Proposition 3.5.1. The following relations hold good: (i) Hfi*jk = (ii) PiHhjk ~ ~"Rhjk' Proposition 3.5.2. Let (M,F) &e a Cartan space and Shzjkt Hhfjk the /¿-curvature tensors of BY and of Berwald connection of its (locally) associated Finsler spacCf respectively. Then we have in £7* Swk = - 2Vih*Rsjki (5.2) where (HihjkY means that the value of Hihjk calculated in (rc,p), y* = ¿^(^p). (Hhijk =9isSh*jk, Hihjk = dhsKfjk)- Let us denote by the h-covariant derivative with respect to Berwald connection BY=(diNijiNij,Q). (/¿-curvature), (5-1) (hv-curvature)
Symplectic Transformation 433 Theorem 3.5.2. Let CT and BT be the Cartan and Berwald connections of respectively. Then (i)GÀ=fi}fc-^[10, (ii)S*tf^ = -2Vfc%. (5.3) Proof: (i) Let us restrict our considerations to the open sets U\U such that Legendre transformation : U* —> U is a diffeomorphism. If we consider the ¿-dual of Cartan connection CT, CT = we have in U P9^C*W ([30], page 114). Taking the ¿-dual of this relation, we get ft? = ft?||0 - &Nit = V^uo, where VSJfc||0 = K?ny>\ pA = g*hTPr- (ii) We have > = Îi+ Gékg*^ + = <Tij||fc - = -2Vfc>. Definition 3.5.1. A Cartan space is called a Landsberg space if V**||q = 0. It is called a Berwald space if W*||ji = 0 = g*uVFk = — ? ¿V**, = Using the ¿-duality between Cartan and Finsler spaces, we can easily prove: Proposition 315.3. A Cartan space is a Landsberg (Berwald) space if and only if every associated Finsler space is locally Landsberg (Berwald) space. The following theorems characterize Cartan spaces which are Landsberg and Berwald spaces. Theorem 3.5.3. A Cartan space is a Landsberg space if and only if one of the following conditions holds: (a) H^k = Ô/fe, (b) P?/ = 0, (c) = 0. Proof: Using the ¿-duality, we get W*||0 — 0 4=> (c) <=> (b). Then (a) 4=^ (b) follows from (5.3). Theorem 5.4. A Cartan space is a Berwald space if and only if one of the following conditions is true: (a) <5/^ = 0,
434 Hrimiuc and Shimada (b) BY is a linear connection (that is Gfk are functions of position only), (c) are functions of position only. Proof: Obviously (a) <=> (b). Now let us prove vijA|[A = O<=^(5Afc = o. The associated Finsler space (17, F) is Berwald space because of Cijk\h = 0 (¿-dual of W*||* — 0), therefore the coefficients of Cartan connection are functions of position only. From = F]k(x,y), y* = we obtain also that are functions of position only. Now’, using (5.3) and again V^kno = 0, we get (a). Conversely, we obtain Gjlkh = 0, and (5.3) yields Taking the ¿-dual of this equation, we obtain dh$jk - jJc||o) = 0 or Ffkh - & jHO-h = 0, where F/kh is the Tw-curvature tensor of the Rund connection of (U, F). From this relation we obtain Cijk\h ~ following the same way as in [30], page 161, and by ¿-dualization we get V**fc||h = 0. Finally, from the above considerations we can easily prove (a) <=* (c). Definition 3.5.2. A Cartan space (M, F) is called locally Minkowski space if there exists a covering of coordinate neighborhoods in which depends on Pk only. Proposition 3.5.4. A Cartan space (M, F) is a locally Minkowski space if and only if every locally associated Finsler space is locally Minkowski space. The following result characterizes the Cartan spaces which are locally Minkowski spaces. Theorem 3.5.5. A Cartan space is locally Minkowski space if and only if one of the following conditions holds: (i)S^* = 0, 6^ = 0, (¡1)^ = 0, V^||fc = 0. Proof: If (M,F) is locally Minkowski space, then Ny = 0 and Slg^k — 0» Therefore (i) and (ii) hold.
Symplectic Transformation 435 If (ii) is true, using the ¿-duality we can easily prove that (M, F) is locally Minkowski space (see also [47]). If (i) is true, Gh'jk = 0 yields that (M, F) is Berwald space and thus Vijk\\h = 0. On the other hand, 8hzjk = 0 and from (ii) and Proposition 3.5.1, it follows Rhjk = 0. • _ In the same time Gf^ = Hjk holds and therefore, we get RtSjk = S^jk — 0 from (5.1). Definition 3.5.3. a) Cartan space (M, F) is said to be of scalar curvature if there exists a scalar function K = K(x,p) such that Hkijkp^XhXk = K{jg*hjg;k - gUg^p^XW (5.4) for every (x,p) G D* and X — (X’) € TXM. b) A Cartan space (M,F) of scalar curvature is said to be of constant curvature K if the scalar function from a) is constant From Proposition 3.5.1, (ii) and (5.4) we easily obtain that (M, F) is of constant curvature K if and only if RijkP^ ~ K F2^, where h*ik — g^—^Pipk is the angular metric tensor of Cartan spaces. Theorem 3.5.6 (i) A Cartan space is of scalar curvature K(xtp) if and only if every associated Finsler space is of scalar curvature y1 = é*H(æ,p). (ii) A Cartan space is of constant curvature K if and only if every associated Finsler space is of constant curvature K. Proof: Contracting (7.2) by p’,p*,X\Xfc, we get 8hijkpi^xhxk = -(XiwYp'pWx* or &WaWXkXh = -(HihjkyitfXhXk'}\ (5.5) (Here the ¿-dual of X = X^x)^ is X° — X*(a?)5i.) But a Finsler space is said to be of scalar curvature K(X,y) if Hi^jky^X^^X1^ = K^g^k — 9ik9hj')yiyjXhXit and if K(x,y) — const., it is said to be of constant K (see [30], page 167). Now, using (5.5), we obtain the proof. Remark 3.5.1. We can get some similar results as in Proposition 3.5.3, Pro¬ position 3.5.4 and Theorem 3.5.6 for a Finsler space, considering Cartan spaces, locally associated to it. Therefore, some nice results in Finsler space can be obtained as the ¿-dual of those from Cartan spaces. 3.6 Applications of the ¿-Duality In this section, we shall give some applications of the ¿-duality between Finsler and Cartan spaces.
436 Hrimiuc and Shimada In terms of the Cartan connection a Landsberg space is a Finsler space such that the hv-curvature tensor P^jk ~ 0 [30]. A Cartan space is called Landsberg if P’Vj* = 0. Using the ¿-duality it is clear that a Finsler space is a Landsberg space iff its ¿-dual is a Landsberg one. In [53] (see also [31], [28]) was proved that a Randers space is a Landsberg space iff &ia = 0 (here stands for covariant derivative with respect to Levi- Civita connection of the Riemannian manifold (M,o^)). For Kropina spaces we have a “dual” of the above result: Theorem 3.6.1. A Kropina space is a Landsberg space if and only if bitk = bifk - bkfi + aikfibj, fta*/,. (6.1) Proof: The Randers metric F&,p) = y/o^PtPj ± Fpi is a Landsberg metric iff (here “H” stands for covariant derivative with respect to the Levi-CiviU con¬ nection of the Riemannian manifold (Af, a^)). Hence, we have the following equivalent statements: The Kropina space is Landsberg <=> its ¿-dual is Landsberg But {*&} - {!k} - - ajko^f«}, where fk := dfc(log6) and { {jfc} are the coefficients of the Levi-Civit& connection of dijt respectively &ij (a>ij = prOij therefore dij and Oij are conformal metrics). So we get bi-k = bifk - bkfi + Oikfjbj, fj := a3*fi which was obtained in [5], [28], [31] by a different argument. We can obtain other properties of Kropina spaces from those of Randers by using the ¿-duality. Theorem 3.6.2. i) Kropina space is a Berwald space if and only if ^7kbi — bifk — bkfi H- Q'ikfibj. (6.2)
Symplectic Transformation 437 ii) Kropina space is locally Minkowski space if and only if the condition (6.2) above holds and also Rihjk — J J {aiikfij + Oijfhk + fmQ‘ikQ'hj}> (6.3) UM where /* := ¿^(logd), f* :® aikfky fij = ^ifj+fifj> Vk stands for the covariant Q derivative with respect to Levi-Cività connection of (M,a»j) and Rihjk is the Riemannian curvature tensor. First of all we need the following Lemma 3.6.1. Let (M,F) be Carton space with Renders metric F(x,p) = yja^pipj + tfpi. Then (i) (M, F} is a Berwald space if and only if V^bi — 0, (ii) (M, F} is Minkowski space if and only if — 0, Rfjk = 0, (6-4) _ o where Vfc stands for Levi-Cività connection of (M^j) and R is Riemannian curvature tensor. Proof: The proof of this Lemma follows step by step the ideas of Kikuchi [28]. For example, here, to obtain that (M, F) is Berwald space on the condition — 0, we use Theorem 3.4.3 and prove that the Cartan connection of this space is CT = ( {j^}, {j]ç} where are the coefficients of the Levi-Cività connection of ôÿ. Proof of the Theorem 3.6.2: We take the ¿-dual of the Kropina metric (4.14) and we get the Hamiltonian (4.18). We have âij = = e2<T%-, a := log2 - logô. Therefore, Riemannian manifolds (M,a^) and (Af,ô#) are conformal and the coefficients of Levi-Cività connections are related as follows: The condition (6.4) is written as (6.2). Also, for the conformal metrics, we have o o Rikjk = Rihjk - sjffik + - OikCrej) +(i*ay - tfatk)amn<7m<Tn-
438 Hrimiuc and Shimada For our position cri == -fa crij = -'Vifj - fifj — —fij and using the above equality, we get (6.3). Finally, we apply the results of the previous section. Remark 3.6.1. (i) The results of Theorem 3.6.1 were obtained by Kikuchi [27] in a different form and also by Matsumoto ([31], [32]) in this form, by using of totally different ideas (see also Shibata [52]). (ii) A Finsler space with a Kropina metric is Berwald space if and only if (6.2) is true. Indeed, it is easily checked that the Cartan space with Randers metric (4.15) is a Berwald space if and only if Va&i = 0 (see [53], [5S] for Finsler spaces with Randers metric). Therefore, we follow the same idea as in Theorem 6.1. Let us now give another example of using of ¿-duality. If (M, F) is Finsler space with P(x,y) = + • • • + On(®)(yn)m}1/’" (m-th toot metric [3],), its ¿-dual is Cartan space having fundamental function F(x,p) = [ (P1/ + --- + 1 (ante)/“* 1 (atCr)/-1 where | = 1. In particular, if ai(a;) = • ■ • = a* (a) = ^(®) — + • ♦ ♦ + (<l>i = const.), we get the ecological metric of Antonelli ([3], [20]): F(x, y) = e^W)"1 + * • * + (j,"r and its ¿-dual is F(x,y) = + • • • + (p„/}^. The geodesics of (M, F), parameterized by the arclength, are just the ecological equations (see [3]). The ¿-duals of these equations have a simpler form: <6'5’ (Hamilton equations), where H = % F2. The solutions of (6.5) are h-paths of (M,F) [20].
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PARTS
Holonomy Structures in Finsler Geometry L. Kozina
Contents 1 Holonomy of Positively Homogeneous Connections 453 1.1 Connections of a Tangent Bundle 453 1.2 Holonomy Group of a Positively Homogeneous Connection .... 454 1.3 Curvature and Holonomy Algebra of a Positively Homogeneous Connection 455 1.4 Homogeneous Holonomy of Finsler Manifolds 457 1.5 Metrizability of Positively Homogeneous Connections 458 2 Holonomies of Finsler V—Connections 463 2.1 A Topological Group and Its Lie Algebra 463 2.2 V-Connections 464 2.3 The V-Holonomy Group and V-Holonomy Algebra 465 3 Holonomies of the Finsler Vector Bundle 469 3.1 Linear Connections of the Finsler Vector Bundle 469 3.2 Osculation of Finsler Pair Connections 470 3.3 hv-Holonomy Groups of the Finsler Vector Bundle 472 3.4 The Mixed Holonomy Groups 473 3.4.1 Invariant distributions of parallel transport 474 3.4.2 Decomposition and reducibility of mixed holonomies . . . 475 4 Holonomies of Special Finsler Manifolds 477 4.1 Berwald Manifolds 477 4.1.1 The characterization of the Berwald connections 478 4.1.2 The generalized de Rham’s decomposition theory 479 4.2 Landsberg Manifolds 481 447
Introduction In this report we intend to collect results concerning the holonomy concept in Finsler geometry. The presentation is based mostly on the works [9, 15, 28, 29, 30, 34, 36, 43]. Other papers dealing with the holonomies of Finslerian structures are listed in the References. Most of the statements herein will be given with proofs. In Riemannian geometry the holonomy groups, which consist, roughly speak¬ ing, of all parallel translations along loops at a point, measure the non-flatness of the structure. They are compact Lie subgroups of the Orthogonal linear groups, and their Lie algebras are spanned by P^-i o R(P^(ti),Pv(v)) involving the curvature mappings Ru,v for all vectors u and v and the parallel translations P# along curves In the Finslerian setting one can use several generalizations of the Levi-Civiti parallel translation, and consequently, several types of holonomy groups can be derived in a Finsler manifold. It is not clear, as yet, which is the most adequate. One approach is to work on the base manifold, and then the connections of the tangent bundle are fundamental tools. Since, in a general Finsler manifold there does not exist a metrical and linear connection, one of the assumptions should be weakened. So we arrive at the notion of positively homogeneous con¬ nection, also called a nonlinear connection in the literature. A detailed study on the holonomy group of positively homogeneous connections was initiated by W. Barthel in [9]. These are, however, not linear groups. They are subgroups of an infinite dimensional diffeomorphism group of positively homogeneous trans¬ formations. For a Finsler manifold, the holonomy group of the canonical pos¬ itively homogeneous connection is a subgroup of the norm-preserving posit¬ ively homogeneous diffeomorphism group. This is described in the main part of Chapter 1. In the last part of the chapter the metrizability problem of positively homogeneous connections is discussed, citing the works [28] and [34]. Another approach in connection theory of Finsler manifolds is when linear connections are considered in the Finsler vector bundle (called the support ele¬ ment bundle, in the classical literature), which can be a pull back bundle over the tangent space, the vertical bundle, or the second tangent bundle. In this case the covariant derivation has the usual property, but all quantities depend on the tangent vectors (classically, on the support elements). This is motivated by the fact that the metrical fundamental tensor of a Finsler manifold is defined on the space of support elements. Chapter 2 presents the holonomy theory of 449
450 Kozina V-connections which is directly associated to the positively homogeneous con¬ nections, based upon the work of T. Okada ([36]). This study applies the theory of principal bundles and principal connections. In Chapter 4 several types of holonomies are introduced and studied in the Finsler vector bundle. We apply here our method of osculation when a Finsler pair connection is approximated by a linear connection along a fixed vector field ([28]). This construction is similar to that of M. Matsumoto ([33]) called Y- linear connection. The osculation is used to establish a relationship between the holonomies of the Finsler vector bundle and those of positively homogeneous connections in the base manifolds. The mixed holonomy groups of Grangier ([IS]) seems to be too restrictive, for the reducibility of the mixed holonomy group implies the reduction of the Finsler manifold to a Riemannian structure. The last chapter is devoted to special Finsler manifolds. The first class, Berwald manifolds admit linear metrical connections, and it enables one to apply the sophisticated theory of linear holonomy. Szab<5 ([43]) utilized this for classification of all positive definite Berwald spaces. In the case of the more general class of Finsler manifolds, called Landsberg manifolds, the holonomy group is a finite dimensional Lie group but not a linear group. This result is based upon a theorem of Ichijyo ([23]). Now we recall, more precisely, some classical results about the holonomies of Riemannian manifolds. This part is quoted mainly from Bryant’s work ([12]). Let M be a connected and simply connected manifold, and let g be a Rieman¬ nian metric on M. Associated to g is the notion of parallel transport along curves. Thus, for each (piecewise C1) curve <p : [0,1] —► M, there is associated a linear mapping P^ : —> T^M, called parallel transport along which is an isometry of vector spaces and which satisfies the conditions P$ — and =» Pps 0 where is the path defined by <p(t) — ^(1 — t) and <£>2<Pi is defined only when <£i(l) = 9^(0) and, in this case, is given by the formula <P2<P1 W = <P2(2t- 1) for 0 < t < for i < t < 1. 2 These properties imply that, for any x 6 Mt the set of linear transformations of the form P^ where p(0) — <p(l) = x is a subgroup Hol* C O(TXM) and that, for any other point y € M, we have Hol^ = P^HolxFp where <p : [0,1] —► M satisfies 9?(0) = x and ^(1) — y, Assuming that Af is simply connected, Hol* is a connected and closed subgroup of SO(T®M). Theorem 0.0.1. (Ambrose and Singer, 1953, [6]). The holonomy algebra is spanned by all elements of the form P^-i o ¿(P^u), Pv(v)) involving the curvature mappings RuyV for all vectors fields u and v at x and the parallel translations P^ along curves <p(Q) — x. Theorem 0.0.2. (de Rham, 1952, [40]). If there is a splitting TXM = Vi 6 V2 which remains invariant under all the action o/Holx, then the metric g is locally
Holonomy Structures in Finsler Geometry 451 a product metric in the following sense: The metric g can be written as a sum of the form g = 51+52 m such a way that, for every point y<zM there exists a neighborhood U of y, a coordinate chart (31,2:2) : U —► x and metrics gi on so that gi = (&). In this reducible case the holonomy group Hol® is a direct product of the form Hol* x Hol£ where Holj. C SO(Vi). Moreover, for each of the factor groups Ho4? there is a submanifold Mi C M so that TxMi = Vi and so that Holj, is the holonomy of the Riemannian metric gi on Mi. From this discussion it follows that, in order to know which subgroups of SO(n) can occur as holonomy groups of simply connected Riemannian mani¬ folds, it is enough to find the ones which, in addition, act irreducibly on &n. Using a great deal of machinery from the theory of representations of Lie groups, M. Berger [10] determined a relatively short list of possibilities for irreducible Riemannian holonomy groups. See also the works of Alekseevski, Brown and Gray ([5], [11]). Theorem 0.0.3. (Berger, 1955, [10]). Suppose that g is a Riemannian metric on a connected and simply connected n-manifold M and that the holonomy group Holx acts irreducibly on TXM for some (and hence every) x G M. Then either (M,g) is locally isometric to an irreducible Riemannian symmetric space or else there is an isometry b : TXM —► so that H = ¿Holz6_1 is one of the subgroups of SO (n) in the following table. Subgroup Conditions Geometrical Type SO(n) U(m) SU(m) Sp(m)Sp(l) Sp(m) G2 Spia(7) any n n — 2m>2 n — 2m > 2 n = 4m > 4 n = 4m > 4 n = 7 n = S generic metric Kahler Ricci-flat Kahler Quaternionic Kahler Hyperkahler Associative Cayley
Chapter 1 Holonomy of Positively Homogeneous Connections In this chapter we give first the basic notions of connection theory: horizontal map and subspaces, parallelism and covariant derivation. Our main goal is to define and study the holonomy group of a positively homogeneous connection. This was done by W. Barthel ([9]). We present his main results, especially the partial extension of the Ambrose-Singer theorem which relates the curvature structure and the holonomy algebra of a positively homogeneous connection. The last section concerns the role of the holonomies in the metrizability problem for a positively homogeneous connection. 1.1 Connections of a Tangent Bundle By a connection of tm we mean a splitting H: 7f*(ta/) —> ttm of the short exact sequence 0 —> Vtm ?TAf 0 where div: ttm —► is given by c?7t(A) = (7rTjw(A),d7r(A)) for A G TTM. H is also called a horizontal map, and its images HZTM — ImP|^jxrirU)Ai are the horizontal subspaces which are complementary to the vertical subspaces: TTAi = Vtat G Pnf, The projections v and h adapted to this direct decom¬ position will be called the vertical and horizontal projections, respectively. Let ip: I —► M a curve in the base space. A vector field Y € X(M) is called parallel along <p if dY (ip) are horizontal vectors, where ip denotes the tangent curve of tp. This means that H(Y o<p,(p) = dY(cp). From elementary calculus it follows that for a given curve <p and a vector z in T^(o)M there exists — at least locally — a vector field Y which is parallel along ip and z = Y(y>(0)). Y(y?(t)) is called the parallel translation of z along ip, and denoted by z). The covariant derivation V: X(M) xX(M) —► X(M) is given as Vx Y = 453
454 Kozina a(v(dZ(X))) for all X, Y -G X(M) where a « pr2 o : Vtm —► tm. It is plain that Y(t) is parallel along <p if and only if V0Z(t) = 0. Denote ¡¿t : TM —> TM the multiplication by t € R in the fibres of tm’• It is said that a horizontal map satisfies the positively homogeneity condition when holds for all z € TM, v G TM and t € R. If the differentiability of If is not assumed at the zero vectors of tm, and H satisfies the positively homogeneity condition, we speak of a positively homogeneous connection (nonlinear connec¬ tion). When H satisfies the positively homogeneity condition and differentiable anywhere, then we get a linear connection. Homogeneous connections arise naturally in Finsler geometry. In fact, in general it is not possible to introduce a linear connection in tm compatible (metrical) with the Finslerian metric, only a positively homogeneous one. In what follows we consider a manifold M of class C7*, and a positively, homogeneous connection on it. Definition 1.1.1. Let tp : [0,1] —+ M be an oriented curve, and 0 : Z —* TM a curve in TM over <p. $ is called a parallel vector field along y? is 0 is horizontal, i.e. its tangent vectors 0 belong to the horizontal subspaces. Then, we say that 0(1) is the parallel transport of 0(0) along 9?. In [9] Barthel showed that for any curve <p : [0,1] -+ M and zq G Tv^M there exists a unique parallel vector field 0 : Z —► TM along the entire curve <p with initial vector zq — 0(0). Then the mapping zo 0(1) is called the parallel transport Pv : Tv(o)M —► T^M along <p- The parallel transport P^ of a positively homogeneous connection is a bijective positively homogeneous map: P^tz) = iPv(z). Considering a local coordinate system (z,y) on tt~ 1 (U), a positively homo¬ geneous connection has 2-index parameters, by which the horizontal subspaces HZTM can be generated: These local parameters are positively 1-homogeneous: Nl{xyty) = tN^(xyy). Then, the covariant derivations of a vector field Y = Yi^7 G X(7r“x(ZZ) in the direction u = ul~ € TPM is given as follows: ^Y=ui 1.2 Holonomy Group of a Positively Homogen¬ eous Connection The holonomy group for positively homogeneous connections is defined as usual for linear connections: It is the group at the point x generated by the par¬ allel translations along all loops at x. This is not a subgroup of the linear
Holonomy Structures in Finsler Geometry 455 group but gives a subgroup of the group of all invertible positively homogen¬ eous differentiable map of the fibre. In general it is neither infinite dimensional diffeomorphism group nor Lie group. In the last chapter we show a class of special Finsler manifolds where the holonomy groups of its canonical positively homogeneous connection are compact Lie groups (see Theorem 4.2.4). The holonomy groups of the positively homogeneous connection H, denoted by Hol£ at x € M is given as follows: Hol£ = : TXM -+ TXM | <p: [0,1] M,^(0) =?(1) = x} It was defined and investigated in details by W. Barthel ([9] ,1963). The holonomy groups at different points of a connected manifold are isomorphic, since they have the usual relationship: where joins the points and xq. We consider all positively homogenous mappings of the tangent fibre TXM: : ZL-M TXM | $ is positively homogeneous: &(Xv) = A#(v), A > 0} This has a natural structure of Banach vector space where the norm is given by 11M = sup uj^Q,z^Q (it is independent from an arbitrarily chosen norm |.| in TXM). The parallel transports along all loops at x € M are bijective positively homogeneous mappings of the tangent fibre, therefore Hol£ is not a linear group except in the case of Berwald spaces (see Chapter 5), however, it is a subgroup of the locally Banach topological group consisting of positively homogeneous bijective (7°° mappings on TXM \ {0}. 1.3 Curvature and Holonomy Algebra of a Pos¬ itively Homogeneous Connection For two positively homogeneous mappings A, B : TXM —► TXM their Lie bracket [A, B] : TXM -+ TXM is defined as follows: (AB](v) = dBv(A(y)) -dAv(B(vï). The covariant derivative of a field A : M —> SS(M) of positively homogeneous mappings is defined at any v € TXM: (yvA)(u) where U e X(M) is a parallel vector field with starting vector u = tZ(7r(v)) in the direction v: VvU = 0. Then VvX : TXM -*• TXM is also positively homogeneous.
456 Kozina In local coordinates, with the notations v = , u — and A(tt) — Al(u)^r we obtain easily (V0A)(u) = ? - ^x,u)N{(x,u) + Nik(x,Ak(u)^ Proposition 1.3.1. Let 93 (M) be two fields of positively homo¬ geneous mappings, and f € C°°(M). Then it holds for any v € M: (1) Vv(fA) = v(/)A + /VvA, (2) Vv[A,B] = [VvB, A] + [B,V„A]. For the proof we refer to [9] (Satz 10). The curvature of a positively homogeneous connection is the mapping X(M) x X(M) X(M) defined as follows: R(X9Y)Z = a(vz[X^Y^ where X^,vz,ot denote the horizontal lift of X € 3C(M) to Z, the vertical projection at Z, and the natural projection VTM —► TM identifying the tangent space of the tangent vector fibre with the fibre itself. Clearly, R is linear and skew-symmetric in X and Yf but not linear in Z, only positively homogeneous in general. It local expression the curvature is given as follows: «^¿^¿>(«*¿1 “ z> - V Proposition 1.3.2. (Ricci identity). Let <p : (-a, a) x (—a, a) —> M be a surface element with <p(0,0) = x, di^(0,0) — u, $2^(0,0) — v, and A : (—a, a) x (—a, a) —> 93 (M) a field of positively homogeneous mappings over the surface element <p. Then Vv V«A(0,0) - VwA(0,0) == [A(0,0), R(u, v)] holds. See the proof in [9] (Satz 12). Definition 1.3.3. Let t£1} = Tid(Hol*) C and for k > 1: ¿*+1) = Then is called the holonomy algebra of the positively homogeneous connection. It is easy to see that fjolg. is a subspace of 93x, and t* C f)ol£ C . Further¬ more, the holonomy algebra l)o£, equipped with the Lie bracket of positively homogeneous mappings above, provides a Lie algebra structure.
Holonomy Structures in Finsler Geometry 457 Proposition 1.3.4. (Barthel, 1963, [9]). For any two non-zero vectors u.v e TXM the mapping Ru,v : z € TXM —> j?(u, v)z e TXM satisfies: Rw € . Proof: Consider a surface element of class C°° ¥: (-a, a) x (-a, a) M>, (a, <p(a, p) lying entirely in a normal neighborhood of the point x e M. Suppose y?(0,0) = x, and u = di<p(0,0), v = ft>^(0,0). Denote the parameter lines by a<p : t € [0,/?] -+ M;t i—► and : * € [0, a] —> w ¥>(<,/?). Then the composition of parallel translations: C(a,£) == P~£ o p~£ o P^ o P^ belongs to Hol£. Now it can be seen that C(a,£) depends smoothly on a and /?. Fixing a, we obtain a curve Ca in Hol£. Its tangent vector at 0 gives an element in t* for any a € (—a, a). Differentiating again, the tangent of at a = 0 gives an element in t~. Lengthy but routine calculation shows that Qcr^(O) — Riv^v), (See for details [9].) □ Theorem 1.3.5. (Barthel, 1963, [9]). For an arbitrary Ui,tt2, • • • G TXM let us construct the mappings h^....uli-TxM^TsM by induction: &) “ R(Ui)U2) b) k > 2. Consider Xi(t),... ,Xk-i(t) parallel vector fields along <p(t) with Xi (0) = t*i, • • • > 1 (0) = Ufc-i. Let Then the vector space rx spanned by all mappings h^1tt,tUh gives a Lie subalgebra of the holonomy algebra For the proof we refer to [9]. 1.4 Homogeneous Holonomy of Finsler Manifolds A function L: TM —> Ris called a Finsler fundamental function in a tangent bundle tm if
45S Kozma 1. L(u)>0 2. L{Xu) = XL(u) VA € R+w € TM 3. L2 is smooth except on the zero section 4. Pv(®,3/) = ' 18 P°sitive definit€ for any (X'V) °- A manifold M endowed with a Finsler fundamental function L is said to be a Finsler manifold. The last assumption implies that the indicatrix ir-{z€TxM = 7r“1(a;) | L(z) = 1} at each point x G M is convex (see [41]). The indicatrix bundle 3tm = (ITM, if, M, S’71-1) is formed with the indicatrices Ix as fibres. Roughly speak¬ ing, a Finsler fundamental function Lp at p G M gives a positively homogeneous norm in the tangent space TPM, There is a canonical positively homogeneous connection [8], called here as Barthel connection given as follows • y) = 'ilj{x,y)yk - ^r(.x,y)^(.x,y)^(x,y)ykyl where (7^(s,yf) are the Christofiel symbols of gij(x,y) with respect to x. In a coordinate-free manner, this connection was constructed by Grifone ([19]), see also in [45]. In that terminology the fundamental lemma of Finsler geometry states that in a Finsler manifold there exists a unique conservative (i.e. metrical), homogeneous and torsion free connection. For a Finsler manifold (M, L) the parallel translation of the Barthel posit¬ ively homogeneous connection HB preserves the length of vectors, i.e. the pos¬ itively homogeneous holonomy group Hol£ is a subgroup of the norm-preserving positively homogeneous bijective transformations {$: TXM -► TXM | pos.hom., L o 0 = L, C^onT^M \ {0}} 1.5 Metrizability of Positively Homogeneous Connections Definition 1.5.1. A positively homogeneous connection is called metrizMe if there is a Finsler fundamental function such that the length of translated vectors along any curve remains constant. By formulae, the metrizability of a positively homogeneous connection means that L o Fp — const.
Holonomy Structures in Finsler Geometry 459 Furthermore, using the covariant derivation of L, it is equivalent to the condi¬ tion: VjtL dxk kdyi = o. We remark that this notion is weaker than that of a variational spray (or variational connection). See [25]. It is known that if a linear connection has a relatively compact holonomy group then it is the Levi Civita connection of some Riemannian connection. In the positively homogeneous case the Barthel connection plays the same role instead of the Levi Civita connection. Theorem 1.5.2. (Kozma, 2000, [30]) . If the holonomy groups of positively homogeneous connection given on an arcwise connected manifold are compact Lie groups then the connection is metrizable. Proof: We use the method of Z.I. Szabd given for the linear case ([43]). In the first step we construct an invariant norm at the point x € M using the Haar biinvariant measure of the holonomy group: choose an arbitrary norm and take the average of all translations of the specified vector: £(*) = f L0(h(z»d^h). AeHoL It is easy to check that we get an invariant positively homogeneous norm at x: W)) = [ „ 5 Lo(h(h(z)))d/x(h) = f L^h(z))dKh) = L(zy JheHol* Jh=hheB.ola In the second step we extend this norm to the whole manifold: where y?-joins t(z) and x. We show that this extension is independent of the choice of the joining curve. If r also joins ?r(z) and x, then r<p is a loop at x € M, so PT$ € Holx, therefore £(*(*)) = L(PT(P9(Pv(z)M = L(PT^Pv(z)m = I(F„(z)), for L is invariant with respect to Holx. □ The question of metrizability has been investigated in several aspects. Namely, a spray (a path space) is called metrizable if the paths of the spray are just the geodesics of some metric space (Riemannian or Finsler space). Then the spray is called variational. On the other hand, a positively homogeneous connection is called metrizable if there is a Finsler fundamental function such that the con¬ nection is metric, i.e. length-preserving [34]. We adopt here this notion. It is open whether the spray of the connection is variational in this case.
460 Kozina The next two results were proved by Matsumoto and Tamassy ([34]). A vector y G TXM is called an eigenvector of the holonomy group Hol*, if there exists a mapping h G Hol* such that h(y) “ Xy, (A > 0), and A is called an eigenvalue of Hol*. If H is metrizable, we see L(x, h(y)) = Lfay), that is L(ir}A3/) — L(xfy), which implies A = 1 by the homogeneity assumption of L. Therefore for the metrizability of a positively homogeneous connection H it is necessary that its holonomy groups has no eigenvalue which is not equal to one. We denote the ray through the vector y 6 M by and the set of rays at a point x € M by The holonomy group Hol* acts on TxAf, and on T*M/R+ as well. For any y G the set P<?(y} C TM means here all the vectors of all the horizontal lifts of curves starting from x G M. Theorem 1.5-3. (Matsumoto and Tamassy, 1980, [34]). If the holonomy group of a two dimensioned positively homogeneous connection H on a connected dif¬ ferentiable manifold has no eigenvalue which is not equal to one and it is trans¬ itive on then H is metrizable in a unique way within homothetic transformations. Proof: The transitivity of the holonomy group means Hol*(?/o^+) — TXM/W~. Therefore Holx(2/o)ri2/IR+ 0 0 for any y G TXM. But Hol*(j/o)nj/R4' cannot con¬ sists of more than one vector, for this would mean the existence of an eigenvalue which is not equal to one. Then we define a function L on M as follows: Let L be positive and constant on each P^y). If £ £ P<p(ya) then y £ Hol* (2/0), however according to the transitivity of the holonomy group n Hol* (2/0) / 0, and there exists a unique A > 0 such that Xy G Hol* (2/0 )< On the other hand the ho¬ mogeneity of the positively homogeneous connection implies P^Xy) = XP^y). Therefore, if we define L(P^(yo)) = co > 0 and require positively homogeneity of L : LtP^Xyf) = XL(P<p(y)), then L is well defined, positively homogen¬ eous and positive on M. Moreover, Pp(y) — c holds for any y G M, thus the space is metrizable. If the value L(xo, 2/0) is changed for an other value, it gives rise to a homothetic transformation of L, and these prove the assertion of the theorem. □ Theorem 1.5.4. (Matsumoto and Tamassy, 1980, [34]). If a positively homo¬ geneous connection H of a 2-dimensional connected manifold is metrizable and there exists an arc E of fix vectors ofTXoM/R+, then H is metrizable in many ways. Proof: Let L be a metric function on M. Then L(P^(?/o)) is constant for every 2/o? and L[x$,y) = 1 is an indicatrix curve 1* at zq. Let us deform I in TXQM into another differentiable curve J* such that I* is met by any ray s/R+ in exactly one vector again, and such that I remains unaltered at the intersection with the rays belonging to the complementary arc S — (T*0M/R+) \ E of E. We define an L* on T*0M such tha£L*(2/) - 1 if y G I* and ¿’(At/) = XL* (2/) (A> 0), and we extend L* over M such that L* is positively homogeneous on M and L* is constant on every P<p(yfi Thus L* is well defined on M, it is positive, and
Holonomy Structures in Finsler Geometry 461 the last two conditions are compatible since S consists of fix vectors and since Z* coincides with I on S. Then H is metrizable with respect to L. Thus L* is a new metric function on M different of L. Since the above deformation may be performed in many different ways, so we get many metrizations of H, □
Chapter 2 Holonomies of Finsler V—Connections This chapter sketches the principal bundle approach of holonomy structures for Finsler connections, based on the work of T. Okada ([36]). First a special topological group and its Lie algebra are defined whose subgroups and subspaces may represent the holonomy groups and algebras of special type of Finslerian connections, called V-connections. These connections are closely related to positively homogeneous connections of a Finsler manifold, for this relationship see [32, §8]. 2.1 A Topological Group and Its Lie Algebra Let Vn be a vector space of dimension n except the zero vector and GL(n, R) be the general linear group of dimension n, We shall denote by G the set of all <7°° mappings a : Vn GL(n,R) which satisfy the following conditions: (1) a(t>) is positively homogeneous of degree 0 with respect to v € Vn, i.e. for any positive number A, a(Av) = a(u). (2) For any v € Vn, a(t/)t/ = v has a unique solution t/ € Vn, i.e. the mapping a(*)* : Vn —► Vn is injective. Then the set G becomes a local Eréchet topological group, whose group structure and distance are defined as follows: (1) The multiplication aob for a,b € G is defined by (ao&)(v) — a(6(v)v)6(b’). (2) The unit element e of G is defined as a mapping whose values are con¬ stantly the unit matrix. (3) The inverse element a“1 of a e G is defined by a“1^) = a(v')”1, where = v. 463
464 Kozma (4) The distance d(a, ft) of a, b € G is defined by d(a, 6) = ||a — 6||, where put W- = .up IDWcWI, P» = H==i |p| = ¿i + i2 + * * • + In¬ Next we shall define the Lie algebra fl of G> which can be naturally defined by virtue of the Lie algebra gl(n, R) as follows: Let flI(n,R) be the Lie algebra of GL(n,R). The Lie algebra $ of the group G is a linear set spanned by all C°° mappings X : V* —> gl(n, R) which are positively homogeneous of degree 0. The bracket operation ins is defined by [A o B]j(v) = Ai(«)B|(V) - Bfc)Afr) + (dAj/dvk)(y)Bk(v>‘ - (^/avfc)(v)4(v)^ for any A, B € fl. It is easily seen that, by defining the distance in fl similar to (4) above, then fl becomes a Fréchet space and G is an open subset of g with respect to this distance. 2.2 V-Connections Here we sketch the theory of V-connections, for further details see [32]. Let M be a C°° manifold of dimension n and L(M) be the linear frame bundle over M, Definition 2.2.1. A V-connection Vv on M is a family of C°° distributions rM in L(Af) parameterized by v € Vn} which satisfies the following conditions: (1) The tangent space L(M)Z to L(M) at z € L(M) is the direct sum of the tangent space L(M)J to the fiber at z and rlv\ i.e. L(M)- = L(M)J+rLvL (2) holds for any g € GL(n,R), where Rg is the right translar tion of the frame bundle. (3) rW is positively homogeneous of degree 0 with respect to v, i.e. holds for any positive number A.
Holonomy Structures in Finsler Geometry 465 Let Mz be the tangent space to M at x E M and 4^ • be the operation of the horizontal lift with respect to fM. The basic vector field corresponding to u E V is defined by (<zu). We denote the differential mapping of the left transformation L- : GL(n, R) —► L(M); g *-> zg, by the same symbol Lz. Then, for A E LZ(A) is called the fundamental vector field corresponding to A, In terms of the canonical local coordinates (sa,zf) of z E L(M), these vector fields are written as BM(«) = ¿zftd/dx“ - Fdba(x°,z^zjd/dz^ LX(A) = A]z?d/dz^ where v = (v*), u — (i?) and A — (Aj). 2.3 The V—Holonomy Group and V—Holonomy Algebra Let x be an arbitrary point of the manifold M and be a piecewise C°° closed curve at x, If we denote by (z) the result of the parallel displacement along <p with respect to I'M, starting from the point z of the fiber on x} then there is a mapping a{z, ♦) : Vn —► GL(n,R), identified with an element of the group G defined in Section 2.1, such that P^(z) = z • o^(z,v). It is easy to show that ay(zg,g~lv) — ^”1av>(z,v)5 holds for any g E GL(n,R). By the composed parallel displacement with respect to I'M, corresponding to the continued by another closed curve we obtain from z the point z • a^^Zy v), where (z, *) is the multiplication of a^z, *) and a^Zy *) in the group G> Similarly, by the parallel displacement along the inverse closed curve <p of the </>, we obtain from z the point z • a”x(z, v), where a“x(z, *) is the inverse element of a^(z, *) in the group G. Therefore, the set Hol]f of the av(z, *) corresponding to all piecewise C°° closed curves <p at x E M is a subgroup of the group G. Definition 2.3.1. The subgroup HolY of G as above given is called the V- holonomy group at z with respect to the V-connection Tv. As Hol^ = ad(g x) o Hol]f holds for any g E GL(n, R), Hol^ is isomorphic with HolY. Accordingly they are called the holonomy group HolY with reference point x = tt(z), where 7r is the projection of the bundle L(M). For two distinct points x and xf of the connected manifold M, the group Hol^ is isomorphic with Holy/, hence they are briefly called the V-holonomy group of M with respect to If a mapping K : (—1,1) x Vn —► fl((n,R); (t, v) »->■ Kt(v)y is (7°° differenti¬ able and positively homogeneous of degree 0 with respect to v, then Kt is called a C°° curve in jj. The tangent vector K to a (7°° curve Kt at K$ is defined by K(v) = ^|t=s0A?(t)(v). Then lim|| = 0 holds, where the notation || || is defined in Section 2.1. That is, K — |t==0AV holds in the sense of the convergence with respect to the distance of the Lie algebra £j.
466 Kozma Theorem 2.3.2. (Okada, 1973, [36]). LetT* be the set of all tangent vectors at e to every C00 curve through e in the V-holonomy group Hol^, and let Tf*1 be the set of all tangent vectors to Tf. Every 7? (p = 1,2,...) and fjoL — are the linear subsets of the Fr£chet space g, and T} C FjoL C t\ (closure of Tf). Moreover, fjoL is a Lie subalgebra of g. Proof: The proof is divided into three steps. (a) Let us first prove that Ti is a linear subset of g. Suppose A,B G 75, then there exist C°° curves at, bt in Hols, such that oq — bn = e, A = ■% |t=sOat and B = ¿|t=06t. It is observed that atobt and a^t for any real number k are also C°° curves in Hol-, satisfying — I dt u~o (at O &t) d 1 zr = A + B, Ut 11—0 Al dt I t=o Ofct« kA, from which A + B € Tl and kA € are obtained. (b) By induction, we show that every and F)oL are the linear subset of g. Assume that TJ fc a linear subset of g. Take A, B € TZ“1, then there exist <7°° curves At,Bt € T? such that A « and B = By assumption At + Bt and kA* are also the C°° curves in T? and (At + Bt) = A + B( ¿1 (fcAt) = feA. dt\t=G dt\t-c Therefore we obtain A + B € 7'?4’1 and kA € Tp+\ As T$ c Tf-1 is obvious, faL is also the linear subset of g. Since lim|| - a|| = 0 holds for any C°° curve At € T}, we obtain 7? c 7L1, where T3 is the closure of_7j with respect to the norm in g. Similarly, 7^_1 c 7J holds. Assume 7J C T*, then 7T+1 C 7L1 is shown from the last result. (c) To prove that Fjolz is a Lie a subalgebra of g, we show by induction that [7J,7J] C Let a3, bt be two C°° curves in Hz such that ao =; bo — A = sl,=0a* andB = iliio6*- Kwe-Put K„{v) = K|t==0(aaoi>toa71)(v), then we obtain K,{v) = = a.«1(v)f)S(a71(v)v)~1ai,(v) + {(daa/dvjffl-1 (t>)w)B(a71 (v)v)ajx (v)®}a7l (®), which implies that K3 is a C°° curve in T* and Kq — B. Since
Holonomy Structures in Finsler Geometry 467 [X o Bj is an element of 7?. Therefore we obtain [7? o 7L1] c 7?. Suppose 4t be a C°° curve in T? and B G 7^» then A = |t=0 At belongs to 7?"1. Now, if [At o B] G T?p is assumed, then we obtain [A o B] € 7?p+1, because _ sl1.«{x‘WBW - B«'4‘W + - ^(«W«.)»} = A(v)B(v) - B(v)A(u) + ^(v)B(v)v - j^(y)A(y)v = [Ao B](v). That is, if \Tf o 7J] C T:p is assumed, then we obtain [77+1 o TZ] C T?p+1. Similarly we obtain pH’"1'1 o 7*4*1] C Consequently, we proved that frz o ijJ c t)x. □ Definition 2.3.3. The Lie subalgebra i)oL of 9 in Theorem 2.3.2 is called the V-holonomy algebra at 3. As fjoL^ — od(p_1) o fjo[z holds for any g G GL(n,lR), i)oLp is isomorphic with FjoL. Therefore they are called the holonomy algebra reference point x = r-(z).
Chapter 3 Holonomies of the Finsler Vector Bundle The Finsler vector bundle 7t*(tm) is an analogue of the Finsler principal bundle used in M. Matsumoto’s theory for Finsler connections ([32]). Since the metrical I y) fundamental tensor gij^y) = - QyiQ^j ' defined on TM, (^homogeneous, the arena for Finslerian connection theory is here the pull back bundle It is easily seen that the Finsler vector bundle is canonically isomorphic to the vertical bundle Vtm< The first section describes the linear connections of the Finsler vector bundle, the second one offers an osculation method which helps to associate a linear connection of the base manifold to any Finsler pair connection if a fixed vector field is given. This osculation enables us to find relationships between the holonomies of the Finsler vector bundle and the holonomy of the osculating linear connections on the base manifold. Section 4.3 gives the notion of (h,v)-holonomy group where the loops are considered in TM, composed from horizontal and vertical curves. The mixed holonomy group of Diaz and Grangier is slightly different, but its reducibility has a strong implication, namely, the reducibility of the mixed holonomy implies that the structure is Riemannian. 3.1 Linear Connections of the Finsler Vector Bundle Following the ideas of M. Matsumoto given for principal bundles and principal connections, we use a vector bundle approach here. Thus the pull back bundle — (TM xm TMyTMjpriiF) is called now a Finsler vector bundle. Definition 3.1.1. A pair (HF, jff) is called a Finsler pair connection where HF is a connection of tt*(tm)» and H is a positively homogeneous connection of We speak of a linear Finsler pair connection, when HF satisfies the 469
470 Kozina corresponding homogeneity condition, too: for all t G R, Z G TM xm TM^ U G where is the multiplication by t G R in the fibres of 7t*(tm). In the linear case the pair is denoted by Definition 3.1.2. For a Finsler manifold (M,L) a Finsler pair connection (KF, H) is called h-metrical if for any ^-horizontal curve : I —* TM (i.e. — (*" ° ^)) the parallel translation with respect to HF preserves the length of Finsler vector fields: V parallel S e X(S o ip) = const.. It can be expressed in the form dL(fi’/'(Z,Wr)) = 0 VZ G TMW g HTM. QT ^g = 0 V X € where g is the Riemannian metric in the Finsler vector bundle induced by L; g..(X y) - On a Finsler manifold there are known several important Finsler pair con¬ nection such as of Cartan, Berwald, Rund, etc. See for details [32] or [45]. We shall also refer to torsion and curvature tensors. For their definitions and properties see also e.g. [32, pp. 72-74.], or [45, page 39]. 3.2 Osculation of Finsler Pair Connections Let us consider a Finsler pair connection (HF, H) of the Finsler vector bundle ’’■'(■’’m)- Let Y 6 X(Af) be a fixed vector field on M. Then the map .ffy defined as HY(ztv) := is a horizontal map for tm , i.e. a connection in tm> where the map fly: TM —> TM Xm TM is given as fry (z) = (Y (%(£)), s). This connection HY is called as an osculating connection of (tfF, H) along Y 6 X(Af) [28]. It is easy to show that if HF is a linear connection in tt* (tm), then any osculating connection HYalong Y G X(M) is linear as well. In order to investigate relationships between the structures at different levels we define a lift and a projection of some sections. Let Z G X(M) be an arbitrary vector field of tm. The section ZT of tt* (tm) defined as ZT(z) = (z, Z(7r(s))) for all z G TM is called as a lift of Z. Let now
Holonomy Structures in Finsler Geometry 471 Y G X(Af) be a fixed, and E € Secir*(rM) an arbitrary section. Then the map Sy = pr2 o S o Y is called as a projection of S along Y G X(M). The following theorem states that the parallelism of sections is hereditary through lifting and projecting of sections with respect to osculation [28]. Theorem 3.2.1. a) If S € Secir*(rM) is parallel along Y op with respect to HF, then Sy is parallel along p as well with respect to Hy. b) If Z 6 X(Af) is parallel along p with respect to HY, then Z^ G Secit*{rM) is parallel along Y op with respect to HF. Proof: a) First calculate dEy (£). Using the relation S o Y = fly o Sy and the parallelism of S along Y o p, we obtain dSy (<£) = dpr2(dS(dy (<£))) — dpr2(dS(y o ^?)) — = dpr2(HF(E o Y o p, (Y o ¥>)•)) = o sp), dYfcp))) = Considering that Y is parallel along p we can continue as follows = O p), H(Y o v», 0))) = HY(Si- O V,<p). This proves the part a). Before we prove part b) note the following simple corollary. If Z € X(M) is parallel along p with respect to HY and S € Sec7r*(rw) is parallel along Yop with respect to HF> furthermore there exists such a to € I that Sy(^(to)) = Z(^(to))> then Sy op = Z op in some neighborhood of to. b) Let S G Sec7r*(Tjvf) be such a section that is parallel along Y o p with respect to HF, and starts from fly (Z(^(0))). We state that Z' oYop — T^oYop. This will give our assertion b), for the parallelism of a section depends only on the value along the curve. In fact, using the previous corollary Sy o p = Z o p, and considering some simple properties of lifting and sinking we get Z^ o Y o p = fly O z Op fly O Sy o p = S O y o p. This theorem shows us that the notion of osculation could be defined in terms of parallelism. Namely, starting from a parallelism structure PF of the Finsler vector bundle fixing a vector field Y G X(M), an osculation parallelism structure PY can be defined as follows: P% ^przoP^ofiy It can be verified that this type of osculation means the same as the previous one. A similar argument is valid for covariant derivations under the next theorem [28]: Theorem 3.2.2. Denote VF and Vy the covariant derivations belonging to HF and HY, respectively. Then for any X G X(M), Z € X(Af) and S G Sec^ir^)
472 K'ozma (^dr(x)S)y — hold. An analogous relationship is valid for curvature structures through oscula¬ tion. In [28] there are given the relationships for horizontal and vertical projec¬ tions. We regard now the metrical aspects of this osculation method. Theorem 3.2.3. If a Finsler pair connection (£fF, H) is h-metrical with respect to a Riemannian metric g in induced by L, then the osculating linear connection along an absolute parallel vector field Y is metrical with respect to the osculating Riemannian metric gY = go fiy. Conversely, if the (v)h-torsion tensor R} — 0, and for any absolute parallel vector field Y the osculating linear connection HY is metrical with respect to the osculating Riemannian metric gY, then HF is h-metrical. Proof: Take a parallel vector field Z € X(M) with respect to HY along a curve <p in M. It is to be shown that the square of the length gY(Z o Z o is a constant function. By Theorem 3.2,1 Z^ € Sec7T*(rAi) is parallel along Y o <p with respect to HF^ and by the assumption Y o is a horizontal curve in TM, therefore L?(Z^ oYoip) = g(2fi oYo<p,Z^ oy oy>) is constant. By the definition of gY and using the relationship fiy o Z — Z^ oY we have py(Z O (faZoip) = g($Y o'Zo<py/3yoZ<>p) = gift oY oip,Z^ oY o<p), which was to be proved. Conversely, for any (V} W) where V e TM x m TM, W e HTM there exists an absolute parallel vector field Y such that (V, W) = (/3y(z), dy(u)) where £ = przty'hv = d7r(W). Then dL(FfF(V,W)) = = dLodfiyo dpr2 o H*\/3y(z), dr(v)) = = d(L o/3y)o Hy(z, v) = dLY o HY (z, v) - 0, where LY (z) = gY(&z). □ 3-3 Holonomy Groups of the Finsler Vector Bundle Given a Finsler 'pair connection for tm, several types of holonomy groups can be considered. Naturally, there are holonomy groups of HF at the level of the Finsler vector bundle 7t*(tm), denoted by Hoe at z e TM generated by parallel translation of HF along arbitrary loops in TM at z € TM. Holf is linear group and contained in Gl(ri).
Holonomy Structures in Finsler Geometry 473 We define now the notion of ht>holonomy groups of (2ZF,i7) as follows: Consider a loop cp at x € M and its horizontal lift 0 starting from z e TM: tto 0 = ^, 0(0) = z. 0 is not necessarily a loop. Join 0(1) and 0(0) with a vertical straight line r in TXM. The parallel translations of HF along composite loops 0 * r generate the hv-holonomy group Hol^at z G TM. Combine now our method of osculation with the notion of holonomy. We can prove the following Proposition 3.3.1. Let us denote by Hol^ the holonomy group atx e M of the osculating linear connection HY of a linear Finsler pair connection (HF,H). Then there is an injection Ho£-Hol$(x). Specially, ifYe X(M) is an absolute parallel vector field of H, then Hol^ is mapped into the hv-holonomy group Holy^. Proof: The injection is given as follows: Take a h G Hol^ generated by the loop 9? at x. Consider now the loop Y o <p at y(z), It determines an element hF of Hol$(x). It is well defined, for if (p generates the same h as <p, then Y o (p does as Y o <p. By Theorem 3.2.1 we have pY pY > k pF pJ7 *tp * Yo^p ~ Therefore this map is an injection. For an absolute parallel vector field Y, the loop Y o <p is a closed horizontal curve, so hF — PFO(f, is contained in Holy^. □ 3.4 The Mixed Holonomy Groups In [18] G. Grangier gave a similar notion of holonomy group Hol™, called as mixed holonomy group. He used the Cartan connection and there the second part of parallel translation was substituted by the canonical isomorph¬ ism {*} x i5} x TXM> which does not depend on HF. Then Hol™ £ Holf. The main result of [18] is that the reducibility of the mixed holonomy group implies that the Finsler space is Riemannian and de Rham decomposition arises. Definition 3.4.1. For all € M we consider the transformations t : v x —► w x between the fibers of the Finsler vector bundle tt*(tm) which is induced by the Finsler parallel transport P$ of the Cartan connection along any continuous and piecewise differentiable curve 0 joining v and w. These mappings are called of the first kind. The transformations of the second kind are defined only for vectors v and w in the same fiber: 7r(v) = flr(w) « x. t: v x TXM —► v x TXM> t(v, u)•—> (w, u).
474 Kozina For any v € M, an automorphism of the fiber v x T^M is called a mixed transformation if it has the form t — f2 o 5 o ti, where if (respectively, 12) is a transformation of the first kind with starting point v (respectively, with endpoint v), and s is a finite composition of transformations of first and second kind, having each starting point equal to the end point of the preceding one. All mixed transformations at v form a group Hol™ called the mixed holonomy group at v. If only null-homotopic.curves are allowed then we get the restricted mixed o holonomy group HolJ*. Naturally it is a subgroup of HolJ1. Moreover, the following reduction theorem is valid. Theorem 3.4.2. (Diaz and Grangier, 1976, [15]). o (1) -if n > 2, then HolJ1 is a Lie group. (2) If n > 3, then Hol^/HolJ1 is isomorphic to Holf/Holf, and Hol^1 is a o Lie group f HolJ1 is its connected Lie subgroup containing the identity. Proof: (1) For two vectors v, w G M in the same fiber TXM, denote the canon¬ ical (affine) parallel transport by ¿(v,w), and the Finslerian parallel transport of the Caxtan connection by P? along a curve $ : I —► TXM joining v and w in TXM. Then by ¿(v, w) o P£ is an element of the restricted mixed holonomy o group HolJ*. (2) In dimension > 3, every punctured space TXM is simply connected, therefore we can define a mapping F: Hol^/Hol? -»• Holf/ffol£ 0 °C- which maps a class i modulo Hol™ to the class t(^) modulo Hol£ of the parallel transport relative to VF along the curve p associated to t F is really an isomorphism. The last step comes in a classical way. □ o The tensorial characterization of the Lie algebra of HolJ1 is still open. 3.4.1 Invariant distributions of parallel transport Theorem 3.4.3. (Diaz and Grangier, 1976, [15]), Let a vector subbundle (dis¬ tribution) D of the Finsler vector bundle be invariant with respect to the parallel transport of'V*'. Then the following assertions are equivalent: (1) T> is invariant with respect to the parallel translations of the Cartan con¬ nection (2) For every x € M there exists a subspace Ex C TXM such that Ex = pr2(Ds).
