TRANSLATIONS OF
akashi Sakai
American Mathematical Society

Selected Titles in This Series
149 Takashi Sakai, Riemannian geometry, 1996
148 Vladimir I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, 1996
147 S. G. Gindikin and L. R. Volevich, Mixed problem for partial differential equations with
quasihomogeneous principal part, 1996
146 L. Ya. Adrianova, Introduction to linear systems of differential equations, 1995
145 A. N. Andrianov and V. G. Zhuravlev, Modular forms and Hecke operators, 1995
144 O. V. Troshkin, Nontraditional methods in mathematical hydrodynamics, 1995
143 V. A. Malyshev and R. A. Minlos, Linear infinite-particle operators, 1995
142 N. V. Krylov, Introduction to the theory of diffusion processes, 1995
141 A. A. Davydov, Qualitative theory of control systems, 1994
140 Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic
systems, 1994
139 I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, 1994
138 Yu. P. Razmyslov, Identities of algebras and their representations, 1994
137 F. I. Karpelevich and A. Ya. Kreinin, Heavy traffic limits for multiphase queues, 1994
136 Masayoshi Miyanishi, Algebraic geometry, 1994
135 Masaru Takeuchi, Modern spherical functions, 1994
134 V. V. Prasolov, Problems and theorems in linear algebra, 1994
133 P. I. Naumkin and I. A. Shishmarev, Nonlinear nonlocal equations in the theory of waves, 1994
132 Hajime Urakawa, Calculus of variations and harmonic maps, 1993
131 V. V. Sharko, Functions on manifolds: Algebraic and topological aspects. 1993
130 V. V. Vershinin, Cobordisms and spectral sequences, 1993
129 Mitsuo Morimoto, An introduction to Sato's hyperfunctions, 1993
128 V. P. Orevkov, Complexity of proofs and their transformations in axiomatic theories, 1993
127 F. L. Zak, Tangents and secants of algebraic varieties, 1993
126 M. L. Agranovskii, Invariant function spaces on homogeneous manifolds of Lie groups and
applications, 1993
125 Masayoshi Nagata, Theory of commutative fields, 1993
124 Masahisa Adachi, Embeddings and immersions, 1993
123 M. A. Akivis and B. A. Rosenfeld, Elie Cartan (1869-1951), 1993
122 Zhang Guan-Hou, Theory of entire and meromorphic functions: Deficient and asymptotic values
and singular directions, 1993
121 LB. Fesenko and S. V. Vostokov, Local fields and their extensions: A constructive approach, 1993
120 Takeyuki Hida and Masuyuki Hitsuda, Gaussian processes, 1993
119 M. V. Karasev and V. P. Maslov, Nonlinear Poisson brackets. Geometry and quantization, 1993
118 Kenkichi Iwasawa, Algebraic functions, 1993
117 Boris Zilber, Uncountably categorical theories, 1993
116 G. M. Fel'dman, Arithmetic of probability distributions, and characterization problems on abelian
groups, 1993
115 Nikolai V. Ivanov, Subgroups of Teichmuller modular groups, 1992
114 Seizo Ito, Diffusion equations, 1992
113 Michail Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations, 1992
112 S. A. Lomov, Introduction to the general theory of singular perturbations, 1992
111 Simon Gindikin, Tube domains and the Cauchy problem, 1992
110 B. V. Shabat, Introduction to complex analysis Part II. Functions of several variables, 1992
109 Isao Miyadera, Nonlinear semigroups, 1992
108 Takeo Yokonuma, Tensor spaces and exterior algebra, 1992
(Continued in the back of this publication)

TRANSLATIONS OF
MATHEMATICAL
MONOGRAPHS
VOLUME 149
Takashi Sakai
Riemannian Geometry
гШП^/я American Mathematical Society
^B^

Editorial Board
Shoshichi Kobayashi
Katsumi Nomizu (Chair)
RlMAN KIKAGAKU
(Riemannian Geometry)
by Takashi Sakai
Copyright © 1992 by Shokabo Publishing Co., Ltd.
Originally published in Japanese by Shokabo Publishing Co., Ltd., Tokyo in 1992.
Translated from the Japanese by Takashi Sakai
1991 Mathematics Subject Classification. Primary 53-01,
53C20, 53C21, 53C22, 53C23, 53C35, 58G25, 35P15
Abstract. The aim of this textbook is to provide to advanced undergraduate and graduate students an
introduction to modern Riemannian geometry that could also serve as a reference. The book begins
with an explanation of the fundamental notions of Riemannian geometry. Special emphasis is placed on
understandability and readability, to guide students who are new to this area. The remaining chapters
deal with various topics in Riemannian geometry, with the main focus on comparison methods and their
applications.
Library of Congress Cataloging-in-Publication Data
Sakai, Τ (Takashi), 1941-
[Rlman kikagaku. English]
Riemannian geometry / Takashi Sakai; translated by Takashi Sakai.
p. cm.—(Translations of mathematical monographs; v. 149)
Includes bibliographical references and index.
ISBN 0-8218-0284-4 (alk. paper)
1. Geometry, Riemannian. I. Title. II. Series.
QA649.S2513 1996
516.3'73—dc20 96-6475
CIP
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them,
are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research.
Permission is granted to quote brief passages from this publication in reviews, provided the customary
acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society. Requests for
such permission should be addressed to the Assistant to the Publisher, American Mathematical Society,
P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-
permissionOams. org.
© Copyright 1996 by the American Mathematical Society. All rights reserved.
Reprinted with corrections 1997.
Translation authorized by the Shokabo Publishing Co., Ltd.
The American Mathematical Society retains all rights
except those granted to the United States Government.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
10 9 8 7 6 5 4 3 2 01 00 99 98 97

Contents
Preface to the English Edition ix
Preface ♦ xi
Chapter I. Preliminaries from Manifolds 1
1. Vector Spaces 1
2. Manifolds 5
3. Vector Bundles and Linear Connection 15
Problems for Chapter I 19
Notes on the References 20
Chapter II. Fundamental Concepts in Riemannian Geometry 23
1. Riemannian Metric 23
2. Geodesies 32
3. Curvature 40
4. Prom the Point of View of the Tangent Bundle 53
5. Riemannian Measure 61
6. Riemannian Submersion and Complex Projective Space 74
Problems for Chapter II 77
Notes on the References 80
Chapter III. Global Concepts in Riemannian Geometry 83
1. Complete Riemannian Manifolds 83
2. Variation Formulas and Jacobi Fields 87
3. Approximation by Finite Dimensional Manifolds
and the Index Theorem 97
4. Cut Locus 102
5. Ambrose's Theorem 112
6. Isometry Group and Holonomy Group 117
Problems for Chapter III 130
Notes on the References 132
Chapter IV. Comparison Theorems and Applications 135
1. Spaces of Constant Curvature 135
2. Comparison Theorems for Jacobi Fields 143
3. Applications of Comparison Theorems 154
4. Toponogov's Comparison Theorem 161
5. Convexity 168
6. Symmetric Spaces 175
Problems for Chapter IV 189

viii CONTENTS
Notes on the References 190
Chapter V. Curvature and Topology
of Riemannian Manifolds 193
1. Curvature and Fundamental Group 193
2. Compact Manifolds of Positive Curvature 201
3. Open Manifolds of Nonnegative Curvature 211
4. Manifolds of Nonpositive Curvature 221
Problems for Chapter V 237
Notes on the References 239
Chapter VI. Isoperimetric Inequality and Spectral Geometry 241
1. The Isoperimetric Inequality 241
2. The Berger Isoembolic Inequality 252
3. Eigenvalue Problem for the Laplacian 262
4. Curvature and Spectrum 275
5. Heat Kernel and Spectral Geometry 282
Problems for Chapter VI 286
Notes on the References 287
Appendices 289
1. Irreducible Decomposition of the Curvature Tensor 289
2. Homogeneous Spaces 291
3. Injectivity Radius Estimate and Closed Geodesies 294
4. Maximum Principle 300
5. Differential Forms 301
6. Gromov's Convergence Theorem
and Collapsing of Riemannian Manifolds 304
Hints and Solutions to Exercises and Problems 323
Chapter I 323
Problems for Chapter I 323
Chapter II 324
Problems for Chapter II 326
Chapter III 328
Problems for Chapter III 329
Chapter IV 331
Problems for Chapter IV 332
Chapter V 333
Problems for Chapter V 333
Chapter VI 335
Problems for Chapter VI 336
Bibliography 339
Index

Preface to the English Edition
This volume is an English translation of my textbook on Riemannian geometry
originally written in Japanese and published in 1992 by Shokabo, Tokyo. I wrote
the Japanese edition mainly because at that time there were no textbooks written
in Japanese that introduced modern Riemannian geometry to advanced
undergraduate and graduate students and that could also serve as a reference. On the other
hand, there are many textbooks and monographs on Riemannian geometry written
in Western languages at various levels and treating a variety of topics. I have
consulted them, and I have been influenced especially by the books by M. Berger and
A. Besse, J. Cheeger and D. G. Ebin, and W. Klingenberg.
Now let me mention the points on which I put emphasis in the present volume.
(1) After reviewing fundamentals on differentiable manifolds in Chapter I, I try
to explain the fundamental notions and results of Riemannian geometry in Chapters
II and III with particular emphasis placed on understandability and readability,
since, in my teaching experience, many students find it difficult to grasp Riemannian
geometry on their first try.
(2) In the remaining chapters, among various topics in Riemannian geometry I
am mainly concerned with the comparison methods and their applications. I take
an approach using Jacobi fields to comparison methods in Chapter IV, and give
their applications to the relation between the curvature and topology, geometric
inequalities, and spectral geometry in Chapters V and VI.
In principle, I faithfully translated the Japanese edition, except for
correcting small errors and adding a few comments on further developments. However,
Appendix 6 on Gromov's convergence theorem and the collapsing of Riemannian
manifolds has been expanded and revised considerably. I also added more
references and notes on the references to each chapter, although they are still far from
being complete.
I would like to express my gratitude to K. Grove, H. Karcher, A. Katsuda, W.
Klingenberg, R. Porter, and W. Tuschmann for useful suggestions and advice. I
also thank K. Shimakawa for helping me with the A^S-WT^i typesetting.
Takashi Sakai
May, 1995

Preface
In this volume we give an exposition of the fundamental concepts and results of
Riemannian geometry, and explain especially the ideas called comparison methods
and their applications, assuming some fundamentals on difFerentiable manifolds.
First we briefly mention the birth of Riemannian geometry. In his "Elements"
(Stoicheia), Euclid (Eukleides) systematically arranged many facts of elementary
geometry that had long been known, taking an axiomatic viewpoint for the first
time. Namely, defining the notions of point, line, plane, angle, etc., and describing
some of the most fundamental relationships among them as the axioms (or
postulates), he systematically deduced, through strict logic, other marvelous geometric
facts (propositions, theorems) based on the axioms. From an axiomatic viewpoint
it had been suspected ever since the age of Euclid that the fifth postulate, which
is equivalent to the statement that for a given line / and a point ρ in the plane
there exists a unique line parallel to / through p, could be proven from the other
axioms. After various attempts over more than 2,000 years, some people began to
suspect that a new geometry might be developed by the denying the fifth postulate
and leaving the remaining axioms as they stand. Janos Bolyai (1832) and N. I.
Lobachevsky (1830) were the first who published their new geometry. Gauss
himself also reached the same conclusion, but did not publish since he feared that false
controversies might be caused by misunderstandings.
The discovery of non-Euclidean geometry brought about serious examinations
of the foundations of geometry and the concept of space. For instance, Gauss
measured the inner angles of a triangle whose vertices where the summits of three
high mountains far apart in Germany, and tried to judge which geometry reflects
the real world.
Under these circumstances G. F. B. Riemann proposed in 1854 an epoch-making
view in his Habilitationsschrift, "Uber die Hypothesen, welche der Geometrie grund-
liegen", submitted to Gottingen University. Namely, instead of taking an axiomatic
viewpoint, he proposed to consider more general "Mannigfaltigkeiten"(manifolds),
which are locally homeomorphic to Euclidean space of a fixed dimension and "spread
out" manifold. Then he discussed how to measure the length of curves, the
distance between two points, the angle between vectors, etc., on a given manifold,
and introduced the notion of a Riemannian metric inspired by the surface theory
of Gauss. Further, Riemann defined the notion of the (sectional) curvature of a
Riemannian metric in terms of the Gauss curvature of a surface. Then he noted
that the sectional curvatue of a Riemannian metric is constant if and only if
figures are freely movable in a manifold without expansion or contraction. He also
pointed out that, for manifolds of constant curvature k, the flat case (i.e., к = 0 )
describes Euclidean geometry, and the negative constant curvature case describes
xi

xii PREFACE
the non-Euclidean geomtery of Bolyai and Lobachevsky. Manifolds of positive
constant curvature correspond to the elliptic non-Euclidean geometry of Riemann. It
was reported that old Gauss, who attended Riemann's lecture, was deeply touched.
Thus a completely new and huge field of geometry opened. Riemann's idea was
first developed by G. Ricci, T. Levi-Civita, and other people as an absolute
differential calculus for tensors, which seemed rather formal. However, such tensor calculus
turned out to provide a needed mathematical tool when Einstein established his
general theory of relativity with a gravitation field in 1916, and Riemannian
geometry was highlighted.
Subsequently Hermann Weyl and Elie Cartan took a more general view of the
connection, and unified Riemann's idea and F. Klein's program interpreting
geometries in terms of transformation groups. S. Cohn-Vossen, W. Blaschke, and others
studied the global properties relating the metric invariants to the topology of the
surface. H. Poincare, G. D. BirkhofF, M. Morse, J. Hadamard, E. Hopf, and
others worked on various properties of geodesies from different standpoints. H. Hopf
studied the global properties of spaces of constant curvature, and E. Cartan
originated and made an extensive study of the symmetric spaces, a remarkable class
of Riemannian manifolds. Through all this essential work Riemannian geometry
was linked to various fields of mathematics (e.g., dynamical systems, calculus of
variations, topology), and it was recognized that the relation between local
properties (e.g., curvature) determined by the metrics and global properties related to
the whole structure of manifolds are important objects of the investigation. Also
the notion of differentiable manifolds was defined rigorously in the terminology of
modern mathematics by H. Weyl and H. Whitney, and the fundamental concepts
of manifolds and Riemannian geometry were consolidated. For instance, H. Hopf
and W. Rinow defined the notion of completeness of a Riemannian metric, through
which the global notions were established.
In the present book, after reviewing fundamentals on differentiable manifolds in
Chapter I, we treat with care some fundamental concepts and results of Riemannian
geometry in Chapters II and III. Especially, we explain the notions of geodesic,
Jacobi fields, and curvature together with many examples in Chapter II, and some
global concepts and results of Riemannian geometry, which are mainly related to
geometry of geodesies, in Chapter III. I hope that the reader may grasp Riemannian
geometry in outline through Chapters II and III.
Modern Riemannian geometry has been developed in many branches from
various viewpoints mainly as geometry on manifolds, and it is impossible to cover all
topics in a textbook. In the present volume we are mainly concerned with the
comparison methods and their applications in Chapters IV, V, and VI. A complete
simply connected Riemannian manifold of positive constant curvature δ is
isometric to the sphere of radius l/y/δ. H. Hopf conjectured that a complete simply
connected Riemannian manifold whose sectional curvature is not necessarily equal
to a positive constant but remains close to a positive constant is still topologically a
sphere. Then Η. Ε. Rauch established this fact in his epoch-making paper in 1951.
M. Berger and W. Klingenberg improved and developed Rauch's idea, and got the
best possible sphere theorem for the case where the ratio of the minimal and the
maximal value of the sectional curvature is greater than 1/4. Through their work
and work of D. Gromoll, J. Cheeger, E. Ruh, K. Shiohama, P. Eberlein, K. Grove,

PREFACE
Xlll
Η. Karcher, and other geometers, great progress has been made in studying the
relation between metrical invariants and global properties of Riemannian manifolds. In
particular, comparison methods, which compare a given Riemannian manifold with
a standard Riemannian manifold of constant curvature in terms of some geometric
invariants, were developed. In Chapter IV we state these comparison methods in a
unified manner in terms of Jacobi fields. Then we apply these methods to the
relation between curvature and topology of Riemannian manifolds in Chapter V, and
to the inequalities among geometric invariants and spectral geometry in Chapter
VI. On the other hand, since the fields treated in Chapters V and VI are still in
rapid progress, we cannot state in detail the front line of current research in this
textbook. However, in Appendix 6 we mention some of M. Gromov's ideas, which
have been one of the main sources promoting the recent development of Riemannian
geometry, and have inspired many excellent young geometers.
On the other hand, we cannot state in detail the applications of dynamical
systems, partial differential equations, etc. to Riemannian geometry, e.g., minimal
submanifold, harmonic map, heat flow, etc. For these topics the reader may consult,
e.g., Hajime Urakawa's book [Ur-2].
I would like to express my gratitude to Professor S. Murakami, who invited me
to write this book, and to Mr. S. Hosoki of Shokabo Publishing Company for his
kind cooperation.
In concluding the preface, I would like to remember the late Professor Shigeo
Sasaki, under whose guidance I began to take an interest in Riemannian geometry.
Professor Sasaki was one of the pioneers of modern differential geometry in Japan,
and emphasized the importance of studying global problems that are also related to
other fields of mathematics. He himself did much pioneering research on
Riemannian geometry. He passed away in the summer of 1987, when I began to prepare the
present book. During the writing I often wished that he were still alive to advise
me, and often recalled his enthusiasm for mathematics and his great personality.
Takashi Sakai
April, 1992

CHAPTER I
Preliminaries from Manifolds
Riemannian geometry is usually developed on smooth manifolds. In this
chapter we review some fundamental notions on manifolds. Since there are many books
on manifolds, for proofs of many results in this chapter we refer the reader to the
references cited at the end of this book. Those readers who are familiar with the
fundamental notions on manifolds may start with Chapter II and consult Chapter I
as needed. However, since here we systematically give some fundamental concepts
and results on manifolds that will be used in this book, it will be convenient to
read through this chapter.
1. Vector Spaces
1.1. We mainly deal in the following with finite-dimensional real vector spaces.
Let V be an m-dimensional real vector space. If we choose a basis {ej™^, V
is isomorphic to the Euclidean vector space Rm :=1{(x1,... ,xm):xl £ R} by
assigning its components to each element of V. Now we review briefly some methods
which produce new vector spaces out of given vector spaces. Fundamental concepts
of linear algebra, such as linear map, subspace, quotient space, direct sum, etc., are
assumed to be known. We denote by dim V the dimension of the vector space V.
(I) (dual space). V* := {a : V —» R; a is a linear map} has the structure of
an m-dimensional vector space and is called the dual space of V. For a basis {e*}
of V we define el £ V* (i = 1,... ,ra) by el(ej) := 6{j (6ц — l,6ij = 0 for г ф j).
Then {e*}·^ forms a basis of V* which is called the dual basis of {е*}^. We have
a natural isomorphism from V onto (V*)*, if we assign to every ν £ V the element
of {V*)* defined as v(w*) := w*(v), w* £ V\
(II) (tensor product). Let V and W be vector spaces of dimension m and n,
respectively. Then the space Нот(У, W) := {φ : V —» W; φ is a linear map} has
the structure of a vector space of dimension mn. In fact, if we take bases {e^}
and {fj} of V and W, respectively, and define φ^ £ Нот(У, W) (1 < г < т, 1 <
j < n) by ^ij(efc) = uikfj, then {φ^} forms a basis of Нот(У, W). Note that
Нот(У, W) is isomorphic to the vector space of real η χ m matrices in this way.
Hom(V*, W), also denoted by V 0 W, is called the tensor product of V and
W. For υ £ V, w £ W we define г>®ги £ V®W by г>®ги(г>*) := v*(v)w. Then any
element of V 0 W may be expressed as a linear combination of elements of the form
v<g>w, and, in fact, {ei0/j}i<i<m,i<j<n forms a basis of V®W. Note that У0И^ is
isomorphic to the vector space {φ : V* χ W* —> Д; φ is a bilinear map} by assigning
to^0^ the bilinear map: (v*,w*) £ V* xF н-> w*(v®w(v*)) = v*(v)w*(w) £ Д.
Further, we obviously have Hom(V, W) = Г0Ж, У0Д ^ V, where " ^ " denotes
an isomorphism of vector spaces. We also note that linear maps / : V —> V\ and
The symbol " := " means that its left-hand side is defined by the right-hand side.

2
I. PRELIMINARIES FROM MANIFOLDS
g : W —» W\ determine a linear map / ® <7 : V ®V\ —» VK ® VKi defined by
(/ ® g){v ® ги) := /(г>) ® р(гу).
(Ill) (tensor space). For a vector space У we define the tensor space of type
(r,s) of V, which is denoted by
r;(v) = y®···®^®^*®···®^,
r times s times
as the vector space
{φ : V* χ · "ХГхУх·· · χ У —» Я; <£ is a multilinear (i.e., linear with
r times s times
respect to each variable) map}.
Its elements are called tensors of type (r, s). Also we set Tq(V):= R. If X{ £ V (1 <
i <r), y* eV* (1 < j < s) are given, then we get an (r, s)-tensor by the following
formula:
χι ® · · · ® xr ® j/J ® · · · ® y*(x*i. ■. ,^;,yb ... ,ys) := Пх*(^г)у^(%·)■
Then we easily see that {eix ® · · · ® eir ® ejl ® · · · ® ejs} forms a basis of Tsr(V),
and dimTJ(y) = rar+s. Thus t £ Tsr(V) may be expressed as
t = У^ *V""Vei, (8> •••(8>eir. ® ел ® · · · ® eJe
in terms of the components. In the present book we shall follow Einstein's
convention that we omit the summation symbol Σ when the same indices (for instance
i\, Jb etc, in the above) appear in pairs, one upstairs and the other downstairs.
For instance, the above equation is written as
t = t\l'-\reu ®--·®βν ®ejl ® --®ejs.
We note that we have canonical isomorphisms ТЦУ)* = Т;(У*) and Т;+;,'(У)
^ Т;(У)®Т;/(У). Then T(V) := 0r,s>o7J(^) carries the structure of an algebra
relative to " ® ". Further, for T^(V) and fixed 1 < к < r, 1 < I < s we have a
linear map С = Cf : Trs(V) -> ΐχΐχ1^), called the contraction, which is defined
as2
Czfc(xi ® · · · ® xr ® yl ® · · · ® У*)
:= y^{xk)x\ ® ··· ® ifc ® ··· ® жг ® y* ® -"yf ® ··· ® y*.
Following Einstein's convention, contraction may be written in terms of the
components as (Cf (£))**'\\'*j~_\ = ^ι· •^•••is-i' wnere upstairs (resp., downstairs) m
appears in the A;-th (resp., l-th) position.
Now let A : V —» W be a linear isomorphism. Then the transpose linear map
A* : W* —» V* defined as А*(ги*)(г>) := ги*(А(г>)) is also a linear isomorphism. Л
and A* induce a linear isomorphism
A®···® A® A*"1 ® --·® A*"1 \V ® ---® У®У* ® --·® У*
-ν / > ν ' > ν '
r times s times r times s times
-> W® •••®W®W*® •••®W*,
2In (1.1) х^, etc. means that the term ж^, etc. should be omitted in the expression.

1. VECTOR SPACES
3
which preserves type (i.e., maps Tl(V) to T£(W)) and commutes with
contractions. Thus a linear isomorphism A : V —» W may be extended to an algebra
isomorphism A : T(V) —» T(W) between tensor algebras. Conversely, any such
tensor algebra isomorphism A : T(V) —» T(W), that preserves type and commutes
with contractions is induced from a linear isomorphism from V onto W. In fact,
A := A \ T£(V)3 is a linear isomorphism from У = T£(V) onto Ж = T$(W).
Setting В := Л | T^V), we have В = A*~x because (Bv*)(Av) = C(Av <g> Bv*) =
A{C{v®v*)) = v*(v) = (j4*-V)(At;) for any υ G V, v* G V\ Note that
A | Т0°(У) : R -> Я is the identity map.
Next let Z) : T(V) —» T(V) be a linear map which preserves type and commutes
with contractions. D is called a derivation of T( V) if Z) satisfies the Leibniz formula
(1.2) D(t®s) = Dt®s + t®Ds.
Again note that such a derivation may be induced from a linear map A : V —» V,
where L> | 7?(V) = -A* and L> | 7#(V) = 0. The set of all derivations of V
obviously has a vector space structure. Moreover, it is a Lie algebra if we define
the bracket operation by [D, D'} := DoD' -D'oD for derivations D, D' (see (2.8)
and (2.9)).
(IV) (exterior algebra). We call the vector space
Ak(V) := {a : Vx--xV —> R; a is a skew-symmetric fc-linear map}
к times
the k-th exterior power oiV* and its elements k-forms. Here α is said to be skew-
symmetric if for any permutation σ of {1,... , k} we have α(χσ(ΐ),... ,xa(k)) =
sgnσ · ct(x\,... , ж*.), where sgn σ denotes the sign of a permutation σ. For instance,
we define
(1.3) x\ A · · · Ax*k(xi,... ,χ^) := det(x*(xj)) for ж*,... ,x£.
Then we easily check that x\ A · · · Λ x*k G Afc(V) and that я*^) Λ · · · Λ £*(fc) —
sgntf · xj Λ · · · Λ x*k. Then {eh A · · · Λ eik; i\ < · · · < ik} forms a basis of Afc(V),
and dimAfc(V) = (™). In particular, we have A°(V) = Д, A1^) = V\ Ak{V) =
{0} (k > m). Further we define for α G Ak(V) and /3 G A'(V) their exterior product
ctAPeAk+l(V)hy
аЛ/3(хь... ,Zfc+z)
:= ^[|jX](SgnCr)a(^(l),··· ,ЗД№а(Н1)>'·· ,^a(fc+i))·
σ
Note that αΛ^ = (-1)ы/?Ла and A*(V) := (B™=0Ak{V) has the structure of
an algebra with respect to " Л ".
Now in the same manner we may construct
Ak(V) := {ξ : V* χ · · · x V\ —» Я;£ is a skew-symmetric fc-linear map}.
к times
Then {e^ Л · · · Л e*fc;zi < · · · < ik} forms a basis of Ak(V), and we may consider
the exterior product ξ Α η G Ak+i(V) of ξ G Ak(V) and η G A/(V) as above. Note
that for / G Hom(V) we may define /* G Hom(Afc(V)) by /*(χι Λ · · · Л хк) :=
f(xi)A-..Af(xk).
3A | T^V) means the restriction of A to T£(V).

4
I. PRELIMINARIES FROM MANIFOLDS
1.2. Let V be an m-dimensional real vector space. An inner product g on V
is defined as a map g :V xV —> R which satisfies
(I. 1) g is a bilinear map;
(I. 2) g(x, y) = g(y,x), x,yeV\
(I. 3) g(x, x) > 0 for all χ G V, where equality holds if and only if χ = 0.
We also denote g{x,y) by (x,y). For instance, R171 carries the canonical inner
product go defined by go((xl, · · · ,#m), (y1, · · · ,УШ)) := ΣΤ=\ χ1νι- Now once an
inner product is given on V we may define the norm ||x|| of a; G V by y/(x,x).
Then from the Cauchy-Schwarz inequality
(1-5) \(х,У)\< \\х\\Ы\
(equality holds if and only if χ and у are linearly dependent), we may define the
angle Z(z,y) (0 < Z(z,y) < π) of i,y(/0)EV by
cosZ(x,y) = (ж/||ж||, y/||y||>.
Now a basis {е*}·™^ is called an orthonormal basis if (e*, ej) = 6{j (1 <i,j< m). In
the following we write simply o.n.b. for orthonormal basis. In this manner we may
define the concepts about measure in terms of the inner product. For instance, the
r-dimensional volume of the parallelotope P(v\,... , vr) := {ΣΓ=ι U Щ', 0 < ti < 1}
spanned by г>1, ... , vr G V (r <m = dim V) is given by y/det((vi,Vj)).
A linear map / : V —> V is called an orthogonal transformation (or linear
isometry) if the equality
</(*),/(»)> = <*,!/> (i,»eV)
holds, and the set of all orthogonal transformations of V forms a group O(V). In
particular, the orthogonal transformation group of (Ят,<7о) is denoted by O(m).
Next in terms of a given inner product we get a linear isomorphism b : V —> У *
defined by Ь(г>)(ги) := (г>,ги). Then we may define the inner product on V* so that
b : V —> V* is a linear isometry. We easily see that if {e^} is an o.n.b. of V then its
dual basis {ег} forms an o.n.b. of V*.
Exercise 1. Set gij = (e{,ej) for a basis {ei} of V. Then show that b(x) =
gijX^e1 for χ = хге{, where we follow Einstein's convention.
We may also define the inner products on TJ(V) and on Ak(V) and Ak(V) from
an inner product on V so that {e^ 0 · · · 0 eir 0 ejl (g> · · · 0 ejs}, and {ег1 Л · · · Л ег*;
ζι < ... < z'fc}, {eix Л · · · Л e{k} are o.n.b.'s, respectively, where {e^} is an o.n.b.
of V. For instance, we have {x\ 0 · · · 0 xk, y\ 0 · · · 0 yk) = Πί=ι(χή2/ί)· Let
v\,... ,vr G V be linearly independent and {ej}[=1 an o.n.b. of the r-dimensional
subspace (v\,... , г>г)д spanned by v\,... , vr. Writing V{ = a\ej, we get
v\ Л · · · Л vr = a\Jl · · · ar3rejl Л · · · Л ejr
= < sgn I . .1 a\jl · · · arjr > e\ Л · · · Л ег
and consequently
IK Л · · · Л vr|| = |det(oi)| = y/det((vi,Vj))
is equal to the volume of the parallelotope spanned by г^,... , vr.

2. MANIFOLDS
5
1.3. We may also consider various geometric structures on a vector space V
besides the inner product. Let ω : V χ V —> Й be a skew-symmetric bilinear
map, namely, a 2-form on V. a; is said to be nondegenerate if its null space Νω :=
{χ G V',u>(x,y) = 0 for any у G V} consists only of the 0-vector, or equivalently
det(o;jj) φ 0 if we express ω as ω = о;^ег Л e·7,ω^ = —uuij.
A nondegenerate 2-form ω on V is called a symplectic form, and V is called a
symplectic vector space.
Exercise 2. Show that symplectic vector spaces are of even-dimension.
Further show that we may choose a basis {ej,en+j}i<j<n of V so that u(ei,ej) =
o;(en+i,en+J) = 0 and a;(eben+j) = 6{j (1 < z,j < n).
Now a subspace И^ of a symplectic vector space V is said to be isotropic if
a; | W x W = 0. For instance, 1-dimensional subspaces are isotropic, and the
dimension of an isotropic subspace is less than or equal to η := dim V/2. To
see this we introduce an inner product on V and define a linear transformation
/ : V —» V by (I(x),y) = uj(x,y). Then / is a linear isomorphism because ω is
nondegenerate. Now for an isotropic subspace W we see that I(W) is orthogonal
to W, and we get 2dimH/ = dimW + dimI(W) < dimV. In particular, we call a
maximal isotropic subspace, which is of dimension n, a Lagrangian subspace.
Now note that Cn := {(z\,... , zn)\ Z{ = X{ + \T^\yi £ С} (or generally a
complex vector space of complex dimension n) may be considered as a real vector space
isomorphic to R2n = {(ж1,... ,xn,yl,... ,2/n)}. We define a linear isomorphism
J : R2n -> Д2п by Jiz1,... , zn) := χ/^φ1,... , zn). Note that we have a matrix
representation
J =
0 -£„
^n 0
where 25n denotes the n-th unit matrix. J is in fact an orthogonal transformation
and satisfies J2 = —E2n- Then u(u,v) := (J(u),v) (u, ν G β2η) defines a
symplectic form on R2n. We easily see that Rn := {(x1,... , xn, 0,... , 0); xl G Д} is a
Lagrangian subspace. Moreover, for any φ G J7(n) := {φ G 0(2n); φο J = J ο φ},
ip(Rn) gives a Lagrangian subspace.
Exercise 3. Verify the above fact. Show that, conversely, any Lagrangian
subspace of Cn = R2n may be written in this form.
2. Manifolds
2.1. Let Μ be a HausdorfF topological space. A pair (υ,φ) of an open set U
of Μ and a homeomorphism φ : U —> Дт from /7 onto an open subset of R171
is called a (local) chart and /7 is called a coordinate neighborhood. If we have a
family Д := {{υα,φα)}α£Α of charts in Μ with UaG^ ^a = ^' ^en we sa^ ^^а^
Μ is an m-dimensional topological manifold with an atlas A. Roughly speaking,
a chart (U,<p) gives a coordinate system or a map on /7, and a manifold Μ may
be described by an atlas consisting of such maps as the globe. Thus topological
manifolds are locally homeomorphic to Euclidean space of fixed dimension, and we
want to apply calculus of several variables, which is a powerful tool in Euclidean
space. However we should note that coordinates depend on the choice of charts.
We say that an atlas A = {(UQ, φα)}αβΛ is of class C°° (or just C°°, or smooth) if
the following holds:

6
I. PRELIMINARIES FROM MANIFOLDS
(2.1) Whenever UaC\Up φ φ, coordinate transformations ψβθφαι : φα{υαΓ\υβ) —»
ψβ{υα^υβ) are C°° maps between open subsets of Rm.
Since φ Q ο φ~l is the inverse of φ β ο φ ~ι, ψβ ο φ~ι is a difFeomorphism and
its Jacobian matrix ί)(<£α ο φ β1) is of rank m everywhere. Let иг (г = 1,... , га)
denote the coordinates in Дт. For a chart (υα,φα) we set ж^ := иг o^Q (г =
1,... ,ra), which are called local coordinates. A topological manifold Μ with a C°°
atlas is said to be a C°° manifold. However, note that there is a large choice of
atlas on a C°°manifold M. We say that a chart (/7, </?) is compatible with a C°°
atlas Λ if <£ ο φα~ι and φαο φ~ι are C°° maps whenever U C\UQ ф Ф- Then all
charts compatible with Л form a maximal atlas containing Д, and their coordinate
neighborhoods form a base for the topology of M.
Now let / : Μ —» R be a real-valued function on a C°° manifold M. / is said
to be of class C°° at ρ £ Μ, if / ο <^α_1 : ipa(Ua) —* Я is of class C°° at φα(ρ),
where (/7α,<^α) is a chart around ρ £ Ζ7α. Note that by (2.1) this definition does
not depend on the choice of charts around p. We denote by T{V) the set of all real-
valued functions defined on an open subset V С М and of class C°° everywhere.
P(V) carries the structure of an algebra with respect to the usual addition and
multiplication of functions. We also denote by F(p) the family of C°° functions
defined on neighborhoods of p. Next a continuous map Φ : Μ —» N between C°°
manifolds Μ and N is called a C°° map if / ο Φ £ Τ(Μ) whenever / £ Τ(Ν). If
a C°° map Φ : Μ —> N is bijective and its inverse Φ-1 : TV —> Μ is again C°°,
we say that Φ is a diffeomorphism and Μ is diffeomorphic to N. In the following,
manifolds are assumed to be of class C°° and connected, and to satisfy the second
countability axiom unless otherwise stated. Such manifolds are paracompact and
admit partitions of unity, which will be given in the following two forms:
(2.2) For an open covering {V^}^gb of Μ we may choose {ρβ)β^Β С F(M) which
satisfies the following:4
(i) suppp/з С V/з and {8\ιρρρβ}β£Β is locally finite. Namely, for any ρ £ Μ
there exists a neighborhood W of ρ such that there are only finite many /?'s
with W Π suppp/з ф ф.
(ϋ) Ρ β > 0 and ΣββΒ Ρβ = ^ (^0Γ Ρ ^ Μ note that X^G# Ρβ(ρ)ls m fact a finite
sum because of (i)).
We call {ρβ}β^Β a partition of unity subordinate to {V^}^G^.
(2.3) For an open covering {^^^ofMwe may choose at most countably many
functions pi £ Т(М) (г = 1,2,...) which satisfy the following:
(i) For each г, suppp* is contained in some λίβ and compact. Further, {suppp^}
is locally finite (this is different from (2.2), where suppp^ is compact),
(ii) Α>0βη<ΐΣ~ιΡ< = 1·
2.2. Recall that smooth curves and smooth surfaces in Euclidean space may
be approximated at every point by tangent lines and tangent planes, respectively,
which are linear objects. To every point ρ of a C°° manifold Μ of dimension m,
we may also assign an m-dimensional vector space TPM, called the tangent space
to Μ at p.
4suppp/3 := closure of {p G Μ;ρρ(ρ) φ 0}.

2. MANIFOLDS
7
Let (α, 6) be an open interval containing 0. A C°° map с : (α, 6) —» Μ with
c(0) = ρ is called a (C°°) curve through p. We want to define the tangent space to
Μ at ρ as the space of "tangent vectors c(0)" to a curve с through p. Although we
cannot define c(0) as in Euclidean spaces, we may consider the directional derivative
Xf := ^ |t=o f(c(t)) of / £ ^(p), which satisfies
X(af + bg) = aXf + bXg, X(fg) = f(p)Xg + g(p)Xf
(2.4)
a,beR; f,geF{p).
Now we define this X as c(0), and call it the tangent vector to с at p. In general,
we call X : ^"(p) —» Д satisfying (2.4) a derivation of ^"(p). Then the space of all
derivations of F{p) forms a vector space if we define as (aX + bY)f := aX/ + bYf
for derivations X, У, and a, 6 £ Д. We denote this vector space by TVM and call
it the tangent space to Μ at p. Take a chart {ΙΙ,φ,χ1). Then for ς £ /7 we define
(d/dO (q)eTqM (t = l,...,m) by
(2-5) έ^/:=έ/0^1(^1(^
where д/диг denotes partial differentiation with respect to the г-th coordinate.
Then {д/дх*(д)}?1 ! gives a basis of TqM for each q £ /7, which will be called
the natural basis. In particular, TPM is an m-dimensional vector space. Note
that c(0) defines an element of TVM, and conversely any tangent vector may be
expressed in this form. Now if we take two charts (J7a, (ра,хга), {Уз, ψ3, xl3) around
ρ, then the Jacobian matrix Ό{ψβ ο φα~ι) = [dx^/dxjQ]i<ij<m of the coordinate
transformation ψβ ο φα~ι : (x^,... , xjj ι—> (χ^,... , χ™) induces the change of
basis of TPM given by
£<»> = Σ §<«'«>-sj«·
We also write d{ instead of д/дхг, when we fix a chart.
Now let ГМ = \JpeM TPM be the set of tangent vectors to Μ and τΜ '■ Τ Μ —»
Μ the map assigning ρ to χ £ TPM. Then it is an important fact that TM carries
a 2m-dimensional C°° manifold structure such that тд/ is a C°° map, and this
indicates that the concept of manifold is natural and useful. In fact, for an atlas
{{UQ, ψα)}αβΛ of Μ set UQ '■= rM~l{Ua). For X £ Uq,tm{X) = ρ we may write
X in the form X = £,г{д/дхга){р) with respect to the natural basis, and we set
ψα(Χ) := (*i(p),... ,<(P),^,... ,£m) G Я2т- Then {(Ua, фа)}аел gives an
atlas for TM. We call TM the tangent bundle of M.
Exercise 1. Let У be an m-dimensional vector space which is difFeomorphic to
Rm. For χ £ V\ define a map ф : У -> TPV by 0(x)/ := ^ \t=o f{p+tx), f £ ^(p).
Show that ф is a linear isomorphism. We denote the inverse of ф by ip : TPV —» У
and call it the canonical identification. Writing χ = хге* with respect to a basis
{e*}, show that ьр{{д/дхг){р)) = е* (г = 1,... ,m).
Now let Φ : Μ —» TV be a C°° map. For ρ £ Μ we may define a linear map
ΌΦ{ρ) : TPM —» Τφ(ρ)7ν, which is called the differential of Φ at p, by
(£»(p)(X))/ := X(/ ο Φ), / £ ^(Φ(ρ))
for Χ £ TpM. Note that this induces a C°° map £>Ф : TM -> TAT. The following
theorem shows that we may see the local behavior of Φ through its differential.

8
I. PRELIMINARIES FROM MANIFOLDS
Theorem 2.1 (mapping theorem). Let Φ : Μ —» N be a C°° map and r the
rank of the differential ΌΦ(ρ) of Φ at ре Μ. Set m = dim Μ, η = dim N.
(1) If r = m(<ri), namely, ΌΦ(ρ) is infective, then we may choose a chart
([/, φ) around ρ and a chart (V, ψ) around Φ(ρ) with respect to which Φ is expressed
in the following form:
ψοΦο^νΓ..,ο-(^Γ- ,um,o,... ,0).
(2) Ifr = n(<m)j namely, ΌΦ(ρ) is surjective, then we may choose a chart
([/, φ) around ρ and a chart (V, ψ) around Φ(ρ) with respect to which Φ is expressed
in the following form:
ψοΦοφ-ι(η\... ,um) = (u\... ,un).
(3) (Inverse mapping theorem). If r = m = n, namely, ΌΦ(ρ) is bijective,
then there exists an open neighborhood U of ρ such that Φ \ U is a diffeomorphism
from U onto an open set Φ(ΙΙ) of N.
In particular, if ΌΦ(ρ) is injective at every point ρ £ Μ, we call Φ : Μ —» Ν
an immersion. For an injective immersion Φ : Μ —» N we may identify Μ with a
subset Φ(Μ) of N. However, in general it is not true that Φ : Μ -> Ф(М) (С Ν) is
a homeomorphism with respect to the relative topology. If this is true then we call
an injective immersion Φ : Μ —> N an embedding. For an immersion Φ : Μ —> Ν
we may choose an open neighborhood U of any point ρ £ Μ so that Φ | U is
an embedding from the mapping theorem (1). Now a subset S of Μ is called a
submanifold of Μ if 5 carries a C°° manifold structure such that the inclusion
map l : S ^-> Μ is an embedding. We call dim Μ - dim S the codimension of
5. For instance, any open subset of Μ is a submanifold of codimension 0. When
an injective immersion Φ : Μ —> N is given, some authors call N an (immersed)
submanifold of N. By virtue of the fundamental results due to H. Whitney, any
m-dimensional manifold (m > 1) may be immersed into il2m-1 and embedded into
R2rn. Moreover, such immersion and embedding may be realized by proper maps.5
Next Φ : Μ —> N is called a submersion if ΌΦ(ρ) is surjective for every point p.
Then from the mapping theorem (2), Φ~ι(ς) is an (m —n)-dimensional submanifold
of Μ for every q £ Φ(Μ), and is called the fiber over g.
Exercise 2. For a C°° curve с : (α,6) —> Μ we define c(£) £ Χφ)Μ by
c(£)/ = -&f(c(t)). Then show that c(£) = Dc(d/dt), where £ denotes the coordinate
of Я.
2.3. Let Μ be a C°° manifold and suppose that to every point ρ £ Μ a
tangent vector Xp £ TPM is assigned. If a map X : Μ —> ΓΜ given by ρ ι—► Xp is
C°°, then X is said to be a ( C°° ) vector field on M. Note that the space X(M)
of all vector fields on Μ forms a vector space (and in fact an ^r(M)-module).
We may define vector fields on an open set U of Μ in the same manner. In
particular, with respect to a chart (υ,φ,χ1) we get the vector fields д/дхг : ρ ι->
(д/дхг)(р) on /7 (г = 1,... ,m). Then any X £ #([/) may be uniquely expressed
as X = Хгд/дхг,Хг £ ^(U). Now we consider vector fields from the following two
viewpoints.
This means that the inverse image of every compact subset is compact.

2. MANIFOLDS
9
(I) A vector field X may be characterized as a derivation of the algebra Τ(Μ).
Namely, if for / <E F{M) we define Xf(p) := Xpf, then Xf G F{M) and X satisfies
the following properties of the derivation.
X(af + bfl) = aXf + &*<?, X(/fl) = fXg + flX/,
( " } a,be Д; f,geF(M).
Conversely, for a derivation X : F(M) —» ^"(M) which saisfies (2.6) we define Xp G
TPM, ρ G Μ as follows. First note that Xf(p) = 0 if / | U = 0 on a neighborhood
/7 of p. In fact, choose a </? G ^"(M) so that </?(p) = 0 and φ \ Μ \ U = 1. Then
we get / ξ φ$, and consequently Xf(p) = φ{ρ)Χί{ρ) + ί(ρ)Χψ(ρ) = 0. Now for
/ G ^"(M) we define Xpf := Xf(p), where / G ^"(M) is an extension of /. Note
that this does not depend on the above choice of /, and we see that Xp G TpM.
Since locally we may write X = (Ххг)д/дхг, ρ ι—► Xp defines an element of X(M).
Now for Χ, Υ G X(M) we define the bracket operation by
(2.7) [X, Y]f = X(Yf) - Y(Xf), f G f(M).
Then we easily see that [X, Y] € X(M) and
[X, У] = -[Y, X], [fX, Y] = f[X, Y] - (Yf)X,
['> [X + Y,Z] = [X,Z} + [Y,Z],
and also (the Jacobi identity)
(2.9) [[X, Y],Z} + [[Y, Z],X\ + [[Z, X],Y} = 0.
Namely, X(M) carries the structure of a Lie algebra with respect to [ , ].
(II) (dynamical systems viewpoint). For a vector field X on Μ and ρ G M,
a curve с : (—δ, δ) —> Μ with c(0) = ρ is called an integral curve of X through p,
if Χφ) = c(t) holds everywhere. Taking a chart (/7, <£, жг) around ρ and writing
£г(£) := xl(c(i)), X = Хгд/дхг, we may get an integral curve through ρ of X by
solving the system of ordinary differential equations
^ = Го^ (t = l,...,m)
at
under the initial condition хг(0) = хг(р) (г = 1,... , m). Thus from the fundamental
theorem of systems of differential equations we see the following: For any ρ G Μ
there exist an open neighborhood U of ρ and an ε > 0 such that we have a unique
integral curve cq(t) through every q G U defined for \t\ < ε. Moreover, cq(t) depends
smoothly on (q, t).
Now taking a different viewpoint, we fix i, | t |< ε, and set φt(q) := cq(t). Then
ψι defines a diffeomorphism from U onto an open set (ft(U) of M, and ψ^φ*. = y?t+s
holds where the both sides are defined. Namely, a vector field X generates a local
one parameter group ψι of local diffeomorphisms, which is also called the flow
generated by X.
Especially for any vector field Xona compact manifold Μ (or more generally X
with compact support), </?* is defined above on all of Μ and for any t G R. Thus φί ο
φ3 = y?t+s everywhere, and X generates a one parmeter group of diffeomorphisms
of M. If we may take such a global flow {(ft}teR f°r X, we sav that X is complete.
For instance, suppose we have a > 0 such that an integral curve с of X through
any point ρ G Μ is defined for | t \< a; then X is complete.

10
I. PRELIMINARIES FROM MANIFOLDS
We note that for a diffeomorphism Φ of Μ and X £ X(M) we get ΌΦ(Χ) £
X(M), which is defined by ΌΦ(Χ)(ρ) := ΌΦ(ρ)Χφ-ΐ(ρ). Then it is easy to show
that ΌΦ([Χ,Υ]) = [ΌΦ{Χ),ϋΦ(Υ)].
Exercise 3. Let {<pt} be the flow generated by a vector field X. For Υ £
X(M), show that [X,Y]P = ft |t=0 Zty-tO^p)). Next let {^s} be the flow
generated by Υ. If X and У are complete, show that we have [X, Y] = 0 if and
only if iptoips = ipso φί for all s, t £ R.
Since [X, У] may be expressed in terms of differentiation using the flow {φι}
of X, we also denote [X, Y] by £χΥ and call it the Lie derivative of Υ by X. In
the same way we may consider the Lie derivative of various geometric objects, e.g.,
tensor fields, by X using {^} (see §3.1).
Now we state the Probenius theorem in terms of vector fields; this theorem
plays a fundamental role in the geometry of manifolds. If to every point ρ of a
C°° manifold Μ a A;-dimensional subspace Dp of TpM is assigned, we say that a k-
dimensional distribution V is given on M. When for every point ρ £ Μ there exist
an open neighborhood U of ρ and X\,... ,Xk £ X(U) such that {Xi(q)}i=\ forms
a basis of Dq at every q £ /7, we call Ρ a C°° distribution (or subbundle of TM).
For instance, a vector field X which vanishes nowhere defines a 1-dimensional C°°
distribution on M. Just like integral curves of X, a submanifold Ν οϊ Μ containing
a point ρ is called an integral manifold of Ρ through ρ if TqN = Dq for every q € N.
Now when does there exist an integral manifold of V through every point of Ml
Theorem 2.2 (Frobenius theorem). Let V be a к-dimensional C°°
distribution on M. We call V involutive, if for any vector fields X and Υ that take values
in V (i.e.,Xp,Yp £ Dp, ρ £ Μ), [Χ, Υ] takes value in V. V is said to be
completely integrable if for any ρ £ Μ there exists an integral manifold N of V through
V-
Then any completely integrable distribution V is involutive, and the converse
is also true. More precisely, if V is involutive, then for any ρ £ Μ we have a chart
(υ,φ,χ1) around ρ with φ(ρ) = 0 and φ{17) = {{и1,... ,иш); \иг\ < а] (а > 0) such
that the submanifold {q £ U; xkJ*~l(q) = £fc-H (г = 1,... , πι — к)} in U is an integral
manifold ofV for any £k+\... ,£m (£ R) with \£к+{\ < а.
We call integral manifolds of the above form slices. We also say that a k-
dimensional completely integrable distribution defines a foliation of codimension
m — к on M.
Remark. For an involutive distribution V on Μ there exists a unique
maximal connected integral manifold of V through any point ρ £ Μ which is in general
an immersed submanifold of M. Here maximal means that it is not a proper subset
of another integral manifold (see [War-3] for more detail).
2.4. Let / : Μ -> R be a C°° function. Then the behavior of levels f~l{t)
as t varies is also affected by the manifold structure. Regarding Df(p) : TPM —>
Т/(р)Д ~ R as an element of (TPM)*, a point ρ £ Μ with Df(p) = 0 is called a
critical point of /, and f{p) is called the critical value. If f~l{t) (ф ф) does not
contain critical points we say that t is a regular value of /. In this case /-1(0 1S a
hypersurface of Μ (i.e., submanifold of codimension 1), as is seen by Theorem 2.1
(2). On the other hand, for a critical value to, /-1(£o) may be rather complicated

2. MANIFOLDS
11
and the topology of /-1 (t) may change when t passes throuh a critical value to. This
may be explicitly analized when critical points satisfy the following nondegeneracy
condition. For a critical point ρ of / we may define the symmetric bilinear form
D2f(p) as D2f{p){u, v) := X(Yf)(p), where Χ, Υ are vector fields on Μ with Xp =
u,Yp = v. Then we easily see that D2f(p) is symmetric with respect to Χ, Υ and
does not depend on the choice of X, Y. We call D2f(p) the Hessian of / at a critical
point p. A critical point ρ is said to be nondegenerate if D2f(p) is nondegenerate,
i.e., if its null space {u G TPM; D2f(u,v) = 0 for any ν G TPM} = {0}. Next, we
call D2f(p) negative definite on a subspace W of TPM if D2f(p)(w,w) < 0 for all
nonzero w G W, and we define the mrfex of a critical point ρ as the dimension of a
maximal negative definite subspace of D2f(p). If we consider the symmetric га х га
matrix [(d2(f о (p~1)/dulduj)((p(p))]i<ij<Tn taking a chart ([/, <£, жг) around the
critical point p, then ρ is nondegenerate if and only if this matrix is regular, and the
index is equal to the number of its negative eigenvalues counted with multiplicities.
Now it is possible to find a canonical form for / around a nondegenerate critical
point p. In fact, the Morse lemma asserts that we may find a chart ([/, φ) around
ρ so that / may be expressed as
к т
(2.10) /o^-1(ui,---,um) = /(p)-5^txi2+ 51 "Л
i=l i=k+l
where к denotes the index of p. Thus nondegenerate critical points are isolated and
the index controls the behavior of / around p. A C^ function which admits only
nondegenerate critical points is called a Morse function. It is known that Morse
functions are in fact generic and any C00 function on Л/ may be approximated by
Morse functions (with respect to the C^ topology).
Now for / : Μ -> Я, we set Ma~ := {p G M;f(p) < a}, Ma := {p G
M; f(p) < a}. Then the behavior of Ma as a increases is described by the following
two fundamental results in Morse theory (see e.g., Milnor [M-l]6).
Theorem 2.3. Let /_1([aȣ>]) be compact and contain no critical points of f.
Then /_1([a,6]) is diffeomorphic to /_1(a) x [a,b], and Ma is diffeomorphic to
Mb. Moreover, the inclusion map ι : Ma ^-> Mb gives a homotopy equivalence. (In
fact, diffeomorphism is given by the flow of the vector field V//|| V/ ||2, where V/
denotes the gradient vector of f with respect to a Riemannian metric on Μ defined
in Chapter II, §1.3).
Theorem 2.4. Suppose that /-1([a, 6]) is compact and contains only one
critical point ρ of index k, which is nondegenerate and in f~l ((a, &)). Then we may take
a k-cell ek (i.e., an embedded closed k-dimensional disk in M) in /-1([a, 6]) such
that ek Π /-1(a) = dek, and there exists a deformation retraction from /_1([a, £>])
onto f~1(a)Uek. Namely, we have a homotopy Η : /_1([a, 6]) x [0,1] —> /-1([a, £>])
with H{q,0) = q,H{q,l) G f~1(a)Uek(q G /_1([M])) ^d H(q,t) = q (q G
f-l(a)Uek, 0<t < 1).
Let / : Μ —> R be a Morse function such that Ma are compact for all a G R.
Then, combining the above theorems, we see that Μ carries a homotopy type of a
CW-complex obtained by attaching fc-cells for every critical point of / with index
k.
See the Bibliography at the end of this book.

12 1. PRELIMINARIES FROM MANIFOLDS
Mb
Ma
Figure 1
Figure 2
Remark 2.5. Suppose that we have a curve с in /_1 ([a, &]) joining two points
in f~1(a) in Theorem 2.4. If the index к of the critical point ρ is greater than 1, then
с is homotopic to a curve in /-1(a) fixing the end points. In fact, first deform с to
a curve C\ in /_1(a)Uefc fixing the end points via the above deformation retraction.
Since к > 2, we may deform C\ slightly so that C\ does not pass through the center
of ek. Then we may deform the part of C\ which is contained in ek along radial
segments from the center to a curve in дек С f~1(a). Thus for a Morse function /,
with Theorem 2.3 we see that any curve in /_1([a^D JommS two points in /_1(a)
may be deformed to a curve in f~1(a) fixing the end points, if α is a regular value
and the indices of critical points of / in in /_1([a, &]) are greater than or equal to
2.
2.5. If a group G has the structure of a C°° manifold such that the map
G χ G —» G defined by (a, b) i-> ab~l is of class С°°, G is called a Lie group. Then
for a G G we have diffeomorphisms of G defined by La
which are called the left translation and right translation by a, respectively. A
vector field X G A'(G) is said to be left invariant if DLa X = X for all a € G.
Denoting by g the vector space of all left invariant vector fields on G, we may easily
see that [X,Y] G 0, if Χ, Υ G 0. Namely, g carries the structure of a Lie algebra
as a subalgebra of X(G). For any vector ж in the tangent space TeG to G at the
identity e, we define the vector field X on G by Xa := DLa(e)x. Then X is in fact
of class C°° and left invariant. Therefore, a map assigning Ie G TeG to I G g
gives a linear isomorphism, and we have dimg = dimG. g is called the Lie algebra
of a Lie group G. Sometimes we define the bracket [x, y] on TeG by [x, y] = [X, У]е
and identify g with TeG.

2. MANIFOLDS
13
Now we give some examples of Lie groups. R™ is an m-dimensional (abelian)
Lie group with respect to addition. A discrete subgroup Γ of rank m of R171 is
called a lattice. Γ may be written as Г = {^2пгег '-Щ € Ζ} with respect to a basis
{e;}^! of R171. Now the quotient group Tm := Дт/Г is a compact abelian Lie
group, called an m-dimensional torus. The Lie algebras of R171 and Tm are given
by Rm with the trivial bracket operation (i.e., [x, y] = 0).
Now let Λ4η(Λ) (resp., Л4П(С)) denote the vector space of all real (resp.,
complex) square matrices of degree n, which carries the structure of a Lie algebra
relative to the bracket operation [A, B] := AB — В A. Note that dimMn(R) = n2
and dimMn(C) = 2n2. In the following we shall give some examples of Lie groups
consisting of matrices. We denote by En the identity matrix of degree n, and the
determinant, trace and transpose of a square matrix A will be denoted by det^4,
traced and *A, respectively. For a complex matrx Л, A stands for its conjugate
matrix.
(2.11) GL(n, R) := {A £ Mn{R); detA φ 0} has the structure of a C°°
manifold as an open subset of Ain(R) and is a (nonconnected) Lie group of dimension
n2, whose Lie algebra gl(n,R) is isomorphic to Mn(R)· Similarly, GL(n,C) is a
(connected) Lie group of dimension 2n2 whose Lie algebra gl(n, C) is isomorphic
to Mn{C). They are called the general linear groups.
(2.12) Let 0(n) := {A £ Mn(R)', l AA = En} be the group of orthogonal matrices
of degree n. Then Ο (ή) is a (nonconnected) Lie group of dimension n(n—1)/2 with
Lie algebra o(n) := {Л £ МП(Я);'Л + Л = 0}. SO(n) := {A £ O(n);det Л = 1}
is a (connected) Lie group and is in fact the identity component of 0(n). They are
called the orthogonal and the special orthogonal groups, respectively.
(2.13) U(n) := {A £ Mn(C); lAA = En} is an n2-dimensional (connected) Lie
group with Lie algebra u(n) := {A £ Mn(C); lA + A = 0}. SU{n) := {Л £
U(n)\ det Л = 1} is a (connected) Lie group of dimension n2 — 1, and its Lie
algebra is given by su(n) := {A £ u(n); trace A = 0}. They are called the unitary and
the special unitary group, respectively. We note that U(n) is isomorphic to the one
given in §1.3.
(2.14) 5L(n, R) := {A £ Л4П(Н); det Л = 1} is a (connected) Lie group of
dimension n2 - 1 with Lie algebra sl(n, Д) := {A £ Л4П(Н); trace Л = 0} and is called
the special linear group. SL(n, C) and sl(n,C) are defined similarly.
(2.15) We put
\En 0
0 -1
K =
eMn+i(R).
Then 0(n, 1) := {Л £ GL(n+1, Д); * Л/f Л = if}, which consists of linear
transformations leaving the Lorentz inner product (χ1)2 Η l· (xn)2 — (xn+l)2 invariant,
is a (nonconnected) Lie group of dimension n(n + l)/2. Note that its Lie algebra
o(n, 1) is given by {U £ Mn+i(R); 'UK + KU = 0}.
Now let g be the Lie algebra of a Lie group G. We denote by ψι the flow
generated by X £ g. Since X is left invariant, if <pt(e) is defined for | t \< ε then
ipt{a) = aipt(e) is also defined for | t \< ε. Namely, X is complete and t н-> <£*(e) is
a homomorphism from R to G, which is called a one parameter subgroup of G. If
we put exp X := <£i(e), then we get a G°° map exp: q —> G, which is called the

14
I. PRELIMINARIES FROM MANIFOLDS
exponential map of G. Note that exptX = <£*(e), because s i-> ipst(e) is an integral
curve of tX. Thus, regarding Tog = g at the zero-vector 0 of g and TeG = g, we see
that Z)exp(O) is the identity map. Then, by the inverse mapping theorem, exp gives
a diffeomorphism from an open neighborhood of 0 in g onto an open neighborhood
of the identity e of G.
Exercise 4. Show that we have exp A = ΣΤ=ο ^V^· f°r the examples (2.11) —
(2.15).
Exercise 5. Show that the flow generated by X £ g is given by t \—> Rexptx-
Now a homomorphism from a Lie group G to a general linear group GL(V)
is called a representation of G over a vector space V. For a £ G, La о Ra~x :
/iGGh aha~l £ G is а С°° group isomorphism of G, and its differential Adga :=
Z)(La о #а_1)(е) at e gives a Lie algebra isomorphism of g = TeG. Then а £ G ι—►
Adga £ GL(q) gives a representation of G, which is called the adjoint
representation. Note that we have
d ' Ad0(exp tX)Y = [X,Y] (:= ad Х(У)).
(2'16) A ,
ас \t=o
In fact, this follows from
d
t=o ai
Adg(exp tX)Ye.
(DRexp(-tx)DLexptx Ye)
t=o
We write just Ada instead of Adga, when there is no fear of confusion.
Exercise 6. Show that exp(Ada(X)) = a · exp X · a~l.
Now let Μ be a C°° manifold and G a Lie group. If we have a C°° map
μ : G χ Μ —» Μ such that μ(α6,ρ) = μ(α,μ(6,ρ)) and μ(ε,ρ) = ρ for all а, 6 £ G
and ρ £ Μ, we call G a Lze transformation group acting on M. Denoting μ(α,ρ)
also by а · ρ for а £ G, we get a diffeomorphism а : ρ ι—► а · ρ of M. In fact, note
that a~l gives the inverse map of a. In particular, we say that G acts transitively
on Μ if for any p,q £ Μ there exists an а £ G such that a · ρ = q. We give an
example of Lie transformation group. Let Я be a closed subgroup of G. Then
Η is an (embedded) submanifold of G and is a Lie group with respect to this
manifold structure. Moreover, the coset space G/H has a C°° manifold structure
such that the canonical projection π : G —> G/i/ is a surjective submersion (see
e.g., [Hel], [Ma], [War-3]). If we define μ : G χ G/H -> G/tf as μ(α,6#) := абЯ,
we get a Lie transformation group G acting on G/i/ transitively. In this case we
also denote the action of a £ G by La.
Conversely, let G be a Lie transformation group acting on M. We set Hp :=
{a £ G; α·ρ = ρ}, which is a closed subgroup of G and is called the isotropy group
of G at p. If G acts transitively on Μ, then it is known that G/i/ is diffeomorphic
to Μ, where a diffeomorphism is given by ai/p ι—► a · p. The manifolds of the form
G/H are called homogeneous spaces, which give many examples of manifolds and
may be studied in detail using the theory of Lie groups and Lie algebras.
Exercise 7. Show that SO(m + l)/SO(m) is diffeomorphic to the sphere
5m := {x £ Hm+1; || χ ||= 1}, and J7(n+ 1)/J7(n) is diffeomorphic to 52n+1.

3. VECTOR BUNDLES AND LINEAR CONNECTION
15
Exercise 8. Let G be a Lie transformation group acting on Μ and X an
element of the Lie algebra of G. Define the vector field X := μ+X on Μ by
Xp := ji |t=o exp£X · p, and show that [μ*Χ,μ*Ϋ]Μ = -μ*[Χ,Ϋ], where [ , ]M
denotes the bracket of vector fields on M.
3. Vector Bundles and Linear Connection
3.1. Recall that the tangent bundle TM of a C°° manifold Μ carries a C°°
manifold structure such that тм '· TM —» Μ is а С°° map. Checking the manifold
structure of TM, we see that тм '· TM —» Μ has the structure of a vector bundle,
defined as follows.
Definition 3.1. τ : Ε1 —» Μ is called a A;-dimensional (real) vector bundle if
the following two conditions are satisfied:
(1) Ε, Μ are C°° manifolds and τ : Ε1 —» Μ is a surjective C°° map. For every
ρ G Μ, τ-1(ρ) is a A;-dimensional (real) vector space.
(2) For every ρ G M, there exist an open neighborhood U of ρ and a diffeo-
morphism Φ^/ : τ-1 (17) —> U x Rk with the following properties:
(i) ρτλ ο Ф^ = τ Ι τ-1 (/У). In particular, τ is a submersion,
(ii) For any q G /7, Φ^7 := pr2 о Фц \ r~1(q) : r~1(q) —» ilfc is a linear iso-
morhism, where pr\ : U x Rk —> U,pr2 : U x Rk ^> Rk denote the canonical
projections.
We call ([/, Фц) a chart of the vector bundle τ : Ε —> Μ. Ε, Μ and τ are called
the £оЫ space, base and projection of the bundle, respectively. τ-1(ρ), ρ G M, is
called the Угбег over p, and is also denoted by Fp(r).
As examples of vector bundles we have tangent bundles, and the product bundle
Μ χ Rk with the projection pr\ : Μ χ Rk —» Μ. Now for vector bundles τ :
Ε —» Μ, σ : Ε —» Μ, we call τ a subbundle of σ if Ε С F,a \ Ε = τ and
τ-1 (ρ) are subspaces of cr-1(p) for all ρ G M. Next for A;-dimensional vector
bundles σ : F —> N and τ : Ε —» Μ, a C°° map Φ : Ε —» Ε is called a bundle
map if Φ maps each fiber a~1(q), q G AT, linear isomorphically onto some fiber
τ-1(<£(<7)), φ(α) G M. Then </? : TV —» Μ is in fact a C00 map. In particular, if
Μ = N and there exists a bundle map Φ which is a diffeomorphism with φ = id^,
then σ and τ are said to be isomorphic as vector bundles. Vector bundles that are
isomorphic to product bundles are called trivial. Now we will construct some new
vector bundles from given vector bundles as in §1.1.
(I) (induced bundle). Let τ : Ε —» Μ be a fc-dimensional vector bundle, and
let a C°° map φ : TV —» Μ be given. Then we have a A;-dimensional vector bundle
φ* τ over TV which is constructed as follows: First set E\ := {(q, v) G TV χ Ε; φ(α) =
τ (υ)}. We define т\ : E\ —» TV and Φχ : Εχ —» Ε1 by τι(ς,ν) := ς and Φι (ς, ν) := ν,
respectively. Obviously we have φοτ\ = τοΦλ. Choose a coordinate neighborhood
V of q G TV and a chart ([/, Φ(/) of τ around y?(g) such that tp(V) С /7. Then we
have £in(VxT-1(/7)) = {(г, (Ф^(г))_1(х)); г G V,x G ilfc}, and we may introduce
a C°° manifold structure on E\ such that Εχ Π (V χ τ-1 ([/)) is diffeomorphic to
V χ Rk and Ει is a submanifold of TV χ Ε. Furthermore, τ1~1(ς) has the structure
of a A;-dimensional vector space by ti(q,V\) + ^(tf, ^2) := (ς,£it>i + £2^2)^ and
Φι : Ει —» Ε is a C°° map. Also note that Φι | r^~1(q) —» r_1(^(^)) is a linear
isomorphism for any ς G TV. Thus if we define Φχ,ν : r1~1(Vr) —» V χ Rk by
$i,v((r?l0) := (^^(r)^))? then η : ΕΊ —» TV is a A;-dimensional vector bundle

16
I. PRELIMINARIES FROM MANIFOLDS
with charts {(У,Ф\у)}. We call r\ the induced bundle of τ via φ : N —» Μ, and
denote it by </?*τ.
Note that Φι is a bundle map. Conversely, if a bundle map Φ : F —» Ε from
a vector bundle σ : F —» TV to a vector bundle τ : Ε —> Μ indudes a C°° map
φ : N —> M, then σ is isomorphic to the induced bundle φ*τ. Further, if a C°°
curve с : [α, 6] —» Μ (or, generally, a submanifold l : N ^> M) is given, we may
consider the induced bundle с*тд/ (resp., £*тм) of the tangent bundle тм-
(II) (Whitney sum). For vector bundles τ and σ we may define their direct
product τ χ σ, which is a vector bundle with the total space Ε χ F, base space
Μ x TV, projection τ χ σ : Ε χ F —> Μ χ Ν and charts (/7 x V, Ф(/ х Фу), where
each fiber (τ χ σ)_1(ρ, ς) is a vector space τ_1(ρ) χ σ_1(ς) = τ-1 (ρ) Θσ_1(ς).
Now for vector bundles τ : Ε —> Μ, σ : F —» Μ over the same base M we may
consider the vector bundle τ 0 σ := Δ*(τ χ σ), where Δ : Μ —» Μ χ Μ stands for
the diagonal map, defined as Δ(ρ) := (ρ,ρ). We call this vector bundle τ 0 σ the
Whitney sum of τ and σ. Note that the fiber ^ρ(τφσ) over any ρ £ Μ is naturally
isomorphic to the direct sum Fp(r) 0 Fp(a).
Exercise 1. Let σι,σ2 be subbundles of τ such that each fiber Fp(t) is the
direct sum of Fp(a\) and Fp(tf2). Show that τ is isomorphic to σ\ 0 σ2.
(III) (tensor product, exterior power). Let τ; : E{ —» Μ (г = 1,2) be vector
bundles over Μ. For each ρ £ Μ we take the tensor product Fp(t\) 0 -Fp(t2) of
vector spaces Fp(ti) = r^l{p), Fp(r2) = r2~l(p) and set Ε := \JpeM Fp(ti)0Fp(t2).
We define the map τ : Ε —» Μ by assigning ρ to elements of Fp(ri) 0 ^(тг). Take
charts ([/, Фг.(/) (* = 1? 2) of Tj so that they have a common coordinate neighborhood
U. Now define Φν : r~\U) -> /7 x (Я*1 0 Я*2) by
ФиЫ 0 v2) := (p, *bP(vi) ® Ф^РЫ)
for г>1 0 г>2 £ ^ρ(τι) 0 Fp(t2). Then Ε —> Μ carries a vector bundle structure
such that ([/, Φ(/) form a system of charts. This vector bundle is called the tensor
product of т\ and r2, and denoted by т\ 0 r2.
We may define similarly the vector bundle Hom(ri,T2) whose fibers are given
by Hom(Fp(ri), Fp(t2)). Note that in this case a chart ([/, Фи) is given by Φυ(ί) =
(Ρ^2/,ρ0/°(^ι/,ρ)~1) f°r / € Hom(Fp(Ti),Fp(r2)). In particular, taking a trivial 1-
dimensional vector bundle e over Μ, we call r* := Hom(r, б) the dual vector bundle
of r. For instance, the dual vector bundle of the tangent bundle тм '· Τ Μ —» Μ
is called the cotangent bundle of Μ and denoted by r^ : T*M —> M. We denote
by {<fa*}™ x the basis of T£M dual to the natural basis {^fr}™ ι of TPM. Further,
for a fc-dimensional vector bundle τ : Ε —» Μ, we may define in a similar manner
its tensor bundle
ΤτΛτ) :=τ0···0τ0τ*0···0τ*
s v * ν ' * ν '
r times s times
and its A;-th exterior powers
Kk(r) := τ* A · · · Λ τ*, Afc(r) := τ Α-·-Ατ.
fc times fc times
In particular, the tensor bundles and exterior powers of the tangent bundle тм of
a C°° manifold Μ are called simply the tensor bundles and the exterior powers of
M, and are denoted by T^(M) and Afc(M), Afc(M), respectively.

3. VECTOR BUNDLES AND LINEAR CONNECTION
17
Now recall that vector fields play an important role in the theory of smooth
manifolds. A vector field X on Μ may be considered as a C°° map X : Μ —» TM
which satisfies тм ° X = idM· For a general vector bundle τ : Ε —» Μ, a C°°
map ξ : Μ —> Ε with τ ο ξ = [ам is called a section of r. Note that the space
C°°(t) of sections of τ carries the structure of an T{M)-module. In particular, we
call sections of the tensor bundle XJ(M) (resp., A;-th exterior power Ak(M)) of Μ
tensor fields of type (r, s) (resp., differential k-forms) on M. Now a tensor field Τ
of type (r,s) is characterized as a map
Τ : X*(M) x · · · x ** (M) x *(M) x · · · x X(M) -> ^(M)
r times s times
that satisfies the condition
(3.1) Τ is ^r(M)-linear with respect to each variable,
where Λ'*(Μ) denotes the ^r(M)-module of all differential 1-forms on M. In fact,
let Τ be a tensor field of type (r, 5), and for a* G Λ'*(Μ) and Xj G Л'(М) define
Γ(αι,... ,αΓ,ΛΊ,··· ,Ха){р) = Γρ(αι(ρ), · · · ,αΓ(ρ),ΛΊ(ρ),... ,Χβ(ρ)).
Then we may easily check (3.1). The converse may be verified by the same argument
given in §2.3 (I). Similarly, a differential A;-form ω may also be characterized as a
skew-symmetric fc-linear map ω : X(M) χ · · · χ X(M) —» F(M) of ^r(M)-modules.
We denote by Т*(М) and ylfc(M) the ^r(M)-modules of tensor fields of type
(r, s) and differential fc-forms on Μ, respectively.
Now we mention the Lie derivative CXT of a tensor field Τ with respect to a
vector field X. Let ψι be the flow of local diffeomorphisms of Μ generated by X.
Then, for ρ G Μ, Όφ^1 = Όψ-t : Τψι^Μ —» TPM is a linear isomorphism and may
be extended to an algebra isomorphism D(pt from the tensor space T(7\f (p)M) onto
T(TPM), which preserves type and commutes with contractions. For Τ G Tsr(M)
we define
d
(CXT){P) := -
{D(pt{TMp))).
t=0
Then Cx preserves type, commutes with contractions, and satisfies the Leibniz
formula CX(T <g> S) = CXT ®S + T® CXS. In particular, for / G Tg(M) and
Υ G 7^(M) we get Cxf = Xf and CX(Y) = [Х,У]. Further, for ω G 7?(Μ) we
have
(Cxu>)(Y) = C(£x(a; 0 У) - a; 0 £*У) = Χ(ω(Υ)) - ω([X, У]),
and so on.
Now for differential forms the exterior differentiation d : Лк(М) —» ylfc+1(M)
is defined for ω G Л*(М) and X0,...,Xke X{M) by
(3.2) *=o
«j
Then d is Л-linear and satisfies ά(ω Λ σ) = άω Λ σ + (—l)fcu; Λ άτ for α; G ylfc(M).
Further, d possesses the fundamental property d2 (:= dod) = 0. A differential form
a; with do; = 0 is called a closed form, and a differential form ω in the form ω = da
is called an exact form (see Appendix 5 for properties of differential forms).

18
I. PRELIMINARIES FROM MANIFOLDS
We remark that we may consider various geometric structures on differentiable
manifolds through tensor fields and differential forms. For instance, if there exists
a closed differential 2-form α on Μ such that ap is nondegenerate at any ρ £ Μ,
then we call α a symplectic form and Μ a symplectic manifold. Note that, if Μ is
a symplectic manifold, then TPM (p £ M) are symplectic vector spaces and dim Μ
is even. For instance, the cotangent bundle T*M of Μ carries a natural symplectic
form (see the Remark in Chapter II, §4.2 (III)).
3.2. Let τ : Ε -> Μ be a vetor bundle and C°°(E) the ^"(M)-module of
sections of r. Now if to vector fields X £ X(M) and sections ξ £ C°°(E) there
correspond Vχξ £ C°°(E) which satisfy
(3.3)
Г Vfx+grt = /Vx£ + 9νγξ, ξ e C°°(E), X,Y€ X(M),
I f,geF(M);
I Vx(£ + η) = νχξ + νχη, ξ,η€ C°°(E), Χ € Χ(Μ);
( Vx(/£) = (Χ/)ξ + /VX£, ξ e C°°(E), f € F{M),
we say that a linear connection is given on E, and V χ ξ is called the covariant
derivative of ξ via X. We note that (Vx£)(p) is determined by Xp and the values
of ξ on a neighborhood U of p. In fact, if ξ vanishes on /7, take an / £ ^"(M) such
that f{p) = 0 and / | Μ \ U = 1. Clearly we have ξ ξ /ξ. Then we get
(VxOOO = (V*(/0)(p) = (Xp/)i(p) + /(P)(V*0(P) = 0.
It is also easy to check the same assertion for X. Namely, Vx£(p) is determined
by the values of Χ, ξ on a neighborhood of p. Now we take a chart ([/, φ, хг) of Μ
around ρ and write X = Хгд/дхг. From (3.3) we have
(VxO(p) = ^Xi(p)(Va/ax.O(p)·
This means that Vx£(p) (also written as Vxp£) is determined by Xp and the values
of ξ on a neighborhood of p.
Now for Χ, Υ £ #(M) we set
(3.4) Я(Х, У)£ := νχνγξ - Vy V*£ - V[x,y]£.
Then R satisfies (3.1), and (R(X,Y)£)(p) is determined by Х(р),У(р) and ξ(ρ).
We call Д the curvature tensor of the linear connection.
Next we consider the induced bundle φ*τ of τ, induced by a C°° map φ : TV —»
Μ. ρ induces an ^(M)-linear map С°°(т) £ ξ ι-> ρ*ζ := ξ ο φ £ 0°°(φ*τ). Then
from a linear connection V on τ we have a linear connection V* on φ* τ determined
by
^γ,ψ'ξ. = *i(Viv(,)y, ξΙ Υ e X(N), q Ε N.
We call V* (also written <p*V) the connection induced from V.
Recall that С°°(тм) = X(M) for the tangent bundle тм, and we may consider
for a linear connection V on тм
(3.5) T(X, Y) := VXY - VYX - [X, Y] (6 X(M)),
which is T(M)-linear with respect Χ, Υ, and therefore defines a tensor field of type
(1.2) on Μ. We call Τ the torsion tensor of V. Finally, we note that a covariant
differentiation Vx on тм may be extended to a covariant differentiation on the

PROBLEMS FOR CHAPTER I
19
tensor bundle T(M) which preservse type and commutes with contractions as in
case of the Lie derivatives (see Chapter II, Proposition 1.3 for more details).
Problems for Chapter I
1. Let {е*}·™^, {/j}jLi be bases of an m-dimensional real vector space V. Let [агА
be the matrix of the change of bases given by fj = al-e{, and [fc£] the matrix given
by /J = b?kek, where {ег},{Р} denote the bases dual to {ei},{/j}, respectively.
Note that we have 6j.a* = <5j. Now for a tensor t £ T^(V) we denote by ?£;;£ and
Pj\\\'ljrs the components oft with respect to {e*} and {fj}, respectively. Then show
that
Conversely, suppose that for any basis {e*} of V we have an rar+s-tuple tl\'"l^s
of real numbers which satisfy (*) for the change of bases. Then show that these
determine a tensor t £ ТЦУ).
2. Let A be an orthogonal matrix of degree m; that is, lAA = Em. Then show
the following.
(1) Suppose m is odd and det A = 1. Then A admits a nonzero fixed point
χ £ Rm, i.e., Ax = x(x^0).
(2) Suppose m is even and det A = — 1. Then again A admits a nonzero fixed
point x.
3. (1) Let Φ : Μ —> N be a C°° map and q £ Φ(Μ). Suppose that for any
ρ £ Φ-1(<7) we have гапк/)Ф(р) = η (:= dim TV). Then show that Φ_1(^) 1S a
submanifold of Μ of codimension n. In particular, for a submersion Φ : Μ —> TV,
Φ-1 (q) is a submanifold of Μ of codimension η for any q £ Φ(Μ).
(2) Show that the sphere Sm{r) := {{x\... ,zm+1) £ Ят+1;Е(ж*)2 = г2}
(г > 0) of radius r carries the structure of an m-dimensional C°° manifold.
4. Show that O(n), SO(n), /7(n), SU(n) carry the Lie group structures, and
determine their dimensions.
5. (1) Set
" 0 -En
ω =
eM2n(C).
[En 0
Then show that Sp(n) := {A £ /7(2n); ιΑωΑ = ω} is a Lie group (called the
symplectic group) whose Lie algebra is given by sp(n) := {A £ u(2n); l Αω + ω A =
0}. Also show that
5p(n) = < д t . ; A £ u(n), В is a symmetric complex η χ η-matrix >
and determine dim5p(n).
(2) Show that Sp(n)/Sp(n - 1) is diffeomorphic to the sphere 54n_1. What is
the fundamental group of Sp(n)?
6. Let A(V) be the space of all Lagrangian subspaces of a symplectic vector space
(ν2η,ω). Then show that A(V) may be identified with U(n)/0(n) and carries the
structure of a C°° manifold of dimension n(n + l)/2.

20
I. PRELIMINARIES FROM MANIFOLDS
7. Show that the m-dimensional real projective space RPm, which is obtained
from Sm = {iG -Rm+1; || χ || = 1} by identifying χ and —x, carries the structure of
an m-dimensional C°° manifold. Show that 50(3) is diffeomorphic to RP3.
8. (1) Define a map Φ from the torus T2 = S1 χ S1 to R3 by
φ(θ, φ) := ((2 + cos0) cos0, (2 + cos0) sin 0, sin0).
Show that Φ is an embedding and illustrate the image of Φ.
(2) Define a map Φ from S2 to R6 by
Ф(х,у,г) := (x2,y2,z2,V2yz,V2zx,V2xy).
Show that Φ is an immersion and induces an embedding from RP2 to R6.
9. A C°° manifold Μ is said to be orientable if we may choose an atlas Λ =
{(υα,ψα)}αβΑ such that the Jacobians det D(ipp ο φ~ι) of all coordinates
transformations φ β ο φ~ι are positive. We say that such charts determine a positive
orientation. Show that the tangent bundle TM of any C°° manifold Μ is
orientable.
10. Suppose that to each fiber Fp(r) = r_1(p) of a vector bundle τ : Ε —> Μ
an inner product gp is assigned so that ρ ι-> gp(£p,VP) belong to F(M) for any
ξ, η £ С°°(т). Then we call g a fiber metnc of r. Show the following.
(1) Let σ be a subbundle of τ and Fp(a)1- the orthogonal complement of Fp(a)
in Fp(t). Then \JpeM Fp(a)1- carries the structure of a vector bundle σχ such that
r = αθα1.
(2) τ is canonically isomorphic to the dual bundle r* = Horn (τ, ε). Show that
a fiber metric g of τ may be extended to fiber metrics of the tensor bundles XJ (τ).
11. Let Μ be a submanifold of Rn and set Ε := {(p,u) e Μ χ Rn; u_LTpM},
where ulTpM means that и is orthogonal to TPM. Let vm be the restriction of
the projection Μ χ Rn —» Μ to E. Show that им carries the structure of a vector
bundle, which is called the normal bundle of Μ. Show also that i*r^n =rw0 z/M,
where ι denotes the embedding of Μ into Rn. Finally, show that if Μ is an oriented
hypersurface of Rn then им is a trivial line bundle.
Notes on the References
For linear algebra, the calculus of functions of several variables, and
fundamental results on ordinary differential equations, which constitute the background for
the theory of differentiable manifolds and geometry of manifolds, we refer to, e.g.,
[Hir-Sm], [Fl], [Sp-1].
§1. For tensor products and exterior products of vector spaces, see, e.g., [War-
3], [Fla], [St], [Ko-No-I].
§2. The notion of differentiable manifolds was established by Weyl and
Whitney ([Whi]). Now there are many textbooks on differentiable manifolds. See, e.g.,
[Abr-Mar], [B-Go], [dR-2], [Hir], [Ko-No I], [Na], [Ma], [Si-Th], [St], [War-3], where
proofs of results not presented in this book may be found. In particular, see [Hir]
for the Whitney embedding theorem. For the proof of the Frobenius theorem and
maximal integral manifolds, see [Ma], [War-3]. For Morse theory, Milnor's classic

NOTES ON THE REFERENCES
21
[M-l] is still a very nice introduction (see also, e.g., [Hir]). For Lie groups and
homogeneous spaces, we refer to [Hel], [Ise-Ta], [Ma], [War-3].
§3. For vector bundles and linear connections see [M-St], [Ko-No I], [Po]. In
recent years symplectic geometry has been playing an important role in many fields
of mathematics including Riemannian geometry. For an introduction to symplectic
geometry see, e.g., [Abr-Mar], [Ar-2], [Dui-1], [Aud-Laf].

CHAPTER II
Fundamental Concepts in Riemannian Geometry
In this chapter we first define the notion of a Riemannian metric g on a smooth
manifold Μ by defining an inner product on each tangent space. Then we may
consider the length of curves and define the distance on Μ as the infimum of the
length of curves joining the given two points. Namely, Μ has the structure of
a metric space through g. Second, we see that a linear connection V, which is
adapted to the given Riemannian metric g and is called the Levi-Civita connection,
is uniquly defined, and we get "differential calculus" for tensor fields and other
geometric objects on M. Then the deviation from the usual differential calculus
in Euclidean space may be measured by the curvature tensor, which is one of the
most fundamental Riemannian invariants and, roughly speaking, locally determines
the Riemannian metric. Geometrically, the curvature tensor appears through the
sectional curvature, Ricci curvature, scalar curvature, etc. Third, we may introduce
a natural measure from a given Riemannian metric. Namely, we may consider the
volume of various figures on Μ and obtain "integral calculus" on Λ/.
In Riemannian geometry we are concerned with these various Riemannian
invariants, and relation between Riemannian invariants and the manifold structure.
Curvature appears in many such problems. For instance, in Riemannian geometry
geodesies correspond to straight lines in Euclidean geometry. If we assign an initial
point ρ e Μ and an initial direction, then there exists a unique geodesic 7 in Μ
emanating from ρ with the given initial direction. The local behavior of 7 on Μ
depends on how Μ is curved, and the global behavior is also related to the manifold
structure. In this chapter we give some fundamental concepts and results of
Riemannian geometry with respect to the curvatures, together with many examples.
We again note that we follow Einstein's convention.
1. Riemannian Metric
1.1. Let Μ be a smooth manifold of dimension m. If to every point ρ £ Μ an
inner product gp is assigned on TPM so that the functions g(X, Y) : ρ 1—► gp{Xp, Yp)
on Μ are of class Cr for all X,7G X(M), then we call g a Cr Riemannian metric on
Μ and the pair (M, g) a Cr Riemannian manifold. Namely, g is a positive definite
symmetric Cr tensor field of type (0, 2) on M. For u, ν £ TPM (resp., Χ, Υ £
X(M)) we also use the notation (u, v) (resp., (X, Y)) for the inner product. Then
we may consider the norm ||u|| of a tangent vector и and the angle Z(u, v) between
u, ν φ 0 in TPM. The set UPM := {u £ TPM\ ||u|| = 1} of unit tangent vectors
at ρ forms the sphere of radius 1 in TPM, and the set UM := LLgm^p^ °f a^
unit tangent vectors is a submanifold of TM of codimension 1 (apply problem 3 of
Chapter I to the function Ε : u £ Τ Μ н-> (u, и) £ Я), called the unit tangent bundle
of M. In the following we consider C°° (i.e., smooth) Riemannian metrics unless
otherwise stated. We also say just Riemannian manifold Μ for short, omitting g.
23

24 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Now let ([/, φ, хг) be a chart on Μ and д{ = д/дхг (г = 1,... , га) the natural
basis. If we set
(1.1) gij = {di,dj), l<ij<m,
the positive definite symmetric square matrix [gij] of degree ra defines a Riemannian
metric on U which is also denoted ds2 = gijdxldx^. First we give some fundamental
examples.
Example 1. Let V be a real vector space of dimension ra with an inner
product ( , ). For ρ eV we denote by tp : TpV —» V the canonical identification1.
Then if we set
(1.2) 9P{u,v) := (lpu, lpv),
g defines a Riemannian metric опУ. In particular, for m-dimensional Euclidean
space R171 = {(x\... ,xrn);xi <E Я}, {д{ = д/дх1}^ forms an o.n.b. of TpM
at every point ρ G R171 with respect to the above metric, which will be called the
canonical Riemannian metric go of Rm.
Example 2. Let (M, g) be a Riemannian manifold and Φ : N —» Μ an
immersion. We set
(1.3) hp{u,v) := 0ф(р)(£>Ф(р)и, ΌΦ(ρ)ν), и, v e TpN.
Then h defines a Riemannian metric on Μ because ΌΦ(ρ) is injective. h is called the
induced metric from g via Φ, and we also denote h = Ф*д. Thus every submanifold
of a Riemannian manifold admits the induced metric.
Let (M, g),(N, h) be Riemannian manifolds. A smooth map Φ : Μ —» Ν
is called an isometric immersion (resp., local isometry) if Φ*/ι = g (resp., Φ*/ι =
g, dim Μ = dim TV). A diffeomorphism Φ : Μ —» N with Φ*/ι = g is called an
isometry, and such Μ, Ν are said to be isometric. In particular, a diffeomorphism
Φ of Μ with Φ*<7 = g is called an isometry of (M, g). From the inverse mapping
theorem, for any local isometry Φ : Μ —» N and for any ρ £ Μ, there exist open
neighborhoods U of ρ and V of Φ(ρ) such that Φ | U : U —» V is an isometry.
Exercise 1. Let π : Μ —» N be a covering map. We have the induced metric
n*h on Μ from a Riemannian metric h on N. Show that deck transformations
of π are isometries of π* ft. Conversely, show that if ρ is a Riemannian metric
on Μ such that all deck transformations of π are isometries, then there exists
a unique Riemannian metric ft on TV such that n*h = g. In this case we call
π : (Μ, g) —> (TV, ft) a Riemannian covering.
Example 3. Let (M{,gi) (i = 1,2) be Riemannian manifolds. On the product
manifold M\ χ Μ2 we may introduce the product Riemannian metric g\ x £2 (or
Pi θ #2) by
(1.4)
(91 X 02)(Pl,P2)((Ul,U2), (171,172)) := (0l)pl(lXi,i;i) + Ыр2(И2,1>2),
where we have used the identification T(pi.P2)(Mi x M2) = TPlΜχ Θ ΤΡ2Μ2.
xSee Exercise 1 of Chapter I, §2.

1. RIEMANNIAN METRIC
25
Example 4. The sum of two Riemannian metrics on Μ is a Riemannian
metric. Let g be a Riemannian metric on Μ and φ G F(M) everywhere positive.
Then we get a Riemannian metric tp2g defined by
(1-5) (v29)P(u, v) = ip2(p)gp(u, υ),
which is said to be conformal to g.
Example 5. Let (M, g), (TV, ft) be Riemannian manifolds and π : Μ —> N a
submersion. The subspace KerDn(p) = Tpn~1(q) (q = π(ρ)) of TPM is denoted by
Vp and called the vertical space at p. Then the orthogonal complement Hp := VJ-
is called the horizontal space at p. Now if Dn(p) : Hp —» T^TV is a linear isometry
for every ρ G M, we say that π is a Riemannian submersion. For instance, the
orthogonal projection of V to a subspace W С V in Example 1, π of Exercise
1, and projections of a product Riemannian manifold onto each component are
Riemannian submersions.
Exercise 2. For a Riemannian metric g on Μ = Μχ χ M2 suppose that
the following condtions (i), (ii) hold: (i) TPl Μι, ΤΡ2 Μ<ι are orthogonal at every
(pi, p2) G Μι χ M2. (ii) The projections щ : (Μχ χ Μ2, 0) -> (Мг, #) (г = 1,2)
are Riemannian submersions. Then show that g = g\ x #2·
Does there always exist a Riemannian metric on a given manifold Μ ? Since
we assume that Μ is paracompact, Μ may be embedded in R2rn (m = dimM)
by Whitney's theorem, and Μ admits an induced metric from (R2rn,g0). We
can also argue as follows. Take a partition of unity {pa} subordinate to an atlas
A= {(υα,φα)}α£Α of M. Then ipa(Ua) are open sets of R171 and admit Riemannian
metrics ga. Defining ftQ as pa · φ^9α on J7Q and 0 outside Ua , we get a smooth
symmetric tensor field of type (0, 2) on M, which gives a Riemannian metic on an
open subset of Ua given by pa > 0. Then g = ΣαβΑ ^<* ls ш ^^ а finite sum on
a neighborhood of every point, and positive definite since there exists an α with
pa > 0 at every point. Thus g is a Riemannian metric on M. Then we see that
there are many (in fact infinite-dimensionally many) Riemannian metrics on Μ
from Example 4. But it is generally difficult to find the Riemannian metrics best
adapted to the given manifold structure. In the following we shall give various kinds
of invariants of Riemannian metrics and study the relation between Riemannian
invariants and their relation to the manifold structure.
1.2. We may consider the length of curves on a Riemannian manifold M. For a
C°° curve2 с : [α, b] —> Μ we define its length L(c) (or Lg(c) ) by
(1.6) L(c)= I \\c(t)\\dt
J a
and its arc-length s(t) by
(1.7) s(t)= f\\m\\dt.
J a
If с is regular (i.e., c(t) / 0,< G [a,6]), the arc-length s = s(t) is strictly
monotone increasing because s'(t) = \\c(t)\\ > 0. Denoting by t = t(s) the inverse
function, we get the parametrization of a curve by arc-length, c(s) = c(t(s)), 0 <
2We may define the length for С ^curves and piecewise С ^curves in the same way.

26 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
s < L(c). Note that in this case we have \\c(s)\\ = \\t'(s)c(t(s))\\ = 1. In general,
a curve is said to be normal if ||c(£)|| = 1, and of constant speed (or parametrized
proportionally to the arc-length) if ||c(£)|| is constant.
Now we may immediately extend the notion of length to piecewise C°° curves.
A continuous curve с : [α, 6] —> Μ is said to be piecewise (regular) C°° if there
exists a subdivision Δ : a = to < · · · < tn = b of [a, b] such that с | [U-i^U] (i =
1, · · · , n) are (regular) C°° curves. If we define the length of a curve с as L(c) =
ΣΓ=ι L(c | [ti-i,U]), then this does not depend on the choice of Δ.
Setting C(I) := {c : / —» M\c is a piecewise С°° curve } for a closed interval
/ and С := \JC(I), we get a functional L on С which assigns the length L(c) to a
curve cGC. We list some fundamental properties of the length.
(L. 1) L(c) > 0, and equality holds if and only if с is a trivial point curve.
(L. 2) For ci £ C([a,6]), c2 £ C([6,c]) with ci(6) = c2(6) we get L(c) = L(a) +
L(c2), where с = ci U c2 G C([a,c]) is the curve joining c\ to c2.
(L. 3) Let 0 : [a, /9] ι—> [a, 6] be a monotone piecewise C°° function. Then we
get L(c) = L(co φ).
(L. 3) follows from the change of variable formula for integration, and means
that the length of a curve is preserved under a parameter transformation φ. For
instance, we have L(c) = L(c_1), where c_1 is the curve given by reversing the
orientation of с : [α, b] —> Μ, namely, c_1(£) = c(a + 6 - t).
Next we introduce the distance on Μ via the notion of the length of curves.
Denoting by Cpq the set of curves in С joining ρ to q, we define
(1.8) d{p, q) := inf{L(c); с <Е Cpq).
Cpq is nonempty since Μ is assumed to be connected. In fact, fixing p, the set of
end points of curves с £ С emanating from ρ is easily seen to be open and closed in
Μ because Μ is locally difFeomorphic to Euclidean space3.
Proposition 1.1. The above d satisfies the following axioms of the distance
function of a metric space:
(D. 1) d(p, q) > 0, where equality holds if and only if ρ = q\
(D. 2) d{p,q) = d{q,p);
(D. 3) (triangle inequality) d(p, q) + d(q, r) > d(p, r).
Moreover, the topology of (Μ, d) coincides with the manifold topology. In
particular, the function d : Μ χ Μ —» R is continuous with respect to the manifold
topology.
Before starting the proof, we set Br(p):= {q £ M;d(p,q) < r} and call it the
metric ball of radius r centered at p. From the above proposition we see that Br (p)
is an open set of Μ whose closure is given by Br(p) = {q £ M; d(p, q) < r}.
PROOF. (D. 2) follows from the fact that I: Cpq -> Cqp defined by 1(c) = c"1
preserves the length. (D. 3) may be derived using (L. 2), and the continuity of d
is clear from the triangle inequality. For (D. 1) it suffices to show ρ = q assuming
that d(p, q) = 0. Take a chart (/7, φ, хг) around ρ £ Μ with φ(ρ) = о (the origin of
R171) and a compact set Κ := φ~1(Βε(ο)). Since и н-> д(и,и) is continuous on the
3If Μ is not connected and p, q belong to different connected components, we define d(p, q) —
oo.

1. RIEMANNIAN METRIC
27
compact set {u = ? {θ/θχ*){ς); q e К, £(C)2 = 1} of ГМ, there exists апй>1
such that
D2 ^ 9(u,u) 1 { д
Then for a curve с G C([a, 6]) in i^ with c(a) = ρ we get
(*) Дс) = / Ι|έ(*)ΙΙ <*< > ^ j[ ^{±Ш<и > ^Mc(b))i
where we set xl(t) = xl(c(t)) and the last inequality in (*) follows from the fact that
in R171 the line segment from о to </?(c(6)) is the shortest curve joining end points4.
Now suppose d(p, q) = 0. Firstly, if с G Cpq does not lie in К we have c(to) G dK
(the boundary of K) for t0:= inf {* G [a, 6]; c(i) 0 /^} and L{c) > L(c \ [0,i0]) >
ε/Д from (*). Thus q belongs to К if d(p, q) = 0. Secondly, if с is contained in
К we get L(c) > ||^(д)||/Д from (*), and consequently 0 = d(p,q) > \\ip(q)\\/R,
namely p= q.
Now we show that the manifold topology coincides with the topology derived
from the distance. It suffices to show the following assertions (i) and (ii) taking a
chart (ΙΙ,φ,χ1) around any ρ G Μ with φ(ρ) = ο :
(i) For any coordinate neighborhood V(C U) of ρ we may choose an r > 0 such
that Br{p) С V.
(ii) For any r > 0 there exists a δ > 0 such that φ~ι{Ββ{ό)) С Вг(р).
Here we only show (ii). (i) may be proved as above. We take K, R as before
and choose 0 < δ < min(s, r/R). For q G φ~ι(Βδ(ο)) take a curve с defined by
c(t) = <£_1(£<£(<7)),0 < t < 1. Then с is contained in K, and we get
Цс) = I \\c{t)\\ dt<R [ \\φ(ς)\\ dt < R6 < r,
Jo Jo
namely, d(p,q) < r and q G Br(p). D
Exercise 3. Give a proof of (i).
Remark. Note that the distance on the unit sphere S2 in (Д3, go) defined by
the induced Riemannian metric differs from the distance on S2 as a subspace of
the metric space R3. As for some properties of the distance defined by the length
of curves, we refer to Problems 2 and 3 at the end of this chapter. It is possible to
define the length for absolutely continuous curves more general than piecewise C1
curves.
1.3. We show that it is possible to introduce a linear connection adapted to
a given Riemannian metric. Recall that a linear connection V : X(M) χ Χ(Μ) —»
X(M) on the tangent bundle тм is characterized by
(1.9) Vx+yZ = VXZ + VyZ, VfxY = f V*r,
(1.10)
VX(Y + Z) = VXY + VxZ, Vx(fY) = (Xf)Y + fVxY
for X, У, Ζ G X{M\ f G ^(M).
See the proof of Lemma 2.7 for a proof of this fact.

28 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Theorem 1.2. Let Μ be a Riemannian manifold. Then there exists a unique
linear connection V on тм satisfying the following conditions:
(1.11)
VxY — VyX = [X, Y] {i.e., the torsion tensor vanishes).
(1.12) Х(У, Z) = (VXY, Z) + (У, VXZ).
Before starting the proof we note the following: Let α : X(M) —> F(M) be
^r(M)-linear, regarding X(M) as an ^r(M)-module. Then there exists a unique
/7 £ #(M) such that a(Z) = (C/,Z),Ze Л'(М). In fact, α is a tensor field on Μ
of type (0, 1) and may be locally expressed as α = ctidx1 with respect to a chart.
Denoting by (gli) the inverse matrix of (<7tj), we see that U = gljctjdi is the desired
one.
PROOF. Adding both sides of the following three equations, which are obtained
from (1.12),
(VXY,Z) + (Y,VXZ)=X(Y,Z),
(VyZ,X) + (Z,VyX) = y(Z,X),
-(vzx,y> - (χ, vzy> = -аду),
we get using (1.11)
(i.i3) <v*y>z> = \ix(Y> ζ) + γ(ζ>χ) - ад γ)
+ {[Χ,Υ], Ζ) - ([Υ,Ζ], Χ) + ([Ζ,Χ], Υ)}.
This implies the uniqueness. To see the existence, we fix X, У £ Λ'(Μ) and define
α : Л'(М) н-> 7"(М) by setting α(Ζ) equal to the right-hand side of (1.13). We
may easily check that α is F{M)-linear. Then from the above remark we have a
unique U £ X(M) such that a(Z) = (U, Z). We define VXY as this /7. Then we
may check (1.9)—(1.12) by direct computations using the properties of the bracket
of vector fields. D
We call the above linear connection V the Levi-Civita connection of p, and VxY
the covariant derivative of У by X. Note that the map (X, У) £ X(M) χ Χ(Μ) i->
V^y £ X(M) is not ^r(M)-linear with respect to У, and V does not define a
tensor field of type (1, 2) on M. However, on an open subset U of Μ we see from
(1.13) that (VxY) | U = VX\u(Y | /7), where on the right-hand side V means the
Levi-Civita connection of the induced Riemannian metric on U.
Now let (ΙΙ,φ,χ1) be a chart with the natural basis {д{ = д/дх1}. Then C°°
functions Tikj (1 < г, j, к < m) on U are given by
(1.14) Vdid3 := ГДА,
and are called the Chnstoffel symbols. Conversely, denoting X \ U = Хгд{ and
У | U = y*ft for X, У £ Л'(М), we may write
(1.15) {VxY) | J7 = (X · yfc + Ι\*,-ΧΎ')&.
In particular, (У*У)Р depends only on Xp and the values of У on a curve in Μ
which is tangent to Xp. So we also denote this vector by VχρΥ. For instance, let
У (t) be a C°° vector field along a C°° curve с : [α, 6] —> Μ, a section of the induced
bundle c*rM. Namely, У : [α,6] -> TM is C°° and У(*) £ Tc{t)M. Then, denoting

1. RIEMANNIAN METRIC
29
VcY also by Ve/QtY (or simply VY when there is no fear of confusion), we have
the following expression for VY with respect to a chart:
dYk
(1.16) VY = Vd/dtY = (— + Γ+ίΑΎήθκ,
where we set Y(t) = Yl(t)di(c(t)), xl(t) = xl(c(t)). In fact, VtY is the covariant
derivative with respect to the induced connection of the Leci-Civita connection via
c.
Exercise 4. (1) Show the following formula for the ChristofFel symbols with
respect to a chart (υ,φ,χ1):
(1.17) Tikj = \gkl{di9ji + dj9il - digij).
In particular, we have T{kj = Tjk{.
(2) Let I\fcj,facb denote the ChristofFel symbols with respect to local charts
(ΙΙ,φ,χ1) and (V, ip,ya), respectively. Verify the following transformation law:
Π Ш f с _ ду^дх^дх^г к , д*хк дус
[ } a b dxk dya dyb l 3 дуадуь дхк'
A vector field Y(t) along a C30 curve с : [a, b] —► Л/ is said to be parallel along
с if Va/дг^ = 0. Writing this condition with respect to a chart, we get
dYl
(1.19) — + T3lkxJYk = 0 (i = 1,... , m),
which is a system of linear ordinary differential equations of the first order. Then
for any initial vector и £ TC^M we have a unique parallel vector field U(t) along
с with /7(a) = u which is defined on the whole interval [a, 6] and of class C°°.
Moreover, the map и ι—► U is linear. In fact, to see these assertions, cover с by a
finite number of coordinate neighborhoods and apply successively the fundamental
theorem of linear ordinary equations to (1.19).
The linear map P(c)l : Tc^a)M i-> TC^M defined by assigning /7(6) to и is
called the parallel translation along с P(c)l is an isomorphism because P(c~l)l :
Tc(b)M i-> Tc(a)M gives the inverse map of P(c)g. In the following we denote
Ρ(€~λ)1 also by P(c)ba. Now we note that parallel translations preserve the inner
product. In fact, for parallel vector fields U(t),V(t) along с we get from (1.12)
jt(U(t), V(t)) = (VC/(i), V(t)) + (U(t), W(i)> = 0
and consequently (/7(6), V(b)) = (/7(a), V(a)). Further we may immediately
extend the above arguments to piecewise C°° curves by applying them successively
to each smooth part. Thus we may identify TpM and TqM for p,q £ Μ using the
Levi-Civita connection. However, note that the identification depends in general
on the choice of the curve с joining ρ to q.
Exercise 5. We may express the covariant derivative in terms of the parallel
translaton. Namely, for Χ, Υ £ X(M) and a curve c(t) with c(0) = p, c(0) = Xp,
show the following:
VXpy = lim^{P(c)<yc(t)-yp}.

30
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Now we may extend the covariant differentiation to tensor fields. Let U be an
open set of Μ and T(U) = 0r s>0 Tsr(U) the ^r(/7)-module of all tensor fields on
U. We denote by С : Tsr(U) -> 7JSi{U) the contractions. Then we have
Proposition 1.3. For X £ X{U) there exists a unique Vx : T(U) ι—> T(U)
with the following properties.
(1) Vx is a derivation. Namely, it is an R-linear map which preserves type
of tensor fields and satisfies for T, V £ T(U)
Vx(T <g> T') = VXT <g> Τ' + Τ <g> VXT', VXCT = CVXT.
(2) Vx 25 pwen 6t/ Vxf = Xf for f £ T(U), and on T0l(U) = X(U) it is
the above covariant differentiation of the Levi-Civita connection.
(3) For an open set V С U we have (VXT) \ V = V X\VT \ V.
Proof. For ω £ 7i°(t/) we get from (1), (2)
(νχω){Υ) = ϋ{νχω ® Υ) = ϋ(νχ(ω ®Υ)-ω® V XY)
= νχΟ(ω ®Y)- ω(νχΥ) = Χ(ω(Υ)) - ω(4χΥ),
from which we get the uniquness. Since the last term of the above equation is
^(/У)-linear with respect to У, it gives a tensor field of type (0,1), which we define
as Vχω. Then (3) clearly holds. Similarly, for ω £ TS°(U), Vχω is given by
(1.20)
(νΧω)№ X.)
S
= Χ ·ω(Χ\,... ,Χ3) -^2ω(Χι,... ,VxXi,.. ,X3)·
i=\
Next, for Τ £ Tsr(U), note that Τ(ωλ,... ,ωτ) £ TS°(U) for ωχ,... ,ωτ £ Ti°(U).
Then VχΤ is given by
(VxT)(u;i,...,u;r)
(1.21) r
= Ух(Г(и,.·· ,^r)) -^Τ(α;ι,... ,Vx£Ji,... ,u;r)
г=1
as before, and we may check (1) by a direct computation, e.g., using a local
expression (1.23) of Exercise 6 (below). D
Next, for Τ £ Tsr(M) we define VT £ Trs+l(M) by
(1.22) VT(X1,--.,Xs,X):=(VxT)(X1,...,Xs)(£T0r(M)).
A tensor field Τ with VT = 0 is said to be parallel For instance, (1.12) means that
the metric tensor g is parallel. Also we may consider the higher order covariant
differentiation Vk : Tsr(M) -> T3\k(M) (k = 2,3, · · ·), which is inductively defined
as Vk = V(Vfc"1).
Exercise 6. Let Τ = T^;;^dxjl ®- · ·<8>άτ'· (8x9^ ®· · -®dir be the local
expression of Τ £ 7^r(M) with respect to a natural basis. Setting (VdkT)(dj1,... , djj
= VkTjl^di, ®---®dir show the following:
(1.23)
r s
T7, T>i\---ir _ о rpix...ir . \ ^ ρ га rpii...l...ir _ \ ^ ρ m /pii гг
Wklji-js -°klji:.js + Z^lk llh is Z^ifc 3s1jl...rn...jsi
ot=\ β=1

1. RIEMANNIAN METRIC
31
where / and m are in the α-th and /3-th positions, respectively.
Remark 1.4. (1) The parallel translation P{c)l : TC^M —» TC^M may
be extended to a linear isometry from Tsr(Tc(a)M) onto Tl(Tc^M). Then the
assertion of Exercise 5 also holds for tensor fields.
(2) Some authors write T>"f.k for VkT]l-)r.
Now we define some fundamental differential operations on Riemannian
manifolds in terms of the covariant derivatives.
Definition 1.5. (1) For / G Τ(Μ) we define the gradient vector V/ £ X(M)
of/by
(1.24) (V/, X) := X/, XeX(M).
With respect to a local coordinate system it is expressed as (V/)* = g^djf.
(2) For / e F{M) we define a symmetric tensor field D2f of type (0,2), called
the Hessian of /, by
(1.25)
D2f(X, Y) = (VxV/, Y) = XYf - (VxY)f Χ, Υ € X(M).
Locally we have D2f{di,dj) = gjkVi(Vf)k = V^·/.
(3) For X € <Y(M) we define the divergence divX of X by
(1.26) divX = trace(y^VyX) (= VtX* = -j=di(VGX')) .
(4) The Laplacian Δ/ € J"(M) of / € T{M) is defined by
Δ/ = -trace£>2/ = -div(V/) = -VJVj/
(L27) = -^<**^/),
where we set G = det(^).
Further properties are given in the following exercises.
Exercise 7. Show that D2f(X,Y) = D2f(Y,X). Prove (1.26) and (1.27).
Exercise 8. (1) Show that V(fh) = fVh+hVf and div(/X) = Xf+fdivX.
(2) Prove the following identities:
(1.28) div(W/) = -hAf + (V/, V/i),
(1.29) A(fh) = hAf - 2(V/, Vh) + /Δ/ι.
A function / with Δ/ = 0 is said to be harmonic. This equation is valid for
C2 functions /. However, it is known that harmonic functions are in fact of class
C°°. Further, for a domain Ω in Μ with (piecewise) smooth boundary ΘΩ and a
given continuous function φ on ΘΩ, we have a unique continuous function / on Ω
which is harmonic on Ω and satisfies the boundary condition / | ΘΩ = φ.
Exercise 9. Show that the exterior differential duj of a differential fc-form ω
is given by
(1.30)
к
аи{Х0,Хи...,Хк) = ^{-1)\ЧХ1и)(Х0,...,Хи... ,Хк).
i=0

32
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Remark 1.6. The Hessian of / defined above coincides with the one given in
Chapter I, §2.3, at critical points of /.
2. Geodesies
2.1. For a C2 curve 7 in a Riemannian manifold Μ the equation
(2.1) Vd/da(t) = 0
means that 7 is auto-pararell and proceeds straight. This property reflects a
property of straight lines in Euclidean geometry, and we call a curve satisfying (2.1) a
geodesic. With respect to a chart (υ,φ,χ1), (2.1) turns into
d2xl ^ i dxj dxk .. ч
(2·2) -ж+г^иг^г=0 ί-1.···.'»).
where we set xl(t) = χι(^(ί)). This is a system of second order (nonlinear) ordinary
differential equations, and the following holds by the fundamental theorem. Let
p0 £ Μ and u0 £ TPoM be given. Then there exist an e > 0 and a neighborhood
U of u0 in Τ Μ such that for any и £ U we have a unique geodesic 7 defined for
|£| < € which satisfies the initial conditions 7(0) = p{= tmu),7(0) = u(£ TPM).
We denote this geodesic by 7n. Then 7n(£) depends on t and u smoothly5. In
particular, geodesies are C°° curves. We note that (2.1) puts a restriction on the
parameter.
Exercise 1. For a geodesic 7 show that (7(£), i(t)) = const, namely 7 is of
constant speed.
Now it is easy to see that for a compact set К of TM there exists an e > 0
such that a geodesic 7n(£) may be defined for | t \< e for any u £ K. Next we note
that for a £ Д the equality
(2.3) 7au(0 = 7u(ai)
holds. In fact, regarding both sides of the above equation as curves with parameter
t, they are solutions of (2.1) with the same initial point тми and the initial direction
au, and they coincide from the uniqueness of the solution.
Now fix p. Since UPM is compact, there exists a δ > 0 such that 7u(t) (u £
UPM) is defined for | t \< 26. Then from (2.3) we see that 7U(1) is defined for
и £ Βδ(ορ), where ov denotes the origin of TPM. Now we define a map expp by
exppU := 7n(l)· Then expp is a C°° map6 from an open set D of TPM containing
Βδ(ορ) to Μ, and will be called the exponential map. We compute the differential
Dexpp(op) of expp at the origin op. First we rewrite (2.3) in the following form:
(2.4) expptu = 7n(0^ exPP °p = P·
Then under the identification of T0pTpM with TPM we get
Dexpp(op)u = —
expptu = 7n(0) = u,
t=o
and Dexpp(op) is the identity. Then by the inverse mapping theorem there
exists an € > 0 such that expp | B€(op) is a diffeomorphism onto an open set В in
°In the case of a Cr Riemannian metric, 7u(0 is of class Cr_1 (resp., Cr+l) with respect to
и (resp., t).
GIn the case of a Cr Riemannian manifold it is of class Cr~l.

2. GEODESICS
33
Μ containing ρ7. Then taking an o.n.b. {е*}·^ of TPM we may define a diffeo-
morphism φ : В —» {ж G um; ||x|| < б} by assigning to ς G Б the components
of (expp | Be(op))_1g with respect to {e;}, namely, φ(α) = (x1(q),... ,хш(а)) if
βχΡρ(Σχ4?)β0 = Q· Therefore, we get a chart (Β,φ,χ1) around ρ which will
be called a normal coordinate system at p. Thus we may construct charts of Μ
concretely in terms of geodesies when a Riemannian metric is given on M.
Remark. expp is not necessarily defined on all of TPM. If expp is defined
on TPM, namely, if all geodesies emanating from ρ are defined for all parameter
values, we say that (M, g) is geodesically complete at p. See Chapter III, §2, for
more details.
Now we somewhat generalize the above argument. There exist a coordinate
neighborhood U around ρ G Μ and a δ > 0 so that exp^u is defined for и G U :=
{u G TM; q = rMu G t/, ||u|| < δ}. We define a C°° map Φ : U -► Μ χ Μ by
Ф(и) := {тми, βχρ^η), q = тми,
and compute the differential ΌΦ(ορ) of Φ at ov G TPM. Let (жг) be a normal
coordinate system at ρ and (хг,С) the coresponding local coordinate system in
TM. Then we get
£>Ф(ор)0/&:* = (д/дх{(р), д/дхг(р)), ΌΦ{ορ)θ/θξι = (0, д/дх\р)),
and consequently гапк/)Ф(ор) = 2m. Then from the inverse mapping theorem we
may choose an open neighborhood V of ρ and e > 0 so that Φ is a difFeomorphism
from a neighborhood V := {u G TM; тд/u G V, ||u|| < e} of op in TM onto an open
neighborhood of (p,p) in Μ χ Μ. Taking an open neighborhood W of ρ in Μ such
that W x W С Φ (1θ, we see the following:
(2.5) Any two points qi,q2 G W may be connected by a unique geodesic 7ςι<72 :
[0,1] —> Μ with £(79ις2) < e, and 7ςις2(0) G TM depends smoothly on gi and ς2·
(Note that the image of 7ςις2 is not necessarily contained in W.)
(2.6) For any q G И^ехр^ | B€(oq) is a difFeomorphism and IV is contained in
expq(Be(oq)). (See also Chapter IV, §5.)
2.2. Now we compute the differential Dexpp(u) of the exponential map expp :
Βδ(θρ) —> Μ at и G Bs(op), which measures the difference between Euclidean
space TPM and a Riemannian manifold M. For this computation (and further for
many differential operations of tensor fields) the following curvature tensor plays
an important role.
Theorem 2.1. For Χ,Υ,Ζ G X(M) define
(2.7) R(X, Y)Z := VXVYZ - VYVXZ - V[x,y]Z (G X(M)).
7In fact В = Ве(р). See Corollary 2.8.

34
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Then R is a C°° tensor field of type (1,3) and satisfies the following:
(2.8) R(X, Y)Z = -R{Y, X)Z,
(2.9) R{X, Y)Z + R{Y, Z)X + R{Z, X)Y = 0,
(2.10) (R{X, Y)Z, W) + <Я(Х, Y)W, Z) = 0,
(2.11) (ВД r)Z, W) = <Λ(Ζ, W)X, У),
(2.12) (V* Д)(У, Ζ) 17 + (Vyfl)(Z, X)J7 + (Vzfl)(X, У)1У = О.
(2.9) and (2.12) are called the first and second Bianchi identities.
PROOF. We may check that R is a tensor field, i.e., ^r(M)-linear with respect
to X, У, Z, by direct computations. (2.8) is obvious. For (2.9) and (2.12) first
compute the left-hand sides according to the definition of the curvature tensor and
its covariant derivative. Then use (1.11) and the Jacobi identity for the bracket
operation of vector fields to show that they vanish. For (2.10) it suffices to show
that (R(X,Y)U, U) = 0 for X, У, U <E X(M) (in fact, substitute U = Ζ + W in
the above equation and expand). From (1.12) we have
да,у)[/,[/) = (vxVy[/, u) - (VyVx[/, u) - (v[x,y]t/, u)
= 1-{XY{U,U)-YX{U,U) - [Х,У]([/,[/)} = 0.
Finally, (2.11) follows from
(R(X,Y)Z, W) = {R{Y,Z)W, X) + (R(Z,X)W, Y)
= -(R{Z, W)Y, X) - (R(W, Y)Z, X) - (R(X, W)Z, Y) - (R(W, Z)X
= 2(R(Z, W)X, Y) + (R{W, Y)X + R(X, W)Y, Z)
= 2(R{Z,W)X, Y) - (R(X,Y)Z, W).
Here the first equality holds by (2.9) and (2.10), the second by (2.9), the third by
(2.8) and (2.10), and the last by (2.9), (2.8), and (2.10). This completes the proof
of the theorem. D
R is called the curvature tensor (field) of (M, g). We also define the curvature
tensor field of type (0, 4), which is denoted again by R, as8
R(X, У, Z, W) := (Я(Х, Υ)Ζ, W).
Exercise 2 (expression of the curvature tensor by components). We set
R(di,dj)dk = Rijkdi and R(dudj,dk,di) = Rijkl.
Then show the following:
{D I Qr I ЯГ' -LP' pm г I pm
JMjk — uiL j к UjL i k ~\- L i ml j k L j ml i fc,
Rijkl = Rijk^gml-
(2.13)
Exercise 3. For a tensor field Τ of type (r, s) on M, show that
(2.14) V2T( ,X,Y)-V2T( ,Y,X) = -R{X,Y)oT,
8Unfortunately the sign of the curvature tensor is different with different authors. However
the sectional curvature, the Ricci curvature, and the scalar curvature, which will be denned later,
are always the same.

2. GEODESICS
35
where R(XP,YP) denotes the derivation of the tensor space T(TPM) extending the
linear map of TPM given by Ζ —> R(X, Y)Z. In local coordinates we have
y * y l Ji---Js y Ly k 3\--3s
(2.15) = y^ /?LI_<aT*i-m-*- - Ν Λ Ft,,, mT11·
= Σ^'"^; - Ε *«*m*j;
Now we return to the computation of Dexpp(u^, ΐλ,ξ G TpM, where we regard
ξ as a vector in TUTPM. For a curve s ь-► ^ + s£ in TPM tangent to ξ at w, we
define a C°° map α : [0,1] x (—e,e) —> Μ by
α(ί, s) := exppt(u + s£).
For each fixed s, the curve cs : t ь-► α(£, s) is a geodesic emanating from ρ and
Co = 7w On the other hand, t ь-► fj(£,0) (:= /}а(£,0) J^) gives a vector field Y(t)
along 7^, and from the definition we get
(2.16) Y(l) = Dexpp(uK, Y(t) = *£>expp(ftx)£.
Z)expp(«)^=r(l)
expP
TpM
Figure 3
Generally when a continuous map α : [α, 6] χ (—с, е) —> Μ is given, we have
a family of curves cs : £ ь-► a(t, s) for s 6 (—α, α), which are called the variation
curves of Co. α is called a variation of Co. If α is of class C°° we may consider the
vector fields 9a/cU := Da(d/dt),da/ds := Da(d/ds) along cs.
Lemma 2.2. Denote Va« (resp., Va«) fa/ V a (resp., V a ) /or s/iori. T/ien
"a^" as at as
/л ^ч ^ да ^ da
/Λ,«χ _^ ^ 9α ^ ^ 9α „,9α 9α49α
(2·18) v£v£^-v£v^ = i?%^aJ·
Proof. With respect to local coordinates we get, from (1.16) and Tfj =Tjhi,
_ да (д2а*гк да>да>\
v&te = {mfc+rij-dT-dF)dk
from which we also get [^, ^|] = 0. (2.18) is clear from this and the definition of
the curvature tensor. □

36
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Now we return to the above Y{t). In this case, since variation curves cs are
geodesies, we get V^|eO. Then setting V = V^_, we get from (2.17) and (2.18)
VW(i) = V£V£U=0^
— „da „ Ιда, лЧ да, Л да, лЧ
= v*u=oV^w + ^(^(M),^(M)j^(M)
= R(<y(t),Y(t))m
We get also У(0) = 0, УУ(0) = V_a_u=o(u + βξ) = ξ· Thus the vector field Y(t)
along 7 satisfies
(2.19) VVr (t) + Д(У (f), 7(*))7(*) = 0,
(2.20) У(0) = 0, VF(O) = ξ.
(2.19) is a system of linear ordinary differential equations of the second order which
may be considered as the linearization of the equation (2.1) for geodesies. If the
initial conditions У(0),УУ(0) are given, there exists a unique vector field Y(t)
along 7 satisfying (2.19) and the initial conditions, which is of class C°°9 on the
interval where 7 is defined. We call such a vector field a Jacobi field along 7. Then
the linearity of the equation implies that the set ^(7) of all Jacobi fields along 7
forms a 2m-dimensional vector space.
Prom the above we see that Dexpp(u)£ = У(1), where Υ is the Jacobi field
along 7 = 7n with the initial conditions Y(0) = 0, VY(0) = ξ. Looking at (2.19),
we see that the curvature tensor controls the behavior of Jacobi fields. Thus,
roughly speaking, it also controls the exponential map (normal coordinates) and
the manifold structure. This principle will be seen in many places of the present
book. Now we give an application of Jacobi fields.
Proposition 2.3 (Gauss Lemma). For и £ TPM suppose that a geodesic 7 (t) =
ju(t) is defined for 0 < t < b. Then expp is defined on an open neighborhood of
{tu\ t £ [0,6]} in TPM, and we get the following:
(1) Dexpp(tu) maps и to *y(t).
(2) // we regard ξ £ TPM also as a vector in TtuTpM via the canonical
identification, then the equality
(Dexpp(tu)t,>y(t)) = (u,0
holds. In particular, \\ Dexpp(tu)u \\ = \\ и ||, and Dexpp(tu)£ _L *y(t) if£±u.
PROOF. The first assertion follows from a fundamental property of the system
of differential equations. (1) follows from Dexpp(tu)u = -^ |s=o expp(t-\-s)u = ^(t).
To see (2) let Y(t) be the Jacobi field along 7 with У (0) = 0, VY(0) = ξ. Then we
get
^(Y(t)n(t)) = (VVy(i), 7(*)> = -<Я(У(*),7(*)Ж*), 7(*)> = 0.
9 For a Cr Riemannian metric it is of class Cr.

2. GEODESICS
37
Namely, (Y(£), *y(t)) is a linear function of t and equal to (ξ, u)t because of the
initial condition. Then we get
<Dexpp(t«K,7(t)) = (jY(t), 7(ί)) = (ξ, и),
which completes the proof of the proposition. D
Note that the above argument shows that a Jacobi field along 7 which is
perpendicular to 7 at two different points is perpendicular to 7 everywhere. From
the above computation (2.16) of Dexpp we may also see the following: Let 7 :
[0,6] —» Μ be a geodesic with 7(0) = p, 7(0) = u. Then for £ > 0, Dexpp(tu) is
not regular (i.e., rankZ)expp(£u) < m) if and only if there exists a nonzero Jacobi
field Υ along 7 such that Y(0) = 0 and Y(£) = О. If this happens we say that
7(£) is conjugate to ρ along 7, and t is called the cojugate value. The number
n(t) := dim{Y <E J(7); Y(0) = 0, Y(£) = 0} = dim Ker£>expp(*u)10 is called the
multiplicity of the conjugate point 7(2).
Lemma 2.4. Suppose j(t) (t > 0) zs no£ conjugate to ρ along 7. Tften for any
ν G TPM and W G ΤΊ^Μ, there exists a unique Jacobi field Y(t) satisfying the
boundary condition Y(0) = v, Y(t) = w.
PROOF. First we show the uniqueness. If we have two Υι, Y2 G ^(7) satisfying
the same boundary codition, then Ζ := Υχ - Y2 G J(j) satisfies Z(0) = 0, Z(£) = 0
and is equal to 0 from the assumption. To see the existence, note that J0{l) :=
{Y G *7(t); ^(0) = 0} is an m-dimensional vector space. The linear map assigning
Y(t) G ΤΊ(ι)Μ to Υ G Jo(l) is injective from the assumtion and consequently is
a linear isomorphism between J0{l) and ΤΊ^Μ. Thus for any w G ΤΊ^)Μ there
exists a Jacobi field Yi (t) with Υχ (0) = 0 and Y\ (t) = w. Now if j(t) is not conjugate
to ρ along 7, note that ρ is not conjugate to j(t) along 7_1. Applying the above
argument to 7_1, we get a Jacobi field Y<i G ^(7) with Υ2{0) = ν, Υ2{ΐ) = 0 for any
υ G TpM. Then Υ = Yi + Y2 is the desired Jacobi field. D
Exercise 4. Let ^j be the components of a Riemannian metric g with respect
to a normal coordinate system around p. Show that
9ij (P) = uj, Sfc^j (p) = 0, Tikj (p) = 0
2.3. Geodesies are defined as curves proceeding straight which share a
property of straight lines in Euclidean geometry. Now straight line segments are
characterized also as shortest curves. In the case of a Riemannian manifold Μ, we call a
curve с G Cpq with L(c) = d(p, q) again a shortest curve joining ρ to q. First we look
for a necessary condition to be a shortest curve. Let с : [α, b] —> Μ be a shortest
piecewise C°° curve joining ρ to ς. A variation α : [α, 6] χ (—€, e) —» Μ of с is called
a piecewise C°° variation in Cp(?([a, 6]), if there is a subdivision Δ of [a, 6] such that
ct I [£i_i,£i] x (—€,€) is C°° and the variation curves {cs} (—e < s < e,c0 = c)
belong to Cpq([a, &]). We call the vector field W(t) := fj(£, 0) along с the variation
vector field of a. Conversely, for a given piecewise C°° vector field W(t) along с
with W(a) = 0, W(b) = 0, we may define a piecewise C°° variation α of с in Cpq
with the variation vector field W by a(£, s) = expc(i)sH/(t).
10That is, the nullity of Dexpp(tu).

38 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Proposition 2.5 (First Variation Formula). Let с G Cpq([a, b}) be a regular
piecewise C°° curve. Take a variation a of с with a variation vector field W as
above. Then11
(2.21)
ds
L(c.) = - I {W(t),VtL^)dt
s=0
* P(*)ll
/(
h\ ( }'\№-0)\\ \\c(U + 0)\\
The right-hand side of (2.21) depends only on the variation vector field W and
is also denoted by DL(c)W. We call it the first variation of L by W.
Proof. The left-hand side of (2.21) is equal to
ι pb ι £\ £\ \ 1/2
ώ1·-0/. \Ж'Ж/ dt
= jib(v&^(t,0),c(i)/||c(i)||^di
= [ {jt(W(t),c(t)/\\c(t)\\) - (W(t), V£(c(f)/||c(i)||)> j dt.
Now for the integral of the first term of the last equation apply the fundamental
theorem of calculus to each subinterval [ti-i,U] and add them. Then we get the
right-hand side of (2.21). D
If a piecewise C°° regular curve с G Cpq satisfies DL(c)W = 0 for the variation
vector field W of any variation of с in Cp<7, we call с a stationary (or critical) curve
of L. If с G Cpq is a shortest curve, then с is stationary. We have furthermore
Proposition 2.6. Let с G Cpq be a piecewise C°° regular curve of constant
speed. Then с is a stationary curve of L if and only if с is a geodesic.
PROOF. Since a geodesic с is of class C°° and of constant speed, we easily
see that it is stationary from (2.21). Conversely, let с be stationary and take a
subdivision Δ so that с | fo-i, *г] is C°°. First take a vector field W\(i) := V a_c(t)
dt
along с and a C°° function / : [a, 6] —> R such that f(U) = 0 (г = 0,... , к) and
f(t) > 0 (t Φ U). Setting W(t) := /(*)Wi(*)> we get a piecewise C°° variation
vector field along с Then from (2.21) we get
0 = DL(c)W = -\ J" № \\Ve. c(t)\\2 dt (/ = ||c(i)||)
and consequently V a c(t) ξ 0 on each [^-i, U]. Next we choose W so that W(U) =
c(U - 0) - c(U + 0) (i = 1,... , к - 1) and W{a\ W{b) = 0. Then from
1 fc-i
0 = DL(c)W =τΣ \\c(U - 0) - c(U + 0)||2
nc(ti — 0) denotes the tangent vector to с | [tt-i,*t] at t = tt and c(ti + 0) denotes the
tangent vector to с | [U, U+\] at t = ti.

2. GEODESICS
39
we get c(ti - 0) = c(U + 0), namely, с is of class C1 on [a, 6] and satisfies (2.1) on
each subinterval. Then с is of class C°° by (2.2). D
Exercise 5. Let p,q,r e Μ satisfy d(p,q) + d(q,r) = d(p,r) and suppose that
there exist shortest normal geodesies 71 and 72 joining ρ to q and g tor,
respectively. Then show that 71 U 72 is smooth at q and defines a shortest normal geodesic
joining ρ to r.
However, we still don't know whether a geodesic 7 G Cpq is a shortest curve
joining ρ to q. Later we will study this problem in detail (see Chapter III, §4), and
here for a geodesic 7 : [0, /] —» Μ we compare the length of 7 with the length of
some other curves from ρ to q := 7(/). We set u = 7(0). A shortest geodesic is also
called minimal.
Lemma 2.7. Let expp be defined on an open neighborhood U of a segment
11—> tu,t G [0, i], m TPM. Lei </? : [a, 6] —» U be a piecewise C°° curve with φ(α) = op
and </?(&) = lu. Setting c(t) := expp ip(t), we get a curve с G Cp<7, q = expp lu. Then
L(c) > £(7) = Ζ|| tx ||. Moreover, if expp is regular on U, then L(c) > £(7) when
C([a,6])^7([0,/]).
PROOF. Setting r(t) := ||^(£)||, we first consider the case where r(t) > 0(t >
a). Writing φ(ί) = r(t)e(t), e(t) G UPM for t > a, we get
φ(ί) = r(t)e(t) + r(t)e(t), <β(ί), έ(ί)> = 0.
Now from the Gauss lemma we get || Dexpp(ip(t))e(t)\\ = 1 and
(Dexpp(tp(t))e(t), Dexpp(tp(t))e(t)) = 0.
This implies that
\\c(t)\\ = \\Όβχρ„{φ(ί))φ(ί)\\ > ||1>ехррЫ0)(г(*)е(*))11
= \m\ =
Therefore, we have
L(c)
*/. \>{t)l
±Mt)\\
dt> ||И6)||-|Иа)||| = /Н = Д7).
Next we turn to the general case and put to := s\ip{t G [a,6]; r(t) = 0}. Applying
the above argument to φ \ [to, 6], we get the desired inequality. If equality holds
under the assumption that expp | U is regular, we have r(t) = 0 for a < t < to and
r(t) > 0, e(t) = 0, r(t) > 0 for t > t0. Then φ(ί) takes the form φ(ί) = r(t)e, e =
u/\\u\\, with r(t) > 0. Prom this the last assertion immediately follows. D
Corollary 2.8. Suppose expp : Be(op) —» Μ is a diffeomorphism onto an open
neighborhood of M. Then for any q (Φ p) G expp(Be(op)) there exists a unique
minimal geodesic 7 parametrized by arc-length from ρ to q, where 7 is given by
7(i) := expp t(expp~lq/ Цехр"1 q\\). In particular, d(p,q) =|| exp~x q ||, anrf
ехррБе(ор) coincides with the metric ball Be(p) = {q G M; d(p,q) < e}.
PROOF. Set / := || expp _1^||,u := expp ~lq/l G UPM. Then 7(4) := expp£u, 0
< t < /, is a normal geodesic joining ρ to q. Take an arbitrary с G Cp(?([a, 6]). If
c([a,6]) С ехррД:(ор), we apply Lemma 2.7 setting φ(ΐ) := expp_1c(t), and get
L(c) > 1/(7). Moreover, if с is parametrized by arc-length with L(c) = £(7), we have

40 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
φ(ί) = tu by the proof of the lemma. If c([a, b]) is not contained in expp B€(op), then
we get L(c) > e > L(y) as in the proof of Proposition 1.1. Thus d(p, q) = L(y) = i,
and 7 is a minimal geodesic. The last assertion is then obvious. D
From the above we see that any sufficiently short arc of a geodesic is a shortest
curve. Next we compare the length of a geodecsic 7 with the lengths of curves с
joining the end points of 7 and close to 7.
Corollary 2.9. Let 7 : [a, b] —► Μ be a geodesic. Suppose that there exist no
conjugate points top = 7(a) along 7, and set q = 7(6). Then there is a neighborhood
U of 7 in Cpq([a, b}) with respect to the compact open topology [i.e., the unifom
convergence topology) with the following proerty: For с £ U we have L(c) > £(7)
and L(c) > L(7) »/7[M]) Φ c([M])12·
Proof. We may write y(t) = expp(t - a)u, и := 7(a). By the assumption,
expp is regular at su, 0 < s < I := b - a, and we may choose an open neighborhood
U of su in TPM so that expp | U is a difFeomorphism. Since the closed interval
[0, 1} is compact, we may choose a subdivision 0 = Sq < S\ < · · · < Sk = I of [0, /]
and open neighborhoods [/» (i = 1,... , к) of [si-ь si]u (C TPM) sathat expp | U{
are difFeomorphisms. Then we have ^y([ti_i, t»]) С exppUi if we set U = α + S{ (i =
0,... , к). Now we may choose e > 0 sufficiently small so that an open
neighborhood U€ := {c £ Cpq([a, 6]); max{d(c(*),7(£)); t £ [a, 6]} < б} of 7 with respect to
the compact open topology satisfies the following: For any element с £ U€ we may
get a variation {cs}o<s<i of 7 in Cpg with C\ = с so that cs([£i_i,£i]) С exppf/i
(г = 1, · · · , к) for 0 < s < 1. Then starting with cs \ [a, ti] we first set (ps(t) =
(expp I Ui)~lcs(t) and define^ | [£»-ъ*г] inductively by <£s(*) = (expp | Ui)~1cs(t).
Since cs(ti) £ exppt/j Π expp[/i+i for 0 < s < 1, we see that <£s(£t - 0) =
φδ(ίί Η- 0) and <^s : [a,6] —► TpM is a piecewise C°° curve joining op to /u. Since
we have exppips(t) = cs(t) from the construction, our assertion follows immediately
from Lemma 2.7. D
3. Curvature
3.1. First we see that, very roughly speaking, the curvature tensor field
R introduced in Theorem 2.1 locally determines the Riemannian metric. Let
(ΒΓ(ρ),φ,χι) be a normal coordinate system around ρ £ Μ. We want to look
for the coefficients of the Taylor expansion of ^(ζ1,... ,£m) at χ = 0. Set
gij(t) = gij(txl,... , tx™), and note that for a fixed χ = (χ1, ... , χ171) the curve
7(£) := (tx1,... , ixm) is a geodesic emanating from p. Now we consider variations
c%i (г = 1,... , m) defined by
аг-(г, s) := {tx\ ... , f (x* + s),... , tom).
Then variation curves of a.{ are geodesies, and its variation vector field Yi(t) =
td/dxl(^(t)) is a Jacobi field along 7 satisfying the initial conditions Yi(0) =
0, VYi(0) = д/дх{(р), as is seen by the argument deriving (2.19) and (2.20). Thus
we have
t29ij(t) - am зд>·
12Passing beyond the conjugate point, we cannot further choose such U (see Chapter III,
Lemma 2.11).

