Nankai Tracts in Mathematics
Vol.2
THE INDEX THEOREM
AND
THE HEAT EQUATION METHOD
Yanlin Yu
World Scientific

Nankai Tracts in Mathematics - Vol. 2
THE INDEX THEOREM AND THE
HEAT EQUATION METHOD
by Y L Yu (Suzhou University, China)
This book provides a self-contained representation of the local
version of the Atiyah-Singer index theorem. It contains proofs of
the Hodge theorem, the local index theorems for Dirac operator
and some first-order geometric elliptic operators by using the heat
equation method. The proofs are up to the standard of pure
mathematics. In addition, a Chern root algorithm is introduced
for proving the local index theorems, and it seems to be as efficient
as other methods.
ISBN 981-02-4610-2
WWW. worldscientific.com
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NANKAI TRACTS IN MATHEMATICS
Series Editors: Shiing-shen Chern, Yiming Long, and Weiping Zhang
Nankai Institute of Mathematics
Published
Vol. 1 Scissors Congruences, Group Homology and
Characteristic Classes
by J. L Dupont
Vol. 2 The Index Theorem and the Heat Equation Method
byY.LYu

Nankai Tracts in IVIathematics - Vol. 2
THE INDEX THEOREM
AND
THE HEAT EQUATION METHOD
Yanlin.Yu
Department of Mathematics
Suzhou University
PR China
Yl^ World Scientific
m Singapore • New Jersey • London • Hong Kong

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The index theorem and the heat equation method / Yanlin Yu
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1 Atiyah-Singer index theorem 2 Heat equation I Title II Senes
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Even if there will be no research results, it is worthwhile
to study the Atiyah-Singer index theorem.
S.S. Chern
PREFACE
The Atiyah-Singer index theorem appeared in 1963. Since then I had never
failed to be astonished by its depth and beauty. After the Chinese cultural
revolution I tried to review my knowledge in topology by reading the Atiyah-
Singer index theorem, as algebraic topology used to be my major during the
early sixties of the 20th century. At that time I did not realize that I could
do something on the Atiyah-Singer index theorem someday. But Professor S.S.
Chern shaped our destiny. It was him who in 1985 said the phrase above,
and encouraged me to give a series of lectures on this theorem in the next
year, at the Nankai Institute of Mathematics. During the preparation of the
lectures I had found a proof of the local index theorem for the Dirac operator
by using the heat equation method, which resulted the main part of the lectures
and a preprint of the Institute of Mathematics, Academia Sinica, No.26 (1986
March). That preprint was subsequently published in [38]. With a supplement
of several basic geometric materials it finally resulted in a book (in Chinese)
published in 1996. This book is essentially the English version of that. So it
might be said that the present book is a result of constant encouragement and
mspiration from Professor S.S. Chern.
This book may be considered as an introduction to the Atiyah-Singer index
theorem and its sebsequent evolution. The first chapter is a preliminary of the
Riemannian geometry, in which the moving frame method is stressed. The
principal bundle connection theory is introduced in Chern's style in order to
meet the needs of latter discussions. Chapters 2 and 3 are devoted to the heat
equation theory on manifolds, and the Hodge theorem following the Milgram-
Rosenbloom's scheme as well. Chapter 4 is for the Chern-Weil theory, which

VI PREFACE
IS a bridge between global and local considerations on manifolds. A Chern
root algorithm is also introduced there. The algorithm is not entirely rigorous
but has the merit of deducing geometric results rapidly. Chapter 5 explains
the Clifford algebra and super algebra, which may be viewed as a basic
knowledge for geometry. And the last three chapters are devoted to the local index
theorems for the de Rham-Hodge operator, the signature operator, the Dirac
operator and the Riemann-Roch operator. We prove them by using either of
the Chern root algorithm (see §4.4) and a strict method (see §7.1). One may
choose one of the two proofs according to his/her preference.
I would like to take this opportunity to express my immense gratitude to
the teachers and friends, who help the formation of this book. I had the fortune
to study mathematics under Professors Z.H. Jiang, W.T. Wu, S.T. Liao, G.L.
Wu and B.J. Jiang, their influence shaped everything I did in mathematics.
Professors Q.K. Lu and S.K. Wang gave strong support to the research leading
to this book. Professor H. Wu taught me several materials written in the book,
and Dr. Y.K. Lau read the whole original manuscript and made a number of
helpful suggestions. Dr. S. Hu and Ms E.H. Chionh checked the manuscript.
Professors A. Rigas, C.L. Shen, A.M. Li, Q. Zhou, W.P. Zhang, Z.Q. Lu, H.T.
Fen, B. Cheng, F.E. Wu and J.W. Zhou help a lot also.
I am also grateful to the Funds of the National Natural Science Foundation
of China and the Chinese Academy of Science, and also to the Department of
Mathematics, Suzhou University for their support during the writing of this
book.
At last I would like to pay my respect to Professors M. Atiyah and LM.
Singer for giving us such a beautiful theorem.
Li He new village, Suzhou Yanlin Yu
November, 2000

CONTENTS
PREFACE V
DEFINITIONS AND FORMULAS ix
CHAPTER 1 PRELIMINARIES IN RJEMANNIAN GEOMETRY.... 1
1.1 Basic Notions of Riemannian Geometry 1
1.2 Computations by using Orthonormal Moving Frame 11
1.3 Differential Forms and Orthonormal Moving Frame Method 14
1.4 Classical Geometric Operators 23
1.5 Normal Coordinates 41
1.6 Computations on Sphere 60
1.7 Connections on Vector Bundles and Principal Bundles 70
1.8 General Tensor Calculus 78
CHAPTER 2 SCHRODINGER AND HEAT OPERATORS 83
2.1 Fundamental Solution and Levi Iteration 84
2.2 Existence of Fundamental Solution 89
2.3 Cauchy Problem of Heat Equation 94
2.4 Hodge Theorem 98
2.5 Applications of Hodge Theorem 106
2.6 Index Problem 112
CHAPTER 3 MP PARAMETRIX AND APPLICATIONS 115
3.1 MP Parametrix 115
3.2 Existence of Initial Solutions 119
3.3 Asymptotic Expansion for Heat Kernel 127
3.4 Local Index for Elliptic Operators 130
CHAPTER 4 CHERN-WEIL THEORY 139
4.1 Characteristic Forms and Characteristic Classes 139
4.2 General Characteristic Forms 150
4.3 Chern Root Algorithm 162
4.4 Formal Approach to Local Index of Signature Operator 165

CONTENTS
CHAPTER 5 CLIFFORD ALGEBRA AND SUPER ALGEBRA.... 175
5.1 Clifford Algebra 175
5.2 Super Algebra 183
5.3 Computations on Supertraces 190
CHAPTER 6 DIRAC OPERATOR 195
6.1 Spin Structure 195
6.2 Spinor Bundle 201
6.3 Dirac Operator 203
6.4 Index of Dirac Operator 207
CHAPTER 7 LOCAL INDEX THEOREMS 213
7.1 Local Index Theorem for Dirac Operator 214
7.2 Local Index Theorem for Signature Operator 231
7.3 Local Index Theorem for de Rham-Hodge Operator 241
CHAPTER 8 RIEMANN-ROCH THEOREM 245
8.1 Hermitian Metric 245
8.2 Hermitian Connection 249
8.3 Riemann-Roch Operator 259
8.4 Weitzenbock Formula 268
8.5 Index Theorem 273
8.6 Riemann-Roch Operator in Complex Analysis 276
REFERENCES 279
INDEX 283

DEFINITION AND FORMULAS
We list several definitions and formulas used in this book below. If one
familiarizes with other definitions and formulas, please compare them with
ours before reading this book.
0.1 Differential forms
The standard pairing between vector fields and differential forms is an
^(M)-linear map
B : A*(Af) X T{TM) - - - T{TM) -^ T{M)
k times
: (a;,Xi,---,Xfe) h-^ a;(Xi, • • • ,Xfe)
satisfying that if a;i, • • •, a;jk G A^(M), then
a;i(Xi) ••• (^i{Xk)
{ui A'"Auk){Xi,'",Xk) =
i^k{Xi) ••• (j^k{Xk)
Or if a E A^(M), (3 E A'(Af), then
(aA/3)(Xi,...,X,+,)
TT
where tt runs over the set of permutations of {1, • • •, fe + /}, and ( — 1)'^ is 1 or
— 1 if TT is an even or odd permutation.