Holonomy Structures in Finsler Geometry 475 (3) For every pair E, 0 G 5ес7г+(тм) vhth S € SecV it follows that T(E, 6) € Sect). (4) T) is invariant with respect to the parallel translations of the Berwald con¬ nection In the proof we use the following observation. Lemma 3.4.4. Let T> be a vector subbundle of the Finsler vector bundle (гм). Then the following are equivalent: (i) D is invariant with respect to a linear connection VF. (ii) For all Y G £(M) E G SecD implies VyS G SecD. Proof: (1) <=> (2) Let x G M and u, v be a pair of nonzero tangent vectors in TXM. Using the notation of the previous theorem we have £(v,u) о P§ g G™. The invariance with respect to Hol£ implies £(v, u) о P£(Dv) =s Duy and so £(u,v)(Pv) = P$(DV) — Pv. Therefore pr2(Pt*) = pr2(Pv)- The converse is trivial. (2) (3) For every pair S,0 G SecTr*(rAf) we have T(S, e) = 2J[i(e), я(Е)]+v£e)s. Since T> is invariant with respect to the parallel transport ofVF7 S € V im¬ plies G SecP. If for a Finsler vector field S G XF(M) = Зестт^т/и) one denotes by s(S) the restriction of ¿(E) to the tangent fiber ТЖМ, it is easy to see that ®(б?тг[г(0),Я(Е)]) = V®^x(S). Therefore (ii) implies that ¿7г[г(0),Я(Е)] G SecP for any E G SecP. So J3) holds. Conversely, for any X, Y G X(TxM) one can construct E, 0 6 XF(Af) with X = ¿(S), Y = ¿(0), and so that if X G Px, then S G SecP. It follows, in virtue if the assumption (iii), that <йф(0),Я(Е)] G P, and consequently V^-У G Dx* (3) «<=> (4) The relation T(E, 0) V^S - VF@)E and the assumption of invariance at parallel transport with respect to D imply that the assertion (3) is equivalent to the following: S G SecP implies G SecP for all 0 G SecTT^Tjtf). Then it follows that S G SecP implies PyS G SecP for all Y G X(M) which is equivalent to (4). □ 3.4.2 Decomposition and reducibility of mixed holonomies The main result of [18] is that the reducibility of the mixed holonomy group implies that the Finsler space is Riemannian and de Rham decomposition arises. Let P* be a nontrivial subspace of v x T^yM which is invariant with respect to НоЦр. Then Pj is invariant with respect to Hol^, too. Denote by D% the orthogonal supplementary subspaces of Pj in v x and translate parallel both of the subspaces with respect to The two distributions w k —
476 Kozma 1,2 are orthogonal. Furthermore, both are invariant with respect to the mixed holonomy groups. It is evident for P1, and the following argument proves this for 7?2: Since l?Hs invariant with respect to Hol171, Q G 2?1 implies T(Q, G) G P1 for all G € XF(M). Thus for any S G Z>2 we have #((Q, G), E) = 0j= 0), ft), consequently, Sei?2 implies T(S, G) G T? for any G € XF(M). Condition (3) of the previous theorem gives the assertion required. In virtue of condition (2) of the previous theorem, for any x € M there exist the subspaces Vk C TXM such that for all v G txM the relation Vk = pr2(Pj) holds (k = 1,2). Both distributions x Vk are C°°-integrable. In fact, both are invol- utive: Let X,Y G X(M) be two vector fields with values in Vk. Then we construct the Finslerian vector fields S,G as follows: S(v) = (t>,X(îr(v))), G(v) = (v, r(*(v))). Since d?r[tf(S),ir(0)] = DW(S)G - PW(e)S is in D*, and dn[H('E),H(&)] ■= we obtain [X,Y] 6 Vk. By Frobenius theorem for all x € M there exist two maximal integral sub¬ manifolds Mi and Ms of M subordinated to the distributions V1 and V2, and a neighborhood U of x such that U — U1 x Ï72, Uk C M*, and for all y G U Vk = dtjb(TyÀ.(M*)) holds, where ¿* : Mfc —► M is the injection, yk is the component of y in Uk, k = 1,2. It can be proved, further, that denoting the restriction of the energy function E := L2 to Mfc: Ek Ek(Xk) = E(dbk(Xk)f Xk G TMk, for any vector field X on U we have the decomposition X = cUi(Xx) + dt2(-^2) and E(X) — ^(X1) + £2(X2). Therefore the manifold is Riemannian. In fact, fixing a nonzero vector X2 G TU2, E^u^X1) = EMdt^X1) + <M%2)) - £21ip(X2) holds, and so the fact that is C°° on M implies that El [¿jx is C°° on TU1. Such neighborhoods cover the entire M, so M is a Riemannian manifold. The argument above proved the following result: Theorem 3.4.5. (Diaz and Grangier, 1976, [15]). (1) Let (M,L) be a Finsler manifold. If there exists a vector v G M such that the mixed holonomy group Hol^ is reducible, then the manifold is Riemannian. If additionally, M is simply connected, and complete, then M is decomposable in the sense of de Rham. (2) Non-Riemannian Finsler manifolds are irreducible (with respect to the mixed holonomy groups.)
Chapter 4 Holonomies of Special Finsler Manifolds In this chapter we present some results on two special classes of Finsler man¬ ifolds. The first one, the class of Berwald manifolds is characterized by the assumption that the canonical Berwald (Barthel) homogeneous connection is a linear connection of the base manifold. Then, as Szabd has shown ([43]), the space is also Riemann metrizable, therefore the method of Riemanniah holonomy can be applied for Berwald manifolds. In Section 5.1 the main classification res¬ ults of Szabd’s work [43] is described. The second section deals with Landsberg manifolds. This class is much larger than the class of Berwald spaces, neverthe¬ less its geometric behavior is fairly understood. Based upon the property that the parallel transport between the tangent fibres is a Riemannian isometry, we can show that the homogeneous holonomy group is a compact Lie group in the case of Landsberg manifolds. 4.1 Berwald Manifolds This section is an outline of Szabd >s work ([43]). We summarize its main results without proofs. Let (M, £) be a Finsler space, where M is a connected differentiable manifold of dimension n and L(x, y), is the fundamental function of the space defined on the manifold M of non-zero tangent vectors. The study is restricted here only to the case when L is positive and the fundamental tensor — \didjLr is positive definite. The connection coefficients G*jk of the Berwald connection are defined by Gi = l9ik(lfdkdrL2-dkLii), <% = №, Gjfe=4'Gl-. If V/c stands for the covariant derivative with respect to the Berwald connec¬ tion, then VfcL = 0, = 0 hold, where — diL. from these identities it 477
478 Kozma follows that the parallel displacement along the curves of M derived from the Berwald connection keeps the length of the vectors invariant, where the length of a vector X 6 TP(M) is defined by |X| = Lp(Xyj. This property is for the Berwald connection of the space characteristic. (We note that the homogeneous connection G^ is also called Barthel connection, see 1.4.) A Finsler space is called a Berwald space if the Berwald connection of the space is a linear connection on the manifold Af, that is, the coefficients Gjk are functions of the position (x*) only. Several characterization are found in [32]. In what follows we shall denote a Berwald space by the triple (M,L,V), where V is the linear connection of the given Berwald space. It is clear that a Riemann space (Af, g) or a locally Minkowskian space is a Berwald space as well. The last type of Finsler spaces can be defined also as Berwald spaces (M, L, V) for which the curvature tensor R of the linear connection V vanishes. 4,1.1 The characterization of the Berwald connections A linear connection V of a manifold Af is called Riemann metrizable if a Riemann space (Af, <?) exists such that V is the Riemannian connection of the space (M.g). V is called Berwald, respectively, strictly Berwald metrizable, if a Berwald space, respectively, a non-Riemannian Berwald space of the form (Af, L, V) exists, that is, the given V is the Berwald connection of the space (M,L). The first basic observation is the following Theorem 4.1.1. (Szabd, 1981, [43]) . Let be a Berwald space. Then the connection V is also Riemann metrizable. The proof of this theorem was adapted for Theorem 1.5.2. The following main theorem characterizes the linear connections, which are strictly Berwald metrizable. Theorem 4.1.2. (Szabd, 1981, [43]) . Let V be a torsion free linear connec¬ tion on a connected manifold M whose curvature tensor R does not vanishes everywhere. The V is strictly Berwald metrizable if and only if the following two conditions are satisfied: 1) V is Riemann metrizable; 2) V is either locally reducible or it is a locally irreducible, locally symmetric connection of rank > 2. The necessity of condition 1) follows from Theorem 4.1.1, and the necessity of condition 2) follows from a theorem of Simons [42]. The sufficiency is simple in the reducible case, namely an invariant norm Lp can be given at a point p G Af as follows = 7|X|2 + yjAbF + |X112^ + - • • + W7 for any natural number s, and X — Xo+XiH hXfc is decomposed with respect to the decomposition TPM — Vo+Vi + * • •+Vk, corresponding to the reducibility
Holonomy Structures in Finsler Geometry 479 of the holonomy group HolP. (See Ambrose-Singer [6].) The indicatrix of this norm is clearly not an ellipse. Then, with the help of the parallel translation Lp can be naturally extended to a fundamental function L on the whole manifold (just as was done in the proof of Theorem 1.5.2). The sufficiency of the locally irreducible case is the most difficult part of the proof. It extensively uses the theory of symmetric Lie algebras. See the original work of Szabo [43]. 4.1.2 The generalized de Rham’s decomposition theory Let V, respectively, V be linear connections on the manifolds M, respectively. M. For every point p € Af x Af the tangent space TP(M x M) of the manifold M x M can be decomposed into the direct sum TP(M x Af) — Tp ф Tp such that 7г*(Тр) 0, and 7г»(Тр) = 0, where the mappings 7f : M x M —► M, 5r: Af x Af —* M are the natural projections. If X is a vector field on M, then let 7f*(X) denote the unique vector field on Af x Af for which тг*7г*(Х) = X and 5гФ7г*(Х) = 0. Next, тг*(Х) is similarly obtained for a vector field X G X(Af). The linear connection V x V on M x Af denotes the Descartes product of V and V, that is, V x V^X7f*r = тГwhere X, Y G X(Af)j~ V x = 0, where X G X(Af), Y G X(Af); V x = 0, where X G X(M), Y G £(M); V x V^x^Y = 7Г*VxK, where X, Y G X(M). Definition 4.1.3. The Berwald space (Af, L, V) is said to be the Descartes product of the Berwald spaces (AfJT, V) and (Af, Z, V) if the following condi¬ tions are satisfied: 1) Af — ~M x Af; 2) V => V x V;. 3) Lp(n*X) = ¿^(Х), where~X G Тад(М) and Lp(?r*X) - ¿^^(X), where X G T^(M), p G Mx M, Note that this Descartes product is the regular one with respect to the product of the manifolds and the linear connections, however, it is not the usual one regarding the metric. In fact, the Finsler fundamental functions on the factor manifolds should be extended onto the product manifold such that it must be parallel with respect to the product of the linear connections. Such a product of the metric can be controlled at an arbitrarily fixed point, since at that point the Finsler fundamental functions on the factor tangent spaces should be extended onto the whole product tangent space such that it is invari¬ ant with respect to the holonomy group on the product manifold. Then this Finsler fundamental function extends onto the whole manifold by parallel dis¬ placement. The Finsler fundamental function after Theorem 4.1.2 shows that this product metric is not uniquely determined even in that case when all the factor metrics are Riemannian ones. Just the usual Descartes product gives in this case a Riemannian metric. All the other product metrics are appropriate non-Riemannian Berwald metrics.
480 Kozina Let (M, L, V) and (Af, L, V) be two Berwald spaces. Consider the Finsler space (M x M^L x L) defined by (Z x L)„(X) = + p € Af x Af, X e TP{M x M), This Finsler space is also a Berwald space, and it is the Descartes product of the spaces (M, L, V) and (M, L, V) in the above sense. Furthermore, we jaote again that there exist infinitely many Berwald spaces of the form (Af x M, L, V x V) which can be considered as the Descartes product of two given Berwald spaces (Af,L, V) and (Af, L, V). For instance, if for every natural number k we define the fundamental function by WQ = -^y(Z x £)2(X) + p G M x M, X € TP(M x Af), then the (Berwald) space (Af x M, V x V) is the Descartes product of the spaces (Af, L, V) and (Af, £, V). It is plain that in a Berwald space (M x Af, L, V x V) (which, can be de¬ composed to the Descartes products of the spaces (AT, L, V) and (M, L, V)) the submanifolds of the form (a, M) and (M, 6), a G Af, b G M, are totally geodesic. The Descartes product of more than two Berwald spaces can be defined by induction. We say that a Berwald space (Af, £, V) is locally decomposable to a Descartes product, if every point p G Af has a neighborhood U such that the restriction of the given Berwald space on U (denoted by (Lf, £, V)) can be decomposed to a Descartes product of Berwald spaces. Definition 4.1.4. A Berwald space (Af, L, V) is said to be locally, respectively, globally symmetric, if the linear connection V is locally, respectively, globally symmetric. In this case the rank of the space is defined by the rank of V. The space (Af, L, V) is called complete, if V is complete, i.e., if every geodesic can be extended to a geodesic xt defined for all —oo < t < oo.* The next theorems have been obtained in [43]. Theorem 4.1.5. (Szabd, 1981, [43]) . A connected Berwald space must be one of the following four types: 1) (Af, L, V) is a Riemannian space 2) (Af, L, V) is a locally Minkowskian space, 3) (M, L, V) is a locally irredu¬ cible and locally symmetric non-Riemannian Berwald space of rank r > 2. 4) (Af, L, V) is locally reducible, and in this case (Af, L, V) can be locally decom¬ posed to a Descartes product of Riemannian spaces, locally Minkowskian spaces and locally irreducible, locally symmetric non-Riemannian Berwald spaces of rank r > 2. Corollary 4.1.6. A Berwald space of dimension 2 is either a Riemannian space or a locally Minkowskian space.
Holonomy Structures in Finsler Geometry 481 Corollary 4.1.7. A Berwald space (M,L,V) of dimension 3 must be one of the following types: 1) (M, £, V) is a Riemannian space; 2) (A7, L, 57) is a locally Minkowski space; 3) (M, L, V) can be decomposed locally to the Descartes product of a 1-dimensional and a 2-dimensional Riemannian space. Theorem 4.1.8. (Generalized de Rham’s decomposition theorem). A connec¬ ted simply connected complete Berwald space (M, L, V) can be decomposed to the Descartes product of a Minkowski space (Mq,I№), simply connected com¬ plete irreducible Riemannian spaces and simply connected complete irreducible globally symmetric non-Riemannian Berwald spaces of rank > 2. Such a de¬ composition is unique up to an order. It is clear from Theorem 4.1.5 that the prototypes of Berwald spaces are the Riemannian spaces, locally Minkowski spaces and locally irreducible globally symmetric non-Riemannian Berwald spaces. Other Berwald spaces can be loc¬ ally described with the Descartes product of these spaces. Hence it is important to determine all locally irreducible globally symmetric non-Riemannian Berwald spaces. These are exactly the homogeneous spaces of the form (G/H,L, V), where any possible G/H is listed in Cartan’s lists (see in Szabd’s work ([43]), or in Helgason’s book [21, page 516—518]), and V is such a G-invariant lin¬ ear connection on G/H as before. The non-Riemannian fundamental function L metrizes the connection V as a non-Riemannian Berwald space. There ex¬ ists infinitely many such non-Riemannian fundamental functions (see Theorem 4.1.2). For any such Berwald space (G/H, L, V) the elements of G act on G/H as isometries of the space, that is, the function L is G-invariant. Conversely, for every G-invariant Finslerian fundamental function L, the Finsler space (G/H, L) is a Berwald space with the Berwald connection V. 4.2 Landsberg Manifolds In the literature, a Finsler space is called Landsberg if its canonical Berwald connection is v-metrical. Here, in the definition of Landsberg spaces we use the Berwaldian Finsler pair connection constructed from an arbitrary homogenous connection 27 in First we show a global process for introducing the Berwal¬ dian Finsler connection in the vertical bundle Vtm which canonical isomorphic tO 7T*(7m). Let H: TM xm TM —► TTM be a horizontal map for rw, and v € TM a fixed tangent vector, x — Denote 27v: TXM —► TTM the map arising from H by fixing v : Hv(z) =* H(z, v), where z 6 TXM. To define a connection for Vtm we are to give a map HB: VTM XtmTTM —► T(VTM). Taking into account that TTM = HTM © VTM, HB is enough to define for vertical and horizontal vectors, and then to extend it linearly for all TTM. First let Z € VTM and U € VTM. Then we choose HB(Z, 17) as an ele¬ ment of the induced vertical subspaces such that ditv(HB(Z, U)) — U. Such an element uniquely exists. Secondly let U € HTM be a horizontal vector, and v — dv(U). Then the horizontal map for the Berwald connection is given as
482 Kozina follows: HB{Z,U) = sodHv{Z), where s: TTTM —> TTTM means the canonical involution for TM. Thus we really get a horizontal map, for Im = V(Ttm)- The connection in Vtm determined by HB is just the Berwold’s connection, for a short computation shows that its local components are F<* = QN* q CS — 0 It is obvious that the Berwald connection in Vtm is linear, for it was defined by a differential map. It is also easy to prove (see [44]), that H satisfies the homogeneity condition if and only if the Finsler pair connection (H3,#) is deflection free. As mentioned, here we give a slightly more general definition of Landsberg spaces than usual. Instead of a Finsler fundamental function L, we suppose that a Riemannian metric g is given in the Finsler vector bundle 7r*(rjvf). Definition 4.2.1. Let (rw»p) be a Finslerian vector bundle, H a positively homogeneous connection and g a Riemannian metric in the Finsler vector bundle H) is called a Landsbergian vector bundle if the Berwaldian Finsler pair connection (H3, H) is h-metrical. Applying the above construction of the Berwaldian connection HB the as¬ sumption can be expressed as dL°sodHv = Q vtTM, where ¿2(u) = Using the covariant derivation of HB it is equivalent to Vug = 0 for any horizontal U. We have the classical notion of Landsberg space if g is derived from a Finsler d2L^ fundamental function L: gij(x,y) = ? and H is the Barthel positively oyvoy3 homogeneous connection given in Section 1.4 is used: H — HB. A series Miff* conditions is known for Landsberg spaces [32, p. 162]. Consider now the parallel translation with respect to H. Due to'[9] it exists for entire curve <p: I —► M. Therefore the parallel translation : TPM —► TgM is a positively homogeneous bijective map if <p joins p € M and q € M. Secondly, a fibre TPM at a point can be regarded as a Riemannian space by g(p, z) fixing the point p. We can prove now the next Theorem 4.2.2. (Ichijyo, 1983, [23]; Kozma, 1996, [29]). (rM,p, H) is a Lands¬ bergian vector bundle if and only if the parallel translation of the positively ho¬ mogeneous connection is an isometry between the fibres as Riemannian spaces for any curve. Proof: Consider the parallel translation Pv^: —► T^M between the points ¥>(0) and <p(t). Pv,(t) is an isometry if dPv(t): T(TV(^M)
Holonomy Structures in Finsler Geometry 483 is. a linear isometry, te. VZ G T^T^yM) — V-TM (TM XmTM)~ g(Z,Z) = g(dP^(Z),dPv^(Z)). Denoting p(Z,Z) = L*(Z), the condition means that L* o dP^ty = const. Note that the generalization of the equality of mixed partial derivatives for the case F: R x M -+ IV is described in the following manner: s o dF(y) — (dF(v))‘ where $: TTN —► TTN is the canonical involution for N, and the dot and d denote the differentials with respect to R and M, respectively. Using this for the case F(t,z) — Ptfi(t)(z): TVM —► TM we get (L* o dP^Y - dL* o (dP^t)y = dr o s o dP^ty Applying the relationship P^ = Hv between the parallel translation and the horizontal map, we continue further as = dL*oscdHv = 0 because of the Landsberg property. □ This theorem was proved by Y. Ichijyo in [22] by another method. As corol¬ lary we obtain a result of [7]: Corollary 4.2.3. For a Landsberg space the volume junction of the indicatrices is constant. There are a lot of interesting results concerning Landsberg spaces [16, 27, 52, 53]. Specially, Landsberg spaces are characterized by the condition that the indicatrix I? at any point p € M is a totally geodesic submanifold of the total space ITM of the indicatrix bundle. In [1] T. Aikou proved that the Landsberg property is equivalent to that the tangent fibres are totally geodesic submanifolds of TM with respect to the Sasaki metric of TM, In the case of Landsberg spaces we can prove using the previous result that the holonomy group is a compact Lie group. Theorem 4.2.4. (Kozma, 2000, [30]). The holonomy group of a Landsberg manifold is a compact Lie group. Proof: Theorem 4.2.2 means that the holonomy group is a closed subgroup of the isometry group of the fibre considered as Riemannian space. On the other hand the indicatrix remains invariant when the holonomies are applied. Take the restriction of the holonomies on the indicatrix. Now the indicatrix at a point x is a compact Riemannian space, therefore its isometry group is a compact Lie group [26]. Thus the holonomy group is a closed subgroup of the compact Lie isometry group, consequently itself is a compact Lie group, too. □
484 Kozma Acknowledgement The author is very much indebted to Professor Peter L. Antonelli for his en¬ couragement, and Professor Zoltán I. Szabd for his advice.
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PART 6
On the Gauss-Bonnet-Chem Theorem in Finsler Geometry Brad Lackey
Contents 1 Topological Preliminary 497 2 The Method of Transgression 499 3 The Correction Term 503 4 Special Cases 505 4.1 Riemannian Geometry 505 4.2 The Chern Connection 505 4.3 A Special Family of Finsler Connections 506 493
494 Lackey When coupled to the Hopf-Poincard index theorem, the Gauss-Bonnet for¬ mula is firmly planted at the crossroads of the researches which form the dis¬ cipline of global differential geometry. Leading away from this central position are the paradigms of invariant theory, index theory, obstruction theory, and K-theory; as well, it has inspired innumerable other individual results. There are two (not entirely distinct) views of the Gauss-Bonnet formula. Historically, the formula gives the relationship between curvature and “angular excess”. The notion of angular excess for a geodesic triangle on a surface is self- evident; as one moves to more general figures in higher dimension, angular excess becomes burdened by increasingly opaque definitions. Perhaps the ultimate result along these lines is Theorem II by Allendoerfer and Weil, [2], which states that for a Riemannian polyhedron P of dimension n, whose boundary consists of polyhedra Pa, we have (-I)V(P) = [ *«) <fo(C) + £ [ [ d<) JP « JP* ’/r(C) where £'(£) is the Pfaffian of the Riemaimian curvature suitable normalized, r(£) is a subset of the unit sphere at < often called the “exterior” or “outer” angle of Pa at Ci the form d£|Pa) is a measure of the curvature of Pa analogous to the geodesic curvature of a curve on a surface, dv is the Riemannian volume form, and is the “interior Euler characteristic” (computed the same way at the Euler characteristic, but only using the interior faces of a triangulation). The modern view of the Gauss-Bonnet formula is that the curvature of a Riemannian manifold reflects the topology of the space. Perhaps the ultimate result along these lines is Theorem I of Allendoerfer and Weil, [2], which states that for a compact and oriented Riemannian manifold without boundary, we have x(M)= [ *(C)dv(C). JM The proof of Theorem I of Allendoerfer and Weil follows immediately from Theorem n, with some straight forward combinatorics. The proof of Theorem II, however, is immensely complicated and involves an equally complicated res¬ ult of Weyl, computing volume of tubular neighborhoods of a Riemannian cell embedded in Euclidean space; this can be found as Lemma 6 in [2]. As no non- Riemannian Finsler space can be isometrically embedded into a Euclidean space (or generally even into a fixed norm linear space), any hope of extending the Gauss-Bonnet theorem to Finsler spaces using the techniques of Allendoerfer and Weil is hopeless. Fortunately, the paper of Allendoerfer and Weil was not the final word in the history of the Gauss-Bonnet formula. The next year, Chern proved the Gauss- Bonnet formula of Allendoerfer and Weil using what we now call the “method of transgression”, [8]. This method is so powerful, it has its own field of study in invariant theory, called Chern-Weil theory. Moreover, the technique is so simple that most authors now call the Gauss-Bonnet formula the “Gauss-Bonnet-Chem theorem.”
Gauss-Bonnet Formula 495 The method of transgression has set the standard for the generalization of the Gauss-Bonnet formula to Finsler manifolds. Let us briefly examine the philosophy behind the technique. The goal of any theorem in Chem-Weil theory is to represent a characteristic class of some vector bundle over a manifold by some geometrically significant object. The Euler class is the subject of the Gauss-Bonnet-Chem theorem. For a Riemannian manifold, the geometrically significant object is the Pfaffian of the Riemannian curvature form. Proof of the Gauss-Bonnet formula for a Riemannian goes as follows: (1) take a unit vector field with possibly isolated singularities; (2) extract small balls about these singularities - then the remainder of the manifold is in some sense trivial, (3) realize the Pfaffian is the derivative of a form, the geodesic curvature, which is only defined on the total space of the sphere bundle; (4) apply Stokes’ theorem and one is left with the integral of this geodesic curvature along the surface of each of these carved out balls; finally, (5) pull-back to the manifold using the unit vector field, and as the radius of a ball tends to zero, this can be none other than the index of the vector field at the singularity (multiplied by the volume of the sphere). Extending the formula to Finsler spaces is entirely natural; while the method of transgression for a Riemannian manifold requires the lifting of the curvature to the sphere bundle, for a Finsler space the curvature form is already a form on the sphere bundle. Lichnerowicz, [11], extended the Gauss-Bonnet formula to a very restricted class of Finsler spaces, now called Cartan-Berwald spaces, by applying the method of trans¬ gression to the Pfaffian of the Cartan curvature. But, only in the case of these Cartan-Berwald spaces does this yield a geodesic curvature that measures the index of the vector field at a singularity. That is, only on a Cartan-Berwald space can this Pfaffian represent the Euler class. The method of transgression can be operated in reverse. Beginning with a candidate for the geodesic curvature, one can exterior differentiate to find an analogue for the Pfaffian of the curvature. This approach is taking by Shen, using Cartan’s connection, in the unpublished work, [13]. Bao and Chem, [3], discovered that the Chem connection offers much better success, but with a price. The Chem connection is not metric compatible, hence the Pfaffian of the curvature is not closed. But, they showed that with a suitable correction term, one gets a geodesic curvature form restricting to a multiple of the volume form on the fibres. Yet, the volume of the fibres is not generally constant on Finsler manifolds, hence one gets a weighted sum of indices of the unit vector field - each index weighted by the volume of the fibre above the singularity. At this point it is not clear that the integral is independent of the section taken. Bao and Chem restrict to the case where the volume of the fibres is constant, hence with suitable normalization, the corrected curvature polynomial does represent the Euler class, [3, Theorem 3]. Lackey, [10], made a simple modification to the technique of Bao and Chem. Normalizing by the typically nonconstant volume of the fibre before applying the method of transgression, one finds a correction term that indeed sums with the normalized Pfaffian of the Chem curvature form to represent the Euler class
496 Lackey for any Finsler space. Unfortunately, the correction term involves the derivative of the volume function. However, in said work, Lackey considers a general torsion-free connection on a Finsler space, and shows that by proper choice of metric-incompatibility one can absorb these extrinsic terms. This gives a correction term that is a polynomial in the curvature and connection forms alone, however the connection is somewhat more complicated than the Chem connection.
Chapter 1 Topological Preliminary Let M be a smooth, compact and orientable manifold without boundary. Let e e Kn(M,Z) be its Euler class. Let x : SM —► M be the (projective) sphere bundle. Our goal is to represent the pull-back, ?r*e € Hn(SM, Z), by differential forms of geometric content. That is, we search for closed forms 2 € Qn(SM) such that for any section : M —► SM, possibly with isolated singularities, Jm where x(M) is the Euler characteristic of M. If {a,*} are the singularities of -0, then by the Hopf-Poincar6 theorem, %(M) = ^mdex,^) As we are taking the limit as e —> 0, we may take any form K € On^1(SM) and have as long as II = (mod dx). But, UMis the boundary of M \ Bc(a?^) (with the wrong orienta¬ tion). So, by applying Stokes’ theorem, xW = -^lim [ m
498 Lackey The final term in this string of equalities is very misleading. The integrand, ^*dH, is not defined at the singularities of ip. This equality states, more pre¬ cisely, that the negative of the Euler characteristic is the residue of ^*dll relative to M. For a Riemannian manifold, we have dlL = 7r*n where Q is the suitably normalized Pfaffian of the Riemannian curvature form, and II is its geodesic curvature form, [8]. In this case, 2 — wherever -0 is defined, and therefore the integrand can be smoothly extended across the singularities of & This has little bearing on the residues however, as the extension will no longer be exact in general. This last statement is a loose paraphrasing of the main result of Allendoefer and Eells, [1], stating that the derived cohomology of the differen¬ tial graded module of differential forms with singularities is isomorphic to the deRham cohomology. In fact, for a Riemannian manifold, it may be more enlightening to say that the (suitably normalized) Pfaffian of the Riemannian curvature form is exact modulo a zero manifold, and that its residue relative to M less any such zero manifold is the Euler characteristic. This is precisely the approach taken by Eells, [9], which includes not only the representation of the Euler class by curvature forms, but also the Stiefel-Whitney classes.
Chapter 2 The Method of Transgression The technique of Bao and Chem, extended by Lackey, is to apply the method of transgression as if our Finsler space were Riemannian; the necessary correction term will automatically be generated in the process. As a technique, the method of transgression is a difference equation in exterior differential forms; therefore we will need the structure equations of our Finsler connection. As usual, let F be a Finsler norm, and w71 = the Hilbert form. Here and later, we use (z, y) as an adapted homogeneous coordinate system of 7T: SM —> M. Let {uP} = {wa,a;n} be an orthonormal coframe of tt*TM over some open domain of SM. We are only considering torsion-free connections, so the first structure equations read: A A wn dwn = A wa - cdnn A wn. Notice the use of left-invariant forms, rather the right-invariant ones preferred by Bao and Chem. It is easy to verify that is a vertical coframe of SM, see [7], for instance. These forms transform in the same way as {wa}. By declaration, we may take to be an orthonormal coframe of SM (the interested reader may want to compare this to the Sasaki lift). The pure depart of A •»• A a/1“1« generates a volume form on each fibre of SM, For a Riemannian manifold, the volume of each fibre is constant: the volume of the Euclidean sphere S’1”1. In many Finsler spaces, for instance Landsberg spaces, the volume is also constant, although it need not be this value, [6]. We will denote Vol‘(x) for the volume of SXM with this metric. The metric-incompatibility of our connection is given by the structure equa¬ tion: WjA + - Mjki where Aijk *= f Cartan tensor. 499
500 Lackey The curvature of our connection, = du?k + u?^ A u?A, has no vy-term as the connection is torsion-free. That is, «S- = Aw‘ + -F% wft A <. Our connection is not metric compatible and therefore is not skew. The fail¬ ure of w, Q to be skew will cause the tensors M, A and their covariant derivatives to appear in several places. We will find the following notation useful: <r>k = uik+uki Y,jk = Q.jk+Qkj. Following Chem, we define the following polynomials for 0 < k < $<fc> := A • • ■ A №*-*“** A A • • • A ■- eax-an-xH“1®2 A • • • A ft“2*-»"»* A n“2*+*n A wQ2*+2„ A • • • A If n is odd we instead take ) ;= 0. Note that = (n — 1)! wxn A • • • A wn_1n; we want II to restrict to the normalized volume form, and so we are really interested in the forms The relationship between these two families of forms is the difference equation “ (dw - w (<” - “ - ■ *w+“ ■ +a“’) • where we use yA“1) := 0. The correction terms appear naturally. These are complicated polynomials in M, A, their covariant derivatives, and the com¬ ponents of d log Vol. For now, we can continue without specific knowledge of their internal formulation. Notice that all the have at least one dir-term, except <I><°\ If there were a nonvanishing yy-curvature, this would not be true, and the method of transgression would fail. Focusing on the case n = 2m, we take (-1)’ n = 1 V 1 2m_1. (m _ 1)! 2^ 2r • (2m - 2r - 1)1! • r! Vol(z) For an odd integer j, we use the notation j!l — 1 »3 * 5 j. In particular, we have 2—1 • (m -1)! = ^5$, and so 11(3?) = ^dVols.M (mod fa) as desired. The exterior derivative is i) 411 Vol(®) L4’»-1-((m-l)!)2 - m—1 (-If - m—a 1 v 2m-i. (m _ i)i 2r • (2m - 2r - 1)!! • r! ~
Gauss-Bonnet Formula 501 Recall that the Pfaffian is defined as Pf(O®j) — €<1..»iwn£1<a A • • - A Qjn-lin. After simplification, we find that Pf(Q^) equals + €ai...ttn_in0'1“2 A • • • A A S“""1«. (2.1) Therefore, •Pf(n^.)+5-] , 1 dll - Vol(x) |_221TO_1 • (m — 1)! • m! where the invariant F is constructed from the 2^ and the extraneous term in (2.1). In the odd dimensional case, n — 2m +1, we define The exterior derivative of this form is where again the 2-terms are complicated polynomials in M, A, their covariant derivatives, and the components of d log Vol. Theorem 2.0.1 ((Lackey, [10])). Let (M,F) be a compact, orientable Finsler n-manifold without boundary, and ip a section of the its projective sphere bundle % : SM —► Af, possibly with isolated singularities. Let (w^) be any torsion-free Finslerian connection with curvature Then, = x(M) where and for n — 2m for n = 2m + 1 Q - < 0 n = 2m n = 2m + 1 The are polynomials in the entries of cr, Q, S, as well as dlog Vol(x) (explicit formulae to follow).
Chapter 3 The Correction Term Computation of the correction term is quite complicated; the interested Reader can consult section 5 of [10] to find details. To give a flavor for what is involved, we reproduce the computation of First, d$<°) = (n — 1) • eai...an_1dw“1n A w“2n A • ■ • A wa"_1n — (n -1) • i(0) - (n - 1) • e(J1...an_1 (w“13 A aP’n) A w“!n A • • • A In this last term, there are only two j-values which can contribute to the sum: j = O!i,n. This term for j = n is just (n - 1) - A3><°\ For j = ai, we note n-1 52 €0«a-«n-iA W^n A w“’n A • • • A w“"-', /3=1 =52(n “ 2)! • A wln a • • •A w"»1 = A $(0) • 3=1 Therefore, Vol(s) C” ‘$(0) + (-¿log Vol (x) + (n - 1) • wnn - wfy A . Specifically, we have — ((n — 1) • wnn - — ¿log Vol(x)) A The computation of for 0 < k < are considerably more difficult. Whether n = 2m or n — 2m +1, for 0 < k < m one has q« n ~ ~1 [(„ -1) • u>nn -w^- dlog Vol(z)j A +fc A • • • A Ajas«:-saa).-i<»a*«s*+i A w“=*+in A • • • A W8"-1, , 503
504 Lackey where jttaA-s^k-xorsfeaat+i _ A ^aat-iaafe Qa2^^ A <Ta~kn + E*2^1* AiJ^n — (n — 2k — l)^2*-1®2^1 A aa2Je“afc+1 + (k — l)Qa2fc-2aa* A <T®2Af-2Qafc-x . In the even dimensional case Q,^ is just the (negative of the) extraneous term in (2.1). Explicitly, Sw = -e1I1...(1,.lfiaiOSA-Afi<'’T3a“-=AS#»-lll. If n = 2m +1, then qW = meai..,a2mnO!1Of2 A • • • A Q^m-aaam-s A [ A a«s™n _|_ A LJOf2mn + A 0-a2™-1^2™ + (m - 1) A
Chapter 4 Special Cases 4.1 Riemannian Geometry The Christoffel-Levi-Civita connection of a Riemannian manifold is character¬ ized by M = A = 0. So, we have and <r5fc vanishing as well. Moreover, Vol(a;) = Vol(Sn“1). Therefore, = 0 for all k, and hence 5* = 0. We recover the classical Gauss-Bonnet-Chem theorem for Riemannian manifolds, as a result. Theorem 4,1.1 ((Chem, [8])). Let (M,p) be a compact, orientable Rieman¬ nian n~ manifold without boundary. Then, if is the Christoffel-Levi-Civita connection with curvature (f^)? where n = 2m 0 for n = 2m +1 4.2 The Chem Connection If u4 / 0, then F is properly Finslerian. The choice Mijk — 0 determines the Chem connection of F; the connection forms and curvatures live d priori only on SM. Theorem 4.2.1. Let (Af, F) be a compact, orientable Finslerian n—manifold without boundary, and a section of the its projective sphere bundle 7r: SM —► M, possibly with isolated singularities. Let ) be the Chem connection with curvature Then, 505
506 Lackey where for n = 2m for n = 2m + 1 and for n = 2m +1. 1 (2m)I We have Q<°> = -dlog Vo!(x) A for 0 < k < we have +fciat...a„_in“’“2 A-'-Ail“2»-’“»*-2 A - (n. - 2k - A a0,2*a2’,+1 + (k - A or“»-»“»*-»] A w“:k+'„ A • • • A Wa*-ln ; and, if n — 2m 4-1 we have QW = 7n€O1...aainirxtts A •1 • A na2m'3“3m“a A (1*^-1 A + (m - 1) Qa2m-ia2m-3 A ^„1-3*2™] Proof: We need only indicate two things. First, as Aabn — 0 for all a, b we have aan = 0 and Son = 0 for all a. The former is immediate, while the latter requires some computation (or see exercise 2.5.4 in [4]). Second, as we have A = 0. If the volume of the indicatrix is constant, then d log Vol(x) vanishes, recov¬ ering the formulae in Bao and Chem, [3]. 4.3 A Special Family of Finsler Connections The Gauss-Bonnet formula strives to represent the Euler class in terms of geo¬ metrically significant objects. One could argue that this requires the integrand of the formula to consist of polynomials only in the connection forms, curvature forms, and covariant derivatives thereof. Our most general formula fails in this regard, as the correction terms contain explicit reference to d log Vol (z). But, the only place where d log Vol (z) explicitly appears in any of the Q poly¬ nomials is within the term (n — 1) wnn — — dlog Vol(z) I. If the connection
Gauss-Bonnet Formula 507 forms are such that this term vanishes, then the integrand of the Gauss-Bonnet formula will consist solely of polynomials in the connection forms, curvature forms, and covariant derivatives thereof. This is what was proposed in Lackey, [10]. For the following, write := — for the Hilbert form and fy* gjk — tjtk for the angular metric tensor. Take constants a, b with a + b = 1, consider the connection with metric incompatibility given by Mjkt fa1 = -J— ((a • tjik - b ■ hjk) ■ dlog Vol(z)) 71 1 For any fixed a, b, such connections have three important properties: 1. It is constructed solely from the Finsler metric. 2. If the volume of the indicatrix is constant then we have the Chera con¬ nection. 3. The correction term F is a polynomial in the connection and curvature coefficients.
Bibliography [1] Allendoerfer, C.B. and Eells, J. Jr. (1957/8) On the cohomology of smooth manifolds, Comm. Math. Helv. 32, 165-179. [2] Allendoerfer, C.B. and Weil, A. (1943) The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53, 101-129. [3] Bao, D. and Chem, S.S. (1996) A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. Math. 143, 233-252. [4] Bao, D., Chem, S.S. and Shen, Z. (2000) An Introduction to Riemann- Finsler Geometry, Springer-Verlag. [5] Bao, D. and Lackey, B. (1996) A Hodge decomposition theorem for Finsler spaces, C.R. Acad. Sei. Paris 323, 51-56. [6] Bao, D. and Shen, Z. (1994) On the volume of unit tangent spheres in a Finsler manifold, Results in Math. 26, 1-17. [7] Chem, S.S. (1943) On the Euclidean connections in a Finsler space, Proc. Natl. Acad. Sei. USA 29, 38-43. [8] Chern, S.S. (1944) A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. Math. 45, 747-752. [9] Eells, J. Jr. (1959) A generalization of the Gauss-Bonnet theorem, Trans. Amer. Math. Soc. 92, 142-153. [10] Lackey, B. (2002) On the Gauss-Bonnet formula in Riemann-Finsler geo¬ metry, Bull. London Math. Soc. 34. [11] Lichnerowicz, A. (1949) Quelques théorèmes de géométrie différentielle globale’, Comm. Math. Helv. 22, 271-301. [12] Shen, Z. (1994) On a connection in Finsler geometry, Houston J. Math. 20, 591-602. [13] Shen, Z. (1996) A Gauss-Bonnet-Chem formula for Finsler manifolds, preprint, available through the author’s homepage, http: //www.math.iupui.edu/~zshen/. 509
PART 7
The Hodge Theory of Finsler-type Geometries Brad Lackey
Contents 1 Elliptic Complexes 521 1.1 The Hodge-deRham Complex 521 1.1.1 Exterior derivative 522 1.1.2 DeRham’s Cohomology 523 1.2 Elliptic Complexes 523 1.2.1 Linear Differential Operators 523 1.2.2 The Laplacian 525 1.3 Elliptic Operators 527 1.3.1 Sobolev Norms 527 1.3.2 Elliptic Operators 529 1.4 The Hodge Decomposition Theorem 531 2 The Weitzenbock Formula 533 2.1 Complete Positivity 534 2.2 Covariant Formalism 536 2.2.1 Torsion on an Elliptic Complex 536 2.2.2 The Adjoint Operator 538 2.2.3 The Weitzenbock Formula 538 2.3 Existence and Uniqueness of a Connection 539 2.4 A Bochner Vanishing Theorem 541 3 Complete Positivity of the Symbol 543 3.1 The Geometric Ratio 543 3.2 Computing the Geometric Ratio 545 3.3 An Example 547 515
Preface The notion of the Laplace operator in Finsler geometry saw rapid development in the 1990’s. Since this is not the topic of this article, we refer the reader to the comprehensive collection of works in this subject, [3]. Rather, this report concerns “Hodge Theory”, which we interpret to mean results derived from the isomorphism between cohomology and the space of harmonic forms. Most of the works given, and cited, in [3] deal only with the Laplace-Beltrami operator on functions, and hence properly belong to the subject entitled, “Potential Theory.” The most general Hodge Decomposition Theorem holds in the context of an elliptic complex of partial differential operators (of a fixed order) between vector bundles equipped with metrics; at present day, this result is considered classical, and reader can find it in the more analytical books on differential geometry - for instance Warner, [12], or Wells, [13]. We will examine this general setting in some detail in the first chapter. In fact, this general setting for the Hodge theorem encompasses Finsler geometry, at least in the context of positive definite metrics defined on the whole of the sphere bundle. At the suggestion of Chem, Bao and Lackey examined the problem of how to formulate the Hodge theorem for Finsler spaces. After some time, they realized that the general Hodge Decomposition Theorem as stated above would apply, if one could generate a metric on each grade of differential forms using the Finsler structure. They accomplished this by: (1) pulling back the forms to the sphere bundle; (2) using the natural inner product there, albeit with a normalized volume; and, (3) with a partition of unity argument produced the metric tensors. This yielded the Hodge theorem for Finsler spaces, [4]. We will examine this technique in the middle of the first chapter. This construction created a difficulty seemingly missed by all the classical authors. In the general Hodge theorem, no assumptions are made about the leading order symbol of the Laplacian - indeed, none are needed beyond ellipti¬ city. Yet, when producing a Weitzenbock formula, by which we mean a covariant expression for the Laplacian, all previous works immediately assume this symbol is the tensor product of a Riemannian metric with the identity operator. This is indeed the case for the classical elliptic complexes on a Riemannian manifold. However, for a Finsler space, this is dramatically not the case. The moral of the construction by Bao and Lackey is that for a Finsler space, the Hodge-deRham complex (that is, the exterior algebra with the exterior de¬ rivative) should be treated as a general elliptic complex, as there is no additional 517
518 Lackey structure preserved by the induced metrics. Specifically, the Hodge-deRham complex of a Riemannian manifold carries a representation of the Clifford al¬ gebra, while that of a Finsler space does not. Therefore, the creation of a Weitzenbock formula for a general elliptic complex was needed, but none exis¬ ted. The generation of a general Weitzenbock formula is the topic of the second chapter. One of the main problems in constructing a Weitzenbock formula is that one needs connections on the vector bundles making up the elliptic complex. Since these bundles have metrics, is it reasonable to consider metric compatible connections; but, mere metricity is insufficient to specify a unique connection. If one takes any metric covariant derivative, one can generate a covariant formula for the Laplacian. Typically this formula will include terms involving the second and first covariant derivative of the section. Lackey postulated, in [9], that the vanishing of the first order covariant de¬ rivative term is a good condition to place on the connection. This condition, after a long computation, reduces to a linear system involving the unknown com¬ ponents of a metric connection. In fact, this system is well-posed and therefore existence and uniqueness of the connection are tied together. The system has a unique solution if and only if the symbol of the Laplacian satisfies an additional positivity condition - beyond ellipticity - that we will call “complete” positivity. The initial analysis of complete positivity for the symbol of the Laplacian appeared, in [10], with the hope that all Laplacians have completely positive symbol. However, this was soon discovered not to be the case. Bao and Lackey, in [5], restricted to the case of a Finsler surface; they produced the “geomet¬ ric ratio,” which is bounded between | and 3 precisely when symbol of the Laplacian is completely positive. Then, in [6], by examining the geometric ratio for a Randers surface, they produce examples of Finsler spaces for which these bounds fail. These explicit constructions form the third chapter. We would like to conclude with citing some other topics that fall under the title of Finslerian Hodge Theory, but are incomplete or too tangential to include here. The Hodge-DeRham complex is one of four of the classical elliptic complexes of Riemannian geometry; the others being the signature complex, the spin complex, and the ^-complex. The signature complex requires the introduction of the Hodge star. For Riemannian manifolds, this object is entirely natural, and compatible with the Clifford algebra structure on the exterior algebra. In [9] (or see [11]), Lackey proposed the extension of the Hodge star to the Finsler case through the equa¬ tion (0, 0) — J where the inner product was that given by Bao and Lackey in [4]. This definition seems satisfactory as the induced spaces of self-dual and anti-self-dual forms are well behaved. Unfortunately, the Hodge-deRham oper¬ ator (that is, the Dirac operator on forms) is not as well behaved, and forms that are harmonic to this complex are not typically harmonic in the usual sense (Lackey calls these forms “signature forms”). Nonetheless, the Weitzenbock formula follows from generality, and the analogue of Lichnerowicz’s vanishing theorem follows: the self-dual curvature is positive, the signature is nonpositive; and if the anti-self-dual curvature is positive, then the signature is nonnegative.
Hodge Theory 519 A notion of the spin complex for a Finsler space is far from clear. The representation of the Clifford algebra given by a Riemannian geometry is a key element in the formulation of the spin bundles; at present there is no replacement for this construction on a Finsler space. Yet, Flaherty has proposed two natural versions of spin structures on the sphere bundle of’a Finsler space. In [7], he introduces “vertical spinors” on odd dimensional Finsler manifolds. The base manifold plays the role of a parameter space for a family of spin structures on its tangent spheres. The notion of “horizontal spinor” is introduced in [8], more in the spirit of this work. Yet, the fibre-wise dependence of the spinors and ensuing Dirac operator is quite evident, linking the theory to the geometry of the sphere bundle rather than that of the manifold itself. Perhaps the most beautiful study in geometry - outside the Gauss-Bonnet theorem - is the Hodge theory of the ^-complex on Kahler manifolds. Oddly enough, the task of extending this to Finsler spaces has never been taken up, despite that there appears to be no immediate complications. The construction of the ¿-complex requires no geometry (in the same way as the Hodge-deRham complex for real manifolds). The Hermitian structure on the projectived tangent space of a complex Finsler space is well known, see Abate and Patrizio for a complete treatment [1]. Although there are many candidates for the Kahler condition on a Finsler space, none are too complicated to analyze in this context, again see [1].
Chapter 1 Elliptic Complexes The elliptic complex supports a bridge between algebraic topology and differen¬ tial analysis. On one side, cochain complexes are the basic elements in homolo¬ gical algebra; the cohomology of these complexes is one of the most fundamental measurements of nontriviality. On the other side, elliptic operators - although defined by local information - carry some global nature of their domain in their kernel and index. The elliptic complex is the object built from these two no¬ tions: a cochain complex where the connecting maps are differential operators carrying some notion of ellipticity. 1.1 The Hodge-deRham Complex The prototypical elliptic complex is the Hodge-deRham complex: the space of differential forms together with the exterior derivative. The general theory of elliptic complexes is a straight forward generalization of this complex. Throughout the following, M is a smooth manifold of dimension n. Depend¬ ing on desired context, M can be quite general. We will demand our manifold be compact, oriented, and without boundary; yet, each of these conditions may be dropped with appropriate assumptions placed upon our forms. If we wish to allow for noncompact manifolds, we need to consider forms which are square- integrable. For non-orientable manifolds, we must work with twisted differential forms, and volume measures. If we wish our manifolds to have boundary, we need to stipulate what sort of boundary conditions our forms will satisfy. For p > 1, we write QP(M) for the space of smooth p-forms on M. As usual, we take Q°(M) to be thé space of all smooth functions. We will often have cause to examine the components of a differential form. Let be a local frame field of M - that is, the are locally defined tangent vector fields, which at every point of their domain form a basis of the tangent space. Let be the dual coframe field. Then, over the domain of the frame field, any p-form 521
522 Lackey G € HP(M) can be written = 12 0h--i,W WÙ|SA--- A w*|x In the final term, the summation convention demands that each of the indices ¿1, * * * ,ip are summed over all possible values; this means that we' must take as totally skew in its indices, which means that each terms is counted pl-times in the sum. Because of the heavy use of components of multilinear objects, we will find a multi-index convention very useful. We will follow the rules: • A multi-index is an ordered tuple of indices I = (zj, • • • ,zp) where each index may take any value in its appropriate range. • The degree of the multi-index I = (i1? ■ • • ,ip) is p, and is denoted |Z|. • The concatenation of two multi-indices is denoted (IJ) and is a multi¬ index of degree ]I] + [ J|. • The summation convention is used when a multi-index appears in a term, one up and one down, then summation over all possible values of the multi-index is implied. For instance, := A - • • A The expression above for the p-form in multi-index notation, then reads 1.1.1 Exterior derivative The exterior derivative can be defined in terms of invariant axioms, but these are just restatements of the coordinate formula: A dz’1 A • • - A dzX p! drf This formula is easy to remember, but not wholly accurate. In order to use the summation convention, we demanded that the coefficient be totally skew in all its indices, which dQ^.^/dx^ is certainly not (unless p = 0). In order to make the formula for the exterior derivative conform to our conventions, we need to introduce the e-tensor: 01 î * • • j jp) an even permutation of (&i, • ♦ •, kp) Ok • • ,jp) an odd permutation of (¿1, • • • ,kp) otherwise
Hodge Theory 523 The form dx3 A dx*1 A • • • A dxtp is already totally-skew, owing to the wedge product. Therefore, dx3 A dx'1 A - • - A dx'? = 1 (p+ 1)! We use this equation to correctly rewrite the above formula for the exterior derivative: d6 = _1 dOj^ • 1 pl (p-hl)I dx31 A • • • A dx***1 A €bJ) 1 p! J dx? (p +1)! dxJ. 1.1.2 DeRham’s Cohomology Clearly, d(dO) = 0: the coefficient involves which vanishes when skew- symmetrized in j <-► k. This shows that the exterior differential forms with the exterior derivative form a cochain complex: 0 -f n°(M) ^(M) fin(M) -> 0. We use the usual notation for cocycles/closed forms and coboundaries/exact forms: Zp = ker(d : 2P(M) -> ^(M)) = {0 6 iF(M) : d0 = 0}, Bp = im(d: - Qp(M)) = {0 6 2P(M) : 0 = #}. Now, Bp C as a vector subspace; the p-th deRham cohomology of M is the quotient space /P(M) = Z* /BP > 1.2 Elliptic Complexes The elliptic complex is the most straight forward generalization of the Hodge- deRham complex. To keep close contact with our main example, we will insist on using covector bundles (although this will have no bearing on the subject until -we consider covariant derivatives). We will also use the term “form” for a global section of one the covector bundles, and even use an index convention that mimics the multi-indices above. 1.2.1 Linear Differential Operators Much as the exterior derivative can be defined invariantly, the notion of a linear partial differential operator can be formulated without coordinates. But, this would require us to introduce the notation of a jet bundle, which is too far afield. Let V and W be covector bundles over a smooth manifold M, and denote their space of global sections by T(V) and r(W), A linear partial differential operator
524 Lackey L : r(V) —* T(Uf) is a linear map that in any coordinate system (x?) of M, and coframes (fr4) and (ez) of V and W respectively, L has the form (W);= £ |S|<i Here, the sum indicates that all multi-indices S whose degree is at most £ are allowed; for a multi-index S = (si, • • • one means ai*1 dxs dx^ ‘ • • dx*h ' Under change of coordinates and change of coframe, one finds two basic invariants for a partial differential operator. The first is its order: the least I such that at least one coefficient (.A5)/4^) is non-zero for |S| = L Second is its symbol (sometimes called “principal” or “leading order” symbol). If (pj) is the induced coordinates of T*M by the coordinate system (a-7), then the object (vL)jA(x,p) = £ (^,s)/(a:)ps |S|=£ is invariant under change of coordinate. As well, this object transforms prop¬ erly under change of coframe (when changing coframe, any term containing a derivative of a change of coframe matrix will have on 6 strictly less than £ deriv¬ atives, and so not appear as a term in the above formula). Therefore, we have <rL(x9p) : Vx —► Wx a linear map. Alternatively, if we write p : T*M \ {0} —► M for the projection, then <?L : p*V —> p*W is a vector bundle homomorphism. Note that we exclude p — 0, as the symbol is always trivial there. Definition 1.2.1. The symbol of a linear partial differential operator L, is the bundle morphism oL. Note that this definition is non-traditional in two ways: many authors in¬ clude coefficients of the lower order terms, and/or would include a factor of i&. The coefficient of the lower order terms are invariants in some sense, but they are not bundle morphisms; moreover, they play on a minor role in the theory and are therefore easier to consider separately. The only reason to include the factor of i* is to have the map L w crL preserve adjoints - that is, after endow¬ ing V and W with metrics, one can compute L*, which is a partial differential operator whose symbol we might wish to satisfy <r(L*) = (crL) . This is not the case, as we have defined the symbol. We have a(L*) = (—l)i(aL)t. We will ultimately restrict to the case I — 1, at which point it is easier merely to track the negative sign rather than complexify. Definition 1.2.2. Let be a collection of covector bundles over a com¬ pact and orientable manifold M, without boundary. A complex over this col¬ lection is a selection of linear differential operators, all of order > IXY”-1) rev”) r(Vp+1) ->•••,
Hodge Theory 525 such that cP = 0. A complex is said to be elliptic if the sequence induced by the symbol map over \ {0}, is an exact sequence. 1.2-2 The Laplacian Let us consider some additional structure on our elliptic complex. We insist 1. the manifold M is oriented with volume form y/G dx, and 2. each covector bundle Vp is equipped with a metric tensor Gp. Note that “y/G dxn is strictly notation; we do not intend to place a metric on M from which the volume form is derived (although one could always to this in principle, there is no unique choice of such a metric). Moreover, the metrics Gp are selected arbitrarily with respect to the volume form on M and each other. In fact, it is not clear how there could be any relationships between these in general. The purpose of introducing these structures is to endow each space of section with an inner product: (9,^= / G^^VGdx. With this done, the connecting maps of the elliptic complex act between inner product spaces. So, we may compute their adjoints. In this context, the adjoint of a linear partial differential operator is again a partial differential operator of the same order. In general, the formula is quite complicated, so we will not reproduce it here. We will consider the adjoint of first order operators in detail, later. The adjoint of the connecting map d: r(Vrp) —► r(V’iH_1) is a partial differ¬ ential operator d* : IXV7^1) —> r(yp). Without introducing an more notation, we will also write the adjoint of d: r(Vp_1) —► r(yp) as d* : T(VP) —> r(VT”1). Definition 1.2.3. The Laplacian of an elliptic complex at degree p is the linear partial differential operator A *= dd* + d*d: T(VP) r(Vp). Clearly, if the connecting maps are of order <£, then the Laplacian is of order 2^, and its symbol is given by <rA = <r(dd* + d*d) = (-l/(<7d(ad)* + (ad)‘ad). Example [(Bao-Lackey)] This example is the main thesis of this article, and is a straightforward generalization of the construction of Bao and Lackey, [4]. Consider a geometric framework that involves a fibre bundle tt : E —► M, and
526 Lackey a metric g defined on We shall suppose that the fibres of E are com¬ pact, and naturally equipped with volume forms» C®, that have been normalized: Js« C« = n = dhn(^) and n + d = dim(.E). Finsler geometry naturally arises in this framework: E = SM, g is the fundamental tensor, and C# is the normalized fibre volume generated by the angular metric, see [4]. Therefore, we will call the construct above a Finsler- type geometry. Note that there is a natural volume form on E induced by this structure: y/gdx AC®» This formula is misleading as stated, as the form G is defined on the fibre Ex, not on the space E. However, any lift of C® (for instance by using a nonlinear connection) will satisfy C® « C® modulo da-terms. Therefore, the expression y/g dx A C® is independent of how we lift to E. In general, forms on E will not have an inner product structure. However, horizontal forms - that is, those having only da-terms - do. For 0» 0 6 QP(M), we define At first, it does not appear that we have the desired framework: a volume form on M and metrics on each grade of exterior differential forms. So we simplify the above expression. Let {/?a} be a partition of unity subordinate to a trivialization of E, and let /i,/2€Q°(M). Then /1/2 y/gdx A Cx /1/2 Í [ Pay/gÇxjdx1 where we have introduced the tensor density on M, ^(S)= Í Vs * Ea This is the canonical volume form for our manifold. This argument carries over to p-forms identically. Namely,
Hodge Theory 527 where the matrix on p-forms is given by the tensor on M, (1.1) In multi-index notation the inner product on p-forms is, {9,4>}p=~, f e^jGIJVGdx. y Di Ins Pl Jm Suppose the metric g depends only on x - that is, it is the pull-back of a Riemannian metric from M to E. As the volume of each fibre of E is one, we have VG = y/g and Gw*"*™? — ghii ...gWP recovering the traditional Riemannian construction. Yet in the properly Finslerian case, • • • G™», in general. There may exist general comparisons between the metrics of different degrees of forms, but these relationships are far from clear. In order to continue the analysis of the Hodge-deRham complex for Finsler-type type geometries, we must allow for arbitrary selection of a metric on each degree of form, which means we may as well study a general elliptic complex whose connecting maps are first order operators. 1.3 Elliptic Operators Our goal will involve a detailed study of the Laplacian of an elliptic complex. In any elliptic complex, the Laplacian is an elliptic operator, hence we may apply the regularity results from this general theory. Since this brand of analysis is tangential to our presentation, we expose some of the constructions used, but refer the reader to say Warner, [12], or Wells, [13], for a more detailed treatment. 1.3.1 Sobolev Norms Let M be a smooth and compact n-manifold without boundary, oriented with volume form VGdx; and, let V be a vector bundle over M of rank d. Let {pa} be a partition of unity subordinate to a covering of M by coordinate charts which locally trivialize V and If G F(V*), then pQ$ is supported in a coordinate chart. Thus we can view as a vector valued function on Rn with support in the compact set dom(pa). That is, we may interpret pa$ € C£°(lRn,Cd). So we may express p^ in terms of its Fourier transform PaV’W = (27r) ? f pa^(p)eix'p dp. (1-2) For fixed x, we have is canonically isomorphic to itself, the term x • p is just the natural pairing between vectors and covectors. Hence the p-variables describe points in 7^Rn. We still have in place the identification between our neighborhood in M and an open set in the pull-back of which is an isomorphism between T*Rn and
528 Lackey T*M. We may as well take the p’s as coordinates of T*M. For concreteness let us write out, for one time only, what the proper expression should be for (1.2) including all the dependence on coordinates. Say : (7 —> is the coordinate chart in question. Then (1.2) would, and perhaps should, read Pa<Kx) = (2ff)-* [ <Fadp. Jt’M We have written as if this function does not depend on x. In some sense it does not - the value is independent of x - however it is only defined for x e dom(pa). Since our partition of unity is locally finite, to any x € M there are only finitely many Pa$(p) defined. If we set po^(p) = 0 when x £ dom(pCK), then it makes perfect sense to speak of the function ^(p) = SaPaV'Cp)- This function is piece-wise constant in x for any fixed p. Summing over cu gives us the expression $(x) - (2?r)“'*' [ $(p)eix‘pdp, where we have suppressed the use of the coordinate charts </>a. The object $ is a section of the bundle p*V we get by pulling back V over p: T*M —► M. This section depends on the coordinate systems used. We define the sth Sobolev norm as Ml= [ i II^(p)IIx(i + ||?||2)^Wg<*c where the norm ||$(p) ||a is that induced by pulling back the Hermitian inner product from V to p* V. Similarly, ||p|| is computed in a local coordinate system by pulling back the Euclidean norm from by our coordinate charts. We require no metric structure on T’M, and suppose none. Although this Sobolev norm depends on the choice of partition of unity and coordinate cover, one can show that changing the partition and cover yields an equivalent norm. Definition 1.3.1. The sth Sobolev space, W*(V), is the topological vector space obtained by the completion of T(V) in || • ||s. Theorem 1.3.2 ((Sobolev embedding)). For s > [§] + k 4- 1, we have w*(v)-+rfc(y). Here, r*(V) is the fc-fold continuously differentiable sections of V. Theorem 1.3.3 ((Rellich compactness)). Fort < s, W*(V) *-»- W*(V) completely continuously. Another important estimate is as follows. Proposition 1.3.4. Let r < s < t and e > 0 be given. Then there exists a constant k > 0 such that ||^||* < + fc||^||? for all e W*(V). We will write W~(V) = W*(V) and W-~(V) = UieK V^(V).