3. CURVATURE 41
Differentiating successively both sides of this equation with respect to t, we get by
induction on к = 1,2, · · ·
fc(fc-l Wfc-2)(i) + 2ktgi<k-1\t) + ί Vfc)(i)
(3-1) * /IfeN
=Σ(ΐ)<ν(*~,)1Γ*(')·ν(,)^(«>>·
On the other hand, taking the successive covariant derivatives of the Jacobi equation
WYi(t) + R{Yi(t)^(t))i(t) = 0 with respect to V = Vjl, we get
fc-2
(3.2) V^Yi(t) + Σ (fc l 2) (^k-2-^R)(V^Yi(t),7(t))7(t) = 0.
In particular, at t = 0, from the initial conditions we get
Yi(o) = o, vy-(o) = д/дх\р), v2Yi(o) = o,
V3l-(0) = -Rtkr(p)xkxld/dxm(p),
V4Yi(0) = -2VJRtklrn(p)xJxkxld/dxm(p),
and so on. Namely, V^Yi(O) may be expressed in terms of the universal
homogeneous polynomials of degree (/ — 1) with respect to ж1,... ,xm whose coefficients
are given by the successive covariant derivatives up to order (/ — 3) of the curvature
tensor evaluated at p. Thus we get
Proposition 3.1. With respect to a normal coordinate system around p, the
coefficients of the Taylor expansion of gij(xl,-.. ,£m) at χ = 0 may be expressed by
the universal polynomials of components of the curvature tensor and its successive
covariant derivatives evaluated at ρ :
(3.3) 9ij(x\ ... ,zm) = δίά + \Rikji{p)xkxl + 0(\\ χ ||3).
Exercise 1. Continue the above Taylor expansion as far as possible.
Now let Φ : (Μ, g) —> (Μ, g) be an isometry, and let V, R denote the covariant
differentiation and the curvature tensor of M, respectively. Then we get
f Db{VxY) = νΌΦ{Χ)ϋΦ(Υ),
\ D$(R(X,Y)Z) = ίϊ{ϋΦ(Χ),ϋΦ{Υ))ϋΦ(Ζ).
In fact, the first equation holds since we may easily check that ΌΦ(Χ), ΌΦ(Υ) £
X(M) н-> ΌΦ(νχΥ) £ Χ(Μ) satisfies the conditions which uniquely characterize
the Levi-Civita connection of g. Then the second equation follows from the above
and the definition of the curvature tensor. Further, for geodesies 7 = 7^,7 = 7u
with initial tangent vectors и £ ТрМ,й := ΌΦ(ρ)η £ Τφ(ρ)Μ, respectively, we get
(3.5) Φ(7(ί)) = 7(*), Ζ>Φ(7(ί))οΡ(7)? = Ρ(7)?οΖ?Φ(7(5)),
where Ρ(ι), etc. denotes the parallel translation. In fact, from the equation
ν0Φ(7(ί))7(*)^(7(ί))7(ί) = £>Φ(7(ί))(ν7(07(ί)) = 0,

42
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
we see that t \—> Φ(^(ί)) is a geodesic satisfying the initial conditions Φ(7(0)) =
Φ(ρ), Ζ)Φ(7(0))7(0) = u, and consequently coincides with 7(2). Next setting W(t) =
ΌΦ(^(ί)) ο P(^)stw for w £ Γ7(β)Μ, we get
%t)W(t) = Ζ)Φ(7(0)(ν7(ί)Ρ(7)» = 0.
Namely, W(t) is parallel along η(ί) and satisfies iy(s) = D$(i(s))w. Since a
parallel vector field along j(t) is uniquely determined by the value at s, we have
W(t) = P(j)t ° /?Ф(7(в))гу. Conversely, we have the following.
Theorem 3.2 (E. Cartan). Let (M,g), (M,g) be Riemannian manifolds of
dimension m, and take points ρ £ Μ, ρ £ Μ. Tofce α linear isometry I : TpM —>
TpM and an e > 0 «г/с/г iftai expp | B€(op) and expp | B€(op) are diffomorphisms
onto B€(p) (C M) and B€(p) (C M), respectively. Then the diffeomorphism Φ :=
exppO/ о exp"1 : B€(p) —» Bc(p) *5 an isometry if and only if the following holds:
For any normal geodesic 7(£), | t \< e, emanating from ρ £ Μ, set η(ί) := Φ(*γ(ί))
and It := P(7)? 0/0 ^(7)0 : Ti{t)M ~> Г^(0М. ГЛсп
(3.6) /*(Д(и,v)w) = Я(/*и,Itv)Itw, u,v,w £ Γ7(ί)Μ.
Moreover, in this case
(3.7) £>Φ(7(ί)) = /t (= P(7)? ° / ° РЬУо).
PROOF. First note that j(t) is a geodesic emanating from ρ with the initial
direction /(7(0)) by the definition of Φ. If Φ is an isometry then we get (3.7) from
(3.5), and (3.6) follows from (3.4). To show the converse, assuming (3.6) it suffices
to see that ΌΦ(α) : TqM —» ΤΦ^Μ preserves the norm at any q £ B€(p). By
Corollary 2.8 there exists a unique minimal geodesic 7 parametrized by arc-length
joining ρ to q, and we may write q = 7(i), $(q) = 7(i), 0 < I < e. Since there exist
no conjugate points to ρ along 7 | (0, б), Lemma 2.4 implies that for any w £ TqM
we have a unique Jacobi field У along 7 with У (0) = 0, У (ί) = w. We consider the
Jacobi field У along 7 with the initial conditions У(0) = 0,УУ(0) = ДУУ(О)).
Then we have Y(t) = ΌΦ(Υ(ή) from the definition of Φ.
Now take o.n.b.'s {ei}, {ёг := /(ег·)} at ρ, ρ and denote by ei(t),ei(t) the
parallel translations of e*, e* along 7,7, respectively. Then {ei(t)}, {ei(t)} are o.n.b.'s
of T7(t)M,T;y(t)M, respectively, and we may express У(£),У(£) and the curvature
tensors as follows:
Y(t) = ήί)β,·(«), У(*) = У<(0*(0.
Я(е<(0,7(0Ж0 = Д?(0^(0, Д(ё<(0.7(0)7(0 = Я? (0^(0·
Then the equations for the Jacobi fields У, У are given by
(3.8) ^(0 + ^(0^(0 = 0, ^у!(0 + Д}(0^(0 = о.
Now applying (3.6) to и = ej(t), ν = w = *y(t) and noting that ej(t) = It(ej(t))
and7(i) = /*(7(i)), we get
(3.9) R)(t) = R)(t), \t\<e.
On the other hand, from the initial conditions we get Уг(0) = У*(0) = 0, ^Уг(0) =
^Уг(0). Thus Уг'(г),Уг'(£) are solutions of the same differential equation (3.8)

3. CURVATURE
43
satisfying the same initial conditions. In other words, Yl(t) = Yl(t). Then w =
Y(l) and D$(q)w = Y(l) have the same components with respect to o.n.b.'s {ei(t)}
and {ei(£)}, respectively. Hence their norms are equal and the proof is complete.13
D
Exerecise 2. Suppose that two isometries Φι, Φ2 : Μ —» Μ satisfy Φι (ρ) =
Φ2(ρ) and ΌΦι(ρ) = ΌΦ2{ρ) for some ρ £ Μ. Then show that Φι = Φ2 (recall
that M is assumed to be connected).
3.2. The concept of the curvature appears not only as the curvature tensor
but also in various forms as follows.
(I) (sectional curvature). For a two-dimensional subspace σ of the tangent
space TPM we choose an o.n.b. {u, v} of σ, and define the sectional curvature Κσ
of σ by
(3.10) Κσ := (R(u,v)v,u) (:= K(u,v)),
which measures how Μ is curved to the direction σ. We may easily check that Κσ
does not depend on the choice of o.n.b.'s of σ.
Exercise 3. For any basis {u, v} of σ, Κσ = К (и, ν) may be expressed as
(3.11) Κσ = (R{u,v)v,u)/ || uAv ||2,
where \\u Av\\2 = \\u\\2\\v\\2 — (u,v)2 is the square of the area of the parallelogram
spanned by u, v.
We note that the sectional curvature (with the inner product) determines the
curvature tensor. In fact, from (3.11) (R(u,v)u,v) = —K(u,v) || и Λ ν \\2 is
determined from the sectional curvature. On the other hand, setting /(α,/З) :=
(R(x + az, у + βνυ)(χ + az), у + /Зги) - (R(y + аг, ζ + βνο)^ + аг),ж + /Зги) for
fixed ζ, ι/, 2 and W, we get
(3-12) {R{Xiy)ZiW) = ^^L{0,0),
and (Д(х, y)z,w) is determined by terms of the form (R(u, v)u, v). It is also possible
to express (R(x,y)z,w) directly (but in a rather complicated manner) in terms of
sectional curvatures (and the inner product).
We may consider the sectional curvature as a real valued C00 function14 defined
on the Grassmann bundle Gm,2(M) of 2-dimensional subspaces of tangent spaces
to M, and sometimes the sectional curvature reflects more geometric properties
compared with the curvature tensor. For instance, if Κσ is constant for all σ £
Gm)2(M) then we call (M,g) (m > 2) a Riemannian manifold of constant curvature.
We may also consider Riemannian manifolds whose sectional curvatures take a fixed
sign or take values in a fixed range, namely, δ < Κσ < Δ.
Lemma 3.3. A Riemannian manifold Μ (m = dim Μ > 2) is of constant
curvature к if ond only if the following identity holds:
(3.13) R(x,y)z = k{(y, z)x - (x, z)y}, x,y,ze TPM; ρ £ Μ.
'See Chapter III, §5, for a globalization of this theorem.
In the case of a Riemannian manifold of class Cr, it is of class Cr~2.

44
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
PROOF. If (3.13) holds, then we easily see by a direct computation that Μ is of
constant curvature k. Converesely, suppose Μ is of constant curvature k. Setting
S(x,y)z := R(x,y)z — k{(y,z)x — (x,z)y} we see that 5 is a tensor of type (1, 3)
which satisfies the algebraic conditions (2.8) — (2.11) of the curvature tensor. Now
from the assumtion we have
(*) (S(x,y)y,x) = 0, x,ye TPM; ρ eM.
In fact, this is obvious if {x,y} is linearly dependent. If they are linearly
independent, then (*) follows from (3.11). Now inserting у + ζ instead of у in (*) and
using (*), (2.8), (2.10), (2.11), we get (S(x,y)x,z) = 0, namely, S(x,y)x = 0 for
any ж, у £ TPM. Again inserting χ + и instead of χ in the last equation we get
S(x,y)u + S(u,y)x = 0. On the other hand, S(x,y)u = —S(y,u)x — S(u,x)y by
(2.9). These two equations imply 2S(u,y)x = S(u,x)y. Replacing x,y we get
S(u,x)y = 0. □
Examples of Riemannian manifolds of constant curvature will be given in the
next subsection. Now we give a geometric interpretation of the sectional curvature
which measures a deviation of Riemannian manifolds from Euclidean space.
Lemma 3.4. In a 2-dimensional subspace σ ofTpM consider a circle of radius
r centered at the origin op, and denote by lr the length of cr, which is the image of
the above circle by expp. Then
,~ . .4 ,. 27ГГ — lr π rr
(3.14) ι|ίδ__ = _Κσ
PROOF. Take an o.n.b {e^} of TPM containing an o.n.b {ei,e2} of σ, and
consider the normal coordinate system with respect to {e^}. Then cr may be
expressed as χλ(ί) = r cost, x2(t) = r sin£, xk(t) = 0(k > 3). Then, if we apply
Proposition 3.1, ||cr(£)|| is expanded in the form
IM*)II2 = Σ 9a^kitW{t)x^t) = r2 /i - jKa + 0(r3)\,
i,j = l ^ '
and consequently
/·2π ρ2π f 2 )l
lr = Jo \\cr(t)\\dt = rj |ΐ--ΑΓσ + 0(Γ3)|
= 2πΓ-^ΑΓσΓ3 + 0(Γ4).
ό
Then the lemma follows immediately. D
(II) (Ricci curvature). Contracting the curvature tensor, we get
(3.15) Ric(:r, y) := trace(2 i-> R(z, x)y),
which is a tensor field of type (0, 2) and is called the Ricci tensor. Note that
K\o,(x,y) is symmetric with respect to x,y. In fact, taking an o.n.b. {e^} of TpM,
we have from the definition and (2.8), (2.10), (2.11)
Ric(x,y) = ^2(R(eilx)y,ei) = ^2(R(ei,y)x,ei) = Ric(y,x).
dt

3. CURVATURE
45
For a unit tangent vector и £ TPM we choose an o.n.b. {ei := u, e2, · · · ,em} of
TpM and define
m
p(u) := Ric(u, u) = )K(u,ej).
3=2
ρ is a smooth function on the unit tangent bundle l/M, and is called the Ricci
curvature. It gives less information than the sectional curvature. However, in
relation to the volume and analytic methods in Riemannian geometry, the Ricci
curvature plays an important role. Note that as in (3.13) the Ricci curvature
determines the Ricci tensor: for x,y £ TP(M) \ {0}
(3.16)
Ι Η^ϊ^ι)* ♦'''-'(и)»*'"-'(та) И
Ric(x,y) = < if У φ -x,
Hi) м2 "»--*■
Exercise 4 (expression of the Ricci tensor by components). We set pi3 =
Ric(di,dj). (1) Show that pjk = Д^тЛ (2) Show that
(3.17) Vipjk - V3pik = Vifyjk1.
Now if the Ricci curvature p(u) of (M,g) is constant on UM, then we call
(M, g) an Einstein manifold. Prom (3.16) note that we have p(u) = с for some
constant с if and only if the equality
(3.18) Ric{x,y) = cg{x,y)
holds for any x, у £ TpM (p £ M). Next we see a geometric meaning of the Ricci
curvature. With respect to a normal coordinate system {хг} around ρ £ Μ we
write g = (gij)· Then we get
Lemma 3.5. For и £ f/pM, r > 0 the following holds:
(3.19) det(9lJ(ru)) = 1 - ^-r2 + 0(r3).
Proof. Recall that gia(ru) = 6г3-^КгЫз{р)ики1+0(г3) holds by (3.3). Then
we get
det(9y (r«)) = sgn ft 4 Π (*«. - у ^afciie (р)Л' + 0(r3)
^ ' a=l ^
2 m / · \
= 1 - J Σ sSn (ΐ m ) oli> · · · RMi° (P)' *' *™m«*«' + ОИ
a=l ^ '
2 m
= 1-^^л' + °(г3)·
a=l
Now since the natural basis {д/дхг} forms an o.n.b. at the center ρ of a normal
coordinate system, we have ΣΤ=ι Rakia^u1 = p{u). D

46 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
(III) (scalar curvature). For ρ £ Μ we define
(3.20)
τ(ρ) = trace Ric(p) = У^р(ег,ег) ({е*} is an o.n.b. of TVM)
and call τ the scalar curvature. With respect to a chart we have an expression
τ = gjkRijkl by components. The scalar curvature gives poor information compared
with other curvatures, but is manageable since it is a smooth function on Μ.
For instance, we may ask whether there exists a Riemannian metric whose scalar
curvature is a given function on Μ.
Proposition 3.6 (Schur Lemma). (1) Suppose that at every point ρ £ Μ
the sectional curvature Κσ equals a constant kp for every plane σ С ТРМ. If
πι := dim Μ > 3, then Μ is of constant curvature.
(2) Suppose that at every point ρ £ Μ the Ricci curvature p(u) equals a
constant cp for any и £ UPM. Ifm>3, then Μ is an Einstein manifold.
PROOF. It suffices to show (ii), because if Κσ = kp holds for any plane σ С
TpM, then we have p(u) = (m — l)kp for any и £ UPM. Carrying out covariant
differentiation on both sides of pij = cgij, we get VkPij = дкс · gij. On the other
hand, from (3.17) we get an equation V/Д' -fc = Vipjk — V'jpik = diC-gjk —djC-gik-
Multiplying both sides of this by gjk and taking the sum, we have g^k^iR\jk =
(m — 1)с?гс, the left-hand side of which is equal to Vi(pijg^) = d{C. Thus we get
(m — 2)d{C = 0, from which our assertion follows, because we assume that Μ is
connected. D
Remark 3.7. If m = dim Μ = 1 we have R = 0. If m = 2, then for σ = TPM
and any и £ UPM, we have Κσ = p(u) = rp/2. Next if m = 3, we take an o.n.b.
{ei}?=i of TPM consisting of eigenvectors of the symmetric bilinear form Ric(x,y).
Then note that Ric(ej, ej) = 0 (г ф j) and
Ric(ei,ei) = {R(ei,ej)ej,ei)) + (u(ei,efc)efc,ei),
where {г, j, k} is a cyclic permutation of {1,2,3}. Thus we get
(3.21)
2{R(ej,ek)ek, ej) = Ric(ej, ej) + Ric(efc, ek) - Шф*, е{)
{{i,ji &} is a cyclic permutation of {1,2,3}),
(R(ej,ek)ek,ei) = 0 (г, j, A: are different).
Exercise 5. Show that 3-dimensional Einstein manifolds are of constant
curvature.
3.3. In this subsection we will compute the curvature tensors for some basic
examples.
(I). Let (V, go) be an m-dimensional vector space with an inner product go
which is considered as a Riemannian manifold (§§1.1, Example 1). A vector field
YonV may be considered as a C°° map Υ : V —» V if we identify TPV (p £ V)
with V. Now for X £ X(V) we define
(3.22)
(VxY)(p):=DY(p)X(eTpV).

3. CURVATURE
47
Then V satisfies the conditions of Theorem 1.2 and gives the Levi-Civita connection
of (V,g0). Take an o.n.b. {ej of V and identify V with Rm. Let {di}f=l be the
natural basis wih repect to the coordinates (хг). Then we have VχΥ = (X · Yl)di
for X = Хгд{, Υ = Yldi G Λ'(ν). With respect to these coordinates we get
Tjlk = 0. Therefore, the curvature tensor R, sectional curvature, Ricci curvature,
and scalar curvature of (V, go) all vanish everywhere. The equation for a gedesic is
given by
d2
^7(07(0=0 & ^2^(0=0 & 7(t) = tu + v{u,veV),
and geodesies are nothing but straight lines. expp : TPV —» V is a difFeomorphism
(in fact an isometry) given by expp и = ρ + *,pu. Also note that Jacobi fields Υ
along a geodesic 7 are given by Y(t) = (at + b)E(t), where E(t) are parallel vector
fields along 7 and a, 6 G Д.
(II). Let (M,g) be a Riemannian manifold and ι : TV ^-> Μ a submanifold of
M. We consider the induced metric h = i*g on TV. For ρ G TV we identify TpTV
with the subspace Dt(p)(TpN) of TPM and take its orthogonal complement ΤΡΝ^.
Then we get the orthogonal decomposition TPM = TpN(&TpN^. We easily see that
TN1- := (J eN TpN1- has the structure of a vector bundle of dimension (ra - n),
which is called the normal bundle of a submanifold TV and denoted by v^. Thus
the restriction of the tangent bundle TM to TV may be written as the Whitney
sum Τ Μ I TV ^ TTV 0 TN±. For и G TpM we denote by uT and u1- the TpTV-
component and the Tp TV -'--component of u, respectively Now let ViV/ be the Levi-
Civita connection of (M, g). Then for X, Г G *(TV), recall that (У^У)(р) (p G TV)
is determined by values of У on a curve in TV tangent to Xp, and we decompose it
into two components as a vector in TPM = TPN Θ ТРЛГ±.
(a) We may easily check that ρ ι-> (V^f У)т(р) satisfies all the conditions of
the Levi-Civita connection VN of (TV, h). Namely, we get
(3.23) V£r = (V^r)T.
(b) We set S{X,Y) := (V$fy)x, which is a symmetric tensor field on TV of
type (0, 2) taking values in TN^. In fact, we may check by direct computations
that S is ^r(TV)-linear with respect to X, Y. Note that
s(x, y) - s{y, x) = (v^fy - v^x)x = [x, y]x = 0.
We call S the second fundamental form15 of TV. We also define ^4ξ : TPTV —» TPTV
for ξ G ΤρΤνχ (ρ G TV) by
(3.24) (Αξχ,ν):=-(8(χ,ν),ξ).
Then ^4ξ is a symmetric linear transformation which is called the shape operator.
The eigenvalues of ^4ξ are called principal curvatures of TV in the normal direction ξ.
Note that ^4ξ can also be given in the following way. Extending ξ to a C°° normal
vector field on a neighborhood of ρ in TV, we take the orthogonal decomposition
Vf ξ = (Vf 0T + (V^O"1 for x Ξ TpTV. Then we have
(3.25) Αξχ = (Vf 0T.
15The signs of the second fundamental form and the shape operator also differ from one
author to the next.

48 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
In fact, for any у £ TPN, we get (Αξχ, у) = -<S(a:,y),0 = -<У^У,0 =
(у, Vf 0 = ((Vf 0T. У>, where X, Г are vector fields on N with Xp = x, Yp = y,
respectively.
Now the curvature tensors RM,RN and the sectional curvatures KM,KN of
the connections Vм, VN, respectively, satisfy the following fundamental relation.
We assume that dim TV > 2.
Proposition 3.8 (Gauss formula). For x, y,z,w e TPN (p £ N),
(3.26) (RM(x,y)z)T = RN(x,y)z +As{y,z)x- As{x,z)y,
(3.27)
RM(x,y,z,w) = RN{x,y,z,w) + (S(x,z), S{y,w)) - (S(y,z), S(x,w)),
(3.28)
KN(x, y) = KM(x, y) + <S(*,*), S(y,y)) - (S(x,y), S(x,y)),
with {x,y} an o.n.b. in (3.28)
PROOF. Let X, etc., be a vector field on N with Xp = x. Then from the
definition of the curvature tensor we get
RM(x,y)z=V?{V»Z + S(Y,Z)}-VM{V%Z + S(X,Z)}
-V^Y]pZ-S([X,Y}p,Z).
Taking the TpTV-component of both sides of the above equation, we have
(RM(x,y)z)T = V?V?Z - V?VNXZ - Vfrfl1pZ + As(v,z)x - As{x,z)y
= RN{x, y)z + AS(y,z)X - AS(x,z)y.
Taking the inner product with w and noting (3.24), we get (3.27). Then (3.28) is
obvious. D
Exercise 6. (1) Let ξ be a C°° normal vector field defined on N and X £
X(N). Show that the normal vector field (V^f ξ)1- defines the covariant derivative
V^£ of a linear connection on the normal bundle ν ν of TV, which satisfies the
Weingarten formula
(2) Verify the Codazzi formula
{RM{X,Y)Z)L = (VXS)(Y,Z) - (VYS)(X,Z),
where
(VxS)(Y, Z) := VJiS(Y, Z) - S(V%Y, Z) - S(Y, V#Z).
Now we give some definitions which will be used later. If the second
fundamental form S vanishes at a point ρ of a submanifold N of (M, g), then N is said to be
totally geodesic at p. N is called a totally geodesic submanifold if it is totally
geodesic at all points. Then every geodesic 7 of TV with the initial direction и £ TPN
is also a geodesic of M, since VM/y(t) = VN/y(t) = 0. Namely, if N is totally
geodesic, then any geodesic 7 of Μ with the initial direction и £ TN is contained
in N. Conversely, a submanifold N with this property is totally geodesic, since

3. CURVATURE
49
5(u, и) = (V^)1- = 0 holds for any и £ TN. Next we define the mean curvature
vector Η οϊ Ν at ρ by
1 1 n
(3.29) H=—trace5= —ys(c<,e<),
г=1
where {βϊ}"=1 is an o.n.b. of TPN and η := dim TV. A submanifold N with Η = 0
is called a minimal submanifold of Μ.
In particular, when TV is a hypersurface of Μ (i.e., dim TV = dim Μ - 1), the
eigenvalues of Av are called principal curvatures of TV, where ν is a unit normal
vector to N. The arithmetic mean of principal curvatures is called the mean curvature
of TV.
Remark. The concept of curvature was first introduced for curves and
surfaces S in R3. Gauss showed in his Theorema egregium that the product of two
principal curvatures of 5 at ρ £ 5, which is expressed in terms of the first and
second fundamental forms, is in fact equal to (U(ei,e2)e2,ei), where {ei,e2} is an
o.n.b. of TPS. Note that (#(ei, в2)е2, ei) is determined only by the first
fundamental form, namely the induced metric on 5, and is called the Gauss curvature of
S at p. For a two-dimensional subspace σ of TPM of a Riemannian manifold M,
take an open neighborhood U (C σ) of op and note that expp U is a surface which
is totally geodesic at p. Riemann defined the sectional curvasture Κσ as the Gauss
curvature of expp U at p. (3.28) shows that this coincides with our definition of
κσ.
Exercise 7. Let / : D —» R be a C°° function defined on a domain D of
Ят_1, and let Mf := {p = (x, f(x)) £ Rm; χ £ D} be the graph of /. which is
considered as a hypersurface of ilm. Then, if Df(x) = 0 at χ £ Ζλ д/дхт is a unit
normal vector to M/ at ж. Show that the second fundamental form of Л// at χ is
given by
Show the following using this fact: For a point ρ of a Riemannian manifold M, и £
f/pM and a symmetric linear transformation A of u-1 := {г> £ TpM; (v,u) = 0},
there exists a hypersurface Ν οϊ Μ around ρ with unit normal и at ρ such that the
shape operator Au is equal to the given A.
(III). Let Sm(p) be the m-dimensional sphere of radius ρ in ilm+1 (m > 2)
with the canonical Riemannian metric ho. We endow STn(p) with the induced
metric go from /io- For ρ £ STn(p) note that £p = ^p is the outward unit normal
vector to 5m(p) at p, and for X £ Л'(5т(р)) we get
where x(i) denotes a curve in Sm(p) tangent to X at £ = 0. Namely, we have
S(x,y) = —-(x,y), and (3.26) implies that the curvature tensor # of (Sm(p),g0)
is given by
(3.30) R{x, y)z = -i{<y, *)x - <x, г)2/}.

50
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
(SW.<70)
Figure 4
Hence (Sm(p), <7o) (m > 2) is of constant curvature 1/p2.
Next a geodesic j(t) of STn(p) emanating from ρ with the initial direction
и 6 UpSTn(p) satisfies the differential equation
0 = V7(0 = (%*)7W)T=7WT·
Namely, we may write i(t) = /(i)ty(O- Then /(*) = <7(*),7(*)>/P
= — (7(i), i{t))/ρ = — I/p, and the above equation becomes
7(0 + ^7W = 0,
which is a linear differential equation with constant coefficients. Solving this, we
get, under the above initial condition,
(3.31) 7(£) = cos - · ρ + sin - · pu,
Ρ Ρ
which is in fact the great circle obtained as the intersection of 5m (p) and the plane
through the origin spanned by ρ and pu. Further, any Jacobi field Y(t) along 7 that
is perpendicular to 7 satisfies WY(t) + p~2Y(t) = 0. For the initial conditions
У(0) = a, VY(0) = b we denote by a(t), b(t) the parallel translation along 7 of α, 6,
respectively. Then we may easily solve the Jacobi equation and get
(3.32) Y(t) = cos - · a{t) + psin - · b{t).
Ρ Ρ
Note that (3.32) hold for Jacobi fields in Riemannian manifolds of constant
curvature 1/p2.
(IV). For a Riemannian manifold (M, ho) and a positive C°° function φ on
Μ, we consider a Riemannian metric g = φ2 ho conformal to ho on M. We denote
by VP,V the covariant differentiation with respect to 0, /io, respectively, and by
Яр, Я the curvature tensor of #, /io, respectively. Then we have the following. We
leave the proof to the reader, since it is just direct computations.
Proposition 3.9. Set f = log<£, and let V/, D2f denote the gradient and the
Hessian of f with respect to ho, respectively. Then
(3.33) V9XY = Vxy + (Xf)Y + (Yf)X - h0(X, y)V/.

3. CURVATURE
51
For the curvature tensor R(x,y,z,w) = (R(x,y)z,w) of type (0,4), we have
(3.34)
R9 = eV | Д + (D2f -df®df + i/i0(V/, V/)ft0) Θ /*>} ·
Яеге, /or tensors ft, A; o/ type (0,2) we rfe/ine
ft Θ к(ж, y, z, w) := ft(x, z)k(y, w) + ft(i/, w)k(x, z) - h(x, w)k(y, z) - h(y, z)k(x, w).
In particularу if φ = c, a constant, then g = c2ho is said to be homothetic to
h0, and we get V9 = V, R9 = c2R and Κ9 = ^Κσ.
Remark. Setting pij = Vidjf - difdjf + \ || V/ ||2 gij, we may write (3.34)
with respect to local coordinates as
(3.35)
{Rg)ijkh = φ {Rijkh - (ho)ihPjk + {ho)jhPik - {ho)jkPih + {ho)ikPjh}·
(V). Let Нш := {(ж1,··· ,zm); χ171 > 0} (πι > 2) be the upper half-space
of Rm with the Riemannian metric fto, which is the restriction of the canonical
Riemannian metric of R™. Take a C^ positive function φ := 1/x171 on Нш, and
consider the Riemannian metric go = <£2fto· Noting that / = - logz™, from (3.34)
we have for the curvature tensor R of g0
(3.36)
R(x, y, z, u) = -g0(x, w)g0{y, z) + g0(x, z)g0(y, w),
namely, (Hm, go) is of constant curvature -1. Prom (3.33) the equation for a normal
geodesic ^(t) is given by
7(i) + 2(7(i) · /Ж0 - Λο(7(«),7(0)ν/(7(ί)) = 0·
If we decompose 7(f) = (x(t),y(t)), x(t) € Rm~l, y(t) € R+, then we get
(3.37)
\x(t)-2(y(t)/y(t))x(t)=0
\y(t)-2y2(t)/y(t) + y(t) = 0.
Figure 5
Now we consider a semicircle in ffm, centered at α € Rm x and of radius
R, which is orthogonal to Rm~l. Take a parameter t of the semicircle so that its

52 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
paramrtric representation t ι—> (x(t),y(t)) is normal relative to go· Then we may
write
[ x(t) = Rcoss(t) · e + a (e is a unit vector of Ят_1),
< y(t) = Rsins(t),
[s(t) = sins(i), namely, s(t) = 2tan-1 e(i~'o),
and this satisfies (3.37). On the other hand, for any ρ e Нш and и е UpH171, we see
by elementary geometry that there exists a unique semicircle with center in Дт_1
and orthogonal to Дт_1 (including straight half-lines x(t) = a, y(t) = e* which are
parallel to the £m-axis), which passes through ρ and is tangent to и at p. Therefore,
such semicircles cover all geodesies of (H171, g0). Since y(t) —> 0 as t —> ±oo because
s(i) = 2 tan-1 e*, we see that geodesies are defined for all parameter values, and g0
is geodesically complete.
We call (Hm,go) the hyperbolic space; it is in fact a model of hyperbolic non-
Euclidean geometry by the upper half-space. Note that if we endow Hm with the
metric g = φ2ho, φ = (ржт)-1, we get a geodesically complete simply connected
Riemannian manifold of constant curvature — p2.
Exercise 8. Let Bp := {x € Дт; ||x|| < p) be a disk of radius ρ endowed with
the Riemannian metric g\ = ip2h0, where ho is the restriction of the canonical metric
of Дт, and ψ is a positive C°° function on Bp defined by ψ(χ) := 2/(p2 - ||ж||2).
Now we consider a diffeomorphism Φ : Нш —» Bp defined by
*(я'у)-Ч|1*11а + (у + 1)а'Н*На + (у + 1)а;еЯ
for (ж, у) G ВТ'1 х Я+ = Нт.
(1) Show that Φ : (Η171,до) —> (Bp,g\) is an isometry, where p0 = <^2^o and
<£ = (pxm)_1. Thus (Bp,gi) is of constant curvature -p2.
(2) Show that geodesies of (Bp,g\) are given by semicircles of Bp (including
straight lines through the origin) which are orthogonal to the boundary dBp.
We call (Bp,g\) the Poincare model of the hyperbolic space.
Figure 6
Exercise 9. Let Y(t) be a Jacobi field along a geodesic *y(t) in a Riemannian
manifold of constant negative curvature -p2, which is perpendicular to ^(t). Then
show that Y(t) may be written as
(3.38) Y(t) = cosh pt · a{t) + - sinh pt · b(t),
Ρ

4. FROM THE POINT OF VIEW OF THE TANGENT BUNDLE 53
where a(t) and b(t) are parallel vector fields along *y(t) with a(0) = У (0) and 6(0) =
vr(o).
4. From the Point of View of the Tangent Bundle
Recall that the Levi-Civita connection V : X(M) χ X(M) —> X(M) of a
Riemannian manifold Μ does not define a tensor field on M. However, we show
that this connection may be described by a linear object on the tangent bundle
Τ Μ of Μ. Such a point of view was implicitly expressed in the Prenet formula of
curves and surface theory of Gauss, but becomes clear after the notion of manifolds
was definitely established and the tangent bundle of a C°° manifold turned out to
be a C°° manifold.
4.1. Let tm : TM —» Μ be the tangent bundle of a Riemannian manifold M.
Let φ := τ^([/), ψ, (x\ ξ*)) be the chart of TM defined from a chart (t/, p, x{) of
M, namely, ^(u) = (хг(р),... ,*m(pU\ · · · ,Г) for u = ^(p),pG I/. Here
we further consider the double tangent bundle of Л/, namely, the tangent bundle
ttm : TTM -> TM of TM. Note that a chart (τ^φ), Φ) of TTM is given for
η := X^iu) + ^^-(u) G TnTM, и e TM, ρ = тд/u by
Then the projection т^л/ : TTM —> TM may be expressed in terms of the above
charts in the form
τΤΜ:(χ\ξ\Χ\ηι)^^1ΛΊ,
and the differential Dtm '· TTM —> TM of tm is given by
Prom these expressions we easily see that ttm and Dtm define the vector bundle
structures on TTM over TM. Now we consider the tangent space TUTM to Μ at
и G TM and set ρ = тл/U. First note that TPM is an m-dimensional submanifold
of Τ Μ as a fiber of tm over p, and TUTPM is an m-dimensional subspace of TUTM,
which coincides with Ker Dtm(v). On the other hand, since TPM is a vector space,
we may identify TUTPM with TPM. With respect to the above chart we have
TUTPM = {(^,е,0,^);(^) е Ят}, and
iu : (x\CM) ^ TnTpM .-> (ж4,ту4) G TpM
gives the identification. We call TUTPM the vertical space of TUTM and denote it
also by К,.
Now we show that we may naturally assign a complementary subspace Hu to
Vu in TUTM by the given Levi-Civita connection. Let X G TPM and take a curve
c(£) which is tangent to X at t = 0. Then the vector field u(£) along с obtained by
parallel translating и along с may be regarded as a curve in TM through u, and
we denote by Xu G TUTM the tangent vector to u(£) at t = 0.
With respect to the above chart, setting
f(0) = f and x*(t) = х{(сЦ)), we get
0 - (V » u(t)Y = C(0) + ГД(р)^(0).

54
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
u\ ^X
u(t)
c(t)
Figure 7
TM
rM
Μ
Therefore, we have
(4.1)
Хи:=(х\е,Х\-Г/к£кХ*)
and Xu does not depend on the choice of c, which will be called the horizontal lift
of X at и e TM.
Now the space Hu := {{х\С,Х\-Г/к£кХ1);(Х{) <Е Rm} of all horizontal
lifts of tangent vectors in TPM at и forms an m-dimensional subspace of TUTM,
which is called the horizontal space of TUTM at u. Obviously, we have VUC\HU = 0,
namely, TUTM = Ниф Vu. Note that, by the above local expressions, и »—► Vu, и »—►
Hu define C°° distributions of dimension m on TM. Now for и е TPM we define a
linear map К : TUTM —» TPM as follows: For η e TUTM we denote by ηυ (resp.,
ηπ) the Vu-component (resp., Я^-component) of η with respect to the decomposition
TUTM = Hu Θ Vu, and define as Κ (η) := tur]v· We call К the connection map of
the Levi-Civita connection.
Proposition 4.1. К : TTM —> TM zs α C°° map16 which satisfies the
following:
(1) The following two diagrams are commutative, and К is a bundle map with
respect to both of the two vector bundle structures of TTM.
TTM
TM
TM
Μ
TTM
dtm[
TM
TM
Ϊ™
Μ
TPM is a linear isomor-
(2) К | TUTPM : TUTPM —> TPM coincides with the identification iu.
(3) Hu = Κ~λ(ορ) (ρ = тми), and DrM{u) : Hu
phism.
(4) For и e TPM and X e X(M) we get K(DX(u)) = VUX, where we
consider X as a C°° map X : Μ —> TM.
PROOF. Noting that the decomposition of (χ\ξ\X\rf) G TUTM into
horizontal and vertical parts is given by
It is of class Cr l if the Riemannian metric is of class Cr.

4. FROM THE POINT OF VIEW OF THE TANGENT BUNDLE
55
we get
(4.2) K((x\^, Χ\η*)) = (χ\η* + ГД№"),
from which we may easily verify (1), (2), (3). To see (4), we set и = il-^(p), Χ =
Х1£т and note that DX(u) = (χ\Χ\ξ\^θ^1). Then we get K(DX(u)) =
{x^&djX1 + Γ/*Χ*ξ'), which is equal to VUX. D
Exercise 1. Conversely, for К : TTM -> ГМ satisfying (1), (2), we define VUX
by (4). Show that V defines a linear connection on TM.
Next we express the conditions (1.11), (1.12) in terms of the connection map
K. First we define a transformation j of TTM by
3(xi,e,Xi,Vi) = (xi,Xi,e,j),
which is in fact independent of the choice of charts. Then we get
(4.3) j(DY(X))-DX(Y) = Lxl[X,Yl X,YeX{M).
In fact, if we set X = Хг-^т and Υ = Y^-^j, the local expression of the left-
hand side of (4.3) is given by (х\Х\0,Х*д,У* - Y'djX1), which is equal to the
right-hand side. Therefore, (1.11) is equivalent to
(4.4) Koj = K.
Next we turn to (1.12). Recall that the unit tangent bundle UM = \JpeM UPM is a
(2m — l)-dimensional submanifold of Τ Μ and is the sphere bundle of тд/ : Τ Μ —>
Μ. In case of the Levi-Civita connection we have
(4.5) Hu С TUUM for и e UM.
In fact, for X e TpM recall that Xu (G Hu) is defined as the tangent vector at
t = 0 to the curve u(t) in TM obtained by parallel translating и G UM, which
remains in UM.
Exercise 2. Show that, conversely, the linear connection defined from К with
(4.5) satisfies (1.12).
Exercise 3. Let X,Y,Ze X(M) and Χ, Ϋ be horizontal lifts of X, У,
respectively. Show the following:
(0
(ii) K([X,Y]Z) = -R(X,Y)Z.
4.2. Now we list some merits of considering the connection on TM.
(I). We may introduce a natural Riemannian metric G on Τ Μ from a given
Riemannian metric g on Μ. In fact, for 77,7/ G TUTM, тми = ρ we set
(4.6) Gft,r/') := j(DrM(4), Dtm(1/)) + «?(*(!?), Κ(η')).
For 77 we identify щ with an element of TpM via Dtm, and 77^ with an element of
TpM via К | TUTPM = lu\ namely, we take η := {ηπ,'Πν) € ΓΡΜ Θ ΤΡΜ. Then
(4.6) can also be expressed as

56
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
and G defines a Riemannian metric17 on TM. We may also consider the induced
metric on UM, which is again denoted by the same letter G. These metrics G are
called Sasaki metrics. Note that, by (4.6), rM ' {TM,G) -> (M,<j), (f/M, G) ->
(M, g) are Riemannian submersions.
(II) (geodesic flow). We consider a vector field S on Τ Μ defined by Su :=
uu,u G TM. Namely, Su G TUTM is determined by the conditions Dtm{Su) =
u, /f 5n = 0, and is a horizontal vector expressed as Su := (u, 0). Also with respect
to a chart, Su may be written as
(4.7) я = (*',£', г.-где^*), « = ^
and is a C00 vector field. S is called the geodesic spray. The (local) flow generated
by S is called the geodesic flow and denoted by фг.
Lemma 4.2. VKe have (f)tu = Ίη(ί), wAere 7n denotes the geodesic of Μ with
7u(0) = тми,7и(0) = u. /n particular, φι leaves UM invariant, and S \ UM is
tangent to UM.
PROOF. Since *yu(t) is parallel along *yu(t), the tangent vector ^%{ί) to the
curve 11—> *y(t) in TM is equal to the horizontal lift ofju(t) at 7U(£), namely Sju(ty
This means that 0tu = 7U(£), and the remaining assertions are clear. D
Remark. The above characterization of geodesies is in fact the process of
reducing the system of second order differential equations
d2xl { dxj dxk _
~αΨ+ jk~M~dT~°
to the system of first order differential equations
dxl { άξι · , fc
We may also characterize Jacobi fields as vector fields on Τ Μ that are invariant
under geodesic flow.
Lemma 4.3. Let и G TPM, and let 7 = ηη be the geodesic with the initial
direction u. Identify TUTM = Hu 0 Vu with TpM 0 TpM, and denote by Y(t) a
Jacobi field along 7 with the initial conditions Y(0) = A, VY(0) = B for (Л, В) G
TUTM. Then
(4.8) £>&(А,В) = (У(*),УУ(*)).
Proof. Let £(s) be a curve in TM with £(0) = u and £(0) = (Л, Б), and
set c(s) := TAf^(s). Then a(t,s) := expc(s) ££(s) gives a variation of 7 consisting
of geodesies, and by the same computations as in (2.19) we see that the variation
vector field fjf(£,0) is a Jacobi field along 7, which satisfies the initial conditions
dct
— (0,0) = c(0) = L>rM£(0) = A
17G is of class Cr_1 if g is of class Cr

4. FROM THE POINT OF VIEW OF THE TANGENT BUNDLE
(Proposition 4.1 (1)). Hence ff (t,0) = Y(t). Further, we have
D<t>t(A,B) = js
M№) = ί
s=0
ds\ ^W(t)·
ls=0
Note that under the above identification, the horizontal part of the last term is
equal to
d_
ds
7ξ(.)(<) = ?£(*,0) = У(«),
s=0 OS
w = (έ^^^έ^)' w{u) = (°'^{p))
D™(UsJi{s){t))
and its vertical part is equal to
κ (£L*w(t))=v-^M)=v*£M)=w(i)·
This completes the proof. D
(III). Next we show that we may introduce a symplectic form α on Τ Μ via
g to apply the theory of Hamiltonian dynamics. In fact, for и G TPM and (Л, В),
(A', B') G TJM, under the identification TUTM ** TPM Θ TpM, we define
α{{Α,Β),{Α',Β')):={Α,Β')-{Α',Β).
Then clearly α is a nondegenerate skew-symmetric bilinear form on TUTM, which
defines a symplectic structure. We give a local expression of α with respect to a
chart of ΓΜ. For и — £*^~т(р) under the above identification we may write
_d_
dx"
and we have
<*(x\e) = 9ijdxi л <ti>j + 9ikTjkιξ1 dx1 Λ dxJ = dx* Λ d(g^3).
Then we easily see that α is a closed differential 2-form on TM and defines a
symplectic structure on TM.
Remark. On the cotangent bundle T*M of M, there exists a natural
symplectic form a* := <&гг Л dr\i, where (жг, 7^) denotes local coordinates of η G T*M,
i.e., 77 = 77г<&сг. Note that the above α is obtained from a* via the identification of
Τ Μ and Τ* Μ under b (p. 4).
Lemma 4.4. Tfte geodesic flow (j)t leaves invariant the symplectic structure a
on TM. Namelyf ф\а. = а.
PROOF. Recall that ф*га(щ^2) = αφφΜι,Όφ^). For (А,В) G TUTM we
have, from Lemma 4.3, D<j)t(A,B) = {Y{t),VY{t)), where Y(t) is the Jacobi field
along 7n with (У(0), VY{0)) = (A,B). Then we may write
a(D4>t(AuBi),D4>t(A2,B2)) = (Yi(t),VY2(t)) - (Y2(t)1VY1(t)).
The derivative of the right-hand side of the above equation vanishes because of the
Jacobi equation and the properties (2.10), (2.11) of the curvature tensor. Therefore,
for any t the left-hand side equals ct((Ai,B\), (A2,B2)), which is the value of the
left-hand side at t = 0. Π

58
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Now if we take a function E(u) = ^\\u\\2 on TM, then the geodesic spray S
satisfies
(4.9) a(S, X) = XE, X G X{TM).
In fact, we set Su = (u, 0),XU = (A,B) and let £(s) be a curve in TM with
£(0) = u, £(0) = Xu. Then clearly the left-hand side of (4.9) is equal to (u, B). On
the other hand, the right-hand side is equal to
2ds
«(*),«*)> = «(0), VAi(0)> = (η,Κξ(0)) = <u,B>.
s=0
We say that 5 is a Hamiltonian vector field on the symplectic manifold TM with
Hamiltonian E. Note that we have Csct = 0, which follows from Lemma 4.4.
4.3. Let TV be a submanifold of a Riemannian manifold (M,g), and let a
geodesic 7 = 7ξ : [0, b] —» Μ emanating perpendicularly from N with 7(0) = ξ G
TpN-L, ρ G AT, be given.
Definition 4.5. A Jacobi field У along 7 is said to be an N-Jacobi field if У
satisfies the initial conditions
(4.10) Y(0) G TP7V, УУ(0) - ΑξΥ{0) G TpTV\
where ^4ξ denotes the shape operator of N with respect to the normal vector ξ.
Note that the set of all 7V-Jacobi fields along 7 forms an m-dimensional vector
space J™.
Before giving geometric meanings of TV-Jacobi fields we prepare from the
viewpoint of §§4.1. Let vN : TN1- -> N be the normal bundle of N. Then for X G X(N)
and a C°° normal vector field ξ (i.e., section of ν χ) we may consider the covariant
derivative V^£ := (V^f ξ)-1 (see §3, Exercise 6), which defines a linear connection
on vN. Now as before we may consider the connection map K^~ : TTN1- —» TN1·
so that ϋΤ±(ί?ζ(Λ')) = V^£ and tf-1 | ΤξΤρΝ^, ξ G ТрЛГ-1, gives the canonical
identification between the tangent space ΤξΤρΝ1- to the vector space TpN1- and
TpN1- itself. For each ξ G TpN1- we set V£ := ΤςΤρΝ1- (= KerDi/N(0), #ξ :=
Ker (Κ1- Ι ΤξΤΝ^). Then we have a direct sum decomposition ΤςΤΝ1- = Ηξ Θ νξ
such that D//7V | Ηξ · Ηξ —* TpN and K^~ : VJ: —» TpN1- are linear isomorphisms.
Thus we may identify ΤξΤΛΓ1- with ΤρΝφΤρΝ1- by assigning (Л, В) G ΤρΝφΤρΝ1-
to r/ G ΤξΤΛΓ-1-, where we put Л = ΌνΝ(η), Β = Κ^-(η). We write this identification
in the form
(4.11) Tj:=(i4,B)N.
Then we may introduce a Riemannian metric on TN1- as the Sasaki metric on TM.
Now by assigning expp ξ G Μ to ξ G TpTV1- we have a C°° map exp-1 from an open
neighborhood of the zero section 0(TN±) := {op G TpN±]p G N] to M, which is
called the normal exponential map of N.
Lemma 4.6. Let Υ be a vector field along a geodesic 7 = 7ξ : [0,6] —» Μ (ξ G
TpN^) normal to N. Then Y(t) is an N-Jacobi field if and only if there exists a
C°° variation a : [0, b] x (—6, e) —» Μ such that the variation curves 11—> α(£, s) are
all geodesies perpendicular to N at t = 0 for fixed s G (—6, e).

4. FROM THE POINT OF VIEW OF THE TANGENT BUNDLE
59
PROOF. //. We may show that Y{t) := §7(^,0) is a Jacobi field just as in
the proof of (2.19). Prom the assumption we have a(0, s) G TV, £(s) := f^(0, s) G
Ta(o,s)N±. Then obviously ξ{0) = ξ, и := У(0) = |f(0,0) G TpTV. On the other
hand, we get
УУ(0) = V^^(0,0) = V^-^(0,0) = (Vu£(s))T + (V„e(S))x
= ^u+(v„e(S))x,
namely, VF(0) - ^У(О) € ΓρΛΓχ.
Onfy г/. For η := (У(0), УУ(0) - Л€У(0))^ € ΤξΤΝ^, take a curve ξ(β) in
ΤΝ1- that passes through ξ and is tangent to η at s = 0. We set c(s) := /^лг£(в)
and take a variation of 7 denned by a(t, s) := exp1- f£(s), which is the desired one.
In fact, У(£) := §j(£, 0) is a Jacobi field along 7 and satisfies the initial conditions
^(0) = ^(0'0) = m = DVNJ] = r(0)'
Vr(0) = V^^(0,0) = (VJlO(0) + (V^0T(0) = K^(0) + ЛсУ(0) = УГ(0).
Hence we get Y{t) = Y{t). D
Remark 4.7. £N := {(;4,B)N G T{TN,Ae TpN, Β-ΑξΑ G TpTV-1} is an
Tridimensional subspace of a symplectic vector space ΤξΤΜ, and we may easily check
that α | £дг Ξ 0. Recall that such a subspace is called a Lagrangian subspace. Now
in general for any Lagrangian subspace С of TUTM we may consider Jacobi fields
along 7n with the initial condition (У(0), УУ(0)) G С.
Next we consider TV-Jacobi fields in relation to the normal exponential map.
Let 7 = 7ξ : [0, b] —» Μ, ξ G ΤρΝ^, be a geodesic normal to TV. The following may
be proved in a manner similar to the previous lemma.
Lemma 4.8. Y{t) is an N-Jacobi field along 7 if and only if there exists
{A,B)N G ΤξΤΝ1- such that Y(t) = Dexp±{t^){A,tB)Nj where we regard(A,tB)N
as an element ofT^TN^, and Y(0) = Л, VY{0) = Β + ΑξΑ.
Exercise 4. Prove the above lemma.
Now for a geodesic 7 = 7ξ : [0, b] —» Μ normal to TV at £ = 0, if there
exists a nonzero TV-Jacobi field Y{t) along 7 with Υ {to) = 0{to > 0), we call
7(^0) = exp-1- £o£ a /oca/ pomi of TV along 7 and ίο its focal value. Prom Lemma 4.8
we see that 7(^0) is a focal point of TV if and only if rank/) exp-1 (£0£) < яг. The
nullity of Dexp±{to£) is called the multiplicity of the focal point. In the following
we list some facts on exp-1 and TV-Jacobi fields which correspondingly hold for exp
and Jacobi fields.
(4.12) For the normal bundle и ν : Τ Ν1- —» TV we identify TV with the zero section
(^{TN-1) via ρ \—► ор and regard AT as a submanifold of TN±. Suppose that the
normal exponential map exp-1 is defined on an open neighborhood Ρ of TV in TN1-.
Then the differential £>ехрх(ор) : T0pTN^ (^ TPN Θ TpTV-1) -> TPM is a linear
isomorphism by the same argument as in §§2.1. The inverse mapping theorem
implies that exp-1 is a difFeomorphism if restricted to an open neighborhood of each
op. Further we have exp-1 op = p, namely, exp-1 | TV = id^v· Then if TV is a closed
submanifold of Μ we may check that there exists an open neighborhood U of TV in

60
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
TTVX such that expx | U is a diffeomorphism onto an open set of Μ (we leave the
proof to the reader). If TV is compact, then there exists an e > 0 such that expx is
a diffeomorphism when restricted to £e(0(TTVx)) = {ξ G TTVX; \\ξ\\ < e}.
(4.13) Let 7 = 7ξ : [0, b] -> Μ (6 > 0), ξ G TpTVx\{op}, be a geodesic normal to TV,
and suppose that 7(6) is not a focal point of TV. Then У G J7^ ■-> У (6) G ΤΊ^)Μ is
a linear isomorphism. Namely, for any w G ΤΊ^)Μ there exists a unique TV-Jacobi
field У along 7 with Y(b) = w (see Lemma 2.4).
(4.14) (Gauss Lemma). Let ξ G TpTVx and {A,B)N Ξ ΤξΤΝ±. Recall that we
may identify (0,£)n Ξ TcTpTVx with ξ via tfx : TcTpTVx -> TpTVx. Then we have
(i) £>ехрх(*0(<М0* = *Ш, and in particular ||£>ехрх(0(0,ОлН1 = ||£||.
(ii) <Ζ?βχρχ(ίΟ(^«)Ν,7ξ(*)> = (Β,ξ)ί.
Exercise 5. Give a proof of (4.14).
(4.15) Let ξ G TpTVx, and suppose that expx is defined on a neighborhood V of
{*& 0 < t < 1} in TAT-1. For any curve t i-> ρ(ί), α < ί < 6, in Ρ С TTVX with
ρ(α) G 0(TTVx), р(Ь) = /ξ, we set c(t) := expx p(i). Then we have
L(c)>L(^) = Mb)\\=im.
Furthermore, if expx is regular on V and c([a, b]) φ 7ξ([0,/]), then L(c) > £(7ξ).
(4.16) Let TV be a closed submanifold of a Riemannian manifold Μ, and suppose
expx is a diffeomorphism on a neighborhood Ы of Ν = 0(TTVx). Then for any
point g G expx (W) there exists a unique minimal geodesic 7 parametrized by arc-
length from a point of N to q which realizes the distance d(g, TV), and 7 is given
by
7(i) := expx(i(expx | U)-1q)/\\(exp± | Η)"1?!!)·
Note that 7 is perpendicular to TV.
(4.17) Let 7 : [0,6] —» Μ be a geodesic emanating perpendicularly from a closed
submanifold TV of M. Suppose there exist no focal points of TV along 7. Put q = 7(6)
and CN,q := {c G C([0,6]); c(0) G TV, c(b) = q}. Then there exists a neighborhood
V of 7 in C;v,q([0,b]) with respect to the compact open topology such that for any
с G V we have L(c) > L(7) and L(c) > L(7) if 7([0,&]) ^ c([0,6]).
Exercise 6. Prove (4.15), (4.16), (4.17) referring to Lemma 2.7 and Corollaries
2.8 and 2.9, respectively.
Now we turn to the fact stated in Remark 4.7. We consider a geodesic 7 with
7(0) = ρ G M, 7(0) = и G TPM, and a Jacobi field Υ along 7. Recall that we
may decompose TUTM = TpM 0 TPM into the horizontal and the vertical spaces.
Now Υ is determined by the initial condition (У(0),УУ(0)) G TJM. Given a
Lagrangian subspace £ of TJM, we call a Jacobi field У with (У(0), УУ(0)) G £
an C-Jacobi field. Then the space J с of all £-Jacobi fields along 7 forms an
Tridimensional vector space, since dim С = m. Now if there exists a nonzero УG J с
with У(£о) = 0(^o > 0), we call 7(^0) an C-conjugate point to ρ along 7. Setting
W{t0) := {У G Jc\ У {to) = 0} we call n(t0) := dimW{t0) the multiplicity of the
£-conjugate point 7(^0)·

5. RIEMANNIAN MEASURE
61
Exercise 7. Let фг be the geodesic flow on Τ Μ and V(to) := {(0,£) G
T^(toyTM} the Lagrangian subspace consisting of vertical vectors. Show that
n(io)=dim(D0to£nV(io)).
Then we have the following.
Lemma 4.9. Let 7(^0) be an C-conjugate point along 7, and take a basis
{Yi}£Li of Jc such that Yi, · · · ,Yn(t0) form a basis ofW(t0). Then:
(1) {VYi{t0),··· ,VYn(t0)(to),Yn(to)+i{to)r·· ^m(to)} forms a basis of
(2) There exists an e > 0 such that {*i(£)}£Li .forms α 6aszs of ΤΊ^Μ for
0< I* — *o| <c
PROOF. Let indices i,j vary in the ranges 1 < г < n(to), n(to) + 1 < j < m,
respectively. {Vl^(£o)}i=i axe linearly independent, because Yi(£o) = 0. Next
note that {Y3 {to)} (j = n(to) + 1, · · · , m) are linearly independent. In fact suppose
(VY5)(to) = 0. Then VY5 G W{t0) and we may write VYa = а%. Since Уь · · · , Ут
forms a basis of J^, we have V = 0. Next we show that (VYi(to), Yj(to)} = 0.
Recall that the geodesic flow фг preserves the symplectic form α of Τ Μ and D(f)tC
is a Lagrangian subspace oiT^TM. Since (Yi(£o), VY*(£o)), (*i(*o), ^*j(*o)) G
D(f>toC, we have
0 = <*((У;(*о), ν^(ίο)), (Yj(to), VlS-(io)))
- (^(io), V^-(io)> - OS(*o), ν^(ίο)) = -<V*i(to), Yj{to)).
This completes the proof of (1). Then since {VY{(to), Yj(to)} forms a basis for
Ty{to)M and
Пт-^-=УУг(*0)
(because Yi(to) = 0; see Problem 11 for Chapter II), we see that for small e > 0,
{Yi{t)/(t - t0),Yj{t)} and consequently {Yi{t),Yj{t)} form a basis of Tl{t)M if
0 <| ί — *o |< с- п
Corollary 4.10. C-conjugate points appear isolated along 7. In particular,
focal points of a submanifold N along a geodesic normal to N are isolated, and so
are conjugate points to ρ along a geodesic emanating from p.
5. Riemannian Measure
5.1. First we see that we may introduce a natural measure on a Riemannian
manifold (M,g). Let Cb(M) be the vector space of real-valued continuous
functions on Μ with compact support, and define the norm of / G Cb(M) as ||/||
:= sup{|/(p)|; ρ G Μ}. In general, for a linear map μ : Co (Μ) —» Я, if for any
compact subset К С Μ there exists a positive number ακ such that
(5.1) |μ(/)| < aK\\f\\, fe C0(M), supp/ С К,
then μ is called a Radon measure on Μ. When Μ is compact, C(M) = Co(M)
is a Banach space with respect to the above norm, and a Radon measure on Μ is
nothing but a bounded linear function on C(M). In particular, a Radon measure
μ is said to be positive if μ(/) > 0 whenever / > 018. Now let g be a Riemannian
For functions f,gonM,f>g means that f(p) > g(p) for all ρ G M.

62
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
metric on M. Take an atlas Λ := {(UQ, ipQ,xlQ)}Q£A and a partition of unity {pQ}
subordinate to {UQ}. For / G Cq(M) we define
(5.2) !/,(/) := Σ [ (Ρα ' / ' V/G^) ο φ»* dxla^ dx™,
where G^ denotes the determinant of a matrix (g\j ) consisting of the components
of g with respect to local coordinates (xlQ). Note that the integrals of the right-hand
side are (Lebesgue) integrals of continuous functions with compact support defined
on open subsets (fQ{UQ) in -Rm, and the sum is in fact a finite sum because supp/ is
compact. We also denote vg(f) by JM fdvg. Now we check that this definition does
not depend on the choice of atlas and partitions of unity. Let В = {(V/з, ψβ, χ1β)}βεΒ
be another atlas and {τβ} a partition of unity subordinate to {V^}. First note that
G^ and G^ satisfy the following transformation formula:
(5.3)
jGW(p) = | detD(Vaorp-l)(^(p))\y/GM(p), p€Uan V0.
Then we have
Τ ί (Tfi'f'y/GW)o1,jldxy..dx2
β Ηβ{νβ)
β
= Σ / (τβ-Pa-f' VOW) οψ-1 dx\ ...dx]
= Σ / fo · Ρα · / · ν^ΐοψ; · I detD(^a o^1)! dx\ ---dx^
αβ Jl>0{v0nua)
= Σ/ (τ/3·Ρα·/·ν/№)θ^1^···<
^β^φα{ϋαηνβ)
(change of variable formula for integrals in R171)
(Pa-f->/GM)o<p-ldxla..'dxZ.
Now we easily see that vg satisfies (5.1) and gives a positive Radon measure on
M. Note that in the case of (Дт,р0), vgo coincides with the Lebesgue measure.
Remark 5.1. More generally, suppose that for any chart ([/, φ) of a C°°
manifold M, a continuous function μα defined on U is given so that {μυ} satisfies the
following for the coordinate transformation of charts (υ,φ), (V,^):
(5.4) μν(ρ) = | det Ό{φ ο ψ-ι)(ψ(ρ))\ μυ(ρ), peUnV.
Then we call {μυ} a density on M. It defines a Radon measure on Μ in the same
manner as above.
Remark 5.2. Recall that for an oriented manifold Μ we may consider the
integral fMu of a differential m-form ω. From a Riemannian metric g on Μ we
may define a differential m-form dM, which is called the volume element, as follows:
for a positively oriented o.n.b. {ei}^Ll we define dM(e\,... ,em) = 1. Then for
a positively oriented chart (/7α,<£α,:τα) we have dM = VG^ dxla Λ ··· Л <£г™.

5. RIEMANNIAN MEASURE
63
Ъ*Ф,'1
*\ /Φ,
Figure 8
Therefore, for an oriented Riemannian manifold A/ we may also write vg(f) =
JMfdMioTfeC0(M).
Now if a (positive) Radon measure is given on A/, we may develop the general
theory of the Lebesgue integral, which we will now briefly review without proofs.
(I) (integrable function, integrable set). For a lower semicontinuous function19
h > 0 we define ^{h) := sup{^(/); / G Cb(M) satisfies / < /ι}, and for any
function / > Owe define &£(/) := inf{i/*(ft); h > f is lower semicontinuous}. A
function / on Μ is said to be integrable if there exists a sequence {/n} С Co (Μ)
such that v*(\f — /n|) —» 0. Then {^(/n)} is a convergent sequence, and its limit
does not depend on the choice of {/n}· We denote the limit by JM f dvg, which we
will call the integral of /. In particular, / G Co(M) is integrable and its integral
coincides with the above vg{f). Next, a subset Л С М is said to be integrable if its
characteristic function χ a20 is integrable. We call fMXAdvg the measure (or
Tridimensional volume) of Л; it will be also denoted by vol Л or volm Л. In particular,
sets of measure 0 are called null sets, and we say as usual that properties that hold
except for null sets "hold almost everywhere".
(II) (measurable set, measurable function). A subset Л С М is said to be
measurable if Κ Π A is an integrable set for any compact set К of M. The family of
all measurable subsets of Μ is closed under the operation of taking the complement
and countable unions (intersections). Generally, a family of subsets of Μ that is
closed under the above operations is called a σ-algebra. For instance, elements of
the smallest σ-algebra containing the family of all open subsets of Μ are called
Borel sets. Borel sets, including open subsets and closed subsets, are measurable.
Any compact subset A is integrable, with vol Л < +oo. Next a function / : Μ —» R
is called a measurable function if the inverse image of any open subset of R is a
measurable subset of M. For instance, bounded lower semicontinuous functions are
19If pn —► ρ then we have lim inf h(pn) > h(p). h may take the value +00.
20χ a is defined as xa(p) = 1 if Ρ € A and xa(p) = 0 if ρ ^ Λ.

64 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
measurable. Also we note that a measurable function is integrable if and only if
v;(\f\) < +00.
Remark 5.3. Take an atlas {(/7;, ψι)}^\ of Μ consisting of countably many
charts with compact closures V{. We may choose open subsets W\ with W{ С U{
so that {W{} forms an open covering of M. Then a subset Л С М is measurable
(resp., a null set) with respect to a Riemannian measure i/g if and only if all Α Π W{
are measurable (resp., null sets). Now Α Π W{ is measurable (resp., a null set) if
and only if φ(Α Π W{) is measurable (resp., a null set) with respect to the Lebesgue
measure of Rm. Therefore, the property that Л С М is measurable (resp., a null
set) does not depend the choice of Riemannian metrics. For instance, submanifolds
of dimension η (η < га) are null sets, and for a C°° map Φ : N —» Μ the set
{p = φ(ς) e Μ; гапк/)Ф(<7) < га} of critical values of Φ is a null set of Μ by
Sard's theorem.
(III). Let Φ : Μ —» N be a difFeomorphism and ft a Riemannian metric on N.
Then we may define a Radon measure Φ*^ on M, called the pull-back of ^, by
ФЧШ-^^оФ"1), feCo(M).
Then we have in fact Φ*^ = ^φ*/ι· Namely, we have the following change of
variable formula:
(5.5) / fd^h= [ /οΦ"1^.
Jm Jn
In fact, for charts ([/, φ), (V,xp) of M, TV, respectively, we denote by Gu, #v the
determinants of the matrices obtained from the components of g = Φ*/ι and ft,
respectively. Then note that we have \fG\j = \ detD(ij> οΦο^_1)ο^|· \fHv ο Φ
and apply the usual change of variable formula. In particular, we get Φ*ν9 = vg
for an isometry Φ : Μ —> Μ.
Exercise 1. Give a detailed proof of (5.5).
For a submanifold TV of a Riemannian manifold Μ, the ra-dimensional volume
of N equals 0 if η := dim TV < dimM, as we saw in Remark 5.3. In this case
it is natural to consider the η-dimensional volume voln N of TV, which is defined
as the measure (i.e., η-dimensional volume) of N with respect to the Riemannian
measure Уф*д of the induced Riemannian metric via the embedding Φ : N —» Μ.
Note that voln N is finite if N is compact. Generally, for a C°° map Φ : N —> Μ
from a manifold TV to a Riemannian manifold Μ we set ft := Φ*#, and for each
chart ([/, <£,£г) of N we set ft^/ := y/det h(di,dj). Then {ftt/} gives a density ин on
TV, and we define the volume vol Φ as JN dv^. For instance, the length L(c) of a
smooth curve с : [α,6] —> Μ given in §1.2 is equal to vole.
Exercise 2. Let D be a domain of the (u, г>)-р1апе and Φ : D —> Д3 a smooth
map. Then show that vol Φ = JD \\xu Λ а^Н^шй;, where we set xu = (хи,Уи,ги),
etc., for Φ(η,ν) = (x(u,v),y(u,v),z(u,v)). Namely, vol Φ is simply the surface
area.
Now if we want to compute the volume of a given set or the integral of a
function, we usually take a chart and reduce to the computation of multiple integrals
in Euclidean space. For instance, take a normal coordinate system (Вг(р),(р,хг)
centered at p. Then expp : Br(op) —> Br(p) is a difFeomorphism, and we denote by

5. RIEMANNIAN MEASURE
65
g the canonical Riemannian metric on TPM defined by the inner product on TPM.
Then, setting g = g \ Br(p), g = g \ Br(op) for short, we get from the definition
(5.6) exp* ug = i/exp;g = ^/det(^ о ехрр)^.
We remark that the absolute value of the Jacobian of expp at χ G Br(p) is given by
д д
Λ···Λ
дх1 дхп
= yjdetgijiexppx).
Now we express points of Br(op) by polar coordinates. Namely, for χ G Br(op)\{op}
let (||x||,x/||x||) G (0,r) χ 5m_1 (S™-1 = UPM) be the polar coordinate of x. We
define a C°° map θ : (0, г) χ S™-1 -> Μ by
(5.7) θ(ί,τχ) :=expptu,
and we set
(5.8) 0(f,u) := r-ydet^(6(i,u)),
which is a C°° function defined on (0, r) x 5m_1. Then we have the following.
Lemma 5.4. (1) θ*ν9 = 0vgo, where go denotes the canonical product metric
on (0,r) x 5m_1.
(2) For u G 5m_1 iei {ei,··· ,em_bem := u) be an o.n.b. ofTpM. Take
Jacobi fields Yi(t) (i = 1, · · · ,m — 1) along the normal geodesic ju with Yi{0) =
0, VYi{0) = ei. Then
e(t,u) = \\Y1(t)A-.-AYrn-1(t)\\
(5.9)
Η
det(< Yi{t),Yj(t) >i<ij<m-i) ·
Proof. We define θ : (0,r) χ 5™-1 -> Br(op) \ {op} by θ(ί,τχ) = tu. Then
we easily get det De(t,u) = tm~l and θ*^ = tm-lu§0. Since from (2.16) the
equalities Yi(t) = tDexpp(tu)ei (г = 1,· · · ,m-l) and7n(£) = Z)expp(£u)em hold,
the absolute value of the Jacobian of expp at tu is equal to
Wd/дх1 л... лд/дхт\\ = \\(Yi(t)/t) л · · · л (ym_i(0/0 a7«(0ll
= ||г1(0л...лгт_1(0ИЛт-1,
from which (5.9) follows. Further, we have
e*vg = e*(exP; ug) = tm~l >/5it^J^0
= ||yi(0 л... л ym-i(0ll^o = *(*>u) "go,
which proves (1). □
In particular, we denote by a;m the volume of the unit ball B\(o) of (Дт,р0),
and by am_i the volume of the unit sphere 5m_1 with respect to the canonical
Riemannian metric. Recall that a;m may be written as 7гт/2/Г(у + 1) using the
Γ-function.

66
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Corollary 5.5. Let Μ be a Riemannian manifold and ρ e M. Suppose that
expp | Br(op) is a diffeomorphism. Then
vol Br (p) = / \ /det gij о exp dx1 · · · dx71
JBr(oP) V
= / dS171-1 [ e(t,u)dt.
Js™-1 Jo
(5.10) "~r(0p)
In particular, a;m = am_i/m for (Ят, <7o)·
PROOF. Applying Fubini's theorem from the theory of the Lebesgue integral,
we get
vo\Br{p)= / άθ*ν9= / e(t,u)dv~go
= [ {[ 0(t, u)dt}dSm-1.
Js™-1 Jo
Note that 0(f,u) = fm_1 in the case of (Ят,^о). □
Exercise 3. (1) Show that
0(t, u) = Г"1 - ^r+1 + 0(tm+2).
6
(2) Show that
/ p^dST-1 = тршт (UPM ^ S™"1).
(3) Show that for и £ UPM
lim ^Ш"1-^Ц) = PM Ит Г™"™ ~ VOl Br (p) uum
t^0 *™+1 6 ' r-o rm+2 6(ra + 2) p'
5.2. In this subsection we give some useful theorems on integration in
Riemannian manifolds. First we state Fubini's theorem, generalized to the case of
Riemannian submersions. Let π : (Μ, g) —» (TV, /ι) be a Riemannian submersion.
Recall that for each q £ π(Μ), π-1 (ς) is an (m - n)-dimensional submanifold,
which carries the Riemannian measure v9q with respect to the metric gq induced
on π~ι(α) from g. For a function / defined on Μ we set fq := / | π-1 (ς). Now if
fq is an integrable function on π-1 (q) with respect to vQq, we set
(5.П) /(*)=/ /*AV
Theorem 5.6. Let π : (M,g) —» (TV, /ι) 6e α surjective Riemannian
submersion. If f is a real-valued continuous function with compact support (resp., an
integrable function ) on M, then f is a continuous function with compact support
on N {resp., fq is integrable for almost all q £ N and f is an integrable function
on N ), and
(5.12) / fdug= ( fdvh (= f duh f fqdugq) .
JM JN \ JN JK-l(q) I

5. RIEMANNIAN MEASURE
67
PROOF. We give a proof in the case of continuous functions with compact
support; the integrable case may be proved in the same manner. Prom the mapping
theorem (Chapter I, Theorem 2.1 (2)), for any ρ G Μ we may take a chart ([/, φ)
with φ(ρ) = о around ρ and a chart (V = tt(U), ψ) with ip(q) = о around q := π(ρ)
so that <p(U) = B?{o) x Bf-n(o), ψ(ν) = B?{o) and
ψοποφ-^χ1,... ,xn,xn+\... ,хт) = (ж1,... ,xn).
Now we take an atlas {(Ζ7α, ψα)}α£Α consisting of charts with the above property,
and choose a partition of unity {pQ} subordinate to the open covering {UQ}· Then
we have
/ fd"g = Σ Ρ*' fdvg.
J μ a JuQ
Now suppose that we have proved the theorem in the case of π : Ua —» Va (a G Л).
Then, noting that supppa С /7Q, we get
V / pa · fdvg = J2 Pa' fdvh = J2 dvh \ pa · fdu9q
a JUQ a JVQ a JVQ λ-ΐ(ς)ΠΓα
= V / dvh \ Pa· fdis9q=y2 dvh \ pa-f dvgq
= dish }du9q = / fdvh.
JN J π-1 (a) JN
Чя)
Therefore, it suffices to show the theorem for π : U —» V assuming that supp f С U.
Now for the natural basis {д/дхг, д/dxa}\<i<n,n+\<a<m of TPM (p G U) and the
natural basis {d/dxl}i<i<n of TqN, we have
^(έ^)) = έω ^ 4^)
0.
Then the horizontal lift & of (d/dxl)(q) at ρ G π χ(ς), which is a vector
perpendicular to π-1 (q) and mapped to (d/dxl)(q) by Ζ)π, may be expressed as
* в(р)-°?£(р).
&E
&Εα
where α·1 is given as follows. Set g^ := (д/дхг, д/дхъ) and <7аь := (д/дха, д/дхь).
Let (<7а6) denote the inverse matrix of (<7аб)· Then we may easily see that af =
gib9ab- Therefore, denoting by Gu(p) the determinant of the matrix of the
components of the metric tensor g with respect to the above natural basis, we have
у/сЦЩ =
д
д
(ρ)
д
Шл-лШа^т(р)л-л&
= ||ξι(ρ)Λ··.Λξη(ρ)|| '
:(р)
(ρ)λ···λ^γ(ρ)
дхп+х
дхт'

68
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Since Dn restricted to the horizontal space Hp is an isometric linear isomorphism
and ϋπ(ξ{) = д/дхг, it follows that
VgU&j =
^γ(?)λ···λ^γ(?)
dx
—— ЫЛ---Л- (p)
n+i v^' /9rm
| dx1 νΊ/ &e
= ^det hi j(q)^detg^{p).
Now we set ж = (χ1,... , χη), ϊ = (χη+1,... , xm) for short, and define
f(x):= / f οφ~λ(χ,χ) ^detgab ο φ-1^,^) dxn+1 . ..dx171.
JB™-n{o)
Since the function in the above integral has compact support, it is uniformly
continuous with respect to χ = (χ, χ), and / is also a continuous function of χ with
compact support. Therefore, if we set I — m - n, the Fubini theorem for R171
implies that
I
fdVg
= / foip 1(x,x)Jdethij oip-1(x)y/detgab οφ~1(χ,χ)άχ1 · · · dx"
JB?(o)xBi(o) V
/B-(O)XB^O)
В-(о)
yldethij{^-l{x)) dxλ · · · dxn
χ / ί°ψ l(x, x)\/det даъ οφ~1(χ1 χ) dxn+l · · · dxn
Jb'{o)
= I f(x)Jdet hij ο ψ-*(χ) dx1 · · · dx171 = \ f dvh,
JB?(o) V JV
3»(°)
which completes the proof of the theorem.
D
Corollary 5.7. Under the assumption of the theorem suppose that the volume
of Μ is finite. Then
vol(M,£)= / νο1η(π l(q))dvh.
JN
In particular, for a Riemannian covering π : (M,g) —» (N,h) of order k, we get
vol(M, g) = к vol (TV, h). Further, for a Riemannian manifold Μ of finite volume
we have
vol([/M,G)=am_lVol(M,<7),
where G denotes the Sasaki metric.
Exercise 4. For the Riemannian product manifold (Μ χ N,g xh), show that
vol(M x N) = vol Μ · vol N.
We give an application of Fubini's theorem. Let / be a proper C°° function
defined on a Riemannian manifold Μ. Then the set of critical values of / is a
null set of R and the set О of regular values is an open subset of R. For t £ 0,
f~l(t) is a compact hypersurface of Μ, and the gradient vector V f(q), f(q) = t, is

5. RIEMANNIAN MEASURE 69
perpendicular to f~l{t). In fact, for any X £ Tqf~l(t) we have (V/, A") = Xqf =
0. Now we set
Ωέ := {ρ £ Μ; f(p) < f}, Vt := vol a,
Γέ := {ρ £ Μ; f{p) = *}, Л, := νοί^χΐγ
Theorem 5.8 (Coarea Formula). For an integrable function и on Μ the
following hold:
(1) Let gt be the induced metric on Tt from g. Then
(5.13) / u\\Vf\\dug = Γ dt [ udvgt.
JM J-oo JTt
(2) t \-► Vt is of class C°° at a regular value t of f such that Vt < +oo, and
(5-14) JtVt=i ΙΙ^ΙΓ1^·
PROOF. Let с be a regular value of / and take an open interval (a, b) С О
containing с. Then we have a vector field X := V//||V/||2 on an open subset
/_1(a, b) of M. Let φι be the flow generated by X. Then we get
£пш) = тх)Ыя)) = 1
and ί{ψο{α)) = с. Thus it follows that /(^_c(g)) = t for q £ /_1(c). Now since / is
proper, a map Φ : /_1(c)x(a, 6) —» /_1((a, &)) defined by Φ (ς, ί) := ipt-c(q) isasur-
jective diffeomorphism, and DQ>((q,t))d/dt = X(^_c(g)) = V//||/||2 is
perpendicular to f~l{t). We consider the induced Riemannian metric Ф*д on /_1(c) x (a> &)·
First we compute its components with respect to a chart (ψ,χ1) (1 < г < m — 1) of
/_1(c). For 1 < г, j < m - 1 we have
(**9){q,t)(d/dx\d/dxi) = g{D4>t-c(q){dldx%D<pt-c(q){dldx?)).
which is equal to the local expression of the metric gt induced on /-1(£) from g
with respect to a chart (ψ ο ip~[lc, хг ο φ^1€), and will be denoted by (^)^(Φ(ς, t)).
Second, note that
(Ф*д)Ш)(д/дх\д/сН) = 0, (Ф*д){дЛ)(д/сН,д/т) = g(X,X) = 1/||V/||2.
Therefore, for the local expression (Ф*д)аь (1 < α, b < τη) of Ф*д with respect to a
chart (ψ := ψ χ id, (жг, £)), we have
v/det(<i>*<7)ai,(<7,i) = ^(Λ)„(Φ(9,0)/Ι|ν/(Φ(ί,ί))ΙΙ·
Then from this fact, (5.5), and Fubini's theorem, it follows that
/ «||V/||di/e=/ («||ν/||)οΦΛ/Φ.9
Jf-1(a,b) Jf-l(c)x(a,b)
= J (иоФ)о ψ-1 Jdet(gt)ij ο Φ ο ψ-ι dx1... dxm-1dt
= / и ο (ψ ο 4>Tlc)~l J&&{дь)га о (^ о ^"Jj"1 dz1... ахш~1<И
7^(/-4c)x(a,6))
Г6
I dt I udvgt.
Ja Jf-Ht)

70
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
In the above we computed the integrals with respect to a fixed chart. To be more
precise, choosing an atlas of /_1(c) and a corresponding partition of unity, take the
sum of the integrals of the above local form. Since the set of critical values of / is
a null set, for an integrable function и on Μ we get
I dt I udvgt = / u\\Vf\\di/g.
J-oc Jf-l{t) JM
Further, for a regular value t we set и = 1/||V/|| on IV Then from
rt+h
Vt+h-Vt=l dt j^dugt
it follows that
rt+h
d 1 f + /* 1 f 1
*Vt=fe 11 dt k wi\\dV9t = k ijvTii ^-
which completes the proof of the theorem. Π
Remark 5.9. If we set Ω* := {p e M\ f(p) > t}, then (1) holds as it stands
and (2) becomes
d_
dt
>V* = -Jr HWir1^.
We remark that the above argument also works for the following case: Let Ω be
a domain in a Riemannian manifold Μ whose boundary 8Ω is a compact smooth
hypersurface. Suppose that a continuous function / : Ω —» R is of class C°° on Ω
and satisfies / | 9Ω = 0. Then the assertions (1), (2) of Theorem 5.8 hold.
Now we state an important integral formula, which is called the divergence
theorem or the Green theorem. First we recall the notion of manifold with
boundary. Let R™ := {(x\... ,zm); xm > 0} be the upper half-space of R171. Then the
boundary of R™ is given by the hyperplane Дт_1 defined by x171 = 0. Now, for an
open subset /7+ of Я+, a real-valued function / on U+ (resp., a map φ : U+ —» Rn)
is said to be of class C°° if / (resp., φ) is a restriction of a C°° function to an open
set U of Дт containing [/+ (resp., a C°° map from U to Rn). Then a (connected)
HausdorfF topological space Μ satisfying the second countability axiom is called an
m-dimensional manifold with boundary if Μ admits an atlas {(t/a, φα)}α£Α such
that
(i) {UQ} is an open covering of M.
(ii) φα : UQ —» R™ is a homeomorphism onto an open subset of Д+.
(iii) If UQr\Up Φ ф, then the coordinate transformation
ψβ ° Ψ*1 : Ψα{υα Π ϋβ) -> φβ{ϋα Π ϋβ)
is of class C°°.
Since for ρ e Μ the property that φα(ρ) G int R™ (resp., Дт_1) does not
depend on the choice of φα, we call ρ an interior (resp., boundary) point of Μ.
Then the set M° of interior points is a normal m-dimensional manifold and the
boundary dM of Μ, which consists of boundary points, is an (m — l)-dimensional
manifold unless дМ = ф. If Μ is compact (resp., orientable), then so is dM.
However, dM is not necessarily connected even if Μ is connected.

5. RIEMANNIAN MEASURE
71
Definition 5.10. Μ is called a compact m-dimensional C°° Riemannian
manifold with boundary if the following conditions are satisfied:
(1) Μ is an m-dimensional С°° manifold with boundary.
(2) Μ is a compact subset of an m-dimensional Riemannian manifold N.
Note that dM carries the induced Riemannian metric as a submanifold.
Theorem 5.11 (Divergence Theorem, Green Theorem). (1) Let X be a C1
vector field with compact support on a Riemannian manifold M. Then
(5.15) / divXdi/p =0.
/ dh
Jm
(2) Let (M,g) be a compact Riemannian manifold with boundary, and denote
by dA the Riemannian measure on dM with respect to the induced metric. Let ν
be the outward?1 unit normal vector field on dM. Then, for a C1 vector field X22
on M,
(5.16) [ dWXdvg= [ {X,v)dA.
JM JdM
PROOF. Let <pt be the flow generated by X. With respect to a chart ([/, φ, χ1)
of Μ we express the induced measure of ip*g in the form
First we give a preliminary lemma:
Lemma 5.12.
dt
άι/φ*9 = divX · dvg.
i=0
Proof of the lemma. First by a direct computation we see that the left-hand
side of the above equation is equal to
(X · detgij)/2y/detgij + y/detg~j —
,>ШК··^
Now recall that for a regular square matrix (atj(t)) with differentiable component
functions we have
-det{aij{t)) = (ау(0ая(0)^(ао-(0),
where (alj(t)) denotes the inverse matrix of (α^(ί)). Therefore, noting that £ \t=o
δφ^/δχ1 = δ{Χα, 8φ$/δχι = 6f, we see that the left-hand side of the equation in
the lemma is equal to
{\{Xkdk9ij)gij + д&Лv/del^dx1... dxm
= {ViX^y/dtf^jdx1... dx171,
which equals the right-hand side. D(Lemma 5.12).
21This means that ехрх(—tv) belongs to M° for sufficiently small t > 0.
22This means that X may be extended toaC1 vector field on TV.

72
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
Then for a compact subset К of M, vt := vo\((pt(K)) = j ,K) dvg = JK άνφ*9
is differentiable at t = 0, and from the above lemma we get
dt
-L
Vt = I div Xdvg
(to be precise we should take an atlas of Μ and a partition of unity).
Now we turn to the proof of (1). Since X has compact support and is complete,
<^ is defined for all £ £ Λ and leaves the complement of the compact support of
X invariant. In particular, ^(suppX) С suppX. Therefore t i-> vol(<^(suppX))
assumes its maximum at t = 0. Applying the above argument to К = suppX, we
conclude that t »-> vol(<^(suppX)) is different iable at t = 0, and we have
/ div X dvg = / div X dvg = —
JM ЛиррХ dt
(vol(y?t(suppX)) = 0.
t=o
Now we turn to the proof of (2). Since Μ is a compact subset of a Riemannian
manifold TV, we may consider the normal bundle vqm of a compact hypersurface
dM and the normal exponential map exp-1 : TdM^~ —» N. We may take an e > 0
so that exp-1 : Be(0(TdM^)) -> AT is an into diffeomorphism ((4.12)). Then from
(4.16) we see that exp-1 ВС(О(Г0МХ)) = Бе(<ЭМ) := {ς £ Ν; d(q,dM) < e}
and for any point q £ B€(dM) there exists a unique normal shortest geodesic
from dM to q that realizes d(q,dM). We denote the unit outer normal vector
at χ £ dM by ι/χ, and define a map Φ : <9M χ (-с, с) -> B€(0(TdML)) by
Ф(х, s) := svx. We also set Φ := ехрх оф : <9Mx (-€, б) -> Ве(дМ). Then Φ, Φ are
diffeomorphisms. Now for the flow ^ generated by X we get (pt(dM) С Ве(дМ) if
|£| is sufficiently small. Therefore, we may define C°° maps 0 : dM χ (-£, e) —» dM
and / : dM χ (-£, б) —» R by the following:
(5.17)
*-1M*)) := (0(M), f{x,t)),<t>(x,t) £ βΜ, /(x,0 £ Д.
Now we take a chart (V,ψ, {xl)i<i<m-i) of dM, and define a chart of B€(dM) as
(*(V x (-€,6)),^ = (ψ χ id) о ф-1, (у*, s)), where we set23 y{ = x{ орп оф"1, s =
pr2 ο φ-1. Then ^(Ф(ж, s)) is the tangent vector to the geodesic s »-> exp-1 s ι/χ,
and we have д/дуг(У(х, s)) = D4ts(x)(d/dxl(x)), where we set Ф5(ж) := exp-1 svx.
Note that the last tangent vector is equal to the value Yi(s) at s of a c?M-Jacobi
field Yi with the initial conditions Yi(0) = д/дх1(х), VYi(0) = Λ„χΥί(0). Now
from the Gauss lemma (4.14) we have g{-§^, J^) = 1, <7(J^, ^r) = 0. Therefore,
denoting by μ(χ,β) the positive square root of the determinant С?ф(ух(-С,б)) of the
components matrix of g with respect to ψ at Φ(χ,δ), we see that μ(χ,δ) is equal
23pri, pr2 denote the projections of дМ х (-e,e) onto the first and second components,
respectively.

5. RIEMANNIAN MEASURE 73
to \\Y\(s) Λ · · · Λ ym-i(s)||. Now from Fubini's theorem we have
/ &V9 ~ / &V9 — \ &V9 ~ / &V9
J\ptM J Μ JiptM\M JM\iptM
= / μ(ζ, s)^1.. .dxrn~1ds
J{{x,s);0<s<ma.x{f{x,t),0}}
- J μ(χ, s)dxl...dxm-ld
J{(x,s); min{/(i,i),0}<s<0}
= / dun Ι |μ(^, s)/yjdethij(x) > ds,
where (hij(x)) denotes the component expression of the metric h induced on dM
from g with respect to the chart ψ. Then from the lemma it follows that
/ div X dvQ = — / dvQ = lim - < / dvg - \ dvg \
J μ dt \t=0 JiftM t^o t IJ^tM J μ J
f fi /,/(x,t) ι 1
= lim / < - / μ(χ, s)/wdet hi3(x) ds > dv^.
г^° JdM (Uo v J
Then, noting that f(x, 0) =0 and
N Q ο Μ ι
μ^' °^ = Idx1^ A'"A dx™-1^ = Vdet Mx)
we get
/ div Xdvg= [ -J-{x,0)dvh.
J Μ JdM W
Thus it suffices to verify ^(x, 0) = (Xx, vx). To see this we fix an χ £ dM
in (5.17) and regard both sides of (5.17) as curves with parameter t. Taking the
tangent vectors to these curves at t = 0, we have
Xx = £>*(*,0) (^(x,0),^(x,0)\ = jt l^oexp^/OMKtx.o
Noting that f*(x,0) £ TxdM, we have §£ (x,0) = (Xr, ι/*). D
Corollary 5.13. Suppose Λ £ Cl(M), / £ C2(M) 24.
(1) If Μ is α Riemannian manifold and hV f has compact support, then
(5.18) / {(V/, Vh) - hAf} dvg = 0.
Jm
In particular, if f has compact support, it follows that JM Afdvg = 0. Next, if
hVf and fVh have compact supports, then
(5.19) / {ΛΔ/ - /Δ/ι} di/g = 0.
Jm
2ACr{M) := {/ : Μ -► Я; / is a Cr function}.

74 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
(2) Let Μ be a compact Riemannian manifold with boundary. Then, with
respect to the outward unit normal vector field ν to the boundary dM,
(5.20) / {(V/, V/i) - hAf} dvg = [ h(vf) dA.
JM JdM
In particular, JM Δ/ dvg = — JdM vf dA and
(5.21) [ {fAh-hAf}dug= [ {h(iyf)-f(vh)}dA.
JM JdM
PROOF. Prom (1.28) it follows that (V/, V/ι) -hAf = div(/iV/). Then (5.18)
immediately follows from Theorem 5.11(1). (5.20) follows from this equation and
Theorem 5.11(2) if we note that (W/, v) = h(vf). (5.19) and (5.21) follow from
the equation /Δ/ι - hAf = div(/iV/ - /V/ι) and (5.18), (5.20). D
We mention that when we apply the Green theorem with respect to the inward
unit normal of the boundary, we should change the sign of the right-hand side of
(5.16).
Exercise 4. Show that any harmonic function on a compact Riemannian
manifold is constant (use (5.18)).
6. Riemannian Submersion and Complex Projective Space
In this section we introduce the complex projective space with the canonical
Riemannian metric from a Riemannian submersion viewpoint. First, for a given
surjective Riemannian submersion π : (M,g) —» (B,/i), we are concerned with
the relation between the curvature tensors of (M, g) and (TV, /ι). Let V, VT be
the covariant differentiations of g, /ι, respectively, and denote by V1- the covariant
differentiation of дъ, which is the induced Riemannian metric on the fiber Fb :=
π-1 (6) over b £ B. In the following /7, V, W,... denote vertical vector fields on
Μ (i.e., vector fields tangent to the vertical space TpFb, b = π(ρ), at every point
ρ £ Μ), and X, У, Ζ,... denote horizontal vector fields. Now a horizontal vector
field X is said to be basic if there exists a vector field X onB such that Όπ(ρ)Χρ =
Χπ(Ρ) (ρ £ Μ)25. Then for a vector field X on В and ρ £ Μ, a horizontal vector
Xp £ TPM is uniquely determined by the condition Όπ(ρ)Χ = Χπ(ρ). Then
ρ ι—► Xp is of class C00, and we get a basic vector field X, which is called the
horizontal lift of X. Note that if Χ, Υ are basic then the horizontal component
[X, Y]T is the horizontal lift of [X, У], and [X, U] is vertical if X is basic and U
is vertical. Also note that we have U(X, Y) = 0 for basic vector fields Χ, Υ and a
vertical vector field U.
Now for £, F £ X(M) we define
(6.1) TEF := (У^^)т + (V^iF1)1,
(6.2) AEF := (VstF^ + (VetFt)x.
Then we may show the following by direct computations.
25In general, X and X which satisfy this relation are said to be π-related. It is easy to see
that if У, Υ are π-related then [X, У], [X, Y] are also π-related.

6. RIEMANNIAN SUBMERSION AND COMPLEX PROJECTIVE SPACE
75
Proposition 6.1. Τ and A are tensor fields of type (1,2) on M, and
(6.3) Ax Υ = i[X, У]х, TuV = Tv U,
where Χ, Υ {resp., U, V) denote horizontal (resp., vertical) vector fields.
Furthermore,
VvV = WbV + TvV, VUX = TUX + {VUX)T,
Vx U = (Vx C/)x + AXU, VXY = Ax Υ + (Vx У)т,
(Vx Y)T is the horizontal lift of VX Υ if Χ, Υ are horizontal
(6.5) . . x
lifts of X, Y, respectively.
In particular, if Τ ξ 0 then fibers are totally geodesic, and if Л ξ 0 then the
distribution defined by horizontal spaces becomes involutive. The next proposition,
which was first found by B. O'Neill, gives the relation between the curvature tensors
R, R-1, RT of V, Vх, VT, respectively. We omit the proof, which follows by direct
computations (see, e.g., [ON], [Bes-2]).
Proposition 6.2. Let X, Y, Z, Z' denote horizontal vectors and U, V, W, W
vertical vectors. Then the following equalities hold:
(R{U, V)W, W) = (R±(U, V)W, W) + (Tu W, Tv W)
-{ΤνΨ,ΤυΨ'),
(R(U, V)W, X) = {(Vu T)v W, X) - ((Vv Τ)υ W, X),
(R(U, X)Y, V) = ((Vx T)u V, Y) - (Tu X, Tv Y)
+ ((Vu Α) χ Y, V) + (Αχ U, AY V),
(R(U, V)X, Y) = ((VV Α)χΥ, U) - ((Vt, A)x Y, V)
( ' ' +(AXV, AY U) - (Ax U, AY V)
+ (TUX,TVY)-(TVX,TUY),
(R(X, Y)Z, U)=- <(VZ A)x Y, U) + (AY Z, Tv X)
-(ΑχΥ,ΤυΖ)-(ΑχΖ,ΤυΥ),
(R(X, Y)Z, Z') = (RT(X, Y)Z, Z') + 2(AX Y, Az Z')
+ (Ax Z, AY Z') - (AY Z, Ax Z%
where at every point ρ G M, RT (Xp, Yp)Zp denotes the horizontal lift of
RT(Dn(p)Xp, D*(P)Yp)(Dn(P)Zp).
Corollary 6.3. In particular, we have for the corresponding sectional
curvatures K, K^, KT of V, V-1", VT, respectively,
(6.7)
K(U, V) = KL(U, V) + \\Tu V\\2 - (Tv U, Tv V),
({U, V) is an o.n.b. consisting of vertical vectors)
K(X, U) = ((Vx T)u U, X) - \\TV X\\2 + \\AX U\\\
(\\X\\ = \\U\\=1)
K(X, Y) = KT(X, Ϋ) - Z\\AX Yf = KT(X, Ϋ) - 1\\[X, У]х||2,

76 II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
where {X, Y} is an o.n.b. of a 2-dimensional subspace in ТВ and Χ, Υ denote the
horizontal lifts of Χ, Υ, respectively.
From the corollary we see that the sectional curvature of В is greater than
or equal to the sectional curvature of the corresponding horizontal section of TM.
This fact is useful in constructing Riemannian manifolds of positive or nonnegative
curvature.
Now recall that the complex projective space CPn is constructed as follows.
We define an equivalence relation " ~ " in Cn+1 \ {0} as follows, where Cn+1 :=
{(z°, z1,... , zn)\ ζ1 Ε С} is (η + l)-dimensional complex Euclidean space:
(z0,...,zn)~(w°,...,wn)^(z°,...,zn) = X(w°,...,wn) (AeC\{0}).
Then an equivalence class [z° : ... : zn] represents a (complex) 1-dimensional
subspace of Cn+1, and the complex projective space CPn := (Cn+1 \ {0})/ ~ is
the space of complex lines of Cn+1 through the origin and carries the structure of
an 2n-dimensional C°° manifold.
Now let 52n+1 := {ζ Ε Cn+1; |z°|2 + · · · + \zn\2 = 1} be the unit sphere of
Cn+1 (= B?n+2) with the canonical Riemannian metric of constant curvature 1.
We define a map π : 52n+1 -> CPn by
(6.8) π(Λ... ,*"):= [z°:...:zn].
Then π is a surjective C°° submersion, and each fiber π_1([ζ° : ... : zn]) =
{е^'(Д... , zn); 0 < t < 2тг} is a great circle of 52n+1.
Now we define for t Ε R a C°° map φ% : 52n+1 -> 52n+1 by ^(z°,... , zn) :=
еУ=й(Д ^ zn^ Then ^0> # ^ гпj and ^^(^o,... , zn) are orthogonal with
respect to the canonical inner product of Cn+1 = Д2п+2, and t »-> ^t(z°,... ,zn)
defines a great circle (i.e., geodesic) emanating from (z0,... ,zn) with the initial
direction \f^l(z°,... , zn), which gives a fiber as above. Note that {^}^я gives
a one parameter transformation group of isometries. In fact, <pt may be expressed
in the matrix form β^~^ιΕη+\ Ε U(n Η-1) and is an orthogonal transformation of
д2п+2 pother we may check that the vector field ξ, ξρ = ^ \t=o ψίΡ, determined
by ipt is tangent to the fibers everywhere with \\ξ\\ = 1.
Now let (ζ0,... ,ζη) Ε 52η+1. Then for tangent vectors Xb, Yb to CPn at
b = [z° : ... : zn] and ρ Ε π-1 (6), there exist unique tangent vectors Xp, Yp
to Μ at ρ such that Dn(p)Xp = Xb, Dn(p)Yp = Yb and Xp, Υρ±ξρ. Moreover
we see that ΧψιΡ = ϋφί(ρ)Χρ holds because the ipt are isometries. We define a
Riemannian metric /io on CPn by
(6.9) /io№, Yb) := Po(^p, УР).
Note that this definition does not depend on the choice of ρ Ε π-1 (6) in the above.
Therefore, π : (52η+1, g0) —» (CPn, ho) is a surjective Riemannian submersion,
and ξ gives a vertical vector field. For a vector field X on £, the above X defines
the horizontal lift of X. Now basic vector fields Χ, Υ are ^-invariant and we get
[ξ, X] = 0, ξ · g0(X, Υ) = 0. In particular, Τ = 0 and
Αχ Υ = g0(Vx У, 0 f = -0о(Г, Vx ξ) ξ
by (6.2). Therefore, Corollary 6.3 implies that for an o.n.b. {Χ, Υ} of a plane σ
of TCPn, the sectional curvature Kj is given by KT(X, Ϋ) = 1 + 3||ЛХ ГЦ2.

PROBLEMS FOR CHAPTER II
77
Now we explain how the above Riemannian metric ho is closely related to the
complex structure of CPn. In general, for a (real) 2n-dimensional C°° manifold
M, a tensor field J of type (1,1) on Μ is said to be an almost complex structure if
Jp = -idrpM holds for any pGM, where we regard Jp : TPM —» TPM as a linear
map. (M, J) is called an almost complex manifold. For instance, any complex
manifold carries an almost complex structure which is the operation of multiplying
by γ/—Ϊ on each complex tangent space regarding as a real vector space. We call a
Riemannian metric h on an almost complex manifold Μ an Hermitian metric if Jp
is a linear isometry of hp at every point ρ £ Μ. For any Riemannian metric /ii on
Μ we have an Hermitian metric h defined by h(x, y) = {h\(x, y) + h\(Jx, Jy)}/2.
Now if the almost complex structure J of an almost complex manifold Μ is parallel
with respect to an Hermitian metric /ion Μ, /ι is said to be a Kahler metric. In
general almost complex manifolds are not necessarily complex manifolds. However,
it is known that an almost complex manifold Μ with a Kahler metric h is a complex
manifold. Then (M, J, h) is called a Kahler manifold.
Exercise 1. On an Hermitian manifold (M, J, h) define a differential 2-form
Ω by Ω(ζ, у) \= h(Jx, у). Show that Ω is closed if VJ = 0.
Now we turn to CPn. Let X be a vector field on CPn and X its horizontal
lift. We define a tensor field J of type (1,1) by
(6.10) JX := Όπ(νξΧ) = Dn{Vx£).
Note that νξ Χ = Vx ξ is a ^-invariant horizontal vector field. Then we have
J2X := Όπ(νξνχξ) = Dn(R{£, Χ)ξ) = -Dn{X) = -X
and J defines an almost complex structure. From ho(JX, Y) = ^o(V^X, Y) =
£go{X,Y) - 9o{X, νξΥ) = -h0(X, JY), we have /i0(JX, JY) = -h0{X, J2Y)
= ho(X, Ϋ) and /io is an Hermitian metric. Further, recalling Τ ξ 0, [ξ, Χ] = 0
and (6.5), we get
(VT J)Y = Vj(Jr) - JVjr = VjZMVcr) - £>ttV€(Vx Г)т
= Dn{(Vx Vc У)т - (VcVx У)т} = £>тг(Я(Л\ ОУ)
= ϋπ(9ο(Χ,Υ)ξ) = 0.
Namely, J is parallel and /io is a Kahler metric, which is called the Fubini-Study
metric on CPn. Also note that AXY = -g0{Y, νχξ) ξ = -h0(JX, Υ) ξ. Thus we
have
(6.11) KT{X, Ϋ) = 1 + 3/i0(JX, Г)2,
where {X, У} denotes an o.n.b. of a plane in TCPn. In particular, sectional
curvatures of CPn satisfy 1 < К J < 4, and Kj = 4 if and only if σ is a holomorphic
plane spanned by X, JX. Finally, note that if we start with the sphere of radius 2
then we get the Fubini-Study metric on CPn with \ < К J < 1.
Problems for Chapter II
1. Let Μ be a Riemannian manifold and с : [α, b] —> Μ а С1 curve. Then show

78
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
that
KOI-шИ*+ »>■«».
h—>o a
2. For a continuous curve с : [α, 6] —» X in a metric space (X, d), we define its
length Ld(c) by
Ld{c) := sup < 2^ d(c(ii-i), Φ0);
Δ : a = to < t\ < ... < tk = b is a subdivision of [a, b] > .
Next, for x, у G X define di(x, у) := inf{Ld(c); с : [a, 6] —» X is a continuous
curve joining ж to у }. Then note that d(:r, у) < di(x, у), and prove the following:
(1) For a piecewise C1 curve с in a Riemannian manifold (M, g) we have
L^(c) = Ld(c), where d denotes the metric defined by (1.8). A continuous curve
с : [α, 6] —» Μ with Ld(c) = d(c(a),c(6)) is a geodesic up to parametrization.
(2) For a Riemannian manifold (M, p) we have d = d{.
3. Let Μ be a Riemannian manifold and C([a, b]) the space of all piecewise C1
curves with the compact open (i.e., uniform convergence) topology. Show that the
functional L : с G C([a, b]) »-> L(c) G Я is lower semi-continuous (i.e., if cn —» с
(uniform convergence) then liminf L(cn) > L(c)).
4. Let Μ be a Riemannian manifold. Show that a chart ([/, <£, жг) with y?(t/) =
{(ж1,... ,хш)\ £^(яг)2 < r2} of Μ is a normal coordinate system if and only if
Ej 9iAx)xJ =χί·
5. Let (r, 0) be polar coordinates of the plane. We define a Riemannian metric g on
the plane by g(d/dr, д/dr) = 1, д(д/дг, д/дв) = О, д{д/дв, д/дв) = /2(r, 0),
where /(г, 0) is of class C2 with /(r, 0) > 0 (r > 0), /(0, 0) = 0 and §£(0,0) = 1.
Then show the following:
(1) Rays 70 defined by 0 = const, are geodesies.
(2) Let V(r) be a parallel vector field along 70 which is perpendicular to 70.
Then F(r) = /(r, 0)V(r) is a Jacobi field along 70.
(3) The sectional (Gauss) curvature at 70(r) is given by
d2/(r,fl)/f,r
6. We set £>n = {ж G Дп; ||x|| < 1}, Sm = {у е Дт+1; ||y|| = 1}. Let
(г, ж), r > 0, χ G 5n_1, denote polar coordinates on Dn. Let ds2l_1, ds2^ be
the canonical Riemannian metrics on 5n_1, 5m, respectively. We consider the
Riemannian metric g on Dn χ 5m given by ρ = dr2 + h2(r)ds2l_1 + /2(r)rfs2n,
where /(r), p(r) are positive real-valued C°° functions. Show that Ricci curvatures
of g are given as follows. Let v, w be unit tangent vectors of (Sn~1,ds2l_1) and

PROBLEMS FOR CHAPTER II
79
(Sm, ds2m), respectively, and set U = d/dr, V = v/h, W = w/f. Then:
(1) Rk(U, V) = Ric(V, W) = Ric(W, U) = 0,
(2) Ric{U,U) = -(n-l)ft-1 -ft" -τη/-1 · /",
(3)
Ric(V, V) = -ft-1 · ft" + (n - 2)ft-2(l - (ft')2) - m ft"1 · Λ' · /-1 · /',
Ric(W, W) = -/-1 ■ /" - (n - Dft"1 · ft' · Г1 · /'
U +(ГО_1)/-2(1_(Л2).
7. Let (M, p) be an m-dimensional Riemannian manifold. Then for the
curvature tensor R, Ricci tensor Ric and scalar curvature r, show that the following
inequalities hold at every point pGM.
(1) ||Ric||2 > τ2/m, where equality holds if and only if Ric = ^ g.
(2) ||i?||2 > 2 ||Ric||2/(m-l), where equality holds if and only if Κσ is constant
for any 2-plane σ in TPM.
8. Let (M, g), (M, g) be Riemannian manifolds and d, d corresponding distances
on Μ, Μ, respectively. Suppose that a surjective map / : Λ/ —► Λ/ preserves the
distances, namely, d(f(p), f{q)) = d(p, q) for any p, q e M. Show that / is a
difFeomorphism from Μ onto Μ with /*<j = g, namely an isometry. What happens
if we drop the assumtion on surjectivity ?
9. Let (M, g) be a Riemannian manifold and Τ Μ the tangent bundle of Μ. We
have C°° distributions V, i/ on TM defined by vertical spaces Vu and horizontal
spaces Hu of TUTM for u G TM, respectively.
(1) Show that V is completely integrable.
(2) Show that Η is completely integrable if and only if (M, g) is flat (i.e.,
Κσ=0).
10. Let (TM, G), (E/M, G) be Sasaki metrics on the tangent bundle and unit
tangent bundle of a Riemannian manifold (M, g). Show that TPM and UPM (p G M)
are totally geodesic submanifolds of Τ Μ and UM, respectively.
11. Let Y(t) be a vector field along a C1 curve с : [α, 6] —> Μ. Show that if
У(о) = 0, then УГ(а) = limt_a ^(0/(* ~ a)·
12. (1) Show that the group G of all isometries of (Rm, go) is generated by
parallel translations τα(α G ilm) and orthogonal transformations A G O(m). Show
that the subgroup of all parallel translations is a normal subgroup of G.
(2) Show that the group G of all isometries of Sm is 0(m + 1), where we
consider 5m as the unit sphere centered at the origin in Дт+1. Show also that if m
is even then any isometry φ G SO(m + 1) of 5m preserving the orientation admits
a fixed point, and if m is odd then any isometry of 5m reversing the orientation
admits a fixed point. What is the isometry group of (ДРт, до)?
13. For ρ > 0, we set H™ := {(*, x) G R χ Ят; ί2 - ||x||2 = 1/p2, f > 0}, which
is a hypersurface of Дт+1 diffeomorphic to R171 as a connected component of the
inverse image of a regular value l/p2 of a map (£, x) \—► £2 - ||:r||2. Note that in the
case m = 2 this is a sheet of the two-sheeted hyperboloid t2 - x\ — x\ = l/p2.

80
II. FUNDAMENTAL CONCEPTS IN RIEMANNIAN GEOMETRY
(1) Define a map и : Η™ — Bp С Rm by u(t, χ) := p2(l + pi)"1^· Then
show that и is a diffeomorphism and its inverse u~l is given by
sPP2-h\r р2-Ы\2
Now /io((^, x), (s, ι/)) := -is + (χ, y> defines a symmetric tensor field of type (0,
2) on Дт+1, whose restriction to H™ will be denoted again by h0. Show that
(u_1)*/i0 is equal to the Riemannian metric g = 4/{p2 - \\y\\2}2 · {^2{dy1)2} on
Вp given in §3, Exercise 8. Thus {H™, h0) is a geodesically complete Riemannian
manifold of constant curvature -p2.
(2) Show that the isometry group of H™, h0) is 0+(l, m) := {a : RxR™ ->
R χ Дт; a is a linear map with a*/io = ho and a(H™) = H™}.
(3) Show that geodesies of (H™, /i0) are obtained as the intersections of H™ С
Дт+1 and 2-dimensional subspaces of Дт+1.
14. Show that connected components of the fixed point set of an isometry of a
Riemannian manifold Μ are totally geodesic submanifolds.
15. Let Μ be a Riemannian manifold and 7 a geodesic on Μ emanating from ρ
with 7(0) = и G UPM. For an o.n.b. {е»}^, ет = u, denote by {e^i)}·^ the
parallel translation of {e*} along 7. Let Yi(t) (i = 1,... ,ra - 1) be Jacobi fields
along 7 with the initial conditions У*(0) = 0, VYi(O) = e». Define a (m-l)x(m-l)
matrix j4(£) by У*(£) = j4(£)e»(£) (г = 1,... ,m - 1). Then show the following :
(1) A"(t) + Ay(t)i4(i) = 0, where Ry(t) denotes a (m - 1) χ (m - 1) matrix
defined by Щ^е^Ь) := R(e{(t), 7(£))7(£). j4(£) satisfies the initial conditions
i4(0) = 0,i4,(0) = £?m-i.'
(2) A(t) = tEm_i - ^Ru - &K + 0(*5), where /ζ(ж) := (Vttfl)(x, u)u.
(3) 5r(p) := expp{x G TpM : ||x|| = r} is a hypersurface of Μ difFeomorphic
to 5m_1 when r is sufficiently small. 7(7·) is a unit normal vector to Sr(p) at 7(7·).
Then the shape operator Αγ(Γ) is given by Л'(г)Л_1(г).
(4) Ay(r) = iEm_i - §Ди - ^K + 0(r3).
16. Let ((7M, G) be the Sasaki metric of the unit tangent bundle of a Riemannian
manifold Μ of dimension m. Let 77 be the dual differential 1-form of the geodesic
spray S and dvQ the volume element of G.
(1) Show that dvc is given by l/(ra - 1)! · 77 Λ (dr?)171-1 up to sign.
(2) Show that the geodesic flow <j>t leaves dvc invariant.
Notes on the References
There are many textbooks on Riemannian geometry; some recent ones are [Bi-
Cr], [M-l], [Gr-K-Me], [Ko-No I], [Ch-Eb], [B-Ga-Ma], [Ga-H-La], [Bes-2], [B-Go],
[K-5], [Sp-2], [dC], [Мог], [Cha-3]. In this chapter we owe a lot to [Gr-K-Me],
[Ch-Eb], [B-Ga-Ma], [Bes-2], and [K-5].
§1. We followed [No], [Ko-No-I], [Gr-K-Me], etc., for the modern expression
for the Levi-Civita connection. However, the classical tensor calculus with indices
still seems to be useful (e.g., for a computation taking traces).
§2. The material treated here is rather standard. We use the method of Jacobi
fields throughout the present book. Corollary 2.9 is due to [Ga-H-La].

NOTES ON THE REFERENCES
81
§3. For Proposition 3.1 see also [B-Ga-Ma], [Sa-1], [Gra, Chap. 9]. (3.12) is
due to [Ga-H-L], and an explicit formula for (R(x, y)z, w) in terms of sectional
curvatures and the inner product is given in [Ch-Eb], p. 16. For more details about
Proposition 3.9, see, e.g., [Bes-2].
§4. The geometry of the tangent bundle was investigated systematically first
by S. Sasaki ([Sas]). For the connection map we followed [Gr-K-Me] (see also [Bes-
2]). I learned a characterization of Jacobi fields from the geodesic flow viewpoint
from W. Klingenberg, and Lemma 4.9 is due to him. There is another important
approach to connections in terms of frame bundles of manifolds, which is useful in
treating general geometric structures on manifolds (see, e.g., [Ко-No I], [St], [Po]).
§5. The main references for this section are [B-Ga-Ma] and [Cha-3]. For the
Radon measure we refer to, e.g., [Schw]. The Fubini theorem, the Coarea formula,
and the Green theorem will be used throughout this book.
§6. Curvatures of Riemannian submersions were first studied by B. O'Neill
([ON]) and play a role in Riemannian geometry as important as that of immersions.
For more details on Riemannian submersions (e.g., Proposition 6.2) we refer to, e.g.,
[Bes-2].

CHAPTER III
Global Concepts in Riemannian Geometry
In this chapter we are mainly concerned with the fundamental concepts of
Riemannian geometry which are related tc the global properties of manifolds. The
behavior of geodesies emanating from a point ρ of a Riemannian manifold Μ is
related not only to the curvature of Μ but also to the global properties of Μ. In
Euclidean geometry straight lines are also distance minimizing curves. However,
in Riemannian geometry, a normal geodesic 7 | [0, t) emanating from ρ is distance
minimizing for small t > 0, but in general there exists a point 7(^1) where the
distance minimizing property first breaks down. The set Cp of these points along
the geodesies emanating from ρ is called the cut locus of p: it is empty in Euclidean
geometry. It turns out that the cut locus inherits the topology of Λ/ to a great
extent. To consider such concepts, geodesies should be defined for all parameter
values. In §1 we treat this condition, which is called completeness, and give the
fundamental Hopf-Rinow theorem.
After that, with respect to the minimizing property of geodesies, we consider the
space С of curves on Μ satisfying some boundary condition, and the energy integral
Ε (or the length L) which is a functional on С Then the geodesies satisfying the
boundary condition are characterized as critical points of Ε on C. For the distance
minimizing property of geodesies, the Hessian of Ε plays an important role. In
fact, M. Morse defined the index of geodesies from this viewpoint in the calculus
of variations, and gave the index theorem which expresses the index of a geodesic
in terms of conjugate points. Further, a generic С has the homotopy type of a
CW-complex which is obtained by attaching a fc-cell to each geodesic of index k.
Therefore, the behavior of geodesies on Μ is essentially related to the topology of
С and consequently also to the topology of M.
In §2 we give the first and second variation formulas for the energy integral E,
and in §3 we treat the Morse theory for С by approximating С via finite-dimensional
C°° manifolds. In §4 we consider the cut locus, and also introduce the concept of
the injectivity radius in relation to the cut locus. The cut locus is also closely
related to the differentiability of the distance function.
Now recall that in Chapter II we stated a theorem of Cartan, which shows how
curvature locally determines the Riemannian metric. In §5 we state a theorem due
to W. Ambrose which is a global version of Cartan's Theorem. In the final section,
§6, we explain about the isometry group and the holonomy group of a Riemannian
manifold, and state the de Rham decomposition theorem, which is a typical global
theorem in Riemannian geometry.
1. Complete Riemannian Manifolds
If we try to study the global behavior of a Riemannian manifold Μ by looking
at how geodesies run on Μ, it is desirable that geodesies may be extended for all
83

84
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
parameter values, and that any two points in Μ may be connected by a minimal
geodesic. Recall that in §2 of Chapter II, we said a Riemannian manifold Μ is
geodesically complete at ρ £ Μ, if all geodesies 7n(£), и £ TPM, emanating from ρ
are defined for all — oo < t < oo. Μ is called geodesically complete if it is
geodesically complete at all points. For instance, (-Rm, go) is geodesically complete, but the
Riemannian manifold obtained by removing one point from (-Rm, go) is not
geodesically complete at any ρ £ Μ. Also note that Hm := {(x1,... , xm); xm > 0} with
the induced metric from (Дт, до) is not geodesically complete at any point, but
is geodesically complete with respect to the hyperbolic metric introduced in
Chapter II, §3 (V). The next fundamental theorem shows that geodesic completeness
reflects nice properties when we try to understand the global behavior of a
Riemannian manifold.
Theorem 1.1 (Hopf-Rinow Theorem). Let (M, g) be a (connected)
Riemannian manifold. Then the following conditions are equivalent:
(i) (M, g) is geodesically complete at a point ρ £ Μ.
(ii) (Μ, g) is geodesically complete.
(iii) For a fixed point ρ £ Μ, the set Br(p) := {q £ M; d(p, q) < r} is compact
for any r > 0.
(iv) For any ρ £ Μ and any r > 0, Br(p) is compact.
(v) (M, d) is complete as a metric space. Namely, any Cauchy sequence of Μ
is a convergent sequence.
Definition 1.2. A Riemannian manifold (M, g) which satisfies one of the
above conditions is simply called a complete Riemannian manifold.
Corollary 1.3. For any two points p, q of a complete (connected) Riemannian
manifold M, there exists a minimal geodesic 7 joining ρ to q (i.e., a geodesic with
Lb) = d(p, <?))·
Corollary 1.4. Let Μ be a compact C°° manifold. Then any Riemannian
metric g on Μ is complete. (The converse also holds. See problem 1 at the end of
this chapter.)
Proof. We will give the proof by showing that (i) —»the assertion of Corollary
1.3 and (i) —> (iii) —► (iv) —► (v) —► (ii) —► (i). First, assuming (i), we consider the
following assertion:
(*)r For any q £ Br(p) there exists a minimal geodesic 7 joining ρ to q.
We show that (*)r holds for any r > 0. To see this, first take e > 0 such that
expp I B2e(op) is a diffeomorphism onto Z?2€(p). Then, by Corollary 2.8 of Chapter
II, (*)r holds for 0 < r < 2e, and Se(p) := dB€(p) is compact. Now let r > e and
q £ Br (ρ) \ B€ (p) be given. Note that there exists a point q £ Se (p) such that
dfa q) = d(p, q) + d(q, q) = e + d(q, q).
In fact, choose piecewise C°° curves cn £ Cpq (n = 1, 2, ...) parametrized by arc-
length such that L(cn) < d(p, q) + ^. Let qn be the first intersection point of cn
with Se(p). Then we have d(p, qn) + d(qn, q) < L(cn) < d(p, q) + ^. Since Se(p)
is compact, we have an accumlation point q of {<7n}, which is as desired because of
the triangle inequality and the inequality obtained by taking the limit of the above
inequalities.
Now let 7 = 7P> $ be a minimal geodesic parametrized by arc-length joining ρ
to q. 7 may be defined for all parameter values, by virtue of the assumption (i). It

1. COMPLETE RIEMANNIAN MANIFOLDS
85
suffices to show that 7(d(p, q)) = q. For that purpose we set
Τ := {t G [0, dip, q)]; dip, 7(t)) = t and
dip, 7(t)) + d(7(t), q) = dip, q) - (**)}
and set t0 := supT. If we see that to = d(p, q), then we have 7(d(p, <?)) = 9 from
(**).
Figure 9
Note that Τ is clearly a closed subset containing [0, e]. In the following we
derive a contradiction assuming that to < d(p, q). Set q[ = 7(£o) and choose a
<5 > 0 such that 26 < d(p, q) —to and B2e(qf) ls a normal coordinate neighborhood.
Since q φ q' we may take a point 91 G Ss(qf) such that
d(</, gi) + d(<?i, q) = d(qf, ρ),
as before. Let 7i be a normal minimal geodesic with 7i(£o) = Qf, 7i(£o + 6) = q\.
Then we get
d(p, ?') + Φ', 9i) = d(p, «0 + d{q', q) - d(qu q)
= d(p, q) -d(qu q) < d{p, qx),
and consequently, by the triangle inequality,
dip, q') + d{q\ qi) = d(p, qx).
Therefore, 7 and 7i make a straight angle and 7 | [0, to] U 7i forms a minimal
geodesic joining ρ to q\ (see Chapter II, §2, Exercise 5). Namely, we get
Ql = 7(*0 + δ), dip, 7(t0 + «)) = ίο + «
and
d(p, Qi) + Φι, 9) = dip, Я') + Φ', 9i) + %b 9) = d(p, q') + d(q', q) = d{p, q).
Then to + <5 G .F, which contradicts the definition of £0- This completes the proof
of (*)r for all r > 0 assuming (i).
Now we show (iii), assuming (*) for all r > 0. For any sequence {qn}%Li С
Br{p) take minimal geodesies 7n joining ρ to qn, parametrized by arc-length. Let
un G UPM be the initial direction of 7n. Since UPM is compact, we have an
accumulation vector и G UPM of {un}. We also take an accumulation value / of
{d(p, qn)}, and we may choose subsequences {иПк} and {qnk} such that wnfc —»

86 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
u, d(p, qnk)-*l- Then (d(p, qnk)) -+ 7u(0 £ Br{p), and {qn} admits an
accumulation point.
Next we prove (iv) assuming (iii). For any q £ Μ and r > 0 we have Br(q) С
Br+d{P,q) (p) by the triangle inequality. Then Br(q) is compact because it is a closed
subset of a compact set Br+d{p,q)(p)·
Now we establish (v), assuming (iv). Take a Cauchy sequence {pn} of (M, d).
Since {pn} is bounded, we may choose an r > 0 such that {pn} С Br(p). Since
Br(p) is compact by (iv), we may choose a subequence {pnk} sucn that Pnk —> q €
Br(p). Then in fact we have pn —> q from a property of Cauchy sequences.
Next we show (ii) from (v). In view of (2.3) of Chapter II, it suffices to show
that any normal geodesic is defined for all parameter values. Let 7 be a normal
geodesic emanating from p. We set Τ := sup{t > 0; 7 is defined on [0, t}} and
show that Τ = +oo. Suppose Τ < +oo, and take any sequence {tn} with tn | T.1
Then
d(7(*n), 7(*m)) < Ц7 I [*n, U) < \tn ~ tm\.
Namely, {"y(tn)} is a Cauchy sequence of Μ and converges to a point q £ M. Take
an open neighborhood W of q which satisfies (2.5) and (2.6) of Chapter II. Then for
sufficiently large η > m we have 7(£n), 7(£m) € И^, and 7 | [tm, tn] is a minimal
geodesic joining 7(£n) to 7(im). Now take a normal minimal geodesic 71 joining 7(£n)
to ρ with 71 (tn) = 7(*n). Then, noting that d(7(i„), ρ) = lim^oo d(7(i„), 7(iz)) =
Τ -tn, we get
d(<7, 7(*n)) + d(7(i„), 7(*m)) = d(^, 7^m))·
Therefore, 7 | [tm, tn] U 71 makes a straight angle at 7(in) = 7i(^n), and 7 may
be defined on [0, T]. Then, considering a normal geodesic emanating from η(Τ)
with the initial direction 7(T), we may extend 7 beyond the parameter value T,
which contradicts the definition of T. Therefore Τ = +oo. Similarly, inf{£ <
0; 7 is defined on [£, 0]} = -00.
Finally, (ii) —> (i) is obvious, and Corollary 1.4 follows from (v). D
Exercise 1. Show that a Riemannian manifold (Μ, g) is complete if and only
if its geodesic spray 5 is a complete vector field on Τ Μ (or UM).
Now we remark that any C°° manifold admits a complete Riemannian metric.
In fact, Μ may be embedded as a closed submanifold of R2rn by Whitney's theorem.
Then the metric induced on Μ from (Д2т, до) is complete.
Exercise 2. Prove the above fact.
We note that we may approximate any Riemannian metric g on Μ by complete
Riemannian metrics with respect to the C°° topology (see [Morr]).
Now we define the diameter d(M) of a Riemannian manifold Μ by
d(M) = sup{d(p, q); p, q £ M}.
If Μ is complete, then d(M) < 00 holds if and only if Μ is compact. In fact, if
Μ is compact then the continuous function d defined on the compact set Μ χ Μ
assumes its maximum. Conversely, if d(M) < +00, then Μ = Bd^M){p) for any
ρ £ Μ, and Μ is compact by Theoren 1.1 (iii).
In the following we assume the completeness of the Riemannian metric when
we consider global properties of a Riemannian manifold.
ltn Τ Τ means that tn(< T)(n = 1, 2,...) are monotone increasing and converge to T.

2. VARIATION FORMULAS AND JACOBI FIELDS
Remark 1.5. In the above proof we deduced Corollary 1.3 assuming (ii) by
considering the boundary value problem for the ordinary differential equation
satisfied by geodesies. On the other hand, from the viewpoint of the calculus of
variations, geodesies are stationary curves of the length function on the space of curves
joining two given points. Prom this viewpoint we may deduce Corollary 1.3
assuming (iv). Let L be the length functional on the space Cpq([0, 1]). Prom the definition
of the distance, there exists a sequence {cn} of Cpq([0, 1]) with L(cn) —» d(p, q). We
may assume that the cn are of constant speed. We show that {cn} has a convergent
subsequence with respect to the uniform convergence topology. First, note that the
cn are contained in a compact subset B2d(p, q) (p) for sufficiently large n. Second,
the cn : [0, 1] —» Μ (η = 1,2,...) are uniformly continuous, since
d(cn(t), cn(t')) < L(cn | [t, t']) = \t- t'\L{Cn) < 2d(p, q)\t -1'\.
Then from the Ascoli-Arzela theorem we have a convergent subsequence cnk —» с,
where с : [0, 1] —» Μ is a continuous curve, with respect to the uniform convergence
topology. Then we have d(p, q) = lim^oo L(cnk) > L(c) > d(p, q). Namely, с is a
shortest curve joining ρ to q and is a miminimal geodesic.2
Remark 1.6. Corollary 1.3 may hold even if (M, g) is not complete, e.g., an
open ball Br(o) of R171 with the metric induced from (-Rm, go)· Also for the case of
compact Riemannian manifolds with boundary, condition (i) is not so meaningful.
However, it is possible to show that (iii), (iv), and (v) are equivalent.
Exercise 3. Let (M, g), (TV, h) be complete Riemannian manifolds, and let
(Μ χ TV, g χ h) be the product Riemannian manifold. Prove the following:
(1) g χ his complete.
(2) For ((pi, (ft), (p2, <Ы) € Μ x TV the following Pythagorean theorem holds:
(1.1) d2gxh((pu tfi), (pa, 42)) = d2g{Pl, pa) + d2h(qu q2).
Exercise 4. Let Μ be a complete Riemannian manifold and TV a closed sub-
manifold of M. Show that for any q G Μ there exists a geodesic 7 joining a point
ρ e N to q with £(7) = <%, TV) (:= inf{d(q, г); г G TV)}), which is called a
minimal geodesic from TV to q (7 is perpendicular to TV at p. See Proposition 2.4 of the
next section).
Exercise 5. Show that a 1-dimensional complete Riemannian manifold is
isometric to either (Д, go) or the circle (51, go) of length /.
2. Variation Formulas and Jacobi Fields
2.1. In Chapter I, §2, we considered the length functional L on Cpq([a, 6]), and
got a necessary condition to be a shortest curve at which L assumes the minimum.
Since L is invariant under parameter transformations, such shortest curves are
determined up to parametrization, and sometimes L is inconvenient. On the other
hand, for с G C([a, b\) we define the energy integral E(c) of с by
(2.1) E(c) = \J^ \\c(t)\\4t,
2See problems 2 and 3 for Chapter II.