DEFINITIONS AND FORMULAS
For u G A*'(M), we also have
Jk+l
0.2 Connection and Curvature (invariant point of views)
Let
V : T{TM) X T{TM) -^ T{TM) : (X, Y) ^ Vx^,
be the Levi-Civita connection, i.e.
(i) V is a connection,
(ii) X(Wi,W2) = {VxWuW2) + {WuVXW2),
(iii) Vx^ - VyX = [X, y], where [X,Y] is the Lie bracket of X and Y.
Define the curvature tensor i2, its covariant derivative S^xR and Aq as follows:
R{X, y, Z) = (vx Vy -VyVx- V[x,y])Z.
{VxR){Y, Z, W) = Vx{R{Y, Z, W)) - RiVxY, Z, W)
-R{Y, S7xZ, W) - R{Y, Z, sjxW),
Ao = ^ D{E^, E^), (see Definition 1.4.2).
0.3 Connection and Curvature (by using orthonormal moving
frame)
Let {El, • • •, jE7n} be an orthonormal moving frame on M", i.e. {E^,Ej) —

DEFINITIONS AND FORMULAS xi
6jj, define
T% = {s7E.E„Ek),
Rtjki = {R{Et,Ej,Ek),Ei) ,
Rtjki^s = {{VE,R){Ei,Ej,Ek),Ei) .
Then
^U ~ 2 i ^ij ~^ ^kj + ^kt } '
Rzjki = ( ^«rjjt - EjT[j^ + 2Jr^j^rj, - 2Jr,\r', - ^c/^r'^^) ,
s s 3
Rijkl.s = EgRijkl — 2_^ ^TtRmjkl — 2_^ ^TJR^mkl — 2_^ ^TkRtjml
m m m
~ / _j ^ slRtjkm 5
m
and
(Bianchi identity I) R^ji^i + Rkiji + Rjkii = 0,
(Bianchi identity II) Rtjki.s + Rsiki.j + Rjski.i - 0.
0.4 Moving frame method ( C art an-C hern)
Define Uij and U^j by
3
R{X,Y)E. = J2^,.{X,Y)Ej,
J
then
r,^=u,j{Ek),
Rijki — —^tj{Ek,Ei) = -Qki{E^,Ej) .
Uij and l^jj are determined by

xii DEFINITIONS AND FORMULAS
J
Bianchi(I) and Bianchi(II) are
"^Uj Afij, =0,
J
0.5 Normal coordinate system
{yi 5 • • • 5 yn } is a normal coordinate system centering at p if and only if
(i)(yi(p),---,yn(p)) = (0,---,0),
(ii) for t = 1, • • • ,n
y^ = Y^9^Jiy)' yj^ where g,j{y) = ( —, —).
Let p{q) be the distance between p and g, and j- be the unit tangent vector
field of geodesies starting at p, then
piQ) = y/yi + "' + yn
and
2 = 1
5p
Let {jE7i, • • •, jE7„} be the orthonormal moving frame compatible with the
normal coordinate system {yir " •)yn}, define {T^j^Htj^Hjik} by solving the

DEFINITIONS AND FORMULAS
following equations
d
- 7 ,H^jE^,
%
V^Ej = y ^ HkijEk.
9yj
Let d = pj-, then
where
a
a,/3
^Jkl = {[VE^ VEj - VEj VE^ - V[E^,EJ]) Ek , El).
0.6 Principal bundle
Let TT : P -^ M be a principal G-bundle, a connection is a set
u = {u^\aeMF{P)}
satisfying
u, g = Ad(^-^)a;, + ^*([G]), \/a G MF(P), \/g : U, ^ G,
where MF(P) is the set of local sections of P (i.e. the set of moving frames),
Ua^ is the definition domain of cr, and [G] is the Maurer-Cartan form of G.
The curvature of u is
defined by
The curvature satisfies
n = {n^\aeMF{P)}
n^ = du^ + -[ua^i^al
n^.g = Ad{g-^)n^,

XIV DEFINITIONS AND FORMULAS
and
The last equality is the Bianchi identity (II).
0.7 Characteristic forms
Let ^ be a real matrix of rank 2n, define 7, (A) by
f det(A/ + ^) = A" + ri{A)X^-^ + . + r^^),
Pf(^) = 2^ J2 €{iuJi,',in,Jn)A,j,'"A^^j^.
For a complex matrix A of rank n, define r^{A) similarly. Then for connection
matrices Q and Ct^ of rank 2n and n respectively, define
P.(0)=r2.(^0)GA4'(M),
f(fi) = pf(ifi)GA2"(M),
c.(fi«) = r.(^fi^) e A2'(M) ® C.
For an antisymmetric il and a conjugate antisymmetric fi*^, the above
definitions imply
1 + Pi(fi) + P2(«) + • • • = det (/ + ^fi) ,
where
1 + ^ c. (ft-^) = det (7 + ^fi^V
He : g£(n, C) ^ g£(2n, R) : ^ + V=l5 ^ (
A -B
B A

DEFINITIONS AND FORMULAS
0.8 Chern roots
Let {^1, • • •, ^n} be the Chern roots, which are the generators of Polyo(T", C
The symmetric polynomials of Chern roots represent characteristic forms by
Definition 4.2.8 (2). These symmetric polynomials are called the Chern root
expressions of the characteristic forms. We list some well-known Chern root
expressions as follows
«i< <«,
^{^'')= J2 ^«i---^«.-
«1< <«8
If the Chern roots are viewed as 2-forms, they are called the new Chern roots.
The new Chern roots satisfy
and
Hence
/ 0 ui
-ui 0
27r
-n =
^nc
Ui
2ir
«<= =
/ 0
\
0 u„
-w« 0 /
Ul
2w
ne (n^) =
-Ui
0 /

XVI DEFINITIONS AND FORMULAS
and a tricky formula
Rtja^ — —^tj{Ea,Ei3) = —'^'K 2_^'^s{Eat Eii){82s-1,%^28,3 — ^2s,1^28-1,3)-
8 = 1
The new Chern roots are used in the Chern root algorithm, where the Chern
root algorithm is a kind of formal computations.
0.9 .F-linear operators
Choose an orthonormal moving frame {jE7i, • • •, jE7n} on an open set U of
M, let
i{Ej) : A^iU) -^ A^-^U) : u ^ i{Ej)u
be defined by
or
(z(^,)a;)(Xi,...,X,_i) = a;(^„Xi,...,X,_i),
k
i{Ej){u,, A • • • A a;, J = X](-^)'~^^J«''*'«i A • • • A S,, A • • • A a;,.
8=1
Define
then
E+=u,,A+i{E,):A*{U)^A*{U),
E;=u,A-i{E,):A*iU)^A*{U),
f E+E++E+E+ = 2S,j
E+E- + E-E+ = 0
e;e;+e;e; = -2S„.

DEFINITIONS AND FORMULAS xvii
0.10 Differential operators
For / : M -^ R G J^{M) with respect to an orthonormal moving frame,
D{X)f = Xf,
D{X,Y)f = (XY-S7xY)f,
D{Xi, ■■■, X,n + l)f = Dx, DiX2, • • • , Xm + l)f
-D{\;/XiX2,X3,--- ,Xm + l)f D(X2,---,Xm, VXi-X'm+l)/,
Let E he a vector bundle over M. For W G T{E),
D{Xi,---,X„+i)W = Dx,D{X2,---,X^+i)W
-Dis7x.X2,X3,---,Xm+i)W D{X2,---,Xm,Vx.X^+i)W,
where D{X)W is a connection on the vector bundle E. And the Laplace-
Beltrami operator is
Ao^J2d{E,,E,).
0.11 Weizenbock formula
(1) Letd + 6: A*(M) -^ A*(M) be define in Definition 1.4.12 or 1.4.18, then
(d + Sf = -Ao+Ir-IY. R.,„E+E+Ei-Er,
where R is the scalar curvature defined by
K = — y ^ Rtjtj'
hj
(2) Let D : r{E) -^ T{E) be the Dirac operator defined in Definition 6.3.1,
then
D^ = -Ao+\r.

xviii DEFINITIONS A ND FORMULA S
(3) Let D : T{E) -^ T{E) be the Riemann-Roch operator defined in Definition
8.4.3, then
1 /ZT ^"
D^ = -Ao +-R - ^^-Y^ Yl ^i(^«' ^^) • (^ ' ^«^^)-
0.12 Heat equation
The following equation
r .d
has a solution
limt_o V47rt H{t, 0) = 1
-s/^a \ _^ v^^a^ ^^ y/^at^ 2i
\87rsmh^ 3 "^^ / 8 2
where a is a constant.
0.13 Local index theorems
(1) Let M be an oriented Riemannian manifold of dim 2n, and let
be the super structure defined in Definition 1.4.12. Let
n = (d + <5)2:A*(M)-^A*(M),
and G{tyq,p) be the fundamental solution for the heat operator ^ + D.
A
limtr G{i,p,p) = (ni • • •nn)(£^i, • * * ,^2n).