Hodge Theory 529 1.3.2 Elliptic Operators Let us return to our differential operator L : r(V) —► F(W), which in a coordin¬ ate system (^*) of M, has the form If we apply this operator to the Fourier integral (1.2), we get |S|<£ v Using the Sobolev norms, we get the following inequality. t In the last step, we used Rellich’s lemma. Thus, L : W*+€(V) —► WS(W) is bounded. Definition 1.3.5. A differential operator, L : r(V) —► F(W), is elliptic if its symbol map crL: p*V -* p*W is a vector bundle isomorphism. Recall that when dealing with the symbol map, we use p: T*M \ {0} —> M, With the zero section removed, ellipticity can be restated as crL(p) is invertible for every nonzero p 6 T*M. Clearly, an operator can only be elliptic if the vector bundles on which it lives are of the same rank. With the additional assumption of ellipticity, we may prove a reverse inequality to the one above. The following is the key regularity estimate for elliptic operators, often called the elliptic or Garding’s inequality, and so we reproduce the proof of this from [12].- Proposition 1.3.6. Let L : T(V) —> F(W) be an elliptic operator of order Then Ms+z<C-([|£V-||, + MW. (1-3)
530 Lackey Proof: Let us denote by L the differential operator which consists only of the ¿-th order terms of L. Thus <rL — crL. As aL is an isomorphism and as M is compact, there exists a constant c > 0 so that ]|<7L(a:,p)£||2 > c||£||2 for all x € M, ||p|| = 1, and £ € p’V*. But aL is homogeneous of degree I in p so that ll<« > clhrilill“ Now, (IlK+llO = [ [ >Z< + H||2] (i + IIpII2)*^^ > [ i kpimis+n<] a+iMiw/Gtto > K [ [ 11^111(1+ i|p||3)s+Z^<to Jm Jt-m = where K > 0 is constant. Thus, IMI2+Z < K' ■ (li^ii2 + |K) < k" ■ (ii^H, + imi,)2 . But we have, ||L^|b <_ ||L^IU + IIC& ~ ¿Mb IPMb + rIMb+*-i f°r 803316 constant r > 0 since L — L is a differential operator of order I — 1. By the estimate above, we get for each small e, MUi < ^hii^+^ik < (eiMi^+fcii^n-)2 for some constant k. So inserting all this into the above, IMUr < c- (||L^||S + er||^||H7 + Mil’ll») • We choose € < jlj, subtract the ||^||«+z term to the left side of this inequality, and set our constant C appropriately to get the desired result. Corollary 1.3.7. Let L be elliptic, and Lip — £ with f € VP’(W). If ip e then ip e VP+£(V). Proof: Let -0 € V7_00(V), that is G W^V) for some t € R. If t > s, then ll^ll^ < c ■ (lirvriu + |MW = c • (Hill, + IMI,). Hence ip € Wtf+£(V) as desired. If t < s then we have . So Iterating, we eventually find that t + > s for some integer n, so we return to the case above.
Hodge Theory 531 Corollary 1.3.8. Let L be elliptic and LiJ) — 0. Then $ 6 r(V). Proof: We have i/) G W°°(V) from above,and W°°(V) — Tfc(V) for all k by the Sobolev embedding theorem. Thus ij) G r(V). Corollary 1.3.9. The kernel of an elliptic operator. L, is finite dimensional. Proof: For ij) G ker(£) and s — 0, inequality (1.3) reads Mk<o- Mo- So, any orthonormal basis of ker(£) is bounded in W£(V). Yet by Rellich’s lemma, WZ(V) <-> W°(V) completely continuously. Therefore, any orthonormal has a Cauchy subsequence in W°(y)3 which can only occur if it is finite. 1.4 The Hodge Decomposition Theorem The following is the general Hodge decomposition theorem for elliptic complexes. The proof we give is a simple modification of that in [4]. Theorem 1.4.1. Let (yp,Gp)pGZ be a collection ofRiemannian covector bundles over a compact and orientable manifold M without boundary, oriented manifold with volume form VG dx. Let —► nv”-1) nv*) 4 riv^1) -* • • • be an elliptic complex of linear differential operators of order H. and define A = d*d 4- dd* to be the associated Laplacians. Then the following hold. (a) HP(M) is finite dimensional; (b) JF>(M) ~ ker(A : r(V*) -+ r(V*)); (c) r(yp) — im d © im d* © ker A where the summands are orthogonal. Proof: Let us write — ker(* : — (V^) —> — (W)) (usually called the har¬ monic p-forms). We claim that is a finite dimensional space of smooth forms. This will follow immediately by corollaries 1.3.8 and 1.3.9 once we es¬ tablish that A is elliptic. Fix a nonzero £ G T*M. For ease of reading, let us denote A = <r(d: r^”1) — r(V*))(i) B = <r(d:r(Vi>)->r(V3>+1))(i) for the two symbols. One can easily see that aA(f) — AA* + B*B. As we are working with an elliptic complex, we have that im A = ker B. Now, V? — im A © (imA)1 — im A © (ker B)x = imA©imB*.
532 Lackey On im A, ker A* = (im A)-1 and ker B = im A imply that AA* is injective and B*B vanishes. Identically, on im B*, we have AA* vanishes and B*B is injective. Therefore <rA($) is injective, and hence an isomorphism, as desired. As we now know that H'S is finite dimensional, (a) follows immediately from (b). To prove (b), let us take 0 E Zp and consider the element 0 + Bp £ Bp as an affine subset of r(V’J>). If we take closures in W°(VP), then & + Bp is a closed and convex subset of the Hilbert space Zp, Therefore, this set contains a unique element of least norm, fa We claim that 3q is in fact harmonic. To this end, let ip e r(V'J’“1). Then by construction, the function /(e) — ||0o + ed0||2 has a unique minimum at € = 0. Yet, /(e) = |IM2 + 2^0,#) + ||#||2. In order to have a minimum at e = 0, we must have 0 = (fa dip) = (d*faip). Yet, 0 € IX?*"1) is arbitrary, and IXV*"1) is dense in W°(yp_1). Thus, d*$Q = 0. We already know that d$o = 0, therefore We note that the map 0 + Bp 0Q is well defined, linear, and injective, by construction. To see surjectivity, just note that if #o is harmonic, then 0q 4- Bp i—► fa Therefore, (b) is established. Finally for (c), note that A is self-adjoint, hence ker A = (im A)-1. Thus, (ker A)-1 — im A. Yet, we have seen that ker A is finite dimensional, so im A has finite codimension and is therefore closed. Hence, W°(VP) = TtfV' ® im ■. If we take a 6 € P(VP) then this decomposition give an rj e H'S and £ € W°(VP) such that 0 = Tj + A£. Yet, as both 0,7? are smooth, so is A£. Thus, by corollary 1.3.8, £ is smooth as well. Therefore 0 = d(d*£) + d*(d£) + 77 where and are smooth. Finally, noting that im d = (ker d“)±> we see that these three terms are orthogonal and live in the appropriate Rummands-
Chapter 2 The Weitzenbôck Formula Let M be a compact and oriented manifold without boundary, with volume form y/G dx, As usual, the volume is completely general. Suppose also that we are given a collection of Riemannian covector bundles (Vp,Gyp€z> and an elliptic complex, —<■ rev”-1) 4 r(v”) 4 r(v₽+1) ->•••• where the connecting maps are now first order partial differential operators. Our goal is to derive a general covariant formula for the Laplacian on any grade of this elliptic complex. The covariant derivatives on each of the vector bundles Vp have yet to be determined. Yet, we will need the assistance of a covariant derivative on M. As M is a volume space, the natural conditions are that the affine connection be torsion^free and volume compatible: Vy/G dx = 0. In other words, we have taken a torsion-free Sl(n)-structure on M. In the end, this connection will have no bearing on the Weitzenbock formula^ As each vector bundle Vp has a metric, it makes sense to require the co¬ variant derivative of Vp be metric compatible: = 0. This does not uniquely determine the connection; there is no notion of “torsion-free” on an arbitrary vector bundle. Therefore, we will need to use the structure within elliptic complex itself to determine the connection precisely. Unnecessary in the previous chapter, we must now take extreme care as to the geometric nature of our objects, so let us consider a connecting map d: r(Vp) -+ r(Vp+1). For 0 G r(Vp), this map will have the local structure +s>’ or with indices inserted, <d^^^+SR!9t. 533
534 Lackey The symbol term, 5: ► -(W ), has£‘(§) e )§® 7§M. That is, B' (§) is a vector-valued matrix on M. We see that such an object carries both the geometric character of the bundle and the manifold. A slightly more complicated object to consider is the symbol of the Laplacian. Since we are now dealing with a elliptic complex of first order operators, cA(p) is a homogeneous quadratic in the components of p. Moreover, the ellipticity guarantees that <rA(p) is invertible when p 0 (in fact, it is easy to see that it is positive definite). A natural construction is to take two covectors £,0 € T*M and polarize the symbol: aA(C, ip) - (<?&(£ + i>)~ aMC “ ’W) • This object is a symmetric bilinear form on T*M with values in End(Vp). To achieve our Weitzenbock formula, we will need a notion of positivity stronger than ellipticity. This “complete positivity” was introduced in [10]. 2.1 Complete Positivity Let (V, G) be a finite dimensional real vector space with an inner product G, and let W be another finite dimensional real vector space. These play the roles of the fibre of Vp and T*M, respectively, in the above. Now let S € VZ*® W*®L(V) be a matrix-valued tensor. Assume that for all £, 0 6 Wf we have S(f, 0) = S(0, £) and that , £) is a positive definite Asymmetric matrix, whenever £ / 0. This tensor is much like the symbol of the Laplacian. Definition 2.1.1. A symbol, S, as above is completely positive if, for any n and collection of n linearly independent vectors, {£*} € W, we have $3) is positive definite as an element of Mn(L(V)) = L(V™). Here, Vn has the usual norm, || (vtt) ||2 = Sa G(v<*’v«)- Note that for n = 1, this is just the usual definition of positivity for the Laplacian. We call this extended notion “completely positive” in analogy with the situation in operator space theory. As a principle submatrix of a positive definite matrix is positive definite, it clearly suffices just to consider the case n — dim(W) in which case {£«} forms a basis of W. One can also easily verify that when S is completely positive for a particular basis of W, then it is for all bases of W. Therefore, complete positivity is a property of S alone. Let us define an bilinear pairing on W ® V by (£ ® v, 0 ® w)s — G(Vi S(£, 0)w) for elementary tensors, and extending bilinearly. Clearly, from the bilinearity of both S and G, this pairing is well defined. Lemma 2.1.2. (•, -)s is an inner product if and only if'Eis completely positive.
Hodge Theory 535 Proof: Bilinearity is built into the definition. Symmetry is also clear: (C ® vt 0 w)s = G(v, S($, ip)w) « v) = G(w, £(-0, £)v) = ® w, $ ® v)e. In order to prove positive definiteness, let us take 22a ®^a € W® V. Without loss of generality, we may assume that the ¿a ax® linearly independent (otherwise we reduce the expression to this case). Then J £a ® Va > ® Vj(?)s = £(?(««, > 0 a 0 a0 as £(£*,&) is positive definite, and = 0 if and only if va = Q. Conversely, if E is not completely positive, then for some choice of ^Vq (not all zero), < 0, a/3 and so (22« ® 22,3 ® v#)s < 0, as well. Therefore, this bilinear form is not positive definite, so not an inner product. Proposition 2-1.3. Suppose E is completely positive. Then, for € Ty®L(V), the formula ^9B,t9C) = ^)C) extends bilinearly to an inner product on W ® L(V). Proof: Again, bilinearity and symmetry are clear, we need only show positivity. To this end, let 22a 6» ® Ba G W ® L(V) be such that the £* are linearly independent. Now note that ^QpB^(£ai&)Bp is a symmetric element of L(V). Let v be an eigenvector (of unit length) of this matrix with corresponding eigenvalue X. Then A = «0 = ^G(Bav,^,^)B0v) a0 = |]5>a®B«v|ll:>0. a As the eigenvectors are complete, the spectrum of 22a/? non" negative. Thus TrcQ^p &)Bp) > 0. If equality holds, then all the ei¬ genvalues of this matrix must vanish; that is 22«/? &)B$ ~ 0- But ^om above, we see that for any v € V, || 22O -Basils = 0* Thus 22a = 0 in VK® V; but the £a are linearly independent, so Bav = 0 for all a. Therefore, Ba = 0, as required.
536 Lackey 2.2 Covariant Formalism We will find several occasions where we will need to “covariantly integrate by parts.” An immediate consequence of the divergence theorem is the following Lemma 2.2.1. Let M be a compact and oriented manifold without boundary, with volume form y/Gdx. Given any connection that satisfies s= d* log y/G, the following formula holds for any global vector fields: Jm This “volume compatibility” in the lemma above is precisely the statement Va/5 dx = 0. We suppose throughout that our manifold is equipped with such a connection. This choice of connection will play no role in the final result, as we will see. 2.2.1 Torsion on an Elliptic Complex We will analyze our elliptic complex at the p-th grade: ► r(Vp-1) r(V”) 4 ny**1) ->•••, To avoid unnecessary notational difficulties, we use the following coordinate formulae: d = M'^i+B-) : r(V”-1)^r(V’), d= (B'^i +5+) : r(V₽)^r(V”+1). Fix a connection on Vp. Write the connection coefficient matrices are in a coordinate neighborhood of M as Since we have chosen covector bundles, tradition has the covariant derivative given by We can express the maps d using the covariant derivative d& = = B'V10 + (B’w('/)+S+)0. We have defined = B^w, + S+. V 1
Hodge Theory 537 ( oo) Identically, we define +5“, in which case dtp — X’ V]0+ T,+ .6. Yet this term is related to the connection on V’p_1. We define the V“00' other torsion term of by the relation d^ = Vi(X^)+^. From this, we easily compute T . Proposition 2.2.2. For all 6 G r(Vp) and</> € r(V'p_1). the following formulae hold: where • 7^=8^ +<$+, . TQ = -5|4l + wj^1 “ «4^ log v^ + S-. We see that a connection induces two types of what we call torsions: 7^. Note that although the covariant derivative on M was used to compute T“, it does not enter into the definition of the torsions. Example Returning to the Hodge-deRham complex, consider a connection on V1 = T*M. This can be considered as an affine connection on M; upon expressing (wy)** = 7*^, we have rr+9 J* J - — (J -J V >11-11« “ e €li-lk7 II “ e V ll-lk 7 Ikll-; • That is, when it is possible to interpret our vector-bundle connection as an affine connection on the manifold, our notation of torsion coincides with the usual notion. Incidentally, Cr-), =7llm -a,iog^. So our other “torsion” measures the failure of the connection to be volume¬ compatible.
538 Lackey 2.2.2 The Adjoint Operator Now we wish to compute a covariant formula for d*-: —* V?. To avoid clutter, we will use indices for the metric and forms of Vp, and R, S. T, • • • for those of Vp+1. Consider, fa, g^r + (Ttys'VGdx = - [ V,(GRSiMB\)s'}e1VGda:+ [ G^R^Cjs'eiVGdx. Jm v 7 Jm V We introduce the transpose, MSJ Gji. Using this notation, we rewrite the above expressions as = fM GIJ [-V,- + ((Tty^ty 6jVG dx, where we have used metric compatibility of the connection: = 0. There¬ fore, ^ = ^•((51)^)+^)^. 2.2.3 The Weitzenbock Formula Our present goal is to compute a covariant form for the Laplace operator, A = d*d + dd\ Before we delve into this, some notation is in order. Let us write (5l)uBy +4<(Xl*)u = a-111 +a-W, where = i((Bl)uBll + (B,l)uB,+Xl(X»)u+X,l(Xl)u) , aA'fe = 1 ((5l)u5“ - (Bl')u5' + A* (.A11 )u - X11 (4')u) • The notation for a A serves a dual role: it is the symmetric part of Su5+jL4u, and it is the symbol of the Laplacian, considered as a matrix valued symmetric bilinear form. The two terms comprising the Laplacian are ^ = -vz (^(¿I'rvu^+Vy -7;-)(Al)uV|t?+7^)(^-))ui. 77 7 7(2.1) and -V,- +(^))L'5IV|0 + (V) 7 (2.2) d*de ~ -Vy
Hodge Theory 539. Putting these two. together and taking the symmetric and skew-part where appropriate, we discover the most general Weitzenbock formula on an elliptic complex, V'J Ad = -V^o-A^Vfc) (2.3) + [v*(*a*)+<4i(V/))u - Vz)(Ai)u + CZ£))uBl - (£|)UV/)] v3-d :1 4“ V'' V' ' ' V - V/ ((S1)“^) +V,>(VU + (39)U7??] “■ Example Returning to Finsler-type geometries, the metric G^lfcl",-7j>fcp is given by formula (1,1), and the exterior derivative by We identify (Bl)jcJ = ^ne can a^JSO compute the symbol of the Lapla¬ cian, see [4], +GC^GBK . If the Finsler space is Riemannian, then this expression reduces to But note that times the Kronecker-e is just the identity on p-forms- 2.3 Existence and Uniqueness of a Connection We have derived a general Weitzenbock formula in (2,3). Yet, this expression is mostly worthless. A “good” Weitzenbock formula would have no linear term in V0. In other words, using formula (2.3), among the metric compatible connec¬ tions on Vp, we search for one for which Vfc(aA'*) + - X-)(A’)U + (7^))u5l - (B'W) = /. At this point, it is not clear what sort of equation this is for the unknown components of the connection (we have already assumed the connection is metric-compatible). Our present goal is to rewrite this equation in a fashion that makes this dependence clear. We will need to introduce some notation representing the metric-compatibility of our connection; to be consistent with the literature, we adopt the same rather
540 Lackey distasteful notation in Lackey, [10]. In index-free notation, the metric compat¬ ibility condition reads: 0 = = (dkGjG'1 - We write Pk — so that Pfc=ww+(WW)‘. Notice, 3fc(L‘) = dk{GLG-1) = GLdk{G-1) + (dkL'f + (dkG)LG-1 = PkL' + ldkltf-L'Pk, and (Pitf = Pk for all k, When dealing with, the metric Gp on Vp let us write P^ = (dj-GyC”1. Re-expressing the equation • Vfc(a^*) + ^(^))u-7;-)(Xl)u + (^))uBl - (Bl)u7J+) = / v V V v in terms of the skew-symmetric part of the connection, is quite a chore, and fortunately a wholly straightforward one. We discover that the equation above expands to crA^c^ - (o-A^w^y = (S+)uBl - (Bl)uS+ +XI(S")U -S-(Xl)u +i [(^(4>)u+xllM')u) pfcw-pj? + - Al(-4“)u +v4ii(^I)u) +(5l)udj|B“- (B»)ud||5l + - (5hjB“)uBI +(3M')(.4II)U - Xll(^|Xl)u - X'iM1)0 + (MM1)“ -(bI)ljp||'/+°o)bI| + (Bii)up||'/+oo)Bi +^P1pr°O)(.4|l)u -Xl^v'~OO)(Xl)lJ] . If r is the rank of Vp as a vector bundle, then the above statement is n * linear equations in the components of - for which consist precisely of n • ’-(r~ ■ unknowns. Another point of interest is that the dependence on the manifold connection T has completely disappeared. Moreover, it is the only the symbol of the Laplacian that determines the solvability of this equation. Let us define the right side of the above previous expression as CL Lemma 2.3.1. Suppose the symbol of the Laplacian is completely positive. Then there exists a unique solution to - (aA'^y =£l (2-4)
541 Hodge Theory among ¿A<s collection of skew-symmetric matrix-valued forms dxk Proof: As this is a well-posed linear system, we need only show that the left side of (2.4) is injective. That is, we show the only solution to <г^кш^ - = 0, (2.5) is d/p) = 0. We multiply (2.5) from the left with wjp^ and take the trace, giving 0 = Tec + (тД^ЦЮ)*) = -2 TrG ((ы^/аД*^). But from proposition 2.1.3, this trace is a norm, hence = 0, as required. Therefore, we have proven the main theorem of this chapter: Theorem 2.3.2 ((Lackey, [9])). Let M be a compact oriented manifold with volume form у/G dxf and Г be any torsion-free connection on M such that = ¿¿log^/G. Take any elliptic complex of first order differential operators on Riemannian covector bundles, (fVp,Gp),d), whose Laplacian A — d*d+dd* has completely positive symbol. Then there exists a unique metric compatible connection on each Vpf such that for 0 € Г(УР) A0 - VJGrA^V^) + H&>0 , where R^ is the curvature endomorphism ofa№>:. rW = ±atiktljk + fakatik + lv, + T^(A>)‘) + (т+у&) +т^(Тад)‘ + (aft)^ • 2.4 A Bochner Vanishing Theorem Recall that the Hodge Decomposition Theorem shows that the cohomology of an elliptic complex is isomorphic to its space of harmonic forms, Continuing the assumptions and notation of Theorem 2.3.2, we have the following fundamental inequality. Lemma 2.4.1. Let 0 € Then {0,H^0} < 0 with equality holding if and only if V0 — 0. Proof: From the Weitzenbock formula, we have, = {'7,е,а^к'7к9) + {9>'В.^&).
542 Lackey If 0 6 then {e, R^e) = -(v^, <r^k v*0). But, as the symbol of the Laplacian is assumed completely positive, this second term is negative definite. That is, (0, < 0 with (0, R<p)0) = 0 if and only if (Vj0,aAi?fcVfc0) = 0 if and only if V0 = 0. With the lemma proven, we are able to extend Bochner’s Vanishing Theorem as follows» Theorem 2-4.2 ((Lackey, [9])). Given the situation as in corollary 2.3,2. if R(*) is positive semidefinite then all harmonic forms are covariantly constant. IfRW is in fact positive definite, then = f. Proof: If R^ is positive semidefinite, then from the fundamental inequality, (0. R^0) = 0 for all harmonic forms. Hence again by the lemma, all harmonic forms are covariantly constant. If R^ is positive definite, then (0,R^0) = 0 implies 0 = 0. Hence — /.
Chapter 3 Complete Positivity of the Symbol As we have just seen, the complete positivity of the symbol of the Laplacian plays a crucial role, not only in the existence and uniqueness of the connection, but also in the generalization of Bochner’s Vanishing Theorem. We wish to examine the necessity of including this positivity assumption. In any of the classical elliptic complexes of Riemannian geometry, the sym¬ bol of the Laplacian is = gij 0 Id. This is clearly completely positive. Therefore, to look for examples where complete positivity may fail require we go beyond Riemannian geometry. This chapter focuses on the symbol of the Laplacian of the Hodge-deRhain complex on a Finsler surface, and can found in the two paper by Bao and Lackey, [5, 6]. 3.1 The Geometric Ratio The geometric ratio arises from a direct computation of the complete positivity condition for the Laplacian on one-forms. To be complete, we show that the sym¬ bol of the Laplacian on functions and two-forms is always completely positive for surfaces. Hence, the bounds on the geometric ratio given are both necessary and sufficient conditions for all the constructions of the previous chapter for Finsler surfaces. Zero-Forms On functions, D°(M), the symbol of A is always completely positive. Virtually by definition, In words, as we are dealing with a rank one vector bundle, the symbol of the Laplacian is a 1 x 1-matrix valued bilinear form; therefore, ellipticity and complete positivity are equivalent concepts. 543
■544 Lackey Two-Forms The same argument works for the complete .positivity of crA on 2-forms. Let us give a direct verification nonetheless. It is convenient to raise one index and express Here, I, Jy K, L are 2-form indices, hence can take the values (12) or (21). Ln particular, we have aAij(12)(12) = G<12)W + Gf(21)b,fr) Recall that we have defined G(yx«)= 1 /■ v G Js*m so that /^11(12X12) aA12(12X12) \ / C22 _Gm\ ^21(12X12) aA22(12)(12) ) ~ U^S ) ' _G12 J , where we introduce the first important object [ det($y)7$&. v<? Js,m Complete positivity of <rA - at the 2-form level - is now apparent. One-Forms It remains to address the issue for 1-forms. Again, it is more convenient to raise one index, and express To make the following expressions more legible, let us punctuate o-A’j7j = aA^A Our matrix aA[n](ii] ^[linw] ^[111(21] aA[12Hul <tA112)1121 irAl12U21l CTAPliM g-aMPu] o-a^I12! crAt22!^) o-AMl22! \ ffAi12H22l o-AP^22] ctAI22)!22] /
Hodge Theory 545 is / GUGU GUG12 GUG12 G12G12-re \ GirG12 G11G22 - 2« G12G12 + e g22g12 gug12 G12G12 + e G11G22 - 2e g22g12 1 \ G12G12 + e GKG12 GMG12 g22g22 J where € := - (det f g* -J det 5“) (3.1) and we introduce the second important object det/5“ := det(^) = G11 G22 - (G12)2, Recall, G^ := -jg Is.mS’3 VH justifying the notation. The symbol of the Laplacian on one-forms is completely positive if and only if the above matrix is positive definite. This is true if and only if all the principal sub-determinants are positive. We reproduce the following table from [5]: size principal sub-determinant 1x1 2x2 3x3 (G11 )2 (G11)2 /det^ I (£U )2 [/<tetps + det/^] [3/det^ - det Jps] 4x4 X[/det^ + det/^]2 [3/det^-det/<fl [3det/p# -/detp*] The lxl and 2x2 sub-determinants are clearly positive. The 3x3 and 4 x 4 are if and only if we have the following inequality involving the “geometric ratio”: 1 /det g» 3 det/g» Equivalently: 1 det j gl 3 < /detg« <3. One of these inequalities is automatic: by convexity of the determinant, / det g^ < det/^. The other inequality, however, is not always true, as we will soon see. Incidentally, the geometric ratio of a Riemannian metric is 1. 3.2 Computing the Geometric Ratio In this section, we examine the algorithm by Bao and Lackey, [6], which can be used to compute the geometric ratio systematically. The idea is, in local
546 Lackey coordinates, express the Finsler function in polar coordinates r, 0 for the y variables: y1 — r cos fa y2 =r sin<£. As we are interested only in the punctured plane, r > 0 and 0 < <j> < 2%. Since F is positively homogeneous of degree one in 0, F(x, y) = r • F(x\ cos <£, sin <£) =: r • e^x;co8 where f (x; cos 0, sin faj := log[F(x; cos fa sin <£)]. As in [6], we adopt the abbreviations J ‘ dfa d<?' Straightforward computations lead to the components of the fundamental tensor: 9u = 012 = 022 — 1 + f2 + ~f - f sin 2$ - (f2 + 5/) cos2^j f cos2<£ - (f2 +sin2^ e2/, l + /2 + |/ + / sin2^+ + cos2<£ (3-2) (3-3) (3.4) Hence, det(py) = (1 + f3 + /) e4/, Vs = Vl + ^ + Ze^. . The volume form dG, the volume Vol (a;), and the normalized volume form £ of SXM are, respectively, de = = Vi+/2+7<^, Vol(x) = = Jl + P+fd#, C = Next, we have
Hodge Theory .547 Using this and glx = ) > et cetera, we have Q11 = Jo27fg22e~2/ Jo’Ci+^+Z)«2'^ ’ G12 __ -f£* 9i2<j~2fd</> Jo’(i+/2+7)e2/^ ’ q22 _ Jo"gne~2/ Finally, the numerator and the denominator of our geometric ratio are, re¬ spectively, detjp* = det(G^), /det^ = The geometric ratio, ^/g* is therefore, (/o2’ 322 ® 2/ (Jo2’ 91! e 25 #) - (J,2* gi2 e_2/ [JT (1 + J2 + 7) df] (ft* e-2/ <#) 3.3 An Example Let us consider a Handers surface F(s,y) := yV + bi(x) y\ As usual, we define b* := bj, whose length with respect to a# is B~ yjoij w. For our Finsler structure to be strictly convex (and nonnegative), we must take B < 1 - see [2]. Fix any point xo € M. If B = 0 at x0, then the geometric ratio is 1 (as the space is Riemannian at this point). So we will assume B / 0 at xo. Let {ei, ea} be an orthonormal basis of TX0M with respect to dx* ® dx3 and let (a*) coordinate system about xo such that: a dx1 Xo = ei, Therefore, at x<^ we have F{x0,y) = vW+W + By1,
548 Lackey or in terms of the polar coordinates r, 0 of the previous section F(x0, y) = r • (1 + B cos 0). Now, to compute the geometric ratio at our point xo, we have f = log(l 4-Bcos0), giving f = f = —Bsin0 1 + Bcos0’ -B(B4-cos0) (1 + Bcos0)2 ’ /2 + i/ = l + ? + i/ = -B(cos04-Bcos20) . 2(1 + Bcos¿)2 ’ 2 4“ B“ 4" 3B cos 0 2(1 + Bcos0)2 ’ 1+?+/ = 1 1 + Bcos0’ The components of the fundamental tensor are: Ph — 1 + B2 4- 3Bcos0 — Bcos3 0, $12 = Bsin30, $22 = 1 + B COS3 <f>. and, y/g — (1 4-Bcos0)3/2. Then, ¿0, Vol(zo), and Cr, satisfy as = ■7-. 1 vi 4- Bcos0 v*-> - c = 1 x Vol(ito) -\/l 4- Bcos0 (3.6) (3.7) The volume Vol(xo) is related to the following complete elliptic integral of the first kind: dfi, ^/1 ~ fc2sin2p Substituting 0 = 2/z, and with some manipulation, (3,7) becomes
Hodge Theory 549 Fortunately, we will not need to know this value explicitly. Next, we find the miraculous formula 27T Vol(zo)’ (3.8) Although this is completely trivial to derive, only the combination y/G • Volfco) enters into the computation of the geometric ratio. Thus, the above elliptic integral above plays absolutely no role in the geometric ratio. All the terms necessary to compute the geometric ratio are: 1 f2* 1 H-Bcos3^ 2tt Jo (1 + Bcos^)2^’ 1 f2?r l + B2 + 3Bcos^-Bcos3^ 2TrJ0 (1 + Bcos^)2 ’ J_ f2” —Bsin3<p 2% Jo (1 + Bcos0)2^ and 1 /*2?r 1 1 / det^ = — f (i + Bcos^J^- We must still compute det / detfG^’). First. <?12 = 0, because the integrand is odd about <j> = ir. The remaining antiderivatives are tedious, but straight forward. The important results are: 1 f2* 1 2?r Jo (l+Bcos^)2^ 1 f2* cos<ft 2% Jo (l+.Bcos^)2^ 1 [2v cos3 2irJ0 (1 + Bcos^)2^ Therefore, G11 = G22 = In particular, the numerator of ou 1 (1-B2)v'l-B2’ —B (l-B^Vl-B2’ 2 - 3B2 - 2(1 - B2)V1 - B2 B3(l - B2)V1 - B2 r geometric ratio is det f = 4 (1 - VI - B2)2 B4 VI - B2
550 Lackey We have already determined its denominator: /detpB = 1 (1 - B2)V1-B2 ’ The geometric ratio is therefore (3-9) Figure 3.1: From [6]. The geometric ratio as a function of B. The ratio is larger than | if and only i£ B <BO& 0.9139497. In order to obey the geometric inequality, we must constrain B so that the ratio is strictly larger than Setting the formula above equal to j and rationalizing, we see that the cut-off value Ba satisfies I2(i-yr^)2 (i-b2)=b^. This gives Bo = ^v/(5-v^)(6+v^)- Numerically, Bo w 0.9139497. The geometric inequality is satisfied at xo if and only if the Riemannian length B of the drift vector at xo is constrained by 0 $ B < Ba := ^y(5-5/3)(6 + A/3). (3.10) This irrational number works for every Randers surface since when B < Ba < 1 the Finsler function is defined globally on TM, strictly positive on TM \ 0, and will have positive definite fundamental tensor. We finish with the concrete example provide in [6]. Let M be the torus of revolution T2 in Euclidean space obtained by revolving the circle (x—R)2+z2 =
Hodge Theory 551 r2 (which lies in the xz-plane) around the z-axis. Here, both r and R are constants, with 0 < r < R. A global coordinate system on T2 has angle f which goes around the z-axis, and angle 9? which parametrizes each (meridian) cross-sectional circle of radius r. We define a Randers surface with the following data: • The underlying Riemannian metric is the one induced by the Euclidean scalar product. Explicitly, (R + rcos^)2d? ® + r2d<p ® . « The underlying 1-fonn is where e is some constant to be stipulated below. computation gives B = •j> + yCoay T and so In particular: • If we set c < (R - r) y (5 - \/3)(6 + V3), then the resulting Randers surface’s Laplacian has completely positive symbol. • If we set e > (R + r) ¿y(o-^(6 + \/3), then the Laplacian of the resulting Randers surface does not have completely positive symbol at any point. • If e is between these two values, then the symbol of the Laplacian will be completely positive in a proper subset of the surface, but not so elsewhere^
Bibliography [1] Abate, M. and Patrizio, G. (1994) Finsler Metrics - A Global Approach, Springer, [2] Antonelli, P.L., Ingarden, R.S. and Matsumoto, M. (1993) The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Press. [3] Antonelli, P.L. and Lackey, B. (1998) The Theory of Finslerian Laplacians with Applications, Math. Appl. 459, Kluwer Academic Publishers. [4] Bao, D. and Lackey, B. (1996) A Hodge decomposition theorem for Finsler spaces, C.R. Acad. Sd. Paris 323, 51-56. [5] Bao, D. and Lackey, B. (1998) A geometric inequality and a Weitzenbock formula for Finsler surfaces, in The Theory of Finsler Laplacians with Ap¬ plications, Math. Appl. 459, 245-275, Kluwer Academic Publishers. [6] Bao, D. and Lackey, B, (1999) Handers surfaces whose Laplacians have completely positive symbol, Nonlinear Analysis 38, 27-40. [7] Flaherty, F.J. (1996) Dirac operators for Finsler spaces, Cont. Math. 196, 59-65. [S] Flaherty, F.J. (1998) Spinors on Finsler spaces, in The Theory of Finsler Laplacians with Applications. Math. Appl. 459, 277-282, Kluwer Academic Publishers. [9] Lackey, B. (1996) A Weitzenbock Formula for Elliptic Complexes, Disser¬ tation, University of Houston. [10] Lackey, B. (1998) A Bochner vanishing theorem for elliptic complexes, in The Theory of Finsler Laplacians, with Applications, Math. Appl. 459,199- 226, Kluwer Academic Publishers. [11] Lackey, B. (1998) A Lichnerowicz vanishing theorem for Finsler spaces, in The Theory of Finsler Laplacians with Applications, Math. Appl. 459, 227-243, Kluwer Academic Publishers. 553
554 Lackey [12] Warner, F.W. (1983) Foundations of Differentiable Manifolds and Lie Groups, Springer. [13] Wells, R.O. Jr. (19S0) Differential Analysis of Complex Manifolds, Springer.
PART 8
Finsler Geometry in the 20th-Century M. Matsumoto
Contents 1 Finsier Metrics 565 1.1 Extremals 565 1.1.1 Homogenous Functions 565 1.1.2 Regularity 567 1.1.3 Wierstrass Invariant 568 1.2 Finsier Metric ; . 569 1.2.1 Fundamental Function 569 1.2.2 Indicatrix 571 1.2.3 Locally Minkowski Space 573 1.3 Randers Metric 574 1.3.1 Rander’s Asymmetric Metric 574 1.3.2 Randers Spaces 576 1.3.3 Examples of Randers Space 577 1.4 (a, /3)-Metric 581 1.4.1 Time Measure on Slope 581 1.4.2 Finsier Space with (a,/^-Metric 584 1.4.3 Flat-Parallel Minkowski Space 587 1.5 1-Form Metric 587 1.5.1 Finsier Space with 1-Form Metric 587 1.5.2 1-Form Linear Connection . . . . 591 1.5.3 T-Minkowski Space 591 1.6 m-th Root Metric 592 1.6.1 Finsier Space with m-th Root Metric 592 1.6.2 Geodesics of m-th Root Metrics 594 1.7 Birth of Finsier Geometry 595 1.7.1 Early Works 595 1.7.2 Historical Materials 599 2 Connections in Finsier Spaces 601 2.1 Frame Bundles 601 2.1.1 Structure of the Frame Bundle 601 2.1.2 Fundamental Vector and Basic Form 602 2.1.3 Tensor Field 605 2.2 Linear Connections 607 559
560 Matsumoto 2.2.1 Connection Coefficients 607 2.2.2 Covariant Differentiation 609 2.2.3 Torsion and Curvature. 611 2.2.4 Ricci Formula and Bianchi Identities 614 2.2.5 Parallelism and the Leci-Cività Connection 616 2.3 Vectorial Frame Bundles 618 2.3.1 Tangent Bundles 618 2.3.2 Vectorial Frame Bundles 620 2.3.3 Distributions in Vectorial Frame Bundles 624 2.4 The Theory of Pair Connections 628 2.4.1 Pair Connections 628 2.4.2 H and V-Covariant Derivatives 631 2.4.3 Torsions and Curvatures of Pair Connection 633 2.4.4 Bianchi Identities of Pair Connections 635 2.4.5 D-and ¿/-Conditions 638 2.4.6 Parallel Displacement 640 2.4.7 Homogeneity of Pair Connection 641 2.5 Standard Finsler Connections ’ 644 2.5.1 Typical Vertical Connections . . 644 2.5.2 Cartan Connection 646 2.5.3 U- and P1-Processes 650 2.5.4 Chern-Rund Connections 654 2.5.5 Berwald Connection 656 2.5.6 Hashigughi Connection 659 2.6 Special Finsler Connections 661 2.6.1 Induced Finsler Structure 661 2.6.2 Induced Finsler Connection 664 2.6.3 Induction of Standard Connections 667 2.6.4 Vector Relative Connection 670 2.6.5 Barthel Connection 672 2.6.6 Cartan Y-Connection 674 3 Important Finsler Spaces 677 3.1 Finsler Space of Dimension Two 677 3.1.1 Berwald Frame and Main Scalar 677 3.1.2 Landsberg Angle and Length of Indicatrix 680 3.1.3 Torsions and Curvatures 684 3.1.4 Two-Dimensional Finsler Space with J(æ) 689 3.1.5 Equations of Geodesics in Two-Dimensional Space .... 693 3.1.6 From Geodesics to the Metric, I 696 3.1.7 From Geodesics to the Metric, II 701 3.2 Riemannian Space and Locally Minkowski Space 709 3.2.1 Deicke’s Theorem 709 3.2.2 BrickelTs Theorem 712, 3.2.3 Szabo’s Theorem 714 3.2.4 Locally Minkowski Space 715
Finsler Geometry in the 20th-Century 561 3.3 Stretch Curvature and Landsberg Space 717 3.3.1 Stretch and Shrink 717 3.3.2 Stretch Curvature Tensor 718 3.3.3 Landsberg Space 720 3.4 Berwald Space 723 3.4.1 Characteristics of Berwald Space 723 3.4.2 Two-Dimensional Berwald Space 726 3.4.3 Я-Curvature Dependent on Position Alone 728 3.4.4 C-Reducibility 732 3.5 Wagner Space 735 3.5.1 Generalized Berwald Space 735 3.5.2 Wagner Space 737 3.5.3 Wagner Space of Dimension Two 739 3.6 Scalar Curvature and Constant Curvature 741 3.6.1 Finsler Space of Scalar Curvature . 741 3.6.2 Stretch Curvature of Space of Scalar Curvature 743 3.6.3 Numata’s and Shibata’s Theorems 745 3.6.4 Isotropy 747 3.6.5 Ricci Tensor 750 3.7 Finsler Space of Dimension Three 753 3.7.1 Modr Frame and Connection Vectors 753 3.7.2 Ricci Identities 759 3.7.3 Main Scalars 761 3.7.4 Curvatures 763 3.7.5 Three-Dimensional Bianchi Identities 768 3.7.6 Semi-C-Reducibility 770 3.8 Indicatrix and Homogeneous Extension. . • 775 3.8.1 Indicatrix as Riemannian Hypersurface 775 3.8.2 Indicatory Tensor and Homogeneous Extension 777 3.8.3 Indicatorization 780 4 Conformal and Projective Change 783 4.1 Conformal Change 783 4.1.1 Geometrical Meaning of Conformal Change 783 4.1.2 Conformal Changes of Landsberg and Berwald Spaces . . 785 4.1.3 Conformally Closed Berwald Space 787 4.2 Conformally Flat Finsler Space 790 4.2.1 Conformally Invariant HMO-Connection 790 4.2.2 Conformally Berwald and Conformally Flat Spaces 793 4.2.3 Conformally Flat Space of Dimension Two 794 4.3 Conformal Change and Wagner Space 796 4.3.1 One-Sided Projective Change 796 4.3.2 4.3.2. Conformal Change of Wagner Space 798 4.4 Projective Change 799 4.4.1 Relation Between two Gi 799
562 Matsumoto 4.4.2 Metrics in Projective Relation 800 4.4.3 Douglas Projective Tensor 802 4.4.4 Weyl Projective Tensor 804 4.4.5 Projective Connection 807 4.4.6 Projective Invariants Q’s 809 4.5 Douglas Space 814 4.5.1 Equations of Geodesics of Remarkable Form . 814 4.5.2 Characteristics of Douglas Space 816 4.5.3 Douglas and Landsberg Space 818 4.5.4 Special Douglas Space 821 4.5.5 Geodesic of Two-Dimensional (a, £)-Metric 823 4.6 Finsler Space with Rectilinear Extremals 827 4.6.1 Projective Flatness 827 4.6.2 Finsler Space with Rectilinear Extremals 830 4.6.3 Rectilinear Coordinate System 833 4.6.4 Berwald Space with Rectilinear Extremals 837 5 Finsler Spaces with 1-Form Metric 839 5.1 Finsler Spaces with 1-Form Metric 839 5.1.1 1-Form Finsler Connection 839 5.1.2 Cartan Connection of Space with 1-Form Metric 843 5.1.3 Two-Dimensional 1-Form Metric 844 5.2 Curvature of Two-Dimensional 1-Form Metric 847 5.2.1 Scalar Curvature of 1-Form Metric 847 5.2.2 1-Form Metric with Constant Main Scalar 848 5.2.3 Locally Minkowski Space with 1-Form Metric 852 5.3 Conformal Change of 1-Form Metric 854 5.3.1 1-Form Cartan C-Connection 854 5.3.2 Conformal T-Flatness 856 5.4 Finsler Space with m-th Root Metric 858 5.4.1 Basic Tensors of m-th Root Metric 858 5.4.2 Cartan Connection of m-th Root Metric 859 5.4.3 Christoffel Symbols of m-th Order 862 5.4.4 Berwald Connection of m-th Root Metric 865 5.5 Stronger Non-Riemannian Finsler Space 867 5.5.1 Existence of Orthonormal Frames 867 5.5.2 T-Tensors of Cubic and Quartic Metrics 870 5.5.3 Strongly Non-Riemannian Cubic and Quartic Metrics 872 5.6 Two-Dimensional m-th Root Metrics 874 ‘5.6.1 Main Scalar of m-th Root Metric 874' 5.6.2 Main Scalar of Cubic Metric 875 5.6.3 Main Scalar of Quartic Metric 877 5.7 Berwald Spaces of Cubic and Quartic Metrics 879
Finsler Geometry in the 20th-Century 563 5.7.1 Berwald Spaces of Dimension Two With Cubic and Quartic Metrics . ; • 879 5.7.2 Berwald Space of Three-Dimensional Cubic Metric 882 6 Finsler Spaces with (a, ^-Metrics 889 6.1 Fundamental Tensor of Space with (a,/3)-Metric 889 6.1.1 Components of Fundamental Tensor 889 6.1.2 Regularity of (a, ^-Metrics ’. .890 6.1.3 Irregular (<*/?)-Metrics 893 6.2 C-Tensors of (ce,/?)-Metrics 894 6.2.1 Generalizations of C-Reducibility 894 6.2.2 Semi-C-Reducibility of (a, /3)-Metrics . 895 6.2.3 C-Reducible Finsler Spaces 897 6.3 Connections for (a, ^-Metrics ; . . . 901 6.3.1 Berwald Connections of (a, ^-Metrics 901 6.3.2 Berwald Spaces with.(a/3)-Metrics 903 6.3.3 Locally Minkowski Space with (a,/3)-Metric 906 6.3.4 Equations of Geodesic of (a,/?)-Metric 907 6.3.5 Generalized Berwald Space with (a, /?)-Metric 910 6.4 Douglas Space with (a,/3)-Metric 913 6.4.1 Condition for Douglas Space 913 6.4.2 Kropina Space of Douglas Type 914 6.5 Two-Dimensional Space with (a/?)-Metric 916 6.5.1 Relation Between Berwald Frames 916 6.5.2 Main Scalar-of (a,/?)-Metric 918 6.5.3 Two-Dimensional Landsberg Space with (a. /3)-Metric . . 920 6.6 Strongly Non-Riemannian (a^)-Metric 924 6.6.1 Riemannian Space with (a, £)-Metric 924 6.6.2 Essential Vector Fields p and Y 926 6.7 Conformal Change of (a,/?)-Metric 928 6.7.1 Conformal Change of Pair (q, /3) 928 6.7.2 IÆ-Connection 930 6.7.3 Randers and Kropina Spaces Conformal to Berwald Space 931 6.7.4 Conformal Flatness of (a, /?)-Metric 934 6.8 Projective Change of (a,/?)-Metric 936 6.8.1 /3-Change‘of (a,/?)-Metric 936 6.8.2 Protectively Related (a, /3)-Metrics 938 6.8.3 Protectively Flat Randers and Kropina Spaces 940 6.8.4 Projectively Flat Kropina Space 942 6.9 Randers Spaces of Constant Curvature 946 6.9.1 The First Condition ♦ 946 6.9.2 The Second Condition 950 6.9.3 The Form of °R 952 6.9.4 RCG Space and RCT Space 952
564 Matsumoto Editors Notes: Throughout this text we make use of the symbol [AIM] to indicate reference to [6] which was an important book, on which the present work is strongly based.
Chapter 1 Finsler Metrics 1.1 Extremals 1.1.1 Homogenous Functions Let TJ be an open region of the real number space IRn = {(tf1,.. . ?arn)} of n dimensions. We shall consider a real-valued function fix1, ...,xn,y1,...,yn) = f(x,y): U x IR" ->■ IR1 of 2n arguments (a*) and (y*), i == 1,... ,n, which is assumed to be of class C2 for x in U and for y unrestricted. Next, let C be a curve [a, 6] —► U in U of the form x1' = x*(t), a t £ b, where xi(t) are assumed to be of class C1 in [a, 5]. Such a curve having the fixed endpoints x(a), x(b) is called admissible. Now we shall consider the integral x = dx/dt, along these admissible curves and the variational problem in parametric form. Then it is well-known that each segment of class C1 of a curve C which affords a minimum to J(C) must satisfy the Euler equation dt dxi = 0. (l.l.l.l) From the point of view of our geometrical problems it is natural to require that the admissible curves be regarded as oriented curves with increasing para¬ meter t and that the integral J(C) is independent of the choice of such para¬ meters. Thus, we deal with a change t = ^(r), t e [c,d] C IR1, of parameter 565
566 Matsumoto having d$/dr > 0 and suppose where a = 4>(c) and b = If we differentiate this with respect to d and pay attention to the arbitrariness of the endpoint x(b), then we get '(*■ f )©-'(•■£)• In particular, we can take the parameter change t — pr with a fixed positive number p and obtain /(x’Sp=/(a:’(S)p)- (tLi-3) Definition 1.1.1.1. A function ^(u1,... >un) = g(u) of n arguments u — (t?) is called positively homogeneous of degree r in u (for brevity, (r) p-homogeneous in u), if the equation g(pu) = prg(u) (1.1.1.4) is satisfied for any positive number p. Thus, (1.1.1.3) shows that the integrand /(x^y) must be (1) p-homogeneous in y. Conversely, it is obvious that (1.1.1.3) implies (1.1.1.2). Thus, we have the Caratheodory Theorem: Proposition 1.1.1.1. The integral J(C) is independent of the choice of para¬ meter along the oriented curve C, if and only if the function /(x, y) is (l)p-homogeneous in Y. Remark: EYom ftx.py) == /(a, y)p we have fi(x,py) =■ /¿(or, y)p, where /¡(x, y) = d/(x, p)/ftr*. Hence the derivatives /¿(x, y) are still (1) p-homogeneous in y. Let us continue to consider an (r) p-homogeneous function g(u) in u. Differ¬ entiate (1.1.1.4) with respect to p and put p = 1. Then we have (^)wi = r5W. (1.1.1.5) Conversely, (1.1.1.5) implies regarding g(pu) as a function h(p) of an argument p, the above shows d(h(p)/pr)/dp = 0, and hence we have h(p) = prh(l), which is (1.1.1.4). Con¬ sequently, we obtain the Euler Theorem on homogeneous functions as follows:
Fmsler Geometry in the 20th-Century 567 Theorem 1.1.1.1. A function g{u) of class Cl is {r) p-homogeneous, if and only if it satisfies the condition (1.1.1.5). Further, differentiating (1,1.1.5) by v?, we have Therefore, Theorem 1.1:1.1 shows Proposition 1.1.1.2 If a function g{u) of class C2 is (r) p-homogeneous in u, then dg/duz is (r — l)p~homogeneous inu. Finally, (1.1.1.4) yields hm g{pu) = 0 for r > 0, while Um g(pu) = g{u) for r = 0. Therefore we have Proposition 1.1.1.3. If an (r)p-homogeneous function g{u) in u is continuous atu = 0, then (1) r > 0 : p(0) = 0, (2) r = 0 : g{u) is constant» 1.1.2 Regularity We considered the Eider equation (1.1.1.1) on the assumption that f{z, y) be of class C2 and (l)p-homogeneous in y. Putting fa = df/dy1 and faj = dfa/dy3, we have Since fa are (0) p-homogeneous in p, we have fajy* = 0. Hence the rank of matrix {faj) must be less than n and we can not rewrite the differential equations Ei{C) of second degree in the so-called normal form. Next, since we get Ei{C)tf = 0 from {dfa/dxtyy* — df/dxi, these n equations are not independent. Now we shall rewrite (1.1.1.1) in terms of the function (1.1.2.1) which is (2) p-homogeneous in p, so that where we put Ft = dF/dy\ Taking the normalized parameter r of C, defined by
568 Matsumoto clearly we have dr/dt = f(x>x) and f(x,x(dt/drf) = f(x,dxfdr) = 1 from the homogeneity. Hence Ei(C) = dFi/dr - dF/dx1 — 0, y = dx/dr, and, if. we put g2p <1L2'2> we obtain a(C) - ri(g) + { (g> - g) - 0, (1.1-2.3) where y — dx/dr. Therefore, if det(Fij) does not vanish, then Ei(C) are n independent equa¬ tions and can be rewritten in the normal form. Definition 1.1.2.1. If det(F^) does not vanish, then the variational problem in parametric form is called regular. 1.1.3 Wierstrass Invariant We treat the equations of extremals in the two-dimensional case as an example. Denote the function f(x,y\ x =* (a;*), y = (p*) as f(x,yypyq) wherep — x and q — y, and partial derivatives of f by subscripts. Then the Euler equations (l.l.l.l) are These are not independent, as shown above. Since f is (l)p-homogeneous in (p, q) and fp and fg are (0) p-homogeneous in (p,q) by Proposition 1.1.1.2, Theorem l.l.l.l shows /ppp + fpqq — fqpP + fqqQ — 0» which imply the existence of a function W(oj,p,p,g) given by fp? _ _ _ Zw _ xir 42 PQ & ' called the Weierstross invariant Further, since fx and fv are still (l)p-homogeneous in (p,g), Theorem l.l.l.l yields fxpP + fx^q = fxi fypP + fygq “ fy Substituting these in (1.1.3.1), it is seen that either extremal equation is reduced to the single equation fxq- fyp + (pq- qp)W = 0, (1.1.3.2) called the Weierstross form of the Euler equation. It is remarked that W(x,p,p, q) is (—3) p-homogeneous inp(q).