88
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
which is also a functional on С ([a, &]). Prom the Cauchy-Schwarz inequality we get
(2.2) L2(c)< 2(6-a)E(c),
where equality holds if and only if ||c(£)|| is constant, namely, с is of constant speed.
In this section we consider the space of curves satisfying some boundary
condition endowed with the functionals Ε or L. Let Μ be a complete Riemannian
manifold and В a closed submanifold ofMxM. We set
Св([а, b}) := {c : [a, b] —» M; с is a piecewise C°° curve
(2'3) with {c{a), c(b)) £ B},
where В gives a boundary condition. For instance, if В = N\ x 7V2, where N\, N2
are closed submanifolds of M, then Св([а, b}) is a family of curves joining points of
Λ/χ to points of N2. If Β = {(ρ, ρ) £ Μ χ Μ; ρ £ Μ} is the diagonal set of Μ χ Μ,
then С в {[a, b}) is the family of closed curves of M.
Now we consider L, Ε as functionals on Св([а, 6]) and look for curves at
which L, Ε assume the minimum. We apply the calculus of variations, which is a
generalization of differential calculus for functions of several variables to infinite-
dimensional spaces. Namely, considering С в ([a, 6]) as an "infinite-dimensional
manifold" we first look for "stationary (critical) curves" as candidates for minimal
curves. Prom this viewpoint the energy integral Ε is convenient, since Ε may be
considered as a "difFerentiable function" on С в ([a, b}).
Remark. It is in fact possible to introduce an infinite-dimensional smooth
manifold structure modeled on a Hilbert space on the space of absolutely continuous
curves с with square integrable ||c(£)||, which includes the space of piecewise C°°
curves, so that Ε is a C°° function. We may also introduce a distance ρ on Св([а, b})
by
(2.4)
/(ΙΙέι(ί)ΙΙ-ΙΜί)ΙΙ)2*
J a
p(cb c2) := max d(c\(t), c2{t)) +
а<£<6
Then E, L are continuous with respect to p.
Here we do not adopt the precise infinite-dimensional manifold approach.
Instead, we give a vector space ТсСв([а, b]) which corresponds to the "tangent space"
to С в ([a, b}) at c, and follow the idea of the calculus of variations. Namely, we
define
(2.5)
TcCB{[a>, b}) := {X : [a, 6] -> TM\ piecewise C°° vector field along
с satisfying the boundary condition (X(a), X(b)) £ T^C^C^B},
which is an infinite-dimensional vector space. Namely, we regard a variation а
of с in С в ([a, 6]) as a "curve" through с in С в ([a, b}), and a vector field along с
as a "tangent vector" to a. Following Chapter II, §2.3, we call an element X £
ТсСв{[а, b]) a variation vector field of с (satisfying the boundary condition B). We
remark that there exists a variation of с which is "tangent" to X.
Lemma 2.1. For X £ ТсСв([а, b}) there exists a variation a. : [a, 6]x(-€, e) —>
Μ of с which satisfies the following conditions and is called a variation of с tangent
toX :

2. VARIATION FORMULAS AND JACOBI FIELDS
89
(1) a is continuous and there exists a subdivision Δ : a = to < t\ < ... <
tk = b such that a \ [U-\, ti] x (—£, e) is of class C°°.
(2) Define curves cs : [a, 6] —» Μ (-e < s < e) by cs(t) := a(£, 5). Then
c0 = с and cs e Св([а, &]).
(3) %{t,0) = X(t).
PROOF. Take a subdivision Δ so that с | [£;-i, U] and X \ [U-\, U] are smooth
and с | [U-i, t{] (i = 1, ... , к) are contained in coordinate neighborhoods. For
г = 2, ... , к — 1 we define α | [U-ι, U] x (-£, б) by a(t, s) := ехрф) sX(£)> which
clearly satisfies (1) and (3). Next take a smooth curve P(s) = (Pi(s), /^(s)) in В
which is tangent to (X(a), X(b)) e Т(с(а)>с(6))£ at 5 = 0. Now in order to define
ct I [to, t\] χ (-€, б), we first extend the vector field X \ [to, t\] χ {0} to a vector
field Υ along [t0, *i] x (-€, б) so that Y(t0, s) = /3i(s) and Y(tu s) = §*(ib s).
Next, considering integral curves of Υ emanating from c(t) = a(t, 0), we get
ct I [to, t\] x (-€, б). By the same argument we get a \ [tk-\, tk] x (—£, б) and
may easily check (1), (2), (3). D
Now we consider the "directional derivative" DE(c)X of the energy integral Ε
in the direction X. For that purpose we take a variation α of с tangent to X and
compute £ \s=0 E(cs).
Proposition 2.2 (First variation formula). Let X G ТсСв([а, b}), and a a
variation of с tangent to X. Take a subdivision Δ of [a, b] such that a | [U-χ, ti] χ
(—€, e) (i = 1, ... , k) are of class C°°. Let cs be variation curves of с defined by
cs(t) = a(t, s) with Co = c. Then we get
(2.6)
d fb k~X
js |.=o E(cs) = -j (V*_c(t), X(t))dt + Σ{Χ&), c(U - 0) - c(ti + 0))
+ (X(6), c(6)> - (X(a), c(o)>.
77ie right-hand side of (2.6) depends only on X, which we abo write DE(c)(X) and
call the first variation of Ε with respect to X.
Proof. We have by a direct computation
d „, χ I fb d Ida da\J± fb I da da\ Jx
Гь Г d Ida da\ Ida _ da\\ J±
Then, noting that ^(i, 0) = c(i), §f(£, 0) = X(i) and applying the fundamental
theorem of calculus to each interval [£*-!, £i], we get the right-hand side of (2.4). D
A similar computation implies

90 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Corollary 2.3. Let с G С в ([a, b}) be a piecewise regular C°° curve. Then for
the length functional L we get
♦gH.»-,gS3f>
Exercise 1. Give a proof of (2.7).
A curve с G С в ([a, b}) is called a stationary curve oi Ε on Св{ [α, 6]) if DE(c) (X)
= 0 holds for any X G ТсСв([а, 6]), which corresponds to a critical point of a
function of several variables. The next proposition generalizes Proposition 2.6 of
Chapter II under a general boundary condition.
Proposition 2.4. 7 G Св([а, 6]) zs a stationary curve of Ε if and only if
7 : [a, 6] —» Μ 25 a geodesic and (7(a), —7(6)) G (Т(7(а)л(5))В)±.
Proof. The "if part may be checked by an easy computation. Conversely,
suppose 7 is a stationary curve. By the same argument as in Proposition 2.6 of
Chapter II we may show that 7 is a geodesic. For any (x, y) G Τ^α^Ί^))Β take an
X G Τί0β{[(ι, b]) with X(a) = x, X(b) = y. Then from the first variation formula
we have
0 = ΌΕ(Ί)(Χ) = (X(b), 7(b)) - (X(a), 7(a)) = - ((x, y), (7(a), -7(6))),
where (, ) in the the last term stands for the inner product of Τ(7(α) Л(6))(М χ Μ).
Since (ζ, у) is arbitrary, we have our last assertion. D
We call a geodesic 7 satisfying the above boundary condition a B-geodesic.
Exercise 2. If Β = Ν\ χ Ν<2, then B-geodesics are geodesies which are
perpendicular to both of Λ/χ, N2- In particular, if Β = {ρ} χ {q} then B-geodesics are
geodesies joining ρ to q. If Б = {(ρ, ρ); ρ G Μ} then B-geodesics are geodesies
satisfying 7(a) = 7(6), 7(a) = 7(6), namely, closed geodesies.
2.2. A B-geodesic 7 is a candidate for a curve at which Ε assumes the
minimum on С в ([a, b}). First we consider a condition for 7 to assume a local minimum.
For that purpose we introduce the "Hessian" of E, following the finite-dimensional
case.
Proposition 2.5 (Second variation formula). Let 7 : [a, b] —> Μ be a B-
geodesic. For X G Τί0β([ο>, Щ) take a variation {cs} of η tangent to X as above.
Then
d2 \ „, ч
a^ E{Cs)
Ui> \s=0
(2.8)
J {(VX(t), VX(t)> - <Я(Х(<), 7(*))7(«), *(<)>}*
+V(7<«),-7(6))(*(<0, X(b)), (X(a), X(b))),

2. VARIATION FORMULAS AND JACOBI FIELDS 91
where A denotes the shape operator of a submanifold В С Μ χ Μ with respect to a
normal vector (7(a), -7(6)). Since the right-hand side o/(2.8) depends only on the
variation vector field X, we denote this by Ό2Ε(η)(Χ, Χ), and call it the second
variation of Ε via X.
Proof. First note that
42
d2 „, ч Г d /„ da da\ J
Then from Lemma 2.2 of Chapter II we get
d I da da\ I __ da da\ I da ^ da\
ds\ π ds ' dt I \ π * ds' dt / \ * ds' π dt /
da da\ In,da da da da
/^т да __ da \
d I da da\ I da da
: Jt \ * dl' ~dt/ " \v* dl' £Ж
9α da da da\ I da da
R{W WW Ts) + V*W v£ a!
Now setting s = 0 we have
-τ 9α
Va-=0
since 7 is a geodesic. Furthermore, §f (£, 0) = *y(t) is smooth, and we get
d2 ' '6
ВД = / {(VX(*), VX(i)> - <ВД*), 7(*)Ж*)< *(*)>}*
j=0 «/α
ds2
Now the final two terms are put together into
-(vA(^(a'o)'^(6'o))'wa)'^(6))!
using the inner product on ^(a), 7(6)) Щ х Л/). Finally, noting that (7(a), —7(b)) £
Τ(7(α)/γ(5))Β± by a property of B-geodesics, and recalling the definition of the shape
operator ((3.24) of Chapter II, §3.3, (II)), we get (2.8). D
Remark 2.6. Let 7 £ Cb{[o>, b}) be a B-geodesic with / := ||7(£)|| > 0, which
is constant. Then we have the following second variation formula for L:
m L{Cs)
(2'9) = Ί Γ{(νΧΧ(0' VX±{t)) " <β(Χ±(ί)' ^»^' X±(Wdt
+ I (AWa)t.m)(X{a), X(b)), (X(a), X(b))),
where X^it) = X(t) — (X{t), η{ί)/1)η{ί)/1 denotes the vertical component of X(t)
with respect to ^(t).

92 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Exercise 3. Verify (2.9).
Now we define a bilinear form Ό2Ε{η) on ΤίΟβ{[ο,, b}) by
(2.10)
D2E(7)(X, Y)
r\(VX(t), W(t)> - (R(X(t), 7(0)7(i), Y(t))}dt
J
J a
+ (AMa)i.m)(X(a), X(b)), (Y(a), Y(b))) ,
which is a symmetric bilinear form because of the properties of the curvature tensor
(Chapter II, Theorem 2.1) and the shape operator (Chapter II, (3.24)). Namely,
Ό2Ε(η) is the Hessian of Ε at 7. Noting that
jt(VX(t), Y(t)) = (VVX(i), Y(t)) + (VX(t), VY(t)),
and integrating by parts, (2.10) may also be written as
(2.11)
ϋ2Ε(Ί)(Χ, Υ)
Г (VVX(i) + R(X(t), 7(0)7(0, Y(tM
-I
J a
fc-1
+ Σ(νΧ(ίτ - 0) - VX(U + 0), Y(U))
i=\
+ (A{i{a)._m)(X(a), X(b)) + (-VX(a), VX(b)), (Y(a), Y(b))) ,
where a = to < · · · < tk = b is a subdivision of [a, 6] such that X \ [U-\, t{] are of
class C00.
Thus a B-geodesic 7 assumes a local minimum if Ό2Ε{η) is positive definite.
First we give a characterization of the null space3 of Ό2Ε(η).
Lemma 2.7. X £ Τί0β{[ο>, b\) belongs to the null space of Ό2Ε(η) if and only
if X is a Jacobi field along 7 which satisfies the following boundary condition:
(2.12)
(-VX(a), ЧХ(Ь)) + АЫа),_т)(Х(а), X(b)) ±.Th{a)Mb))B.
Proof. Let X e T7CB([o, b}) belong to the null space of D2E(j) and take a
subdivision Δ of [a, 6] so that X | [it-ь it] are smooth. First take a C°° function
/(i) on [a, 6] such that f{U) = 0 (г = 0, ... , к), f(t) > 0 (t φ h). We define a
vector field Y(t) along 7 by Y(t) := f(t)(VVX(t) + R(X(t), 7(i)h(0) (t φ U) and
Y(ti) = 0 (г = 0, ... , к), which belongs to TyCB([a, b\). Then from (2.11) we get
0 = D2E(7)(X, Y) = - / /(i)||VVX(i) + Λ(Χ(ί), i(m(t)\\2dt,
Ja
and X(£) satisfies the Jacobi equation WX(t) + Я(Х(£), 7(0)7(0 = 0 on each
interval [U-U U]. In particular, we get WX(U - 0) = -Д(Х(^), 7(f»))7(*i) =
3The subspace {X G T7Cs([a, 6]); D2£(7)(X, Y) = 0 for any У G T7Cs([a, 6])}.

2. VARIATION FORMULAS AND JACOBI FIELDS
93
WX(ti + 0) for г = 1, ... , к - 1. Next choose а У G T7CB([a, 6]) so that
Y(a), У (Ь) = 0, У fo) = VX(ti - 0) - VX(ii + 0) (t = 1, ... , к - 1). Then
fc-l
0 = ϋ2Ε{Ί){Χ, Υ) = Σ IIVX(*i - 0) - VX(U + 0)||2,
i=l
and consequently VX(tt - 0) = VX(i» + 0). Therefore, X is a C2 vector field
and satisfies the Jacobi equation, which implies that X is in fact C°°. Finally, for
any (x, y) G T(7(a)>7(b))B we may take а У G Τί0β([ο>, b]) satisfying the boundary
condition Y(a) = x, Y(b) = y. Then we get
0 = Ό2Ε(Ί)(Χ, Υ)
- ((-VI(a), VX(b)) + AWa)..m)(X(a), X(b)), (x, y)) ,
which shows that X satisfies the boundary condition (2.12). Conversely, if X G
Τί0β([ο>, b]) is a Jacobi field along 7 satisfying (2.12), then X clearly belongs to
the null space of Ό2Ε(η) from (2.11). D
Corollary 2.8. (1) If В is totally geodesic, for instance Β = {(ρ, q)} or the
diagonal set of Μ χ Μ, then
(2.13)
D2Eb)(X, Y)= f {(VX(t), VY(t)) - {R(X(t), 7(i))7(0, Y(t))}dt.
Ja
Thus the null space of D2E(f) is given by the space of Jacobi fields X along 7 with
X(a), X(b) = 0 in the case of В — {(ρ, q)}, and the space of periodic Jacobi fields
along the closed geodesic 7 in the case where В is diagonal.
(2) IfB = NixN2, then
(2.14)
D2E(7)(X, Y) = J {(VX(t), VY(t)) - (R(X(t), 7(0Ж<), Y(t))}dt
+ (Ai{a)X(a), Υ (a)) - (AmX(b), Y(b)).
Therefore, the null space of Ό2Ε(η) is given by the space of all Jacobi fields along
7 which are simultaneously N\- and N2-Jacobi fields.
Exercise 4. Give a proof of Corollary 2.8. Verify what happens in case of the
length functional L.
2.3. By the above argument, the nullity (i.e., dimension of the null space) of
Ό2Ε(η) on Τί0β{[ο>, b}) is finite. For a B-geodesic 7, the dimension of the maximal
subspace of Τί€β([ο>, b\) on which Ό2Ε(η) is negative definite is called the index of
7 and denoted by indB7. This invariant measures the obstruction for Ε (or L) to
assume a local minimum at 7, and plays an important role. It is known that inde7
is finite. In the following we study inde7 m detail in the case when Β = Ν χ {q),
where q G Μ and N is a submanifold of M. First we give some preliminaries.
Recall that for В = N x {q}, B-geodesics are geodesies 7 : [a, b] —> Μ which

94
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
emanate perpendicularly from N and end at q. Ό2Ε{η) on ΤίΟβ([ο>, b]) is given by
(2.15)
D2E(j)(X, Y) = J {(VX(<), VY(t)) - (R(X(t), 7(t))7(0, Y{t))}dt
+ {A^t)X(a), Y(a)).
Then elements of the null space of ϋ2Ε(η) on Т^СвЦа, b}) are TV-Jacobi fields along
7 with X(b) = 0. Now corresponding to (4.7) of Chapter II we get the following.
Lemma 2.9. Let 7 : [a, 6] —» Μ be а В-geodesic with Β = Ν χ {q}.
(1) // there exist no focal points of N on 7([a, &)), then Ό2Ε(η) is
positive semidefinite on Τί0β([ο>, b}). Namely, Ό2Ε(η)(Χ, X) > 0 for any X G
T7CB([a, 6]).
(2) // there exist no focal points of N on 7([a, &]), then Ό2Ε{η) is positive
definite. Namely, Ό2Ε(η)(Χ, X) > 0 for any X G ΤΊ€Β{[α, &]), X ^ 0.
PROOF. Let {Y\, ... , Ут} be a basis of Jf. First we show that for X G
Т7Св([а, 6]) we may choose piecewise C°° functions fl(t) (i = 1, ... , m) so that
X may be written as X(t) = P^Y^t) for both cases (1), (2). In fact, by (4.13)
of Chapter II, {Уг(0} forms a basis of ΤΊ^Μ for a < t < b in case (1) and
for a < t < b in case (2), and the assertion is clear for these Vs. For t = a
we choose a basis {Z\, ... , Zm} of J if- consisting of iV-Jacobi fields so that
{Ζχ(α), ... , Zn(a)} forms a basis of ΤΊ^Ν and Ζσ(α) = 0 (α = η + 1, ... , m).
Then we get (VZa(a), Zk(a)) = 0 (fe = 1, ... , n) from the initial condition. Now
from Chapter II, Lemma 4.9, {Zfc(i), Za(t)/(t - a)} (1 < к < η, η + 1 < а < m)
forms a basis of ΤΊ^Μ if £ — а > 0 is sufficiently small, and
lim Za(t)/(t -a) = VZQ(a)
t—*Q
(Problem 11 for Chapter II). Thus the above vector fields may be extended to
(piecewise) С°° vector fields along 7 on [a, a + e] which form a basis of ΤΊ^Μ, t G
[a, a + e]. Therefore, we may write
X(t) = gkWk(t) + ga(t)(Za(t)/(t - a)),
where gk{t), gQ(t) are (piecewise) Cx functions. Noting that X(a) G T7(a)7V,
we get <7a(a) = 0 and gQ(t) = (t - a)gf(t) with (piecewise) C°° functions gf.
In particular, we get X(t) = gk(t)Zk(t) + gf (t)ZQ(t). Since we may write Z{ =
ol\Yj (ol\ G Д, 1 < г, j < m), our assertion holds for t = a. In the same manner we
may write X(b) = /{(6)У;(6) in case (1). Next we show that for X G CB([a, b})
(2.16)
Ό2Ε(Ί)(Χ, X) = [ ((fnWW, (/'")'№(*)> * (1 <iJ< m).
J a
After this we get D2E(<-y)(X, X) > 0 for any X G ΤΊ0Β([α, 6]), which completes the
proof of (1). If equality holds in the above inequality, then we get /г = const (1 <
г < m) and X is an iV-Jacobi field. Namely, X belongs to the null space of Ό2Ε(η)
on Τί€β{[ο>, b\). Since 7(6) is a focal point of TV if and only if the dimension of
this null space is positive, the proof of (2) is completed. Now we turn to the proof
of (2.16). First note that Yu ... , Ут are iV-Jacobi fields and (Yi(t), VY^t)) (i =

2. VARIATION FORMULAS AND JACOBI FIELDS 95
1, ... , m) belong to a Lagrangian subspace of ΤΊ^)ΤΜ. Therefore,
(VYi(t), Yj(t)) - (Yi(t), VYtf)) = 0 (1 < t, j < m).
On the other hand, noting that
(VYi(t), VYj(t)) = |(νκ4(ί), Yj(*)) - (Wy^i), Yj(t))
= jt(VYi(t), Yj(t)) + (R(Yi(t), 7(0Ж0, >S'(0>,
we get
(VX(i), VX(t)> = <ν(/'(0*ί(0), V(/'№(*))>
= <(ή'(0*(0, (Я№(0> + Я0Л0№(0, vi$(0>
+ urywHt)+(n'wrmvYit), y^))
= {(fym(t): (п'ти)) + ^{гтчтъъ^, Yjit))}
+лояо<я(>т -кожо, эд>·
Then the left-hand side of (2.16) is equal to
' {(VX, VX) - {R(X, 7)7, X)}di+ (Ау(в)А:(а), Х(а))
' (Cf )X (/,')'^·>λ + Lf (0/j(0<vim ЗД>]«
./а
/
J a
+ (Ai{a)X(a), X(a)).
Now noting that
[.T(i)/J(0<V*i(0, >j(0>]a = -fW4")(Ana)Yi(a), Yj("))
= - (Ау(в)л:(о), л-(о)>
(because X(b) = 0), we see that the right-hand side of the above equation is equal
to the right-hand side of (2.16). D
Now we consider a space of vector fields along 7, defined as
ΤΊεΝ([α, b}) := {piecewise C°° vector field X along 7 with X(a) G iV}.
Note that we do not require the condition X(b) = 0, and consider only the initial
condition. Now the right-hand side of (2.15) gives a symmetric bilinear form on
Τί0ν([(ι, &]), which will be called the index form of an iV-geodesic 7 and denoted
also by IN(X, Y).
Lemma 2.10. Let 7 : [a, 6] —» Μ be an N-geodesic of Lemma 2.9, and suppose
there exist no focal points of N on 7([a, &]). Then for any X G Т7Сдг([а, b]) there
exists a unique N-Jacobi field Υ G J^ with Y(b) = X(b). Moreover, /лг(У, Y) <
In{X, X), where equality holds if and only ifY = X.
PROOF. Prom (4.13) of Chapter II we have a unique Υ G Jj^ which
satisfies Y(b) = X(b). Prom the proof of the previous lemma we may write X(t) =
Ρ(ί)Υι(ί), a<t<b, where {Yi)T=i is a basis of J7N. Then, as before,
IN(X, X)= f <(/*)% (п%)* + Гт*(Ь)(Ч¥г(Ь), Y3(b)).
J a

96
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Noting that Υ = fl(b)Y{, we see that the last term of the right-hand side of the
above equality is equal to (VY(b), Y(b)). Therefore,
IN(X, X) > (W(6), Y(b)) = IN(Y, Y).
Equality holds if and only if (fl)'Yi = 0. This means that fl = const (i =
1, ... , ra), namely, X = Y. D
We give an application of this lemma.
Lemma 2.11. Let N be a submanifold of Μ and η : [α, +οο) —» Μ a geodesic
emanating perpendicularly from N. Let 7(6) be the first focal point of N along
7. Then for any t > b, 7 | [a, t] cannot be a minimal geodesic from N to 7(6),
namely L(j | [a, t]) > d(N, 7(i)). In particular, geodesies emanating from ρ are
not minimal beyond their first conjugate value.
Figure 10
PROOF. Since 7(6) is a focal point of TV, we have a nonzero iV-Jacobi field Υ
along 7 with Y(b) = 0. Prom Chapter II, §2.1, we may choose sufficiently small
0 < € < t — b and δ > 0 so that for any t0 < s < t\, 7(s) and 7(^1) are not conjugate
to each other along 7, where we put to = b — <5, t\ = b + e. Now apply Lemma 2.10
to the case where TV is a point 7(^1) and the geodesic is (7 | [to, £i])-1 instead of
7. Then we get a Jacobi field X along 7 | [t0, t\] with X(t0) = Υ (to), X(ti) = 0,
which minimizes the index form ΙΊ(α) m the class of vector fields along 7 | [£0, t\]
satisfying the same boundary condition.
Now we define a vector field Ζ along 7 | [a, t\] by Z(t) = Y(t) (a < t <
t0), Z(t) = X(t) (to < t < ti), and a vector field Ϋ by Y(t) = Y(t) (a < t <
b),Y(t) = 0(b<t<t1). Then
0 = IN(Y I [a, 6], Υ | [a, 6]) = IN(Y \ [a, tx], Υ \ [a, tx])
= IN(Y\ [a, i0], У I [a, to}) + Il{tl)(Y \ ft>, *i], ^ I [*o, *i])
> /N(r Ι [α, ίο], У I [a, to}) + /7(ίι)(* Ι [*ο, *ι], Χ Ι [ίο, *ι])
= Jn(Z| [α,ίι], Z| [a, ii])
= Ό2Ε(Ί)(Ζ, Ζ) (Ζ e T7CNx{7(il)}([a, h})).
Here we note that ΙΊ^) is given by
/7(tl)(y Ι [ίο, ti], Ϋ Ι [ίο, ti]) = / '{(Vr, УУ) + (Д(У, 7)7, У>}Л.

3. APPROXIMATION BY FINITE-DIMENSIONAL MANIFOLDS
97
Since Ϋ satisfies Y(t0) = X(to), ^(*ι) = ^(*i) = 0 and is not differentiable at
t = 6, Υ is different from X. Therefore, we get the strict inequality
J7(tl)(* I fob til * I fob ii]) > I«u){Y I [*o, hi Υ Ι [t0, ίι]).
Now take variation curves 7S of 7 | [a, t\] generated by Z. Then ^ |s=o E(j3) =
0, jp \s=o Ε(*ys) < 0; and this implies that for a sufficiently small s > 0
L(7) = L(7o) = λ/2(*ι - α)Ε(Ίο) > y/2(tx - а)Е(Ъ) > Цъ)
which means that 7 | [a, £1] (and consequently 7 | [a, t}) is not distance minimizing,
namely, L(7 | [a, ^]) > d(N, 7(*i))· D
We give the index theorem for geodesies which refines the above result in the
next section.
3. Approximation by Finite Dimensional Manifolds
and the Index Theorem
3.1. In this subsection we are concerned again with the case Β = TV χ {ρ},
where TV is a closed submanifold of a complete Riemannian manifold Μ and q G M.
For С в '·= Св([0, 1]) = {с : [0, 1] —> М; с is a piecewise C^ curve with c(0) G
TV, c(l) = q] with the energy integral E, we approximate Св by finite-dimensional
manifolds to apply Morse theory. Let a (> 2d(TV, q)) be a positive number, and set
C£ := {c G CB; £(c) < a2/2}, C£~ := {c G CB\ E(c) < a2/2). We choose an r > 0
so that the following hold:
(1) For any point ρ of the compact set Ba{q), Br(p) is convex in the sense of
(2.5) and (2.6) of Chapter II. _
(2) For the compact subset К := TV Π Ba(q) of TV, exp-1 is a diffeomorphism
if restricted to an open set containing ΒΓ(θχ) := {ζ G TpTVx; ρ € Κ, \\ξ\\ < г}.
Next we fix a subdivision Δ : 0 = to < t\ < · · · < tk = 1 such that ^+1 — U <
r2/a2 (i = 0, ... , к - 1). Now we define a subset C%(A) of CB as
(3.1)
C%(A) := {c G C#; (i) с | fo, ii+i] (1 < г < к - 1) are minimal geodesies;
(ii) с | [to, £1] is a minimal geodesic joining TV to c(ti)}.
We define C^~(A) similarly. For с G C% note that d(c(£), (?) < L(c) < a, namely,
cCBa№ Then
d(c(^), c(ti+i)) < L(c I [it, t<+1]) < y/2{ti+i-U)E{c) < r
(i = 1, ... , к - 1), and similarly d(TV, c(t\)) < r. Therefore, we have unique
minimal geodesies joining c(t{) to c(^+i) and TV to c(t\), respectively. Joining them,
we get a curve in С β (Δ). Furthermore, Cq(A) is a strong deformation retract4 of
C%. In fact, if we define Η : C% χ [0, 1] -> C% by
Я(с,-)(«) := ίσί(<)' '^«^** + -(^-«*)'
lc(t), *< + e(*i+i - *i) < * < *<+i,
4For 0 < s < 1 there exists a homotopy Hs : CB — C£ such that Я0 = id, Hi : CB ^ CB(A)
and Я6 | C%(A) = id.

98
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
where σ\ denote unique minimal geodesies from c(^) to c(U + s(t{+i —U)) (from N
to c(to + s(t\ - to) when г = 0). Then {Hs} with Яя(с) = Η (с, s) gives a desired
homotopy.
Exercise 1. Verify the above.
Now we note that C%~(A) carries the structure of a C°° manifold. In fact,
consider the direct product of (k — 1) copies of Μ and a function Ё on it defined
by
(3.2) £.(Pl,...|Pfc_0.= -|___ + ^__:_|>
where we set q = pk- Now we set
^ := {(Pi, · · · , Pfc-i) e Μ χ . · · χ Μ; Я(рь ... , pfc_i) < α2/2},
Λ<Γ := {(pi, ... , pfc-i); Β(ρι, ... , pfc_i) < a2/2}.
Then Λ1£~ is an open subset of Μ χ · · · χ Μ and Ё is a C°° function on Μ%~.
This follows from
d(pi, iV), d(pi,pi+i) < y225(pi, ... , pfc_i)max|ii+i — *»| < r
and (2.5), (4.16) of Chapter II. For a generic α, ΛΊ£ is a submanifold with boundary
with М^~ as its interior. Now we define a map Φ : Ai% Э (pi, ... , Pfc-i) ·—> с £
Сд(Д) by assigning a curve с such that с | [to, ti] is a minimal geodesic from TV to
pi and с | [ti, U+i] are minimal geodesies joining pi to Pi+i. Then we may check
that Φ is a homeomorphism with respect to the topology on Св defined in the
Remark in §2.1, and Ε ο Φ = Ё. It is easy to see that E~l([0, b2/2\) is compact
for 0 < b < a, and Φ maps M^~ onto С%~(А). Therefore, we may introduce a C°°
manifold structure on C%~(A) so that Φ is a difFeomorphism and Ε is a proper C°°
function on it. The next lemma shows that it suffices to consider C%(A).
Lemma 3.1. (1) ТСС^_(Д) = [Υ e TCCB\ Υ | [*o, *ι] is an N-Jacobi field,
Υ | [t{, U+i] are Jacobi fields along с \ [ti, U+ι] (i = 1, ... , к — 1)}.
(2) 7 £ Cq~(A) is a critical point of Ε if and only if η is a B-geodesic, namely,
a geodesic emanating from N perpendicularly and ending at q.
(3) Let η be a critical point of E. Then, for Υ, Ζ G T7C%~{A),
fc-1
(3.3) D2E(<y)(Y, Ζ) = J2(VY(U - 0) - VY(U + 0), Z(U)).
г=1
(4) The null space of Ό2Ε(η) | Т7С^~(Д) coincides with the null space of
Ό2Ε(η) | Τί0β, which is given by [Y G Jj*\ Υ(1) = 0}. Moreover, inde7 ™ equal
to the index of Ό2Ε(η) | Т7С^~(Д). In particular, inde7 is finite.
PROOF. (1) follows immediately from a characterization of (TV-)Jacobi fields,
Lemma 2.4 of Chapter II, and (4.13). To see (2), first note that for с G C%~(A) we
have from (2.6)
fc-1
DE(c)(Y) = J2(Y(U), c(U - 0) - c(U + 0)) - (У(0), с(0)).
г=1

3. APPROXIMATION BY FINITE-DIMENSIONAL MANIFOLDS 99
Then (2) follows by the same arguments as in Proposition 2.4. (3) follows from
(2.11) and the properties of TV-Jacobi fields.
The first assertion of (4) may be verified by using (3.3) as in Lemma 2.7. For
the second assertion it suffices to see that Ό2Ε(η)(Χ, X) > 0 holds if X G ΤίΟβ
is orthogonal to ΤΊ€α^~ (A) with respect to Ό2Ε(η). In fact, suppose that for any
ГеВД-(Д) we have
fc-l
0 = Ό2Ε{Ί){Υ, Χ) = Σ(νΥ(ίτ - 0) - VY(U + 0), X(U)).
i=l
Let Ϋ e T7CaB~{A) be given by the condition Y(U) = X(U) (i = 1, ... , к - 1).
Then we get Ό2Ε(Ί)(Ϋ, Υ) = Ό2Ε(η)(Ϋ, Χ) = 0. Setting Ζ := Χ - У, we have
Z(U) = 0(t = 1, ... , k). Now Ό2Ε(Ί)(Χ, Χ) = Ό2Ε{Ί){Ζ, Ζ) > 0 follows by
applying Lemma 2.9 to each 7 | [ti, U+ι] (i = 0, ... , к - 1). D
3.2. Let 7 be a critical point of Ε | Ca^~(A). Recall that 7 is said to be
nondegenerate if the Hessian Ό2Ε{η) | Т7С^~(Д) is nondegenerate (i.e., its null
space consists only of 0). This condition is equivalent to the fact that q = 7(1) is
not a focal point of TV along 7 by Corollary 2.8. On the other hand, q is a focal point
of TV if and only if q is a critical value of the normal exponential mapping exp-1 of TV
(i.e., q = exp-1 ξ with rank Dexp-1 (ξ) < m: see Chapter II, Lemma 4.8). By Sard's
Theorem the set of such critical values is a null set of Λ/. Therefore, for a given TV,
for almost all q G Μ it follows that any geodesic emanating perpendicularly from
TV and ending at q is nondegenerate. For this q, critical points of Ε on C^~(A)
are nondegenerate and Ε is a Morse function. Applying Morse theory, we see
that C%~(A) (and also C^~, which is a deformation retract of C^~(A)) carries the
homotopy type of a CW-complex obtained by attaching a λ-dimensional cell to
each critical point 7 G C%~(A) of index λ. Therefore, the indices of geodesies are
closely related to the topology of C%~(A) and also the topology of M.
Morse expressed the index of a geodesic in terms of focal (conjugate) points in
the following way.
Theorem 3.2 (Morse index theorem). Let Μ be a Riemannian manifold and
TV a submanifold of M. Let 7 : [0, 1] —> Μ be а В-geodesic for В = TV χ {q},
namelyf a geodesic emanating perpendicularly from TV and ending at q. Denote by
7(51), ... , 7(sfc) (0 < 5i < · · · < Sk < 1) the focal points of TV along 7 | (0, 1),
which appear isolated. Let n(sj) (j = 1, ... , к) be the multiplicity 0/7(5^). Then
к
(3.4) indB7 = ^n(5j),
i=i
namely, the index 0/7 is equal to the number of focal points of N along 7 | (0, 1)
counted with multiplicities.ъ
PROOF. Recall that £ := {(У(0), Vr(0)); Y is an TV-Jacobi field along 7} С
Ιγ(ο)ΤΜ is a Lagrangian subspace, and η(ί) is a focal point of TV if and only if
W(t) := ЩгС Π Vv(t) φ {0}, where its multiplicity n(t) is given by dimH^)
(Chapter II, §4). For focal values 0 < si < ... < s^, we define an injective linear
map ζ : ®W{sj) -> ΤΊ€Β as follows: for ξ = (Y(sj), VY(sj)) G W(sj) with
Y(Sj) = 0, C(0 is defined as a broken Jacobi field Ϋ given by Y(t) = Y(t) {0<t<
5If N = {p}, then the index of 7 is given in terms of conjugate points.

100
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Sj), Y(t) = 0(sj <t< 1). Then subspaces C(W(sj)) for different values of Sj are
linearly independent, and for all Si, Sj we have
(3.5) £>2£(7)(C(W(Si)), С(Щ^)) = 0,
which easily follows from (2.11) if we note that W(si) С D<j)SiC. Therefore, setting
W := ©*=iC(^(sj)), we see that dimW = EjLin(si)· 0n the other hand>
taking a sufficiently large a > 0 and a sufficiently fine subdivision Δ of [0, 1] such
that si, ... , Sk are included in the set of subdividing points, we see that inde7
is equal to the index of Ό2Ε(η) on ΤΊ0αΒ~{Δ) (Lemma 3.1) and W С T7C^"(A).
Now we denote by T_ (resp., T+) the direct sum of eigenspaces corresponding to
negative (resp., nonnegative) eigenvalues of Ό2Ε(η) | Т7С^~(Д). Then we have
ΤΊ0%~(Α) = Τ_ Θ Τ+ and indB = dimT_. Let p_ : ΤΊ0%~(Α) -> T_ be the
orthogonal projection with respect to the above direct sum decomposition; we show
that p- | W is injective. In fact, suppose that p~(X) = 0 for some X e W\ {0}
with X = X. +X+ £ T-®T+. Then X = X+, and we have 0 = £>2£(7)(X, X) =
D2E(f)(X+, X+) > 0. Since Ό2Ε(η) is positive semidefinite on T+, X = X+
belongs to the null space of D2E(j) and is a smooth iV-Jacobi field along 7, which
contradicts the fact that X £ W\{0} is not smooth at some parameters. Therefore,
we have Ker(p_ | W) = {0}, namely, dimW = dimp_(W), and it follows that
πκΐβ 7 = dimT_ > dimp_(W) = $^j=i n(sj)·
Next suppose dimT_ > dimp_(>V). Then there exists a nonzero X £ 71
which is orthogonal to p_(>V) with respect to Ό2Ε(η). Note that X is orthogonal
to >V, because X is orthogonal to T+ with respect to Ό2Ε(η). In particular, for
Ϋ £ С(Щ^)) (j = 1, ... , fc) we get from (3.3)
0 = Ό2Ε{Ί){Χ, Ϋ) = (Vr(Sj), X(Sj))
and X(sj) is perpendicular to (VY(sj)\ (Y(sj), VY(sj)) £ W(sj))r. Therefore,
as in the proof of Proposition 2.9, using Lemma 4.9 of Chapter II, we may write
X(£) = fl(t)Yi(t) with piecewise C°° functions fl(t), where [Y\, ... , Ym} is a basis
of J7N. Then we get Ό2Ε(η) (Χ, Χ) > 0 by (2.16), which is a contradiction, and
the proof of the theorem is complete. D
Remark 3.3. (1) The nullity of D2E(/y), which is also denoted by nullity7,
is given by n(l) (Corollary 2.8, (2)).
(2) In the above, the geodesic 7 is parametrized on [0, 1]. However, this is not
essential and the index theorem holds in the same form for geodesies parametrized
on any interval [a, b].
(3) For a general boundary condition £, it is not so elementary to express
the index inda7 in terms of some appropriate notion of "conjugate points" with
respect to some initial condition. As in the previous theorem, the notion of focal
points is natural when Β = Ν χ {q}. When В = N\ x N2 or Β = {(ρ, ρ); ρ £ Μ}
we have the results due to W. Ambrose and W. Klingenberg, respectively. J. J.
Duistermaat developed a general theory to express indB7 in terms of £-conjugate
points and the correction term, when any Lagrangian subspace С of T^yTM is
given (see [Mo], [Am-2], [ΚΙ], [Κ-3], [Dui-2]).
Remark 3.4. The approximation of С β by finite-dimensional smooth
manifolds given in the first part of this subsection may be developed also in the case
Β = Λ/χ χ 7V2, where Λ/χ, N2 are compact submanifolds of M. Also let Μ be
a compact Riemannian manifold, and consider the space of closed curves С в with

3. APPROXIMATION BY FINITE-DIMENSIONAL MANIFOLDS 101
Β = {(ρ, p)\ ρ G Μ}. In this case take an r > 0 so that Br(p) satisfies (2.5) of
Chapter II for all ρ G Μ (to be more precise, take the convexity radius min{r(p); ρ G M}
defined in Chapter IV, §5). Next, for a given a > 0 choose a subdivision Δ
of [0, 1] so that max(£i+i — U) < r2/a2. We define as in (3.1) the subspace
Cq~(A) := {с G C^~; с | [U, U+ι] (i = 0, ... , к — 1) are minimal geodesies} of
C^~. Then Сд~(Д) carries the structure of a C^ manifold on which Ε is a proper
С°° function, and the result corresponding to Lemma 3.1 holds.
Now we are concerned with the relation between the curvature and the
index. Let Μ, Μ be complete Riemannian manifolds of dimension m. Suppose
normal geodesies 7 : [0, I] —» M, 7 : [0, /] —» A/ are given. We take an
isometric linear isomorphism I : ΤΊ^Μ —» T^^M such that /7(0) = 7(0), and define
J* : T7(o)M —» T^(t)M by J* := P(7)? 0/0 Ρ(τ)ο· which is also an isometric linear
isomorphism. Then we have V о It = It о V, where V, V denote the covariant
derivatives with respect to *y(t) and 7(2), respectively. We set ρ = 7(0), q = 7(i),
etc. Now we compare the indices ind7,ind7 of 7. 7 as critical points of Ε on
Cpq(M), Cpq(M), respectively.
Lemma 3.5 (M. Morse, J. J. Schoenberg). Suppose that for any plane section
a(t) of ΤΊ(ι)Μ containing j(t) and any plane section a(t) of T^^M containing
7(£), we have Κσ^ < K^(t) for a^ t £ [0. /]· Then ind~y < ind7. Further, if
Ka{t) < K&(t) /0Γ α^ t £ [0, i], ^en ind7 + nullity 7 < ind7.
PROOF. For X G Τί€ρ4{Μ) we define X G T^CM{M) by X(f) = /fX(f). Then,
if Κσφ < K&(t), from the second variation formula we get
D2E(7)(X, X) = f {(VX(i), VX(f)> - K(X(t), j(t))\\X(t) Л 7(*)||2}<Й
JO
> [ {(VX(*), VX(i)> - ВД), 7W)II*W Λ7(«)ΙΙ2}Λ·
Jo
Then for a subspace Ы С ΤΊΟρη(Μ) on which Ό2Ε(η) is negative definite, we see
that Ό2Ε(η) is negative definite on the subspace W := {X\ X G W}. Therefore we
have ind7 > ind7. The second assertion may be proved in the same way. D
Proposition 3.6. Let 7 : [0, /] —» Μ be a normal geodesic.
(1) Suppose for all plane sections σ of TM containing 7 we have Κσ < Δ.
Then there exist no conjugate points to 7(0) along 7 | [0, /] for I < π/yfK, where
we take π/yfK = +00 when Δ < 0.
(2) Suppose Κσ > δ (> 0) holds for any plane section σ of Τ Μ containing 7.
/// > π/у/б, then ind7 > m — 1 (m = dim M).
PROOF. Recall that Jacobi fields in a space of constant curvature к are
explicitly given in Chapter II, §3. Then we may easily see that there exist no conjugate
points along any geodesic 7 when к < 0. If к > 0, then any Jacobi field Υ(t) along 7
such that Y(0) = 0 and perpendicular to 7 may be written as Υ(t) = sin y/kt · E(t),
where Ε is a parallel vector field along 7. Since a Jacobi field along 7
vanishing at two points is perpendicular to 7, it follows that the first conjugate value is
equal to π/y/k and its multiplicity is m — 1. Therefore, in this case the index of
7 I [0, /], / > π/y/k, is greater than or equal to m — 1 by the index theorem. Then
the assertions of the proposition are clear from Lemma 3.5. D

102 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Corollary 3.7 (S. B. Myers). Let Μ be a complete Riemannian manifold, and
suppose that the sectional curvatures Κσ satisfy Κσ > δ (> 0) everywhere. Then
Μ is compact, and d(M) < π/\/δ.
PROOF. We may assume that dim Μ > 2. Suppose we have two points p, q £
Μ with / := d(p, q) > π/\/δ. Then for a minimal geodesic 7 : [0, /] —» Μ joining
ρ and q we get ind7 > m — 1 > 1, which contradicts the minimality of 7. D
Corollary 3.7 will be generalized in the next chapter, where we will treat the
comparison theorems on the curvature and Jacobi fields in more detail.
4. Cut Locus
4.1. Let Μ be a complete Riemannian manifold. Then for ρ £ Μ, expp is
defined on all of TPM, and there exists a minimal (i.e., distance realizing) geodesic
segment joining any two points of Μ. However, in general geodesic segments are
not necessarily minimal, and there might exist many minimal geodesies joining two
given points. For instance, for any compact Riemannain manifold no geodesic of
length greater than the diameter can be minimal, and for (5m, go) any great
semicircle joining ρ and its antipodal ρ is minimal. On the other hand, any sufficiently
small arc of a geodesic is a shortest curve between the end points of the arc. Thus,
denoting by 7n : [0, +00) —» Μ the geodesic emanating from ρ with the initial
direction и £ UPM, it is natural to introduce the following quantity:
(4.1)
t(u) := sup{£ > 0; 7n I [0, t] is minimal, namely, d(p, ju{t)) = t}.
Obviously 0 < t(u) < +00, and if t(u) < +00 then it is the last value of t such that
7n I [0, t] is minimal. First we give a fundamental property of t(u).
Proposition 4.1. (1) Suppose t{u) < +00. Then Τ = t(u) if and only if
7n I [0, T] 25 a normal minimal geodesic and at least one of the following conditions
(a), (b) holds:
(a) 7n(T) 25 the first conjugate point of ρ along jUf
(b) There exists a vector ν £ UPM, ν фи, such that ju(T) = 7^(T).
(2) t : UM Зин t(u) £ R+ U {+00} 25 a continuous function.6
Proof. (1) If ηη{Τ) is a conjugate point to ρ along 7^, then ηη cannot be
minimal beyond this value, by Lemma 2.11 or the index theorem. If (b) holds, then
for any € > 0, ηu \ [0, Τ + e] is not minimal. In fact, it suffices to show this for
sufficiently small e > 0. Take a minimal geodesic β joining ην(Τ — e) to 7n(T + e).
Then we get
2e = d(lu(T + 6), 7tt(T)) + d(7t,(T), Ίυ(Τ - б))
>d(7u(r + 6),7t,(r-6)).
This follows from Exercise 5 of Chapter II, §2, and the fact that 7n | [Τ, Τ + e]
and (7^ I [T — €, T])_1 do not make a straight angle at 7U{T) = 7u(T) because
7n(T) φ Ίυ(Τ). Therefore, we have
ЦЪ | [0,T- e] U β) = Τ - ε + d(7v(T - б), 1и(Т + е)) < L{lu \ [0, Г + б]),
6 A fundamental system of neighborhoods around +00 in Ди+{оо} is given by {(r, +00); r >
о}·

4. CUT LOCUS
103
Figure 11
namely, 7 | [0, Τ + e] is not minimal, which shows the "if" part of (1). Next set
Τ = t(u). First, 7n I [0, T] is minimi since d(p, 7u(T)) = limtTTd(p, ^u(t)) = T.
Second, assuming that q := ηη(Τ) is not conjugate to ρ along 7^, we prove assertion
(b). Take a neighborhood U of Tu in TPM such that expp | U is a difFeomorphism.
Then for sufficiently large η with qn := 7n(T+ 1/n) G expp /7 we take minimal
geodesies ηη parametrized by arc-length joining ρ to qn, and set un := 7n(0) G Z7PM.
Since /7PM is compact, we may assume that un —» г> G /7PM by taking a subsequence
if necessary. Then we get 7„(T) = lim7Un(T + 1/n) = ηη{Τ) (see the following
Lemma 4.2). If i> = u, then for sufficiently large η it follows that d(p, qn)un G U and
expp(T+ l/n)u = exppd(p, qn)un, and therefore we have (T+ l/n)u = d(p, qn)un,
which contradicts the fact that Τ + 1/n > d(p, qn). This means that ν φ и, and
the proof of (1) is complete.
Now we prepare a lemma for the proof of (2).
Lemma 4.2. A sequence {^η} of geodesies is said to converge to a geodesic
7 = In if Pn := 7n(0) -> p, un := %{0) -> u G ГрЛ/. Г/геп ->„(*„) -> 7(0
whenever tn —» £. Further, if ηη are normal minimal geodesies joining pn = 7n(0)
ίο ςη = 7n(^n) such that ηη —» 7, £n —» /, ί/ien η is a minimal geodesic joining ρ to
9 = 7(0·
The first assertion of the lemma holds because a geodesic 7U(£) depends
continuously on the initial direction и and the pameter t. The second assertion is also
clear, because d(p, ς) = limn^oo d(p„, g„) = lim^^ tn = I = £(7 | [0, I}).
Now we turn to the proof of (2). It suffices to show that t(un) —» t(u) when
(Pn, ^n) —* (p, ^) in the unit tangent bundle UM. Let Τ be any accumulation value
of {t(un)}, including +00. By Lemma 4.2, 7^ | [0, T] (if Τ = +oo then ηη | [0, t] for
any t > 0) is minimal, and Τ < £(u). If Τ = +oc we are done. Assuming Τ < +oo,
we write Τ = lim£(un), taking a subsequence if necessary. By Proposition 4.1 (1)
at least one of the following possibilities (a), (b) holds:
(a) t(un) is the first conjugate point to pn along 7Un,
(b) there exists vn G UPnM, vn φ un, with 7un(t(un)) = ъп(Кип))-
Note that at least one of (a), (b) holds for infinitely many n. If this happens for
(a), we may choose infinitely many unit vectors {wn}^=1 which belong to the kernel
Ker D expPn (t(un)un) and are contained in a compact subset of UM. Therefore, we
may take a convergent subsequence whose limit w is contained in Kei\Dexpp(Tu).
Namely, the rank of Dexpp(Tu) is less than m and 7n(T) is the first conjugate point

104
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
to ρ along 7n.7 If (b) holds for infinitely many n, we may assume that vn ι—> ν £
/7PM, taking a subsequence if necessary. If ν φ u, then (b) of (1) holds for T. Next,
if ν = и we see that ju{T) is conjugate to ρ along 7n. In fact, suppose the contrary.
Then expp is regular at tu £ TPM and consequently the map Φ : TM —» Μ χ Μ
is regular at Tu, where Φ is given by Ф(и) := (ρ, exppu). Therefore, Φ is a
difFeomorphism if restricted to an open neighborhood U of Tu in TM. Prom ν = и
we see that (pn, £(u„)un), (pn, έ('Μη)^η) belong to U for sufficiently large η and are
different. On the other hand, we get Φ(ί(ηη)ηη) = ${t(un)vn) from (b), which is a
contradiction. Therefore, we have (a) or (b) for T, and so Τ = t(u). Since Τ is an
arbitrary accumlation value of {t(un)}, we have \imt(un) =T. D
Remark. If Μ is compact, then t(u) < +00 for any и £ UM, and vice versa.
Definition 4.3. Let Μ be a complete (connected) Riemannian manifold and
ρ e M. If t(u) < +00 for и £ UPM, we call t(u)u (resp., exppt(u)u) the tangent
cut point (resp., cut point) of ρ along *yu. The sets Cp := {t(u)u; и £ UPM, t(u) <
+00}, Cp := expp Cp are called the tangent cut locus and the cut locus of p,
respectively. Further setting Tp := {tu; 0 < t < t(u), и £ UPM}, we call Ip := exppIp
the interior set at p.
The sets Cp, Cp, Tp are related in the following way:
Lemma 4.4. (1) 1РГ)СР = φ, Μ =Ip\J Cp, and Ip = M.
(2) Ip is a maximal domain containing op £ TPM, on which expp is diffeo-
morphic.
(3) Cp is a null set of M, and dimCp < m — 1.
PROOF. (1) By Corollary 1.3, any point q £ Μ may be joined from ρ by
a minimal geodesic, from which the second and the third assertions are obvious.
Next suppose exppu = ехррг> for и £ Tp, ν £ Cp. Then two minimal geodesies
7ϋ/||ϋ|| I [0, INI], ъ/\\у\\ I [0, И|] intersect, and we get ||u|| = £(u/||u||), which
contradicts the definition of Tp.
(2) Zp is a connected (in fact star-shaped) open set and contains no tangent
conjugate points. Hence expp is regular on Jp. We may show that expp | Zv is
injective by the same argument as above.
(3) Since t(u) is continuous, it follows that Cp is a null set and dim Cp = m — 1
if it is not empty. Since expp is of class C°°, Cp = expp(Cp) is also a null set. Then
dim Cp < m — 1 follows from a result of dimension theory.
Finally, we show the maximality of Tv asserted in (2). In fact, otherwise we
have a point и £ Cp which is not a tangent conjugate point of p, and a neighborhood
U of и in TPM such that expp is a difFeomorphism on Tp U U. Then we may easily
see that expp(Z7 \ϊρ) is contained in Cp, which contradicts dimCp < m — 1. D
From Lemma 4.4 (1) we see that for any point q £ M\CP there exists a unique
normal minimal geodesic joining ρ to q. In particular, if Μ is compact then Tv
is an m-dimensional open disk whose boundary dCp is homeomorphic to 5m_1.
Therefore, we see that Μ may be obtained from the cut locus Cp of ρ by attaching
an m-dimensional disk via the map expp : Cp —» Cp. Further note that the cut
7This argument also shows that the function UM Э и ι—► to(u) G R U {-(-00}, the first
conjugate value to тми along 7^, is continuous.

4. CUT LOCUS
105
locus Cp is a strong deformation retract of Μ \ {p}. In fact, for 0 < s < 1, a
homotopy defined by
(4.2)
( expp[ {s · ^exp"1 q/\\ exp"1 q\\)
H = I +(1-5)||ехр;Ч||}ехр;Ч/||ехр;Ч||]
I when ς £M\(CP U {ρ})
( q when q e Cp
gives the desired strong deformation retraction (see the following Exercise 1).
Exercise 1. Setting H(q, s) := Hs(q), show that Η : Μ \ {ρ} χ [0, 1] ->
Μ \ {ρ} is a continuous map with Щ = id, H\ : Λ/ \ {p} —» Cp, #s | Cp = id.
Exercise 2. If ς is a cut point of ρ along 7. show that ρ is a cut point of q
along 7_1.
Thus the cut locus Cp inherits to a considerable extent the topology of M.
First we give some examples.
Example 1. Let (5m, go) be the sphere of radius 1 in Rm+1. Then all normal
geodesies 7 emanating from ρ £ Sm pass through the antipodal point ρ (:= —ρ) of
ρ at the parameter value π, and they never intersect before π. Therefore Cp = {p},
and ρ is the first conjugate point to ρ along all 7.
Example 2. Recall that the m-dimensional real projective space RPm is
obtained from 5m by identifying points ρ and their antipodal points p, and the deck
transformation J of the two-fold covering map π : 5m —> ДРт is given by /(p) = p.
Since / is an isometry of (5m, go), RPm carries a Riemannian metric h0 such that
π is a local isometry and is of constant curvature 1. Then normal geodesies
emanating from π(ρ) £ RPm are images of normal geodesies ηη (и £ TpS171) emanating
from ρ via π. Then for u, ν £ /7PM, u/v, and π > £ > 0, we get from p_Lu, г> the
following:
π(7^(0) = π(7υ(0) ^ cos^ 'P + sm^ ' u = —cost -p — sin£ · г>
Φ> 2 cos £ ■ ρ = — sin t · (и + г>) Ф> £ = π/2, г> = — и.
Namely, for 0 < t < π, π ο ηη intersects with π ο ηυ(ν φ и) only when ν = —u,
£ = 7г/2, and the first conjugate value along any geodesic is equal to π. Therefore
Cp = π{7η(π/2); и £ /7PM}, which is the image of a great sphere 5m_1 of 5m via
π, and is an (m — 1)-dimensional projective subspace.
Example 3. For (Дт, до), {Нш, до) there exists only one normal minimal
geodesic joining two given different points, and Cp = φ for any point p.
Example 4. Let Γ be a lattice in R171 and consider an m-dimensional torus
Tm = Дт/Г. Then elements of Г give deck transformations of the universal
covering π : Rm —» Rm/T via parallel translations, which are isometries of (Дт, <7o)·
Therefore, we get a flat (i.e., constant curvature 0) Riemannian metric ho on
Tm such that π is a local isometry. Setting ρ = π(ο), we may assume that
Rm = ТрТш, π = expp, because straight lines of R™ through 0 are mapped to
geodesies of Tm through ρ via π. Since we have no conjugate points (Proposition

106 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
3.6), we get
Xp = {ν e R171; \\v\\ < \\v - 7|| for any 7 G Г \ {o}},
and this is an open convex subset bounded by hyperplanes which pass through
\η (7 £ Γ \ {о}) and are perpendicular to 7. Then the cut locus Cp is given by
π(θϊρ). In particular, if Г = Zm, namely, if Г has a basis consisting of o.n.b.
{ej}·^, then Cp = dXp is the boundary of a cube obtained by parallel translating
each coordinate plane X{ = 0 by ±\ei, and is in fact the union of the subsets defined
by χι = ±^, — ^ < Xj < ^ (j φ г) with respect to г = 1, ... , га. Therefore, Cp is
the union of ra tori J*™-1.
Exercise 3. For a flat torus (T2 = Я2/Г, h0), where Г is given by Г :=
((1, 0), (αχ, α2))ζ, determine Cp and Cp.
4.2. Now we are concerned with the relation between the topology of Μ and
that of Cp.
Proposition 4.5. Let Μ be a compact (connected) та-dimensional Riemann-
ian manifold.
(1) The inclusion map ι : Cp ^-> Μ induces isomorphisms l* : П{(СР, q) —»
7Гг(М, q) (1 < г < m — 2) and an epimorphism l* : 7rm_i(Cp, 0) —» 7rm_i(M, 0)
between the homotopy groups.
(2) For ί/ie homology and cohomology groups we get isomorphisms
1. : Щ(СР, Z) -* Щ(М, Z), l* : tf*(M, Z) - tf*(Cp, Ζ)
for г φ га, m — 1. If Μ is orientable, we get isomorphisms l*, l* also for г = m — 1.
// Μ 25 nonorientable, then we have the following exact sequences:
0 -+ Ζ -+ Нт.г(Ср, Ζ) - tfm_i(M, Ζ) - 0,
(4'3) о -+ н171-1 (м, ζ) -+ я^1^, ζ) -+ ζ -+ ο.
FtnaHj/, tfm(CP, Ζ) ^ 0 ^ tfm(Cp, Ζ).
PROOF. (1) For a continuous тар /ι : (5\ *) —» (Μ, q) (q e Cp), we may
assume that ρ £ h(Sl) for г < га — 1, by moving /г slightly via a homotopy if
necessary. In fact, first approximate h by a smooth map /г and note that h(Sl)
is a null set. Therefore, we may choose a point ρ £ М£г) and € > 0 such that
В := Be (p) is a normal coordinate neighborhood and contains p. Then by deforming
h(Sl) Π Β along geodesic rays emanating from ρ to c?B fixing the outside of £, we
get the desired homotopy. Now taking the deformation retract Hs of Proposition
4.4, we see that H\ о h : (5\ *) —» (Cp, ς) is homotopic to h. Therefore l* :
7Гг(Ср, ς) —» 7Ti(M, g) is surjective. Next suppose that h : (5\ *) —» (Cp, #) (1 <
г < m — 2) satisfies £*[/i] = 0.8 namely, h may be extended to a continuous
map h : (B ,*)—*· (Μ, ς), where В is a closed disk with dB% = Sl. If
г Η-1 < га — 1, then by the same argument as above, h is homotopic to a map from
(£ , *) to (Cp, ς) which is an extension of h. Then we get [h] = 0, and l* is
inject ive.
(2) We show only the case of the homology group. The case of the cohomology
group may be treated in the same way. In the following we omit the coefficient group
[h] denotes the homotopy class of h.

4. CUT LOCUS
107
Ζ. First we note that
H*(Ip, Cp) = H*(Ip, Xp \ {op}) (Cp is a strong deformation retract of Xp)
= H*(Xp \ Cp, Xp \ (Cp U {op}) = Η*(ϊρ, Xp \ {op}) (excision theorem)
^H^(M\ Cp, Μ \ (Cp U {ρ}) (expp : Xp -> Μ \ Cp is a diffeomorphism)
= H*(M, Cp) (excision theorem).
Recalling that Tp is homeomorphic to a closed disk and Cp = dXp is homeo-
morphic to the sphere 5m_1, we have Hi(Xp, Cp) = О (г φ га), Нш(Хр, Cp) = Ζ.
Then we also get Hi(M, Cp) = О (г φ га), НШ(М, Ср) = Ζ. Now we consider the
homology exact sequence corresponding to Cp ^-> Μ ^-> (M, Cp):
• · · - tfi+i(M, Cp) - Я,(Ср) - Ή(Μ) - Я,(М, Ср) - Я,_х(Ср) -, · · ·
and get Hi(Cp) = Щ(М) ϊοτ i φ т, т — 1. For г = m — 1 we have an exact
sequence:
—> Ят(М) ^> Ят(М, Cp) —> Ят_!(Ср) —. Ят_1(М) -^ 0.
If Μ is nonorientable then we get Hm(M) = 0, and (4.3) follows when we note that
НШ(М, Cp) = Z. If Μ is orientable, then the fundamental class [M] of M, which
is a generator of H^M), is mapped to a generator of Ят(Л/, Ср) via (*?m? and we
get Ят_1(Ср) = Ят_1(М). Finally, Hm(Cp) = 0 follows from the exact sequence
0 <* НШ^{М, Cp) —, Ят(Ср) -^ Ят(М) ^ Ят(Л/, Cp) ^ Ζ
by the same argument as above. D
Exercise 4. Suppose Μ is a compact Riemannian manifold with dim Μ > 2.
Then show that Μ is simply connected if and only if Cp is simply connected.
By Proposition 4.1, any cut point along a geodesic either is the first conjugate
point or appears before it. We denote by Qp the set of first tangent conjugate
points along geodesies emanating from p, and set Qp := exppQp, which is called
the first conjugate locus of p. We give the following result of F. Warner ([War-2])
as an application of Morse theory.
Proposition 4.6. Let Μ be a complete simply connected Riemannian manifold
and ρ e M. Suppose that for any w £ Qp the multiplicity n(w) of w as a conjugate
point is greater than or equal to 2. Then Cp = Qp.
Proof. For и £ UPM we denote by to(u) the first conjugate value to ρ along
7U. It suffices to show that ηη \ [0, t] is minimal for any 0 < t < to(u). We set
q := 7n(£). Then we may choose a sequence qn —> q so that the qn are not conjugate
to ρ along any geodesic joining ρ to qn, and therefore the energy integral Ε is a
Morse function on Cp~n(A), where Δ is a subdivision of [0, 1] and a > 0 (see §3.2).
Now note that expp is a diffeomorphism on a sufficiently small neighborhood of
tu in TpM because of t < to(u). Since и н-> to(u) is continuous, we may choose
un £ UPM —> u, tn —> t (0 < tn < to(un)) so that qn = ^Un (tn). If we can show that
7nn I [0, tn] (= 7innn I [0, 1]) are minimal geodesies, then ηη \ [0, t] is minimal by
Lemma 4.2. We derive a contradiction by assuming that 7n := 7tnu„
| [0, 1] -» Μ
is not a minimal geodesic joining ρ to qn. Take a minimal geodesic δη € Cpqn from

108
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
ρ to qn. Then the indices of 7n, δη are equal to 0 by the index theorem. Since Μ
is simply connected, we have a homotopy Hs (0 < s < 1) in СРЯп from jn to <5n.
Taking an a > 0 with a2 > 2тах{Е,(Я5); 0 < s < 1}, we have a homotopy Я5
from 7n to <5n in С := Cp~n(A) by Exercise 1. Note that critical points of Ε \ С
are nothing but geodesies joining ρ to qn. Since Ε is a Morse function on С and
ind7n = Ίηάδη = 0, by the Morse lemma (Chapter I, (2.10)) the following holds:
There exists an e > 0 such that for any homotopy Hs from jn to δη in С we have
E(HSl) > Ε{Ίη) + €, E{HS2) > Ε{δη) + € for some su s2 e [0, 1].
On the other hand, indices к of critical points of Ε are either 0 or greater
than or equal to 2 by the assumption and the index theorem. Then С carries the
homotopy type of a CW-complex obtained by attaching a A;-dimensional cell to
each geodesic of index k. We consider Hs as a curve in С and remove к (> 2)-cells
by a homotopy fixing the end points, and we get a homotopy Hs from jn to δη
with E(HS) < max{£(7n), Ε(δη)} + € for any e > 0. (See Chapter I, Remark 2.4.
Note that the critical points of index 0 do not affect the process stated there.) This
contradiction completes the proof. D
Remark 4.7. In general, the relation between Cp and Qp is rather
complicated. In fact, A. Weinstein constructed on any compact Riemannian manifold Μ
other than S2 a Riemannian metric g such that Qp Π Cp = φ for some ρ e Μ (see
[We-2], [It]).
Exercise 5. For 52 show that CpDQp φ φ for any point ρ with respect to
any Riemannian metric on 52.
Exercise 6. Let Μ be a compact Riemannian manifold. Show that Μ is
simply connected if Qp = Cp.
Now we define the distance function dp : Μ —> R to ρ by dp(q) := d(p, q).
Then dp is a continuous function, and we consider the relation between the cut
locus Cp and the differentiability of dp.
Proposition 4.8. (1) dp is of class C°° on Μ \ {Cp U {p}}, and its gradient
vector Vdp(q) at q £ Μ \ {Cp U {p}} is given by
(4-4) (Vdp) (q) = Wdp(ff)),
w/iere 7P<7 denotes a unique minimal geodesic from ρ to q parametrized by arc-length.
In particular, ||(WP) (q)\\ = 1.
(2) Suppose that there exist at least two normal minimal geodesies (say, 71, 72)
joining ρ to q. Then dp is not differentiable at q. Note that such q belongs to Cp.
Proof. (1) Since exp'1 : M\(Cp\J{p}) -> TpM\{op} is an (into) diffeomor-
phism and dp(q) = \\ exp"1 q\\, we see that dp is of class C°° at q G M\ {Cp U {p}}.
Next, for any X e TqM, take a smooth curve c(s) tangent to X at q = c(0).
We may assume that c(s) £ Μ \ (Cp U {p}) when \s\ is sufficiently small, and get
a smooth variation α of 7P<7 by taking minimal geodesies 7pc(s) £ C([0, d(p, q)])
joining ρ to c(s). Then from the first variation formula (2.6) we get
— d(p, c(s)) = (X, jpq(d(p, q))),
aS \s=0
namely, (Vdp) (q) = %q{d{p, q)).

4. CUT LOCUS
109
Figure 12
(2) Suppose that dp is differentiable at a point q satisfying the assumption.
Set / := d(p, q), X = (Vdp)(q), and take an e > 0 so small that qe := 71 (/ — e)
belongs to a convex neighborhood W centered at q in the sense of Chapter II, (2.5).
Now for any F G T^M we set c(s) := expq s У, and let aes be the normal piecewise
C°° curve obtained by first proceeding along 71 from ρ to q€, and then proceeding
to c(s) along a minimal geodesic from qe to c(s). Then setting les = L(aes), we get
l\ > dip, c{s)) and /§ = dip, c(0)). It follows that £ \s=0 les > £ \s=0 d(p, c{s)),
and the first variation formula implies that
(4.5) (У, X) < I
П =(Y,ii{dp(q))).
s=0
Applying the same argument to 72, we get
(4.6) (У, X) < (Y, b(dp(q))).
Setting Υ = -j(dp(q)) in (4.6), we now have (X, 72(^(9))) > 1· Similarly, setting
Υ = l2{dp(q)) in (4.5), we have
1 < (X,b(dP(q))) < (ii(dp(q)),UdP(q))).
Since 71, 72 are unit vectors, it follows from the Cauchy-Schwarz inequality that
ii{dp(q)) = l2(dp{q)), namely, 71 and 72 coincide. This is a contradiction, and the
proof is complete. Π
Remark 4.9. It is known that the set of points q satisfying the assumption of
(2) is dense in Cp.
Next, for the Hessian of dp we get the following from the second variation
formula.
Lemma 4.10. Let q e M\ {Cp U {p}} and и G TqM be given. Take a normal
minimal geodesic 7 : [0, dp(q)] —> Μ joining ρ to q. Let X(t) be a Jacobi field
along 7 satisfying the boundary condition X(dp(q)) = u, X(0) = 0 (see Chapter
IIf Lemma 2.4), and let Χ^(ί) := X(t) — (X{t), 7(£)Ж0 be the Jacobi field that
is the vertical component of X(t) with respect to 7. Then
(4.7) D2dp(q) (и, и) = (VX±(dp(q)), X±(dp(q))).

no
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
PROOF. Let r(s) := expqsu be the geodesic tangent to и at q. Take minimal
geodesies 7S : [0, d(p, q)] —» Μ joining ρ to r(s), which define variation curves of
7. Then the corresponding variation vector field is simply X. Now, noting that τ
is a geodesic and X is a Jacobi field, from the second variation formula (2.9) we get
d2
D2dp(q) (u, u) = j^ |e=0 £(7s)
= A(V^(0, VA^i)) - (Я(^(0, 7(0)7(0, XHmdt
Jo
= [(νχχ(ί), χχ(ί))]ί, = (vxx(0, xx(0>,
where we have set I = dp(q). □
Remark 4.11. Recall that for Jacobi fields Χ, Υ along 7 vanishing at t = 0
we have (VX(<), У(*)> = (Χ(ί), VF(0>· Therefore, when u,vG TqM, denoting by
X, У the Jacobi fields along 7 with X(0) = У(0) = 0, X(dp(q)) = u, У(йРЫ) = ν,
we obtain
(4.7)' D2dp(g) (u, V) = (VX^(rfp(g)), Y±(dp(q))).
In particular, we see that Vdp(q) = 7(rfp(g)) belongs to the null space of the Hessian
D%(q).
Exercise 7. Let / : Μ \ Cp —» R be defined as f(q) := ^dp(g)2. Show that
/ is of class C°°, and that for и € TqM
(4-8) ||V/(9)|| = rfp(g),
(4.9) D2f(u, u) = dp(9) (VA-(dp(g)), X(rfP(g))),
where X is as in Lemma 4.10.
Now we estimate the size of coordinate neighborhoods over which normal
coordinate systems are valid in relation to the cut locus. This plays an important role
when we study global properties of complete Riemannian manifolds.
Definition 4.12. Let Μ be a complete Riemannian manifold. For ρ e Μ we
define the injectivity radius ip(M) at ρ as
sup{r > 0; expp | Br(p) is a difFeomorphism}.
We call i(M) := inf{zp(M); ρ £ Μ} the injectivity radius of Μ.
Proposition 4.13. (1) ip(M) = d(p, Cp), and ρ \—► ip(M) is a continuous
function from Μ to R+ U {+00}.
(2) For и £ UPM denote by to(u) the first conjugate value to ρ along the
normal geodesic *yu, and set t0(p) := mm{t0(u)] и £ UPM}. Denote by /0(p) the
minimum of the lengths of nontrivial geodesic loops at ρ (i.e., geodesies emanating
from ρ and ending at ρ which are not a point curve), and set lo(p) = +00 if there
exist no such geodesic loops. Then ip(M) = min{£0(p), lo{p)/2}.
PROOF. Prom the definition we may easily check that ip(M) = min{£(u); и £
UPM}, where t(u) is a continuous function on UM defined by (4.1). On the other
hand, we have Cp = expp{t(u)u; и £ UPM} also from the definition, and ip(M) =
d(p, Cp) is obvious. Next we prove (2). First, we have ip(M) < min{£0(p), /o(p)/2},
because the distance realizing property of geodesies breaks down after the first

4. CUT LOCUS
111
conjugate value or the parameter value when two minimal geodesies emanating
from ρ intersect. Second, to see the reverse inequality, we may assume that
ip(M) < +00 and take а и £ UPM with t(u) = ip(M). If t(u) is equal to the
first conjugate value to(u), then we clearly get t(u) = t0(p). If t(u) < to(u),
then by Proposition 4.1 there exists a v £ UPM (ν φ и) such that 7u(t(u)) =
jv(t(u)) =: q. When t(u) is the first conjugate value to ρ along ην we again get
t(u) = to(p). Therefore we may assume that t(u) < to(v). Then expp is
regular at t(u)u and t(u)v, and we may take disjoint open neighborhoods {7, V of
t(u)u, t(u)v, respectively, so that expp is difFeomorphic on these neighborhoods.
Since ju(t(u)) = jv(t(u)) = q, if we show that 7u(£(u)) = -ην(ί(ύ)), then 7 :=
(7U I [0, t(u)]) U (ηυ Ι [0, t(u)})~1 is a geodesic loop based at ρ with 1,(7) = 2ip(M).
which implies i0(p) < 2гр(М). So suppose -yu(£(u)) 7^ -7^(11)). Then there
exists a vector w e UqM, that is different from — 7u(£(u)), — 7r(£(u)) and makes
acute angles with both of these vectors. Setting 6(s) = expq sw, we consider
curves w\(s) := (expp | U)~16(s), гй2(з) = (expp | V)~16(s) in TqM, that pass
through t(u)u, t(u)v, respectively. By the Gauss lemma, ii)i(0), ^2(0) make acute
angles with —u, —v, respectively. Therefore, for suficiently small s > 0, we have
wi{s) С U П Bt{u)(op),w2{s) С V Π Bt{u){op). Namely, ||ii)i(s)||, ||гу2(з)|| <
t(u). On the other hand, since 7w1(s)/\\w1(s)\\{\\m(s)\\) = 7iD2(s)/||iZ)2(5)||(||^2(s)||) =
<5(s), it follows that t(u) > ||u>i(s)|| > i(ii)i(s)/||iZ;i(s)||) , which contradicts the
choice of u. Therefore we get ηη[ί{μ)) = —%(t(u)), and the proof of ip(M) >
min(io(p), lo{p)/2) is completed.
Finally we show the continuity of ρ ι—► гр(М). Suppose pn —> p. It suffices
to show that for any accumulation value Τ (including +00) of {in := iPn(M)}^=1
we have Τ = гр(М). First, we show that ip(M) > T. We may assume that
гр(М) < +oo and take и £ UPM with i(u) = ip(M). For a subsequence {n^} with
гПк —> Τ choose unfc £ /7Pn (M) such that иПк —> u. Since ί is continuous, we get
t{unk) (> ^nfc) —* t{u) and consequently Τ < ip(M). Second, for рПк note that by
(2) at least one of the following holds:
(a) iUk = t0(pnk).
(b) There exists a geodesic loop jnk with length 2гПк.
In fact either (a) or (b) holds for infinitely many k. If this happens for (a), then as
in Proposition 4.1 we see that Τ = limznfc is the first conjugate value to ρ along ju
if it is finite, and we get ip(M) < T. If the above happens for (b) and Τ is finite,
then 7nfc converges to a geodesic loop at ρ with length 2T (taking a subsequence if
necessary), and again we get гр(М) < Т. П
Corollary 4.14. Let Μ be a compact Riemannian manifold. Then i(M) > 0.
Further, setting t0 := min{io(^); и £ UM} and denoting by Iq the minimum of the
length of nontrivial closed geodesies of M, we have г(М) = min{£o, fo/2}.
PROOF. The first assertion is clear because t : UM —> R is continuous on the
compact set UM, and i(M) < min{£o, fo/2} is easily seen as in the proposition.
Take a point ρ with ip(M) = i(M). If ip(M) = t0(p), then we get t0 = to(p).
If ip(M) < £o(p), then we get a geodesic loop 7 at ρ with length lo(p). Set
q = 7(гр(М)). If q is conjugate to ρ along 7 | [0, ip(M)} or (7 | [гр(М), 2гр(М)])"\
then we get г(М) = /o(p)/2 = to. Otherwise, we show that 7(0) = 7(/o(p))
holds, which implies that 7 is a closed geodesic and i(M) > Iq/2. In fact, note
that ρ is not conjugate to q along geodesies 71 := (7 | [0, ip{M)])~l and 72 :=

112
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
7 | [zp(M), 2ip(M)] which join q to ρ (see the proof of Lemma 2.4 in
Chapter II). Now if 7(0) φ 7(/ο(ρ)), then the tangent vectors to 71, 72 at ρ satisfy
7i(zp(M)) φ —72(гр(М)). Therefore, we get iq(M) < ip(M) by the same argument
as in the proof of (2) of the previous proposition. This contradicts the choice of p,
and the proof of the corollary is complete. D
Figure 13
Note that i(M) may be equal to zero for a complete but noncompact Riemann-
ian manifold Μ (see Figure 13). If we can estimate г(М) for compact Μ in terms of
geometric invariants, then we will have an estimate for the uniform size of normal
coordinate neighborhoods of Μ, which gives restrictions on the topology of Μ (see
§3 of the next chapter).
5. Ambrose's Theorem
In this section we give a theorem due to W. Ambrose, which is a global version
of Theorem 3.2 of Chapter II. First we give some preliminaries. A continuous curve
7 : [0, /] —» Μ in a Riemannian manifold Μ is called a once broken geodesic if
there exists a t0 £ [0, /] such that 7 | [0, t0] and 7 | [to, I] are geodesies. Now let
Μ, Μ be complete m-dimensional Riemannian manifolds and ρ e Μ, ρ e M. Let
/ : TPM —» TpM be a linear isometry. Then for any once broken geodesic 7 in Μ
we may define a once broken geodesic 7 in Μ by
~m.= fexPp*/(7(0)), 0<t<to,
7U' \exp5(to)(i-io)(/io(7(*o + 0))), to<t<l,
where Ito : ΤΊ^Μ —> T^^M is a linear isometry given by
(5-1) /to:=P(7)?0o/oP(7)£\
Then note that we may define It : ΤΊ^Μ —> ΤΊ^Μ for 0 < £ < / in the same
manner as in (5.1).
Theorem 5.1 (W. Ambrose). LetM, Μ be complete m-dimensional
Riemannian manifolds and I : TpM —> ΤρΜ a linear isometry. Suppose that Μ is simply
connected and, for any once broken geodesic 7 : [0, /] —► Μ in Μ,
/4(Л(«, «И = J2(Jt(u), /,(«))/,(«;)
/or ant/ и, ν, w e ΤΊ^Μ, 0 <t < I,

5. AMBROSE'S THEOREM
113
where R, R denote the curvature tensors of Μ, M, respectively. For any minimal
geodesic 7 : [0, /] —» Μ define a geodesic 7 by 7(2) = exp^t/(7(0)), and define a
map Φ : Μ —» Μ by Φ(7(£)) := ^(t). Then Φ is well-defined and a C°° Riemannian
covering. In particular, if Μ is also simply connected, then Μ and Μ are isometric.
Proof. Take o.n.b.'s {e{} of TPM and {ё{ := Ι(β{)} of TPM. We identify
TPM, TpM with R™ via the above o.n.b.'s. For χ £ Rm we denote by ηχ (resp.,
Ίχ) the geodesic in Μ (resp., M) emanating from ρ (resp., p) with the initial
direction x. Let г (χ) be the minimum of the injectivity radii at 7X(1), 7X(1) of
M, M, respectively. We set О := {(χ, у) £ R171 x Rm; \\y\\ < г(х)}, which is an
open set of Rm χ R™ because of the continuity of χ ι—► i(x) (Proposition 4.13). For
(x, y) £ О set
(5.3) exy := exp7x(1) P(7*)?i/, ёху := ехрЫ1) Р(7*)?У·
Figure 14
Note that ex : Вцх)(о) —> Μ, ёх : Вг(х)(о) —» A/ are (into) difFeomorphisms.
For (χ, у) G О we get a once broken geodesic 7x.y of Л/ by joining geodesies
t^lxit) (0 < t < 1),
t -> ex((* - 1)2/) = exp1x(1)(i - 1)Р(ъ)?У (1 < * < 2).
We may define ηχ,υ in Μ in the same manner. Note that exy = 7x,y(2), exy =
7x,y(2). Now we define a relation (хь yx) ~ (x2- 2/2) on О by
(5.4)
f(l) eXlj/i =eX2j/2, eXlyi=eX2y2,
(2) /о(Р(7х2,У2)2 oP(7ll.(1)«) = (Р(7х2.У2)3 оР(%1>У1)»)о/,
which is an equivalence relation as is easily seen. Let X := О/ ~ be the quotient
space and π : О —> X the canonical projection. Denoting by [x, y] the equivalence
class of (x, y), we define mappings e : X —» M, ё : X —» Μ by
(5.5) e([x, y]) := exy, e([x, y]) := exy.
For the proof of the theorem, first we introduce a complete Riemannian manifold
structure on X such that e, e are local isometries. We start by proving two lemmas.
Lemma 5.2. (1) Let (x, y) £ 0. Tften ί/iere exzs£ open neighborhoods U of
exy and U ofexy such that exoe~l is an isometry from U onto U.

114
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
(2) If (xi, У\) ~ (x, y), then we may choose the above U and U so that eXl о
e~^ : U —» U is an isometry, and in fact coincides with ёх о e~l.
Proof of lemma. Let о be the origin of ilm, and recall that ex : Вцх)(о) —»
Μ, ёх : В{(х)(о) —» Μ are difFeomorphisms. Now apply the assumption (5.2) of
the theorem to a linear isometry Ix := Ρ(ίχ)® о / о Р("ух)1 : T7x(i)M —» Τ7χ(ΐ)Μ;
we see that the assumption (2.6) of Cartan's Theorem (Chapter II, Theorem 3.2)
is satisfied. Therefore, Fx := exp7x(i) oIx oexp"1,^ = ex oe~l is an isometry from
ex{Bi{x)(o)) ontoex(Bi{x)(o)), and we get DFx(ex(z)) = P(%,z)2oIx °P{lx,z)\ at
exz, ζ G £j(x)(o) (Chapter II, (3.7)). Now take any connected open neighborhood
О С Вцх)(о) of у. Then (1) follows on setting U = ex(0), U = ex(0). Next,
suppose (xi, 2/i) ~ (x, 2/), and take as above connected open neighborhoods O, 0\
of у, ух, respectively, so that eT(0) = eTl(Oi) := /7. Then U is an open
neighborhood of exy = eXl2/i, and Fx, FXl are isometries from U into M. We have
Fx(exy) = exy = eXxyx = FXl(eXlyl) = FXl(exy), and DFx(exy) = DFXl(eXlyi)
because of (5.4) (2) and the above expressions for DFX, DFXl. Hence the
isometries FX and FXl coincide (Chapter II, §3, Exercise 2). Note that, in particular,
MO) = FX(U) = FX1(U) = eXl{Oi). О (Lemma 5.2)
Lemma 5.3. Suppose (χχ, y\) ~ (x2, 2/2)· Then there are open neighborhoods
Oi С Bi(x.)(o) of yi (г = 1, 2) which satisfy the following conations:
(1) e~^ oeXl maps 0\ diffeomorphically onto O2.
(2) ё~2! о ёХ1 also maps Ο ι diffeomorphically onto O2, and in fact coincides
with e~l о eXl.
(3) For Zi eOi(i = 1,2), (xi, z\) ~ (хг, z2) if and only if eXlz\ = eX2z2.
Proof of lemma. By Lemma 5.2 we may take Oi, 02 so that eXl (0\) = eX2 (O2)
:= /7, and FXl, FX2 coincide as isometries from U onto U := eXl(Oi) = e~X2(02).
Then (1) and the first part of (2) follow easily, and from ёХ1 о e"1 = eX2 о е"1 we
get the second part of (2). To prove (3) it suffices to show that (xb z\) ~ (x2, Z2)
follows from eXl z\ = eX2z2 (z{ G Oi). By virtue of (2) we have e~2l oeXl (z\) = e~2l о
eXl(zi) = Z2, namely, eXlz\ = eX2z2- Further, from DFXl(eXlZi) = DFX2(eX2z2) it
follows that
PiTlxi^l ° Ixi ° РЬхг.гг)2! = P{lx2iz2)\ ° 'x2 ° Ρ(Ίχ2,ζ2)1
Noting that 7ar1>2l (1) = 7rr2.z2(l)> e^c-7 and rewriting the above equality, we get
I ° (^(7х2>г2 )l о P{lxi ,SI )§) = (P(7x2)22 )g о P(jXl ,21)») ο /,
which shows that (χι, Ζ\) ~ (x2, z2). D (Lemma 5.3)
Now we introduce a topology on X. For any χ G ilm and any open set О С
Bi(x)(o), we set О := {[χ, у]; 2/ G О}, which is defined to be an open set of X. For
open sets 0\ С ВцХ1)(о), 02 С Б^Х2)(о) we show that 0\ Π Ο2 may be written
as the union of above O's, which implies that {0} forms a base for the family of
open subsets of X. Suppose [x, y] G 0\ Π 02. Then there exist 2/г € Oi(i = 1, 2)
such that (x, y) ~ (xi, 2/1) ~ (x2, 2/2)· We may choose open neighborhoods О С
^г(х)(^),0/1 (с Οι), 02 (С 02) of 2/, 2/ь 2/2, respectively, so that Lemma 5.3 holds
for the pairs (O, 0[) and (O, 02). Then it follows that ex(0) = eXl (0[) = eX2(0'2),
and for ζ G О there exist zi e 0[(i = 1,2) such that ex(z) = eXl(z\) = eX2(z2).

5. AMBROSE'S THEOREM
115
We get (x, z) ~ (xb z\) ~ (x2, 22) by Lemma 5.3, and so О С 0[ Π 02. Next we
see that X has a countable base for open sets. In fact, taking a countable dense
set {xk}^Li of R™ and countable bases {Ok.j}^L\ for ВцХк)(о), we may check
that {Ofc,j}i<fcj<00 is a countable base for X. Now we see that X is a HausdorfF
space. Namely, for [χι, yi] ^ [x2, 2/2] we show that there exist open neighborhoods
Oi С Bi(x.)(o) (г = 1,2) with Οι Π 02 = 0. Suppose the contrary. Then we get
sequences yn -> 1/1, zn -> y2 with [xb yn] = [x2, 2n]. Namely, we have
(-I-J ^х\Уп = ^i2^n and cXlyn = €.χ2ζη,
(2) / о (Р(7х2,гп)§ о Р(7Х1 ,„„)§)) = (Р(Чха.,п)§ ο Ρ(7χ1>νη)§) ο /.
Letting η —» οο, we get (χι, у ι) ~ (x2, 2/2) from (1), (2), which is a contradiction.
Furthermore, X is arc wise-connected, because for any [x, y] € X a, curve in X,
obtained by joining two curves t i-> [x, (1 - t)y] (0 < t < 1), £ ι—► [(2 - £)x, 0] (1 <
£ < 2) together, joins [x, у] to the fixed point [0, 0].
Now we show that X carries a manifold structure. For [x, y] € X take a
neighborhood О С Bi(x)(o) obtained from Lemma 5.3 by setting (xi, yi) = (x2, y2) =
(x, y). We define a surjective map ix : О —> О as ^х(г) = [χ, ζ]. Since ex | О is a
difFeomorphism, i,x is injective by Lemma 5.3 (3) and in fact is a homeomorphism
by the definition of the topology for X. Further, for e as in (5.5), e \ О : О —> ex(0)
is bijective, because of the above choice of О and Lemma 5.3 (3). Since e ο ιχ — ex
and ex is a difFeomorphism from О onto ex(0), e | О is a homeomorphism onto
a coordinate neighborhood ex(0) of Μ. This implies that X is an m-dimensional
topological manifold.
Now for [xi, yi], [x2, y2] G X take neighborhoods О; of уг (г = 1,2) as above
and suppose 0\ Π 02 ^ 0. Then (Ог, е"1 о е | Ог) (г = 1,2) are charts around
[χ*, yi], and the coordinate transformation
(β"1 ο e I 02) о (е-1 о e | ΟΟ"1 : β^^ίόι Π 02)) -» е£{£Фх П 02))
is equal to e^1 о eXl, which is a difFeomorphism by Lemma 5.3. Therefore, X
carries an m-dimensional C°° manifold structure such that e : X —> Μ is a local
difFeomorphism. Note that e is surjective, since Μ is complete and there exists a
minimal geodesic joining ρ to an arbitrary point q G M. e is an immersion, and we
consider the induced Riemannian metric on X from M, which is again complete.
In fact, since e is a local isometry, any geodesic of X emanating from a fixed point
[0, 0] is given in the form ί н> [ίχ, 0] and may be defined for all real numbers t.
Theorem 1.1 (1) implies that X is complete. Next we see that ё : X —> Μ given in
(5.5) is also a surjective local isometry. In fact, surjectivity follows as above, and
for [x, y] G X choose an open neighborhood О С Вцх)(о) and the corresponding
OCX. Then
ё | О = (ёх о е^1) о ex ο ^,"1 = (ёх о е"1) о е | О,
and our assertion follows since ёх о e~l is an isometry by Lemma 5.2.
Now the theorem follows immediately from the following Theorem 5.4. In fact,
e : X —> Μ is a Riemannian covering, and consequently an isometry because
Μ is simply connected. Therefore ё о e~l : Μ —» Μ is a local isometry and a
Riemannian covering due to Theorem 5.4. Finally, it is clear from the definition
that eoe"1(7(i)) = 7(0- D

116
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Theorem 5.4. Let (M, g), (AT, ft) be m-dimensional (connected) Riemannian
manifolds, and assume that (M, g) is complete. Then any local isometry π :
(Μ, g) —» (AT, /ι) 25 a Riemannian covering.
PROOF. Since π is a local isometry, geodesies of Μ are mapped to geodesies of
N via π, and, for ρ £ Μ and q = π(ρ), Dn(p) : TPM —» ΤςΑΓ is a linear isometry.
First, we show that N is also complete. For a geodesic % of N with the initial point
q £ N and the initial direction ν £ UqN, consider the geodesic ηη of Μ with the
initial point ρ and the initial direction и := Dn(p)~l(v) £ Z7PM. Then 7n may be
defined for all parameter values since Μ is complete, and therefore so is % = no^u.
Next we show that π is surjective. In fact, let q\ £ N be given. We have a minimal
geodesic 7 : [0, /] —» AT joining ρ to (71. Denoting by 7 a geodesic of Μ emanating
from ρ with the initial direction Ζ)π(ρ)_1(7(0)), we get π(7(/)) = η (I) = q\. Now
for any q £ AT, take 0 < г < ^(АГ) and consider a metric ball Br(q). We set
7г_1(^) = {ра}а€Л (с М) and show that:
(i) π | Br(pa) : Br(pQ) —> Br(q) is a (surjective) difFeomorphism, in fact an
isometry.
(ii) n-\Br(q))=[JQeABr(pQ).
(iii) Br(pa) Г1 Br(p0)= φ if a ? β.
Then Br(pa) are connected components of n~1(Br(q)), and Br((?) is an evenly
covered neighborhood of π. It follows that π : Μ —» AT is a covering map.
First we show (i). Since π is a local isometry and maps geodesies emanating
from pQ to geodesies emanting from q, the following diagram is commutative:
£>π(ρα)
£r(0pJ > #rK)
expPa I |exPq
Br{Pa) > Br{q)
π
Then since expPa is surjective and expq oDn(pQ) is bijective, it follows that
π | Br(pQ) is a bijective smooth map. Moreover we have π*ft = #, which implies
that π is an immersion. Therefore, π | Br(pa) is a difFeomorphism from Br(pQ)
onto Br(q).
Next we show (ii). Obviously Ua^(Pa) С n~1(Br(q)) by (i). Conversely, let
P\ £ π-1(£Γ((7)). For (71 := π(ρι) we may write q\ = expqv, ν £ Br(oq), and
7(£) = βχρς(1 — i)v, 0 < £ < 1 is a geodesic in N joining q\ to q. Then, taking a
geodesic 7 emanating from pi with the initial direction Dn(pi)~1(^(0)), we have
7(1) = pQ for some a, because π ο 7(1) = 7(1) = q. Note that from £(7) < r we
get pi £ Br(pQ).
Finally, we see that (iii) holds. Suppose pi £ Br(pQ) Π Br(pp). We take a
minimal geodesic 7 (resp., σ) in Br(pQ) (resp., Br(pp)) joining pa (resp., p^) to
p. Then the geodesies π ο 7 and π ο σ are geodesies in Br(q) from ρ to π(ρι)
with length less than r. Since r < iq(M), these geodesies are minimal and in fact
coincide. Therefore we have 7 = σ, which implies a = β. D
Now recall that (Ят, #о), (Sm, £o), (#m, 9o) given in Chapter II, §3.3, (I),
(III),(V), respectively, are complete simply connected m-dimensional Riemannian
manifolds Μ of constant curvature к = 0, ρ2, -ρ2, respectively. They are the
most standard Riemannian manifolds. Now let N be any complete m-dimensional

6. ISOMETRY GROUP AND HOLONOMY GROUP
117
Riemannian manifold of constant curvature k. Then, by (3.13) of Chapter II, the
assumption (5.2) of Theorem 5.1 is satisfied for Ν, Μ : = N. Therefore, we get
Corollary 5.5. Let N be a complete Riemannian manifold of constant
curvature k. Then the universal Riemannian covering Μ = N of N is isometric to one
of the above canonical Riemannian manifolds of constant curvature k.
6. Isometry Group and Holonomy Group
6.1. The set /(M, g) of all isometries of a Riemannian manifold (Μ, g) forms a
subgroup of the group of all difFeomorphisms of M, and carries the natural compact
open topology. Further, it may be shown that 7(M, g) has the structure of a Lie
group with respect to the above topology, and acts as a Lie transformation group
on Μ (for the proof of these facts we refer to [No-Ко-1], [Ko-3]). Now for an element
X of the Lie algebra of I(M, g) we define a vector field X on Л/ by
d
(6.1) Xv :=
dt
φάν),
t=o
where φί denotes the one parameter group of ДМ, g) generated by X. Since
ip\g = g, X satisfies
(6.2) Cxg = 0,
or equivalently
X · g(Y, Z) = g([X, У], Z) + g(Y, [X, Z]), У, Ζ G X(M).
Note that obviously X = 0 if X = 0.
In general, a vector field X on Μ which satisfies (6.2) is called a Killing vector
field. For such X the (local) one parameter subgroup φί of (local) difFeomorphisms
of Μ generated by X consists of (local) isometries, because d/dt(^*tg) — ^*tCxg —
0.
Lemma 6.1. (1) X is a Killing vector field if ond only if
(6.3) (VyX, Z) + <VZX, У) = О, У, Ζ e X{M),
i.e.f if and only if VX is skew-symmetric as a linear map X(M) зУи VyX G
X{M).
(2) If X is a Killing vector field, then X is a Jacobi field along any geodesic
ofM.
(3) If X is a Killing vector field, then for U, V G X(M) we have
(6.4) (VuVX)V + R(X, U)V = 0.
PROOF. (1) follows from the following: for У, Ζ G X(M)
Cxg = 0& Х(У, Z) = ([X, У], Z) + (У, [X, Z])
^ (V*y, Z> + (У, VXZ) = (VXY - VyX, Z> + (У, VxZ - VZX>.
Next note that the local one parameter group φι generated by a Killing vector
field X consists of local isometries, and for any geodesic 7, {^7} is a variation of
7 consisting of geodesies whose variation vector field is X \ 7. Therefore X \ 7 is a
Jacobi field (see Chapter II §§2.2), which completes the proof of (2).
Now we show (3). The φί are isometries, and
(*) Dtpt(VuV) = ν0φιυϋΨίν, U,Ve X(M).

118
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Differentiating (*) with respect to t at t = 0, we get
CxVuV = VCxuV + VuCxV.
Then by the definition of the curvature tensor and the equality £χΥ = VχΥ —
VyX = [X, У], (*) is equivalent to
Я(Х, U)V = VVuVX - VuVyX,
which is nothing but (6.4) D
Remark 6.2. (1) The space of all Killing vector fields on Μ forms a subal-
gebra of the Lie algebra X(M) of vector fields on Μ.
(2) A diffeomorphism of Μ which satisfies (*) is called an affine transformation
of M. The set of all affine transformations of Μ also carries the structure of a Lie
group with respect to the compact open topology. Then a vector field on Μ which
is defined from an element of its Lie algebra satisfies (6.4). In general, a vector field
on Μ satisfying (6.4) is called an infinitesimal affine transformation and satisfies
Lemma 6.1 (2).
Corollary 6.3. Let (M, g) be a complete Riemannian manifold. Then any
Killing vector field X on Μ is complete as a vector field, and the Lie algebra of all
Killing vector fields on Μ is isomorphic to the Lie algebra of I(M, g).
PROOF. First we show that if an integral curve xt of X through ρ is defined
for 0 < t < a, then xt can be defined for 0 < t < a. In fact, from (6.3) we have
X(X, X) = 0, namely, ||χ*|| = с along xt. It follows that for 0 < t, t' < a
d(xt,xt>)<\ J \\xt\\dt\=c\t-t'\.
Then by the completeness of Μ the limit lim*ja Xt exists, and this completes the
proof of the first assertion. The second assertion is clear from this. D
Let Μ be a complete Riemannian manifold. Recall that an isometry φ G
/(M, g) is distance preserving relative to the distance d on Μ induced from the
Riemannian metric g, and, conversely, any distance preserving map from (M, d)
onto (M, d) is an isometry (Problem 8 for Chapter II). Let {φη} be a sequence
of distance preserving transformations of (M, d). Then {φη} is equicontinuous,
and if {φη{ρ)} is convergent for some ρ G M, then, by the Ascoli-Arzera theorem,
{ψη) has a convergent subsequence in /(M, g) (recall that Μ is assumed to be
connected). Furthermore, if φη —► φ with respect to the topology of /(M, g),
then φη(ρ) -> φ (ρ) and Όφη(ρ) -> Όφ(ρ), since φη(βχρρίν) = exp¥,n(p) t Όφη(ρ)ν
converge to φ{βχρρ tv) = exp^(p) t Όφ(ρ)ν.
Now let O(M) be the set of all o.n.b.'s of Μ and π : O(M) —► Μ the map
which assigns ρ to an o.n.b. of TPM. Then 0(M) carries a C°° manifold structure
of dimension m + ra(ra - l)/2 such that π is a C°° map as in the case of TM,
and further π : O(M) —► Μ is a principal bundle over Μ with the structure group
0(m).
Lemma 6.4. Fix an o.n.b. (p, {ei}^) at ρ G M. Then a C°° map Φ :
/(Μ, g) —> O(M) defined by Φ (φ) := (φ (ρ), {Dipfa)}^) is an embedding, and
Ф(/(М, д)) is a closed submanifold of O(M).

6. ISOMETRY GROUP AND HOLONOMY GROUP
119
PROOF. Suppose that Φ (φ) = Φ (ψ) for φ, ψ G /(Μ, g). Then φ(ρ) = ψ (ρ) and
Όφ(ρ) = Dip(p), namely, φ = ·φ (Chapter II, §3, Exercise 3). Hence Φ is injective.
Next we show that ΌΦ is injective. Let X be an element of the Lie algebra of
/(M, g) and X the corresponding vector field on M. Suppose that ΌΦ(Χ) = 0; we
will show that X = 0. To see this, note that the tangent space to O(M) at (p, {e^})
is isomorphic to ГрМ0о(ГрМ), where o(TpM) denotes the vector space of all skew-
symmetric linear maps of TpM. Now Xv = 0, since ΌΦ(Χ) = 0. Extending e^ to
vector fields on a neighborhood of ρ and denoting by φί the one parameter group
of isometries generated by X, we get
Όφ^α) ^ -Cxei{p) = VetX.
t=o
Then from Lemma 6.1 (1) we have ΌΦ(Χ) 9* (0, (VX)(p)) = 0, namely, VX(p) =
0. On the other hand, by Lemma 6.1 (2) X is a Jacobi field along any geodesic
7 emanating from ρ which satisfies the initial conditions X(0) = Va/a^(0) = 0.
Therefore, X vanishes along any 7 and X = 0. Thus Φ is an injective
immersion. To see that Φ is an embedding, it suffices to show that, for {φη}^=ι, φ С
7(M, g) with φη{ρ) —► ψ(ρ), Βφη(ρ) —► Όφ(ρ), the <^n converge to y? uniformly
on any compact subset K. Otherwise, there exist e > 0 and {гПк} С К such
that d(<£nfc(rnJ, ¥>(гп*)) - €· We write r"fc = exPPlnkunk {unk G /7PM, Znfc =
d(i>, rnfc))· We may assume that lnic —► /0, unfc —> u0(G UPM) and rnfc —> r0 (G Ji),
taking further subsequences if necessary. Then we get
Vnk {rnk) = exP<^nfc (p) Znfc (D<^* (pW*)
-> exp^(p) /0(£V(pH) = (p(expp/oUo) = <p(r0),
which implies € < Птп_эс %fc(4)» ^(rnJ) = <%?(г0), v?(r0)) = 0, a
contradiction. Finally, we show that Ф(/(М, д)) is a closed subset of O(M). Suppose
Ψη(ρ) —► 9 and Όφη(ρ) —► Л G 0(TPM) for a sequence {y?n} С O(M). Now we
may choose a subsequence {<£nfc} which converges to φ G /(M, #) with respect to
the compact open topology as noted above. Then D<pnk(p) —► Dip(p) = Л, and
the <£n converge to φ with respect to the compact open topology by the above
argument. Π
Let IP(M, g) := {φ G /(M, g); φ(ρ) = ρ} be the isotropy group of 7(M, g)
at ρ G M. Then any sequence {φη} С /P(M, ρ) admits a convergent subsequence
because φη(ρ) = Ρ, and /P(M, g) is a compact subgroup of /(M, 0). The same
argument implies that /(M, p) is compact if Μ is compact.
Now if /(M, 0) is big, then (M, 0) posseses a large amount of symmetry. For
instance, if /(M, g) acts transitively on M, namely, for any p, q e Μ there exists an
isometry φ such that φ(ρ) = q, then (Μ, 0) is said to be homogeneous. In this case
we may study in detail the structure of (M, g) through the theory of Lie groups.
Let Iq(M, g) be the identity component of /(M, g). Then I$(M, g) is a connected
Lie group with the same Lie algebra as /(M, g).
Exercise 1. Let (M, g) be a homogeneous Riemannian manifold.
(1) Show that (M, g) is complete.
(2) Show that Io(M, g) also acts transitively on M.
Now from Lemma 6.4 we get dim/(M, g) < dim O(M) = m(m + l)/2. We
consider the most symmetric case where dim/(M, g) = m(m + l)/2. First note
d_
dt

120 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
that Ф(1о(М, д)) is a connected open and closed nonempty subset, and coincides
with a connected component of O(M). In particular, if an o.n.b. {eJatpGM and
an o.n.b. {fi} at q £ Μ belong to the same connected component, then there exists
a φ £ Io(M, g) such that φ(ρ) = q, Όφ{ρ){βι} = {fi}. Suppose m := dim Μ > 2
and let σι, σ<ι be arbitrary plane sections of TPM, TqM, respectively. Since we may
choose o.n.b.'s {e*}, {fi} of TVM, TqM, respectively, so that they belong to the
same component of O(M) and satisfy σι = (eb ег)я, σ2 = (/ι, /г)я, we have a
<£ £ /(Μ, ρ) such that Όφ(ρ)σ\ = σ2, and therefore Κσι = Κσ2. Namely, (Μ,g)
is of constant curvature, say <5, and homogeneous. We show that if δ < 0 then Μ
is simply connected. Assuming the contrary, let α £ πι (Μ, ρ) be different from
the identity. Then we may take a normal geodesic loop 7 at ρ which represents
a. In fact, consider the universal Riemannian covering π : Μ —> Μ. Then 7 is
obtained by projecting via π a minimal normal geodesic 7 of Μ joining ρ £ π-1 (ρ)
to the image p\ of ρ under the deck transformation corresponding to a. If δ < 0,
Μ is isometric to R171 or Нт with the Riemannian metric of constant curvature
δ (Corollary 5.5). Then by §4, Example 4, 7 and consequently 7 are uniquely
determined. On the other hand, take a unit vector и £ TPM, и ф 7(0), and
o.n.b.'s {e^}, {fi} with ei = 7(0), /1 = и at p, which belong to the same connected
component of O(M). It follows that there exists a φ £ Iq(M, g) such that </?(p) =
ρ, Ζ)</?(ρ)7(0) = u. Noting that ψ may be connected to the identity by a continuous
curve in Io{M, g), we see that ^07 is a geodesic loop at ρ which also represents a.
Since φο*γ is different from 7, we have a contradiction, and Μ is simply connected.
Now we turn to the case δ > 0. If Μ is simply connected, Μ is isometric to
the sphere S™ of constant curvature δ. If Λ/ is not simply connected, then the
same argument as above implies that there exists only one element α £ πι (Μ, ρ)
different from the identity, and any geodesic emanating from ρ is a geodesic loop
of the same length representing a. Therefore πι(Μ, ρ) = Ζ2, and in the universal
cover Μ = 5™, the image of ρ £ π-1 (ρ) under the deck transformation determined
by α is the first conjugate point along geodesies emanating from p, namely, the
antipode of p. Therefore, Μ is isometric to the real projective space of constant
curvature δ. As for the isometry groups of these model spaces, see Problems 12
and 13 for Chapter II. Summing up, we have
Proposition 6.5. Let (M, g) be a complete Riemannian manifold. Then
dim ДМ, g) < m(m + l)/2,
where equality holds if and only if (Μ, g) is isometric to one of the following
Riemannian manifolds of constant curvature:
(1) (Rm, go), (2) (Hm, 90), (3) (Sm, g0) or (RPm, g0).
As we saw above, Riemannian manifolds with large symmetry are rather
restricted, and generically /(M, g) is small.
Proposition 6.6. Let (M, g) be a compact Riemannian manifold.
(1) Suppose the Ricci tensor of Μ is negative definite everywhere. Then any
Killing vector field on Μ is equal to 0, and /(M, g) is a finite group.
(2) Suppose the Ricci tensor of Μ is negative semidefinite everywhere. Then
any Killing vector field on Μ is parallel.
(3) A compact homogeneous Riemannian manifold whose Ricci tensor is
negative semidefinite everywhere is isometric to a flat torus.

6. ISOMETRY GROUP AND HOLONOMY GROUP
121
PROOF. Let X be a Killing vector field. Then from (6.4) we get
R(U, X)X = {VuVX)X = Vu{VxX) - {VX){VuX)
= Vu(VxX)-(VX)2U,
and from the definitions of the Ricci curvature (Chapter II, (3.15)) and the
divergence of vector fields (Chapter II, (1.26)) it follows that
Ric(X, X) = div(VxX) - trace(VX)2.
Then from the Green Theorem (Chapter II Theorem 5.11) we have
(*) / {Ric(X, X) + trace(VX)2}^ = 0.
Jm
Now note that VX is a skew-symmetric linear transformation of the tangent space
to every point, and we get an inequality trace(VX)2 < 0. where equality holds if
and only if VX = 0. To see (1) from (*) and the assumption, note that we have
Ric(X, X) = trace(VX)2 = 0, namely, X = 0. This implies that dim/(M, g) = 0,
and /(M, g) is a finite group since it is compact. Under the assumption of (2) we
again get Ric(X, X) = trace(VX)2 = 0, which implies that VX = 0.
Finally we show (3). Io{M, g) acts transitively on M. and its Lie algebra
consists of Killing vector fields. If Χ, Υ are Killing vector fields, then we have
VX, VY = 0 and consequently [X, Y] = VXY - VyX = 0. Therefore /0(M, g)
is a compact connected abelian Lie group. Now note that the isotropy subgroup
of Io{M, g) at every point ρ consists only of the identity, because Io(^I, g) acts
transitively and is abelian. Therefore, we may assume that Λ/ itself is a compact
abelian Lie group and left translations are isometries. By (2). Killing vector fields
are parallel and orbits of one parameter transformation groups are geodesies. Thus
the exponential mapping from the Lie algebra m of the Lie group A/ to A/ coincides
with expp, where ρ corresponds to the identity. We endow m = TPM with the
canonical (i.e., flat) Riemannian strucuture obtained from gp. On the other hand,
for any и e m = TVM, transitivity implies that there exists a Killing vector field
X with Xp = u, which is parallel. Then the curvature tensor vanishes everywhere
from the definition. Hence expp : m —► Μ is a Riemannian covering by Theorem
5.4, and an epimorphism with respect to the group structures. Then the kernel Γ
of expp is a discrete subgroup of m of rank dim M, since Μ is compact. Therefore
Γ is a lattice and Μ = m/Γ is a flat torus. D
Exercise 2. Let X be a Killing vector field and set f(p) := (Xp, Xp), which
is a C°° function on M. Suppose ρ is a critical point of /. Then show that, for p,
the orbit ψι (ρ) of the one parameter subgroup ψι of isometries generated by X is
a geodesic through p.
6.2. Let (M, g) be a Riemannian manifold and fix a base point pGM. Then
the parallel translation P(c) along a piecewise C00 loop с based at ρ gives a
linear isometry of TPM that does not depend on the choice of orientation preserving
parameter transformations of the curve. For loops Ci, c<i based at ρ we denote
by c\ U C2 the loop joining them. Then we get P(c\ U C2) = Pfa) ° P(c\) and
P(c~l) = Pfc)-1, where c~l denotes the loop obtained from с by reversing the
orientation. The trivial point curve {p} corresponds to the identity. Therefore
H(p) := {P(c); с is a piecewise C°° loop based atp} is a subgroup of the
orthogonal transformation group 0(TpM) of TpM, which is called the holonomy group of

122 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
(M, g) at p. Also we set #°(p) := {P{c); с is a piecewise smooth loop at ρ ho-
motopic to the point curve {p}}, which is a subgroup of #(p), called the restncted
holonomy group. Note that for P(c) £ H°(p) we have a homotopy {cs}0<s<i
joining с to {p} consisting of piecewise smooth loops based at p, and P(cs) depends
continuously on 5. In fact, parallel translations P{cs) are defined in terms of a
system of ordinary differential equations, and solutions depend continuously on the
parameter 5. Therefore, P(cs) joins P(c) to the identity, and H°(p) is contained in
the special orthogonal group SO(TpM) preserving the orientation. Further, note
that for the above с and any loop C\ at p, C\ U с U cj"1 is also homotopic to {p}.
Namely, #°(p) is a normal subgroup of H(p).
Now we consider the relation between the holonomy group and the
fundamental group. To a piecewise smooth loop с based at p, we assign P(c~1)H°(p) £
H(p)/H°(p). If с is homotopic to c', we have P{c') oPfc"1) = P(c~l Uc') <E H°(p).
Therefore the above map с н-> P(c~1)H°(p) depends only on the homotopy class of
с and induces an epimorphism
(6.5) Ф: тп(М,р)-Я(р)/Я0(р),
which is called the holonomy endomorphism. Since πι(Μ, ρ) consists of at most
countably many elements (we assume that Μ satisfies the second countability
axiom), so does H(p)/H°(p).
Exercise 3. For p, q£ Μ take a piecewise C°° curve C\ joining ρ to q. Then
show that a map that assigns P(ci) о Р(с) о P^1) £ H(q) (resp., H°(q)) to
P(c) £ H(p) (resp., #°(p)), gives an isomorphism between H(p) (resp., #°(p))
and H(q) (resp., H°(q)).
Therefore, the (restricted) holonomy group does not depend on the choice of a
base point p, and is also denoted by H(g) (by H°(g)).
Exercise 4. (1) Let (M, g) be the universal Riemannian cover of (M, g).
Then show that H(g) = H°(g) is isomorphic to H°(g).
(2) For the Riemannian product (Μι χ Μ2, g\ x #2) of (Mi, g\), (M2, 02),
prove the following:
H(gi x 92) = H(9l) x Я(^2), H°(9l x 00 9* Я°(0О χ Я°Ы.
The purpose of the present subsecton is to give the de Rham theorem on the
decomposition of a Riemannian manifold with respect to the holonomy group. First
we give some fundamental facts to help with the understanding of the holonomy
group (for the proof of facts not proved here we refer to [No-Ко-1], [Sal]).
(6.6) #°(p) is a connected subgroup of 0(TpM) and in fact a Lie subgroup, and
#°(p) is a compact subgroup of SO(TpM). Moreover, #(p) carries the structure
of a Lie group whose identity component is H°(p).
What is the Lie algebra of H(g) and H°(p)? We denote by h(p) the Lie algebra
of #(p) for ρ e M. First recall that for x, у £ TPM, the linear transformation
R(x, y) : TPM Э ζ ι—► R(x, y)z £ TPM is skew-symmetric by a property of the
curvature tensor (Chapter II, Theorem 2.1), and defines an element of the Lie
algebra o(TpM) of 0(TpM). We show that R(x, y) £ h(p). Take vector fields
Χ, Υ defined on a neighborhood of ρ such that Xp = x, Yp = у and [X, Y] = 0.

6. ISOMETRY GROUP AND HOLONOMY GROUP
123
Let ipt, ψ3 be the (local) flows generated by X, У, respectively; note that φι, ψ3
are commutative. We also denote the integral curves of Χ, Υ through q £ Μ by
ας, /?ς, respectively; we have aq(t) = 4>t{q), etc. Then if for и £ TPM we set
u(i, s) := P(^t(p))2oP(ap)?u, we get Ve/e^(i, β) = 0 and Vd/dtu{0, 0) = 0. We
also set
r(i, s) := Ρ(βρ)·0 о Р(аФЛр)У0 о P(^t(p))° о Ρ(αρ)°,
which is in fact a parallel translation along a loop at ρ because tps0lPt{p) = ^ft^sip),
and belongs to H°(p) because this loop is homotopic to {p}. Now using Exercise 5
of Chapter II, §1, we have
V aV a u(0, 0) = lim — .
Note that the left-hand side of this equation is equal to R(y< x)u. Therefore, if we
take \fi instead of s = t and set rt := r(y/i, уД) £ H°(p) in it, we get
dt
rt = -R{x, y), i.e.. R{x. y) £ h{p).
\t=o
Next, for q £ Μ take a piecewise Cx curve c\ joining ρ to q, and consider
for ж, у £ TpM an element P(ci)"1 о fi(P(ci)x. P(ci)y) о P(Cl) of о(ГрЛ/), which
will be seen to belong to h(p). In fact, at q we take the above rt corresponding to
P(ci)x, P(c\)y and get
dt
(P(d)-1 о rt о p(Cl)) = -Ρ(^)-1 о Д(Р(с1)х, P(ci)y) о P(Cl),
lt=0
which belongs to /i(p). Conversely, W. Ambrose and I. M. Singer got the following
fundamental result.
(6.7) h(p) coincides with the subalgebra of o(TpM) generated by {P(ci)_1 о
R(P(ci)x, P(ci)y) о P(ci); x, у £ TpM, C\ is a piecewise C°° curve emanating
from p}.
Exercise 5. Show that the holonomy group of (5m, g0) (m > 2) is isomorphic
to SO{m).
Now we note that the structure of the holonomy group of a Riemannian
manifold is closely related to the various geometric structures on the manifold. Here we
give the following simple examples.
Lemma 6.7. Let (M, g) be an m-dimensional Riemannian manifold.
(1) Μ is orientahle if and only if H(p) С SO(m).
(2) (M, g) is a Kahler manifold if and only ifm = dim Μ is even and H(p) С
J7(m/2).
Proof. (1) If Μ is orientable, there exists a parallel nonzero m-form ω on
Μ which determines the positive orientation on TpM at every point ρ £ Μ. For
any P(c) £ H(p) we have P(c) ωρ = ωρ\ namely, P(c) preserves the orientation and
belongs to SO(m). Conversely, suppose that H(p) с SO(m), and choose an m-form
ωρ at ρ which determines an orientation on TPM. Then for q £ Μ take a piecewise
C°° curve Ci joining ρ to q, and set ωη := Ρ(θ\)ωρ, which is in fact independent of
the choice of c\, since elements of SO(m) leave ωρ invariant. Therefore we get a
parallel nonzero differential m-form ω on Μ, which implies that Μ is orientable.

124 III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
(2) If (M, g) is a Kahler manifold, there exists a parallel tensor field J of type
(1,1) on Μ such that at every point ρ e M, Jp is an isometry satisfying J% = -id,
regarding Jp as a linear transformation of TPM. Now recall that an orthogonal
transformation of TPM belongs to U(m/2) if and only if it commutes with Jp.
Since J is parallel, for P(c) e H(p) we get Jp(P(c)x) = P(c)Jpx, χ G TPM. Hence
P(c) e U(m/2). Conversely, suppose that m is even and H(p) С U(m/2). Since m
is even, we may choose an orthogonal transformation Jp of TPM with j£ = —id. For
q e Μ take a piecewise C°° curve C\ joining ρ to q, and define a linear isometry Jq
ofTqMas the parallel translation of Jp along ci, namely, Jq = P(c\)o JpoP(ci)_1,
which is independent of the choice of c\. In fact, let c<i be another piecewise
smooth curve from ρ to q. Then we get P(c2 U cf *) о Jp = Jp о Р(с2 Ucj"1), since
P(c2 Ucf1) <E H(p) С U(m/2). Therefore it follows that
РЫ о JP о P{c2yl = P(ci) о Jp о P(d)-1,
and we get a parallel tensor field J of type (1,1) on M, which defines a Kahler
structure on Μ. Π
Exercise 6. Show that Μ is orientable if Μ is simply connected.
Symmetric spaces play an important role in the classification problem of the
holonomy groups (see Chapter IV, Remark 6.14).
Now we turn to the de Rham decomposition theorem. Since the holonomy
group H(p) of a Riemannian manifold Μ at ρ e Μ is a subgroup of the group
of orthogonal transformations of TPM, it follows that TPM is decomposed into
irreducuble invariant subspaces D9, which are mutually orthogonal. Namely, we
have a decomposition TPM = D0®Di®· · -фДь, where D0 := {x G TPM\ P(c)x =
χ for allP(c) e H(p)} denotes the subspace on which any element of H(p) acts as
the identity, and Д (г = 1, ... , к) are irreducible invariant subspaces of TPM
with respect to the action of H(p). In general, we call a Riemannian manifold Μ
irreducible if H(p) acts irreducibly on TPM.
Now suppose Λ/ is a complete simply connected Riemannian manifold. In
the following, we show that Λ/ may be expressed as a Riemannian direct product
of Euclidean space M0 and irreducible Riemannian manifolds M{ (г = 1, ... , к)
according to the above decomposition of TPM.
First as preparation we consider the following situation: Let Μ be a (connected)
Riemannian manifold and ρ e M. Suppose that TPM = D' 0 D" is orthgonally
decomposed into nontrivial invariant subspaces D', D" with respect to the action
of H(p). We define from D' a distribution V on Μ as follows. For q £ Μ take a
curve с G Cpq and define a subspace D'(q) as the parallel translation of D' along
с For another c' e Cpq we have P(c')~l о Р[с) е Я(р), which leaves Df invariant.
Therefore, the above definition does not depend on the choice of curves in Cpq, and
we get a distribution V on Μ. Taking a basis of D'(q) and parallel translating it
along minimal geodesies emanating from ρ on a normal coordinate neighborhood
around q, we get a C°° local basis of V around q, and V is a C°° distribution. In
exactly the same manner, we have another C°° distribution V" = {D"(q); q £ Μ}.
Clearly, we get TqM = D'(q) 0 D"(q) at any q <E M.
9D is said to be invariant if elements of H(p) leave D invariant. D is said to be irreducible
if any invariant subspace of D is either {0} or D itself.

6. ISOMETRY GROUP AND HOLONOMY GROUP
125
Lemma 6.8. (1) Let X G X(M). If Υ e X(M) takes values in V {resp.,
£>"), so does VXY.
(2) V, V" are completely integrable. Let Μ', Μ" he maximal integral sub-
manifolds through q G Μ ofW, V"', respectively. Then M', M" are totally geodesic
{immersed) submanifolds of Μ. Μ', M" are complete with respect to the induced
metric if Μ is complete.
(3) There exist open neighborhoods V, V, V" of q in Μ, Μ', M", respectively,
such that V is isometric to the Riemannian direct product V x V".
PROOF. (1) This clearly follows from Chapter II, §1, Exercise 5.
(2) If Χ, Υ G X{M) take value in £>', then [X, Y] = VXY - VYX also takes
value in V by (1), and V is involutive. Therefore, from the Probenius theorem
V is completely integrable, and there exists a maximal integrable submanifold
M' through q. For the second fundamental form, from (1) we have S(X, Y) =
(VxY)1- = 0, and M' is totally geodesic. Therefore, geodesies of M' are also
geodesies of Μ, and we may easily show that Μ' is complete if Μ is complete. The
above argument also works for V".
(3) Since V is completely integrable, we may choose a chart ([/', φ', (χ1, ... ,
χ171 , у171 +1, ... , ym)) around q such that <p'(q) = o, and slices of Uf given by
ya = const (a = ra' + 1, ... , ra) are integral submanifolds of V', where we set
ra' = dimM' (see Chapter I, Theorem 2.2). Similarly, for V" we may choose
a chart (U"', φ", (у1, ... , ym', xm'+1, ... , xm)) around q such that <//'(<?) = o,
and slices уг = const (i = 1, ... , ra') are integral submanifolds of V"'. Note that
dim M" = m-rn!. In the following, indices г, j (resp., a, b) are assumed to move in
the range 1 < г, j < ra' (resp., ra' + 1 < a, b < ra). Then д/дхг (resp., д/дха) are
vector fields taking value in V (resp., V"). Since V and V" are orthogonal at each
point of M, taking e > 0 sufficiently small we may assume that (V, (x1, ... , χ171))
with V = {r e U'DU"; \xl{r)\ < e (I = 1, ... , ra)} defines a local coordinate system
of Μ around q. Then V := {qf G V\ xa(q') = 0}, V" := {<?" G V; xl(q") =
0} are integral submanifolds of P', V" through q, respectively. V, V" are open
sets of M', M", respectively and we may assume that V is difFeomorphic to V x
V". Now we show that, with respect to the above chart (V, (x1, ... , xm)), the
components of the metric tensor g satisfy the following: gij(xl, ... , хш) depend
only on x1, ... , хш' and gab(xl, · · · , zm) depend only on xm'+1, ... , xm. In fact,
we get
9 / д д \ I д ^ д \
gxa w \ ш*дх1' dxJ / \ dxl' *^ dx·? /
τ^ dxa' dxi j \дхг' ~м дха
which equals 0, since Vd/Qxtd/dxa G P", 9/dxj e V, etc. Similarly, we may
check that -^gab = 0. On the other hand, we have clearly gia{xl, ... , хш) = 0.
Therefore (gij), {даь) define Riemannian metrics on V, V", respectively, and (V, g)
is isometric to the Riemannian product (V7, g') x (V", p"). D
Now for a; G ΤςΜ, we write χ = (x;, x") according to the orthogonal
decomposition TqM = D'(q) Θ D"(q). Then, from the above, for x, y, ζ G X^M we get
Я(х', y')^ G D'(^), Я(х", y")^" G D"(^). Further denoting by Я', Я" the
curvature tensors of V, V", respectively we have Я(х', у')ζ' = Я'(х', у')*', Я(х", у")2;"

126
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
= R"{x", y")z" and it follows that
(6.8) R(x, y)z = (#(*'■ y')z\ R"{x\ y")z"),
since V = V χ V" is a Riemannian product and for r = (rf, r") e V we have
D'r=Tr'V', D'l = Tr„V".
Lemma 6.9. Suppose Μ is simply connected. Let H'(p) (resp., H"(p)) denote
the normal subgroup of H(p) consisting of elements which fix all vectors of D"
(resp.j D'). Then H(p) may be written as the direct product of H'{p) and H"(p).
Proof. #'(p), Η"(ρ) are normal subgroups, because for any loop с based at ρ
the parallel translation P(c) leaves Df and D" invariant. Also we may easily check
that elements of H'(p) commute with elements of H"(p), and Н'(р)Г)Н"(р) = {е}.
Therefore, to show that H(p) = H'(p) χ H"(p) it suffices to see that for any loop
с at p, P(c) may be expressed as a product of elements of H'(c) and H"(c). First,
for q e M, taking a neighborhood V = V x V" of q in Lemma 6.8, we consider
loops с at ρ of the following form: Take a curve c\ joining ρ to q and a loop c<i at q
in V, which may be written as c2 = c2 x c2f, where Co ? Cn are loops at q of V, V",
respectively. Then с is defined as с = c\ UC2 UcJ"1, and loops of this form are called
lassos because of their shape.
Figure 15
Now for c' = a U c'2 U cj-1 and c" = cx U ci>' U cj-1, noting that V = V x V"'
is a Riemannian product, we see that P(c'2) (resp., P(cf2f) fixes every vector of
D"{q) (resp, £>'(<?)), and we get P{c') e H'{p), P{c") <E H"{p). Since P(c) =
P(c') о Р(с"), our assertion holds for lassos.
Now let с : [0, 1] —► Μ be an arbitrary loop based at p. Since Μ is simply
connected, we have a homotopy Η : [0, 1] x [0, 1] —» Μ from с to the point curve
{p} such that #(£, 0) = c(i), #(*, 1) = p, #(0, s) = #(1, s) = p. Take sufficiently
fine subdivisions 0 = t0 < t\ · · · < tk = 1, 0 = s0 < s\ < · · · < sk = 1, so that
H([ti-i, U] x [sj_i, Sj]) (1 < г, j < A;) are contained in coordinate neighborhoods
V{j which satisfy Lemma 6.8 (3).
Now we define loops c\j based at H(ti-\, Sj-i) in Vij by joining the following
four arcs : t <E [U-U U] i-> #(£, Sj_i), s G [sj-i, Sj] »-> Я(*», s), £ G [^-i, *»] »->
#(^ Η- U-ι - t, Sj), and s e [sj_i, Sj] »-> H(U-i, Sj_i H- Sj - s). Next we set
cij (0 := ^(^ sj-i)> * € [0> ^-i], which are curves joining ρ to H(U-i, Sj-i).
Then we get lassos Qj = c\V U c·^ U [c·^]-1, and P(cij) are written as elements of
H'(p) χ #"(p). On the other hand, P(c) may be expressed as the product of these
P(cij) (see Figure 16) and therefore written as an element of H'(p) χ Η"{ρ). D

6. ISOMETRY GROUP AND HOLONOMY GROUP 127
Figure 16
Next we choose for q £ Μ an open neighborhood U = U' x U" (Riemannian
direct product) of q, so that U' (resp., [/") is an integral submanifold of V (resp.,
V") through q. Fix a vector u"q £ D"(q)\ then for any r' £ U' we get u" £ D"(r) by
parallel translating u^ along a curve с in [/', which does not depend on the choice
of c, because U = Uf x [/" is a Riemannian direct product. Therefore, we get a
C°° parallel vector field u" on Uf which takes value in V". For s £ R we define a
map F™" : [/' —» Μ by ^"(r) = expr su", which is a C°° map, and the following
holds.
Lemma 6.10. (1) Let sq > 0. If we take Uf (Э q) sufficiently small, then
for any 0 < s < So, F™ is an isometry from U' onto an open set Fsu" [U') of an
integral manifold ofV through expq su^, and
(6.9) DFf(r)(w) = Ρ(ξΓ)>, w £ TrV.
where £r denotes the geodesic with the initial direction u'r' given byξr(s) := expr s u'r'.
Furthermore, from r \—> d/dsF" (r) we get a parallel vector field defined on Fsu (Uf)
that takes value in V" and is denoted by u"(s). Then
(6.10) Fs%i2 = *£'<"> о Fjf (0 < Sl. s2 < ai + s2 < s0).
(2) Let X(t) be a parallel vector field along a curve c(t) in U'. If we define
X(t, s) := P(£c(t))®X(t), then t \—> X(t, s) is a parallel vector field along a curve
t~Ff(c(t))inFf(U').
PROOF. First suppose that sq is sufficiently small and ξη \ [0, so] is contained
in U = U' x U". Then Ff(r) = (r, ^(s)), and Ff is an isometry from U' onto
Uf x {fq(s)}. Thus the assertions of the lemma clearly hold. In the general case we
cover ξς([0, so]) by a finite number of U{ = U[ x U" with the above properties, and
repeat the above process. If we take the first U' small, then F™ (0 < s < sq) is
a difFeomorphism from U' onto an open subset Fsn (Uf) of an integral submanifold
of V through £s(q), and we get the assertions of the lemma. D
Remark. The above argument may be applied also for an open subset U" of
an integral submanifold of V" and a parallel vector field u' on U" taking values in
V'. Setting G™ (r) = exprtu'r, we have the assertions similar to those of Lemma
6.10 for Gi.

128
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Now we return to the first situation, and suppose that Μ is complete and
simply connected. We consider maximal integral submanifolds M', M" of W, V"
through p, respectively. We will show that Μ is isometric to the Riemannian direct
product Μ := Μ' χ Μ".
For ρ := (ρ, ρ) G Μ' χ Μ", we denote by J the canonical identification between
TPM = D'{p)®D"{p) and ΤΡ(Μ' χ Μ") =* Ζ?'(ρ)Θΰ"(ρ). In the following, for any
once broken geodesic 7 of Μ through ρ and once broken geodesic 7 of Μ which is
defined in terms of J as in §5, we show that the assumption
(6.11) R(Itx, Ity)Itz = ItR{x, y)z, x, y, z, G Tl{t)M,
of the Ambrose Theorem is satisfied. First we consider a geodesic 7 : [0, /] —» Μ
emanating from ρ with the initial direction 7(0) = (u', u"), v!', и" ф 0. Let 7'
(resp., 7") be a geodesic of M' (resp., M") emanating from ρ with 7'(0) = v!
(resp., 7~//(0) = u"). Then η(ί) = (Υ(ί), 7~"(0) is a geodesic in M. Let ж =
(x;, x"), у = (yf, y"), ζ = (zf, z") G ΤΊ(^Μ denote the decomposition of tangent
vectors with respect to Ό'(η(ί)), Ό"(η(ί)). Then, noting (6.8), it suffices to check
(6.11) for x', у', ζ' G Ό'{Ί{ί)) and χ", у", ζ" G I>"(7(*))-
In Lemma 6.10, for u" := P^'^u" we defined an isometry FsUt from a
neighborhood [/' of η'(ί) in M' onto FsUt ([/'). We prove the following equations for
0 <£ </:
(6.12)
f F*"(Y(*)) = 7(0 = {y(0W, Oif^y(«))-?(«) = РП7(*),
|-| F5(7,W)=Pr-27(0 = iy(0W,
OS \s=t
{{DFfyxx' = μην*') = Ρ(Ϋ)? όρη ο/ο ρ(Ίγ0χ' (χ' e d;(()),
wherepr\, pr2 denote the orthogonal projections onto V', D" or ХЧ(t\M', Τ' -„tt\M",
respectively.
To see this we set Τ := sup{*i G [0, /]: (6.12) holds for 0 < t < ti}. Since for
ρ there exist neighborhoods /7, U'. U" of ρ in Μ, Μ', M", respectively such that
U = U' x /7" is a Riemannian product, we have Τ > 0. Suppose that Τ < I. Take a
neighborhood У of 7(T) in M, and open sets V, V" in integral manifolds D', D",
respectively, containing j(T) such that У = V x V" is a Riemannian product.
Choose / - Τ > e > 0 so that η([Τ - £, Τ + б]) С V. Then for Τ - e < t < Τ + e we
may write η{ΐ) = (^'(ί), η"(ί)), where 7', 7" are geodesies through η(Τ) in V7, V",
respectively. Denote by u'^ the parallel translation of i"(t) along 7;(£). Then from
the properties of Riemannian product manifolds we get for Τ < t < Τ + ε
(DF^T(i(t))rlJ = P{i)J ο Ρ{Ίγτ{χ% χ' G /Г(7(0).
Since (6.12) holds for t < T, we get ξ+(τ){Τ) = 7(T), fy(T)(T) = pr27(T) =
7"(T) by continuity. Then applying Lemma 6.10 and taking a sufficiently small
neighborhood U' of η'{Τ), we see that F?* (0 < s < Τ + б) is defined on /7'
and satisfies the assertions of Lemma 6.10. Note that we have (DF^T)~1xf =
Ρ{Ί')τ °pr\oIo Ρ(Ί)1 χ\ χ1 g D'(7(T)). Now for Τ < t < Τ + δ, where δ > 0 is

6. ISOMETRY GROUP AND HOLONOMY GROUP
129
a small number satisfying ΐ'([Τ, Τ + δ]) С U'', noting that F^T(y(t)) = y(i), etc.,
we get
*5(Ϋ(0) = *Й- ° *f(Y(i)) = KMl'(t)) = 7(<),
Л*?"(У(0)У(<) = (Л*Й- ° ^" И*) = ^"Jr(7'(<))7'(<) =pri7(*),
*5(тЧ*)) = i»vK<), DF(uJr(7'(i))7'(i) - ΡΠ7(ί)
_0_
as
ала
[0*f ]" V = [^'«ГМ^Й·]-1^ = [DF*]-1 о P(7')f ο P(7)<~(z')
= P(i>)Jo[DF?}-1oP(1yT(x')
= Ρ(γ)Γ ο P(y)° ορτ,ο/ο P(7)J о P(7)<-(*')
= Р(7')? ορη ο / о Ρ(7)^(α;') = рг^Лх')·
Therefore, we see that (6.12) holds also on [Τ, Τ + <5], which contradicts the
definition of T, and so Τ = /.
Then, recalling that Ftu is an isometry, we see that
priltRtf, y')z' = (DFfr'R'ix', y')z'
= BfdDFf")-1^, (ЯР^'Г УХОД*")-1*'
= R'(pnltx', vrxhy'){WxItz').
Similary, considering G instead of F, we have
pr2ItR(x", y")z" = R"(pr2Itx", pr2Ity")(pr2Itz").
Then (6.11) follows directly from the above. (6.11) is trivial in the case where v! or
u" is equal to 0. Also in the same manner it is not difficult to check (6.11) for once
broken geodesies, which will be left to the reader as an exercise. Therefore, from
the Ambrose Theorem we see that Φ : Μ —► Μ' χ Μ" is a Riemannian covering.
By definition, we get Φ | M' = id^', Φ | M" = idA/«. Now note that M' is simply
connected. In fact, for any loop с in Mf we have a homotopy {Hs} in Μ from с to
a point curve, since Μ is simply connected. Then pr\ о Ф(Н8) gives a homotopy in
M' from с to a point curve. Similary, M" is also simply connected, and therefore
Φ is an isometry by the Ambrose Theorem.
Now we state the de Rham decomposition theorem.
Theorem 6.11. Let Μ be a complete simply connected Riemannian manifold.
Then Μ is isometric to the Riemannian direct product M$ χ M\ χ · · · χ Μ*., where
Mo is Euclidean space with the canonical Riemannian metric and M{ (г = 1, ... , к)
are complete simply connected irreducible Riemannian manifolds. Moreover, this
decomposition is unique up to order, and the holonomy group of Μ is the direct
product of holonomy groups of Mi (г = 1, ... , к).
PROOF. Take a point ρ e M, and consider an orthogonal irreducible
decomposition of Tp Μ = Do 0 D\ 0 · · · Θ Ό^ into invariant subspaces with respect to
the holonomy group H(p) stated as before. Now we may define the C°°
distributions T>i from Di and let Mi be the maximal integral submanifods of T>i through

130
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
p, which are complete totally geodesic submanifolds of Μ. Then the above
argument with induction implies that Μ is isometric to the Riemannian direct
product of M{ (i = 0, ... , k), where Mi are simply connected, and Mi (i = 1, ... , k)
are irreducible. Further the subgroups Hi(p) of #(p) which consist of elements
fixing all vectors belonging to Dj (j φ Ϊ) are normal subgroups, and we have
#(p) = #o(p) x ··· x Hk(p) by virtue of Lemma 6.9. Note that #*(p) is
isomorphic to the holonomy group of Mi (see Exercise 4). In particular, H(p) fixes all
elements of Do = TPM$. We decompose D0 into mutually orthogonal 1-dimensional
subspaces. Then the corresponding maximal integral submanifolds are isometric to
R because they are simply connected, and we see that Mo is isometric to Rd.
Finally, the uniqueness of the above decomposition of Μ up to order follows from the
following lemma.
Lemma 6.12 Under the assumption of the theorem, the decomposition TPM
= Do 0 · · · 0 Dk is uniquely determined up to order.
Proof of the lemma. First we note that Do is determined uniquely as the
maximal subspace on which P(c) \ Do = id/)0 for any P(c) £ H(p). Therefore, it
is sufficient to show that any irreducible invariant subspace Ε {φ {0}) which is
orthogonal to Do coincides with some Di (i = 1, ... , k). For χ £ Ε, χ φ 0, we
write
χ = χι Η h Xi Η \-Xk {xi £ Di, г = 1, ... , к),
and consider the action by a* £ Hi(p). We get
aiX = £! + ··· + aiXi + · · · + Xk £ E.
Therefore it follows that Jbi {JLiJb'i — JL U>i J^
£ Ε Π Di. We consider subspaces
Si := (xi - diXi\ a{ £ Hi(p))R (I <i<k) of ЕП D{. If 5< = {0} for all 1 < г < fc,
then χ is fixed by Hi(p) (0 < г < к); that is, it is fixed by all elements of H(p) and
belongs to Do, which is a contradiction. Thus there exists an г such that Si Φ {0},
which implies 57 = Ε = Di because Ε and Д are irreducible. This completes the
proof of the lemma and also the theorem. D
Corollary 6.13. Let Μ be a complete simply connected Riemannian manifold,
and Μ = Mo x Λ/ι x · · · x M^ the de Rham decomposition. Then for the isometry
group we get Io{M) = /o(^o) x · · · x Io(Mk)- Furthermore, φ £ I(M) maps V0 to
T>o and acts on {Dj}f=1 as a permutation group.
Problems for Chapter III
1. Suppose that any Riemannian metric on a C°° manifold Μ is complete. Then
show that Μ is in fact compact.
2. Let Ω be a domain in a Riemannian manifold Μ with smooth boundary and ν
the outward unit normal vector field to the boundary Ν := 0Ω. For a C°° function
и on N with compact support, we set at(p) := exp-11 u(p) vv and Nt := {ctt(ρ); ρ £
Ν}. Then {Nt} is a variation of N, and we consider the domains Ω* bounded by
Nt with Ω0 = Ω. Prove that
d \ f d \ f
— νοΙΩί = / udA, — volm-iNt = (m - 1) / η(ρ)η(ρ)άΑ,
ατ \t=o Jn at \t=o Jn
where η(ρ) stands for the mean curvature of TV at ρ £ TV and dA denotes the
induced Riemannian measure on TV.