DEFINITIONS AND FORMULAS
(2) Let M be an oriented Riemannian manifold of dim 2n, and let
A;(M) = A+(M) + A_(M)
be the super structure defined in Lemma 1.4.17. Let
U = {d + 6f:Kl{M)-^Kl{M\
dt
and G{t,q,p) be the fundamental solution for the heat operator ^ + D.
Then
limt^rG(.,p,p)=|(n^)}(^..-.^-).
(3) Let M be an oriented Riemannian manifold of dim 2n with a Sptn{2n)-
structure, and
T{E) = r{E-^) + T{E-)
be the superstructure defined in Definition 6.2.1. Let
D = Z)2 : r{E) -V r{E),
where D is the Dirac operator. And let G{t,q,p) be the fundamental
solution for the heat operator ^ + D, then
limtlG(.,p,p) = |(nii|^)}(^--.^3„).
(4) For a Kahler manifold M of complex dimn, let
r{E) = TiE^^^"^) + riE""^^)
be the super structure defined in Definition 8.4.3. Let
D : T{E) -^ T{E)
be the Riemann-Roch operator and □ = D^, and let G{i^q,p) be the
dt
- D2
fundamental solution for the heat operator ^ + D, then
s — l 2

CHAPTER 1
PRELIMINARIES IN RIEMANNIAN GEOMETRY
The readers of this book are assumed to have an acquaintance with
elementary differential geometry, which contains the curve and surface theory in R^
and the Prenet moving frame method, and with elementary differential
manifold (tangent vector, Jacobi identity, differential form and exterior differential).
Such background materials could be found easily elsewhere, for example [40]
or the first three chapters of [13].
1.1 Basic Notions of Riemannian Geometry
Gauss proved a very famous, and important theorem (Egregium
theorem) in 1827, and Riemann delivered a historic Habilitation address in
1854. Those are two great events, which gave birth to differential geometry or
Riemannian geometry. The core of the events displays a process from the first
fundamental form to the curvature. The process is very mystic indeed so that
several even great mathematicians were busy with it in more than fifty years
after Riemann died. They found a notion of connection also while clearing the
process. Nowadays the first fundamental form is called the Riemann metric.
By a precise language the above process contains
(1) the definitions of Riemann metric, connection and curvature,
(2) an algorithm to get a Levi-Civita connection and curvature in sequence
from the Riemann metric,
(3) Bianchi identities, which show relations among components of curvature.
Before we explain the process let us recall the basic knowledge about
differential manifolds. By using the definition of differential manifold we introduced
the notions of C°°-functions, a tangent vector space at every point x E M.
Denote the set of all C°°-functions by !F{M)^ and the tangent vector space at
X e M by TxM. Define TM = UxTajM with a projection tt : TM -^ M :v \-^ x,

2 PRELIMINARIES IN RIEMANNIAN GEOMETRY
where v G T^M. Call TM the tangent bundle of M. A (tangent) vector field X
of M is defined to be a section of TM, it is to say that X is a map X : M -^ TM
with a property wX = 1 : M -^ M. We denote the set of all C^-sections of TM
by r(TM). It is easy to introduce some algebraic operations in T{M),TxM
and r(TM), which are
• : R X Ta^M -> T^M : (a, v) ^ av,
+ : r(TM) X r(TM) -> r(TM) : (X, Y)^X + Y,
• : T{M) X r(TM) -^ r(TM) : (/, X) ^ fX.
Moreover, we also introduced two analytic operations
• : r(TM) X T{M) -^ r(TM) : (X, /) ^ Xf
[•,.] : r(TM) X r(TM) -> r(TM) : (X, Y) ^ [X, Y],
and checked the following properties
{f + g)X = fX + gX, (fg)X = f(gX), f{X+Y) = fX + fY
X{f + g) = Xf + Xg, X{fg) = (Xf) ■ g + f{Xg),
{X + Y)f = Xf + Yf, {gX)f = g{Xf),
[X, Y] = -[Y, X], [Xi +X2,Y] = [Xi, Y] + [X2,Y],
[fX,Y]^f[X,Y]-iYf)X
[[X, Y],Z] + [[Z,X], Y] + [[r,Z],X] = 0 (Jacobi identity).
Now we are going to explain the process initiated by Gauss and Riemann.
We take a view from the outset of this book that all the manifolds to be used
are compact, smooth, and without boundaries. Let M be a manifold
of dimension n, a metric g is an assignment, which assigns to every tangent
space TxM an inner product
g^{;-):T^M xT^M^R

1 1 Baste NoUons of Rtemanman Geometry 3
such that for any local coordinates (aji, • • •, ajn) if we let
^.,(.) = ^.(^,^),
then Qij is a C^-function with respect to the coordinates of x, where ^ is the
tangent vector at the point x of the aj,-curve, and 1 < 2,J < n. For (tangent)
vector fields X, Y", a function g{X, Y) is defined by
g{X,Y){x) = g4X,,Y,),
where X^ = X{x). Sometimes we denote g{X,Y) by {X,Y).
Definition 1.1.1 Suppose M is a manifold, a map
V : T{TM) X T{TM) -^ T{TM) : (X, Y) ^ Vx^,
IS called a connection if it satisfies
(i) Vx^ is ^(M)-linear with respect to the variable X, that is to say, for
X, W, Xi, X2 G T{TM) and a function / G T{M),
VfxW = f\7xW,
(li) Vx is a derivative operator when X is fixed, i.e.
Vx{Wi + W2) = VxWi + VxW2
Vx{fW) = {Xf)W + fs7xW.
Definition 1.1.2 A manifold with a metric is called a Riemannian
manifold. For a Riemannian manifold M, a connection V is called a Levi-
Civita connection if it also satisfies
(iii) X{Wu W2) = iVxWu W2) + {Wu Vx^2),
(iv) \7xY -\7yX = [X, y], where [X,Y] is the Lie bracket of X and Y.
Theorem 1.1.3 (Fundamental Theorem for Levi-Civita
connection) For a Riemannian manifold there exists a unique Levi-Civita
connection.

4 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proof By using equalities (iii), (iv) repeatedly of Definition 1.1.2 in
turn on the quantity (Vx^?^) we have
{s7xY,Z)^'^^X{Y,Z)-{Y,VxZ)
^':l^X{Y,Z)-{Y,[X,Z])-{Y,VzX)
^=^X{Y,Z)-{Y,[X,Z])
-Z{Y,X) + (^7zY,X)
(«v) («««)
^=^X{Y,Z)-{Y,[X,Z])-Z{Y,X)
+([Z, Y],X) + Y{Z, X) - {Z, [Y, X])
-{Z,VxY),
consequently
(VxY, Z) = ^{-{X, [Y, Z]) + X{Y, Z)
+{Y,[Z,X])^Y{Z,X)
+{Z,[X,Y])-Z{X,Y)},
SO the uniqueness is true. As soon as we verify that the left hand side of the
above equality satisfies conditions (i)-(iv) in Definitions 1.1.1 and 1.1.2, the
truth of the theorem becomes self evident.
Definition 1.1.4 Suppose V is a connection on M, let a map
R : r{TM) X r(TM)x r{TM) -^ r{TM)
be defined by
R{X,Y,Z) = (Vx Vy -VyVx- V[x,y])Z.
We call the above map i2(-, •, •) the curvature of V- Usually we denote
R{X,Y,Z) by R{X,Y)Z.