Finsler Geometry in the 20th-Century 569 1.2 Finsler Metric 1.2.1 Fundamental Function We consider an n-dimensional smooth manifold AP. Let Af* = TxMn be the tangent (vector) space at a point x of Mn and TMn its tangent (vector) bundle. The projection map 7rr : TAP —► AP is defined by irT(y) = x for y e Mx. For a local coordinate system {Uy (a*)} in a coordinate neighborhood U of Mn a tangent vector y at a point x = (x4) is written as yi{d/dxz}x, and hence we get a local coordinate system (tc), (z*, j/1)} in the coordinate neighborhood tt71(17) of TMn, called a canonical coordinate system of TMn. Thus TMn is regarded as a 2n-dimensional smooth manifold. Let {C7, (a?)} and (if, (a7*)} be two coordinate systems of AP having the put Xi = dx1,/dxi and for the inverse JCr « dxi/dxf. Then the following rela¬ tions are obtained between the corresponding canonical coordinate coordinates (aj\F) and (x^F).: (1.2.1.1) For instance, according to the classicjd definition a tensor field T of (l,2)-type in AP is a collection on n3 smooth functions T^k (s), called the components of T, in each coordinate neighborhood U such that in the region UC\U its components Tjk(x) and T^(x) satisfy the transformation law Trat(,X)=T^k(x)XriXi^. (1.2.1.2) We shall extend the notion of tensor field to a collection of smooth functions in each canonical coordinate neighborhood which satisfy the transformation law (1.2.1.2). Such a tensor field is called a vectorial tensor field. Ci. §2.3.2. (In [AIM] [6] this is called a spray tensor field.) For instance, the equations (1.2.1.1) shows that the collection (j/*), as func¬ tions of (x,y), is a vectorial tensor field of (l,0)-type, called the supporting element. Differentiation of (1.2.1.2) by yq gives (1.2.1.3) This shows that the partial derivatives by yi of the components of a vectorial tensor field of (r, s)-type constitute a vectorial tensor field of (r, s + l)-type. Assume that a vectorial scalar field y) is given in Mn. Then we define the notion of length of a curve. Let xi =■ xl(t), a t b, be the equations of a segment of a curve C in a coordinate neighborhood U. The length, s of the segment is given by the integral (1.2.1.4)
570 Matsumoto The manifold Mn equipped with such a notion of length is called an n-dimensional Finsler space with the fundamental function L(x,y) (or metric function, or Lag¬ rangian) and denote by F7* = (Mn,L(x,y)'), if L fulfills the conditions below (Finsler 1918; also Rund, 1959; Laugwitz, 1965; Asanov, 1985; Matsumoto, 1986). For our differential-geometric purpose we must assume the differentiability for L(x,y) with respect to the arguments хг and y*. Unless otherwise stated, wTe shall assume that L(x, y) is of class (7°° (or (7°°) in x and, in accordance with our manifold Mn being of class (or C°°). Farther, according to the discussions in §1.1.1, we shall stipulate the following four properties: (1) The length of any oriented curve does not depend on the choice of para¬ meter. That is, the fundamental function L (x, y) is (1) p-homogeneous in y. (2) The length integral (1.2.1.4) gives rise to the regular variational problem. That is, the Finslerian spray tensor field (<7^), defined by Q~F r2 = F=y, (1.2.1.5) has non-zero determinant. The differentiation symbols di = d/dxi and di = d/dy* are used throughout. We shall denote by the inverse of the matrix (5^). The tensor field g is called the fundamental tensor or metric tensor. Its components gij[x,y) and g^(x, y) are (O)p-homogeneous in у and we have yi = = diF = L(diL), (1.2.1.6) y)s/V = 2F(x, y) = L2(x, у). (1.2.1.7) If gij(z,y) be continuous at у = 0, then Proposition 1.1.1.3 shows that gij are functions of position x alone and (1.2.1.7) is reduced to a Riemannian metric L\x,y) — gij^y^. This, together with the homogeneity, leads us to the following assumption: (3) In each tangent space Mx we have a region M* such that L(x,y) is dif¬ ferentiable in y\ where M* does not contain у = 0 and is a positively conical region, that is, consists of non-zero tangent vectors у G M* for which py G M* for any p > 0. Thus M* = U M* is the domain D of definition of L. An important special case is TM\{0}. If we put t =5 yxlL(x,y), then (1.2.1.7) shows gij{x,y}tl^ = 1, that is, the Finslerian vector field I = (£*) has unit length with respect to the fundamental tensor g. However, owing to the existence of the Finsler metric L{x,y), it will
Finsler Geometry m the 20th-Century 571 be convenient to introduce the absolute length |y| of a tangent vector y* as the value of L(xt y). If L(x, y) is positive, then we have from the homogeneity, and thus I = (¿*), called the normalized supporting ele¬ ment, has the unit absolute length. This leads us to the fourth assumption: (4) The fundamental function L(x, y) is positive-vcdued for any y belonging to the positively conical region MJ, This positivity assumption is desirable, because the integral (1,2.1.4) gives the length of the curve. Further, the positive definiteness of gij(x,y)v*vi desirable owing to (1.2.1.7) and |v|y = is called the length of v relative to y. But, it is well-known that the Riemannian metric appearing in the theory of general relativity for instance is not positive definite and, as will be seen later on, the positive definiteness of gij(x, y)v1^ gives rise to the exclusion of some interesting examples of Finsler metrics. Likewise, we do not require the symmetry property of L : L(x,y) — L(x, — y). Consequently, throughout the following, we shall be concerned with Finsler , metrics satisfying the four assumptions above, and these will be sufficient for our purposes. (1.2.1.6) can be written as yi/L = gijt? = fyL. Thus, ¿i = Vi/L = diL. (1.2.1.8) 1.2.2 Indicatrix The extremal of the length integral (1.2.1.4) is called a geodesic of the space. From (1.1.2,3) it follows that a geodesic is a curve given by the differential equations (1.2.2.1) where s is the normalized parameter, that is, the arc-length, and (1.2.2.2) which are quite important quantities in Finsler geometry. We now introduce the Christoffel symbols 7^(2:, y) constructed from gij (s, y) with respect to x : (1.2.2.3) Then we get Z&fay) = 7oo = 'YjkVW- (1.2.2.4)
572 Matsumoto Throughout this book the index 0 is used to show transvection by yl. For instance, (1.2.1.6) and (1.2.1.7) may be written as yi— gw and L2 — poo- On the other hand, we have the Christoffel symbols constructed from p^ (x, p) with respect to y which give rise to the C-tensor: Cjikfay^gihCljk _ (dkgij + fygik - digjk) 2 (1.2.2.5) 2 * It is easy to show that Cjik are symmetric in i, j and k and satisfy Cijkfx, y)yk — CijQ = 0 by Theorem 1.1.1.1. Each tangent space Mx may be regarded as an n- dimensional Riemannian space with the.Riemannian metric gij(xiy)dyldyj where x=(xT) is fixed. Then, are the Christoffel symbols of Mx which are constructed from the Riemannian metric above. This is a very special type of Riemannian space, since lijk of a general Riemannian space are not symmetric in all indices. As has been seen, for y with L(xyy) > 0 we have <L2'2'6> The set Ix — {y G Mx\L(xyy) = 1} is called the indicatrix at the point x, It is a hypersurface of Mx and may be written in parametric form yi = p^u*), a = l,...,n — 1. Ix has the induced Riemannian metric SaßW = 3ij (®, B* = (1.2.2,7) Differentiate L(xyy(u)} = 1 by ua. On account of (1.2.2.6) we get ABX-0, (1.2.2.S) which implies that £» are covariant components of the normal vector of Ix. If we introduce the angular metric tensor hijfay) =9ij=L(didjL\ (1.2.2.9) then (1.2.2.7) may be written as Saß («) = fey (s, y(u)) B^B£. (1.1.1.7') Thus the Riemannian metric gaß of Ix may be regarded as the induced one from the angular metric tensor h.
Finsler Geometry in the 20th-Century .573 Since djL is (0)p-homogeneous in y, (1.2.2.9) shows hijix.y^y1 = 0, so that the rank r of the matrix (hy) must be less than n. If r is less than n — 1} then (1.2.2.9) implies a contradiction to the regularity: rank of (pij) equals n. Therefore we have Proposition 1.2.2.1. The components hij of the angular metric tensor h con¬ stitute the matrix of rank n — 1. A non-trivial solution (v*) of the system of algebraic equations 7^ = 0, i = l,...,n, must be proportional to y1. Consider the Taylor expansion of the function L2(x, I) at a point ¿o of lx * L*(x,t) = L2(x,£0) + [SiX2]^ - 4) where [... ] denotes that £ = 4) is assumed in the expression. From the homo¬ geneity of L we have [ftLVo = 2I2CMo), [¿Wo = &*21- Hence the expansion is reduced to L2(x,£) = gi$(x&)£W +... and the homo¬ geneity yields i2(x,i/) = +... ; (1.2.2.10) For yc e Ix the set Ix(yo) = {y^ijfayoWyi = 1} is called the the osculating indicatrix at y^ Therefore it follows that the indicatrix Ix is osculating with the osculating indicatrix Ix(^o) at each point yo- Ix(sfo) is a quadratic hypersurface with the origin y = 0 as its centre. If the space F71 is Riemannian, then the indicatrix coincides with its osculating indicatrix at every point. 1.2.3 Locally Minkowski Space We are concerned with the fundamental tensor gij(x,y) of an n-dimensional Finsler space F71 = (AP, L(z, 3/)) and the two extreme cases: g^ depends on x alone and on y alone. The dependence on x alone is nothing but a characterization of Riemannian metric and Cij* — 0 from (1.2.2.5). Since Cij* are components of the C-tensor, the dependence on x alone does not depend on the choice of coordinate system. On the other hand, the dependence on y alone is characterized by dkpij = 0. Then (1.2.1.7) shows that g^ depend on j/ alone, if and only if L depends on y alone. But dzL and d^gy do not constitute tensor fields and hence we must state as follows: Definition 1.2-3.1. A Finsler space Fn w (Afn, £(x, y)) is called a locally Minkowski space, if Mn is covered by coordinate neighborhoods in each of which
574 Matsumoto L does not depend on the arguments (z£). Such coordinates (x1) are called adapted. It is well-known that a real vector space V is called a Minkowski (Banach) space, if a Minkowski (Banach) norm is defined on V. A Minkowski (Banach) norm M of a point v of V is a real-valued function with the following properties: (1) |(v|| 0, and ||v|| = 0 if and only if v == 0, (2) ||®1+V2|| S ||vi|| + IM> (3) ||pv|| = p||v|| for any p > 0. If V is n-dimensional and we put v = viei in a base (e^), i = 1,.. . ,n, of V, then the norm ||v|| is represented as a real-valued function ||v|| = /(v*) of n arguments v\ Consequently, the tangent space Mx at a point x of a Finsler space F71 » (Mn,L(x,y)) can be regarded as a Minkowski space with the norm function L(x,y), though, rigorously speaking, L(x,y) must also satisfy the first two of the three conditions above. If L does not depend on x = (xl) in an adapted coordinate system (U, (z*)), then (CZ, L(j/)) may be regarded as a domain of a Minkowski space. We shall consider locally Minkowski space again in §3.1.3, where some coordinate-free characterizations of such space will be given. 1.3 Randers Metric 1.3.1 Rander’s Asymmetric Metric We quote at length the first two sections of G. Randers’ (1941) original paper: 1. Introduction. In the geometry of affinely connected spaces the metrical and the affine properties are completely independent. The characteristics of the space described by the curvature tensor are derived from the definition of parallel displacement, or covariant derivation. Only when we want to compare vectors of different directions, the need for a metric arises. We are, however, quite free in the choice of metric. That is, we can choose at will measuring units in every direction at each point of the space. The locus of the end points of all the unit lengths radiating from a certain point zq. is called the indicatrix. In Euclidean geometry this locus is a (hyper-) sphere around zo- In Riemannian geometry the indicatrix is a quadratic surface around xq with coefficients equal to the fundamental tensor g^, which already exists at each point by the definition of the Riemannian parallel displacement. There would be nothing to prevent the choice of another tensor, h^, to define the metric. As far as the application to physical space-time goes, however, there is no indication of the need for a new tensor, because there is no need for twenty independent potentials. However, the Riemannian metric has one
Finsler Geometry in the 20th-Century 575 property which does not seem quite appropriate for the application to physical space-time, and that is the perfect symmetry between opposite directions, for any coordinate interval. Perhaps the most characteristic property of the physical world is the unidirection of time-like intervals. Since there is no obvious reason why this asymmetry should disappear in the mathematical description it is of interest to consider the possibility of a metric with asymmetrical properties. 2. The eccentric metric. It is known that many reasons speak for the neces¬ sity of a quadratic indicatrix. The only way of introducing an asymmetry while retaining the quadratic indicatrix, is to displace the center of the indicatrix. In other words, we adopt as indicatrix an eccentric quadratic (hyper-) surface. This involves the definition of a vector at each point of the space, determining the displacement of the center of the indicatrix. The formula for the length ds of a line-element dx* must necessarily be homogeneous of first degree in dx*. The simplest “eccentric*’ line-element possessing this property, and of course being invariant, is ds = k^dx* + (g^dx^dary2, (1.3.1.1) where is the fundamental tensor of the Riemannian affine connection, and is a covariant vector determining the displacement of the center of the indicatrix. If a space of Riemannian affine connection is given, we are, as mentioned earlier, completely free in our choice of metric, and consequently free in our choice of the vector This vector, therefore, does not describe any properties of the Riemannian space considered, but only the properties cf the units chosen for measuring intervals. To change from one vector field k^ to another, only means to change from one system of asymmetrical units to another. By each of the measuring systems we can determine certain paths in the space, defined by the condition 6 i ds = 0. (1.3.1.2) We may now divide the variety of arbitrary vector fields k^ into classes giving the same paths (1.3.1.2). If we only allow changes of units within each class, the vector k^ has attained a certain significance besides describing the units chosen, namely it defines a set of paths. The only change of kp which will not affect the path by (1.3.1.2) is the addition of a gradient vector, X = + (1-3.1.3) The change (1.3.1.3) will according to (1.3.1.1) result in the addition of a total differential dt/> to ds. Hence ds^ds+fj^dx* (1.3.1.4) and this addition does not affect Bq. (1.3.1.2). The change of units corres¬ ponding to the transformation (1.3.1.3) will be called a k transformation. The fundamental difference from the gauge-transformation of Weyl should be no¬ ticed. Weyl’s transformation is change on units at different points, while the k transformation is a change of units in different directions at the same point.
576 Matsumoto 1.3.2 Randers Spaces Definition 1.3.2.1. In an rz-dimensional smooth manifold Mn a Finsler metric r(z5^) = a(x,y) + p(x,y) is called a Randers metric^ where a(s, y) is a Riemannian metric {aijtx^y1^}1^2 in Mn and /3(a;, y) is a differential 1-form bitxjy* in Mn. In the following we shall admit even a so-called pseudo-Riemannian indefinite metric Oi(x,y). The Finsler space F71 — (Mn, L = a + /3) with a Randers metric is called a Randers space. We first consider a geodesic. Let r be the Riemannian length, that is, dr = a(a;,d2:) and put yi = da?/dr. Then a geodesic as given as (l.l.l.l); S(C) = {if> Sa-I , fd(^) _ W_\ = 0 dxi J 1 dr dxi J From a2 = aijtxjy'yi we have da _ akiy* da 2°°fc dyk ~~ a ’ dxk ~~ a ’ where 7^(3?) — Ofr7i* are Christoffel symbols constructed from %■. Then we a a have the above first {• • ♦} as in the form of (1.1.2.7): The second {♦ ■ •} is easily written as 2Jio where we have put Therefore, we have differential equations of geodesics in the form (1.3.2.1) (1.3.2.2) where we put Fj (a?) = a^Frj and (air) is the inverse of the matrix (%•)• It is obvious from (1.3.2.1) that the k transformation does not affect the equations (1.3.2.2), as Randers mentioned. We are concerned with the problem of positivity of a Randers metric, on the assumption of positive definiteness of a. Suppose that L is positive: {ay(s)yV}1/2 > ~bi(x)y\ Vy‘^0. If we take y* = —S’ (= — airbr) in the above, then we have {a^btb^2 > btB1.
Finsler Geometry in the 20th-Century 577 If we denote by b the length of the vector bi with respect to a, then the above shows b > b2, that is, 1 > b. Conversely, if 1 > b holds, then the well-known inequality |ayBV| < W * 0 gives IWI < w?}1/2, which implies L = a + p > 0, 0. Next, suppose again that L is positive for Vi/ / 0. Then L(x, y) x L(x, — y) > 0, which is written as ~ (W)2 = (ay - mW > o. Hence dij — fybj is positive definite. Conversely, if —bify is positive definite, then we get L(x, y) x L(x, —y) > 0. Since L(x,y) -I- L(x, —yj — 2a(x, y) > 0, both L(x, y) and L(e, —y) are positive. Therefore we have Theorem 1.3.2.1. A Randers metric L = a + fi is positive-valued for any y, if and only if (1) the length of bi with respect to a is less than 1, or (2) Oij — bibj is positive definite, provided that o? is positive definite. If we put Yi = Oiryr and kij = Oij - (Yi/a)(Yj/oi) (the angular metric tensor of the Riemannian a), then the fundamental tensor gij(x,y) of the Randers space is written in the form + + + (1.3.2.3) and it will be shown in §6.1 that det (ffy) = + det (ay). (1.3.2.4) Consequently, the Randers metric is regular, provided that L is non-zero. 1.3,3 Examples of Randers Space We consider the equation L(x, y) = 1 of the indicatrix Ix at a point x of a Finsler space F* = (Mn, ¿(s,y)). Definition 1.3.3.1. Let f(x,y) — 0 be the equation of the indicatrix Ix. Then the fundamental function L(x, y) of F™ is given as an implicit function satisfying f(x,y/L) = 0. The method of getting L from — 0 is called Okubo’s method. In fact if we put L = 1 in f(x, y/L) = 0, then we get the equation f(x, y) — 0 of Ix< This L is homogeneous of degree one in y because f(x, (py)/(pL)} = 0
578 Matsumoto for any non-zero p. We must, of course, choose L from f(x, y/L} = 0 which satisfies the conditions for the fundamental function. This method has been communicated from K. Okubo to the author. Example 1.3.3.1. We shall show a geometrical example of a Randers metric of dimension two. Let E2 be a Euclidean plane with an orthonormal coordinate system (x,y). At an arbitrary point P(x,?/) of it = E2 — {0} we define the indicatrix curve Ip (Fig. 1.3.3.1) such that Tp is a circle with the center O = (0,0) and radius e OP, where e(x, y) is a positive-valued function. Then Ip is given by the equation (1) u2 4- v2 — (14- e)(x2 + ?/2), in the current coordinates Since the tangent space of E2 at P can be identified with E2 itself, we may put u = x 4- x and v = y.+ y* Then (1) is written as (2) e(x2 4- y2) - 2(xx 4- yy) - (±2 + ÿ2) = 0. Now we apply Okubo’s method to (2): Replacing (x,ÿ) by (x/L^y/L) we get (3) e(x2 4- y*)L2 ~ 2{xx 4- yy)L - (±2 4- y2) = 0. This algebraic equation for L has two solutions, one is positive and the other negative. We choose the positive solution where we put (1.3.3.1) Thus, we obtain a Randers surface (tt, L) where the Randers metric L is given
Finsler Geometry in the 20th-Century 579 by (4). From A (as, y, da, dy) = ¿(log y/x2 + y2) it follows that, provided that e = const., Fij defined by (1.3.2.1) vanishes and (1.3.2.2) shows that this Randers space has the same geodesics as that of the Riemannian space (7r, ^P2 + A2) (but is not Riemannian)! Of. 6.9. Example 1.3.3.2. We shall give another example of Randers metric in tt. At a point P(z, y) we define the indicatrix Ip (Fig. 1.3.3.2) such that Ip is a conic section with the focus P, the directrix g through O and orthogonal to OP, and eccentricity e(x,y). Then Ip is given by the equation (1) y/(u -x)2 + (v - ^)2 = e(xu + yv) \/x2 + y2 ’ in the current coordinates (-u,v). It is noted that in case e > 1, Ip is assumed to be on the positive side of xu + yv. Putting u — x + x and v = y + y, we get from (1) (2) y/(x2 4- y2)(x2 4- y2) — e(x2 + y2 + xx + yy). Then Okubo’s method leads us to the fundamental function (3) where p and A are given by (1.3.3.1). Thus we obtain a two-dimensional Randers space (tf, L) with the L above. It is obvious that this Randers space has the
580 Matsumoto same geodesics as that of the Riemannian space (?r,p/e). We are again concerned with geodesics of the Randers planes given in the two Examples above. Let us deal with them in the polar coordinates (r, 0) where x = rcos0 and y — rsin0. Putting (u,v) — (r,0), (1.1.3.2) gives the equation of geodesic: L„ - LBu + (uv + w) = 0. (1.3.3.2) Proposition 1.3.3.1. Assume that the fundamental function L(r,fyu,v) is written as f(01r/r') which is p-homogeneous function f (21,22) ¿n (21,22). Then the geodesics are logarithmic spirals r = aeM with constants a and 6, provided that f has non-zero second derivatives. Proof: Denoting /a = df/dza, a — 1,2, we have Iu = 7> i« = o, ¿».-/22/r3, in> = -A2(J). Hence (1.3.3.2) in this case is written as fl2UV2 + (vik — UV)f22 — 0. The homogeneity gives /21V -F fyfa/r) — 0 and, in consequence, the above is reduced to u/u = rjr + v/v, provided that uvfa 0. Then two integrations lead us easily to the equations of logarithmic spirals. In terms of (r, 0) we get (1.3.3.1) in the form (a) P2 =(£)* +(¿A (b) A=£. (1.3.3.3) Therefore we have Corollary I.3.3.I. If e is a constant, then the geodesics of the Randers planes given in Example 1.3.3.1 and 1.3.3.2 are logarithmic spirals with the origin 0(0,0) as the pole.
Finster Geometry in the 20th-Century 581 The Figure I.3.3.3 shows three spirals Cq(—), Ci(—) and ) of the set Cn = C(an,6n),n = 0,:±:l,±2, fromP(ri,0i) to Q(r2»fe)» where ti = exp(on^i + bn), r2 = exp (on(02 + 2n?r) 4- bn). Figure 1.3.3.3 Ref S. Ho jo [55] and M. Matsumoto [109], [110]. There is an extensive modem literature on logarithmic spirals occurring in organic nature. The clas¬ sic work, On Growth and Form, by D’Arcy Thompson, will introduce the reader to a myriad of creatures, from shell animals to mountain goats, which display precisely this spiral pattern. Whereas, there is good reason to model physiolo¬ gical growth via some minimization criterion, in particular, some Finsler metric, just how such a choice may be made is still unknown. No progress has been made since publication of Thompson’s great work, over 80 years ago. 1.4 (a, /3)-Metric 1.4.1 Time Measure on Slope We shall quote P. Finsler’s letter to M. Matsumoto in 1969, here translated from German into English by Matsumoto. In astronomy we measure distance with time, in particular, in light-years. When we take a second as a unit, the unit surface is a sphere with the radius of 3000,000 km. To each point of our space is associated such a sphere; this defines the distance (measured in time) and the geometry of our space is the simplest one, namely, the Euclidean geometry. Next, when a ray of light is considered as the shortest line in the gravitational field, the geometry of our space is a
582 Matsumoto Riemannian geometry. Furthermore, in an anisotropic medium the speed of the light depends on its direction, and the unit surface is no longer a sphere. Now, on a slope of the earth’s surface we sometimes measure distance with time, namely, the time required to walk,such as seen on the guide posts. Then the unit curve, taking a minute as a unit, will be a shorter distance on an uphill road than on a downhill road. This defines a general geometry, although it is not exact. The shortest line along which we can reach a goad, for instance, the top of a mountain, as soon as possible, will be a complicated curve. Matsumoto gave an exact formulation of a Finsler surface described in the second paragraph of this letter (for the case of a slope and a downhill walk) as follows: Consider a plane tt, indicated by the quadrangle AB CD in Figure 1.4.1.1, inclining to the horizontal plane ABEF at an angle e. Suppose that a per- son, starting from a point 0 of 7r, walks on 7r along a straight road OU at an angle 3 with the direct downhill road OX. If the person is able to walk v meters per minute on a horizontal plane, he arrive at a point Qt, after t minutes where OQt = vt. Actually, the earth’s gravity g • sine acts on him; the component (g ■ sine) sin6 perpendicular to OU is cancelled by his muscles. Thus, another component (g-sine) cos 3 in the direction OU pushes him QtPt = (i23/2) sine cos 0 ahead in t minutes. D F B A Figure 1.4,1,1 In Figure 1.4.1.1, GH is normal to the plane 7r and HPt is orthogonal to OU, and then GPt is orthogonal to OU. Thus QtH = (QtG) sine and QtPt = (Q*G) sine cos 0. Consequently, he can walk the distance OPt = vt + (^/2) sine cos 0 along
Finster Geometry in the 20th-Century 5S3 OU in t minutes. The result is that his velocity along OU is equal to v + (tg) sins cos 3, which certainly depends on t. But, according to our experience, we walk along a road of constant gradient at a certain constant speed with some resistance (friction) of the ground and our own control. Yet, it is known in mechanics that a body falling in the air reaches a constant speed called the terminal velocity due to the resistance (friction) of air. Therefore, it seems natural to suppose he walks OPt^i = v + (p/2) sins cos 0 meters in a minute and this velocity remains unchanged hereafter. Consequently, we may state the Principle. With respect to the time measure, a plane with an angle e of inclin¬ ation can be regarded as a Minkowski plane; its indicatrix curve is a “limaQon” (Figure 1.4.1.2) given by r = v + acos0, (1.4.1.1) in the polar coordinates (r, 0) whose pole is the origin O and the initial line is the direct downhill road OX. where a = (g/2) sins. Next, suppose that we have an orthonormal coordinate system (x,y7z) in an ordinary space; the (a?, y)-plane is the sea level, z (^ 0) shows the altitude above sea level, and a slope of a mountain is regarded as the graph S of a smooth function z = f(x,y) of two arguments. The plane 7rp tangent to S at a point Pfay, ffay)) is spanned by two vectors. Bi := d(OP) dx = (i,o,A), = (0,1, A). Suppose that the plane ABCD of Figure 1.4.1.1 is the origin is now
584 Matsumoto the point P. Let ei and ez be unit vectors on OX and OY of Figure 1.4.1.1. respectively, that is, ei is the direct downhill vector and e2 the horizontal vector on 7TP. Then we have ^1 “ fyi fx 4" Zy )i = fzi 0), where we have put p_1 = 1 + f£ + f~ and ç”1 = y]fl + fy • Next, any vector on ttp is written as a linear combination xei +yez, and the limaçon (1.4.1.1) is written in form x2+y~ = v\Z®2 + ÿ2 4- ax. (1.4.1.2) Since (a?, y) may be regarded as a local coordinate system of the two-dimensional manifold S, any vector of np is written as xBi 4- yBz. Thus we have xei 4-ÿes = ¿Bi 4- yBz, w’hich implies * = - (<ify)y, y = + (qZxJît Therefore we get xfx 4- yfv = —xpjq and s2 4-ÿ2 + (i/x + ÿfy)2 = x2 + y2. We consider a = (p/2) sine. Since the unit normal vector £3 of S is equal to p(—fx9 —fyj 1), we have cose = p, sins = y/l—p2 — p/q and a = pg/2q. Then ax = -(g/2)(xfx +ÿ/y). Finally (1.4.1.2) is written as i2 + ÿ2 + (xfx + ÿfy)2 = vy]x2+ÿ- + (ifx + ÿfyf* - (I) (xfx + ÿfy), and Okubo’s method leads us to the fundamental function £(x’y’i,ÿ) = w = 2 ’ (L4-1-3) a2 = ®2 + V2 + (xfx + ÿfy)2, 0 = ifx + ÿfy. (1.4.1.4) Consequently, a slope of a mountain can be regarded as a two-dimensional Finsler space with the fundamental function (1.4.1.3). Note that a2 of (1.4.1.4) is the induced Riemannian metric of S as a surface in an ordinary 3-space. Ref M. Matsumoto [97], §16; [100]. T. Aikou, M. Hashiguchi and K. Yamau- chi [3]. 1.4.2 Finsler Space with (a,/?)-Metric Generalizing the Randers metric and the slope metric, we can state the
Finsler Geometry in the 20th-Century 5S5 Definition 1.4.2.1. The fundamental function L of a Finsler space F* = (M,£) is called an (a,/3)-metric, if L is a (1) p-homogeneous function of two arguments o(®. y) = {ay (®)yV}1/2, V) = bit^y*, where a is a Riemannian fundamental function and ß a differential 1-form. The space Pn = (M, a) is called the associated Riemannian space and the covariant vector field bi the associated vector field. Ref. M. Matsumoto [86], [106]. In the following we shall denote by 7}k(z) the Christoffel symbols constructed from that is, the connection coefficients of the Levi-Civitä connection of Rn, and by the comma (,) the covariant differentiation with respect to this connection. Example 1.4.2.1. We are again concerned with the plane — E2 — {0} and orthonormal coordinate system- (x,p) on it. At an arbitrary point (P(x,y) of 7Г the norm ||PP|| from P to a point R(u,v) is to be defined by ||PP|| = PR/OH (Figure 1.4.2.1), where PR is the Euclidean length and OH is the one of perpendicular to PR. From the area of ДОРР we have OH = \yu — xv\/y/(u — x)2 + (v — y)2. Then, if we put и = x + x and v = у+ y, then we get ||PP|| = (x2 + y2}/yx — Consequently, we have the Finsler metric 2 L(x,y,i,y) = , (1.4.2.1) where p is defined by (1.3.3.1) and (xy - yx) (x2+y2) * (1.4.2.2) So, we obtain an (a,/?)-metric L = or/ß where a = p and ß = |/x|. Since we have p/x, y, dx, dy) — d(Arctan(p/a;)), we get д = 0, (1.4.2.2') similar to (1.3.3.3). Thus, Proposition 1.3.3.1 shows any geodesic of (?r,L) with L given by (1.4.2.1) is a logarithmic spiral with 0 as the pole. The indicatrix of this Finsler plane consists of two circles which contact the line OP at P and have the diameter equal to the length of OP, because we have OH = PQ which implies ||OQ|| = 1.
586 Matsumoto Ref. W. Wrona [171]. Figure 1.4.2.1 Generalizing the above (a, /?)-metric we have a class of interesting (a, ^-metrics defined as follows: Definition I.4.2.2. The (a, ^-metric L ==• a2//? is called a Kropina metric and L = am+1/3~m (m / 0, —1) a generalized m-Kropina metric. Ref. V.K. Kropina ([79],VKK2); M. Matsumoto [86]; C. Shibata [148]. Example 1.4.2.2. R.S. Ingarden [6] exhibited a Finsler space in his theory of thermodynamics as follows: Let X be a manifold of thermodynamical states of dimension n and x — (x1,,.., xn) a coordinate system in it. We introduce X* = R1 x X and = (/2 = 0,1,...,«), y* •- (y**) = (y°,!/)> y° '■= V = di/du, y := dz/du, where u is a parameter. The Finsler space Fn+1 — (X*,D,L) is defined as follows: D is defined by v > 0 for each x € X and L : D —#• R1 is given by . := 1 I dx,idxf^ fx'=z 2v
Finsler Geometry in the 20th-Century 5S7 Where S(z,z/) is the relative entropy (information gain between states x and a/). F*14-1 is said to be thermodynamic geometry or thermodynamic space-time with thermodynamic time t = x° and thermodynamic time-direction component v^=yQ = dt/du. This metric can be written in the form dt which may be regarded as a special case of a Kropina metric. For more detail on the mathematical theory of this metric, see C. Shibata [148]. In case of the canonical distribution of an ideal gas we have n = 1 and L(t,x,v,y) = (W V for a certain constant K. It may be called a parabolic Finsler plane. Note: this metric ds of X* is unusual as a Kropina metric, because this a2 = gij(x)dx%dx? is a Riemannian metric of the n-dimensional X, but not of the whole (n + l)-dimensional X’. 1.4.3 Flat-Parallel Minkowski Space We now introduce a special class of Finsler spaces F71 = with (a, £)-metric. Assume that the associated Riemannian space — (m,a) is locally flat, that is, has vanishing curvature tensor. Then it is well-known that there exists a covering by local coordinate systems {U, (z*)} of M, called adapted coordinate systems, in each of which the fundamental tensor &ij has constant components, so that all the Christoffel symbols 7^ in Rn vanish. Further as¬ sume that the associated vector field bi is parallel in R”, that is, the covariant derivatives vanish. This means that all the components bi are constant in the adapted coordinate systems (a;*). Consequently, both a and /3 are functions of y* alone and Fn is locally Minkowski, independent on the form of the func¬ tion L(a,£). Therefore, we get a special class of locally Minkowski spaces with (a, /?)-metric as follows: Definition 1.4.3.1- A Finsler space with (a, /?)-metric is called flat-parallel Minkowski, if the associated Riemannian space is locally fiat and the associated vector field bi is parallel in jR*. 1.5 1-Form Metric 1.5.1 Finsler Space with 1-Form Metric An n-dimensional smooth manifold M is called completely parallelizable, if there exist n smooth vector fields ba, a = 1,... ,n, which are linearly independent at every point. Let b# be components of ba in a local coordinate system. Then the
Matsumoto det(b^) does not vanish and, in consequence, we have n linearly independent covariant vector fields a®, whose components constitute the inverse matrix (af) of (b^). Thus, we get n linearly independent differential 1-forms = a?(x)dx\ The converse is also true; n linearly independent differential 1-forms dx) = af(x)dxi induce n linearly independent contravariant vector fields whose components constitute the inverse matrix (b^) of (a^). Definition 1.5.1.1. Assume that an n-dimensional smooth manifold M.admits n linearly differential 1-forms aa(x,dx) = a?(x)dx\ a — l,...,n. A Finsler metric L(a?) is called a 1-form metric, if L(a?) is a (1) ^homogeneous function of n arguments Thus, each tangent space Mx of a Finsler space F71 = (M, L(aa)) with 1- form metric may be regarded as congruent to the Minkowski space Vn with the norm L(va). Ref. M. Matsumoto and H. Shimada [126]. Y. Ichijyo [57] have defined the notion of {V, H}-manifold based on the Minkowski norm and linear Lie groups leaving the norm invariant. His {V, manifold is essentially same as a Finsler space with 1-form metric. If we put F = L2/2 and denote by the subscripts a, . of F the partial derivatives with respect to a?, aP,..., then we have the fundamental tensor and the C-tensor Cijk of a Finsler space with 1-form metric as follows: = (1.5.1.1) Thus, det(Ftt0) 0 must be assumed. Example 1.5.1.1. Some special 1-form metrics have appeared from the stand¬ point of mathematics, physics and biology as follows: (1) Berwald-Mo&r metric. £(®,y) = (yV...»n)1/n, defined in a local coordinate neighborhood, given by A. Moor [130]. Its generalization ¿(a;^) = (a1a2...an)1/n was given by G.S. Asanov [8]. (2) A special Randers metric L = {(a1)2 + • • • + (a**)2}1/2 + fca1 was given by Y. Ichijyo [57], where k is a constant.
Finsler Geometry m the 20th-Century 589 (3) An m-root metric was studied by H. Shimada [152] and, P.L. Antonelli and H. Shimada [7]. It is called the ecological metric [6]. Example 1.5-1.2. We shall pay attention to the Finsler metric L = (K/2(y2/v) in Example 1.4.2.2. Its indicatrix is the parabola with the origin (y, v) = (0,0) as the vertex. See R.S. Ingarden and L. Tainassy [65], [66]. By generalizing their idea, M. Matsumoto [105] showed a geometrical Finsler metrics as follows: Let E2 be a Euclidean plane having an orthonormal coordinate system (xy y) with the origin O and <5 (a, y) a positive^ valued smooth function on tt = E2 ~ {(?}. For an arbitrary point P(x,y) of tt we take two points Fi, Fs and two straight lines A» A (Figure 1.5.1.1) as follows: (1) PFi and PP2 are orthogonal to OP and their lengths are equal to <5(OP). (2) A (resp. Fo) is parallel to OP and through Fo (resp. Fi). We define the indicatrix I? at P as two parabolas c*, i = 1,2, having the focus Fi and the directrix A respectively, and the Finsler plane as thus obtained is called the parabolic Finsler plane of the first kind. The parabola ci is given by the equation {xu + yv - (x2 + J/2)}2 + 4$(or 4- y2)(yu - xv) = 0,
590 Matsumoto in the current coordinates, putting u = x + x, v = y + y as in Example 1.3.3.1. Consequently, Okubo’s method gives the fundamental function L(x,y,x,y} - (x2 -i- y2)(xy - yx). Similarly the equation of C2 gives another fundamental function which is obtained from the above L by changing {xy - yx) for {yx — xy). Consequently, the fundamental function of the parabolic Finsler plane of the first kind is given by 12’ Lx{x,y,x,y) = (1.5.1.2) where A and /x are differential 1-forms defined by (1.3.3.1) and (1.4.2.2) respect¬ ively. If we define the indicatrix at P as two parabolas c;, i =* 3,4, (Figure 1.5.1.2) which are given from ci,cs by rotating in a 90-degree arc, then we obtain the parabolic Finsler plane of the second kind with the fundamental function L2(x,y,x,y)-^^-, (1.5.1.3) where a1 and a2 are as above. In these parabolic Finsler planes the function ${x, y) is called the density at the point P(x, y). Proposition 1.3.3.1 shows that if the density is constant, then the geodesics of these Finsler planes are logarithmic spirals with the origin O as the pole. Figure 1.5.1.2
Finsler Geometry in the 20th-Century 591 1.5.2 1-Form Linear Connection We shall consider a Finsler space F* — (Af, L(a®)) with 1-form metric. Then the equations (1.5.2.1) determine the quantities T^(x) uniquely as = b\dka?(x). (1.5.2.2) It is obvious from (1.5.2.1) that these £}*(:&) constitute a linear connection T1 with respect to which both n covariant vectors aa and n contravariant vectors ba are parallel. This connection T1 is called the 1-form linear connection. We are concerned with a geodesic of F*. From (1.5.2 J) we have for F = L2/2 djF = F«agr§, d,F (= %) = Faa?, difyF = + Fjfyjal = SjA-rSi + ykV^i, and (1.2.2.2) gives 2Gj = PjaFqq + yk^jQ ~ 3/&Fqj?. The 1-form linear connection F1 has the torsion tensor Tjk = rj* - TJj given by (1.5.2.3) ♦ and hence we obtain 2^(3?, y) = FJo + Too, (1.5.2.4) (1.5.2.4) where T]k = girgj9Tfk. Therefore we have the equations of geodesic in the form g+{rJ1W+Ii^)}(£)(£)-o. (X.5.2.6) 1.5.3 T-Minkowski Space It follows from (1.5.2.3) that the 1-form linear connection T1 is without torsion, if and only if we have dkaf = djak, a = 1,..., n, that is, all the covariant vector fields a® are locally gradient, that is, we have locally n functions x* satisfying dyr* = a®. Then (x®) may be regarded as a local coordinate system because of det(a^) / 0, in which y* — = a®. Consequently, the metric L(a®) can be written as L{ya\ Thus the space is a kind of locally Minkowski space. Definition 1.5.3.1. A Finsler space F” = (M,L(a®)) with 1-form is called T-Minkowski, if all the covariant vector fields (a®) are locally gradient vector fields, that is, all the 1-forms a® = af (x)dx{ are differentials of functions locally. Example 1.5.3.1. We are again concerned with the parabolic Finsler planes with the 1-form metric L(ax,a2) in Example 1.5.1.2. We have the components a® as a\ ~xz, ai = yz, a% = -yz, = xzf
592 Matsumoto where z 1 — 4$(z2 + у2), and b{ = 4Sx, bj = 46y, bl = -4Sy, Й = 4&c. Hence the 1-form linear connection Г1 is given by where r2 = or + y2. Consequently we get the torsion tensor of Г1 : Therefore the planes are T-Minkowski, if and only if the density 6 is constant. 1.6 m-th Root Metric 1.6.1 Finsler Space with m-th Root Metric We consider an n-dimensional Finsler space F71 = (M, L) with such a metric L(z< y) that Lm = ahi..jt(x)yhyi ...yk, (1.6.1.1) where are components of an m-th covariant symmetric tensor, that is, Lm is an invariant homogeneous polynomial in y* of zn-th degree. Definition 1.6.1.1. The Finsler metric L given by (1.6.1.1) is called an m-th root metric. In the case m = 2 the metric L is Riemannian, and in the cases m — 3 and 4 these metrics are called cubic and quartic, respectively. Example 1.6.1.1. We have two well-known cubic metrics: n =2, 71 — 3, W = {уг)3 + (У2)3, (L2)3 = (y1)3 + (y2)3 + (У3)3 - Wy3- These metrics have been considered by P. Humbert [56] as generalizations of Euclidean metrics. Example I.6.I.2. In his quartic metric theory of gravity, I.W. Roxburgh [142] has payed special attention to the strongly spherically symmetric metric ds4 =* Adt4 + Bdtrdsr + Cdx4, where <hs2 = cfe2+dy2+dz2 and A, B, C are functions of the Newtonian potential U = m/r, r2 = x2 + y2 + z2. For the later use, we shall reduce it to a two- dimensional quartic metric £4 = Co(yx)4 + 6c2(yx)2 (ÿ2)2 + c4(y2)4,
Finsler Geometry in the 20th-Century 593 with the coefficients Ci(æ), i — 0,2,4. Now we are concerned with F71 = with m-th root metric L, given by (1.6.1.1). Differentiating (1.6.1.1) by y* and then we obtain ...yk, m(m — l)Lm"2^ + = m(m - 1)0^..^(x)^*... yh. Consequently, we get (a) €¿ = 0», (b) hij = (m-“l)(av - o»^), (1.6.1.2) where we put From (1.6.1.2) it follows that the fundamental tensor gij = hij +&i£j is given by 9ij = (^ - l)aij - (m - tydidj. (1.6.1.4) Definition 1.6.1.2. A Domain D of a Finsler space with m-th root metric is called regular^ if the matrix (a^) is non-sirujular at every point of D. Example 1.6.1.3. We deal with the cubic metrics Li and L? of Example I.6.I.I. which imply det(a^) = yxy2/L~, Hence Li is regular, if y^y2 / 0. which imply det (a#) = -L3/4 “ “(y14-^4-y3){(y2-y3)2 + (y3-j/1)2 + (i/1 - JZ2)2}/S. Hence ¿2 is regular in the domain where y14- y2 4- y3 / 0. We shall restrict our consideration of Finsler spaces with m-th root metric to a regular domain in the sense above. Let (aij ) be the inverse of the matrix (uty). (1.6.1.4) leads to where p is a scalar and a1 = aîrar. The condition gucg^ “ on p yields = {(m -1) - m - 2)a2}, (1.6.1.5)
594 Matsumoto where a2 — an?. Then & — g^tj — gijaj is written as ? = (l/(m-l) -f-pcrja’. Multiplying by ¿i = a», we get 1 = (l/(m - 1) + pa2)a2. This together with (1.6.1.5) yields a2 — 1 and p = (m - 2)/(m -1). Thus we get (»)* = «*, (b) g* = . (1.6.1.6) Ref. This terminology was introduced by R.S. Ingarden [147] in his Ph.D. Thesis, where he was the first to realize the importance of Finsler ideas on Randers’ work. J.M. Wegener [168] has first considered Finsler spaces with cubic metric of dimension two and three. According to his paper [169], the paper above was written as the third part of his thesis on Finsler spaces in March 1935 to the German University in Prague, the referee being L. Berwald. In 1979, M. Matsumoto and S. Numata considered the theory of cubic met¬ rics and H. Shimada [152] immediately extended their theory to the theory of m-th root metrics. Recently, M. Matsumoto and K. Okubo [123] and Matsumoto [113] have developed systematically the theory of m-th root metrics which was founded by Wegener and Shimada. They indicated in 1996 an interesting metric of a three-dimensional Berwald space conformal to the metric L2 of Example 1.6.1.1 which Wegener failed to find. 1.6.2 Geodesics of m-th Root Metrics Now we consider a generalization of Christofiel symbols Tj/c = g^r{ik,r} of a Riemannian metric L2 — gik(x)yiyk given by (2.2.5.3) to m-th root metrics. The are defined by 2{i&, r} — dkgir 4* diQkr drgik* (1.6.2.1) We deal with, for instance, a cubic metric Lz = ahi^x^y^ and put = dfiOijp 4" di<ty* 4" djGhip ** with a number N3, which we shall determine conveniently such that N2 = 2 holds. Then we get №[{pM, j} 4- (pv\ h} 4- {pjh, ¿J] = Sdpahij 4- dhdijP 4- ditty* 4- = 4dpahij 4- If we decide N3 = 4, then we get dpdhij « {phi, j} + {pij, h} -I- {pjh, 3} - {hij, p}, (1.6.2.2) ► 4{hij,p} = d№ijp 4- ditty* 4* djafaip — dpCL^ij. (1.6.2.3) Similarly, if we treat of a quartic metric L4 == ahijk&yhy'yiy16, then it is observed that 6{hiyfc,p} = dhdiji* 4* •" + dkdhijp dpQhijki
Finsler Geometry in the 20th-Century 595 leads to dpOhijk - {phij, *} + ••• + {pkhij, j} - 2{hijk,p}. Thus it is conjectured that 2(7x1 — dh.On.,.jkp 4" • • ♦ + &kQ>hi...jp dpQ‘hi.,.jk) (1.6.2.4) will be suitable definition and then we get dpthi^k = [phi.. J, £} + ••• + {pkhi..., J} —(m — 2'){hi...jk,p}. (1.6.2.5) For a cubic metric, we have from (1.2.2.2) 2Gi = 2Gr(2ari — a^Oi) Multiplying by a* = r. we get 2arGr = (däaPqT')ypygyrysi and hence Consequently, we obtain (1.6.2.6) Finally the equations of geodesic (1.2.2.1) can be written as It is remarkable that the {pgr, 2} do not contain dx*/ds. 1.7 Birth of Finsler Geometry 1.7.1 Early Works [40] Paul Finsler (1951): Uber Kurven und Flachen in allgemeinen Räumen, Dissertation, Göttingen, 1918. Verlag Birkhäuser, Basel. P. Finsler (1894-1970) entered in the University of Göttingen in 1913. The director of his thesis [40] was Prof. C. Carathéodory. It was emphasized later by Finsler that his thesis was written under the influence of Carathéodory’snotion of geometrization of the theory of variations. After some preliminaries, the arc-length of a curve x = a?(i) is defined by the integral
596 Matsumoto corresponding to our (1.2.1.4). That is, our L(x,y) is Finsler’s In his thesis Finsler considered only “curves and surfaces of a generalized space, not of a Euclidean space”. [23] Ludwig Berwald (1919/20): Ueber die erste Krümmung der Kurven bei allgemeiner Massbestimmung, Lotos Prag 67/68, 52-56. L. Berwald (1883-1942) entered in the University of München in 1902. He became a full professor of the German University of Prague in 1922. When he wrote [23] in Prague, 1920, he was a lecturer still. In [23] he considered the theory of curves in a generalized space in the sense of Finsler. [40] was quoted in [23]. [154] John Lighten Synge (1925): A generalization of the Riemannian line¬ element, Trans. Amer. Math. Soc. 27, 61-67. This was written when J.L. Synge was in the University of Toronto and began with the statement as follows: In a manifold of N dimensions and coordinate system xi, let P(a;i) and Qtz1 + dxl) be two points with infinitesimal coordinate differences. Our fundar mental postulate is as follows: Postulate. The points P and Q define an invariant infinitesimal line-element ds, expressible as a function of z1, s2,..., xN, dx^dx2,..., Obviously ds must be homogeneous of the first degree in the differentials, and we write (1.1) ds? = Fix1,. ..,2^; dxl,...,dxN), where F is homogeneous of the second degree in the differentials. We shall in general write (1.2) F(^,...,xJV;e1,...,^)=F(a:;i). The further essential postulate in the differential geometry of Riemann is (1.3) F(x; dx) = gijtfafdx?. where gij are functions of the coordinates only. In the present paper, I wish to develop the more obvious deductions from (1.1), without assuming (1.3). Finsler [40] was quoted in [154]. The fundamental tensor fa is defined by , _ 1 PFfax) /ij 2 dxidtf ’ It is noted that Synge’s F differs with Finsler’s F: In our symbol Synge’s F is equal to L2.
Finsler Geometry in the 20th-Century 597 On account of (1.2.2.4), the equations (1.2.2.1) of geodesics are written in the form which coincides with Synge’s (3.5), while the parallel displacement defined by Synge is X* + C^XW +^kxiik = 0. It seems rather strange that 5* is contained in the above. [160] James Henry Taylor (1925): A generalization of Levi-Civit&’s parallelism and the Frenet formulas, Trans. Amer. Math. Soc. 27, 246-264. [161] J.H. Taylor (1925): Reduction of Euler’s equations to a canonical form, Bull. Amer. Math. Soc. 31, 257-262. [161] begins with the statement as follows:. As a by-product of the preparation of an earlier paper by the author, [160], (Dissertation, University of Chicago, 1924), two useful methods of solving for the second derivatives in Euler’s equations associated with the problem of minimiz¬ ing an integral were discovered. In this paper, these two methods are presented in detail. Taylor’s length integral is written as Thus Taylor’s F, which coincides with Finsler’s F, was quoted in [160]. He intro¬ duced the function f = F2/2 and the fundamental tensor is = d2f Taylor’s parallelism is the same with Synge’s, containing the second deriv¬ atives of the curve along which a vector is displaced. [154] was presented to the American Math. Society, Dec. 30, 1924 and pub¬ lished in voL‘27, 61-67. On the other hand, [160] was presented to the Society, April 19, 1924 and published in the same vol. 27, 246-264. Thus “presented” and “published” are reversed for these papers and this resulted, in a loss to Taylor. Hence Taylor was forced to write a note as follows: I regret to say that adequate reference has not here been made to the paper, [154]. When I returned the proof sheets of my paper, I was aware that Mr. Synge had written a paper on the same general subject, but the scanty and indirect account I had of his paper did not indicate much overlapping. It was only upon publication of Mr. Synge’s paper that an adequate account of his results became available to me. The same remarks apply to §4 of my paper [161].