PROBLEMS FOR CHAPTER III
131
Next suppose that N minimizes the (m —l)-dimensional volume among (ra-1)-
dimensional submanifolds of Μ that are smooth boundaries of domains with the
same volume as that of Ω. Then show that the mean curvature of N is constant.
(Here we need not assume that Ω and N are connected.)
3. Let / : N —» Μ be an immersion from a compact manifold N with boundary
dN into a Riemannian manifold M. A C°° map F : (-£, ε) χ Ν —► Μ is called
a variation of / if /t := F(£, ·) are immersions from N to Μ with /0 = / and
ft \ dN = f \ dN for t e (-€, б). Let X(p) := DF(0, p)0/di be the variation
vector field along N. Denoting by V(t) := JN dvj*g the volume of the immersed
submanifold determined by /t, prove that
V'(0) = n [ (H,X)dvf.g,
JN
where η = dim N and Я denotes the mean curvature vector.
Suppose V'(0) — 0 holds for any variation F of /. Then show that the mean
curvature vector of / vanishes everywhere and / is a minimal immersion.
4. Let π : Μ —► Μ be a Riemannian submersion.
(1) Let ρ G M, and let 7 be a geodesic of A/ emanating from π(ρ). Show that
there exists a unique (locally defined) geodesic in Μ emanating from ρ which is a
horizontal lift of 7.
(2) Let 7 be a geodesic in M, and suppose that 7(0) is a horizontal vector.
Then show that 7(2) is a horizontal vector for any t, and 7 := π ο 7 is a geodesic
in M.
(3) Show that Μ is complete if Μ is complete. How about the converse?
5. Let Μ = (CPn, ho) be complex projective space with the canonical Riemannian
structure, with 1 < Κσ < 4. We consider geodesies of Μ emanating from ρ :=
(1 : 0 : ... : 0). For и G UPM we denote by ηη the geodesic emanating from ρ with
the initial direction u.
(1) Let Yw be a Jacobi field along ju satisfying Yw(0) = 0, VYw(0) = w, and
w(t) the parallel translation of w along ηη. Show that if w = J и then Yw(t) =
^sin2£ · w(t), and if w ±u, J и then Jw(t) = sint · ги(£).
(2) Let ν G t/pM. If ν is linearly independent of (u, Ju)r, then show that
the minimum positive value of t such that *yu(t) = 7υ(£) is given by ί = π. If ν G
(u, Ju)r, then £ = π/2 is the minimum positive value of t such that ^v{t) = 7u(£)·
(3) Show that the cut locus Cp of ρ coincides with the first conjugate locus of
ρ and is given by Cp = {7η(π/2); и G UPM}. Show that Cp is a totally geodesic
submanifold of Μ isometric to (CPn~l, ho). Compute the volume vol(CPn, ho).
6. Let Μ, Ν be complete Riemannian manifolds and ρ G M, q G N. Show that
the cut locus of (p, q) in the product Riemannian manifold Μ χ Ν is given by
(cpxiv)u(Mxg.
7. For a point ρ in the torus in R3 with the induced metric from R3 (see Figure
17), guess the cut locus Cp of p.
8. Let Μ be a complete Riemannian manifold and ρ G M. For 0 < r < ip(M)
we consider the distance hypersphere Sr(p) '·= {q G M; d(p, q) = r}, which is
difFeomorphic to the sphere 5m_1. For и G UPM we denote by 7n the geodesic with
the initial direction u. Then 7n(r) is the unit normal vector to Sr(p) at 7u(r), and

132
III. GLOBAL CONCEPTS IN RIEMANNIAN GEOMETRY
Ρ
Figure 17
we denote by S^u^ the second fundamental form of Sr(p) with respect to 7u(r).
Show that £vtt(r)tt(y(r), Z(r)) = -(Y(r), VZ(r)> (= -(VF(r), Z(r))), where У, Ζ
are Jacobi fields along 7n with У(0) = Z(0) = 0.
9. An isometry / of a Riemannian manifold Μ is called a Clifford translation if
М3рь4 d(f(p), p) G R is constant. Show that for (Ят, р0), Clifford translations
are just the usual translations. Next let π : Μ —> Μ be a Riemannian covering
and Γ the deck transformation group of π. Show that if Μ is homogeneous, then
Γ consists of Clifford translations.
10. Show that A G 0(m + 1) is a Clifford translation of (5m, <jo) if and only if
there exists a complex number A with |A| = 1 such that all eigenvalues of A are
equal to either A or A.
11. Let (M, g) be a Riemannian manifold and (TM, G) the tangent bundle of
Μ with the Sasaki metric. Suppose that the geodesic flow фь on Τ Μ consists of
isometries of G. Then show that (M, g) is of constant curvature 1.
12. Show that the set of points q G Cp that satisfy the condition of Proposition
4.8 (2) is dense in Cp.
13. Give a proof of Corollary 6.13.
14. Let Μ be a complete Riemannian manifold and N a closed submanifold.
Define the cut locus C\ of N, and check whether properties similar to those in the
case of С ρ hold.
Notes on the References
In this chapter we again owe a lot to the articles given in the Notes on the
References in Chapter II.
§1. The notion of completeness for Riemannian metrics was first introduced
by Hopf and Rinow ([Ho-Ri]), and Theorem 1.1 gives a foundation for the study of
global properties of Riemannian manifolds.
§2. Here we considered the variational formulas under a general boundary
condition. It is possible to introduce an infinite-dimensional manifold structure
modeled on a Hubert (or Banach) space on the space of curves, and develop the
Morse theory (see, e.g., [K-4, 5], [Pa-3]). For the calculus of variations on more
general manifolds of maps, we refer to, e.g., [Ur-2].
§3. Here we follow the argument of [M-l] for the approximation of path spaces
by finite-dimensional manifolds. For the proof of the Morse index theorem we follow
an idea due to W. Klingenberg ([K-3]), which is related to symplectic geometry.

NOTES ON THE REFERENCES
133
As for the Morse index theorem under more general boundary condition, see, e.g.,
[Am-2], [K-3], [Kl]. For the relation between the Morse index and the Maslov index
in the general calculus of variations, we refer to [Dui-2], [Bo-2], [Ed], [Ar-1].
§4. The notion of the cut locus was first introduced by Poincare ([P]) and
J. H. C. Whitehead ([Wh]), and S. B. Myers studied the detailed structure of cut
loci in compact analytic surfaces ([My-2]). Then Klingenberg considered again the
cut loci and the injectivity radius in connection with the sphere theorem ([K-l, 2]).
For the fundamental properties of cut loci, we followed [K-l], [Ko-4], [War-2]. In
particular, Propositions 4.5 and 4.6 are due to [War-2], and Proposition 4.8 is due
to [Bi], [Wol]. Proposition 4.13 and Corollary 4.14 are due to Klingenberg. For the
detailed structure of cut loci in general Riemannian manifolds, see [Buc-1, 2], [It],
[Gl-Si], [Suga], [We-2].
§5. In this section we followed Ambrose's fundamental paper [Am-1], which
generalized a similar result for symmetric spaces, and has some applications in §6
and Chapter IV (see also [Hi] for the case of the linear connection). If we allow
general broken geodesies with many break points, the proof becomes considerably
simpler (see, e.g., [Ch-Eb]).
§6. For more details on the isometry groups and more general automorphism
groups of geometric structures, we refer to [No-Ko I, II], [Lie], [Ko-3]. Myers and
Steenrod ([My-Ste]) first proved that the group of all isometries of a Riemannian
manifold carries the structure of a Lie group (see [Pa-2], [Ко-2] for generalizations).
Proposition 1.6 is due to S. Bochner ([Boc]).
For the holonomy group, we refer to [No-Ko I], [Bes-2], [Sal]. (6.6) is due to
A. Borel, A. Lichnerowicz, and H. Yamabe. (6.7) is due to W. Ambrose and I.
M. Singer ([Am-Si]). Lemma 1.7 treats the very special case of G-structures (see,
e.g., [St], [Ko-No I] for more details). The de Rham decomposition theorem was
first proved in [dR-1]. See also [Ko-No I], [Wu-1], [Meu] for proofs. As for the
detailed structure of the (restricted) holonomy group of a Riemannian manifold
M, M. Berger ([B-l]) showed that H°(p) acts transitively on UPM, if H°(p) is
irreducible and Μ is not locally symmetric (see also Chapter IV, Remark 6.14),
using E. Cartan's classification of orthogonal representations of Lie groups. Later
a direct proof was given by J. Simons ([Sim]).

CHAPTER IV
Comparison Theorems and Applications
Riemannian manifolds of constant sectional curvature give most standard
models of Riemannian manifolds. In this chapter, we are concerned with the methods
of investigating properties of Riemannian manifolds by comparing with those of
model spaces. If the sectional curvature or the Ricci curvature of a Riemannian
manifold Μ is bounded below (or above) by some constant, then we may get much
information on Jacobi fields along geodesies in Μ by comparing with those in the
corresponding model Riemannian manifold of constant curvature. In §2, we state
a comparison theorem on Jacobi fields in a unified manner, and give some
applications. In §§3 and 4 we continue to state applications of comparison theorems to
various situations, e.g., geodesic triangles, geodesic hinges, lengths of curves,
volumes, and the Hessian of the distance function, etc., which will play an important
role in Chapters V and VI. In particular, the Toponogov comparison theorem in
§4 for geodesic triangles in a Riemannian manifold whose sectional curvature is
bounded below is a global version of the Rauch comparison theorem in §2, and is a
typical theorem in global Riemannian geometry. In §5 we explain convexity, which
is also one of the fundamental concepts and plays an important role in Chapter V.
To begin with, in §1 we briefly mention Riemannian manifolds of constant
curvature. E. Cartan extended manifolds of constant curvature to a wider class of
Riemannian manifolds, those whose curvature tensors are parallel. These symmetric
spaces carry a rich geometry, and may be studied by various methods including the
theory of Lie groups and Lie algebras. In §6 we state some fundamental properties
of symmetric spaces from the viewpoint of Riemannian geometry.
1. Spaces of Constant Curvature
1.1. Recall that (Я™, <?0), (Sm(l/p), <?o), (i*m, 9o = Λ) (φ = (ρ*™)"1),
given in Chapter II, §3.3 (I), (III), (V), are complete simply connected Riemannian
manifolds of constant curvature δ = 0, ρ2, —ρ2, respectively. On the other hand,
by Corollary 5.5 in Chapter III any complete simply connected m-dimensional
Riemannian manifold of constant curvature δ is isometric to one of the above model
spaces. They are most standard Riemannian manifolds, and correspond to
Euclidean or non-Euclidean geometries. In this section we give some fundamental
properties of Riemannian manifolds (Μ, g) of constant curvature. First note that
in such Μ, we may move figures without expansion or contraction, as was pointed
out by Riemann. More precisely, a Riemannian manifold (M, g) is of constant
curvature if and only if the following property, which is called the axiom of free
mobility, holds: For any two points p, p' of Μ and any o.n.b.'s {ξ*}·™^, {£t'}£Li
at p, p', respectively, there exists an isometry from a neighborhood of ρ onto a
neighborhood of p'\ which maps ρ to p' and {&} to {^}. In fact, suppose Μ is
of constant curvature. We take a linear isometry J : TPM —> TP>M' defined by
135

136
IV. COMPARISON THEOREMS AND APPLICATIONS
Ι ξι := ξ'{ (г = 1, ... , га), and apply the Cartan theorem (Chapter II, Theorem 3.2)
to (Μ, ρ), (M', p') and I. Note that the assumption (3.6) of the theorem is satisfied,
since the curvature tensor of a Riemannian manifold of constant curvature is given
by (3.13) of Chapter II. Conversely, suppose the axiom of free mobility holds, and
let σ С ΓρΜ, σ' С ΤΡ>Μ' be arbitrary plane sections of TM. Let {ξι, &}, {£J, &}
be o.n.b.'s of σ, &\ respectively, and extend them to o.n.b.'s {&}£ι, {^}ϋι °f
TPM, TP'M'', respectively. Then we have an isometry φ from a neighborhood of ρ
onto a neighborhood p' such that Όφ(σ) = σ'. Therefore we get Κσ = Κσ>, and
Μ is of constant curvature.
Further let M™ be a complete simply connected m-dimensional Riemannian
manifold of constant curvature δ. Let p, p' e Μ and o.n.b.'s {&} of TPM and {ξ'{}
of TP'M' be arbitrarily given. Then we apply the Ambrose theorem (Chapter III,
Theorem 5.1) as before, and there exists an isometry of Μ which maps ρ to p' and
{&} to {£г·}. Therefore, the dimension of the isometry group I(M™) is equal to
m+(m— l)/2 = m(m+l)/2, which is the maximal dimension of the isometry groups
of m-dimensional Riemannian manifolds. In particular, I(M™) acts transitively on
M™, i.e., M™ is a homogeneous space.
Next we review the behavior of a Jacobi field Υ along a normal geodesic 7 :
[0, +00) —► A/. Let {e^t)}^ be a parallel field of o.n.b. along 7. Suppose Y(t)
is perpendicular to 7, and write У(£) = Σίί(ί)βί(ί). Then from the equation for
Jacobi fields (Chapter II, (2.19)) and (3.13) of Chapter II, we get f"{t) + 6fi(t) = 0.
Now let ss(t) be the solution of the differential equation
(1.1) f"(t) + 6f(t) = 0, /(0) = 0, /'(0) - 1.
Setting cs(t) = s'6(t), we obviously get c'6(t) = —6ss(t) and
<£'(*) + 6c6{t) = 0, c6(0) = 1, ci(0) = 0.
Then denoting by Ε\(ί), Ε2{ί) the parallel translations along 7 of ^(0), VF(0),
respectively, we see that the Jacobi field Y(t) is given by
(1.2) Y(t) = c«(i)Ei(i) + 86(t)E2(t).
Finally we note that a Jacobi field tangent to 7 is written in the form (at + b)7(£),
where α = (W(0), -y(0)>, b = (У(0), 7(0)>.
Exercise 1. Show the following:
I (sin \/6t)/Vs, δ > о,
ss(t) = lt, 6 = 0, cs{t) = {
[(smh^/W\t)/y/\f\, 6<0,
COS
1, * = 0,
[cosh^/ϊφ, (5<0.
Exercise 2. Let Μ be a Riemannian manifold of constant curvature δ. Then
show that the principal curvatures of the distance sphere Sr(p) (0 < r < ip(M))
are equal to cs(r)/ss(r) =: со^(г).
Next we explain that a Riemannian manifold Μ of constant curvature δ satisfies
the following axiom of plane: Let W be any A;-dimensional subspace of TPM, ρ G M.
Then S := expp W Π Be(op) is a A;-dimensional totally geodesic submanifold of M,
where 0 < e < ip(M). To see this we may assume that к > 2. Let и е W
be a unit tangent vector and ν e W. Then a Jacobi field Y(t) along 7^ with
Y(0) = 0, W(0) = ν is tangent to 5 for |*| < €. By (1.2), the parallel translation

1. SPACES OF CONSTANT CURVATURE
137
of υ along 7n is also tangent to 5. Therefore we get Tlu^S = P{^u)°tW, and the
parallel translation £(t) of a normal vector ξ to S at ρ along ηη is also normal to S
at 7u(£)· We show that S is totally geodesic at 7n(£). Let a(r, s) := exppr(u + su)
be a variation of 7n, whose variation curves as are geodesies emanating from ρ
and contained in 5. For ξ £ W1- we denote by £(r, s) the parallel translation of ξ
along as, and note that £(r, s) is a normal vector field to S along a. It suffices to
show that Vd/ds£(r, 0) = 0. Now obviously Vd/ds£(0, 0) = 0, and Vd/drVd/dsi =
#(§7, §j)£ is equal to zero by virtue of Chapter II, (3.13). Therefore we get our
assertion.
In particular, in the case of a complete, simply connected M™ of constant
curvature <5, if δ < 0 we have ip(M) = +oo and the above argument holds for
r = +oo. Namely, S = exppW is a complete simply conneted totally geodesic k-
dimensional submanifold of M™ and isometric to М£. S is called a A;-dimensional
subspace of M™. In the case where δ > 0 we may take M^ = S™, which is a
hypersphere centered at the origin and of radius l/y/δ in ilm+1. Note that in
this case we have zp(5m(<5)) = n/y/δ. Now we identify W as a A;-dimensional
subspace of ilm+1 which is orthogonal to ρ £ S™. Then S = exppW is nothing
but a great sphere of 5^, which is obtained as the intersection of the (k + 1)-
dimensional subspace (W, p)r1 with 5m(<5). 5 is totally geodesic and a complete
simply connected Riemannian manifold of constant curvature δ if к > 2.
Exercise 3. Show that in the case of the real projective space RPm the above
5 is a A;-dimensional projective subspace.
Exercise 4. Suppose m = dim Μ > 3. Show that Μ is of constant curvature
if and only if the following axiom of plane holds: For any ρ £ Μ and any (2-
dimensional) plane Η of TPM, exppU is a totally geodesic submanifold of M.
where U is a sufficiently small open neighborhood of op in Я.
1.2. Again let M™ be an m-dimensional complete simply connected
Riemannian manifold of constant curvature δ. Let pi, p2* Рз be three points on Л/™ which
do not lie on a geodesic segment. Let 7^+2 be a minimal geodesic joining p, to рг+ь
where г ξ 0 (mod. 3). Then from the axiom of plane we see that the geodesic
triangle Δ(ριΡ2Ρ3) is contained in a two dimensional complete simply connected
totally geodesic submanifold. If δ < 0, or δ > 0 and the perimeter / of Л(р\р2Рз)
is less than 2п/уД, then this geodesic triangle is uniquely determined.
Figure 18
(W> p)r means the subspace which is spanned by W and p.

138
IV. COMPARISON THEOREMS AND APPLICATIONS
If I = 2π/\β(δ > 0) then Δ(ριρ2ρ3) is either a great circle or a biangle
consisting of two half great circles joining two of {pi}- Now for these geodesic
triangles we have trigonometry in Euclidean or non-Euclidean geometry. Let U+2 =
d(pi, Pi+i) denote the length of the side 7^+2, and let a* = /i(pi-ipiPi+i) denote
the angle between 7;-i(0) and —7^+1(^+1), called the (inner) angle of Δ(ρχρ2ρ3)
at the vertex pi. We recall some fundamental formulas:
(1.3) (Law of Cosines). For (5 = 0,
ll = ll+i + i?+2 - 2ί<+ιί<+2 cos α*.
For <5 > 0,
cos y/bli = cos \ff)li+\ · cos y/6li+2 + sin y/δΐΐ+ι · sin \ίδΙι+2 cos a;.
For δ < 0,
cosh >/H't = cosh у/Щи+х · cosh д/Щ/г+2 - sinh ^/ψ\Ιί+λ · sinh y/\6\li+2 cos a;.
(1.4) (Law of Sines). For (5 = 0,
ii/sinai = /2/sina2 = /3/sina3.
For (5 > 0,
sin sinai = sin Sin Q2 = Sin sin аз.
For (5 < 0,
sinhy^/i/ sinai = sinh y/\6\l2/ sma2 = sinh y/\S\l3/ sina3.
(1.5) (Sum of the inner angles of a triangle). Let S be the area of Δ(ριρ2ρζ).
Then
αχ + a2 + a3 - π = δ S.
In fact, (1.5) is a special case of the Gauss-Bonnet formula.
Exercise 5. Let Δ(ριρ2ρ3), Δ(ρ[ρ,2ρ,3) be geodesic triangles in M™. Suppose
that l2 = 1'2, h = /3, and further l2, l3 < n/y/δ when δ > 0. Then show that
di > αϊ if and only if /1 > l\.
1.3. In this subsection we state some fundamental facts on complete Riemann-
ian manifolds of constant curvature, which are also called space forms.
First, we consider a complete flat (i.e., constant curvature 0) m-dimensional
Riemannian manifold M. Then the universal Riemannian covering of Μ is isometric
to R™ with the canonical Riemannian structure. We denote by π : R™ —> Μ the
covering pojection (see also Chapter V, §1 for the covering space). Let α be an
element of the fundamental group πχ(Μ, ρ). Then α may be also considered as a
deck transformation of π, which is an element ra(a) of the isometry group M(m)
of (-Rm, go) with π о га(а) = π, and the deck transformation group is a discrete
subgroup of M(m) acting freely on R171. Now M(m) is the semidirect product of the
orthogonal group 0(m) and the group of parallel translations. We denote by Γ the
subgroup of πι (Μ, ρ) consisting of elements ra(a) which are parallel translations.
Bieberbach showed that if Μ is a compact flat manifold, then this Γ is a free abelian
normal subgroup of rank m and is of finite index in πχ(Μ, ρ). Therefore, Γ may
be identified with a lattice in TVM = ilm, and Μ is finitely covered by a flat torus

1. SPACES OF CONSTANT CURVATURE
139
Rrn/T. Also two compact flat m-dimensional Riemannian manifolds are affinely
diffeomorphic if and only if their fundamental groups are isomorphic, and for given
m there exist only finitely many equivalence classes of compact flat m-dimensional
Riemannian manifolds with respect to affine diffeomorphisms. For m = 2, 3 the
complete classification of compact flat Riemannian manifolds is known (see J. A.
Wolf [Wo-1], Chapter 3, and L. S. Charlap [Char] for these facts).
Next we consider the holonomy group H(p) and the restricted holonomy group
#°(p) of a flat complete Riemannian manifold Μ at ρ £ Μ. Prom the Ambrose-
Singer theorem (6.7) of Chapter III, we have H°(p) = {e} and dim#(p) = 0.
Namely, the parallel translation along a loop с based at ρ depends only on the
homotopy class of c. On the other hand, we have the holonomy endomorphism
Φ : πι (Μ, ρ) —» Η(ρ)/Η°(ρ) = Η (ρ), which assigns P(c_1) £ Η (ρ) to a loop с
based at ρ (Chapter III, (6.5)). In the flat case, regarding a = [с] е πι(Μ, ρ)
as a deck transformation m(a) £ M(m) of the universal Riemannian covering
π : R171 —» M, we can interpret ra(a) as follows. Set ra(a) = (r(a), £(a)), where
r(a) £ 0(m) and t(a) £ R™ denote the rotation and translation part of ra(a),
respectively. Take the line segment / which joins ρ £ π-1 (ρ) to pi := m(a)p.
Then a geodesic loop с of Μ based at ρ obtained by projecting / via π represents
a, and we get / = t(a). Now note that a vector of TpM which is obtained by
parallel translating и £ TVM along с is equal to the image Dn(p')up', where йр>
denotes the parallel translation of up := Dn(p)~1u £ TpR™ to pi in R171. On the
other hand, from π ο m(a) = π we get Dn(p) = Dn(p') о Dm(a) = Dn(p') о r(a).
It follows that we may identify Φ(α) with r(a). Since Φ is surjective we have
H(p) = {r(a); α £ πι (Μ, ρ)}. In particular, when Μ is compact we see that the
kernel of Φ is given by the above Γ, and H(p) = πι(Μ, ρ)/Γ is a finite group by
the Bieberbach theorem.
Now we consider the positive constant curvature case. In the following, we
may assume without loss of generality that Л/ is a complete Riemannian manifold
of constant curvature 1. Then the universal Riemannian cover of Μ is isometric
to 5m, which is the unit hypersphere in um+1, and Μ is compact by the Myers
theorem. The deck transformation group Γ of π : 5m —► M, which is isomorphic
to the fundamental group πχ(Μ, ρ), is a finite subgroup of the isometry group
0(m+ 1) of 5m, and any element of Γ different from the identity acts without fixed
points. Conversely, for any subgroup Γ of 0(m + 1) satisfying these conditions,
π : 5m —► 5т/Г gives a Riemannian universal cover of a compact Riemannian
manifold 5т/Г of constant curvature 1. Such T's have been completely classified by
J. A. Wolf ([Wo-1], Chapter 7). The real projective spaces correspond to Г = {±id},
and here we give examples where Г are cyclic groups. We set m = 2n — 1 and define
the following matrix Τ in SO(2n)\
cos 2πρι /q — sin 2πρι /q
sin 2πρι /q cos 2πρι /q
0
cos 2npn/q — sin 2npn/q
0 8ΐη2πρη/<7 cos2npn/q
where p\ = 1 and the natural numbers Pi(2 < г < n) are relatively prime to a
natural number q > 2. Then the cyclic group T(q; p2, ... , pn) generated by Τ is
of order q, and Tk (1 < к < q — 1) have no fixed points on 5m. Therefore, we

140
IV. COMPARISON THEOREMS AND APPLICATIONS
get compact Riemannian manifolds L(q; p2, · · · , Pn) ·= S2n~l/T(q\ p2, ... , pn)
of constant curvature 1, which are called lens spaces. Since they give examples of
spaces which are homotopy equivalent but not homeomorphic to each other, they
are also important in topology. The relation between the Riemannian invariants
and the topology of manifolds is one of the main themes in recent Riemannian
geometry; it will be treated in Chapter V. Here we give a relation between the
diameter and the fundamental group as an introduction to the subject.
Let Μ be a compact Riemannian manifold of constant curvature 1. If Μ is
simply connected, then Μ is isometric to the standard sphere and its diameter
d(M) is equal to π. It is not difficult to see, using spherical geometry, that Μ is
simply connected if d(M) > π/2 (see also more general argument in Chapter V,
§2). Therefore, if Μ is not simply connected, we get d(M) < π/2, and the question
is when Μ takes the maximal diameter π/2. Recall that Μ may be expressed
as 5т/Г, where Γ is isomorphic to the fundamental group G of Μ and may be
considered as a subgroup of 0(m+1). Namely, Γ gives an orthogonal representation
of G. Then there exist irreducible orthogonal representations {σι, ... , σΓ} (1 <
r < 2n) of G over Ят+1 such that Г is expressed as Γ = (σι Θ · · · Θ ar)(G). The
fundamental group G is said to admit a fully reducible orthogonal representation if
r > 1, namely, if Γ has a nontrivial invariant subspace in Ят+1. Real projective
spaces and lens spaces provide such examples. Then we have
Proposition 1.1. Let Μ be a complete Riemannian manifold of constant
curvature 1. Then the fundamental group G of Μ admits a fully reducible orthogonal
representation if and only if d(M) = π/2.
PROOF. Suppose d(M) = π/2 and take points p, q with d(p, q) = π/2. We set
Ap := {r <E M; d(p, r) = π/2} and Ap := k~1(Ap) С 5m. Note that Лр, Ap φ φ.
First we show that any geodesic 7 joining x, у e Ap, whose length / is less than
π, is contained in Ap. Otherwise, there exists an interior point ^(t) of 7 such that
d(p, 7(£)) = d(p, 7) < π/2, where 7 is parametrized by arc-length. Since at least
one oft, l — t is less than π/2, we may assume that £(7|[0, t]) = t < π/2. A minimal
geodesic τ joining ^(t) to ρ is perpendicular to 7 at *y(t) by the first variation
formula, and we have L(r) < π/2. Now we lift r, (7|[0, £])-1 to geodesies in 5m
emanating from */(t) £ π-1 (7(f)) С 5т with respect to the covering projection
π : 5m —► Λ/. Then by the Law of Cosines, we see that the length of the minimal
geodesic σ of 5m joining the end points of the above lifted geodesies is less than π/2.
Therefore, projecting σ via π, we have d(p, χ) < π/2, which is a contradiction. In
particular, by the assumption on the diameter, Ap is a connected compact convex
subset2. Then Ap is a A;-dimensional topological manifold with boundary whose
interior is a totally geodesic smooth submanifold of Μ (for details see the structure
theorem for convex sets, Theorem 5.5 of the present chapter). Next we show that
dAp = φ in our case. Suppose dAp φ φ to the contrary, and take an η e Ap
with d(n, dAp) = max{d(z, dAp)\ ζ e Ap}. Let δ be a shortest geodesic loop of
Μ based at η which is not homotopic to a point curve. Then δ is obtained by
projecting a minimal geodesic in 5m joining ni,n2 £ π_1(η) via π. If L(6) = π,
then ήι, n2 are antipodal points, and we have G = π\(Μ) = Ζ2 from the definition
of δ. Therefore, Μ is isometric to the real projective space and G admits a fully
reducible orthogonal representation. We consider the case where L(6) < π. Then
2If x, у G Ap then minimal geodesies joining x, у are contained in Ap (see §5 for details).

1. SPACES OF CONSTANT CURVATURE
141
δ С Ap from the above, and δ is contained in the interior of Ap because of convexity
(see Lemma 5.4 for the precise argument).
Now there exists a point r (φ η) of δ which satisfies d(r, dAp) = d(6, dAp).
Take a normal minimal geodesic ψ : [0, b] —» Ap with ψ(0) = г, гр(Ь) G дАр, which
realizes the distance d(r, dAp). Next take a parallel vector field У along ψ so that
V(0) is tangent to δ at r, and a variation α : [0, b] x (—£, e) —» Μ whose variation
vector field is V. Since ^ is a minimal geodesic joining an interior point of the
convex set Ap to its boundary, it follows that a(6, s) G Л£3 (—е < s < e) (see
Lemma 5.7 for the precise argument). On the other hand, applying the second
variation formula, we get
D2E(6)(V, V) = - [ K(V(t), 6(t))\\V(t)\\2dt < 0,
Jo
which contradicts the fact that ψ is a minimal geodesic joining δ to dAp. Therefore,
we have dAp = φ and Ap is a compact totally geodesic submanifold of M. Set
к = dim Ap. Now for Ap we may show as above that any geodesic of length less
than π joining two points of Ap is contained in Ap. If к = 0, then Ap consists
of antipodes of 5m and is invariant under the action of G. If к > 1. then Ap
is connected. Otherwise, connected components of Ap are /г-dimensional great
spheres of 5m, and there exists two points in Ap such that they belong to different
connected components and their distance is less than π. This contradicts the above
property of Ap, and so Ap is a great sphere Sk which is invariant under the action
of G. Therefore, a (k + l)-dimensional subspace (0 < к < m - 1) of ilm+1 spanned
by Ap is an invariant subspace of Г, and G admits a fully reducible orthogonal
representation.
Next suppose G has a fully reducible representation. Namely, we have
irreducible orthogonal representations {σι, ... , ar} (r > 1) of G over Rm+l such that
Γ = (σι 0· · ·θσΓ)((7). We show that d(M) = π/2. Note that an orthogonal matrix
A belonging to Γ is written in the form
e (σι θ···θσΓ)(β), αϊ <Ε σι(β), degai = d.
Now it suffices to show that there exists two points p, q e Μ with d(p, q) = π/2.
We consider RTn+1 as a direct sum of representation spaces of σ; (г = 1, ... , r),
and let π : 5m —> Μ be the universal covering of Μ. We take a geodesic 7U in Μ
emanating from ρ = π(1, 0, ... , 0) with the initial direction
и := £>π(0, ... , 0, yd+u ... ,ym+i) € t^pA/,
and show that 7n is minimizing up to the parameter value π/2. In fact, ηη is given
by
11-> 7r(cosi, 0, ... , 0, i/d+isin*, ... ,2/m+isin£).
If 7U(£) is a cut point of ρ along 7n for t G [0, π/2], then there exists a unit
vector ν = Ζ)π(0, 22, ... , zm+i) G /7pM different from и such that 7U(£) = 7v(t)·
Namely, there exists an A G (σι 0 · · · θ ar)(G) different from the identity matrix
A =
αϊ 0
0 α
3A superscript "c" means the complement.

142
IV. COMPARISON THEOREMS AND APPLICATIONS
which satisfies
cost
Z2 sint
Zm+i Sill*
Qi 0'
0 ά
cost
0
0
j/d+i smt
Ут+ι sint
A =
<*1
0
(счеагЮ).
Now denoting by an the (1, l)-component of ab we get an cost = cost from the
above equation. Suppose an = 1. Then (1, 0, ... , 0) £ 5m is a fixed point of Л,
which is a contradiction. Therefore we have cost = 0, namely t = π/2, and this
completes the proof of the proposition. D
Exercise 6. Let Μ be a compact Riemannian manifold of constant curvature
1. Show the following: If m = dim Μ is odd, then Μ is orientable. If m is even,
then Μ is either simply connected and isometric to the unit sphere, or π\{Μ) = Z2
and Μ is isometric to the real projective space of constant curvature 1.
Now we consider a complete Riemannian manifold Μ of negative constant
curvature —1, which is also called a hyperbole manifold. In this case the universal
Riemannian covering of Μ is isometric to H171, and its deck transformation group
Γ is a subgroup of the isometry group of Ηrn which acts properly discontinuously
and freely on Нш (see also Chapter, V §1).
We give examples of hyperbolic manifolds in the case of m = 2. Let Έ9 be a
compact orientable surface of genus g > 2. By the theory of surfaces, topologically
Σ9 may be obtained by identifying the sides of a regular 4<?-sided polygon Ρ (see
Figure 19 in the case of g = 2). Now the vertices of Ρ are identified to one point
under the above identification, and if the sum of inner angles of Ρ is equal to 2 π
then we get a smooth surface.
Figure 19
Note that if we realize Ρ in the Euclidean plane this is impossible (if g = 1 then
the sum of inner angles of a square in R2 is equal to 2π, and the above identification
gives a flat torus). However, in H2 it is possible to give a regular 4<?-sided polygon
Ρ satisfying the above condition whose sides are geodesies, as in Figure 19. In fact,

2. COMPARISON THEOREMS FOR JACOBI FIELDS
143
denoting by α and S the sum of inner angles of Ρ and the area of P, respectively,
from the Gauss-Bonnet formula we get
a = 2(2g-l)n-S.
Note that if S —► 0, then α —► 2(2g - 1)π (> 2 π), and if vertices go to infinities,
then α —► 0. Therefore, we may construct a regular geodesic 4p-sided polygon in
H2 with α = 2 π because of continuity. Namely, Σρ (ρ > 2) carries a hyperbolic
manifold structure. Further, it is known that hyperbolic structures on Σ9 are not
unique and the space of such Riemannian metrics of constant curvature -1, where
we identify two Riemannian metrics if they are mapped by a difFeomorphism of Έ9
homotopic to the identity, is of dimension 6(g — 1) (Teichmuller space). Thus in
the two dimensional case, generic surfaces carry hyperbolic structures. It is not so
elementary to give hyperbolic manifolds in the higher dimensional case. However,
recently great progress has been made by W. Thurston and others in relation to
the study on three dimensional manifolds, and hyperbolic non-Euclidean geometry
plays a more and more important role (see [Th], [Ra]).
2. Comparison Theorems for Jacobi Fields
2.1. In this section, we compare Jacobi fields along geodesies satisfying some
appropriate initial conditions in two complete Riemannian manifolds Μ. Μ.
Let N be an η-dimensional submanifold of Μ, η : [0. +эс) —► Μ a normal
geodesic emanating from ρ := 7(0) G N perpendicularly to X with the initial
direction и := 7(0) G ΤΡΝ^. Let to(N) be the first focal value of .V along 7. We
set
K(t) := max{Ka; σ (С Tl{t)M) Э 7(f)}.
(2.1) k(t) := mm{Ka; σ (С Tl{t)M) Э 7(0}·
We consider linearly independent TV-Jacobi fields Y\. Y2, ... , Yr of J^ :=
{Y G Jn\ y(£)J_7(£)}. Recall that Y{ (i = 1, ... , r) are determined by the
initial conditions
Yi{0) := Ai G TpN, VYi{0) = AuYi(0) + BU B{±TpN,
where Au denotes the shape operator of N with respect to the unit normal vector
и to N (Chapter II, §3.3). By Chapter II, Lemma 4.8, we may write Yi(t) =
Dexp-L(tu)(Ai, tBi)N. We set U{(t) = A{ + tBi, and for 0 < t < t0(N) consider
the following function:
(2.2) f(t):=log{\\Yl(t)A---AYr(t)\\/\\Ui(t)A.-.AUr(t)\\},
which is independent of the choice of basis of (Y\, ... , Yr) я,4 as is easily checked.
Next note that Нтцо f(t) = 0. In fact, changing the basis of (Yi, ... , Yv)r if
necessary, we may assume that As+i = · · · = Ar = 0 and {A\, ... , A3, Bs+i, · · · , Br}
are linearly independent, where we set
а := dirndl, ... , Ar)R = dim{r(0);r G (Yu ... , Yr)R С J^}.
4(v\, ... , vr)n denotes the subspace spanned by v\, ... , vr-

144
IV. COMPARISON THEOREMS AND APPLICATIONS
Then, noting that limtioYs+i{t)/t = V7s+i(0) = Bs+i, etc., we get
lim||y1(i)A.--Ayr(i)IIAr-i
lim
цо
у1(4)л...лу.(*)л^^л...л^
= ||i4i Л · · · Л As Л Bs+i Л · · · Л Br\\ = lim \\Ui(t) Л · · · Л Ur{t)\\/tr~s
from which our assertion clearly follows.
Also, in Μ we may consider the corresponding notions N, 7, Л, K(t), k(t), etc.
If we take linearly independent Jacobi fields Yi, ... , Yr G J^ on Μ and define /(£)
in the same manner as in (2.2), we get similarly Итцо /(£) = 0·
Now we assume that
dim Μ > dim Μ, dim TV = dim TV := η
and compare f(t) and f(t) under the following assumption either on the sectional
curvature
(2.3) k{t) > K{t) (0<t< t0{N), t0(N))
or on the Ricci curvature
(2.4) p(7(t)) > рШ) (0<t<to(N),to(N)).
For that purpose we set
g(t):= ^{]0g\\Y1(t)A---AYr(t)\\-]og\\Y1(t)A---AYr(t)\\},
and look for the condition that g(t\) > 0 holds for a fixed t = t\ (0 < ti <
to(N), to(N)). The idea of the following argument is to use the index form to
check the sign of g(t\). Namely, as in Chapter III, §2.3, on the space
T7C^([0, ti\) := {X; piecewise C°° vector field along 7 | [0, tx]
with X(0) G TpN, X(t)±<y(t)}
we consider the index form
IN(X, X) = Γ{(νΧ(ί), VX(t)) - K(X(t), 7(ί))ΙΙ*№Ι|2Κ
JO
+ <Aix(o),x(o)>.
Then Lemma 2.10 of Chapter III gives a fundamental property of In, and if Υ is
an 7V-Jacobi field we get
(2-5) IN(Y, Y) = (Vr(ti), Y(t!)).
Note that for Μ we may also consider the corresponding T^C^([0, t\\) and J^.
Now we estimate g{ti) in the following steps:
1.° We set Vi := (Yi(ii), ... , Уг(^)>я. Then {Yi(*i)} are linearly
independent because 0 < t\ < t0(N) (Chapter II, (4.3)). We may assume that {Y{ti)}$=1
forms an o.n.b. of V\. In fact, take 7V-Jacobi fields Zb ... , Zr such that {Zi{ti)}ri=l

2. COMPARISON THEOREMS FOR JACOBI FIELDS
145
forms an o.n.b. of V\. Then from (4.13) of Chapter II, we may write Yi(t) = a{Zj(t),
where (α\) is a constant r χ r matrix. It follows that
d_
dt
log||yiA--.Ayr(i)|| =
log{|detaJ|||Zi(i)A..-AZr(i)||}
t = ti
logHZxiijA-.-AZriOll·
d_
dt
t = ti
2.° If {Yi(t\)} is an o.n.b., we have
log||yi(i)A-.-Ayr(i)||
= Σ<ΥΊ(ίι) A ..· A VYi(ti) A ..· A yr(ix), ?,(и) A ■·. А Уг(^)>
r r
= χ;<ν^(ίι), ri(ii)> = £w, u).
3.° Now suppose we may choose a linear isometry ^ : TpM —► TPM that
satisfies the following conditions:
(2.6) .(4(0)) = 7(0), l(TpN) = TPN.
(2.7) iP(^4^i),---Mti))R = Pb)ol(yi(ti) Yr(ti))R.
Next we set Vx :=< Yi(*i), ... , yr(*i) >я. Then the vector fields И^(£) :=
P(7)° о t о Р(7)^(У<(0)(* = 1> — , Ο along 7 belong to T,C^([0, ^]), and
{H^C^i)}Γ=ι forms an o.n.b. of V\ by 1° and the assumption (2.7). Note that we
may assume that Yi(t\) = Wi{t\) (i = 1, ... r) by considering a change of basis of
(Yi(ti), ··· , Уг(^1))я as in step 1°. In particular, {Yi(ti)}^=1 forms an o.n.b.
of Vi, and we have W*(0) = ιΥ-(Ο).
4.° Then, by the same argument as in 2° and applying Lemma 2.10 of Chapter
III, we get
(2.8)
~dt
log||Yi(t) A · ·· ЛУГ(*)|| = γ,ΙΝ(Υί, Yi) < Y,In(Wu Wi)
ti t=l t=l
= Σ {jf «VWi w·VWiit)) - Km> ^(*))ΐι^(*)ΐι2} ^
+ (AuWi(0),Wi(0))\.
Now note that_||Wi(t)||_= ||U(t)||, ||VWi(t)|| - l|VU(t)||, and K(i(t), W4(t)) >
k(t) > K{t) > K(y(t), Yi(t)), where the second equality sign follows from VWi(<) =

146
IV. COMPARISON THEOREMS AND APPLICATIONS
P(7)° о L о P{^)l{VYi{t)). Then we see that
the left-hand side of (2.8)
<
J2 A<vm ч?м)-к(ш Yi(t))\m)f}dt
r
+ J(4(ty,(0)),tyi(0)}.
1 = 1
Therefore, if we can construct a linear isometry l : TpM —> TPM satisfying
(2.6), (2.7) and
Г Г
(2.9) Σ(Αη(ιΫί(0), iYi(0)) < 53(Лй(У;(0)), iYi(0)),
i=l i=\
then it follows that
d
dt
d
dt
\og\\Y,{t)A---AYr{t)l
\t = ti
namely, g(t\) > 0. However, the two conditions (2.6), (2.7) are not compatible in
general, and (2.9) is related to the eigenvalues of the second fundamental forms of
TV, N. In the following we list the cases (I)-(IV), where the above conditions (2.6),
(2.7) and (2.9) are satisfied.
Exercise 1. Let А, В be symmetric linear transformations of a Euclidean
vector space V. We denote by Атах(Л) (resp., Ат,п(Л)) the maximal (resp., minimal)
eigenvalue of A, and A < В means that (Ax, x) < (Bx, x) holds for any vector
χ e V. Then show that Атах(Л) < \m\xx{B) if and only if a~lAa < В for any
orthogonal transformation a of V.
Case (I): dim Μ > dimM(:= m), dim AT = dim TV = 0, k(t) > K(t), and
1 < r < fh — 1. In this case, we have Au, Ац = О, and (2.6) and (2.7) are
obviously compatible. Also note that we may choose Υί, Y{ (i = 1, ... , r) so that
А{ = 0, A{ = 0, namely, Щг) = tBu Ui(t) = tB{.
Case (II): dim Μ = dim Μ = m, dim N = dim N = m - 1, k(t) > K(t), 1 <
r < m — 1, and ХтАХ(Аи) < \тт{Ай). In this case (2.6) automatically holds, and
we may choose an ι to satisfy (2.7). Then from the assumption on eigenvalues of
the second fundamental forms of TV, TV, respectively, we get l~1 Aul < Ац by virtue
of the above Exercise 1, and (2.9) holds. Also note that we may choose Υί, Ϋ{ so
that Ui(t) = Au В{ = 0, Ui(t) = Au B{ = 0 (i = 1, ... , r).
Case (III): dimM = dimM = m, dimN = dimN = n, k(t) > K{t), r =
m — 1, and we may arrange eigenvalues of Au, Au in the form Xi(Au) < Xi(Au)
(i = 1, ... , n). In this case, any ι : TpM —» TPM with ^(7(0)) = 7(0) automatically
satisfies (2.7). Further, if we choose ι so that it maps a unit eigenvector of Au with
eigenvalue Xi(Au) to a unit eigenvector of Au with eigenvalue \i(Au), then we
get l~1Aul < An, and (2.6), (2.9) are also satisfied. Note that we may choose

2. COMPARISON THEOREMS FOR JACOBI FIELDS
147
{Ya}™=i so that Ui(t) = Au Bi = 0(z = 1, ... , n), Uj{t) = tB0, A, = 0(j =
n+1, ...,m — 1), and similar facts also hold for Y{.
Case (IV): dim Μ = dim Μ = m and Μ is a complete simply connected
Riemannian manifold of constant curvature δ. Suppose that the Ricci curvature of
Μ satisfies p(7(£)) > (m — 1)<5, and r = m — 1. Moreover, as for TV, TV, suppose
that either one of the following conditions (a) or (b) holds:
(a) dim TV = dim TV = 0.
(b) dim TV = dim TV = m - 1. Further, we assume that Ац = Aid^^5 for
some constant A, and traced < (m - 1)A.
Now we show that g(t\) > 0 holds also in the case (IV). We treat here only the
case (b) for TV, TV, and leave the proof of the easier case (a) to the reader. Since r =
η = m-1, we may choose ι so that (2.6), (2.7) are satisfied. On the other hand, since
Μ is of constant curvature δ and TV is totally umbilic at p, we see that TV-Jacobi
fields Yi{t) (i = 1, ... , r) along 7 may be written Yi{t) = (cs{t) + \ss(t))Ei(t),
where E{ are parallel vector fields along 7. Note that from step 1° we may assume
that {Ei(t)} are mutually orthogonal and have the same length. The {Wi(£)} also
share the same property. Then, by the definition of the Ricci curvature, the last
term of (2.8) is equal to
/'
Jo
\ΙΣ(™Μ' ™W)>) -Ρ(7(*))«ί)1|2} dt + traceЛи||Щ0)||2
^ jf { (EWiW, ν£(ί)> J - ptf(t))IIU(*)ll2| dt + (m - 1) λ ||Уг(0)||
*-l f /-ίι
«уШ vw) - кт), £(*))||£(ί)||2)Λ
τη — Ι ,
г=1
+ (Лйу;(о),у;(о))|
log||Yi(*) Л ■ ■ · ΛΫ·„_!(ί)||,
which implies the desired inequality g(t\) > 0. Further note that in this case we
may take Ui(t) = Ai: Bi = 0 (г = 1, ... , m - 1), and Щ in a similar form. In the
case of (a), we may assume that Ui(t) = tBi, At = 0, etc.
On the other hand, note that from the above choices of Ui, Ui we get for each
case of (I)-(IV)
|{log \\Ui(t) A ■ ■ ■ Ur(t)\\ - log \\Ui{t) Л · · · Л Ur(t)\\} = 0.
Summing up, we have
Proposition 2.1. In the above situation, suppose one of (I)-(IV) holds. Then
we get the following:
In this case N is said to be totally umbilic at p.

148
IV. COMPARISON THEOREMS AND APPLICATIONS
(1) F: ty-+\\Yi(t)A··· AYr(t)\\/\\Yi(t)A--AYr(t)\\ is monotone increasing
on (0, t0{N)). In fact, F'{t) > 0, and in particular F(t) > F(0) for0<t< t0{N).
(2) For0<t <t0{N),
f(t) := \\Ϋχ(ί) a ..· лyr(*)ll/IIUW л... л Ur(t)\\
> \\Yx(t) A ..· Л УГ(*)||/||СМ*) Λ .·. Λ Ur(t)\\ := f(t).
(3) t0(N)<t0(N).
PROOF. First suppose t < t0{N), t0(N). Then (1) follows from {logF(t)}' =
g(t) > 0. (2) follows from limt|0 f(t) = Птц0 f(t) = 0 and
{log/>)-log/(*)}'
= ^(0-{bg||^i(0A---Af7r(0l|-bg||t/iWA...A[/r(0ll},=^)>0.
To see (3), suppose to the contrary that t0(N) > t0(N). Then (2) holds for 0 < t <
to(N) and also for t = to{N) by continuity. Then for any case of (I)-(IV) we may
choose Yu ... , Yr so that f(t0{N)) = 0. Then f(t0(N)) = 0, and t0(N) is a focal
value of N along 7. This contradicts the fact that to(N) is the first focal value. D
Remark 2.2. Suppose equality holds for some 0 < Τ < t0(N) in (1) or (2).
Then g(t) = 0 for all t £ [0, T], and the following holds corresponding to (I)-(IV),
if we change bases of (У*)я? (^)я when necessary.
Cases (I), (II): k(t) = К{Ш *#)) = К(Ш Ш) = K(t) and Y^t) =
P(7)? о t о P(7)o(^i(i)) (г = 1, ■ ■. , r) for 0 < ί < Г. Further, in case (II), we get
Amax(iu) = \т\п{Ай) '·= A. Since equality holds in (2.9), it follows that
Атах(Ли)||^(0)||2 = (Au(lYi(0)), LYi(0)) = \тш(Ай)\\Щ0)\\2.
Therefore, Yi{0) = iYi(U) is an eigenvector of AU with eigenvalue A, and the same
assertion also holds for Υί(0).6 Further note that k(t) is the minimal eigenvalue of
the symmetric linear transformation ν н-> R(v, 7(£)Ж0 °f 7W"1? and Yi(t) is its
eigenvector with eigenvalue k(t). Since the Yi are Jacobi fields, we have
V(VYi(t) A Yi(t)) = VVYi{t) A Yi{t) = -k{t)Yi{t) A Y^t) = 0,
which means that VY{(t) A Yi{t) are parallel along 7. The same holds also for Yi·
Now considering the initial condition for Jacobi fields, we get Yi(0) = 0 in case (I)
and VYi(0) = AU Yi(0) = AYi(0) in case (II), where we have B\ = 0 because Yi is an
7V-Jacobi field along 7 perpendicular to 7 for a hypersurface N of M. Therefore,
it follows that VYi(0) Λ Υ{(0) = 0 and consequently VYi{t) A Y{{t) ξ 0 in either
case. Namely, Yi(£)/||Yi(£)|| are parallel along 7, and similarly Yi(£)/||Yi|| are also
parallel along 7.
Case (III): We have k(t) = K(>y(t), Yi{t)) = Ktf(t), Y-(t)) = K{t\ Yi(t) =
P(7)? о t о Р(7)&(ВД) (г = 1, ... , r) for 0 < t < Т.
Case (IV): (a) We have p{j(t)) = (m - 1)6 and У*(*) = s6{t)Ei{t), where
2?г(£) are parallel vector fields along 7. Then R(Ei(t), 7(£))7(£) = 6Ei(t), and
ВД = fc(t) = (5. (b) Wehavep(7(i)) ξ (m-l)<5 and Y^t) = (c6(t) + X8e{t))Ei(t).
Finally, we remark that the above proposition also holds for general (constant
speed) geodesies 7, 7 with ||7(0)|| = ||7(0)||.
6For the maximal (minimal) eigenvalue λ of a symmetric matrix A suppose that (Au, u) =
A||w||2 for u/0. Then и is an eigenvector of A with eigenvalue λ.

2. COMPARISON THEOREMS FOR JACOBI FIELDS
149
2.2. In the previous subsection we treated a comparison theorem for Jacobi
fields in a unified manner. Now we apply the above proposition to useful concrete
cases. First we give the following, which may be inferred from (I), (II) setting
r = 1, and is originally due to H. E. Rauch for (1) and M. Berger for (2).
Theorem 2.3 (Rauch comparison theorem R.C.T.(I)). LetM, Μ be complete
Riemannian manifolds and Υ (resp., Y) a Jacobi field along a normal geodesic 7
(resp., 7) of Μ (resp., M) emanating from ρ (resp., p), which is perpendicular to
7 (resp., 7). Denote by to(p) (resp., io(p)) the first conjugate value to ρ (resp., p)
along 7 (resp., 7).
(1) Suppose dim Μ > dim Μ and k(i) > K(t) (0 < t < to(p)). Then we have
the following assertions.
a) t0(p) >t0(p)·
b) Assume that Y(0) = 0, Y(0) = 0, ||W(0)|| = ||УГ(0)||(^ 0). Then
F(t) := ||^(£)||/||У(£)|| is monotone increasing for 0 < t < to(N), and
,,im (W(t), Y(t)) (VY(t). Y(t))
( ' ^ (Y(t), Y(t)) - (Y{t), Y(t)) '
(2.11) \\Y(t)\\ > \\Y(t)\\.
Further, if equality holds for some Τ £ (0, to (ρ)) in (2.10) or (2.11), then equality
holds for allO <t <T, and K(t) = K(^(i), Y(t)) = Κ(η(ί), Y(t)) = k(t) and
Y(t)/\\Y(t)\l Y(t)/\\Y(t)\\ are parallel along 7 | [0, T] and 7 | [0, T], respectively.
(2) Assume dim Μ = dim Μ. Let N (resp., N) be a hypersurface of Μ (resp.,
M) and 7 (resp., 7) a normal geodesic of Μ (resp., M) which emanates from
ρ £ N (resp., ρ £ Ν) perpendicularly to N (resp., N). Set и := 7(0), й := 7(0)
and assume that
k(t) > K(t) (0<t< t0(P)), Хтах(Аи) < Хтт(Ай).
Then we have the following:
a) t0(N)>t0(N).
b) Let Y(t) (resp., Y(t)) be an N-Jacobi field (resp., N-Jacobi field) along 7
(resp., 7) perpendicular to 7 (resp., 7). // ||У(0)|| = ||У(0)|| (ф 0), then F(t) :=
II^WII/II^WII is monotone increasing for 0 < t < t0(N), and (2.10), (2.11) hold.
Further, if equality holds for some Τ £ (0, to(N)) in (2.10) or (2.11), then
equality holds for allO <t <T, and K(t) = K("r(t), Y(t)) = Κ(η(ί), Y(t)) = k(t),
and Y(t)/\\Y(t)\l ^(^/11^(011 are parallel along 7 | [0, T], 7 | [0, T], respectively.
Moreover, У(0)/||У(0)||,Уг(0)/||Уг(0)|| are eigenvectors of Au, Ай with the
eigenvalue Хт\п(Ай) = Атах(Лп), respectively.
Remark 2.4. In (1) of Theorem 2.3, if we do not assume that Υ, Υ are
perpendicular to 7, 7, respectively, but assume the following condition b)' instead of
the condition in b), then (2.11) still holds:
b)r ^(0), 1^(0) are tangent to 7, 7, respectively, and
11^(0)11 = 11^(0)11, ||Vy(0)|| - ||Vf(0)||, («, W(0)> = (й, УУ(0)).

150
IV. COMPARISON THEOREMS AND APPLICATIONS
Similarly, in (2) further suppose that TV, N are totally geodesic at p, p,
respectively. Then if we replace the condition in b) with the following condition b)', we
still have (2.11):
b)r Let У, У be Jacobi fields along 7, 7 such that УУ(0), УУ(0) are tangent
to 7, 7, respectively, and ||У(0)|| = ||У(0)||, ||УУ(0)|| = ||УУ(0)||, (и, У(0)> =
(й, У(0)> hold.
Exercise 2. Prove the above remark (decompose У into the tangential and
vertical components with respect to 7).
Next we apply the previous theorem to compare the length of curves in Μ, Μ.
Theorem 2.5 (R.C.T. (II)). Let Μ and Μ be complete Riemannian
manifolds.
(1) Suppose dim Μ > dim Μ and fix ρ G Μ, ρ G M. Choose r > 0 so that
expp I Br(op) is an immersion and exp^ | Br(Op) is an embedding. Now suppose
that for any plane section σ at any point of Br(p) С Μ and for any plane section
σ at any point of Br(Op) С Μ, we have the inequality Κσ > Κσ for the sectional
curvatures. Take a linear isometry I : TpM —» TPM. Now for any piecewise
C°° curve с : [0, 1] —» Br(p) = expp(Br(Op)) of' M, we get a piecewise C°° curve
с : [0, 1] —» Br (ρ) in Μ defined by с := expp ol о exp^"1 oc. Then L(c) < L(c).
(2) Suppose dim Μ = dim Μ. Let 7 : [0, /] —» M, 7 : [0, /] —» Μ 6e normal
geodesies and Ε, Ε parallel unit vector fields along 7, 7, respectively, which satisfy
(E(t), *y(t)} = (E(t), 7(t))· Suppose we have Κσ > Ко for any plane sections σ, σ
of Μ, Μ, respectively. Now let f(t) : [0, I] —» Д+ бе α C°° function such thatf(t) is
not greater than the first focal value to(Nt) 0} Nt along a geodesic s н-> exp7^) s E(t),
where Nt denotes a hypersurface βχρΊ^{ιν G ΤΊ^Μ\ w±E(t), \\w\\ < e} (e > 0 is
small). Nt is perpendicular to E(t) and totally geodesic at *y(t). If с, с in M,M
are defined by c(t) := exp7^) f(t)E(t), c(t) := exp7^) f(t)E(t)f respectively, then
L{c) < L(c).
PROOF. (1) We define a(t, s) := exp sci(t), a(t, s) := exppSCi(£), where
dt,
we set c\(t) = exp^ c(£), ci(t) = Ic~\(t)
цс)= /Ίΐ^(ί,ΐ)
Jo W™ I
. Since
\dt, L(c)= /Ί|^(*,1)|
1 Jo II <« 1
it suffices to show that
(2.12)
|£<«
, i)
<
If HI
1 ^ II
for each t. Now for a fixed i, Y(s) := §*(*, s), y(s) := ff (i, s) are Jacobi
fields along geodesies at : s и o(t, s), Qt : s 1—► ά(£, 5), which satisfy the
initial conditions У(0) = 0, У(0) = 0, УУ(0) = έι(ί), УУ(0) = ci(i), respectively.
Since / is a linear isometry and we have ||ci(£)|| = ||ci(£)||, ||ci(£)|| = ||ci(£)|| and
(c\(t), ci(t)) = (c~i(t), c~i(t)}, it follows that the assumption b)' of Remark 2.4 (1)
is satisfied. Therefore (2.12) holds for each t.
We may prove (2) in the same manner. We set a(t, s) := expΊrt^sf(t)E(t)
and a(t, s) := exp~(^ s f(t)E(t). It suffices to show (2.12) for each t. In fact, note
that Jacobi fields Y(s) := ^r(t, s), Y(s) := ^r(t, s) satisfy the initial conditions

2. COMPARISON THEOREMS FOR JACOBI FIELDS
151
ΤΡΜ=ΤβΜ
λ
ехрэ\
exp/ \ Ρ
м{ ""Μ Л
L(c)^L(?)
Figure 20
У(0) = Ή*), У(0) = 4(0, Vr(0) = f'{t)E{t). УУ(0) = f(t)E(t). respectively.
Hence they satisfy the assumption b)' of Remark 2.4 (2). D
Remark 2.6. As an application of the above theorem we consider the
following situation: Suppose the sectional curvatures Κσ of a complete m-dimensional
Riemannian manifold Μ satisfy Κσ < Δ everywhere. We denote by Мд the
m-dimensional complete simply connected Riemannian manifold of constant
curvature Δ. We take r > 0 such that expp : Br(op) —> Br(p) is a diffeomorphism and
r < 7γ/λ/Δ if Δ > 0. We fix a linear isometry / : TPM —» ТрМд . Now suppose a
minimal geodesic 7 : [0, 1] —► Μ joining qu q2 e Br{p) is contained in Br{p), and
set q{ := expp(I(exp~l <&)), 7 := exp^ о J о exp"1 07. Then
(2.13) d{qu q2) > d{qu q2).
In fact, from R.C.T. (II) (1) we get d{qu q2) = L(7) > Щ) > d(qu q2).
Next suppose that equality holds in (2.13) and exp"1 ^ (г = 1,2) are
linearly independent. We set ci(t) := exp'17(f), c\(t) := Ici(t) and a(t, s) =
exppsci(i), a(t, s) = exppsci(t). First note that 7 is also a geodesic of M.
Since Мд1 is of constant curvature, it follows from the axiom of plane (§1.2)
that ά(ί, s) spans a geodesic triangle (pqi q2) with sides 7, 71, 72, where 71 (s) =
ά(0, s), 72(s) = ά(1, s) are geodesies in Μ joinings to qu q2, respectively. Further,
in this case equality holds in (2.12) in the proof of Theorem 2.5 (1), and it follows
that
*(^<.).£<«,.>)-*(£(...>, £(...)).*
by the argument of the case where equality holds in Theorem 2.3 (1). Therefore,
a(t, s) also spans a geodesic triangle 5 = {pq\q2) of constant curvature Δ with
sides 7, 71, 72. Moreover, S is totally geodesic.

152
IV. COMPARISON THEOREMS AND APPLICATIONS
We note that the similar facts also hold for the case of the equality sign in
R.C.T.II (2).
Exercise 3. Show that S in the above remark is totally geodesic by verifying
that minimal geodesies δ joining two sufficiently close arbitrary points n, r2 £ S
are contained in 5.
The comparison theorem is very useful when the sign of the sectional curvature
is fixed or the sectional curvature function is bounded below or above.
Theorem 2.7. Let Μ be a complete Riemannian manifold.
(1) Suppose Κσ < Δ holds everywhere. Let η be a normal geodesic emanating
from ρ with the initial direction и £ UPM, and Υ a Jacobi field along 7 and
perpendicular to 7. Set yA(t) := ||У(0)||сд(£) + ||У||'(0)вд(0· ^ei *o be the minimal
positive value of t such that yA(t) = 0. Then, for 0 < t < to,
(2.14)
(Y(t), VY(t))yA(t) > (Y(t), Y(t))y'A(t), \\Y(t)\\ > yA(t).
(2) Suppose Κσ > δ everywhere. Let η be a normal geodesic emanating from ρ
with the initial direction и £ UPM, and Υ a Jacobi field along 7 and perpendicular
to 7. Suppose У (0), УУ(0) are linearly dependent. Denote by to the first conjugate
value to(p) to ρ along 7 if У(0) = 0. // У(0) φ 0, t0 denotes the first focal value
to{N) along 7 of a hypersurface N through ρ such that и is a unit normal vector to
N at ρ and its shape operator Au is given by Au = (||y||'(0)/||y||(0))id (see Chapter
II, §3, Exercise 7). Then for 0 <t<t0 andy6{t) := \\Y{0)\\c6{t)-\-\\Y\\'(0)s6(t) we
have
(2.15)
<У(0, Y(t))U(t) > (Y(t), VY(t))y6(t), \\Y(t)\\ < y6(t).
PROOF. If У (0) = 0 in (1), we take a Jacobi field Y(t) = sA(0 E(t) along a
normal geodesic 7 in Μ = M£\ where E(t) is parallel along 7 with ||£(0)|| = ||УЦ'(0).
Then we get У(0) = 0, ||У||'(0) = ||У||'(0). Noting that ||У||'(0) = ||УУ(0)||,
etc., we get ||УУ(0)|| = ||УУ(0)||, and our assertion follows from (2.10), (2.11). If
У(0) = 0 in (2), then by a similar argument we have (2.15).
Now suppose У(0) φ 0. First we treat the case (1). Take a hypersurface N
through ρ such that и is a unit normal vector to N and Υ is an iV-Jacobi field. In
fact, take a linear symmetric transformation A of u1- (C TPM) such that А У(0) =
УУ (0), and consider a hypersurface N through ρ with normal vector и whose shape
operator Au is given by A (Chapter II, §3, Exercise 7). On the other hand, on Μ =
Мд take a point ρ and й £ ЩМ. Further, take a hypersurface N in Μ through ρ
such that и is a normal vector to N and Ай = Aid, A = ||У||/(0)/||У||(0). Then the
first focal value of N along 7 = ju is given by the above to in (1), since an N-Jacobi
field along 7 is written in the form {cA{t) + XsA(t))E{t) = {yA(t)/\\Y\\(0))E(t)
with parallel E(t). Now in the argument of §2.1, we change the role of Μ, Μ and
note that we may choose an ι so that it satisfies (2.6), (2.7), because r = 1. In
our case we have (AuW(0), W(0)) = A ||^(0)||2 = А ||У(0)||2 = (УУ(0), У(0)> =
(Au У(0), У(0)). Therefore, (2.9) also holds changing the roles of Μ, Μ. Then we
may prove (2.14) in the same manner as in the proof of Proposition 2.1.
Finally we prove (2) assuming that У (0) φ 0. By the assumption of the theorem
we may write УУ(0) = АУ(0), and we take a hypersurface N through p such

2. COMPARISON THEOREMS FOR JACOBI FIELDS
153
that и is a normal vector to N and Au = Aid. Note that we have in fact A =
ΙΙ^ΙΓ(0)/||*ΊΙ(0), and Y is an TV-Jacobi field. On the other hand, in M = M%
we take a geodesic 7 and a hypersurface N with the same property as in Μ, and
consider an TV-Jacobi field Y(t) = ys{t)E(t) along 7, where E(t) is a parallel vector
field along 7. Then, applying Proposition 2.1 (II), we get (2.15). D
Corollary 2.8. Suppose sectional curvatures Κσ of a complete Riemannian
manifold Μ satisfy δ < Κσ < Δ for all plane sections σ.
(1) Let η be a normal geodesic in Μ and Y(t) a Jacobi field along 7 which
satisfies Y{0) = 0 and is perpendicular to 7. Then
ss(s)/s6(t) < ||У(в)||/||У(0И < sA(s)/sA(t), 0<s<t< тг/л/Δ,
where we interpret π/y/A = +00 when Δ < 0.
(2) Let и G TPM, 0 < ||u|| < n/y/Δ. Then for ν G TPM, v±u,
sA(\\u\\)/\\u\\ < \\Dexpp(u)v\\/\\v\\<s6(\\u\\)/\\u\\.
(3) // Κσ < 0 everywhere, then for any point ρ £ Λ/ and any geodesic 7
emanating from ρ there exist no conjugate points to ρ along 7. // Κσ < Δ holds
everywhere for a constant Δ > 0, then for any point ρ and any normal geodesic 7
emanating from p, the first conjugate value to (p) to ρ along 7 is greater than or equal
to π/\/Δ. On the other hand, if Κσ > δ holds everywhere for a positive constant
δ, then the first conjugate value to ρ along any normal geodesic 7 emanating from
any ρ G Μ satisfies ίο (7) < π/\/δ. In particular, no geodesic segment of length
greater than π/\/δ can be minimal Therefore, Μ is compact and has diameter
d(M) < π/y/S.
PROOF. (1) Applying Theorem 2.7 (1) and noting that yA(t) = \\Y\\'(0)sA(t)
and ys(t) = \\Y\\f(0)ss(t) in this case, we obtain (1) from the fact that t \->
log{\\Y(t)\\/yA(t)} (resp., t !-► log{||y(i)||/!/$(0}) is monotone increasing (resp.,
monotone decreasing).
(2) Apply (1) to a Jacobi field Y(t) along 7n with У(0) = 0, УУ(0) = v/||m||,
and note that Y(\\u\\) = Dexpp{u)v (Chapter II, (2.16)). Then (2) follows from (1)
by setting t = \\u\\ and letting s —> 0.
(3) follows from (2) by the argument before Lemma 2.4 of Chapter II. D
Remark. Note that such an estimate for the norm of ||Z)expp(u)|| (or the
norm of Jacobi fields) does not follow directly from Proposition 3.1 of Chapter II
because of error terms.
Now we apply the above comparison theorems to estimate the Hessian D2dp of
the distance function dp to ρ in terms of the upper and lower bound of the sectional
curvature.
Lemma 2.9. Let Μ be a complete Riemannian manifold whose sectional
curvature satisfies δ < Κσ < Δ. Suppose 0 < г < min(zp(M), π/2\/Δ). Then for
q G Br(p) and ulSJdp (q), we get
SA{dp{q)) s6{dp(q))
Further, the gradient vector Vdp (q) of dp belongs to the null space of D2dp (q).

154 IV. COMPARISON THEOREMS AND APPLICATIONS
PROOF. For given u, ν £ TqM there exist Jacobi fields Χ, Υ along a unique
normal geodesic 7 : [0, I] —> Μ (/ = dp(q)) joining ρ to q such that X(0) =
0, Y(0) = 0, X(0 = u, У(0 = v. Now if we set u = Vdp (<?), then from
Chapter III, (4.7)', we get D2dp(q)(u, v) = 0 for any ν £ ΤςΜ; namely, Vdp(q) belongs
to the null space of D2dp (q). Next if u±.Vdp (0), then the above Jacobi field X is
perpendicular to 7 and we get, again from (4.7) of Chapter III,
D2dp(q)(u, u) = <VX(0, ВД>-
Now we apply Theorem 2.7. In this case we have 2/д(0 = ||-X'||/(0)s^(i), ^(0 =
||Χ||'(0)δ6(0, and [t follows that
сд(0/*д(0 · <u, u) < (VX(0, ВД> < <*(0M0 · <u, u>.
Note that the left-hand side is positive if / < π/2\/Δ and w^O. D
Exercise 4. Under the assumption of Lemma 2.9, set f(q) := \dp(q)2. Then
show that for ulSJdp (q)
(2.17) ^g« (u, tt> < ?Ц^Л < ££iiMi (u, u).
Also show that D2f(Vdp, Vdp) = 1.
3. Applications of Comparison Theorems
3.1. We continue to state applications of comparison theorems for Jacobi
fields. Let Μ, Μ be m-dimensional complete Riemannian manifolds and ρ £
Μ, ρ £ Μ. Let 7 be a normal geodesic in Μ emanating from ρ with the
initial direction и £ UPM, and let ίο (7) denote the first conjugate value to ρ along
7. By Lemma 5.4 of Chapter II, the Jacobian j(t, u) = Jdet(gij(expptu)) of the
exponential mapping expp at tu £ TPM is also given by 6(t, u)/£m_1. Recall that
6(t, u) may be written as follows: Let {ei, ... , em_i, em = u} be an o.n.b. of TPM
and take Jacobi fields Yi(l<i<m — 1) along 7 which satisfy the initial conditions
y.(0) = 0, VVi(0) = e». Then we have 0(f, u) = \\Yi{t) Λ · · · Л ym_i(f)||, which is
independent of the choice of o.n.b. {е*}.
Now take a linear isometry / : TPM —> TPM and let 7 be a normal geodesic
emanating from ρ with the initial direction й := I u. Similarly, considering Jacobi
fields Yi (1 < г < m - 1) along 7 in Μ with УД0) = 0, УУ»(0) = ё» := /еь we get
0(f, u) = ||Yi(i) Λ --- Λ ... ym_i(i)||- Now from Proposition 2.1 (I), (IV) we get
the following Bishop volume comparison theorem.
Theorem 3.1. (1) Suppose dim Μ = dim Μ andk(t) > K(t) {0<t< £0(7))·
Then t0(7) > to (7), and
a) £ 1—► 9(t, u)/6(t, u), £ н-> j(t, u)/j(t, u) are monotone increasing for 0 < £ <
*o(7)-
b) 0(i, fi) > 0(f, u), j(i, u) > j(t, u) (0<t< t0(u)).
If equality holds for some t = Τ (< £0(7)) ^ a) or b), ί/ien equality holds
for any 0 < t < Γ, and ii /oiiowe ίΑοί k(f) = tf(f), 11^(011 = 11^(011- Further,
Yi(t)/\\Yi(t)\l Yi(t)/\\Yi(t)\\ are parallel along 7 | [0, T], 7 | [0, T\, respectively.

3. APPLICATIONS OF COMPARISON THEOREMS
155
(2) Let Μ be an m-dimensional complete Riemannian manifold of constant
curvature δ. Suppose the Ricci curvature ρ of Μ satisfies p(7(£)) > (m _ 1)# (0 <
t < ίο(7))· Then
a) 11—> 6(t, u)/s™~l(t), t \-► tm~lj(t, u)/s™_1(£) are monotone decreasing for
0<t< i0(7)·
b) e{t,4)<s™-\t),j{t,u)<s™-\t)/t™-' (0<f<i0(7))·
// equality holds in a) or b) for some t = Τ (< £o(7))> £Леп equality holds for
all 0 < t <T, and we may write Yi(t) = ss(t) Et(t) (i = 1, ... , m — 1), where the
E{(t) are parallel along 7. It follows that k(t) = K(t) = δ.
(3) Let Μ be an m-dimensional complete Riemannian manifold whose Ricci
curvature satisfies p(u) > (m - 1)δ for all и G UM for some δ > 0. Then ίο(7) <
π/νί, where n/y/δ is the first conjugate value of Λ/. In particular, Μ is compact
and d(M) < n/y/δ. Furthermore, applying this to the universal Riemannian cover
of M, we see that the fundamental group of Μ is finite (S. B. Myers theorem).
Corollary 3.2. Let Μ, Μ be m-dimensional complete Riemannian manifolds.
(1) Suppose Κσ > Κσ for arbitrary sectional curvatures Κσ of Μ and K& of
M. Let ρ G M. For 0 < r < ip(M), take a metric ball Br(p) in Μ and a metric ball
Br(p) in M. Then vol Br(p) < vol Br(p), and equality holds if and only if Br(p) is
isometric to Br(p).
(2) Suppose the Ricci curvatures of Μ satisfy p(u) > (m — 1)6 for any и G Л/.
Then for any 0 < r (< n/y/δ), where π/у/б is assumed to be +oc if δ < 0. we
have vol Br(p) < vr(6). Here vr(6) denotes the volume of a ball of radius r in the
m-dimensional complete simply connected Riemannian manifold Μ = Μ™ and is
independent of the center. If equality holds, then Br (p) is also of constant curvature.
In particular, if δ > 0, then vol Μ < vol S™, where equality holds if and only if Μ
is isometric to the sphere 5^ of constant curvature δ.
Proof. (1) Let / : TpM —> TpM be a linear isometry. Prom Theorem 3.1 we
get 0(t, u) > 0(t, u) (u G UPM, 0 < t < t0(u)). Then our inequality follows easily
from
vo\Br(p) = [ dt [ ()(t, u)dSm-\ vo\Br(p) < [ dt [ 0(t, u)^"1"1
Jo Js™-1 Jo Jsrn~1
(Chapter II, (5.10)). If equality holds, then we get ip(M) > r, and 6(t, u) = 6(t, u)
holds for any и G Up and 0 < t < r. Furthermore, we see that Φ := expp o/oexp"1 :
Br{p) -> Br{p) is a diffeomorphism, and Ζ)Φ(7(ί))7(0 = 4(0, ^^(7(0)^(0 =
Yi(t) (г = 1, ... , m - 1). Prom Remark 2.2 (I) it follows that Y-(t) (resp. Y{{t)) are
mutually orthogonal and ||Υί(£)|| = ||Vi(i)|| (г = 1, ... , m - 1). Therefore, Φ is an
isometry.
Finally we remark that (1) does not necessarily hold for any r > 0. In fact,
consider the sphere Μ of constant curvature 1 and the real projective space Μ of
constant cutvature 1, and take r > π/2.
(2) For ρ G Μ and и G UpM, let t(u) denote the distance from ρ to the cut
point of ρ along ηη (Chapter III, §4). Noting that t(u) < π/\/δ (which is assumed
to be +00 when δ < 0), for и G UpM, t > 0 we define
#(«,«):=<-'-" '"?!' МОЛ'""40' '-'^'
v ; Лп t>t(u), w l0, t>n/VS,

156
IV. COMPARISON THEOREMS AND APPLICATIONS
where we note that s™_1(0 is nothing but 0(t, u) corresponding to M™, and does
not depend on и £ UpM™. Then applying Proposition 2.1 (IV) (a) and noting
that Yi = ss(t)Ei(t), where E{(t) denotes the parallel vector field along 7/n with
Ei(0) = ei, we get for any t > 0
0(f, u) < w(t).
It follows that
volBr{p) = [ dt [ 0(i, u)dSm~l < [ dt [ w{t)dSm-1 =vr{S).
Jo Js™-1 Jo Js™-1
Next suppose equality holds. Then we have t(u) > r (u £ UPM), and 6(t, u) = w(£)
holds for any и £ UPM, 0 <t <r. Then, by Remark 2.2 for Proposition 2.1 (IV)
(a), we may write Yi(t) = ss(t)Ei(t). Therefore, defining Φ just as in (1), we get
ΌΦ(Ε{(ί)) = Ei(t), and again Φ is an isometry.
Finally, suppose δ > 0. Then from Μ = Βπ,^(ρ), we obviously have vol Μ <
vol5™. If equality holds, then we have t(u) = π/у/б for any и £ UPM and Φ :=
exppO/oexp"1 : Bn/^{p)(C M) -> Bn/^-6(p)(C SJ1) is an isometry. Further, along
any normal geodesic ηη emanating from p, any Jacobi field Υ(t) with Y(0) = 0 takes
a form ss(t)E(t) with parallel E(t), and π/y/δ is the first conjugate value along ηη
with multiplicity (m - 1). Therefore, for any и £ UPM we see that the (m - 1)-
dimensional subspace perpendicular to и at π/у/б · и in TpM gives the null space
of Dexpp(n/y/6 · u), and the boundary sphere of Βπ,^(ορ) in TPM is mapped
to a point q via expp. It follows that Μ is homeomorphic to a sphere. On the
other hand, Βπ,^(ρ) is of constant curvature δ and Μ = Βπ,^(ρ) U {#} is also of
constant curvature δ by continuity. D
M. Gromov has remarked that the above corollary may be extended in the
following form, which is useful in applications.
Theorem 3.3. Let Μ be an m-dimensional complete Riemannian manifold
and suppose p(u) > (m — 1)δ for any и £ UM. Then, for any 0 < r < R,
(3.1) vol ВR(p)/vol Br (ρ) < νΗ(δ)/ντ(δ).
Proof. First note that we may assume that r < π/y/δ if δ > 0. In fact,
otherwise both sides of (3.1) are equal to 1, because t(u) < π /y/δ for any и £ UM.
Next for 0 < s < r, r < t < R. applying case (IV) (a) of Proposition 2.1 (1), we see
that t ι-» s™~1(t)/0(t, u) is monotone increasing for 0 < t < t(u). Then, recalling
the definition of θ and w, we get for 0 < s < t
0(f, u)w(s) < 0(s, u)w(t).
Integrating both sides of the above inequality with respect to 0 < s < r and
r < t < R in order, we get
(3.2) J e(t,u)dt I I w(t)dt< J e(s,u)ds/ j w(s)ds.
Now note that Br{p) = {expptu; 0 < t < г (if t(u) > r), 0 < t < t(u) (if t(u) <
г), и £ UPM}. Then from (5.2) of Chapter II and the above (3.2), we get

3. APPLICATIONS OF COMPARISON THEOREMS
157
volBR{p) - volBr(p) = fsn-г dS™-1 frR0(t, u)dt
vR(6)-vr(6) a^ f* w(t)dt
= —^— / I [ 9{t,u)dt/ [ w(t)dt\ dSm-1
Oim-l JSm-1 [Jr Jr J
< —i— / ( / 0(s, u)da/ [ w{s)ds\ dSm-1
Oim-l Js™-1 [JO J0 )
= vol Br(p)/vr(6),
from which (3.1) follows easily. D
Corollary 3.4. Under the assumption of the theorem we have the following:
(1) ForO < η < r2 < r3
(volВгз(р)-volBr2(p))/(vr3(6)-vr2(6)) < volBri(p)/vri(6).
(2) Setd:= d(p,q). Then for η < d + r2
(vri{6)/vd+r2{S)) · volBr2(q) < volBri(p).
(3) If ri + r2 < d := d(p, q), then
vr2{S)/{vd+ri (δ) ~ Vd-ri (δ)} · volBri (p) < volВГ2(q).
Exercise 1. Give a proof of Corollary 3.4.
Theorem 3.5 (Cheng maximal diameter theorem). Let Μ be a complete
Tridimensional Riemannain manifold whose Ricci curvatures satisfy p(u) > (m — 1)6
everywhere for some δ > 0. Then Μ is compact, and d(M) < π/у/б. If the
diameter d(M) is equal to n/y/δ, then Μ is isometric to the sphere S™ of constant
curvature δ.
PROOF. We only need to check the case of equality. Take two points p, q e Μ
with d(p, q) = d(M) = π/уД. We set R = π/\/δ and choose 0 < r < R. Then
Br(p) Π BR-r(q) = φ. On the other hand, from (3.1) it follows that
vol Λ/ = volBR(p) vR{6)
[ ' ) volBr{p) vol Br{p) ~ vr{6)'
Therefore, we have vol£r(p) > ^Ul vol M, and the same argument applied to
q implies that vol£#_r(<7) > υ*~ή\ vol M. Then, noting that vR(6) = vr(6) +
vR-r(6) for the sphere 5^ and R = π/у/б, we get
vol Μ > vol Br(p) + vol ВR-r(q) > volM,
and so equality holds in (3.3), and Br(p) U BR-r(q) = M. Further recalling the
proof of Theorem 3.3, we see that if equality holds in (3.1) then equality also holds
in (3.2). If we consider the case where r = ro := π/2\/δ, then we get t(u) > n/2y/6
for any и е UPM U UqM. For any normal geodesic 7 emanating from p, we have
7(7*0) £ dBro(p) and there exists a unique normal geodesic 7' of length 7*0 joining
7(7*0) to q. Then 7 U 7' is a once broken geodesic, which is a shortest curve of
length R joining ρ to q and makes a straight angle at 7(7*0). Hence any normal

158
IV. COMPARISON THEOREMS AND APPLICATIONS
geodesic emanating from ρ passes through q at the parameter value R. It follows
that t(u) = R for any и G UPM (also for any и G UqM, by the same argument).
On the other hand, since equality holds in (3.2), we also have the equality sign
in Proposition 2.1 (1) (IV)(a), and we get 0(i, u) = s™_1(0 for any и G UPM, 0 <
t < R. Further, Jacobi fields У along ηη with У(0) = 0 may be written in the
form Y(t) = ss(t)E(t), where E(t) are parallel vector fields. Then taking a linear
isometry / : TpM —► TPS™, we see as before that Φ := expp о/ о exp"1 : Br(p) —»
Br(p) is an isometry. Now since 7u(i?) = q for any и G Z7PM, Φ may be extended to
a homeomorphism from Μ onto 5™. On the other hand, Br(p) and consequently
Μ = Br(p) U {#} is of constant curvature δ and Μ is isometric to the sphere, since
Μ is simply connected. D
Remark 3.6. In particular, if Μ is a complete Riemannian manifold whose
sectional curvatures satisfy Κσ > δ (> 0) everywhere and its diameter d(M) is
equal to π/\/δ, then Μ is isometric to the sphere S™ of constant curvature δ
(V. I. Toponogov's maximal diameter theorem [To-2]).
Next we give an estimate for the Laplacian of the distance function dp to ρ G Μ
in terms of the Ricci curvature as an application of a comparison theorem.
Proposition 3.7. Let Μ be an m-dimensional complete Riemannian manifold
and suppose the distance ball Br(p) centered at ρ is disjoint from the cut locus of
p. If the Ricci curvatures satisfy p(u) > (m — 1)δ evrywhere on Br(p), then
(3.4)
&dp(q) > -(m - 1) c6{d(p, q))/s6(d(p, q)) for q G Br(p) \ {p}.
PROOF. Prom the definition we get Adp(q) = — trace D2dp(q). We have a
unique normal minimal geodesic 7 : [0, I] —> Μ from ρ to q with / = d(p, q). Take
an o.n.b. {ei}^ of TqM so that em = 7(i) (= Vdp(q)) and note that em belongs to
the null space of D2dp(q) (Chapter III, Remark 4.11). Let Yi(t) (i = 1, ... , m — 1)
be Jacobi fields along 7 with Υί(0) = 0, Yi(l) = e^. Then the Yi are perpendicular
to 7 and we get from Chapter III, Lemma 4.10,
m-l
Adp(q) = -traceD2dp(q) = - ^ D2dp(q)(eu e»)
m-l
= -Σ(4Υί(1),Υί(1)).
On the other hand, setting и = 7(0), we see that t 1—► 9(t, u)/s^_1(i) (0 < t <
I) is monotone decreasing by Theorem 3.1 (2), and its logarithmic derivative is
nonpositive. It follows that
{iM,
0(t,u) < (m-l)c6(t)/a6{t).
For the above Yu ... , Ут_ь θ(ί, и) differs from \\Yi(t) Л · · · Л ym_i(i)|| only by a
constant factor. Since {Yi(l)} is an o.n.b. we get
d_' m~1
dt
^(^••.л^йЦ^^Щ^С))·
г=1

3. APPLICATIONS OF COMPARISON THEOREMS
159
Therefore, we have
m_1 ( d \ Ί
Σ (УВД, Yi(l)) = \jt\_ 0(t, u) I /9{t, u)<(m- l)c6(0/M0,
г=1 ^ It—Ζ )
from which (3.4) follows. D
3.2. Next we give a result due to E. Heintze and H. Karcher ([He-Ka]) which
generalizes Theorem 3.1 (M. Maeda ([Mae]) also got a similar result).
Theorem 3.8. (1) Let N be an η-dimensional submanifold of an m-dimen-
sional complete Riemannian manifold Μ. Let 7 := ηη be a geodesic emanating
from ρ £ N with the initial direction и £ UpN1-, and let to(N) denote the first
focal value of N along 7. Denote by η the mean curvature of N with respect to
the normal vector и to N, namely, η = Σ^ί/η, where the A* (1 < г < η) are
principal curvatures (i.e., eigenvalues of the shape operator Au of N). Suppose
k(t) >6(0<t <t0(N)). Then
IdetiDexp^ituW™—1 < (c6(t) + τ,ΜΟΓ^Γ""1^
[ ' } 0<t< t0(N).
(2) Let N be a hypersurface of Μ and 7 = 7n, и £ UpN1- (p e N) a normal
geodesic perpendicular to N. Suppose ρ{^{ί)) > (m - 1)6 (0 < t < to(N)) for the
Ricci curvatures. Then
(3.6) |det(Dexp-L(iu))| < (c6(t) + r/s6(i))m_1, 0 < t < t0{N).
Proof. Let Μ = Μ™ be the simply connected space form of constant
curvature <5, and fix ρ £ Μ й £ UpM. To see (1), we take an n-dimensional sub-
manifold N through ρ such that й is a normal vector to N and the shape
operator Au has Aj (1 < г < η) as principal curvatures. Now choose JV-Jacobi fields
{Уа}(1 < a < m - 1) along 7 so that the following holds: For 1 < г < η,
{A{ := Υΐ(0)}γ=ι forms an o.n.b. of TPN consisting of eigenvectors Л, of the
shape operator Au with eigenvalues Aj, and Bi := VYi(O) - ХгАг = 0. For
η + 1 < j < m - 1, we have Aj := Yj(0) = 0, and {Bj := УУ^О)}^^ forms an
o.n.b. of YpN-Ln^(0)±. Then {Ya}T=i forms a basis of J\b)· From Chapter II,
Lemma 4.8, we have Dexp±(t u)(Aa, tBa)y = Ya{t)~ and it follows that
|detDexp-L(iu)| = {{Dexp^it u)(Au 0).v Л · · · Л Dexp±(tu)(An, 0)N
Л Dexp±(tu)(0, £„+i)iV Λ · · · Λ Dexp±(t u)(0, Bm-i)N\\
= ||yi(0 л... л Yn(t) л yn+1(0 л... л Ym-i{t)\\/tm-n-1.
Now setting 7 = ju for Μ, we get by the same argument as above a basis {Уа}^1
for Jfj(l) in the following form: For 1 < г < η,
where the Ε1* (1 < г < η) are parallel along 7, and {E{(0)} forms an o.n.b. of TpN
consisting of eigenvectors 2^(0) of Ац with eigenvalues Aj. For n+l<j<m— 1,
y,(0 = βί(ί)£,·(0,

160
IV. COMPARISON THEOREMS AND APPLICATIONS
where Ej are parallel and {£^(0)} forms an o.n.b. for TpN1- D^O)1. Then the
assumption of Proposition 2.1 (III) is satisfied, and it follows that
||ϊΊ(0 л · · · лym_i(i)|| < \\Yi{t) л··· лym_i(i)||
η
= s?-n-1{t)Y[{cs(t) + \iss(t))
г=1
for 0 < t < to(N). Then (3.5) follows immediately from the above via the inequality
on the arithmetic and geometrical averages.
(2) follows from Proposition 2.1 (IV) by the same argument. D
Exercise 2. Give a proof of (2) in the above theorem.
Now we show that it is possible to give a lower bound for the injectivity radius
i(M) of a compact Riemannian manifold (M, g) in terms of upper and lower bounds
for the sectional curvatures, lower bound for the volume and upper bound for the
diameter. Such an estimate is originally due to J. Cheeger ([Ch-1]), and was later
improved by Heintze and Karcher ([He-Ka]) in the following form.
Theorem 3.9. Let Μ be a compact Riemannian manifold.
(1) Suppose Κσ > δ holds for any plane section σ ofTM, where δ is a constant.
Then for any nontrival simple7 closed geodesic с of Μ, we have
L{c) > 2π(νο1 M/am) · (s6(min(d(M), тт/^)))1"™,
where am = vol (5m, go), and by assumption π/2\/~δ = +oo if δ < 0.
(2) Suppose δ < Κσ < Δ holds everywhere for some constants <5, Δ. Then
i(M) > ηήη{π/\/Δ, 7r(volM/am) · (s6(min(d(M), тг/^)))1"™},
where π/y/A = +oo when Δ < 0, as before.
PROOF. We consider с as a one-dimensional totally geodesic submanifold of
M. For a unit vector и £ Τ cL normal to c, by Proposition 2.1 (III) the first
focal value of с along ηη cannot be greater than π/2\/δ, which is the first focal
value of a one dimensional totally geodesic submanifold in the corresponding M™.
Therefore, the maximal domain I of Tc1 containing the zero-section, on which
the normal exponential map exp-1 of с is a difFeomorphism, is contained in D :=
{ξ e Гсх; \\ξ\\ < I := min(d(M), π/2\/δ)}. On the other hand, denoting by G the
Sasaki metric on Τ с1- defined in Chapter II, (4.11), we see that тТс± : Tc1^ —> с is
a Riemannian submersion. By Theorem 3.7 (1) and the Fubini theorem we have
vol(M, g) = vol(exp-L I) < [ |detZ?expJ-(0|d^G
JD
= / dc / IdetDexp-^OI^GiD,
Jc JDS:=- ~
DnTc(s)c±
< [dc [ dt [ (c6(i)56(0m_1Am"2)im"2d5m-2(l)
Jc Jo J{xeDs;\\x\\=t}
= I dc J am-2C6(t)s?-2(t)dt = L(c)s6(l)m-lam-2/{m - 1)
= L{c)s6{l)m-lam/2^
7This means that с has no self-intersection.

4. TOPONOGOV'S COMPARISON THEOREM
161
where the last equality sign will follow from Exercise 3. Then (2) follows from
Chapter III, Corollary 4.14. D
Exercise 3. Let am denote the volume of the m-dimensional sphere Sm of
constant curvature 1. Show that
am = 2nam.2/(m - 1), am_iam = 2(2тг)т/(т - 1)!
Finally we note that as another application of comparison theorems it is possible
to estimate the principal curvatures of the parallel hypersurfaces Nt := {exp-11 u;
и G UN^} of a submanifold TV in a Riemannian manifold Μ (see Problem 5 at the
end of this chapter).
4. Toponogov's Comparison Theorem
Here we state an important comparison theorem which is due to V. I. To-
ponogov and is a global version of the Rauch comparison theorem. This theorem
provides a powerful tool to investigate the structure of Riemannian manifolds whose
curvature is bounded below, and plays an important role in Chapter V.
Definition 4.1. (1) A geodesic triangle Л(р\Р2Рз) of a Riemannian manifold
Μ is a figure consisting of three distinct points pi, p2, рз called the vertices and
three minimal geodesies Τ{ joiningp^+i to p*+2 (i = 1,2,3 (mod 3)) called the sides.
The angle between the tangent vectors to 7^-1 and 7"^ at pi is called the angle
of А(р\р2Ръ) at pi and denoted by a* = Z(pi-iPiPi+i) °r ^Pi (see Figure 18 in
§1). The perimeter / is defined as / = /1 + l2 + /3, where we set U = £(7*). Also,
in the definition of a geodesic triangle Л(р1р2рз), if two sides 72, 73 are minimal
geodesies and the side 71 is a (not necessarily minimal) geodesic segment with
/1 < l2 + /3 = d(pi, p3) + d(pi, P2)? then we call such a figure a generalized geodesic
triangle.
P^ > τ
Figure 21
(2) A geodesic hinge (p; 7, r) in Μ is a figure consisting of a point ρ G Μ called
the vertex and minimal geodesic segments 7, τ emanating from ρ called sides. We
denote by α the angle between the tangent vectors to 7 and τ at p, which is called
the angle of the geodesic hinge (p; 7, τ). If 7 is minimal but τ is not necessarily
minimal, we call (p; 7, r) a generalized geodesic hinge.
Theorem 4.2 (Toponogov comparison theorem). Let Μ be a complete
Riemannian manifold whose sectional curvatures satisfy Κσ > δ everywhere for some
consrant δ. Denote by M% the 2-dimensional complete simply connected
Riemannian manifold of constant curvature δ.

162
IV. COMPARISON THEOREMS AND APPLICATIONS
(1) (T.C.T. (I)) For a generalized geodesic triangle А(р\Р2Рз) suppose that
72, 7з are minimal and l\ = £(71) < π/\/δ.8 Then the perimeter I < 2n/y/6,
and there exists a geodesic triangle Δ(ρ\Ρ2Ί>ζ) in M$ with the same side lengths
L(7j) = L(7i) (г = 1, 2, 3) and satisfying a2 > a2, a3 > a3.
/// < 2n/y/6, the above А(р\р2Рз) is uniquly determined up to congruence {i.e.
isometry of M$). Further if there exists a geodesic triangle А(р\Р2Рз) of perimeter
2π/\/δ(δ > 0) in M, then Μ is isometric to the sphere 5™ of constant curvature
δ.
(2) (T.C.T.(II)) For a generalized geodesic hinge (p;7, r) suppose that L(t)
< n/y/δ. Let (p; 7, f) be a geodesic hinge in M% such that £(7) = £(7), L(f) =
L(t) and its angle a is equal to that of (p; 7, τ). Let q, q (resp., r, f) denote the
end points of η, η {resp., τ, f), respectively. Then d(q, r) < d(q, f).
Remark 4.3. (1) The assumption /1 < l2 + /3, *i < ττ/ν^ in T.C.T. (I)
guarantees that we may construct a geodesic triangle Л(р1^2Рз) in Af| with sides of
lengths ίχ, Ι2, h- Similarly, under the assumption L(r) < n/y/δ we may take a
minimal geodesic segment in M% emanating from ρ of length L(r), and
consequently construct a geodesic hinge (p; 7, f) with the same angle and side lengths
as (p: 7, τ). Note that under the assumption Κσ > <5, minimal geodesies in Μ are
of length less than or equal to π/у/б.
(2) T.C.T. (I), (II) are equivalent. First we assume (I) and let (p; 7, r) be
a generalized geodesic hinge. Setting / := L(r) (< n/y/δ) and Τ := sup{£0 G
(0, i]; T.C.T. (II) holds forO < t < f0}, it suffices to show Τ = I to verify (II). First
we show that Τ > 0. For t > 0 such that τ | [0, t] is minimal, we consider geodesic
triangles A(pqr(t)) in Μ and A(pqs) in M^ with the same side lengths. Then by
T.C.T (I) we may assume that Z(qps) < Z(qpr(t)). Since Z(qps) < Z(qpf(t)) holds
for triangles A(pqs), A(pqf(t)), we have d(q, f(t)) > d(q, s) = d(q, r(t)) according
to §1, Exercise 5, and therefore Τ > 0. Second, suppose Τ < I. Then we have
d(q, t(T)) = d(q, f(T)), and for geodesic triangles A(pqr(T)) in Μ and A(pqf(T))
in Ml, we get Zf(T) < Ζτ(Τ) from (I). Therefore, for any sufficiently small e > 0
and A{qr{T)r{T + e)), taking a geodesic triangle A{qf(T)s') in M]· with the same
side lengths, we have Z(qf(T)sf) < /.(ςτ(Τ)τ(Τ+ε)) < /.(qf{T)f(T+e)). It follows
that d(q, f(T + e)) > d(q, s') = d(q, τ(Τ Η- б)), which contradicts the definition of
T. Therefore Τ = I.
Conversely, suppose (II) holds and let А(р1р2рз) be a generalized geodesic
triangle. If its perimeter / is less than 2n/y/6, then the corresponding geodesic
triangle А(р1р2Рз) in M% is uniquely determined up to congruence. By the same
argument as above, from (II) and Exercise 5 of §1 it follows that c*2 > c*2, аз > аз-
If / = 2-к/у/Ь and U < n/y/δ (г = 1, 2, 3), we get in the same manner c*2 > 0.2 =
7Г, а3 > ά3 = π, and in particular a2 = a3 = π. If /1 or /2 is equal to π/у/б,
then Μ itself is isometric to S™ via Theorem 3.5 and our assertion clearly holds.
If / = 2π/\/δ and /3 = n/y/δ, then the corresponding geodesic triangle in M% is
a biangle and is not uniquely determined, but we may choose a biangle in Mj so
that the conditions on the angles are satisfied. Finally, it will be shown during the
proof of the theorem that the case I > 2п/уД cannot occur.
For the proof of the theorem the following lemma, due to H. Karcher, plays an
important role.
8In the following if 6 < 0 we assume π/ν^ = +οο.

4. TOPONOGOV'S COMPARISON THEOREM
163
Lemma 4.4. Let (p; 7, r) be a geodesic hinge such that 7, τ : [0, 1] —► Μ
are minimal geodesies. Set ρ = 7(0) = τ(0), q = 7(1), r = r(l) and consider the
geodesic triangle A(pqr). Let I be its perimeter and suppose 4e := 2n/y/6 — I > 0.
Suppose Κσ < Δ (Δ > 0) on a compact domain containing Bi/2(p), and take
a corresponding geodesic hinge (p\ 7, f) in M™ as in Т. С. Т. (II), and set q =
7(1), f = f(l). Then there exists к(б, <5, Δ) > 0 such that if d(p, r) < к(б, <5, Δ)
then d(q, r) < d(q, f), and T.C.T (II) holds in this case. Furthermore, we may
take κ = π/2\/Δ if6<0.
Proof. Let E(t) be the parallel translation of f{0)/\\f(0)|| along 7 in M6m.
We may consider a shortest geodesic с : [0, 1] —> Μ™ joining f to <?, which is
not necessarily parametrized proportionally to arc-length, but may be expressed as
c(t) = exp^(t) f(t)E(t), where / is smooth on [0, 1) and 0 < f(t) < π/уД. This
is clearly possible from the axiom of plane if δ < 0. and we see that in the case
δ > 0 it is again possible if we take к(б, <5, Δ) < e. In fact, since d(p, q) = d(p, q) <
1/2 = n/y/δ — e and d(q, f) < n/y/δ — €, d(p, f) < €, it follows that the perimeter
of A(pqf) is less than 2n/y/6 — 2e and all angles of the geodesic triangle A(pqf)
are less than π. Then by the axiom of plane с may be expressed in the above form
with 0 < f(t) < n/y/δ, and f(t) is smooth for 0 < t < 1. On the other hand, we
take a parallel vector field E(t) along τ in Μ with E(0) = /r(0)/||f(0)||, and note
that (E(t), 7(£)) = (E(t), 7(£))· Now we consider a curve с in Л/, which is defined
as
c(t):=explWf(t)E(t),
and joins r to q. Now suppose 0 < f(t) < π/2\/Δ. Then in Theorem 2.5 (2) the
Figure 22
assumption on the focal value is satisfied, and we get
d{q, f) = L(c) > L{c) > d{q, r)
and consequently T.C.T. (II) holds in our case. Therefore, it suffices to estimate
f(t). We only consider the case where δ > 0 and leave the easier case δ < 0 to the
reader as an exercise. Note that the angle α between E(t) and 7 is constant. Let
β denote the angle with vertex q between 7 and c, and set a(t) := d(q, c(t)). Then

164
IV. COMPARISON THEOREMS AND APPLICATIONS
from the Law of Sines we get
(4.1) sin VSfU) = ^— sin y/Sa(t).
sin α
We will show that f(t) is less than or equal to π/2\/Δ, if we take к sufficiently
small. First, suppose d(q, f) < π/2\/δ. In this case from (4.1) and the Law of Sines
we have
sin V6f(t) = y=— sin \if>d(p, r) < sin \ίδά(ρ, r)
sin y/6d(q, f)
< sin \Γδκ{ε, <5, Δ).
Noting that /(1) = 0 and f(t) < π/2>/Δ for t close to 1, we have f(t) < π/2>/Δ if
к is sufficiently small. Second, suppose π/2\/~δ < d(q, r) < π/у/б — e. In this case
differentiating both sides of (4.1), we have α(ί0) = π/2\/δ for t = to < 1, where
f(t) assumes the maximum. Therefore, again from the Law of Sines it follows that
sin/3 siny/6d(p, г) ът\Дж(е, <5, Δ)
sin
y/swo) = ^ = Muv;"^'^ <
sin α sin Vt)d(q, r) sin\/ie
If we choose к so that the last term here is less than sin(v^ · π/2\/Δ), we get
f{t0) < n/2y/A. □
Exercise 1. Suppose δ < 0. Show that if d(p, r) < π/2\/Δ, then f(t) <
π/2\/Δ.
In the above lemma, we note that the geodesic hinge (p; 7, f) is contained in
a totally geodesic submanifold M| of complete simply connected M™ of constant
curvature δ by the axiom of plane.
Remark 4.5. Suppose the angle α of the geodesic hinge (p; 7, r) in Lemma
4.4 satisfies 0 < α < π and d(q, r) = d(q, f), i.e., the equality sign holds in the
conclusion. Then we show that we may span a totally geodesic triangle A(pqr)
of constant curvature δ with 7, τ as two sides. We use the notation of Theorem
2.5 (2). In this case с is also a shortest curve joining r to q, and in the argument
of Theorem 2.5 we have equality in (2.12) for surfaces α, ά. Therefore we have
equality in the corresponding Proposition 2.1 (II). Then from Remark 2.2 we get
К
(да да\ _
and да/dt are Jacobi fields along geodesies at, where f^/Ц^Ц are parallel vector
fields along at. It follows that α defines a totally geodesic surface of constant
curvature δ which spans the geodesic triangle A(pqr) (see Remark 2.6). Further, ά
also spans a geodesic triangle Δ with vertices p, q, r in M™. Since the perimeter of
Δ is less than 2π/\/2\, it is uniquely determined up to congruence and contained in a
2-dimensional totally geodesic submanifold of M™. If we choose a linear isometry
/ : ΤξΜ —► TqM so that f(l), c(l) are mapped to f(l), c(l), respectively, then
expq Ι(βχρ^λ Δ) gives a desired geodesic triangle Л(р<7г).
Exercise 2. Show that in Remark 4.5 geodesies in Δ^τ) joining q to r{t)
are minimal geodesies in Μ.

4. TOPONOGOVS COMPARISON THEOREM
165
Exercise 3. Show that when equality holds in Remark 4.5, A(pqr) is an
embedded geodesic triangle.
Proof of Theorem 4.2. If δ > 0 and d(M) = π/y/δ, then Μ is isometric
to the sphere of constant curvature δ by the maximal diameter theorem, and the
asserton is clear. Therefore, in the following we may assume that d(M) < π/\/δ if
δ > 0. Let Δ(ριρ2ρ3) be a generalized geodesic triangle in M, and / its perimeter.
We prove T.C.T. (I) in the following three steps.
1° (Case where δ < 0, or δ > 0 and I < 2п/\Д). Set 4e := 2π/\/δ - i, which
is considered as +oo when δ < 0, and suppose that the sectional curvatures satisfy
Κσ < Δ everywhere on a compact subset С := Bi/2{q). Let к = к(б, <5, Δ) be a
positive number given in Lemma 4.4. Take a subdivision 0 = ίο < *i < - - - < ifc = 1
of [0, 1], so that for 71 : [0, 1] —» Μ we have d(ri,ri+i) < к (г = 0, ... , k - 1),
where we set rt = 7i(^)> r0 = p2, rfc = Рз·
Figure 23
For г = 1, ... , k, take minimal geodesies δι joining Τι to p\\ we show by
induction on г that the assertion of T.C.T. (I) holds for generalized geodesic
triangles А{тр\р2Т{). Note that Λ(ρχρ2Τι) are contained in С and the perimeters
/(i) satisfy /(j) ^ ^ because of the triangle inequality. We may easily check that
d{pi, P2) + d(pi, ri) < L(7i | [0, £;]). For г = 1, consider a geodesic triangle
^{PiP2^i) in M| with the same side lengths as the triangle Δ(ριΡ2Τ\). Applying
Lemma 4.4 to the geodesic hinges (p2; 7^"1, 71 | [0, ti}) and (n; <5b (7 | [0, £i])-1),
we compare them with the corresponding hinges in M| and A{piP2f\). Then from
§1, Exercise 5, we get
AP\V2Ti) > Z(pip2n), Z(pirip2) > Z(pifip2),
which completes the proof for г = 1. Next suppose the assertion holds for г - 1.
We consider the geodesic triangles А(р\Р2Гг-\), A{p\fi-\fi) in M| with the same
side lengths as ^(ριρ2π_ι), A(p\ri-iri), respectively. Then from the induction
hypothesis we get by the same argument as in г = 1
(4.2)
Z(pip2n_i) > Z(pip2f;_i), Ζ(ριη_ιρ2) > Ζ(ρι^_ιρ2),
Ζ(ριη_ιη) > Ζ(ριπ_ιη), Z(plriri-l) > Z(plfifi-i).
Then we get a quadrilateral (p\P2^i-\Ti) in M| as in Figure 23, which is a convex
quadrilateral. In fact, it is convex at the vertex f^-i (i.e., Z(p2fj_ifj) < π), since

166
IV. COMPARISON THEOREMS AND APPLICATIONS
by (4.2)
^(р2Гг-\Гг) = Z(j5ifi-ij52) + ^(Ρΐή-ΐή)
< Z(piri-ip2) + /(ριΓί-ΐΓ») = π.
It is convex at pi, because we have
d(p2, n-i) H- d(n-i, n) < Фь P2) + <*(рь ή)
from the triangle inequality. Now extending the side p2ri-i to a geodesic 71, let
f\ be a point on 71 proceeding from r^—1 by length d(fi-\, Г{). Then in M$, we
get d(pi, fi) < d(p\, г[) using §1 Exercise 5. Next, comparing Л(р1р2г[) with a
geodesic triangle A(p\p2r") in M| with the same side lengths as Л(р\р2Гг), we get
Z(piP2r;) = Z{pip2ri-1) > Z(pip2r<-i) = Z(pip2f·) > Z(pip2f").
Changing the role of the above two geodesic triangles, we also get
Ζ-(ρ\Τιρ2) > Z(pif"p2)
and the assertion also holds for г.
2° (Case where δ > 0 and / = 2π/\β). Note that d(pi, p2), d(pb p3) <
π/у/б, since we have assumed d(M) < n/y/δ. Now we have a point r := 71 (£1) on
71 such that
d(pb pa) + L(7i I [0, U]) = L(7i | [tu 1]) + d(pi, Рз) = π/ν^,
because / = 2π/\/δ. We may also assume that d(pi, r) < n/y/δ. Then, as
Figure 24
in Step 1°, we may easily check that d{p\, p2) + d(p\, r) > £(71 | [0, t\\), and
d(pi, Г) + ^(Рз> Pi) ^ ^(71 I [^ь 1])· Then we may take geodesic triangles A(p\p2f)
and Л(р1грз) in M| as in Figure 24 with the same side lengths as the
generalized geodesic triangles A(p\p2r), A(p\rp3) in M, respectively. Then from
1°, we have Z(pifp2) < Δ(ρ\τρ2), Z(pifp3) < Z(pirp3). We will show that
/-(p\rp2) + Z(pirp3) = π, and consequently
Z(pifp2) = Z(pirp2), Z(pifp3) = Z(pirp3).
In fact, otherwise we extend the minimal geodesic joining p2 to f beyond f, and
on it we take a point pf3 with d(p3, f) = d(p3, f). Because Z(pifps) < Z(pifp'3),
we have d(pi, pf3) > d(p\, p3). Then we get in M| a geodesic triangle A(p\p2p3)
whose perimeter is greater than 2π/\/δ, a contradiction. Therefore we get in M|
a geodesic triangle Δ(ρ\ρ2ρ3) with the same side lengths as A(p\p2p3) and such
that Zp2 < Zp2, Zp3 < Zp3. Now by the Law of Cosines in spherical trigonometry,

4. TOPONOGOV'S COMPARISON THEOREM
167
if Z/(7i) = d(p2, p3) < π/у/б, then the geodesic triangle ^(pip2p3) satisfies Zpi =
π (г = 2,3) and in fact is a great circle. Note that in this case we have also
Zp2 = Z-Ръ = π in A{p\piPz)· If -Ц71) = tt/v^, by the similar argument we
see that ^(pip2p3) satisfies Ζρι = π and is a biangle consisting of two half great
circles joining p2 and р$. Moreover, if £(71) < π/y/δ the above argument implies
d(pi, r) = d(pb f) = π/у/б. Therefore, if there exists a geodesic triangle in Μ
whose perimeter is equal to 2-к/у/б, then we have d(M) = π/у/б, and Μ is isometric
to the sphere of constant curvature 6.
3° (Case where б > 0 and / > 2-к/у/б). We show that this case does not occur.
Let r := 71 (t\) be the first point on 71 such that
d(pi, P2) + d(Pi, r) + L(7i I [0, h}) = 2п/у/б.
We may assume that d(pb P2), d(pi, r) < π/\/δ as before. On the other hand,
£(7i I [0, *i]) < £(71) < tt/V^ because / > 2-к/у/б. Therefore, applying Step 2° to
a generalized geodesic triangle А(р\р2г), we see that Zp2 = Zr = π, and рз lies on
a minimal geodesic joining r to pi. Then
2тт/л/« < d(pb pa) + Цъ) + <*(рь Рз)
= d(pi, pa) + L(7i | [0, ti]) + d(r, pi) = 2π/ν^.
which is a contradiction. D
Exercise 4. Let Л(р1Р2Рз) be a generalized geodesic triangle in Μ and
Л(р1Р2Рз) а corresponding geodesic triangle in M| with the same side lengths.
Take points η = 7ι(£ι), η = 7ι(£ι) on 7, 71, respectively, so that L(ji \ [0, fi]) =
-^(71 I [0» ^1])· Show that d(pi, n) > d(pi, n).
Remark 4.6. Suppose that in T.C.T. (II) the angle α satisfies 0 < α < π
and the perimeter / = £(7) + L(t) + d(g, r) is less than 2-к/у/б. Now we consider
the case where the equality d(q, r) = d(q, r) holds in T.C.T. (II). In this case we
show that we may span a totally geodesic triangle of constant curvature б such that
p, q, r are vertices and 7, τ are two sides. To see this, note first that we may assume
d(M) < π/у/б if б > 0. Take a point η = τ(ί0) on the geodesic segment τ joining
ρ to г and similarly a point η = f(£o), where (p; 7, f) is a corresponding geodesic
hinge in M|. Taking a minimal geodesic from ς to f, we get a geodesic triangle
A(pqr), which has the same side lengths as a generalized geodesic triangle A(pqr)
and is uniquely determined up to congruence because / < 2-к/у/б. Prom Exercise 4
we have d(q, n) > d(q, ή). On the other hand, applying T.C.T. (II) to the hinges
(ρ; τ | [0, ίο], 7), (ρ; τ I [0> *o], 7)? we get the reverse inequality. It follows that
d(q, n) = d(q, n) and Z(qnr) = Z(qnr) by a similar argument. Now A{pqr) and
A(pqr) have the same side lengths, and we may apply the argument of case 1° of the
proof of T.C.T. (I). For subdividing points Г{ of τ take the corresponding points fi of
f. Then from the above we get d(q, Г{) = d(q, Г{). For г = 1, from the case where
equality holds in Lemma 4.4 (Remark 4.5) it follows that we may span A(qpri)
with a totally geodesic surface of constant curvature <5, and get Z(rr\q) = Z(rr\q).
Next, comparing the geodesic triangle A(qrir2) in Μ with the geodesic triangle
^(^1^2) in M| and noting that d(q, r2) = d(q, f2), we see that we may span
A(qr\r2) with a totally geodesic triangle of constant curvature б via Remark 4.5.
Further, the parallel translation X(t) of —τ(ί\) along a minimal geodesic σγ joining
q to Γχ in A(qpri) and the parallel translation Y(t) of f(ti) along σγ in A(qr\r2)

168
IV. COMPARISON THEOREMS AND APPLICATIONS
satisfy X(t) = —Y(t), since these geodesic triangles are totally geodesic. Therefore,
A(qpr\) and A(qr\r2) are joined smoothly along σ\. Now repeating this process
successively for г = 2, 3, ... , к we may construct a totally geodesic surface of
constant curvature δ which spans A{rpq). Finally we remark that the above totally
geodesic surface is embedded if τ is a minimal geodesic (see Exercise 3).
5. Convexity
As in Euclidean geometry, the concept of convexity (e.g., convex set and convex
function) plays an important role also in Riemannian geometry. Since in a Riemann-
ian manifold Μ geodesies joining two given points are not necessarily unique, the
situation is somewhat complicated.
Definition 5.1. Let С (φ φ) be a subset of a Riemannian manifold M.
(1) С is said to be strongly convex, if for any points p, q £ Μ there exists a
unique normal minimal geodesic 7 joining ρ to q in Μ, and 7 is contained in C.
(2) If for any point ρ £ С, the closure of C, there exists an e(p) > 0 such that
С П Д:(р)(р) is strongly convex, then С is said to be locally convex.
(3) С is said to be totally convex, if for any points p, q £ С all geodesies joining
ρ to q in Μ are contained in C.
As is seen in the following Theorem 5.3, any metric ball in Μ with sufficiently
small radius is strongly convex. However, for large radii the convexity of metric
balls generally is not guaranteed. By definition, if С is strongly convex or totally
convex, then it is locally convex. If С is locally convex, so is С We will see later
that total convexity is related to the global properties of M, and the structure of
locally convex sets may be analyzed in detail (see Theorem 5.5).
Exercise 1. Let (52, go) be the unit sphere in R3 and N the north pole. For
metric balls Br(N), prove the following facts:
(i) Br{N), Br{N) are strongly convex for 0 < r < π/2, but Br(N) is not
locally convex for π > r > π/2.
(ii) Βπ/2(Ν) is strongly convex. Βπ/2(Ν) is not strongly convex but is locally
convex.
(iii) Br(N) is totally convex if ond only if r = π (i.e., Br(N) = S2).
Remark 5.2. (1) Suppose Μ is complete and for any two points of Μ there
exists a unique normal geodesic joining them. Then the three conditions in
Definition 5.1 coincide.
(2) The intersection of a family of strongly convex sets in Μ is again strongly
convex if it is nonempty. The same fact also holds for totally convex sets in a
complete Riemannian manifold. If the intersection of finitely many locally convex
sets of Μ is not empty, then it is locally convex.
Exercise 2. Prove Remark 5.2.
First we see that metric balls centered at any point ρ in a Riemannian manifold
Μ with sufficiently small radii are strongly convex. We set
(5.1)
r(p) := sup{r > 0; any metric ball contained in Br(p) is strongly
convex and any geodesic segment contained in Br(p)
is a minimal geodesic joining its end points},
and call it the convexity radius at p.

5. CONVEXITY
169
Theorem 5.3. 0 < r(p) < +oo for any ρ e M, and ρ £ Μ н-> r(p) £ R+ U
{+00} is continuous. Further if r(p) = +00 holds at some ρ £ Μ, tfien r(q) = +00
/or euen/ point q £ Μ.
Proof. We first show that r(p) > 0 for any ρ £ Μ. Let гр = гр(М) be the
injectivity radius at p. For 0 < R < ip, take a compact set A := Br(p) and set
г := πήη{ζς; q £ Л},
if := тах{Ка; σ С Х^М (ς £ Л) is a 2-dimensional subspace}.
Then note that г > 0 since the injectivity radius function is continuous. Now
for 0 < r0 < min{z/2, π/2\/Κ, R}, where we take π/2\/Κ = +oo if К < 0, we
show that Bro(p)(c A) satisfies the desired properties in (5.1). First, note that
for any two points q\, q2 in Bro(p) we have d(q\, q2) < 2r0 < г. Therefore, any
normal geodesic segment in Bro(p) emanating from q\ is contained in Bi(q\), and
is a unique normal minimal geodesic joining q\ to the end point because г < iqi.
Second, to see that any metric ball B€(q) contained in Bro(p) is strongly convex, it
suffices to show that for any 0b q2 £ B€(q) there exists a unique normal minimal
geodesic in Μ joining qY to q2 and contained in B€(q). Uniqueness may be proved
as above. As for the existence, we set V := {(<7i, q2) £ B€(q) x B€(q); a minimal
normal geodesic segment joining <7i to q2 is contained in B€(q)}. We show that
V = Be(q) x Be(q). Note that for (qu q2) £ Be(q) x Be(q) we get d(qu q2) < г and
minimal geodesies joining <7i to q2 depend continuously on q\, q2. Therefore, V is
open in B€(q) x Be(q). Next let (<7i, q2) belong to the closure of V in B€(q) x B€(q).
and take a (unique) normal minimal geodesic 7 : [0, /] —► Μ joining <7i to q2. л
is the limit of a sequence of normal minimal geodesies 7^ (k = 1, 2, ...) in Be(q)
joining 0i to ^2 , where {(0^ , q2 )} is a sequence of points in V converging
to (01, 02). Then 7 is contained in B€(q). Suppose 7 is not contained in Be(q).
Then a function [0, /] Э t \—> d(q, ^(t)) £ R+ assumes its maximum e at some
0 < to < I, and a minimal geodesic cto joining q to 7(^0) is perpendicular to ~ at
7(^0)· We consider a variation {ct} of cto consisting of minimal geodesies joining q
to 7(£). Let У(£) be its variation vector field, which is a Jacobi field along cto with
y(0) = 0, У(б) = 7(£o)· Note that У(£) is perpendicular to cto even^'here. Then
since € < Го < г/2, п/2у/К, we get from Lemma 2.9
£>2dg(7(<o))(7(io), 7(«o)) - (W(e). У(е)> > сА-(е)/5л(е) > 0,
which contradicts the fact that t ·—► 6?ς(7(ί)) assumes the maximum at t = £0-
Therefore 7 С B€(q), and У is a closed set in B€(q) χ Be(q). Since clearly V φ φ
and Β€(ς) x Be(0) is connected, it follows that V = B€(q) χ B€(q) and consequently
r(p) > r0 (> 0). Next, from the definition we have r(q) > r(p) - d(p, 0), and the
continuity of r follows. It is also trivial from the above inequality that if r(p) = +00
at some ρ £ Μ, then r = +00. D
Now let С be a connected locally convex set of Λ/, and let us consider the
structure of C. In the definition of local convexity we may assume that e(p) < r(p).
We consider (embedded) submanifolds of Μ that are contained in C. For instance,
any point of С is such a submanifold. Let A;(0<A;<m)bethe maximal
dimension of such submanifolds. We consider the family {NQ}aeA of all fc-dimensional
submanifolds of Μ that are contained in C, and set N := \JaeA NQ(C C). First
we show that N itself is an embedded submanifold of M. Let ρ £ N. Take an

170
IV. COMPARISON THEOREMS AND APPLICATIONS
NQ containing p, a coordinate neighbrhood U (C B€(p)/2(p)) of ρ in NQ, and a C°°
chart φ : U —> Rk'. Next take 0 < <5 < e(p)/2 so that the normal exponential
map expx | Ns{U) : Ns(U) —*· Μ is a difFeomorphism onto an open subset Т$([/)
in M, where we set N6(U) := {v G TqN^', \\v\\ < δ, q e U}. If we can show
that T$([/) Π Ν = U, then /7 is an open subset of N with respect to the relative
topology and gives a coordinate neighborhood. Suppose that there exists a point
qi = exp1- ν G Ts(U) Π Ν, ν € TqNa, that does not belong to U. The minimal
geodesic joining q\ to q is perpendicular to U. Therefore, taking a sufficiently small
neighborhood U' С U of ς, we see that the set of all minimal geodesic segments
joining elements of U' to q\ forms a (A;+l)-dimensional submanifold contained in C.
This contradicts the definition of k, and the family of all (£/, φ) of the above form
defines a C°° atlas for N. Namely, N is an embedded A;-dimensional submanifold of
Μ contained in С Furthermore, for the above ρ and /7, minimal geodesies joining
ρ to q G U are contained in С Π Bc(p), and are in fact contained in N by the same
argument as above. Therefore, N is totally geodesic.
Now we consider the closure N of N. The following lemma is useful.
Lemma 5.4. Let С be a connected locally convex subset and В := Вф)(р)
a metric ball centered at ρ G С Π N. Take q G Β Π TV, qi G В П С, and set
e = d(q, q\). Then for a minimal normal geodesic 7 : [0, e] —> Μ joining q to q\
we have 7QO, e)) С Ν, and in particular qY G N. Further suppose that q\ £ N.
Then for €0 (> e) such that 7 | [0, €0] is contained in £e(p), we have 7(e) £ С for
se (б, £o].
В
N
Figure 25
Proof. For e < s < €0 we set ^ := 7(s). Take a hypersurface W (С В) in TV
that passes through q and is transversal to 7 at (7. From the strong convexity of
Be(p) (р)ПС, it follows that minimal geodesies joining q' to points in W are contained
in C. If we take W sufficiently small, then for U := {£a; G Tq>M\ expq, χ G
lV,||x||<€o?0<i<l}, exp^z f/ is a smooth A;-dimensional submanifold of M
contained in C. Therefore 7([0, s)) С N. Setting s = £, q' = qu we get the
first assertion. As for the last assertion, note that if 7(s) e С (s e (б, £о]), then
<7i = 7(e) G N4 which is a contradiction. D
Theorem 5.5. Let С be a connected locally convex closed subset of a Riemann-
ian manifold M. Then there exists a connected (embedded) k-dimensional totally
geodesic submanifold N of Μ such that С = N.

5. CONVEXITY
171
Proof. If we take the above N for a given C, then N (с С) is a closed subset
of C. We show that N is an open subset of С with respect to the relative topology
of C. Let ρ G AT. Then any point ς G Pe(p)(p) Π С is contained in TV by the previous
lemma, and N is an open subset of С Since Ν φ φ and С is connected, we get
N = C. Finally we see that N is connected. Because С is connected and locally
convex, for p, q G TV we may join ρ, ς by a broken geodesic с in C, whose corners
will be denoted by ρ = po, pi, ... , pk = q. By Lemma 5.4, a minimal geodesic 71,
which is an arc of с joining ρ to pi, is contained in N. except possibly for ρχ. Again
by Lemma 5.4, by taking a minimal geodesic joining 7i(l - δ) to p2, where δ > 0
is sufficiently small, we get a broken geodesic joining ρ to p<i which is contained in
TV, except possibly for p2. Repeating this process, we get a broken geodesic joining
ρ to q which is contained in TV, and N is connected. D
Remark 5.6. Under the assumption of Theorem 5.5. it is possible to show
that С = N is a A;-dimensional topological manifold with boundary (Problem 8 in
Chapter IV). However, dC := C\N is not necessarily a smooth manifold (consider,
e.g., convex polygons in the plane). We call dC the boundary of a convex set C,9
and N the interior of C.
Now let С be a closed connected locally convex subset, and for ρ G С define
the tangent cone C(p) as
C(p) := {v G TPM \ {op}; ехрр£г>/||г>|| is contained in Л"
^ ' ' for some t G (0, e(p))} U {op}.
By the definition, tv e C(p) if ν G C(p) and £ > 0. We set C(p) := (С(р))д.
the subspace of TPM generated by C(p). If ρ G N, obviously C(p) = C(p) =
Tp7V. We consider the case where ρ G <9C. Take q e Ν Π Бе(р)(р) and a minimal
geodesic 7 joining ρ to q. For г> G C(p) \ {op}, c(s) := exppsv is contained in
TV for some s > 0, and therefore contained in TV for all sufficiently small s > 0
because of Lemma 5.4. Now the parallel translation ws of c(s) G TC^N along a
minimal geodesic joining c(s) to ς is contained in TqN', since TV is totally geodesic.
Therefore, Ρ(7)1; = lims_^0^s also belongs to TqN. Namely, P(7)C(P) С Х^ЛТ and
C(p) С P(7~1)(T<77V). Similarly, for ρ and points r in an open neighborhood (in
N) of the above c(s) G N, the set of parallel translations along 7 of the initial
directions tangent to minimal geodesies joining ρ to r forms an open neighborhood
of Р(7)г> in TqN and contained in P(7)C(P). Namely, C(p) \ {op} is an open subset
of C(p). Then, noting that dim C(p) = dim С(р) = dim TV, we get
(5.3) 0(ρ) = Ρ{Ί-λ)Τ4Ν.
Further, by Lemma 5.4 we have 7(0) G C(p), -7(0) £ C(p), and it follows that
Cip) с C(p) (p e 0C).
Lemma 5.7. Lei С be a connected locally convex closed subset. For ρ G dC
suppose that there exist a q G N and a normal minimal geodesic 7 : [0, /] —> С
joining q to ρ with I (:= £(7)) = d(q, dC). Then C(p) \ {op} coincides with an open
half-space Η := {ν G C(p) \ {op}; Z(v, -7(0) < ττ/2}·
9dC may be empty; in this case С is a totally geodesic submanifold of M.

172
IV. COMPARISON THEOREMS AND APPLICATIONS
Proof. Choose 0 < s < I with d(p, 7(3)) < e(p)/2. Then 7 | [s, I] is a
minimal geodesic joining 7(s) to dC, and Z?/_s(7(s)) Π 9C = {p}. We denote
by vr (s < r < I) the parallel translation of ν £ Я, ||г>|| = 1 along 7-1 to 7(r).
Then we get vr £ T7(r)7V. Since г> £ Я, there exists an € > 0 such that the geodesic
emanating from 7(7*) with the initial direction vr is contained in B/_s(7(s)) up to the
parameter value e by the first variation formula, and therefore contained in N. Then
exppev £ B/_s(7(s))n7V, and in fact exppev £ TV, because B/_s(7(s))ndC = {p}.
It follows that Я С C(p). Next we derive a contradiction assuming ν £ Η and
ν £ C(p) \ {op}. Since C(p) \ {op} is an open subset, we may assume that the above
ν satisfies Z(v, -7(0) > π/2 from the beginning. Then since —v £ Я С C(p) \ {op}
for a geodesic 7^ with the initial direction г>, we get 7t,(-e) £ AT for sufficiently
small €. Applying Lemma 5.4 to ην \ [—6,0], we see that ηυ is not contained in N
beyond p. This contradicts ν £ C(p) \ {0P}, and the proof is complete. D
In general, an open half-space Я of С(р) (p £ dC) is said to be a supporting
half-space of С at ρ if C(p) С Я. In the case of Lemma 5.7, a supporting half-space
of С at ρ is uniquely determined. On the other hand, for an arbitrary ρ £ dC
and any e > 0, take a point q £ N with d(p, q) < e/2. Then for p' £ dC with
d(<7, dC) = d(q, p'), a minimal geodesic 7 joining ς to p' satisfies the assumption
of Lemma 5.7 and d(p', p) < e. Therefore, the set of points of dC that admit a
unique supporting half-space is dense in dC. Then, by a limiting argument, for any
ρ £ dC we have a supporting half-space of С at p. Further, we may verify that for
ρ £ дС. С (ρ) \ {ορ} coincides with the intersection of all supporting half-spaces of
С at ρ (see Problem 9 for Chapter IV).
Proposition 5.8. Let Μ be a complete Riemannian manifold and С a compact
totally convex subset with dC = φ. Then the inclusion map С ^-> Μ is a homotopy
equivalence.
Sketch of Proof By Theorem 5.5, С is a compact totally geodesic submanifold
of M. As was seen in Chapter III, §3, the space Μ := Cq~xC = {c : [0, 1] —>
M; piecewise C°° curve with c(0), c(l) £ С andE(c) < a2/2} is homotopy
equivalent to a finite-dimensional manifold Μ(Δ) consisting of broken geodesies.
Critical points of Ε | Μ (A) are geodesies of Μ joining points of С which are
perpendicular to С at end points. Since С is totally convex, they are trivial point
curves consisting of points of C, and will be identified with С itself.
Therefore, by considering the flow generated by — VE in Ai(A) we get a strong
deformation retract of M(A) onto a tubular neighborhood U of C, where С is a
strong deformation retract of U. Now let / : (Dk, dDk) —> (M, C) be a piece-
wise C°° map, where Dk denotes the fc-dimensional unit interval of Rk.
Regarding Dk = Dk~l χ [0, 1], we set f{x){t) = f(x, t), and get a continuous map
/' : (Dk~1, dDk~l) —> (M, C) for some a > 0, which is homotopic to a map
f" : Z)fc_1 —> С Since the reverse process is possible, for the relative homotopy
groups we get nk(M, C) 9* 7rfc_i((Ja>0 Μ С) £* π^Ο^χ) ^(Δ)» С) = 0{к> 1).
Then our assertion follows from results of algebraic topology. D
Next we shall consider convex functions.
Definition 5.9. A real-valued function / defined on a complete Riemannian
manifold Μ is said to be a convex function if / is convex when restricted to any
geodesic 7 of M, which means that / ο η[ία Η- (1 - t)b) < t /(7(a)) + (1 - t)f(^(b))

5. CONVEXITY
173
holds for any a, b £ R and 0 < t < 1. Next, / is said to be a strongly convex function
if for any compact subset К of Μ there exists a <5 > 0 such that for any normal
geodesic 7 emanating from a point ρ £ К we have f("y(t)) + /(7(—£)) ~~ 2/(p) > <5£2
for 0 < £ < <5. Strongly convex functions are convex.
Exercise 3. Show that for a C°° convex (resp., strongly convex) function /,
its Hessian D2 f is positive semidefinite (resp., positive definite) at every point of
M. Show that, for a convex function /, /_1(-oc, a] and /_1(-oo, a) are totally
convex subsets for any a £ R.
It is known that convex functions are locally Lipschitz continuous and in
particular continuous. Now we give an example showing that the existence of a convex
function imposes a restriction on the manifold structure.
Proposition 5.10. Let Μ be an m-dimensional complete Riemannian
manifold. Suppose there exists a strongly convex C^ function f on Μ such that
/-1((—oc, i\) is compact for any t > 0. Then Μ is diffeomorphic to Rm.
Proof. We may assume that m > 2. First, note that / assumes its minimum
since /-1(—00, t] (t £ R) are compact. Suppose that there exist different points
Pi, V2 at which / takes its minimum. Then / assumes its minimum at all points
of the geodesic segments joining p\ to p2. which contradicts the strong convexity.
Therefore, / attains its minimum at a unique point p. Next, note that / assumes
its minimum at a critical point q of /. In fact, for any geodesic ~y emanating from
q, convexity implies that ^f(j(t)) > 0. Summing up, / has a unique critical point
ρ at which / assumes its minimum. We may assume that /(p) = 0.
Second, by the strong convexity the Hessian D2f(p) of / at ρ is positive definite
and ρ is a nondegenerate critical point. By the Morse lemma, we may choose a chart
(£/, ф, хг) around ρ so that φ(ρ) = ο and / may be written as f(q) = ^{xl(q))2 for
q £ U. Now choose an e > 0 such that B€(o) С <t>(U) and take a Riemannian metric
/ion Μ that satisfies h(d/dx\ d/dxj) = % on В := {q £ U; \\</>{q)\\ < e/2}. In
the following we use this Riemannian metric. On the other hand, we set S := {u £
Rm, \\u\\ = e/2} and define a diffeomorphism Fi : 5 x (0, +00) -> Rm \ {0} by
Fi(u, t) := Vt/\\u\\ · и = 2yft/e · u.
Third, we consider a C°° vector field X := V//||V/||2 on Μ \ {p} and let
ψι be the flow generated by X. For q £ Μ \ {p}, t i-> i/Jt(q) is an integral curve
of X, and from ft f(ipt(q)) = (V/, X) = 1 it follows that /(Vi(tf)) = * + f(o)-
Therefore, as / is proper, ^t{q) is defined for t £ (a, +00), where a = —f(q) and
limtia tptiq) = P- Then Μ \ φ[β) is divided into two connected components, of
which one is a bounded set containing ρ and the other is unbounded. We remark
that for the above q there exists a β £ (α, +οο) such that ipe(q) € 0(5). Now we
define F2: S x (0, +00) -> Μ \ {p} by F2(u, i) := ^-еу^Ф'1^))- Note that on
В we have X = ^2(хг/(2^2(хг)2))д/дхг. Integral curves of Χ \ Β are the images
of rays emanating from the origin under φ-1 and are parametrized by Σ(χ1)2- We
get limt|_e2/4^(g) = ρ for q = ф~1(и), и £ 5, and F2(u, £) = ф~1(2уД/е · u) for
0 < t < €2/4. Therefore, F2 is well defined and a C°° map. F2 is injective by virtue
of a property of integral curves of vector fields, and also surjective by the above
remark. On the other hand, X is transversal to ^(0-1(5)), and it follows that the
rank of DF2 is equal to m. Namely, F2 is a diffeomorphism. Finally we define a
map F: Rm -> Μ by
F(o):=p, F|(Hm\M) = F2oFf1.