1 1 Baste Notions of Rtemanman Geometry 5
Exercise 1.1.5 Prove that the above R{X^Y)Z is ^(M)-linear with
respect to all the variables X, F, Z.
Remark 1.1.6 What we have done above is the main part of the
process which had been dealt with by many great mathematicians (Gauss, Rie-
mann and their successors) for more than fifty years. It shows that to crystallize
the notions of the metric, the connection and the curvature from the mass
computations in calculus is very hard. Here, of course, we had missed some points
in the process, e.g. when M embedded in a big space, the curvature defined
above can express the curved shape of it. The proof of these points depends
on a method for getting the integrability conditions, which is similar to that
for getting the Bianchi identities below.
Remark 1.1.7 The notion of connection was invented by the successors
of Gauss and Riemann. Nowadays it becomes very important. The role it plays
in mathematics may be viewed in three ways. Firstly it is an intermediate
station from the metric to the curvature. Just like a marvelous helpful line
m Euclidean geometry it clears the computations in the process. Secondly
its explanation of parallel transformation induces a notion of derivatives in
manifolds, then brought the calculus into geometry. Thirdly there is at least a
quotient space of a set of several connections, which is a priceless mathematics
treasure. It aroused a great progress in topology.
Now we are going to introduce the Bianchi identities. In order to do so
we need the derivatives of the curvature first. For X G V{TM) define a map
SjxR ' T{TM) X T{TM) x T{TM) -^ T{TM)
:{Y,Z,W)^{s7xR){Y,Z,W)
by
{s7xR){Y,Z,W) = S7x{R{Y,Z,W))-R{s7xY,Z,W)
-R{Y,S7xZ,W) - R{Y,Z,S7xW).
It is easy to see that {syxR){Y^Z^W) is ^(M)-linear with respect to all the
variables X, Y", Z, W.
Theorem 1.1.8 Suppose M is a Riemannian manifold, V is the Levi-
Civita connection, then for any X, Y^Z^WE r(TM),
(i) (Bianchi I)
R{X, y, Z) + R{Z, X, Y) + R{Y, Z, X) = 0,

6 PRELIMINARIES IN RIEMANNIAN GEOMETRY
(ii) (Bianchi II)
{VxR){Y, Z, W) + {SJzR){X, y, W)
+(Vyi2)(^,^,^) = 0.
Proof The key trick of the proof is to use the Jacobi identity
[X, [r, ^]] + \z, [X, Y\\ + [y, \z, x]] = o.
From
[x,[r,^]] = Vx[r,^]-V[y,z]^
[^, [X, y]] = Vz Vx i" - Vz Vy ^ - V\x,y\Z.
[y, [^, X]] = Vy Vz ^ - Vy Vx ^ - V{z,x\Y.
it follows that
0 = [X, [y, z]] + {z, [X, y]] + [y, {z, x\\
= VxVyZ-S7yVxZ- S7[x,y]Z
+ Vz VxY -VxVzY- V[z,x]Y
+ Vy VzX -VzVy X - V[Y,z]X
= R{X, y, Z) + R{Z, X, Y) + i2(y, Z, X),
The first Bianchi identity is proved. Further we have
(Vxi2)(y, Z, W) = Vx{R{Y, Z, W)) - RiVxY, Z, W)
-R{Y, sjxZ, W) - R{Y, Z, sjxW)
= S/x Vy S/zW - Vx Vz VyW - Vx V[y,z] W
- VvxY VzW + Vz VvxY W + S7[vxY,Z]W
- Vy VvxzW + sj^xz VyW ■\- V[y,vxZ]^
-S/yS/zS/xW-V \Jz Vy Vxl^ + V[y,z] Vx ^-

If we let
1 1 Baste NoUons of Rtemanman Geometry
a{X, Y, Z, W) = Vx Vk VzW - Vx Vz VyW
-VyVzVxW + \Jz Vy VxW^,
/3(X,y,Z,W) = - Vx VvyzW + Vx Vvzr W^
+ Vz Vv^y^ - Vy Vvx^ W^)
7(X, y, ^, W^) = - Wxy V^W^ + Vvx^ Vy W^
+ Vv«^ VxW^ - "^vzY Vx W,
6(X, y, Z, W) = V[vxY,z]W - V[vxZ,Y]W,
then
iVxRW, Z, W) = a(X, Y, Z, W) + /3(X, Y, Z, W)
+j{X,Y,Z,W) + 6iX,Y,Z,W).
It is easy to see that
a(X, y, Z, W) + a{Z, X, Y, W) + a{Y, Z, X, W) = 0,
(3{x, y, z, w) + f3{z, X, y, w) + /3(y z, x, w) = o,
7(X, y, Z, W) + y{Z, X, Y, W) + y{Y, Z, X, W) = 0,
and
s{x, y, z, w) + s{z, X, y, w) + 6{y, z, x, w)
= V[X,[Y,Z]]~[.[Z,[X,Y]]~[.[Y,[Z,X]]W.
Therefore the Jacobi identity also implies the second Bianchi identity. The
proof is complete.
It is plainly that Theorem 1.1.8 had not yet listed a complete set of algebraic
relations among the curvature and its derivatives. Some important relations
will appear as corollaries of Theorem 1.2.2 in the next section.
In the above discussions we introduce the notions of Riemann metric, the
Levi-Civita connection, the curvature, the derivatives of the curvature and
the relations among them. All we have done were formulated in a so-called
invariant point of view, which means that all notions and relations do not
depend on the choice of reference systems. A reference system used to be a
local basis of M, i.e. {cri, • • • ,crn}, where cr, are sections of TM defined on
an open set U of M, such that for any point x £ U^ {cri(aj), • • • ^(Tn{x)} is a
basis of the vector space T^M, Plainly there always exists a local basis in any

8 PRELIMINARIES IN RIEMANNIAN GEOMETRY
small open set. By the local basis one can express the geometric notions and
relations by quantities and formulas. As far as a geometric notion or a relation
is concerned, the corresponding quantity or the formula will not be invariant
when the local basis is changed. In this way the local basis method is different
from the method of the invariant point of view. Usually we do not like to take
the invariant point of view for complicated computations, the reason may be
that the geometric spaces, e.g. r(TM), used to be R-linear spa<;es of infinite
dimension, while as .7^(M)-modules they are of finite dimension locally.
Now we choose a local basis {cri, • • • ,(Jn}, then any vector field X can be
expressed as
n
1
where /, is a function defined on the open set U. The corresponding
quantities of the vector field X is (/i,***?/n)- Similarly the geometric notions
g^sy^R^syR can be expressed by quantities g%j^T^j^Rijki^Rijki,s respectively,
which are defined by
g,j = g{(Tt,aj) = {(Tt.aj),
s
Rtjkl = {R{o't,o-j,ak),o-i),
Rtjkl,s = ((V<r,i2)K,^j,^Jk),^/),
where (g^^) is the inverse matrix of (gij)- It is not difficult to check that the
above quantities satisfy
k
R{at,aj,ak) = J^Rijkig ^os,
{S7<TrR){o'j,crk,o-i) = / ^Rjkls.ig^^o'm'
s,Tn
Lemma 1.1.9 Suppose {cti, • • • ,cryi} is a local basis, with respect to
which we express Riemann metric, curvature and its derivative by gtj^Rtjki
and Rijki,s respectively. Then
R.jkt = (T.T^u - '^j^Tk + r;*r™ - rf,r™ - c/.r^,)g^i,
■tinjklyS — ^s-^tjkl ~ ^ St ^rnjkl ~ '^ sj ^rnkl ~ I sk-^^J^^ ~ ^ si ■'^^jkm 5

1.1 Baste Notions of Rtemanman Geometry 9
where the repeated indices imply addition has been employed.
Proof From the definition we easily get the first equality. Due to the
equalities
Rtjkl,s = ((V<r,i2)(o-,,0-j,0-jk),0-;)
= crs{{R{crt,crj,ak)),cri) - {R{cr%,crj,crk),\/(T,(^l)
•^ ^sJ^ijkl ~ J- si^'^ijkl ~ J- sj^^'mkl ~ J- sk^^J""^^ ~ ^ si ^^3^""^
we get the second. The proof is complete.
Before we express the relations among the geometric notions by formulas
of quantities, we need to know the brackets of the local basis. For a local basis
{o-i,---,o-n} define Cf^ by
k
Then the Jacobi identities are expressed as
X;(Qcl + cue], + c;^cl) = 0 vi,J, k, I.
s
Theorem 1.1.10 For the local basis we have the following relations
among the quantities of the geometric notions
(0 ^tj = h'^ { (^i9jk + (^j9zk - (^k9zj - gziClk + gjiCl^ + QkiClj },
(ii) (Bianchi identity I) R^j^^i + R^^^^i + R^^^.i - 0,
(iii) (Bianchi identity II) R^Jkl,s + Rsiki.j + Rjski.i = 0.
Proof By using an equality in Theorem 1.1.3
(Vxr,Z) = ^{-(X, \Y,Z]) + X{Y,Z)
+{Y,[Z,X])+Y{Z,X)
+{Z,[X,Y])-Z{X,Y)]

10 PRELIMINARIES IN RIEMANNIAN GEOMETRY
we easily get (i). Theorem 1.1.8 implies (ii),(iii).
As in Theorem 1.1.8 here we had not listed some algebraic relations among
the curvature and its derivatives (see Corollary 1.2.3).
Definition 1.1.11 For a coordinate system {aji, • • •, a^n} the local basis
{ ai~' *" ' af~} ^^ called a natural local basis or a natural frame.
Formula 1.1.12 With respect to a natural frame {^f-, • • •, af~}' ^^^
computation formulas are as follows
fi Pi
^ dXi' dxj' dxjc ' dxi
R„ki,s - ((V..i?)(^, ^, ^), ^),
p p pm p pm p jym p pm p
Theorem 1.1.13 With respect to a natural frame
(i) rj^ = Ig'" {^^g,, + ^g., - ^g,,}, Tf^ = T'^„
(ii) (Bianchi identity I) R^ju + Rkiji + Rjkii - 0,
(iii) (Bianchi identity II) Rtjhi.s + Rsihi.j + Rjski.i = 0.
As before here we had not listed some algebraic relations among the
curvature and its derivatives (see Corollary 1.2.3).