598 Matsumoto [162] J.H. Taylor (1927): Parallelism and transversality in a sub-space of a general (Finsler) space, Annals Math. 28, 620-628. Presented to the Amer. Math. Soc. April 16, 1927. This was written at the University of Wisconsin. In the Introduction we find the following words which are worth remembering: The space here considered is. an n-dimensional space for which an integral of the form evaluated along a curve C is taken as the definition of the arc length of the curve. The geometry of sudi a space was first considered by P. Finsler [40], accordingly I have called such a space a Finsler space. Thus, the notion of Finsler space was bom in 1927. [24] L. Berwald (1925): Über Parallelübertragung in Räumen mit allgemeiner Massbestimmung, Jahresber. Deutschen Math.-Ver. 34, 213-220. This is the extended reproduction of Berwald’s lecture given at the meeting of the Math. Soc. Germany, Sept. 25, 1924. [40] is quoted in [24]. His theory of general spaces (Finsler spaces) in this stage, given in [24], was developed in the complete version in [25]: [25] L. Berwald (1926): Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus, Math. Z. 25,40-73. This paper was presented to the Soc. April 16,1925. Berwald gave a remark at the end of the introduction as follows: The present paper was finished before Synge’s [154] was published. In [154] the fundamental tensor gik^dx) and the notion of angle between two vectors with respect to the line-element of a curve were defined for a general space. These notions had been in [24]. Mr. Synge considered also a parallel displace¬ ment of arbitrary vector field along a curve, but his notion is different from the parallel displacement treated in [24] and the present paper. (Added in proof.) The remark stated above about [154] also applies to J.H. Taylor’s [160], which seems to have been published while [25] was printing. In [160] the notion of length of a vector with respect to a line-element appears. Taylor’s paper [160] was sent for publication earlier than [154] and [25]. In [25] the arc-length is defined by the integral 4 and the parallel displacement of £ by
Finsler Geometry in the 20th-Century 599 with the coefficients (xdx}-- ( > - ddxjg,k ■ What complicated symbols were used! The functions are defined by the equations of geodesics x'H « s'), = 2(9*, in our symbols. The problem of how to determine <£* from F(x, dx) was treated in No. 9, but in very complicated discussions because of det^F/ddx^dx^) = 0. As a consequence, [25] is hard to read; (compare with [161]). Finally, the covariant derivatives of is defined by in his complicated symbols, owing to 'which a theory of curvature could be considered by Berwald. 1.7.2 Historical Materials M. Matsumoto [115]: A history of Finsler geometry, Proceedings of the 33rd Symposium on Finsler geometry, Oct. 21-24, Lake Yamanaka, 71-97. . This was originally written as Appendix of Matsumoto’s monograph (1986). A revised one will be published in Tensor N.S. M. Pinl [138]: In memory of Ludwig Berwald, Scripta Math., 27, 193-203. This is English translation of “Casopis pro pSstonvanf matematiky, roc. 92 (1967), Praha, 229-239”. We have Berwald’s important posthumous paper: [31] L. Berwald (1947): Ueber Finslersche und Cartansche Geometrie IV, Ann. Math. 48, 755-781. With Editor’s footnote: Excerpt from a letter from Dr. Henry Loewig, dated May 11,1946: “In 1939, Berwald lost his post as a professor of mathematics at the German University of Prague because he was regarded as a “non-aryan”. On the 26th of October,* 1941, he was deported to Lodz (Litzmannstadt) Ghetto. Since he did not return after the end of the war, we must suppose that he is not alive any longer.” The author was told in 1971 by Prof. H. Takeno, The University of Hiroshima, the address of Dr. H. Loewig: At that time he was a professor of at the Uni¬ versity of Alberta, Edmonton, Canada. The author send a letter to Loewig immediately and got his answer: Thank you for your letter dated October 25 about L. Berwald. You are asking me to tell you details of Berwald’s life. You will find such details in the paper “In Memory of Ludwig Berwald” by Max Pinl, published in the
600 Matsumoto Scripta Mathematica, Vol. 27, pp.‘193-203. This paper also contains a list of his publications. Professor Loewig continues: After this paper has been published I discovered one serious mistake and two misprints in it and pointed this out to the author. I am enclosing a copy of my correction. Professor Max PinTs address runs as follows: Stammheimer Str. 34-35, 5 Köln-Riehl 60, Federal Republic of Germany. I am writing a letter to him to send you a reprint of his paper “In Memory of Ludwig Berwald”. One of Loewig’s correction: Berwald became a full professor in 1927. [159] L. Tamässy (1970): Ottö Varga in memoriam 1909-1969, Publ. Math» Debrecen 17, 19-26. We find some interesting lines in [159] as follows: After a year O. Varga left the Vienna Polytechnic for Prague’s old Charles University. On concluding his university studies, he obtained his doctorate and his habilitation at a quite young age, in 1933 and in 1937, respectively. Under the influence of his eminent teacher at Prague University, Professor L. Ber¬ wald, the interests of Ott6 Varga soon turned towards differential geometry, in particular towards Finsler geometry, then in its early development. We remember two papers. Both of which are abstracts from their theses. Referee: Prof. Berwald. These are interesting in relation to Berwald and Varga: [163] 0. Varga (1936): Beiträge zur Theorie der Finslerschen Räume und der affinzusammenhängenden Raume von Linienelementen, Lotos Prag 84, 1-4. [169] J.M. Wegener (1936): Untersuchung über Finslersche Raume, Lotos Prag 84, 4-7. The papers [169] consists of three sections as follows: I. Two- and three-dimensional Finsler spaces, II. Hypersurfaces as transversal surfaces of a family of extremals, Ut. Two- and three-dimensional Finsler spaces with a fundamental function L = yy3aiuxfiz'kxf*.
Chapter 2 Connections in Finsler Spaces 2.1 Frame Bundles 2.1.1 Structure of the Frame Bundle We consider a smooth manifold M of dimension n. Let Mx be the tangent vector space of M at a point x. A frame z at x is by definition a base of Mx as a vector space, that is, a set (za), a = 1,..., n, of n linearly independent vectors at x. Let L be the set of all frames at all points of M. We shall define the mapping %£ : L —► M, called the projection such that itl(z) — x is the origin of z. The set of all frames with the origin x is written as tt^1 (x) and called the fibre over x. We are concerned with a local coordinate system {CZ, (a?)} in a coordinate neighborhood U of M. Every tangent vector za of a frame z = (za) at x = (x*) € U is written as which gives a local coordinate system {^^(tZ), of L, called the ca¬ nonical coordinate system of L induced from {CZ, (a;*)}. As a consequence, L is regarded as an (n2 +n)-dimensional smooth manifold and then a fibre over x is an n2-dimensional closed submanifold of L with the coordinates (5*). Let GL(n, R\ simply written as G(n) throughout the following, be the gen¬ eral real linear group which consists of all non-singular real n-matrices g — (<?£). G(n) is a group having the multiplication r : (5, h) e G(ri) xG(n) -+gh= tfhfi € G(n), where 9 = (9$)> 601
602 Matsumoto For an n-dimensional real vector space V with a fixed base (ea) we have the left action of G(ri) on V as follows: i : (g, ®) € G(n) x V -r gv = (pf e V, (2.1.1.1) where 9 - (9b), v = ^aea: This action is effective: If gv = v for all v, then g = I (identity), but is not free: There exists g / I such that gv = v for some v. For a fixed g 6 G(n) we have the mapping g£ : v € V gv € V. (2.1.1.2) We define the action of G(n) on L as follows: 0: («, p) € L x G(n) -♦ zg = (x*, ^p*) € L, (2.1.1.3) where z=(xi,zi), g = (g>). This action is free: zg~ z necessarily implies g = I because of det (x£) / 0. For a fixed g € G(n) we have the mapping 0s-.z = (x1, zl)eL-^zg= (a?, zjp£) e L. (2.1.1.4) This is called the right-translation of L by g € G?(n). The fibre (x) ~ {zg, g € Gr(n)} for a fixed point z of this fibre. Consequently, the manifold M may be regarded as the quotient space L/G{ri). Thus we obtain a collection ¿(Af) = {£,757,, M, G(n)} with the action /3 of C?(n) of L. It is easy to show that L(Af) satisfies the conditions of a smooth principal bundle. Definition 2.I.I.I. The principal bundle — {L, M, G(n)} is called the frame bundle of a manifold M. L and M are called the total space and the base manifold respectively, and <7(n) the structure group, as well as the fiber» 2.1.2 Fundamental Vector and Basic Form Let L~ be the tangent vector space at a point z of the total space L of the frame bundle L(Af). L~ has a subspace Lvz = {X 6 (X) = 0}, the kernel of the differential %£ of the projection %£,. L” is called the vertical subspace of Lx and X € LZ is vertical. If we fix a point z — (z*,z*) of L and x = then we get from /3 the mapping z0-g& G(n) —»zg — (x*,z^) e^(x). (2.1.2.1)
Finsler Geometry in the 20th-Century 603 Consequently the differential -fi' gives the mapping of the tangent vector space G(n)c of G(n) at the unit e onto LZ. Since <?(n)e may be identified with the Lie algebra G'(n) which consists of all real n-matrices A ~ (Af), we get a tangent vector field Z(A) of L which is defined at a point z by Z(A)S = (2.1.2.2) Definition 2.1.2.1. The tangent vector field Z(A) of the total space L of the frame bundle £(M) is called the fundamental vector field, corresponding to A € G'(n). Z(A) is vertical. In fact, (2.1.2.1) gives (2.1.2.3) where Hence (2.1.2.2) yields for A = (4*) 2(A). (2.1.2.4) Since the matrix (^) is non-singular, z*A$ = X£ has the unique solution (j4J) for any X%, and hence each vertical subspace L* is spanned by n2 fundamental vectors. The mapping h G G(n) —► ghg^1 G G(n) for a g G <?(n) induces the mapping ad(g) of G(n)' onto itself, defined by A = « -> ad(g)A = (tfAfo-1®. ad(g) is called the adjoint representation of p. Proposition 2.1.2.1. A right translation /3g ads on a fundamental vector field Z{A} as ^(Z(A)) = Z(ad(<7-1^). In fact (2.1.1.3) gives ^.fA A A (A tfL Ps Yd#’dsU A’ dziJ (2.1.2.5) Then (2.1.2.4) shows
604 Matsumoto Next we define a mapping a : (2, v) G x V -* zv = z^va € Ma?, . (2.1.2.6) where z = (®*, x£), v — vaea. For a fixed v eV we get the mapping av: z € —► zv € Mx. It is easy to show (zg)v — z(gv), that is, a# 0 Pg = (2.1.2.7) For a fixed z G Tr£1(®) we get the mapping za : v G V —► zv G Mx. Then (z$)v = z(gv) leads to (2.1.2.8) The inverse za”x : X G Mx —► z~xX G V is important: ;<T': X = X’ e Mx -f z~*X = e V. (2.1.2.9) Definition 2.1.2.2. The V-valued differential 1-form 6 = zOT1 o on the total space L is called the basic form on £. It is obvious that 6 vanishes on vertical vectors. From (2.1.2.9) we get = (z-^X^, which implies ^ = {(2_1)?^}ea- (2.1.2.10) Proposition 2.1.2.2. The basic form 0 satisfies 9*Va = SFl9. In feet, we observe for X = X'd/dx' + X^d/Qx^ tgaT^X) = = p_1{ia“1(X)}.
Finsler Geometry in the 20th-Century 605 2.1-3 Tensor Field We deal with an n-dimensional real vector space V with a fixed base (ea). Let V* be the dual space of V, that is, the set of all linear mappings V —► R. It has the dual base (ea) such that ea(e&) = , or ea(v) = va for v — vaea. Similarly to i of (2.1.1 J) we have a mapping on V* as c : (g,v*) G G(n) x V* -+ gv* = G V*, (2.1,3d) where v* — t>aea. 9 = (9b), This can be defined by the relation = v*(iTx®)> (2J.3.2) for any v e V. Corresponding to a of (2d,2.6) we can define <2* : (xf, v*) e x F —> zv* — € (Mr)*, (2.1.3.3) where a = 7ri,(z), « = (^,2*), This can be defined by the relation v* = vaea. (^*)(zv) = v*(v), (2d.3.4) for any v eV. For a fixed v* we have the mapping This satisfies zg(v*) = z(pv*), that is, (2J.3.5) aj. o/?s =ajtr- Generalizing these V and V* we obtain the tensor space VJ of (r, s)-type which consists of all multilinear mappings R. The mappings f and f also are generalized to (2.1.3.6)
606 Matsumoto Example 2.1.3.1. We consider : (g,w) € G(n) x Vf, For vjf « vioea, vj = ^2aea, v = vaeo and g = ($J) we have ?w(vj, V2,«) = w(s_1 vj, vj, p_1 v), where we have from (2.1.3.1) and (2.1.1.1) g_M = Vlbffae°. ff“1« - If we put w = w^e^, then w(ff~1v;,g-1V2,g-1v) = wtc('uirgS)(v2sg’){(g-1)fgt}. Consequently we obtain Now we shall be concerned with a manifold M and its tangent vector space Mr. Let {U, (a;*)} be a coordinate system where Ulz. Then Mr has the natural base (d/dx^x. The dual base is written as (dx1-)^ Thus we get the notion of the tensor spaces (Mr) J of.(r,s)-type with the base • ; - The components of a tensor field T with respect to the base above are called the components of T with respect to (x*). We consider a tensor field T of (2,l)-type, for instance: For the later use in Finsler geometry, T is regarded as a mapping T -.zeL-tT^vf, (2.1.3.7) Ts(vl,V2,v) = T^zv^zv^Zv), where x = ^(2). If we put Tx = T^‘{z)e.^b, then the above is written as T?(z) = T^^z-^z-^. (2.1.3.8) Or we can write in the form We observe for a T € V^1 Tza(v*,v) = Tx((zg)v*, (zg)vj = Tx{z(gv^,z(<gv)) = T.(gv*,gv) = g~lfx(y*,v). (2.1.3.8T)
Finsler Geometry in the 20th-Century 607 Thus, in general we get T opg = g~1T. (2.L3.9) Proposition 2.I.3.I. A tensor field T of fas)-type on a smooth manifold M is regarded as a M*-valued function T on the total space L of the frame bundle L(M) satisfying (2.1.3.9), We have to show the inverse. Let T be a VT'-valued function on L satisfying (2.1.3.9). We define by the right-hand side of (2.1.3.8’). These seem to be functions on L. But (2.1.3.9) shows T^(zg) = T^fa), so that are constant on every fibre and hence are functions on the base space M. T is called the tensor function of (r,s)-type on L. Compare (2.1.3.9) with Proposition 2.1.2.2. 2.2 Linear Connections 2.2.1 Connection Coefficients An r-dimensional distribution 2? of a smooth manifold N is by definition a correspondence x € N -+ Dx c Nx where Dx is a closed subspace of constant dimension r. For instance, in the total space L of the frame bundle L(M) of an n-dimensional manifold M we have the vertical distribution Lv : z 6 L Lz, which is of di¬ mension n2, the kernel of the basic form 8 and spanned by the fundamental vectors Z(A)z. Definition 2.2.1.1. A distribution T : z € L —* P- c Ls of the total space L of the frame bundle (M) is called a linear connection in L or on M, if the following two conditions are satisfied: (1) Ls = Pz ® Lz (direct sum), (2) №)=r„, at every point z of L and any g of G(n). Tz is called the horizontal subspace of Ls and X e Pz is horizontal. If a linear connection T is given, then we can write X = + eLz, Xh€Vz, XveLvz, uniquely. Xh and Xv is called the horizontal and vertical parts respectively. We have ?r£(X) = ^(X^). Conversely, we can define 4 : X € Mr -> £z(X) € Ts, x = tt(z), 7r£o4(X) = X.
60S Matsumoto lz is called the lift with respect to T From the condition (2) we have $»4=4,. (2.2.1.1) Definition 2.2.1.2. For a linear connection T we have a C?(n)'-valued differ¬ ential 1-form w on L by (1) w(2(A))=A, (2) w(rs) = 0. w is called the connection form. From Proposition 2.1.2.1 we have u;o^(Z(A)) = ^(^(ad^“1^)) = ad^“1)^. On the other hand, for X € T. we have fi'g(X) e Vzg and hence w(/?'(X)) = 0. Consequently we have . ° = ad^“1)^. (2.2.1.2) The following proposition is essential for a linear connection. Proposition 2.2.1.1. A linear connection F gives the connection form w by (1) above and (2.2.1.2). Conversely, if a -valued differential 1-form cv satisfies these conditions, then (2) above defined the linear connection T whose connection form is w. We shall write the connection form w in a canonical coordinate system (^,4). Put W?(«) = + Wb&tyil!*.. Then (2.1.2.4) gives the condition cu(Z(j4)) » A in the form wg?(^) = A°1 for y any A**, which implies w^z^ — Thus we get <CW - (s-1)?«- (2.2.1.4) Next, for Xz = Xi{d/dxi)z + Xl(d/dz^z, we have from (2.1.2.5) = xi(ft/dxi)zg + xM/dz[)za, and hence o ^(xs) « w^i(zg)Xi + On the other hand, (2.2.1.3) leads to = (9-1)ae{w^)Xi + w^(z)Xie}g^
Finsler Geometry in the 20th-Century 609 On account of arbitrariness of (X*, X*) the above leads to (1) = (ff_1)?«Si(^, (2) wg(zg)g% = (2) is only a consequence of (2.2.1.4). (1) is rewritten as ^gYaiv^zg^zg)-1^ = The left-hand side obtains from the right-hand side by changing z for zg. There¬ fore have the same values at z and zg, and hence these are functions of (s’) alone. Finally, we obtain the expression of wz = (irf(z)) as "№) = + №(x)d^}. (2.2.1.5) The local functions rj.y(x) are called the connection coefficients of T. 2.2.2 Covariant Differentiation Suppose that a linear connection T is given on a smooth manifold Af. We treat a tangent vector field X of M. From rcfL o£z(Xx) = Xx it follows that 4(Xx) is written in the form Then u;(^(Xx) j = 0 and (2.2.1.5) lead to Xj + z^Tj^X* = 0. Consequently we have Definition 2.2.2.1. For a given linear connection T and v 6 V we have a horizontal vector field B(v), which is given at a point z 6 L by B(v)- =* zv = <*(z, v). This is called a basic vector field, corresponding to v G V. From (2.1.2.6) and (2.2.2.1) we get (2.2.2.2) V = Z = (x\zî). Since zîya = X* has the unique solution v°, we have
610 Matsumoto Proposition 2.2.2.1. Given a linear connection T in the total space L of L(M) we have the direct sum As = e Lvz. The horizontal subspace F- is spanned by the basic vectors B(v)z and the vertical subspace by the fundamental vectors Z[A]z. Equations (2.2.1.1) and (2.1.1.7) lead to ' = /% o iz(zv) = €xi(zp(ff_1v)), which shows &sB(y) = B^v). (2.2.2.3) For the basic form 3 we have 0(B(v)z) — -a”1 O7r£(^-(zv)) = -a”1 (,a(v)) = v. Consequently the pair (3,w) is the dual of (^(?l),B(v)) in the following sense: <W))=0, 0(B(v))=v, w(Z(^L)) = A, w(B(v)) = 0. k 7 We should recall here Propositions 2.1.2.1, 2.1.2.2, and equations (2.2.1.2) and (2.2.2.3). Since Z(A) and B(v) are tangent vector fields in L, these operate on a tensor function T. First we shall see the operation of Z(A) : Z(A)T = —A*T. (2.2.2.5) Here the operation of G(n)' on ® defined, for instance, as A-w- (Atvite - WfrA* - w£iAe)e^, (2.2.2.6) ' 4 = (Ag)€G(n)', w = <4=€V21. For instance, taking Tx = T* (z)e£ € Vi1, from (2.1.2.4), (2.1.3.S) and paying attention to we have On the other hand, the operation of B(v) is closely related to the linear connection F.
Finsler Geometry in the 20th-Century 611 Definition 2.2.2.2. Given a linear connection T on M, the covariant derivative VT of a tensor field T on M is defined by VT(v) = B(v)T9 where T and VT are tensor functions, corresponding to the tensor fields T and VT respectively» We deal with a tensor field T — Tf(x)(d/dxh)<8dxi of (l,l)-type for brevity. Then we have f==(r*)$(x)44 Equation (2.2.2.2) yields B(v)f = + r&z? - I$T£)e* where di = d/dx\ Hence, if we put T^d^+T^-Ttr^ (2.2.2.7) then we obtain VT(v) = (2.2.2.S) Consequently (2.1.3.8) leads to (2.2.2.9) 2.2.3 Torsion and Curvature We deal with the Lie brackets of tangent vector fields on the total space L. In general, for two tangent vector fields X — Xi{d/dxi') and Y — Yi[d/dxi) on a coordinate neighborhood U of a smooth manifold N, the Lie bracket [X, Y] is defined as the tangent vector field [X, yj = (xW - ~. It is easy to show that for functions f and g on U we have [/X, pY] = /<?[*, Y] + {f • X(i)}Y - {p • Y(f)}X. (2.2.3.1) Now we are concerned with the Lie brackets of tangent vector fields on the total space L of the frame bundle L(M) equipped with a linear connection T. First we shall show . [Z(A\ Z(B)] = Z([A, B]) + Z(Z(AjB) “ Z(Z(B)A). (2.2.3.2) This is not related to T. The bracket [A, B] is such that A = (AJ), B = =
612 Matsumoto We put (2.1.2.4) at z = as Z(A) = A$Zb, Then it is easy to show [Zb,Zt\ = 5bZi-S*Zb. Thus (2.2.3.2) will be shown by (2.2.3.1). Next we are concerned with ¿(A) and B(v)> On account of (2.2.2.2J we put B(v) = vaBa, Ba = z\Bi. (2.2.3.3) First we consider [Z(A), B(v)]. We have [Zb, Be] = 6bBa and [^(A),B(t>)J = [ASZb,veBe]. Then (2.2.3.1) implies [Z(A), B(»)] = B(A • v) + B(Z(A)v) - Z(B(v)A), (2.2.3.4) where A • v has been defined by (2.2.2.6). Secondly (2.2.3.1) yields = = z^Bi.Bj] + zM^Bj - We can easily show Bi(z%) = —and where the symbol (i/j) denotes the interchange of indices i,j of the preceding terms. Consequently we obtain [Be,B6] = i&Bc + .R^( /pc _ nri -¿(-—Ixc rpk r>k pfc ¿ab ±ij'i'a,zb\z Jkj k ji> Finally we consider [B(t>i), B(v2)]. Since this can be written as [V^B*, v^B^j we get = B(f(vi,«2)) +Z(R(v1,v-i)') (2.2.3.S) + B(B(vi)V2) - B(B(V2)1>1),
Finsler Geometry in the 20th-Century 613 where f(v!,®2)=^c^ea€V ’ = (^vM) e G(n)'. ’ The equation (2.2.3.5) is called the structure equation of the linear connection. If vi and vi are constant, then (2.2.3.5) is reduced to (2.2.3.7) We have to show that T and R in (2.2.3.7) are tensor functions of (1,2) and (l,3)-type, respectively. From (2,2.2.4) it follows that for fixed Vi and v2 0[B(vi),B(v2)] =fz(vi,v2)- Then Proposition 2.1.2.2 shows B{v2)]i = By (2.2.2.3) the left-hand side is rewritten as The right-hand side is equal to Thus we get and for v* € V* we have Txg{v’, vi, va) = v' @zg(vi, «2)) = V*(ÿ-1fs(ffVl,PV2)) = ($«•)(?! (ffVl,S«2)) which shows the tensor property of T by (2.1.3.9). Next we consider R. (2.2.2,4) gives w[B(vi),B(v2)k = BS(V1,V2). Then (2.2.1.2) leads to cjo^[B(vi),B(v2)]- == od(p“'1u;[B(vi),B(^)]^.
614 Matsumoto The equations (2.2.2.3) and (2.2.3.5) lead the left-hand side to w[B(p_1vi), B(g~ -g = R^g^v^g'^), and the right-hand side is equal to ad(g~1Rz(y\1V2\ Hence we have = ad(ff_1)^3(vi, v2). (2.2.3.8) 2i(vi,V2) = (■^04^1^2) 6 G(n)' is naturally regarded as R € V31 as follows: For u* € V* and vo € V we define B(u-,«0,^1, V2) = R(vi, V2)(u’,«o) p 9) = R^at'UdVgvi V%. Then we observe <»i(ff_1)-R(Vl,«2)(«‘>Vo) = {(S_1)e-R?a(>ffc}“<rfo1Jxw2 = R((vi,V2)(gu,,gvQ)'). Consequently (2.2.3.8) yields vo, vi, V2) = ¿^(vi, v2)(ti*, vo) = V0) = Rz (gu*, $ v0, pvi, ¿Vo), which gives Rzg =» g^Ra, the tensor property of R. The tensor fields T and R, corresponding to the tensor functions T and R, are called the torsion tensor and the curvature tensor of the linear connection T under consideration. 2.2.4 Ricci Formula and Bianchi Identities Let K be a tensor function corresponding to a tensor field Given a Jinear connection T, we get the covariant derivative VK of K by VJf = B(v)K. We shall find the commutation formula of covariant differentiation. Considering (B(vi), B(v2)]K, (2.2.3.5) and (2.2.2J5) yield V(VK(v!))v2 - V(Vtf(v2)vi) = B(vi,vs) • K - VK(f(vi,v2)). (2.2.4.1) We are concerned with a tensor field K of (l,l)-type, for instance. On account of (2.2.2.6), the above may be written in the form Tfh vsh tst nh rs-h ryr r^h rpr fn n 4 a\ This gives the law of commutation of covariant differentiations, called the Ricci formula.
Finsler Geometry in the 20th-Century 615 Next we have the Jacobi identity [[*, + [M*] + [i^ X], Y] = 0 for three tangent vector fields Xi Y, Z on a smooth manifold. For the case of the total space L of the frame bundle L(Af) with a linear connection T, we have such identities of four type: [[Z(Ai), Z(A2)], Z(As)] + (1,2,3) = 0. (2.2.4.3) [[Z(Ai),Z(A2)),.B(v)] + [[Z(A2),B(V)], 2(40] + [[B(v), Z(A)], Z(A2)] =0, [(B(vi), B(t>3)], Z(A)] + [[B(«2), Z(A)], BM] + [(Z(A), B(Vl)], Bfa)] = 0, [[B(vi), B(v2)], B(t*)] + (1,2,3) = 0, where A’s G G(n)f and v’s G V are fixed and the symbol (1,2,3) denotes the cyclic permutation of subscripts 1,2,3. These identities are reduced to trivial identities except the last. We shall prove this fact. From (2.2.3.2) it follows that the first is only reduced to the trivial identity [[Ai, A2]. A3] = 0. On account of (2.2.3.4), the second is reduced to [Ai, A2p v = Ai • (A2 -v)-A2' (Ai ■ v), which is trivial from the definition of [Ai, A2] and (2.2.2.6). Next, on account of (2.2.3.4) and (2.2.3.5), the third is written as [B(T(V1,-U2)) +Z(R(v1,v2)),Z(A)] -= 0. This is divided in the horizontal and vertical parts as follows: •Z(A)T(vi, ®s) + A • T(«1, V2) + T(A • «2, vi) - T(A • vi, t>2) = 0, ~ ~ ~ ~ (2.2.4.4J Z(A)jR(vi,V2) - [B(vi,V2),A] +B(A-V2,Vi) -^(A-Vi,V2) — 0. We have ^(A)f(vi,^) = (Z(A)f}(vi,V2) = “(A • f)(vi,f2) = -(A^, - f‘ AJ - 2%A>'^a € V, and it is observed that ”2e<‘ = A ' ^(vl>®2)> ^cg-AfVl^a ~ T(A • Vl, V2). Hence the first of (2.2.4.4) is satisfied. Similarly the second of (2.2.2.4) is satis¬ fied, if we pay attention to = -(A-5)(U1,V2) = -<A&bIg - j^Ag'- <SAJ - 6 <?(n)',
616 Matsumoto and -(4%, - %/BAfr>№ = [Kfa, %), A]. Similarly to the case above, the fourth of (2.2.4.3) is equivalent to the fol¬ lowing two identities: VT(wi,®2,t>3) +T(y3,T(vi,v2y) - B(v3,«i,«2) + (1,2,3) = 0, VH(vi, vs, V3) + + (1,2,3) — 0. Here we have VT («1,-02, V3) = B(«3)f («1,«2) = and (2.2.3.9) shows B(«3, V1,V2) = € v. Consequently (2.2.4.5) are written in the components with respect to (s*) as Tij,k + “ -ftfeij + (»> J> *) = 0, (2 2 4 6) R*ij,k + ^ikr^tj + (M> — 0- These are called the Bianchi identities of the torsion tensor and the curvature tensor respectively. 2.2.5 Parallelism and the Leci-Civita Connection We consider the frame bundle L(M) with linear connection T. Let (7° : I = {t|0 t 1} —► L be a smooth curve. Then we get a smooth curve C = o C° : I -► M. C is called the projection of (7° and conversely (7° is said to cover the vector tangent to C°. If it is horizontal at every point of C°, then C° is said to be horizontal. Definition 2.2.5.1. A curve C° — {z(t)} of L is called a lift of a curve C = {a;(t)} of M if (7° is horizontal and covers <7. Then the frame z(t) is called to be parallel along C, We discuss the lift (7° = {a1 (t), z*a (t)} by using a canonical coordinate system (x\ zj). {x*(t)} is the projection C of (7°. The vector field tangent to (7° is given by /dxl\ d fdz*\ d (df) a? + Cdr) ajp
Finsler Geometry in the 20th-Century 617 Then (2.2.2.1) shows that C° is a lift of C, if and only if dx'/dt = Xi and dz^/dt = -z£T£/(o:(i))X< Therefore we obtain f+«,«<)) (^)=<>. (2.2.5.1) as the differential equations satisfied by parallel frame field along a curve {^(¿)}. Definition 2.2.5.2. A vector field given along a curve C of M is called parallel along C, if it has constant components with respect to a frame field which is parallel along C. It will be clear from (2.2.5.1) that this notion of parallelism does not depend on the choice of the parallel frame field. In fact, if z(t) is parallel along C and Xi = z*ya with constant v’s, then (2.2.5.1) gives rise to T + X‘W‘C) = 0' (2.2.S.2) which are differential equations of parallel vector field {Xi(t)} along {^(t)}. Let a tangent vector field X — Xid/dxi be defined on the base manifold M. Then X is parallel along a curve C = {z*(t)}, if and only if where (Xj) are components of the covariant derivative VX of X. Thus Xj = 0 is the condition for X to be parallel along any curve. Generalizing this notion to a tensor field of any type, we are led to Definition 2.2.5.3. A tensor field T on the base manifold M is called parallel, if the covariant derivative of T vanishes identically. Now we are concerned with a Riemannian manifold (M,p) with the funda¬ mental tensor g — (^j(x)) : ds2 = gijdz'dx?. The covariant derivative Vg with respect to a linear connection T has the components 5»j,A = &k9ij “ ^ijk — Definition 2.2.5.4. We shall introduce a new symbol = Aijkh. + Ajkih “ Afcijhi for a set of quantities Aijkh- To construct from Aijkh is called the Christoffel process. Applying the Christoffel process to we get
618 Matsumoto where Tijk = 9^rT[k. Therefore, if we impose the conditions gt^k = 0 and Tfk = 0, then we obtain rik = | 9^ (dk9ir + dtSkr - dTgjk^ • (2.2.5.3) Thus the linear connection T is uniquely determined from the Riemannian metric 9- Definition 2.2.5.5. Let (M,g) be a Riemannian manifold with the funda¬ mental tensor g. A linear connection (2.2.5.6) is uniquely determined by the following two axioms is called the Levi-Civitd connection: W)Vg = 0, (2)T = 0. As it is well-known, the Levi-Civitd connection is decisively dominant in the differential geometry of Riemannian spaces. In particular, the parallel displace¬ ment of tangent vectors does not change the length or angle: dktgaXW') = = o. 2.3 Vectorial Frame Bundles 2.3.1 Tangent Bundles Let Mx be the tangent vector space at a point x of an n-dimensional smooth manifold M and T the set of all tangent vectors at all points of M. The mapping kt :yeT—>xeMis called the projection of T where x is the origin of y. (®) = Mr is called the fibre over x. Let {^(rr2)} be a local.coordinate system of M. Then a point y € Mr? x — (s*), is written as yt(d/dxi)x, and hence we get a local coordinate sys¬ tem (^,2/*)} of T, called a canonical coordinate system of T. As a consequence, T is a 2?i-dimensional smooth manifold. (See [6] for more detail.) If we take a frame z = (x1, z*) G 7t£1(3;), then on account of the mapping a defined by (2.1.2.6) a point y G vr?1^) is written as zv, v € V. Thus every fibre ttJ1^) is isomorphic to the vector space V. Definition 2.3.1.1. The collection T(M) = (T, kt, M, V, G(n)) is called the it tangent bundle over M. T, M and V are called the total space, the base manifold and the standard fibre respectively, while G(n) is the group of the tangent bundle. G(n) acts on V as i of (2.1.1.1) and on the total space L of the frame bundle L(M) as of (2.1.1.3). Thus we get the mapping 77: ((«, v),5) € (I x V) x G(n) e L X V, (2.3.1.1) = £(*>$)> g~'v = ttg \v).
Finsler Geometry in the 20th-Century 619 Hence we obtain the quotient space (L x V)/G(n) with respect to 77. A point of (L x V)/G(ri) may be written as zv because of (2.1.2.7) and (2.1.2.8). Con¬ sequently the total space T of T(M) may be regarded as this quotient space. Therefore, T(M) is a so-called associated bundle of the frame bundle L(M) with the standard fibre. The subspace Tjj = {X £ Ty|7Ty(X) = 0} of the tangent vector space Ty at a point y £ T is called the vertical subspace and X £ is called vertical Since 7Ty is constant on a fibre 7Ty1(x),7^ is the tangent vector space of this fibre. In a canonical coordinate system (z\ a vertical vector X is written as W/d^. Next we deal with za : v £ V —► zv £ ttJ1^), x = 7tl(z). Since the base (ee) of V is regarded as fixed, we have the global coordinate system (va) of V where v = vaea, and the natural based (d/dvay)u of the tangent vector space Vu at a point u of V. Definition 2.3.1.2. The tangent vector space Vu of V at a point u can be identified with V by the mapping : w = w°ea - w„ = * e Vu. w is called the paraHei vector field on V, corresponding to w £ V. The differential za' of zot gives the mapping Vu Tsu, Since tvt °z a is constant, we have Hence zo'(X) is vertical. Thus we get the mapping tal o Iu : w € V ->■ z'awa € (Tzv.)v, (2.3.1.2) w = waea, z = (xi,zia).. Definition 2.3.1.3. The mapping : Mx —► is defined as follows: X £ Mx is regarded as a point X G 7r^x(o;) and hence we get -o_x(X) = z~xX = v £V for z £ 7r£x(z). Then ¿J(X) — -offy). Also, tyX) is called the vertical lift of X. The vertical lift does not depend on the choice of z £ ir£x(o;). In fact, we can easily verify by (2.1,2.8) and 3g(w) = gw. In coordinate systems we have (2.3.1.3)
620 Matsumoto which shows <2-3-L4> 2.3.2 Vectorial Frame Bundles We consider the frame bundle L(M) and the tangent bundle T(M) over M. Then the projection ttt • T M induces from L(M) another principal bundle over T, usually written as x£(T(M)). № F(M) > L(M) 1 > M Figure 2.3.2.1 Definition 2.3.2.1. The induced bundle (£(Af)) is called the vectorial frame bundle of M and denoted by F(M) = {F,fix,T,<J(n)}. We exhibit the structure of F(M). The base space T of F(M) is the total space of the tangent bundle T(M). The total space F of F(M) is defined as F = {(SG 2) e T X i|’Tr(y) = W2,(2)}. That is, a point u — (j/, z) € F is a pair of a tangent vector y eT and a frame z € L at a same point x e M. Thus we have a canonical coordinate system of F, where (a1,#*) and (a?*,^) are canonical coordinates of y and z respectively, induced from a local coordinate system {U, (z*)} of M. The projection : u = (y, G F —► y G T and we get another mapping 7T2 : u — (y, z) G F —► z G L, called the induced mapping. We have the relation (Figure 2.3.2.1): 7TT O 7T1 = TT£ o 7T3. (2.3.2.1) The subspace F£ = {-¥ e Fu|Tri (X) = 0} of a tangent vector space Fu is called the vertical subspace and X G F£ is called vertical. Similarly to /? and /3ff in the case of L(M), we define the action of G{ri) on F as follows: p: (it = (y,z),g) € F x <?(n) -f ug = (y,zg) e F, (2.3.2.2)
Finsler Geometry in the 20th-Century 621 where zg — /3{z} g). Then we get the right translation of F by g: ps:u - (y, z) €.F —*ug = (y, zg) 6 F (2-3.2.3) Hence we hâve the obvious relations: (a) 7T1 O pg = m, (b) 7T2 O pg = /3g o 7T2. (2.3/14) On the other hand, we get firom p vp : 9 £ ^(n) —+ ug € F> which gives rise to a fundamental vector field F (A) of F(M) : Similarly to (2.1.2.2) and (2.1.2.4), we define r(A)„ = B//(A) = 4^(^)w, A=(A?). (2.3.2.5) The vertical subspace F£ is spanned by n2 fundamental vectors. Similarly to Proposition 2.1.2.2, we can easily show p^(F(A))=F(ad(5-1)A). (2.3.2.6) Let us recall the basic form 9 of L(Af), given by Definition 2.I.2.2. It induces in F(M) a V’Valued 1-form Qh =z 0 o= za 1 o 7Tj, o Tri = 1 o ir'T o Trj, (2.3.2.7) called the h-basic form. It is observed that r Of/g = 6 Offs = e O^j = g~l0o'Ki2 —g^0h. Thus we have 9hop'g^g-10h, (2.3.2.8) Prom (2.1.2.11) we get 0h = (z-^dx^. (2.3.2.9) Now we are concerned with a tensor field appearing in Finsler geometry. The components in a coordinate system (a?) are functions of position (a;*) as well as direction (y*) in general. Thus a Finslerian vector field of an n-dimensional manifold M has n components which are, however, functions of 2n variables (x1,!/4). To rationalize such strange fact, we shall recall the notion of tensor function in §2.1.3. Definition 2.3.2.2. A vectorial tensor field T of (r, s)-type on a manifold M is a V/-valued function on the total space F of the vectorial frame bundle F(M) which satisfies Topg^g^T,
622 Matsumoto for any g € G(ri). Let us deal with a vectorial tensor field T of (l,l)~type, for instance. T is written as T = Tfe^ e Vi1 in the base (e*) of VJ. The condition for T is K(ug) = Then we get which shows that T^u) = ztt(u)(z~l}t, (2.3.2.10) are equal to Tj(ug). Hence Tj are functions on the base space T of the vectorial frame bundle F(M), and thus we get the so-called components Tj^y) of T in the coordinates (s*). Owing to (2.3.2.5), the essential equation (2.2.2.o) is easily generalized to vectorial tensor field T : F(A)T = -A-T. (2.3.2.11) We omitted to put (~) on a vectorial tensor field, and shall omit “vectorial” of vectorial tensor field in the almost all cases in the following. A tensor function T, corresponding to an ordinary tensor field T, may be regarded as a vectorial tensor field, because T o tto satisfies (T O 7T2) O pg = T O O 7T2 = ^“1(T O 7F2)- Definition 2.3.2.3. A vectorial tensor field e : u = (y, z) € F -* z~Ty € V is called the supporting element This e is certainly a vectorial tensor field of (1,0) -type. Because we have s(y,z) = yaea, ya = (2.3.2.12) Thus the components of s in (□:’) are nothing but Proposition 2.3.2.1. The mapping I = fabc): = (y,z) e F -> (z.z"1^) e L x V, yields an isomorphism of the total space F of F(M) with the product LxV. Because we have the inverse Z”1 of I as I-1: (z,v) € L x V - (zv, z) e F. (2.3.2.13) Theorem 2.3.2.1. A tangent vector X € Fu, u = (y,z), is uniquely given by each one of the pair (X-^Xz) or the pair (X2.X3) as follows:
Finsler Geometry in the 20th-Century 623 (a) Xi =5 ttj (X) and X2 = ^(X) such that X = Gfli-VCXi - <X2)), V = s(u). (b) X¡ and Xz = ¿(X) such that X = (r1)'(X2>X3). Proof: (2) is obvious from Proposition 2.3.2.I. We shall show (1). It is enough to prove that From Tn = a o I we have 7FÍ = a' o it2 + got o ¿r', v = e. (2.3.2.14) Hence we get Xi = a¿(X2) 4- -¿/(Xa). Corollary 2.3.2-1. A tangent vector X 6 FL vanishes, if (1) 7rl(X)=^(X) = 0, or (2) %l(X)=e'(X)=0. We shall apply (a) of Theorem 2.3.2.1 to a fundamental vector field F(A). X1=w'1(F(A))=0, X2 = tt'(F(A))=Z(A). To find X3, we consider a V-valued function <j>: x = (x¿) £ TV —► ^®(a;)ea € V on a manifold TV. The differential <// is 4>'(X)~x\d^} (A)*, v = ^). Hence we have ¿'(X)=IV(X(¿)). (2.3.2.15) See Definition 2.3.I.2. Consequently (2.3.2.11) yields e' (F(A)) = Iv (F(A)e) = Iv(-A • e) = -A • £■ There we obtain F(X) = (r1)'(¿(A), -ÁT). (2.3.2.16)
624 Matsumoto 2.3.3 Distributions in Vectorial Frame Bundles We shall give two distributions of F(M) as follows: Definition 2.3.3.1. In the tangent vector space Fu of the total space F of the vectorial frame bundle F(M), (1) the subspace F£ = {X e FU|^(X) = 0} is called the induced-vertical subspace. (2) the subspace F£ = {X € o %i(X) = 0} is called the quasi-vertical subspace. (2.3.2.1) gives immediately the inclusion relations F^CF^. Definition 2.3.3.2. A tangent vector field Y(v) = (Z_1)'(Otv), v € V, on the total space F, is called the induced-fundamental vector field. From (b) of Theorem 2.3.2.1 it follows that the equations (a) T^y(v) = 0, (b) £/y(v) = v, (2.3.3.1) characterize y(v), and hence y(v)u € F*. Since any X € F^ can be written in the form (I“1)'(0, X3), X3 € K(u)> the induced-vertical subspace is spanned at every point by the induced-fundamental vectors. Proposition 2.3.3.1. An induced-fundamental vector field Y(v) satisfies (1) 7rir(v) = .a'(v)= at u = (y, z). (2) Y(v)s = v, (3) p;y(«) = y(5-^). .Proof: (1) is shown by (2.3.2.14) and Definition 2.3.I.3. Next we have (2) from (b) of (2.3.3.1) and (2.3.2.14). We consider (3). By (2.3.2.4) and (2.3.3.1) we have A{p'syW} =03° = 0, and by the tensor property of s and (2.3.1.3) we have «'{/’P'M} = (s_1e)'y(v) = Hi'®«'1'!’) = 9~*v- Hence (2.3.3.1) shows (3). Proposition 2-3.3.2 We have the direct sum
Finsler Geometry in the 20th-Century 625 Proof; We first consider X € F£FiF^ Xi = ^(X) = 0 and X% = ^(X) = 0. Hence Corollary 2.3.2.1 leads to X — 0. Next we deal with X e F£, u = (j/,z), that is, TtrfaiCX’)) = 0, which shows 7r|(X) e Hence we have vG V such that ttJ/X) = -a'(v), and Z = X — y(y) is vertical on account of (1) of Proposition 2.3.3.I. Therefore we obtain X = K(v) + Z, Z(v) € F* and Z G F%. Consequently F£ is spanned by induced-fundamental vectors Y(v)u and fun¬ damental vectors F(A)U. We shall write Y(y) in a canonical coordinate system (zi,yi,zia). First we have Z_1 : (z = v = («“)) -u = (xi,yi = 4«°, 4)- Hence the differential (I-1)' is {^k> a!*-} -* <va(w) + A}5 (Z-1)' : Consequently Y{v) » (7 1/(0,v) is written as y W» = 4«“ > U=(y,z). (2.3.3.2) Since an induced-vertical vector X satisfies 713 (X) == 0 and 7r2 : y\ z%) = (x\ z*), we have X = X*(d/dy') and (2-3.3.2) implies X = y(v), v = (3"'1)®Xiea. From (2.3.3.2) it is obvious that I^(^i),y(v2)] = 0, (2.3.3.3) and (2.3.2.5) gives iww = [4^ (^), 4^(3^)] Consequently (2.3.3.2) and (2.2,2.6) lead to (2.3.3.4) Definition 2.3.3.3. A horizontal connection Vh in the vectorial frame bundle F(M) is a distribution ueF^F^cFv satisfying (1) Fu - r£ e F<, (direct sum), (2) p'(r£) = Give a horizontal connection rA, Proposition 2.3.3.2 yields the decomposition of every tangent vector space Fu of F as Fu=r£©z£®J£, (direct sum). (2.3.3.5)
626 Matsumoto Then, by the projection tti : F —> T we get the decomposition of every tangent vector space of T as follows: Ty = fli(Fj) (Fj). (1) of Proposition 2.3.3 J gives 7Ti(jF£) = Ty at u = (?/, z)< Therefore we obtain the direct sum T9 = *i(r£) © T^, u = (y, z). (2.3.3.6) The condition (2) shows that ^(Fj) does not depend on the choice of u G More generally that (2.3.3.6) we put Definition 2.3.3.4. A spray connection N in the tangent bundle T(M) is a distribution y G T —> Nv C TVi satisfying Ty=Ny® (direct sum)1. X G Ny is called horizontal with respect to N. By the symbol we denote the vertical component of X € Ty with respect to N Definition 2.3.3.5. With respect to a spray connection N a V-valued 1-form 0V is defined on the total space F of F(M) as (a’% = I-1 o (-a-1)' o vN ° where u = (y, z) and v = z~xy. 3V is called the v-basic form with respect to N. By xot o Iv(u) = ly{zu), ueV<we have another expression of 0^ as (^')u = xa~1o(^)~1ovjvo7ri, u= (y,z). (2.3.3.7) Since (2.1.2.9) gives isa_1 = g^oT1, (2.3.3.7) shows (2.3.3.S) To write 3V in a canonical coordinate system, we put 0* = (0*)*ea, (0V)‘ = 0tW + 0^)d2/i + 0?bdzJ. Given a vertical vector X = X*(O/dzl) 6 FJ, we have (0V)°X = 3fbXi = 0, which implies 3fb — 0. Next, given an induced-vertical vector X = Xi(d/dyi) G Ft. we have (0*)aX = 3^ and (2.3.3.7) gives (0V)°X = Hence 0*) = (z“1)®. Finally, we treat (2.3.3.8) similarly to the case of the connection form in §2.2.1. Then we get e?(«S) = GTX)^(«). which shows that N? = zjflf are functions of (a;’, if’) alone. Therefore we obtain fl* = (0*)aeo, (ev)a = (z-1)? (dy* + Wjdxi). (2.3.3.9)
Finsler Geometry in the 20th-Century 627 The locally defined functions Nfa.y) are called the coefficients of the spray connection N. We deal with a horizontal vector X — Xi{d/dxi)y + X® (d/dy^y € Ny. Owing to vw(X) = 0, (2.3.3.9) yields X& + N^X? = 0. and hence we have P-3-3-1») Definition 2.3.3.6 A vertical connection Fv in the vectorial frame bundle F(M) is a distribution u e F —* T£ C F£, satisfying (1) F<=r*®FZ (direct sum), (2) ¿(1^) = T^. From Proposition 2.3.3.2 and (3) or Proposition 2.3.3.1 it follows that the induced-vertical distribution ueF->Fj;CFf[$z kind of a vertical connection. Definition 2.3.3.7. The induced vertical distribution F* is called the flat vertical connection. F* is actually flat, because F* is spanned by the induced-fundamental vector fields y(v) and we have (2.3.3.3), The conditions for rv are quite similar to those for a linear connection. In fact, I” is a connection in the principal bundle {F^.TTi.Tr^C^.GCn)}, with the base space and the total space F(x) = {ti e F|ttt o 7Ti (ti) = æ}, x 6 M. Its right-translation by g € G(n) is pg itself. The tangent vector space F(x)u is nothing but FJ. Proposition 2.3.3.3. We put F(x) as above. A vertical connection Tv is such that the restriction rv|F(x) is a connection in the principal bundle Denote the lift with respect to Fv by : X € 7r71(m)2/ —► £^(X) € F£, u = (y, z). Similarly to Definition 2.2.2.1, we have a basic vector field = % q sa'(v) = % o %(zv), (2.3.3.11) where is the vertical lift. This satisfies In fact, (2.3.1.3) and (2.1.2.S) show Pg ° 0 »<*'(?) = ^g° = ^g°=ga'(S~1v) (2.3.3.12)
628 Matsumoto We consider the expression of Bv(v) in a canonical coordinate system. Since (1) of Proposition 2.3.3.1 and (2.3.3.11) show that Y(v) - Bv(v) is vertical, we may put K(v) = Bv(y) + F(i7(v)), (2.3.3.13) where Î7(v) G (7(n)'. Then (c) of Proposition 2.3.3.1, (2.3.3.12) and (2.3.2.6) yield 4(m.) =y(ff-1v)us which gives Îf(ÿ_1u) = od(s~1)Z7(v), similarly to (2.2.3.S). Hence, if we put U(u',v0,v) = U(v)(u*,v0) = U&wfr, u* « voea, vo = Vq ea, v — vaea, then we obtain the Vÿ-valued functions Ugc satisfying the tensor property: Uug - S_1UU and having the components The tensor U is called the vertical tensor field of Tv. From (2.3.2.5) and (2.3.3.13) we obtain (2.3.3-14) Finally we shall recall the flat vertical connection F*. Proposition 2.3.3.4. The induced-fundamental vector fields Y(y) are v-basic vector fields of the flat vertical connection F*. Its vertical tensor field U vanishes identically, 2.4 The Theory of Pair Connections 2.4.1 Pair Connections We have prepared for defining the connections which are needed to develop the differential geometry of Finsler spaces. Now we shall state the fundamental notion of the connection as follows; Definition 2.4.1.1. A pair connection (T\rv) on a smooth manifold M is a horizontal connection FÂ and a vertical connection Fv in the vectorial frame bundle F(M). For a given horizontal connection we have the direct sum decomposition FU = I*®F%,
Finsler Geometry in the 20th-Century 629 as in Definition 2.3.3.3, and for a given vertical connection P we have the direct sum decomposition *2 = ^, as in Definition 2.3.3.6. Consequently, for a given pair connection (P,P) we have the direct sum decomposition F„=r^®r:@2^. (2.4.1.1) Then the sum ru = r£@q;, (2.4.1.2) gives rise to a connection V : u e F —>TU E Fu in the vectorial frame bundle F(M). In fact, (b) of Definitions 2.3.3.3 and 2.3.3.6 give = (2.4.1.3) Next, the horizontal connection Th gives rise to a spray connection N by (2.2.3.6): (a) Ty = Ny e TJ, (b) = <(r£). (2.4.1.4) Let £u: X eTy-> £V(X) € ru be the lift with respect to F. Then (2.2.4.2) and (2.4.1.4) show (a) W) = I* (b) 4(7J) ~ TS. (2.4.1.5) Thus we obtain the pair (T, N) from the pair connection (P,P). Now we shall consider the inverse direction: Suppose that a pair (T, 2V) of a connection T in the vectorial frame bundle F(M) and a spray connection N in the tangent bundle T(M) be given. We denote by the lift with respect to T and define P and P by (2.4.1.5). These are horizontal and vertical connections respectively. Because we have (a) Fu = T, e (b) Ty = Ny® T£, (2.4.1.6) from which we get (2.4.1.1). Further o — ¿ug yields p^(F£) = and p^FJ) — T^. Consequently we get the pair connection (P,P). The correspondence above of (P,P) and (V,N) is obviously one-to-one. Therefore we can state Theorem 2.4.1.1. Defining a pair connection (P; ,P) in F(M) is equivalent to defining a pair (F, 2V) of a connection T in F(M) and a spray connection N in The correspondence between them is given by (P;, P) - T = P e P, = <(P), (r, JV) -> = ¿(2V), P = £(P). We consider a smooth manifold Af equipped with a pair connection (P, P) or equivalently the pair (F, 2V). We have the basic vector field B^v) of P by
630 Matsumoto (2.3.3.11), which is now called v-bosic vector field. Tv is spanned by v-basic vector fields at every point of F. (2.3.3.11) implies Bv(v)u = 4 o ?y(zv). (2.4.1.7) Let us define another basic vector field Bh(y) as Bk(v)u = 4 ° ^(^), u = (y, (2.4.1.8) where two lifts appear: ev:Mx^Ny, 4:T9 = ^©7;^rS®ri. Bk(v) is called the h-basic vector field. It is obvious that Vk is spanned by these vectors at every point of F. We have the equation (2.3.3.12) for B^v) and also (2.4.1.9) because we have p'x(Bh(v)u) =tu3 °4((~s)(i' Xv})-Bk(g xv)„s. We have the h-basic form Qh defined by (2.3.2.7) and the v-basic form Gv defined by Definition 2.3.3.5 with respect to N.. Further we introduce the con¬ nection form <jj of T. This is a C?(n)'-valued 1-form given by (a) w(F(A)) = A, (b) w(r) = 0. (2.4.1.10) Similarly to the case of a linear connection, we have the equation satisfied by w : w op' = (2.4.1.11) We shall write w in a canonical coordinate system. We put w = wf ) c> = <ífci + Víb(í)dyi + Wm «fee- We are concerned with (2.4.1.9) and (2.4.1.11). then we get first wgf(-u) = and r$* = (^^XkW. uik - are functions of (x*\ y*) alone. Consequently we obtain “’?(«) = + ^(rjfcdr* + Ujkdyk)}. Now, (b) of (2.4.1.10) leads to (2.4.1.12) rtt 9 (2.4.1.13)
Finsler Geometry in the 20th-Century 631 Then (b) of (2.4.1.5) shows er=- Since rj is spanned by Bv(v), we understand that Ufa are just the vertical tensor field which appears in (2.3.3,14). We consider Bh(v) = zv — ziaya'(d/dxi'} and (2.3.3.10) leads to Therefore (2.4.1,13) gives _4F*(2j)}, (2.4.1.14) where *we put F^-U^. (2.4.1.15) The triad (Bh(v), Bv(v), B(j4)) of tangent vector fields on F is the dual of (r\r<',w) of differential 1-forms on F in the following sense: ek(Bh(v}') =v, F>(Bh(v'j) =Q, W(BA(«))=0, ^(B’(v)) = 0, ^(B”(v)) =V> = 0, Ôh(F(Aj)=0, r(F(A))~0, u(F(A)) = A. (2.4.1.16) These are easily shown by their definitions. We have the set of functions (Fjk>Nj,Ufac) of variables («*,?*) : These are called the connection coefficients of the pair connection, although (CTJA) are still components of the vertical tensor field. It is remarked that we have the relation (2,2.4,13). According to it, it seems that Fjk are more essential than 1%. The latter are not connection coef¬ ficients. 2.4.2 H and V-Covariant Derivatives Similarly to the case of a linear connection, we define the notions of covariant differentiations of a tensor fields with respect to a pair connection (rA,rv) as follows: Definition 2.4.2.I. The h and v-covariant derivatives VhT, VVT of a tensor field T are given respectively by VhT(y) = B^vjr, WT(v) = BV(T). First we have to show their tensor property. Let T be of (l,l)-type, that is Tvp(iT,ui) = Tutgv^gvr), t? € V*> vi € V.
632 Matsumoto Then we have By (2.4.1.9) we have = ^-8ft(sv)u{Tui;(«*,vi)} = a pg} = Bh(gv)u{Tu{gu*,gvTy} = = p-l(VAT)„(«*,v1,«), which implies VftT ° Pg — g~1'VllT, the tensor property of V&T. We consider the components of covariant derivatives in a canonical coordin¬ ate system. Let T be, again, of (l,l)-type. Then (2.3.2,10) and (2,4.1,14) lead to VfcT(v) = 4v»{* - z^F*(—)} where we put 6i = di-Nrdr. (2.4.2.1) Thus we get = [(z-^z&t* - *«Q^]eg = + ziF^(z-^k(S-^ -zlF^z-^Me^ = zlv^z-'faWiT? + - T*F&e*. Consequently, we obtain VhT(v) = T^-,e v°eb, Tk;c = (z-1)^ zizi, (2.4.2.2) Th = Stf + TTFh _ r^pr, 6i is called the 6-operator which is defined with respect to the spray connection N. Similarly we obtain the v-covariant derivatives on account of (2.3.3.12) and (2.3.2.10) as follows: VvT(v)=Ti“:cVce») T^c= (*" W 44 (2.4.2.3) T? fy? + TTty - T^.