174
IV. COMPARISON THEOREMS AND APPLICATIONS
Then F is a bijective map and F \ R171 \ {0} is a diffeomorphism onto Μ \ {p}. It
suffices to check that F and F~l are C°° on neighborhoods of 0 and p, respectively.
To see this, note that Fi(u, t) = 2y/i/e · u, and F2(u, t) = ф~1(2\/1/е · u) for
0 < t < 62/4. Then we get 0 о F(x) = χ for χ £ В, and our assertion clearly
holds. D
Remark 5.11. The above result is due to R. Greene and H. Wu ([Gre-Wu-1]).
In the case of Euclidean space, any convex function / : R171 —» R may be
approximated by C°° convex functions. For a general complete Riemannian manifold Μ,
it is known that for any strongly convex function / and for any e > 0 there exists
a strongly convex C°° function /0 such that
sup{|/(p)-/0(p)|;p£M}<e.
Therefore, Proposition 5.10 holds without assuming C°° regularity of /. On the
other hand, in general it is not known whether convex functions on complete
Riemannian manifolds may be approximated by convex C°° functions in the above
sense. For more details on convex functions and the topology of manifolds
admitting convex functions, see [S-2].
Next we will treat Busemann functions. Let Μ be a complete open (i.e., non-
compact) Riemannian manifold with dim Μ > 2. A geodesic 7 : [0, 00) —» Μ
emanating from ρ parametrized by arc-length is called a ray emanating from ρ if
d{"f{t), 7(s)) = I* ~~ sl f°r an* t, s > 0. Similarly, a normal geodesic 7 : R —> Μ is
called a line if any arc of 7 is a minimal geodesic segment joining end points. Since
Μ is noncompact, for any ρ ζ Μ there exists a ray emanating from p. In fact,
choose a sequence {pn} (n = 1, 2, ...) of points such that d(p, pn) —> +00 and take
minimal geodesies ηη parametrized by arc-length joining ρ to pn. Let и £ UPM
be an accumulation vector of a sequence {7n(0)} С UPM. Then we may easily see
that the geodesic 7 = ηη with the initial direction и gives a ray emanating from p.
Now for a ray 7 in Μ, we define the Busemann function b1 as
(5.4) by(q):= lim (t - d(g, 7(t))).
Note that £ н-> t-d(q, *y(t)) is monotone increasing because of the triangle inequality,
and bounded since t - d(q, 7(f)) = d(p, 7(£)) - d(<7, 7(f)) < d(p, ς). Therefore, as
t —> +00, £ - d(<7, 7(£)) is uniformly bounded on any compact set К of Μ and
uniformly converges to 67 on K.
Moreover, b1 is Lipschitz continuous on M, since for q\, q<i £ Μ we have
IMfli) " МЫ1 < t Hm^ |%i, 7(0) " d(02, 7(0)1 < %ь Ы·
For a given ray 7 and q £ M, take a sequence rn —> +00 (n —> +00) and normal
minimal geodesies joining q to 7(rn). If {7n(0)} С UqM converges to и £ UqM,
taking a subsequence if necessary, then ηη is a ray emanating from ς, which is called
an asymptote of 7. In general, there may exist many asymptotes of 7 emanating
from q.
In the next chapter, the convexity of Busemann functions will play an important
role in studying the structure of complete open manifolds whose curvatures are of
fixed sign. Sometimes we also consider
(5.5) b-(q):=t]imjd(q,7(t))-t),
which differs from 67 in the sign.

6. SYMMETRIC SPACES
175
6. Symmetric Spaces
6.1. In §1 we treated Riemannian manifolds of constant curvature, which most
standard Riemannian manifolds are. E. Cartan considered a class of Riemannian
manifolds whose sectional curvatures Κσ remain constant when plane sections σ
are parallel translated along curves on the manifolds. These Riemannian manifolds
carry rich geometric structures, which may be studied in detail using the theory of
Lie groups and Lie algebras. We begin with a definition.
For a point ρ in a (connected) Riemannian manifold M, an isometry sp of Μ
which fixes ρ and satisfies Dsp(p) = — id^M is called a geodesic symmetry at p.
Then, for a geodesic 7 emanating from p, sp о 7 is also a geodesic emanating from
ρ and we get sp(7(£)) = 7(—t). Therefore, ρ is an isolated fixed point of sp and we
get Sp = idM because of Chapter II, §3, Exercise 2.
Definition 6.1. A Riemannian manifold Μ is called a (Riemannian)
symmetric space, if for any ρ £ Μ there exists a geodesic symmetry at p.
Exercise 1. Show that ilm, 5m, H™ with the canonical Riemannian
structure are symmetric spaces.
Let Μ be a symmetric space. Then for a geodesic 7 emanating from ρ we get
s7(a)7(£) = 7(2a — t). For t £ R define
(6.1) Pi(7) := S7(i/2) OS7(0)>
which is an element of G := I$(M). Clearly Pt{l)(p) = 7(£).
Lemma 6.2. Let Μ be a symmetric space. Then we have the following'.
(1) Μ is complete and ps(l)(l(t)) = 7(t + s) for any geodesic 7 in M.
(2) Όρ8(η): ΤΊμ)Μ —»T7(t+s)M coincides with the parallel translation P/+s(7)
along 7.
(3) G = Io(M) acts transitively on M, namely, A/ is a homogeneous
Riemannian manifold.
(4) For a geodesic 7, t £ R 1—► Pt(l) £ G defines a one parameter subgroup of
the Lie group G. Therefore, the geodesic 7 is given as an orbit of the one parameter
subgroup pt (7) ·
PROOF. (1) Suppose a geodesic 7 is defined on [0, o]. Then the geodesic
t £ [0, a] —» pa (7) (7(0) passes through 7(a) at t — 0, and its tangent vector at
t = 0 is given by Dpa(7)(7(0)) = -DsMa/2)(7(0)) = -f |e=o 7(<* - *) = 7И-
Therefore 7 may be extended to [0, 2a]. Repeating this process, we see that *y(t)
is defined for all t > 0 and Λ/ is complete. It is also clear from the above that
Ρα(7)(7(*))=7(* + α).
Next we show (2). For a parallel vector field X(t) along 7(2), 11-> Z?s7(a)X(£) is
parallel along a geodesic £ 1—> *y(2a — t), and we get Z?s7(a)(X(a)) = —X(a) &t t = a.
Therefore Ds^a)X(t) = —X(2a - t), and applying this to ps(7) = s7(s/2) ° s7(o)>
we get Dps(j)X(t) = X(t + s), which proves (2).
To see (3), for any two points p, q of Μ take a normal minimal geodesic 7 :
[0, I] —> Μ joining them. Then pi(j)p = q, and we have an isometry /7/(7) which
maps ρ to q.
Finally, we show that pi+s(7) = pt{l) °Ps{l)· Both sides are elements of
G and map ρ = 7(0) to η(ί Η- s). On the other hand, both Όρί+8(η)(ρ), and

176
IV. COMPARISON THEOREMS AND APPLICATIONS
Dpsii) ο Όρι{η/)(ρ) coincide with P(7)?+s. Therefore, pt+sb) = Pt{l) ° Psb) by
virtue of Exercise 2 from Chapter II, §3. D
We call a one parameter subgroup t i-> pt{l) determined by a geodesic 7 as
above a transvection. For χ G TPM, the transvection determined by the geodesic ηχ
is also denoted by pt{x)· The Killing vector field X determined by a one parameter
subgroup pt (x) defines an element of the Lie algebra of G.
Proposition 6.3. (1) Let Μ be a symmetric space. Then the curvature
tensor R of Μ is parallel, namely, VR = 0.
(2) For a Riemannian manifold M, VR = 0 holds if and only if the sectional
curvatures Κσ remain constant when we parallel translate plane sections σ along
any curve in M.
(3) For a Riemannian manifold Μ, VR = 0 holds if and only if a geodesic
symmetry sp defined locally at any ρ G Μ by
(6.2) Sp(exppu) = expp(-u), и G Br(op) (0 < r < гр(М)),
is an isometry.
(4) Suppose a complete simply connected Riemannian manifold Μ satisfies
VR = 0. Then M is a symmetric space.
Proof. (1) It suffices to show that
(6.3) (VuR)(x, y)z = 0, for any u, x, y, ζ e TPM, ре М.
Since the geodesic symmetry sp is an isometry and leaves VR invariant, we get
-(VuR){x, y)z = Dsp((VuR)(x, y)z)
= (VDSp(x)R)(Dsp(x), Dsp{y))Dsp{z)
= (V-uR)(-x, -y){-z) = {VuR){x, y)z,
from which (6.3) follows. Next, suppose VR = 0. Then for the parallel translation
P(c)® along a curve с we have
(6.4) P(c)°(R{x, y)z) = R(P(c)°tx, P(c)°y)(P(c)°z).
Since parallel translations are linear isometries, it follows that for an o.n.b. {x, y}
of a plane section σ of TC^M
K(P(c)°tx, P(c)°ty) = (R(P(c)°tx, P(c)°ty)P(c)°ty, P(c)°tx)
= K(x, y),
which shows the "only if" part of (2). Conversely, the "if" part may be proved
by noting that (6.4) holds under the assumption, since the curvature tensor is
determined by sectional curvatures and the inner product (see Chapter II, (3.12)).
Now we show (3). If sp defined by (6.2) is an isometry, then we get (6.3) by
the same argument as in (1). Suppose VR = 0. Then (6.4) holds, and we apply
the Cartan Theorem (Chapter II, Theorem 3.2) to / : TpM -> TpM defined by
Iu = —u. Since It = —Ρ(7)έ_έ, the assumption (3.6) of the Cartan Theorem is
satisfied. Then Φ of the Cartan Theorem is an isometry and equal to sp. Finally,
(4) follows by applying the Ambrose Theorem (Chapter III, Theorem 5.1) to the
above / and showing that Φ of the theorem is equal to sp. D

6. SYMMETRIC SPACES
177
Remark 6.4. A Riemannian manifold which satisfies (2), (3) of the above
proposition is called a locally symmetric space. Symmetric spaces are locally
symmetric. However, the converse does not hold in general. For instance, compact
Riemannian manifolds of constant negative curvature and lens spaces with
Riemannian metric of positive constant curvature are locally symmetric spaces that
are not symmetric spaces (see Lemma 6.11 (2) and 6.2 (III)). On the other hand,
Riemannian universal coverings of complete locally symmetric spaces are symmetric
spaces.
Now we are concerned with the behavior of Jacobi fields on a (locally)
symmetric space Μ. Let У be a Jacobi field along a geodesic 7 = 7U with the initial
direction и G TpM. Suppose Υ is perpendicular to 7. Setting u1- := {x G TVM\ (x, u) =
0} we define a linear transformation Ru oiuL by Rux := R(x, u)u. Prom properties
of the curvature tensor, we get in fact RuuL С и1- and Ru is symmetric. Let A be
an eigenvalue of Ru and e G u1- a corresponding eigenvector. We denote by E(t)
the parallel translation of e along 7. Then Y(t) := cx(t)E(t), Z(t) := s\(t)E(t)
are Jacobi fields along 7. To see this, recalling that сд, s\ were given in §1 (1.1)
and noting VR = 0, we get
Va/atVd/dtY(t) + R(Y(t), -уЮЖО
= -Xcx(t)E(t) + cx(t)R(P(7)°te, P(7)?u)(P7?u)
= -\cx(t)E(t) + cx(t)P(7)°t(Rue)
= {-\cx(t) + \cx{t)}E{t)=0,
and the same computation works for Z. Therefore, for eigenvalues Αχ, ... , Am_i
of Ru and an o.n.b. {ег·} of u1- consisting of the corresponding eigenvectors,
t ч Yi(t) = cXi(t)Ei(t), Zi{t) =3Xl(t)Ei(t)
form a basis of the vector space Jj- of all Jacobi fields along 7 which are
perpendicular to 7. The above facts hold also for locally symmetric spaces.
Now we consider Jacobi fields obtained by one parameter transformation groups
of isometries. Recall that the restrictions of Killing vector fields to geodesies 7 are
Jacobi fields (Chapter III, Lemma 6.1). In case of symmetric spaces, we get the
following.
Lemma 6.5. Let Μ be a symmetric space.
(1) Let Y(t) be a restriction of a Killing vector field Υ on Μ to a geodesic 7,
and denote by X the Killing vector field obtained by the transvection pt := Pt(7(0)).
Then Y(t) satisfies the initial conditions Y{0) = ΥΊ(ο) and Va/at^(0) = [^\ ^7(0)·
(2) Let Z(t) be a Jacobi field along 7. Then Z(t) satisfies Vd/QtZ(0) = 0
if and only if Z(t) is the restriction of a Killing vector Ζ on Μ obtained from a
transvection ps(Z(0)) to 7.
PROOF. (1) Let gs be the one parameter subgroup of I0(M) generated by Y.
Then Y(t) = £ |s=0 д*ЬШ and с1еаг1У Y(Q)
= У7(0). For the transvection

178
IV. COMPARISON THEOREMS AND APPLICATIONS
Ps(7(0)), Dps gives a parallel translation along 7, and we get
d
vd/9tnt) = ds
p(7)!+%<t+.)
d_
ds
Dp-s(Yyit+s)) = [X, y]7(t).
\s=0
(2) We first show the "only if" part. For a transvection ps = ps(Z(0)), W(t) :=
-§; \s=o Ps(7(0) 1S a Jacobi field along 7 and satisfies И^(0) = Z(0), Vd/dtW(0) =
Va/dsU=o^Ps(7(0)) = 0> because of Lemma 6.2 (2). Therefore, we get W(t) = Z(t).
The "if" part may be proved by the same computation. D
Corollary 6.6. (variational completeness). Let Μ be a symmetric space and
7 : [0, +00) —» Μ a geodesic in M. For a conjugate point 7(^0) (t0 > 0) to 7(0)
along 7, there exists a transvection ps = ps(x), χ φ 0 such that ps(7(0)) = 7(0)
and ps(7(£o)) = 7(^0) hold for all s £ R.
PROOF. For a conjugate point 7(^0) *° 7(0) there exists a nonzero Jacobi field
Y(t) = sx(t)E(t) with У (ίο) = 0, if we consider a basis of J7X of the form (6.5).
Then from sa(*o) = 0, we have A = (nn/t0)2 for some η £ Z+. It follows that
c\(tf0) = 0, namely, VY(to) = 0 for some 0 < t'0 < t0. Then ps = ps(Y(tO)) satisfies
the assertion of the corollary, if we apply Lemma 6.5 (2) to t'0. In fact, У is the
restriction of the Killing vector field obtained from ps to 7. Since s 1—► ps is a one
parameter group of isometries, we get
^ps(7(0))=Z)ps(7(0))£
р5(7(0))=£>р5(7(0))У(0)=0
ls=0
for all s. Therefore, ps(7(0)) = 7(0) for all s £ R. Similarly, we get ps(7(£o)) =
7(*o). □
Proposition 6.7. Let Μ be a symmetric space. Denote by X the Killing vector
field defined from a transvection pt{x) for χ £ TVM. Then
(6.6) Я(я, y)z = -[[X, У], Z]p.
PROOF. Let 7 be a geodesic with the initial direction x. Then for a Killing
vector field W, we have from the proof of Lemma 6.5 (1)
(6-7) Vd/dtWlit) = [X, W]l{ty
Since Y(t) = Yy(t) 1S a Jacobi field along *y(t), we get
R(x, y)x = Vd/dtVd/dtY(0) = Vd/dtlt=0[X, У]7(0
= [X, [Х,У]]7(0) = -[[Х,У],Х]р.
Then noting that 3R(x, y)z = R(x + z, y)(x +z) - R(y+ z, x)(y + z) + R(y, x)y +
R(z, x)z — R(x, y)x — R(z, y)z, we have (6.6). D
6.2. In this subsection we investigate a symmetric space (M, g) from the
viewpoint of Lie group theory. We fix ρ £ Μ. Recall that the identity component
G = Iq{M) of the isometry group of Μ acts transitively on Μ as a Lie
transformation group, and the isotropy group Η := {he G; h(p) = p} is compact. Then the
quotient space G/H carries the manifold structure such that a map Φ : G/H —> Μ
defined by Ф(дН) := g(p) is a diffeomorphism. In particular, we get a Riemann-
ian metric Ф*д on G/H induced from g, which is invariant under left translations

6. SYMMETRIC SPACES
179
Lk (Lk(aH) := kaH), к £ G. In fact, we have L*k о Ф*д = Ф*(к*д) = Φ* д. Now
using the geodesic symmetry sp of Μ at p, we define a reflexive automorphism σ
(σ2 = idc, σ Φ idc) of G by
(6.8) a(k) := sp о к о sp.
Then the fixed point set F := {k e G; a(k) = k} of σ is a closed subgroup of G,
and we show that F0 С Η С F, where F0 stands for the identity component of
F. In fact, let ht (0 < £ < 1) be a continuous curve in F with /i0 = e. Then we
get sp о ht о sp = /it and in particular sp(ht(p)) = ht(p). Since ρ is an isolated
fixed point of sp, it follows that ht(p) = p, namely, ht £ Η (0 < t < 1). Next, for
ft G Я we have σ(/ι)(ρ) = ρ, Ό(σ(Κ))(ρ) = -{Dsp о £>ft)(p) = Dft(p). It follows
that σ(/ι) = ft, namely, h e F.
Now we state the above facts in terms of the Lie algebra g of G. Put θ :=
Da(e) · β —» β· Then θ is a reflexive automorphism of g with eigenvalues ±1.
Therefore, setting
«ι == {* e 0; Θ(Χ) = -X},
( ' ' ъ-.= {xeg-, θ(Χ) = X},
we have 0 = m + i) (vector space direct sum).
Lemma 6.8. With respect to the above decomposition g = m + f)? we have the
following:
(1) [ϊ), ϊ)] С ϊ), [ϊ), m] С m, [m, m] С ϊ), and ϊ) 25 the Lie algebra of H.
(2) m may be identified with the tangent space to G/H at #, and ΌΦ :
m —> TPM pzi;es α vector space isomorphism. Define an inner product Q on m
6j/ Q(X, У) := (ΌΦ(Χ), ΌΦ(Υ))Ρ. Then Q is Adgtt-invanant, and
(6.10) Q(&dz(x), y) + Q(x, *dz(y)) =0, ijGm,zG(),
where a,dz is defined in Chapter I, (2.16).
(3) // we identify g with the Lie algebra of Killing vector fields on M, we have
the following isomorphisms.
m = {X; Killing vector field defined by a transvection pt(x) for χ £ TPM}
= {X; Killing vector field with VX(p) = 0},
ϊ) = {Y; Killing vector field with Yp = 0}.
PROOF. (1) Since θ is an automorphism of g, the first three formulas are
clear. Let exp : g —> G be the exponential map for the Lie group G. For X £ g,
the following shows that i) is the Lie algebra of H.
Θ(Χ) = X&expt Θ{Χ) = exp tX (t £ R)
<& a(exptX) = exptX (t e R) & exptX £ F {t £ R)
oexptx eH (te R).
(2) ΌΦ : g —> TpM is surjective, and Кег/)Ф = ί). It follows that ΌΦ \ m is
a linear isomorphism from m onto TPM. Next, for ft £ Я we have DΦ(AdhX) =
Dh ο ΌΦ(Χ) and therefore
Q(Adft(X), Adft(y)) = <Ζ?Φ(ΑάΛ(Χ)), DΦ(Adft(У)))p
= <Ζ?Λ(Ζ?Φ(Χ)), £>Л(£>Ф(У)))Р
= <Ζ?Φ(Χ), ДФ(У)>р = Q(X, У),

180
IV. COMPARISON THEOREMS AND APPLICATIONS
from which (6.10) follows by differentiation.
(3) For χ G TPM we denote by X the Killing vector field determined by a
transvection ps(x)· Then noting that sp о s7(t) о sp = s7(-t), we have Θ(Χ) =
£ \s=o <r(Ps(x)) = ts \s=o {P-s{x)) = -X Namely, X G m, and X is uniquely
determined by the value Xp = x. Therefore, χ \-+ Χ gives the first isomorphism.
Next let φί be the flow generated by a Killing vector field Υ. Then note that
Yp = Q&4>t(v)=v(teR)&4>te я,
and the assertion on ϊ) follows. Finally, to show the second isomorphism note that
for Χ, Υ G m determined by x, у G TPM, respectively, we have VyX = [Y, X]p = 0
because of (6.7) and [m, m] С ϊ). Then the second assertion follows from Lemma
6.5 (2). D
Now since i) is the Lie algebra of a compact Lie group #, there exists an
Ad Я-invariant inner product ( , ) on i). In fact, take a normalized Haar
measure dh on Η with JH dh = 1, and an inner product β on ϊ). Then (x, y) :=
JH β(Α.άϊιχ, Adhy)dh is the desired inner product. Now define an inner product
Q on g so that m and f) are orthogonal and Q | m (resp., Q \ i}) is given in Lemma
6.2 (2) (resp., as above). Then this Q is also AdЯ-invariant and again satisfies
(6.10).
Now let Μ be a simply connected symmetric space and apply the de Rham
decomposition theorem (Chapter III, §6). Then we have the Riemannian direct
product decomposition of M, and the direct product decomposition of the isometry
group
Μ = M0 x Mx χ · · · χ Mfc, /0(M) = /o(M0) x · · · x /0(Mfc),
where Mo is a Euclidean space and Мг· (г = 1, ... , к) are irreducible simply
connected Riemannian manifolds. Since geodesies of Μ emanating from ρ =
(po, Рь · · · 7 Pk) are decomposed into geodesies of Мг· (г = 0, ... , к) emanating
from pi, it follows that the geodesic symmetry sp of Μ at ρ is also decomposed as
sp = sPo χ sPl χ · · · x sPk, and Мг· (г = 0, ... , к) are also symmetric spaces.
We first treat irreducible symmetric spaces. For a Lie algebra g, define the
Killing form В of g by B(x, y) \— trace(ada; о adi/), x, у G g. Then we may easily
see that β is a symmetric bilinear form on g. Further, for any automorphism θ of
g we get
£(ad2(x), y) + B(x, bdz(y)) = 0, Β(θχ, ву) = В{х, у).
A Lie algebra g is said to be semisimple if β is a nondegenerate bilinear form.
Theorem 6.9. Let Μ be an irreducible symmetric space with dim Μ > 2. Let
ρ G M, and let д = m + ϊ) be the decomposition of the Lie algebra д of Io(M) given
by (6.9). Denote by [m, m] the subalgebra off) generated by {[x, y] G i); x, у G m}.
Then we have the following:
(1) д is a semisimple Lie algebra, and [m, m] = i).
(2) The Lie algebra of the holonomy group of Μ may be identified with i).
(3) Μ is an Einstein manifold and its sectional curvatures Κσ satisfy either
always Κσ > 0 or always Κσ < 0. Further, under the identification of TPM with
m, we have Ric(x, y) = — ^B(x, y).
PROOF. Let χ e i), у G m. Then ad χ ο ad у maps ϊ) (resp., m) into m (resp.,
i)). Therefore we have B(x, y) = 0, and ϊ), m are orthogonal with respect to B.

6. SYMMETRIC SPACES
181
Similarly, for x, у G i), adx о ady maps i) (resp., m) into i) (resp., m). We denote
by В | i), 03 | m the restrictions of β to ί) χ ϊ), m χ m, respectively. First we show
that В | i) is negative definite. Take an o.n.b. {fa}a=i °f Ь w^h respect to Q and
iGi). Denoting by B\ the Killing form of i), we have
Бх(х, ж) = ^Q(adxoadx(/a), /a) = - ^Q(adx(/a), adx(/a)) < 0.
Next, for the trace of ad χ о ad ж, take an o.n.b. {е*}^ of m and compute
m
V^ Q(adx о adx(e*), e*).
г=1
For χ G i), u G m, we denote by X, U the corresponding Killing vector fields on Μ,
respectively. Then identifying m with TPM, we get from (6.7) and Chapter I, §2,
Exerecise 8, [u, x]0 = — VUX. It follows that
Q(adx о adx(e,). ег) = Q([[e*, χ], χ], е») = -(V^.^X, e»)
Recall that w н> УД is a skew-symmetric linear map of m, since X is a Killing
vector (Chapter III, Lemma 6.1). Let (Αι3) be the matrix representation of the
above linear map with respect to {e,}. Then we get from the above
^Q(adxoadx(ei), ег) = ^^AXJAJX = -^,Ab - °"
г i.j i.J
Now suppose equality holds. Then we have VX(p) = 0. Xp = 0. and therefore
X = 0, namely, χ = 0. Summing up, we see that В \ i) is a negative definite
symmetric bilinear form.
Second, we consider β | m. Q \ m, В \ m are both ad i)-invariant symmetric
bilinear forms on m, and Q \ m is positive definite. On the other hand, the Lie
algebre i)(p) of the holonomy group at ρ is generated by {R(x, y)\ x, у £ m} in
the case of symmetric spaces because of Chapter III (6.7), and therefore generated
by ad[m, m] (c i}) by Proposition 6.7. We set i}± := [m, m]. Then since Μ is
irreducible, adi}± acts irreducibly on m. Since Q \ m, В \ m are ad ^-invariant, we
have В \ m = A Q \ m for some A £ R. In fact, for an eigenvalue A of В \ m with
respect to Q \ m, the null space of£|m — AQ|misa nonzero ad ^-invariant
subspace and coincides with m by the irreducibility. If A = 0, we have В \ m = 0.
Then for x, у е m, we get B([x, у], [х, у}) = В(х, [у, [х, у]]), the right-hand side
of which is equal to 0 because x, [y, [x, y]] G m. On the other hand, since [x, y] €t)
and β | i) is negative definite, we get [x, y] = 0, namely, i}i = 0. It follows that Μ is
flat, which contradicts the irreducibility. Therefore, λ φ 0 and В is nondegenerate,
namely, g is semisimple.
Third, we show that i) = i}i which completes the proof of (1), (2). Put дх =
ί}± + m. Then gx is an ideal of g and therefore semisimple. Let α be the orthogonal
complement of gx with respect to Б, which is also an ideal of g contained in i). For
χ G а, у G m we get [x, y] = 0, and (6.7) implies VX(p) = 0. On the other hand,
Xp = 0 since χ G i). It follows that α = 0, namely, i}i = i).
Finally, we prove (3). We identify m with TPM and note that В \ m =
A Q | m (λ Φ 0). Then for an o.n.b. {x, y} of a plane σ С m, we get
#σ = #(x, у) = (Я(х, y)y, x) = -Q(ad([x, y])y, x)
= --£(ad([x, y])y, x) = -B([x, у], [х, у]),

182
IV. COMPARISON THEOREMS AND APPLICATIONS
which is of fixed sign, because [x, y] £ i) and В | i) is negative definite. Next we
consider the Ricci curvature. For χ £ m, we get
Ric(x, x) = ^2(R(x, вг)еи х) = т^^(к e»], [ж, e»])
i=l
1 m
= —— Y^B(adx о adze*, e*) = — Y^Q(adx о adx(e*), e*),
г=1
where {ег·} is an o.n.b. of m. Now since ad χ (χ £ m) is a skew-symmetric linear
transformation of g which maps i) (resp., m) into m (resp., i)), for / := ad χ о ad χ
we easily see that trace/ | i) = trace/ | m = ^ trace/. It follows that
(6.11) Ric(x, x) = --B(x, x) = --Q(x, χ), χ £ m,
and Μ is an Einstein manifold. We remark that in the proof of Ric = — \B | m we
need not assume irreducibility. Π
Now we make the following definition:
Definition 6.10. Let Μ be a symmetric space and g = m + i) the
decomposition (6.9) of the Lie algebra g of the isometry group G = h(M).
(1) If [m, m] = 0, namely В | m = 0, then we call Μ a symmetric space of
Euclidean type.
(2) If g is semisimple and В | m is negative (resp., positive) definite, then Μ
is said to be of compact type (resp., noncompact type).
Then from the de Rham decomposition theorem, a simply connected symmetric
space may be expressed as the Riemannian direct product of Euclidean space and
simply connected irreducible symmetric spaces of compact and noncompact types.
Note that a symmetric space of Euclidean type is flat, since for x, у £ m we get
Я(х, у) = -ad [χ, у] = 0.
Exercise 2. Show that a symmetric space Μ of Euclidean type may be written
as a Riemannian direct product Rk χ Trn~k (0 < к < m), where Тш~к is a flat
torus.
Let Μ be a symmetric space, and let Μ = Mq χ M\ χ · · · χ Mk be the de
Rham decomposition of the universal Riemannian covering Μ of M. We note
that if g is semisimple then Μ is free of the Euclidean factor. In fact, in the
decomposition g = m + fj of the Lie algebra g of the isometry group of Μ, m
is isomorphic to m since their elements are given by transvections determined by
geodesies. For the direct product decomposition g = g0 Θ · · · Фд^ of the Lie algebra
д corresponding to the de Rham decomposition, each factor is an ideal of д and we
get [mo, mo] = 0, [m0, §<] = 0(г = 1, ... , к), [m0, fjo] С m0. Since Killing vector
fields on M, which are elements of g, may be lifted to Killing vector fields on M,
we may consider д as a subalgebra of д. Now if there exists a nonzero element
χ £ mo (С д), then we have B(x, g) = 0, which contradicts the fact that g is
semisimple.
Now suppose Μ is of compact type. Then for the de Rham decomposition
Μ = M\ χ · · · χ Mk of M, each factor Mi is irreducible and we get from Theorem

6. SYMMETRIC SPACES
183
6.9
B\m(x, x) = -2Ric^(x, χ) = -2^Ric^(zb x{)
i
= / v A^ Q\Xj, Xi),
i
where χ = χ ι + · · · + Xk denotes the decomposition of χ with respect to the de
Rham decomposition of rii, and A; denotes the value of A for Mi determined by
Theorem 6.9. Since В | m is negative definite, we have A* < 0 for all i. Therefore,
the sectional curvatures of Μ are nonnegative everywhere and its Ricci curvatures
are positive everywhere, in fact greater than or equal to min(—Aj/2). These
properties of curvatures also hold for M, and the Myers theorem (Theorem 3.1 (2), and
Chapter V, Theorem 1.1) implies that Μ is compact and its fundamental group is
finite. Also the isometry group I(M) of Μ is compact.
The same argument implies that for a symmetric space of noncompact type,
the sectional curvatures are nonpositive everywhere and the Ricci curvatures are
negative everywhere. For the fundamental group of symmetric spaces we get
Lemma 6.11. (1) The fundamental group of a symmetric space Μ is an
abelian group.
(2) A symmetric space Μ of noncompact type is simply connected.
PROOF. Let 7 : [0, I] —» Μ be a geodesic loop based at ρ Ε Μ parametrized by
arc-length, and pt the transvection determined by 7. Then from 7(0) =7(0 = Ρ we
get pt 7(0) = pt (7(0)7 namely, η(ί) = *y(t+l) (t £ Д), which means that 7 is a closed
geodesic. Therefore, for any element α of the fundamental group πχ(Μ, ρ)10 of Λ/,
there exists a closed geodesic 7 which represents α and passes through p. We use the
notation α = [7]. Let δ (resp., e) be a closed geodesic which represents β £ πχ(Λ/. ρ)
(resp., the product a · β). Then the geodesic symmetry sp at ρ maps ->. δ. e to
7_1, <5_1, €_1, respectively. Therefore, denoting by (sp)* the endomorphism of
7Γι(Μ, ρ) induced from sp, we have {sp)*(a · β) = [sp(e)] = [б-1] = (α · 3)'1. On
the other hand, we get (sp)*(a · β) = (sp)*a · (sp)*/3 = a-1 · 3~l = (3 · a)-1· It
follows that α · β = β · α.
Next suppose that Μ is of noncompact type. We derive a contradiction
assuming that there exists a nontrivial closed geodesic η in Λ/. We consider a Jacobi
field along 7. Since the Ricci curvature p(^>(0)) < 0. we have a negative eigenvalue
A of Ru (u = 7(0)). Then from (6.5) we may choose a parallel vector field E(t)
along 7 such that Y(t) = coshy/\X\t E(t) is a Jacobi field along 7. Note that
11*4*) II -* +°° as t -> +00. On the other hand, since W(0) = 0, Y(t) is obtained
from a transvection pt of Μ by Lemma 6.5 (2), and is a restriction of a Killing
vector field. Therefore, ||У(£)|| is bounded since 7 is a closed geodesic, and we get
a contradiction. D
Summing up, we get
Theorem 6.12. (1) Let Μ be a symmetric space of compact type. Then Μ
is compact and its fundamental group is a finite abelian group. The isometry group
I(M) of Μ is compact and its Lie algebra g is semisimple. The sectional curvatures
of Μ are everywhere nonnegative and the Ricci curvatures are everywhere positive.
10As for the fundamental group and covering space, see also Chapter V, §1.

184
IV. COMPARISON THEOREMS AND APPLICATIONS
(2) Let Μ be a symmetric space of noncompact type. Then Μ is simply
connected. The sectional curvatures of Μ are everywhere nonpositive and the Ricci
curvatures are everywhere negative. The Lie algebra gofG = Iq(M) is semisimple,
and the isotropy group Η of G at ρ £ Μ is a maximal compact subgroup of G.
For the proof of the theorem it only remains to show that Я is a maximal
compact subgroup of G in (2). Let К be a compact subgroup of G which contains Я.
Now Μ is a complete simply connected nonpositively curved Riemannian manifold,
namely, an Hadamard manifold, which will be treated in detail in Chapter V, §4.1.
Prom Theorem 4.8 of Chapter V it follows that К admits a fixed point q £ Μ. If
we write q = g(p), g £ G, then the isotropy group of G at q is given by gHg~l and
we get Я С К С gHg~l. It follows that Η = К, since dim Я = aim К and the
number of connected components of Я and К is the same.
Now we construct symmetric spaces from (connected) Lie groups G with Lie
algebra g. Suppose that there exist an involutive automorphism σ {φ id) of G
and a closed subgroup Я of G, such that Ad^ is a compact subgroup of GL(q)
with F0 С Я С F, where F := {a £ G; σ(ά) = a} is the closed subgroup of
fixed elements of σ with the identity component F0. Then θ := Da is a reflexive
automorphism of g, and if we define m, i) as in (6.9), we get a vector space direct
sum g = m + f) with [i), m] С m, [m, m] С ϊ). Prom Я С F we get AdЯ(m) = m.
Since AdЯ is compact, there exists an inner product g on m such that
(6.12) g(Adhx, Adhy) = g(x, y) for any h £ Я, х, у £ т.
If such a Lie group G with Я, σ, g satisfying the above conditions is given, we call
(G, Я) a Riemannian symmetric pair. A Riemannian symmetric pair (G, Я) is
said to be effective (resp., almost effective), if normal subgroups of G contained in
Я consist only of the identity (resp., are only discrete subgroups).
Exercise 3. Let (M, g) be a symmetric space. Then for G = /o(M), the
isotropy group Я at ρ £ Μ, and σ in (6.8), show that (G, Я) is an effective
Riemannian symmetric pair.
Conversely, for a Riemannian symmetric pair (G, Я), let Μ := G/H be the
quotient manifold and ρ := Η the origin. We define an action of a £ G on Μ
by the left translation La(kH) := akH. Then La gives a diffeomorphism of M,
and a i—► La is injective if (G, Я) is effective. Further, for the natural projection
π : G —» Μ = G/H, its differential Dn gives an identification between m and TVM,
and (DLh)(p) coincides with Adm/i for h £ Я. Now from an AdЯ-invariant inner
product g on m, we define a Riemannian metric on Μ, which is denoted again by
<7, as follows:
i6 13x 9a-P(x, У) :=g(DL-lx, DL~ly),
ap = LaH, x, у £ Ta.pM, ae G.
This is well defined, and elements of G act on (M, g) as isometries. We show
that (M, g) is a symmetric space. First we determine the geodesic symmetry sp
at ρ £ Μ. Let sp be defined as a · ρ \—> σ(α) · ρ (α £ G). If α · ρ = b · ρ, then we
have b~la £ Я С F, namely, σ(6)-1σ(α) £ Я. Therefore, sp is well-defined and is
a diffeomorphism of Μ. sp is reflexive because so is σ. Since tangent vectors to Μ

6. SYMMETRIC SPACES
185
at a · ρ may be written in the form DLa(x), χ £ m, we have
sp(aexptx · p) = a(aexptx) ρ = a(a)exp(—tx) ρ
and therefore Dsp(DLax) = —DLa^x. It follows that sp is an isometry of Μ and
Dsp(x) = —x (x £ m = TPM), which shows that sp is the geodesic symmetry at
p. Second, at a · ρ we may easily see that sa.p := La о sp о L~l is the geodesic
symmetry at a · p, and (M, g) is a symmetric space.
Now we give some examples of symmetric spaces.
(I) Let Я be a compact (connected) Lie group. First note that Я carries a bi-
invariant Riemannian metric which is invariant under left translations La{Lab :=
ab) and right translations Ra(Rab := 6a). In fact, on the Lie algebra i) of Я there
exists an Ad Я-invariant inner product ( . ). and we define an inner product on
TaH (a £ Я) as (x, y)a := (DL~1(x), DL~1(y)). Then we easily see that α ι—► ( , )α
defines a Riemannian metric on Η which is left invariant. Further, for x, у £ TaH
we get
(DRb(x), DRb(y))ab = (DL^DRb(x). DL^DRb(y))
= (Adb-^DL-'x). Adb-^DL-'y))
= (DL-\x).DLZl(y)l^{x.y)a.
which shows that the above Riemannian metric is also right invariant. Now we
show that Я is a symmetric space. First, the geodesic symmetry se at the identity
e is given by se(a) := a-1. In fact, we obviously have si = id# (se φ id#) and
Ds€(e) = — idfj. Furthermore, for x, у £ TaH
(Dse(a)x, Dse(a)y)a-i = (DLa о Dse(x), DLa о Dse(y))
= (Dse о DR~\x), Dse о DR-\y))
= (DR^(x),DR^(y)) = (x,y)a,
which implies that se is a geodesic symmetry at e. Second, the geodesic symmetry
sa at a £ Η is easily seen to be given by sa = Laoseo L^1, and Я is a symmetric
space.
Exercise 4. For a compact Lie group Я we set G = Η χ Я, σ(α, 6) =
(6, α), Δ = {(α, α); α £ ί}}, m = {(ζ, —χ); χ £ ί}}, and define a Riemannian
metric g by g((x, —x), (y, —y)) := 4(x, y). Then show that (G, Л) is a Riemannian
symmetric pair and the corresponding symmetric space is isometric to the above
Я.
Exercise 5. Show the following for a compace Lie group Я.
(1) Geodesies emanating from e are given by t \—► expfo, χ £ ί).
(2) Let X, У be left invariant vector fields on Я. Then
vxr = \[x, η <ВД W *> = JII[Α-, *ΊΙΙ2
Я(Х, r)X = -J(adX)2r.
(II) (Grassmann manifolds). We denote by F one of the fields R of real
numbers, С of complex numbers, or Η of quarternions. Corresponding to F = Д,
C, Η we set G = SO(p + q), SU(p Η- ς), Sp(p Η- ς), respectively. We denote by ao

186
IV. COMPARISON THEOREMS AND APPLICATIONS
Ш :
the diagonal matrix whose first q elements on the diagonal are equal to —1 and the
remaining ρ elements are equal to 1. We define a reflexive automorphism σ of G as
σ(α) = аоаай1 (a G G). Then the set F of the fixed points of σ is given by
F = {[o °] € G; α e °{q) (ort7(9)'5p(9))' β e °(i>) (orC7(p)'5p(<7))} ·
Then for the vector space direct sum decomposition g = m + i) with Я := F we get
. = < \_t _ n ; zisagxp matrix over F > .
Finally we define an inner product on m by
f-±trace(xy) (if F = Д),
(x, y) := < -2 trace (xy) (if F = C),
[-trace (xy) (if F = H),
which is in fact Ad H-invariant. Namely, (G, H) is a Riemannian symmetric pair.
The corresponding symmetric spaces Μ = G/H are given as follows: Fp+q :=
{'(ii, ... , χρ+ς); Xi G -F} carries the structure of a vector space over F, where
the scalar multiplication is given by multiplying a e F from the right to each
component. We denote by Gp.q{F) the space of all p-dimensional F-subspaces of
jpp+q Then the above G naturally acts transitively on Gp,q(F) from the left. Take
a base point о := (eg+i, ... , ер+я)р, where we set e* = έ(0, ... , 0, 1, 0, ... , 0) (1
lies in the г-th position). Then we may easily check that the isotropy group of G at о
is given by Η and Gp,q(F) = G/H is a C°° manifold, which is called the Grassmann
manifold over F. In particular, G\,n(F) is called the η-dimensional projective space
over F, which consists of one dimensional subspaces of Fn+1. We note that Gi?n (R)
is isometric to the η-dimensional real projective space of constant curvature 1, and
G\.n(C) is isometric to the complex projective space with the Fubini-Study metric
(see Chapter II, §6) of complex dimension n. Furthermore, G\,n(H) is also of
positive sectional curvature.
(Ill) Now let Μ be a symmetric space of positive sectional curvature. Then
Μ is compact. Let и e UPM and consider a linear map χ G и1- (С ТрМ) н->
R(x, u)u G uL which is symmetric with positive eigenvalues, and therefore is
a linear isomorphism. It follows that for any у G u1- there exists an χ G u1-
with R(x, u)u = —y. We denote by X, U the Killing vector fields obtained from
transvections determined by x, u, respectively, and consider the Killing vector field
Ζ := [X, [/], which belongs to i) and satisfies Zp = 0. Then from (6.6) we get
[Z, U]p = —[/7, [X, U]]p = —R(x, u)u = y. Let φι G Η be the one parameter
subgroup generated by Z. Then we get y?t(p) = ρ and ^ |i=0 D(f-t(p)u = y.
Since у G u1- is arbitrary, it follows that the orbit {Dip(p)u; φ G H} of и under the
action of the isotropy group Я at ρ contains an open neighborhood of и in UPM,
and therefore coincides with UPM since it is open and closed in UPM. Namely, for
any v, w G UPM there exists an isometry φ G Η such that φ(ρ) = ρ, Όφ(ρ)ν = w.
It follows that the tangent cut locus of ρ (and consequently of arbitrary points of
M) is a hypersphere of TPM. We normalize the Riemannian metric on Μ by a
homothety so that the above sphere is of radius π. Then we get d(M) = i(M) = π,
and cut points of ρ are either always the first conjugate points, or always are not
the conjugate points to p, along all geodesies emanating from p.

6. SYMMETRIC SPACES
187
Case 1 (all cut points are not conjugate to p). In this case, as we saw in
Chapter III, Proposition 4.13 and Corollary 4.14, all geodesies emanating from ρ are
closed geodesies of length 2π, and the cut locus Cp of ρ is a connected component
of the fixed point set of the geodesic symmetry sp. Therefore, Cp is a totally
geodesic hypersurface of Μ by Problem 14 for Chapter II. Now let и G UPM, and
let Αχ, ... , Am_i be the (positive) eigenvalues of Ru. Then t = 2π is a conjugate
value to ρ with multiplicity m - 1 along 7n, and Jacobi fields Zi(t) = s\t(t)Ei(t)
satisfy Zi(2n) = 0, namely, sin2\/Xin = 0. On the other hand, the first conjugate
value is greater than π, and it follows that A; = 1/4 (г = 1, ... , m — 1). Namely, Μ
is of constant curvature 1/4, and is isometric to the real projective space because
d(M) = i(M) = π.
Case 2 (all cut points are the first conjugate points to p). In this case, for
any geodesic ηη (и G UPM) emanating from p, the first conjugate value is equal to
π and its multiplicity is equal to a constant к (> 1). Then the maximal eigenvalue
of Ru is equal to 1 for the same reason as above. Let Eu be the (k + l)-dimensional
subspace of TpM spanned by и and the eigenspace of Ru with eigenvalue 1. Note
that for ν G EunUpM the maximal eigenvalue of Ry is equal to 1 (see footnote 6,
in Remark 2.2). Then the set {v G Eu; \\v\\ = 1, ехрр7гг> = 7и(тт)} is a nonempty
open and closed subset of EunUpM by the proof of Corollary 6.6, and therefore
coincides with EunUpM. It follows that for any ν G Eur\UpM, an arc in EUDUPM
joining ν to —v is mapped to the point q = 7η(π). Then t i-> expp tv (0 < t < 2π)
is a geodesic loop at q, and is in fact a closed geodesic by the proof of Lemma
6.11. It follows that all geodesies in Μ are closed geodesies of length 2π, and
as in Case 1 we see that the cut locus Cp of ρ is a totally geodesic submanifold
of Μ as a connected component of the fixed point set of the geodesic symmetry
sp. If 1 is the only eigenvalue of Ru, then Μ is of constant curvature 1 and
simply connected, since the cut locus and the first conjugate locus coincide (Chapter
III, Exercise 6). Namely, Μ is isometric to the sphere of constant curvature 1.
Next, suppose that there exists a positive eigenvalue A of Ru with A < 1. We
show that A = 1/4. In fact, let w be an eigenvector of Ru with eigenvalue A
and recall that 2π is a conjugate value to ρ along ^u of multiplicity m - 1, since
all geodesies of Μ are closed geodesies of length 2π. Therefore, the Jacobi field
Y(t) = s\(t)W(t) satisfies Υ{2π) = 0, where W(t) denotes the parallel vector field
along 7n with W(0) = 0. It follows that sx(2n) = 0, and we get A = 1/4. In
this case Μ is a simply connected Riemannian manifold whose sectional curvatures
satisfy 1/4 < Κσ < 1. Now we define an equivalence relation ~ in UPM = 5m_1 by
и ~ ν Ф> Eu = Ev Ф> expp nu = expp πν. Then, by the above, equivalence classes
are A;-dimensional great spheres Sk of UPM, and {πχ; χ G Sk} is mapped to one
point of Cp via expp. Thus we have a foliation of 5m_1 with leaves Sk. It is known
that к is equal to 1, 3, or 7, and such symmetric spaces are isometric to one of the
complex projective spaces, the quaternionic projective space given in (II), and the
Cayley projective plane (see, e.g., [Hel], [Ka-2]).
(IV) Next we give an example of a symmetric space of noncompact type. Set
P(n, R) := {x G SL(n, R); lx = χ and χ is positive definite}, which is the space
of all hyperellipsoids centered at the origin in Rn with volume 1. Now the group
G = SL(n, R) acts on P(n, R) by a · χ = axla for a G SL(n, R), χ G P(n, R).
Then for the unit matrix En G P(n, R) we have G · En = P(n, Я), namely, G acts
transitively. We also easily see that the isotropy group Η at ρ := En e P(n, R) is

188
IV. COMPARISON THEOREMS AND APPLICATIONS
given by SO(n, R). Я is a maximal compact subgroup of G, and their Lie algebras
are given by g = {X £ gl(n, Я); trace X = 0}, i) = {X <E gl(n, Я);'Х = -X},
respectively. The Killing form on g is given by B(X, Y) = trace XY and is a
nondegenerate symmetric bilinear form on g. Then the orthogonal complement m
of i) with respect to В is given by m = {Y £ gl(n, Я); lY = У, traced = 0}.
Now В is positive definite on m and negative definite on i). We give an involutive
automorphism σ by σ(Χ) = —lX. Then σ | i) = idf,, σ | m = -idm, and therefore
(Χ, Υ) := —Β(Χ, σ(Υ)) = tra,ce(XlY) defines an AdЯ-invariant inner product
on g. It follows that (G, H) is a symmetric Riemannian pair and P(n, R) =
SL(n, R)/SO(n) is a symmetric space. Finally, for orthonormal vectors Χ, Υ £ m,
noting that [X, Y] £ i), we get
K(X, Y) = <Я(Х, Y)Y, X) = <[У, [X, У]], X)
= -Я([У, [X, У]], σ(Χ)) = Я([У, [X, У]], Χ)
= -Я([Х,У],[У,Л']) = -||[Л',У]||2<0.
Remark 6.13. In general, for a semisimple Lie group G without compact
factor whose center is finite, and a maximal compact subgroup #, G/H is a symmetric
space of noncompact type with nonpositive sectional curvature.
Remark 6.14. As stated in Theorem 6.9, the Lie algebra of the holonomy
group of an irreducible symmetric space Μ is given by the Lie algebra i) of the
isotropy group Η of G = Iq(M) at ρ £ Μ. If Μ = G/H is simply connected, then
Η is connected and H(p) = H. For a general irreducible Riemannian manifold Μ,
Μ. Berger showed that either its restricted holonomy group H°(p) acts transitively
on UPM or Μ is locally symmetric. In the latter case H°(p) is the isotropy group
of the isometry group Io(M).
Let Μ be a symmetric space and ρ £ Μ. Let g = m Η- ί) be the decomposition
of the Lie algebra of G = Iq{M) given in (6.9). Recall that i) is the Lie algebra of
the isotropy group Η at p, and m may be identified with TPM. Now a maximal
abelian subalgebra α contained in m is called a Cartan subalgebra. It is known
that Cartan subalgebras are conjugate to each other with respect to the adjoint
representation of Η on m, and m = [j^eH Ad ft a, where Ho denotes the identity
component of H. Therefore, the dimension of Cartan subalgebras is a constant,
which is called the rank of Μ, and is equal to the dimension of maximal flat totally
geodesic submanifolds in Μ. In particular, a symmetric space Μ is of rank 1 if and
only if Μ is of positive or negative sectional curvature according as Μ is compact or
noncompact. Now the more detailed strucutres and properties of symmetric spaces
may be analyzed by using the so-called root system of (G, H) with respect to a
Cartan subalgebra α (see [Hel] for details).
Here we only give a result due to the author as an example: Let Μ be a compact
symmetric space and ρ £ Μ. Let α be a Cartan subalgebra with respect to the
above decomposition g = m + i). Then T(G, H) := {x £ а; ехрЯ £ χ} is a lattice
in a, and we have
Theorem 6.15. The tangent cut locus Cp of ρ in Μ is determined by the
tangent cut locus Ca of ρ in the flat torus a/T(G, Я). That is, Cp = Ad#o(Ca)·
From the above it follows that the cut locus Cp coincides with the first conjugate
locus if Μ is simply connected ([Cr]), and it is possible to determine the detailed

PROBLEMS FOR CHAPTER IV
189
structure of the cut loci of compact symmetric spaces in terms of the root systems
(see [Sa-3, 4], [Nai], [Та]).
Problems for Chapter IV
1. (1) In the sphere (5m, go) of constant curvature 1 show that the volume of
a metric ball Br(p) (0 < r < π) is given by vol Br(p) = am-i JQr sinm_1 tdt.
Show also that volrn-\dBr{p) = am_i sinm_1 г (0 < r < π).
(2) In the m-dimensional complete simply connected Riemannian manifold of
constant curvature —1 show that volBr(p) = am_i /0rsinhm-1 tdt.
Show also that volTn-idBr(p) = am_i sinhm_1 r.
2 Let Μ be a Riemannian manifold and ρ £ Μ. Suppose the sectional curvature
Κσ satisfies δ < Κσ < Δ for any plane stction σ of TPM, ρ £ Μ. Let R be the
curvature tensor of type (0,4) and define the tensor Rk of type (0,4) by
Rk(x, y, z, w) := fc{(y, z)(x. w) - (x. z)(y, w)}.
Then show that for x, y, z, w £ UPM we get the following:
(1) |Я(ж, ι/, ζ, w) - Щл+6)/2{х. У- ζ. w)\ < 2(Δ - δ)/Ζ.
(2) |Д(ж,у, ζ, u;)|<|max(A-6).
3. Let Д be the curvature tensor of a Riemannian manifold Л/. We define R £
Нот(Л2тл/) by (R(x Л у), ζ Λ w) := (Д(х, i/)u'. 2). which is called the curvature
operator.
(1) Show that Д is symmetric with respect to the inner product on A2(TPM)
induced by the Riemannian metric.
(2) Let rmax (resp., fmin) denote the maximal (resp., minimal) eigenvalue of
R. Then show that the sectional curvatures Κσ satisfy fmm < Κσ < rnmx.
(3) Suppose that Κσ satisfy δ < Κσ < 1 for all plane sections σ. Show that
R is positive definite if δ > 1 - 3/(2[m/2] H- 1), where m = dim M.
4. Let Μ be a complete Riemannian manifold and Κσ < 0 everywhere. Then for
a Jacobi field Y(t) along a geodesic 7 with Y(0) = 0, show that
||W(0)|| < ||y(i)||/i (i>0).
5. Let Μ, Μ be m-dimensional Riemannian manifolds. Let N (resp., N) be a hy-
persurface of Μ (resp., M) through ρ £ Μ (resp., ρ £ Μ) with unit normal vector
field ν (resp., v). Consider the parallel hypersurfaces Nt := ex.O^{tvq; q £ N}, Nt
:= exp^fovq-, q £ N} of N, TV, respectively. Denote by Атах(Л1/р) and \m\n(ApP)
the maximal principal curvature of N with respect to vv and the minimal principal
curvature of N with respect to vp, respectively. Now suppose k(t) > K(t) holds
along geodesies η{ί) := expptvp, ^(t) := exppti>p emanating from p, p,
respectively, and also Атах(Л1/р) < \тт{Айр)· Then show that
Amiix(Ay(0) < Xmin(A~{t)) for 0 < t < t0(N).
6. Let Μ be a complete Riemannian manifold with Κσ > 0, and 7, σ : [0, +oo) —>
Μ normal geodesies emanating from p. Suppose 7 is a ray, namely, ^(7(0), 7(s))

190
IV. COMPARISON THEOREMS AND APPLICATIONS
= s (for all s > 0), and α := Z(7(0), σ(0)) < π/2. Then show that
lim d(a(0), a(t)) = +00.
t—>+oc
7. Let Μ be a Riemannian manifold. For / : Μ —» Я, we set E1/ := {(p, a) £
Мхй; /(ρ) < α}. Then / is a convex function if and only if Ef is a totally convex
set of the Riemannian product Μ χ R.
8. Let Μ be a closed connected locally convex set of a Riemannian manifold M.
Show that Μ is a topological manifold with boundary.
9. Let С be a closed connected locally convex set. For ρ £ дС show that the
tangent cone C(p) \{op} coincides with the intersection of all supporting half-spaces
of С at p.
10. Let (M, g) be a symmetric space.
(1) Show that a complete totally geodesic submanifold 5 of Μ is a symmetric
space with respect to the induced Riemannian metric.
(2) Let 5 be as in (1). Under the identification TPM = m (p £ 5) we denote by
η the subspace of m determined by TPS. Then show that [[n, n], n] С п. Conversely,
if a subspace η of m satisfies this condition, show that S := expp η is a complete
totally geodesic submanifold of M.
11. Let G be a compact Lie group with a bi-invariant Riemannian metric (§6.2
(I)). Show that the multiplicities of conjugate points along any geodesic are even.
12. Determine the cut locus of a lens space L(q; P2, ... , pn) with constant
sectional curvature 1 at the point χ := π(1, 0, ... , 0). Determine all closed geodesies
of L(q;p2, ... , p„).
Notes on the References
§1. As for the axiom of free mobility and the axiom of plane, we refer to, e.g.,
[C-2], [La]. See [Char], [Wo-1], [No-Ko I] for more details on flat manifolds. In
particular, for a geometric proof of the Bieberbach theorem see also [Bus-2], which
follows an idea of M. Gromov for almost flat manifolds (see Appendix 6). For
spherical space forms, the complete classification is known ([Wo-1]; see also [Gi-2]
for other topics). Proposition 1.1 is due to [Sa-S]. Recently J. McGowan ([Mc])
obtained a sharp diameter estimate for all spherical space forms. For hyperbolic
space forms, hyperbolic non-Euclidean geometry (see e.g., [Bea]) plays an important
role. In particualr, for the Teichmuller space of a compact orientable surface of
genus g we refer to a recent book [Bus-3] of P. Buser. For the three dimensional
case see [Th]. [Ra] is a recent text book on general hyperbolic manifolds.
§§2, 3. A comparison theorem for Jacobi fields was first obtained by Η. Ε.
Rauch ([R-l, 2]) to prove his sphere theorem, by applying the Sturm-Liouville
comparison technique in ordinary differential equations. Since then comparison
methods in Riemannian geometry have been generalized and developed extensively
by many authors. See, e.g., [B-4], [War-1], [Ka-1, 3] [Kas-1], and the textbooks
[Gr-K-Me], [Ch-Eb], [K-5]. In §2.1 we followed H. Karcher's approach ([He-Ka],
[Ka-1]) to a unified treatment of Jacobi field comparison theorems. There is also
another comparison technique using the Riccati type equation instead of the Jacobi
equation, which is not treated here (see, e.g., [Esc-He-2], [Ka-3], [G-8], [In]).

NOTES ON THE REFERENCES
191
In §3 we are mainly concerned with volume comparison theorems. Note that
to get the volume estimate from above we only need a lower bound for the Ricci
curvatures. M. Gromov gave Theorem 3.3 ([G-5]), which is a rather simple
generalization of the Bishop comparison theorem ([Bi-Cr], [Gu]). However, this turned
out to be a powerful tool to investigate the structure of complete Riemannian
manifolds whose Ricci curvatures are bounded below. For instance, Theorem 3.5 was
first proved via spectral geometry by S.-Y. Cheng ([Che-1]). See also [S-3], [Ka-3],
[Ito]. The general injectivity radius estimate, Theorem 3.1, was first obtained by J.
Cheeger ([Ch-1]) via T.C.T. Then E. Heintze and H. Karcher improved the result
using their volume comparison theorem ([He-Ka], see also [Ma]).
§4. There are now many proofs for the important Toponogov comparison
theorem (e.g., [To-1], [Gr-K-Me], [B-4], [Ch-Eb], [K-5], [Ka-3], [S-3]). Our proof is
based on [K-5]. See also [Ka-3] for a proof based on the Riccati comparison
technique, and [Bu-G-Pe] for T.C.T. in more general Alexandrov spaces. We also refer
to [Ab] for a generalization of T.C.T. and its applications.
§5. The basic references for this section are [Ch-Gr-1] and [Ch-Eb], to which
we owe much. See also [Wal]. For convex functions and Busemann functions we
also refer to [S-2] and [Gre-Wu-1].
§6. Symmetric spaces were introduced and extensively studied by E. Cart an
([C-l]), and have played an important role in Riemannian geometry. In fact, some
fundamental notions in Riemannian geometry were established in modern
terminology in the attempt to understand Cartan's work. Helgason's textbook ([Hel])
systematically treats various aspects of symmetric spaces. See also [Bes-2], [Ise-Ta],
[No-Ko II], [K-5], [Wo-1]. Variational completenes (Corollary 6.6) is originally due
to R. Bott, who used Morse theory of the path space to study the topology of
compact Lie groups and symmetric spaces ([Bo-1], [Bo-Sam], [M-rl]). For symmetric
spaces of positive curvature (i.e., symmetric spaces of compact type of rank 1) see
also [Cha-1], [Ka-2], and for symmetric space of noncompact type see, e.g., [E-l].
See [Cr], [Sa-1, 2], [Та], [Nai] for cut loci in compact symmetric spaces.

CHAPTER V
Curvature and Topology
of Riemannian Manifolds
In this chapter, we study the relation between the metrical properties and
topological properties of Riemannian manifolds after the preparations made in the
previous chapters. Recall that complete simply connected Riemannian manifolds of
constant curvature are the most standard Riemannian manifolds whose properties
are well understood. Now comparing a given Riemannian manifold Μ with one of
these model spaces via comparison theorems, we may investigate geometric
properties of Μ and get information on the topology of M. In the first section, we are
concerned with the fundamental group of Riemannian manifolds from the above
viewpoint. In fact, we will see that the sign of curvatures has a great influence on
the structure of the fundamental group. In §2, we treat the structure of compact
Riemannian manifolds of positive curvature in detail. The so-called sphere theorem
(Theorems 2.1, 2.7) is one of the most celebrated results in global Riemannian
geometry, and has been the driving force for further developments in this field. In §3,
we are concerned with complete noncompact Riemannian manifolds of nonnegative
(also positive) curvature, and we state the fundamental theorems due to J. Cheeger
and D. Gromoll. In the final section, §5, we turn to the Riemannian manifolds
of nonpositive curvature. Although their geometric and topological properties are
different from the positively curved case, the comparison theorems and convexity
play an important role again.
1. Curvature and Fundamental Group
Let Μ be a (connected) C30 Riemannian manifold and πι(Μ, ρ) the
fundamental group of Μ with base point p. To study the relation between the metrical
properties and fundamental group of M, it is useful to consider the universal
Riemannian covering π : Μ —> Μ. We recall some fundamental facts on the covering
spaces (for details see e.g., [Si-Th], [Wo-1]). We call a diffeomorphism μ of Μ a
deck transformation (or covering transformation) of M, if μ satisfies π ο μ = π.
Then the set of all deck transformations of Μ forms a group Γ, which is called the
deck transformation group. Γ acts freely on Μ and properly discontinuously on Μ.
Namely, if μ £ Γ admits a fixed point then μ is the identity, and for any ρ £ Μ
there exists a neighborhood U of ρ such that we have at most finitely many μ £ Γ
with μϋπϋ φ φ. Further, Γ acts simply transitively on each fiber π_1(ρ), ρ £ Μ,
and is isomorphic to πι(Μ, ρ). In fact, let a loop с : [0, 1] —> Μ based at ρ and
ρ £ 7г_1(р) be given. Take a subdivision Δ : 0 = to < · · · < tk = 1 of [0, 1] and
open neighborhoods Ui of c(ti-\) (i = 1, ... , к) such that U{ are evenly covered
193

194 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
by π1 and с([^-ь U]) С U{. Then we construct a lift с of с emanating from ρ
as follows: Let U\ be the connected component of π_1(/7ι) containing ρ and set
с | [£0, t\] := (π | f/i)-1 о с | [ίο, £i]· Next let U<2 be the connected component of
7T_1(^2) containing c(t\) and set с | [t\, £2] ·= (тг | t^)-1 °c \ [t\, £2]· Repeating
this process successively, we get a desired lift с of c. Then a map μ defined by
ρ н-> c(l) gives a deck transformation, which depends only on the homotopy class
of c. It follows that [c] £ πχ (Μ, ρ) ι—► μ £ Γ gives an isomorphism. If Μ = Μ/Γ is
compact, we say that Γ acts uniformly on Μ.
Now suppose (M, 0) is complete. Then the metric g induced on Μ via π is
again complete. In fact, for a given й £ TpM we may easily see as above that
the geodesic 7 with the initial direction й is a lift of a geodesic 7 in Μ with the
initial direction и := Dn(u) £ TPM (ρ = π(ρ)). Since 7 is defined for all parameter
values, so is 7. Also note that deck transformations μ are isometries with respect
to g, because μ*# = (πο μ)*g = n*g = д. Therefore, we may study the structure of
7Γι(Μ, ρ) through Γ, which is a discrete subgroup of the isometry group /(M, g).
First we recall a theorem of S. B. Myers.
Theorem 1.1. Let (M, g) be an m-dimensional complete Riemannian
manifold. Suppose the Ricci curvatures of Μ satisfy p(u) > (m — Ι)δ everywhere for
some positive constant δ. Then Μ is compact and π\ (Μ, ρ) is a finite group.
PROOF. Prom Theorem 3.1 of Chapter IV, Μ is compact and d(M) < n/y/δ.
Since (M, g) is locally isometric to (M, g) and complete, (M, g) satisfies the
assumption of the theorem. Therefore Μ is compact. Then тг~1(р) is finite since it
is a discrete compact subset of Μ, and so is Γ. D
Next we consider the case where the Ricci curvatures are nonnegative
everywhere. First we give some preliminaries. A group G is said to be finitely generated
if G has a finite set of generators. Let {g\, ... , gk} be generators of a finitely
generated group G. For g £ G, the word-length 1(g) of g is defined as the minimal
number s such that g may be expressed as g = g\x · · ·g\s (ei = lor — 1) in terms
of generators. Set 7(s) := §{g £ G; 1(g) < s}. Now G is said to have
polynomial growth if there exist a positive constant a and a positive integer к such that
7(s) < ask holds for any s > 1. G is said to have exponential growth if there exists
a constant a > 1 such that 7(s) > as holds for any s > 1.
Exercise 1. Show that the definition of polynomial and exponential growth
does not depend on the choice of generators. Let G be a free group generated by к
elements. Show that 7(s) = {k(2k - l)s - l}/(k - 1). If G is a free abelian group
generated by к elements, what is 7(s)?
Lemma 1.2. The fundamental group of a compact manifold Μ is finitely
generated.
PROOF. It suffices to show that the deck transformation group Γ of the
universal cover π : Μ —> Μ is finitely generated. Introduce a Riemannian metric g on
Μ and set d = d(M). Then the induced metric g = n*g is complete. Fix a p £ Μ
and take a subset S := {a £ Γ; d(p, ap) < 2d + e} of Γ for e > 0. Then S is finite
because Γ is discrete, and we show that S generates Γ. In fact, for α £ Γ take a
^his means that π is a homeomorphism onto Ui when restricted to each connected
component of -K~l{Ui).

1. CURVATURE AND FUNDAMENTAL GROUP
195
minimal geodesic 7 : [0, 1] —> Μ joining ρ to ap. Then 7 := π ο 7 is a geodesic
loop in Μ based at ρ = π(ρ) and represents an element of πι(Μ, ρ) corresponding
to a. Now choose a sufficiently fine subdivision Δ := 0 = to < · · · < tk — 1 so
that d(^(ti-i), *у(и)) < e (i = 1, ... , k). Denoting by ηι minimal geodesies in Μ
joining ρ to 7(^1), consider loops η{ := η^χ U7 | [ii-ь U] U7"1 based at p. Then
Ь(ъ) <2d + e, and 7 is homotopic to 71 U · · · U 7*.. We define a* £ S (i = 1, ... k)
inductively as follows. Set c*o = id^. Then the end point of the lift of 7* to Μ
emanating from oti-ip may be written as ai(ai-ip) with a* £ S. Since a = a^o- · ·οαι,
it follows that S generates Γ. D
Remark 1.3. Since S is finite, it is possible to choose a sufficiently small
€ > 0 so that {μ £ Γ; 2d < d(p, μρ) < 2d + ε} = φ. Therefore, we may take
{a £ Γ; d(p, ap) < 2d} as a set of generators for Γ.
Next for a compact Riemannian manifold Μ we consider the interior set Xp
of ρ £ Μ (see Chapter III, Definition 4.3). Recall that expp : XP(C TpM) —> Ip
is a difFeomorphism. Let π : Μ —> Μ be the universal cover. For ρ £ π-1 (ρ)
we set D := exppo(Dn(p))~1(Xp). Then D is a domain in Μ containing ρ and
coincides with the set of end points of the lifts in Μ emanating from ρ of unique
normal minimal geodesies joining ρ to points q £ Xp. Now from the properties of
the interior set, we easily see that
(1.1) 7Γ-1(ΙΡ)= [JctD, αΌηβΌ = φ(α^β),
and π I aD : aD —> Xp is a difFeomorphism. Therefore, Xp is evenly covered by
π. We have Μ = \Jaer cxD. In fact, for any point q £ Μ take a minimal geodesic
7 : [0, 1] —> Μ joining ρ to q. Then the end point of the lift 7-1 of 7-1 in Λ/
emanating from q may be written as ap, a £ Γ. Since we have 7([0. 1)) С aD, it
follows that q = 7(1) £ aD. Such a D is called a fundamental domain of π. D is
compact, and clearly volD = vo\D = volM.
We show that there is a close relation between the curvature and growth of
the fundamental group of a Riemannian manifold. The following result is due to J.
Milnor ([M-3]).
Theorem 1.4. Let Μ be a compact Riemannian manifold.
(1) Suppose the Ricci curvatures of Μ satisfy p(u) > 0 everywhere. Then
7Γι(Μ, ρ) has polynomial growth.
(2) Suppose the sectional curvatures Κσ of Μ are negative everywhere. Then
π ι (Μ, ρ) has exponential growth.
PROOF. (1) Take 5 = {μ £ Γ; d(p, μρ) < 2d} as generators of the deck
transformation group Γ of the universal Riemannian cover π : Μ —> Μ, where
ρ £ 7Γ-1(ρ), ρ £ Μ. If α £ Γ satisfies 1(a) < s for a positive integer 5, then we
clearly have d(p, ap) < 2ds. Now let {αχ, ... , a7(s)} be the set of elements α £ Γ
with 1(a) < s. We estimate 7(s). For 0 < e < ip(M), note that any two metric balls
of {Be(aip)}]=i are disjoint. In fact, suppose q £ Be(aip) Π Be(ajp) (г ф j). Then
projecting two minimal geodesies joining αφ to q and ajp to q, respectively, via π,
we get two geodesies in Μ joining ρ to q of length less than e (< ip(M)), which is

196 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
a contradiction. Therefore, noting that U7=i Ве(аФ) С B<2ds+e(p), we have
7(s)
Σ νθ1 Be (aiP) ^ Vo1 B2ds+e (Ρ) ·
i=l
Now since a* are isometries of (M, g), we have from the Bishop-Gromov inequality
(Chapter IV, Theorem 3.3)
volВе(аф) = volΒ6(p) > / ε γ
VOl B2ds+e(Ρ) VOl B2sd+e(P) " \ 2srf + € /
Summing up, we get 7(s) < ((2ds + e)/e)m, and the proof of (1) is complete.
(2) Since Μ is compact, we may assume that Κσ < —a2 holds everywhere for
some a > 0. We set d = d(M), e = ip(M) and take a set S := {μ G Γ; d(p, μρ) <
2d Η- б} of generators for Г. Let D be a fundamental domain of π : Μ —> Μ
determined from 2p. Now for α G Γ and q G aD we join ρ to £ by a minimal
geodesic on which we take points ρ = q\, q2, · · · , qs+i = q such that d(qi, qi+i) <
e (i = 1, ... , s). We may choose βι G Γ (/?ι = id, β3+ι = a) so that ξι G АД and
we set o.i = β^1 ο βί+1 (г = 1, ... , s). Then
d{p, OLip) = ά{βφ, βι+ιρ) < d(/3»j5, &) + d(qu qi+i) + d(qi+u ft+ip) < 2d + £,
namely, a* G 5. Since a = a\ o· · oas, it follows that 1(a) < s. Now let q G Bst(p).
If £ G αί), then we take £i, ... , qs+\ on a minimal geodesic joining ρ to q so
that they equally divide the geodesic and therefore d(qi, qi+\) < e. It follows that
1(a) < s and Bse(p) С Uz(a)<sa^· Now noting that i(M) = +oo for Μ (see
Theorem 4.1 for details), we apply the Bishop theorem (Chapter IV, Corollary 3.2
(1)) and get (see Problem 1 for Chapter IV)
vo\(Bse(p)) > vse(-a2) (:= νο1Βθ6(ρο, M™q2))
fse /sinhaA™-1 J
= am-17o V—^—У
Then we may verify that vol(£S6(p)) > c\eC2S holds for some positive constants
ci, C2 and all sufficiently large s. On the other hand, we have volBse(p) <
vol((jl{a)<saD) = vol(UZ(Q)<saD) = 7(s)vol£> = 7(s)volM. It follows that
7(s) > ci/volM · eC2S, and the proof of the theorem is complete. D
Exercise 2. Let Μ be a complete Riemannian manifold whose Ricci
curvatures are nonnegative everywhere. Then show that a finitely generated subgroup
of πχ (Μ, ρ) has polynomial growth.
Next we show that, for a compact Riemannian manifold Μ, in every nontrivial
free homotopy class of a loop с in Μ there exists a closed geodesic 7 that minimizes
the length in the free homotopy class of с
Let μ G Γ be a deck transformation of π determined by c, and consider a
conjugacy class [μ] (μ φ e), which corresponds to the free homotopy class of c.
Note that for a fixed q G Μ the set {d(q, μξ)); ξ G π-1 (ς)} (С R+) depends only
on the conjugacy class [μ] and has a positive minimum. In fact, for £0 G π-1 (ς) any
ξ G 7г-1(<7) may be written as αξο (a G Г), and we have d(q, μξ) = d(a%, μα%) =
d(qo, α~λμα(ξο)), and the number of α~ιμα (resp., q), such that d(%, α~λμα(%))
(resp., d(q, μξ)) is less than a given R > 0 is finite because Γ is discrete. Now

1. CURVATURE AND FUNDAMENTAL GROUP
197
let / be a function on Μ defined by f(q) := inf{d(q, μξ)\ q £ n~1(q)}. Then / is
continuous. To see this, take an evenly covered neighborhood U and a connected
component U of π_1(/7). For a £ Γ we define a function fa : U —> Я by /Q(g) :=
ίί((π | at/)-1 (ς), μ((π | aU)~l(q))), which is obviously continuous. As before, for
a given R > 0, if we take U sufficiently small, there are only finitely many α £ Γ
such that fa(q) < R(q £ U). Since /(ς) = minQGr{/Q(g)}, it follows that / is
continuous. Now Μ is compact and / assumes its minimum /, which is positive
because Γ acts freely.
Figure 26
Now take qo £ Μ and q0 £ n~l(qo) with / — f(qo) and d(%, μ%) = I. Let 7 :
[0, 1] —> Μ be a minimal geodesic joining qo to μς0· We have 7(1) = μ(7(0)) = μ%.
Further, we see that 7 and μ ο η make a straight angle at μξο· In fact, otherwise,
for € > 0 sufficiently small, a minimal geodesic joining 7(e) to μ(7(£)) is of length
less than /, which contradicts the choice of £0· Therefore Ζ)μ(7(0)·)7(0) = ^(1) and
7 := π ο 7 is a closed geodesic through q0. By construction. 7 belongs to the free
homotopy class of с and minimizes the length in this free homotopy class. Summing
up, we get
Lemma 1.5. (1) Let Μ be a compact Riemannian manifold which is not
simply connected. Then there exists a closed geodesic which minimizes the length in
every nontrivial free homotopy class of a loop.
(2) Let Μ be a compact Riemannian manifold which is not simply connected.
Then there exists a closed geodesic which minimizes the length in the class of ho-
motopically nontrivial closed curves in M.
To see (2), let la denote the length of a closed geodesic 7Q which minimizes
the length in a given nontrivial free homotopy class α of loops in Μ. Then /0 ·=
inf{/Q; α is a nontrivial free homotopy class} is positive because la > 2i(M) > 0
(Chapter III, Corollary 4.14). Now take a sequence lak converging to /0 and a
sequence 7Qfc of corresponding closed geodesies. Since Μ is compact, jak converges
to a closed geodesic 7 of length /0 (taking a further subsequence if necessary). Then
70 is a desired closed geodesic.
Exercise 3. Show that 70 is not homotopic to a point curve and simple (i.e.,
without self-intersection).
Next, for the structure of the fundamental group of a positively curved
Riemannian manifold, the following theorem due to J. L. Synge is fundamental.

198 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Theorem 1.6. Let Μ be a compact even-dimensional orientable Riemannian
manifold whose sectional curvatures are positive everywhere. Then Μ is simply
connected.
Proof. Suppose Μ is not simply connected. Then from Lemma 1.5 (2) we
have a closed geodesic 70 : [0, /0] —> Μ parametrized by arc-length which minimizes
the length in the family of homotopically nontrivial closed curves. Then the parallel
translation Ρ := P{jo)f0 along 70 leaves 70(0) = 70(^0) invariant and therefore
defines a linear isometry of the (m — 1)-dimensional subspace V := 70(0)x of
T7(o)M, where m = dimM. Since Μ is orientable, Ρ \ V defines an element of
SO(m - 1) with respect to a positively oriented o.n.b. of V. Since m - 1 is odd, Ρ
admits 1 as an eigenvalue and its eigenvector is a fixed point of Ρ (see Problem 2
for Chapter I). Namely, we have a nonzero periodic parallel vector field X along 7.
Recall that 70 is a critical point of the energy integral Ε on Св([0, /]), where Β is
the diagonal set of Μ χ Μ and Χ £ Т7оСв([0, /о]). Namely, we have DE(jo) = 0.
Now from the second variation formula (Chapter III, (2.13)) it follows that
Ό2Ε{Ί){Χ, X)
= [ °{(VX(t), VX(t)) - (ВД*), X(t))X(t), i(t))}dt < 0
Jo
because of the curvature assumption. Let je(t) := expyo^eX(t) be variation
curves of 7 generated by X. Then from the above we have L("y€) < у/21{)Е{^е) <
y/2loE(fo) = L(jo) for sufficiently small e > 0, which is a contradiction because
the 76 are homotopic to 70. □
Corollary 1.7. Let Μ be a compact even-dimensional Riemannian manifold
of positive sectional curvature. Then either Μ is simply connected or πχ(Μ, ρ) =
Ζ'2·
Exercise 4. Prove Corollary 1.7.
Now we apply the above idea to the injectivity radius estimate.
Corollary 1.8 (W. Klingenberg). Let Μ be a compact simply connected even-
dimesional Riemannian manifold whose sectional curvatures satisfy 0 < Κσ < Δ
for some positive constant Δ. Then the estimate i(M) > π/\/Δ holds for the
injectivity radius of Μ.
PROOF. By Chapter IV, Corollary 2.8 (3), the first conjugate value along any
normal geodesic in Μ is always greater than or equal to π/y/K. Then if i(M) <
π/\/Δ, there exists a closed geodesic 70 of length 2i(M) by Chapter III, Corollary
4.14. Since Μ is simply connected and therefore orientable, applying the above
argument of the theorem we get a periodic parallel vector field X along 70 and
perpendicular to 70. Then variation curves 76 of 70 generated by X are C°° closed
curves of length L(je) < £(70) = 2i(M) (e > 0). We may assume that *y€(t) φ 0
for sufficiently small e > 0. Let qe be a furthest point on 76 from pe := 76(0).
Since d(pe, qe) < г(М), there exists a unique normal minimal geodesic ae joining
qe to pe. Then considering a variation of ae consisting of normal minimal geodesies
joining points on 76 near q€ to p6, and applying the first variation formula, we see
that σ6(0) is perpendicular to 76 at qe since je is smooth. Now q = 70 (г(М)) is
the unique furthest point on 70 from ρ = 7o(0). Therefore lim6^o qe = q, and we

1. CURVATURE AND FUNDAMENTAL GROUP
199
may assume that σ6η(0) converge to ν £ UqM, taking a sequence en —> 0. Then
the normal geodesic σ emanating from q with the initial direction ν is a minimal
geodesic joining q to ρ and is orthogonal to 7о(г(М)). Then the argument in the
proof of Proposition 4.13 (2) of Chapter III implies the existence of two minimal
geodesies joining ρ and a point in a neighborhood of q of length less than г(М),
which is a contradiction and completes the proof. D
Remark 1.9. For an odd-dimensional compact Riemannian manifold Μ of
positive sectional curvature, πχ(Μ, ρ) is finite but its order may be arbitrary large
(consider, e.g., lens spaces). However, S. S. Chen has conjectured that any abelian
subgroup of 7Γι(Μ, ρ) is cyclic. If Μ is a compact Riemannian manifold of negative
sectional curvature the conjecture is known to be true (see §4 for this and the
fundamental group of manifolds of negative or nonpositive curvature).
Finally we mention the estimate for the first Betti number b\(M). Applying
the Green theorem, S. Bochner obtained the following:
Theorem 1.10. Let Μ be a compact orientable Riemannian manifold with
nonnegative Ricci curvature, namely, p(u) > 0 for any и £ UM. Then b\(M) <
m(:= dimM), where equality holds if and only if Μ is isometric to a flat torus.
Further, if p(u) > 0 everywhere, then b\(M) = 0.
PROOF. By the Hodge-Kodaira theorem the first cohomology group Я1 (M, R)
is isomorphic to the vector space Hl(M) of harmonic 1-forms on M, and we have
&i(M) = dimH^M).2 Now the Lapalacian Δα of a differential 1-form α on Μ is
given by
(1.2) (Δα)< = -9lkVi Vkal + pxl*mgml.
Then applying the Green theorem (Chapter II, Theorem 5.11) to a harmonic 1-form
a, we get
0= / (a, Aa)dM=- [ (ЧкЧкаг)аЧМ + / раа{а1аМ
Jm Jm Jm
= [ (Vfca*)(Vfca;)dM+ / paototdM (a* = дисц).
Jm Jm
Therefore if p(u) > 0 everywhere, for any harmonic 1-form α we get
(1.3) Va = 0, Ric(att, a*) = 0,
where att denotes a vector field on Μ defined by g(a\ X) = α(Χ), Χ £ A'(M),
namely, (ай)г = gijctj = аг with respect to a local chart. Now suppose p(u) > 0
everywhere. Then the Ricci tensor is positive definite and we have att = 0, namely,
any harmonic 1-form а = 0. It follows that b\(M) = 0. Next, in the case of
nonnegative Ricci curvature, by (1.3) any harmonic 1-form α satisfies Va = 0,
namely, α is parallel and is uniquely determined by the value ap €TPM* at ρ £ Μ.
Therefore, b\(M) = dimH^M) < m = dimM. Finally, suppose dimH^M) = m
and take a basis {/?i, ... , /3m} of Hl(M) whose elements form an o.n.b. of TPM*
at every ρ £ Μ. Put X{ := β\ (г = 1, ... , m), which are also parallel and form
an o.n.b. of TPM at every ρ e M. Since VXX{ = 0 (X <E *(M)), it follows that
[Xu Χό] = VXtXj - VXjXi = 0, R{XU Xj)Xk = 0 (г, j, к = 1, ... , m), namely,
the sectional curvature vanishes everywhere and Μ is flat. Note that the X{ are also
2See Appendix 5 for differential forms.

200 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Killing vector fields. Let φ\ denote the (global) flow generated by Xi, which consist
of isometries of M. We have φ\ ο φ^ = φ{ ο φ\ (s, t G Я), because [X{, Xj] = 0.
Now we define an action Φ of R171 on Μ by
(1-4) Ф(*ь...,*т,р):= φΙο-'-οφ^ίρ),
where (£i, ... , £m) G Дт, ρ e Μ. Namely, the abelian Lie group Rm acts on Μ
as an isometry group of Μ, and we show that the action is transitive. In fact,
we fix a ρ G Μ and define a C°° map Фр : Дт —» Μ as Φρ(ί) := Φ(ί,ρ) for
t = (ίι, ... , £m). Then we get ΌΦρ(ί) ^- = Χ{(Φρ(ί)), and Ζ?Φρ(ί) is a linear
isometry. On the other hand, X := Y^UXi is a parallel vector field on Μ and
the integral curve φ?(ρ) through ρ is a geodesic. Now for any point q G Μ take
a minimal geodesic 7 : [0, 1] —» Μ from ρ to q, and let Xp = Y^t%Xi{p) be the
initial tangent vector to 7. Then for a parallel vector field X = Y^UXi, we get
q = φ*(ρ) = φΙλ о · · · ο φ1^ (ρ), since s ι—> y?Jti о · · · о <^™^ (ρ) is an integral curve
of X through ρ and is a geodesic emanating from ρ with the initial direction X.
Therefore, Φ acts transitively. Now the isotropy group Η := { t G Дт; Φρ(ί) =
ρ} at ρ is a closed subgroup of Дт and is discrete since dim Μ = dim Дт. If the
rank η of Я is less than m, then Η spans an η-dimensional subspace of Rm and
Μ = Дт/Я cannot be compact. Therefore, Η is of rank m and is a lattice of Дт.
It follows that Μ is isometric to a flat torus Rm/H. D
Now we estimate 61 (M) when the Ricci curvatures satisfy p(u) >{m — 1)5 (<5 G
Д). Let h : πι(Μ, ρ) —> H\(M\ Ζ) be the Hurewicz homomorphism, which is
defined by assigning to a loop с at ρ the one-cycle determined by с Then /1 is
an epimorphism whose kernel is given by the commutator group G of πι(Μ, ρ).
Now suppose we may choose elements 71, ... , 7^ of πι(Μ, ρ) so that the normal
subgroup (71, ... , 7&) generated by 7j's is of finite index in πι(Μ, ρ). Then the
subgroup /f of H\(M, Z) generated by /1(71), ... , ^(7^) is also of finite index. It
follows that
(1.5) 61 (M) = rank#i(M, Z) = ranktf < k.
Now the problem is how to choose such 7;'s. For that purpose we consider the
universal Riemannian covering π : Μ —» Μ, and identify a 7 G πι (Μ, ρ) with
the deck transformatin determined by 7. We fix an element ρ G 7г_1(р) and set
||71| = d(7p, p). Then we have the following:
Lemma 1.11. For any e > 0 there exist {71, ... , 7^} С πι (Μ, ρ) with the
following properties:
(1) ||7i||<2d(M) + 6 (l<t<fe).
(2) ||7Γ4ΊΙ>« (i#j).
(3) (71, ... , 7*;) 25 of finite index in πι(Μ, ρ).
PROOF. There are only finitely many 7;'s which satisfy (1). We take a maximal
subset {71, ... , 7fc} of πι(Μ, p) which satisfies (1),(2), and show that this set
also satisfies (3). Suppose the contrary, i.e., setting Λ := (71, ... , jk), which
is a normal subgroup of πι(Μ, ρ), we have #(πι(Μ, ρ)/Λ) = +oo. We consider
a Riemannian covering π' : Mf := Μ/Λ —» M. Then the deck transformation
group of π' is isomorphic to πι(Μ, ρ)/Λ and Μ' is noncompact. Choose a p' G
(π')_1(ρ). Then there exists a point q' G M' with d(p', q') = d(M) + €, since M'
is complete but noncompact. On the other hand, we have d(p, n'qf) < d(M) and
there exists a μ' G πι(Μ, ρ)/Λ such that ά(μ'ρ', q') < d(M). Therefore, we get

2. COMPACT MANIFOLDS OF POSITIVE CURVATURE
201
d(p\ μ'ρ') > d{pf, qf) - d{qf, μ'ρ') > e and d(p\ μ'ρ') < d(pf, q') + d{q\ μ'ρ') <
2d(M) + e. Now take a representative μ £ πι (Μ, ρ) of μ' G πι (Μ, ρ)/Λ so that
ΙΗΙ = Им-1 II = d(p, μρ) = d(p', μ'ρ') < 2d(M) + ε. Then for any 7» G Л, we have
1Иг|| = Il7f Vl = dbiP, μ~1ρ) > d{p', (jx')~V) = d{p', μ'ρ') > 6, since π' is
distance decreasing, and μ-1 satisfies (1),(2). This contradicts the maximality of
Λ. D
Now we give an estimate for the order A; of Λ in terms of the volume. In fact,
take € = 2d(M) in the above lemma and consider {Ве/2(ъР)}{=1 т М, which are
mutually disjoint by (2). Prom (1) we get \J^=1 Bd{M)(jip) С B5d{M)(p). It follows
that
к
^vol£d(M)(7ip) <volB5d(A/)(j5).
г=1
Since the 7* are isometries of Μ, the Bishop-Gromov comparison theorem implies
k< VOlff5d(M)(p) < V5d(M)(S)
volBd{M)(p) ~ vd{M)(6) '
Thus we have the following result, due to M. Gromov.
Proposition 1.12. Suppose the Ricci curvatures of a compact Riemannian
manifold Μ satisfy p(u) > (m — 1)6 everywhere. Then
(1.6) h(M)<v5d{M)(6)/vd{M)(6)
holds. Furthermore, if δ < 0, then
(1.7) h(M)< / sinh171-1 sds/ / sinn™-1 dds.
Exercise 5. Give a proof of (1.7).
2. Compact Manifolds of Positive Curvature
Recall that an m-dimensional complete simply connected Riemannian manfold
whose sectional curvatures are a positive constant δ is isometric to the hypersphere
of radius l/y/δ in RTn+1. H. Hopf asked whether a complete simply connected
Riemannian manifold Μ of dimension m is topologically similar to 5m if its sectional
curvatures are not equal to a positive constant but vary only little from a positive
constant. Η. Ε. Rauch obtained an epoch-making result with respect to this
problem in 1951 ([R-l, 2]). Let (M, g) be a complete simply connected Riemannian
manifold whose sectional curvatures Κσ satisfy
(2.1) {0<)δ<Κσ<Α.
Note that such an Μ is compact. Rauch showed that Μ is homeomorphic to the
sphere if δ/A is greater than |. Subsequently the value of δ/A was improved by
M. Berger and W. Klingenberg, and we now have the following best possible result
([B-2], [K-l, 2]; see also [B-ll]).
Theorem 2.1 (Sphere theorem). Let (M, g) be a complete simply connected
Riemannian manifold of dimension m whose sectional curvatures satisfy (2.1).
Suppose δ JA > \. Then Μ is homeomorphic to S171.

202 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Before starting the proof of the theorem we give some preliminaries. We say
that a Riemannian manifold Μ is δ-pinched if Μ satisfies (2.1) for Δ = 1. Note that
we may assume Δ = 1 without losing generality by considering the Riemannian
metric y/Ag if necessary. For the proof we essentially use comparison theorems
stated in the previous chapter and the injectivity radius estimate. In fact, such
methods were developed to attack the problems of determining the structure of
(5-pinched manifolds, or more generally the relation between the metrical properties
and topological properties of Riemannian manifolds.
For p, q G Μ we denote by min(p, q) the set of minimal geodesies parametrized
by arc-length joining ρ to q. The next lemma is due to M. Berger ([B-2]).
Lemma 2.2. Let (M, g) be a compact Riemannian manifold and ρ G M.
Suppose that q G Μ is a furthest point from p, namely, d(p, q) = max{d(p, r); r G M}.
Then for any и G UqM there exists a 7 G min(p, q) such that Z(u, ~7(d(p, q))) <
π/2.
PROOF. We may assume that Κσ > —к2 (к > 0) holds everywhere. Take a
geodesic t(s) emanating from q with the initial direction u. Now choose 7S G
min(p, t(s)) and set as := Z(r(s), -7s(d(p, r(s))). From T.C.T. (II) we have for
sufficiently small 5 > 0
coshA;ii(p, q) < coshkd(p, t(s)) coshks + smhkd(p, r(s)) sinhks -cosas.
Then, noting that d(p, q) > d(p, t(s)), we get
- cosh kd(p, q) · sinh — < sinh kd(p, t(s)) cosh — · cos as.
Δ Δ
Now letting 5 —> 0, we may assume that some sequence {7Sn} converges to 7 G
min(p, q) as sn —> 0. Then ctSn —> Z(u, — 7(d(p, q))) =: c*o, and cosao > 0 follows
from the above inequality as sn —> 0. D
The following injectivity radius estimate is due to W. Klingenberg and T. Sakai
([K-Sa]) and to J. Cheeger and D. Gromoll ([Ch-Gr-3]).
Theorem 2.3. Let (M, g) be a complete simply connected Riemannian
manifold satisfying (2.1). Suppose δ/Δ > 1/4. Then i(M) > π/y/A.
The proof is presented in Appendix 3.
Remark 2.4. Note that when Μ is of even dimension we get our assertion
assuming only (2.1) (see Corollary 4.8). In general, i(M) > π/y/A does not hold if
we only assume (2.1). In fact, for Berger spheres with δ/A < 1/9 we have i(M) <
π/y/A. Further, for any e > 0 we have examples of 7-dimensional M's which are
(5-pinched with δ/A < ^§7 and i(M) < e. As for these remarks, see Appendices 2
and 3. These examples show that the classification problem of compact Riemannian
manifolds of positive curvature is very difficult.
Now we turn to the proof of the sphere theorem. We may assume that Δ = 1,
namely, Μ is a compact simply connected Riemannian manifold such that
(2.2) δ<Κσ<\, δ > \.
Then from Theorem 2.3 and Chapter IV, Corollary 2.8(3), we have π/2\ίδ < π <
d{M) < π/уД.

2. COMPACT MANIFOLDS OF POSITIVE CURVATURE
203
Lemma 2.5. Under the above assumption take p, q £ Μ with d(p, q) = d(M)
(> п/2уД). Then for any r e Μ we have either d(p, r) < π/2\/δ(< π) or
d(q,r) <n/2V6(<n).
PROOF. It suffices to show that for r £ Μ with d(p, r) > π/2\/δ we have
d{q, r) < π/2π\/δ. For τ £ min(p, r), by Lemma 2.2 there exists a 7 £ min(p, q)
such that α := Z(f (0), 7(0)) < π/2. Then from T.C.T. (II) we get
cos \fdd(q, r) > cos \fdd(p, r) cos \fdd(p, q)
+ sin \fdd(p, r) sin \fdd(p, q) · cos α
> cos \fdd(p, r) cos \fdd(p, q) > 0,
namely, d(<7, r) < π/2\/δ. D
By Theorem 2.3, expr : £π(οΓ) —» Μ is a diffeomorphism onto an open subset
Втг(г) of Μ for every r e M, and from Lemma 2.5 it follows that Βπ(ρ) U Βπ(ς) =
Μ. Therefore, Μ is the union of two embedded m-dimensional open disks. Then
Μ is homeomorphic to the sphere according to a theorem of differential topology
(see [Rus], p. 50). In our case we may construct a homeomorphism from Μ onto
5m in a rather elementary fashion, which we will now do.
Lemma 2.6. Under the assumption of Theorem 2.1. let % be an arbitrary
geodesic emanating from ρ with 7(0) = и £ UPM = Sm_1. Then there exists a
unique 0 < t\(u) < π/2\/δ which satisfies d(p, ~ju(ti(u))) = d(q, %(*i(u))). and
и £ UPM i-> ti(u) £ Я+ is a continuous function.
PROOF. The existence of t-[(u) is clear, since d(p, 7^(0)) = 0 < d(q, 7U(0)) =
d(p, q) and d(p, 7η(π/2ν^)) = π/2\/ό > d(g, ηη(π/2\β)) by Lemma 2.5. We
show the uniqueness. Suppose there exist (0 <)t\ < £2 (< π/2\/δ) such that
d(p, 7u(*0) = d(g, 7u(i<)) = U(i = 1, 2). Then we have
*2 = d(p, 7и(*г)) = d(p, 7u(*i)) H- d(7u(ii), 7n(^))
= d(g, 7u(*i)) + d(7u(ii), 7u(*2)) < <% 7и(*г)) = t2
and the equality sign holds in the last inequality. Then ju(U) lies on a minimal
geodesic joining q to 7n(£2), by Chapter II, §2, Exercise 5. This implies that q
coincides with p, which is a contradiction. Finally, we show the continuity of t\ (u).
Suppose un £ UPM —» u (n = 1, 2 ...), and let t0 be any accumlation value of
{ti{un)}· Taking a subsequence {иПк} with ti(unk) -> £0, we get d(p, 7u(*o)) =
limfc^ood(p, 7unfc(*iKifc))) = limfc-oo d(g, 7u„fc (*iKifc))) = d(q, 7u(M) from the
continuity of the distance function. Then we have to = t\ (u) from the uniqueness
of ti(u), and therefore ti(un) —» £i(u). D
Proo/ 0/ Theorem 2.1. Prom Lemma 2.6 we see that the map u £ /7pM i->
7u(*i(w)) £ Μ is a homeomorphism from 5m_1 onto {r £ M; d(p, r) = d(q, r)}.
Now we construct a map Φ from the unit sphere 5m onto Μ as follows. Let ρ £ 5m
and £ the antipodal point of p. Let ρ, ς £ Μ be as above. We choose a linear
isometry / : 7>5m -> TPM, and for u £ /7p5m we define
(expp{2t · ί! (Λ2)/π · lu), 0<t<n/2
expq{2{n - t)/n · exp-1(7/u(ii(/u))},
7r/2<t< π.

204 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Now we regard p, q as the north and south pole, respectively. Then the northern
hemisphere {expp tu; й G /7p5m_1, 0 < t < π/2} is mapped by Φ homeomorphically
onto {r G M; d(p, r) < d(q, r)}, recalling the definition of t\(u) and the fact that
t\(u) < π(< г(М)). Next, in the second case of the above definition, note that
Ίΐύ^ι(Ιΰ)) = expp(£i(/u)/u) is contained in Bn(q) and exp"1 : Bn(q) —» Bn(oq) is
a diffeomorphism. It follows that the two definitions in (2.3) coincide on the equator
grn-ι . ^ _ π^25 and φ maps 5m_1 homeomorphically onto {r G M; d(p, r) =
d(q, r)}. Finally, in the southern hemisphere π/2 < £ < π, the south pole q =
ехррпй is mapped to q, and Φ maps the southern hemisphere homeomorphically
onto {r G M; d(p, r) > d(q, r)}. Therefore, Φ : 5m —» Μ is a homeomorphism. D
Now we give a generalization of the sphere theorem in terms of the curvature
and diameter.
Theorem 2.7. Let Μ be an m-dimensional complete Riemannian manifold
whose sectional curvatures satisfy Κσ > δ everywhere for some δ > 0. Note that
d(M) <n/y/6 by the Myers theorem. Then
(1) If d(M) = n/y/δ, then Μ is isometric to the sphere S™ of constant
curvature δ.
(2) If d(M) > π/2\/δ, then Μ is homeomorphic to the sphere.
(1) is nothing but the Toponogov or Cheng maximal diameter theorem (Chapter
V, Theorem 3.5). Under the assumption of (2), M. Berger first proved that Μ is a
homotopy sphere, using Morse theory for the path space. Then K. Grove and K.
Shiohama ([Gro-S]) proved (2) more directly. Namely, taking points p, q G Μ with
d(p, q) = d(M), they considered the behavior of the distance function dv (dp(r) :=
d(p, r)) to p, and they discovered the notion of critical points of dp which turned
out to be very useful. M. Gromov applied the notion of critical points of dv to
various situations.
Now we consider the distance function dv to any point ρ G Μ of a general
complete Riemannian manifold. As stated in §4 of Chapter III, dp is smooth on
Μ \ ({p} U Cp). On the other hand, ρ is the unique point at which dp assumes its
minimum 0. Now we give the following definition.
Definition 2.8. q φ ρ is said to be a critical point of dp (or critical for p), if
for any и G UqM there exists a 7 G min(p, q) which satisfies
(2.4) Z(u, -7(d(p, <?))) < π/2.
We also call ρ a critical point of dp, for convienience.
For instance, for a compact Riemannian manifold Μ and ρ G M, a point q
which is furthest from ρ (that is, satisfies d(p, q) = max{d(p, r); r G M}) is a
critical point of dp by Lemma 2.2. If q φ ρ is critical for p, then by definition
there exist at least two minimal geodesies joining ρ to q. Therefore, q belongs to
the cut locus Cp and dp is not differentiable at q (see Chapter III, Lemma 4.8).
Nevertheless, we use the notion critical, since if r is noncritical for ρ then it is
possible to allow a neighborhood of r to come nearer to ρ by an isotopy of Μ, as is
seen in the following.

и
)
Ρ
Figure 27
Suppose r € Μ is noncritical for p. Then we may find xr € UrM and 0 < ar <
π/2 such that
(2.5) Z(xr, -y(d(p, r))) < ar
for any 7 £ min(p, r). In fact, by definition there exists а и € UrM which satisfies
Z(u, -7(d(p, ς))) > π/2 for any 7 £ min(p, q), and
inf{Z(u, -7(d(p, ς))); 7 € min(p. r)} > π/2
since min(p, r) is compact. It suffices to set xr := —u. Further, we may choose a
sufficiently small neighborhood Ur of r such that a unit vector field X on Ur
obtained by parallel translating xr along normal minimal geodesies from г to arbitrary
points η of Ur satisfies
(2.6) Z(Xn, -y(d(p, η))) < αΓ.
In fact, suppose the contrary. Then there exist r^ —» r, 7^ € min(p, r^) (A: =
1, 2, ...) such that Z(Xrfc, —7fc(d(p, rfc))) > a^. Taking a subsequence of {7^}
which converges to some 7 € min(p, r) and considering the limit, we have the
contradiction Z(Xr, ~7(d(p, r))) > ar. Then the following holds.
Lemma 2.9. Let ρ € Μ and take concentric metric balls Bt2(p) С Bei(p)
centered at p. Suppose there exist no critical points of dp on A := Bei(p) \ Bt2(p).
Then for any open neighborhood U of Bei(p), there exists an isotopy of Μ which
maps B€l(p) into B€2(p) and fixes the points outside U.
PROOF. Since any point г of a compact set A is noncritical for p, we may
take a pair (J7r, Xr) which satisfies (2.6). We may choose finitely many such pairs
{{Ui, Xi)}iLi so that Ui С U and {Ui} is an open covering of A. Then we put α :=
max α* (< π/2), where the а* (г = 1, ... , Ν) are determined by (2.6) corresponding
to (Ui, X{). Take a partition of unity {pi} subordinate to {Ui} and note that
supppj С Ui. Now extend piXi to a vector field on Μ by setting 0 outside Ui and
consider a vector field X = Σΐ=ι piXi on M. Then X vanishes on Μ \ О (О :=
U^ Ui). Setting a = cos a, for r e A, 7 e min(p, r) we get
{Xr, -7(d(p, r))> = £>(г)(ВД, -7(d(p, r))>
> ^pj(r)cosa = o(> 0).

206 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
In particular, Xr φ 0 on A. We set b := min{||Xr||; r G A} > 0 and β :=
max{pr||; r G A}. Then it follows that (Xr/||Xr||, -η(ά(ρ, r))> > b/β and there
exists 0 < α0 < π/2 such that Z(Xr, -η(ά(ρ, r))) < a0 for any 7 G min(p, r), r G
A X is complete since X has compact support. Let <pt be the global flow on Μ
generated by X, and note that the initial directions of minimal normal geodesies
joining r G A to <£*(r) converge to Xr/||Xr|| as t —» 0. It follows that we may choose
a sufficiently small δ > 0 so that for any r G Л, £ G (0, δ] the angle between the
initial direction of the minimal normal geodesic joining r to <pt(r) and —7(d(p, r))
is less than c*o, where 7 G min(p, r) is arbitrary.
Now we show that for a given /0 > 0 there exists a δι > 0 such that
(2.7) d(p, r) - d(p, <pt(r)) > 6cosa0 · t
for r G Л with d(p, r) > i0 and 0 < t < 6\. We show this for any 0 < В < b cos a0
instead of 6cosao. In fact, otherwise there exist tn J, 0 and rn £ A —> r with
d(p, rn) > ^0 for some /0 > 0 such that d(p, rn) - d(p, iptn(rn)) < Btn. Now
we may assume that Κσ > —к2 on a compact domain BR(p) for R ^> e\. Then
we apply T.C.T. (II) to geodesic hinges consisting of normal minimal geodesies
cn G min(rn, p) and cn G min(rn, ^tn(r„)), and get
(2.8)
cosh Ы(р, r„) - coshfcd(p, ψιη{τη))
> 2 sinh — —-^—— < sinh kd(p, rn) cosh — —-^—— · cos a0
^ Ι Δ
-C08hfcd(p,rn)riDhfcd(r"'^-(r"))}.
Then dividing the both sides of the above inequality by ktn and letting η —» +oo,
we get from
d(p» rn) - d(p, ¥>tn(r„)) < Btn and lim d(
)/<» = PMI,
n—>+ос
the inequality BsinhA;d(p, r) > b cos a0 sinh kd(p, r), which is a contradiction.
Now from (2.7) it follows that <£я(В€1 (р)) С Bt2(p) for sufficiently large Я > 0.
In fact, first note that t 1—> d(p, ipt{r)) is monotone decreasing, by (2.7). Suppose
ipt(B€l(p)) <£ Bt2{p) for all t > 0. Then there exist £n —» +00, rn G B€l(p) with
d(p, iptn(rn)) > €2 (0 < t < tn). For an accumulation point r0 G B€l(p), we have
d(p, ^i(ro)) > ^2 for all £ > 0, which contradicts (2.7) with /0 = e2. Then y>R gives
a desired isotopy. D
Now we get a sphere theorem corresponding to Reeb's theorem (see [M-l]) for
smooth functions.
Proposition 2.10. Let Μ be a compact Riemannian manifold and ρ G M.
Suppose dp has only two critical points p, q. Then Μ is homeomorphic to the
sphere.
PROOF. By the assumption and Lemma 2.2, we have a unique point q G Μ with
d(p, q) = max{d(p, r); r G M}. Take i(M) > e2 > 0 so that £€2(p) Π Bt2(q) = φ.
Now we may choose an e\(< d(p, q)) such that d(p, q) — 61 > 0 is sufficiently small
and B€l(p) D Μ \ B€2(q). In fact, otherwise we get a sequence {гп}™=1 С М
with d(<7, rn) > €2 and d(p, rn) —» d{p, q). Then an accumulation point r of {rn}
is a furthest point from ρ different from q, and we get a contradiction. Now for

2. COMPACT MANIFOLDS OF POSITIVE CURVATURE
207
A := Bci(p) \ B<l2{p) the assumption of Lemma 2.9 is satisfied, and we have an
isotopy ipR of Μ such that <^я(В€1(р)) С Bt2(p). It follows that if r £ B€2(q) then
we have у>д(г) <E φΕ{Μ\Β,2{α)) С у>д(Ве1(р)), namely, Ββ2(ς[)υ^1(Ββ2(ρ)) = Μ.
Therefore, M may be covered by two embedded disks and is homeomorphic to the
sphere, as remarked before. D
Now we turn to the proof of Theorem 2.7. It suffices to show that for p, q £ Μ
with d(p, q) = d(M), the critical points of dp are given by {p, q}. Suppose there
exists another critical point q\ of dp. We may assume d(p, q) < π/y/δ, and first we
consider the case where d(q, qi) < п/2у/б. Then for σ £ min(^i, q), there exists a
7i £ min(tfi, q) such that α := Ζ(σ(0), 7ι(0)) < π/2. It follows from T.C.T.(II)
that
cosVod(p, q) > cos у/б d(p, q\) cos у/б d(q, q\)
+ sin y/δ d(p, qi) sin у/б d(q, q\) · cos a.
Noting that d(p, qi) < d(p, q), d(q, qi) > 0 and d(p, q) > п/2у/б, we get
0 > cos у/б d(p, q) (1 - cos у/б d(q, qi))
> sin у/б d(p, q\) sin у/б d(q, q\) · cos α > 0,
which is a contradiction. Second, suppose d(q, q\) > п/2у/б. By the same argument
as above we have
0 > cos y/6d(p, q) > cos у/б d(p, qi) cos у/б d(q, qi).
Therefore cos у/б d(p, q\) > 0, namely, d(p, q\) < п/2у/б holds. Then, again noting
that d(p, q) > d(q, qi), we have from (2.9)
0 > cos y/6d(p, q) · (1 - cosV^^p, #i)) > 0,
which is again a contradiction. D
Next we consider the 1/4-pinched case. The next result is due to M. Berger
([B-2]).
Theorem 2.11 (Rigidity theorem). Let Μ be a complete simply connected
Riemannian manifold whose sectional curvatures Κσ satisfy (0 <) б < Κσ < Δ
everywhere. Suppose б/A > 1/4. Then π/y/A < d(M) < π/у/б, and
(1) If d(M) = π/у/б, then Μ is isometric to the sphere of constant curvature
6.
(2) If d(M) > π /у/б, then Μ is homeomorphic to the sphere.
(3) // d(M) = π/\/Δ, then Μ is isometric to a compact simply connected
symmetric space of positive sectional curvature.
Remark 2.12. In (3), Μ is isometric to either the sphere of constant curvature
Δ or one of the complex projective space, quarternionic projective space, Cayley
projective plane with the canonical Riemannian metric whose sectional curvatures
satisfy Δ/4 < Κσ < Δ (see Chapter IV, §6.2, (III)). Therefore, the assumption
6/A > 1/4 (resp., d{M) > п/2у/б ) in Theorem 2.1 (resp., Theorem 2.7) is best
possible. For a generalization of the above theorem we refer to [Gr-Gro].
In Theorem 2.11, (1) follows from the Toponogov maximal diameter theorem,
and (2) follows from Theorem 2.7, since d(M) > π/yfA > п/2у/б. In the following

208 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
we give a proof of (3) assuming Δ = 1. Then we get d(M) = i(M) = π because of
Theorem 2.3 and the assumption of (3). It follows that for any point ρ G Μ the
tangent cut locus Cp of ρ is a hypersphere centered at the origin in TPM of radius
π.
Lemma 2.13. Under the assumption of (3) the cut locus Cp of any ρ G Μ is
a (connected) totally geodesic submanifold of M. Any normal geodesic emanating
from ρ is a closed geodesic of length 2π, and intersects Cp perpendiculary at the
parameter value π.
Proof. Cp is clearly connected. We show that Cp is a totally geodesic sub-
manifold. If Cp consists of just one point, our assertion is obvious. Let x, у G Cp\
we show that any normal geodesic τ : [0, /] —» Μ joining χ to у with length less than
2π is contained in Cp. In fact, otherwise there exists a point r := τ (to) (0 < to < I)
which realizes the distance d(p, r([0, to])), and d(p, r) < π. Then, by the first
variation formula, 7 G min(p, r) is perpendicular to τ at r = τ (to)- Since / < 2π, at
least one of to and I — to is less than π, and we may assume to < η without loss of
generality. We apply T.C.T. (II) to the geodesic hinge (r; τ-1 | [0, to], 7_1) with
vertex r. Noting that Κσ > 1/4, d(p, r) < π, we get
cos(d(p, x)/2) > cos t0/2 · cos(d(p, r)/2) > 0,
Figure 28
namely, d(p, χ) < π, which is a contradiction, and τ is contained in Cp.
Furthermore, any minimal normal geodesic 7* joining ρ to r(t), 0 < t < /, is perpendicular
to τ at the parameter value π. Applying T.C.T.(II) to a geodesic hinge (r(t);
τ | [£, t + б], 7^_1) with vertex r(£) for small € > 0, we get
0 = cos(d(p, r(i + e))/2) > cose/2 · cos(d(p, r(t))/2) = 0
and equality holds in T.C.T. (II). From Chapter IV, Remark 4.6, we may span
τ | [ί,ί + б], 7^-1 with a totally geodesic triangle Δ(ρ, r(t), r(t + e)) of constant
curvature 1/4, and we get a Jacobi field У in Μ along 7* which may be written
in the form Y(s) = sin | · E(s), where E(s) is a parallel vector field along 7^ with
Ε(π) = f(t). Taking the limit as t —> 0 (resp., t —> /), we have a minimal normal
geodesic ηχ G min(p, ж) (resp., 7y G min(p, 1/)) perpendicular to τ and a Jacobi
field along jx (resp., 7y) of the above form.
Now for x, у G Cp take a minimal normal geodesic τ joining a; to 1/. Since
L(t) < π, τ is contained in Cp, and therefore Cp is a locally convex closed set.

2. COMPACT MANIFOLDS OF POSITIVE CURVATURE 209
By Chapter V, Theorem 5.5, Cp is a topological manifold with boundary, and
its interior TV is a A;-dimensional totally geodesic С°° submanifold. To prove our
assertion it suffices to show that dCp = φ. Suppose dCp φ φ. Then by Lemma
5.7 of Chapter IV there exists a q G dCp such that (Cp)^ \ {op} is a half-space
#, where (Cp)(g) denotes the tangent cone at q. Let τ : [0, /] —» Μ (Ι < π) be
a normal minimal geodesic joining q to a point r e N, and note that r((0, /]) С
ЛГ, f (0) G (£7ρ)(ς) \ {op} = H. Now take a geodesic 7 G min(p, q) perpendicular
to τ at q and a Jacobi field Υ along 7 such that Y(s) = sin | · ^(s), where E'(s)
is a parallel vector field along 7 with £7(π) = f(0). Let at be variation curves of 7
generated by the Jacobi field —Y(s) along 7, which are normal geodesies emanating
from p. We have д/dt \t=o ««(π) = -У(π) = -f(0). Noting that at(n) G Cp, we
have — f(0) G (£ρ)(ς) = Я, which is a contradiction. Therefore, Cp is a totally
geodesic submanifold of M. Further, for any q G Cp and ж eTqCp we get a Jacobi
field Y(s) = sin I · E(s) along any 7 G min(p, ς) as above with Ε (π) = χ. Recall
that the tangent cut locus Cp is the hypersphere in TPM of radius π. It follows
that a C°° map expp : Cp —> Cp has rank к := dimCp at every point и G Cp and
is a submersion. Therefore, the kernel Mu of Dexpp(u) at u G Cp is of dimension
m - к - 1. By the Gauss lemma, Λ/*η is contained in TUCP and u »-> Mu defines an
(m — k — l)-dimensional distribution N, which is easily seen to be involutive. Then
the maximal integral manifold S of Μ through и is mapped via expp to one point
q = expp и of Cp, and an embedded submanifold.
Now noting that 7ξ(1) = expp£ = q for ξ G 5, where 7ξ denotes a geodesic
emanating from ρ with the initial direction ξ, we define a C°° map Φ : S —►
/7ςΜ Π T^Cp1 by Φ(ξ) := ^7ξ(1). Then Φ is an immersion. In fact, for w G Τξ5
take a curve ξ3 in 5 with £0 = w· Then for a Jacobi field У(£) := 9/9s |s=o (7ξβ (0)
we have Y(l) = 0 and УГ(1) = Vd/dtlt=ld/ds \s=0 Ы.(t)) = Vd/da\e=0(W)) =
πΌΦ(ξ)(ιυ). Thus ϋΦ(ξ) is injective, and on the other hand we have dimS =
dim(/7gMnTgCpL) = m-k-l. It follows that Φ(5) is an open subset of UqMnTqC~.
We may easily check that Φ(5) is also a closed subset. Therefore, if к < m — 1.
then /7ςΜ Π TgCp1 is connected and Φ(5) = /7ς η Τ4Ορ. In particular, for any
ж G /7ςΜ Π TqCp there exists a normal geodesic 7 emanating from ρ such that
7(π) = x. It follows that any normal geodesic emanating from ρ returns to ρ at the
parameter value 2π, and is a closed geodesic because г(Л/) = π.
Next suppose к = m — 1. In this case any point of Cp is not a tangent
conjugate point, and for any normal geodesic 7 emanating from ρ there exists a normal
geodesic σ emanating from ρ different from 7 which satisfies 7(71-) = σ(π). Because
i(M) = π, 7 U σ~ι is a closed geodesic of length 2π (Chapter III, Corollary 4.14),
and our assertion holds. D
Now we turn to the proof of Theorem 2.11. We examine the behavior of Jacobi
fields along a normal geodesic 7 emanating from p. Set q = 7(π) G Cp. We
denote by J^ij) the vector space of Jacobi fields Υ along 7 with ^(0) = 0, which
are perpendicular to 7, and note that dimj^-^) = m - 1. Then the subspace
JXjA := [Y g J7'(jL(7); Y(t) = sin | · E(t), where £■(£) is a parallel vector field along
7 with Ε'(π) G TqCp} is of dimension /г, by the previous lemma. On the other
hand, if к < m - 1 then q is conjugate to ρ along 7 with multiplicity m - к — 1, and
Jx := {Z G J'oL(7); Z(7t) = 0} is a subspace of dimension m-k-l. Let Ζ G J7i.

210 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Take the sphere 5m(l) of constant curvature l,pG 5m(l), and a linear isometry
/ : TPM —» Tp5m(l). Let 7 be a normal geodesic in 5m(l) emanating from ρ with
the initial direction /7(0), and define a vector field Ζ along 7 by
Z(t):=P(7)?o/oP(7)J(Z(t)).
Then Ζ vanishes at £ = 0, π, and from Κσ < 1 we get
0 = ϋ2Ε(Ί)(Ζ, Ζ)
= f {<vz(o, νζ(θ) - (R№), 7(0Ж0, ^(0)K
7o
> f {(VZ(0, VZ(0) - <Ζ(ί), Ζ(ί))}Λ = D2Etf)(Z, Ζ) > 0.
./о
Note that the last inequality holds since Ό2Ε{η) is positive semidefinite on the
tangent space T^Cg to the path space С β of 5m(l), where we set Β = {ρ} x{q}, q =
7(1). In particular, we get D2E(j)(Z, Z) = 0, and Ζ belongs to the null space
of Ό2Ε(η). It follows that Ζ is a Jacobi field on 5m(l) and may be written as
Z(t) = s'mtE(t), where Ё is a parallel vector field along 7. Therefore Ζ e J\ itself
also may be written as Z(t) = sintE(t), where E(t) is a parallel vector field along
7, by the definition of Z.
Now we define a symmetric linear transformation R(t) of ΤΊ^Μ by R(t)u :=
i?(u, 7(£))7W· Then for 0 < t < π, from the Jacobi equation we see that Y(t) (Y e
J1/4 \ {0}) is an eigenvector of R(t) with eigenvalue 1/4, and Z(0 {Z e Ji) is an
eigenvector of R(t) with eigenvalue 1. Therefore, (Y{t), Z(t)) = 0 and jit)1- are
spanned by mutually orthogonal {^(0; Υ € *7i/4} and {Z(0; Ζ G J^i}.
Now we show that Μ is locally symmetric. It suffices to show that the (local)
geodesic symmetry sp : Βπ(ρ) —> Βπ(ρ) at ρ G Μ is an isometry. Recall that
Dsp(p) = -id. Let r (^ p) G Βπ(ρ) and take a (unique) normal geodesic 7
joining ρ to r, namely, 7(0 = г (0 < t < π). Let w G '"HO"1· Then there exist
7 G J1/4 and Ζ G Ji such that w = Y(t) + Z(0- Further we may write Y(t) =
sin %Ei(t) and Z(0 = sintE2(t), where ΕΊ, E2 are parallel along 7. Now we have
Dsp(r){Y(t) + Z(0) = Y(-t) + Z(-0, because У(0) = Z(0) = 0 and by the
definition of sp. It follows that
Dsp(r) (sin -Ei(t) + smtE2{tU = -sin-£i(-0 - sintE2{-t),
from which we get ||Z?sp(r)iu|| = \\w\\. Since Dsp(7(0) = _7(_0ϊ we see tnat
Dsp(r) : ΤΊφΜ —> ΤΊ(_ήΜ preserves the norm, and therefore is a linear isometry.
Since Μ is simply connected and locally symmetric, Μ is a symmetric space of
positive curvature. This completes the proof of the theorem. Finally, we note that
in (3) we have к < m - 1, because otherwise Μ is isometric to the real projective
space of constant curvature 1/4 (see Chapter IV, §2, Example (III)). □
Remark 2.14. For another proof of the sphere theorem (Theorem 2.1) see
[Esc-2], which is based on M. Gromov's idea. Since on S™ (m > 7) there are
many exiotic differentiable structures (i.e., not diffeomorphic to the standard C°°
manifold structure), it is natural to ask whether a δ (> l/4)-pinched complete
simply connected Riemannian manifolds Μ is diffeomorphic to the sphere with the
standard manifold structure. After the pioneering works of D. Gromoll and Y.
Shikata, there are many contributions, due to K. Shiohama and M. Sugimoto, E.

3. OPEN MANIFOLDS OF NONNEGATIVE CURVATURE
211
Ruh, K. Grove, H. Karcher and E. Ruh, H. C. ImHof and E. Run, Y. Suyama,
etc. ([Gr-1], [Shik], [Sug-S], [Gro-Ka-Ru], [ImH-Ru], [Suy]). See also the excellent
survey article [S-3] by K. Shiohama. It is now known that Μ is diffeomorphic to
the standard sphere if δ > 0.654 ([Suy]).
As a generalization of the sphere theorem, M. J. Micallef and J. D. Moore [Mic-
Mo] showed by using harmonic maps that a compact simply connected Riemannian
manifold Μ is homeomorphic to the sphere if the sectional curvatures Κσ of Μ
satisfy min Κσ/ max Κσ > 1/4 for σ С TPM at every point ρ G Μ (or if its curvature
operator is positive definite at every point of Μ). Ε. Ruh ([Ru-2]) also proved that
there exists a 6(m) with 1/4 < δ(τη) < 1 such that a compact Riemannian manifold
Μ of dimension m is diffeomorphic to one of the spherical space forms of constant
curvature 1 if Κσ satisfies minΚσ/max.Ka > 6{m) at every point of Μ (see also
[Ni], [Hu]). W. Seaman ([Se]) showed that 4-dimensional compact oriented 0.188-
pinched Riemannian manifolds are homeomorphic to either S4 or CP2 by using
the Bochner technique.
It still seems very difficult to classify compact Riemannian manifolds of
positive sectional curvature. It is yet not known whether S2 x S2 admits a Riemannian
metric of positive sectional curvature, although the Riemannian product metric of
standard Riemannian structure of S2 is of nonnegative sectional curvature. Also it
is not known whether there are exotic spheres admitting a Riemannian structure of
positive sectional curvature, although D. Gromoll and W. Meyer ([Gr-Me-2])
constructed on one 7-dimensional exotic sphere a Riemannian metric of positive
sectional curvature almost everywhere. On the other hand, N. Hitchin ([Hit]) showed
that some exotic spheres cannot admit a Riemannian metric of positive sectional
curvature (in fact, of positive scalar curvature).
Remark 2.15. For a compact Riemannian manifold Μ with positive Ricci
curvature, R. Hamilton showed that if dim Μ = 3 then Μ is diffeomorphic to one
of the three-dimensional spherical space forms, by deforming a given Riemannian
metric to the metric of constant curvature 1 via solving a heat equation ([Ha-
1]). As a pinching version of the Bishop theorem (Chapter HI, Corollary 3.2 (2)),
G. Perelman ([Pe-1]) proved that there exists an e(m) > 0 such that a compact
Riemannian manifold with Ricci curvature p{u) > m — 1 and volume vol (M) >
am -б(га) is homeomorphic to Sm (K. Shiohama ([S-l]) first obtained such a result
under the assumption of a lower bound for the sectional curvature). On the other
hand, if we replace the assumption on the volume by one on the diameter we need
some additional conditions to show that Μ is homeomorphic to the sphere (see
[An-l], [Ot-1]).
We remark that for a compact Riemannian manifold with nonnegative Ricci
curvature such that the Ricci curvature is positive definite at one point, it is possible
to deform the Riemannian metric to a Riemannian metric of positive Ricci curvature
([Au-l], [Eh]).
Finally, for manifolds of positive scalar curvature, we refer to [Gr-2], [G-L],
[Hit], [Scho-Y-1], [Ro-Sto].
3. Open Manifolds of Nonnegative Curvature
In this section we are concerned with complete noncompact Riemannian
manifolds of nonnegative or positive curvature. Recall that for any point ρ of a complete
noncompact Riemannian manifold, there exists a ray 7 emanating from p. Then

212 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
we get the Busemann function (Chapter IV, §5)
67(<7):= lim {t-d(q,7(t))},
t—*+oc
which plays a fundamental role in the following.
3.1. In this subsection we study the structure of complete open (i.e., noncom-
pact) Riemannian manifolds whose sectional curvatures satisfy Κσ > 0 everywhere.
The following arguments are due to the pioneering work of J. Cheeger and D. Gro-
moll ([Ch-Gr-1]).
Proposition 3.1. Let Μ be a complete noncompact Riemannian manifold of
nonnegative sectional curvature and 7 a ray on M. Then the Busemann function
b1 determined by 7 is a convex function.
PROOF. Recall that t - d(q, ^(t)) converges unifomly to 67(g) on compact
subsets of Μ as t —» +00, and b1 is Lipschitz continuous. Therefore, to see the
convexity it suffices to show that for any normal geodesic on Μ and for any a > 0
(3.1) 267(c(0)) < 67(c(a)) + 67(c(-a)).
Figure 29
Now we set x(t) := d(-r(t), c(a)), y(t) := d(-y(t), c(-a)), z(t) := d(7(i),c(0)),
and we denote by a(t) the angle at c(0) between the tangent vectors to с and a
minimal geodesic 6t joining c(0) to ^(t). Applying T.C.T. (II) to the geodesic hinges
(c(0); c, 6t), (c(0); c_1, 6t), we get
x(t) < z(t)y/l + a2/z2(t) - 2acosa(t)/z(t),
y(t) < z(t)y/l + a2/z2(t) + 2acosa(t)/z(t).
Now note that
r.h.s. -l.h.s. of (3.1) = lim {2d{c(0), -y(t)) - d(c(a), ^{t)) - d{c(-a), -y(t))}.
t—>+ос

3. OPEN MANIFOLDS OF NONNEGATIVE CURVATURE
213
= lim
t—>+oc
Since z(t) —► +00 as t —► +00, we get from the above inequalities
r.h.s. -l.h.s. of (3.1) > lim z(t) h - y/l + a2/z2(t) - 2acosa(t)/z(t)
->/l + a2/z2(t) + 2a cos a{t)/z{t)\
{-a2/z(t) + 2ocosa(i)}/{l + y/l + a2/z2(t) - 2acosa(t)/z(t)}
-{a2/z(t) + 2acosa(f)}/{l + y/l + a2/z2(t) + 2acosa(<)/z(*)}l
= 0,
which completes the proof of the proposition. D
Now we fix a point ρ of a complete open Riemannian manifold Μ of nonnegative
curvature. For t > 0 we set Ct := Π7{^71((""00' *])}» wnere the intersection is
taken over all rays 7 emanating from p. Note that В^(р) С Ct holds because
67(<7) < d(p, q). Then we have
Lemma 3.2. Ct is a compact totally convex subset of Μ and satisfies the
following:
(1) Ift2 > tl9 then Ct2 D Ctl and Ctl = {q e Ct2; d(q, dCt2) >t2- *i}· In
particular, dCtl ={?G Ct2; d(q, dCt2) = t2 - ti}.
(2) (Jt>0Ct = M.
(3) pedC0.
Proof. Let q\, q2 € 6~1((-oo, i\). Since 67 | с is a convex function for a
geodesic с : [0, 1] —► Μ joining q\ to q2, we have b1(c(s)) < (1 - s)/(c(0)) +
s/(c(l)) < t. Therefore, 6~1((-oo, i\) is totally convex, and so is Ct, which is
the (non-empty) intersection of such 6~1((-oo, £])'s. Note that ρ e Ct{t > 0).
because b1(p) = 0. Now suppose Ct is noncompact. Then there exist {pn}^=1 С Ct
with d(p, pn) —> +00. Take minimal normal geodesies ηη joining ρ to pn and an
accumulation vector и e UpM of {7n(0)} С UpM. Then 7U is a geodesic ray
emanating from ρ and contained in a closed subset Ct. On the other hand, we have
blu (7u(s)) = 5, and if we take s > t we get a contradiction. Namely. Cf is compact.
Next we prove (1). The first assertion is obvious. Suppose q e Ct2. namely,
b1{q) < t2 holds for any ray 7 emanating from p. If q £ Cfl. for any г G c?Ci2 we
may choose rn —» r and rays 7n emanating from ρ with 6~.n(rn) > t2. Then, since
<%, r„) > d(q, 7„(i)) - d(rn, 7„(ί)) (ί > 0), we have
d(q, rn) > # lim {(f - d(rn, 7n(i))) " (' " <*(*· %(')))}
It follows that d(q, r) > t2 - tu namely, d(q, dCt2) > t2 - tx. Next, assuming
that Ctl С {q e Ct2\ d(q, dCt2) > t2 - t\}, we will derive a contradiction. Then
there exists a q e Ct2 such that d(q, dCt2) > t2 - t\ but q & Ctl. We have
670(g) > *i for some ray 70, and t - d(q, 70(f)) > *i for sufficiently large t > 0,
namely, ς e Bt_tl(70(i)). On the other hand, since q e Ct2, for any t > t2
we get t - d(q, 70(i)) < 670(g) < *2, i.e., q & Bt-t2bo{t))- For the above t, on
a minimal normal geodesic τ joining 7o(£) to q we take a point r := r(£ - t2).
Then £ - d(r, 7o(*)) = ^2 implies that 67o(r) > £2 and г cannot be an interior
point of Ct2. Therefore d(q, r) > d(q, dCt2) > t2 - tu and on the other hand

214
V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
d(q, r) = d(7o(£)> я) ~ (£ — £2) < (t — t\) — (t — t2) = t2—t\, which is a contradiction.
This completes the proof of the last assertion of (1).
Next we prove (2). For q £ Μ take t > d(p, q). Then for any ray 7 emanating
from ρ and s > 0, we have s - d(q, 7(s)) = d(p, 7(s)) - d(q, 7(s)) < d(p, q) < t,
namely, b1(q) < t and q £ C*.
As for (3), ρ £ Co is obvious. For a ray 7 emanating from ρ take a sequence
{7(*n)} (*n 1 0) and note that &7(7(*n)) = tn > 0. Then η{ίη) £ C0 and ρ £ 5C0,
since limn^+oc 7(£n) = p. D
From the above lemma, we get a family {C*}, £ > 0, of compact totally convex
subsets which exhausts Μ. Now from {C*} we construct a compact totally convex
subset 5 without boundary, which is a totally geodesic submanifold of Μ (see
Chapter IV, §5). The existence of such 5 gives a strong restriction on the topology
of M. For instance, Proposition 5.8 of Chapter IV implies that Μ is homotopy
equivalent to the compact submanifold 5. We begin with preliminaries.
Lemma 3.3. Let Μ be a complete Riemannian manifold of nonnegative
curvature and С a totally convex (or, more generally, connected locally convex) closed
subset of Μ with dC φ φ. Then the distance function ψ : С -+ R+ to dC
defined by ψ(ρ) := d(p, dC) is a concave^ function in the sense that for any normal
geodesic 7 : [a, b] —» С we have
(3.2)
ip(^f(aisi + a2s2)) > aiV(7(*i)) + «2^(7(^2)),
0<ai,a2, a\ + a2 = 1, S\,s2 £ [a, 6].
Furthermore, suppose 7p(^(s)) = / on some closed interval [a, b]. Let τα : [0, /] —» С
be a minimal normal geodesic joining 7(a) to dC such that ά(η(ά), dC) = I. Let
V(s) be a unit parallel vector field along 7 | [a, b] with V(0) := fa(0). Then for
any s £ [a, b], [0, /] £ t \—► exp7(s) £^(s) pwes a minimal normal geodesic rs joining
7(s) ίο dC. If we define a map φ : [a, 6] χ [0, 1} —> С by </?(s, £) := exp7(s) tV(s),
then φ spans aflat totally geodesic rectangle in C.
Cs
tit)·
rs
Ma)
Figure 30
3This means that ρ ι—► — d(p, dC) is a convex function.