1 2 Computattons by ustng Orthonormal Moving Frame 11
1.2 Computations by using Orthonormal Moving Frame
It is well known in the elementary differential geometry that Prenet frame
is very elegant and very useful for getting geometric invariants from the tedious
computations when dealing with curves in R^. One of the characteristic
features of Frenet frame is that it is a family of orthonormal frames. It gives us a
feeling that an orthonormal frame field may be more convenient to use than the
general local basis {cri, •••, cTyj} in §1. A local orthonormal basis {Ei, -" ^En}
is called a family of orthonormal frames or an orthonormal moving frame,
if it satisfies
where
' \ 0, iit^j.
As we did in the local basis case, for an orthonormal moving frame {Ei,-- - ,En}
we know g^J = {E^.Ej) = S^j, and we define Cf^, rf^, R^ju, Rtjki.s, by
C;^^ = {[E,,E,],E,),
T^j = {VE.Ej,Ek),
Rijki,s = {{VE,R){Ei^Ej,Ek)^Ei)'
then by Lemma 1.1.9 and Theorem 1.1.10 we have
Formula 1.2.1
(i) g,j = S,j
(ii) rf,:.i{Cf,+Q^ + C^J,
(iii) R^jki = ( E^r^j^ - EjT^j^ + T^j^r,, - r,\rj, - C'jT[j^ ),
(iv) Rtjkl,s = ^sRijkl — ^Tt^jkl — ^Tj^tmkl " ^Tk^jml — ^^jRtjkm^
and
(v) (Bianchi identity I) Rijki + Rktji + Rjkti = (^,
(vi) (Bianchi identity II) Rijki,s + Rstkl,j + Rjskl,t = 0.

12 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Theorem 1.2.2
(vii) rf,--r^„
(viii) Riju = —Rtjik-) Rijki.s — —Rijik.si
(ix) Rijkl = —Rjtkh Rijkl,s = —Rjikl,s^
(x) R^jki = Rkhj^ Rijkl.s — Rkhj^s-
Proof Due to Cf^ = -C^, and formula 1.2.1 (ii),
SO (vii) is true, (viii) is true, due to (iii), (iv) and (vii). By definitions we have
R{X, y, Z) = -R{Y, X, Z), S7wR{X, Y,Z) = - sjw R{Y, X, Z),
which imply (ix). The proof (x) is as follows. The following computations
Rijkl = —Rktjl — Rjktl
= Rktlj + Rjkh
= —Rlkij — Rtlkj — Rljki — Rkljt
= ^Rkhj + Rzljk + Rljik
= ^Rkhj - Rjiik
= '^Rklij — Rtjkh
Rijkl.s = '" = 2Rkhj,s — Rtjkl,s
imply the assertion (x). Thus the proof is complete.
Corollary 1.2.3
(i)
{R{X,Y)Z,W) = -{R{X,Y)W,Z),
{s7vR{X,Y)Z,W) = -{vvR{X,Y)W,Z).

1 2 ComputaUons by ustng Orthonormal Movtng Frame 13
(ii)
{R{X,Y)Z,W) = {R{Z,W)X,Y),
{s7vR{X,Y)Z,W) = {vvR{Z,W)X,Y),
(iii) With respect to a natural frame, the formulas similar to (viii)-(x) in
Theorem 1.2.2 also hold, i.e.
■*Mjkl — ~-t('tjlk^ ^jkl,s — ~^tjlk,s^
JMjkl — ~J^j%kh ^jklyS — ~^j%klyS^
Rtjkl = Rkhj^ Rtjkl^s = Rkhj^s-
Proof Check (i) by using the definitions. In order to prove (ii) it is
sufficient to check (ii) when X,Y,Z,W,V G {jE7i, • • •, jE7n}, which is just the
assertion of Theorem 1.2.2(x). (iii) may come from (i) and (ii). We omit the
details here.
Remark 1.2.4 In Corollary 1.2.3, (ii) may not be easy to prove directly.
From this fact we may see the advantage of using an orthonormal moving frame.
The same advantage may be seen in the following definition.
Definition 1.2.5 Let M be a Riemannian manifold. For any function
/ : M -^ R G T{M) define a function Aq/ : M -^ R such that if {jE7i, • • •, jE7n}
is an orthonormal moving frame, then
where Z>(-, •) is the second order derivative operator
D : T{TM) X T{TM) x T{M) -^ T{M) : (X,y,/) ^ D{X,Y)f,
defined by
D{X,Y)f = {XY-^xY)f.
Remark 1.2.6 We must prove that the above Aq/ is well defined, that
is to say, for any two orthonormal moving frames {jE7i, • • •, jE7„ } and {jE^i, • • •, jE?n}
% %
in the common neighborhood on which two frames are defined.

14 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Definition 1.2.7 Let M be a Riemannian manifold. Define the m-th
derivative operator
D : r(TAf) X • • • X r(TM) xT{M) -^ T{M)
' r '
m times
: {Xi,'" ,Xm,f) ^ D{Xi,"' ,Xm)f,
such that
D{X)f = Xf
D{Xi, " ' ,Xm + l)f = XiD{X2r " ,Xm + l)f
-D{\/XiX2,X3,' •• ,Xm~^l)f D{X2, '" ,Xm,VXiXm + l)f'
It is easy to check that D{Xi, • • • ,Xm) is .7^(M)-linear with respect to the
variables Xi, • • •, Xm-
1.3 Differential Forms and Orthonormal Moving Frame Method
In §2 we display the process from the first fundamental form to the
curvature and the Bianchi identities by using a reference system of orthonormal
moving frames. In this section we will do the same with the same reference
system, meanwhile we also apply differential forms. Although no new results
can be obtained, but a new formulation for the process is given. Such a new
formulation is understood as an orthonormal moving frame method, which
can be generalized to a moving frame method in §7. Now let us begin by
recalling the basic knowledge about differential forms.
Let M be a differential manifold, and A^{M) be the set of all differential
Ai-forms, and
a*{m) = Y1a\m).
k
In A*(M) there are several algebraic and analytic structures, they are
(1) addition
+ :A^{M) xA^{M)-^A^{M) : {ui,U2) ^ u;i+u;2,
(2) multiplication
A : A^(M) X A\M) -^ A^+'(M) : (a;i,a;2) ^ a;i Aa;2,

1 3 Dtfferenttal Forms and Orthonormal Moving Frame Method 15
(3) exterior differential
d : A^(M) -^ A^+^(M) lu^du,
with the properties that
(i) the addition operation is commutative and associative, i.e.
{ui + U2) + a;3 = a;i + {u2 + a;3),
(ii) the multiplication A is skew-commutative and associative, i.e.
a;i Aa;2 = (-l)*^'a;2 Aa;i, Va;i G A*^(M), a;2 G A'(M),
(a;i A 0^2) A 0^3 = a;i A (0^2 A 0^3),
(iii) d{du) = 0,
(iv) (+,A) is distributive, i.e.
(a;i + 0^2) A 0^3 = a;i A 0^3 + 0^2 A 0^3
(*^3 A (a;i + U2) = a;3 A a;i + a;3 A U2,
(v) {d,+) satisfies
d(ui + 0^2) = dui + du2,
(vi) (d. A) satisfies
d{ui A U2) = (dui) A a;2 + (-l)*'a;i A {du2), \fui G A*^(M),
(vii) A^(M) = T{M); and for /, gf G A^(M), fAg is the ordinary multiplication
of two functions; moreover the constant functions 0 and 1 are the unit
elements of the addition and the multiplication respectively,
(viii) any differential form can be restricted to any neighborhood U; if {aji, • • •,
Xn} is a coordinate system in U, then as a .7^(?7)-module, A*{U) is
generated by {dxi, • • • ,dajn}, and
df — -—dxi H (- -—dxn.
OXi OXn