Finsler Geometry in the 20th-Century 633 Definition 2.4.2.2. For a tensor field T the O-covariant derivative is given by V°T(t>) = Y(v)T. According to Proposition 2.3.3.4, V0T(t>) is the v-covariant derivative in the pair connection (r\ F*) with the flat vertical connection. Then (2.3.3.13) together with (2.3.2.11) gives V'TCv) = V°T(v) + Ufa) * T. (2.4.2.4) (2.3.3.2) yields immediately V°T(v) = Tb -c v'et TS-C = z[zi, (2.4.2.5) 7*.,= 3,3*. We deal with the covariant derivatives of the supporting element e. Since (2) of Proposition 2.3.3.1 gives VM*)=^ A'M’ (2.4.2.6) and (2.3.3.13) gives Vefa) - v + Ufa) • e, + j/%, (2.4.2.7) it may be better to write Ufa) • £ in the form Ufa v). Definition 2.4.2.3. The deflection tensor field D of a pair connection (P\rv) is given by Dfa) = V*e(v). Owing to (2.4.2.2) the components of the deflection tensor D are given by (2.4.2.8) 2.4.3 Torsions and Curvatures of Pair Connection We consider the Lie brackets of basic vector fields on the total space F of the vectorial frame bundle F(M) with a pair connection r\rv), and introduce the torsions and curvatures, as in §2.2.3. If we put then we have h and v-basic vector fields Bhfa)y B^fa) in the forms z^vaBj, zfaaB? respectively. Thus, for constant vj and v% eV (2.2.3.1) leads to
634 Matsumoto Bt(v2)] and [Bv(ui)rBv(v2)] are written in the similar forms. We have = ^ + «i^SJ. A+F«B;> Bi(&a) = B?(ziva) = -¡faW*, where we have put = 6iNi - (»/A Pij = fyN? ~ ffi, (2.4.3.1) RkMj = Kfo + U^, Kfa = 6^ + F&F*- - (i/j), (2.4.3.2) Phii = Jty " Ufc* +U^, = d^..- (2.4.3.3) = &^i + U^Uk - (i/j). (2.4.3.4) Then we obtain (a) [Bh(vi)JB'l(îi2)] =Bh(T(v1,V2))+Bv(R1(vuv2)) (2.4.3.5) +F(R2(v1,Vz)), (b) [Ba(vi),B*(v2)] = Bfc(J7(«1)V2)) +B’'(Pl(vi,v2)) + F(P2(vltV2)) (c) [B*'(vi),Bt'(v2)] = ^(S^v^vz)) + F(S2(v1,v2)), where the components of R1, P1, R2, p2 and & are R^P*, R^., P^ and as defined above and that of T and S1 are V ^=^~-FfcP Sjfc = U}fc-^. (2.4.3.6) It is remarked that U is the vertical tensor field of T< The equations (2.4.2.S) is called the called as follows: T ... (h) h-torsion tensor, U ..»(h) hv-torsion tensor, S1 ... (v) v-torsion tensor, R2 ... h-curuature tensor, P2 ... hv-curvature tensor, structure équations and T,..., S2 are R1 ... (v) h-torsion tensor, P1 ... (v) hv-torsion tensor, S2 ... v-curvature tensor.
Finsler Geometry in the 20th-Century 635 Remark: In §§2.4.3 and 2.4.4 the following abbreviation symbols are used: (i/J), (1/2), A(i2]: interchange of 1,2 (¿, J) and subtraction, (i, &)> (1>2? 3), S(i23), S(ijfc) : cyclic permutation of 1,2,3 (i, J, fc) and sum¬ mation. The structure equations give rise to the commutation formulae of h and ü-covariant differentiations. We apply them to a tensor field K and obtain (a) Vk{V^(«2)}-(l/2) = VaA'(T(vi,v2)) + VK(B1(vi,«2)) -^(«1,%) -K (b) Va{V“A(^)} - 5T{V^Cvi)}^) = +V"A'(P1(vi,v2))-P^vJ-K, (c) ViVKWXvi) - (1/2) = V^S1^)) - S®(vi, va) • A., (2.4.3.7) 2.4.4 Bianchi Identities of Pair Connections We are concerned with the Jacobi identity [X, [y, Z]\ + [y, [Z,X]] + [Z, [X, yj] = 0, satisfied by the Lie brackets. Applying this identity to the h and t^basic vector fields, we get the following four classes of identities: We shall write, for instance, Bh(y2) as J?2 f°r brevity. +(1,2,3) =0, [B£, [B2, BJJ] + [B2, [*3, #]] + [BJ, [Bi, B2fcl] = 0, [Bi,[B?,Bj]] + [B?,[BS,Bfl] + [Bj,[Bi,Bi]J =0, [B?,[^,BJJ]+(1,2,3) =0.
636 Matsumoto For instance, we shall deal with the first class in detail. (1) of (2.4.2 Ji) leads to [в^,вл(т(2,з))] + [в^в^л^з))] + ^^(^(2,3))]+(1,2,3) = 0, where T(2,3) = *уз) and so on are used for brevity. On account of (2.4.2.S) together with (2.2.3.1) we have [Bf, Bh (T(2,3))] = Bh (T(l, T(2,3)).+ Bv (Я1 (1, T(2,3)) + ^{#(1, T(2,3))} + Bft{Vft (T(2,3)) («х)}, [В£,В*(Я1(2,3))] = Bk(U(l,Rx(2,3))) + BV(P1(1, R42,3))) + P{P2(l,B1(2,3))} + B’{Vh(B1(2,3))(v1)}, [Bi,F(B2(2,3))] = -Bh(R?(2,3) -V1) + F{Vft(B2(2,3))(vi)}. To get the last equation, we used [Р(А),ВЛ(«)] = Bh (A • v) + Bft(P(A>) - F(Bk(y)A), which is easily verified by (2.2.3.1). Consequently, the identity of the first class is divided in Г\ Tv and F^-components as follows: ■ E(123){T(l,T(2,3))+VftT(l,2,3) + ?7(l,B1(2,3)) -A2(l,2,3)} =0,' + P1(1,B1(2,3)) + VW(1,2,3)} = 0, > Ъ^){#(1,Т(2,3')) + P2(1,B1(2,3)) + VhR?(l,2,3)} = 0„ (2.4.4.1) where we used the notation VfcT(l, 2,3) = Vh (T(vi, V2)) (t<s), ^(l, 2,3) = Я2 («2, Vj-VL In terms of the components these identities (2,4.3.1) are written in the forms + T^,k +!% - R^k} = 0,' S(Wfc) = 0, > + FtirRjk + = (a) (b) (c) (2.4.4.2)
Finsler Geometry in the 20th-Century 637 Next, we consider the second class. In a similar way to the above we obtain V’TXl, 2,3) - £7(T(1,2),3) + A(12]{T(1, £7(2,3)) + Vfc£7(2,3,1) +17(1, Px(2,3)) - P2(l,2,3)} = 0, V* P1 (1,2,3) - P1 (T(l, 2), 3) + S1 (3, P1 (1,2)) - P2(3,1,2) + A[12]{Pl(l,£7(2,3)) + P1(1,P1(2,3)) - VfcPx(2,3,l)} =0, VVP2(1,2,3) — P2 (T(l, 2), 3) + S2 (3, P^l, 2)) + A[12]{P2(1, £7(2,3)) + P2(1,P1(2,3)) + V^W)} = 0. In terms of the components, these are written as (a) T^.k-U^ + ^{T^UTk + E7^ + £7^.-pAfc} = 0, (b) P*j 'fc ~ ~ Pktj (■>444) + A[y]{P££7ft + P&.P& + P^} = 0, (c) Rtij+ SikrRij + A(tf]{P&r£7fc + PiirPjk + -fjyfc;»} = Next we consider the third class, which gives £7(3, S1 (1,2))—£2(3,1,2) + A[12I {£7(£7(3,2), 1) - V"£7(3,2,1)} = 0, VS1 (1,2,3) + P1 (3, S1 (1,2)) + A[12] {P1 (£7(3,2), 1) - S1 (1, Px(3,2)) - V’P1^ 2,1) + P2(l,3,2)} = 0, VhS2 (1,2,3) + P2 (3, Sl (1,2)) + A[i2] {P2 (£7(3,2), 1) - S2 (1, P\3,2)) - V’P2(3,2,1)} = 0. (2.4.4.5) The first of (2.4.4.5) is nothing but the equation (2.4.2.4) in terms of the v-covariant differentiation. On the other hand, the second and third are written
638 Matsumoto in terms of components as follows: (a) S^+^ + A^P^ -S^-P^!i + ^} = 0) (2.4.4.6) 0») S^ + P^ + ^{P^U^ - - Pehkj;i} = 0. Finally the fourth class yields only two identities as follows: S(!23) {s1 (1, Sx(2,3)) + VS1 (1,2,3) - S2(l, 2,3) = 0, S(123) W, Sl(2,3)) + V’Ss(l,2,3)J = 0. In terms of components they are written as (a) S(y*){S^+^:ft-S^} = 0, (b) £(yfc){St^fc + S^:fc} = 0. These are quite similar to (2.2.4.6) of a linear connection. All of these identities satisfied by torsion tensors and curvature tensors are called the Bianchi identities for torsion tensors and curvature tensors of the pair connection. 2.4.5 P-and [/-Conditions We define a mapping for a fixed v eV 7rv = o^ o 772 : u = (y.z) € F —> zv eT. By the supporting element e : u= (y^z) e F —► z^y € V we get t = av o tto. v = e, (2.4.5.1) Then (2.3.2.13) can be written as fli = t' + o s'. We apply this mapping to the h and v-basic vector fields Bh(v) and B^(u). Since (2.3.2.15) leads to sfBh(v) = VAs(v), ?Bv(u) = V^v), in the symbol given by Definition 2.3.1.2, we obtain (1) r^Bh(v} =r'B/l(v) + 5a'(V/l£(v)). (2) 7rJBv^) = T,Bv(v) + 2a'(Vv£(v)). (S.4.5.2)
Finsler Geometry in the 20th-Century 639 Remark: In a canonical coordinate system we have r : y\4) - (?, 41?), t? = (s'W- d d \ (d n , „ ’ dy*’ dzi) 0>V Consequently (2.2.4.12) and (2.3.3.14) give rtsW - sM(gjr). Now we are concerned with (a) of (2.4.5.2). According to Definition 2.4.213, it is written as ^Bh(v) = t'B^v) + -a' (D(v)). (2.4.5.3) Consequently, according to (2.4.1.4) we get Proposition 2.4.5.1. We consider a pair connection (r\ Tv) and equivalent pair (r, 2V). Then r,{rh) coincides with the spray connection N, if and only if = 0, that is, the deflection tensor field D vanishes identically. Definition 2.4.5.1. If the deflection tensor field D vanishes identically, then the pair connection is said to satisfy the D-condition, From (2.4.2.S) it follows that the D-condition is written locally as yrFrj = N]. (2.4.S.4) Next we deal with (b) of (2.4.S.2). From (2.3.3.11) its left-hand side is equal to -¿/(v). On the other hand we have (2.4.2.7). Thus (b) of (2.4.S.2) is written in the form r'Bv(v) = --¿(Ufa v)). (2.4.5.S) Proposition 2.4.5.2. /(T*) vanishes, if and only ifVve = 6, that is, vertical tensor field U satisfies U (s, v) = 0 for any v EV, Definition 2.4.5.2. If the vertical tensor field U of a pair connection satis¬ fies U(s,v) — 0 for any v E V, then the pair connection is said to satisfy the U-condition, From (2.4.2.7) it follows that the D-condition is written locally as iTC^O. (2.4.O.6) Now we deal with a pair connection satisfying both the D and D-conditions. Applying the Ricci identities (2.4.2.7) to K « e, we have the following remark¬ able fact:
640 Matsumoto Theorem 2.4.5.1. For a pair connection satisfying the D- and U-conditions, the torsion tensors of (v)- type are obtained from the curvature tensors as fol- lOWS: 7 0 7 0 -H (vi,v2) = v2), P (vi,v2) = P2(e,vi,t^), *S,1(vi,V2) — S2(e, vi,v2). In terms of the components the above are written in the forms Then (b) of the Bianchi identities (2.4.4.2), (2.4.4.4) follow from (c), and (a) of (2.4.4.6), (2.4.4.S) follows from (b). 2.4.6 Parallel Displacement Let <7* be a smooth curve: I = {t|0 t < 1} —► F in the total space F of the vectorial frame bundle F(M). Then we get the curve C — o C* in the total space T of the tangent bundle T(M). C* is said to cover C. Similarly, let C : I T be a smooth curve in T. Then we get the curve Co = ttt o (7 in the base manifold M. C is said to cover Co. Given a pair connection (r\r*) or equivalently the pair (T, N), if a curve C* of F is tangent to T at every point, then C* is called V-horizontal. Similarly, if a curve C of T is tangent to N at every point, then C is called N-horizontal. If a curve C" of F covers a curve C of T and is T-horizontal, then C* is called a F-lift of C. Similarly, if a curve C of T covers a curve Co and is ^horizontal, then C is called a N-lift of Cq. Let C* — ^¿(t)) be aT-lift of C = (s*(T),j/’(t)). Then (2.4.4.13) shows ^l=Xi *£ = X® dt ’ di ’ ^--z^Xi + UijXW}. Hence we obtain the differential equations of the T-lift as ^ + <4{rU«)(^) +^(«10 -«■ (2-4-61) Next, let C = (x* (t),y*(t)) be a N-lift of Co = . Similarly to the above we obtain the differential equations of the N-lift by (2.3.3.10) as follows: (2A6-2) Definition 2.4.6.I. Let C* = («(<)>y(t),2(t)) be a T-lift of a curve C — (x(t), j/(t)) of T. Then the frame field z{t) along the curve Co = (s(i)) of AT is called parallel along Co vhth respect to the vector field y(t) given along Co-
Finsler Geometry in the 20th-Century 641 Hence (2.4.6.1) gives the differential equations of such a parallel frame field a(t). We deal with two special two cases as follows: Fist, suppose that the curve C of T is JV-lift of Cq. Then, substituting from (2.4.6.2) and paying attention to (2.4.1.5), we have (a) ^ + ^(®-y)^ = 0, (2.4.6.3) (b) = Secondly, suppose that C is vertical, that is, x(t) = zo is fixed, then (2.4.6.1) is reduced to + z*U'kj(x0, y) = 0. (2.4.G.4) In the case (2.4.6.3), the curve C* = (#(t),y(t), s(t)) is tangent to the hori¬ zontal connection while in the case (2.4.6.4), the curve C* = (azo, y(t), z(t)) is tangent to the vertical connection F. Definition 2.4.6.2. Let v(t) be a vector field given along a curve Cq — (a?(t)) of the base space Af. If v(t) has constant components with respect to a parallel frame field z(t) along Co with respect to a vector field j/(t), then v(t) is called parallel along Cq with respect to y(t\ Hence v(t) — vi(t)d/dxi, vl(t) = z*(i)va with constant and (2.4.6.1) gives the differential equations of such v(t) : + U^y) = 0. (2.4.C.5) It is specially interesting when y(t) is a JV’-lijft of Cq. Then (2.4.6.3) yields (a) + = (2.4.6.6) 09 2.4.7 Homogeneity of Pair Connection We define a transformation of T as hp : y e T py € T, for p € K+. This induces a transformation of F by u =($/,£) e F pu = (py,s) € F, which is called a homogeneous transformation of F by p. We have
642 Matsumoto (a) 7F1 o Hp = hp o %!J (b) TT2 o Hp — 7T2, (2.47.1) as the characteristic properties of Hp. ks it has been stated in §1.1, a real-valued function f(xyy) on T is called (r) p-homogeneous in y\-if fohp=prf holds for any positive number p. Definition 2.4.7.I. (1) A vector-valued function f on F is called (r) p-homogeneous, if f°Hp = prf holds for any p > 0, (2) a tangent vector field X of F is called (r) p-hoomgeneous, if Hp(X) = prX holds for any p > 0, (3) a vector-valued 1-form a on F is called (r) p-homogeneous, if ao Hp = pT a holds for any p > 0, (4) a distribution D on F is called p-homogeneous, if Hp(D) = D holds for any p>0, (5) a distribution D on T is called p-homogeneous, if hp(D) « D holds for any p>0. In canonical coordinate systems (æ*, y*) of T and (a:*, z*) of F, we have hptây*) = (x^py*), Hp{x\y\z\) = Hence we have (a) hfp(di,di) = {diypdi), (2.47.2) (b) ^(ft,ft,^) = (âi,pÔi,^)î where df — d/dz&- Proposition 2.4.7.1. (1) The supporting element £ is (1) p-homogeneous, (2) a fondamental vector field F(A) is (0) p-homogeneous, (3) the h-basic form № is (0) p-homogeneous, (4) a induced-fundamental vector field Y (v) is (1) p-homogeneous. Proof: (1) (2.3.2.12) gives e(pu) — ps(u). (2) (2.3.2.5) gives HtfWj = 0 = ^{A) = F^.
Finsler Geometry in the 20th-Century 643 (3) (2.3.2.7) and (2) of (2.4.7.1) yield (4) (2.2.3.1) shows «3 °H'p(Y(v')) = ’I2(yW) = °> s' off'(y(v)) =ps'(Y(v)') = pv. Hence we have H£(y(a)) = Y(pv) = pY(y). Now we are concerned with the homogeneity of a pair connection. Definition 2.4.7.2. A pair connection (r\ Tv) is called p-homogeneous, if both Th and rv are p-homogeneous distributions. Theorem 2.4.7.1. A pair connection (F\TV) is p-homogeneous, if and only if one of the following three conditions holds: (1) both T = + Fv and N — ^(P*') are p-homogeneous, (2) anyh and v-basic vector fields Bh(v) andB^iy) are(O) and (1) p-homogeneous respectively. Then the vertical tensor field U is (-1) p-homogeneous. (3) The connection form w and v-basic form are (0) and (1) p-homogeneous, respectively. Proof: (1) if both I* and Fw are p-homogeneous, then T = + Tv is ob¬ viously p-homogeneous and (1) of (2.4.7.1) gives hp(N) — N. Conversely, the p-homogeneity of (F, JV) implies h'P * *y = H'po£u = ^po h'p. (2.4.7.3) Hence we have HJ(T*) = O 4(^) = ip* o h^Nv) = ¿Mv) = r^, and similarly = P' because of h'p(rT') = Tv. ■ (2) (2.4.1.7) and (2.4.7.3) give H'p o £u o£y(zv) = ip» o o £y(zv) = £₽« o £w(«v) = BA(v)pu. On the other hand, (2.3.3.13) yields H'p(Y(v)v) =PY(y)pv = p{Bv(y)pu + • =j3J(b«(w)„)+h;(f(d-wj), which together with (b) of Proposition 2.4.7.1 leads to = pB^v)^, U(v)pu =p-'U(v)„.
644 Matsumoto The latter shows that U is (-1) p-homogeneous. (3) is shown by the duality (2.4.1.16). Next, (2.2.4,12) and (2.3.3.14) yield W/W = pNfa), F*(pu) = F*(u). Therefore we have Corollary 2.4.7.1. A pair connection (r\v) is p-homogeneous, if and only if the connection coefficients are (0), (1) and (-1) p-homogeneous functions respectively. Definition 2.4.7.3. A pair connection (T\ P) is said to satisfy the Us-condition, if y(s) is v-horizontal, that is, Yfa 6 P. From (2.3.3.2) we have y(s)u = yid/dyi at u = (s*,?/1,^). We have a relation between the homogeneity and ¿/o-condition. In fact, from the trans¬ formation Hp we get a curve Hu : p G -* pu € K for a fixed u G F. From Hi : (d/dp)p y^d/dy^ we get Consequently we have Proposition 2.4.7.2. A pair connection (T\P) satisfies the ^-condition, if and only if Hi(d/dp} is v-horizontal at p = 1. From (2.3.3.13) and CZ(s)(u’,u) =« Ufa\v,£) we have Theorem 2.4.7.2. A pair connection (T\P) satisfies the Uz-condition, if and only if the vertical tensor field U satisfies Ufa e) = 0 for any v G V. Thus the fact = (2.4.7.4) is the Ui-condition. Recall the ¿/-condition: tfU^fa y) = 0 in §2.4.5. 2.5 Standard Finsler Connections 2.5.1 Typical Vertical Connections We consider an n-dimensional Finsler space F* = (M,L(x,y)) with the fun- damental function Lfay). We shall construct a pair connection in F* from Lfay) with the object of studying differential-geometric properties. Such a pair connection is called a Finsler connection of P.
Finsler Geometry in the 20th-Century 645 As it has been observed in §2.2.5, a Riemannian space has the Levi-Civita connection which is the most reasonable from all the viewpoints. On the other hand, we have had some Finsler connections from the author’s standpoint and his purpose. The present section is devoted to giving Finsler connections useful for various purposes. As it has been mentioned in §1.2.2, the tangent space Mx at a point x of M is regarded as a Riemannian space with the Riemannian metric P'vfaoi x$ = (a?o) being the coordinate of x. Consequently it is natural to introduce in Mx the Levi-Civit& connection ^x constructed from gij(xQ9y) with respect to (y*). According to (1.2.2.5), the connection coefficients of yx are nothing but the components C^x^y) of the C-tensor. Therefore Definition 2.2.5.5 leads to Proposition 2.5.1.1. Assume that the vertical connection Vv of a Finsler connection FT satisfies (1) v-metrical: Vvg — 0, (2) (v) v-torsion S1 — 0. Then the components Ujk of the vertical tensor field U are uniquely determined as the components Cjk of the C-tensor. ' In this case FT satisfies the U and ¿Zn-conditions, that is, jA^ = C^ = 0, (2.5.1.1) which are consequences from the (0) p-homogeneity of (x, y). Further we have the v-curvature tensor S2 of the form (2.5.1.2) because we have (2.4.3.4) and gktdjChi = djChti— . where the first term is symmetric in i,j. On the other hand, we pay attention to the fact that are components of the vertical tensor field C7. Thus the vanishing of does not depend on the coordinate system. Further Proposition 2.3.3.4 leads us to Proposition 2.5.1.2. We have the Finsler connection FT = (r\r*) whose vertical connection T0 is the fiat vertical connection F\ (1) The vertical tensor field U vanishes, (2) the v-covariant differentiation is nothing but the partial differentiation by y\
646 Matsumoto (3) the (v) v-torsion tensorS1 and the v-curvature tensorS2 vanish identically. It seems that (F\ F*) is a very convenient connection, but Definition 2.4.2.2 shows V’Sy = V°iy=2C'yfe. (2.5.1.3) 2-5.2 Cartan Connection We consider the horizontal connection of a Finsler connection (r\rv), and pay attention to the axioms of the Levi-Civit& connection: (1) ^metrical: Vhg == 0, (2) (h) h-torsion T — 0. But (2-2.4.12) shows that the axioms (1) and (2) do not uniquely determine rA, because we have many unknown functions, not only F%ki but also N%. Con¬ sequently we pay attention to the deflection tensor D whose components are given by (2.4.2.S). Therefore, adding to (1) and (2) we have (3) deflection tensor D = 0, that is, = 0. Thus we shall consider a Finsler connection satisfying (1), (2) and (3). First, on account of (2.4.2.2), (1) is written in the form Then, similar to the case of the Levi-Civita connection, the Christoffel process (Definition 2.2.S.4) leads immediately to the beautiful form ^ik = 1^9ir + $i9kr - Sr9ik)> But we have unknown functions Nj yet in the ¿-operator. In terms of the ^-operator the above is rewritten as &ijk ~ 'Yijk CijrNrk OkjrNJ + CikrNj-) (2.5.2.1) where 7y* ® 1/2 (dig^ + - dy&fc). Transvecting with yi (2.5.2.1) yields Foja = 70jffc ^kjr^Q‘ (2.5.2.2) Here and throughout the following, the subscript 0 denotes the multiplication by and summation (i.e. transvection with y*). Further, transvecting the above yields FOjo — 7ojo,, that is, ^00 ~ T&p Since the condition (3) gives F$k - from (2.4.2.8), the above shows 2Vg = 7^ and hence (2.5.2.2) is written as 2V£ — 7^ — Oj/yoo- Thus, the spray connection has been determined. Then (2.5.2.1) gives the definite form of
Finsler Geometry in the 20th-Century 647 We recall G* given by (1.2.2.4). Then it is easy to show that N] = Gj- (= - Cj^So, (2.5.2.3) where are the Christoffel symbols constructed from ^(x, y) with respect to x\ Therefore we have Definition 2.5.2.1. Let Fn = (M, L(x,y)) be a Finsler space with the fun¬ damental function L(x,y) and g = the fundamental tensor. The Cartan connection CV = (r\rv)‘of Fn is a Finsler connection which is uniquely de¬ termined by the system of axioms: (1) h-metrical : Vàp = 0, (2) (h) h-torsion T = 0, (3) deflection tensor D — 0,, (4) v-metrical : = 0, (5) (v) v-torsion S1 — 0. The horizontal connection Th is determined as Fy* (= gjrF^) of (2.5.2.1) and Nj of (2.5.2.3). The vertical connection T* is determined by the Christoffel symbols constructed from Pv(x,y) with respect to that is, the components Cjk of the C-tensor. Ref, E. Cartan [35]. The system of axioms (1),...,(5) as above was first given by M. Matsumoto [85], and was discussed in detail by T. Aikou and M. Hashiguchi [2]. The h and v-covariant differentiations with respect to the Cartan connection GT are denoted by (j, |) respectively throughout the following. The commuta¬ tion formula (2.4.3.8) of these differentiations are written as (a) K^k - (j/k) = «¡fy - K№ijk - K^k, ■ (2.S.2.4) (b) = K?P?jk - K*PTik - K^jk - (c) - 0*) = - K$srjk. Since CT satisfies the D- and ¿/-conditions, Theorem 2.4.5.1 shows (a)^ = ^.fc! (b) P& = P0%. . (2.5.2.S) The components of the torsion and curvature tensors of CT are specially given by (a) = (2.5.2.6)
648 Matsumoto (b) (c) + (d) P*k = F$k-C^+CtrPrk, (e) S^ = CrfcC^-(JA), where the tensors K^k and Fhk are defined as (a) K*k = SkF* + ]%F* - (j/k), (2.5.2.7) (b) *& = M- It is noted that the tensor K is defined similarly to the form of the curvature “tensor R^k of the Levi-Civit& connection. Applying (2.5.2.4) to the fundamental tensor we get the identities satis¬ fied by the covariant components Rhijk = 9irP^jk and Phijk = dirPhjk ‘ vs (a) Rhijfe “ —‘P'ihjki (b) Pfrijk — ~Pihjk’ (2.5.2.3) Now we are concerned with the Bianchi identities of CT. First (2,4,4.2) are written as (a) SWfc){^.fc-C^fc} = 0, (2.S.2.9) (b) S(W{^ + ^} = 0, (c) s(W<t+4^} = o. As it has been remarked at the end of §2.4.,5 that (b) is a consequence of (c). Next, (2.4.4,4) are written as (a) AM{C^ + C^-^ft} = 0, (2.5.2.10) (b) + R^CFjk + P^k} = 0 (c) |* + + Ali31 {P*kli + R^CJk + P&%} = 0. (b) is also a consequence of (c). Next, (2.4.4.6) are written as (a) Wli + P&j + P&qk} = 0, (2.5.2.11) (b) S*jlk + Afo-]{P^b- + ~ S^} = 0. Finally (2.4.4.S) are reduced to (a) S(W{S<%} = 0, (2.5.2.12)
Finsler Geometry in the 20th-Century 649 (b) S(W{^I4=O. Now, if we rewrite (a) of (2,5.2.9) in the covariant components, then we get by (2.5.2.S) ^(ijk){Rhijk + Cjk-ftrj*} = 0, where Rijk = girRjk — Roijk- By transvecting with we have S(W{^fc} = 0. (2.5.2.13) We deal with (a) of (2.5.2.10). If we write it in the covariant components Phijk = 9vrPhjk and Pijk = girPjk ~ Poijk> then {Qij/ett •+ ^hi^rjk 4" Phijk} = 0* Applying the Christofiel process with respect to (h, i,j), the above yields Phijk — Cjkiih Cjkhn 4“ ChjPrik ~~ C'ijPrhk' Transvecting with yh and then 3A we have P^k — CjkitQ—CijPrOk and P^k — 0. Consequently we obtain the interesting forms of Pijk and Phijk as follows: (a) Pijk = Cijki0, (2.5.2.14) (b) Phijk = A^]’{CJ^Crt/ciO Crjkhii}* (b) of (2.5.2.14) shows the following identities: (a) B(w>){P^} = 0, (2.5.2.15) (b) {Phijk} = (c) — SjwjiO, (d) PhiQk — PhijO — 0. We shall return to (a) of (2.5.2.11). In terms of covariant components it is written as ~ 4" Pihkj 4" PhriC'kj} = 0» Apply again the Christofiel process in (7m, j). Then we obtain Phijk — 4- PikrCjh} (9 5 2 16) = 4- PhjrCik}. Thus we .have the interesting equations on Phijk and Pijk as (b) of (2.5.2.5) and (2.5:2.16). Finally we shall list some important formulae for the later use: (a) 1^0, fc£ = 0, 4^0, (2.5.2.17) (b) L\i = £i, 4b = kt,
650 Matsumoto where hij — gij — is the angular metric tensor (1.2.2.9) (a) hijlk = 0, (2.5.2.18) (c) \,.fc = 2Cijk - . Example 2.5.2.1. We shall show a little application of the Cartan connection. We have the notion of concurrent vector field; it is defined by in a Riemannian space with the Levi-Civita connection. In a Finsler space with the Cartan connection was defined by S. Tachibana [158] as follows: A tangent vector field X*(s) is called concurrent, if (1)X< = -^ (2) X^,=0. From X1 = Xi(x) and (2) we have X*|j = 0. Here we shall define it as follows: A contravariant vector field Xi(x,3/) is called concurrent, if (a) X£ = -(5J, (b) X^O. The condition (a) implies that a concurrent X* does not vanish at each point. We pay attention to the Ricci identities (2.5.2.4). These yield immediately (a) X’X,* == 0, (b) X%k + = 0, . (c) X%A = 0. (2.5.2.19) (b) and (b) of (2.5.2.S) give XrXzPrijk == 0. Hence (b) leads to (2) above and then (c) is satisfied by (e) of (2.5.2.6). Next we observe Xi I, = 0 = djXi + XrC< = d^X\ x-iiy = ^xq, - djXi+xrc£ - ^x- Therefore both Xi and X» are functions of position alone. Proposition 2.5.2.1. A concurrent vector field X* of a Finsler space with the Cartan connection is defined by Then both X2 and Xi = giTXr are functions of position (¡s’) alone and satisfy XiCik = 0 and (2.5.2.19). 2.5.3 U- and P1-Processes We are concerned with two Finsler connections FT and ‘FT on a manifold M. Let and (Bh(y)f B°(v)) be respective h and v-basic vector fields.
Finsler Geometry in the 20th-Century 651 From (2.4.1.S) we have - Bh(y)) — 0. Hence 'BE(v) - Bh(y) is quasi-vertical by Definition 2.3.3.1 and Definition 2.3.3.6 shows the existence of Dh(v) e V and AA(v) € G(nf such that fBK(v) Bh(y) + Bv(DA(v)) + F(Ah(v)). (2.5.3.1) From (2.3.2.6) and (2.3.3.12) and (2.4.1.9) we can prove that Dh and Ah are tensor fields of (1,1) and (1.2)-type, respectively. Applying (2.5.3.1) to the supporting element £ and paying attention to Defin¬ ition 2.4.2.3, (2.4.2.7) and (2.3.2.11), we .have 'D(v) = D(v) + Dh(v) + U&D\v)} - Ah(s,v). (2.5.3.2) Next, (2.3.3.11) shows 7r£ (z('Bv(v) -B^(v)) = 0 and hence we have Av(v) 6 G(n)' such that '^(v) = Bv(v) + F(A°(t>)). (2.5.3.3) A” is a tensor field of (l,2)-type and (2.3.3.13) shows Y(v) = 'B”(v) + Fftr(v)) = Bv(v) + F(?(v)), (2.5.3.4) which implies • 'U(v) = U(y)-Av(y). (2.5.3.5) Thus Av is simply given by (2.5.3.5). Definition 2.5.3.1. (A\ Av^Dh) is called the set of the difference tensors of FT from 'FT. Further we construct [/B/l(u)/Bv(v)] by substituting (2.5.3.1) and (2.5.3.3). Then, comparing Fà and I* components, we get . 'P^vi, va) + Dâ(,ÎZ(vi,v2)) = F1^!,^) + Sl(Dh(vi),V2) - V^MXva) - Av(va) ♦ + Ah(vl)^-F(Av(v2))DA(v1). By paying attention to = D*(A*(vi,v3)) - A’(P*(vi),V2), from (2.2.2.6), the above is written as 'P1 (Vi > V2) = p1 (yi,vz) + s1 v2) + Aft(v2, Vi) g. ' We first pay attention to the horizontal connections. Proposition 2.5.3.1. The Finsler connection 'FT has the same horizontal connection with a Finsler connection FT, if an d only if the difference tensors of FT are {Ah,AviDh) = (0,Av,0).
652 Matsumoto We shall denote by FT (Ph) the set of Finsler connections having the same rh. Thus, for FT, 'FT G Fr (1^) we have 'Bh(y) = Bh(y) and (2.5.3.4) shows they have the common Bv (v) + F (U(v)) which is equal to Y(y). These common fBh(y) — Bh(y), 'B’'(v) = B’’(t,) + P(UW) ( = r(v)), (2'5‘3a) give a unique Finsler connection, because it is obtained from FT e FT•(I*) by the difference tensors (A\ Av, Dh) = (0, U, 0). Definition 2.5.3.2. The Finsler connection 'FT which is obtained from FT G FT (rA) by the difference tensors (Ah, Dh) = (0, ¿7,0) is called the Finsler -connection. The process of constructing it from FT is called the U -process. By constructing the structure equations of the Finsler connection from (2.5.3.7), we obtain its tensors as follows: = 'T = T, 'C7 = 0, ■'R1=R1i 'P^P1, 'S1 = 0, 'S2=0, =P2(«1,«2) + VAiZ(v2,V1) - UtPXv!,»!)) (2.5.3.8) = P(vi,t>2), = P2(«i,«2) - = KiyuVï), where F and K are defined by (2.4.3.3) and (2.4.3.2), respectively. The most remarkable point is, of course, '[7 = 0. Next, we are concerned with the spray connection N. Equations (2.5.3.1) and (2.4.1.4) show Proposition 2.5.3.2. A Finsler connection 'FT has the common spray con¬ nection N with a Finsler connection FT\ if and only if the difference tensors = (A\Av,0). If we put Dh — 0 in (2.5.3.6), then we get ^(va.vi) = 'P1^,^) - P^vi, V2). Using the Kronecker’s S : 6(v) = v, the above may be written as Ah(y) = 'P^u.J) - P^M)- Thus (2.5.3.1) gives N(v) = B\v)-F(P\v,6)')> Y(v) = Bv(y)+F(U(y)), k“' ' ' are common to all the Finsler connections having the same spray connection N. This set of Finsler connections is denoted by FT (77). There exists the Finsler connection whose (Bh (v), Bv (v)) are (N(v),y(v)) : Its difference tensors (j4h,Av,DÂ) from a FT € FT(N) are given by (A^(v), A* (v), Dh(vfi = (-Px(v, <5), U(y), 0).
Finsler Geometry in the 20th-Century 653 Definition 2.5.3.3. A Finsler connection obtained from a FT € FT(JV) by the difference tensors (A*(v),A*(v),D*(v)) = ( — Px(v>^)i^(v)jO) is called the Finsler N-connection. Since we have defined the (/-process. we shall here pay attention to -P1 (v, <J) only and put Definition 2.5.3.4. To construct the Finsler connection *FT from a FT e FT(N) by the difference tensors (A*(v), Av(v),Dh(v)) = ( — P1^, 5), 0,0) is called the Pl-process. Let (*Bh(u), *Bv(v)) be the h and v-basic vector fields of *FT above. Then we have = Bh(v) - FfP^v,¿)), *B*(v) = Bv(v). (2.5.3.10) By (2.5.3.2) and constructing the structure equations of *FT, we obtain the torsions and curvatures of *FT as follows. *D(v) = D(v) + Px(v,s), U, *BX = Px, *T(vi, V2) = T(vi, v2) - A^P1^)}, +Px = 0, *BX=SX, *S2 = 52, •P2(vi,^) =P2(v1,t;2) + P1(tr(v1,v2),^ I-(2-5*3‘n) *P2(vi,v2) = P2(vi, V2) +P1(T(vi,V2),<5) + A[12] {P1 fa, P1 (vi, ty + V^P1^!, <5, v2)}. From the above it follows that a remarkable property of • FT is *P1 — 0. To construct the Finsler //-connection, we must construct first the *FT by the P1-process and next the '* FT by the (/-process : '(‘^(e)) = *Bh(v) = Bh(y) - = 2V(t>), '(*Bv(v)) = *Bv(v)+F(*U(y)') = B’(v) + F(U(v)) = K(v). On the other hand, the first is the tZ-process and next the P1 -process : * (Bk(yï) = 'Bh(y) - F(Pl(v,3)) = Bh(v) - F(Px(v, 5)) = N(v), *('B“(v)) = 'Bv(v) = Bv(v) + F(I7M) = Y(v). Therefore we have Theorem 2JL3.1. The Finsl&r N-connection can be constructed from a FT € FT (7/) by the P1-process and next the U-process, or equivalently by the U-process and next the P1-process.
654 Matsumoto 2.5.4 Chern-Rund Connections Let be the connection coefficients of the Cartan connection CT given by Definition 2.5.2.1. Definition 2.5.4.1. The Finsler connection CRT which is obtained from CT by the ^/-process is called the Chem-Rund connection. Thus the connection coefficients of CRT are (Fjk,65,0) and the covariant differentiations are denoted by Equation (2.5.3.8) gives the deflection tensor D = 0 and torsion and curvatures of CRT : 7 = 0, ¡7 = 0, Sl = Û, P2 — F, R2 = K, S2 = 0, and P1 and P1 coincide with that of CT. The Ricci commutation formulae (2.4.3.8) are reduced to (a) ~ 07*) = ^jk-- K?Krjk - K&RTjk, (2.5.4.1) (b) K^k - = KÏF& - K!iFrk - K^k, (2.5.3.8) yields (a) K^k=R^jk-C^ (3.5.4.2) (b) ^k = P^ + c^~c^k. Since CRT satisfied the D- and ^-conditions, Theorem 2.4.5.1 shows (a) yiK^k = R^k, (b) ^ = 7*, (2.5.4.3) which are also obvious from (2.S.4.2). We are concerned with the Bianchi identities of CRT. The first class (2.4.4.2) is written as (a) E(w{K^} = 0, (2.3.4.4) (b) {Æy,fc + PirKjk} = (c) S(W{K?y*+F^} = 0. The second class (2.4.4.4) is written as (a) F^-F^O, (b) R^.k - 4- {Pfa + P^k} = 0, (c) Kfa.k + + F&Prk} = 0. (2.5.4.S)
Finsler Geometry in the 20th-Century 655 The third class (2.4.4.6) is reduced to (a) Ate-j{F^-P^}x=0, (2.5.4.6) (b) F^-F^O. (a) of (2.5.4.3) and (b) of (2.S.4.6) are obvious from (2) or (2.5.2.7). CRT is, of course, ¿-metrical, but not v-metrical : gtj.k — 2Cijk* Then, applying (2.5.4.1) to we have (a) K^k + Kihjk + = 0, (2.5.4.7) (b) Fhijk H“ Rihjk “1“ 2@hirPjk = ZCftiktj^ where K^k = 9irK£jk and Fhijk = Ref In H. Rund [145], R2 and K of our notation appear on p. 101 and p. 97. In his paper [143], Rund introduced the connection coefficients Pjk by means of which a parallel displacement of a vector field Xi is given by d*Xx = —PjkXidzk (in his notation), Fjt = 7jt - Cjr7*o- According to his opinion, the idea to introduce the above parallelism is Minkowskian, while Cartan’s is Euclidean. A. Kawaguchi, the reviewer of Rund [143] in Math. Rev., wrote: Although the ideas and methods in this paper are interesting and may be a contribution to the theory of Finsler spaces, the introduction of the last one of the four conditions for seems to the reviewer to be incomplete, because the covariant differential Dgy has never been defined, although the author, defined that of a tensor aij(x) whose components depend only on the position xi but not on the direction y\ In Rund [144] he modified the above and obtained new connection coef¬ ficients Pfi. See p. 59 of Rund [145]. These, however, coincide with F*k. of CT, as indicated by the reviewers E.T. Davies in Math. Rev, and A. Deicke-W. Siiss in Zentralblatt. On the other hand, in 194S S.-S. Chem [36] has introduced a remarkable connection in Finsler geometry by means of some connection 1-forms. In 1994 M. Anastasiei [5] wrote: That connection remained outside of the mainstream of the development of Finsler geometry in the next decades. It was only briefly treated in the monograph by H. Rund [145], and not at all in that of M. Matsumoto [97]. Chem came back to his connection in 1992, in a paper with D. Bao, its extraordinary usefulness in treating global problems in Finsler geometry was shown. Anastasiei then shows that Chem’s connection coincides with the Rund con¬ nection. Since this Finsler connection was first introduced by Chern, it is quite natural that it bear his name. However, Chem has rather graciously suggested that it be called the Chem-Rund connection.
656 Matsumoto Although the name “the Rund connection” has been a current word since around 1980, we use the name “the Chem-Rund connection” in the present monograph. But we have some confusion of symbols. In D. Bao, S.-S. Chem. and Z. Shen [20] the Chem-Rund connection is throughly used. See p. 39, their and Bjitf of p. 52, (3.3.2) are nothing but Kjk^ and of our (2.5.2.7)* 2.5-5 Berwald Connection Here, starting from the Cartan connection CT, we get a Finsler N-connection introduced by Definition 2.5.3.3 as follows: Definition 2.5.5.1. The Finsler ^connection BT constructed from the Cartan connection CT is called the Berwald connection. According to Definition 2.3.5.3, the Berwald connection BP is obtained from CT by the difference tensors ( - JPa(v, 5), 0(v), 0), or from CRP by the P1-process. The h and v-basic vector fields Bh(v)iBv(y) are equal to N(y) and Y(y) respectively, given by (2.5.3.9). From (2.4.1.15) and (2.2.2.5) we have Hence, if we put ~ Gj (¿) - ZiG§ (^) }, (2.5.5.1) then we have BP =* (G!^ where G^Ft + P^. (2.5.5.2) In this equation, it is noted that the (v) hv-torsion P1 = (P£) of CT is sym¬ metric in i>j. From (2) of (2.5.2.6) we are led to the relation Then, from (2.5.2.3) we have the very simple expression of the connection coef¬ ficients of BP as Proposition 2.5.5.1. The connection coefficients (Fjk1 Ujk1 Nj) of the Berwald connection BP are.given by (Gjfc,O,G£) where ^ = 9^, G^k=9kG}. (2.5.5.3)
Finsler Geometry in the 20th-Century 657 Since BP is obtained from CRT by the /^-process, the relations (2.5.3.11) give the torsion and curvature tensors of BP in terms of CRT : the surviving torsion tensor is only (a) of (2.5.2.6): = (2.5.5.4) The ^-curvature tensor S2 vanishes obviously and the h and hv-curvature tensors are written as (a) h-curvature H : H*k = K*k + Aw{P*lk + P^P/^}, (2.3.5.5) (b) hv-curvature G : G$jk = F§k + J* fc. The h and v-covariant differentiations with respect to BY are denoted by (;, •) throughout the following. Then the Ricci commutation formulae are writ¬ ten as (a) K^k - (j/k) = KTH\k - K?Hrjk - K&fy, (2.5.5.6) (b) K^.k - = K^G^k - K'Gfa, (c) K?M-(J/*)=O. We have simple forms of the curvature tensors H and G as follows: (a) H*k = 5kG% + (%G*k - {j/k} = (2.S.5.7) (b) G^k=9kG^. The former is given by (2.4.3.2) and (2.5.3.4). The latter is given by (2.5.2.7), (2.5.5.5) and (2.3.5.2) as G^k = dk(F$+P& = dkG^. Now we are concerned with the Bianchi identities. They are reduced to only the following (a) = (2.5.5.S) (b) S(yA){^} = 0, (c) S(W{^;fc+G^fc} = 0, (d) Stij'k + ^[v]{Gtjk'ii } = °> and = R!^ which has been given by (a) of-(2.5.5.7). The Berwald connection BP is not ^-metrical : g^k = 2C^. It is also not /^metrical, because (2.S.5.2) easily yields 9ijik = —ZPijk = —ZCijw (2.5.Ô.9)
658 Matsumoto We apply (b) of (2.5.5.6) toyi = giryr- iij'k ~yi<ktj — ~yr@ijk' Since we have Vi>j = (Stryr)',j = -2Pir>^r = 0, yi-kij ~ {Si^y ~ 9ikij = %Pikj> we obtain the relation between the (v) /w-torsion tensor P1 of CT and the Av-curvature tensor G of BP as H%k ~ ^yrGijk' (2.5.5.10) It is remarked that we have the identities y'Gijk = ^Gijk = ^Gijk = 0- (2.5.5.11) Also (b) of (2.5.5.7) shows that G^k are symmetric in the subscripts.. We have (a) of (2.5.5.5); the relation between R2 of CRT and that of BP. Further, from (a) of (2.5.4.3) and (a) of (2.5.5.5) we get Rhijk = Rhijk “ C'hirRjk 4" + PhjrPik}' (2.5.5.12) Hence (2.5.2.S) leads to Rhijk Rihjk — (Rhijk PhkrRij 4“ (2.5.5.13) We are interested in constructing the Berwald connection from the axiomat¬ ical standpoint, just as the Cartan connection. Theorem 2.5.5.1. The Berwald connection BP = (Gj&,0, Gj) is a Unique Finsler connection satisfying the system of axioms as follows: (1) L-metricol: — 0, (2) (h) h-torsion T = 0, (3) (v) hv-torsion P1 = 0, (4) deflection tensor D = 0, (5) vertical tensor IT = 0. Proof: We must first show (1): L\i = 0. In CT we have 9&k = o, vij = 0, (¿2),fc « (ffijliV).* - o, and hence = 0: Ai = diX-(drL)GT = £;<=0.
Finaler Geometry in the 20th-Century 659 Now we shall find the Finsler connection (Fjk,Ujk = 0,Nj) satisfying (1) - (4). Putting F = L2/2, we have from (1) OiF = LdiL = L(NTdrL) = yrNr, y^diF = y^yrNT) = (g^NT + yrdjNT)y\ By means of (2) - (4) we observe . yryidjNr = yTNT = djF. Therefore (1.2.2.2) gives Gi = i (y'djdrF - %F) = 1 which implies & = 1/2 N^yr and Gj = № = 1 1 Nj = Nj. Finally (3) implies Fjk = djNfc = G^. Ref. Theorem 2.5.5.1 was given by T. Okada [135]. The detailed discussion of Okada’s system of axioms was given by T. Aikou and M. Hashiguchi [2]. 2-5.6 Hashigughi Connection As it has been shown, from the Cartan connection CT we obtained the Chem- Rund connection CRT by the ¿/-process and next the Berwald connection BT by the pi-process. Further we have Theorem 2.5,3.1. Thus the diagram shown by Figure 2.5.6.1 holds and we get the following remarkable connection: CT ——> CRT jp1 1 1 HV ► BY u Figure 2.5.6.1 Definition 2.5.6.1. The Finsler connection HV constructed from the Cartan connection CT by the Px-process is called the Hashiguchi connection. Consequently (2.5.3.10) and (2.5,3.11) give all the information on the Hashiguchi connection HT :
660 Matsumoto The connection coefficients of NT are (Fjk+Pjk, Cjki Gj) where (J^fc, Cjk, GJ) are those of CT. Hence (2.5.5.2) shows Kr»(Gj*,Cjk,Gj). (2.5.6.1) Next(2.5.3.11) gives *D = 0 because of D = 0 and Pjkyk - 0 of CT, and further 'P = 0, *T = 0, ‘R1 = R1 : *^ = (7:0}*, *Px = 0, ‘S^O, *SZ = S2-. (2.5.1.2), *p~ : ’j^fc = +j$|k, (2-5’6‘2) *R? : + J$|fc}. On the other hand, BV is obtained from HT by the ¿/-process. Consequently (2.5.3.8) gives S'th. ♦ p/i i xt/x . ^ijk — *ijk + ? rrh * ph sih nr **ijk (2.5.6.3) Proposition 2.5.6.1. The four Finsler connections ST, GT, CRT and HT has the common spray connection N = (GJ) and their (v) h-torsion tensor H1 — (*&). R$k = 5kGj - (j/k), 6k = dk- Gidr, is regarded as the curvature tensor of N. We shall show that R1 may be regarded as the curvature tensor of N. On account of (2.3.3.10), N is locally spanned by n tangent vector fields If N is integrable, that is, there exists an n-dimensional subspace S : yz = ^(m) of the 2n-dimensional total space T of the tangent bundle T(Af) such that Xf) are tangent to S, then the tangent vectors of S must be Hence we have dtf = -Gl(x,y(x)). Conversely, if this system of differential equations are completely integrable, then we have a subspace S : y^ziy^yo) which are solutions of the system and S is an integral manifold of N\ The integrability of the system is given by - (i/k) =Rik = 0..
Finsler Geometry in the 20th-Century 661 Thus R1 may be called the curvature of N, We have (a) of (2.5.3.7) and further show = (2.5.6.4) (b)^ = (l/3){^-(j/fc)}. First we have from (a) of (2.5.3.7) H^k = R^k. (2.5.6.5) This is also obvious because BV satisfied the D and {/-conditions, so that The¬ orem 2.4.5.1 gives (2.5.3.5). Then (a) of (2.5.5.8J yields Rik = SjOk~SkOi- (3.5.6.6) Now, on account of (a) of (2.5.3.7) we have = Rjk + y^ = ^ + Hfrk. Then (2.3.6.6) leads to (b) of (2.3.6.4). Theorem 2.5.6.1. Let FT — (FJ^., a Finsler connection and (;,:) the h and v-covariant differentiations with respect to FT. Then the Finsler connection *FT = (*Fik,*Ujk,N$) is h and v-metricd^ where *Fik = F}k + ¡Fgri-,k /2, = U*jk + girgrj -.k /2. The proof is easily obtained by the direct calculation. The method above to obtain the metrical Finsler connection is called the Kawaguchi process. Corollary 2.5.6.1. The metrical Finsler connections obtained from KT, CRT and FT by the Kawaguchi process coincide with the Cartan connection CT. This is obvious from (2.3.5.2) and (2.3.5.9). Ref, The notion of the Hashiguchi connection was communicated by M. Hashiguchi to the author in 1969. The Kawaguchi process was given by A. Kawaguchi [69]. 2.6 Special Finsler Connections 2.6-1 Induced Finsler Structure We-consider an n-dimensionai Finsler space F71 — (AP, L(a,y)) with the-fun¬ damental function L(x, y) and a hypersurface AP”1 of AP which is given by the parametric equations AP 1 : x* — a = 1,... ,n - 1.
662 Matsumoto The matrix consisting of the projection operators — dxi/duCi is assumed to be of rank n — 1. Then = (B*) is regarded as n - 1 linearly independent vectors tangent to Mn-1 and any vector X1 tangent to Mn~l is expressed in the form X* = , where X* are components of this vector in the coordinate system (uft) of M””1. • To introduce a Finsler structure in M71“1, the supporting element yi at a point of (tia) of Mn^ is to be taken as tangent to Hence we may write (2.6.1.1) This (va) is thought of as the supporting element in Af*“1 at the point(ua). Denote yi of (2.6.1.1) by Then v.) - L(x(u),y(u,v)) (2.6.1.2) gives rise to the fundamental function of. Mn”\ induced from the one of the ambient space. Thus we obtain the (n — l)-dimensional Finsler space F71“1 = called the Finslerian hypersurface of F". In the following, we shall use the notation Also, dp = d/du? and dp = d/dv^ are written from (2.6.1.1) as — Bp&t + Bçpdi, dp = Bpd. From the induced metric L* we get c _ à L a a- &<dpL*) r a — UclIj*, gap — » Oa'fy — 2 1 of Fn_1. Paying attention to dpB^ = 0, (2.6.1.2) yields = C^^CiikB^B3^. (2.6.1.4) At each point (ua) of Mn 1 we get the unit normal vector J5’(u, v) with respect to v — (va), defined by (a) gij^u^y^v^B^^B3 - 0, (2.6.15) (b) gii(x(iu),y(u,v))BiB^ = 1. The normal vector Bi obviously depends on the supporting element y(u, v) and hence it should be said that we have the normal cone generated by Bi at the point (ua). From (2.6.1.1) we have y} — girB^vQ and then (a) of (2.6.1.5) gives yiBi — 0. This together with (2.6.1.4) show's that the angular metric tensor h# = g^ —£z£2 satisfies hi^B^h^ h^B^B3 =ti, hi^Bi = l. (2.6.1.6)
Finsler Geometry in the 20th-Century 663 The matrix (B^, B*) is non-singular as it is easily verified from (2.6.1,5) and we have its inverse matrix (B?,Bi): = 5i, BiB? = 0, and further B’.B® + B’Bj = <5J. (2.6.1.8) We use the relations Bf = g^gijB^ and Bi gijBi, Let us deal with a tensor field X — y)) of Mn, for instance. We get the projection Xa^ = XijkB^B^B^ on Mn_1 of X and further Xa=XijkB^Bk, x = xijkBiB^Bk. Then we have the relations among them as follows: (a) XiikB^B^Xa^Bl+X^Bk, (2.6.1.9) (b) XijkB* = X^Bf + XaB,, (c) XijkBiBk = XaB? + XBt. To show them, we pay attention to the first terms of the right-hand sides of (2.6.1.9): For instance, on account of (2.6.1.8) we observe X^B? = (X^B* B$Bfc)B? = XihfcB*($-BJBft)Bfe = ^fcBi-XaBJ. Thus (b) is shown. Now, from the C-tensorC^jt and the vertical tensor Uijk of F71 we define MaP --- CijkB^Bk, Ma = c^b*. 6 x Ug = B30UjkBfBk, Ug = B3gUjkBiBk. ' Differentiation (2.6.1.5) by we have 2Maja + gijB*adpB3 = 0, M? + gijB^dpBi = 0. Hence we get fyB’ - —2M£Bj - MpB*, (2.6.1-11) which shows the dependence of the normal vector BJ on the supporting element v*. Next we get on account of (2.6.1.9) dpB? = dg^G^B3,) = (-20^ + 2ga''CijkB$)B>l = -2C^B* + 2^(C^jBf + M^Bi) = 2M$Bi.