3. OPEN MANIFOLDS OF NONNEGATIVE CURVATURE
215
Proof. For s £ (a, b) let tj : [0, /] —» С be a minimal normal geodesic from
7(5) to dC. Setting α := /(7(5), τ?(0)), it suffices to show that
(3.3) ^(7(s)) < I - (s -s) cos α = V>(7(s)) - (s - s) cos α
on some interval (s, 5 + <5). In fact, considering 7_1, (3.3) holds on (s - <5, s + δ).
Then on this interval ^ о 7 is bounded above by a linear function / - cos a- (s — s),
and locally may be expressed as a minimum of a family of linear functions. Since
linear functions are concave and so is ^07, our assertion follows. Now to see (3.3),
first we consider the case where α > π/2. Let ξ be the orthogonal projection of
7(5) to fj(O)-1- and £(t) the parallel translation of ξ along tj : [0, /] —» C. Then
||£|| = cos(a - π/2). Now we consider a curve
cs(t):= expT_{t)(s-s)£(t), 0<t<l.
Note that cs(t) is contained in С for small t. On the other hand, д/ds \s=s cs(l) =
ξ(1) is orthogonal to the minimal geodesic tj joining 7(5) to dC, and a property
of the tangent cone (Chapter IV, Lemma 5.7) implies that the end points cs(l) are
not contained in the interior N of С for sufficiently small s - s > 0. Therefore
d(cs(0), dC) < L(cs), and, on the other hand, L(cs) < l by the Rauch comparison
theorem R.C.T.(II) (2). Next apply T.C.T. (or R.C.T.(II) (1) when s is close to
s) to a geodesic triangle (7(s), 7(5), cs(0)). Note that the angle of this geodesic
triangle at the vertex 7(5) is equal to α — π/2 and the side lengths of this angle are
given by s — 5, (s — s) cos(a — π/2). Then the opposite side of the corresponding
hinge in the Euclidean plane is of length (s - s) sin(a -π/2), and T.C.T (II) implies
d(7(s), c5(0)) < —(s - s) cosa. Namely, d(7(s), dC) <l — (s — s) cosa.
Second, we consider the case where α < π/2. In this case take a minimal
geodesic σ joining 7(s) to tj, and let Tj(to) be the end point of σ at which 7 and tj
are perpendicular. Then the first case implies that d(^(s), dC) < d(rj(to), dC) =
I -10. Next, applying T.C.T.(II) to the geodesic hinges (7(5); 7 | [5, s], tj \ [0, t0})
and (ts(£o); cr~\ (Ts I [0, to})'1), where the angles at vertices 7(5), 7?(£0) are given
by α, π/2, respectively, we get
d2(Tj{t0), 7(5)) < ^o2 + (s - s)2 ~ 2io(s - s) cosa,
(s-s)2<d2(Tj(t0),7(s)) + t02.
Then it easily follows that t0 > (s - s)cosa and d(7(s), dC) < I - (s - s) cosa,
which proves (3.3). Finally, suppose we have ^(7(si)) = ^(7(5)) = / for some
si > s. Then from the concavity we get ^(^(s)) = l(s < s < s\). Furthermore, the
first variation formula implies that α = π/2. Then, by the equality case of R.C.T.
(II) (2) (Chapter IV, Remark 2.6), the curves cs are geodesies from 7(s) to dC
of length /, namely, they realize the distance d(7(s), dC), and span a flat totally
geodesic rectangle φ. Π
Now we state the theorem of J. Cheeger and D. Gromoll ([Ch-Gr-1]).
Theorem 3.4. Let Μ be a complete noncompact Riemannian manifold of non-
negative curvature. Then Μ contains a compact totally geodesic submanifold S with
dim 5 < dimM, which is totally convex. Furthermore, Μ is diffeomorphic to the
normal bundle of S.
Proof. By Lemma 3.2 the Ct (t > 0) are compact totally convex subsets of
Μ whose boundaries dCt are nonempty because dimC* = dimM. Now we set

216 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
С = Ct (t > 0), and for r > 0 consider
Cr := {ρ € С; d(p, дС) > r}, Cmax := f]{Cr; Cr φ φ}.
If Cr φ φ then Cr is totally convex. In fact, let 7 : [a, b] —» С be a geodesic with
7(a), 7(6) G Cr. For t G [a, 6] we write t = a\a + a26, ab a2 > 0, c*i + a2 = 1,
and get for ψ(ρ) = d(p, dC)
^(7(0) > «ι^(7(α)) + «2^(7(6)) > r
by (3.2). Namely, 7 | [a, 6] С Cr and Cr is totally convex. Next let 7*0 be the
maximum of ψ on С Then clearly Cmax = {p G C; d(p, <9C) = r0}. Therefore,
points of Cmax are equidistant to dC, and by Lemma 3.3 any geodesic 7 in Cmax
is perpendicular at every point of 7 to a minimal geodesic from that point to dC.
Then we get dimCmax < dimC. Now we set С (I) := Cmax, and if dC{\) = φ we
are done. Otherwise, we set С(2) := C(l)max by the above procedure. Thus we
get a sequence of compact totally convex subsets С D С (I) D С (2) D · · · such that
dimC(z + 1) < dimC(z). Therefore, for some к either dC(k) = φ or dimC(fc) = 0,
namely, C(k) is one point. In either case the first assertion of the theorem is proved.
Now we turn to the second assertion. We consider the distance function ds
to S and show that there exist no ds-critical points in Μ \ 5, where the notion of
ds-critical points is defined in exactly the same manner as in Definition 2.8 taking S
instead of p. To see this, by the construction of 5, note that any point r G M\S lies
on the boundary dC of a totally convex subset С whose interior contains 5. Take
a supporting half-space Η = {ν G C(r); ^.(vq, ν) < π/2} of С at г with vq G C(r)
and C(r) С Я. Now if r is ds-critical, then there exists a normal minimal geodesic
7 from 5 to г such that Z(vo, —7(d(r, 5)) > π/2. Note that 7 | [0, d(r, 5)) lies in
the interior of C. On the other hand, from the convexity, 7(d(r, 5) — e) cannot lie
in the interior of С for sufficiently small e > 0, and we get a contradiction.
Therefore, Μ \ S is free of ds-critical points and for any r G Μ \ S there
exists a unit vector ur G TrM such that Z(ur, -7(d(p, 5))) > π/2 for any normal
minimal geodesic 7 from 5 to г (note that the above ur corresponds to — xr in
(2.5)). Then a desired difFeomorphism may be constructed as in Lemma 2.9 (taking
€1 = +00, €2 = 0), as we shall now sketch. By averaging the above ur's we may
construct a smooth vector field Υ on Μ \ 5, which is in fact —X in Lemma 2.9,
with the following property :
AYr, -7(d(p, S))) > π/2 + 0(d(p, 5))
for any 7 G min(5, r), where 0(r), г > 0 is a positive continuous function. Now
take an e > 0 so that the normal exponential map exp-1 of 5 is a difFeomorphism
from B2e{os) := {υ G i/(S); ||v|| < 2e} onto B2e(S)(C M). Since for г G £2e(5)
there exists a unique normal minimal geodesic 7 from S to r, we may further
assume that Yr = 7(d(r, 5)), which is in fact a gradient vector of ds (see Chapter
III, Proposition 4.8), for r belonging to an open neighborhood of B€(S). Also we
may assume that \\Y\\ = 1 everywhere. Let ipt be the flow generated by Y. Then
<Pt(r), r G Μ \S may be defined for all t > 0, and t i-> d((pt(r), S) is strictly
increasing (compare with (2.7)). In fact, for any R > 0 there exists a £(#) > 0
such that d((pt(R)r, S) > R for any point r G Bc(5) \ 5. Set AT = (exp-L)~1{r G
M; d(r, 5) = б}, and note that if d(r, S) = e then ipt(r) may be defined for t > -£,
and we have φί(βχρ± ν) = exp±(l + t/e)v for υ G TV and -£ < £ < 0. Then we get a
difFeomorphism Φ from Nx (-£, +00) onto M\S defined by Φ(ν, £) := (^(exp^ г>).

3. OPEN MANIFOLDS OF NONNEGATIVE CURVATURE
217
Clearly Ν χ (-£, +oo) may be identified with v(S) \ 5, so that Φ | Ν χ (-£, 0)
is nothing but exp^. Therefore, Φ may be extended to a diffeomorphism between
i/(5) and M. D
The above S is called a soul of M. Note that S may be written as S =
C(k — l)max for some C(k - 1), and unless S consists of one point we may span a
flat totally geodesic rectangle using a geodesic segment contained in 5. Therefore,
we get the following theorem of D. Gromoll and W. Meyer ([Gr-Me-1]).
Corollary 3.5. Let Μ be a complete noncompact Riemannian manifold of
positive sectional curvature. Then the soul S constructed above consists of one point,
which is called a simple point, and Μ is diffeomorphic to Rm.
Cheeger and Gromoll conjectured that the assertion of Corollary 3.5 holds
under the following weaker assumption: Μ is a complete noncompact Riemannian
manifold with nonnegative sectional curvature such that the sectional curvatures
are positive at a point of A/. The conjecture was recently solved by G. Perelman
([Pe-2]).
Next we give some examples.
Example 1. Let Μ be a cylinder in R3. which is a complete noncompact flat
surface. Then the circles perpendicular to the axis of Μ are souls of Μ. Next let
Μ be an elliptic paraboloid of revolution in R3. which is a complete noncompact
surface of positive curvature. Then the vertex of Λ/ is the unique simple point.
Example 2. Let TV be a complete Riemannian manifold of nonnegative
sectional curvature and G a compact Lie group. Recall that G carries the structure
of a symmetric space of compact type with nonnegative sectional curvature. Then
the Riemannian product Ν χ G is also of nonnegative curvature. Suppose a closed
subgroup Η of G acts on TV as an isometry group. Since Η acts on G as right
translations, Η acts on G x TV freely and isometrically by h(g, p) := (gh~l, hp).
Then we may endow Μ = (G x N)/H with a Riemannian metric such that
the canonical projection π : G x N —» (G x N)/H is a Riemannian
submersion. Then, by the O'Neil formula for Riemannian submersions, Μ is also of
nonnegative curvature (in fact, Riemannian submersion is one of the main tools
for producing metrics of positive or nonnegative curvature). In particular, we set
G = SO(m + 1), Η = SO(m),N = Дт, and let elements of Η act on R171 as
orthogonal transformations. Then Μ = (SO(m+ 1) χ R^/SOfa) may be
identified with the tangent bundle TS™ of the sphere 5m = {SO{m + 1) χ {o})/SO{m).
Therefore, TS™ carries a Riemannian metric of nonnegative curvature, and if m > 2
then 5m (C TS™) is the unique soul of TS171.
3.2. In this subsection, we are concerned with a complete noncompact
Riemannian manifold Μ whose Ricci curvatures are nonnegative everywhere. In the
following the Busemann function b1 again plays a fundamental role and turns out
to be a subharmonic function. We begin with the definition.
Definition 3.6. Let φ be a real-valued function on a Riemannian manifold N.
(1) A real valued function / is said to be a C°° support function of φ at ρ G M,
if the following two conditions are satisfied:
(i) / is a C°° function defined on a neighborhood W of ρ with f(p) = φ(ρ).
(ϋ) /(<?) <¥>(<?), Я eW.

218 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
(2) φ is said to be a subharmonic function, if for any point ρ £ N and for any
€ > 0 there exists a C°° support function /P} € of φ at ρ such that the Laplacian
Δ/ρ>€ of /p>c satisfies Δ/Ρ>€(ρ) < с.
The following maximum principle is a fundamental property satisfied by sub-
harmonic functions.
Theorem 3.7. Let N be a Riemannian manifold and φ a continuous
subharmonic function on N. If φ assumes a maximum, then φ is constant.
The proof is given in Appendix 4.
Now we turn to a complete noncompact Riemannian manifold Μ with nonneg-
ative Ricci curvature. Let 7 be a ray on Μ and b1 the corresponding Busemann
function. Recall that for any ρ £ Μ there exists a ray ην emanating from ρ and
asymptotic to 7 (see Chapter IV, §5). Now for t > 0, q £ Μ we set
(3.4) blpM-= Mp) + *-<%7p(0)-
Then £>7p,t(p) = b7(p), and
blp.t(Q)-b1{q)= lim {* + <%, 7(5)) - d(p, 7(e)) - %, 7p(0)}
= lim {t+ d(g, 7(e))-<*(P, 7(e))-d(9, 7.(0)}
= lim {<%, 7(s)) - d(7.(t), 7(e)) - d(q, 7e(t))} < 0,
s—>+oc
where the 7S are minimal normal geodesies joining ρ to 7(e) such that 7S —» 7P
as 5 —> +00. On the other hand, since ην is a ray and t > 0, we may choose a
neighborhood W of ρ so that W is disjoint from the cut locus of 7P(£) (^ p). Then
q £ IV 1—► с?(^, 7p(0) is smooth and 67p5i given by (3.4) is a C°° support function
of 67 at p. Now under the assumption on the Ricci curvature, from the estimate
for the Laplacian of the distance function to 7P(£) (Chapter IV, Proposition 3.6),
we get Δ67ρ,ί(ρ) < (m — l)/t. Since t > 0 may be arbitrarily large, we have
Proposition 3.8. Let Μ be a complete noncompact Riemannian manifold with
nonnegative Ricci curvature. Then the Busemann function b1 corresponding to any
ray η of Μ is subharmonic.
Now we state the following fundamental splitting theorem of Cheeger and Gro-
moll ([Ch-Gr-2]). The proof presented here is due to J. H. Eschenburg and E.
Heintze ([Esc-He-1]).
Theorem 3.9. Let Μ be a complete Riemannian manifold of nonnegative
Ricci curvature. Suppose Μ contains a line 7. Then Μ may be decomposed as
a Riemannian direct product Μ = Μ' χ R.
PROOF. Recall that a normal geodesic 7 : (—00, +00) —> Μ is called a line
if d(7(s), 7(0) = I* "" sl (^ s € Щ- If Μ contains a line, then Μ is noncompact.
Setting 7+(£) := 7(i), 7~(0 := 7(-*) (* > 0), we get two rays 7+, 7"
emanating from 7(0). Now we denote by 6+, b~ the Busemann functions determined by
7+, 7~, respectively, which are subharmonic functions by Propostion 3.8. Since 7
is a line, we get for ρ £ Μ
(b+ + b~)(p) = lim {2f - dip, 7(t)) - <f(p, 7H))} < 0
r—>+oc
and (6+ + 6~)(7(£)) = 0 on 7. Therefore, by the maximum principle, 6+ + 6~ = 0.
Let 6pjt := 67±p,i be C°° support functions of 7^= at ρ as before. Then b^t < b+ =

3. OPEN MANIFOLDS OF NONNEGATIVE CURVATURE
219
—b~ < —b~t, and the equality signs hold at p. Then b± is of class C1 at p, and
V6±(p) = V6p>t(p). Further, ||V6±|| ξ 1 by a property of the distance function
(Chapter III, Proposition 4.8). Next we show that b± are harmonic functions.
Take a metric ball В centered at ρ with sufficiently small radius, and harmonic
functions hr*1 which are equal to b± on the boundary of B. Applying the maximum
principle to subharmonic functions b± - h±, we get b± < h±. On the other hand,
0 = 6+ + b~ < /i+ + /i~, and we apply the maximum priciple to /i+ + h~. It
follows that h+ + h~ = 0 and b± = /i±. Therefore, b± are harmonic on В and C°°
functions. Now we prepare the following lemma.
Lemma 3.10. Let f be a C°° function on a complete Riemannian manifold
with ||V/|| = 1, where V/ stands for the gradient vector field of f. Then integral
curves o/V/ are lines of M.
Proof of the Lemma. Let 7 : (a, b) —» Μ be a maximal integral curve of V/.
For a < s < t < b take any piecewise smooth curve с : [0, 1] —» Μ joining 7(5) to
7(£). Then we get
L9(c) = J1 \\c(t) \\dt > jT |(c(i), V/>|di > IjJ1 |/(c(t))di|
= |/(c(l))-/(c(0))| = 1/(7(0)" /(7(0)1
'/
t
6(u),Vf)du\ = t-s = Lg(7\[s,t}).
It follows that 7 | [5, i] is a minimal normal geodesic joining 7(5) to 7(2). In
particular, ^(7(0), 7(i)) = t, and if b < +00 then limt]bl{t) exists because of
completeness of M. Then we may extend 7 beyond 6, which is a contradiction.
Hence b = +00, and similarly a = -00. D(lemma).
Now, applying the lemma to b±, we see that integral curves of V6^ are lines.
At any point ρ G Μ take an o.n.b. {ei, ... , em_i, em := Vb±(p)} and consider the
field of o.n.b. {Ei^)}^ obtained by parallel translating the above o.n.b. along the
integral curve ηρ of V6± through p. Since Em(t) = V6±(^p(i)) and VЕтпЕг = О,
we get
0 < Bic(Em(t), Em(t)) = ^<ВД№< Em(t))Em(t). Et(t))
t = l
m
= Y,{-(VEmVE,Em, Ei){t) - <VrEiEmSm. ЕШ
i=\
(m \ m
^(V£,£;m, Et) - Σ (V£t£m, Ej){VEjEm, Ei)(t)
= £m(i)(A6±) - P26±(7(i))ll2 - -||ί?26±(7(ί))Ι|2 < 0,
where D2b± denotes the Hessian of b± (Chapter II, Definition 1.5) and is a
symmetric tensor field of type (0, 2). Therefore D2b± = 0, and the gradient V6± of
b± is a parallel vector field. In particular, V6± is a Killing vector field, and we
denote by φί the one parameter transformation group of isometries generated by
V6+. Then the action of Όφ^ρ) on TPM coincides with the parallel translation
along t i-> ift{p) = 7p{t)- Now M' := (6+)_1(0) is a hypersurface of M, because

220 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
||V6+|| = 1. In fact, M' is totally geodesic since its normal vector field V6+ is
parallel. Note that b+(ipt{p)) = t for ρ G M', and (6+)_1(0 = ψι(Μ'). Now we
define Φ : Μ' χ R —» Μ by Φ (ρ, t) := <pt{p)- Then we may easily check that Φ
is a difFeomorphism, and in fact an isometry, since Dipt coincides with the parallel
translation. D
Now as an application of the splitting theorem we study the structure of the
fundamental group of a compact Riemannian manifold of nonnegative Ricci
curvature ([Ch-Gr-2]). Recall that in the case of positive Ricci curvature, πι(Μ, ρ) is a
finite group by the Myers theorem.
Theorem 3.11. Let Μ be a compact Riemannian manifold with nonnegative
Ricci curvature.
(1) The Riemannian universal covering space Μ of Μ may be decomposed as
a Riemannian direct product M = NxRk(0<k<m = dim M), where N is
a compact simply connected Riemannian manifold of nonnegative Ricci curvature
and Rk denotes the Euclidean space with the canonical Riemannian metric.
(2) There exists a finite normal subgroup Φ of the fundamental group π\ (Μ, ρ)
of Μ such that π* := πι (Λ/, ρ)/Φ is isomorphic to a crystallographic group (i.e., a
discrete and uniform1 subgroup of the group of motions of Rk). By the Bieberbach
theorem, π* contains a free abelian normal subgroup of rank к and of finite index.
PROOF. Let π : Λ/ —» Μ be the universal Riemannian covering, and consider
7Γι(Λ/, ρ) as the deck transformation group Γ. Then, by Theorem 3.10, Μ may be
decomposed as a Riemannian product Μ = Ν χ Rk (0 < к < га), where N does
not contain any line. First we fix (p, q) G Ν χ Rk and consider any isometry φ of
M. Note that φ maps lines through ρ to lines, and lines through any fixed point
form a A;-dimensional space. Therefore, ψ maps {ρ} χ Rk to some {ρ'} χ Rk, and
also maps Ν χ {q} to some Ν χ {qf}, since Ν χ {q} is orthogonal to {ρ} χ Rk.
Let πι : Ν χ Rk —» TV, π2 : Ν χ Rk —> R be the canonical projections, and set
πι(φ)(χ) := πχ(φ(χ, q)), χ G N, n2(y) := π2(ρ(ρ, 2/)), J/ G ilfc. Then the above
argument implies that π\(φ), ^2{φ) are isometries of TV, Rk, respectively. We show
that π\(φ)(χ) is independent of the choice of q. In fact, for any q' G N take a C°°
curve c(t) joining q to q'. Then we have -^πι(φ(χ, c(t))) = (£>πι ο Zty)(0, c(£)),
which is equal to 0 since Όφ(0, c(t)) is tangent to ilfc. Therefore πι(^?(χ, q)) =
π\(φ(χ, q')). The same argument implies that ^2{φ){ν) does not depend on the
choice of p. Namely, we may write φ(χ, у) = (π\(φ)χ, π2(φ)ν)·
Now to verify (1), it suffices to show that N is compact. Since Μ is compact,
we may take a compact fundamental domain К for the deck transformation group
Г. Then the image of πι (if) by πι(Γ) coincides with TV, because T(K) = M. Now
suppose N is noncompact and take a ray 7 : [0, +00) —> N. Choose φη G Γ
so that πι((^η)_1)7(η) G πι (if), and set ηη{ί) := πι((^η)_1)(7(η + £)), -η <
£ < +oo. Then {7„} is a sequence of rays in N such that the 7n(0) belong to a
compact set πι (if), and admits a convergent subsequence. We may assume that
Vn '·= 7n(0) —> ν G /7PM, taking a subsequence if necessary. Then we may easily
see that ην : (—oo, +00) —> TV is a line, and we have a contradiction.
Now we prove (2). Since the isometry group I(N) of a compact Riemannian
manifold is compact, the kernel Φ = кег7г2 of π2 : Г(С I(Ν) χ i(ufc)) -> i(flfc)
4 A subgroup π* of the group of motions of Rk is said to be uniform if Rk/n* is compact.

4. MANIFOLDS OF NONPOSITIVE CURVATURE
221
is a finite normal subgroup of Γ. Now we consider a Riemannian covering Μ =
NxRk -> Μ/Φ 9£ NxxRk and the isometry group /(Μ/Ψ) ^ 1(^)х1(11к). Note
that the projection тг2 : ДМ) x ДЯ*) -> /(Я*) maps π* := Г/Ф (С ДМ/Ф))
isomorphically onto a discrete and uniform subgroup of I(Rk). As for the last
assertion, see Chapter IV, §1. D
For general complete Riemannian manifolds of nonnegative Ricci curvature,
the structure theorem as Cheeger-Gromoll theorem for manifolds of nonnegative
sectional curvature is not known. However, R. Schoen and ST. Yau have shown
that a complete noncompact 3-dimensional Riemannian manifold with everywhere
positive Ricci curvature is difFeomorphic to R3 ([Scho-Y-2]). Finally, we give
Proposition 3.12. Let Μ be a complete noncompact Riemannian manifold
with nonnegative Ricci curvature. Then vol Μ = +зс.
PROOF. It suffices to show that Нтд_^+эс vol Вr(p) = +oo. We apply the
Bishop-Gromov comparison theorem (Chapter IV. Corollary 3.4 (3)). Fix r > 0.
Take a point q £ Μ with d(p, q) — d > r. and set d = r + r\. Noting that
B2ri+r{v) Э Bri(q), we get
vol£2ri+r(p) > vri(0)/{vri+2r(0) - ι·Γι(0)} · volBr(p).
If we set R = 2r\ + r, it follows that
volBR{p) >(R- r)m/{(R + 3r)m - (R - r)m} ■ volBr(p).
Then, since limR^+oc{R - r)m/{{R + 3r)m - (R - r)m)} = +эс. our assertion
follows. D
4. Manifolds of Nonpositive Curvature
4.1. In this section, we are concerned with complete Riemannian manifolds
whose sectional curvatures are everywhere nonpositive or everywhere negative.
They are called manifolds of nonpositive curvature or manifolds of negative
curvature, respectively. Again convexity plays an important role. First, for a Jacobi
field Y(t) along a geodesic 7 we set f(t) := (Y{t), Y{t)). Then from the Jacobi
equation we get
f"(t) = 2{(УУ(<), Vr(i)> - (WW, 7(0)7(0, Y(t)))
= 2{<w(0, vy(0> - K(Y{t), 7(0)11^(0 л7(0112} > ο.
Namely, f(t) is a C°° convex function. Then critical points of / are minimal points,
and if f(ti) = f(t2) = 0 for some t\ < t2, then / | [£1? t2] = 0, namely, Y(t) = 0.
Therefore, if a nonzero Jacobi field Y(t) along a geodesic satisfies Y(to) = 0, then
Y(t) φ 0 for any t Φ to, and there are no conjugate points along any geodesic.
Similarly, if a nonzero Jacobi field Y(t) satisfies VY(to) = 0, then f'(to) = 0, and
/ assumes a positive minimum at t = to- It follows that Y(t) φ 0 everywhere.
Namely, along any geodesic 7 there exist no focal points (of a hypersurface which
is perpendicular to 7 and totally geodesic at 7(0)·
Exercise 1. For a Jacobi field Y(t) along 7, show that t i-> ||V(i)ll ls a^so
convex.
Theorem 4.1 (Hadamard-Cartan Theorem). Let Μ be α complete
Riemannian manifold of nonpositive sectional curvature. Then the following hold:

222 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
(1) For any ρ G M, expp : TpM —> Μ is a covering map.
(2) Suppose further that Μ is simply connected. Then, for any point ρ G Μ,
expp : TpM —> Μ is a diffeomorphism, and for any two points p, q G Μ there exists
a unique normal geodesic joining ρ to q, which is in fact minimal (i.e., distance
realizing).
PROOF. It suffices to show (1). Since there are no conjugate points along
any geodesic emnating from p, expp is regular at each point of TpM. Therefore,
g := exp* £ defines a Riemannian metric on TpM, and expp is a local isometry
from (TPM. g) onto (M, g). We show that g is complete. For и G TPM, the line
t ι—► tu (—00 < t < +oo) in TPM is mapped under expp to the geodesic *yu(t), and
therefore is a geodesic with respect to g. Since и is an arbitrary vector of TpM,
(TpM, g) is geodesically complete at op, and complete by the Hopf-Rinow theorem.
Then our assertion follows from Theorem 5.4 of Chapter III. D
Remark 4.2. Let Μ be a complete Riemannian manifold. Then a point
ρ G Μ is called a pole if expp : TpM —» Μ is a regular map. Then we have
the assertions (1), (2) for a pole p. For a complete Riemannian manifold Μ of non-
positive sectional curvature, its universal covering is difFeomorphic to Euclidean
space and its homotopy type is determined by πι (Μ, ρ) (such an Μ is called a
Κ (π; 1) space). In particular, nk(M, ρ) = 0(k >2).
A complete simply connected Riemannian manifold of nonpositive sectional
curvature is called an Hadamard manifold. Then the universal Riemannian covering
space X of a complete Riemannian manifold Μ of nonpositive curvature is an
Hadamard manifold, and we get Μ = Χ/Γ, where Γ is the deck transformation
group of π : X —» Μ and isomorphic to the fundamental group of Μ. Therefore,
to study manifolds of nonpositive curvature, it is useful to study metrical properties
of Hadamard manifolds ant their isometries. Note that, by Theorem 4.2 (2), for
subsets С С X the notions of strong convexity, local convexity and total convexity
agree in an Hadamard manifold. A subset С of an Hadamard manifold X is simply
called a convex set, if for any points p, q € С the geodesic joining ρ to q, which is
unique up to the parametrization, is contained in C.
First we give some fundamental properties of an Hadamard manifold X.
Proposition 4.3. Let X be an Hadamard manifold and d : Χ χ X —> R
the distance function. Then d is a convex function with respect to the product
Riemannian metric.
PROOF. Any geodesic 7 of Χ χ Χ may be written as η{ί) = (7ι(£), 72(0)»
where 71, 72 are geodesies of X. We may assume that 71 ^72. Then 71, 72 cannot
intersect at two points by Theorem 4.1. First, suppose 71 (ίο) Φ 72(^o) and take
geodesies at : [0, 1] —> Μ joining 71 (t) to 72(i) for t in a neighborhood of t0. Then
f(t) := L(at) = d(7i(£), 72(0) ls a C°° function of t. As in the case of the second
variation formula (Chapter III, Remark 2.6), noting that 71, 72 are geodesies, we

4. MANIFOLDS OF NONPOSITIVE CURVATURE
223
get
/"(ίο) = ^ |t=t0 L{et)
= yj[1{(Vy±(S),Vy±(S))
- K(Y±(s), at(s))\\Y±(s) Aat(s)f}ds > 0 (I = ||<7to||),
where Y(s) denotes the Jacobi field along ato obtained as the variation vector
field of the variation {at} of ato consisting of geodesies, and Y1- is the vertical
component of Υ to ato. Second, suppose 71 (ίο) = 72 (£o)· Then by the above we
get ^-d(7i(0> 72(0) ^ 0 except for £0, and f(t) assumes its minimum 0 at t = t0.
It follows that f(t) is convex. D
Remark 4.4. Let W be a strongly convex subset of a complete Riemannian
manifold of nonpositive curvature. Then the distance function d : W x W —» R is
convex, by the same argument as above.
Exercise 2. Let A be an Hadamard manifold. Show the following:
(i) Let Я be a closed totally geodesic submanifold of A. Then X Э ρ ι—>
d(p, Η) is a convex function.
(ii) Let μ be an isometry of X. Then ХЭри d(p, μ(ρ)) is a convex function,
(iii) The convexity radius rp(X) = +00 for any ρ e X.
Proposition 4.5 (Comparison theorem for triangles). LetX be an Hadamard
manifold and А(р\р2Рз) a geodesic triangle. Denote by 7* geodesic segments joining
Pi+ι to Pi+2, and set h := L(7i), <*i := ^(7i-i, ~7ΐ+ι(0)), where i ξ 1, 2, 3 (mod
3). Then
(4.1) U2 >/г+12+/г?+2 -2/г+1/г+2С08аг,
(4.2) αϊ +α2 + α2 < π,
w/iere if equality holds in (4.1) or (4.2), ί/ien we may span aflat and totally geodesic
surface with the geodesic triangle Л(р\р2Рз).
PROOF. Since we have i(X) = +00 for an Hadamard manifold A', we may apply
the Rauch comparison theorem (II) (Chapter IV, Theorem 2.5) to the geodesic
hinge (pi\ 7~Д, 7г+г) in A and a corresponding hinge in (Дт. до) with the same
side lengths Z^+i, li+2 and the same angle a,·. Then (4.1) immediately follows. The
assertion on equality follows from Chapter IV, Remark 2.6. D
Exercise 3. Give a proof for (4.2) and the assertion on equality.
Corollary 4.6. Let o^ (i = 1, 2, 3, 4) be the angles of a geodesic quadrilateral
{P1P2P3P4) in X with vertices Pi. Then Σι=ι α» — %π. If equality holds, we may
span aflat totally geodesic surface with the quadrilateral (P1P2P3P4)'
Exercise 4. Give a proof of Corollary 4.6.
Now we give some examples of manifolds of nonpositive curvature.
Example 1. Symmetric spaces of noncompact type are Hadamard
manifolds. In particular, (-Rm, go) with the flat (i.e., Κσ = 0) Riemannian metric,
and (Hm, go) of constant negative curvature, are Hadamard manifolds.

224 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Example 2. Let Μ, Ν be Riemannian manifolds of nonnpositive curvature.
Then the Riemannian direct product Μ χ TV is again of nonpositive curvature.
However, note that Μ χ TV is not a Riemannian manifold of negative curvature even
if Μ, Ν are of negative curvature. Let π : Μ —» Μ be a Riemannian covering.
Then Μ is of negative (resp., nonpositive) curvature if and only if Μ is of negative
(resp., nonpositive) curvature.
Example 3 ([Bi-ON]). Let (M, g), (TV, ft) be Riemannian manifolds and / a
positive C°° function on Μ. Now we define a Riemannian metric G on Ρ = Μ χ Ν
by
(4.3) σ = π^+Κ//)2·π^Λ,
where пм : Ρ —» Μ, π^ν : Ρ —> Ν are canonical projections. A Riemannian metric
on Ρ of the form (4.3) is called a warped product of p, ft. Then we have
Lemma 4.7. (1) If g, ft are complete, so is G.
(2) Lei g be complete. Then G is a Riemannian metric of negative curvature
if and only if the following conditions (a), (b), (c) hold.
(a) Either dim Μ = 1 or Μ is of negative curvature.
(b) f is a positive strictly convex С°° function on M.
(c) Either dim N = 1 or one of the following holds: N is of negative
curvature (when f assumes a minimum) or N is of nonpositive curvature (when f does
not assume a minimum).
PROOF. Let {n}^, r* := (pi, qi) G Ρ = Μ χ TV, be a Cauchy sequence with
respect to G. Take curves c^ : [0, 1] —> Ρ joining r* to r? of length less than
2dc(ri, rj), which may be written as cij(t) = (dij(t), eij(t)). Then
d<?(Pi, ft) < Lg(dij) < LG(cij) < 2dG(ru rj)
and therefore {pi) is a Cauchy sequence, which is convergent with respect to dg
since (Μ, ρ) is complete. Next note that, by the above, {dij} is contained in a
fixed compact set К of M. Let m > 0 be the minimum of / restricted to K. Then
Lh(etj) < 2dc(ri, rj)/m, and it follows that {qi} CiVisa Cauchy sequence and
convergent with respect to ft. Then {ri} is a convergent sequence in P, and Ρ is
complete.
Now we may compute the curvatures of (P, G), since тгм : (Ρ, G) —> (Μ, ρ) is a
Riemannian submersion. Let σ be a plane section at (p, q) € Ρ and {(ж, t/), (u, г>)}
С Т(р^Р an o.n.b. of σ. Then we get the following (we leave the computation to
the reader):
(4.4)
Κσ = Kg(x, u)\\x A u\\g2 + f2(p){Kh(y, v) - ||Vs/(p)||2}||y Λ v\\h2
- f(p){\\v\\h2Dlf(x, x) - 2h(y, v)D2gf(x, u) + \\y\\2hDg2f(u, «)}.
Then we may verify the assertion of (2) from (4.4), using the following remarks. If /
is a C°° convex function, then the critical points of / are minimum points (Chapter
IV, §5). Also, if g is complete and / is a positive С°° strongly convex function that
does not assume its minimum, then inf{||Vp/(p)||; ρ G M} = 0, where Vgf (resp.,
D2gf) stands for the gradient vector (resp., Hessian) of / with respect to g. D
Now we turn to Hadamard manifolds. For a subset A of an Hadamard manifold
X, the minimal convex subset ch(A) which contains A is called the convex hull of

4. MANIFOLDS OF NONPOSITIVE CURVATURE
225
A. Now for a compact subset К С X we consider metric balls containing К and
denote by p(p) the minimal radius of metric balls centered at ρ containing K,
namely, p(p) = max{d(p, к); к £ K}. Then from p(p) - p(q) < d(p, q), we see that
ρ ι-> p(p) is a continuous function on X. We will show that ρ assumes a minimum
at points in ch(K).
In general, for a closed convex subset С of X and pGl, there exists a point of С
which realizes the distance d(p, C) = inf {d(p, q)\ q £ C}, since С is closed. Suppose
there exist two such points q\, q2 (<7ι φ Ч2) with d(p, qi) = d(p, q2) = d(p, C).
Then the geodesic τ joining q\ to q2 is contained in C, and for the geodesic triangle
A{PQiQ2) the angles at the vertices q\, q2 are greater than or equal to π/2 by the
first variation formula. The sum of the inner angles of A{pq\q2) is greater than
π, which contradicts (4.2). Therefore, for ρ £ X there exists a unique point of С
realizing the distance d(p, C), which is called the foot of the perpendicular from ρ
to С and denoted by ncP- Now we note that nc is distance decreasing, namely,
d(p\, P2) > d(ncPi, 7ГСР2)· To see this we may assume that pb p2 & C, ποΡι φ
KcP2- Then take geodesies 71 : [0, 1] —» X joining ncPi to p\ and 72 : [0, 1] —» X
joining 7ГсР2 to p2· We set /(£) := d(7i(£), 72(0)· Let 7 be a normal geodesic
joining ncPi to 7TcP2- Then, by a property of feet of perpendiculars, the angles
between 71 (0), 7(0) and —7(1), 72(0) are obtuse or right. By the first variation
formula, /'(0) > 0. On the other hand, f"(t) > 0 by Proposition 4.3. Therefore,
f(t) is monotone increasing.
Now we turn to the case where С := ch(K), and apply the above argument.
We get d(p, k) > d(ncP, к), к £ К, and therefore p(p) > p{ncp)- Since metric
balls in X are convex and ch(K) is compact, ρ assumes its minimum at a point
of ch(K). Now if ρ assumes its minimum at two points ρ, ρ of ch(K), then along
the geodesic joining ρ top the distance function dk to a point к £ К is convex.
It follows that for the middle point q of this geodesic segment we have d(q, к) <
\{d(p, к) + d{p, к)) < \{ρ{ρ) + p{p)) = p{p) for keK. Then p{q) < p{p) since К
is compact, and we have a contradiction.
Summing up, for a compact subset К of Χ, ρ assumes its minimum at a unique
point of Qh{K), which is called the center of K. We give an application, due to E.
Cart an, of this concept.
Theorem 4.8. Let X be an Hadamard manifold and G a compact subgroup of
the isometry group of X. Then there exists a fixed point ρ of G, namely, g(p) = ρ
for any g £ G.
PROOF. The orbit К := {g{q)', g £ G} of q £ X under the action of G is
compact. Let ρ be its center. Since К is invariant under any isometry g £ G, we
get p{g{p)) = max{d(k, g{p))\ к £ К} = p(p). It follows that g(p) = ρ because of
the uniqueness of the center. D
The above results correspond to those of elementary geometry. We give another
example, which is related to the Steiner theorem in elementary geometry. In the
Euclidean plane take a triangle Δ ABC. Steiner's problem asks for a point Ρ which
minimizes the sum of the distances PA + PB + PC to the vertices of the triangle.
The answer is as follows: If /Л, ZB, ZC are less than 2π/3, then there exists a
unique point Ρ in the interior of AABC such that ZAPB = ZBPC = ZCPA =
2π/3, and this Ρ is the desired point. If one of ZA, ZB, ZC is greater than or
equal to 2π/3, then the corresponding vertex is the desired point.

226
V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
Now we consider n(> 3) points pi, P2, · · · ? Pn in an Hadamard manifold X,
which are distinct and are called vertices. We want to look for a point which
minimizes the sum of the distances to p\, p^, ... , pn · Such a point is called a
Steiner point, and the set of Steiner point is called the Steiner set. A vertex pi is
said to be a Steiner vertex if
Σ cos z (PjPiPk) < γ-,
j<fe(i.MO
where Z(pjPiPk) denotes the angle at the vertex Pi between the geodesies joining pi
to pj and pi to pk- We say that pi, ... , pn are completely degenerate, if pi, ... , pn
lie on a geodesic in X. Then we have the following.
Proposition 4.9. The Steiner set С of vertices p\, ... , pn is given as follows.
(1) If p\, ... , Pn are not completely degenerate, then С consists of the one
Steiner point.
(a) Suppose there are no Steiner vertices. Then there exists a unique point
ρ that belongs to the interior of the convex hull of {pi, ... , pn} and satisfies
Σ cos APjPPk) = 2".
In fact, ρ is the Steiner point.
(b) Suppose pi is a Steiner vertex. Then Pi is the only Steiner vertex and
is the Steiner point.
(2) Suppose pi, ... , pn are completely degenerate and are on a geodesic 7. We
may assume that Pi = 7(^1) with t\ < £2 < · · · < tn. Then
(a) If η is odd, then С = {pn+i }.
(b) If η is even, then С = 7([£§, £§+1]).
PROOF. First recall that the distance function dp is a convex function and
proper, namely, ^"^[O, r]) is compact for any r > 0. Since the cut locus Cp of any
point ρ G X is empty, the distance function dp to ρ is smooth onI\ {p} (Chapter
III, Proposition 4.8) and its gradient vector Vdp(q), q G X \ {p} is given by 7(/),
where 7 : [0, /] —> X denotes the normal geodesic joining ρ to q with / = d(p, q).
As in Proposition 4.3, its Hessian D2dp(q)(x, χ), χ £TqX, is given by
D2dp(q){x, x)
= I {(vr^W, vyH*)) - KiyHt), 7(0)11^(0 Л7||2}л,
Jo
where Υ denotes the Jacobi field along 7 with Y{0) = 0, Y(l) = x, and Y1- denotes
the orthogonal projection of Υ to 7-1-. It follows that D2dp(q) is positive semidefi-
nite. Then for an element χ of the null space of D2dp(q), the above Y1- is parallel
along 7, and ^ξΟ because Y(0) = 0. Namely, the null space of D2dp(q) is a
one-dimensional subspace of TqX generated by 7(/).
Now we set / = £3"=1 dPi and our problem is to determine the Steiner points
at which / assumes its minimum. Since / is also convex and proper, the Steiner
set С is a compact convex subset of X. С is contained in the convex hull of
{Pi, · · · , Pn}, since the projection to a closed convex set is distance decreasing. If
the vertices Pi are completely degenerate, then our problem is reduced to an easy
line geometry, which is left to the reader. We treat the case (1) in the following. For

4. MANIFOLDS OF NONPOSITIVE CURVATURE
227
q £ X \ {pi, ... , pn}, / is C°° at q. For the gradient vector of / we have Vf(q) =
ΣΓ=ι X{Pi, q), where X(pi, q) denotes the (unit) tangent vector at q to the normal
geodesic 7* joining pi to q. Further, we have D2f(q)(x, χ) = ΣΓ=ι D2dPi (q)(x, x) >
0, and if D2f(x, x) = 0 for an χ £ X^X (ж φ 0) then a; belongs to the null space
of D2dPi(q) (i = 1, ... , n). Therefore, X(pi, q) are linealy dependent on x. This
means that pi, ... , pn, q lie on a geodesic, which is a contradiction in case (1).
Therefore, D2f is positive definite at any q £ X \ {pi, ... , pn} in case (1).
Next we show that С consists of only one point. In fact, suppose q\, q<i £
С (<7ι Φ Я2) and take the geodesic segment 7 joining q\ to #2· Then, by the convexity
of /, / is constant on 7 and 7 is contained in C. Now take a point ς on 7 different
from all the pi (i = 1, ... , n), and note that for the tangent vector χ at q to 7
we have D2f(x, x) = 0. This contradicts the strong convexity of / at q, and so С
consists of the one point q.
Now suppose q is different from the vertices p*. Then / is C°° at q, and we
get Υ^1=λ X{Pi, q) = 0. On the other hand, if this condtion is satisfied, then q is
a critical point of a convex function / and is a minimal point of /. Now we give
a characterization of this condition. We set X{ := X(pi, q), у = ΣΓ=ι Xi and note
that the X{ are unit vectors. Then we have
Lemma 1. у = 0 <Ф T,j<k{j^i)(x3i xk) = -^ (t = 1, ... , η).
Proo/ of Lemma 1. From (3/, y) = Σι(χΐι χι) + 2Σ7<*:(:Ε.7> ж&) and (!/» я») =
{xu Xi) + Ei^t(xi» χύι we Set
(4.5)
(t/, y) -2{xi, y) = (n-
2)+ 2 J] <*i»**>»
and (=>) is obvious from this. Conversely, suppose (y, y) = 2(x{, y) (i = 1 n).
Summing over г, we get n(y, y) = 2{y, y), and у = 0 because η > 3. D
Now the assertion (a) of (1) is obvious. Next we consider the case where
/ assumes its minimum at a vertex p*. We show that this happens if and only if
\\z\-\ Ι-2ί-ι+2»+ιΗ l· zn\\ < 1, where we set zk := X{pk. Pi) {к Φ 0· First we
show the "if " part. For any geodesic 7 emanating from px. we set F(t) := /(7(0)·
Then from the Cauchy-Schwarz inequality we have
UmF'(i) = lim(V/(7(*)), Ή0> = (W) + J>, W
\ кфг I
> 1
кфг
>0.
Then, by the convexity, F(t) is monotone increasing and / assumes its minimum
at pi, since 7 is an arbitrary geodesic emanating from p*. Next suppose that
\\z\ + · · · + Zi-ι + Zi+i H- · · · + z„|| > 1. Take a geodesic 7 emanating from p*
with 7(0) = - EMi VII Екфг **ll» and set F(0 := /(7(0) ^ before. Then
limF'm
no
7(0), 7(0) + Σ^
кфг
Σ**
кфг
<0

228
V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
and there exists a t > 0 with F(t) < F(0) = f(pi). In other words, / cannot assume
its minimum at Pi. Now assertion (b) of (1) is clear from the following lemma.
Lemma 2. Let Z\, ... , ^_ι, ζί+\, ... , zn be unit vectors. Then
||*i+ ··· + *_!+Ζί+ι + ··· + ζ„|| < 1
if and only if
Σ <**■**>< -("-2)/2.
3<к(з.кфг)
Proof of Lemma 2. We set ζ = zx + · · · + ^_ι + ζί+λ + ··· + *„. Then we get
i^< 3<к{з,кфг) 1фг
and consequently 1 - {z, z) = ~{n-2)-2Y2j<k(j,k^i)(zJi zk), from which Lemma
2 is clear, and the proof of Proposition 4.9 is complete. DD
Exercise 5. Show that Proposition 4.9 implies Steiner's theorem in the case
of a Euclidean triangle.
Two normal geodesies 71, 72 : R —» X are said to be parallel, and we write 71 ||
72, if a := sup{d(7i(f), 72), d(72(0> 7i); t e R} < +00. We show that if 71 || 72,
then we have a flat totally geodesic embedding φ : R x [0, a] —» X with 7^) =
</?(£, 0), 72 (ί) = </?(£, α), changing the initial point of 72 if necessary. To see this, first
note that 11-> d(7i(f)» 72), * ·-> й(7г(0» 7i) are c°nvex and bounded, and therefore
they are constant. Further we may easily check that d(^i(t), 72) = d(^2(t), 71) = a.
Now let π72 : 7i —> 72, π7ι : 72 —* 7i be maps assigining the feet of perpendiculars.
Then perpendiculars are orthogonal to both of 71, 72 by the first variation formula.
We have π72οπ7ι = id72, π7ι οπ72 = id7l. Therefore, for t\ < t2 all the vertex angles
of the geodesic quadrilateral with vertices 71(^1), π72(7ι(£ι)), π72(7ι(£2)), 71(^2)
(see Figure 31) are equal to π/2, and Exercise 4 implies that this quadrilateral
spans a flat totally geodesic surface which is isometric to a rectangle in R2. Now
change the parameter of 72 so that π72(7ι(0)) = 72(0), and denote by 7* : [0, a] —»
X (t e R) the normal geodesies joining 71 (t) to 72(0· Then we define φ(ί, s) :=
7t(s), t e Д, s e [0, a], which gives a desired embedding.
Ί
a
я dc
Λτ,(7ι(/ι)) ^2(7Ί(/2))
■^•л
Figure 31

4. MANIFOLDS OF NONPOSITIVE CURVATURE
229
Lemma 4.10. Let 7 : R -+ X be a normal geodesic in X and Ρ a closed
convex set consisting of normal geodesies parallel to 7. Then there exists a closed
convex set С such that Ρ is isometnc to RxC.
Proof. Let {7s}s€S be the set of normal geodesies in Ρ parallel to 7. Since
Ρ is convex, there exists a totally geodesic fc-dimensional submanifold N of X
contained in Ρ and satisfying Ν = Ρ (Chapter IV, Theorem 5.5). Then the above
argument implies that ΛΓ э 7.(t) -> 7,(t) defines a unit parallel vector field on
N. Now we consider a (k - l)-dimensional distribution i/x on N which assigns the
orthogonal complement of 7s in TqN to q = 7.(f) e N. Then v± is involutive, since
q~vq is parallel. Namely, u1- is completely integrable, and integral submanifolds
of i/x are totally geodesic. Now let q 6 N, 7si. 7i2 с Л*. Then
(4·6) s^sw^^w.
In fact, let δ be a normal geodesic joining q to π,.,(ί). which is contained in the
maximal integral manifold Я of v1- through q by the" argument before Lemma 4.10.
Similarly, a normal geodesic joining ?,€Я(о points jt^ foi) is contained in Я
and so is 7Γ7ίι ο «5. On the other hand, π,Μ ο δ is contained in ->Sl by definition. It
follows that π7η ο «5 is a trivial point curve, and (4.6) holds. Note that (4.6) holds
also for q e Ρ and 7si ,7j!cP by continuity.
Tr.,°Tr.,(r(<i)) лф(12))
Figure 32
Now for ρ := 7(0) we set С := {тг7»; s e 5}, and define Φ : ЛхС-> Р by
ф(*> ^7.(Р)) := п-у.Ы*))· Note that С is a closed convex set because of (4.6). We
may easily see that Φ is bijective and maps Rx ЫС diffeomorphically onto N.
Finally, from (4.6) we get
<*2Κ51(7(ίι)),7Γ752(7(ί2)))
= ^2Ksl(7(ii)), π7ί2 οπ75ι(7(ί1)))2 + <ί2(π752(7(ί2)), π^ 0π75ι(7(ί1)))2
= ^K1(P),T7s2(p)) + d2(7(i1),7(i2)),
which shows that Φ is an isometry preserving the distance. We may also easily
check that N is isometric to the Riemannian direct product R χ int С D
4.2. Next we consider the set of points at infinity for an Hadamard manifold
X, and give a compactification of X. We begin with a definition.
Definition 4.11. Let 7, 7' : [0, +cc) -» X be normal geodesies which are
rays. If there exists a positive constant D such that d(7(i), 7'(0) < D for any
t > 0, then 7, 7' are said to be asymptotic, and we write 7 ~ 7'.

230 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
For instance, in (Дт, go), 7 ~ 7' if and only if they are parallel. Next let
(B171, go) be the Poincare model of the simply connected space form of constant
curvature — 1. Then geodesies are circles orthogonal to 5m_1 = дВш. For a ray 7
in (Bm, go), we denote by 7(00) the point of 5m_1 = дВш at which 7 intersects
5m_1. Then 7 ~ 7' & 7(00) = У (00) <ί=> d(*y, 7') = 0.
For a general Hadamard manifold X. the notion of asymptotic rays cleary gives
an equivalence relation on the set of all rays in X. We call an equivalence class of
7 a point at infinity determined by 7, which will be denoted 7(00) as in Вш. Now
let 7 be a ray and ρ any point in X. We show that there exists a unique ray 7P
emanating from ρ which is asymptotic to 7. Uniqueness follows from the following.
Let 71, 72 be different rays emanting from p, and set α := Ζ(7χ(0), 72(0)) > 0.
Then by (4.1) we have
d2(7i(0> 72(0) > 2*2(1 - cosa) -> +00 (f -> +00).
Next, to show the existence, let *yt be rays emanating from ρ that join ρ to 7(2).
Then, for a fixed £, from the convexity of the distance function we get for 0 < s < t
d(7(e), 7t((d(p, 7(*))/<)*)) < ^d(7(0), 7*(0)) < d{p, 7(0))·
Now choose fn —> +00 so that 7t„(0) —» u € i/pM, and set 7p := 7u. Then
d(7(s), 7p(s)) = lim d(7(e),7t„((d(p,7(*n))/*n)e)<d(P.7(0))
η—»+οο
for s > 0, and therefore 7 ~ 7P. Thus we may identify the set X(oo) of points
at infinity with UPM = 5m_1, and we introduce a topology on X(oo) and X :=
X U X(oo) by the following two conditions: Let ρ e X he fixed.
(4.7) Let {pn} be a sequence in X. Then pn —» 7(00) means that d(p, pn) —> +00
and 7PPn(0) —> 7p(0), where 7PPn denotes the ray joining ρ to pn, and 7P denotes
the ray emanating from ρ determined by 7(00).
(4.8) 7n(oo) -> 7(00) means that (7n)P(0) -> 7P(0).
Note that (4.7) does not depend on the choice of p. In fact, first, for q e X
we get d(q, pn) > d(p, pn) — d(p, q) —> +00. Second, for a sequence {7ςΡη} of
rays in X we have 700 = 7q for a limit ray 700 = lim^+oo 7qPn(fc) of any convergent
subsequence {7qPn(fc)} of {7ςΡη }. In fact, for t > 0, convexity of the distance function
implies, as before,
d(7p(*), 7oc(*)) = к^хЛ^РРгЧк)(^ 7ςρ„(*,(*))
and 7oo ~ 7P; that is, 700 = 7ς· It follows that limn^+00 7ςΡη = 7ς, and (4.7) holds
for q.
Exercise 6. Show that (4.8) also does not depend on the choice of ρ G X.
By the definition of the above topology, X(oo) is homeomorphic to the sphere
5m_1 and X is homeomorphic to the m-dimensional closed disk В .
Now we consider Busemann functions corresponding to rays in an Hadamard
manifold X (see Chapter IV, §5, (5.4)). Let 7 be a ray in X. We adopt b~ defined

4. MANIFOLDS OF NONPOSITIVE CURVATURE
231
by
(4.9) b~(p):= lim (d(p, 7(t)) - i),
' t—>+oo
so that b~ is a convex function. Recall that b~t(p) := d(p, ^y(t)) — t is a convex
function (Proposition 3.4).
Lemma 4.12. Busemann functions h = b~ may be characterized by the
following condition:
(4.10) h is a convex C1 function whose gradient vector field V/ι satisfies \\Vh\\ = 1
everywhere.
Proof. Prom the first variation formula, we have ξ · b~t = — (%7(ί)(0), ξ) for
ξ £ UPX. The right-hand side of this equation converges locally uniformly with
respect to ρ as t —» +00. In fact, from (4.1) and the triangle inequality, we get
l(W)(°)> 0 - <-W)(o), 01 < IIW)(°) -7г7(*)(°)Н
< coe-^iP, 7(0) + d2(p, 7(5)) - (t - s)2)/{2d{p, 7(0) · rf(p, 7(5))}
< m~-i ί * " <Г' + S~1)d{p' 7(Q)) + Г '«"'^fo ^(°)) ]
\ (1 + *-Ч(р, 7(0)))(1 + S"4(p, 7(0))) J '
and we note that the last term of the above formula converges to 0 locally uniformly
with respect to ρ as s, t —» +00. It follows that b~ = Ит^^+00 b~ t is differentiable,
and we get V6" = -7P(0) since ξ · 6" = - lim^+oo(7p7(i)(0), ξ) = -(%(0), ξ).
Note that ρ н-> %(0) is continuous. Then b~ is C1, and we have ||V6~|| = 1. We
also get
(4.11) b-(7p(t)) = -t + b-(p\ ρ eX.
In particular, b~ — b~ = b~ (q) is a constant function for p, q £ X. 6~ is convex
since it is a limit of the convex functions b~ t.
Conversely, suppose h satisfies (4.10). By the proof of Lemma 3.10, integral
curves of V/ι are normal geodesies and in fact lines. We denote by ap(t) the integral
curve of V/ι which passes through ρ at t = 0, and by σρ(—эс) the point at infinity
representing a ray σ~ι : t £ (0, +00) i-> crp( —<)· Now we show that σρ(—00) =
ση(—oo) for any p, q £ X. First we consider the case where /i(p) = h(q) =: с
Set f(s) := d(ap(s), aq{s)). Then by Proposition 4.3 we have f"(s) > 0. Let
—t < 0. Since h(ap(—t)) = h(aq(—t)) = с — t and h is convex, the geodesic
segment 7 joining σρ(—t) to aq(—t) is contained in /i_1(—00, с — ί]. Now Я :=
/ι-1 (с — ί) is a hypersurface of X and σρ, aq are perpendicular to #. Therefore,
the angles /(7(0), σρ(—t)), Δ{—η{ά(σρ{—ί), aq(—t)), &q(—t)) are obtuse or right
angles. From the first variation formula, it follows that f'(—t) > 0 and d(p, q) =
/(0) > f(—t) = d(ap(—t), aq(—t)), which means that σρ(—00) = aq(—00). Second,
we consider the general case where h(q) = h(p) + a. In this case, take q\ := aq(—a)
instead of q and note that h(q\) = h(p). Then we have
d(ap(-t), aq(-t)) < d(ap(-t), aq(-a - t)) + \a\
= d(ap(-t), aqi(-t)) + \a\ < d(p, qx) + |o|,
which again implies that σρ(—00) = σς(—oo). Now it suffices to show that h = Ь~_г
σρ
for ρ £ X with h(p) = 0. Since rays σ"1, σ~ι are asymptotic, the corresponding
Busemann functions differ only by a constant, and we have V6~_i (q) = Vb~_1(q) =

232
V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
&q(0) = Vh(q) for any q £ X. On the other hand, we get h(p) = b _i(p) = 0, and
σρ
therefore h = 6~_i. □
σρ
For a Busemann function h on X, the level hypersurfaces Ht := Λ_1(0 are
called horospheres, and /i-1((—oo, £)) are called horoballs. For ρ £ X the restriction
of /ι to σρ is a linear function of t, and the line σρ intersects every horosphere Ht
at a unique point nt(p).
For instance, in the case of (Дт, #о) Busemann functions are affine functions,
and horospheres (resp., horoballs) are hyperplanes (resp., open half-spaces). In
the case of (H171, go), horospheres of a Busemann function b~ corresponding to a
geodesic ray η{ί) := (0, ... ,0, el) are given by hypersurfaces xm = const (> 0).
Lemma 4.13. (1) nt(p) may be characterized as a unique point of Ht which
satisfies d(p, 7rt(p)) = d(p, Ht) = \h(p) — t\.
(2) For any q £ Ht, q' £ Ht> we have d(q, Ht>) = d{q'', Ht) = \t - t'\.
Proof. Since h(ap(s)) = s+h(p), we get nt{p) = &p(t—h(p)) and d(p, 7Tt(p)) =
\t — h(p)\. Next let q £ Ht satisfy d(p, Ht) = d(p, q). Take a normal geodesic
Ipq · [0, I] —> X joining ρ to q. Then from
* = M7m(0) = Λ(ρ) + / (V/l(7P<7(5)), 7M(e)>de,
we get, via the Cauchy-Schwarz inequality,
d(p, π4(ρ)) = μ - Λ(ρ)| < / (V/ι, 7Μ>|ώ < / = d{p, q).
Jo
Therefore, the equality signs hold in the above inequalities. Then V/ι, %q are
linearly dependent. It follows that 7P<7 is an integral curve of V/ι (or — V/ι), and
we get q = nt(p), which completes the proof of (1). (2) follows from d(q, Ht>) =
d(q,nt,(q)) = \h(q)-t'\ = \t-t'\. Π
By Lemma 4.13, we may consider horospheres as the distance spheres centered
at the points at infinity. Next we compute the second fundamental form of
horospheres. Let 7 : [0. +oc) —► X be a ray. Then a Jacobi field Υ along 7 is said to be
a stable Jacobi field if t £ [0. +oc) 1—► ||У(£)|| is bounded. Note that this condition
is equivalent to the fact that t \—► ||У(£)|| is monotone decreasing, because of the
convexity (See Exercise 1).
Proposition 4.14. Let η be a ray emanating from ρ £ X.
(1) For any ν £ TPX, there exists a unique stable Jacobi field Yv along 7 which
satisfies Yv(0) = v, and the map ν £ TPX 1—> Yv(t) £ ΤΊ^Χ gives a vector space
isomorphism.
(2) Yi,(£)_l_7(£) Φ> Yv(t) is tangent to the horosphere Ht of b~ at ^(t).
(3) The second fundamental form of Ht with respect to the normal vector η(ί)
is given by
(4.12) (AmYv(t), Yw(t)) = (VYv(t), Yw(t)) = (Yv(t), VYw(t)).
PROOF. (1) Uniqueness follows from the fact that any Jacobi field Υ φ 0 along
7 with Y(0) = 0 satisfies ||У(£)|| —> +oo (t —» +00) because of convexity. To see the
existence, for n = 1, 2, ... take Jacobi fields Yn along 7 with Уп(0) = ν, Уп(0) = 0

4. MANIFOLDS OF NONPOSITIVE CURVATURE
233
(Chapter II, Lemma 2.4). Then Jacobi fields Yn — Уш vanish at t = 0 and the
Rauch comparison theorem (see Problem 4 for Chapter IV) implies that
||УУп(0) - Vrm(0)|| < ||yn(f) - Ym(t)\\/t.
On the other hand, t \—► ||Υ™(£)|| is monotone decreasing on [0, m] because of
convexity, and it follows that
||УУп(0) - Vrm(0)|| < \\Yn(n) - Ym(n)\\/n < \\v\\/n (n < m).
Therefore, {VYn(O)} is a Cauchy sequence and converges to a limit w G TPX. Let
Yv be the Jacobi field along 7 with Yv{0) = v, VYv(0) = w. Then ||У„(*)|| =
limn^+oc ||УП(£)|| < ||v||, and Yv is a desired stable Jacobi field.
(2) is clear from 7(i) = -Vb"(7(i)). As for (3), from (2) and (3.25) of Chapter
II we get
(Ay{t)Yv(t), Yw(t)) = (VW)Yv(t), Yw(t))
and the proof of the proposition is complete. D
4.3. In this subsection we are concerned with the properties of isometries
of an Hadamard manifold X, and we apply them to study the structure of the
fundamental group of X. We begin with a definition.
Definition 4.15. Let X be an Hadamard manifold and μ an isometry of X
(resp., convex subset С of X). Recall that άμ : X (resp., C) —» R defined by
άμ(ρ) := d(p, μ{ρ)) is a convex function.
(1) If άμ assumes the minimum 0, namely, admits a fixed point, then μ is said
to be elliptic.
(2) If μ assumes a positive minimum, then μ is said to be hyperbolic.
(3) If μ does not assume a minimum, then μ is said to be parabolic.
An isometry which satisfies (1) or (2) (i.e., assumes a minimum) is called a
semisimple isometry.
Let μ be a semisimple isometry. Then the minimal set ηιίη(μ) := {pGl;
άμ(ρ) = πήη{άμ(ς); g G X}} is a closed convex subset. Further, the fixed point set
πιίη(μ) of an elliptic isometry μ of X is a complete totally geodesic submanifold of
X (Problem 14 for Chapter II). Note that an isometry μ of X maps asymtotic rays
in X to asymptotic rays and therefore induces a homeomorphism of X(oo).
Exercise 7. Let (Η2, go) be the upper half-plane model of the simply
connected 2-dimensional space form of constant curvature — 1. Set H2 = {z =
χ + у\Г-^\ У > 0}. Then g(z) := (az + b)/(cz + d) (a, 6, c, d <E Д, ad - bc> 0)
defines an isometry of (H2, go). Show that g {φ id) is hyperbolic, elliptic, or
parabolic if and only if (a — d)2 + 46c is positive, negative, or equal to 0, respectively.
Proposition 4.16. Let μ be an isometry of an Hadamard manifold X {resp.,
a closed convex set С of X). Then we have the following'.
(1) μ is elliptic Φ> there exists a point pGl (resp., ρ G C) such that К :=
{μι(ρ)\ I G Ζ} is a bounded subset.
(2) Let μ be hyperbolic. A geodesic 7 : R —> X (resp.C) is said to be an axis
of μ, if μ(Ί(ϊ)) = 7(£ + £0), ^0 = mindM, holds for any t G R. μ leaves 7 invariant.
Then two axes 71, 72 of μ are parallel and πιίη(μ) may be expressed as the union of
axes of μ. Further, πιίη(μ) may be decomposed into a Riemannian direct product
πήη(μ) = W x R, where W is a closed convex set of X {resp., C), and μ leaves

234 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
this decomposition invariant In fact, μ = (idw, i~(to)), where r(£0) denotes the
parallel translation t —» t + to in R.
(3) Let μ be parabolic. Then there exists a point ζ G X(oo) at infinity which
is fixed by μ. μ maps horospheres determined by ζ to themselves.
Proof. (1) (=>) is clear if we take a fixed point of μ as p. Conversely, the
center of a μ-invariant compact set К gives a fixed point of μ.
(2) Let ρ G πήη(μ). πήη(μ) is μ-invariant and takes a geodesic segment 7
joining ρ to μ(ρ). Then, for the middle point χ of 7, we get from convexity
άμ(χ) < {d{p, μ{ρ)) + ά{μ{ρ), μ2(ρ))}/2 = άμ{ρ\
where in fact equality holds because ρ G πιίη(μ). Therefore, χ G πήη(μ) and
d(x, μ(ρ)) + ά(μ{ρ),·μ(χ)) = άμ(ρ) = άμ(χ) = d(x, μ(χ)).
It follows that two geodesies joining χ to μ(ρ) and μ(ρ) to μ(χ), respectively,
coincide, and μ leaves 7 invariant. Then μ | 7 is a hyperbolic isometry of R and may
be written as μ(^(ί)) = η{ί + to) (t G R). Namely, min(μ) consists of axes of μ.
Next let 71, 72 be axes of μ. Then for η G Ζ we have
d(7i(* + "to), 72(* + nt0)) = ά(μη(Ίι(ί)), μη(72(0)
= d(7i(*),72(*)), 0<*<*0·
Then iGiin d(7i(i), 72(0) ls a bounded convex function and is constant.
Therefore 71 || 72. Since πήη(μ) is convex, by Lemma 4.10, πιίη(μ) may be written as
W x R, where W is a closed convex set. Since {w} χ R is an axis of μ, the last
assertion on μ is also clear.
(3) Suppose μ is parabolic. By definition, for a fixed point ρ G X we may
choose {pn} С X (resp.,C) such that d(p, pn) —» +00 and limn^+00 d(pn, μ(ρη))
= inf {d(<7, μ(<7)); ς G X (resp., C)}. We may assume, taking a subsequence if
necessary, that Pn —> ζ e X(oo). To see that μ(ζ) = ζ, it suffices to show that ^(7) ~ 7,
where 7 denotes a ray emanating from ρ determined by z. In fact, we get
d(7(*), M7(*))) = lim d(7ppJ0.M7ppn(*)))
η—>+эс
= J^^PPn^ 7μ(ρ)μ(ρ„)(0)
- Лтэс^Рп' P(Pn))/d(P' Pn) + (d(p, pn) - i)d(p, M(p))/d(p, pn)}
= d(p, μ(ρ)).
Next we prove the assertion on horospheres. First note that 6~7 ο μ = b~. Prom
7 ~ μ(7), we get b~y = b~ + 6~7(p) = 6~ + 6~(μ_1(ρ)) with ρ = 7(0). It follows
that
b-(q) = t * ь;ш) = ь^(мЫ) - ь;^-\Р)) = t - ь;^-1{р)),
and we get μЯί = Ht-i, I = b'^'1^)). D
Exercise 8. Show that μ is elliptic (hyperbolic or parabolic) if and only if
μ1* (к G Ν) is elliptic (hyperbolic or parabolic).
Lemma 4.17. Let X be an Hadamard manifold and С С X a [nonempty)
closed convex subset of X. Let Л be a set of semisimple isometries of С that are
mutually commutative. Then S := Γ\α£Δ min(a) is a nonempty Α-invariant closed
convex subset. Further, S may be decomposed as a Riemannian direct product

4. MANIFOLDS OF NONPOSITIVE CURVATURE
235
5 = S\ x Rk so that any α £ Δ preserves this decomposition and may be written
as a= (ids1, a'), where Ы is a parallel translation in Rk.
PROOF. We give a proof of the lemma by induction on the dimension of C.
The assertion is obvious if dim С = 0. So, first, suppose Л contains an elliptic
isometry β (β Φ id). Then min(/3) is a Л-invariant closed convex set, because Δ is
commutative. Note that α £ Л acts on min(/3) as a semisimple isometry. In fact,
for ρ £ X, let 7rmin(0)p be the foot of the perpendicular from ρ on min(/3). Then
πτη\η{β)<χ{ρ) = &(кт\п(р)Р), since τηιη(β) is invariant under a. On the other hand,
the map assigning the foot of the perpendicular is distance decreasing, and dQ(p) >
da{^mm(p)P)' Namely, dQ assumes its minimum on min(/3). Now if dimming) =
dimC, then we have min(/3) = С and β = id, which is a contradiction. Therefore,
we get the assertion of the lemma by applying induction to min(/3).
Second, suppose Δ contains a hyperbolic element β. Let 7 be an axis of β. For
any element α £ Л we get βαη = αβη = cry, and αη is also an axis of β. Therefore,
c*7 || 7, and dp is Л-invariant because of commutativity. It follows that τηιη(β) is a
Л-invariant closed convex subset consisting of axes of β which are mutually parallel.
Then, by Lemma 4.10, min(/3) may be decomposed as min(/3) = С x Д, where С
is a closed convex subset. Now any α £ Л maps the axis 7 to an axis αη parallel to
7, and preserves the above decomposition. Therefore, we may write α = (αϊ, c*2),
where ct\ (resp., 0.2) is an isometry of C" (resp., R). Now α £ Л acts on min(/3) as
a semisimple isometry (for the same reason as above), and so is a\. On the other
hand, Lemma 4.16 implies that β acts on min(/3) as β = (idc, τ(£ο))· Then 0.2 is
a parallel translation in Д, since it commutes with the parallel translation τ (to).
Therefore, min(a) Π πύη(β) may be decomposed as min(ai) x R. Now we denote
by ρ ι an endomorphism assigning a semisimple isometry αϊ of С to α £ Л. Then
(С, Р1(Л)) satisfies the assumption of Lemma 4.17 and dim С < dim С By the
induction hypothesis we have Γ\αι€Ρι(Δ) mm(ai) = S\X Rl and αϊ = (ids;» т(^о))·
It follows that Пае л ™η(α) = Г\а1еР1(Л) πώι(αι) χ R = Si x Rl+1. The assertion
for а £ Л may be easily verified. D
Now we may state an important result on the structure of the fundamental
group of a compact Riemannian manifold of nonpositive curvature, which is due to
D. Gromoll and J. A. Wolf ([Gr-Wo]) and H. B. Lawson and S. T. Yau ([L-Y]).
Theorem 4.18. Let Μ be a compact Riemannian manifold of nonpositive
curvature and π : X —> Μ the Riemannian universal cover of M. We identify the
fundamental group of Μ with the deck transformation group Γ of π. Then for a
(nontnviat) solvable subgroup Δ of Γ, there exists a Δ-invariant complete totally
geodesic flat submanifold Ε of X which is isometric to (Rk, g0) (k > 1).
PROOF. Let а £ Л (α φ id). Since M is compact and α admits no fixed points,
dQ assumes a positive minimum by Lemma 1.5. Namely, α is hyperbolic. First,
suppose Л is an abelian subgroup. Then 5л := Пае л mm(a) 1S a Л-invariant
closed convex set of X. By Lemma 4.17, Sa may be decomposed as a Riemannian
product Sa = S\ x Rk (k > 1), and a e Δ acts on 5л as а = (id^, a'), where οι
acts on Rk as a parallel translation. Therefore, Ε := {s} χ Rk is a Л-invariant
complete totally geodesic flat submanifold on which Л acts discretely as a group of
parallel translations. Then Л is isomorphic to Zl (I < k).
Next we consider the case where Л is a general solvable group. By definition,
we have a finite series of subgroups Δ = Δ0 D Δ1 D - - D Лп_1 D Δη = {id}

236 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
of Л such that Лг+1 С Лг are normal subgroups and Δ1 /Лг+1 are abelian (г =
0, ... , η—1). In the following, we show by induction on η that the closed convex set
Sa := Паелтш(а) ls Л-invariant and may be written as a Riemannian product
Sa = S\ x Rk, where а £ Δ acts on 5л as а = (id^, a'), and {а'; а £ Л} acts
discretely and freely on Rk as an isometry group. If η = 1, then Л is abelian
and the assertion follows from the above. For Л1, by the induction hypothesis we
have a decomposition S^1 = S[ x Rl, where S^ is Δ ^invariant and the action
of Δ1 may be expressed as above. Since Δ1 is a normal subgroup of Д we see
that 5/\i is Л-invariant. In fact, for а £ Л we get α(5^0 = &{Γ\β£Δι mm(/^)) =
Γ\β£Δι min(a/3a_1) = 5/\i. Then a £ Л preserves the above decomposition of S^i,
since for ρ £ 5J, {ρ} χ ilz is spanned by lines which pass through ρ and are invariant
under the action of Δ1. Now Δ/Δ1 is abelian, and we consider its action on S[.
Prom the first case we get a Riemannian direct product decomposition S^/δ1 —
5" χ Rl , where Δ/Δ1 preserves this decomposition and acts trivially (resp., as
parallel translations) on 5" (resp., Rl ). Then we may easily check that 5л =
S" x (Rl x Rl) and the action of Δ preserves this decompostion with the desired
properties. Then Ε = {ρ} χ Rk, Rk := Rl χ Rl is a desired submanifold. D
Now we give applications of the above theorem. First we state a result on the
fundamental group of manifolds of compact negative sectional curvature due to A.
Preissmann ([Pr]) and W. Byer ([By]) .
Theorem 4.19. Let Μ be a compact Riemannian manifold of negative
curvature. Then any (nontnvial) solvable subgroup of the fundamental group of Μ is an
infinite cyclic group.
PROOF. Let Л be a solvable subgroup of Г and consider the decomposition
5л = S\ x Rk in the proof of Theorem 4.18. Then, from the assumption on the
curvature, we have к = 1. Therefore, any a ζ Δ has a common axis 7 on which Л
acts discretely and freely as parallel translations, i.e., a(7(£)) = j(t-\-t(a)). Let ao
be an element of Л such that |£(ao)| = min{|£(a)|;a £ Л \ id}. Then any a e Δ
may be written as а = а§ {к £ Ζ) on the axis. Since ctQka fixes the origin of the
axis, we get а = ak. D
Theorem 4.20. ([Y-l]) Let Μ be a compact Riemannian manifold of nonpos-
itive curvature. Suppose the fundamental group of Μ is solvable. Then Μ is flat
and isometric to Rm/T.
PROOF. Applying Theorem 4.18 to Γ = πι (Μ, ρ), we get a Γ-invariant
complete totally geodesic submanifold Ε of the universal Riemannian cover X of M,
which is isometric to (Rk, go). Note that for ρ £ Ε geodesies joining ρ to η(ρ) (7 £
Γ) are contained in E. If TPE = TPX, then we get Ε = X and the proof is
complete. Suppose TPE С TPX. Then for a normal geodesic с of X emanating from
ρ perpendicularly to E, c(0) is the unique foot of perpendicular from c(t) to the
closed convex subset E. Then the image с := n(c) of с under the covering map
π : X —> Μ is a geodesic in Μ emanating from ρ = π (ρ). We show that there exists
no cut point of ρ along c. In fact, otherwise let c(£0) be the cut point. Since Μ is
of nonpositive curvature and free of conjugate points, there exists a 7 £ Γ (7 Φ id)
such that d(c(*0), 7(^(0))) = d(c(*0), c(0)) = d(c(*0), E). Then 7(c(0)) (^ c(0)) is
also the foot of the perpendicular from c(to) to E, a contradiction. On the other

PROBLEMS FOR CHAPTER V
237
hand, Μ is compact and therefore there exists a cut point of ρ along c. Therefore,
TPE = TPX. D
Remark 4.21. Let Μ be a compact Riemannian manifold of nonnegative
curvature. We give another example in which the assumption on the fundamental
group determines the structure of M. Suppose πι(Μ, ρ) has a trivial center and is
decomposed into a direct product πι(Μ, ρ) = G\ χ G<i. Then Gromoll and Wolf
([Gr-Wo]) and Lawson and Yau ([L-Y]) showed that Μ may be decomposed as a
Riemannian product Μ = Μχ χ M2 with G{ = 7r(Mj, pi) (i = 1, 2).
Remark 4.22. Let Μ be a compact Riemannian manifold. Comparing
Theorems 4.19 and 4.20, we see that we have the different conclusions on the structure
of the fundamental group of Μ according to whether Μ is of nonpositive curvature
or of negative curvature. Also, for an Hadamard manifold X, it is known that the
metric structures of the set X(oo) of points at infinity are different according to
the above assumptions (see [Ba-G-Schr]).
The behaviors of geodesies are also different according to the above
assumptions. For instance, suppose Μ is a compact (or more generally complete and of
finite volume) Riemannan manifold of negative curvature with finite lower bound —a2
for its sectional curvatures. Then there exists a normal geodesic 7 in Μ such that
7(£) (t £ R) is dense in UM (in fact, the geodesic flow is ergodic in UM). Further,
the notion of rank may be defined for complete Riemannian manifolds of nonpositive
curvature corresponding to the rank of (locally) symmetric spaces as follows: For
и £ UM we denote by rank(u) the dimension of the vector space of parallel Jacobi
fields along the geodesic ju, and define rank (M) := min{rank(u): и £ UM}.
Now suppose Μ is a complete Riemannian manifold of nonpositive curvarture
with finite volume and finite lower bound for its sectional curvatures. Then W.
Ballmann, K. Burns and R. Spatzier showed the following: If the universal
Riemannian covering Μ is irreducible and rank(M) > 2, then Μ is isometric to one
of the Riemannian symmetric spaces of noncompact type. On the other hand, if
the rank of Μ equals 1, then Μ has similar properties as in the negatively curved
case. For instance, there exists a normal geodesic 7 in Μ such that i(t) (t £ R) is
dense in UM (see [Ba],[Bur-Sp]).
Finally, we remark that the existence of a Riemannian metric of negative Ricci
curvature does not restrict the manifold structure except for the 2-dimensional
case. In fact, for any manifold Μ with dim Μ > 3 it is now known that there
exists a complete Riemannian metric of negative Ricci curvature (see [Gao-Y] for
the compact three-dimensional case and [Lo] for the general case). Also, for any
manifold Μ (dim Μ > 3), there exists a complete Riemannian metric of negative
constant scalar curvature (see [Au-2], [Lo]).
Problems for Chapter V
1. Let Μ be a compact Riemannian manifold and π : Μ —» Μ the Riemannian
universal cover. We identify the fundamental group πι(Μ, ρ) with the deck
transformation group Γ of π. Now we fix a point ρ £ π-1 (ρ) and define a norm of 7 £ Γ
by ||7|| := d(p, 7(p)). Take a set 5 := {7 £ Г; Ц7Ц < Sd(M)} of generators for Γ
(see Lemma 1.2). We define a second norm ||7||aig of 7 £ Γ as the word-length of 7
with respect to 5. Now prove that
d(M)||7||alg<||7ll<3d(M)||7||aIg.

238 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
2. For the situation of Problem 1, for a positive number r we define
N(r) := Jt{7 £ Г; ||7|| < г}, Л(Г) := liminf logN(r)/r,
r—*+oo
h{M) := liminf log(vol£r(p, M))/r.
Then prove the following:
(1) Л(Г), h(M) do not depend on the choice of ρ £ M.
(2) h(M) > Λ(Γ) (If we take Si := {7 £ Г; Ц7Ц < 2d(M)} as a set of
generators and let ||7||*lg be a norm defined by the word-length with respect to Si,
then, setting, Nx(r\ := tt{7 Ξ Γ; ||7llaig < г}, МП := liminfr_+00 logWi(r)/r,
weget/i(M) >2d(M)Mr)).
3. Let Μ be a complete Riemannian manifold and V, W compact submanifolds of
M.
(1) Suppose that Μ is of positive curvature and V, IV are totally geodesic.
Then show that V DW φ φ whenever dim V + dim W > dim M.
(2) Suppose that the Ricci curvatures of Μ are positive everywhere and V, W
are minimal hypersurfaces. Then show that V C\W Φ φ.
4. Let Μ be an orientable compact Riemannian manifold and / : Μ —> Μ an
isometry.
(1) If dim Μ is even and / preserves the orientation, then / admits a fixed
point, i.e., a point ρ with f(p) = p.
(2) If dim Μ is odd and / reverses the orientation, then / admits a fixed point.
5. Let Μ be a compact hypersurface of (Дт+1, до) (m > 2) with the induced
Riemannian metric. Show the following.
(1) Suppose the sectional curvatures of Μ vanish nowhere. Then Μ is of
positive curvature.
(2) Under the assumption of (1), Μ is difFeomorphic to the sphere Sm (show
that the map which assigns to ρ £ Μ the unit outer normal vector to Μ at ρ is a
covering map from Μ to Sm).
6. Let (CP2n~l, h0) be the complex projective space of odd complex dimension
(> 3) with the Fubini-Study metric whose sectional curvatures satisfy 1/4 < Κσ <
1. Let φ be an isometry of CP2n~l defined by
ψ[ζ\ '■■'■ z2n} := [Ξ„+ι : · · · : *2n : -*i · · · : -zn]
in terms of homogeneous coordinates. Then φ is free of fixed points, and φ2 = id.
Let Μ := CP2n~l/{φ, id} be the Riemannian manifold obtained from CP2n~l.
Show that d(M) = π (= d(CP2n-1)).
7. Let Μ be a complete noncompact Riemannian manifold of nonnegative
curvature and С a compact totally convex subset of C. Suppose a normal geodesic
7 : (—00, +00) —» Μ is contained in C. Show that there exists an a > 0 such
that 7 С дСа, where Ca is as given in the proof of Theorem 3.4. Next let S be
a soul of Μ. Show that for a geodesic 7 which intersects S transversally we have
limt-.ioo d(7(i), S) = +00.

NOTES ON THE REFERENCES
239
8. Let μ be ал isometry of (Дт, <7о)· Show that 7 is semisimple and min(/x) is an
affine subspace of Rm.
9. Let Μ be a complete Riemannian manifold of nonpositive curvature and Br(p)
a (strongly) convex metric ball. For s 6 (0, 1) define a map ps : Br{p) —► Bsr(j>)
by Ps(q) := exppisexp"1^)). Then show that d{p3(x)y p3{y)) < sd(x, у), х, у G
Br(p).
10. Let ЛГ1? X2 be Hadamard manifolds and set X = X\ χ X2, which is an
Hadamard manifold with respect to the Riemannian product metric. Then a
normal geodesic 7 of X may be written as 7(2) = (71 (αχέ), 72(^2*)), where 71, 72 are
normal geodesies in X\y X^ respectively, and αχ, α2 > 0 satisfy a\ + a^ = 1. Now
let #t, i^t» (t = 1, 2) be horospheres determined by rays 7 : [0, +00) —► X, 7» :
[0, +00) —► Xi (г = 1, 2), respectively. Show that Ht = {(j?if p2); Pi € #ti (г =
1, 2),αι*ι+α2*2=*}.
11. Let X be an Hadamard manifold and X{oo) the set of points at infinity. X is
said to be a visiblity manifold if for any different points z, w G X(oo) there exists a
geodesic 7 : Λ —► X such that 7(+oo) = z, 7(—00) = w. Show that an Hadamard
manifold X is a visibility manifold if the sectional curvatures K„ satisfy K„ < —a2
everywhere for some positive constant a.
Notes on the References
§1. For general results on the covering space and deck transformation group
we refer to, e.g., [Si-Th], [Wo-1], and in this section we owe [B-8] and [G-6] a lot.
The relation between the fundamental group and the curvature of a Riemannian
manifold was first studied by J. L. Synge ([Sy]) and S. B. Myers ([My-1, 3]). Mil-
no^s later work [M-3] stimulated recent results on the subject (see [Wo-2], [G-4]).
Corollary 1.8 is due to [K-l] and played an important role in the sphere theorem.
Bochne^s technique to prove Theorem 1.10 is also a powerful tool to investigate
the relation between the curvature and topology of Riemannian manifolds through
harmonic forms. For further details in this field, we refer to [Yano-Bo], [Gol], [Po],
[Be-2], [Wu-2]. For the estimate for the first Betti number of noncompact complete
Riemannian manifolds of nonnegative Ricci curvature, we refer to [An-2].
Gromov gave an estimate for the first Betti number 61 (M) of a compact
Riemannian manifold Μ from above in terms of the diameter and the lower bound
of Ricci curvatures ([G-6]). We followed his argument in Proposition 1.2. It is
also possible to show that there exists an e = e(m, d) > 0 such that if the Ricci
curvatures of a compact m-dimensional Riemannian manifold Μ satisfy p(u) > — e
everywhere and d(M) < d, then b\(M) < m (see also Appendix 5 and [Ga-1] for an
analytic proof). Gromov conjectured that if b\(M) = m, then Μ is homeomorphic
to a torus (see [Ya-1], [Cou-Ro] and Appendix 6). Very recently the conjecture was
settled by T. H. Colding ([Col-1, 4], [Ch-Col-2]).
§§2 and 3. The textbooks [Gr-K-Me], [Ch-Eb], [K-5], [Ga-Hu-La], [dC], [Cha-
3] treat the subjects of §§2 and 3. There are also many survey articles on the
subjects, which will be briefly mentioned in the following. We suggest that the
reader tackle these articles, according to their interest, after reading this chapter.
For general topics on the relation between the curvature and topology, see [G-6],
[G-8], [Gre], [Gro-1], [Me], [Sa-5]. The notion of critical points for the distance

240 V. CURVATURE AND TOPOLOGY OF RIEMANNIAN MANFIOLDS
function, which was introduced in [Gro-S], [G-5], plays an important role, and we
refer to the survey articles [Ch-5], [Gro-2] for more details and further applications.
Next as for manifolds of positive (nonnegative) curvature, we refer to the recent
paper [Gr-2] (see [S-3] for the difFerentiable sphere theorem). See also [MO-Ru] for
the comparison theorem for symmetric spaces.
§3 is mainly based on fundamental papers by J. Cheeger and D. Gromoll ([Ch-
Gr 1, 2]; see also [Esc-He-1]). Proposition 3.12 is originally due to S. T. Yau ([Y-3]).
The proof presented here is taken from [Ch-G-Ta], which is an application of the
Bishop-Gromov volume comparison theorem.
Now we add some remarks on the Ricci curvature. As noted before, the Bishop-
Gromov volume comparison theorem is one of the main tools for studying manifolds
Μ whose Ricci curvatures are bounded below. U. Abresch and D. Gromoll ([Ab-
Gr]) introduced the notion of excess: For p, q € Μ the excess function E(r) is
defined as E(r) := d(r, p) + d(r, q) - d(p, q) (> 0), and under the assumption
p(u) > 0 they gave an estimate E(r) < 4{h(r)/(s(r) - /i(r))}™_1Mr)> where we
set s(r) := min{d(p, r), d(q, r)} and h(r) denotes the distance from r to a minimal
geodesic joining ρ to q. Then they applied this (and the notion of critical points
of the distance function) to show that, e.g., a noncompact complete Riemannian
manifold with nonnegative Ricci curvature and sectional curvature bounded below
is difFeomorphic to the interior of a compact manifold with boundary under an
assumption on the diameter growth (see also [Ch-5], [Pe-1]).
For examples of manifold of positive (nonnegative) sectional (Ricci) curvature,
we refer to [BB-2], [Ch-4], [Sha-Ya-1, 2].
R. Hamilton ([Ha-1]) considered the following heat equation for a one parameter
family of metrics g(t), t > 0 on a compact m-dimensional Riemannian manifold Μ :
d~tgii = mT9ij-2pij'
where #(0) is an initially given Riemannian metric g. In the case m = 3, if g
is of positive Ricci curvature, he showed that the above equation may be solved
for alH > 0, and дж := lim^oc g(t) is a Riemannian metric of positive constant
sectional curvature (see [Ha-2] for the case m = 4, and also [Ni] and [Hu] for the
higher dimensional case). Such analytic methods (see, e.g., [Au-3], [J], [Bou], [Y-4])
also provide a powerful tool in Riemannian geometry. For instance, the problem of
finding a Riemannian metric of constant scalar curvature in the conformal class of
a given metric on a compact manifold (the Yamabe problem) was finally settled by
R. Schoen by solving a nonlinear elliptic equation (see [Scho], [Au-3]).
See [Col-2, 3, 4], [Ch-Col-1, 2] for very important recent studies on the structure
of spaces with Ricci curvature bounded below.
§4. Basic references for this section are [Ba-G-Schr], [Ch-Eb], [K-5]. Example
3 is due to Bishop and O'Neill ([Bi-ON]), who first systematically studied the
convexity of manifolds of nonpositive curvature. Proposition 4.9 is taken from [N-
Sa-Mo]. The topology for the space X(oo) of points at infinity and the notion
of visibility manifolds was introduced in [E-ON]. We owe a lot to [Ba-G-Schr] for
our treatment of Busemann functions and isometries of Hadamard manifolds. For
more details on the structure of the fundamental group of manifolds of nonpositive
curvature see e.g., [Ba-E], [Schr]. Finally we recommend that the reader consult P.
Eberlein's survey articles [E-l], [E-2], [E-Ham-Schr] after finishing this section (see
also [Shi]).

CHAPTER VI
Isoperimetric Inequality and Spectral Geometry
Let с be a simple closed curve in the Euclidean plane. Then the length /
of с and the area A of the domain V enclosed by с are subject to the relation
Ζ2 > Απ A, where equality holds if and only if с is a circle. Namely, the domains
of maximal area enclosed by simple closed curves of fixed length / are disks, and
the above inequality is called the isoperimetric inequality. This is in fact one of the
most famous inequalities between various geometric invariants. We will begin this
chapter by studying some geometric inequalities among Riemannian invariants on
the measure which are related to the isoperimetric inequality.
Next we will treat the eigenvalues of the Laplacian Δ on a compact Riemannian
manifold M. In fact, Δ is the most fundamental partial differential operator on M,
and its eigenvalues are related also to physical problems such as how the tempara-
ture varies on Μ with time when a heat source is given at a point, or how Μ sounds
when we vibrate Μ (regarding it as a membrane). Therefore, eigenvalues are also
closely related to the shape of Μ, namely, Riemannian invariants of Μ. In fact, we
may estimate eigenvalues of Δ in terms of Riemannian invariants, and ideas from
the comparison methods and the isoperimetric inequality play an important role.
1. The Isoperimetric Inequality
1.1. Let Ω be a bounded domain in Euclidean space Rm (m > 2) with smooth
boundary 9Ω. Then between the m-dimensional volume vol Ω of Ω and the (m— 1)-
dimensional volume volm_idi? οίθΩ, the following relation, called the isoperimetric
inequality, holds:
(1.1) voU-idi? > imivoli?)1^,
and equality holds if and only if Ω is a metric ball Br(p) of Rm. Here 7m is
defined as follows: Denoting by am_i (resp., a;m) the volume of (m-1 )-dimensional
unit sphere (resp., the volume of the unit metric ball В in Дт), we set 7m :=
/ (m —l)/m
ctm-ι/ωϊη
The isoperimetric inequality has a long history, and many proofs are known
(see, e.g., [Os-1,2]). In the following, we give a sketch of the proof given in [B-10]
and based on an idea of M. Gromov. We may assume that vol Ω = иош, since both
sides of (1.1) can be multiplied by cm_1 via a homothety of the scale factor c. We
fix an o.n.b. {e*} and let В be the unit metric ball centered at the origin. We
define a map / : Ω —» В as follows. For ρ G Ω take the hyperplane H\ (p) through ρ
perpendicular to e\ and defined by ж1 = α1. Ω is divided by H\(p) into two pieces,
i?+ := {x e Ω; xl > a1} and i?~ := {x G i?; x1 < a1}. Then we take a hyperplane
#i(p) parallel to Ηχ(ρ) so that νο1ί?+/νο1£+ = νοΙβ'/νοΙΒ", where Б+, В~
are pieces of В devided by #i (p) as above. Next set Ωλ := Hi (ρ) Π Ω and take the
241

242 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
(m-2)-dimensional (affine) subspace #2(2?) in H\(p) passing through ρ and
perpendicular to e2. We choose an (m —2)-dimensional subspace H2(p) contained in H\(p)
perpendicular to e2, so that for B\ := H\(p) Π Β we have volm_ii?+/volm_iZ?i~ =
volm_ii?f/volm_iBf. where i?f, B^ are defined as above. Repeating this
process m times, we get Hm(p) = {p}, and the corresponding #m(p) also consists of
one point, which is defined to be f(p). Then / is surjective and its г-th component
fl(xi, ... , Xrn) depends only on x\, ... , X{. Therefore, we get dp/dx^ = 0 (j > г),
namely, the Jacobian matrix Df = [dfl/dxj] is triangular. Now from the definition
of / and the Fubini theorem, we see that / is measure preserving and det Df(p) = 1.
Denoting by λι(ρ), ... , Am(p) the diagonal elements of Df(p), we get Xi(p) > 0
and X\(p) · · -Am(p) = 1. On the other hand, we have ||/(p)|| < 1, ρ G β, because
ί(Ω) С В.
Now denoting by ν the outward unit normal vector field to ΘΩ and regarding /
as a vector field on i?, we get from the Green theorem (Chapter II, Theorem 5.11)
J^ a\vfdugo = 1впШ, Ня)) dA, div/(p) = £ У~(р) = J>(p).
Then, noting that div/(p) = Υ2Κ(ρ) > ^(Πϋι Мр))1/ш = m because of the
inequality between the arithmetic and geometric means, we have
(*) ωτη = mvoli? < / div/di^0 = / (f(q), u(q)) dA < volm_i(<9i?),
where the last inequality follows from the Cauchy-Schwarz inequality. Recalling
that raa>m = am_i, we get volm_idi? > am_i under the assumption that voli? =
ujm, namely, (1.1).
Now we check the case where equality holds. First, note that in this case we
have \\{p) = · · · = Am(p) = 1 for ρ e Ω, and (/, v) = 1 on ΘΩ. It follows that
f(q) = v(q), q £ ΘΩ. Recalling that, for ρ £ i?, Hm-i(p) is a line parallel to em,
we show that Hm-i(p) intersects ΘΩ at exactly two points. In fact, otherwise there
exists ap £ Ω such that Hm-i(p) intersects ΘΩ at q\, q^ (q\ φ qi), and the segment
of Hrn-\{p) between q\ and q2 does not intersect Ω. Then from the definition of /
we have f{qi) = /(^2), and this point belongs to the interior of B. On the other
hand, we have ||/(<7i)|| = ||/(<72)|| = 1, which is a contradiction. Second, if equality
holds in (1.1), then equality holds in (*) for an arbitrary o.n.b. {е*}, and therefore
we may choose em and consequently Hm-i(p) in arbitrary directions. Then Ω is a
convex domain in Дт, since any segment joining two points of the boundary 3Ω is
contained in Ω.
Third, if equality holds, we have dfl/dxj = 0 (j > i) and A* = df1 /дхг = 1.
Therefore, we may assume that
(**) f(x\ x\ ... , xm) = (x\ x2 + g2{xl), x3 + g3(x\ z2), ...)
by parallel translating Ω if necessary. Then from vol Ω± = vol B± and the coarea
formula (Chapter II, Theorem 5.8) we have volm_ii?i = volm_i£i. Hence, by the
method of constructing /, for Ω\ and /1 : J?i —> Bi, which is the restriction of
/ to i?i С Дт-1, the equality sign holds in an inequality corresponding to (*)
after the normalization volm_ii?i = ωτη-\. Moreover, this holds for successively
defined i?j, fi : Ω{ —» B{ (г = 1, ... , m — 2). In particular, for i?m_2, which is a
section of Ω by a plane parallel to (em_i, ет)я, and for /m_2 : Ωτη-2 -+ #m_2,
the equality sign holds in an inequality corresponding to (*), and hence (**) holds.

1. THE ISOPERIMETRIC INEQUALITY
243
Я-1Ы
Figure 33
In this two-dimensional case, we may easily check that i?m_2 is a disk. Since we
may take (em_i, ет)я as arbitrary 2-planes, it follows that a (nonempty) section of
Ω by any 2-plane is a disk whose radius is not greater than 1. Since / is surjective,
we get a section of Ω which is a disk D of radius 1. We denote by q\, q<i antipodal
points of D. Then sections of Ω by arbitrary planes D containining q\, q<i are disks
of radius 1, and Ω is a ball of radius 1 centered at the middle point of the segment
joining qx to q2.
Remark 1.1. In general, for any compact (m- l)-dimensional submanifold Η
without boundary in RN and any m-dimensional manifold Ω of RN with boundary
3Ω = Η which minimizes the volume among m-dimensinal submanifolds bounded
by #, the isoperimetric inequality (1.1) holds. Moreover, equality holds if and only
if Η is an (m — l)-dimensional sphere contained in an m-dimensional affine subspace
of RN. This general isoperimetric inequality was recently proved by F. J. Almgren
(see [Alm-2]).
Next let (5m, g0) be the sphere with constant curvature 1 and Ω a domain in
Sm with smooth boundary. Take a metric ball В in Sm such that vol Б = voli?.
Then it is known that
(1.2) volm_i<9tf > volm_i<9£,
and equality holds if and only if Ω is congruent to B.
Furthermore, the same isoperimetric inequality also holds for the hyperbolic
space (Hm, g0) (see [Bu-Z]).
1.2. If we want to consider the isoperimetric inequality of the form (1.1) in a
(connected) Riemannian manifold M, we should represent a constant coresponding
to 7m in terms of geometric quantities of M. In this subsection, we consider the
following isomperimetric quantities in an m-dimensional (m > 2) compact
Riemannian manifold (M, g) without boundary, and estimate them in terms of geometric
invariants of Μ; this estimate will be applied in §4. In this subsection a domain Ω
in Μ means an open submanifold of Μ which is not necessarily connected.
Definition 1.2. (1) For β <E (0, 1) we set Wp := {Ω С Μ; Ω is a domain
with smooth boundary ΘΩ with voli? = βνοΙΜ}. Then Wp Φ φ. Now we define
the isoperimetric function Η(β) (= Η(β, g)) as
(1.3)
hifi) := inf{volm_i0i2/volAf; Ω <Ξ Wff).

244 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Then clearly /i(/3) > 0, and we get /i(/3) = /i(l - β) because 8Ω = д(М \ 12), and
Λ(0):= lim0^oh{P) = O(=h(l)).
(2) The Cheeger isoperimetric constant hc(= ftc(M, #)) is defined as
(1.4)
hc := inf{volm_i9i?/voli7; β С М is a domain with smooth
boundary such that 2 voli? < volM}.
(3) We define the isoperimetric constant 1(M, g) as
(1.5)
I(M, g) = inf{(volm_i0i2)m/(voli2)m_1; β С М is a domain
with smooth boundary such that 2 vol Ω < vol M}.
Exercise 1. Prove the following equalities:
hc = inf{M/J)/min(/J, 1 - β)} = η*£{Η(β)/β; 0<β< 1/2},
Z(M, p)1/m = (volM)1/m inf{/i(^)//^(m-1)/m; 0 < /3 < 1/2}.
Exercise 2. Denote by Η0(β) the isoperimetric function Η(β) for (5m, <70)·
Show that
h0(i sin™-1 tdt/ f sinm-4dn =sinm"1r/ / sin™"1^.
(Use Problem 1 for Chapter IV and (1.2).)
In the following we estimate these constants in terms of the Ricci curvature
and the diameter d(M) of (M, g). First we state some fundamental properties of
the isoperimetric function.
Proposition 1.3. (1) Η(β) is a continuous function.
(2) Η{β) ~ i^volM)"1/™/^™-1)/™, where Η(β) ~ €β{™-ΐ)/τη means that
lim^o^(^)/^(m"1)/m=c
PROOF. Note that vol(M, c2g) = cmvol(M, g) and ft(/3, c2#) = /i(/3, #)/c for
a positive constant c. Therefore, we may assume that vol Μ = 1 if necessary.
(1) Since A(volBc(p)/volM) < volm_i<9£e(p)/volM -> 0(c -> 0), we have
lim^^o Λ(/9) = 0, and also lim^^i /i(/3) = 0. Namely, Η(β) is continuous at β = 0, 1.
Since Μ is compact and its sectional curvatures are bounded, from the Rauch
comparison theorem (Chapter IV, Corollary 2.8 (2)) there exists an α > 0 such
that for any ρ G Μ and any 0 < e < a
&mem < volBe(p) < lu>mem,
lotm^e™-1 < volm_!dSe(p) < \am-\em-1.
On the other hand, for Ω С М we get by the Fubuni theorem
/ vol(Be(p)ni2)di/p(p) = / vol(Be(q))diyg(q) > Lm6mvoli?,
Jm Jn 4
and so for € > 0 there exists a p G Μ such that
vol(Be(p)ni2) > fu;m£m(vom/volM).
Now to see the continuity of Η(β), it suffices to show that the inequality
\h{fi) - h{0)\ < cm{(/3' - β)I minCi?, 1 - /?)}(—D/-, 0 < β < β' < 1,

1. THE ISOPERIMETRIC INEQUALITY 245
holds for some positive constant cm, when β' — β is sufficiently small. Let Ω £ Wp
and set e := (4(/?' - β)/3β/ωτη)1/τη (< α). Then, as was shown above, we may take
a ρ £ Μ such that
vol(Be(p) Π β) > fu;m£mvom > (/?' - β)νο\Ω/β' = β' - β.
Then we have
vol(i2 \ Be(ρ)) = vol ί2 - vol (Be(ρ) Π Ω) < β,
and we may choose 0 < ε' < e so that vol(i? \ B€> (ρ)) = β.
Now, by the definition of Η(β'), for any η > 0 there exists an Ω G W/3/ such
that Ιι(β') < volm_i<9i? < Η(β') + r/, and we get the following:
M/J) < volm_ie(i2\Be,(p)) < volm_iei2 + volm_ieBe,(p)
</i(^) + r7+fam_1(6,)m-1.
Since 77 is arbitrary, this gives us
Κβ) < Η{β') + fa™-^')™-1 < Λ^;) + fem-l*m-i
= /1(Ю + ^{(^-/^)/^}(т"1)/т,
where cm is a positive constant depending only on m. On the other hand, noting
that Η(β) = h(l— β) and considering 1 — /?', 1 — /? instead of /9, /?', respectively, we
have
ΛΟ^) < Л(/?) + cm{(^' - /3)/(l - /J)}^-»/"»,
which proves the continuity of /i(/^).
(2) First we show that for any e > 0 there exists a positive number ρ =
p(M, g, e) (< z(M)/2) with the following property: For any domain Ω with smooth
boundary in Μ such that i? is contained in a normal coordinate neighborhood
#2p(p), Ρ G M, we get
(1.6) voL-iefi/ivolfl)^-1»^ > (1 - 6/2)7m.
In fact, as in (1), from the Rauch comparison theorem we have
volm_i<9f2 > (1 - £/4)volm_i<9i2, (уоШ)(т"1)/т < (1 + c/4)(voll?)im-1)/m,
where Ω = exp"1 (i?) С TPM, and the volumes of Ω and dfi are considered with
respect to the inner product of Rm = TPM. Then (1.6) follows from the isoperimetric
inequality (1.1) in Rm.
Now we turn to the general case. Let p\ pi be a maximal set of points of Μ
such that Bp/2(pi) (г = 1, ... , /) are mutually disjoint. Note that |Ji=1 Bp(pi) =
M. In fact, otherwise there exists a point p/_i e Μ such that d(pz+i, Pi) >
p(i = 1, ... , /), which implies that Βρ/2(ρι~ι) Π Bp/2{Pi) = φ (г = 1, ... , /). This
contradicts the maximality. Therefore, we have
ι ι
Σνο\Βρ/2{Ρι) < volM < 5^volBp(pi).
i=l г=1
Now suppose the sectional curvatures Κσ of Μ satisfy δ < Κσ < Δ and vol Μ = 1.
Then from the Bishop comparison theorem (Chapter IV, Corollary 3.2) we may
estimate the number / as follows:
(1-7) KM^))-1^^^))"1·

246 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Now set B\ := Br(pi) and apply the coarea formula (Chapter II, Theorem 5.8)
to the distance function to pi. Then we get
r2p
vol {B\p Π Ω) > / volm_ ι (8Β\ Π Ω) dt
Jp
and by the mean value theorem there exist ρ < r(i) <2p(i = 1, ... , /) such that
volm_i(<9£[(i) Πβ)< vol β/ρ.
Taking the sum, we get
ι
(1.8) ^volm_i(<9£[(i) Π β) < ΙνοΙΩ/ρ.
г=1
Now let 6 be a connected component of M\(Ji=1 сШ[ . Then i?fl& is contained
in some Bi % , and we denote by Ω' the union of such Ω Π b 's for all connected
components 6. Then from (1.8) we get
νο1™_ι<9β' < volm_i&f2 + 21 vol Ω/p.
On the other hand, applying (1.6) to each Ω Π b and taking the sum, we have
νοΙη-^Ω' > (1 - €/2)7m · Х^оЦЯПб))1^
6
^{Ι-ε^η^νοΧΩ)^.
Prom these two inequalities we get
(1.9) voU-id^Avom)11^ > (1 - €/2)7m - 2/(volf2)-/p.
Now we turn to the proof of (2). Applying (1.9) to any Ω e Wp, we have
(νο1™_ι<9β/νο1Μ) · (νοΙΜ)™β^ > (1 - e/2)^m - 2/(volf2)-/p.
Letting β = vol i?/vol Μ —» 0, we obtain
lim Η(β)(νο\Μ)^βλ^ > 7m,
since € > 0 is arbitrary. As for the reverse inequality, taking an e > 0 for ρ G Μ
such that vol (B€(p)) = β vol Μ for sufficiently small β > 0, we get
Η{β){νοΙΜ)^β1-^ < volm^dB^/ivolBtip))11^
m-l
-iam-i/W71 =7m (£->0)
and this completes the proof of (2). D
Now from a result of geometric measure theory due to F. Almgren ([Alm-1]),
for each β £ (0, 1) it is known that there exists a compact hypersurface N of Μ
with the following property: N is the boundary of a domain Ω in Μ which satisfies
voli? = β vol Μ and volm_idi?/volM = Η(β); namely, there exists a domain Ω
which realizes Η(β). To be more precise, N is not necessarily a smooth hypersurface.
However, for ρ £ Μ \ N let q be the foot of the perpendicular from ρ to N. Namely,
q e N satisfies d(p, q) = d(p, N). Then N is smooth in a neighborhood of q. In

1. THE ISOPERIMETRIC INEQUALITY
247
/ ~x dt
(1.10) ±
dt
particular, the set N of points of N at which N is smooth is an open dense subset
of TV.
Next we note that the mean curvatures η of N at points of N with respect to
the unit outward normal vector ν are equal to a constant. In fact, let ρ £ N and
и an arbitrary C°° function supported in a neighborhood of ρ contained in N. We
consider variations Nt of N defined by at(q) := exp-1 tu(q)vq. Then the part of N
consisting of singular points is left fixed by the above variations, and Nt divides
Μ into two domains i?t (with Ω0 = Ω) and M\Qt, whose volumes are given by
β(ί) vol Μ and (1 — β(ί)) vol Μ, respectively. The first variation formula for the
volume (Problem 2 for Chapter III) implies that
j Ι Γ
vol Qt = u(p) dA,
t=o Jn
volm-iNt = (m - 1) / η(ρ)η(ρ) dA.
t=o Jn
In particular, considering the variations Nt defined from и with JN и dvg = 0, we
have volm_i7V < νο^-ιΛ^, and Problem 2 for Chapter III implies that η is equal
to a constant (we remark that we need not assume Ω is connected).
On the other hand, if the integral of и is not equal to 0, we have β'{0) φ 0.
Suppose /3'(0) > 0. Then (1.10) implies that
limsup{fc(/J(f)) - Η(β)}/(β(ί) - β)
< lim(volm_iJVt - volm_i7V)/(voli7i - voli?) = (m - 1)η
and we have an estimate for η:
(1.11) η > —ί— limsup{/i(/3 + б) - /i(/3)}/e.
m — 1 e^o
Note that the mean curvature of N with respect to the unit inward normal is equal
to — η at regular points.
1.3. We begin with an estimate of the Cheeger isoperimetric constant due to
S. Gallot ([Ga-2]).
Theorem 1.4. Let Μ be α compact Riemannian manifold of dimensioin m (>
2), and suppose that the Ricci curvatures of Μ satisfy p(u) > (m—l)6 for a constant
δ. Let d = d{M) be the diameter of M. Then
(1.12) hc(M)>U c^-^dtl ,
where cs is given in Chapter IV, §1.
PROOF. Since ΙϊΐΩβ^0^β)/β = +oo by Proposition 1.3 (2), and /i(/3) is
continuous, there exisits 0 < β < 1/2 at which /i(/3)//3 assumes its minimum, which is
equal to the Cheeger constant hc. For this β there exists a hypersurface N of M,
that realizes /i(/3) and divides Μ into two domains, Ω and Μ\Ω. Recall that the
mean curvatures of N at regular points belonging to N are equal to a constant η.
Then from Theorem 3.8 (2) of Chapter IV we get
rP fd-p
voli2<volm_iN / Jv{t)dt, vol(M\i2) < volm_iAT / J-r,{t)dt,
Jo Jo

248
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
where we have set ρ := max{d(x, TV); χ G Ω} and ^(t) = [cs(t) -\-η ssit)}7^-1 (a+
:= max(a, 0), a G β). We easily see that max{d(x, TV); χ G M\i7} < d — p, where
d is the diameter of Μ. We may assume that η > 0 by interchanging i? and Μ\Ω
if necessary. First we prepare the following lemma.
Lemma 1.5. Suppose a, b > 0, a + 6 = d. Then
a a rb \ pd/2
Jn(t)dt, / ,/_,,(*) d* J < / J0(0*.
Proo/ o/ Lemma 1.5. Note that the first (resp., second) integral on the left-
hand side is monotone increasing (resp., decreasing) with respect to a. Therefore,
the left-hand side assumes its maximum at some a = α(η), for which we have
(1.13)
/ Jr,{t)dt= J
JO JO
7(*)A.
Note that α(η) < 6(r/). Note also that we may assume cs(t) + ^(i) > 0 on
[—6(77), 0(77)] by taking a, 6 small if necessary.
Differentiating both sides of (1.13) with respect to 77 and noting that α + b =
d, J-n{t) = Jv(~t), we get
a'fa) =
m — 1
Μη)
λμ,»+^«,ι>U *-"■«-·)*-Л *'м,<о«>.
./0
where </?(£) = 5,5(t)/(cs(t) +775,5(0)· Now we may easily check that φ(ί) is monotone
increasing. Next we set
Η(η)
ρα(η) *b{rj)
:= / Jr,{t)dt+ / ,/_„
ν/θ ν/θ
(ί)Λ
and compute Η'(ή). Then from the above, we get
Η\η)
rHv)
2(m-l)
= -J4(a(v)) / (-V(-O)^(-*) rfi + J-v(Hv)) / ¥>(<)Л(0 di-
Jo Jo
га(~п)
Now note that -φ(-ί) > φ(ί) > 0, Jv(t) > Jv{-t), 6(77) > 0(77), and ^(α(η)) >
J-V(b(rf). Then we see that the right-hand side of the above equation is
< Jn(a(rf)
< φ(α(η))^(α{η))
ρα(η) Μη)
Jo Jo
na{rj) гЪ(г))
l Jrj(t)dt- / Jrj(-t)dt
Jo Jo
= 0.
It follows that #'(77) < 0, and Я(77) assumes its maximum at 77 = 0. Namely,
o(0) = 6(0) = d/2 and Η (η) < tf(0) = 2 f*/2 J0{t) dt. On the other hand, from
(1.13) we get ^{η) Jv(t) dt = \Η{η) < f*/2 J0(t) dt. D(Lemma 1.5)

1. THE ISOPERIMETRIC INEQUALITY 249
Now we return to the proof of the theorem. We have
( rp rd~p
hc = vol N/ min(vol 12, vol Μ \ Ω) >
>{fj2 c^-\t)dt\ ,
mm
/o
Jr,(t)dt, I J-rj{t)dtl
which completes the proof of Theorem 1.4. D
Remark The above inequality is known to be best possible if δ < 0 or δ =
π2/ά2.
Now we give an estimate of the isoperimetric constant due to P. Berard, G.
Besson, and S. Gallot ([Be-Bess-Ga-1], [Ga-2]).
Theorem 1.6. Let Μ be a compact Riemannian manifold of dimension m (>
2) with d(M) = d. Let (5m, go) be the sphere of constant curvature 1, and set
h0(P) = h(p,(Sm,9o)).
(1) Suppose the Ricci curvatures satisfy p(u) > m — 1 (u £ UM). Then
(1.14)
цр) > ι r'\ '-"" ^1/m
/ fd/2 Ϊ 4m
ltdt I cos"1"1 tdt \ (>1).
M0) yjo
(2) Suppose p(u) > 0 (u € UM). Then
ltdt\
(3) Suppose p(u) > -(m- l)(uG UM). Then
> min I /
(116) \ .. „, , I/-'
/*π/ /* m-1
/ sinm~4<ft/ / созп^гЫг,
Уо Уо
J Г sin™-1 tdt/ f
d m-x |
cosh^~ 2Ы£
PROOF. In the following, δ is equal to one of 1, 0, or —1 according to whether
we are considering part (1), (2), or (3) of the theorem, respectively. First, suppose
h/ho assumes its minimum at βο £ (0, 1). We show that h is differentiable at
βο and Η'(βο) = (/ι//ι0)(Α)) · ^ό(Α)), using the same idea as in (1.11). Setting
с = Jq sinm_11 dt, note that
(1.17) h0 (- / sinm"4rfn = -sinm_1r
by Exercise 2, and Η$(β) is differentiable. Recall that there exists a domain Ω which
realizes h(/?o) such that the mean curvatures of TV := ΘΩ at regular points are equal
to» a constant η. Considering variations Nt of N given by at(p) := exp^ tu(p)vp (p G
N) as before, we get (1.10). Setting f{t) := Η(β(ί)) - (/ι//ι0)(Α))Μ/?(*))> from the

250 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
definition of β0 we see that f(t) assumes its minimum 0 at t = 0. Noting that
β(ί) = volf^/volM and using (1.10), we get
4P{t)) <volrn-ldnt/vo\M = h(P0) + |(m-l)r/ / udA/ vol Μ \t + o{t\
ho№)) = MA)) + hb(Po)F(0)t + o(i)
= MA)) + { Wo) / udA/ vol Ail ί + o(0,
and therefore
0 < /(0 < / udA J vol Μ · {(m - 1)7/ - (/i/M(A)) · MA))}* + o(i)·
Λν
Since £ may take positive and negative values, and f(t) assumes its minimum 0 at
t = 0, the coefficient of t in the last term is equal to 0. It follows that /'(0) = 0
and
(1.18) (m - 1)η = (А/Ло)(Д>)' Wo) = Л'(Д>)·
Now we divide Μ into domains i? and Μ\Ω, and get as before, by the Heintze-
Karcher volume comparison theorem (Chapter IV, Theorem 3.8 (2)),
rp rd-p
voli2<volm_iN / Jv(t)dt, volM\i2< volm_iAT / J-V(t)dt.
Jo Jo
It follows that volM < volm-ιΝ fp_dJr](t)dt. Now we set v(i) := (c6(t), ss(t))
and w := {Η0(β0)1^τη~1\ ηΗ0(β0)ι^τη~ι>}), which are considered to be vectors in
R2. Then from the last inequality we get
(1.19) l^TTW\lP W\v)Tldt.
f^oyPO) Jp-d
Next we estimate ||ги||. We set β0 = \ /Qr sinm_11 dt and note that MA)) =
(m - l)cotr holds by (1.17). Then from
i1 + (S)}№}2/M) = 1
and (1.18), it follows that
< - max < 1
с
Wo)Y\ 2
Лш) Гстах1Чм^)] г
In the following we only give a proof of assertion (1). Assertions (2) and (3)
may be verified by the same idea. For (1), since v(t) = (cost, sini) and w is a
constant vector, it follows that
fP fd/2 pd/2
/ <v(0, w/IHI>+_1 dt ^ / cosm-4<ft = 2 / cosm-4<ft.
Л>-с* У-с*/2 Jo

1. THE ISOPERIMETRIC INEQUALITY
251
COSe(t) = <v(t\w/\\w\\>
Figure 34
Next suppose /i(A))/MA>) < 1- Then from HP-1 - llc and (L19)> we Set
1<Ш-.2
rd/2
/ cos;
Jo
m-ltdt
( r/2
\2L см
tdt} < 1,
which is a contradiction. Hence /ι(Α))/^ο(Α)) > 1· By the same argument,
1 <
ГМЛГГ fd
|.MA>)J Jo
cos
m-1
tdt/
f
Jo
cos
т"ЧЛ,
namely, (1.14).
Second, from Proposition 1.3 (2) we have
l/m
lim
β
im A(/j) = {vol(5m, p0)/volM}1/m > Ι ΓmF^tdtl [ sin"1"1**!
'^° ^o [Jo / Jo )
( r , fd/2 Λ1'™
>< / sirT-Hdt / cosm_4d^
where the last term is equal to the right-hand side of (1.14). This completes the
proof of (1.14). Finally, suppose equality holds in (1.14). Then h(P0)/h0(Po) = 1
and d = π. The Cheng maximal diameter theorem (Chapter IV, Theorem 3.5)
implies that Μ is isometric to (5m, go). D
Finally, we estimate the isoperimetric constant T(M, g). We give only a sketch
(for details see [Ga-2]). First suppose p(u) > 0(u £ UM). In this case we may
verify that F(/3) := ϊι(β)/β{τη-1)/τη is monotone decreasing on [0,1/2]. Then from
Theorem 1.4 we have
(1.20)
/i(/?)//?(m-1)m > h{\) / \ · 2-1/™ > 2-l'mhc > 2l~^/d.
Next suppose p(u) > — (m — 1) (u £ UM). We consider the minimum of F(/3).
It suffices to consider F | [0, ^]. First, suppose F(/3) assumes its minimum at
/?o € (0, ^). Take a domain Ω which realizes h(/?o) such that the mean curvatures
of Ν = 8Ω are equal to a constant η at regular points of N. As in the proof of
Theorem 1.3, we consider
f(t):=h№))-(4fo)/hm )£(*)"

252 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Then as before / assumes its minimum at t = 0, and η = ^-volm_iC?i?/voli2. Now
we again apply the Heintze-Karcher theorem, and get
fd
volM < vo\rn-\dQ / (cosh*+ 77 sinh i)™-1^·
Jo
Therefore, noting that hc < volm_idi?/voli?, we obtain
F(/?0) = volm_i<9f2/{(volf2)^(volM)~ }
> (mr/)1^ ί ί (coshi + r/sinhi)m_1^ J
> /i^" I / ( cosh t + — sinh A dt\
On the other hand, as in (1.20) we have
F(\) ^2~™Нс>2~™ / / cosh™"1 tdt
and by Proposition 1.3 (2) and the Bishop comparison theorem
F(0)=7m(volM)-- >7m ( / sinhm-4dM
Summing up, we have from Exercise 1
Theorem 1.7 (S. Gallot). Let (M, g) be a compact Riemannian manifold.
Then the following estimate holds for the isoperimetric constant T(M, g).
(1) // p(u) > 0 (u <E UM), then
(1.21) T(M, g)l/Tn > 21-1/m(volM)1/m/rf(M).
(2) Ifp{u)>-{m-l){ueUM),ihen
(1.22)
Г fd(M) / ι \ ГП — 1 "j _ m
/ ( cosht -\—- sinht) dt\
\Jo V m J
I(M, g)™ > (volM)- · /ιΓ7
2. The Berger Isoembolic Inequality
In this section, in connection with the isoperimetric inequality we state the
following isoembolic inequality due to M. Berger ([Bes-1]). We substantially follow
the argument due to J. L. Kazdan ([Kaz]).
Theorem 2.1. Let (M, g) be an та-dimensional compact Riemannian
manifold. Then
(2.1) vol(M,^)>{z(M,^)Mm.am,
where i(M, g) denotes the injectivity radius and am = vol(5m(l), go). Moreover,
equality holds if and only if (M, g) is isometric to an m-dimensional sphere of
positive constant curvature.

2. THE BERGER ISOEMBOLIC INEQUALITY
253
For the proof of the theorem it is necessary to consider the unit tangent bundle
tm '· UM —» Μ and the geodesic flow. We begin with some preliminaries.
Let 7n be a geodesic in Μ emanating from ρ e Μ with the initial direction
и e UPM. Take an o.n.b. {е*}^ of TPM such that em = и and Jacobi fields
Yi(t) (i = 1, ... , m - 1) satisfying the initial conditions У*(0) = 0, VYi(O) = e*.
Then 0(£, u) := \\Y\(t) Л · · · Л ym_i(£)|| does not depend on the choice of the o.n.b.
{e{} and is a smooth function on R+ χ UM (see Chapter II, (5.10), Lemma 5.4,
for the definition of θ(ί, и)). For 0 < r (< г(Л/)) we have
vo\Br{p)= f dSm~l [ 6{t,u
Jues™-1 Jo
)dt.
Let ф3 be the geodesic flow on UM. We shall see how 0(t, ф8и) is expressed in
terms of Jacobi fields.
Let {βϊ(ί)}^1 be the parallel translation of {ег} along 7n, and Yi(t; s) (i =
1, ... , m — 1) Jacobi fields along ηη satisfying Yi(s: s) = 0, VYi(s; s) = e{(s).
Then we may write Yi(t\ s) = £3jl^ aji{t- s)ej(t)- and consider the η χ η matrix
A(t\ s) = [dji(t; 5)], where we put η := m — 1. Then we have
(A"(t:s) + R(t)A(t:s)=0,
' [A{s; 5)=0, A'(s;s) = En,
where the prime stands for the differentiation with respect to t, En is the identity
matrix and R(t) = [Rji(t)\ denotes the symmetric η χ η matrix given by
m-l
R{*i(t), 7u(0)7u(0 = Σ RjiWejit) (г = 1, ... , n).
j=l
Then from
0(i, 0su) = ||yi(i + s; s)A---AYn(t + s\ s)\\ = |detA(i + 5; s)|
we get
(2.3) β(ί, 0su) = I det A(* + 5; s)\.
Now we turn to the proof of Theorem 2.1. Note that the injectivity radius
of a compact Riemannian manifold (M, g) is positive, and both sides of (2.1) are
multiplied by the same factor cm after a homothetic change c2g. Therefore, we may
assume that г(М, g) = π (= г(5т(1), <7о))> and we wisn t° show that vol(M, g) >
otrn = vol(5m(l), g0). From this assumption we get
Bs{tm(u)) Π Βπ.8{τΜ{Φηη)) = φ (0 < 5 < π/2)
and therefore
vol(M, g) > volBs(rM(u)) + volВп-8(тм{ФпП)).
We integrate both sides of the above inequality on /УМ with respect to the measure
vq given by the Sasaki metric G on UM. Recall that the geodesic flow ф3 preserves
the measure vq (Liouville's theorem; see Problem 16 for Chapter II). Then from
the Fubini theorem we get
am_ivol2(M, g) = vol(M, g)vol(UM, G)
> I vo\Bs(TM(u))di>G+ / νοΙΒπ-8{τΜ(φπύ))άι/0
Jum Jum

254 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
= f dvg\ vo\Bs(p)dSm-x + f νοΙΒπ-8(τΜ(φπη))άφ*πν0
Jp€M JUPM JUM
= am.1 f {vol Bs(p)+vol Bn_s(p)}dvg (UPM = Sm~l)
J рем
for 0 < 5 < π/2, and it follows that
vol(M, gf
> I dvg\ ( dt ( 0(t, ^dS™-1 + / dt [ 9{t, ujdS™-1 \
J рем [Jo J up μ Jo J up μ J
= J dt J 0(i, u) dvG + / dt 0(i, и) dvG
Jo Jum Jo Jum
= I dt I 0(ί, φπ-8η)άι/α + / dt 0(i, <t>su)dvG.
Jo Jum Jo Jum
Integrating both sides of this inequality with respect to 5 over [0, π/2], and noting
that det A(t + 5; 5) > 0 for 0 < t < π, we have
(2.4)
\vol(M, я)2
>
f ( /·π/2 rs /"τ/2 ρ-π—s \
l dvG\ ds 0(ί, 0π_5η) dt+ ds 0(i, ф8и) dt \
Jum [Jo Jo Jo Jo J
= / dvG\ ds 0(i, ф8и) dt \
Jum [Jo Jo J
(change the variable π — 5 —» 5 in the integral of the second line)
= / duG\ I ds I det A(t + s; s) dt\
Jum [Jo Jo J
= / dvG\ I ds I det A(t + s; t) dt\ ,
Jum I Jo Jo J
where the last integral is obtained from the previous one by changing t and 5 and
applying a formula for iterated integrals.
The following analytic inequality plays an essential role in the proof of Theorem
2.1.
Proposition 2.2. Set η = m - 1. Suppose that A(t; s) and R(t) satisfy (2.2)
and for any s £ (0, π), A(t + 5; t) is nondegenerate (i.e.,det A(t + s; t) φ 0) on
0 < t < π. 77ien
/•π ртг — s /*π/2
(2.5) ds det ^(* +5' t)dt > π sinn 5 ds.
Уо Л) Уо
Equality holds if and only if R(t) = £7n and A(£; 5) = sin(£ — s) · En.

2. THE BERGER ISOEMBOLIC INEQUALITY
255
Now suppose Proposition 2.2 holds. Then Theorem 2.1 may be proved as
follows. Prom (2.4) and (2.5) we get
%ol(M, g)2 > dvG\ I ds I det A(t + 5; t) dt \
2 J им [Jo Jo J
>π / dvc I smn sds = πνο1(Λ/, g)am-i / sinn sds.
Jum Jo Jo
It follows that
Λ7Γ/2
vol(M, g) > 2am_i · / smn sds = vol(5m, g0) = am,
Jo
and the proof of (2.1) is complete. Next suppose equality holds in (2.1). Then for
any и £ UM we get R(t) = En along 7U, which means that R(x, u)u = χ for any
χ £ TTM(U)M with (x, u) = 0. Therefore, Λ/ is of constant curvature 1. Next we
show that Μ is simply connected. In fact, otherwise let 7 be a shortest geodesic
loop based at ρ e Μ that represents a nontrivial element of πι (Μ, ρ) (see Chapter
V, Lemma 1.5). Then the length / of 7 satisfies / < d(Sm) = π if we consider
the universal Riemannian covering π : Sm —► Λ/. Then we have i(M, 0) < π/2
(Chapter III, Corollary 4.14), which is a contradiction. Therefore, Μ is simply
connected and isometric to the sphere of radius 1.
Now we return to the proof of Proposition 2.2. We begin with some
preliminaries.
Lemma 2.3. SetS+ := {B £ Mn(R)\ В is a positive definite symmetric nx
η matrix }, which may be considered as an open convex set in Дп(п+1)/2. Then
the function F : <S+ —> R given by F(B) := (deti?)-1 is a strongly convex C^
function.
PROOF. It suffices to show that the Hessian D2F is positive definite, namely,
for В e S+ and any nx η symmetric matrix А (Ф 0) we have /"(0) > 0, where we
set f(t) := F(B + tA). Set C(t) = B + tA and note that
^-(detC(t)) =detC(t)-tmce(AC(t)-1), 4r
at dt
Then it follows that
f'(t) = jidetCit))-1 = -(detC(i))"1 •trace(AC^)-1),
f"(t) = (detC(0)_1(trace(AC(0-1)2 - (detC{t))~l · trace (а^-С{Ь)~Л ,
and
/"(0) = (detBy^itraceAB-1)2 + trace^-MB"1)}.
We may easily check that trace(AB-1 AB~l) > 0 if Α (φ 0) is symmetric and В is
a positive definite symmetric matrix. D
Lemma 2.4 (Jensen inequality). Let Ω := [α, 6] be α closed interval and μ a
positive measure on Ω. Set μ(Ω) = /Ω άμ. Then, for Β : Ω —> Ω,
(2.6)
{det QT B(r)dM(r) / μ(Μ < ^(detB(r))-1 ф(г) / μ(Ω),
\-1 _ D-1 Л D-l
С(г)"1 = -£-М£-
t=o

256 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
where equality holds if and only if B(r) is equal to a constant matrix almost
everywhere.
PROOF. For B, B$ £ <S+ we get from Lemma 2.3 and the Taylor theorem
F(B) > F(B0) + DF(B0)(B - B0).
Now substituting В = B(r), B0 = Jn B(r) άμ/ μ(Ω) in the previous inequality and
integrating with respect to μ on Ω, we get
/ (det B(r))~l άμ > μ(Ω) /det ( ( B(r) άμ / μ(Ω) j | ,
and equality holds if and only if B(r) = Bq almost everywhere. D
Now the proof of Proposition 2.2 will be divided into several steps.
1°. First note that for solutions A(t\ s) and A(t) := A(t\ 0) of (2.2), we have
(2.7) ιΑ(ί)Α'(ί) = *Α'{ί)Α(ί)
(lA stands for the transpose of A), and
(2.8) A(t; s) = A(t) · / [^(r)A(r)]"1 dr · *А{з).
In fact, to see (2.7) we set f(t) := ιΑ(ί)Α'(ί) - 1А'^)А(г), and check that f'(t) = 0
holds (by (2.2) and the symmetric property of the curvature tensor). From the
initial condition A(0) = 0, we get f(t) = 0. Next, let B(t\ s) denote the right-hand
side of (2.8). Then B{s\ s) = 0, B'{a\ s) = A{s)[tA{s)A{s)}-1 έΑ(δ) = Εηι and
we may verify that B(t; s) satisfies (2.2) by a direct computation. Then from the
uniqueness of the solution we get A(t\ s) = B(t\ s).
2°. We set φ(ί) := {det A(t)}^n and note that φ(0) = 0 and
^(0) = lim{deti4(0An}1/n = lim{det(i4(i)A)}1/n = L
φ(ί) > 0 (0 < t < π). We show that, for any 0 < s < t < π,
(2.9) {det A{t\ s)}1/n > φ(ί)φ{8) ί ^"2(r)(ir.
Furthermore, equality holds for 0 < s < t < π if and only if A(r) = φ(τ)Εη for
s <r <t To see this, set B{r) := [1А(г)А(г)}-\ Then B(r) e 5+ for 0 < r < π,
and we get
{det A(t\ s)}1/n = φ(ί)φ{8) /det [ / B(r)dr] \
from (2.8). Now let ^(r) = ip~2(r)dr be a positive measure on Ω = [s, t], and apply
Lemma 2.4 to ip2{r)B{r). Noting that μ(Ω) = fi<p~2{r)dr and {detB(r)}"1 =
<^2n(r), we get
det[ / B(r)dr\ ·μ(Ω)η< / {detB(r)}"1 · φ-2η-2{r)dr/ μ(Ω)
LJs J Js
= j φ-2(Γ)άΓ/μ(η) = 1,
where equality holds if and only if φ2 (r)B(r) is equal to a constant matrix. (2.9)
easily follows from this inequality. Next suppose that equality holds for 0 < s < t < π

2. THE BERGER ISOEMBOLIC INEQUALITY 257
in (2.9). Then 'A(r)A(r) = φ2(τ)Εη, since limrj0 >l(r)/V(r) = En. Differentiating
both sides of this equation, from (2.7) we get 2lAA' = l A! A + *AA' = 2ψφ'Εη.
Because lA = φ2 Α~λ, it follows that φ Α = φ' A, namely, (Α/'φ)' = 0. Then,
considering the initial condition, we get A(r) = φ(τ)Εη.
3°. Now note that
ρπ /»π — s Г7г/2 rs
/ ds det A(t + s; t)dt= / ds det A(t + π - s; t)dt
ν/π/2 JO JO Jo
We divide the interval [0, π] of integration in the left-hand side of (2.5) into [0, π/2]
and [π/2, π]. Then from the above equation and (2.9), we get
(*)
fTr/2 ρπ-s ( ps+t \ n
dt
/•π/2 ρπ-s r ps+t \ '
left-hand side of (2.5) > ds <4>{t + s)(p{t) / φ~2(τ) dr \
/•π/2 ps r ρί+π-s \ n
+ / ds <(p(t + n-s)(p(t) / <£~2(r)dr> <ft.
Recalling that we should have φ(ί) = sint when equality holds in (2.5), we define
u{t), 0 < t < π, by φ(ί) = smteu^\ and set
(2.Ю) ϊ ,_, ч __.„._,__,_ , ,w_2 2
v(r) = u(t) + u(s + i) - 2u(r)
/i(r) = sint sin(s + £)/sin2 г
for t < r < max(s + t, π - s + t) < π. Note that u(£) may diverge for t = π, and
ν, /ι are determined after s, t are given. Then we may write
/•π/2 /·π-δ f ps + t \ n
(*) = / ds / ^ / ft(r) e"(r> dr \ dt
/•π/2 /·δ ζ' ρπ—s+t ^ η
+ / ds \ h(r) eu(r) dr ^ Λ.
Now we set
(2.11)
ρπ-s r ps+t \ n
fx(s):=J IJ A(r)eAu(r)drJ dt
ps f ρπ-s + t *\ n
+ \ h(r)eXv{r)dr\ dt (0 < A < 1)
and get (*) = /0 fi(s)ds. On the other hand, noting that Jt s h(r)dr = sins,
which is obtained by the change of variable и = tan r, we see that
ρπ-s r ps + t \ n ps r ρπ-s + t \ n
/o(*)=/ \ h(r)dr\ dt+ I h(r)dr\ dt
ρπ — s ps
= / sinn sdt+ sinn sdt = π sinn s.
Jo Jo
Hence for the proof of (2.5) it suffices to show that /0 f\(s)ds > /0 fo(s)ds.
For this purpose we investigate the behavior of f\{s) (0 < A < 1) with respect to A.