16
PRELIMINARIES IN RIEMANNIAN GEOMETRY
It is worthwhile to mention that one can define the differential forms
without knowing anything about the notions of the tangent vector or the tangent
vector field (see [40] or [13], for example), although there exist pairing
relations between A*(M) and T{TM). In fact, there are two pairings prevalent in
mathematics early or late. Now we introduce one of them, which was adopted
by most mathematicians nowadays. The pairing
B : A*(M) X T{TM) x • • • x T{TM) -^ T{M)
' y '
k times
: (a;,Xi,---,Xjk) i-^ a;(Xi, • • • ,Xjk)
is determined by the following conditions
(1) a;(Xi, • • •, Xjk) is .F(M)-linear with respect to the variables a;, Xi, • • •, X^;
(2)ifa;i,---,a;fc G Ai(M), then
^i(Xi)
(a;i A • • • A (^k){Xi, • • •, Xk) -
^k{X^)
^h{Xk)
The above pairing implies the following formulas:
(3) If a G A^(M), /3 G A'(M), then
(aA/3)(Xi,...,X,+0
~ j^ X^(-l)''<^(^7r(l), • • • , X^(fe))/3(^7r(fe+l), • • • , ^7r(fe+0)'
TT
where tt runs over the set of the permutations of{l,---,Ai + /}, and ( —l)'^
is 1 or —1 depending on whether tt is an even or odd permutation.
(4)Ifa;G A^(M),then
(da;)(Xi,...,X,+i)
+ ^(_l)«+^a;([X, ,X,],Xi,...,je„...,i,,...,X,+i),
i<3

1 3 Differential Forms and Orthonormal Moving Frame Method 17
here the notation X, means that the term X, is discarded.
Now let us explain the orthonormal moving frame method. For an n-dim
Riemannian manifold, choose an orthonormal moving frame {jE7i, • • •, jE7n}, and
let {ct;i, " ' ,Un} be the coframe dual to {Ei, • • •, jE7n}, i.e. it satisfies
u;^{EJ) -8,j, \/iJ = l,---,n.
Then define a set of 1-forms u^j and 2-forms Ct^j by solving the following
equations
R{X,Y)E,=J2^J^(^^^)^J'
J
Lemma 1.3.1 We have
where
r,j = {\/E^EJ,Ek), Rijki = {R{Ei,Ej)Ek,Ei).
Proof The lemma follows from
a
= {Ve,E,,E,) = TI^,
Cttj{Ek,Ei) — y^^{Ctaj{Ek,Ei)Ea,Et)
a
= {R{Ek, Ei)Ej,Et) = Rkiji
= —Rkhj-
Sometimes we denote
/ ^11 ••• ^In \
\ Unl • •• (^nn /

18
PRELIMINARIES IN RIEMANNIAN GEOMETRY
I 1^11 • • • Q.in \
9.^ = 9.=
\ ^nl
(1)
Lemma 1.3.2
^nn )
(2) Ua = dUa -^U(r ^UJ(T
{i)ii A'.U -^ 0{n) is a map, where U is the domain on which the orthonormal
moving frame is defined, and 0{n) is the orthogonal group, then
U(r A — A~^ 'Ufy ' A-\- A"^ ' dA.
{A)n^A^A-^'il^'A,
Proof Prom
u,j{X) = iVxEj.E,) = X{Ej,E,) - {Ej,S7xE,)
= -{Ej,S7xE^) = -Uj,{X)
it follows that the second equality in (1) is true. For two vector fields X^Y £
r(TM), we have
{du,){X,Y) = Xu,{Y) - Yu,{X) - u,{[X,Y]);
{-J2^,Au,,){X,Y) = J2(^,,{X)u,{Y)-u,,{Y)u,{X)),
J J
Denote the right-hand sides of the above two equalities by a«, /3«, then we have
Y,<^.E. = 5](Xa;.(r) - Yu.{X)-u,{[X,Y]))E,
= Vx{J2'^,{Y)E,) -Y^u;,{Y) Vx E.
2 2
- Vy iJ2'^.iX)E,) + Y,'^.iX) vy E, - [X,Y]
2 2
= VxY-Y,'^,iY)u,,.iX)Ej
- Vy X + '^w,{X)w„{Y)Ej - [X,Y].
hJ

1 3 Differential Form,s and Orthonorm,al Moving Frame Method 19
By
\7xY - VyX - [X, Y] = 0, oj,j = -wj,
we get
hence a, = /3,, (1) is proved. For the simplicity we denote u^ by w, then
Vx vk <^ = Vx(o- • w(y)) = (vxo-) • w(y) + o- • Xw(r)
= a-{w(X)a;(y) + Xa;(y)},
thus
R{X, Y)a = (Vx Vr - Vr Vx - V[x,y] V
= (T • {w{X) ■ w{Y) + Xw(r) - u{Y) ■ u{X)
-Yu>{X)-u;{[X,Y])]
= <T-{{d^){X,Y) + {whw){X,Y)},
consequently
n = cZw + w Aw,
(2) is proved. The following computations
= a-{u„{X)-A + XA]
= (o- • A){A-^ ■ u}„{X) -A + A-^- XA]
= (a ■ A){{A-^ -oj^-A + A-^- dA){X)},
together with the definition
\7x{(^-A) = {a-A)u„AiX)
imply (3). Since R{X,Y)Z is :F(M)-Unear,
R(X, Y){<T ■ A) = {R{X, Y)<t) ■ A = (o-ft<,(X, Y)) • A
= {a ■ A){A-^ ■ n4X,Y) ■ A).

20
PRELIMINARIES IN RIEMANNIAN GEOMETRY
Hence (4) is proved, and the proof is complete.
Cartan Lemma 1,3.3 Given {Ei, - -- ,En} and {ct;i, - -- ,Un} as above,
the following system of equations has a unique solution {^«j},
1 ^
Proof First let us prove the uniqueness. Suppose the equation has two
solutions {O^j} and {^,j}, then let
Y^Uj AaJ^ rzO,
J
{a,j} satisfies
which implies
where {cKjjjk} satisfies
k
Applying the above two equalities about aijk in sequence we have
It implies a,jjk = 0. Hence the uniqueness is proved. Similarly applying the
equalities
{^ijk — ^kjt -^ fjkii
^ijk = —^jiki
on O^jTz in sequence we get O^j^ ^s linear functions of ftj^. It is easy to check that
those functions have the following property that if {fijk} with antisymmetric
indexes j, k satisfies
J<k

1 3 Dtfferenttal Forms and Orthonormal Moving Frame Method 21
then the 2-forms
k
satisfy the equations in Lemma 1.3.3. Therefore the lemma is proved.
Lemma 1.3.4 The notations are given as above, then we have
(i) X^o;, Al^,. = 0;
J
(ii) dn^J = ^{^tk Aukj - i^tk A l^jkj).
k
Proof Because
J J
= ^^k ^(^kj Aa;^, + ^(^j A(l^j, - ^cjjk ^(^ki)
J,k J k
J
dn,j = d{du,j + Y^i^tk ^^kj) = X^CC^'^^aJk) Aa;jkj - u^j, Adu^j)
k k
= ^i^zk -^i^ti A i^ik) A Ukj
k I
k I
= ^{^tk A Ukj - u^k A ilkj)
k
the lemma is true.
Lemma 1.3.5 Lemma 1.3.4 implies Bianchi(I) and Bianchi(II). More
exactly,
(i) {J2^J A nj,){Ek, Eu Em) = R^klm + Rz mkl
3
(ii) {dil^J - ^{il^s A Usj - U^s A l^,j)}(£^Jk, Eu Em) =
s
y^kUj^m, T ^mktj,l i ^lTmj,k)'