664 Matsumoto Similarly we have the following two equation: fyB? - d?Bi = MpBi. (2.6.1.12) Therefore we have the frame B*) and its dual (Bf f Bi). Among B^. B? and Bit the three, except the first, depend on va as it is shown by (2.6.1.11) and (2.6.1.12). 2.6.2 Induced Finsler Connection Now we are concerned with a Finsler space F1” = (Mn, L(z, ?/)) equipped with a Finsler connection FT = (FJfc, Nj). To define the notion of induced Finsler connection in a hypersurface Mn_1, we first consider the absolute differentials Dyi of the supporting element y* and DX* of a Finslerian vector field X*(x,y), defined by Dy* - dy* + Nj(x, y)dx3, DXi=dXi + {Tijk(x,yjda;k + Untidy*}. These come from (2,4.6.2) and (2.4.6.1) respectively. From the infinitesimal viewpoint: Du* = 0 and DX* = 0 imply the parallel displacements of y* and X*, respectively. Definition 2.6.2.1. The induced Finsler connection IFF of a hypersurface Mn_1 of a Finsler space Fn — (Mn,L(xjy)) with a Finsler connection FT is a Finsler connection such that, in the infinitesimal viewpoint, the parallel displacement with respect to IFF is obtained from that with respect to FT by the projection on Mn_1 : That is, Dva — BftDyi and DXa — B“DXZ where y* = Biva and Xi = BÎX*. Remark: We have the Finsler hypersurface F^1*1 = (Mn_1,L*(u,v)) with the induced Finsler metric L* in the last section, so that F^-1 has a Finsler connection FT* which is defined from L*. FT* is called the intrinsic Finsler connection. Thus F™-1 has two Finsler connections, one is the induced connec¬ tion IFF and the other is the intrinsic Finsler connection. However, as it will be seen, they do not coincide with each other in general. Let us denote the connection coefficients of IFF by (Fj^U^Nf). Then, on account of (2.6.1.1), the condition Dva == B^Dy* is written as dva + Ngdu0 = BfiB^dul3 + + NjB’pdu13'), which implies Nf = BfÇB^ + JVjs£). (2.6.2.1) Similarly the condition DX“ = BfDX* yields two equations as follows: = B?{B^ + BÿyjkBk + (2.Ô.2.2) (2.6.2.3)
Finsler Geometry in the 20th-Century 665 If we put = + (2.6.2.4) then this together with (2.6.2.1) gives B^ + = N^B^ + H7B\ (2.6.2.5) the normal component of Bq^+NJB^ is called the normal curvature vector. Consequently, (2.6.2.2) and (2.4.1.15) lead to = Bf{B^ + B^B* + l^B%)}. (2.6.2.6) Theorem 2.6.2.I. The induced Finsler connection I FT = (Fgyi Nfj) of a hypersurface of a Finsler space with a Finsler connection FT — (F}k, U^Nj) is given by (2.6.2.6), (2.6.2.3) and (2.6.2.1). The (h) h and (v) v-torsion tensors T*,SJ of I FT are given from (2.6,2.6) and (2.6.2.3) as follows: T,: = B?B%rjkBk + - U“H0, : S?py = Bf$ikBpBk, where U? is defined by (2.6.1.10). Next, if we put = Bi{B^ + B^kBk + lPjkBkHy)}, then this together with (2.6.2.6) leads to B^ + B’^B* + i&B%) = + H^B*. This is regarded as the so-called Gauss equation in the theory of hypersurface of a Riemannian space. is called the second fundamental h-tensor of F71-1. It is, however, remarked that is not a symmetric tensor in general. Next, differentiating (2.6.2.5) by v& and substituting from (2.6.2.9), we have B^ + = (dpN« - 2M^Hy)Bi + (d0H^ - MpHJB*. Hence, this together with (2.6.2.9) yields the (v) /intorsion P,1 and the other as follows: Pi ■ P^u = B?I^BkB^ + (2M£ - UfiHy, (2.6.2.10) -Hfr = B^B^ + (Mp - U0)Hy, (2.6.2.11) where U’s are defined by (2.6.1.10). Prom (2.6.2.6) we get = B^B^ + ^kBk 4- UikBkHJ}. (2.6.2.7) (2.6.2.S) (2.6.2.9)
666 Matsumoto This together with (2.6.2,1) yields the deflection tensor D* as follows:. D, : DZ, = + UlkBkH^, (2.6.2.12) We need the h and ^-covariant differentiations with respect to the induced Finsler connection IFT. First we introduce the relative h-covariant derivative. For a tensor Yffi such as with Latin and Greek indices, we define it by xri®, X„via i vka T?i via 7?k YiP »7 - ¿ky - ^*¿7 (2 6 2 13v i •c'a isia jnd ' + rj7?2\y7 - YjS where ¿7 » — N$d$ is the ¿-differentiation with respect to the induced spray connection (NJ) and are mixed connection coefficients given by JF^ = (2.6.2.14) which has appeared in the left-hand side of (2.6,2,9). The relative v-covariant derivative is defined by y~ia . — A yia .t y karri -yiarrk *j/3 .7 — CfyYj0 + Y^ Ufy - Yk^Ujy i‘V^TTQ viarrS T XjpUfry where are also the mixed connection coefficients given by üir=lw From (2.6.1.3) we have Sfi = Biff8i + BiH0di. Hence, for Y*, for instance, we get 1?7 = Yffi + Y?jBiH^ yi = YiBi. ■'f lgDT For the projection operators Bj we have (2.6.2.15) (2.6.2.16) (2.6.2.17) (2.6.2.1S) 23^7 = 1^, Bß^ = U^rBi, (2.6.2.19) where Upy = is to be called the second fundamental v-tensor, corresponding to ZT^. We are now concerned with the metrical conditions. From (2.6.1.2), (2.6.2.17) and paying attention to B'ti = 0, we have L.-a = £aL, in the form L..a=L;iBi. Next, from (2.6.1.4), (2.6.2.18) and (2.6.2.19) we get 9<xßrt ~ (SijtkBk + 9i)-.kBkH^B^Bp. (2.6.2.20) (2.6.2.21) Similarly we have 9aß,.'t — 9ij-kB^tBgBk. (2.6.2.22)
Finsler Geometry in the 20th-Century 667 2.6.3 Induction of Standard Connections We consider the induced Finsler connections of a hypersurface F71"1 = in a Finsler space Fn = (MTOfL(æ,ÿ)) which are constructed from the standard Finsler connections BT, CT, CRT and BT. Let us first consider the Cartan connection CT and the induced Finsler con¬ nection ICT. We are concerned with the system of axioms in Definition 2.5.2.1. Since gij.tk = gijih = 0 in CT, (2.6.2.21) and (2.6.2,22) give gaßn - gaß.k = 0 in CT, (2.6.2.21) and (2.6.2.22) give gaß-y « gapn = 0 in ICT. Next (2.6.2,12) and Ujk = Cjk give D* = 0. Also (2.6.2.7) leads to = 0, but T* does not vanish. In fact, we have Proposition 2.6.3.1. The induced Cartan connection ICT of a hypersurface of a Finsler space with the Cartan connection CT is such that (1) h-metricfd, (2) (h) h-torsionT^ = M£B7 - M“Bß, (3) deflection D* = 0, (4) v-metrical, (5) (v) v-torsion Si — 0, where Mßy =; gay^ß — ^B^B^Cijk and Hß is the normal curvature vector. Remark: Since IL does not vanish in general, ICT is not the Cartan connec¬ tion CTm, in general, which is the intrinsic Cartan connection constructed from L*(w,v). From (2.6.2.S) and (2.6.2,4) we have Hßo — Hß = MßH$. By multiplying by v7, (2.6.2.11) together with the above leads to dßHs = 2Hß + MßH<). (2.6.3.1) Now consider the condition T* = 0 : MjfHy = M*Hß. Assume that Hß / 0. Then we have ha satisfying Mf = hFHß. The symmetry of Mßa implies the existence of ha (= gaßh^} — hHa, and hence Mßa « hHßH^. From Moa = 0 we get hH$ — 0. h = 0 implies Mßa = 0. Bb = 0 together with (2.6.3.1) shows Bß == 0, contradiction. Consequently we have Theorem 2.6.3.1. The induced Cartan connection ICV of a hypersurface S of a Finsler space with the Cartan connection CT coincides with the intrinsic Cartan connection CT* constructed from the induced Finsler metric L*. if and only if S satisfies (1) Maß = 0, or (2)Ba —0.
668 Matsumoto Ref O. Varga [164], [166]. M. Matsumoto [96]. Now we are concerned with the Chem-Rund connection CRT and the in¬ duced connection I CRT. The vertical tensor U vanishes and hence (2.6.2,3) implies U* — 0. Fjk and Gj of CRT coincide with those of CT and are determ¬ ined by “^metrical”, T1 = D = 0. Thus (2.6.2.12) shows D. — 0 and (2.6.2.7) gives T* ® 0. (2.6.2.22) gives ga^ = Therefore we have Proposition- 2.6.3.2. The induced Chem-Rund connection ICRT of a hyper¬ surface of a Finsler space with the Chem-Rund connection CRT is such that (1) “ 21^afiR'r (2) (h) h-torsion T* == 0, (3) deflection D* — 0, (4) vertical tensor 17* = 0. Thus, the important h-metrical condition does not hold by ICRT. Since Paj3;7 “ 0 and (2) - (4) determine CRT, we have Theorem 2.6.3.2. The induced Chem-Rund connection ICRT coincides with the intrinsic Chem-Rund connection CRT* constructed from the induced Finsler metric L* if and only if (1) Ma/? = 0, or (2) B* = 0. Next, we deal with the induced Berwald connection IBT of a hypersurface. The Berwald connection BT is determined by the system of axioms stated in Theorem 2.5.5.1. First L*;o! = 0 holds by (2.6.3.20), T* — 0 by (2.6.2.7), £>* = 0 by (2.6.2.12). Next Pj are given by (2.6.2.10). Consequently we have Proposition 2.6.3.3. The induced Berwald connection IBT of a hypersurface of a Finsler space with the Berwald connection BV is such that (1) L*-metrical, (2) (h) h-torsionT* — 0. (3) deflection D* = 0, (4) (v) hv-torsion (5) vertical tensor 17* = 0. Note that the most remarkable fact, P} = 0, of the Berwald connection, does not hold. Theorem 2.6.3.3. The induced Berwald connection IBT of a hypersurface coincides with the intrinsic Berwald connection constructed from the induced Finsler metric £*, if and only if
Finsler Geometry in the 20th-Century 669 (1) . or (2) Ha=0. Finally we consider the induced Hashiguchi connection IHT of a hypersur¬ face. If we pay attention to Ujk = for HF, then (2.6.2.7) gives — MpHy — Also (2.6.2.10) gives P* as in the case in I BY. Therefore we have Proposition 2.6.3-4. The induced Hashiguchi connection IHT of a hypersur¬ face of a Finsler space with the Hashiguchi connection HF is such that (1) L.-metrical, (2) (h) h-torsion Tfa = M$Hy - M°H0i (3) deflection D* — 0, (4) (v) hv-torsion (5) vertical tensor U* = C-tensor. Theorem 2.6.3.4 The induced Hashiguchi connection IHT of a hypersurface coincides with the intrinsic Hashiguchi connection HT* constructed from the induced Finsler metric, if and only if (1) M^ = 0, or (2) Ha = 0. Now, we consider the notions of path of a Finsler space F71 and that of a Finsler hypersurface F1"1, In a Finsler space F* with a Finsler connection FT, a curve C = (s’(i)) of the base manifold M is a path with respect to FT, if the tangent vector field (j/* ® dx'/dt) otC is parallel with respect to the spray connection N, that is, Dy1 — dy* + N](x, y)dx? - 0. A path in F”“1 is also defined by Dv* = 0. We have Dv01 — BfDy* in Definition 2.6.2.1. On the other hand, BiDyi = Bi{B^dvP + BfaP + N}B^du0) = Bi(Bi0 + NjB^)d^. Hence (2.6.2.4) shows BiDy' — HpdvP. Hence we have Dy1 = DvaBia + HaduaB\ (2.6.3.2) Definition 2.6.3.1. A hypersurface F"_1 with the induced Finsler connection IFF of a Finsler space F" with a Finsler connection FT is called a hyperplane, if each path of the ambient space Fn on Fn^1 is a path of F71-1. That is, D^dx'/dt) — 0 implies D(dua/dt) = 0 necessarily. Thus (2.6.3.2) shows Ha — 0 always.
670 Matsumoto Theorem 2.6.3.5. A hypersurface F*1“1 with the induced Finsler connection IFF is a hyperplane, if and only if the normal curvature vector Ha vanishes identically. Ref. The definition of hyperplane as above is the Definition 1 of a hyperplane of the first kind of M. Matsumoto [96). In this paper we have the definitions of ftj/pezpZanes of two other kinds. 2.6.4 Vector Relative Connection We consider a Finsler space F71 — (M, L(z, y)) with a Finsler connection FT = (r\ru) = (I\2V). Suppose that M admits locally a non-zero tangent vector field Y = (y£(ir)). Y(x) is regarded as a cross-section of the tangent bundle Y :x e M -* y (x) € T. Hence we get the mapping Tf: z € L -> (y(x), z) eF, x = kl(z). This satisfies (a) 7ri o = y o 7TX,, 7f2 ° t? = identity, (2.6.4.1) (b) i7o^ = P5oi?, 'n°z0 = rj(z)P> In the respective canonical coordinate systems (s*, z\) and (a?*, y\ z^) of T and F, we have ^>4) = (»Sv4 ~ > 5^) = (^F + W{w)' 3^)’ and the dual rf of the differential rf is given by rT(dx\dy\d£) = (^,(^^^,¿4). (2.64.2) Now we deal with the connection form with respect to T and the v-basic form with respect to I*, given by (2.4.1.12) and (2.3.3.9) respectively. From them we obtain the differential forms ^-^(u), *(**)= 77*(0*)> on L. From (2.4.1.12) and (2.6.4.2) we have *cu = *^{d/dg^\ with X“= + *rjk(x) = + U*h(x,Y^}dkYh. From (2.4.1.11) and (2.6.4.1) we have Ws = ad^~x)*u, *u(Z(A)) = A. (2.6.4,3)
Finsler Geometry in the 20th-Century 671 Thus -*w is certainly a connection form on L from Proposition 2.2.1.1 and hence we obtain the linear connection *T(Y) whose connection form is +cv. That is, the horizontal subspace *TZ « {X € Lz|*w(X) =0}. On the other hand, we have ’(0V) = 0°ea with 6a = + Nj(xyY(x)). (2.6.4.4) It is easy to show that o and hence the coefficients Yj constitute a tensor field. Or, directly we have = Yi-Ni{xX) + Y*F№,Y} = Yf + D}{x,Y)y where the last term is the deflection tensor. Therefore Yj(x) is a tensor field. The connection coefficients *Tjfc of (2.6.4.3) can be rewritten on account of (2.4.1.15) in the form *Vijk^rjk(x,Y) + U;r(x,Y')Y^. (2.Ô.4.5) Definition 2.6.4.1. The linear connection *T(Y) with the connection form *o> is called the connection associated to the Finsler connection FT by a tangent vector field Y(x). The tensor *Y — (Y^(x)) is called the v-basic Y-tensor field. Definition 2.6.4.2. Given a vectorial tensor field T : F —► we have a tensor field *T = T o rç. This is called the Y-tensor field associated to T by Ytf). The condition *T o fi'g = g~uT for a tensor field is easily verified. In the components Tfa,y)> for instance, of (l,l)-type, we have *Tfa) — Tj (x, Y(x)). We denote by (,) the covariant differentiation with respect to T. For a tensor field *T above, we have from (2.6.4.4) d^ = [dkii+dr^-N^)]Y, where • ]y shows that we put y — Y(x) in [♦ • • ]. Hence we may write *2^(œ) = s/)]y. It is easy to show %k = [^+^]Y.. (2.C.4.6) We shall find the torsion tensor and the curvature tensor of the connection *r(Y). First (2.6.4.S) gives directly ‘Tjk = Tjk{x, Y) + U^x, Y)Yk - U^x, Y)Yf, (2.6.4.7) where T and U are torsion tensors of FT and Y£ are components of the v-basic V-tensor field T
672 Matsumoto To find the curvature tensor ’ R, we first get dkY} - (j/k) = [r*>+ {(/£ + 2^)17 - (j/fc)}]y, which gives V/k - OA)' = + {4y7 - (i/fe) } + Wy* ] Y• (2-6-4-8) where R, P and S are torsion tensors of FT. Next we have for a Y-tensor field *Xi = Xz{x^ Y} + yi r Yi J- Y* Vr i Y* V7" + XtrYr/k+Xfr:sYrY^Y. Consequently, we have *X^k - (j/k) = [(xifc - (j/fc)) + {(X’,r - X^)Y£ - (j/k)} +{x^-(r/s)}YJYi +^fe"07fc)}]y- Therefore the Ricci identities (2.2.4.2) and (2.4.3.S), together with (2.6.4.7) and (2.6.4.S) yield *&hjk = + {P^YZ - (j/k)} + SiraYTYk>]y, (2.S.4.9) where R, P and S are curvature tensors of FT and * Y = (1J) is the v-basic Y’tensor. Proposition 2.6.4.1. The torsion tensor *T and the curvature tensor *R of the connection associate to a Finsler connection FT by Y(x) are given by (2.6.4.T) and (2.6.4.9) respectively. 2.6.5 Barthel Connection We continue the theory of vector relative connections. Thus, assume that the underlying smooth manifold M of a Finsler space F” = (M, L(®, y)} admits a non-zero smooth tangent vector field Y(x) and define the notion of Y-tensor field *T =s TqT} associated to a tensor field T of M by Y(x): Definition 2.6.5.1. The Y-tensor field *g(z) = (gijfaY)) associated to the fundamental tensor field *g(x>y) * Finsler space Fn = by Y(x) is called the Y-Riemannian metric and the space (M,* g) is the Y-Riemannian manifold. On account of (2.6.4.6) we have
Filler Geometry in the 20th-Century 673 Proposition 2.6.5.1. If a Finsler connection FT of Fn is h and v-metrical, then the connection *r(y) associated to FT by Y(x) is metrical with respect to the Y-Riemannian metric *g. We have considered the Cartan connection CT which is h and v-metrical. Thus the connection *r(Z) associated to CF by Y(xf is metrical with respect to *g. Definition 2.6.5.2. The connection *r(y) associated to the Cartan connection CT by Y(x) is called the Barthel connection induced by K(æ) and denoted by Theorem 2.6.5.1. The Barthel connection *BFfY) induced byY(x) is metrical with respect to the Y -Riemannian metric *g and has the torsion tensor “Tjk{x) = CjT(x,Y)Yk — Therefore *BT(y) is not the Levi-Ci vit à connection of the y-Riemannian manifold (M, *p) in general. We observe the absolute differential Du* of a vector field v* with respect to the Barthel connection *Br(y) : Dv* = dvi + ■j'Tijkdxk. (2.6.5.1) This is linear in In fact, on account of (2.5.2.1), (2.5.2.3) and (2.6.4.5), we have ^jk = - CjrGl ~ + CjkrG^ + Cpfly, (2.6.S.2) Gj = froj “ Cjr7ooly- In particular, as to the vector field y(s) itself, we have Y^Cj^x^Y) = 0, and hence Proposition 2.6.5.2. The absolute differential DY of the vector field Y(x) with respect to the Barthel connection *BT(Y') induced byY(x) is given by DY* = dY*+Y^^k(x,Y)-C*kr(x,Yyf0j(x,Y)}dxk. The coefficients which is inside of {• • *} is not linear in Y, though it can be simply constructed from *<?. Ref The origin of the notion of linear connection T(y) was R.S. Ingarden [61], in which we find DY1 = 0 of Proposition 2.6.5.2, as it was given by Barthel [21]. This DYi = 0 was again published in 1994, and further revised by In¬ garden, Matsumoto and Taméssy [64]. See Ingarden and Matsumoto [63].
674 Matsumoto 2.6.6 Cartan Y-Connection The problem of the Barthel connection raises the new question: How to get the Levi-Cività connection with respect to the K-Riemannian metric 'g as the connection associated to a Finsler connection FT by a given vector held Y(z)7 Thus, we have to pay attention to (2.6.4.7). As *T vanishes, if and only if we have This leads us to Theorem 2.6.6.1. In a domain D of the underlying manifold M of a Finsler space F* = (M,L(æ,y)) which admits a non-zero vector field Y(x), a Finsler connection is uniquely determined by the system of axioms as follows: (1) h-metrical : Vhg — 0, (2) (h)h-torsionTiT^ (3) deflection tensor D — 0, (4) v-metrical : = 0, (5) (v) v-torsionS1 = 0, where Lq (z, y) = L(x, y)/L(®, Y(s)) and Yf(x) = diYi + Nj (x, Y(x)). Why the Lq7 It is desirable on account of Corollary 2.4.7.1 that T^k(xty) is (0) p-homogeneous. Since we have Ujk = Cjk by (4) and (5) and Yj, do not depend on y*, (UjrY£ ~ V^YJ) are (-1) p-homogeneous. So this gap of homogeneity can be avoided by putting Lq and that Lq(x,Y) = 1 in [T^Jy. Now we shall give the proof of Theorem 2-6.6.1. Let us recall the theory in §2.5.2. First, we have = Fjk + Fjik. If we put Tijk = gjTTfk and Atjk = t (2A61) then the Christoffel process leads to Fijk = 'Yijk ■“* CijrNk + OikrNj + Aijk. (2.6.6.2) Transvecting by yi and next by yki we have Ftyk = CijrNç 4- AojA, Fojo = 70j0 + Aojo* Since we get Nj = from the condition (3), we get = Too+-Aqq and hence Nj = - c?rT5o) + (4/ - cjM. (2.6.6.3)
Finsler Geometry in the 20th-Century 675 So Nj is determined and (2.6.Ô.2) gives Fjk. We sum up with Theorem 2.6,6.2. A Finsler connection is uniquely determined in a Finsler space F™ = (M, L(x, y)) by the system of axioms as follows: (1) h-metrical, (2) (h) h-torsion T is given, (3) deflection tensor D = 0, (4) v-metrical. (5) (v) v-torsion S1 0. The connection coefficients (Fjk,Ujki Nj) of the connection above are given by (2.6.d2), (2.6.6.3) and UJk = C]k. In the case of Theorem 2.6.6.1, we have Aijk « L0(CikrYf - CikrYT), Xjo = 0. (2.Ô.6.4) which leads to Wj(^2/)=7j,-C?r(75o-W). Since yor = {daYT)Ya + N$(x, Y) and NÇ(x, Y) = 7oo(*> Y). Consequently we obtain Nfe, K) = -fax, y) + Cjr(z,y)(^y’-)y\ (2.Ô.6.5) Therefore we have Definition 2.6-6.1. The Finsler connection, determined by the system of ax¬ ioms in Theorem 2.6.6.1, is called the Cartan Y-connection and denoted by CTT. Its connection coefficients ÇFjkiUjk)N^) are given by (2.6.6.2J, (2.6.6.S) and Ujk = C^. Thus, we have the main result: Theorem 2.6.6.3. The connection *T(Y) associated to the Cartan Y-connection CYL byY(x) is the Levi-Cività connection *7(Y) of the Riemannian Y -manifold In conclusion, we obtain (1) the Barthel connection *BT(y) associated to the Cartan connection CT by à^(rc)5 and (2) the Levi-Cività connection *7(K) associated to the Cartan ^-connection CYT by Y(x). (1) cT-*Br(y), (2) oyr->*7Cn. It seems that the original CT is simpler than CTT, while the result *7(K) is certainly simpler than *BT(y).
676 Matsumoto Example 2.6.6-1. We consider the Barthel connection *ST(y) induced from a concurrent vector field Y^x), which is treated in Example 2.5.2.I. Then (2.6.4.4) gives in CT yj = 9jY* + YrF^(x,Y) = Y$ = -<5j, and hence (2.6.4.3) leads to *T*jk(x) = Fjk(x,Y)— Cjk(x,Y). Thus the torsion *7^. = 0, and hence Theorem 2.6.5.1 shows that the Barthel connection is just the Levi-Civita connection of the Y -Riemannian space. Fur¬ ther we observe Yfa =djYi + Yr^rirj^djYi + Yr^j-C^ as it has been shown in Example 2.5.2.1. Therefore Y(x) is really a concurrent vector field of the Y -Riemannian space. We shall find the curvature tensor *7? of the y-Riemannian space. From (2.6.4.9) *R is of the form - Pfa) + S^Jy, and (2.5.2.15) leads to Further (b) of (2.5.2.19) yields TTP^k(x,Y) + Cjk(x,Y) = Pjk(x,Y) + C}k(x,Y) = 0, and hence Cjkl0(x,Y) — —Cjk(x,Y) from (a) of (2.5.2.14). Consequently, we *Shjk>o(.x> = ■'4-b*] io]y = -2Sijk(xyY). Therefore we obtain the final form of *R?fljk as ^fc(*) = [^fc-sU]y-
Chapter 3 Important Finsler Spaces 3.1 Finsler Space of Dimension Two 3.1.1 Berwald Frame and Main Scalar We shall deal with Finsler spaces of dimension two in the characteristic way. First we define the orthonormal frame field. According to Proposition 1.2.2.1, the components hij of the angular metric tensor h of a two-dimensional Finsler space F2 constitute the matrix (hy) of rank one, and hence det (^) ~ hnh22 - (h12)2 = 0. (3.1.1.4) If hu = /i22 « 0, then (3.1.1.1) gives hu = 0, that is, contradiction hij — 0. Thus, we may assume hij / 0 and take the.sign s — ±1 of hu. Then shn = (mi)2 gives a non-zero mi uniquely up to the sign. Next, ehi2 — mim2 gives m2 and (3.1.1.1) leads to sä22 = (m2)2. Consequently we have (mi,m2) and the sign s, satisfying hij » emimj, i,j = 1,2. (3.1.1,2) Since we have g% = fytj 4- hij from (1.2.2.9), gij is written as gij = ¿i£j -b smimj. (3.1.1,3) The sign e is called the signature of F2. From hijtf = 0 it follows that (3.1.1.2) gives rrij^ = 0. Hence m* — g^rrtj constitutes the orthogonal frame (¿*,m’), called the Berwald frame, and now we get the co-frame (£ifmi). The equation (3.1.1.3) gives gijrn? = £mi{rn,jm^, which implies rrijm? =s e. (3.1.1.4) Hence (¿Sm*) is a orthonormal frame. (3.1.1.3) yields % = rtj + sm>, gij = W + emW. (3.1.1.5) 677
678 Matsumoto If we put h = — £2mx), k = e/(£i?7i2 — ¿2^1), then the sets of equations (Fmi = 0, rr^mj = s) and (W = 0, yield respectively mim' = e) (mi, m2) = h(—¿2,^1), (mx,m2) = ¿(—¿2,^i)> (3.1.1.6) and (3.1.1.4) shows hk = e. Then (3.1.1.3) leads to g — ( = det(p.^)) == e/k2 = eh2 = h/k. (3.1.1.7) Farther, we obtain (m1,m2')/h = (3.1.1.8) We denote di by (.¿) and find and Differentiating Lp = yi by y\ we get and (3.1.1.5) leads to Lfy = em'mj. Also, differentiating L.i = by y^ we get L.i,j =hij/L — and (3.1.1,2) leads to Lti.j — smimj. Consequently, we have Lfy = L^.j « emimj. (3.1.1.9) Next, we deal with the C-tensor Cijk- From CijkVi == Gijktf = Cijkyk = 0 we can put LCijk = Irnimjrnk* (3.1.1.10) The scalar I as thus defined is (0) p-homogeneous and called the main scalar of F2. Remark: We wrote above uehn = (mi)2 gives uniquely within the sign? In fact, the orientation of rm is not determined by (3.1.1.2). If we take the inverse orientation of m<, then (3.1.1.10) shows that the sign of I is changes. In the following it will be seen that I2 is more essential than I itself. Differentiating fyn1 = 0 and gijrrfmj = e by yk, (3.1.1.9) and (3.1.1,10) lead to (Lm^)€i = -mic, = -Im^ = 0 is obvious from the homogeneity of m’, and hence the above gives Lm*k = —(& + From &mi = 0 and rr^mi = e we have similarly I/mi,k = — (¿i slmi}mk. Therefore, we obtain Lrn^ = Lmi.j = — (£ — (3.1.1.11) The equations (3.1.1.9) and (3.1.1.11) lead to L(^mj - £jmi) k = el(£imj — ^rn^mk- (3.1.1.12)
Finsler Geometry in the 20th-Century 679 Now, let S(x,y) be a (r) p-homogeneous scalar. Then = rS, and hence S.i, may be written as LS.i - rS£i + S-pMi. Then as thus defined, is a (r) p-homogeneous scalar. Throughout the following we shall deal with homogeneous scalars in yz of degree 0. Consequently, we have the above in the form LS.i = S.2mi. (3.1.1.13) Example 3.1.1.1. We treat of a tangent vector field of the underlying manifold M2 of a Finsler space F2. Let us put vx(a?) = v1? + Both v1 — vz&i and v2 — €vlmi are (0) p-homogeneous. By (3.1.1.19), (3.1.1.11) and (3.1.1.13) we have LdjV* = Q = v1\2 + v1£mzmj + tr j2 rnjm' — v2(t + which implies v;2 = tr, v22 == —£(vx — Iv2). This is obviously necessary and sufficient for to be functions of position alone» Similarly, a covariant vector field Wi = wi^ + wqjth is Wi(x), if and only if W1;2 = W2, W2;2 = —e(wi + Iwq). These conditions show that if one ofv1 andv2, or one of wj and W2 vanishes, then the other vanishes also. From (3.1.1.3) and (3.1.1.5) it follows that g = det(p^) is written as g — s(£im2 — ^rai)2» V9 = elf1™2 — ^m1)2. On the other hand, we have in the general dimensional case 9-i = = (2C/jhW* = 2Cig, where Ci = g^Cijk = Then it follows from (3.1.1.10) that Lg.i — 2eglmi. (3.1.1.14) (3.1.1.15) (3.1.1.16) Example 3.1.1.2. In the two-dimensional case a tensor field T = (ti<7) of (0,2)-type can be written as Tij = + TWiTOj + Tztrmtj + Tnmimj. It is easy to show* that det (T^) « (TnTaa - Ti2T2i)(fimj - tyni)2, and hence (3.1.1.14) gives det (Tij) = £g(TnT22 - Ï12T21).
680 Matsumoto Similarly, we get det (T*) = (TnT22 - 7I2721). If Tij is skew-symmetric, then Tn = T22 = 0 and T2i = — Ti2. Hence may be written as Tij — T(£i7Jlj ¿jTTtijy with a scalar T, Example 3.1.1.3. If a two-dimensional Finsler space F2 admits a concurrent vector field, then F2 is a Riemannian space. In fact, Example 2.5.2.1 and (3.1.1.10) give X'(Imimjmk) = 0, which im¬ plies I = 0 or — 0. The latter leads to Xi = 0 on account of Ex¬ ample 3.1.1.1, a contradiction. 3.1.2 Landsberg Angle and Length of Indicatrix From (1.2.1.8) it follows that the first vector I of the Berwald frame (€,m) is given by = d{L. On the other hand, we consider the differential equation LdiO = mit (3.1.2.1) From (3.1.1.11) we have L2djd{0 = —tjtni — (€$ — which is symmetric in i.j. Hence (3.1.2.1) is completely integrable and hence it gives a function 0(z, y) with the parameters (a;1) • 0 is called the Landsberg angle. Since the Jacobian d(Lt 6) _ (€i7n2 - €2?ni) W,i/2)” L ^u’ from (3.1.1.14), the pair (L,0) may be regarded as a coordinate system in the tangent space (M2)x and called the Landsberg polar coordinate system. Remark: We consider a Euclidean plane with the orthonormal coordinate sys¬ tem (x9 y). A polar coordinate system (p, is given by z = pcos<£, y = psin<£. Then we get This corresponds to (3.1.2.1). For a scalar S{x,y} we have from (3.1.2.1) 4*00
Finsler Geometry in the 20th-Century 681 (3.1.2.2) Thus (3.1.1.3) gives g = («№)-*■ Assume that S be (0) p-homogeneous, and we get W ^ = <7 dL °’ de s-2' From §1.2.1 and (1.2.2.5) it follows that a tangent space Mx of a Finsler space E* = (M, L(xy p)) is regarded as a Riemannian space with the metric ds2 = gij(x^y)dyidy^t The components Cjk of the C-tensor are Christoffel symbols of Mx. Consequently, the v-covariant derivative in CT : is the covariant derivative o/X1 in the Riemannian space Mx. We restrict our consideration to the two-dimensional case and use the Lands¬ berg polar coordinate system (L, 9) — (pa), a = 1,2. Then we have <fe2 = 9^a^b, gab = gy (|^) (|p). Since the Jacobian matrix d(y1ty2)/d(L19) is the inverse of d(Li9)/d(y1^y2), we get dL e’ de Therefore, we have Pn P12 922 = (ifc) = S'v(^r) =gi^{Lm?} =0, ~ 9ij (^0 = ^rnx)(LmP) = sL2. Thus in the Landsberg polar coordinate system, we get dx2 = di2+eL2<W2. (3.1.2.3) Now we consider the theory of curves in the Riemannian space having the positive signature e = +1. Let y* = ^(s) be the equation of a curve C and (t’,7?) the unit tangent and normal vectors of C respectively. Then f= Kn\ (3.1.2.4) where K 0 is the curvature of C. In the Berwald frame (A m) we put t' — F cos a + n? sin a, nz = smoi-^vn^ cos a. (3.L2.5)
682 Matsumoto Along C we have dLfds = L,^ and dfi/ds = 6,iti) so that (3.1.2.5) gives dL — = cos a. ds dti _ sin a ds~ L (3.1.2.6) On account of (3.1.1.9) and (3.1.1.11) we get similarly d& _ m*sina dm* _ (€* + Im^sina ~ds ~ L 9 L ‘ Hence we have zsin2a\^ < zsinax, _ , . , fdot\ z =-(—X+(—)(cosa - Is^m + fe)n • Therefore, on account of (3.1.1.10) we have (3.1.2.4) in the form dot since ds L (3.1.2.7) Proposition 3.1.2.1. In the Landsberg polar coordinate system (L,0) of M2, we have the complete system of differential equations (3.1.2.6) and (3.1.2.7) of a curve of M2, where I and K are the main scalar and the curvature of the curve respectively. In the following we apply the theory above to the indicatrix C : y* = y*(s) where L(yi(s)) » 1 (§1.2.2). Then (3.1.2.6) and (3.1.2.7) are reduced to da cos a = 0, ~ =z since, — + since = K. ds ds Thus we obtain, a — p s-1' *->■ Further, we use the polar coordinate system (p, (f>) of M; : y1 = pcos0, y2 — psunj). We shall write ds of <7 in terms of dtj>. C is given by L(pcos0,psin^) = pL(cos 0, sin 0) = 1. Putting #(0) = L(cos <£, sin <£), we have p = 1/«£>(<£). Hence we have C : y1 — cos0/£, y2 = sin</>/&. If we put = £.i(cos0?sm<£),
Finsler Geometry in the 20th-Century 683 then the homogeneity implies #1 cos# + #2 sin# — Thus, along C we have dy1 {-#sin# —cos#(—#isin#+^gcos#)} d# ~ & _ {—$ sin^ + (& — $2 sin#) sin 0 — #2 COS2 #} - $2 = _*2 £2 ’ Similarly, we get dj^/d# — #i/&2 so that we obtain = $“4{pn($2)2 - 2^1*2 + 522(4?l)2}<ty2. Since = -¿¿(cos#, sin#) and (pn,-912,922} = g^22,912,9X1}, we get ds2 = (p/#4)d#2. Therefore we proved Theorem 3.1.2.1. In the two-dimensional Riemannian space M2 with ds2 = tfijOr^d^dy2 , the indicatrix of M2 is a curve of curvature K = 1 and its arc¬ length s is equal to the Landsberg angle 0 upto an additive constant, ds for the indicatriz is written as L = L(xi,z2;y1,y2'), (jAs/2) = (cos&sM)- The Euclidean angle a has the domain 0 a 27r, but the domain of the Landsberg angle 9 is not necessarily equal to 2%, as it will be shown by the following examples. Example 3.1.2.1. We are concerned with a Renders space F2 with L — a+/3, where the Riemannian metric a2 = aij(x}dxidx^ is assumed to be positive- definite. On account of Theorem 1.3.2.1, the length B = y/a&bibj of bi with respect to a is less than 1, if and only if 9ij(x)y}yiy^ is positive-valued. Thus B < 1. From (1.3.2.4) we have zZ\3 9 = (-) det (°v)- Let (y^y2} be an isothermal coordinate system: a(x>y} = a(x} y/(3/1)2 + (y2)2 with some positive function a(x), that is, Oij — a26ij. Hence, in (jz1,^/2) =• (cos#,sin#) we get
684 Matsumoto Therefore the total length Li of the indicatrix is equal to WéW+GrW}1/2 ?w/2 = 2 / {(1 + 6cos<£)-1/2 + (1 — b cos^)-1/2}^ Jo where b — + (&s)2/a. The value of this elliptic integral is more than 2ir, because (l + a;)_1/2 + (l-a;)_1/2 >2, 0 < x < 1. Ref. M. Matsumoto [93] , and K. Okubo [137]. Example 3.1.2.2. On a Euclidean plane with a orthonormal coordinate system (jz1, jz2), we consider the Minkowski metric i=(!y1lJ, + |y2lp)1/p, which-is a p-th root metric (§1.6.1). In the first quadrant we have 9n = (iZ1)'-2«!^ + (p - l)(sz W2"2*, 512 = (2 - P)(y1y2)p^2^2~2pi 922 = (y2)p~2{(y2)p + (p - l)(2/x)p}L2_2p. Thus, we have (p- l)(y1?/2)p_2L4“2p and g^d^dy^ is positive-definite, if and only if p > 1. In (51, jz2) — (cos<£, sin 0) we get yfg _ •'/jT— 1 (cos sin 0)fr/2)~x L2 (cos? <p + sin* 0) Putting T = tan<£, the total length L\ of the indicatrix is equal to r a ! T f°° 4ttVP“ Ï {(ThF)? Thus we have Li 2tt and the equality holds only ifp = 2. 3.1-3 Torsions and Curvatures We treat first a two-dimensional Finsler space F2 with the Cartan connection CT. Let S be a scalar and we put Sii — S,i 4-5,2 LS|i = Sîi £i + S;2 m». (3.1.3.1) The coefficients (5,i, 5,2 ) and (S*i, S;z ) are called the h and v-scalar derivatives of 5, respectively. Since we have Sii = diS-(drS)Gri, S\i = diS.
Finsler Geometry, in the 20th-Century 685 the same results hold in the Berwald connection BP. Thus we get S, 1 = S^, S,2 = S;i = LSk?, S.2 = sLSkmi Throughout the following, our discussions will be restricted to (0) p-homogeneous scalars. Then we get S;i = 0 and (3.1.1.13) holds. We consider the covariant derivatives of the Berwald frame (&m). In CT we have = 0 and g^k — 0» and hence = 0. Thus ^7 — 0, — 0, miij = 0. Next, we obtain, for instance, + mr(IrnrTnlmj)' Hence (3.1.1.9) and (3.1.1.11) lead to Llz\j = Llj\j = erriim^ Lm'lj = —rmy, Lmi\j = —fhmj. (3.1.3.3) (3.1.3.4) Brom (3.1.3.4) we get LtyiTTij — 0. (3.1.3.5) To consider the covariant derivatives in the Berwald connection Br, we first treat Cijkih' Brom (3.1.1.10), (3.1.3.1), (3.1.3.3) and (2.5.2.14) we get (a) LCijk,h = (Ijth + (3.1.3.6) (b) Ctffcio « Cijw = Pijk - I^rriiTnym.k. Thus (2.5.S.9) leads to = 2Zfi77lt772j‘77lA, (3.1.3.7) where (;) is the h-covariant differentiation in BT. Now we deal with L£* = yx. From L.ti = 0 and = 0 we get — 0. Next A = girF and (3.1.3.7) lead to = 0. Next, differentiating — 0 and .gtfmhmf = e, (3.1.3.7) leads to = 0 and — I,i mj. Thus we get ® From rrti = girml we get rrivj — Consequently, we obtain — 0j — 0? — si a m*mj9 = —el^mirrij. (3.1.3.8) Brom (3.1.3.8) we have (¿¿m, - = sI^iTrij - (3.1.3.9)
686 Matsumoto Now we shall consider the torsions and curvatures, by treating commutation formulae of covariant differentiations» First of all, we have Theorem 3.1.3.1. In the two-dimensional case, the v-curvature tensor S2 of the Cartan connection vanishes identically. This remarkable fact is obvious from (3.1.1.10) and (5) of (2.S.2.6). Now, we have from (3.1.3.2) that 5,1 = 5^, 5,1,2 ~ S,2 — eSijm?) 5,2.1 = (^5|j^)lfr Then, from (2.Ô.2.4) it follows that 5,1,2 ~~ 5,2,1 = . The h-curvature tensor Rhijk of CT is skew-symmetric in (h,ï) and (j,A), and ■ hence Proposition 3.1.1.2 shows the existence of a (0) p-homogeneous scalar R such that (a) Rfoijk — £i<fàh){fjrnk — £/¡.771^), (b) Rtfk ~ y^Rhijk = eLRmi{fjm,k fck'mj)* R is called the scalar curvature or the Gauss curvature of Berwald. Then we get 5,1,2 “■ 5,2,1 ~ -RS*. Next, for a (0) p-homogeneous scalar S we have 5,ia = sL(Sn&)\jTn? = sLSii\^mj + 5,2 5;2,i = (£¿5^).^ = eLS\jdrn?r. Hence (2.S.2.4) leads to 5fX;2 - 5;2,i - + 5 2 = -sLS|r^^+S,2. The first term of the right-hand side vanishes from (3.1.3.6). Next we have 5,2î2 — = LS^rrfrn? - e5,i, 5;2>2 = e{eLS\jm?\irr^ = LS^tm'rn?. Hence we get from (2.S.2.4) 5,2:2 - 5;2,2 = £(5,i|j - 5|^)mW - sS 1 = L(^SirCT. - S^rrtmi - s5,i = -eI5,2 - eZjS.2 - s5,i.
Finsler Geometry in the 20th-Century 687 Therefore we obtain the commutation formulae*. (a) S,i,2 — 5,2.1 ~ (b) S,1;2 - 5*2.1 = 5t2? (c) St2-2 — S^2t2. “ “£(5,1 + IS,2 + ■f,15;2). Next, (3.1.3.6) and (2.5.2.16) yield LPhijk = 7,i (4^ - (3.1.3.10) (3.1.3.11) As a consequence, it is seen that P/iijk is symmetric in J, k. Finally we deal with Chij\k- L(LChij)\h = LtkChij + On the other hand, (3.1.1.10) and (3.1.3.4) lead to = L^Imhrriimj^k = 4" £iC>hkj “F fcjC'hik)' Hence, if we define Thijk = ^Chij |fc 4- ¿kChij 4- ¿jChik 4- tiChjk 4- ¿hCijk t (3.1.3.12) then in the two-dimensional case we get LThijk = i&mhrntmjmk* (3.1.3.13) The tensor T^ijk is called the T-tensor. It is completely symmetric in the general dimensional case. Because we have according to the definition of the above shows Ctajl* — Proposition 3.1.3.1. The main scalar I is a function of position alone, if and only if the T-tensor vanishes identically. In fact, I = I(x) is I|i = 0, so that (3.1.3.2) show’s I.3 = 0. Ref The T-tensor was introduced by H. Kawaguchi [71]. Simultaneously M. Matsumoto [87] found it in the two-dimensional case. Now we are concerned with the Bianchi identities (2.3.2.9) - (2.5.2.12) of CT. Among them, the first three and the last two identities are clearly reduced
688 Matsumoto to trivial identities in the two-dimensional case, (a) of (2.5.2.10) is essentially rewritten’in the form (2.5.2.16). (b) of (2.5.2.10) and (a) of (2.5.2.11) are con¬ sequences of (c) of (2.5.2.10) and (b) of (2.5.2.11) respectively. Thus, we have to consider only two. (c) of (2.5.2.10) and (b) of (2.5.2.11). (b) of (2.5.2.11) is reduced to The first term is symmetric in i, j from (3.1.1.10) and (3.1.3.11). Next (3.1.3.1) gives from (3.1.3.4) and (3.1.3.5) I'~Rmhkj h = which is also symmetric in i, j. Therefore, this identity is also trivial. Now (c) of (2.5.2.10) is written in the form LÄynhtjIfc + {RmhirfLCjfc') + LPfrihir^jk I'Prnhjkii ~ (VZ)} ~ From (3.1.3.10), (3.1.3.11) and (3.1.3.5) we have IsRmhij\k Rmhir(LCjk) ~~ (VJ) LPmhirPjk ~ (*/Î) ~o, I'Rmhjkii ~ {i/j) = Zti,i(€mT7Z7i, Therefore, this identity is written in the form sR’2 + RI + /, i,i = 0. (3.1.3.14) Summarizing the above, we have Proposition 3.1.3.2. In the two-dimensional case, the torsion and curvature tensors of the Cartan connection CT are written as = eLRm^tjmk - Cjk = L^Im^m^ Pl:Pjk - ^lrr^mjmk, R2 : ^hjk = ¿Whin* - &mh) (fyn* - P2 : ^hjk = (.^m1 - S2 = Q. The main scalar I and the scalar curvature R are in the relation (3.1.3.14). The scalar derivatives satisfy the Ried identities (3.1.3.10).
Finster Geometry in the 20th-Century 689 We now consider the Berwald connection BY and its h and /incurvature tensors H and G. From (2.5.3.7), we have R^k = and hence (3.1,3.10) gives immediately Hkhij = £{R(^kmh - ¿Wk) + R^mkmhy^imj - tyroi). (3.1.3.15) Thus HkMj consists of the skew-symmetric and symmetric parts in k, h. Next (2.5.S.2) and (2.5.5.T) give Giijk — &kFhj + ?hj-k' From (2.5.2.7) and (2.5-2.6) it follows that the first terms above is equal to Phjk = Phjk + “ ChrPjk' The second term is written as P^\k - p^clk + PijCL + Since the term LCihTP^k of the first and the similar three terms of the second are of the same form we have Consequently, we get = {-21,1 + ( A1j2 + (3.1.3.16) We now consider the h-curvature tensor K of the Chern-Rund connection CRT. (2.5.2.6) gives jzi z?® "or — ^hjk - ^hr^jk- Therefore, we obtain K^k « {sR^m* - Fmh) — Rlmhm^^jmk - ¿rfrij). (3.1.3.17) Proposition 3.1.3.3. In the two-dimensional case, (1) the h and hv-curvature tensors of the Berwald connection BY are written as (3.1.3.15) and (3.1.3.16), and (2) the h-curvature tensor of the Chern-Rund connection CRT is written as (3.1.3.17). 3.1.4 Two-Dimensional Finsler Space with I(x) The present section is devoted to the consideration of two-dimensional Finsler spaces with the main scalar I -which depends on a position x alone, that is, /52 = 0. First, we put 2v^ff (3.1.4.1)
690 Matsumoto which is (2)p’homogeneous. Now put D — log# and denote by subscripts of B and D the derivatives by yi. Then Di == Bi/B is (-l)p-homogeneous and (3.1.1.15) leads to LDX (3.1.4.2) The purpose of the following is to find the expression of Bijk in terms of I. From Bi = BDi we get Bij = B(DiDj + Bijk — B[DiDjDk 4- {DiDjk 4- (i, J. k)} 4- Dijk]* We shall use the following simple symbols for brevity: £ij ^i^jt (*»)« — £imj 4- tjMi, rriij = miTtij £ijk — £i£j£ki (££m)ijk (£mm)ijk = Zirrijmk + (i, j, ty, . ™ijk — mimjmk- Then (3.1.1.9) and (3.1.1.11) lead to Ltij.k = E(trn)ijmk> Lrrnj.k = {-(¿m)ij 4- Zsimijjmki L(£m)ij,k - 4- el(tm)ij 4- Zem^mk. Now, from (3.1.4.2) we obtain L2Dij = -2£ij 4- €l(bn)ij 4- (2s - el;2 and hence = B{2£y ~ s/(^)v + e(2 -1^}. Consequently, Example 3.1.1.2 and (3.1.4.1) show 4det(B^) =: 4e - -I2. (3.1.4.3) Next we have L3Dijk = 4^-fc - 2eI(Um)ijk - 2(2s - el;2 -12) 4" (61 “ 2s/2 — 4ZZ;2 ~ sLt2\2)^ijk’ Since is (0) p-homogeneous. we have Bijkft — Bijk^ — Bijk^ o and hence Bijk are proportional to mijk> Therefore B^k = 0, if and only if Bijk^rr^m^ — 0. We get LDirrt = -I, tfDijTrf'm? = 2s - eZ;2— A tfDijkmWm* = 6sl - 2I2 - 4eZI.2 -1;2;2. Thus, rfBijkmWm* = -B[en,2 4- (3.1.4.4)
Finsler Geometry in the 20th-Century 691 Therefore (3.1.4.3) and (3.1.4.4) show Proposition 3.1.4.1. If the moan scalar I is a function of position x alone, then Bijk = 0 and 4det(Bv) = 4e — Z2. It is obvious that B^k = 0 if and only if B is a quadratic form in y* : 2B = éy(x)»V- Then Bij = We shall classify the spaces under consideration in the four classes as follows: (i)S = l, /*>4, 25 = j371 (ii) £ = 1, I2 = 4, 2B = 02, (iii) e = 1, I2 < 4, 2B = ^2 + 72, (iv) s = -1, 2B = 0-y, where /3 — Pi(&)y* and 7 = Qi(x)yi are 1-forms in y*, independent to each other: Pi<to-P2<h. ^0. (i) : 2bij = piqj +pjqi and 4 det (5^) « -(pigs -P2$i)2 = 4 - Z2. Putting r2 = Z2 - 4, we get r = p\qz - P2Qi- (ii) : bij - piPj. We put r == p^ -pzqi. (iii) : bij - piPj 4- qiqj and 4det(dy) = 4(pi$2 -P2£i)2 = 4 - Z2. Putting 4r2 = 4 — Z2, we have r = pi$2 — p^qi* (iv) : bij = piqj +pjqi and 4 det (5^) = -(pigs -P2$i)2 = -(4 + Z2). Putting r2 = Z2 + 4, then we have r = piq2 —pzqi* In all the cases we have r » piq$ — p2Qi- Now (3.1.1.11) and (3.1.2.1) lead to di log y/eg — sl(x)di0. By integration, we get y/eg = j(x)exp(eI0) with some function j(x). Thus (3.1.4.1) gives L2 = (2JB) exp(sZ0). (3.1.4.5) We shall find 0, To do so, we take (£,7) as the variables instead of (y1,^2). From the elements (dB/dy\ d^/dy*) = (p^ qt) of the Jacobian r = dffi^/dty^y2} we get the inverse Jacobian ^(yx,y^)/^(/?, 7) of the elements /V _ /ft -ft \ \dp ’ dp)~\r 1 r )’ dy1 Qy2} _ / _pz pi &y ’ dy) ~ \ r’r
692 Matsumoto Consequently, (3.1.2.1) gives — = (—A -i- (dy2'\ 00 \0yiJ\00J't\0ys)\00) _ (miqs - nt2gi) rL We have (3.1.1.6) and (3.1.1.7): (mi, m2) = h(—Z2^1), hr = eg. Thus, we may put h = y/sg. Consequently, M ... fh'X -. ri ^7 7 00 \rL)q' rL2 2tB‘ Similarly we obtain 7 W = P dp~ 2rB' &y~ 2rB ’ (i) and (iv): {dO/dp,dO/&y} = (—l/r/3, 1/ry) and integration leads to <-(-w with some function «(x). Consequently (3.1.4.5) yields *=w?r By including g(x) and k(x) in P and 7, we finally get L2 = P*f(l /P)1^ • (ii) : (dO/dP, dO/dy) — (—^/rP2, 1/rP} and integration leads to L3 = i02 exp { (g) + «}, with some «(x). By including y(x), «(x) and r(x) in P and 7, we have finally, L2 = p2exp(Iy/p). (iii) : (dO/dP, dO/dy) s= (— 7/rQ32+72), P/r(p2+y2)) and hence integration gives 0=(1) Aretan (2)+k, with some w(x). By including j,« and r in p and 7, we obtain L3 - (02 + t2) exp { (1) Aretan }. In this case, we have 4r2 — 4 — I2. We rewrite 2r as r and then L2 = (J32 +- 72) exp{(2Z/r) Arctan (7//?)}, r = y/^ — T2. Theorem 3.1.4.1. Two-dimensional Finsler spaces having the main scalar I = I(x), that is, the vanishing T-tensor, are classified the four classes as follows:
Finsler Geometry in the 20th-Century 693 (i) £ = .1, I2 > 4 : Zr = r = (ii) s = 1. Z2 = 4 : L2 = /32 exp(l7//3), (iii) £ = 1, Z2 < 4; £2 _ (yj2 +<y2) exp{(2Z/r) Arctan (7/^), r = V4 — Z2, (iv) e = — 1; L2 = r = VZ2 + 4, where /3 and y are independent 1-forms in y\ In (i) and (iv) both of L2 may be written in the form L2 = F-?, s + t = where s and t are such that (i) s < 0 < is (iv) 0 < s < t, st (3.1.4.6) (3.1.4.7) Ref. L. Berwald [26], [30]. It seems that the case (iv) was first considered by G.S. Asanov [10]. According to Theorem 3.1.4.1, the four forms of L2 can be defined, provided that /3 does not vanish Therefore, Corollary 3.1.4.1. If a two-dimensional Finsler space has the main scalar I = I(x) and its fundamental function L can be defined in any non-zero yi) then the space must be a Riemannian space. In fact, only “(iii), I = 0” is possible. 3.1.5 Equations of Geodesics in Two-Dimensional Space We have had the equations of geodesic in the general form (3.1.5.1) with the parameter s = arc-length, where &(x,y) are defined by In the two-dimensional case, we have (3.1.5.1) in a single equation (1.1.3.2), called the Weierstrass form: hxq ~~ fsyp 4" (p^ qpfW — 0, (3.1.5.2)
694 Matsumoto where (s*) = (x,y)> (y*) = (dx*/ds) = (p, g) and W is the Weierstrass fonction: ?" pq p Now, let us write (3.1.5.2) in terms of y1 = dy/dx and y" = d?y/dx2. To do so, we define the associated fundamental function A(x, y, z) as A(x, y,z) = L(x,y,l,z), £(*> W>P> 9) = A(x, y, ^p, (3.1.5.3) where the parameter t is assumed to be p — dxjdt > 0. Then (3.1.5.2) can be written as ' Azzy" d-A^tf + Axz-Aj, = 0, z = y', (3.1.5.4) which is called of the Rashevsky form. We consider (3.1.5.1) again.. From pq — qp = p3p" and (3.1.5.1), that is, p = — 2G1, q = — 2G2, we obtain the equation of geodesic in the form y" = {G1 (x, y; p,q)q- G2 (x, y, p, q)p}. Since G^ix.y^p.q) are (2)p-homogeneous in (p,ç), we get ^{x^p.q) — p^G^X) y, 1, j/). Thus the above is rewritten as y" = 2{G\x,y, 1, VW - G2(x, y, 1,3/)}. (3.1.5.5) Proposition 3.1.5.1. In the two-dimensional case we have the equations of geodesics as follows: (1) (3.1.5.2), (p,q) = (dx/ds, dy/ds). (2) (3.1.5.4), with (3.1.5.3), (3) (3.1.5.5), y = dy/dx. y" = &y/dx2. We shall deal with the functions Gi(x>y\p, q) in detail. If we put Li = diL and Lty = diL. then Zf}» = Li — L{r]Gri = 0, From these equations we obtain La (=W) = ^r(T, _ hr* Grl . £«ü) = £ *■ • (3.1.5.6)
Finsler Geometry in the 20th-Century 695 The latter gives - (7) - - (^) № - Hence we have M (= ¿1(2) - ¿2(1)) === - mi^2) 1 and then (3.1.1.6) leads to M = {sh/L2)mryiG^. Thus we get 2mrGrr = eML2/h. This and (3.1.5.6) yield the following expression of 2G* in the Berwald frame (1, m): 2Gi = Lo£i+(^)mi. By (3.1.1.7) and (3.1.1.8) this is rewritten in the form Finally, we have from the definition of W and (3.1.1.6) W,32 = i?₽ = ^=(J)(»ni)2 = (J>2(*2)2 W This together with (3.1.1.7) gives WL3 = g. (3.1.5.7) Consequently we have Proposition 3.1.5.2. In the two-dimensional case the functions G* appearing in the equations of geodesics (3.1.5.1) are given by 2LG1 = Lop - 2L& = Loq+ (^)LP, where Lq = Lxp 4- Lyq, M = Lxq — Lyp and W is the Weiersbrass function. Ref. M. Matsumoto [111]. The following example is also given in that paper. Example 3.1.5.1. We consider a two-dimensional fundamental function r / > _ iVQ\ (*2P + xyq + y2^ \ L(x,y,p,q) = ).