258 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
4°. Differentiating f\(s) with respect to λ, we get
A.
dXJ
ρπ-s f ft+s -\ " ! ζ- ft+s \
fx(s)=n I h(r) eA"(r) dr \ I h(r)v(r)eXv{r)dr\dt
s ( ρπ—s+t
+ П
Ш
h(r) eXv{r) dr
г и
π — s+t
h(r) v(r) eXv(r) dr \ dt
r> dr\
and
^fx(s)=n(n-l)£ S(At>s(W-2[Jt h(r)v(r)eXv^dr} dt
+ n Г '(Λ,.(λ))η_1 J / * h(r) v2(r) eXv{r) dr\ dt
+ n{n-l) j" 8(Λ>π_,(λ))"-2|^π S h(r)v(r)eXv^dr}
+ пГ ί(\π-5(λ))"-1{Γ S h(r)v2(r)eXv^dr\dt >0,
dt
where we have set At,s(X) := /t h(r) eXv^ dr.
Therefore, we get /i(s) > /o(s) +d/d\ |a=o /a(s) by the Taylor theorem, and
it suffices to show that
(2.12)
_d_
dX
/a(*) = 0.
a=o
Furthermore, suppose /i(s) = fo(s) for all 0 < s < π/2; then f\(s) does not
depend on A and we get d2 f\(s)/d\2 = 0, which implies ν = 0, namely, и = const.
It follows that </?(£) = Cisint Considering the initial condition, we have, finally,
φ(ί) = sini.
5°. We set k(s) := f*~s dt Jt s h(r) v(r) dr and note that
ndX
fx(s) = sin""1 s{k{s)+k{n-s)}.
λ=0
Then to see (2.12), it suffices to show that k(s) + к (π - s) = 0. To compute k(s),
we put
ψ®
Г/2 u(r)
Jt sin2 r
dr

2. THE BERGER ISOEMBOLIC INEQUALITY
259
and get
ρπ — s / pt+s \
k(s)= {u(s + t) + u(t)} I h(r)dr)dt
-f
Jo
= sins
-2/ \sm(t + s)smt [ ^-dr)
Jo I Jt siirr /
dr )dt
sms{u(t + s) + u(£)} d£
/•7Γ—S
■2 / sin(£ + s) sin£{?/>(£)-i/>(£ + s)}<ft
Ι ί τχ(ί)ΛΗ- / гх(«)л|
U π — s /»π
sin(i + s) sin ί ^(ί) dt + / sin ί sin(s - ί) ^(ί) Λ
}
Similarly, we have
k(n - s) =sins < / u(t)dt+ / u(t)dt>
- 2 < / sin(s - t) sin t ψ(ί) dt+ sin t sin(s + t) φ{ί) dt > ,
and therefore
k(s) + k(n - s) = 2 sin s < / u(t) dt - sin 2t v(t) dt I.
Figure 35

260
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Now let Z?i, D2 be the domains in the (t, r)-plane shown in Figure 35. Then
}
-II
Λ/2 Sin Г Jr
/»π ρπ
u(r) dr + I u(r) dr = u(r) dr,
Jtt/2 Jo
. . sin2£ · u(r) , ,
dtdr- II Y-^-dtdr
D2 sin r
sin 2tdt - / . V dr I sin It dt
/О Лг/2
Γπ/2
/О ./π/
and it follows that /c(s) + k(n - s) = 0. Therefore, the proof of (2.5) is complete.
Finally, suppose equality holds in (2.5). Then /0 /1 (s) ds = /0 f0(s) ds, and
it follows that /i(s) ξ /o(s), since we have shown /i(s) > fo(s) in 4°. Therefore
^ο(^) = sin£, also as shown in 4°. Further, if equality holds in (2.5), then equality
holds in (2.9) for every 0 < s < t < π, and we have A(t) = smtEn. In particular,
in the equation (2.2) we have R(t) = En, and on solving (2.2) we obtain A(t\ s) =
sin(t — s) En. This completes the proof of Proposition 2.2. D
We give an application of our isoembolic inequality without proof. If all
geodesies of a compact Riemannian manifold (Μ, g) are closed geodesies of length
/, we call (M, g) a Ci-manifold. For instance, symmetric spaces of positive
curvature are C/-manifolds (see Chapter IV, §6 (III)). Further, it is known that there are
many C\-Riemannian metrics on 5m (m > 2) that are not of constant curvature (see
[Bes-1], [Ki]). Now let (M, g) be a C/-manifold. Then the geodesic flow φ3 on the
unit tangent bundle UM of Μ defines a fixed point free S^action, whose orbits are
identified with closed geodesies of length /. Therefore, the orbit space UM/S1 may
be considered as a 2(m — l)-dimensional manifold C(M, g) of all closed geodesies
of Μ. Let η denote the 1-form on UM which is the dual of the geodesic spray with
respect to the Sasaki metric G. Then the volume element vG of UM is given by
(rn-iy.7) л (d??)™-1 (see Problem 16 for Chapter II), and we get a symplectic form
ω on C(M, g) by the condition άη = π*ω. From the Fubini theorem we get
vol([/M, G) = ?-i— / η Λ (Α,)™"1
(га - 1)! JUM
_ 1 [ (f Л ^m-l _ I f
(m - 1)! JieC(M,9) У/тг-1(7) / (Ш ~ 1)! Jc(M,g)
Now we set
j(M,g):= ( /"Vr1
Jc(M,g)
= (m- l)!vol([/M, G)/lm = (ra - l)!am_ivol(M, д)/Г
which is known to be an even integer, by results of A. Weinstein. Further, if Μ
is homeomorphic to the sphere 5m, then for a Cz-metric g on Μ, A. Weinstein
and С. Т. С. Yang showed that j(M, g) = 2. Therefore, we have vol(M, g) =
2/m/{(ra - l)!am_i} in this case ([We-4], [Yan]).

2. THE BERGER ISOEMBOLIC INEQUALITY
261
A compact Riemannian manifold (M, g) is called a Wiedersehens manifold if
the cut locus Cp of any ρ e Μ reduces to one point, say q. It follows that all normal
geodesies 7 emanating from ρ reach q and meet again at a fixed parameter value
1/2 independent of p, and hence the German name "Wiedersehens". In particular,
i(M) = d(M) = 1/2, and all geodesies are closed geodesies of length /. Namely,
a Wiedersehens manifold is a Cz-manifold homeomorphic to the sphere. We may
normalize the Wiedesehens metric so that / = 2π, namely. i(M) = π, and get
vol(M, 9) = 2(2πΓ/{(πι - 1)!α^} = am
by the above and Chapter IV, §3, Exercise 3. Then equality holds in Theorem 2.1
(2.1), and the Wiedersehen manifold (M, g) is isometric to the sphere of positive
constant curvature. In fact, this was proved for m = 2 by L. Green, and remained
a long-standing open problem for general m. Finally, M. Berger settled the
problem using his isoembolic inequality. See [Bes-1] for details about C/-manifolds,
Wiedersehens manifolds, and the related Blaschke conjecture.
There are various kinds of inequalities among geometric invariants related to
the measure, and we give one more such inequality. Let g be a Riemannian metric
on a 2-dimensional torus Τ2, and set
sys(T2, g) := mi{Lg(c);c ^ 0, namely, с is a homotopically
nontrivial piecewise C1 closed curve}.
Then Charles Loewner proved the following isosystolic inequality:
(2.13) Area(T2, <?)/sys2(r2, g) > л/3/2,
where equality holds if and only if (T2, g) is isometric to a flat torus defined by
R2/T with an equilateral lattice Г.
We sketch the proof of (2.13). First, (T2, g) gives a complex structure on T2,
and the Uniformization Theorem tells us that g is conformal to a flat metric go.
Namely, there exists a positive C°° function φ on T2 such that g = φ2go- On
the other hand, the isometry group G of (T2, go) is a compact Lie group that
acts transitively on T2. In fact, translations of R2 define isometries of (T2, go).
Now we introduce the normalized Haar measure dG on G with volG = 1, and
average φ by the action of G. Namely, we consider a function φ defined by φ(ρ) :=
faeG vfa'P) dG. It follows that
Jp(b-p)= / (p(ab-p)dG= / ip(Rba · p)R*b dG = / φ(α· p)dG = φ{ρ).
JaeG JG JG
Then φ is constant, since G acts transitively on T2. On the other hand, from the
Cauchy-Schwarz inequality we get
φ2{ρ)< [ dG- [ <p2(ap)dG= [ (a*ip)2dG,
JG JG JG
where equality holds if and only if ψ is constant. Now g := φ2go is again a flat
Riemannian metric, and we have
Area(T2,£)= / φ2 dvgo < [ dvgo [ (a»2 dG = f dG f (a>)2 dugo
JT2 JT2 JG JG JT2
= I ^2<2^о=Агеа(Г2,0),
Jt2

262 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
since a £ G is an isometry of (T2, go). On the other hand, we get
sys(T2, g) = inf | J Tp(c(t)) dt- с ^ 0 I
= inf < J dt J φ(α · c(t)) dG = Lg(a · c) dG\ с + 0 \
>inf{Lp(c);c^0}=sys(T2,^),
where с is parametrized by arc-length with respect to go. It follows that
Area(T2, <?)/sys2(r2, g) > Area(T2, ^)/sys2(T2, g)
= Area(T2,^o)/sys2(T2,^o),
and therefore (2.13) may be reduced to the flat case.
Now let Г be a lattice of R2 such that (T2, go) is isometric to β2/Γ, and e\ a
nonzero vector in Γ with minimal length. Next choose β2 £ Γ with minimal length
among vectors in Γ linearly independent of e\. Then {ei, ег} forms a basis of Г, and
||ei|| < ||e2|| < ||ei+e2||, ||ei-e2|| by definition. Now, setting α := Z(eb e2), h :=
||ei ||, and /2 := ||e2||, from the Law of Cosines we get
/i2 + /22 - 2/i/2 cosa > /22 > /i2, /i2 + /22 + 21 ih cosa > /22 > I2.
Therefore, it follows that —\ < cosa < ^. Now note that Area(T2, go) =
/i/2sina, l\ = sys(T2, go), and
Area(T2, p0)/sys2(T2, go) = l2sma/h > >/3/2.
Here equality holds if and only if ||ei|| = ||e2|| and α = π/3(θΓ2π/3), which means
that Γ is an equilateral lattice.
Pu's inequality in the following exercise may be proved in the same manner
(see [Pu], [B-5, 6, 7] for more details).
Exercise 1. For a Riemannian metric g on the real projective plane ДР2,
show that Area(ilP2, g)/sys2(RP2, g) > 2/π, where equality holds if and only if
g is a Riemannian metric of positive constant curvature.
Finally we remark that M. Gromov considered a class of m-dimensional
compact manifolds Μ called essential manifolds (roughly speaking, manifolds whose
m-dimensional topology is controlled by 1-dimensional toplogy, e.g., aspherical
manifolds) and showed that there exists a positive constant cm, depending only
on m, such that vol(M, p)/{sys(M, g)}™ > сш for any Riemannian metric g on an
m-dimensional essential manifold Μ ([G-6]).
3. Eigenvalue Problem for the Laplacian
3.1. Let Μ be a Riemannian manifold. Recall that in Chapter II, Definition
1.5, we defined the Laplacian Δ/ of a C°° function / on Μ by
(3.1) Δ/ = -div(V/) = -trace D2/.
With respect to a chart (/7, φ, хг), (3.1) may be written as

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
263
where we set G = det(^j). In particular, when Μ = (Rm, go) we have Δ/ =
~~ ΣΤ=\ d2f/dxl2, which coincides with the usual Laplacian up to the sign. Now
for general Μ, if we take a normal coordinate system around ρ G M, from (3.2) we
get
Δ/(Ρ) = "Σ
d2f
дхг2
(ρ)
at p. Further, take an o.n.b. {e*} of TPM and let 7* = η€% (г = 1, ... , га) be a
geodesic emanating from ρ with the initial direction е*. Then we may also write
(3.3)
Δ/(ρ) = -£
dt2
/Ы*))·
t=0
In fact, let (хг) be the normal coordinate system determined by {е*}. Then we
have xi(7i(t)) = 6{jt, and therefore
d2
dt2
t=o
/(-»(*)) = 0(p).
The next lemma tells how to compute the Laplacian for some special functions.
Lemma 3.1. Let f be a real-valued C°° function defined on Br(p) (0 < r <
ip(M)) that depends only on the distance dp to p. Namely, f may be expressed as
f(q) = (f{dp(q)), where φ is a C°° real-valued function. Then for q G Br(p) \ {p}
we have
(3.4)
Δ/(ί)
->
<->+$>4
dp(q) and и = expp lq/\\ expp 1 q\\ G UPM. Recall that 6(r, u) is given
where r
in Chapter II, (5.8), and Θ' stands for differentiation with respect to r
PROOF. Let 7 = ηη be a geodesic emanating from ρ with the initial direction
u. Then q = 7(r). Choose an o.n.b. {e^} of TPM such that e\ = u, and let
ei(t) (i = 1, ... , m) be the parallel translation of e* along 7. Then for geodesies
Ίί = 7ei(r) emanating from 7(r), we get
dt2
fb№ = <
έ=0
t=0
(i = l)
dp(7<W)}
dp(7i(0) (*>2).
έ=0
Now for г = 2, ... , m, we consider variations of 7 which consist of geodesies as :
[0, r] —> Μ joining ρ to 7i(s). Then the corresponding variation vector fields
are Jacobi fields Z{(t) with Z{(0) = 0, Ζ\(τ) = e^. Note that 7 is a Б-geodesic
in Cb([0, r]) for the boundary condition Β = {ρ} χ 7*. It follows from the first
variation formula (Chapter III, Corollary 2.3) that ^ |i=o dp{^i{t)) = 0. Moreover,
from the second variation formula (Chapter III, (2.9)), noting that Z{ are Jacobi
fields and 7; are geodesies, we get
^ I dP(7i(*)) = (VZi(r), Zi(r)).
\t=o
dt2

264
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
On the other hand, let Yi(t) be Jacobi fields along 7 satisfying the initial conditions
y.(0) = 0and VYi(0) = e;, and note that 0(i, u) = \\Y2{t) А · · · AYm{t)\\ by Lemma
5.4 of Chapter II. We may write Z{ = ajYj and get
\\Z2{t) A · · · Λ Zm(t)\\ = det{aij) 0(t, u).
Since {Zi(r)}^L2 forms an o.n.b. of i±{r), we have det(a^) = l/0(r, и). It follows
that
771 ι ι
\\Z2(t) Λ · · · Λ Zm(t)\\ = в'(г, и)/в(г, и),
771
ΣΦΖΜ, Zi(r)) =
2 = 2
therefore
^ d2
1=2
d
~ dl
t=o
fbi(t)) = ^(r)-ff(rtu)/e(r,u),
which implies (3.4). D
Exercise 1. Let Φ be an isometry of M. Show that Δ(Φ*/) = Φ*Δ/, where
φ*/ = /οφ.
Remark 3.2. Δ is a second order linear partial differential operator, and the
term containing the second order partial differentiation with respect to local
coordinates is given by —дг^д2/дхгдх^. Therefore, the principal symbol of Δ is given
by £ ·—► gl-*iiij, which is a function on T*M. Namely, Δ is an elliptic operator
since [glj] is positive definite (see, e.g., [Au-3]). In fact, Δ is the most fundamental
differential operator on Riemannian manifolds.
Now we turn to the eigenvalue problem for the Laplacian Δ, and first we briefly
explain the background. We put a heat-source at a point p0 of a Riemannian
manifold Μ, and consider how the heat diffuses on Μ as time elapses. We denote
by u(p, t) the temperature of a point ρ £ Μ at the time t. Then for any domain V
of Μ and any fixed t, the infinitesimal rate of the change of heat flux
dt Jpev
t)dvg
is proportional to the integral of the temperature gradient on the boundary dV with
respect to the unit outer normal vector field v, namely, Jdv(4u, v) dA.
Normalizing the proportional constant, from the Green theorem (Chapter II, (5.20)) we
get
— / u(p,t)dvg= I (Vu,i>)dA=- / Audvg.
<ft Jv Jdv Jv
Since V is arbitrary, we have the heat equation (or diffusion equation)
(3.5) Δη+ —=0.
Further, in the case where the heat is absorbed at the boundary ΘΩ of a domain
Ω containing p0, we have the boundary condition u(p, t) = 0 for ρ £ ΘΩ. Now
we try to solve (3.5) by separation of variables, setting u(p, t) = f(p) g(t). We get
Af(p)/f(p) = —g'(t)/g(t), which is equal to a constant A. Namely,
Δ/ = A/, g' + \g = 0.
Therefore, it is fundamental to look for / £ F(M) with Δ/ = A/, and we consider
the following eigenvalue problem.

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
265
Definition 3.3. (I) Let Μ be a compact (connected) Riemannian manifold
without boundary. A real number A is said to be an eigenvalue of Δ if there exists
a C°° function f on Μ which is not identically zero and satisfies Δ/ = A/. Such
an / is called an eigenfunction of Δ corresponding to the eigenvalue A. For an
eigenvalue A, the vector space of eigenfunctions with eigenvalue A is called the
eigenspace corresponding to A.
(II) Let Μ be a compact (connected) Riemannian manifold with boundary
дМ (дМ is not necessarily connected). We denote by M° the interior of Μ. If
there exists a C°° function / on M, not identically zero, with / | дМ = 0 and
Δ/ = A/ on M° for a real number A, then A (resp., /) is called an eigenvalue
(resp., eigenfunction) of Δ with respect to the Dirichlet boundary condition. For
instance, we may consider the Dirichlet eigenvalue problem for a compact domain
Ω with smooth boundary in a Riemannian manifold.
Remark 3.4. Instead of the heat equation ди/dt + Au = 0, we may also
consider the wave equation d2u/dt2 + Au = 0. If we try to solve the wave equation
by separation of variables, again eigenvalues and eigenfunctions of Δ appear. Under
the Dirichlet boundary condition, if we regard Μ as a vibrating membrane, then
the eigenvalues of Δ may be considered as the frequences of the normal modes of
the membrane. Mark Kac asked a famous question about the relation between the
shape of Μ and the eigenvalues of Δ: "Can one hear the shape of a drum?" (see
[Kac]).
3.2. Next we introduce some function spaces which are necessary for the
eigenvalue problems (I), (II). Let Μ be as in Definition 3.1 (I) or (II), and let vQ
denote the Riemannian measure induced from a given Riemannian metric g. Now
L2(M) (= L2(M, vg)) := {φ : Μ —» Я; a measurable function with /Λ/ φ2 dvg <
+00} is a Hubert space with respect to an inner product (φ, ψ)0 := Jyi^vdiyg.
where we identify two functions which are equal almost everywhere. We denote by
Ρ0(Μ) the space of C°° functions with compact support in M°. Then Τ(Μ) and
Т0(М) are dense in L2(M). We need ^*0(Л/) for the Dirichlet eigenvalue problem
(II). Now for φ, ψ £ F(M) we consider an inner product
(<P, Ψ)ι := (v>, Ψ)ο + (Vp, V^)0 ((Vp, V^)o := / (Vp, V</>) dug)
JM
and consider the completions Hl(M, g), НЦМ, g) of 7"(M), F0(M) in L2(M) with
respect to the norm || ||i, respectively. They are called the Sobolev spaces. One of
the reasons why we consider these spaces is the following Sobolev theorem: The
inclusions
Hl0(M,g)^H\M,g)^L2(M)
are continuous. Moreover, the inclusions
(НЦМ, 9), || у ^ (L2(M), II Ho), (Hl(M, 9), II HO - (L2(M), II ||o)
are compact operators, which means that they map bounded sets in the domains
to relatively compact sets in L2(M) (see, e.g., [Au-3]).
Since it is known that spaces H^(M, g), Hl(M, g) do not depend on a
Riemannian metric g, we can also write them as Я^(М), Η1 (Μ), respectively, for
short. We also mention the following properties: For φ £ H^(M) (or Hl(M))

266 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
we may consider its gradient vector Vip in the distribution sense: V<£ is a
measurable vector field on Μ with (V<£, Vip) := /M(Vy?, Vip)dvg < +00, and it
satisfies (φ, /)ι = (φ, /)o + (V<^,V/)o for any smooth function /. In particular,
we have \\φ\\ι2 = |M|o2 + II^HIo· Also, in the case of (II) it is known that if a
φ G H\M) Π C°(M) satisfies φ\ΘΜ = 0, then φ G Hl0(M).
In the situation of Definition 3.1, we take F(M) (resp., {/ G F(M); f \ dM =
0}) in case (I) (resp., (II)) as the domain of definition of Δ, which will be denoted
by ,F*(M) in a unified manner.
Lemma 3.5. Δ : .Τ7* (Μ) —> L2(M) is a positive and symmetric linear map.
In particular, the eigenvalues A of A in Definition 1.3 are nonnegative.
Proof. From Chapter II, §1, (1.29), we get /Δ/ = |Δ(/2) + (V/, V/).
Integrating both sides over Μ, we obtain
(Δ/, /)o = tlV/Ho2 - \ I a\v{Vf2)dug = HV/llo2 - / div(/V/)*/9,
Δ J Μ J Μ
and the second term of the right-hand side vanishes in both cases (I), (II) on
applying the Green theorem (Chapter II, Theorem 5.11). Therefore (Δ/, /)o > 0,
which proves the positivity of Δ. Next note that again Chapter II, §1, (1.28) implies
that
(Δ/, /1)0 - (/, Δ/ι)0 = / div(-/iV/ + fVh)dvg, /, h G JF,(M).
JM
The right-hand side of the above equation again vanishes via the Green theorem.
For instance, in case (II), denoting by ν the outer unit normal vector field to dM,
we get
(Δ/, /1)0 - (/, Δ/ι)0 = / (f-vh-h.vf)dA = 0.
JdM
Hence Δ is symmetric and the eigenvalues A of Δ are nonnegative. In case (I), we
have A = 0 if and only if its eigenfunction / satisfies V/ ξ 0, namely, is constant.
In case (II) the eigenvalues are positive because of the boundary condition. D
Next we consider the eigenvalue problem from the viewpoint of quadratic forms.
For φ G ^ч(М) we consider the Rayleigh quotient defined by
(3.6) ВД := / ||VH|2 dug I f ψ2 dug {ψ ψ 0).
JM I Jm
Note that if we denote Hl(M) or H^(M) by Η\(Μ) according to which case, (I)
or (II), we have, the Rayleigh quotient R(ip) may be defined for φ G Η\{Μ). Then
we have the following.
Lemma 3.6. Set μι := inf{#(<£); φ G Я+1(М)\{0}}. Then the set Ε ι
consisting of 0 and all нЕЯ] (Μ) \ {0} satisfying R(u) = μι forms a finite-dimensional
(nontrivial) subspace of Η*Χ(Μ). Further, и G E\ is a weak solution of the
eigenvalue problem Au = μιη, in the sense that /M(Vu, Vip)dvg = μι \Μηφάν9 for
any ψ G Hl(M). и is in fact an eigenfunction of A corresponding to the eigenvalue
μι.
PROOF. Clearly μχ > 0. Take a sequence {φη}™=ι С Hl(M), \\φη\\ο = !>
such that R((fn) —> μι- Taking sufficiently large n, we may assume that ||</?n||i =
ll^nllo + Κ(ψη) < μι + 1 + en < μι + 2 (en -> 0). Namely, {φη} is bounded in

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
267
the Hilbert space Hl(M), and therefore there exists a subsequence {φηι} which
converges weakly to some ν G H\{M). On the other hand, by the Sobolev theorem
{φηι} is relatively compact in L2(M), and we may choose a subsequence {un} of
{φηι} such that {un} converges strongly to some и G I?(M) in L2(M). Since
Hl(M) <-^> L2(M) is continuous, {un} also converges weakly to ν in L2(M) and we
get ν = u, ||if||o = 1. Now from the Cauchy-Schwarz inequality we have
Κ, φ)\ < К»? \\φ\\2 < (μ, + 1 + €„)M|?. * g Hl(M)
and taking the limit it follows that (υ< φ)2 < (μι + 1)||^||?. In particular, we have
|M|? < μι + 1, and consequently μι < Д(г) = {||r||? - ||r||g}/||r||02 < /ц. Namely,
we have E\ := {г> G #*(M); ν = 0 or Я(г) = μι} / {0}. Now we show that
(3.7) uG^i^ (</?, u)i = (μ! + 1) (f. u)o for any <p G Я*(Л/).
In fact, 4= follows easily on setting φ = и and using the above argument. To see
=>, note that t ι—> Я(п + £</?) assumes its minimum at t = 0 for any </? G H\(M).
Taking the derivative at t = 0, we get (Vu, V</?)o = μι(^, <^)o, i.e., the right-
hand side of (3.7). Then from (3.7) it is obvious that Ε ι is a vector space. Now
{u G E\\ ||u||0 = 1} is bounded in #*(M), since \\u\\l = (μι + 1)Η§ foruGEi,
and therefore a compact subset in L2(M). It follows that E\ is finite-dimensional.
Finally, we rewrite (3.7):
(3.8)
ue Ει <=> / (Vu, Vip) dvg = μι ηφάν9 for any φ G H\(M).
Jm Jm
If u is of class С°°, then from Corollary 5.13 of Chapter II the right-hand side of
(3.8) becomes JM Δη · φάν9 = μι JMuipdug. This means that и G E\ is a weak
solution of the eigenvalue problem. Then, by the regularity theory for elliptic partial
differential equations (see, e.g., [Au-3]), и is in fact a C°° function. Namely, the
above μι is an eigenvalue of Δ, and E\ is the eigenspace with eigenvalue μι. Note
that for the Dirichlet eigenvalue problem, if и G Jr(M) Π Hq(M), then и | дМ = 0
and и satisfies the boundary condition. D
Next let Li (resp., Hi) be the orthogonal complement of E\ in L2(M) (resp.,
H*X(M)). Then, noting (3.7), we have an embedding Hi ^-> Li, which is again
compact. Setting μ2 := ini{R(u); и G Яь и ф 0} and using the same argument as
in the proof of Lemma 3.6, we see that there exists a finite-dimensional subspace
E2 of Li (resp., Hi) such that R(ip) = μ2 for any φ e E2\ {0}, and the following
holds:
ue E2 <=> (u, y?)i = (μ2 + 1) (гх, y>)0 for any ^ G Ηλ.
Further, и £ E2 is of class C°° and satisfies Δη = μ2η. Namely, μ2(> μι) is an
eigenvalue of Δ with the eigenspace E2. Repeating this process, we have a sequence
0 < μι < μ2 < · · · of eigenvalues of Δ and a sequence {Ei}^.x of eigenspaces which
are mutually orthogonal. Since L2(M), H\(M) are of infinite dimension, such a
sequence of eigenvalues is infinite. Now we show that lim^+oo βϊ = +°o. In
fact, if μι < μ (г = 1, 2, · · ·) for some μ, then we have {щ}^ С L2(M), which are
orthonormal. Because \\щ\\1 < l + μ, we see by the Sobolev theorem that {ui}^zl is
relatively compact in L2(M), which is a contradiction. A similar argument implies
that {μι} is discrete.
Next suppose the closure Ε of Ε := 0°^ Εχ in H\(M) does not coincide with
H\(M). Then, considering μ := inf R(u) on the orthogonal complement of E, we

268
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
see that μ is greater than μι (г = 1, 2, ...), which is a contradiction. Namely,
0°^! Ei is dense in Η\(Μ), and consequently also dense in L2(M). Therefore, for
an o.n.b. {ujg! of 0г°=1 Еи апУ / £ L2(M) may be written as / = £Si α^.
Finally, we note that any eigenvalue A of Δ is equal to one of the above μι. In fact,
suppose Α φ μι (г = 1, 2, ...) is an eigenvalue of Δ, and и denotes an eigenfunction
corresponding to A. Since we may assume that A > 0, we get
(u, щ)о = τ(Δ^ иг)о = т(гх, Δμ»)ο = у (u> u0o·
Therefore u_L 0^ Ei, which implies u = 0, a contradiction. The dimension of the
eigenspace corresponding to an eigenvalue A is called the multiplicity of A. Summing
up, we have the following.
Theorem 3.7. For the eigenvalue problem (I) (resp., (II)) of the Laplacian A,
eigenvalues are nonnegative (resp., positive) and their multiplicities are finite. If
we arrange the eigenvalues according to size as 0 < μι < μι < · · · < μη < · · · , then
{μη} are discrete and μη | +oo. Eigenspaces E{ (i = 1, 2, ...) corresponding to
μι are mutually orthogonal with respect to the inner product of L2(M). Moreover,
0~! Ei is dense in Η*λ{Μ) and L2(M).
Eigenvalues of Δ are also written as (0 <) Ai < X2 < · · · < An < · · ·, each
eigenvalue being counted as many times as its multiplicity. Then we have an o.n.b.
{фг} of L2(M) such that фг is an eigenfunction corresponding to the eigenvalue A*.
It follows from the above that any / G L2(M) may be written as a Fourier series
/ = ΣΖι α>ιΦι, a>i = (/, 0г)о, and we get
(3.9)
(feL'(M)),
£>2 (feH.l(M)).
= 1
In fact, for / G L2(M) we have ||/ - £-=i агфг\\02 = (/, /)o - ΕΓ=ι *2 - 0 (r ->
+00), which implies that ||/||o2 = Σ)ί=ια*2· Next for / G Η\(Μ), noting that
||/ - ΣΙ=ι аг^г||1 -> 0 (r -> +00), we see that
(ν(/-Σα^)' ν(/-Σα^
Γ
= (V/, V/)0 - 2^аг(У/, V0i)o + 51 ЫЪФФи V^)o
г=1 i,j = l
converges to 0 as r —> +00. Now from Chapter II, (5.20), we have (V/, V0;)o =
(/, Афг)0 = Аг(/, Фг)0 = Агаг, (V0<, V0j)o = (0<, A0j)o = Ai5<j, considering the
boundary condition when we treat the case (II). It follows that
r
(V/,V/)o-^Aia<2^0 (r^+oo),
г=1
namely,
R(f) = (v/, v/)0/(/, /)o = f>ai2 /Σ"*2·
г=1 / г=1

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
269
Exercise 2. Show that ~\k = ini{R(u); и G H\(M) \ {0} is orthogonal
to (0i, ... , 0£_ι)ΛίηΖ/2(Μ)}, where (0b ... , фк-\)н denotes the subspace of
L2(M) spanned by 0b ... , фк-\. Next show that any и G #i(M) \ {0} that
satisfies u_L(0i, ... , фк-ijR and R(u) — A& is an eigenfuction of Δ corresponding to
the eigenvalue A^.
Proposition 3.8. Let Ai < A2 < · · · < λη < · · · be eigenvalues of the
eigenvalue problem (I), (II), counting multiplicities. Then:
(1) (Max-min theorem)
λ^ = sup ini{R(u); и φ 0, u±Fk-i inL2(M)},
Ffc-i
where the supremum is taken over all (k — 1)-dimensional subspaces Fk-\ of H\(M)
(2) (Min-max theorem)
Afc = inf sup{#(u); u G Lk\ {0}},
where the infimum is taken over all k-dimensional subspaces Lk of H*1(M) (or
•F.(M)).
Proof. (1) First, taking Fk-\ := (o\ #/c-i)h, the inequality A^ <
right-hand side follows easily from Exercise 2. Conversely, let Fk-\ С Я*1 (Л/)
(or .Τ7*(Μ)) be an arbitrary (к - l)-dimensional subspace. Then for the
orthogonal complement F^_1 of Fk-\ in L2(M), we have Fj^_l Π (o\ ok)R φ {0}.
In fact, otherwise the orthogonal projection onto F^_i is injective when restricted
to Ek := (0i, ... , фк)в., which contradicts dimE* > dimF^-i. Therefore, there
exists а и = Σι=ι агФг (ф 0) with u^F^-i, and it follows that
R(u) = Σα^'(ν^' νφ^° /Σαί2 = Σ^α*2 / Σα*2 - **>
i,j Ι г г=1 / г=1
namely, A^ > right-hand side.
(2) Taking Lk = Ek and noting that R(u) < Xk for w G u, we have A^ >
right-hand side. On the other hand, for any fc-dimensional subspace Lk of Η*λ(Μ)
(or ^(M)), the orthogonal projection π : Lk —> Ek-\ to Ek-\ in L2(M) cannot
be injective, as before. Hence there exists а и G Lfc Π Ек-\^, и ф 0. Writing
u = £^J*Lfc а^0г we again have #(u) > A^, namely A^ < right-hand side. D
Recall that in case (I) we have Ai = 0, and the corresponding eigenspace consists
of constant functions, namely, its multiplicity is equal to 1. As for the Dirichlet
eigenvalue problem (II), we have Ai > 0 because of the boundary condition. In
the following we denote by Ai = Ai(M) the least positive eigenvalue, which is also
called the first eigenvalue of Δ and equal to \2 (resp., Ai) in case (I) (resp., (II)).
Remark 3.9. For a linear operator A of a Hilbert space #, if A - Χ'\άΗ has
no bounded inverse operator, then A is called a spectrum of A. In particular, if
Кег(Л - Aid) φ {0} then A is an eigenvalue of A. In our eigenvalue problem
(I), (II) it is known that all spectra are eigenvalues. We denote by Spec(M, g) :=
{Ai < A2 < ... < An < ...} the set of eigenvalues of Δ, counting multiplicities,
which is also called the spectra of (M, g). However, when Μ is not compact, there
might appear spectra which are not eigenvalues.

270
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Now we consider the multiplicity of X\(M) for the Dirichlet eigenvalue problem
(II).
Lemma 3.10. Let φ be an eigenfunction corresponding to X\(M) for the
Dirichlet eigenvalue problem (II). Then φ \ M° is either always positive or always
negative.
Proof. It is known that \φ\ £ Я*1 (М), and we may assume that V|0| is equal
to V0, 0, —V0 at points where φ > 0, φ = 0, φ < 0, respectively. Then we have
Щ\ф\) = Щф) = Ai(M). It follows that \ф\ is also an eigenfunction and belongs to
T(M0) Π C°(M), by Lemma 3.6. On the other hand, because Δ(|0|) = Λχ|0| > 0,
and by the maximam principle (see Apendix 4), \ф\ assumes its minimum 0 exactly
at the boundary dM. Namely, \ф\ > 0 on M°. D
Corollary 3.11. In the eigenvalue problem (II), the first eigenvalue X\(M) is
simple, namely, its multiplicity is equal to 1.
Proof. In fact, otherwise we may take eigenfunctions φι, 02 of X\(M) which
are orthogonal in L2(M) and take positive values on M°. Then we get 0 =
Sm Φ1^2 άν9 > 0, a contradiction. D
Next we consider an o.n.b. {φι} of L2(M) consisting of eigenfunctions with
eigenvalues Ai < A2 < · · · of the eigenvalue problem (I), (II). Then connected
components of Μ \ φι~ι(ϋ) are called the nodal domains of the eigenfunction φι.
R. Courant showed that the number of nodal domains of φ{ is less than or equal
to i. We briefly explain this fact, assuming that boundaries of nodal domains
are smooth. Suppose we have nodal domains Vi, ... , VJ, VJ+i, We consider
functions г/jj (j = 1, ... , г), defined to be equal to ф^ on Vj and 0 outside Vj.
Then i/jj £ Hl(M), and (i/>i, ... , ф{)я is an г-dimensional subspace of Hl(M).
By a similar argument as in the proof of Proposition 3.8, it follows that {i/>j)r Π
(0ь · · · 7 Φι-ι)η Φ {0}> and there exists а и = Σ*=1 dji>j φ 0 which is orthogonal
to 0i, ... , фг-\. Then from Exercise 2 and the Green theorem we get
Xi < R(u) = £ j <Ъ2 jy (V^, V^> dug\ j Σ Ι α;2 jy Ъ
dvQ > = Xi
and и is an eigenfunction of Δ with the eigenvalue A*. On the other hand, и \ VJ+i =
0 and и vanishes on the whole Μ, a contradiction.
In particular, we see that φι corresponding to Ai is of definite sign and φι
corresponding to X2 has exactly two nodal domains.
3.3. In this subsection we give some fundamental examples.
(I) (Sphere (5m, g0)). We may consider (5m, g0) as the unit sphere in Дт+1
centered at the origin o. For / £ ^(Дт+1) and its restriction / | Sm £ 7"(5m) we
have the following: We denote by r the distance to the origin о and by д/dr the
radial differentiation, namely, directional differentiation with respect to the unit
outer normal vector to 5m. Then
5m = Asm(f I 5m) - j£ I 5m - m |£ | 5m.
choose an o.n.b. {е*}·^ of TpS171 and let ^(t) :=
be a geodesic emanating from ρ with the initial direction e^.
(3.10)
In fact,
cos t · ρ
(Δ*'"+Ι)/
at ρ
+ sin t
£
■ e<
gm
be

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
271
Identify ρ with the unit normal vector to Sm at p, and consider the coordinate
system (ж1, ... , хш, r) with respect to the o.n.b. {ei, ... , em, p]. Then we get
2
dt2
t=0
/ЫО) = 0(Р)-^(Р) « = 1 m),
and (3.10) easily follows from (3.3).
Now for an ft £ ^(S171) define a C°° function / on a neighborhood of 5m by
f(x) ·= rk{x)h(x/\\x\\). Then clearly / | Sm = ft, and we apply (3.10). It follows
that
(3.11) ASTnh = {ARTn+1)f \Sm + k{m + k- l)ft.
Therefore, if / is a harmonic homogeneous polynomial of degree к on um+1, then
/(x) = rk(x)h(x/\\x\\) for ft := / | 5m, and ft is an eigenfunction of Asm with
eigenvalue A;(m + к — 1).
Now we denote by 7\ the vector space of homogeneous polynomials of degree
к on um+1, and by Hk the subspace of Vk consisting of homogeneous harmonic
polynomials of degree к. For / £ Vk we set / := / | Sm, Vk := {/ £ ^"(5m); / <E
Vk) and Н^ := {/ £ TV, / £ H*}. Then, by the above, elements of H^ are
eigenfunctions of Asm with eigenvalue k(m + fe — 1). In the following we see that
0£LoHfc is dense in L2(M), which implies that all eigenvalues are exhausted by
k(m + к - 1), к = 0, 1, .... Note that 0£1OP* is a subalgebra of 7"(Дт+1), on
which we introduce an inner product by (P, Q) := /5m PQdvQQ.
Lemma 3.12. With respect to the above inner product, we have the orthogonal
decompositions
V2k = П2к θ r2H2k-2 θ · · · Θ r2fcH0,
V2k+i = H2k+i Θ r2H2k-i Θ · · · Θ r2fcWi.
Note that here Ho = Vo = {constant functions} and H\ = V\ = {homogeneous
polynomials of degree 1}.
Proof. It suffices to show that
(*)fc Pfc+2 = Hfc+2 θ r2Pfc (k = 0, 1, 2 ...).
We prove (*)fc by induction on fc. For Ρ £ H2, we get from (3.11) and the Green
theorem
and clearly H2±r2Vo· Next note that if Ρ £ V2 is orthogonal to r2P0, then
/STO Pdvgo = 0. On the other hand, from (3.11) we have
Дят+1Р = ASTnP - 2(m + 1)P.
Integrating both sides of this equation, we find that JSm ARrn+1 Ρ dvgo = 0. Since
ΔβΓη+1Ρ is constant, it must be 0. Therefore Ρ £ H2, and (*)0 is proved.
Now suppose (*) is proved for 0, 1, ... , к — 1. Then
Vk = Hk® r2Hk-2 Θ · · · Θ rkH0 (or rk~lHi).

272
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
For Ρ G Hk+2, Q € /Hk-2i (k — 21 = fe, ... , 1, or 0) we get by a similar argument,
using Lemma 3.5,
ι ρ r2i+20) _ (fc-2Q(m + fc-2f-l) 2l+2
^r W" (fe + 2)(m + fe+l) ^'Γ W'
Namely, (P, r2l+2Q) = 0 and Wfc+2-Lr27V Next suppose Ρ <Ε Pfc+2 is orthogonal
to r2Vk- Then for any Q G Hk-21 we have, by (3.10) and Lemma 3.5,
(Дят+1Р, r2lQ) = [ As'nPQdvgo - (к + 2)(m + fc + 1) / PQ^0
Ρ · ASTnQdvgo = (fe - 2l)(m + fc - 2/ - 1)(P, r2Z+2Q) = 0.
/,
/5"
Then ΔΗ™ Ρ G Р/с is orthogonal to Vk by the induction hypothesis, and therefore
equal to 0. Namely, Ρ G Hk+2, and the proof of the lemma is complete. D
Now we recall the Stone-Weierstrass theorem in functional analysis (see, e.g.,
[Re-Si]): Let V С Т(М) be a subalgebra satisfying the following conditions (i), (ii),
where Μ is a compact manifold:
(i) V contains constant functions.
(ii) V separates points, namely, for any different points p, q G Μ there exists
an / G V such that f(p) φ f(q).
Then V is dense in T[M) with respect to || ||o-
We apply this result to 0fc>o Hk (C 7"(5m)). By Lemma 3.12, 0fc>o Hk =
0fc>oPfc is a subalgebra and satisfies (i). As for (ii), note that we may choose a
linear function (coordinate function) / G H\ such that f(p) φ f(q). Therefore,
the direct sum 0fc>oHfc is dense in ^"(5m) (and consequently in L2(5m)), and we
have the following.
Proposition 3.13. The eigenvalues of the Laplacian o/(5m, go) are given by
Xk := k(m + fe — 1), fe = 0,1, The eigenspace of Xk is given by Hk. In
particular, the multiplicity of Xk is equal to
m + fe\ /m + к — 2
fe У V k-2
Proof. It suffices to verify the multiplicity of A^, which is equal to
ι. ~~, τ „/ 1. ^ 1. ^ fm + k\ fm + k — 2\
dimHk = dimHk = dimPk - dimPk-2 = ( fc j-i fc _ J
because of Lemma 3.12. D
(II) (Flat torus (Tm, $г) := (Дт, 0О)/Г). Let Г be a lattice of Rm and
{xb ... , xm} a basis of Г. Then Г* := {y G Дт; (у, χ) e Ζ for any 3; G Γ} is
again a lattice of Дт, which is called the rf^a/ lattice of Г. Note that Г* carries a
basis {ж*} satisfying (x*, a^·) = 5^; (1 < г, j < m). Now elements of Г act on Дт
as parallel translations, which are isometries of (-Rm, go)- Since the Laplacian ARn
of Дт commutes with the action of Г (Exercise 1), we see that the eigenfunctions
of Δ (= Дтт) are given by C00 functions / G ^(Дт) \ {0} satisfying ARm f = A /
for some constant A and are invariant under the action of Г.

3. EIGENVALUE PROBLEM FOR THE LAPLACIAN
273
Proposition 3.14. The set of eigenvalues of the Laplacian of a flat torus
(Tm, <7r) is {4n2\\y\\2; у G Г*}. The eigenfunctions corresponding to у = 0 are
constant functions, and the eigenspaces corresponding to y, —y G Г* \ {0} are spanned
by Фсу(х) := cos2tt(:e, у), фу(х) := sin27r(:r, y).
Proof. It is easy to see that фс фу (у G Г*) are invariant under the action of Г
and satisfy ARm0J = 4π2||ι/||2 0J, ΔΗ"0^ = 4π2||ι/||2 фсу. We denote the functions
on Tm induced from 0£, фу by the same letters, and show that the subspace V
of P(M) spanned by {1, фу, фу; у G Г* \ {0}} is dense with respect to the L2-
topology. Again we apply the Stone-Weierstrass theorem. Clearly (i) holds, and
by the addition formula for trigonometric functions V is a subalgebra. Next let
χ φ χ' (modΓ) in Дт, and suppose фу(х) — ocy{x'), osy(x) = фу(х') hold for any
у G Г*. Then (x - x', y) G Ζ for any у G Г*. It follows that ι-χΈΓ, which is a
contradiction, and V separates points of Tm. Finally, {1, фс фу\ у G Г* \ {0}} are
linearly independent by the following exercise. D
Exercise 3. Show that {фсу + \/^\osy; у G Г'} are linearly independent over
C.
Remark 3.15. In Proposition 3.14, the problem of determining the
multiplicity of an eigenvalue A of Δ reduces to a problem in number theory, which
asks how many points of the dual lattice Γ* lie on the hypersphere centered at
the origin of radius y/X/2n in Дт. With respect to this we may also pose the
following interesting problem: For given lattices Γι, Γ2 in Дт, are grx and gy2
isometric if Spec(Tm, gVl) = Spec(Tm, дТ2)1 This is true for m = 2 (see
Problem 11 for this chapter), but there are counterexamples for m = 8, 12, 16,... by
J. Milnor ([M-2]) and others. In general, we may pose the following isospectral
problem: Are two compact Riemannian manifolds (Mi, g\), (M2, £2) isometric if
Spec (Mi, 0i) = Spec(M2, 02)? Many counterexamples have been found, which are
related to number theory through the fundamental group (see [Su-1], [Ik], [Ej], [Ur-
1], [Ku] etc.). An excellent survey article [Be-3] on the isospectral problem may
be helpful. In particular, С Gordon, D. L. Webb and S. Wolpert ([Go-We-Wol])
constructed pairs of nonisometric simply connected domains in the Euclidean (and
hyperbolic) plane that are Dirichlet isospectral, and negatively answered Kac's
original problem. In these examples Sunada's idea ([Su-1]) plays a fundamental role.
See also [Bro] and Remark 5.4.
Now we ask how the eigenvalues are distributed. For that purpose we set
N(X) := tt{Afc; A& < A}, which is the number of eigenvalues less than or equal to A,
and see how N(X) behaves as A —> +00. In case of (Tm, gr) we may easily verify
the following asymptotic formula of H. Weyl.
Proposition 3.16. For aflat torus (Tm, gr)
(3.12) N(X) - u;mA™/2vol(:r\ gr)/(2n)m (A -> +00),
where иош denotes the volume of the unit ball in R171.
Proof. In this case N(X) = ${y G Г; ||y|| < y/X/2n}. Let P(y) := {z =
o?Xj*\ \o? — ώ?\ < ^ (j = 1, ... , m)} be a parallelotope centered at у = n^Xj* G Г*.
Note that the P(2/)'s, у G Г*, are mutually disjoint, and their closures P(y), у G Г*,

274
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
which are congruent to each other, cover R171. Namely, P(y) is a fundamental
domain of the universal covering π : R171 —» Ят/Г*. Now for у G Г* we set
d := max{%, χ); χ G 0P(y)}. Then P(y) С Pr+d(o) if у G Br(o) Π Γ*, and
in particular, UyeBr(o)nr* -Р(у) С Pr+d(o)· On the other hand, for any point
ζ G Br-d{o) there exists a i/ G Γ* such that ζ G P(?/), and we have d(y, 6) <
d(y, z) + d(z, 6) < r. It follows that Br-d{o) С {JyeBAo)nr* Р(у) С ~Br+d(o).
Therefore, setting r ~ y/\/2n and considering the volume, we have
(3.13) (\/λ/2π - d)mum < N(X)vo\P(y) < (\/λ/2π + ii)ma;m.
Now recall that
vo\P(y) = yjdet(xi*, Xj*) = {yjdet(xi, Xj)}'1.
On the other hand, vol(Tm, gr) is equal to the volume of {ajXj; 0 < a3 < 1}, a
fundamental domain of the Riemannian universal cover π : Дт —> Дт/Г. Hence
vol(Tm, ^r) = ^/det(xi, χ,·) and volP(y) = vol(Tm, gr)-\
Then, dividing both sides of (3.13) by Am/2 and letting A -> +oo, we get (3.12). D
(III) We consider the eigenvalue problem (II) for Μ = [0, αϊ] χ · · · χ [0, ат] С
Дт. Let к = (fei, ... , fcm) G Z!p be an m-tuple of positive integers, and set
<t>k(x\...,xrn) = l[sm(^nxi),
which is an element of ^ч(М). Then we easily get Афк = Σϋιί^71"/^)2 ' ^fc·
Namely, the 0^ are eigenfunctons of Δ with eigenvalues 5^(fe^/az·)2. By the theory
of Fourier series, {Σ^^π/αί)2; A; G Z^} cover all eigenvalues of (II). In this
case again we may check tht
(3.14) N(X) ~ a;mAm/2volM/(2^m
by a similar argument.
Remark 3.17. Note that Μ of (III) is not a manifold with smooth boundary.
However, dM is a picewise smooth boundary consisting of smooth pieces. In such
a case the above assertion on the Dirhiclet boundary problem (II) again holds
(except for the smoothness property of eigenfunctions at the singular points of the
boundary).
Now it is generally impossible to get explicitly all eigenvalues of the eigenvalue
problems (I), (II) for a given (M, g). However, it is possible to get some qualitative
properties of eigenvalues. For instance, the above asymptotic formula of Weyl holds
for any eigenvalue problems (I), (II) (see, e.g., [Be-1]).
(IV) Let M™ be the m-dimensional simply connected space form of constant
curvature <5, and consider the eigenvalue problem (II) for a metric ball Br(p), where
we assume that r < π/у/б if δ > 0. We note that the eigenfunction φ for the
first eigenvalue Xi(Br(p)) depends only on the distance ρ to the center. In fact,
let <7i, <72 € Br(p) satisfy dp(qi) = dp(q2). We may take an isometry Φ of M™
such that Φ(ρ) = ρ, Φ(^ι) = Φ(^)· To see this take (unique) minimal normal
geodesies 7^ (г = 1, 2) joining ρ to q\, and choose an isometry Φ with Φ(ρ) =

4. CURVATURE AND SPECTRUM
275
ρ, Ζ)Φ(ρ)7ι(0) = 72(0) (see Chapter IV, §1.1). Then, by Exercise 1, Φ*(φ) is also
an eigenfunction for the first eigenvalue Ai(M), and Φ*(φ) = φ by Corollary 3.11.
Therefore φ(ςι) = φ(Φ(ςι)) = ф(я2), and our assertion holds. We also note that
Xi(Br{p)) does not depend on p, since all Br(p), ρ £ Μ™, are isometric. For more
details on the eigenvalue problems (II) for Br(p) in M™, see, e.g., [Cha-2].
4. Curvature and Spectrum
In this section we are concerned with the relation between eigenvalues (mainly
the first eigenvalue) of the eigenvalue problems (I), (II) and other Riemannian
invariants.
4.1. We begin with the following Lichnerowicz-Obata theorem.
Theorem 4.1. Let Μ be a complete m(> 2)-dimensional Riemannian
manifold such that the Ricci curvatures ρ satisfy p(u) > (m — Ι) δ for all и £ UM,
where δ is a positive constant. Then Μ is compact (Myers theorem) and for the
first eigenvalue of the eigenvalue problem (I), we have X\(M) > δπι, where equality
holds if and only if Μ is isometric to the sphere of constant curvature δ.
PROOF. First we show that for / £ F(M):
(4.1) -^Δ(||ν/||2) = ||D2/||2 - (V/, ν(Δ/)> + Ric(V/, V/).
In fact, the left-hand side is equal to
= gkl9ijVkdif ■ Vidif - д*А(Ы) ■ ^f = ||D2/||2 - gijA(dJ) ■ fyf.
From (2.15) of Chapter II and the definition of the Ricci tensor, we get
-A(dif) = g^kVtdJ = gklVkVid,f = gkl{ViVkdlf - dmf Rfu)
= Vitf'Vkdrf) + Ric(V/, di) = -diAf + Ric(V/, $)·
It follows that
-gijA(dif) ■ d^ = -(V/, V(A/)) + Ric(V/, V/),
and we get (4.1). In particular, if / is an eigenfunction satisfying Δ/ = Αι/, Αι =
Ai(M), then
~Δ(||ν/||2) = ||D2/H2 - AJV/H2 + Ric(V/, V/).
We have Ric(V/, V/) > (m - 1)<5||V/||2 by assumption. Next note that
||£)2/||2 > (trace D2 f)2 /m, where equality holds if and ony if D2 f is aconstant
multiple of g (see Exercise 1 below). On the other hand, from trace/)2/ = — Δ/ = — \\f
we get
-^Δ(||ν/||2) > λ2 f2/m - λ! IIV/II2 + (m - 1)*||V/||2.
Integrating the both sides of the above inequality over Μ and using the Green
theorem, we obtain
{Xi-(m-l)6}· [ \\Vf\\2dug/ / f2dug>\\/m.
JM JM

276
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Now, by a property of the Rayleigh quotient, the left-hand side of the above
inequality is equal to {Ai — (ra — 1)<5}λι, and we get Ai > dm.
Next we consider the case where equality holds: Ai = dm. Then all the
above inequalities are in fact equalities, and Ric(V/, V/) = (m — 1)(V/, V/).
Moreover, we may write D2f = фд for some φ £ F(M). Taking the trace of the
last equation and noting that / is an eigenfunction, we see that φ = —6f. Therefore,
D2f = -6f-g.
Now let / assume its maximum μ at a point ρ £ Μ, and let 7 be any normal
geodesic emanating from p. Note that μ > 0, because JM f dvg = 0. Then
|(/("(0)) = (V/,7(i)>,
4(/(7(*))) = (V,((,V/, 7(0> = D2f№), 7(0) = -«/(7(0),
dt2
d^
dl
/(7(0) = 0.
\t=0
Namely. h(t) := f("y(t)) satisfies an ordinary differential equation h"(t) + 6h(t) = 0.
Solving this under the initial conditions /ι(0) = μ, /i'(0) = 0, we get f{^f{t)) =
μ cos y/dt. Since 7 is arbitrary, we have f(M) = [—μ, μ]. We denote by S the set
of points at which / assumes the minimum —μ. S is compact, and there is a q £ S
with d(p, q) = d(p, S). Now for a minimal geodesic 7 : [0, /] —> Μ joining ρ to q we
have /(7(0) = -μ, f(7(t)) > -μ(0<ί < /), and it follows that l = π/уД. Then
(π/уД >)d(M) > d(p, q) = п/уД, and therefore d(M) = π/y/δ. Now the Cheng
maximal diameter theorem (Chapter IV, Theorem 3.5) implies that Μ is isometric
to the sphere of constant curvature 6. Conversely, for the m-dimensional sphere of
constant curvature δ we have Ai = 6 m by Proposition 3.13 (see also Problem 1 for
this chapter). D
Exercise 1. Let (V, g) be an m-dimensional Euclidean vector space and h
a symmetric bilinear form on V. Show that \\h\\2 > (trace/i)2/m, where equality
holds if and only if h = с g for some constant с
Next we state a fundamental result of J. Cheeger ([Ch-3]) on the estimate of
Ai(M) from below for a general compact Riemannian manifold.
Theorem 4.2. Let Μ be an m(> 2)-dimensional compact Riemannian
manifold. Then for the first eigenvalue X\(M) of the eigenvalue problem (I) we have
λι(Μ) > hc2(M)/4, where hc(M) denotes the Cheeger isoperimetric constant {see
Definition 1.1 (3)).
Proof. Let / be an eigenfunction of Δ with eigenvalue Ai = Ai(M). Prom
the Sard theorem we may take an arbitrary small e > 0 so that 0 is a regular value
of f€ := f + €. We may also assume that νο1/€_1(0, +οο) < vol/"1 (-00, 0), by
considering -fe if necessary. We set M+ := /e_1(07 +°°) and note that
div(/eV/e) + /e(A/) = (V/e,V/e>
(Chapter II, §1, (1.28)). Then from the Green theorem we get
Ai
SM+We\\2dvg JM+f2dug

4. CURVATURE AND SPECTRUM 277
Therefore, it suffices to show that
/л,+ ЦУЛ112^9>^с2(м)
/м+Л2<Ч - 4 ·
In the following we set / = /e for brevity. Noting that V(/2) = 2/(V/), by the
Cauchy-Schwarz inequality we have
{/ HV(/2)||^9} <*f fdvg-\ ||ν/||2ώ/β
and consequently
/ \\Vf\\2dug/ f f2dug>\\( \\V(f2)\\dvg/ [ f*dug\ .
JM+ I Jm+ 4 Um I Jm )
Now we apply the coarea formula (Chapter II, Theorem 5.8 and Remark 5.9) to
h := /2. We set Qt := {p e M+; h(p) > t}, Tt := 5i2t = {p G M+; /ι(ρ) = ί} and
Vt := voli?^, i4t := уо1ш_1Г^. Note that Tt are smooth hypersurface for almost all
t, and from the choice of M+ we have At > hc(M) Vt. Therefore, it follows that
/ || V/i|| dug = [ Atdt> hc(M) [ Vt dt
Jm+ Jo Jo
= -hc(M) Г\^-Vtdt = hc(M) [ tdt [ \\Vh\\-ldi/gt
Jo dt J0 JTt
= hc(M) [ dt [ h\\Vh\\~l dvgt
Jo Jvt
= hc(M) / hdvg,
Jm+
where we have used the identity -^ Vt = - Jr ||V/i||_1 dv9t from the coarea formula.
Namely, JM+ ||V(/2)|| dvg / JM+ f2 dvg > /ic(M), and the proof of the theorem is
complete. D
Now from Theorem 1.4 we have the following:
Corollary 4.3. Let Μ be a compact m-dimensional Riemannian manifold such
that p(u) > (m - 1)6 for all и е UM. Then
λ ( rd(M)/2 ) "2
\i(M)>-{ / cr-\t)dt\
4
μ
Remark 4.4. Let Μ be a compact Riemannian manifold with smooth
boundary. Set hc(M) := inf{volm_idi?/voli?; Ω is an open (not necessarily connected)
submanifold with smooth boundary in Μ }. Then for the eigenvalue problem (II)
we may show that X\(M) > /ic2(M)/4 by a similar argument (see, e.g., [Cha-2]).
It is also known that these inequalities cannot be improved in general (for this fact
and more details on the Cheeger inequality, see [Bus-1]).
4.2. Here we estimate Ai from above. As for the eigenvalue problem (I), we
have Ai(M) = inf {Д(/); / G Hl(M) \ {0}, JM f dvg = 0} , and we want to find

278
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
a nice /, which is called a test function, to estimate R{f). We follow a similar
strategy also for the eigenvalue problem (II). First we state a result due to S. Y.
Cheng ([Che-1]).
Theorem 4.5. Let (M, g) be a complete Riemannian manifold of dimension
m (> 2) whose Ricci curvatures satisfy p(u) > (m — 1)<5, и £ UM, for some real
constant δ. For a metric ball Br(p) (0 < r < ip(M)), denote by \\(Br(p)) the first
eigenvalue of the Dirichlet eigenvalue problem (II). Then
Ai(Br(p))<Ai(r;«),
where \\(r; δ) denotes the first eigenvalue of a metric ball of radius r in the m-
dimensional simply connected space form M™ of constant curvature δ. Moreover,
equality holds if and only if Br(p) is isometric to a metric ball of radius r in M™.
Proof. First, recall that for a metric ball Br(p) in M™, any eigenfunction /
corresponding to the first eigenvalue Ai (r, δ) of the eigenvalue problem (II) depends
only on the distance dp to ρ (see (IV)). Therefore, by Lemma 3.10 we may write
/ = φ ο dp, where ψ is a positive C°° function. Next, note that 0(s, u) of Chapter
II, (5.8) for M™ does not depend on и £ UM, and we denote it simply by 0(s).
Then from Lemma 3.1 we have for 0 < s < r
φ"(8) + θ'(3)/θ(3) ■ <p'(s) + λχ (r, δ)φ(8) = 0,
(4-2)
¥>(г) = О, ¥>'(()) = 0.
Now we take f(q) := ip(dp(q)) as a test function on a metric ball Br(p) in Μ,
which satisfies the boundary condition / | dBr(p) = </?(r) =0. Then we get
(4.3) Xi(Br(p))< [ IIV/II2^,/ / f2dug.
JBr{p) JBr(p)
Because V/ = <p'Vdp, and by Chapter III, (4.4), we have ||V/||2 = (φ')2 (except
for p). From Chapter II, Lemma 5.4, it follows that
/ || V/||2 dug = [ dS™~1 Ι\φ'(8))2θ(8, и) ds,
JBr{p) JS™-1 JO
[ f2dvg= ( d5m_1 / </?2(s)0(s, u)ds.
JBr{p) JS™-1 J0
We integrate the first integral in the above by parts, and get
/ ||V/||2 dug = / dSm~l Γ[{φ(φ'θ)Υ - φ(φ"θ + ψ'θ')} ds
JBr(p) Js™-1 Jo
= [ dSm-1 [ <p(s){-ip"(s)-e'{s,u)/e{s,u)-<p'{s)}e(s,u)ds,
Jsm~1 Jo
where 6'(s, u) denotes the partial derivative with respect to s. Now by the Bishop
comparison theorem (Chapter IV, Theorem 3.1) we have {6(s, u)/e(s)}' < 0 (0 <
s < r). On the other hand, note that 4>'(s) < 0. In fact, by (4.2) we have
(e(s)(p'(s))' = -Ai(r, δ)φ(8)θ(8) < 0. Since φ'(0) = 0, we easily see that tp'(s) < 0.
It follows that
φ'(s) ■ θ'(3, „)/*(*, и) > ψ'(8) ■ θ'(3)/θ(3)

/.
4. CURVATURE AND SPECTRUM 279
and from (4.2) we get
ιιν/ιι2^,
bap)
< [ dSm-1 [ φ(3){-φ"(3)-θ'(3)/θ{3)-φ'(3)}θ(3,η)ά3
Js™-1 Jo
= [ d5m_1 / Ai(r;%2(s)0(s, u)ds = Xl(r; δ) [ f2 dug.
Js™-1 Jo Jbt{p)
With (4.3) this implies that Xi(Br(p)) < Ai(r; δ). Finally, suppose equality holds.
Then {0(s, u)/e(s)Y = 0, and so equality holds in the Bishop volume comparison
theorem. Then Br(p) is of constant curvature δ. Because r < гр(М), it follows that
Br(p) is isometric to a metric ball in M™. D
Next we estimate from above the positive A;-th eigenvalue Xk{M) (counted with
multiplicities) of the eigenvalue problem (I).
Let Μ be a compact Riemannian manifold with p(u) > (m — 1)<5, и £ UM.
For a given e > 0 take a maximal set of points {Pi}i=Si of Μ such that Be(pi)
are mutually disjoint. Then, as is seen in the proof of Proposition 1.3 (2), we
have Ui=i ^2с(Рг) — Л/ and N(e) > vol М(г>2е(<5))-1 by the Bishop comparison
theorem (see (1.7)). On the other hand, for г = 1, ... , N(e) we denote by φι
an eigenfunction with the first eigenvalue \\(Be(pi)) of the Dirichlet eigenvalue
problem (II) for Bc(pi). We may assume that \\фг\\о = 1· We define fi £ Hl(M) so
that it is equal to φι on Be(pi) and equal to 0 outside Be(pi). Then the subspace
Ln(€) '·= (fi)R of Hl(M) is of dimension N(e). Note that (fj, Д)о = Sjk and
(V/j, V/fc)0 = Xi(Be(pj))6jk < Ai(€; 6)6jk hold with respect to the L2(M)-norm.
Now note that for the eigenvalue problem (I) we have Ai = 0, A^(M) = A^+i,
and apply Proposition 3.8 (2) to get
AN(c)_i(M) < sup{#(u); и φ 0, и £ LN(e)}
ί Л(с) 1
= sup < (Vu, Vu)0/(u, u)0; и = ^ аг/, > < Ai(c; (5).
It follows that
(4.4) A[volM/(V2e(6))]_i(M) < Ai(c; <5),
where we used the Gauss symbol [x] (:= maximal integer less than or equal to x).
From this we may estimate λ^(Μ) as A;—» +oo. In fact, since
lim€2Ai(€; (5) = Ai(l; 0)
e—>0
(see Problem 9 for this chapter) and also
\imv2€(6)e-m = 2mui(0)
by Chapter II, §5, Exercise 3, (3), it follows that
lim Afc(M)(volM)mfc-£ = \[тХ[у/оШ/У2е(6)].1{М)(у2е{б))^
к—kx> €—*0
< lim Me; δ)(ν2((δ))% = c(m) 0=^(1; 0)^(0))^).
€—►0

280 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Remark 4.6. We may show that Theorem 4.6 holds generally for Br(p) (0 <
r < d(M)) in the same way. Then we may estimate Xk(M) from above in terms of
k, ra, the infimum of Ricci curvatures, and vol Μ (or d(M)). It is also known that
Afc(M) may be estimated from below in terms of k, the infimum of Ricci curvatures,
and the diameter. For more details see papers by S. Y. Cheng, S. T. Yau, P. Li,
and S. Gallot ([Che-1], [Li-Y], [Be-1], [Ga-2], [Y-2]).
Exercise 2. Let Μ be a compact Riemannian manifold such that p(u) >
(m - 1)<5, и £ UM. Show that Xk{M) < Ai(d(M)/2fe; δ).
4.3. Next we use the isoperimetric inequality to estimate the first eigenvalue.
Theorem 4.7. Let Μ be a complete m-dimensional (ra > 2) Riemannian
manifold whose Ricci curvatures satisfy p(u) > m — 1 for all и £ UM. Let Ω С М
be a (connected) domain with smooth boundary, and take a metric ball Ω* in the
sphere (Sm. go) of constant curvature 1, which satisfies the relation
vol β/vol Μ = vol β*/νο1 (5m, g0).
Then for the eigenvalue problem (II) we have Χ\(Ω) > Χ\(Ω*), and equality holds
if and only if Μ is isometric to (5m, go) and Ω is isometric to Ω*.
Proof. In the following g0 is also denoted by g* for convenience. Let / be
an eigenfunction of Δ corresponding to the eigenvalue Χ\(Ω). We may assume
that / | ΘΩ = 0, / > 0 on Ω. Set Ωί := {ρ <Ε Ω; f(p) > t}. Correspondingly,
in Sm take metric balls Ω\ centered at po, which is the center of i?*, such that
vol β*/vol 5m = vol βέ/νο1 Μ.
Now we define a function /* on Ω* which depends only on the distance to po
and may be considered as a "symmetrization" of /. Namely, we set f*(x) := t if
χ e di?t*. Then /* | 8Ω* = 0, f*(po) = max/, and /* is monotone decreasing
along radial directions. We also note that Ω\ = {χ £ i?*; f*(x) > t}. In the
following we compare the Rayleigh quotients of /, /*, where the coarea formula
(Chapter II, Theorem 5.8 and Remark 5.9) plays an important role.
First, we note that
^voiA = -Jr liv/ir1^,, |vom; = -J \\vrVdug;
for almost all £, where we set Tt := f~l(t) = di?i etc. Then we have
/ \\Vf\\-ldv9t=a [ HVnr1^, a = volM/vol(5"\0O).
Again we apply the coarea formula to ||V/||2, and
[ \\Vffdvg= Γ dt f ||V/||div
JQ JO JTt
Second, from the Cauchy-Schwarz inequality we have
(volm_!rt)2 - (ji dvg)j <f^ \\Vf\\dvgt-J IIV/ΙΓ1^,,
where equality holds if and only if || V/|| is constant on Γ*. Note that this last fact
holds for /*, and the equality sign holds in the above inequality for /*. On the

4. CURVATURE AND SPECTRUM
281
other hand, if we set a(t) := volf^/volM, by the definition of the isoperimetric
function we have
volm_!rt > vol Μ · Λ(α(*)), νοΙ^Γ* = vol(5m, go)h0{a{t)).
Now, since h(a(t)) > h0{a(t)) by Theorem 1.6 (1), it follows that
I ||V/||di/e, ^(νοΙ,η-χΓΟ2/ / IIV/ΙΓ1^,
>(νο1Μ)2Λ2(α(ί))/ / IIV/ΙΓ1^,
>(volM)2(volm_1n/vol(5m,ff0))2/{a / ||V/'Ц"1 dvg;}
>a(vo\m^Tl)2/ f HVrir1^. = af ||V/*||di/9;.
Jr; Jr;
Integrating both sides of the above inequality with respect to t, by the coarea
formula we get
f \\Vffdug>a ί \\Vr\\2dug..
Jn Jn*
Third, again from the coarea formula it follows that
f f2dvg = fXt2dt f IIV/ΙΓ1^, =a [Xt2dt [ llVrir1^,.
Jn Jo Jvt Jo Jr;
= <*[ r
Jn·
2dvr.
Summing up, we get the desired inequality Ai(i?) = R(f) > R{f*) > Ai(i7*).
If equality holds, then we have h(a(t)) = ho(a(t)), which implies d(M) = π (see
Theorem 1.6 (1)), and therefore Μ is isometric to (5m, go)· Further we have
νο^-ιΓί = vol Μ · h(a(t)), and each Qt is isometric to a metric ball in (5m, go)
(see Remark 1.1). D
Remark 4.8. C. Faber and E. Krahn showed the following: Let Ω be a
(connected) domain in Rm and take a metric ball В С Rm with vol Ω = vol B. Then
we have Ai(i?) > \i(B), where equality holds if and only if Ω is a metric ball (see
[Кг]). This fact may be proved in the same way appealing to the isoperimetric
inequality for Rm. Theorem 4.7 is due to P. Berard and D. Meyer ([Be-Mey]). We
remark that similar estimates hold in the situation of Theorem 1.6 (2), (3). In the
case where p(u) > 0 it is also known that
(4.5) λι(Μ) > A!(£m)/m2 · /ic2(M),
where Bm denotes the unit ball in Rm (see [Ga-2]).
Exercise 3. Let (M, g) be a compact Riemannian manifold. Then for an
eigenfuction φ of Δ corresponding to the eigenvalue Ai(M), we have exactly two
nodal domains i?i, i?2 of φ. We assume that they have smooth boundaries.
(1) Show that X\(M) — Χι{Ω{) (г = 1, 2), where Χι{Ωι) denote the first
eigenvalues with respect to the Dirichlet eigenvalue problem.
(2) Suppose p(u) > m — 1, and deduce the Lichnerowicz-Obata theorem from
(1) and Theorem 4.7.

282
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Finally, we mention that we may also consider the eigenvalue problem with the
Neumann boundary condition for a compact Riemannian manifold Μ with smooth
boundary dM. Namely, the problem of determining eigenvalues A and eigenfuctions
/ of Δ under the boundary condition vf \ dM = 0, where ν denotes the unit normal
vector field to dM.
5. Heat Kernel and Spectral Geometry
5.1. As we stated in §3.1. the eigenvalue problem for the Laplacian is closely
related to the heat equation (3.5). Let Μ be a compact Riemannian manifold
(without boundary). We try to solve
Δτχ(ρ, t) + du(p, t)/dt = 0
under the initial condition u(p, 0) = f(p). For that purpose it is useful to consider
the heat kernel (or the fundamental solution of the heat equation) e(p, q, t).
Namely, e : Μ χ Μ χ (0, +οο) —» R is said to be a heat kernel if e is continuous,
of class C1 with respect to t, of class C2 with respect to the first variable ρ G M,
and satisfies
(5.1)
de
Δρβ(ρ, q, t) + — (p, q, t) = 0,
lime(p, q, t) = <5ς(ρ),
where 6q denotes the Dirac measure. The second equation of (5.1) means that for
any continuous function / on M, we have
lim / e(p, q, t)f(p)dvg(p) = f(q).
Physically, e(p, q, t) means the temperature of ρ at the time £, where at t = 0
the initial temperature distribution is concentrated at the point q with total
temperature 1. Once we have a heat kernel e(p, q, t), then if the initial temperature
distribution is given generally by f(q), then
e(p, 9, t)f(q)dvg(q)
(5.2) U(p, t) — I ^V^, y, i,;j y4; wgy
JM
is a solution of the heat equation with the initial condition /, namely, the
temperature of ρ at the time t. Therefore, it is fundamental to consider the existence and
uniqueness of the heat kernel. In the case of (ilm, go), although it is noncompact,
a heat kernel is given by
e0(p, q, t) := (4^)"m/2 exp(-||p - q\\2/4t).
Exercise 1. Verify that e0 is a heat kernel of (Rm, go)-
Now for a compact Riemannian manifold M, the heat kernel is closely related
to the eigenvalues and eigenfunctions of Δ.
Theorem 5.1. Let Μ be a compact Riemannian manifold. Then there exists
a unique heat kernel e(p, q, t). Furthermore,
(1) e : Μ χ Μ χ (0, +oo) —> R is of class C°° and satisfies e(p, q, t) =
e{q,p, t).

5. HEAT KERNEL AND SPECTRAL GEOMETRY
283
(2) Let 0 = Ai < A2 < Аз < · · · be the eigenvalues of the Laplacian of Μ
counted with multiplicities, and {φι}^χ the o.n.b. of L2(M) consisting of eigen-
functions φι of Δ corresponding to the eigenvalue A*. Then
00
(5.3) e(p, q, t) = ^ехр(-А^<^(р)<^(?),
3 = 1
where the right-hand side converges on Μ χ Μ χ [α, +οο) for any a > 0 with
respect to the Ck-topology (namely, the partial derivatives up to order к uniformly
converge) for к = 1, 2,
First suppose the existence of the heat kernel is guaranteed. We have the
Fourier expansion e(·, q, t) = £*/;(?, Ь)фг, ft{q. t) = /л/0г(р)е(р, q, t)dvg(p).
Since e is of class C1 with respect to t and Μ is compact, ft is difFerentiable with
respect to t, and we may change the order of the differentiation and integration.
Then from the Green theorem we get
^аГ1 = JM Φ^Ρ)9€{Ρ^ l) d^(P) = " /v о,(р)Дре(р. ς, t) dug(p)
= -A» / фг{р)е(р, q, t) dug(p) = -Jtft(q. t).
JM
Solving this equation under the initial condition ft(q, 0) = фг{а), we have ft(q, t) =
exp(—Ai£)0i(g). It follows that ^,^1ехР(~^)фг(р)фг^) has the sum e(p, q, i)
with respect to the Z/2-norm, when q, t are fixed. Then we may choose a subsequence
{ik}^=\ of MSi so tnat Σΐΐι ехР(-^)Фг(р)Фг(я) pointwise converge to e(p, ς, *)
for all (ς, t) and almost all p, as к —» +oo. Then from
CX)
(e(-, ς, */2), e(-, <?', i/2))0 = Х>хр(-Аг*)<Ш<М<г')
г=1
we see that Υ^^=ι^^(—\^)φι(α)φι(^) pointwise converges on Μ χ Μ χ Д, and
is equal to β(ς, ς', t). Namely, assuming the existence of the heat kernel, we may
prove (5.3), and consequently the uniqueness of the heat kernel. Obviously, e is
symmetric with respect to p, q. As for the existence of the heat kernel, we only
give a brief sketch (for details see, e.g., [B-Ga-Ma], [McK-Si], [Cha-2], [Su-2]).
Let d be the distance function on Μ. Following the case of Euclidean space,
first we consider (47r£)~m/2exp(—d2(p, q)/^t). However, for a general compact
Riemannian manifold, this cannot be expected to be a heat kernel. Instead, on
U€ := {(p, q) e Μ χ Μ; d(p, q) < б}, where e > 0 is less than the convexity radius
of Μ, we consider
Sk{p, Q, t)
= (4^)-m/2exp(-d2(p, q)/4t){u0(p, q)+tux(p, q) + · · · +tkuk{p, q)}
and determine Uj (j = 0, 1, 2, ...) so that they satisfy
(X + j^j Sk = (47r)-m/2ifc-m/2exp(-<f2(p, q)/4t)Apuk.
Then these Uj are determined successively with respect to a normal coordinate
system around ρ starting with u0(p, q) = {det(^j(exp_1 q))}~*· Next take a C°°

284
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
function η : Μ —» [0, 1] which is equal to 1 (resp., 0) on J7c/4 (resp., outside /7e/2),
and define Hk : Μ χ Μ χ Я+ -> Я by
Я^ ξξ ry5it (on t/c), Я^ = 0 (outside U€).
Then Hk is called a parametrix of the heat equation and possesses properties similar
to those of the heat kernel. The heat kernel itself is constructed from Hk by the
convolution product. In particular,
(5.4)
e(p, q. t) = (4^)-m/2exp(-d2(p, q)/4t) | 7/(d(p, <?))£>Ч(Р, q) + o(tk)
{ j=o
Setting ρ = α in (5.4), we get
e(p, p, 0 ~q (4^)"m/2{u0(p, p) + iui(p, p) + ■ · · + tkuk{p, p) + · · · },
which means that
к
(47rt)m/2e(p, p, 0 = 5>,-(p, p)^" +o(tfc).
i=o
Then it is an important fact that, with respect to a normal coordinate system
around p, Uj (p, p) may be expressed as a polynomial of the curvature tensor and
its successive covariant derivatives evaluated at p.
Corollary 5.2. Z(M, <7, £) (= Z(t)) := Σ^α exp(-A^) converges, and
(5.5)
Z(i) ~ (47ri)"m/2(ao+ai* + a2* + ···)» ^ = / щ(р, ρ)άν9(ρ).
(5.5) follows from (5.3) on setting ρ = q and integrating over Μ. Then
а* (г = 0, 1, 2, ...) are expressed as the integrations of polynomials of the
curvature tensor and its successive covariant derivatives over Μ. In principle, it is
possible to compute а*. For instance, u0(p, p) = 1, and therefore a0 = volM.
However, it is difficult to give concrete expressions of а* (г = 1, 2, ...). In fact,
(5.6)
a0 = vol(M, g), αχ = - / rdvg,
02 = ά /u(2||jR"2"2||Ric"2+5t2) d/y9'
were computed by M. Berger, Η .P. McKean, and I. M. Singer, and аз was computed
by the author. Note that from аз there appear covariant derivatives of the curvature
tensor. For further development of computations of a* see [Gi-3].
5.2. If we want to know the structure of a compact Riemannian manifold
Μ from its spectra, the а; (г = 0, 1, 2, ...) give the useful information. In fact,
from Corollary 5.2 and (5.6) we know the dimension of Μ and its volume from
Spec(M, g). We give a simple example in which аь а2 also appear.
Proposition 5.3. (1) Let (Mo, go) be a compact 2-dimensional Riemannian
manifold of constant curvature k, and (M, g) a compact Riemannian manifold.
Suppose that Spec(Mo, go) = Spec(M, g). Then Μ is also a compact 2-dimensional
Riemannian manifold of constant curvature k.

5. HEAT KERNEL AND SPECTRAL GEOMETRY
285
(2) Let (M, g) be a compact Riemannian manifold, and suppose Spec(M, g) =
Spec(52, go)· Then (M, g) is isometric to (52, go)·
Proof. (1) First note that dim Μ = 2, and therefore ||#||2 = 2||Ric||2 = τ2,
from which we also have a2(M, g) = -^ fM τ2 dvg. In particular, for (M0, go) we
get a2(M0, go) = j^k2vo\(Mo, go)· On the other hand, from the Cauchy-Schwarz
inequality it follows that
a2(M,g) > 1 (/м^^р) / vol(M, g) = ^{(ба^М, <г))2/а0(А/, д)}
= —{(ба^Мо, до))2/а0(Мо, д0)} = — · 4fc2vol(M0, go) = a2(M0, go).
By assumption we have the equality sign in the above Cauchy-Schwarz inequality,
and therefore τ = const., namely, (M, g) is also of constant curvature, which is
equal to к because αι(Μ, g) = ai(M0, go)·
(2) We may apply (1) setting к = 1, and it suffices to see that Μ is
simply connected. Otherwise, its universal cover Μ is isometric to (52, go). Then
we have vol(M, g) = ao(M, g) = ao(52, go) = vol(52, go) = volM, which is a
contradiction. D
Exercise 2. Suppose Spec(RP2, go) = Spec(M, g). Show that (M, g) is
isometric to {RP2, go)·
Remark 5.4. Flat 2-dimensional tori are determined by the spectra, and so
is the Euler characteristic of a 2-dimensional compact Riemannian manifold (see
Problem 11 for this chapter). On the other hand, for surfaces of higher genera it
is known that there exist two 2-dimensional isospectral Riemannain manifolds of
constant curvature —1 which are not isometric; this was proved by M. F. Vigneras
and P. Buser (see, e.g., [Vi], [Bus-3]). It is also known that for a compact
Riemannian manifold (M, g), if Spec(M, g) = Spec(5m, go), then (Л/, g) is isometric
to (5m, go) up to m < 6 (S. Tanno [Tan]).
Finally, we note that the heat kernel is also related to the curvature. For
instance, suppose the Ricci curvatures of a compact Riemannian manifold (M, g)
satisfy p(u) > m - 1, и е UM. Then
(5.7) vol(M, g) e(p, p, t) < ωτη e0{t).
where e0(t) denotes the heat kernel esm(p, p, t) of (5m. g0), and does not depend
on the choice of ρ because of the homogeneity (see, e.g.. [Be-1]).
Remark 5.5. As for the eigenvalue problem (II). we may also consider the heat
kernel with respect to the Dirichlet boundary condition. There have been various
attempts to construct the heat kernel for noncompact Riemannian manifolds (see,
e.g., [Cha-2], [Ch-G-Ta], [Su-2] and references there).
The heat equation is closely related to Brownian motion, and an approach
from probability theory has been found to be effective (see, e.g., [Mol]). On the
other hand, with respect to the wave equation d2u/dt2 + Au = 0, we may again
construct the fundamental solution. In this case, methods of symplectic geometry
are effective in connection with the geodesic flow. However, they are beyond the
scope of the present book (see, e.g., [Dui-Gu]).

286
VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
Problems for Chapter VI
1. Let Μ be a noncompact m (> 2)-dimensional Riemannian manifold, and set
S(M):=inf|{^||V/||^|m/{|w|/|^^} ;/eJ-0(M)|,
which is called the Sobolev constant of M. On the other hand, in this case we define
the isoperimetric constant 1{M) of Μ by
I(M) := inf{(volm_iai2)m/(vom)m_1; β is a bounded (not
necessarily connected) domain with smooth boundary}.
Show that T(M) = s(M).
2. Let А С Rm be a compact domain with smooth boundary. For the convex hull
ch(A) of A show that vol(ch(,4)) < ωτηά(Α)τη/2τη. Check the case where equality
holds.
3. Let (M, g) be a compact Riemannian manifold and с > 0 a positive
constant. Consider the Riemannian metric g = c2g homothetic to g. Show that
Δ* = с-2Д*, Xi(g) = c-2Xi(g) and e*(p, q, t) = cTme'(p, q, c~2t).
4. For the product Riemannian manifold Μ χ Ν of compact Riemannian manifolds
Μ and N, show that Spec(M χ Ν) = {A + μ; λ e Spec(M), μ e Spec(iV)}.
5. Let π : Μ —> Μ be a Riemannian covering, where Μ is compact. What
is the relation between Spec(M, g) and Spec(M, g)l In particular, determine
Spec(ilPm, go) of the real projective space of constant curvature 1.
6. Let (CPn, h0) be complex projective space with the Fubini-Study metric.
Determine its spectra.
7. Let i?i С i?2 be domains in (Hm, go) with smooth boundary. For the Dirichlet
eigenvalue problem (II), show that λ^(1?ι) > λ^(]?2) {к = 1, 2, ...).
8. Let Μ be an m-dimensional compact manifold, and go, g Riemannian metrics
on Μ. Suppose a~lgo < g < ago for some constant a > 1. Show that
a-(m+1)Afc(9o) < Afc(ff) < am+1\k(g0).
In particular, if Riemannian metrics {gn}^=i on Μ converge to a Riemannian
metric ρ with respect to the C°-topology, then Xk(gn) —> ^k(go) as η —» oo.
9. Let Μ be an m-dimensional complete Riemannian manifold and B€(p) a metric
ball of radius e centered at ρ in M. Let Xk{Be(p)) be the A;-th eigenvalue with respect
to the Dirichlet eigenvalue problem (II). Then show that e2Xk(B€(p)) converges to
the A;-th eigenvalue A^(l : 0) of the Dirichlet eigenvalue problem for a unit metric
ball of (Дт, go) as e -> 0.
10. Let (M0, go) be a compact 3-dimensional Riemannian manifold of constant
curvature k, and (M, g) a compact Riemannian manifold. Suppose Spec(Μ, g) =
Spec(Mo, go)· Then show that (M, g) is of constant curvature k. In particular,
show that Spec(M, g) = Spec(53, g0) implies that (M, g) is isometric to (53, g0).
11. (1) Let (Τ2, 0Γι), (Τ2, 0r2) be flat tori obtained from lattices Гь Г2 of Я2,

NOTES ON THE REFERENCES
287
respectively. Suppose Spec(T2, gri) = Spec(T2, gr2), and show that Гь Г2 are
congruent.
(2) Let Mi, M2 be compact 2-dimensional Riemannian manifolds. Show that
the Euler characteristics χ(Μι), χ(Μ2) are equal if Spec(Mi) = Spec(M2).
12. Let (Tm, gr) be a flat torus and Γ* the dual lattice of a lattice Γ in Rm.
(1) Show that the heat kernel e of (Tm, gr) is given by
е(тг(р), π(<7), t) = (4^)"m/2 £exp(-||p - q - </||2/4*),
yer
where π denotes the universal Riemannian covering π : Дт —» Tm.
(2) Show the following Poisson summation formula:
(47rt)-m/2vol(Tm,sr)X;exp(-||y||2/4t)= Σ βχρ(-4π2||^||2ί).
ует у*€Г*
13. (1) Let Φ : Μ —» Дп be an immersion, and let Μ carry the metric induced
from the canonical Riemannian metric of Rn. Denote by Η the mean curvature
vector field of Φ, and set Δ Ψ := (ΔΦ1, ... , ΔΦη). Show that Δ Ψ = ra#, where
m = dim Μ. In particular, Φ is a minimal immersion if and only if the coordinate
functions Фг are harmonic functions on M.
(2) Let Sn be the unit sphere in Дп+1, and let Φ : Μ —» Sn be an immersion.
Suppose Μ carries the metric induced from (5n, go). Show that Φ is a minimal
immersion if and only if ΔΦ = тФ.
Notes on the References
§1. As general references for the isoperimetric inequality in Euclidean space
we refer to [Os-1,2], [Bu-Z], [Ban], [B-10]. See [Schm], [Bu-Z] for the isoperimetric
inequality in spheres and hyperbolic spaces with constant curvature.
J. Cheeger introduced his isoperimetric constant hc in [Ch-3] to estimate the
first eigenvalue of the Laplacian from below (see also [Bus-1], [Cha-2]).
M. Gromov ([G-3]) gave the following isoperimetric inequality and applied it to
estimate the first eigenvalue of the Laplacian from below: Let (Μ, g) be a complete
m-dimensional Riemannian manifold whose Ricci curvatures satisfy p(u) > m — 1,
and Ω С М a, domain with smooth boundary. Let (5m, go) be the sphere with the
canonical Riemannian metric of constant curvature 1. Take a metric ball i?* С 5т
such that /3voli?* = voli?, where β = vol(M, p)/vol(5m, g0). Then volm_i<9i? >
βνολγη-ιδΩ*, with equality if and only if (M, i?, g) is isometric to (5m, i?*, go).
Gromov's idea stimulated works of S. Gallot, P. Berard, G. Besson, and D. Meyer
([Ga-2], [Be-Bess-Ga-1], [Be-Mey]). See also Theorem 4.7 in §4.
In §§1.2 and 1.3 we are greatly indebted to S. Gallot's excellent article [Ga-2].
§2. See [Cro-1] as for the isoembolic inequality for domains in a
Riemannian manifold, and applications. C. Croke also obtained a pinching version of the
isoembolic theorem ([Cro-3]).
Related to the isosystolic inequality, the reader may find many important
results and ideas in Gromov's paper [G-7]. However, even for the 2-dimensional case,
except for the torus, real projective plane and Klein bottle, we do not know
explicitly the extremal metric for the isosystolic inequality, although great progress

288 VI. ISOPERIMETRIC INEQUALITY AND SPECTRAL GEOMETRY
has been made (see [G-9], [Bav], [B-12]). We also recommend [Bu-Z] for many
geometric inequalities not treated here.
§3. We have also many nice textbooks on the eigenvalue problem of the Lapla-
cian ([B-Ga-Ma], [Cha-2], [Be-1], [Gi-1], [Su-2]). We owe a lot especially to [B-Ga-
Ma], [Cha-2], [Be-l]. As noted in Remark 3.4, M. Kac's article ([Kac]) attracted
many geometers to spectral geometry, which was initiated by M. Berger and by H.
P. McKean and I. M. Singer ([B-Ga-Ma], [McK-Si]).
We follow a variational approach to the eigenvalue problem of the Laplacian
(see [Au-3], [Be-1]). For some fundamental results on functional analysis (L2-space,
Sobolev space, Sobolev embedding theorem, Stone-Weierstrass theorem, etc.) we
refer to, e.g., [Re-Si]. As for the regularity properties of solutions of elliptic
differential operators, see [Au-3] and the references given there.
As for the spectra of (Sm, g0) and (Tm, gr), we follow [B-Ga-Ma], [Cha-2]. See
also [Cha-2] for the Dirichlet eigenvalue problem for disks in the simply connected
space forms. For more details on the Weyl asymptotic formula consult, e.g., [Be-
1]. See [Bess-1, 2], [Co-2], [Bando-Ur] for the multiplicities of eigenvalues of the
Laplacian. For the nodal domains, we refer to, e.g., [Che-2], [Don-Fe].
§4. A. Lichnerowicz and M. Obata first studied the relation between the
curvature and the spectrum (Theorem 4.1, [Lie], [Ob]). For a pinching version of the
Lichnerowicz-Obata theorem, we refer to [Cro-2]. Next came J. Cheeger's
fundamental Theorem 4.2 ([Ch-3]). Cheeger also gave an upper bound for λχ(Μ) in
terms of the diameter and the lower bound of the Ricci curvatures, which was
improved by S. Y. Cheng using Theorem 4.5 ([Che-1]; see also [Li-Y]). For instance,
it is possible to show that Afc(M) < m2k2n2/d(M)2 for a compact m-dimensional
Riemannian manifold Μ of nonnegative Ricci curvature. On the other hand, it is
known that X\(M) > π2/d(M)2 for a compact Riemannian manifold Μ of
nonnegative curvature (see [Zh-Ya], [Li-Y]).
§5. The heat kernel for a compact Riemannian manifold was first constructed
by S. Minakshisundaram ([Mi]). See also [McK-Si], [B-Ga-Ma], [Cha-2], [Gi-1], [Su-
2] for more details. The coefficients a» (i = 0, 1, 2 ...) of the Minakshisundaram-
Pleijel expansion formula
oo
Z(t) = Σβ~χ^ = (4nt)~^ ^ait1
i=0
([Mi-Pi]) give useful information on the relation between the curvature and the
spectrum (see [B-Ga-Ma], [McK-Si] for the computation of a0, ab a^, [Sa-1] for аз,
and [Gi-3] for further developments). For the relation between an upper bound of
Z(t) and a lower bound of eigenvalues, we refer to, e.g., [Don-Li]. See also, e.g.,
[Cha-2], [Dui-Gu] for the zeta function approach, Z(s) = Σ°1χ A~s. As for the
relation between the geodesic flow and the Laplacian, see, e.g., [Co-1]. [Dui-Gu].
Consult [Be-B] for references on spectral geometry up to 1982.
Finally, I would like to recommend that the reader look at M. Berger's detailed
survey [B-13] on Riemannian geometry, through which we may grasp main ideas,
motivations and trends of recent Riemannian geometry.

Appendices
1. Irreducible Decomposition of the Curvature Tensor
Let (Vm, g) be an m-dimensional Euclidean vector space and G = 0(m) the
orthogonal group. Recall that the curvature tensor R of the Levi-Civita connection
on a Riemannian manifold Μ satisfies the following algebraic conditions at each
tangent space V = TVM to M:
(i) R(x, y, z, w) = -R{y, x, z, w) = -R(x, y, w, z),
(ii) R(x, y, z, w) + R(y, z, x, w) + R(z, x, y, w) = 0,
(iii) R(x, y, z, w) = R(z, w, x, y), x, y, z,w e V.
We denote by С the subspace of the tensor space T$(V) = V* <g> V* ® V* <g> V*
consisting of (0,4)-tensors R which satisfy the above conditions (i), (ii), (iii). Now
recall that an inner product g on V is extended to the inner product g on the tensor
product TS0(F), and φ <E G acts on T%(V) by
(ipT)(xu ...,*,):= T^Ori), ... , φ-ι(χ.)), Τ e T°(V).
Furthermore, a permutation σ of {1, ... , s} acts on T®(V) by
(<jT)(xu ... , xs) := Τ(χσ(1), ... , χσ(β)), Τ e T?(V).
Now let Pk(C) be the vector space of G-invariant homogeneous real polynomials
of degree A; on С For instance, we fix a permutation σ of {1. 2 4A:} and take
an o.n.b. {ei}7^! of V. For R £ С we define
Pa(R) := Σ σ(ϋ®·-·® Д)^. etl el2,, ehk).
Then Ρσ defines an element of Pk(C). and in fact is obtained from the components
(Д ® · · · ® R)ii...uk by first contracting indices гащ and ζσ(2), and continuing to
contract 2k times up to contracting indices ia(4k-i) and г0[щ. We may easily check
that the above definition does not depend on the choice of o.n.b., and that Ρσ is
G-invariant.
Example 1. We define ρ : С —> S2(V) := {α : V xV —> R; α is a symmetric
bilinear map} by p(R)(x, y) := Σϋι ^(еь я> 2Л ег)> which corresponds to the
Ricci tensor. Next we define t(R) := tracep(R) = ^ · R(e{, ej, ej, e^). Then τ
defines an element of P\(C) and corresponds to the scalar curvature. Furthermore,
IIPlI2 = 9ij9klPikPji, \\Щ\2 = gijgkl9rnngpqRikmPRjinq are elements of P2{C).
Now according to the Weyl theorem in representation theory, Pk(C) are
generated by Pa's, where σ are permutations of {1, ... , 4Α;} ([Wey]). However, because
of properties (i), (ii), (iii), they are not linearly independent in general.
289

290
APPENDICES
Example 2. PX(C) = <т)я, P2(C) = (r2, ||p||2, \\R\\2)r. If dim V = 2, then
ЦДЦ2 = 2||p||2 = τ2, namely, dimP2(C) = 1. If dim V = 3, then we have ||Я||2 =
41И12-т2 (see Chapter II, Remark 3.7), and therefore dim P2{C) = 2. If dim V > 4,
then dim P2(C) =3.
In the following, we consider the irreducible decomposition of С with respect
to the orthogonal action of G. First, we give R0 £ С by
R0(x, y, z, w) := g(y, z)g(x, w) - g(x, z)g(y, w).
Note that p(R0) = (m - \)g. Next we set W := p_1(0)· Then clearly W С т_1(0);
let 7?, be the orthogonal complement of W in r-1(0). Then we have the following
(for the proof see, e.g., [Bes-2], p. 47, [B-Ga-Ma]).
Theorem 1.1. Suppose dim У > 4. Then С may be expressed as an orthogonal
direct sum C = W(BTl(BR-R0 of irreducible G-invariant subspaces. Further, ΊΖ
is given by 1Z = {k £ С; к £ S2(V) with trace к = 0}, where к is defined by
fc(x, y, z, w) := p(j/, ^)fc(x, u;) + p(x, и;)к(з/, ζ) - g(y, w)k(x, z) - g(x, z)k(y, w)
forkeS2(V).
Remark 1.2. If dimV = 2, then С = RR0. If dimV = 3, then С = 1l®RR0.
Remark 1.3. For R £ С its VV-component W(R) is called the Weyl conformal
curvature tensor corresponding to R, and written as
(1.1) W{= W(R)) = R- p{R)/(m - 2) + r(R)/(m - l)(m - 2) · R0.
In terms of the components we have
Wijki =Rijki 7>{9jkpu + guPjk -Qjipik - 9ikPji)
(1.2) mT Z
+ {m-\){m-2){gjkgil~gik9jl)'
Remark 1.4. A Riemannian manifold (M, g) is said to be conformally flat, if
for any point ρ £ Μ there exist an open neighborhood U and / £ F(U) such that
the metric e2* g is a flat metric on U. Then its Weyl conformal curvature tensor W
vanishes everywhere. The converse also holds if dim Μ > 4. It is also known that
a compact simply connected conformally flat Riemannian manifold is conformal to
the sphere with the canonical Riemannian metric of constant curvature 1 (for more
details see, e.g., [Bes-2], [Kui]).
Remark 1.5. Let V2n be an oriented Euclidean vector space of dimension
2n and {ei}2™! a positively oriented o.n.b. of V. For R £ С we set Rijki =
R(ei, ej, е^, e/), and define a 2-form Ω^ by Ω^ := ^R{jkiekAel, where {ег} denotes
the dual basis of {e^}. Then
2^Sgn σΩσ(χ)σ(2) Λί1σ(3)σ(4) Λ ·· · ΛΩσ(2η-1)σ(2η),
σ
where the sum is taken over all permutations σ of {1, 2, ... , 2n}, defines a 2n-form
on V and may be written in the form C(R)el Λ · · · Λ e2n. A computation gives
^(^) = 2^ У ^Sgna SgnT Ra(\)a(2)r(l)r(2) ' ' ' Rg(2n-l)a(2n)r(2n-l)r(2n)^
σ, τ
and C(R) is an element of Pn(C).

2. HOMOGENEOUS SPACES
291
Now let M2n be a compact oriented Riemannian manifold of dimension 2n.
Prom the curvature tensor R of M, we have a smooth function C(R) on Μ by the
above procedure on each tangent space TPM. Then S. S. Chern ([Chern]) showed
that for the Euler characterictic χ(Μ2η) (:= ΣΐΙοί-1)* dim#fc(M, Д)) of M2n
the following generalized Gauss-Bonnet theorem holds:
(1.3) χ(Μ2η) = о(~1)П, / C{R)dM.
In particular, we have
(1.4) x(M2) = — I rdM (Gauss-Bonnet formula),
47r J μ
(1.5) X(M4) = ^-2j^r2- 4||p||2 + ||Д||2} dM.
See [Sa-1] for χ(Μ6).
2. Homogeneous Spaces
Suppose a closed subgroup G of the isometry group of a Riemannian manifold
(M, g) acts transitively on M. Then the isotropy group Η := {h £ G; /i(p) = p}
at ρ £ Μ is compact, and Μ is diffeomorphic to the quotient space G/H. Μ is
called a homogeneous space. Let g be the Lie algebra of G. In the following, for
X e g the Killing vector field obtained from the one parameter transformation
group exptX e G generated by X is again denoted by X (see Chapter III, §6.1).
However, note that [X, Y]m = ~[X, Y]q (Chapter I, §2, Exercise 8) under the
above identification. Then the Lie algebra i) of Η is given by {X £ g; Xp =
0}. Further, Ad0(#) С GL(g) is compact, and there exists an Ad0(#)-invariant
subspace m complementary to i) in g. Namely, we have a vector space direct sum
decomposition g = m + i).
Now let π : G —» Μ = G/H be the canonical projection. Then Dn(e) \ m
gives an identification between m and TPM. Hence we may introduce an inner
product ( , ) on m, which is isometric to gp under the above identification and
Adm(#)-invariant. Here we set Adm(#) := {Ad0/i : m —> m; h e H}. Conversely,
let G be a Lie group, and Η a compact subgroup of G that does not contain a
nontrivial normal subgroup of G. Suppose an Ad0(#)-invariant subspace m of g,
which is complementary to i), is given. Then for an Adm^)-invariant inner product
( , ) on m there exists a unique Riemannian metric g on Μ = G/H such that G
acts on Μ as an isometry group via left translations, and ( , ) is induced from
g as in the above manner. Recall that symmetric spaces give examples of such
homogeneous spaces (Chapter IV, §6). Here we are concerned with the curvatures
of general homogeneous spaces. We omit the proofs, since they are straightforward
computations using Chapter II, (1.13), and Chapter III, (6.2), (6.3) (see, e.g., [Bes-
2], [Bus-Ka]).
Proposition 2.1. Identify Xp £ TPM with the unique Killing vector X £ m
that takes the value Xp at p. Then, for Χ, Υ £ m,
(2.i) (VxK)(p) = -\ix, Y\m + u(x, y),

292
APPENDICES
where [X, Y]m denotes the m-component of [X, Y]g and U : m χ m —» m is given
by
(2.2) ([/(X, У), Ζ) = ±{<[Z, X]m, Y) + <[Z, У]т, X)}.
Proposition 2.2. Lei X, У £ m. Tften ί/ie curvature tensor R satisfies
(2.3)
(Д(Л·, У)У, Χ) = ||C/(X, У)||2 - (U(X, X), U(Y, Υ)) - fU[X, Ш|2
- ί([Χ, [X, Y]e]m, Υ) - i([F, [У, Х]в]т, X).
Corollary 2.3. Let G be a Lie group. Let g be a left-invariant Riemannian
metric on G, which means that left translations are isometries. Let ( , ) be the inner
product on the Lie algebra g of G induced from g, and define (ad X)* : g —» g by
((ad Χ)Ύ,Ζ):= (Υ, (ad X)Z). Then
(2.4) (VxY)e = -±{[X, Y]g + (adX)*Y + (adY)*X},
(2.5)
(R(X, Y)Y, X) = i||(adX)*Y+ (adY)*X||2 - ((adX)'X, (adY)*Y)
- !ii[^ γ]β\\2 - №> ix' yU'y) - И(у' ty'χ^ χ)-
Now a homogeneous space (M, g) is said to be naturally reductive if U = 0.
In this case geodesies through ρ may be given by t »—► exptX · p(X £ m), and for
Χ, Υ £ m we get
(2.6) <ВД У)У, X) = i||[X, Г]т||2 + ([[X, Г]ь, X]m, У).
Further, if there exists an Ad0(G)-invariant inner product Q on g such that m
is the orthogonal complement of f) with respect to Q and gp = Q \ m, then (M, p)
is called a normal homogeneous space. In this case, it follows that
(2.7) (R(X, Y)Y, X) = ±\\[X, Y}mf + \\{X, У]„||2
and the sectional curvatures are everywhere nonnegative.
Next recall the injectivity radius estimate for compact simply connected even-
dimensional Riemannian manifolds of positive sectional curvature (Chapter V,
Corollary 1.8). We give examples showing that the same estimate does not hold for
the odd-dimensional case. Let (CPn, ho) be the complex projective space with the
Fubini-Study metric whose sectional curvatures satisfy 1 < Κσ < 4 (see Chapter II,
§6). Then for the diameter and injectivity radius we have d(CPn) = i(CPn) = π/2.
Distance spheres St{p) := {q £ CPn; d(p, q) = t} are diffeomorphic to the sphere
52n_1 for 0 < t < π/2, and coincide with the cut locus Cp = CPn~l for t = π/2
(see Problem 5 for Chapter III). Now we compute the sectional curvatures of
St(p) (0 < t < π/2) with respect to the metric gt induced from /i0. For that
purpose, take a geodesic ηη emanating from ρ with the initial direction и £ UpCPn,
and consider the second fundamental form S of St (p) with respect to the unit
normal 7n(0· Let v(t) be the parallel translation of ν £ UpCPn along ηη. Then from
the 5*(p)-Jacobi field equation (see Problem 5 for Chapter III), we get
S(Vi(t), V2(t)) = -COtt (l>i, 172), Vi, V2 -L JU,
S(vi(t), v2(t)) = 0, vi±Ju, v2 = Ju,
S(v(t), v(t)) = -2cot2£, ν = Ju.

2. HOMOGENEOUS SPACES
293
Then, by the Gauss formula, the sectional curvatures satisfy
{δ :=) - 1 + 1/ sin21 < Κσ < 3 + 1/ sin21 (:= Δ).
On the other hand, a map assigning 7η(π/2) G CPn~l to q = 7n(£) £ 5*(p) is
a Riemannian submersion from St(p) onto CPn~l, and fibers over 7η(π/2) are
closed geodesies с of length π8ΐη2£ in St(p). It follows that if smt < yfbfb, then
«/4 = (l-sin2*)/(l+3sin2*) < 1/9 and L(c) < 2тг/\/Д namely, г(5*(р)) < π/y/Z.
Therefore, the injectivity radius estimate (Chapter V, Corollary 1.8) does not hold
for the odd-dimensional case, at least for δ/Δ < 1/9. We note that if we write
CPn = U{n + l)/(/7(l) x /7(n)), then G := /7(1) χ U(n) acts transitively on
St(p) as an isometry group, and (St(p), 9t) is isometric to a normal homogeneous
space obtained from the Killing form of U(n + 1). The (St, 9t) are called Berger
spheres, since they were discovered by M. Berger ([B-3]) when he classified all simply
connected normal homogeneous spaces of positive sectional curvature. The above
form of Berger's spheres is due to A. Weinstein ([We-3]).
N. R. Wallach and S. Alott ([Wa-Al]) obtained the following 7-dimensional
examples when they tried to classify all simply connected homogeneous spaces of
positive sectional curvature. Let p, q be relatively prime positive integers, and
consider in 5/7(3)
T(p, q)
\ехр(2пу/=1рв) О О
0 exp(2ny/^lqe) 0
0 0 ехр{-2пу/-Цр + д)в\
0eR
which is a family of subgroups isomorphic to 51. We consider the homogeneous
spaces M(p, q) = 5/7(3)/T(p, q), which are simply connected. We get
H4(M(p, q); Z) = Zr, r = p2 + pq + q2, for the cohomology group. Namely,
there are infinitely maniy different topological types among M(p, q). It is possible
to define a family of Ad(T(p, ^))-invariant inner products ( , )t ( — 1 < t < 0) on
the Lie algebra of 5/7(3) so that the homogeneous Riemannian metrics on M(p, q)
obtained from ( , )t are of positive curvatrure. Η. Μ. Huang ([Hua]) computed the
sectional curvatures of homogeneous Riemannian metrics on Μ (г, г + 1) for t = — |,
and showed that the range of sectional curvatures of Μ (г, г + 1) converges to the
range of sectional curvatures of M(l, 1), which is given by 2/37 < Κσ < 29/8,
as г —» +oo. On the other hand, take a maximal torus T2 of 5/7(3) containing
T(p, q) and consider a fiber bundle M(p, q) —> 5/7(3)/T2. Then each fiber is a
closed geodesic of M(p, q), and we may compute its length:
i(M(p, q)) < n^3/{8(p^+pq + q^}.
Namely, we get a family of simply connected compact homogeneous Riemannian
manifolds of positive curvature such that the sectional curvatures are in a fixed
range, but the infimum of their injectivity radii is equal to zero (see also [BB-
l],[Esc-l],[Kre-St], [Wa]).
We give one more example. Let G be a Lie group and g its Lie algebra. We set
g1 = [g, g], gfc = [g, gfc_1] (k = 2, ...). We call g and G nilpotent if there exists а к
such that gfc = {0}. In this case, it is possible to construct inner products ( , )q (0 <
q < 1) using the nilpotent structure such that ||χ||ς —> 0(q —> 0, χ £ g) but
\\[x, y]q\\ < C|kllq||2/||g (С а positive constant independent of q). By Corollary 2.3,

294
APPENDICES
the sectional curvatures of left invariant Riemannian metrics gq on G determined
from ( , )q remain in a fixed range as q —» 0. If we consider a compact quotient space
Μ = G/t by a discrete group Γ of G, then the diameter of (G/Γ, gq) converges
to 0 as q —» 0, while maxa \Κσ\ remains bounded. Compact manifolds that admit
such a family of metrics are called almost flat manifolds (see [G-2], [Bus-Ka], and
Appendix 6).
3. Injectivity Radius Estimate and Closed Geodesies
We first give a proof of Theorem 2.3 from Chapter V, namely,
Theorem 3.1. Let A/ be a compact simply connected Riemannian manifold
of odd dimension ra(> 3). Suppose the sectional curvatures Κσ of Μ satisfy \ <
Κσ < 1· Then the injectivity radius satisfies i(M) > π.
By Chapter III, Corollary 4.14, and Chapter IV, Corollary 2.8 (3), it suffices
to show that the length of every (nontrivial) closed geodesic in Μ is greater than
or equal to 2π. We derive a contradiction assuming that there exists a closed
geodesic σι with 0 < L(a\) < 2π. Note that for the energy integral we have
0 < Ε (σι) < 2π2. Since Μ is simply connected, there exists a homotopy consisting
of closed curves Hs (0 < s < 1) joining a point curve σο to σ\. Choose a positive
number a so that a > maxo<s<i E(HS). Let Б С Μ χ Μ be the diagonal set and
consider the space Св := {с G [0, 1] —> Μ; (c(0), c(l)) G B} of piecewise C°° closed
curves in M. Recall that we may approximate С в by a finite-dimensional manifold
(Chapter III, Remark 3.4). Namely, for the above a take a subdivision Δ of the unit
interval / := [0, 1] so that Λ := Οβ~(Δ) has the structure of a finite-dimensional
manifold such that the energy integral Ε : Λ —» R is a proper C°° function, and
Л is a deformation retract of C%~ := {c G Св', Ε (С) < α2/2}. Note that σ G Л is
a critical point of Ε \ Λ if and only if either σ is a trivial point curve or a closed
geodesic with 0 < £(σ) < α2/2. Furthermore, the index and the nullity of the
Hessian Ό2Ε(σ) \ ΤσΛ are equal to the index and the nullity of D2E(a) \ ТаСв,
respectively. In the following we work on (Л, Е). Л carries a Riemannian metric
as an open subset of the direct product of copies of Μ. Let VE be the gradient
vector of E. We denote by <pt the flow generated by — VE. Then φί is defined for
all t > 0. We have for с G Л
^E(^(c)) = -||VE(^(c))||2<0,
and ψι leaves invariant critical points of E.
Now let Η := {Η : / —> Л; Η is a continuous curve in Л joining σ0 to σι} be
the set of homotopies between σ0 and σ\ consisting of closed curves in Μ. Then Η
is nonempty and is a ф-family, which means that φ^Ή с Ή for t > 0. We define the
critical value к of the ^-family Η as к := mfnen maxo<s<i E(HS). Then the set К
of critical points of Ε with Ε-value к is compact. Setting Лк~ := {с G Л; Е(с) <
к}, we get the following lemma of Lyusternik and Schnirelmann ([Ly-Schn]).
Lemma 3.2. For any open neighborhood W of К there exists an Η eH such
that Η(I) С Лк~ U W. In particular, К ф ф.
PROOF. Suppose to the contrary that there is an open neighborhood W of К
such that for any Η G Η there exists an 5 G / with Hs & Λκ~ U W. Then by
the definition of к we may choose H^ G Η and sn G / (n = 1, 2, ...) so that

3. INJECTIVITY RADIUS ESTIMATE AND CLOSED GEODESICS 295
maxs£l{E(H{sn))} < к + еп (en j 0) and cn := φ^{Η{£) satisfy E{cn) > к, cn £
W. Then, noting that
к < E(cn) = E(H£>) + J ^ jtE{Vt{H^))dt
<к + еп- / tn||VE(¥>t(ffW))||2di,
JO
we get
JO
Now we may choose 0 < tn < у/ё^ so that bn := ^tn(#i„ ) satisfies ||ν£7(6η)||2 <
y^ and к < E(bn) < к + en. Since the bn are closed curves in a compact Μ
consisting of broken geodesies the number of whose vertices are constant, we may
choose a convergent subsequence bnic —» b G A. It follows that b G K. Then
сП(с G W for sufficiently large n^, and we get a contradiction. К ф ф is clear, since
we may take W = φ when Κ = φ. D
Now as the first step of the proof of the theorem, we show that κ = 2π2.
Suppose first that к < 2π2. Then there exist an Η £ Η and δ > 0 such that
E(HS) < 2(π - (5)2, ее/. Note that L(HS) < 2(π - (5). Let τΜ : TM -> Μ
be the tangent bundle of Μ and consider a map Φ : TM —» Μ χ Μ defined
by Ф(и) := (tm(u), expTM(n)U). Then the first conjugate value in Μ is greater
than or equal to π because Κσ < 1, and therefore Φ is regular on an open set
U0 := {v e TM; \\v\\ < π} of TM. Now we set φ) := #s(0) and show that we
may lift the Hs (s G /) to closed curves Hs in TC^M continuously on s so that
Hs(0) = oc(s) G TC(S)M and Φ(#δ(£)) = (c(s), Hs(t)). For that purpose we set
J76 := {uGTM; ||v|| <π-<5},
s0 := supfs' G /; we may lift Hs (0 < s < s') to Hs cU6n Tc{s)M
so that the above conditions are satisfied}
and show that So = 1· Since σο is a point curve, clearly sq > 0· Suppose so < 1,
and note that we have the lift Hs of Hs for 0 < s < so· Next we want to lift
t »-> Hso(t) to a closed curve in TC(S0)M. Set, as before,
t0 := sup{^ G /; we may lift HSo \ [0, i'] to HSo \ [0, i'] С Tc{so)M so that
the above conditions are satisfied and Hs \ [0, t'} (0 < s < s0)
uniformly converge to HSo \ [0, t'] as s Τ s0}.
Since Φ is a diffeomorphism on {v G ΓΜ; ||v|| < б} for sufficiently small e > 0, we
have £o > 0. If to < 1 we may get a contradiction similarly using the fact that Φ is
a local diffeomorphism, and therefore to = 1. It follows that for 0 < s < so we may
lift Hs continuously on s to closed curves Hs in Us. We show that HS0(I) С Us. In
fact, otherwise there exist the first point Hso(ti) and the last point HSo(t2) (t\ < t2)
which satisfy ||i/so(£)|| = π — δ. Then from the Gauss lemma and the argument in

296
APPENDICES
Chapter II, Lemma 2.7, we get
L(HS0)> [°\\(HS0(t))r\\dt+ f ||A50(i))r||di
J0 Jti
>\\Hio{t1)\\ + \\Hao(t2)\\=2n-26,
where for ξ £ TuTpM (u φ 0), ξΓ denotes the radial component of ξ. This
contradiction implies that HSo(I) С Us- Then, as before, using the fact that Φ is a local
diffeomorphism, we may lift Hs to Hs С Us with the above properties beyond so·
Namely, we get so = 1 and may lift #i to a closed curve #i in Us - On the other
hand, since H\ = σι is a closed geodesic, its lift in TC^M emanating from oc^ is
a line segment and cannot be closed. This contradiction implies that κ > 2π2.
Next, to see that κ < 2π2, we need some preliminaries. Let K' be the set of
critical points of Ε with 22-value к such that their indices are less than or equal to
1. Then we have the following modified Lyusternik-Schnirelmann lemma.
Lemma 3.3. For any open neighborhood W of K' there exists an Η €Η with
H(I) С Лк~ U W. In particular, Κ' φ φ.
PROOF. 1°. We first assert the following: for any σ e К \ К' we may take
an arbitrary small arcwise-connected closed neighborhood V(a) of σ in AK := {cG
Л; E(c) < к}. In fact, we may take V(a) so that for Co, C\ £ V(a) (со Ф C\) there
exists a continuous curve ω : I —» V(a) joining Co to c\ such that ω((0, 1)) С Лк~.
To see this, recall that σ is a critical point of Ε with Ε-value к whose index k is
greater than or equal to 2. Let T~, X^, T+ be the direct sum of eigenspaces of
the Hessian D2E(a) with negative, 0, and positive eigenvalues, respectively. Then
we have the orthogonal decomposition ΤσΛ = T~ 0 Xj Θ Τ+. Next take a normal
coordinate system (exp"1, ΙΑ(σ)) around σ, and for e £ U(a) decompose the normal
coordinate ξ = £(e) = exp"1 e of e into
ξ = Γ + ξ° + t e Τ- Θ Γ° Θ Г+ = ΤσΛ.
Denoting by — X{ (1 < г < k = тавсг (> 2)) (resp., Xj (k + 1 < j < /)) the negative
(resp., positive) eigenvalues of D2E(a), we have the Taylor expansion
(*) Ε(βχρσοξ)=κ-±ΣΧ^η2 + \ Ε W)2 + 0(||£||2),
since σ is a critical point with E-value к. Now we consider a map
*:(c,e,t)»(%f,e,t)
and note that Dct(o) has rank dimvl, since (Όα(ο)η , 7/ ) = ϋ2Ε(σ)(η , η' ). It
follows that α is a diffeomorphism when restricted to a small open neighborhood
of o, and if we set cT^O, ξ°, ξ+) := (Λ(ξ°, ξ+), ξ°, £+) we get Λ(0, 0) = 0. Now
we define
Ψ(η~, e°, e+) := exMtT + Λ(ξ°, £+), ξ°, £+)
and get a chart (^_1, ^(σ)) around σ, where we take ΙΑ(σ) smaller if necessary.
Then the following hold for ψ:
= Λί(Λ(ξ», £+), ξ°, ξ+) = 0 (!<<<*).

3. INJECTIVITY RADIUS ESTIMATE AND CLOSED GEODESICS
297
д2(Еоф)^_ c0
(2) For sufficiently small positive a, 6, —^—^-(η , ξ°, ξ+) I is neg-
ative definite on Ща) := {ψ(η~, £°, £+); \\η~\\ < α, ||£° + ξ+ΐ:<"&}·
(3) There exists an e = ε(σ) > 0 such that for sufficiently small a = α(σ) and
b = 6(σ) > 0, we have Pa,6 := {^(r/", £°, £+); ||τΓ|| = α, ||£° + £+|| < 6} С
^(а)ПЛ"-е.
(3) may be verified by considering Ε о ^(ry~, ξ°, ξ+) in (*) and noting that
/i(0, 0) = 0. Since к = тав(т is greater than or equal to 2, Vaj> is arcwise connected.
Now let ^(*Г, ζ°, £+) £ Λκ~ Π W(a) be any element with r/" ^ 0. Then note that
the curve xp(s) := ψ(3η~, ξ°, ξ+), 1 < s < α/\\η~\\, remains in AK~(lW(a). In fact,
for E(s) := E(7p{s)) we get from (2) E"{s) < 0, 0 < s < α/ЦтГЦ, and E'(s) < 0
holds for s > 0 because we have £"(0) = 0 by (1).
Now we take a, b > 0 so that (2), (3) hold, and show that V(a) := W(a) Π Л*
is as desired. Suppose c0, ci G ν(σ). We may assume that ry~(co), f?~(ci) ^ 0.
Then keeping ξ°, ξ+ fixed we move ry~(co) (resp., r/~(ci)) along a radial line joining
η~ = 0 to ry~(co) (resp., r/~(ci)) until it reaches \\η~\\ = a. Then we get a curve
from Co (resp., C\) to a point in Ρα.6· Next, we take a curve joining these points in
Va,b which is contained in AK~. Joining the above curves together, we get a curve
ω from c0 to c\ in V(a) such that u;((0, 1)) С Лк~.
2°. Next, for σ' G /i' take an arc wise-connected closed neighborhood W(a')
of σ' in Л that is contained in W, and set ν(σ') := W(a') Π Лк. With ν(σ), σ G
Κ\Κ', constructed in 1°, we get an open covering {ν(σ)2; σ G K\K', ν(σ')1; σ' G
/i'} of a compact set К of Л*, where ν(σ)\ etc. denotes the interior. We may
choose finitely many Vi, ... , Vn from these open sets so that they cover K. Further,
we may choose closed neighborhoods Vi, ... , Vn so that V/ D V3, (j = 1, ... , n)
and Vj has the same property as Vj. Since V3; Π Κ (j = 1, ... , π) are compact,
we may choose open neighborhoods Uj of Vj Π Κ in Л so that U3 Π Л* С Vj, and
Vi П Vj ^ 0 whenever ZYi Π ZYj ^ 0.
Now we apply Lemma 3.2 to W := \J™=lUi (D K) and get an H' G Η such
that #'(/) С AK~ \JW. Then #'(/) \ AK~ may be divided into finitely many pieces
S with the following property: S may be covered by a chain {UiJ}J=1 such that
Z4, Π Wij+1 ^ 0(j = 1, ... , / - 1). Then from Vis Π Vij+1 φ φ we may take a
sequence {xq, ... , ж/} of points in Л so that
x0 G Vtl П 5, Xj G V^ П Vij+1 (j = 1, ...,/- 1), z/ G Viz П 5.
If Vij is a neighborhood corresponding to σ e К \ Kf, then by the argument of
step 1° we may join x3-\ to x3 by a curve in Л*~ П V^ (possibly except £j_i and
Xj). If V^ corresponds to σ' e K', we join xj_i and Xj by a curve in VV^ С W.
Then, leaving the part of H'(I) which is contained in AK~ fixed, we get by the
above procedure a curve H" G Η that joins σο to σ\ and is contained in AK~ U W
except for possibly a finite number of points on H". However, these points are
either regular points of Ε or critical points in К \ K' of Ε with index > 2, by
construction. Then we may deform H" by applying φί (resp., the deformation
described in 1°) to the above regular points (resp., critical points in K\ K') to get
a desired Η eH contained in AK~ U W. D
Lemma 3.4. The critical value к of Η is equal to 2π2. Furthermore, there
exist α σ G Κ', and a sequence {сд:}^=1 in AK~ converging to σ such that closed

298
APPENDICES
curves Cfc in Μ may be lifted to closed curves Ck in Uo := {v G TM\ ||v|| < π} as
in the proof of κ> 2π2.
Proof. We derive a contradiction on assuming that κ > 2π2. Then the critical
point σ of Ε in Λ is a closed geodesic in Μ with L(a) > 2π. Since Κσ > \ and
m = dim Μ > 3, the index of σ considered in ίσ(ο),σ(ΐ)(Μ) is greater than or
equal to 2 by Chapter III, Proposition 3.6. Therefore, its index in Св (and A)
is again greater than or equal to 2. Namely, we get Κ' = φ, which contradicts
Lemma 3.3. Thus we get κ = 2π2 and Κ' φ φ. Next, for any open neighborhood
W of K', by Lemma 3.3 there exists апЯЕН with Η (I) С Лк~ U W. Now
we set so ·= sup{s' G I; #([0, s']) С AK~ \ W'}. Then we have s0 > 0, since
we may assume that σ0 $. W and #So G Л*~ П cW. On the other hand, since
L(HS) < 2π (0 < s < so) and #0 = σ0 is a point curve, we may lift HSo to a closed
curve #So in Uo Π Tc(So)M as in the proof of κ > 2π2. Now take Wns so that they
converge to K' and consider the corresponding #So's. Since K' is compact, we may
choose a subsequence {c^} consisting of the above HSo so that {с*.} converges to a
ae/f. □
Now we turn to the proof of the theorem. If we show that σ (G K') in Lemma
3.4 is of index greater than or equal to 2, then we get a final contradiction and the
proof is complete. So let σ(£ο) be the first conjugate point to σ(0) along σ : / —> Μ.
Then we have t0 > \, because Κσ < 1 and by Chapter III, Proposition 3.6. On
the other hand, since there exists a sequence {c^} of Lemma 3.4 converging to σ,
it follows that to = \, and σ(^) is a conjugate point to σ(0) along σ \ [0, \]. By
the same argument we see that σ(1) (= σ(0)) is a conjugate point to σ(^) along
σ | [^, 1]. Note that ind^cr = 1 because of the index theorem and the fact that
Now let Y(t) be a Jacobi field along σ \ [0, |] with У(0) = У(^) = 0, ||УУ(0)||
= 1. We consider the sphere S™ of constant curvature 1 and take a linear isometry
/ : Τσ(0)Μ -> Тр5^, pGSf1. We define a vector field У along a geodesic σ(ί) :=
expptl(a(0)) (0 < ί < |) in 5^ by y(i) := Ρ(σ)? о / о Р(а)^(У(«)). Then for a
space С := {с : [0, ^] —► Μ; c(0) = σ(0), c(^) = σ(^)} of curves in M, we have
from the assumption that Κσ < 1
0 = D2E(a | [0. i])(y, Y)
= Γ {l|Vr(i)||2 - (2π)2Κ(σ(ί), y(i))||y(i)||2} Λ
7o
> [2{\\VY(t)\\2-(2n)2\\Y(t)\\2}dt = D2E(a\[0, ί])(Ϋ,Ϋ)>0.
Jo
Next, note that, on 5]71, D2E(a \ [0, ^]) is positive semidefinite and У belongs to
the null space of the above Hessian. Namely, У is a Jacobi field, and we may write
Y(t) = sin2ntV(t), where V is a parallel vector field along σ \ [0, ^]. Therefore, У
itself also may be written as У (t) = sin27r£ F(i) with a parallel vector field V along
σ | [0, ^]. Note that V, V are perpendicular to σ, σ, respectively. We extend У to
a piecewise smooth vector field ξ along σ : [0, 1] —> Μ by setting it equal to 0 on
σ | [^, 1]. Applying the same argument to σ \ [^, 1], we get an η G ΤσΑ such that
η | [0, \\ = 0 and 77 | [^, 1] is a Jacobi field vanishing at the end points. Now we
set X := (ξ, η)л С ΤσΛ and get D2E(a) \ X = 0 by virtue of the second variation

3. INJECTIVITY RADIUS ESTIMATE AND CLOSED GEODESICS 299
formula. On the other hand, since ξ, η are not smooth, they do not belong to the
null space λί of Ό2Ε(σ) on ΤσΛ. Hence dim Λ' Πλί < 1. If Χ Πλί = {0}, then
we may easily see that the orthogonal projection pr_ of ΤσΛ onto T~ is injective
when restricted to X. Then it follows that тав& = dimX^yl" > dim Λ' = 2, which
is a contradiction, and we get dim Λ' Πλί = I. Recalling that λί is the space of
periodic Jacobi fields along σ, we see that there exists a nontrivial periodic Jacobi
field along σ in the form αξ + bη(a, b e R), which is perpendicular to σ. Prom
the construction of ξ, η it follows that there exists a periodic parallel vecor field
Ζ {φ 0) along σ, which is perpendicular to σ. Then we get
D2E(a)(Z, Z) = [\\\VZ(t)\\2 - (2n)2K(a(t). Z(t))\\Z(t)\\2} dt < 0.
Jo
Since Μ is simply connected and therefore orientable, the parallel translation
Ρ(σ)? belongs to SO(Ta^M). Namely, Ρ{σ)\ leaves invariant a subspace A :=
(σ(0), Ζ(0))χ and preserves the orientation. Since dim A is odd, Ρ{σ)*\ \ A admits a
nonzero fixed point. Then we get a second nontrivial periodic parallel vector field X
along σ which is perpendicular to σ and Z. We have as above D2E(a)(X, X) < 0,
and therefore indea > 2. This is our final contradiction, and the proof of the
theorem is complete. D
The above proof is due to W. Klingenberg and the author ([K-Sa]). For another
proof see [Ch-Gr-3]). In the above the critical value к of a </?-family plays an
important role, which was first introduced to show the existence of closed geodesies
on a compact Riemannian manifold via the calculus of variations. We give a typical
example, originally due to [Ly-Fe].
Theorem 3.5. On any compact Riemannian manifold Μ there exists at least
one {nontrivial) closed geodesic.
PROOF. By Chapter V, Lemma 1.5, we may assume that Μ is simply
connected. Since Ят(М, Ζ) = Ζ, there exists 0 < k < m such that
πχ(Μ, *),... ,7rfc(M, *) = 0, 7rfc+1(M, *) <* #fc+1(M, Ζ) φ 0.
Then we have a continuous map / : Sk+l —> Μ which is not homotopic to a constant
map. First, from / we will construct a continuous map F : (Bk, dBk) —> (C#, C%),
where Bk denotes the A;-dimensional closed unit disk, Св stands for the space of
closed curves in Μ given in Theorem 3.1, and C°B is a subspace of Св consisting
of point curves. For that purpose we identify Bk with the half greatsphere {x =
(x°, ... , xk+l) e 5fc+1; x° > 0, x1 = 0} of the unit hypersphere 5fc+1 in Rk+2.
Now to ρ G Bk we assign a small circle cp, which is obtained as the intersection of
Sk+1 and the plane through ρ parallel to the (x°, z^-plane, and parmetrized on
[0, 1] proportionally to arc-length. Note that cp is a point curve for ρ G dBk. Then
we define F by F(p) := / о cp G Св-
Second, take a positive a with a2 > 2max{E(F(p)); ρ G Bk} and a subdivision
Δ of I = [0, 1] such that Λ := Cq(A) is a strong deformation retract of C%.
Let Hs : С β —> С % (0 < s < 1) be a homotopy which defines the above strong
deformation retraction (see Chapter III, §3.1), and set F(p) := Hi oF(p) G A. Let
ψι (t > 0) be the flow on Λ generated by -VE. Then y>t oF : Bk —> Л is homotopic
to F, and we conversely construct /* : Sk+l —> Μ (£ > 0) as follows: For q G d£fc
we set ft(q) = q. For ς G 5fc+1 \aBfc, note that ρ G Bk\dBk and s e I are uniquely

APPENDICES
Bk
Ρ
x1
Figure 36
determined, so that we may write q = cp(s). Then we define ft(q) := (<^oF(p))(s).
The map ft is continuous, and /0(g) = Η ι о f(q). Therefore, ft is homotopic to
/. Now Η := {Η = ^o F(Bk) С Л; t > 0} is clearly a (^-family, and if we show
that its critical value к := infio{max£ | φίο ο F(Bk)} is positive, then we have a
closed geodesic of 22-value к by Lemma 3.2. Suppose к = 0. Then for any с > 0
there exists a i() > 0 such that Ε \ φίο ο F(Bk) is less than e. For e sufficiently
small, closed curves ft0 о cp (p £ Bk) in Μ are contained in a normal coordinate
neighborhood around ft0(p), and it follows that fto : 5fc+1 —> Μ is homotopic to a
constant map. This contradicts the fact that fto is homotopic to /. D
Remark 3.6. The existence problem of closed geodesies has a long history,
and there are many deep results. For the above proof and further results on this
problem we refer to [K-4, 5] (see also [Bang], [Fra] for recent developments).
Remark 3.7. Very recently, U. Abresch and W. Meyer ([Ab-Me-1]) obtained
the injectivity radius estimate i(M) > π for the almost ^-pinched case: namely,
there exists an e > 0 such that the injectivity radius estimate i(M) > π holds for
any complete simply connected Riemannian manifold Μ whose sectional curvature
satisfies \ — e < Κσ < 1, where e is of order 10 ~6. Using this estimate and a
new comparison method, they improved the sphere theorem: For the above €, an
odd-dimensional compact simply connected (| — €)-pinched Riemannian manifold
is homeomorphic to the sphere ([Ab-Me-2]).
4. Maximum Principle
Theorem 4.1. Let Μ be a (connected) Riemannian manifold and f : Μ —» R
a subharmonic function. Namely, f is continuous, and for any ρ £ Μ and e > 0
there is а С°° support function fp,e of f at ρ which satisfies Δ/Ρι€ < e (see Chapter
IV, Definition 3.6). Then f cannot assume its maximum unless f is constant.
PROOF. Suppose / assumes its maximum at ρ £ Μ. It suffices to show that
/ is constant on an open neighborhood of p. If not, we may take a sufficiently
small closed ball В centered at ρ whose boundary dB is different from d'B:= {q £
dB; f(q) = f(p)}. Then there exists a φ £ F(M) such that φ(ρ) = 0,φ\&Β<
0, φ is positive at some point of дВ \д'В, and V<^ Φ 0 on B. In fact, considering
В as a normal coordinate neighborhood centered at p, if д'В С дВ Π {χ1 < 0},
then φ equal to ж1 in a neighborhood of В is as desired. In general, since дВ ф

5. DIFFERENTIAL FORMS
301
д'В, take a diffeomorphism Φ of Μ that maps В onto В and satisfies Ф(р) = ρ,
Ф(д'В) С дВ П {ж1 < 0}. Then φ, which is equal to ж1 о Ф on a neighborhood of
B, is as desired.
Now we set h := βαφ - 1 (α > 1). Note that h(p) = 0 and
Δ/ι=-(α2||ν^||2-αΔ(^)βα^.
Since || V<^||2 assumes its positive minimum on compact B, we may take a sufficiently
large α so that Δ/ι < 0 on B. Next, as h \ д'В < 0, we may choose an open
neighborhood U of &B such that (/ + r//i)(g) < /(p), q G U, for any η > 0. On
the other hand, since /(p) — f assumes its positive minimum on a compact subset
dB \ U, we may take a sufficiently small η > 0 so that f(p) — f > 77/1 on SB \ /7.
It follows that
(/ + Vh)(q) < f(p) = (/ + ΐϊΛ)(ρ), 9 6 βΒ.
Namely, f + ηϊι assumes its maximum at an interior point p\ of B. Let /Pl<€ be a
support function of / at pi. Then ψ := fPl .6 +77h is a support function of /+77/1 and
assumes its maximum at ρχ. Therefore, we have Δ^(ρι) = — traceD2ip(p\) > 0.
On the other hand, note that
Αφ(ρλ) = Δ/Ρι,6(ρι) + τ/Δ/ι(ρι) < б + 77Δ/ι(ρι).
Since 7/Δ/ι(ρι) < 0, the right-hand side of the above inequality is negative if we
take € > 0 sufficiently small, and we get a contradiction. D
Remark 4.2. In particular, a C2 function / : Μ —> R which satisfies Δ/ < 0
(resp., Δ/ > 0) on Μ cannot assume its maximum (resp., minimum) unless / is
constant. Such a maximum principle was given by E. Calabi ([Ca]), and the above
proof is due to J. Eschenburg and E. Heintze ([Esc-He-1]; see also [Bes-2]).
5. Differential Forms
Here we state, without proof, some fundamental results on differential forms.
5.1. Let Μ be an m-dimensional oriented C°° manifold and Лк(М) the F(M)-
module of C°° differential forms of degree к on M. Then Л(М) := 0^=o Лк(М)
carries the structure of an algebra by the exterior product (Chapter I, (1.4)), and
we have the exterior differentiation d : Лк(М) —► Лк+1(М) which satisfies d2 = 0
(Chapter I, (3.2)). Then a sequence
г(м) = л°(м) ЛлНм)^··^ лк{м)
-i Лк+\М) Λ · · · Λ ЛШ(М) -> {0}
is called the de Rham complex of M. Let Fk(M) := {ω G Лк(М)\ du> = 0} be the
space of closed fc-forms. Note that dAk~1(M) С Fk(M). Then the vector space
HpR(M) := Fk(M)/dAk~l(M), which measures the obstruction for (5.1) to be an
exact sequence, is called the A;-th de Rham cohomology group of M.
Now let с = Σ TiGi be a singular fc-chain in M, where r* G R and σ* := (Sfc, y?i)
are singular fc-simplices consisting of the standard fc-simplex Sk in ilfc and Cx maps
Уг · ^ —> M. Then, for a given a; G ylfc(M), a linear map
/ω:=Σ>/5/ϊ'

302
APPENDICES
defines a singular fc-cochain Φ (ω). Denoting by δ the coboundary operator for
singular cochains, the Stokes theorem
(5.2) Ι ω = I <L· (ω e Лк~1(М), с is a singular fc-chain)
Jdc Jc
implies that δΦ(ω) = Φ(άω). Therefore, Φ induces a homomorphism
Φ, : HkDR(M) - Hk(M; Д),
where Hk(M\ R) := {φ; φ is a singular fc-cocycle with<5<^ = 0}/{<5^; ψ is a
singular (k — l)-cochain} denotes the singular A;-th cohomology group of Μ, and
Φ* is in fact a surjective isomorphism (de Rham theorem). Thus the analytic
cohomology of differential forms is related to the geometric singular cohomology.
Now when a Riemannian metric g is given on M, it is possible to represent
the de Rham cohomology in terms of harmonic forms determined by the
Riemannian metric g. First, we define a linear operator * : Лк(М) —» Am~k(M): For or-
thonormal vectors {xk+i, · · · , Xm} in TpM, take orthonormal vectors {χχ, ... , Xk)
so that {χί)Υ^\ forms a positively oriented o.n.b. of TPM. We define (*a)p £
Атп-к(ТрМ) by the condition
(*a)p(:rfc+i Л · · · Л xm) = ap(x\ Л ... Л хк), a <E Лк(М).
Then * : Ак(ТрМ) —> Am_fc(TpM) is a linear isomorphism, and we have * * ap =
(-l)Hm-k)ap (Qp G Afc(TpM)). Therefore, we may define * : Лк{М) -> Лт"А:(М)
pointwise, and get β Λ *α = (α, /?) rfM, *1 = rfM, where dM denotes the volume
element of (M, g).
Next, we define δ : ylfc(M) -> Л*Г"1(М) by
5a := (-l)^(fc+1)+1 * rf * a (a G Л^М)).
Note that we have (<5ct)i2...tfc = —^l^u2...ik with respect to local coordinates. We
may easily check that δ2 = 0, and δ commutes with the action of isometries of M.
Further, we get
δ/ = 0, Sdf = -div V/ = Δ/ (fe F{M)).
In particular, when Μ is compact we may define an L2-inner product ( , ) on
Лк{М) by
(5.3) («,/?):= / αΛ*/?= / (α,β)άΜ.
Jm Jm
Then <5 is the adjoint operator of d, namely, (α, άβ) = (<5α, /3).
Now we define Δ : Лк(М) —> Л/с(М) by Δ := d<5 + <5d, which is called the
Laplacian acting on fc-forms. Δ is a linear partial differential operator that coincides
with the Laplacian given in Chapter VI when restricted to F(M). We may also
write Δ = (d + <5)2, and Δ commutes with d, <5, * and the action of isometries. If
Μ is compact, Δ is self-adjoint with respect to ( , ). Now α £ Лк(М) is said to
be harmonic if da = δα = 0. Then Δα = 0. In the case where Μ is compact,
if Δα = 0 then da = δα = 0, since (Δα, α) = (da, da) Η- (<5α, <5a). Then the
following fundamental Hodge-Kodaira theorem holds.
Theorem 5.1. Let Μ be a compact oriented Riemannian manifold. Then the
vector space Hk(M) := Ker(A | Лк(М)) of harmonic k-forms is finite-dimensional,

5. DIFFERENTIAL FORMS
303
and we get the orthogonal decomposition
(5.4)
Лк(М) = Hk(M) Θ АЛк(М) = Пк(М) Θ d{Ak~l{M)) Θ 6{Лк+х{М))
with respect to (,), and Hk(M) S HkDR{M) Э* Hk{M; R).
5.2. Harmonic forms are also related to the curvature. For a tensor field Ζ of
type (0, s) we define the tensor field divZ of type (0, s — 1), called the divergence
of Z, by
m
(divZ)p(t7i, ... , υ3-ι) = 5^(VClZ)p(eI-, rb ... , ve_i),
i = l
where {e*} is an o.n.b. of ΤΡΛ/. Note that this definition does not depend on the
choice of {e*}. In particular, we get diva for a £ Лк(М), which is a tensor field of
type (0, к — 1) in general. Next, for α £ AK(M) we define
(5.5)
m к
ρ(α)(ϋι, ... , ν*) := 5I5I(^(et, Vj)a)(vu ... , ^_b e», vj+i, ... , vk),
г=1 j = l
which is a tensor field of type (0, k). Then we have the following Weitzenboeck
formula for the Laplacian Δ :
(5.6) Δα = -div Va + p(a),
(5.7) (Δα, a) = ^Δ||α||2 + ||Va||2 + (p(a), a).
In particular, when к = 1 we get (Δα)* — — pZfcV/Vfcai + Рг/Сктрт/, where ρ*/
denotes the Ricci tensor of Μ. Applying this Weitzenboeck formula, we may get
information on the topology of Μ via harmonic forms under the assumption on the
curvature. This approach is originally due to S. Bochner (see Chapter V, Theorem
1.10). We give other examples. Recall that the curvature operator R £ Нот(Л2гм)
determined from the curvature tensor defines a symmetric linear transformation of
A2(TPM) at every point ρ £ Μ (see Problem 3 for Chapter IV). Now suppose
all eigenvalues of R are nonnegative at all points. Then it is possible to show
that (p(a), a) > 0 for all α £ Ak(M). Now suppose Μ is also compact. Then,
integrating both sides of (5.7) for a harmonic /г-form a, we see by the Green theorem
that all harmonic forms are parallel. If all eigenvalues of R are positive at every
point of a compact Riemannian manifold Μ of dimension m, then we may show
that Hk(M) = 0 (0 < к < m), namely, Μ is a homology sphere.
Next, we are concerned with the estimate of b\(M) (see Chapter V, Theorem
1.10 and Proposition 1.2). Suppose the Ricci curvatures of a compact Riemannian
manifold Μ satisfy p(u) > — (m — l)k2 (u £ UM) for some positive constant k.
Then for a harmonic 1-form α on Μ we get from (5.7)
/ ||Va||2^s<(m-l)fc2/ ||α||2^.
JM JM
On the other hand, we have the so-called Kato's inequality ||V||a|| || < ||Va||,
and it follows that Д(||а||) < (m — \)k2 for the Rayleigh quotient R. If we set
h(a) := \\a\\ - ^м 1м НаН dv9, we have IM h(<a>> dv9 = °-

304
APPENDICES
Now suppose η := b\(M) = dimW^M) > m = dimM. Then it is possible to
find a0 G Hl(M) with
1 /2
/ ||а0||^/ЛоТм<С(т,п)(/ Ы|2 di/Л ,
jm Kjm )
where C(m, n) (> 0) depends only on m, n, and is monotone decreasing with
respect to n. Furthermore, C(n, n) = 1, and limn_+0c ^(m, n) = 0. We consider
h = h(a0) for this a0, and for the first eigenvalue X\(M) of the Laplacian acting
on functions we get
λι(Μ)< / \\Vh\\2dug/ f h2dvg<(l-C2(m,n))-\m-l)k2.
On the other hand, recall that by Chapter VI, Corollary 4.3, we have
ι f rd(M)/2 ) ~2
\i(M)>-{ / cosh™'1 ktdt}
о
It follows that there exists a positive constant 6(m), depending only on the
dimension, such that if min{p(u); и G UM} · d2(M) > -6(m), then η := &i(M) < m.
Remark 5.2. For the proofs of the results in §5.1 we refer to [dR-2], [War-
3], [Po]. For the Weitzenboeck formula and the Bochner technique we refer to
[Yano-Boc], [Po], [Wu-2]. See also the Appendix by J. Dodziuk in [Cha-2]. For
the relation between the curvature operator and the topology, see [Ga-Mey]. The
above estimate of the first Betti number is due to [Be-2], [Ga-2].
For a compact oriented Riemannian manifold Μ, we may consider the
eigenvalue problem for the Laplacian Δ acting on Лк(М), and the existence and
uniqueness of the heat kernel are guaranteed also in this case (see [Gi-1], [Pat], and
[At-Bo-Pat] for a generalization to the heat equation and the index theorem).
6. Gromov's Convergence Theorem
and Collapsing of Riemannian Manifolds
6.1. In Chapters IV and V we compared Riemannian manifolds Μ with
model spaces in terms of some Riemannian invariants, and studied metrical and
topological properties of Μ. More generally we may ask whether topological types
of Riemannian manifolds, whose Riemannian invariants satisfy certain conditions,
are finite. For instance, for a given positive integer m and positive numbers v, d,
consider the following family of Riemannian manifolds:
(6.1)
Mrn(d, v) := {(M, g); compact m-dimensional Riemannian manifold
with \Κσ\ < 1, vol (M, g) > v, d(M) < d},
where Κσ, vol (M) and d(M) denote the sectional curvature, volume and diameter,
respectively. We assume that \Κσ\ < 1 for simplicity. This is equivalent to assuming
that the absolute values of the sectional curvatures are bouded above by a fixed
constant. Riemannian manifolds belonging to Mm(d, v) are said to have bounded
geometry. A. Weinstein showed that the number of homotopy types of manifolds
belonging to Mm(v, d) is finite ([We-1]), and J. Cheeger ([Ch-2]) showed that the
number of diffeomorphism types of manifolds belonging to Mm(v, d) is finite (for
m φ 4). It is an important fact that we have a uniform estimate i(M) > i0(> 0)

6. GROMOV'S CONVERGENCE THEOREM
305
from below for the injectivity radius (or convexity radius) of (M, g) G Mm(v, d)
(see Chapter IV, Theorem 3.8). Then we may also estimate, uniformly from above,
the number of metric balls of radius i0 that are diffeomorphic to a disk and cover
Μ eMm{v, d).
Gromov also introduced a distance on the space of compact Riemannian
manifolds (or, more generally, compact metric spaces), and considered the convergence
of a sequence of Riemannian manifolds, compactness properties, and the
behavior of metrical invariants under this distance. To explain this, first we recall the
Hausdorff distance on the space of subsets of a fixed metric space (Z, d), where d
denotes the distance function.
For nonempty subsets Χ, Υ С Ζ, the Hausdorff distance between Χ, Υ is
defined as
dzH(X, Y) := inf{€ > 0; Be(X) D Υ and Be(Y) D X}
(6.2) Г Ί
= max < sup d(x, У), sup d(y, X) > ,
Uex yev J
where we set Be(X) := {z G Z\ d(z, X) < б}, etc. We also denote dfj by ddH. We
consider SUB(Z) := {X С Ζ, Χ is a, compact nonempty subset} endowed with
the distance dfj.
Proposition 6.1. (SUB(Z), dfj) is a metric space. Moreover, if (Z, d) is
complete or compact, then so is (SUB(Z), dfj).
PROOF. It is easy to see that (SUB, d^) satisfies the axioms of a metric
space. Next suppose (Z, d) is complete, and let {Xn} С SUB(Z) be a Cauchy
sequence. To see that {Xn} is convergent, it suffices to show that {Xn}
admits a convergent subsequence. By taking a subsequence if necessary, we may
assume that dzH(Xn, Xn+i) < 2~n_1, namely, dzH(Xn, Xm) < 2~n for m > n.
Now set Xoc := {limn_oc #n; Xn € Xnwith d(xn, xn+\) ^ 2~n-1}, noting that
{xn} is convergent since it is a Cauchy sequence and Ζ is complete. Xx is
easily seen to be nonempty. We may also easily check that X := X^ is compact
(it is a closed and totally bounded subset of a complete metric space Z) and
\\mdzH(Xn, X) = limdfjiXn, Хж) = 0. Namely, (SUB{Z), dzH) is complete. Next
suppose Ζ is compact. To see that SUB(Z) is compact, it suffices to show that
it is totally bounded. For any e > 0, take an €-net N = {x\, ... , Xk) of Ζ (i.e.,
Ui=i Be(xi) = Z). Then the family of all (nonempty) subsets of N forms an €-net
of SUB(Z). D
For metric spaces X, У, Gromov introduced their distance as follows ([G-4, 6]):
(6.3)
dH(X, Y) := inf{4(/(X), g(Y)); f : X -* Z, g : У - Ζ are
isometric (i.e., distance preserving) embeddings}.
A distance δ on the disjoint union X II У is said to be admissible if the inclusions
X,Y^>XUY are isometric embeddings. Then it is not difficult to check that
(6.4)
dH(X, Y) := inf{d6H(X, У); δ is an admissible distance on X U Y}
and show that d# defines a distance on the family MET := {isometry classes of
compact metric spaces}. d# is again called the Hausdorff distance.