22 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proof Recall the equality
(aA/3)(Xi,..-,X,+0
- j^ Zj(-l)''«(^7r(l), • • • , ^7r(fe))/3(^7r(fe+l), ' ' ' , X^(k+l)),
TT
where a G A*^(M), /3 G A'(M). Then we have
{J2^J^^J^) {Ek^Ej.Em) = J2''Ji^k)^J^{El,Em)
J J
+ J2''j(^rn)^AEk,Ej) + J2''j(^^)^J^(^rn.Ek)
J J
= — 2_^ ^jkRjilm — 2_^ ^jmRjtkl — 2_^ ^jlRjtmk
J J J
— ~J^kilm ~ -ttrmkl ~ J^Umk ^ J^iklm i J^imkl i JMlmk'
Further we have
{dQtj){Ek,Ei,Em) = Ek{^ij{Ei,Em)) - Ei{Ctij{Ek,Em)) + Em{^%j{Ek,Ei))
—Q.ij{[Ek^Ei\^Em) + Uij([Ek^Em\^Ei) — Uij([Ei^Em\^Ek)
— —EjcRtjim + EjR^jkjm — EmRijkl + 2_^ ^klRiJsm — 2_^ ^Ik^ijsm
s s
-Z^^kmRiJsl + Z^^mk^^jsl + 2^^lmRiJsk " / ^^ml^jsk
s s s s
s s
-\-Rkrmj,l + / ^ ^izRkmsj + / ^ ^ijRkmts
s s
— RkUj,Tn — / J ^rmRklsj ~ / ^ ^ mjRkUs',
s s
Y^{n„ Aw,j -w., An,j)}{Ek,E,,E,n) = Y,{-R,,kiTtnj + R>,kir'^s
s s
iJ^iskm^ Ij ■ttsjkm^ ig -ttislm^ %j "T -tCsjlm ^ ks S'
Then the lemma is easy to prove.
Exercise 1.3.6 If {cri, -- - ^cTn} is a local basis, {a;i, • • • ,0;^} is its local
dual basis, and
j<k

1 4 Classical Geometric Operators 23
then
[E„E,] = J2fijEk.
1.4 Classical Geometric Operators
Let M be a Riemannian manifold of dim n, and
V : T{TM) X T{TM) -^ T{TM)
be the Levi-Civita connection. A connection on A*{M)
V : T{TM) X A*(M) -^ A*(M)
can be defined such that
Vx{(^i Aa;2 A • •• Auk) = (Vx^*^!) Aa;2 A • • • Aa;jk
+a;i A {S7x^2) A-- Auj, -\ f- a;i A a;2 A • • • A {syx^^k),
and
{Vxi^^){Y) = X{u,{Y)) - u,{s7xY), \/Y G r(TM).
This connection is also called Levi-Civita connection. The connection may be
viewed as a first order derivative operator
Di-y : T{TM) X A*(M) -^ A*(M) : {X,u) ^ Vx^ = D{X)u.
Definition 1.4.1 The second order derivative operator
D(-, •)• : T{TM) X r{TM) x A*(M) -^ A*(M)
:(X,y,a;)H-.Z)(X,y)a;
IS defined by
Z>(X,y)a; = D{X)D{Y)u - DCVx^^^,
and the higher order derivatives are defined by the same formulas as in
Definition 1.2.7. I.e.
D{Xu ..., Xm+i)(^ = Dx,D{X2, • • •, Xm+i)(^
-D{syxiX2,X3,"' ,Xm^i)u D{X2r",Xm,VXiXm+i)u.

24 PRELIMINARIES IN RIEMANNIAN GEOMETRY
It is easy to check that the k-th. derivative D{Xi, -" ,Xk)u is .F(M)-linear
with respect to all variables Xi, • • •, Xj^,a;, and
R{X,Y)u = D{X,Y)u - D{Y,X)u,
where
R{X,Y)u = (vx Vy - Vy VxV[x,y]V-
Definition 1.4.2 Define a second order differential operator
Ao:A*(M)^A*(M)
by letting
where {jE7i, • • •, jE7n} is an orthonormal moving frame. And call Aq Lap lace-
Belt rami operator.
From the differential forms we get a sequence
with a property dd = 0. This sequence is called de Rham complex. For
a coordinate system {aji, • • •, ajn}, by using the properties of differential forms
listed in §3 we can get a local expression for df, where / G A^{M). If
then
df = V ^'^ '"''dx^Adx^^ A"'Adx^^.
^-^ ox.
However we have not got a local expression for d yet.
Lemma 1.4.3 Let {Ei^- -- ^E^} be an orthonormal moving frame, and
{ct;i, • • •, Ct^yi} be its dual frame. Then
Proof By using the notations for orthonormal moving frame mentioned
in §3, for any a G A*^(M), we only need to check the equality da = X^'*^* A
S/e^ol. Let
^ = d-Y^u,As/Er' A^(M) -^ A^+^(M).

1.4 Classical Geometric Operators 25
It is easy to see that $ is .7^(M)-linear, and
$(a A /3) = $(a) A /3 + (-l)^a A ^(/3) Va G A^(M).
So in order to prove the lemma it is sufficient to check
^{ui) = 0, or dut = ^2^J ^ VEj^i-
J
Because
j,k
3y^
and
3 J k
the proof is complete.
The (Riemann) metric
(•, •) : T{TM) X T{TM) -^ T{M)
can induce a metric
(•,-):A^(M)x A^(M)-^.F(M)
such that for any a,,/?, G A^(M),
(«i,/3i) ••• (ai,/3jk)
(ai A---Aajk,/3i A---A/3jk)
(ajk,/3i) ••• (ajk,/3jk)
It is easy to check that the new metric exists uniquely, and we call it Riemann
metric too. Moreover the original Riemann metric can induce a measure dv on

26 PRELIMINARIES IN RIEMANNIAN GEOMETRY
M (see [36], p. 194 for details). By using the Riemann metric and the measure
we define an inner product
((.,•)) :A\M) xA\M)^K
such that for a,/3 G A^{M),
((«,/3))= / {a,f3)dv.
Jm
It is easy to see that
(1) ((a,/3)) is R-linear with respect to the variables a and /3;
(2) {{a^a)) > 0, and the equality sign holds if and only if a = 0.
The inner product can be extended to an inner product on A*(M), such that
for a e A*^(M), (3 G A\M) and k ^ I, define ((a,/3)) = 0.
By using the inner product we can introduce the conjugate operator d*
of d. A common knowledge tells that for a differential operator there always
exists uniquely a conjugate operator. So now the question is how to express
d*. Hodge introduced a star homomorphism * : A^{M) -^ A"~*^, which
will help indeed. If M is an oriented manifold, choose an orthonormal moving
frame {jE7i, • • •, jE7n} which has the same orientation as M. Let {a;i, • • •, a;„} be
the dual coframe, then we introduce
Definition 1.4.4 For
«i< <«fc
define
Jl< <Jn-k
where 6(zi, • • • ,ijk, ji, • * * ^Jn-k) is 1 or —1 or 0 depending on whether (ii, • * * 5^?
3i,' " fjn-k) is an even permutation of (1, • • •, n) or an odd permutation or
otherwise respectively.
For the above definition we need to prove that the homomorphism * is well
defined. In fact, if we choose another orthonormal moving frame {Ei, • • •, jE7n},
then there is a map from the common domain J7, on which two frame fields are
defined, to the rotation group
A={A^j):U-^SO{n)

1 4 Classical Geometric Operators 27
such that
(Eir",En) = {Ei,"',En)'A.
The above equality implies
where A* is the transpose of A. We extend the definition of /j^ ^j^ for the
general Ccise of indexes such that it is antisymmetric with respect to the indexes.
Then we have
^ = Yj 7^, ^J^^, A • • • A CJ,,
«1< <«fc
"^ j^ X^ /n-^fe^jin •••^jfeu'^^ji A---Aa;j,,
therefore
Consequently *a; can be expressed as
*a; =
j^j/ _ j^xj X^ /n u<^(n^ • • •, ife, .71, • • •, 3n-h)(^3i A • • • A a;^^,,
•6(21, • • • , 2fe,^l, • • • ,3n-k)Aj,ii, • • • Aj^_^ii^_^uJ(i, A • • • A CJ^_, ,
where
a = (ai,---,ajk), /3 = (/3i, • • • ,/3n-Jk)-
Comparing with the expression of *a; with respect to (E*!, • • •, £■„), i-e.