696 Matsumoto We get Lq = 2xypg + (x2 + y2)q2 + ~Xy^ + , 2LG1 = -x2p\ 2LG2 = (^)(p= + ^)(2p+ +^. On account of (3.1.5 .5) we obtain the differential equation of a geodesic as sw" + (i/)2 + i = o> and integration leads to the finite equation . (x~a)2+y2 = ft2, where a and b (> 0) are arbitrary constants. 3.1.6 From Geodesics to the Metric, I Suppose that a two-parameter family of curves O'(a, b) :y- ffa а, &), (3.1.6.1) be given in the (ir,3/)-plane, or in a coordinate domain of a local coordinate system (z, y) of a smooth manifold M of dimension two. Our problem is: to find the Finsler metric L(s,y;i,y) the geodesics of which are given by C{abb). Here we pay attention to the equation of geodesic of the Rashevsky form (3.1.5.4); A(C) — Assy/f + АухУ* + Asch — Ay — 0, Z — yf. Now our problem is to find A{xyy^z) such that A(C) = 0 is satisfied by у of (3.1.6.1). From (3.1.6.1) we have (a) z (= У) = /z(x; a, 6), (b) z' = fxx{x\ а, Ь). (3.1.6.2) Solving the parameters (u,b) from (3.1.6.1) and (a) of (3.1.6.2), we obtain a - a{x.y.z), b — fiix^z). (3.1.6.3) Hence (b) of (3.1.6.2) is rewritten in the form / = fxx(x\ a, p} = u(x. y, z). Next, differentiating the identity Л(С) = Azzu + AyZz + Axz — Ay — 0 (3.1.6.4)
Finsler Geometry in the 20th-Century 697 by z and putting B = Az£, we obtain Bx + ByZ + B£u + Bu£ = 0. (3.1.6.5) To solve (3.1.6.5) for B, we shall recall a theorem on partial differential equations as follows: Let B), i = 1,..., n, and R(x, B) be given functions of xi and B. We shall find the general solution of a first order quasi-linear differential equation To do so, we construct the auxiliary equations and find n independent solutions fi(x,B) — Cj, c’s being constants. Then the given equation (a) shows the existence of a functional relation among ft, that is, ^(/i,..., /„) = 0 gives the general solution for an arbitrary function <1*. Now we return to our problem. The auxiliary equations (b) of (3.1.6.5) are written as From the first two equations dy/dx = z and dzjdx = u we get the solutions (3.1.6.1) and (a) of (3.1.6.2), that is, (3.1.6.3). Next dx — —dB/Buz yields the third solution B = c/CZ(z; a, 5), c = const, where iT(z;a,6) = exp< / u£(x. Consequently (3.1.6.5) shows the existence of a functional relation among a = a, b — 0 and c = B V where V is defined by V(x,y,z') = U(x;a,0). Therefore, this functional relation may be written in the form (3.1.6.7) where H is an arbitrary function of (a,/3). Then, if we construct (3.1.6.8) we obtain A^y^z) in the form A(x, y, z) - A* (x, y, z) + zD(x, y) + C(x, y), (3.1.6.9)
698 Matsumoto where D and C can be arbitrarily chosen, but this A must satisfy A(C) = 0 (3.1.6.10) It is easy to verify that the right-hand side of (3.1.6.10) is independent of z. Therefore we have Theorem 3.1*6.1. The associated fundamental function A(x, z) whose geodesics are given by (3.1.6.1) is (3.1.6.9), where (1) -4*(s,3/,3) is given by (3.1.6.8), and Cfay) and D(x,y) are arbitrarily chosen such that (3.1.6.10) holds. (2) B(x,yiz') in (3.1.6.8) is found as follows: (i) Wefinda(x,y,z) andß(x,y,z) o/(3.1.6.3)/rom (3.1.6.1) and(a) of (3.1.6.2), (ii) U(x;a9b) is (3.1.6.6), Vfoy,*) — U(x;a90) and B is (3.1.6.7), where H may be arbitrarily chosen. Example 3.1.6.1. We consider the family of straight lines y » ax + b as (3.1.6.1). Then we have Ci = x, ß = y- zx> u = 0, U ~ 1 = V. Hence B = Hfay ~ zx). According to the formula we have If we put H? = ÖH /dfy then we get Hence Dx — Cy = 0, which implies the existence of a function E(x, y) satisfying C = Ex and D = Ey. Therefore we have
Finsler Geometry in the 20th-Century 699 Thus (3.1.5.3) leads to Theorem 3.1.6.2. Let F2 be a two-dimensional Finsler space with the funda¬ mental function L{xy y\x, y) whose geodesics are given asy = ax + b. Then L is written in the coordinates (x, y) as L(x,y;x, y) = x [ (z - t)H(i, y - tx)dt + Exx -I- Eyy, Jo where H and E are arbitrarily chosen. Remark: It is obvious that H and E as above must be chosen such that L satisfies the conditions of a Finsler metric. Example 3.1.6.2. We consider the family of spirals r = exp(a0 + 5) in the polar coordinates (r,0). If we put x — 0 and y « logr, then the spirals are written as y — + b. Therefore the result of Example 3.1.6.1 can be applicable to this case without any modification. Example 3.1.6.3. To find a surface of revolution having the minimum area leads to the variational problem of the integral = J y>/l + (y/)s dx, y g 0, and the extremals for it are catenaries :y = a cosh We shall apply Theorem 3.1.6.1 to this C(a, &). Then y a = —=== , ï/ï + «2 (1+32) u = -, V p = x - a log (z + Vl + z2 ), U = cosh V = 1 + z2. Thus B « -ff (a,/?)/(! + z2). Here, let us chose « a. Then we have and Dx — Cy = 0. If we chose C — D = 0,, then we obtain the original integrand A(x, y, z) = j/\/l + z*. Example 3.1.6.4. We are concerned with the well-known linear differential equation y" + P(x)yf + Q(x)y = R(x). (3.1.6.11)
700 Matsumoto Let tz(z) and t>(x) be independent solutions of the homogeneous . y" + P(x)yf + Q(x)y = 0, and w(x) an arbitrary solution of (3.1.6.11), Then the general solution of (3.1.6.11) can be written in the form y = au(x) + bv(%) + w(x), (3.1.6.12) with two arbitrary constants (a, 6). Recall that u(rr) and v(x) are independent, if and only if Wo = ui/ — vu' / 0. Now our problem is to find a Finsler metric whose geodesics can be written as (3.1.6.11). Thus we apply Theorem 3.1.6.1 to (3.1.6.12): z (= 2/0 = ud + bvf + w'. Hence we obtain _ (y'y -vz + B) (u'y - uz + j4) a~ Wo ’ P~ Wo where we put A = uw' — wu', B — yw' — wv'. Next we have = — a(Puf + Qu) — b(Pif 4- Qv) + (R — Fu/ — Qw) = —Pz -Qy + Rf which is obvious from (3.1.6.11). Thus U(x; a, b) — exp ( - j Pdxj = y(x, y, z), and B(x, y, z) — H(a, /3) Pdx). Let us chose H = 1; we get B{x,y. z) = exp (y Fete), A^(x,y, z) = exp ( j Pdzj. Therefore we have A(x, y, z) = (y) exp ( y P<fa) + C(x, y) + D(x, y)z< where C and D must satisfy Dx — Cy = exp (Jpdx)(Qy-R). Then we may take D = 0 and C = exp(J Pdx)(Ry — Qtf2/2). Finally we obtain the fundamental function L(x,y,p,q) = exp^f Pdxj + +Exp + Eyq.
Finsler Geometry in the 20th-Century 701 If we take E = 0, then this L is a kind of Kropina metric. See Defini¬ tion 1.4.2.2 and Example 4.6.3.I. Ref. The problem treated in the present section is the so-called inverse problem of variational calculus. Theorem 3.1.6.1 has been seen in Darboux (Leçons sur la théorie des surfaces, III, Nos. 604, 605). M. Matsumoto [101]. 3.1.7 From Geodesics to the Metric, II The present section is devoted to a related inverse problem of variational cal¬ culus: namely, when the family of curves C(ayb) is given by the parametric form C(a, 5): x — 0(t; a, &), y = ipfa a, 6). (3.1.7.1) Throughout the present section we use the symbols = d<p/dt and ip — dip/dt. To deal with homogeneous functions, we introduce the auxiliary parameter c (> 0) such that x = <P(ct; 0,6), y — ip(ct*, a. &). (3.1.7.2) This yields p » dx/dt = ¿c, q = dy/dt = ipc. Hence we have 2 = 0(rf; a, b), - = a, &). (3.1.7.3) c c We get four equations (3.1.7.2) and (3.1.7.3). FYom three equations among these equations we solve (a, 6, ct) as functions of x. y and one of (p/c, g/c), and substitute into the remaining equation. For instance, we now suppose that from p ♦ x = <£(ci;a, 6), y = ^(c£;a,&), - = 0(ci;a,&) c we can solve a, 5, ct as a = tt*(®,y,2) b = ct = r’(x,y,^y Then the remaining equation is written in the form £ = V(x,y,^, Now we solve c from (3.1.7.6) and obtain c = y(x,y,p,q). Consequently, we obtain a = a* (®, ÿ, ^) = a(x, y,p, q), (®,y, = 0(x,y,p, «), (3.1.7.4) (3.1.7.5) (3.1.7.6) (3.1.7.6') (3.1.7.7) t = 7 = r(x,y,p,q).
702 Matsumoto We have to pay attention to the homogeneity of q, 7 and t in (p, q). In fact, for a positive number k we have from (3.1,7.6) (feg) (Ac) and (3.1.7.6’) implies kc = ^(x^y^kp^kq). Hence 7 is (1) p-homogeneous in (p, q), Then (3.1.7.7) shows that a, 0 and r are (0), (0) and (-1) p-homogeneous, respectively. Next, from (3.1.7.3) we get the following functions P and Q : P = = P(x, y,p,q), q = ^(tt; a, 0)72 = Q(x, y,p, q). (3.1.7.8) It is obvious that P and Q are (2) p-homogeneous in (p, g). Now our problem is to find £(z, y;p, g) such that the equation (3.1.5.2) of a geodesic in the Weierstrass form W(C)=Lx^£w + PW = 0, R = pQ-qP, becomes the identity [W] = 0 in (x, p, p, q). It is remarked that R is (3) p-homogeneous in (p,g), while W is (-3) p-homogeneous. Let us differentiate [W] = 0 by p and g. For instance, differentiating by p, we have From Lpp — Wq2 and Lpq — —Wpq, we have £*7*7 - Lypp = -(Wxp + Wyq)q. Further, on account of the homogeneity of P, Q, R and W, we have Ppp + Pq<l = 2P, Q?P + Qg<l =2Q, RpP + Rrf = 3P, Wpp+WQg = -3W. From these relations it is shown that [W]p and [W% can be written as [W]p = -g[W]o and [W]g = p[W]o where we put [Wjo = W*p + Wyq •+ WpP + WqQ + (Pp 4- <?g) W. Consequently, instead of the two equations (W][ = [W]ç = 0, we have the single quasilinear differential equation [W]o = 0. Now we apply again the theorem, which was used to solve (3.1.6.5), to [W]o = 0. The auxiliary equations are now written as dx _ dy _ dp _ dq __ dW
Finsler Geometry in the 20th-Century 703 If we put the above to be equal to dt, then «-Ç. a-* *-£, p q F Q give the four solutions. Introducing the fourth integration constant to» these solutions are written as x = <£(c(t + to); a, &), U= + to); a, &), 2 ~$(c(t + toy,a,b), | =^(c(t +to); <*,&)• (3.1.7.9) The remaining equation dt = —dW/(Pp + Q9)W is written as dW . . — = {Pp(^,^,c^,c^) + Q9(^,^,c0,c^)}di. Thus» owing to the homogeneity of P and Q, if we put u — c(t + to) and define. TT, II, U by V,P, 5) == P? + Qq, n(u; a, b) = tt(^ ^), (3.1.7.10) U (u; a, b) = exp { J II(u; a, b)du J, then we get the fifth solution W(c(t + t0);a,&) = ^, (3.1.7.11) with the fifth integration constant d. From (3.1.7.9) and (3.1.7.11) we solve for a, &, c,to and d, and then obtain a = a, & = /3, c = 7j io = r - i, d = W V, where V is defined as Vfay,P,<fi = (3.1.7.12) It is noted that V is (0) p-homogeneous in (p, q). Consequently, the differential equation [VK]o = 0 shows the existence of a functional relation among a, £,7, r—t, and WV. It must be, however, remarked that in our problem the following two conditions of W should be requested for the relation: (i) W does not contain t explicitly and hence the relation may be written as WV^aM (ii) W is (-3) p-homogeneous in (p,$).
704 Matsumoto Since and 7 are (0), (0), (0) and (1) p-homogeneous, respectively, for a positive number k we have H* (a, P'k-f') -W (x, y, kp, kq)V (x, y, kp, kq) = {?^^}V(x,y,p,<l). If we put k = 1/7 and Jf*(a,/3,1) = then we finally obtain W{x,y,p,q) = ^^-, (3.1.7.13) where H is, of course, arbitrarily chosen. From Lpp — Wg3 and Lqq = Wp3 we may put L*1 = ej(J‘wdp)dp, L‘2=p?f(Jwdq)dq, (3.1.7.14) where the integrations must be done such that , i — 1,2, are (1) p-homogeneous. Remark: If we were to take 3nr 4- 2xy, for instance, then we would get 4- 2xy)dx - x3 + x2y 4- c(p), where the function c(y) of y could be arbitrarily chosen, so that the above is not necessarily homogeneous in (x, p). We will show how to obtain the homogeneous function of degree r +1 by the integration of a homogeneous function of degree r as follows: Suppose that f(x,y) is a given function homogeneous of degree r in (x,p). From f(x, y) = yTf(x/y, 1) we have If we put h{z) — f f(zy l)dz, then h(z) is homogeneous of degree 0, so g(x, y) = yr+1h(x/y) is obviously homogeneous of degree r 4-1. Now, for instance, we obtain L in the form L = LJ 4- pC*(a?, y, q) 4- P*(a?, y, g). Since C* and Z>* must be (0) and (1) p-homogeneous in q respectively, we may put C* = C(x,y) and P* = gP(a?,p). Therefore we obtain L(x.y;p,q) = L#i (a, y;p, q) 4- pC(x,y) 4- qD(x> p), ¿ = 1,2. (3.1.7.15)- Finally, substituting the above in [W] = 0, we get the condition for C and D as Cy - Dx = (Lt)xq - (L;)w + RW. . (3.1.7.16)
Finster Geometry in the 20th-Century 705 It is remarked that the right-hand side of the above does not depend on (p, g), as easily verified by making use of (Li)XJW7 — pgWx, (£*)ypp = g2^, etc. Theorem 3.1.7.1. The fundamental function L(x^y:p,q) whose geodesics are given by the parametric equations (3.1.7.1) is obtained in the form (3.1.7.15), where L^i — 1,2, are given by (3.1.7.14) and C(x, y) and D(x, y) are arbitrarily chosen such that (3.1.7.16.) are satisfied. W(x,y;p,q) in (3.1.7.14) is found as follows: (i) a*,ß* and 7* of (3.1.7.5) are given from (3.1.7.4), for instance, and y of (3.1.7.6’). from (3.1.7.6). (ii) a.ß and i are defined by (3.1.7.7), and P and Q by (3.1.7.8). (iii) II and U are given by (3.1.7.10), and V by (3.1.7.12). (iv) W is given by (3.1.7.13) where H is an arbitrary Junction. Example 3.1.7.1. We shall deal with the well-known Brachistochrone prob¬ lem, that is, to determine for a heavy particle, the curve of steepest descent between two given points in a vertical plane. This leads to the two-dimensional Riemannian space with the metric y, p> 1} = ’ y>Q’ vy and the curves are its geodesics, called cycloids: x = a(t — sint) +5, y = a(l— cost). We shall apply Theorem 3.1.7.1 to this family of cycloids. We start from three equations: x = ^(ct; a, b) = a{ct — sin(ci)} + 5, y — iKct; a, b) = a{l — cos(ct)}, | = ^(ct; a,b) = a sin(ct), because the remaining equation p/c =. a{l — cos(ct)} has the same right-hand side as the second equation. Then a*, and so on of (3.1.7.7), are given by . _ {(g/^ + y2} 2y ß“ =x-^a*r* + ~, c (3.1.7.7) = Arctan r 2y(g/c) 1 L{(g/c)2-y2}J‘
706 Matsumoto Since the remaining equation p/c = y yields c (= 7) = p/p, we have r = (2) Arctan {(^}- Then we get sin(r7) =5^7 cos(r7) 2sin(ct) ~ {1—cos (ci)} ’ n U = {l~cos(ct)}3, V ={5^sy}2. Consequently we obtain W = H(a,/?){(p2 + g2)22/3/4p7)}. Since this W can be -written as W = H^ff^cry/p3), we may replace by the new H(a, /3\ according to the arbitrariness of H. Finally we obtain W = H(a,^). On the other hand, the original Riemannian metric L gives W = (p2 + 42)“3/2/, and hence this W is obtained from the above W where H is chosen as H = (2a)"3/2. Example 3.1-7.2. We consider the family of semicircles (X — a)2 + j/2 = &2, y > 0, which has been treated in Example 3.1.5.1. Instead of this equation, we treat its parametric form ic = a + 6cost, p = 6sint, and apply Theorem 3.I.7.I. Then, from three equations x = a + 6cos(ct), p = &sin(c£), -=5cos(c£), we get a = x—-, c Then we have &=; v^ + Gz/c)2, ct =. Arctan V = v'y2 + («/c)2, r* = Arctan (^j) > and the remaining equation is p/c — —y. Then 7 = —p/y and a = a: + 7’ ^=g)Vp2+?2, r= (|) Arctan (?).
Finsler Geometry in the 20th-Century 707 Next we obtain P (=-bcPcosfcty = Q ( = -&<?sin(ct)) =-y. Hexice we have U = SinUj 7Г COXU sin-u ’ Since this W can be written in the form H(a,0)0y2/p?, we may write W as p3 If we choose H = 1/g3, then we get W = (p2 + r)"3/2^ corresponding to the well-known L = y/p2 + q2/y> that is, the family of the semicircles are geodesics of the Riemanniah metric ds2 = (dor + dy^/y2 of constant curvature -1. In Example 3.1.5 J we had W = <loty2/p\ Example 3.1.7.3. We are concerned with the family of parabolas by =(x-a)2, y,b>0, on the semiplane JR2 having the vertex (a, 0). This can be written in the para¬ metric form t2 Then we apply Theorem 3.I.7.I. (ci)2 x « 4>(ct;a, &) - a + ci, p = ^(ct; a, b) = ——. c ’ c b * From the three equations except p/c = 1, we get a* = x - 2V (g/c)’ (9/c) ’ and the remaining equation p/c — 1 gives c = p = 7. Hence, we get
708 Matsumoto So we obtain p =o, Q=g, ir = *, n=£, u = exp{if(j)du} =u2, V = (^®)2. Consequently, we have H(q,0) H(q,0) 0yp? ' Thus we use £$ of (3.1.7.14): Let us choose #(a,/3) — fi*. We obtain . r* . .7»—l_.2zi—1 ^-2 —27» - Z 1 1 h2—y P T > n r1» 2 ’ In particular, we treat of the case n — 0. Then C and D may be chosen to be zero and we get This is of the type (i) of Theorem 3.1.4.1. Applying Proposition 3.1.5.2, we obtain G1'=0, G2 = -^-. 4y Next, we treat the exceptional case n = 1, that is, H = 0. We have Hdq}<*9 = 4yp2(log |p| - log |ff|) = 4yp2 log | ^ |, where the term log |p| is added for the homogeneity of Thus we get £;==4plog[^|. Since R = p<rf'2y and W = 4/pç2, (3.1.7.16) gives Cy — Dx — 2/y, and hence we choose C = 0 and D — — 2xjy. Therefore, we obtain (b) L = 2plog|*|+.^. It is easy to show that the Finsler spaces with the metric (a) and (b) have the same geodesics and there are given by W'-(/)2 = o,
Finsler Geometry in the 20th-Century 709 which is the differential equation defining the family of parabolas we considered. Further, Gi of the metric (b), denoted by *<?*, are a little complicated, but it is easy to show that .G^G + Py*, F=—, p where G* are of the metric (a). That is, (a) is projective to (b). See §4.4.2. Ref M. Matsumoto [100], [108], [110]. 3.2 Riemannian Space and Locally Minkowski Space 3.2.1 Deicke’s Theorem Let F™ = (Ai, £(æ,ÿ)) be a Riemaxmian space with a Riemaimian metric Then (1.2.1.5) is written as F=y = did, F = Thus (1.2.2.5) lead immediately to Cijk = 0. Conversely, we have n _ r* __ U - Uijk - —~ 2 ’ which implies that F is a quadratic form in (#*) and hence L(x, y) is a Rieman- nian metric. Therefore we have Theorem 3.2.1.1. Among Finsler spaces, the class of all the Riemannian spaces is characterized by the vanishing of the C-tensor. The condition C^k = 0 means that the vertical connection Fv of the Cartan connection CT is flat, that is, P = F* of Definition 2.3.3-7. There will be many authors having bitter experiences such that on some desirable and interesting conditions the Finsler space under consideration all too soon has been reduced to a Riemannian space. Almost all of such simple results must have not been published. The following is a typical example of such results which are not worth writing. Proposition 3.2.1.1. If the C-tensor is v-recurrent, i.e.f there exits a vector field Xi{x^y) such that the v-covariant derivative ChijU of Chij with respect to CT is of the form Chij\k — ChijXk, then the space is Riemannian.
710 Matsumoto Proof: From Chij‘Uh' = 0 and the assumption, we have = o = - cikj = -c*,, which shows that the space is Riemannian. We have, however, three important theorems which assert the space to be reduced to a Riemannian space. These are never all too soon> but quite astonish¬ ing theorems. Because the proofs of them exceed the contents of our treatment, we shall show only an outline of the proofs. Theorem 3.2.1.2. If the torsion vector At = LCi, Ci = CJ., of a Finsler space F71 = (M, X(z, y)) vanishes identically, then the space F” is Riemannian, provided that L be positive-valued and C4 differentiable for any non-zero y*. Ref This theorem was first proved by A. Deicke [37] by applying affine dif¬ ferential geometry, and F. Brickell [32] gave another proof based on an inequality between geometric and arithmetic means and E. Hopf’s maximum principle for elliptic differential operators. The importance of the class of Finsler spaces with A{ = 0 had been noticed by many authors. For instance, W. Barthel [21], B. Su [153], J. Wegener [169]. In 1934 E. Cartan spent about three pages of his monograph to treat such spaces and showed a close relation between them and his hyperareal spaces. In [129] A. Modr discussed a duality of such Finsler spaces and Cartan spaces (hyperareal spaces) with A1 = 0. L. Berwald already stated ([31], p. 769): Any two-dimensional Finsler spaces with Ai = 0 is Riemannian [from (3.1.1.10)], although it seems that he had noticed it when he introduced the main scalar in 1927 [26]. A geometrical meaning of Ci is as follows: We are concerned with g = det(gijf Then & Vg = (2J5 ')gjk9i3ik = Vg^Cjki = y/gCi, which implies Ci = &(log Jg). (3.2.1.1) As a consequence, g is a function of position alone, if and only if Ai = 0. Thus, in a space with Ai = 0, the concept of the volume-element y/gdsi... dxn can be delined, as in a Riemannian space. We shall sketch the proof of Theorem 3.2.1.2, given by Brickell. Denote by Rn the n-dimensional number space of points (y*) and by Rfi the space R* with the point (0) removed. We consider the fundamental function L = L(xq, y) with x* = x^ fixed as a function on Rq. Suppose that L is (a) positive, (b) differentiable of class C4,
Finsler Geometry in the 20th-Century 711 (c) (l)p-homogeneous. Then the matrix g of elements g*j = dtdj(L2/2) is positive definite, as Deicke has showed. Lemma 1. Let ycty be two points in jSJ. Then Trg^1(yo)g(y') n. Next we introduce the elliptic differential operator A = gijdidj, where g'i denotes the element of the matrix g_1. Lemma 2. The matrix Ag is positive definite. Then Aghh § 0 for each h. Since g^h is (O)p-homogeneous, it attains a maximy-m on Thus E. Hopf’s theorem shows that g^h is constant on . Lemma 2 now implies that Aghk = 0 for all h, k and, as before, Hopf’s theorem shows that ghk is constant on Eft. Now we describe E. Hopf’s theorem. In a number space U = (x*) of dimen¬ sion > 2, we consider an elliptic differential operator th<9 dx^dxk dx* * where and h£(x) are continuous functions on U, and the matrix (g^k) is supposed to be positive definite. Then Theorem (E.Hopf). If a function # of class <? on U satisfies E(&) 0, and if® attains a maximum on U, then $ must be constant If E($) 0 and if<& attains a minimum on J7, then $ must be constant. Proposition 3.2.1.2. Let F™ — (M, L(x, 3/)) be a Finsler space with the fun¬ damental function L(x, y) = cix^y2... yn')1/n, where c(x) is an arbitrary positive-valued function. The torsion vector A* ofF71 vanishes identically. Proof: The fundamental tensor is given by s“~^ HT' where i and j are not summation indices. The det(^) = c2"^—l)”_1n_n, which does not contain y1,. ..,yn.
712 Matsumoto Ref, Soon after Deicke’s paper, [37] was published, A. Modr published a paper [130] as a supplementary note to a paper [129] and showed the interesting example stated as Proposition 3.2.1.2 which is due to L. Berwald. His example is not a counterexample of Theorem 3.2.1.2, but asserts the necessity of the assumptions of L in this theorem. 3.2.2 BrickelFs Theorem We shall turn our attention to the second important theorem. The fundamental function L(x, y) of Finsler spaces treated in this theorem must satisfy the four assumptions as follows: Fl. L(x,y) is defined and positive for all non-zero y\. F2. L(x, — y) = y), called the symmetry. F3. The fundamental tensor g is positive-definite. F4. L(x,y) is differentiable at any non-zero y*. Theorem 3.2.2.1. If the v-curvature tensorS2 — (Slhjk) of the Cartan connec¬ tion CT of an n (> 2)-dimensioned Finsler space F71 = (M,£(z,3/)) vanishes identically, then Fn is a Riemannian space, provided that L satisfy the assump¬ tions (F1-F4). As it has been shown in Theorem 3.1.3.1, any two-dimensional Finsler space has S2 = 0. Next, in the case of general dimension, S2 is the curvature tensor of the tangent space with the Riemannian metric gij(xoiy)dyzdy^, as in §2.5.1. Ref This theorem was proved by F. Brickell [33], and R. Schneider [147] gave another proof from an isoperimetric inequality. A. Lichnerowicz [S3] showed a generalization of the Gauss-Bonnet formula in a Riemannian space to a class of Finsler spaces, called the Berwald space by him, which are characterized by S2 = 0 of CT. (We do not, however, use the name “Berwald space” for such a Finsler space. Of. §3.4.1.) In [70] A. Kawaguchi concluded that a Finsler space with S2 — 0 is Rieman¬ nian, although it seems that his viewpoint is rather intuitive. D. Laugwitz [S2] showed that S2 vanishes for any two-dimensional Finsler space and stated in 1965 the conjecture that a Finsler space ofnfe 3) dimensions with S2 = 0 will be Riemannian, Now we shall sketch BrickelFs proof. First the four lemmas are given as follows: (1) Let f(y) be a real-valued function of class C1 on Rfi, where R$ denotes Rn with the origin removed. f(y) is (O)p-homogeneous, if and only if (W=0- :
Finsler Geometry in the 20th-Century 713 (2) Let f(y) be real-valued function of class C1 on Rft and fyf are (O)p-homogeneous. Then f =s g + c, where g is (l)p-homogeneous and c is a constant. (3) Let Hq be a Riemannian space defined on RJ with a positive-definite Riemannian metric gij(y)dyidyj where gij = d{dj(L2/2). We also give R” and Rq the Euclidean metric (efy1)2 H F (dyn)2 and denotes these spaces by EF and Eq respectively. Suppose that Hq has zero curvature and n > 3. Then there is an isometry y —► Y(y) = (Y'£(y)) of Hq onto Eq which is differentiable of class C3, (Inhomogeneous and such that Y* satisfy Y^djdkY^ = 0. (4) Suppose that L2 is (2)-homogeneous. Then the isometry Y given in (3) is (l)-homogeneous. (1) and (2) are rather trivial, and proof of (3) is based on the well-known isometric immersion of Hq into Eq. Next, the Laplace operator A on E* is considered. (5) Suppose that Y* are real-valued functions on Eft of class C2, (l)-homogeneous and satisfy Y* AY* = 0. Then they are homogeneous linear functions. Finally the functions Yx of (3) are considered. From (4) it is seen that they are (l)p-homogeneous and thus satisfy the hypotheses of (5). Therefore, they are homogeneous linear functions. Consequently the proof is complete. It should be emphasized that the most essential point of the above proof is to show the linearity of Y*. In fact, (Y’) is then regarded as an induced coordinate system of the tangent space. See yr — X^yi of (1.2.1.1). It is noticed by careful examination of Brickell’s proof that the condition (F2) is not made full use. He wrote a note on Theorem 3 of a paper [34] and pointed to the question of whether the symmetry condition imposed on £ is necessary. It is known that it can be omitted that the real vector space V has dimension 3 and L is analytic^ as H.F. Munzer showed. However, it should be remarked that the conditions (Fl, F4) can not be removed. This is confirmed by an interesting example due to S. Kikuchi [73] as follows: Proposition 3.2.2.I. Let = (M,£(z,£)), n = 2k 4- r be a Finsler space of n dimensions with a fundamental function L(p,y)z- = {£1(1, y\y2)}2 + • • • + {Lk(x, y2*"1, y2*)}2 n + zL p,g=2Jf+l
714 Matsumoto where La(z;S/2a”1)3/2a), a = 1,..., k, are {\)p-homogeneous functions in j/2®"1 and y2a, and gpq are functions ofxz alone- The v-curvature tensor S2 and T of this space vanishes identically. This is obvious because of the fact that S2 vanishes for the two-dimensional case. This L can not be defined at a point (y*) = (0,0, y3,..., yn). 3.2.3 Szabo’s Theorem The notion of the T-tensor has been introduced in §3.1.3 dealing with the torsions and curvatures of two-dimensional Finsler spaces. It was found by H. Kawaguchi and M. Matsumoto almost simultaneously. H. Kawaguchi [71] considered Finsler spaces with the vanishing Zw-curvature tensor P2 of CT (Landsberg spaces in §3.3.2), and showed that if a Finsler space F71 is conformal to a Minkowski space with the vanishing T-tensor, then F71 is a Landsberg space. He (1978) proved that a Finsler space F", n > 2, with T = 0 is such that the indicatrix at each point x of F71 is a locally symmetric Riemannian space under the metric induced from the metric of Fn. On the other hand, M. Matsumoto [87] considered a generalization of the transformation theory of Finsler spaces and showed that a two-dimensional Finsler space F2 admits the strictly isometric V-rotations of the maximal order I, if and only if the T-tensor vanishes identically. He [89] studied the Finsler spaces’of dimension three with T = 0 and Bp = g~L^L2/2 (cf. (3.1.4.1)) of quadratic form of yii and then gave a conjecture: There may exist no non- Riemannian Finsler space of dimension more than two with T = 0, provided that the metric satisfies some of the four „assumptions given in §3.2.2. M. Hashiguchi [47] showed the close relations between conformal changes of metric and the T-tensor. See §4.1.2. He proved: The Ziv-curvature tensor P2 is invariant under any conformal changes, if and only if the T-tensor vanishes identically. A Landsberg space remains to be a Landsberg space by any conformal changes, if and only if the T-tensor vanishes. In 1981 Z. Szabd [157] proved M. Matsumoto’s conjectures: Theorem 3.2.3.1. If for a Finsler space F7* with positive definite metric tensor field gijy the tensor field T vanishes, then the space is Riemannian. Here we shall show another proof given by S. Kikuchi [75]. His and Szabd’s proofs are both with the aide of Deicke’s Theorem 3.2.1.2 at the final stage, and therefore L must be C^-differentiable for any non-zero 3/, in particular. Z. Szabd recognized n = 2 as an exceptional case as in Deicke’s Theorem, but we have showed Corollary 3.I.4.I. From (3.1.3.12) and T = 0 we have Ty’/s BCj |A: + 4” &kCj ~ 0, T = Tjkg*k = LCj\kgjk = 1(0^ - CrC&g^ = 0.
Finsler Geometry in the 20th-Century 715 (3.2J.1) shows Cj — djcr, a = log y/g. Therefore we have a is (O)p-homogeneous. Now E. Hopfs Theorem is applied and o is reduced to constant, that is, Cj = 0. Thus, according to Deicke’s Theorem, our space must be Riemaxmian. The proof above leads to Corollary 3.2.3.1. If Fn, n > 2, has T = ghig^kThijk = 0, then F71 is a Riemannian space, provided that the metric is positive definite and C4-differentiable for any non-zero y1. 3.2.4 Locally Minkowski Space We have given the definition of locally Minkowski space in §1.2.3. The man¬ ifold M of a locally Minkowski space F™ = (M, L) is covered by coordinate neighborhoods in each of which L does not depend on the arguments (x*). Such coordinates (a:1) are called adapted. Since L is independent of (x1), G* vanish identically from (1.2.2.2). Con¬ versely, if G1 = 0 hold, then we have which implies diL =s 0. So we have the Proposition 3.2.4.I. A Finsler space F71 = (M, L) is a locally Minkowski space, if and only if M is covered by coordinate neighborhoods in each of which Gi vanishes identically. Such a coordinate system is adapted. We have the transformation law of connection coefficients G^k for a coordin¬ ate change (x*) -+ (xa): GfaXt = G^X^Xck + dkX^ X“ = dixa. (3.2.4.1) Let (xa) be an adapted coordinate system. Then = 0 and we get the system of differential equations for (x a, X *) Six* = X dkX$ = G^Xf, (3.2.4.2) Here Gjk are independent of yi, Because — &d@bc = 0 and ^bcd 8X6 components of the Tw-curvature tensor of BT, so that we have G^k = 0, which is nothing but dkGij = 0. The integrability condition of (3.2.4.2) is ^ + ^rfc-(VA) = 0, the left-hand side of the above are components of the ¿-curvature tensor Hjkh of BT. Thus we obtain
716 Matsumoto Theorem 3.2.4.1. A Finsler space if locally Minkowski, if and only if the h and hv-curvature tensors H and G of the Berwald connection BT vanish identically. Corollary 3.2.4.1. Let F™ = (M, L) be a locally Minkowski space. Then the system (3.2.4.2) of differential equations for gives an adapted coordin¬ ate system (xa). If (a;2) of (3.2.4.2) is also an adapted coordinate system, then Cjk vanish also, and hence we have d&X J = 0. The converse is also obvious. Therefore we have Corollary 3.2.4.2. In a locally Minkowski space, the transformation between two adapted coordinate systems is affine. If we consider'Finsler spaces with CT or CRT connections then the locally Minkowski, property is stated as follows: Theorem 3.2.4.2. j4 Finsler space is locally Minkowski, if and only if (1) R2 = ^7hC = 0 in the Cartan connection, (2) K = F = 0, or K = —0 in the Chern-Rund connection. Proof: In an adapted coordinate system we have L = £(jf), so that Gi = 0, = 0. Thus (2.5.2.1) gives F*k =* 0 and (2.5.2.T) shows K = F = 0 and R2 = 0. ; Chijlk = dkChij = 9kd^^ - 0. Conversely, K = F = 0 show Fjk = Fjk(x) and d^+F^-i-O. This implies we have a coordinate system in which — 0, and hence Nj — 0. Thus (2.5.2.1) shows gij ~9ij(y), so that L2 — gzjViyj — L2(y), which implies that this (a?*) is adapted. K = F = 0 immediately yield Chi^k = dkChij = 0. Conversely, VhC = 0 implies P2 = 0 and F = 0 from (d) of (2.5.2.6). Ref. Corollary 3.2.4.2 has been known since the early period of Finsler geometry, because O. Vargo [165] mentioned it. So far as the author knows, the proof was first published by E. Heil [54]. Corollary 3.2.4.1 has been given by S. Bacsd and M. Matsumoto [15]. Theorems 3.2.4.1 and 3.2.4.2 were mentioned by L. Berwald [27] and by E. Cartan [35], but neither provided any proof. It seems that the proofs of them were first shown by H. Rund in his monograph. In Theorem 3.2.4.1, H and G are h and Jw-curvature tensors of BF, and in Theorem 3.2.4.2, K and F are also h and hv-cxirvature tensors of CRT, but
Finsler Geometry in the 20th-Century 717 the hv-curvature tensor P2 of CT does not play any role in this theorem. The condition VhC = 0 is stronger than P2 — 0, as it is shown by (2.5.2.14), but it is natural to conjecture: If a Finsler space satisfies R2 — P2 =0, then the space is locally Minkowski, provided that the metric satisfies some of the four assump¬ tions given in §3.2.2? This is an unsolved problem. 3.3 Stretch Curvature and Landsberg Space 3.3.1 Stretch and Shrink We consider a Finsler space F™ = (M, L(x, t/)) equipped with a Finsler connec¬ tion FT = Ujk)> 113 denote the fundamental tensor by gij and the h and v-covariant differentiations with respect to FT by (; >:) respectively. We are concerned with an infinitesimal circuit of M which consists of four points P(x), Q(x+dix), R(x+dix+d2(x+dix)), S(x+dzx), and a vector field v = (^) which is given along the circuit and transformed parallel with respect to a parallel supporting element y, Thus (2.4.6.6) yields dvi + v^da? - 0, dyi + Njdrf = 0. (3.3.1.1) Then, for a function /(az, £/), we have df = + (djfldy* = (3.3.1.2) where di is the ¿-operator (2.4.2.1). We deal with the fundamental tensor ghi: dgtn - (Sjghijdxl = (ghi.j + griF^ + ghTFij)dx?. For the covariant components vi = of v dot = + gijdv^ = faij* + grjF&v1 dxk. Thus, for the length V of we have dV2 = dtviK1) = ghi.jVhvlc№.
718 Matsumoto Consequently we obtain along PQR djdiV2 = d2(ffhiyvAvidi«’) = SkiShi^daX^v'diX3' - gri.tj(vrF^kd2Xh')vidix:i - 9hr-jvh(vr Fikd*xk)d,.x? + ghi-jV^dzdix? = (Skghi-.i - gn-.iFkk - Sferu-f?fe)«'lv’di®,d2®* + ghiijVhv*d2dix? = (ghi-j-'k + ghi^FJky^d^dax^ + g^v^dad-L^. Therefore, if we put (d2di - did2)V2 = -1 Ehijk^v^d^dsxi - dzaPdiz**), then we obtain (3.3.1.3) where T is the (h)h-torsion tensor of FT. Thus the “stretch and shrink” of the length of parallel vector field along a circuit is given by the tensor S. 3.3.2 Stretch Curvature Tensor It is obvious from (3.3.1.3) that if the Finsler connection FT is /¿-metrical, then the length of parallel vector field along a circuit is invariant, though its direction may be changed. Consequently, if we are concerned with the Cartan connection CT or the Chem-Rund connection CRT, then (3.3.1.3) is of no interest. On the other hand, the Berwald connection BT = (Cj^GpO) and the Hashiguchi connection HT » are not /¿-metrical and have the common (Gjki (?}) and hence the common S. Definition 3.3.2.1. We consider a Finsler space F71 with the Berwald connec¬ tion or the Hashiguchi connection. The tensor S given by (3.3.1.3) is called the stretch curvature tensor of F*. Then, from (2.5.5.6) it follows that the stretch curvature tensor S is written (3.3.2.1) where Hhijk = 9irHhjk- Equation (2.5.6.3) gives another expression of S : ^hijk — Rhijk 4" *Rihjk‘ (3.3.2.2)
Finsler Geometry in the 20th-Century 719 It is interesting to get the expression of S in the Cartan connection CT. From (2.3.2.2), using (2.S.6.2) and (2.5.2.S), we obtain ^hijk = Phik\j)‘ (3.3.2.3) If we pay attention to the relation between Fjk of CT and Gjk of BT, then (3.3.2.3) is rewritten in the form ^hijk = Phiktf)* (3.3.2 *4) This leads us to the more interesting expression of S. (2.5.5.8.d) and (2.5.5.7,a) give Rtyfik + Gtjk,i ~ &eik;j ~ Oj which implies yh^j.i.k + yhOij^i - yh.G&k.j = 0, and (2.5.5.10) leads to yhRtyt.k ~fyjk',i + = 0. Hence (3.3.2.4) yields ^hijk = yr-HJfc.fc.i » (3.3.2.5) Proposition 3.3.2.1. The stretch curvature tensor S is written as (3.3.2.1) in BF, (3.3.2.2) in BT, (3.3.2.3) in CT, and (3.3.2.5). Ref. The notion of the stretch curvature was introduced by L. Berwald [24], but it had been buried in oblivion, until C. Shibata [149] brought it before the public. Though the tensor S is certainly significant in a Finsler space equipped with nonmetrical connection, we have very interesting expression (3.3.2.5) of S from the standpoint of Proposition 2.5.6.1. Now we consider a two-dimensional Finsler space and find the expression of the stretch curvature tensor S. Since (3.1.3.6) gives Phijtk — (I.i.iA + f^^k^rnhTnimjy (3.3.2.3) implies ^hijk ~ ¿kTrij)* (3.3.2.6) From (3.1.3.14) and (3.3.2.6) we have Proposition 3.3.2.1. If the scalar curvature R of a two-dimensional Finsler space vanishes, then its stretch curvature tensor vanishes.
720 Matsumoto Next (3.1.1.13), (3.1.1.4) and (3.1.1.16) show L(R2gfi — 2Rg(R.t2 + eRI)mi — -2sRgItljmi‘ Thus we get L(R Vlsi ).i = -i Visi Äi,i»nv (3.3.2.7) This and (3.3.2.6) yield Theorem 3.3.2.1. In a two-dimensional Finsler space F2, Fi/lFl depends on a position alone, if and only if F2 has vanishing stretch curvature tensor. Ref Therefore the total curvature can be defined only in a two-dimensional Finsler space with £ = 0, as L. Berwald indicated [25], p. 65-66. See D. Bao and S.S. Chem [19], p. 238. From (3.1.3.14) it follows that l,i ,i = 0 is equivalent to sR;2 -hRZ = 0, which can be written on account of (3.1.2.2) as ^dlog|fl|\ k 96 )' (3.3.2.8) If the T-tensor vanishes, then (3.1.2.2) and (3.1.3.13) show dl/dQ = 0. Therefore we obtain Theorem 3.3.2.2. Let F2 be a two-dimensional Finsler space with vanishing stretch curvature tensor. (1) Its main scalar I is given by the scalar curvature R as (3.3.2.8) where 0 is the Landsberg angle. (2) If its T-tensor vanishes identically, then the scalar curvature R is written as R = exp(A0 + p) where A and p are functions of position. Remark: All two-dimensional Finsler spaces with zero T-tensor are given by Theorem 3.I.4.I. 3.3.3 Landsberg Space The present section we consider the hv-curvature tensor P2 of a Finsler space with the Cartan connection. Definition 3.3.3.1. A Finsler space F71 is called a Landsberg space, if its Tiv-curvature tensor P2 of the Cartan connection CT vanishes identically. Thus (2.5.2.S) shows that the (v)hv-torsion tensor P1 of a Landsberg space vanishes identically, and the inverse holds on account of (2.5.2-16).
Finsler Geometry in the 20th-Century 721 We have the following expression of P1 : (3.3.3.1) where 6r — dr — G^ds and (; ) is the h-covariant differentiation in BI\ ’ Next we have (2.5.5.9). On the other hand, (2.5.5.10) shows that a Landsberg space is characterized by yrG^ ~ 0* Further (2.5.S.2) shows G^k for a Landsberg space. Next (2.5.2.14) shows that Chij»k is symmetric for a Landsberg space and the inverse is true from Chij,0 = Chinjyr = 0. Therefore we have Theorem 3.3.3.1. A Finsler space is a Landsberg space, if and only if one of the following holds: (1) In the Cartan connection CT, (a) the (y)hv-torsion tensor P1 = 0 (b) = 0, (c) Chijik is symmetric. (2) In the Berwald connection BY, (a) BT is h-metrical, (b) Cijktf = 0, (c) the hv-curvature tensor G satisfies yrG^k — 0. (3) (a) CT = HT, (b) CRT = BP. Proposition 2.5.5.T and (2.5.5.T, b) show that G^ is symmetric in the subscripts and satisfies (2.5.5.10). Thus, in a Landsberg space the covariant Ghijk = 9iTGrhjk satisfies y Ghijk — y^Ghijk — if Ghijk — ykGhijk — 0. For a Landsberg space we deal with the /¿^-curvature tensors F and G of CRT and BT, respectively. Expressions (2.5.2.6,d) and (2.5.5\5) show
722 Matsumoto Theorem 3.3.3.2. A Finsler space is a Landsberg space, if and only if (1) F*k = in CRT, (2) Gfo = in BT. From (3.3.2.3) we have Proposition 3.3.3.1. In a Landsberg space the stretch curvature tensor S vanishes identically. Ref We first find the name “Landsberg space” in L. Berwald’s early papers [24], [27]. He payed attention to (2.5.S.9) and mentioned that a Finsler space is called a Landsberg space when Br is /i-metrical. Now, concerned with two-dimensional Landsberg spaces, we see that (3.1.3.11), or (3.1,3.16) shows Proposition 3.3.3.2. A two-dimensional Finsler space is a Landsberg space, if and only if the main scalar I satisfies = 0, that is, V*Z(c) = 0. We shall recall the notion of concurrent vector field. Then (b) of (2,5.2.19) gives immediately Proposition 3.3.3.3. If a Landsberg space admits a concurrent vector field, then it is reduced to a Riemannian space. According to Proposition 3.3.3.1, the condition “vanishing stretch curvature” is weaker than the condition “Landsberg space”. We shall discuss another weaker condition. It is obvious from (3.1.3.6) and (3.1.3.11) that every two- dimensional Finsler space satisfies Pijk = Fhijk = XhOijk — XiChjfri where A = A(x. y) is a scalar and Xh = X^x, y) a covariant tensor. Theorem 3.3.3.3. We consider a Finsler space F71 satisfying (1) Pijk = ACyifc, and (2) Phijk = Pfcikj- Then Fn is a Landsberg space, or has vanishing v-curvature tensor S2 of CT. Proof: (2.5.2.15) shows that (2) is equivalent to (2’) ShijktQ = 0. On the other hand, by differentiating h-covariantly (2.5,2.6, e) and substituting from (1), we get ShijkiQ ~ 2AShijTj. Thus, if S2 does not vanish, then F71 is a Landsberg space.
Finsler Geometry in the 20th-Century 723 Corollary 3.3.3.1. 1/ the hv-curvature tensor P2 of the Carton connection CT of a Finsler space Fn can be written in the form Pfrijk — XfoCijk ~~ then F71 is a Landsberg space or has vanishing v-curvature tensor S2 of CT. This follows because this condition leads to the symmetry Phijk = Phikj and yhPhijk — yKXh.Cijk> that is, Pijk = the second condition of The¬ orem 3.3.3.3. Ref. The theorem above was proved by M. Hashiguchi [45]. A Finsler space satisfying the condition (1) is called *P-Finsler by H. Izumi [68].. 3.4 Berwald Space 3.4.1 Characteristics of Berwald Space We have the important functions GP^y) which appear in the equations of geodesic (1.2.2.1) and are (2)p-homogeneous. From them we construct as (2.5.2.3), (2.5.5.3) and (2.5.S.7): &jk = G^-d^dkG*. From (2.5.5.2) and (1.2.2.4) we have Gjkt^y^y11 = Fjk(3>>y)y:’vk = = 7oo(a:>y)> where 7^(2:, y) are Christoffel symbols constructed from the fundamental tensor with respect to xi as (1.2.2.3). For a Riemannian space, we have We have the equations of geodesic as in BT and PT, and in CT and CRT. Definition 3.4.1.1. A Finsler space is called a Berwald space, if G^k are functions of position alone, that is, the Berwald connection BF is linear. Though G^k are not components of a tensor field, it is obvious that a Berwald space is characterized by the tensorial equations dh.Gjk — ^hjk “ 0 where
724 Matsumoto G^ = 0 means that GL(x, y) are homogeneous polynomials in of degree two. Consequently, Theorem 3.4.1.1. A Finsler space is a Berwald space, if and only if one of the following holds: (1) G'fay) are quadratic polynomials in y\ (2) the hv-curvature tensor G of the Berwald connection BV vanishes -Identic- ally. Thus (2) of Theorem 3.3.3.1 shows Proposition 3.4-1.1. If a Finsler space is a Berwald space, then it is a Lands¬ berg space. Now consider (2.3.5.5): Since for a Berwald space this gives F = 0 because of G = 0 and Pl = 0 from Proposition 3.4.1.1. That is, Fjk are functions of position alone from (2.5.2.T). Conversely, if F =* 0, then (2-5.4.3) shows P1 = 0 and the above implies <7 = 0. Therefore, Theorem 3.4.1.2. A Berwald space is characterized by either one of the fol- (1) Fjk are functions of position alone, that is, the h-connection of CT or CRT is linear, (2) the hv-curvature tensor F of CRT vanishes identically. We shall state one more characterization of a Berwald space. For a Berwald space we have F = P1 = 0, and hence (2.5.2.16) implies P2 = 0. Hence (2.S.4.2, b) gives =s 0. Conversely, VhC = 0 implies P2 = 0 from (2.5.2.14) and P1 — 0 from (2.5.2.5). Thus F = 0 from (2.5.4.2, b). Consequently we have Theorem 3.4.1.3. A Finsler space is a Berwald space, if and only if the C-tensor satisfied ^hC = 0 in GT. Thus a Landsberg space and a Berwald space are characterized by Landsberg: VaC(s) = 0, Berwald: VhC = 0. Ref. When L. Berwald [25] proposed the notion of a Berwald space, he called it “affinely connected space”. V. Wagner was the first to use “Berwald space”. See the footnote of p. 81 of H. Rund [145].
Finsler Geometry in the 20th-Century 725 To show a remarkable example of a Berwald space, we first pay attention to one of the commutation formulae (2.4.3.8): ijfe 'k'3 ¿rjk X-trUjk Hence the Finsler connection FT under consideration is such that the h and ^covariant differentiations commute with each other, if and only if Curvature Pfo - F*k - U^j + U^k = 0, (h)hv-torsion U*k = 0, (v)fa;-torsion Pjk = dkNj - Fgy = 0. Thus the conditions are written as t& = 0, F$k(=dkF&=Q, F^d^. From U = 0 it follows that the v-covariant differentiation (:) coincides with the partial differentiation (•) by p*. Further, for the deflection tensor P} = y^F^j - Nj = yhdhNj - Nj> Hence, if the spray connection of FT is (1) p-homogeneous, then D vanishes. Therefore, Theorem 3.4.1.4. The h and v-covariant differentiations with respect to a Finsler connection FT = (Fjfc, commute with each other if and only if t^ = o, ^fe = jjfc(«) = w Further, if Nj are (1) p-homogeneous, then the deflection tensor D vanishes. For such a connection FT we observe that = ~9ij№ = Hence = 0 implies g^ = 0. On the other hand, we have &k = SijtkÿV, if yifc (= Dk) = 0. Therefore, Theorem 3.4.1.5. If the h and v-covariant differentiations with respect to a Finsler connection FT commute with each other and FT is L-metrical, then FT is h-metrical. Further, if the spray connection of FT is (1) p-homogeneous, and FT is h-metricoL, then FT is L-metrical. Example 3.4.1.1. Let F* — (M, L(a, fty be a Finsler space with (a, /?)-metric, and R? « (Àf,a) the associated Riemannian space (§1.4.2). We denote by 7 =
726 Matsumoto ('/Jfci®)) Levi-Cività connection of Rn and by *7 = (7^,7^ 0) the induced Finsler connection. *7 satisfies all the conditions stated in Theorem 3.4.1.4 and 7j0 are (Inhomogeneous. Since 7 satisfies ct;» = 0. we have Thus, if bi(x) is parallel in Fn, then *7 if ¿-metrical. According to Theorem 3.4.1.5., *7 is ^-metrical and obviously satisfies (1), (2) and (3) of Definition 2.5.2.I. Con¬ sequently, 7^(2) must coincide with Fjk of CT and (1) of Theorem 3.4.1.2 leads to Theorem 3.4.1.6. Let Fn =- (M,£(a,/?)) be a Finsler space with (a,ff)-metric where /3 = bi(x)yt. If bi(x) is parallel in the associated Riemannian space R* — (M,a), then F* is a Berwald space. Ref M. Hashiguchi and Y. Ichijyo [50] Example 2. % being parallel" is, of course, a sufficient condition for the F” to be a Berwald space. See Defini¬ tion 6.3.2.1. Theorem 3.2.4.2 on a locally Minkowski space can be now expressed in other words: Theorem 3.4.1.7. A Finsler space is locally Minkowski, if and only if either one of the following three are satisfied: (1) it is a Berwald space with R2 — 0, (2) it is a Berwald space with-R1 — 0, (3) it is a Berwald space with H = 0, where R2 and H are the h-curvature tensors of CT and BT, respectively, and R1 the (v) h-torsion tensor of CT. Proof: (2) (2.5.2.S) leads to R1 = 0 from R2 = 0. Conversely, R1 = 0 gives R2 = K from (2.5.4.2) and H — 0 from (2.5.5.7,a), and = 0 leads to H — K from (2.5.5.5,a). (3) is obvious from Theorem 3.2.4.1. 3.4.2 Two-Dimensional Berwald Space According to (3.1.3.6) and Theorem 3.4.1.3 we have Proposition 3.4.2.1. A Finsler space of dimension two is a Berwald space, if and only if the main scalar I satisfies Iti = 0, that is, Ai = As = 0 in the notation of (3.1.3.2).
Finsier Geometry in the 20th-Century 727 Then, from (3,1,3,10) we have RI& = 0. If the scalar curvature R vanishes, then the space is locally Minkowski because of Theorem 3-2.4.2. If I# — 0? then Proposition 3.4.2.1 implies I = const, thus Theorem 3.4.2.1. All two-dimensional Berwald spaces are classified as follows: (1) 72 = 0, I const.. (2) fl = 0, I — const. (3) I — const. Berwald spaces belonging to (1) or (2) are locally Minkowski, while those belong¬ ing to (3) are not so. We have Theorem 3.1.4.1 which lists the Finsier spaces with € = Z(s). Con¬ sequently, two-dimensional Berwald spaces belonging to (2) and (3) are given as follows: Theorem 3.4.2.2. Two-dimensional Berwald spaces with the constant main scalar I are classified as follows: (1) e = l, I2 >4: L2=ßy(-y/ßy/\ t = (2) e = 1, J2 = 4: L2 = ß2 exp(fy/ß), (3) e = 1, I2 < 4 : L2 = (/J2 + y2) exp{(2//r) Arctan^/W, r=>/4 —J2, (4). £ = -1: L2 = r = VI2+ 4. where /3 and j are independent 1-forms in y*. Theorem 3.4.2,3. If a two-dimensional Finsier space F2 is a Landsberg space and has the vanishing T-tensor, then F2 is a Berwald space with the constant main scalar I. Proof: Proposition 3.3.3.2 shows that F2 has 14 = 0 and (3.1.3.13) shows ¿2 = 0. Then, applying (3.1.3.10, b) to I, we have <2 — 0- Thus the proof is complete. Remark: As the Theorem above, it is known that a Landsberg space is re¬ duced to a Berwald space, if it satisfies some additional conditions. See The¬ orems 4.2.3.3, 4.5.3.1, 5.1.3.2, fi.4.4.2, 6.2.3.2, 6.5,3.1 and 6.5.3.2, and Corol¬ lary 3.7.6,1. Those are called the Reduction Theorems of Landsberg space. As a consequence, we have an open problem: Find an explicit fundamental metric function of Landsberg space which is not of Berwald space. See M. Matsumoto [112].