306
APPENDICES
Remark 6.2. (1) For homeomorphic metric spaces X, У, define their Lip-
schitz distance as
dL(X,Y):= inf{| logdil/| + | log dil/~x|;
/ : X —» Υ is a homeomorphism},
where dil/ := sup{d(f(xi), f(x2))/d(x\, X2); x\, ^2 G X(x\ φ яг)}· Then d/,
satisfies the axioms for a pseudometric, and if Χ, Υ are compact, then db(X, Y) = 0
implies that X, У are isometric (Ascoli-Arzela theorem). The Lipschitz distance
has been effectively used in pinching problems (see [Shik], [Ka-1]). On the other
hand, the Gromov-Hausdorff metric d# turns out to be useful in comparing not
necessarily homeomorphic Riemannian manifolds in terms of metric invariants.
(2) There are several equivalent versions of the Gromov-Hausdorff distance.
The following is due to K. Fukaya: A map / : X —» Υ between metric spaces,
which is not necessarily continuous, is said to be an e-Hausdorff approximation if
(i) |d(/(xi), f(x2)) ~ d(xu x2))\ < e for xu x2 G X,
(ii) Be(f(X))DY
Then if we define
d#(X, Y) := inf {б > 0; there exist £-Hausdorff approximations
( ' / : X -> Υ and g : Υ -> X},
we have §d# < d# < 2d#, and d# defines the same topology on MET as d#.
However, note that du does not satisfy the triangle inequality. Also, we have
dH(X, У) < max{rf(X), d(Y)} · |exp(rfL(X, Y)) - 1|.
The following is a fundamental compactness property of {MST, d#} ([G-4]).
Theorem 6.3. {Λ1£Τ, d#} is a complete metric space. A subset X С MET
is said to he uniformly compact, if it satisfies the following conditions:
(1) There exists a d > 0 such that the diameter d(X) < d for any X G X.
(2) For any e > 0 there exists К = K(e) such that any X G X carries an e-net
consisting of at most К elements.
Any uniformly compact X is precompact with respect to d#, namely, its closure
X is compact.
PROOF. We sketch the proof of the second assertion. Let {Xn} С Л' be a
sequence, and construct a compact metric space Ζ so that Xn (n = 1, 2 ...) are
isometrically embedded in Z. For that purpose, we set ei := 2~г~1 (г = 1, 2, ...)
and take K\ := K{e{) as in (2). Next we set A{ := {a^ = (feb ... , ki); 1 <
kj < Kj (j = 1, ... , г)} for г = 1, 2, · · ·, and define Pn : Ai —» Xn inductively as
follows: Pn(k\) := x£ G Xn, where {x[n , ... , я^ } is an €i-net of Xn guaranteed
by (2). Next choose an €2-net {я^д, · · · , ж^,к2}(с Χ η) of B£l(x^ ) and define
^((*i, to)) := x%]k2 {l<k2< K2), and so on.
Let pi : -Ai+i —» A; be the canonical pojection, and note that
d(4(Pi(ai+1)), l'+1(ai+i)) < e< + ei+1 (t = 1, 2, ... )·
Now we set Л := (J^i Л,, and let F := {ft : Л —> Д; вираел |ft(o)| < +00} be the
Banach space of maps ft : A —► R endowed with the sup-norm. We define a subset
Ζ := {ft € F; |ft(ai)| < d, |ft(a) - h{Pi(a))\ <e{ + ei+1 for a € Ai+1}

6. GROMOV'S CONVERGENCE THEOREM
307
of F. Then Ζ is closed and totally bounded, and therefore compact. Next we
define maps fn : Xn —» Ζ by fn(x)(a) := d(x, 1гп(а)), a G -4$. Then we easily
see that fn is distance preserving. By Proposition 6.1, {fn(Xn)} admits a
convergent subsequence fnk(Xnk) —» X G SUB(Z) with respect to d^, and therefore
limdtf (Xnfc, X) = 0. Finally, note that completeness of {MET, dH} follows
directly from the second assertion. D
Remark 6.4. When we consider a family of noncompact metric spaces, it is
more convenient to consider the following pointed Hausdorff distance: Let (X, p)
and (Y, q) be metric spaces with base point; Br(p\ X) := {x G X; d(x, p) < r},
etc., denotes the (closed) metric ball of radius г in X centered at p. Then we define
(6.7)
dp<H((X,p),(Y,q))
:— inf{e > 0; there exists an admissible metric δ on XUY such
that 6{p, q) < e, B1/({p; X) С B({Y; (XII Υ, δ)), and
B1/t(q;Y)cBt(X;(XUY,6))}.
Now let MSTq := {(X, p)\ proper metric space with base point} endowed with
the metric dp,#, where proper means that Br(X, p) is compact for any r > 0.
Then a subfamily X С MS To with {Br(p, X); (X, p) G X} uniformly compact
for each r > 0 is precompact with respect to dP;# ([G-4]).
Now we give some examples of Hausdorff convergence:
1°. Let (X, d) be a compact metric space. For e > 0 we denote by eX the
metric space (X, ed). Then eX —> {one point} as € —> 0 with respect to d#-
2°. Set Xn := |J{fc/2n € [0, 1]; fc = 0, 1, ... , 2n}. Then Xn -> [0, 1] with
respect to d# as η —> oo.
3°. Let (M, p) be a Riemannian manifold, and let d denote the distance
obtained from g. Then for ρ G Μ we have ((M, p), rd) —> ((TPM, op), do) with
respect to dp,# as г —> oo, where do denotes the distance obtained from the inner
product gp. In fact, for fr := exp£ : Bi/re(op; (TPM, d0)) —> Bi/r£(p; (M, d)) we
have dil/r, dil/"1 —> 1 as r —> oo for a fixed € > 0 (see Corollary 2.8 of Chapter
IV). Multiplying by r in both of the above metric spaces, we see that
dL(B1/e(op- {TPM, d0), Bl/e(p, (M, rd))) -* 0
as r —> oo. Then our assertion follows from Remark 6.2 (2).
4°. Let (M, g) —> (Β, /ι) be a surjective Riemannian submersion between
compact Riemannian manifolds with connected fibers. Define a family of Riemannian
metrics gt (t > 0) on Μ as follows:
<7t(i7, i;') := t2g(v, v'), v, v' G TpM are vertical,
^(i;? h) := 0, г; G TPM is vertical, h G TPM is horizontal,
\9t{h, hf) := g(h, h'), ft, ft' G TPM are horizontal.
Then (M, pt) —» (B, ft) with respect to d# as ί —> 0. For instance, let π :
(52n+1, p) -> (CPn, ft) be the Hopf fibering. Then (52n+1, pt) are isometric to
the Berger spheres given in Appendix 2, and converge to (CPn, ft) with respect
to du as t —> 0. Note that the absolute values of the sectional curvatures of gt
are uniformly bounded as t —> 0. We have a similar phenomenon for the Wallach
example given in Appendix 2. On the other hand, what happens when t —> oo?

308
APPENDICES
Let D be the 2n-dimensional distribution on 52n+1 defined by horizontal vectors.
Then T(D), the space of C°° vector fields on 52n+1 taking values in D, generates
Γ(Τ52η+1) as a Lie algebra. Define a metric dc on 52n+1 by
dc(x, y) := inf{Lg{c)\c : [0, 1] —» 52n+1 is a piecewise
C°° curve with c(i) G D).
Then dc(x, t/) is finite, namely, χ is joined to у by a horizontal curve. Now we get
(52n+1, &) -> (52n+1, dc) with respect to dH as ί -> oo. (52n+1, dc) is homeo-
morphic to (52n+1, #). However, its Hausdorff dimension is equal to 2n + 2 (see
[Mit]).
Remark 6.5. What kind of properties of metric spaces are inherited under
the Hausdorff convergence ?
(1) X G MET »-> d(X) is a continuous function on MET with respect to dH,
where d(X) is the diameter of X.
(2) For a metric space (X. d), recall that we may define the length L(c) of a
continuous curve с by
L(c) := sup{^ d(c(fi_i), с(^)); 0 = ί0 < · · · < tn = 1},
which might be infinite (see Problem 2 for Chapter II). Then di(x, y) := inf{L(c);
с is a continuous curve joining χ to y} defines a new distance on X which clearly
satisfies d < d{. A metric space (X, d) is said to be a length space if d = d{. Now
if a sequence {Xn} of length spaces converges to a complete metric space X with
respect to d#, then X is also a length space (see [G-6]).
(3) Suppose homogeneous metric spaces (Xn, d) (resp., homogeneous pointed
metric spaces (Xn, pn)) converge to a compact metric space X (resp., proper
pointed metric space (X, p)) with respect to d# (resp., dp,//)· Then X (resp.,
(X, p)) is homogeneous ([G-4]).
Now we return to Riemannian geometry and consider the following family of
Riemannian manifolds:
(6.8)
Srn(d) := {M;M is a compact m-dimensional Riemannian manifold
such that p(u) > -(m - 1), d(M) < d} (C MET),
where p(u) denotes the Ricci curvature. Then Gromov ([G-6]) showed the following
Precompactness Theorem.
Theorem 6.6. <Sm(d) is precompact with respect to d#, г.е., its closure is
compact in MET.
PROOF. It suffices to show that <Sm(d) is uniformly compact. For any e > 0 we
estimate the number of points of a maximal £-discrete set Λί = {pu p2, ... }, which
satisfies Be/2(Pi) Π Be/2(Pj) = Φ(ΐ φ j)· Prom the Bishop-Gromov comparison
theorem (Theorem 3.3 of Chapter IV), we get
vol Μ _ vol Вd(pi) щ_
vo\Be/2(pi) Уо\Ве/2(р{) ~ v€/2'
where vr denotes the volume of a metric ball of radius r in the m-dimensional
hyperbolic space of constant curvature -1. It follows that
vol Μ > Σ vo1 Be/2{Pi) > W' vol Μ ^,

6. GROMOV'S CONVERGENCE THEOREM
309
namely, N = ЛЛГ is finite and satisfies N < Vd/ve/2- Since we have \JBe(pi) = Μ
because of the maximality, {pi, ... , p^} forms an £-net of M, and this completes
the proof of the theorem. D
Remark 6.7. (1) A metric space X belonging to the boundary of <Sm(d) may
not be a Riemannian manifold and may admit singularities. However, X is a length
space, and we may show that the Hausdorff dimension of X is less than or equal to
m by a similar argument.
(2) In the same manner, we may show that the family of all complete
Tridimensional Riemannian manifolds whose Ricci curvatures are bounded below by a
fixed constant is precompact with respect to dp.#. The Gromov-Hausdorff distance
was first introduced by Gromov to show that if a finitely generated group has
polynomial growth, it contains a nilpotent subgroup of finite index ([G-4]).
For more details on (pointed) Hausdorff distance we refer to [G-6], [Pet], and
[F-l, 4].
6.2. Now we are concerned with Mm(d, v) introduced in (6.1). M. Gromov
gave a compactness theorem for Mm{d, v) in [G-6]. Since it was not easy at first
to follow his argument in full detail, several authors gave proofs of the Gromov
theorem, and applications ([Kat], [Peters], [Gre-Wu-2], [Pugh], [Kas-2], [Ch-G-1],
[F-4]). In the following we explain the Gromov compactness theorem, dividing it
into a diffeomorphism theorem ([G-6], [Kat]) and a convergence theorem ([G-6],
[Peters], [Gre-Wu-2], [Kas-2], [Ch-G-1], [Nik]). As we see below, we only need
standard comparison technique for the proof of the diffeomorphism theorem. However,
for the convergence theorem we need more elaborate harmonic coordinates.
Theorem 6.8 (Diffeomorphism Theorem). For any e > 0 there exists α δ =
<5(m, d, v, e) > 0 such that for any Μ, Ν e Мт(а, υ) with dH(M, Ν) < δ the
following assertions hold:
(1) Μ and N are diffeomorphic.
(2) dL(M,N)<e.
PROOF. We give a sketch of the proof. We embed Μ e Mrn(d, v) into a
Hilbert space. (Gromov embedded it into a fixed RN. It is also possible to use
Whitney's embedding technique [Kas-2], [Pugh]. See also [F-3] and [Be-Bes-Ga
2] for embedding in terms of eigenfunctions of Laplacian. Here we follow Fukaya
[4].) Recall that we have a uniform estimate of the injectivity radius i(M) >
z0(m, d, v) > 0 for Μ e Mrn(d, v). Take 0 < μ < min{z0/2, 1}. Then we have a
normal coordinate system on Βμ(ρ; Μ), and the distance function dp to ρ is smooth
on Βμ(ρ\ Μ) \ {ρ}. Moreover, for the gradient vector Vdp and the Hessian D2dp
we have \\Vdp{q)\\ = 1, \\D2 dp(q)\\ < dp(q)/2 + l/dp(q) forqe Βμ{ρ; Μ)\{ρ} (see
Chapter III, Proposition 4.8, and Chapter IV, Lemma 2.9).
1°. Define a smooth map I : Μ -> L2(M) by I(p)(q) := h(d(p, q)), where
L2(M) denotes the Hilbert space of measurable functions on Μ with L2-inner
product, and h : [0, oo) —» [0, 1] is a cut-off function such that h(t) = 1 for 0 < t <
μ/З, h(t) = 0fort> 2μ/3, -4/μ < h'(t) < 0 for μ/З < t < 2μ/3, \h'{t)\ > 2/μ for
4μ/9 < t < 5μ/9, and \h"{t)\ < 4/μ2. Note that DI(u)(r) = h'{dr(p)) (Vdr(p), u)
for и e Tp Μ, and therefore
ЦЯ/(«)ИЬ= / {h'(dr(p))}2(Vdr(p),u)2dug(r).
Jm

310
APPENDICES
We easily see that J is an embedding. Furthermore, using the volume comparison
technique and the properties of ft, we may show that the operator norm ||ΖλΓ||
satisfies
(6.9) Citm^"1 < ||D/|| < 02(τη)μ^-\
where C*(m) stands for positive constants depending only on m. Then if we denote
by ам (resp., ам) the distance on Μ induced from the Riemannian metric (resp.,
induced from the metric on L2(M) via J), it follows that
(6.10) Οι(τη)μ^~1άΜ < dM < 02{πί)μ^~ιάΜ'
2°. Next we estimate the principal curvatures (i.e., eigenvalues of the second
fundamental form, up to sign) of I{M) <-► L2(M). Let TM1- := {(ρ, ξ) e Μ χ
L2(M); ξ±ΌΙ(ΤρΜ)} be the normal bundle of J(M), and exp-1 : TML -> L2(M)
the normal exponential map given by exp±(p, ξ) = Ι (ρ) + ξ. Recall that for a unit
normal vector η G TM1- at ρ G M, the second fundamental form Sn is defined by
5n(u, t;) = -{Du n, DI(v)), ν e TPM,
where ( , ) denotes the L2-inner product (see Chapter II, §3.3). Now let A be an
eigenvalue of Sn with a unit eigenvector u. Take a geodesic x(s) in Μ tangent to
и at ρ = ж(0), and set c(s) := I(x(s)). Then
A||c(0)||22 = 5n(u, u) = -HO), DI(u)) = (n, c(0)),
where n(s) is a normal vector field along c(s) with n(0) = n. Namely, we get
A = (n, c(0))/||c(0)||22. By (6.9), ||έ(0)|β2 > Οι(τη)μ^-1. On the other hand,
||c(0)||L2< (maxlftn + maxlft'l. max ||L>2 <*Γ||ρ}{νο1(£2μ/3(*; Hm(-l)))l *
l г€В2м/з(р) J
< С(т)мт/2"2.
It follows that |λ| < 0^1(τη)μ~πι^2 for a positive constant Сз(т). Now recall that
Dexp±(ln) is singular if and only if exp-L(/n) is a focal point of /(M), and A is
a principal curvature with respect to η if and only if 1/A is the focal distance of
I{M) along a geodesic t ■-> exp-^p, in). Therefore,
(6.11)
exp-1" | ВСз(т)мт/2(/(М); L2(M)) is a local diffeomorphism.
Next we show that there exists an R = Οβ(πι)μπι/2 such that exp-1 is a
diffeomorphism when restricted to Br(I(M)\ L2(M)). First setting C^{m) :=
C3(m)/C2(m), note that the following holds:
(6.12)
There exists a positive constant C^(m) such that if ам{р, q) >
04(πι)μ, then \\I(p) - I(q)\\L2 > 05(τη)μ^2.
(6.12) may be shown again using the volume comparison technique and the
properties of ft. In particular, by (6.10), if dM{p, q) > C3(m)Mm/2, then ||J(p)-J(g)||L2 >
C5(m)M-/2.
Now we set R = Οβ(πι)μτη^2 with Ce{m) := ^ тт{Сз(т), Сь(т)}, and show
that exp-1 is injective when restricted to Br(I(M), L2(M)). In fact, suppose η =

6. GROMOV'S CONVERGENCE THEOREM
311
expx(p, ξ) = expx(g, η) for (ρ, ξ) φ {q, η), \\ξ\\^, \\η\\„ < R. Then
\\Ι{ρ)-Ι(0)\\ν<Ο<,(πι)μ™Ι\
and therefore ам(р, я) < С3(т)дт/2 because of (6.12). Take a shortest curve с :
[0, 1] —» /(Μ) with respect to ^м joining /(p) to /(ς). Then we have a homotopy Η
between two segments 11—> exp-^p, ££), ехрх(^, ί?7) consisting of segments joining
n to c(s) in L2(M). Then Η is contained in £Сз(т)м- 2(J(M);L2(M)). By (6.11),
a standard lifting argument implies that (ρ, ξ) = (q. 77), a contradiction.
3°. Now let TV G Mm{d, v) satisfy dH(M. TV) < δ, and take a <5-Hausdorff
approximation φ : Μ —» TV, which we may assume to be measurable. Define
Γ : N -> L2(M) by /'(?)(r) := h{d(q, y(0))· Then again Г is an embedding
and satisfies (6.9). First, note that taking δ sufficiently small (e.g., less than
C6(mW(2C2(m))), we have /'(TV) С BR{I{M); L2{M)). In fact, for any q G TV
take ρ e Μ such that ά(φ(ρ), q) < δ. Then
l|/'(9) - I(p)\\l> < \\I'(q) - I'(<p(p))\\l> + ¥'{ψ{ρ)) ~ Цр)\\ь* < 2C2(m)Mf-^.
Therefore, we may define a C^-map / : TV —» Μ by /(ς) := J-1 ο Ρ о (ехр-1)-1 о
/'(ς), where Ρ : ΤΜΧ —» /(Μ) is the canonical projection. Note that
ЦДЛ?)) - /'(9)IU* < 2ft(m)Mm/2-1«.
Next, we show that / is a diffeomorphism, if we take μ and δ sufficiently small.
(1) f is regular. Otherwise, there exists a unit vector υ G T^TV such that
Df(v) = 0, namely, DI'(v) is orthogonal to DI(Tf(q)M). To get a contradiction it
suffices to show the following: For any e\ > 0 there exists a δ = <5(m, μ, €ι) such that
if ан(М, TV) < δ we may find a unit vector и G Tf^M with ||ΖλΓ'(ι;) - Z)J(u) ||L2 <
€i. To verify this, set <7i := βχρ^μί;) G TV and take p, p\ G Μ such that
dN&(p), q), άΝ{φ{ρλ), qx) < δ. Now for any г G #§μ(ρ; Μ) \ Βιμ(ρ; Μ) we
compare the geodesic triangles Δ\ = Л(г, /(ς), pi) in Μ and Δ2 = Л(</?(г), q, qi)
in TV. Note that ||J(p) - /'(ς)||ζ,2 < 2δ02(πι)μηϊ~1, as above. Similarly, we get
\\1(Р)-Ц/Ш\ь* < 4δ02(τη)μ?-1. Then, as in (6.12), we get dM(p, f{q)) <
С7(т)6тдт+1) and it follows that side lengths of Ль Δ2 are of order μ, while
2
the differences of the lengths of the corresponding sides of Δχ, Δ2 are of order im .
Therefore, first taking μ sufficiently small so that the Δι (г = 1, 2) are almost
Euclidean triangles (the Rauch comparison theorem), and then taking δ very small
compared with μ, we conclude that the angle at the vertex q of Δ2 and the
angle at the vertex f(q) of Δ\ are arbitrary close by R.C.T. Now let и G Tj^M
be the unit tangent vector to the normal geodesic joining f(q) to p\ at f(q).
Then the above argument implies that, for r G Βζμ(ρ) \ Βιμ^, DI'(v)(r) =
Η'(άφ(Γ)(ς))(νάφ(Γ)(ς),υ)*ιιά DI(u)(r) = h'(dr(f(q)))(Vdr(f(q)),u) are
arbitrary close to each other, and our assertion follows.
In particular, if we denote by Π : Tjt^I'(N) —» Tj(f(q^I(M) the orthogonal
projection, we get \\DI'(v) - T\DI'(v)\\L2 < ex for any ν G UqN.
By the above argument we may also show the following: There exists a linear
isometry Φ : T^TV -> Tf{q)M such that \\DV(v) - DI($(v))\\L2 < £1 for υ G UqN,
if dH(M, TV) <δ.
Also we may easily show that for any e2 > 0 there exists a δ > 0 such that the
above / : TV —» Μ is an €i-Hausdorff approximation when d#(M, TV) < δ.

312
APPENDICES
(2) / is injective. First note that / : Μ —» N is a covering map. Now suppose
f(qi) = /(#2) ·= P· Then ^лг(<7ь ^2) < ^2 (< ^o/2), and we take a minimal geodesic
7 joining q\ to ^2- Then /07 is a loop at ρ contained in a contractible Z?26l (ρ; Μ).
Since / is a covering map, 7 must be a closed curve, namely, q\ = q<i.
4°. f is an almost isometry. To see this, first we show that for any €3 > 0 there
exists a δ = <5(m, μ, e) > 0 such that if d#(M, Ν) < <5, then for ξ = DI'(v), ν £
Uq(N), we have
(6.13) 1 - сз < \\DP о Дехр-'-ГЧОН WIKIU» < 1 + сз.
In fact, note that for ξ there exists a unique /(M)-Jacobi field У(£) along the line
segment t .-> exp-^/fo), tn), η := (exp-L)-1(J/(g))/||(exp-L)-1(//(9))llL«) e TM^,
joining /(/(<?)) to /'(<?) such that Υ (I) = £,/:= ||/'(</) - /(/(?))||ι.*. Then У(0) =
DP о D(exp±)~1 (ξ), and we may write У'(0) = AnY(0) + B, where An denotes the
shape operator of J(M) <—► L2(M) with respect to the normal vector n, and В is
the orthogonal component of У'(0) to I(M). Solving the Jacobi equation, we get
Υ (I) = l(AnY{0) + В) + У(0). Now note that У(/) - ПУ(/) = Ш. Then it follows
that
||r(0)||La ||У(0)|и» У(0) ,_x
l|y(/)IUa - \\iAn(Y(o)) + Y(o)\\L* ~{ tnAn(\\Y(0)h*)U2*
< 1 + &{τη)δμ-\
llyWH*a <1-UU ί Г(0) Ml 4- ||Ш|1^
||r(0)||ia - " п1П0)1к*'"* IIH0)||l»'
<l + C8(m)iM + *ЩЩ*
and (6.13) follows easily. Prom this we have
\\DI\v) - DI(Df(v))\\L, = \\Y(l) - y(0)||L2 < e4.
On the other hand, we get \\DV(v) - 01(Ф(у))\\Ь2 < ex in step 3°. Therefore,
taking δ > 0 sufficiently small and noting (6.9), it follows that \\Df(v) - Ф(у)\\Ь2
is arbitrary small for any ν £ UqN. D
Note that the precompactness theorem and the diffeomorphism theorem
immediately imply the Cheeger finiteness theorem.
Now we turn to the convergence theorem. Recall that a function / defined on
a domain Ω in Rm is said to be α-Lipschitz continuous if its Ca-norm
||/||c- := supfl/Or) - f(y)\/\x - y\a; х,у€П(хф у)}
is finite. / is said to be of class C1,Q if its Cla-norm
||/||Ci.o := H/IIco + ||£>/||Co + ||Γ>/||σ-
is finite, where ||/||c°? etc., denotes the sup-norm. A Riemannian metric g on Μ
is said to be of class Cla, if the components of g are of class Cla with respect to
local charts of some atlas of M.
Theorem 6.9 (Convergence Theorem). Let {(Mn, gn)}^Li be a sequence in
Mrn(d, v), and let 0 < a < 1 be given. Then there exist a subsequence {(Mni, gni)}
and a C°° manifold Μ with a Riemannian metric g on Μ of class Cl'Q such that
the following hold:

6. GROMOV'S CONVERGENCE THEOREM
313
(1) For sufficiently large i, MUi is diffeomorphic to M, and g is a limit of
{/пг9пг} with respect to the Cl,Q topology (0 < a' < 1), where fn% : Μ —» Mn. are
diffeomorphisms.
(2) lim^oc dL(M, MnJ = 0.
By the precompactness theorem we may assume, taking a subsequence if
necessary, that (Mn, gn) converge to a compact metric space X with respect to d#. Then
the diffeomorphism theorem implies that the Mn are diffeomorphic to a fixed
compact C°° manifold Μ for sufficiently large n\ let fn : Μ —» Mn be diffeomorphisms.
Set g^ := f^gn\ we want to apply the Ascoli-Arzela theorem to g^n\ For that
purpose we need the component expressions {(#")} of g^ with respect to local
charts of some atlas, and the uniform estimate of some appropriate norm of (g\™).
Normal coordinate systems give uniform estimate only for the C°-norm (R.C.T.),
and the following harmonic coordinate systems are better for our purpose: A local
chart ([/, φ) of a Riemannian manifold is said to be a harmonic coordinate system,
if the coordinate functions are harmonic. The following fundamental result is due
to Jost and Karcher ([J-Ka]).
Theorem 6.10. For a positive integer m, and г0 > 0, 0 < α < 1, there exist
R = R(m, г0) > 0 and С = С(т, г0, α) > 0 with the following properties: For
any compact m-dimensional Riemannian manifold (M, g) with \Κσ\ < 1, г (Μ) >
го, and for any ρ £ Μ, there exists a harmonic coordinate system {ft1, ... , ftm}
defined on BR(p; M) such that ||^j||ci.a < C, where gij = g(d/dh\ d/dhj). Also,
\\gij — 6ij\\ < СR™. Next, let f be a harmonic function on Вц(р: (М,д)). Then
on BR/2(p\ (M,g)) we get the Schauder estimate
df
dU
d2f
dtedhi
<qi/llco.
CQ
\CQ
Then we get a uniform estimate of the C2,Q-norm of coordinate transformations of
harmonic coordinates.
Now we turn to the proof of the convergence theorem. Since (M, g^) are
d^-close to each other, we may choose {pk)k=\ c Μ that forms an r-net for all
g(n) (0 < r « R), where N is uniformly estimated in terms of m, d, v, r. Let
ftj^ : Вц(рь\ g^) —* Rm be harmonic coordinate systems with respect to g(n\
We may assume that
/4n)(B3r(Pfc; 5(n))) => B2r(o) D h[n){Br(pk-gM)).
Then, by Theorem 6.10 and the Ascoli-Arzela theorem, sequences of induced
metrics <7Jj,n) := g^bdxadxb (1 < к < Ν) on B2r converge to Riemannian metrics
gk := gkabdxadxb of class Cl,Q with respect to the Cla -topology, taking a
common subsequence {щ} for all к if necessary. Next, consider the coordinate
transformations
#M := h\n) о (h^y'\B2r(o) (BP(pfc; $<**>) Π Br(Pl; 9{n)) Φ Φ).
Then by a similar argument, we see that H^ converge to a C2a-diffeomorphism
Hk,i '· B2r{o) —* Rm with respect to the C2a -topology, taking a further
subsequence if necessary. Then the Hk.i's give isometries between (B2r(o); gk)) and

314
APPENDICES
(Z?2r(o); 9ι)· From {{B2r{o), gk)} we get a Riemannian structure g of class C1,Q on
Μ via the identifications by Hk,i- Then (M, p) is a d^-limit 0f {(M, gn)}·
See also [Peters], [Gre-Wu-2] for an intrinsic proof using the center of mass
technique. See [Kod] for the case of compact manifolds with boundary.
Remark 6.11. (1) We cannot expect that the limit metric is of class C2.
(Consider, e.g., a cylinder with two spherical caps in R3 endowed with the induced
metric g, which is diffeomorphic to S2. g is of class C1'1 and may be approximated
by C°° Riemannian metrics with bounded geometry with respect to di.) On the
other hand, if we assume uniform upper bounds for the C°-norm of the г-th covari-
ant derivatives of the Ricci tensors of (Mn, gn) (г = 1, ... , к) in the convergence
theorem, then the limit metric is of class C1+fc (see [Kas-2]). See also [Bem-MO-
Ru], [Band], [Ch-G-1] for the approximation of a Riemannian metric with \Κσ\ < 1
by uniformly smooth Riemannian metrics.
(2) In the convergence theorem we may show that the distance function d of
the limit metric is of class C1*1 on (7\Δ, where U С Μ χ Μ is an open neighborhood
of the diagonal Δ (see [G-6], [Pugh]).
(3) Note that the diffeomorphism theorem does not hold for the pointed Haus-
dorff distance. However, the convergence theorem for the pointed Hausdorff
distance holds in the following form with a slight modification in the proof: Let
{(Mn, pn)} be a sequence in Л1р,т(г0) := {(Μ, ρ); m-dimensional pointed
complete Riemannianian manifold with \Κσ\ < 1, iPn(M) > го}. Then there exist a
pointed C°° manifold (Μ, p) of dimension m with a Riemannian metric of class
С1'**, and a subsequence {(МПг, рПг)} which converges to (M, p) with respect to
Now we briefly explain applications. As mentioned above, the Cheeger finite-
ness theorem follows directly. As applications to the pinching problem, first note the
following: There exists 0 < δ = δ(τη) < 1 such that a compact simply connected
Riemannian manifold (M, g) with δ < Κσ < 1 is diffeomorphic to the standard
sphere. In fact, otherwise we have a sequence of compact simply connected
Riemannian manifolds (Mn, gn), which are not diffeomorphic to the standard sphere,
in Mm(d, v) for some d, v. A limit manifold (M, g) turns out to be of class C°°
in this case because of the strong homogeneity and Hubert's 5th problem, and
therefore is isometric to the sphere of constant curvature 1. Then (Mn, gn) are
diffeomorphic to the standard sphere for large n, a contradiction. Note that δ depends
on the dimension (see Remark 2.14 in Chapter V for better results). However, Otsu,
Shiohama and Yamaguchi ([Ot-S-Ya]) obtained the following version of the sphere
theorem, first showing that dH(M, 5m) is small and then constructing an
embedding of Μ into Дт+1 directly: There exists an e = e(m) > 0 such that if a compact
m-dimensional Riemannian manifold Μ satisfies Κσ > 1 and vol Μ > a;m — e(m),
then Μ is diffeomorphic to the standard sphere.
The following result is due to M. Berger ([B-9], see also [Dur]): There exists
a δ = δ(2η) > 0 such that a compact simply connected Riemannian manifold
of dimension 2n with \ — δ < Κσ < 1 is either homeomorphic to a sphere or
diffeomorphic to one of the compact simply connected symmetric spaces of rank
one.
Finally, we mention the negatively pinched case. M. Gromov ([G-l]) showed
the following: There exists a δ = <5(m, d) > 0 such that a compact m-dimensional
Riemannian manifold Μ with -1 < Κσ < -1 + δ and d(M) < d is diffeomorphic

6. GROMOV'S CONVERGENCE THEOREM
315
to a compact Riemannian manifold of constant curvature —1. In fact, in this case
we have vol Μ > v(m) (> 0). To see this, the following Margulis lemma plays
an essential role: Suppose a complete simply connected Riemannian manifold Μ
of dimension m > 2 satisfies \Κσ\ < 1 and i(M) > 1. Then there exists an
б = e(m) > 0 such that for any discrete subgroup Γ of isometries of Μ and any point
pGM, Ге(р) contains a nilpotent subgroup of finite index, where Te(p) denotes the
subgroup of Γ generated by {7 e Γ; d(p< 7Q?)) < e} (see [G-l]. [He]). Applying this
to the universal covering of a compact negatively curved manifold Λ/ and using an
argument given in Chapter VI, Theorem 4.19, we can get a uniform estimate for the
injectivity radius of M. Further, the condition that d(M) < d may be replaced by
vol Μ < V. In fact, for a compact Riemannian manifold Λ/ of dimension m (> 4)
with — 1 < Κσ < 0, the diameter is bounded above in terms of the volume (see
[G-l]). On the other hand, the assumption on the diameter or volume is necessary
for m > 4 because of some examples of Gromov and Thurston: For any δ > 0, there
exist compact manifolds with Riemannian metrics with -1 < Κσ < -1 + <5, which
cannot admit a Riemannian metric of constant negative curvature ([G-Th]).
Remark 6.12. (1) Dropping the assumption on the upper bound for the
sectional curvatures, K. Grove and P. Petersen considered the class M^d, v) := { M;
compact m-dimensional Riemannian manifold with Κσ > — 1, vol Μ > ν, d(M) <
d}, and proved the finiteness theorem for the homotopy types and the diffeomor-
phism types of manifolds belonging to M^d, v) ([Gro-Pet-1], [Gro-Pet-Wu]). Note
that in this case we have no uniform estimate for the injectivity radius, and a limit
length space X need not be a smooth manifold (consider, e.g., the boundary of
a convex polyhedron in ilm+1). However, such a limit space X is a length space
of finite Hausdorff dimension, and carries nice geometric properties. For instance,
the Toponogov comparison theorem holds, and we can say that the sectional
curvatures of X are greater than or equal to -1 in this sense, although the sectional
curvature itself need not be defined. Such a space is an example of Alexandrov
spaces: A complete locally compact length space X of finite Hausdorff dimension is
called an Alexsandrov space of curvature greater than or equal to A;, if the following
Alexandrov convexity holds: For any geodesic triangle A(pqr) in X, take a geodesic
triangle A(pqr) with the same side lengths in the complete simply connected space
form Ml of constant curvature k. Then for any s on the side qr and s on the
side qr with d(s, q) — d(s, q), we have d(p, s) > d(p, s) (see Exercise 4 in Chapter
IV, §4). Recently, the geometry, topology and structure of singularities of
Alexandrov spaces have been studied extensively (see, e.g., [Ale], [Bu-G-Pe], [Pe-3], [S-4],
[Gro-3], [Ot-Shio]). See also [Gro-Pe-2, 3] for the applications of limit spaces to the
geometry of manifolds whose curvatures are bounded below.
(2) M. Anderson and J. Cheeger ([An-Ch-2]) considered the class 7£m(z0, V)
:= {M; compact m-dimensional Riemannian manifold with p(u) > -1, i(M) >
z0, vol Μ < V} and showed that this class is precompact with respect to the CQ-
topology for any 0 < α < 1 (see also [Gao-1]). The main ingredient in the proof is
the uniform estimate of the radius of metric balls on which harmonic coordinates
may be introduced. They also proved the diffeomorphism type finiteness
theorem for the class {M; compact m-dimensional Riemannian manifold with \p(u)\ <
m - 1, vol Μ > ν, d(M) < d, JM \\Щ\ш^аид < Л}, where ||Я|| denotes the norm
of the curvature tensor ([An-Ch-1]). In this case we have no uniform injectivity
radius estimate. However, a limit space may carry at most a finite number of

316
APPENDICES
(orbifold) singularities, and the detailed analysis of singularities implies the above
result (see also [Band-Kas-Nak]). We refer to [Yang], [Gao-1,2] for the convergence
of Riemannian manifolds with integral bound on curvatures, and applications.
(3) If we consider a family Mm(d) '·= {(Af, g); compact Riemannian
manifold with Κσ > — 1, D(M) < d}, then we do not even have a homotopy type finite-
ness theorem (consider, e.g., lens spaces of constant curvature 1). However, M. Gro-
mov ([G-5], [Ch-5]) showed that there exists an explicit positive constant c(ra, d)
such that for the sum of Betti numbers we have Σ rank Hi (Μ; Κ) < c(m, d) for
any Μ e Mm(d) and any field K.
6.3. In this subsection, we are concerned with the collapsing phenomena of
Riemannian manifolds (as general references we refer to [Pan], [F-4, 5], [Tus-1]).
Recall that for a compact Riemannian manifold (M, g), the (M, eng) converge
(or collapse) to a point as en —► 0 (Example 1°). However, since Ka(eng) =
e~1Ka(g), the absolute values of the sectional curvatures are not bounded as en —» 0,
unless (A/, g) is flat. In general, let {(Mn, gn)}%Li be a sequence in Mrn{d) :=
{(A/, g)\ compact m-dimensional Riemannian manifold with \Κσ\ < 1, d(M) < d}.
Then by the precompactness theorem we may assume that {(Mn, gn)} converges
to a length space X, taking a subsequence if necessary. If dim .AT < m, then we
say that (Mn,gn) collapse onto X (this is equivalent to the fact that the infimum
of the injectivity radii i(Mn) is equal to 0). For instance, consider a flat torus
Τγ := Дт/Г, where Γ = (ei, ... , ет)я is a lattice. For e > 0 take lattices
Г6 := (ei, ... , e*;, ee^+i, ... , £ет)я- Then flat tori Tpe := Rm/T€ collapse onto
a A;-dimensional flat torus as e —> 0. Also from Example 3°, (normalized) Berger
spheres collapse onto complex projective space.
Similarly consider a sequence of 3-dimensional spherical space forms {M{ =
S3/ri}?ll, where I\ С SO(4) are cyclic groups generated by
\R{l/rn) 0
[ 0 Щкг/щ)
where we set
о/т _ [cos 2tt0 — sin 2π0ΐ
W~ |_8ίη2π0 cos2^J *
The M{ are lens spaces. Now if {щ} —» +oc while {ki} is bounded, then the
Mi collapse onto S2 (taking a subsequence if necessary), where S2 may admit
singularities in general. If щ, ki —» +oc, the Mi collapse onto the interval of length
π/2. We may show that the length spaces belonging to the boundary of the family
of all 3-dimensional elliptic space forms with respect to d# are homeomorphic to
either S2 or the interval J.
We give one more example. Let a torus Tk act isometrically on (M, g) £
Μ mid), and suppose the isotropy group of Tk at every point ρ e Μ is not equal to
Tk. Take a one parameter subgroup I : R^>Tk that is dense in Tk. Then the
vector field ν οώ Μ defined by the flow ψί{ρ) := I(t)p vanishes nowhere. For e > 0
define a new metric ge by ge(v, v) := eg(v, v), ge(v, w) = 0 for w±v, and g€(w, w) :=
g(w, w) for w±.v. Then (M, ge) collapses onto M/Tk as e —► 0, where the curvatures
of (M, ge) are uniformly bounded.
Let {Mn} be a sequence of compact Riemannian manifolds with uniformly
bounded curvatures, and suppose {Mn} d#-converge to a compact length space
X. We do not know much about the metrical or topological relation between
((Пг, ^г) = 1),

6. GROMOV'S CONVERGENCE THEOREM
317
Mn and X in general. W. Tuschmann ([Tu-2]) showed that there exist surjective
homomorphisms Φη : π\(Μη) —> X for sufficiently large n.
Now a fundamental question for the collapsing phenomenon is the following:
When does a sequence of Riemannian manifolds in Aim(d) collapse onto a point?
The following almost flat manifold theorem of M. Gromov and E. Ruh is one of the
most striking results in Riemannian geometry ([G-2], [Ru-3]).
Theorem 6.13. There exists an e = e(m) > 0 with the following property:
For any compact Riemannian manifold Μ G Mm(e) there exist a simply connected
m-dimensional nilpotent Lie group N and a discrete subgroup Γ of the semidirect
product Aut(TV) oc N such that TV Π Γ is of finite index in Γ and Μ is diffeomorphic
to Ν/Γ. Namely, the fundamental group Γ of Μ is a finite extension of a lattice
in N by an affine group of N, with respect to a connection for which left invariant
vector fields are parallel. Such an Ν/Γ is called an infra-nilmanifold.
In particular, it follows that a compact m-dimensional Riemannian manifold Μ
with max \Κσ\·ά(Μ)2 < e(m)2 is an infra-nilmanifold. If we normalize the metric so
that d(M) = 1, then the above condition means that max \Κσ\ is sufficiently small
(i.e., < e(m)2). On the other hand, note that infra-nilmanifolds carry Riemannian
metrics with max \Κσ\ ■ d(M)2 < e for any e > 0 (see the last remark in Appendix
2), and such metrics normalized to satisfy \Κσ\ < 1 collapse onto a point. We give
a sketch of the proof of the above theorem following [G-2], [Bus-Ka], [Ru-3] (see
also [F-4], [Gha-1, 2] for other proofs).
The assumption of almost flatness means that if we normalize the metric so
that d(M) = 1, then expp | BR(op; TPM) —► Μ is regular for very large R > 0
at arbitrary ρ G M. Now note that the above theorem generalizes the Bieberbach
theorem (Chapter IV, §1) for the flat case. If Μ is flat, then expp : TPM —► Μ is
a Riemannian universal covering and the deck transformation group Γ = πι (Μ; ρ)
is a discrete subgroup of the affine group 0(m) oc Rm. Further, every element
of Γ is uniquely determined by a geodesic loop at p, and the rotational part of
the corresponding deck transformation is given by the parallel translation along
7. The essential point of the proof of the Bieberbach theorem is to show that if
the rotational part r(*y) of 7 G Γ is small, i.e., ||r(7)|| := max{Z(x, r(j)x)\ χ G
UPM} < \, then 7 is in fact a translation (i.e., ||r(7)|| = 0). It follows that
Λ := Γ Π R™ is a normal subgroup of Γ, and Η := Γ/Λ is a finite group of order
< 2 · (47Г)^(^-1)/2 ьу the volume estimate.
In the almost flat case, for an arbitrary fixed point ρ G M, we consider the
metric g* induced on Br(op\ TpM) from g via expp. Next take a p with 0 < ρ <
Я/2, and take an orthonormal frame field χ G Bp{op\ TPM) i-> u(x) by parallel
translating an o.n.b. u0 of TPM along radial segments. Now we set
(6.14)
np := {φ : Bp(op; TP(M)) —► В2Р{ор\ ТРМ))\ φ is an isometric
embedding with φ(ορ) G Bp(op; TPM)), βχρροφ = expp},
which may be identified with Bp(op; TPM) Π exp"1^}) = {short homotopy class
of geodesic loops at ρ of length < p). Here, short homotopy means that loops
constituting the homotopy are of length < p. We easily see that every short homotopy
class is represented by a unique geodesic loop at ρ of length < ρ by a lifting
argument. For φ G Πρ, we get a geodesic loop α given by the image of the segment
from op to φ(ορ) under expp. Conversely, to a geodesic loop α : [0, 1] —► Μ at

318
APPENDICES
ρ with \a\ (:= L(a)) < p, we assign a local isometry φα £ Up determined by the
condition that φα(ορ) = ά(1) and Όφα(ορ) = Dexp~1(a(l)) oDexpp(op), where ά
denotes the lift of α to TPM emanating from op via expp. Now for α, β £ Up with
\a\ + |/?| < ρ we may consider the product β * α := ψβ ο φα, which is represented
by the geodesic loop in the short homotopy class of α U β. ΐΙρ is not a group (it is
a so-called pseudogroup) and does not determine the fundamental group of Μ in
general. However, in the almost flat case, it turns out that Up determines the
fundamental group for some ρ by careful investigation of the above product structure
of Πρ.
Now for a £ Πρ we denote by r(a) £ 0(m) = 0(TPM) the parallel
translation along the geodesic loop a, which is equal to u~l ο Όφ~ι о и(а(1)) under
the above identification. Recall that we have a bi-invariant distance d(A, B) :=
max{Z(Ar, Bx)\ χ £ 5m_1} = ||A_1B|| on 0(m). In the flat case, the above
r : Пэс —► 0(m) is a homomorphism. In the almost flat case, r : Up —* 0(m) is an
almost homomorphism in the sense that rf(r(/3)or(a), τ(β*α)) is sufficiently small.
Now the first ingredient in the proof of the almost flat manifold theorem is the
following analogy to the Bieberbach case. The proof itself becomes technically long
and difficult because of the error terms caused by almost flatness (see [Bus-Ka]).
Lemma 6.14. For any small θ > 0 there exists an e = e(m, Θ) > 0 such that
the following: Let (Μ, g) be an m-dimensional compact Riemannian manifold with
d{M) = 1 and max \Κσ\ < e2. Set w = w(m) := 2(14)τη(τη"1)/2. Then there exists
a p(> 104w) such that the following assertions hold.
(1) If α, β e UP satisfy d{r(a), r(/3)) < 0.47, then d{r{a), r(/3)) < Θ. The
relation ~ defined by a ~ β «=> d{r{a), r(/3)) < 0.47 «=> a~l * β <E Tp := {a <E
Πρ; ||r(a)|| < 0.48} is an equivalence relation.
(2) Η := Up/ ~ /ias ί/ie structure of a group, where the product in Η is
given by the product of representatives. Η is a finite group whose order is less than
or equal to the above w. Every element of Η is represented by an a £ Up with
\a\ < 2w.
Note that for each [a] £ #, {τ(β)\ β ~ a} is contained in a small convex
neighborhood in 0(m). Then we may consider its center ωο([α]) £ 0(m), and we
have an almost homomorphism uuq : Η —* O(m). Then by the center of mass
technique ([Ka-1], [Bus-Ka]) it is possible to deform uuq to a homomorphism ω :
Η —► O(m) that is close to ωο and therefore injective. We consider Я as a subgroup
of O(m).
Ruh ([Ru-3]) considered the principal orthonormal frame bundle O(M) of M,
and its reduction to the subbundle Q with the structure group H. In fact, for q £ Μ,
the subset {(Dexpp(x)u(x))/i; χ £ exp"1^), /ι £ H} of the fiber 0(Μ)ς = 0(m)
of O(M) over q is divided into ЦЯ equivalence classes with respect to the above
equivalence relation " ~ ", where each equivalence class is contained in a small
convex neighborhood of 0(m). Then we may consider the center of each equivalence
class with some appropriate weight, and get a subset Qq of centers in O(M). Η
acts on Qq simply transitively, since centers are preserved by the action of H. Then
Q = U Qq ls tne desired subbundle. Note that Q determines a flat metric linear
connection Df on M, since Я is a finite group. Ruh showed that the sup-norm of
the torsion of D' is very small, because of the assumption of almost flatness. Also
note that π : Q —► Μ is a covering map. Denoting by D the linear connection on

6. GROMOV'S CONVERGENCE THEOREM
319
Q induced from D', we see that Η acts on Q as an affine transformation group,
and we have a /^-parallel frame field ω on Q.
The second ingredient in the proof is to construct a flat Я-invariant metric
linear connection D0 on Q near D, such that the torsion Τ of D0 is parallel and its
sup-norm ||T|| is sufficiently small. This was established by Ruh, solving a partial
differential equation via the Nash-Moser iteration method. The proof is difficult,
and we refer to [Ru-3] and [Gha-1]. Note that D0 descends to a linear connection
on M.
Then it turns out that Q is a nilmanifold, namely, there exist a simply connected
m-dimensional nilpotent Lie group N and a discrete subgroup Λ of N such that
Q = N/A. In fact, let Μ be the universal covering space of Q with the induced
connection D0 from D0, and consider
η := {X\ 7Ti(Q)-invariant parallel vector field on Μ }
= {parallel vector field on Q },
which is a vector space of dimension m. Now for Χ, Υ £ η we define [X, Y] :=
T(X, Y) £ n, where Τ stands for the parallel torsion of D0 with sufficiently small
sup-norm. Then this defines a Lie algebra structure on η because of the Bianchi
identity and the parallel torsion. Note that any X £ η is a complete vector field on
Μ, and we have a simply connected Lie group TV, with Lie algebra n, that acts on
Μ as a Lie transformation group through flows 4>t{X) generated by X £ n. Then
we may show that, for a fixed e £ M, expp : TeM —► Μ is a diffeomorphism by
solving the Jacobi equation and using the fact that Τ is parallel. Also we may easily
see that F : Μ —► N defined by F(expe X) := φ\(Χ), X £ η, is a diffeomorphism.
Therefore, we may assume that Μ itself is a Lie group with the identity e.
Now let A be the affine transformation group of M, and note that any α £ A
is uniquely determined by g := a(e) and A := Da(e) £ 0(TeM, TgM), since Д, is
a flat connection with the parallel torsion. Then setting A\ := {a £ Л; Da(X) =
X for any X £ η } and Λ2 := {a £ Л; a(e) = e}, we get A = A2 ос Ль where Д1
is a normal subgroup of A isomorphic to the Lie group Μ = N.
Now we turn to the universal covering π : Μ —> Μ with the deck
transformation group Γ, which acts on Μ preserving D0. Then Λ := Γ Π Αι is equal to
{a £ Γ; r(a) = id}, and therefore is a normal subgroup of Γ that is isomorphic
to the deck transformation group of πι : Μ —* Q. The kernel of the canonical
projection Γ —► A/A\ is equal to Λ, and its image is isomorphic to Я, which is
the deck transformation group of the covering Q —► Μ. Therefore, Γ is a group
extension of Λ by H. Since Q = Μ/Λ is compact, we see that Λ is a lattice in M.
Finally we show that Μ is a nilpotent Lie group. In fact, this follows from
the following Zassenhaus-Kazdan-Margulis lemma ([Rag]): Let TV be a Lie group.
Then there exists a neighborhood U of the identitity such that for any discrete
subgroup Λ of TV, Λ Π U is contained in a connected nilpotent Lie subgroup of N.
Now we apply this to the above Μ and Λ. Note that the above exp : η —► Μ is
a diffeomorphism, and we have ||[x, y}\\ < C||x||||j/||, x, у £ η, for sufficiently small
C. It follows that we may choose U so that it contains a fundamental domain of
the lattice Λ. Then the dimension of the nilpotent subgroup containing Λ Π U is
equal to m, and Μ itself is nilpotent. D

320
APPENDICES
Now we briefly mention some generalizations of the above theorem. K. Fukaya
([F-2, 4]) considered the case where {Mn}ncL1 С Mm(d) collapses onto a compact
Riemannan manifold N. Consider, as in the proof of the diffeomorphism theorem,
I : N —* Z/2(7V), In : Mn —* L2(N). Roughly speaking, we may expect that for
sufficiently large n, In(Mn) lies in a tubular neighborhood of I(N), and get as
before maps fn : Mn —* TV, which are regular and therefore fibrations. Since we
have no uniform estimate from below for the injectivity radii of the Mn, we need
some modifications, and the proof of the following fiber bundle theorem ([F-2]) is
more elaborate.
Theorem 6.15. Suppose {Mn}n<L1 С Мш{а) collapses onto a compact Rie-
mannian manifold N. Then, for sufficiently large n, we have fibrations fn : Mn —*
N such that
(1) Fibers of fn are infra-nilmanifolds Gn/Tn {defined as in Theorem 6.13)
and the structure groups of fn are contained in Aut(rn) oc C(Gn)/(C(Gn) ΠΓη),
where C(Gn) denotes the center.
(2) The fn are almost Riemannian submersions in the sense that there exist
en —► 0 such that the inequalities
exp(-en)<\\Dfn(v)\\/\\v\\<exp(en)
hold for horizontal vectors ν of Mn.
The pointed Hausdorff distance version also holds: Let {Mn} be a sequence in
М.ш := {(Μ, g)\ complete m-dimensional manifold with max \Κσ\ < 1}. Let N be
a complete Riemannian manifold of dimension less than m such that max | Κσ \ < 1
and the injectivity radius i^ > с > 0. Then, if Mn —► N with respect to dp,#,
the same conclusions as before hold. See also [Gha-MO-Ru] for a local structure
theorem in terms of nilmanifolds.
On the other hand, T. Yamaguchi ([Ya-2]) considered the case where the
sectional curvatures of Mn are only bounded below, i.e., Κσ > — 1. Then under the
same situation we still get fibrations fn : Mn —» N for large n's, and they are
almost Riemannian submersions. Moreover, the (real coefficient) first Betti number
b\ of the fiber Fn is less than or equal to dim Μ — dim TV, and a finite covering of
Fn fibers over a b\-torus. In particular, there exists an e = e(m) > 0 such that if
a compact m-dimensional Riemannian manifold Μ satisfies min Κσ · d(M)2 > -e
and b\ = m, then Μ is diffeomorphic to the torus.
W. Tuschmann [Tu-1] considered the case of collapsing onto a compact flat
manifold. See also [Ot-2] for the collapsing phenomenon where the limit spaces are
complete Riemannian manifolds.
With full use of collapsing theory and various versions of Margulis' lemma, K.
Fukaya and T. Yamaguchi obtained the following remarkable almost nonpositive
(nonnegative) manifold theorems, which settle Gromov's conjectures ([F-Ya-1, 2]).
Theorem 6.16. For each positive integer m, there exists a positive e := e(m)
such that if a compact m-dimensional Riemannian manifold Μ satisfies
maxKa · d(M)2 < €,
then the universal covering space of Μ is diffeomorphic to Euclidean space.
In fact, they obtained more detailed information on the structure of such an
Μ (see [F-Ya-1]).

6. GROMOV'S CONVERGENCE THEOREM
321
Theorem 6.17. (1) There exists a positive e = e(m) such that if a compact
m-dimensional manifold satisfies min Κσ · d(M)2 > —e, then the fundamental group
7Γι(Μ) of Μ is almost nilpotent, namely, π\(Μ) cotains a nilpotent subgroup of
finite index.
(2) (Generalized Margulis Lemma). There exists a positive e = e(m) such that
for any complete m-dimensional Riemannian manifold Λ/ with Κσ > — 1 and any
ρ G M, the image of the inclusion homomorphism n\(Bc(p)) —► πχ(Μ) is almost
nilpotent.
There is another approach to the collapsing phenomenon, due to J. Cheeger and
M. Gromov. Let {(Mn, gn)} be a sequence of complete m-dimensional
Riemannian manifolds with s\xp\Ka(gn)\ < 1. Suppose the injectivity radii of (Mn, gn)
go to zero as η —► ос, and take pn G Mn with δη := iPn(Mn) —► 0. Now
consider the rescaled manifolds {(Mn, hn := 6~2gn)}, and note that iPn(hn) = 1 and
\Κσ((Μη, /ιη))| < <5n ""* 0. Then, by the precompactness theorem and the
convergence theorem, we may assume that ((Mn, /in), pn) converge to a flat pointed
Riemannian manifold (X, p) with respect to the pointed HausdorfF distance, taking
a subsequence if necessary. Then X is difFeomorphic to the normal bundle of a soul
5 of X, which is a compact flat totally geodesic submanifold of X. Therefore, by the
Bieberbach theorem a finite cover of 5 is a torus, and a finite cover of X admits an
isometric action of a torus. By the convergence theorem, the metric ball Вц(р, X)
is d^-close to BR(pn, (Mn, hn)). Therefore, we have a neighborhood U of pn in
(M, gn), its finite cover f/, and an action of a torus Tk on f/, where the dimension
of the torus may change if the base point varies. Cheeger and Gromov introduced
the notion of F-structure, which indicates how these local actions are compatible
to each other ([Ch-G-2]). They showed that there exists a critical positive number
6 = e(m) with the following property: Let Μ be a complete Riemannian manifold
with \Κσ\ < 1. Then there exists an open set ί/cM, which is roughly speaking the
thin part of Μ, such that there exists an F-structure of positive dimension on /7,
while ip(M) > e for ρ G M\U. For more details on the collapsing phenomenon and
such T-, F- and TV-structures we refer to [Ch-G-2], [Ch-F-G]. Fukaya's excellent
survey articles ([F-4, 5]) will help the reader to understand the theory of HausdorfF
convergence of Riemannian manifolds and various applications.

Hints and Solutions to Exercises and Problems
Chapter I
§1. Exercise 2. m is even because 0 φ det(ujij) = det(uji) = (—1)™ det(ωij).
Next introduce an inner product ( , ) on V and define a linear isometry J : V —» V
by (Jx, y) = ω(χ, у). Then J is skew-symmetric and admits a 2-dimensional
invariant subspace σ := (ei, /i := Jei/||Jei||)/i· Consider the orthogonal complement
of σ and apply induction on m.
Exercise 3. In fact:
W С Сп = Д2п is a Lagrangian subspace
«=> dim W = n, W-LJW
ф=> For an o.n.b. {х{}?=1 of W, {χι, Jxi) gives an o.n.b of Cn = R2n
Ф=> For a = (0,... , 1,... , 0) G Rn and /» := Je» (г = 1,... , η),
the element ^ € 0(2n) given by the condition that
ψβΐ = Ж», V/г = «/^г belongs to U{n).
§2. Exercise 3. For the first assertion, compute ^ \t=o Όψ-ί{Υφι^) · f for
/ G ,Γ(ΛΊ). For the second, show that [X, У] = О «=> £><^У = У, ί G Д, and that
the flow of DiptY is given by 5 ι—► <pt ° 'Фз ° <£-£·
Exercise 4. Set </?(Л) := Σ^ i4fc/fc!. Show that p((i + s)A) = <p{tA)(p(sA)
and ^ |i=0 ^(^) = Ά· Next note that if AB = В A, then ехр(Л + В) =
exp A expB. For instance, to see (2.12) note that A + lA = 0 implies *(exp Л) ехр Л
= exp(* A + A) = e.
Exercise 7. Show that SO(m + 1) acts transitively on 5m and the isotropy
group Hp of SO{m + 1) at ρ = '(1,0,... ,0) G 5m is isomorphic to SO(m). The
other part may be treated in the same manner.
Exercise 8. Use the equality [X, Y]p = -^ \t=o exp(Ad(exp(-£X)y) · ρ and
(2.16).
Problems for Chapter I
2. A admits a nonzero fixed point if and only if A admits 1 as an eigenvalue.
Then note that
det(£m -A) = det{bA A- A) = det{lA - Em) det A
= (-l)mdet^det(Em-Л).
3. (1) follows from Theorem 2.1 (2). To show (2), apply (1) setting Φ(χ) :=
||z||2-r2,:rG Hm+1.
323

324
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
V2n9*Cn, ω =
4. We do only the case of 0(n). Let Φ : Mn{R) -> Sn(R) := {A G
Mn(R)· lA = A} « H"(*+i)/2 be defined by Ф(А) := 'Л Л. Then show that
at Л G O(n) we get
гапк£>Ф(Л) = ϋί^-til = dim Sn(R)
and apply Problem 3. We may also use the exponential map exp (§2, Exercise 4).
As for the dimension, we have
dimO(n) = dimSO(n) = П^П~1\ dim U(n) = n2, dim SU(n) = n2 - 1.
5. (1) For exp : u(2n) —> U(2n), take an open neighborhood £7 of 0 G u(2n)
so that exp \U is a diffeomorphism. Then show that exp maps sp(n) Π /7 diffeomor-
phically onto Sp(n) Dexp(/7). We have dimSp(n) = 2n2 H- n.
(2) Consider the action of Sp(n) on 54n_1 (C C2n). Show that Sp(n) is simply
connected by induction on n.
6. By Exercise 2 of §1 we may take
Γ 0 -Enl
Then by Exercise 3 of §1, U(n) acts transitively on Λ(η) as a Lie transformation
group.
7. For <£ G 50(3), φ Φ e, there exists an a G S2 С R3 such that </? is a
rotation of angle 0 (0 < θ < 2π) around the axis Ra. We set φ := (α,0). Then
(α', 0') and (α, 0) determine the same element of 50(3) if and only if a' = α, θ' = θ
or a' = —α, θ' = 2π — θ. To the identitiy matrix £3 there correspond all elements of
the form (a, 0), (α, 2π) (a G S2). Now we identify S2 with the great sphere x4 = 0
in S3 С R4 and take the unit vector e = (0, 0, 0, 1). We define a map Φ : 50(3) —>
ЯР3 by Φ(α, 0) := π (sin | -α + cos f e), where π : S3 —> RP3 denotes the canonical
projection. Show that Φ is a diffeomorphism. We may also give a proof using
quaternions. In fact, consider the set Q := {q = а+Ы+cj+dk; a2+b2+c2+d2 = 1}
of unit quaternions, which may be considered as S3. Now to q G Q assign the
element 7 G 50(3) defined by η(χ) = qxq~l,x G Я, where Η denotes the field of
quaternions. Then show that q 1—► 7 defines a twofold covering group.
8. Torus in R3 (see Figure 38, below (on page 330)). (2) Compute the rank
of the Jacobian matrix.
9. For an atlas Λ = {(UQ, φα)} of Μ take the atlas A = {(UQ, Ψα)} on page
7. Then the Jacobian of any coordinate transformation of Λ equals the square of
the Jacobian of the corresponding coordinate transformation of A. Note that Μ is
orientable if and only if there exists a C°° m-form on Μ that vanishes nowhere.
10. (1) Take sections {si}^=l on a coordinate neighborhood U of Μ that form
a basis of σ_1(ρ), ρ G /7, and extend them to sections {si}f=l so that they form a
basis of r_1(p). Then apply the Gramm-Schmidt orthogonalization procedure.
(2) The b defined on page 4 gives an isomorphism between τ and τ*.
11. Since τ μ is a subbundle of the induced bundle l*trti of the embedding
ι \ Μ <-^> Rn, we may apply Problem 10. Note that for an oriented hypersurface
Μ of Rn we may construct a global unit C°° normal vector field.
Chapter II
§1. Exercise 1. For a deck transformation φ we get <£*(π*/ι) = (π ο φ)*1ι = n*h.

CHAPTER II
325
Exercise 5. Let {е*}·^ be an o.n.b. of TPM. We denote by {ei(t)} the
parallel translation of {e*} along c(t), and set Yc(t) = Yl{t)ei(t). Show that both
sides are equal to Уг(0)ег·
Exercise 9. Use (3.1) of Chapter 1 and the equality
(VXiuj)(X0,... ,XU... ,Xk)
= X{ · ω(Χ0,... ,Xi,.. · , Xk)
+ Σ(-1)* -ωφχ,Χι-νχ,Κ,Χο,... ,Хи... ,Xj,...,Xk).
i<j
§2. Exercise 4. Let {x1} be the normal coordinate system given by an o.n.b.
{e,} of TPM. For и 6 TPM we denote by Yi(t) the Jacobi field along 7„ with
Yi(0) = 0, Wi(0) = a. Note that
— (exPptu) = —.
Then show that
/ 4 / d I (*ί(*)> *i(*)> «
9ij{P) = <*, ej), u-9ij = jt |t=o V V ^2 M " = О.
We may also argue as follows. With respect to the normal coordinate system we
write 7u(£) = (a4, ... , am£), и = Σαιβι. Since the 7u's are geodesies, we have
Yikj(p)ala? = 0 for any (аг), which implies that Tikj(p) = 0 and consequently
dk9ij{p) = 0·
Exercise 5. 71 U72 is a broken geodesic parametrized by arc-length, which is
a stationary curve of L in Cpr. Then Proposition 2.6 implies that 71 U 72 is C°°.
§3. Exercise 1. Indeed,
9ij = 6ij + ^Rikji(p)xkxl + -^Rikji;m{p)xkxlxm
1 1 f{
+ gi(6ftfcjZ;mn(p) + —Rkiiu{p)Rrn3nu{p))xkxlxrnxn + · · · ·
Exercise 2. Show that {? <E Μ; Φι(?) = Φ2(?), £>Φι(?) = D<&2(q)} φ Φ is
open and closed.
Exercise 5. At any point ρ £ Μ, Ric = p# implies that (i?(x, 2/)i/, ж) = § for
any o.n.b. {ж, у} by Remark 3.7.
Exercise 7. Setting
we get
*£.£>-<*w=" a
Kdxv dxi \ v* ν dxldxi dx™
(г, j = 1, ... ,m — 1). Next take an o.n.b. {е*}^, em = u, of TPM, and let {x1}
be the normal coordinates determined by {е*}. Denote by {Aij) the matrix
representation of A with respect to {e*}™^1· Now take a c°° function /(ж1,... ,xrn~l)
with
/(o) = 0, g(o) = 0, ^o) = Aij.

326
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
Then N = M/ is the desired hypersurface.
§4. Exercise 3. With respect to a normal coordinate system on Μ, we get
[X, Y]z := (0, Z\ XY* - YX\ -{X · Tjik)ZjYk + (Y · r/fc)ZjXfc).
Exercise 5. Let Г (t) be the Jacobi field along 7ξ with Y(0) = 0 and VY(0) =
ξ. Note that the left-hand side of (i) is equal to Y(t) = ί^(ί). (ii) may be proved
in a similar manner.
§5. Exercise 3. (1) Use the formula
y/det9ij{t,u) = 1 - ^-t2 + o(t2).
(2) Take an o.n.b. {e*} consisting of the eigenvectors of the Ricci curvatures
with eigenvalues p\. Then we may write p(u) = Y^PiU2, τ = Σρί, where we set
и = Σηίβΐ· Note that
/ «tdS™-1 = -Ctm-l = U>m.
yu2+...+u2i=1 m
Problems for Chapter II
1. Take a normal coordinate system {x1} around c(t) and write c(s) = (xl(s)).
Then
d(c(t), c(t + ft)) = ^/5><(* + ft))2
with х*(Ь) = 0.
2. (1) We may assume that с is of class C1. Since ||c(£)|| is uniformly
continuous on [a, b], for any e > 0 we may take a subdivision Δ of [a, b] such that
5^d(c(fi_i), c(U)) > Σ l|c(*i-i)ll(*i - *t-i) - €.
Thus we have Ld(c) > Lg(c). The reverse inequality is obvious.
(2) Take a sufficiently fine subdivision Δ of [a, 6] so that c(ii-i), c(£i) are
connected by unique shortest geodesies C; : [U-ι, U] —» M. Then show that
c([ii-i, U}) = Ci([U-i, U]). Next, taking another subdivision and using Exercise
5 of §2, show that the broken geodesic (J C{ is in fact a geodesic.
4. {хг} is a normal coordinate system Ф> <7ij(o) = $ij and £ н-> (fol) are
geodesies Ф> <7ij(o) = 6{j, Tikj(tx)xlx^ = 0 Ф> <7ij(o) = uj, ^{<7u(to)^} = 0 Φ>
gij(tx)xi = хг. For (=>) use the Gauss lemma.
5. (1) Use the argument of Lemma 2.7.
(2) д/дв are Jacobi fields perpendicular to radial geodesies with (д/дв, д/дв)
= /2(r, 0), and j£q are parallel.
(3) Use the Jacobi equation.
6. Such computation of the Ricci curvatures gives nice examples of Riemannian
manifolds of positive Ricci curvature with interesting properties (for more details
see, e.g., [Sha-Ya-1,2], [Ot-1]).
7. Let {ei}^ be an o.n.b. of TPM consisting of the eigenvectors of Ric(p) with
the corresponding eigenvalues {рг}. Then ||Ric||2 = p\ Η hp^, τ = p\ Λ bpm.
(1) Use the Cauchy-Schwarz inequality.
(2) ΣρΙ < (m - 1) Ei^(R(ej, efa, ef)2 < ^||Д||2.
8. Let ρ G Μ, ρ := f(p) G M. Take normal coordinate neighborhoods U :=
Be(p), U := Be(p) and show that f(U) = U, where we use the fact that / is
surjective. Next define a map F : UPM —» /7pM by F(u) := -^ \t=o f(lu(t)), where

PROBLEMS FOR CHAPTER II
327
f(7u(t)) may be verified to be a normal geodesic joining ρ to /(7u(e)). Extending
F to a homogeneous map from TvΜ to TpΜ, which will be denoted again by F, we
see that F preserves the norm and satisfies / о expp = expp oF. Then it suffices to
show that F is a linear isomorphism. To see this, set α := Z(u, v), u, ν £ UPM.
Checking that
С08а = !То Ж '
we know that (F(u), F(v)) = (u, v). Then F is easily seen to be a linear map. It
follows that / is of class С°° and Df(p) = F, i.e., f*g = g.
Even if dim Μ = dim Μ we need the assumption that / is surjective. For
instance, for Μ = Д2\{(п, 0); η = 1, 2, ... } consider the map F(x, y) := (x—1, y).
If dim Μ = dim Μ and Μ, Μ are complete, then the surjectivity follows from the
assumption (for more details see [Pa-1], [Hel]).
9. (1) The TPM, ρ £ Μ, are integral submanifolds.
(2) Use Theorem 2.2 of Chapter 1 and Exercise 3 (ii) of §4.
11. Let {et(t)} be a pararell field of o.n.b. along c(t), and write Y(t) =
Yl(t)ei(t). Show that both sides are equal to Yi(a)ei(a).
12. (2) Let {ej, {f3} be o.n.b.'s of Tp5m, TqS™, respectively. Then {p, ej,
{q, fj} are o.n.b.'s of ilm+1, and there exists a unique element of 0(m + 1) that
maps {p, ei} to {q, fj}. The second assertion follows from Problem 2 of Chapter
II. The isometry group of (RPm, go) is also given by 0(m + 1).
13. (1) Compute the Jacobian matrix of u. Note that (ho)ij = Sij — χιχϊ/t2.
(2) Show that for any two o.n.b.'s of (H™, ho), there exists a unique element
of 0(1, m) that maps the one to the other.
14. Let F be a connected component of the set of fixed points of φ, and set
Vp := {u £ TPM; Όφ(ρ)ιι = и}. Then expp Vp is contained in F. Show that for
a normal coordinate neighborhood U centered at ρ we have F Π U = expp Vp Π /7,
and dim Vq (q £ U) is constant (for more details see [Ko-1]).
15. (1) follows from the Jacobi equation.
(2) A"(0) = О, Л<3)(0) = -Ru, Л(4)(0) = -2VUR , etc.
(3) Y(r) := A(r)x is a Jacobi field along 7 tangent to 5r(p). Show that
Ау(г)У(г) = VY(r) = A'(r)x (for more details see, e.g., [Dj-Van]).
16. (1) Recall that the volume element of 5m_1 С Дт is given by
Σ(-1ΓЧЧ1 A · · · Л de* Л · · · Λ άΓ,
where (£г) denotes the coordinates of Дт. Now for и £ /УМ take a normal
coordinate system around ρ = тм(и). Let (x\ ξ1) be the corresponding coordinate
system in TM. We have gijC£j = 1 on UM. Also, at и = ΣΓ^|τ € t^M we get
r/ = ^ftf, A7 = - Σ άχϊ Λ d? ·
Therefore, at u, η Λ (dr/)m_1 is equal to
(m - l)!^1 Л-ЛатЛ (^i"1)*"1^1 Λ · · · Λ d£* Λ ■ ■ ■ Λ <0
up to the sign, which is (m — 1)! times the volume element of UM at u.
(2) is clear, since фг leaves η and άη invariant.

328
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
Chapter III
§1. Exercise 3. (1) Show that any geodesic 7 of Μ χ Ν may be uniquely written
as η(ί) = (7(t), 6(t)), where 7, δ are geodesies of M, TV, respectively, and vice versa.
Furthermore, 7 is minimal if and only if 7, δ are minimal.
(2) In the above, note that ||4(*)l|2 = II7WII2 + \\Щ\\2-
Exercise 4. Let {pn} CiVbea sequence such that d(q, pn) —» d(p, N). Show
that {pn} admits an accumlation point ρ £ N. A minimal geodesic joining ρ to ς
is what we want.
§2. Exercise 3. Note that
<v^, vx-) - (v.,.* v.,.*) - (ν*.* £)>·
Exercise 4. If В is totally geodesic, we get
D2L(7)(X, Y) = \J {(VXX(0, Vy^W) - ДО^Ю, 7(*))7(*), У'МЖ
If B = JVi x 7V2, we get
D2L{^){X, Y) = the right-hand side of the above equality
+ у{(^(а)ВД, У(а)) - (AmX(b), Y(b))}.
§4. Exercise 3. See Figure 37. The segments with the same arrow are identified
to each other under expp (ρ = π(ο)), and give the cut locus Cp.
(<7h a2)
Figure 37
Exercise 4. For m > 3 the assertion follows from Proposition 4.5 (1). For
m = 2, first suppose Cp is simply connected. Then also Μ is simply connected, by
Proposition 4.5 (1). Next suppose Μ is simply connected. Then Μ is homeomor-
phic to 52. If Cp is not simply connected, then Cp contains a simple closed curve
c, and we may write Μ \ с = D\ U D2, where D\, D2 are disjoint union of domains
in 52. We may assume that ρ £ D\ and take a point q e D2. Then a minimal
geodesic 7 joining ρ to ς intersects с С Ср, which is a contradiction.
Exercise 5. Suppose Cp C\Qlp = φ. Then Cp is a closed curve, and the
argument of Exercise 4 works.
Exercise 6. Suppose Μ is not simply connected, and consider its universal
Riemannian cover π : Μ —» Μ. For different points p, q € π~λ(ρ), take a minimal
geodesic 7 joining ρ to q. Consider the cut point of ρ along the geodesic loop 7 = ποη
at p, which is conjugate to ρ along 7. Then there also appears a conjugate point to

PROBLEMS FOR CHAPTER III
329
ρ in the interior of 7, which is a contradiction.
§6. Exercise 1. (2) For ρ G M, I$(M, g) · ρ is a connected open and closed
subset of M.
Exercise 2. At a critical point ρ of /, we have V χρΧ = 0 because VX is skew-
symmetric. On the other hand, tpt leaves X invariant. Then we get V^ {p)X =
ϋφί(νΧρΧ) = 0.
Exercise 5. Prom R(x, y)z = — (x, z)y + (y, z)x, we see that
1. /,-./ \ m lf, m(m — 1) , . .
dim(#(:r, y);x, у G TpM)r = —^— = dimo(m).
Exercise 6. Use Lemma 6.7 (1).
Problems for Chapter III
1. We show that any sequence {рп}^=1 С М admits an accumlation point.
With respect to a Riemannian metric g. take normal minimal geodesies ηη : [0, ln]
—» Μ joining a fixed point ρ to pn. We may assume that ln —» +00, pn £ Cp, ηη —»
7· Then 7 : [0, +00) —► Μ is a minimal geodesic joining ρ to 7(2) for any t > 0.
Now we take open neighborhoods U, V (V С U) of the set [0, +00)7(0) in TPM such
that expp | U is a diffeomorphism and the sets [0, /n]7n(0) are contained in У for
sufficiently large n. Take a positive C30 function <p on Л/ so that φ(βχρρ ν) = e~"v"
on {expp v\ ν G V, ||v|| > <5}. Then show that {pn} is a bounded sequence with
respect to a (complete) Riemannian metric g\ := </?2<7 (see [No-Oz] for more details).
2. Let {ej be an o.n.b. of TpN(p e N). Let Y{ (i = 1, ... , η = m - 1) be ΛΓ-
Jacobi fields along a normal geodesic ηνρ perpendicular to N that satisfy the initial
conditions Yi(0) = e*, VYi(0) = AUpe{. Then note that the Riemannian measure
dAt of Nt with respect to the induced metric at ^Up(tu(p)) is given by
and we get
lYiituip^A-'-AYm-^tuipmdA,
-£| ||У1(0л---ЛУт-1(0|| = (т-1)т,(р).
Therefore, considering the normal exponential map, we get
vo\nNt= [ \\Yi{tu{p))A.-AYm-i(tu(p))\\dA,
JN
Г ftu(p)
νο1Ωί = νο1Ω+/ dA / ||Yi(s) Λ · · · Λ Ym-i(s)\\ ds,
JN JO
which implies the first assertion. Next suppose that fNηudA = 0 for any и with
JNudA = 0. To show that 77 is constant, set и := r/ — ^j^ fNηdA and use the
equality case of the Cauchy-Schwarz inequality. Here we need not assume that Ω
and N are connected.
3. Put gt = ft 9 and g = g0. Then with respect to a local coordinate system
(x*) of N we have dvQt = J(p, £)ώ/ρ, J(p, t) = {det(gt)ij}l/2/{detgij}l/2. From
this we get

330
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
On the other hand, assuming that {д/дхг(р)} is an o.n.b. of TP7V, check that
d
dt
J(P, t) = -£>(«/&:*&>), д/дх'(р)), X±{p)) + divXT(p).
\t=o ι
Then note that
д д
?<·(
дх*{рУ дх\р)
,X^(p)\ = -{nH{p),X{j>))
and apply the Green theorem.
4. For a curve 7 of M, show that £(7) > 1(107). To verify (1), for a small
arc of 7 take a vector field X in a neighborhood of the arc such that X^t) = 7(£)·
Then consider the horizontal lift X of X in Μ, and take an integral curve 7 of
X through p. Then 7 is a horizontal geodesic in M. (2) follows from (1). In (3)
the converse does not hold in general. Take the Riemannian product of a complete
Riemannian manifold and a noncomplete Riemannian manifold, and consider the
projection to the complete factor.
5. (1) Use the fact that if w = Ju, then Κ(ήη(ί), w(t)) takes the maximal
value 4 of the sectional curvatures, and R(w(t), ju(t))ju(t) = 4w(t) (see (6.11) of
Chapter II). The similar argument also works for the case w±.u, Ju.
(2) Consider the Riemannian submersion π : (52n+1, go) —> (CPn, /i0) of
Chapter II, (6.8). Let t н-> %(t) = (cost, щ sint) and %(t) = (cost, vismt) £
Cn+1(z = 1, ... ,n) be the horizontal lifts of 7^, ηυ (ν φ и) emanating from
(1, 0, ... , 0) e 52n+1, respectively. Then look for the value of 0 < t < π for
which π ο ηη(ί) = π ο %(t).
(3) Prom (1) and (2) we have
CP = {7и(тг/2); и e UPM} = {(0.4,:...: un); (tib ... , un) e S2^1}
Finally, the volume is given by vol CPn = πη /η\.
6. Note that a geodesic 7 in Μ χ TV may be written as η(ί) = (7i(£), 72(£))>
where 71, 72 are geodesies in M, iV, respectively. Then j(t) is a cut point of 7(0)
if and only if at least one of 71 (t), 72(0 is a cut point of 71 (0), 72(0), respectively.
7. See Figure 38. Cp is given by the thick lines.
Cffl
Figure 38
8. Let a(t, s) be a variation of 7U generated by Y. Then
^Μ№).^»-(ν£^|)(Γ.0)-(|,ν4|)(Γ,0)
= -<y(r),W(r)>.
9. Let G be the isometry group of Μ and A the normalizer of Γ in G. First
show that if Μ is homogeneous then A acts transitively on M. Next show that

CHAPTER IV
331
the identity component В of A is contained in the centralizer of Г, and В also acts
transitively on M. For more details on Clifford translations, see, e.g., [Wo-1].
10. Let е6уСГТ (6 £ R, \b\ < π) be an eigenvalue of A. Then for the
corresponding eigenvector χ £ 5m the spherical distance between χ and A x is |6|.
11. Recall that for a Jacobi field Y(t) with У(0) = Л, VF(0) = B we get
D<j)t{A, B) = (Y(t), VY(t)) (Chapter II, Lemma 4.3). Then if the </>t are isome-
tries, we get (VY{t), VY(t)) + (Y(t), Y(t)} = canst, for any Jacobi field Υ on M.
Differentiating both sides, we get (Y — R(Y, 7)7. VY) ξ 0 by the Jacobi equation.
12. Suppose there exists a neighborhood U of qQ = expp uQ (uQ £ Cp) such
that we have only one normal minimal geodesic joining ρ to any q £ С/ПСр, and
derive a contradiction.
13. First note that 7\(p)A/ = 0i=o Όφ(ρ)(Όί(ρ)), where we have further
ϋφ(ρ)(Όο(ρ)) = ϋο(φ(ρ)), and components Όφ(ρ)(Όί(ρ)) (г = 1, ... , к) are
irreducible. Then the last assertion follows from Lemma 6.12. Next let φ £ Io(M).
Take a C°° curve ^ (0 < t < 1) in I(M) joining the identity to φ. Setting
r(t) := 4>t{p), show that for χ £ Όι(ρ)(φ 0), Dtpt(p)x cannot be orthogonal to
Р(т)?Д(р) = Di(vt{p)), which implies that Όφ{ρ)(Όι(ρ)) = Όι{φ(ρ)). Using this
fact, show that if we write
ψ(Ρο, · · · , Pk) = (φο(Ρο, · · · , Pit), · · · , ^fc(Po, · · · , Pk))
then ifi(po, · · · , Pit) depends only on pi. Then y? 1—► ((/?o, · · · , <^fc) gives an
isomorphism between Iq(M) and Iq(M) χ · · · χ /ο(Μ)·
Chapter IV
§1. Exercise 2. Apply Problem 8 for Chapter III to Y(t) = s6{t)E(t).
Exercise 4. Suppose Μ satisfies the axiom of plane. Let {u, v} be orthonor-
mal vectors in TPM. Then by the Gauss formula we may write R(u, v)u = ku^vv.
Show that ku,v does not depend on {u, v} in TPM, and apply the Schur lemma.
Exercise 5. Use the Law of Cosines.
Exercise 6. Apply Problem 12 (2) for Chapter II to the deck transformation
group of the universal Riemannian cover of Μ.
§2. Exercise 2. Let Y(t) = YT{t) + Ух(*) be the decomposition of Y(t) into
tangential and vertical components with respect to *y(t). Note that YT(t) =
{\\Y(0)\\ + (u, VF(0)) t}7(t), etc. Then apply Theorem 2.3 to Y1- and Y±.
§3. Exercise 1. (1) follows from
{volBr3(p)-volBr2(p)}/(vr3(6)-vr2(6)) < volBr2(p)/vr2(6) < volBri(p)/vri(6).
(2) and (3) may be proved in a similar manner.
Exercise 3. In the proof of the theorem, if Μ = (5m, g0) and с is a great
circle, then the equality signs hold in all the inequalities. Hence we get am =
2narn-2/(m- 1).
§4. Exercise 1. We only show the case δ = 0: We have
f(t) = a(£) sin/3/sina = a(t)/d(q, r) · d(p, r) < d(p, f).
Exercise 2. Join q to r(t) by a minimal geodesic in Μ and construct geodesic
triangles A(qpf'(t)) and A(qf,(t)r) in M™ with the same side lengths as A(qpr(t))
and A(qr(t)r), respectively. We take a point r" with d(f'(t), r") = d(r(t), r) on

332
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
a geodesic which extends the side from ρ to f'{i) beyond f'(t). Then show that
d(q, r) = d(q, f") = d(q, f'), and use this fact.
Exercise 4. Suppose d(p\, n) < d(p\, n). For a geodesic triangle A(pip2n') in
M™ with the same side lengths as those of Δ(ριΡ2η), we get Z(pip2n) > Z(pip2nf).
Extend a normal geodesic joining p2 to n' beyond n'. We take p'3 on the geodesic
moving forward from n' by length d(n, ps). Then we get Ζ(βιή'ρ'3) > Z(pinps),
and consequently d(pi, ps) < d(pu p'3) by T.C.T. On the other hand, comparing
Л(р1р2Рз) and Δ(βιΡ2Ρ3), from the relation between the side length and angle we
get d(pi, P3) < d(pi, ps) = d(p\, рз), which is a contradiction.
§6. Exercise 1. Use the axiom of free mobility.
Exercise 2. By Problem 9 for Chapter III the deck transformation group of
the universal covering π : Rm —» Μ consists of translations.
Exercise 4. Define Φ : G —> Η by Φ(α, b) = ab~l. Then Φοσ = δ6οΦ, and
ΌΦ : g —> ϊ) satisfies ΌΦ(χ, у) = χ - у. It follows that /)Ф|т(:г, —х) = 2ж.
Exercise 5. Noting that left invariant vector fields are Killing vector fields,
we get the formula for VχΥ by applying (6.2) of Chapter III to (1.13) of Chapter
II.
Problems for Chapter IV
1. Use Chapter II, (5.10).
2. First note that
\R(x, y, y, x) - Д(л+*)/2(я, 2/, t/, x)\ < —£— IN|2|M|2·
Then use the equalities
4R(x, y, y, z) = R(x + z,y,y,x + z) - R(x - z,y,y,x- z),
6R(x, y, z, w) = R(x, y + z,y + z,w)- R(y, x + z, x + z,w)
- R(x, y- z,y- z,w) + R(y, χ- ζ,χ- z, w).
3. (3) Let f be an eigenvalue of R with an eigenvector ω. Then, setting
Ro '·= R - R(i+6)/2, we get Rquj = τ$ω with f0 = f - (1 Η- δ)/2. Let 2k be the
rank of the 2-form ω, and write ω = Σί=ι ω^ί A ev (i* = i + fe), where {е*}^ is
an o.n.b. Then
к
1^ulRo(el, ег>, er, e0) = ϊ0ωό (j = 1, ... , k).
It follows that |f0| < (1 - δ) [|(fc - 1) + \] , к < [f ]. Use this fact for the proof
(see, e.g., [Bou-Ka] for more details on the curvature operator).
4. Apply Theorem 2.7 (1), noting that ||У||'(0) = ||Vr(0)|| and s0(t) = t.
5. Let w G U~f(t)Nt (resp., w G U^(t)Nt) be an eigenvector of A^) (resp.,
A~,t)Nt) corresponding to the maximal (resp., minimal) principal curvature A
(resp., A). Take an N (resp., 7V)-Jacobi field Υ (resp., Ϋ) along 7 (resp., 7) such
that Y(t) = w (resp., Y(t) = w). Then note that VY(t) = \Y(t), VY{t) = \Y{t),
and apply Theorem 2.3 (2) to Y(t) and (||У(0)||/||У(0)||)У(*).
6. Applying T.C.T. (II) to a geodesic hinge (p; 7 | [0, s], σ \ [0, i\) and using
the triangle inequality, show that for a fixed t we have d(a(0), a(t)) > tcosa by
letting s —> +00.

PROBLEMS FOR CHAPTER V
333
7. For a geodesic 7 : [0, 1] —» Μ joining ρ to q, consider a geodesic η(ί) =
(7(<),/(р)(1-0 + /(^)тМхЯ _
8. Recall that С = N. For ρ G Ν \ Ν take a, q e Βφ)/2(ρ) Π Ν and set
U := {ν G t/дЛГ; expg si; G Ν \ AT for someO < s < e(p)/2}. Then U is an open
subset oiUqN. Note that the above s is uniquely and continuously determined from
v, and write s = f(v). Define a map Φ : [0, 1] x U —» С by Φ(£, ν) := expg tf(v)v.
Then Φ is a homeomorphism onto a neighborhood of ρ G С
9. See [Ch-Gr-1], Proposition 1.8 (p. 420).
10. (1) Use (6.6) and the Gauss formula.
(2) Suppose [[n, n], n] С η. First show that g' := η + [η, η] is a subalgebra of
9. Let G be the connected Lie subgroup of G with the Lie algebra g', and #' the
isotropy group of G' at p. Then S := G' ρ may be identified with G'/H' and defines
a submanifold expp η of M. Now for a geodesic t н-> expp to = expfo · ρ (χ G m) in
M, show that iGn^ expp tx e S (t e R).
11. For a compact Lie group G we have #(:r, y)x = —\{гах)2у. In the
argument of (6.5), Jacobi fields along ηχ are determined in terms of the eigenvalues
and eigenvectors of у ι—► R(x, y)x. Since ad ж is a skew-symmetric linear map of 9,
we see that the nonzero eigenvalues of (ad ж)2 always appear in pairs.
12. See [Sa-2].
Chapter V
§1. Exercise 1. In the case of a free abelian group with к generators, we have
7W = E!Lo2i(i)G).
Exercise 4. If Μ is not orientable, apply Theorem 1.6 to the orientable double
covering Μ of M.
§4. Exercise 2. (i) and (ii) follow from the second variation formula as in the
proof of Proposition 4.3. (iii) follows from Theorem 4.1.
Exercise 3. Take a triangle Л(р1Р2Рз) in R2 with the same side lengths as
Δ(ριρ2Ρ3). Then apply (4.1) and the relation between the side length and angle
(§1, Exercise 5).
Exercise 4. Decompose the quadrateral (P1P2P3P4) into geodesic triangles
Л(р\р2Рз) and Δ(ριΡ3Ρ4). Then apply Proposition 4.5.
Exercise 7. If с φ 0, consider the descriminant Ό = (α - d)2 + 46c of the
quadratic equation cz2 - (a - d)z -6 = 0, and classify the cases according to the
sign of D. We may treat the case с = 0 similarly.
Exercise 8. Suppose μ1* is elliptic, and let ρ be its fixed point. Then apply
Proposition 4.16 to {μι(ρ); I G Ζ} = {ρ, μρ, ... , μΙς~1ρ} to see that μ is elliptic.
Next suppose μ1* is hyperbolic, and let 7 be its axis. Show that ^(7) С min^fc)
and μ leaves the decomposition min (μ1*) = W x R invariant. Therefore, μ may be
decomposed as (μ', τ (to)), and we can check that μ' is elliptic.
Problems for Chapter V
1. Obviously, Ц7Ц < 3d(M)||7||aZ0. Next, for 7 G Γ take a normal minimal
geodesic с : [0, /] —* Μ joining ρ to 7р. For г = 1, ... , ко := [l/d(M)\ we set
pi := c(id(M)). Then we may choose a 7* G Г such that d(pi, Ί~ιρ) < d(M). It
follows that 72, 7i+i ο η~ι (г = 2, ... , к0 - 1), and 7 ° 7^ ^ S. Now deduce that
ΙΗΙαΙ* < ко < h\\/d{M).

334
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
2. (2) Note that {Βε(ηρ)\ η G Γ} are mutually disjoint for 0 < e < г(М),
and ve := volΒε(ηρ) does not depend on 7. On the other hand, if Ц7Ц < r then
ВеЬР) С Br+e(p). It follows that N(r) < vo\Br+e(p)/ve. Use this fact.
3. We establish (2). (1) may be proved similarly. Suppose V Π W = φ, and
let 7 : [0, /] —► Μ be a normal minimal geodesic that realizes the distance d(V, W)
with 7(0) G V, 7(/) G W. Note that 7 is perpendicular to both V and W. Let
{ej}£L2 be an o.n.b. of T7(0)V, and let {^(0} denote the parallel translation of
{e{} along 7. Then {Ег{1)}^2 forms an o.n.b. of T^W. Prom the second variation
formula we get
D2E(j)(Eh Ei) = - f (R{Ei(t), ШШ Ei(t))dt
Jo
+ (АшЩ0), Ei(0)) - (JL,(l)Ei(l), Щ1)),
and therefore
m pi
Υ^Ό2Ε{Ί){Ε^ Ε{) = - p(7(0) * < 0.
Namely, D2E(/y)(Ei, E{) < 0 for some г. This contradicts the minimality of 7 (for
more details, see [Pr]).
4. Suppose / := minpGM d(p, f(p)) > 0. Take a point p0 with d(po, f(po)) = I
and join po to /(po) by a normal minimal geodesic 7. Then a linear isometry
φ := Р(*у)10 о Df(p0) of 7(C))-1- (С TPoM) admits a fixed vector ζ G UPoM because
of the assumption and Problem 2 for Chapter I. Let X(t) be the parallel translation
of χ along 7, and note that 7 is a B-geodesic with respect to a boundary condition
Β := {(ρ, /(ρ)); Ρ £ Μ}. Show that Ό2Ε(η) < 0 by the second variation formula.
5. Set /(p) := ^||p||2, and let q be a furthest point of Μ from the origin
o. Then Vf(q) = 0, and D2f(q) < 0, namely, D2f(q) is negative semidefinite.
Then, at q, for the second fundamental form S with respect to the outer unit
normal vector g/||g||, we get 5(x, x) = {-(ж, χ) + D2f(q)(x, χ)} < 0. Namely, 5 is
negative definite at q, and the Gauss formula implies that the sectional curvatures
of Μ are positive at q.
Next we show (2). We take a C°° unit normal vector field ν on a compact
hypersurface Μ of ilm+1. Regarding ν : Μ —► 5m, we get Dv(p)x = AUpx. Since
Μ is of positive curvature, the eigenvalues of AUp are of definite sign, and it follows
that ν is a regular map. Show that ν is a covering map.
6. Prom Theorem 2.7 we get d(M) < π. On the other hand, show that a
geodesic emanating from π((1 : 0 : ... : 0)) with the initial direction
(0, W2, · · · , Wm, 0, Wm+2, · · · , W2m)
is a minimal geodesic up to the length π.
7. For the first part, note that if 7 С С then s н-> d(7(s), c?C) is concave and
bounded below, and therefore it is a constant.
8. f(x) := d2(x, ηχ) is convex, and critical points are minimal points. Now
writing ηχ = Ax + a (A G O(ra), a G Ят), we have
ζ G min(7) Ф> (Л - £т):с + а±1т(Л - 25m).
On the other hand, note that Ят = Кег(Л - Еш) Θ Im(A - Еш). We denote by a2
the second component of a with respect to the above orthogonal decomposotion.
Then
χ G min(7) <&(A- Em)x + a2 = 0.

CHAPTER VI
335
9. Take a normal minimal geodesic 7 : [0, /] —» Μ joining χ to y, and set
7(i) = exp"1(7(i)), 7β(ί) := expp(s7(i)). Then from R.C.T. we get
d(pa(x), ps(y)) < ЦЪ) = [ \\Dexpp(sif(t))\\dt < s [ \\>y(t)\\dt = sd(x, y).
Jo Jo
10. We have
b~(p)= lim (d2(p,7(s))-s2)/2s
' s—> + oc
= lim {d2(p1,-n(als))-a21s2)/2s + lim (d2(p2, 72(<*2s)) - a22s2)/2s
s—>+oc s—»-+oc
= αι&~(ρι)+α2&~2(ρ2).
11. Let ρ, σ be rays emanating from pGl that determine the points z, W
at infinity, respectively. We denote by dk (k = 1, 2, ...) the length of the
perpendicular from ρ to 7^, where 7^ is a normal geodesic joining p(k) to σ(&). Now
consider a triangle Л& consisting of geodesic segments joining ρ to points of 7^.
Then, by the volume comparison theorem, the area of the part of Ak consisting of
points whose distances to ρ are less than or equal to dk is greater than or equal
to Θά1/2{θ := Ζ(σ(0), p(0))), which is the area of a corresponding sector in the
Euclidean plane. On the other hand, by the Gauss-Bonnet formula, the area of Ak
is less than or equal to π/α2. It follows that {dk} is bounded, and {7^} admits a
convergent subsequence as к —» oo. Then a limit geodesic 7 is the desired one (for
visibility manifolds, see [E-ON]).
Chapter VI
§3. Exercise 2. Take и = фк- Then R(u) = Xk and и ±(ψι, ... , <Pk-i)ii·
Conversely if u± (φι, ... , фк-\)я., we may write и = Σ^ αίΦί·> an(^ we Set
ОС ι ОС
R{u) = YJiai2 ^а{2 >\к.
i=k ' i=k
For the second assertion, check the case where equality holds.
Exercise 3. Note that
ехр(2тг(:г, у) y/^ϊ) = фсу(х) + у/=1фау(х).
Suppose
η
0 = Σα^ еМ2ф, Уз)у^) (Уз ^ Γ*).
Multiplying by exp(-27r(:r, yi)>/—1) on both sides of the above and applying Δ,
we have
η
Then show that otj = 0 by induction on n.
§4. Exercise 1. Let Ai, ... , Am be eigenvalues of ft. Then we have ||ft||2 =
λι2 Η l· Am2, and trace ft = Αι Η h Am. Apply the Cauchy-Schwarz inequality.
Exercise 2. Let p, q e Μ satisfy d(p, q) = d(M). We take a normal minimal
geodesic 7 : [0, d(M)\ —» Μ joining ρ to ς. For г = 0, ... , k, consider metric balls
Bi centered at *y(id(M)/k) with radius d(M)/2k, and apply the argument in (4.4).
Exercise 3. (1) Let φ be an eigenfunction of Δ corresponding to Ai(M).
Then φ | Ω{ (г = 1, 2) are eigenfunctions of the eigenvalue problem (II), and

336
HINTS AND SOLUTIONS TO EXERCISES AND PROBLEMS
λι(Μ) > Ai(i?i). On the other hand, extending eigenfunctions ψι
corresponding to Ai(i?j) to functions on Μ so that they vanish outside i?*, we get X\(M) <
R(il>i) = Xi(ni).
(2) We may assume that voli?i < volM/2. Denoting by 5™ the northern
hemisphere of 5m, we get Χλ(Μ) = Χι(Ωλ) > Ai(i7f) > λι(5™, go) = m.
§5. Exercise 1. To check the first equation of (5.1), use Lemma 3.1. Next, recall
that J^° exp(—r2)dr = y/π. Applying the formula for the change of variable of
integration, we get
fx ( r2\ ι
/ exp dr = ν 4πί
7-oc V *t J
for t > 0. Then show that, for a bounded continuous function /,
/;
lim(47rf)"i / e~£ f(r + a)dr = f(a).
The second equation of (5.2) follows from this.
Exercise 2. (Λ/. g) is also of constant curvature 1. Show that the universal
Riemannian covering π : S2 —» Μ is of order 2 because vol(M, g) = vol(52, #o)/2.
Problems for Chapter VI
1. Let Ω be a bounded domain in Μ with smooth boundary. For an e >
0, we consider a function fe on Μ defined by fe{p) = 1 {p £ Ω, d(p, ΘΩ)> б),
fe(p) = d(p, ΘΩ)/ε (ρ £ β, d(p, ΘΩ) < б), and fe(p) = 0(ρ^Ω). Then show that
lim / ff* dug = vol β, lim / ||V/e|| dvg = νοΙ^^ΘΩ),
e+° J м e-t° Jm
from which s(M) < T(M) follows. To prove the reverse inequality, first applying
the coarea formula to / £ Jr0(M), we have
/ \\Vf\\dug = Г volm-J-Wdt^IiM)* Γν^{ί)άί,
Jm J-oc Jo
where we have set V(t) := vol{p £ M; |/(p)| > t}. On the other hand,
/ \f\—*dvg= dvg \ ^-^t^dt=-— t^V(t)dt
Jm Jm Jo m - 1 m-l J0
Now set
F(i):= {/ ^(ί)*Γ \ G(i):= ^Γϊ / ^^(*)Λ-
Then note that F(0) = G(0) = 0, and show that F'(i) > G'(i)· Namely, we
get F(oo) > G(oo), which gives the desired inequality (see, e.g., [Cha-3] for more
details).
2. We may assume that Л is a convex set with d(A) < 1. Then show that
Β := {(χ — у)/2; χ, у £ A} is symmetric with respect to the origin o, and vol Б >
vol Л, d(B) < d(A). Since В is contained in a ball of radius 1/2, our assertion
follows (see [Bu-Z], p. 93).
3. Note that g^ = с2д^, gij = c~2gij, dv§ = crndvg, and V^ = V9.
4. For / £ Jr(M), g £ F(N) we define / χ g £ T(M χ Ν) by (/ χ g)(p, q) :=
f{p)g(q). Show that
AMxN(fxg) = (AMf)xg + fxANg

PROBLEMS FOR CHAPTER VI
337
and (fxg;fe ^(M), g e F(N))R is dense in L2(MxN) by the Stone-Weierstrass
theorem.
5. φ is an eigenfunction of ΔΜ with eigenvalue А Φ> тр := φοπ is a,n
eigenfunction of ΔΜ with eigenvalue A invariant under the action of the deck transformation
group. We have μ^ΑΡ™, g0) = 2k(2k + m - 1), and its multiplicity is given by
Г2+Л-Г2+^2-2)(^ = 0,1,2,...).
6. Let π : (52n+1, go) —» (CPn, h0) be the Riemannian submersion given in
Chapter II, §6, and recall that 51 acts isometrically on 52n+1. Then show that the
spectra of (CPn, ho) are given by eigenvalues of Δ5 Π such that the corresponding
eigenfunctions are SMnvariant. If the sectional curvatures of h0 satisfy 1 < Κσ < 4,
then μ*. = 4k(n + k) with multiplicity n(n + 2k){n(n + 1) · · · (n + к — l)/k\}2 (for
more details, see [B-Ga-Ma]).
7. Use the min-max theorem (Proposition 3.8).
8. Show that a"(m+1)R9o(f) < Rg(f) < am+1Rgo(f). and use Proposition 3.8
(see, e.g., [Band-Ur] for more details).
9. Take a normal coordinate system around ρ and recall that
9ij{x) = 6ij + -Rikji(p)xkd + o(\\x\\2).
Then (Be(p), g) is isometric to (Bi(p), ge), where we set {ge)i3{x) = e2(he)ij(x) and
(he)i:j(x) = gij{ex). Then show that R9e(f) = t~2Rh((f)· We have e2R9t{f) ->
Rh0(f), where h0 denotes the canonical metric of Rm. Use Proposition 3.8.
10. We have ||Я||2 > ||Ric||2, where equality holds if and only if (M, g)
is of constant curvature (see Problem 7 for Chapter II). Show that a2(M, g) >
^a2(M, g)/ao(M, g), and apply this fact.
11. (1) See [B-Ga-Ma], p. 149.
(2) This is clear from the equalities
αλ(Μ) = - / τάν9 and χ(Μ) = — / rdvg.
b J μ 47Γ J м
12. (1) We may show this in a similar manner as in §5, Exercise 1. Note
that the definition of e does not depend on the choice of p, q. To check the second
equation of (5.1), extend the integral over R171 by parallel translating a fundamental
domain of π : R171 -> Tm by the action of Γ.
(2) Use Proposition 3.14 and Theorem 5.1.
13. (1) Take an o.n.b. {ej •^=1 of TPM and let 7* be geodesies in Μ emanating
from ρ with the initial directions e* (г = 1, ... , m). Then
m H2
ΔΦ(ρ) = -Σ, & l<=° *bW) = -Σ^.(0)7, = -£S(7*(0), 7i(0)) - rnH,
г=1
where D denotes covariant differentiation of the Levi-Civita connection of (Rn, go)·
(2) Let V be the covariant differentiation of the Levi-Civita connection of
(Sn, go), and argue as in (1). We denote by Η the mean curvature vector of
Μ ^ 5n, and get
ΔΦ(ρ) = -Σ |ϊ| φΜ*)) = -Σ^.(0)7ί
Ιί=ο
dt2
г=1
= - Σ<ν-ν.(0)7ί - * (Ρ)} = тН + тФ(р)
(for more details, see [Так], [L]).

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Index
A
adjoint representation, 14
affine transformation, 118, 319
infinitesimal —, 118
almost complex manifold (strucuture), 77
almost flat manifold, 294, 317
Ambrose theorem, 112, 129, 136, 176
Ambrose-Singer theorem, 123, 181
angle
— between vectors, 4, 23
— of a hinge, 161
— of a triangle, 138, 161, 223
arc-length, 25
asymptote, 174
asymptotic, 229
atlas, 5, 20, 70
axiom
— of free mobility, 135, 332
— of plane, 136, 137, 164
axis, 233
В
base space, 15
basic vector field, 74
Berger
— isoembolicinequality, 252
— sphere, 202, 293, 316
Betti number, 199, 201, 316
Bianchi identity, 34, 319
Bieberbach theorem, 138, 317, 321
Bishop comparison theorem, 154, 278
Bishop-Gromov comparison theorem, 156, 157,
196, 221, 309
Borel set, 63
boundary, 70, 171
bounded geometry, 304
broken geodesic, 112
bundle map, 15
Busemann function, 174, 212, 218, 230
С
canonical identification, 7, 53
canonical Riemannian metric
— of Ят, 24
— of 5m, 49
— of Hm. 51
Cartan subalgebra, 188
Cartan theorem, 42, 136, 176
center. 225. 233
change of variable formula, 64
chart. 5. 15
Cheeger
— finiteness theorem. 304, 312
— isoperimetric constant, 244, 247, 276
— theorem, 160
Cheeger-Gromoll theorem. 215. 218
Christoffel symbol. 28
Clifford translation. 132
С ι -manifold, 260
coarea formula, 67. 277. 280
Codazzi formula, 48
codimension, 8, 10
collapse, 316, 320, 321
compactification, 229
complete
— Riemannian metric. 84, 119. 130
— vector field, 9, 13
completely integrable, 10, 79, 125, 229
conformal 25, 50
— curvature tensor, 290
conformally flat, 290
conjugate point, 37, 40, 61, 99, 101, 178, 190
first —, 102, 107, 149, 187
C- —, 61
conjugate value, 37
first —, 96, 102, 104, 149, 152
connection map, 54
constant curvature, 43, 46, 50, 51, 117, 135
constant speed, 26, 32, 88
contraction, 2
convex function, 172, 190, 212, 221, 222,
223, 231
strongly —, 173, 174, 255
convex hull, 224

354
INDEX
convex set, 140, 222, 229
locally —, 168, 171, 190
strongly —, 168
totally —, 168, 172, 190, 213, 215
convexity radius, 168, 223
coordinate neighborhood, 5
cotangent bundle, 16, 57
covariant derivative (differentiation), 18,
28, 29, 48
critical point, 10
— of dp, 204, 206, 216
critical value, 10
— of v?-family, 294, 300
curvature operator, 189, 303
curvature tensor, 18, 34, 41, 43, 79, 122, 189,
289
— of a conformal metric, 50
— of a Riemannian submersion, 75
— of a symmetric space, 176,178
— of a submanifold, 48
curve
C°° —, 7
piecewise C°° —, 26
cut locus, 104, 106, 132, 188, 208
tangent —, 104, 188
cut point, 104, 187
tangent —, 104
D
deck transformation group, 138, 139, 193,
220, 235
density, 62, 64
derivation, 3, 7, 9, 30
diameter, 86, 140, 157, 204, 207, 247, 280,
304. 308
diffeomorphic (diffeomorphism), 6
differential form, 17, 301
differential of a map. 7
Dirichlet boundary condition, 265
discontinuous, 193
distance, 26, 77
— function, 26, 108. 109, 153, 204. 214.
222
distribution, 10, 54, 124, 209, 229
divergence, 31, 71
— theorem, 71
dual
— basis, 1
— vector space, 1
— vector bundle, 16, 20
Ε
effective, 184
almost —, 184
eigenfunction, 265, 270
eigenspace, 265, 272, 273
eigenvalue
— of Laplacian, 265, 268, 269, 279
first — of Laplacian, 269, 270, 275, 276,
280, 304
Einstein manifold, 45, 46, 180
Einstein's convention, 2
embedding, 8
energy (integral), 87
Euclidean space, 46, 79, 116, 138, 238, 239
Euler characteristic, 285, 287, 291
exponential growth, 194, 195
exponential map
— of a Lie group, 14, 121, 292
— of a Riemannian manifold, 32, 36, 65,
104, 153
normal —, 58, 59, 72, 159, 310
exterior
— algebra, 3
— differentiation (differential), 17, 31, 301
— product, 3
— power, 3, 17
F
fiber, 8, 15
— bundle theorem, 320
— metric, 20
first (tangent) conjugate locus, 107, 108, 131
first variation, 38, 89
— formula, 38, 89, 90, 108, 208, 225, 263
flat, 79, 138, 235, 236, 321
flow, 9, 17
focal
— point, 59, 61, 94, 95, 96, 99
— value, 59, 143, 152
foliation, 10, 187
foot of a perpendicular, 225
form, 3
Frobenius theorem, 10, 125
Fubini-Study metric, 77, 186
Fubini theorem, 66, 68, 73, 160, 253
fully reducible, 140
fundamental domain, 195, 196, 220, 274
fundamental group, 122, 139, 140, 183,
194, 195, 220, 235, 236, 317
fundamental solution, 283, 285
G
Gauss
— curvature, 49
— formula, 48
— lemma, 36, 39, 60, 72, 209
Gauss-Bonnet formula, 138, 143, 291
general linear group, 13
geodesic, 32, 39, 40, 50, 51, 80, 90, 96, 131,
175, 185
B- —, 88

INDEX
355
closed —, 90, 160, 187, 197, 208, 260, 299
geodesic flow, 56, 57, 80, 253
geodesic hinge, 161
generalized —, 161
geodesic spray, 56, 58, 86
geodesic symmetry, 175, 176, 179, 184
geodesic triangle, 137, 161, 223
generalized —, 161
geodesically complete, 33, 52, 84
gradient vector field, 31, 50, 68, 73, 219, 231
Grassmann manifold, 43, 186
Green theorem, 70, 74, 121, 199, 242, 267,
275, 276
Gromov
— convergence theorem, 312
— precompactness theorem, 308
Η
Hadamard-Cartan theorem, 221
Hadamard manifold, 222
Hamiltonian vector field, 58
harmonic
— coordinate, 313
— form, 199, 303
— function, 31, 74, 219, 271, 287
Hausdorff
— approximation, 306
— convergence, 308
— distance, 305
pointed — distance, 307
heat
— equation, 264, 282
— kernel, 282, 285
Heintze-Karcher theorem, 159, 247, 250, 252
Hermitian metric, 77
Hessian
— of energy integral, 91, 92
— of function, 11, 31
Hodge-Kodaira theorem, 199, 302
holonomy endomorphism, 122, 139
holonomy group, 121, 126, 130, 139, 188
restricted —, 122
— of symmetric space, 180
homogeneous space, 14, 119, 120, 136, 175,
291
naturally reductive —, 292
normal —, 292
homothetic, 51, 286
Hopf-Rinow theorem, 84
horizontal
— lift, 54, 74, 75, 131
— space, 25, 54
horoball, 232
horosphere, 232, 234, 239
hyperbolic
— manifold, 142
— space, 52, 79, 142
hypersurface, 10, 49
I
immersion, 8
minimal —, 287
index
— form, 95, 144
— of critical point, 11, 12
— of geodesic, 93, 98, 99, 101, 296
induced
— bundle, 16, 28
— connection, 18, 29
— metric, 24, 47, 64
injectivity radius, 110, 111, 160, 198, 202,
252, 293, 294, 305
inner product, 4, 23
integrable function (set), 63
integral curve, 9
integral submanifold, 10, 229
maximal —, 10, 128, 209
interior set, 104, 195
involutive (distribution), 10, 75, 209, 229
irreducible, 124, 129, 188
isometric, 24
— immersion, 24, 287
isometry, 24, 42, 43, 52, 64, 233, 238, 264
— group, 79, 117, 119, 120, 130, 136, 178,
184
elliptic —, 233
hyperbolic —, 233
local —, 24
parabolic —, 233, 234
semisimple —, 233, 234, 239
isomorphism
— of vector bundles, 15, 20
— of vector spaces, 1
isoperimetric
— constant, 244, 252, 286
— function, 243, 249
— inequality, 241, 243, 245
isosystolic inequality, 261, 262
isotropic, 5
isotropy group, 14, 119, 178, 184, 187
J
Jacobi identity, 9
Jacobi field, 36, 37, 47, 50, 52, 56, 78, 92,
117, 131, 149, 152, 153, 177, 189, 209,
312
— of symmetric space, 177
C- —, 60
N- —, 58, 59, 93, 143, 149
stable — 232
Jensen inequality, 255

356
INDEX
К
Kahler manifold (metric), 77, 123
Killing
— form, 180, 293
— vector field, 117, 118, 120, 177, 178,
180, 186, 291
Klingenberg estimate of injectivity radius,
198
L
Lagrangian subspace, 5, 19, 60, 99
Laplacian, 31, 74, 158, 262, 263
lattice, 13, 105, 138, 200, 272
Law of Cosines, 138, 140
Law of Sines, 138, 164
left
— invariant, 12, 292
— translation, 12
length (of curve), 25, 78. 150
lens space, 140, 177, 190, 316
Levi-Civita connection. 28
Lichnerowicz-Obata theorem, 275. 281
Lie
— algebra, 9, 12. 118, 179. 180, 291
— derivative, 10, 17
— group, 12, 19, 117, 178, 292
compact — group, 180, 185, 190, 197
— transformation group, 14, 117, 178
line, 174, 218
linear connection, 18, 28, 54, 55, 319
Lipschitz distance, 306
local coordinate system, 6
locally symmetric space, 177, 210
Μ
manifold
C°° —, 6
— of nonnegative (positive) curvature, 183,
198, 201, 211, 221, 238
— of nonpositive (negative) curvature, 184,
195, 221, 236, 237
— with boundary, 70
mapping theorem, 8, 67
Margulis lemma, 315, 320, 321
max-min theorem, 269
maximal diameter theorem, 157, 165, 204,
276
maximum principle, 218, 219, 300
mean curvature, 49, 131, 159, 247
— vector, 49, 131, 287
measurable function (set), 63
measure, 63
metric ball, 26, 39, 64, 155, 307
min-max theorem, 269
minimal geodesic, 39, 84, 102, 103, 222
minimal submanifold, 49, 131, 238, 287
Morse
— function, 11, 12, 99
— index theorem, 99
— lemma, 11, 108, 173
— -Schoenberg theorem, 101
— theory, 11, 12, 99, 107
multiplicity
— of a conjugate point, 37, 60, 107
— of an eigenvalue, 268, 270
— of a focal point, 59, 99
Myers theorem, 102, 155, 183, 194
N
natural basis, 7
net, 305, 306
nilpotent Lie algebra (group), 293, 317
nodal domain, 270, 281
nondegenerate
— critical point, 11, 99
— geodesic, 99
— 2-form, 5
normal (curve), 26
normal bundle, 20, 47, 59, 72, 215
normal coordinate system, 33, 37, 41, 78,
110, 263
null set, 63
null space, 5, 92, 98, 110
О
o.n.b. (orthonormal basis), 4
one parameter group of (local) diffeomor-
phisms, 9
one parameter subgroup of a Lie group, 13
O'Neill formula, 75
orientable, 20, 62, 106, 123, 142, 324
orthogonal transformation (matrix), 4, 13,
19, 198
Ρ
parallel, 29, 30, 228
— translation, 29, 31, 121, 139, 175
partition of unity, 6, 25, 62
0-family, 294, 300
((δ)-) pinched, 202
Poincare model (of hyperbolic space), 52,
230
point at infinity, 230
Poisson summation formula, 287
polar coordinate, 65, 78
pole, 222
polynomial growth, 194, 195, 309
principal curvature, 47, 49, 159, 189, 310
projection (of vector bundle), 15
projective space, 186, 187, 207

INDEX
357
complex —, 76, 131, 186, 238, 286, 292
real —, 20, 105, 140, 186, 262, 285, 286
proper (map), 8, 68, 308
R
Radon measure, 61, 63
rank
— of a manifold of nonpositive curvature,
237
— of a symmetric space, 188
Rauch comparison theorem
(R.C.T. (I), (II)), 149, 150, 215, 223, 244
ray, 174, 189, 229, 230
Rayleigh quotient, 266, 268, 276, 280
regular curve, 25, 38
regular value, 10
de Rham
— theorem, 302
— decomposition theorem, 129, 180, 182
representation, 14, 140
Ricci curvature, 44, 45, 66, 144, 155, 156,
157, 159, 183, 184, 194, 195, 218, 220,
221, 247, 249, 252, 275, 278, 280, 289,
308
Ricci tensor, 44, 79, 120, 121, 180, 289
Riemannian
— covering, 24, 68, 113, 116, 117, 122,
132, 139, 193, 220, 286
— manifold, 23
— manifold with boundary, 70, 265
— metric, 23, 24
— (direct) product, 24, 68, 87, 122, 129,
131, 218, 224, 237, 286
— submersion, 25, 56, 66, 74, 76, 131, 217,
224
— symmetric pair, 184
right translation, 12
rigidity theorem, 207
S
Sasaki metric, 56, 58, 68, 79, 132, 160, 253
scalar curvature, 46, 66, 79
Schur lemma, 46
second fundamental form, 47, 49, 132, 146,
232, 310
second variation, 91
— formula, 90, 98, 110, 141, 198, 222, 263
section, 17
sectional curvature, 43, 44, 48, 75, 101, 144,
149, 150, 152, 153, 154, 155, 160, 176,
183, 184, 201, 212, 221, 304
semisimple Lie group, 180, 183, 184
shape operator, 47, 58, 80, 91, 143
shortest curve (see also minimal geodesic),
37, 39
simple point, 217
slice, 10
Sobolev space, 265
— constant, 286
— theorem, 265
soul, 217, 238, 321
space form, 138
special
— linear group 13
— orthogonal group, 13
— unitary group, 13
spectrum, 269. 284
sphere, 14, 19, 49, 79, 105, 123, 139, 157,
252, 270, 275, 280, 285
sphere theorem, 201, 210, 211
Grove-Shiohama —, 204
stationary curve. 38, 90
Stone-Weierstrass theorem, 272, 273
subbundle. 10. 15
subharmonic function, 218, 300
submanifold, 8. 19
submersion. 8. 14, 19
support function. 217, 301
supporting half-space, 172, 190
symmetric space. 175, 185
— of compact type, 182, 183, 207
— of Euclidean type, 182
— of noncompact type, 182, 183, 184, 187,
188
symplectic
— form, 5. 18. 57. 260
— group, 19
— manifold, 18
— vector space, 5. 19
Synge theorem, 197-198
Τ
tangent
— bundle, 7, 20, 53. 79, 132
— cone, 171, 190
— space, 7
— vector, 7
tensor, 2
— bundle, 16
— field, 17, 30
— product, 1, 16
— space, 2
Toponogov comparison theorem
(T.C.T. (I), (II)), 161, 202, 206, 208, 212,
215
torsion tensor, 18, 28
torus, 13, 20, 105, 121, 131, 138, 199, 261,
273, 286
total space, 15
totally geodesic (submanifold), 48, 75, 79,
136, 151, 168, 171, 190, 208, 215, 235
totally umbilic, 147
transitive, 14, 120, 121, 175

358
INDEX
transvection, 176
U
uniformly, 194
— compact, 306
unit tangent bundle, 23, 55, 253
unitary group, 5, 13, 19
universal covering space, 117, 155, 193, 194
V
variation, 35, 37, 88, 131
— of a curve, 35
— vector field, 37, 88, 131
piecewise C°° —, 37
variational completeness, 178
vector
— bundle, 15
— field, 8
— space, 1
vertical space, 25, 53, 74
visibility manifold, 239
volume, 4, 63, 64, 68, 131, 160, 221, 241,
243, 244, 252, 280, 284, 304
— element, 63, 80
— of a metric ball, 65, 155, 156, 189
W
warped product, 224
Weingarten formula, 48
Weitzenboeck formula, 303
Weyl asymptotic formula, 273, 274
Whitney
— sum, 16, 20, 47
— theorem, 8, 25, 86
Wiedersehens manifold, 261
word-length, 194, 237

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