28 PRELIMINARIES IN RIEMANNIAN GEOMETRY
we need to prove
I, J
= 6(ai, • • • ,afe,/3i, • • • .Pn-k)'
The proof is as follows
-Afiai * * * -Afiafc -^n^i * * * -^n^n-h
\ An ••• Ain
= 6(ai,---,ajk,/3i,---,/3n-jk) • j *:
= 6(ai, • •' ,ak,(3u '' • ,/3n-Jk)-
So the definition of * is well defined.
Lemma 1.4.5 For an oriented Riemannian manifold M of dimn, the
Hodge star homomorphism * satisfies
(l)fora,/3GA^(M), / G A0(M) = :F(M),
*(a + /3) = *a + */3, *(/a) = / • (*a);
(2) ** = (-1)^^+^ : A^(M) -> A^(M);
(3)fora,/3GA^(M),
a A */3 = («,/3) * 1;
(4) for any vector field X G r(TM),
Vx* = *Vx,
where V is Levi-Civita connection.
The proof of Lemma 1.4.5 is easy, so we omit it.
Definition 1.4.6 If M is an oriented Riemannian manifold of dimn,
define

1 4 Classical Geometrtc Operators 29
Remark 1.4.7 For non-orientable Riemannian manifolds there is no
star homomorphism, but a careful thought shows we can still define *d*. The
check is left as an exercise.
Lemma 1.4.8 For an oriented Riemannian manifold of dim n, the map
6 satisfies
(1) S{a + p)=Sa + 6(3, \/a,(3 G A\M);
(2) 66 = 0;
(3) *6 = {-Ifd^ : A^{M) -^ A"-*^+i(M),
(4) for a e A^{M), /3 G A*^+i(M),
(da,/3) * 1 - (a, <5/3) * 1 = d{a A */3).
Proof (1), (2) and (3) are trivial, so we only prove (4). (3) implies
*<5/3 = (—l)*^+^d*/3, thus due to Lemma 1.4.5 (3) we have
{da, /3) * 1 — (a, <5/3) *l = daA*/3 — aA *<5/3
= d{a A */3) - (- If a Ad^p-aA^6l3
= d(aA*/3).
Thus (4) is proved.
Proposition 1.4.9 If M is a Riemannian manifold, then operator 6 is
the conjugate operator of d, i.e. we have
{{dui,U2)) = {{(^i,6u2)) \lui,U2 G A*(M).
Proof If M is an oriented Riemannian manifold, then by Lemma 1.4.8
(4) we have
{{dui,U2)) - {{(j^i,6u2)) = / {dui,U2)dv- / {ui,6u2)dv
Jm Jm
= l{dui,U2)^l- I{(^i,6u2)^l
Jm Jm
= /d(aA*/3) = 0,
Jm

30 PRELIMINARIES IN RIEMANNIAN GEOMETRY
where M means M with the orientation. So 6 is the conjugate operator of d in
this case. If M is non-orientable, we need to modify the proof. The first way is
to use a standard 2-fold covering spa^e M of M. Because this covering space
is always an oriented manifold, the above argument yields
{{dDi,U2))^ - {{uu6u2))^ = 0, VSi,S2 G A*(M).
Here for a;i,a;2 G A* (M), let cSi,U2 be the pullba^ks of a;i,a;2, then
{{dx^l,U2))M= -((c2Si,S2))^,
{{ui,Su2))m= -((Si,<5a;2))^,
thus the proof is complete. The second way is to express {dui,U2) — {(^i^Su2)
as a divergence of a vector field, i.e. there exists a vector field X G T{TM)
such that
{du)i,U2) — {(jt^i^Su2) = divX,
and then by a formulas in Exercise 2 on p. 199 in [36]
divX ' dv = 0,
Jm
the proof follows immediately. For the existence of X one may refer to the
following exercise.
Exercise 1.4.10 Let M be a Riemannian manifold, X G T{TM). We
can define a map
div : r(TM) -^ :F : X H-> div(X),
such that for any orthonormal moving frame field {jE7i, • • •, jE7n},
n
div{X) = Y,{VEX,E.).
Prove
(i) the map div is well defined,
(ii) For uuU2 G A*(M), let X = Y^{ui,i{Ej)u2)Ej, then X G V{TM), and
3
{dui,U2) — {(j^i^6(jj2) = divX,

1 4 Classtcal Geometrtc Operators 31
where the !F{U)-\inea.i map
i{Ej) : A\U) -^ A^-\U) : u ^ x{Ej)u
is defined by
{i{E,)a;)iXi,---,X^_i) = u;{E„Xi,---,X^_i),
in which Xi, •■ •,Xk-i G r(TU). In other words, i{Ej) is uniquely determined
by
(i) It is a !F{Uy\mea.v map.
k
(ii) t{Ej){ui A • • • A a;fc) = ]^(-l)*-^<5j,a;i A • • • A S, A • • • A a;^,
«=i
where S, means a;, discarded.
Lemma 1.4.11 Suppose the assumption is the same as in Lemma 1.4.3,
then
^^ -J2i{Ez)VE.
«=i
Proof Both S and — 2_]'^{^^)VE^ do not depend on the choice of the
1=1
orthonormal moving frame, so we check the equality in a special
orthonormal moving frame. For any p G M, we choose an orthonormal moving frame
{El,'" ,En} such that
^tj\p = 0^ V2,j = l,---,n.
(Such kind of frames can be found, the compatible frame in §5 is a good
candidate.) For any / G T{M) and a multiple index (ii, • • •, 2a;), let a —
/a;,j A • • • Ao;,^, then
6a\p = (-1)-"+"+' Y, *d[fe{zi,---,tk,Ji,---,3n-k)'^j,A---Awj^_J\p
Jl< <Jn-k
= (-1)"'+"+' E <^U■■■,^k,Jl,■■■,Jn-k)
Jl< <Jn-k
•*[(#Mi A---Aw_,_J|p
Jl< <Jn-k «

32 PRELIMINARIES IN RIEMANNIAN GEOMETRY
If we let
then we continue the above computations as follows
/ 1 \nA;+n+l
•*(a;, Aa;^^ A • • • Aa;n-A:)|p
'€{iJL)ui^ A---Aa;,^_Jp
/_1 \nA;+n+l
therefore
(::;),(,, _i).i:(^./K"-^'(-i)'"-
•6 (IJ) 6 (zL J) o;,^ A • • • A a;,^_ Jp
« /l<..</fc_l \ ^^ /
= -E(^''^)(-i)'"''^'i^---^^''^'--^'^'^ip
n
= - Y^{Eif)i{Ei)u^, A • • • A a;, Jp
Izzl
n
= -E^(^') V^. (Ml A • • • Aa;,J|p,
/=i
Hp = {-Y^K^i)s/Ei)a\p^
1=1
the proof is complete. In the above computations the notation e I j means
1 or —1 or 0 depending on whether I is an even permutation of L or odd
permutation or otherwise respectively. S,^ means that the factor o;,^ is discarded.

1.4 Classtcal Geometrtc Operators 33
Let
A;=even
A;=odd
then it is easy to see that the operator d+ 6 can be restricted to two operators
Di = d + S: A^^^(M) -^ A^^^(M).
Definition 1.4.12 Let M be a Riemannian manifold, the operator
Do = d + S: A^^^^(M) -^ A^^^(M)
is called de Rham-Hodge operator.
Due to Proposition L4.9 d + S : A*(M) -^ A*(M) is a self-conjugate
operator, Dq = Z>i, Z>i = Z>o- Now we try to get a formula for {d-\- <5)^, hence for
DqDq and DlDi. In order to do so we are going to find local generators for
the algebra
Hom^(M)(A*(M),A*(M)).
Let {jE7i, • • •, jE7n} be an orthonormal moving frame defined on an open set U.
Define
E+=u;,A+i{Ej):A*{U)^A*{U),
Lemma 1.4.13 The maps E^ satisfy
C E+E++E+E+ = 26,j
e+e; + e;e+ = o
[ e:e; +e;e: = -26,,.
The proof is a simple check, so is omitted.
Lemma 1.4.14 For $ G Hom:F(M)(A*(C^), A*(C^)) and X G T{TM)
define Vx^ G Hom^(M)(A*(C^), A*(E^)) by
(Vx^)^ = Vx^^-^ Vx^*^, Va;GA*(C/').

34 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Then
V£?z^f ^X^^u^i
Proof As we pointed out before, the equality
k
implies
k
Now we prove that the above two equalities imply
n
k=zl
n
k=il
In fact for any u G A.*{U),
{Ve^{(^j/\))(^ = \7eX^j^)(^ - K A) V£7. ^
A; A;
thus the above second equality is true. For the first equality we only need to
check in the case of a; = a;,i A • • • A o;,^. Because
(VeA^j))^ = VeA^j)^ - K^j) Ve, ^
m
= V£7.{X^(-l)*"^<^ju'*^nA---AS,,A---Aa;,^} -%{Ej)\/e.(^
S-l
= Y^ {-iy-^Sj,^u^^ A • • • A S^E.i^ti A • • • A C,, A • • • A o;,^
l<l<s<m
+ E {-ly^Sj.^Ut^ A • • • A S,, A • • • A \7e,(^ii A • • • A o;,^
m>l>s>l
-i{Ej) Ve. i^

1 4 Classtcal Geometrtc Operators 35
= Y^i{Ej){u^^ A"'A\7e^(^zi A-'-Ao;,^)
m
- Y.i-^y~'^k ('^n ^ • • • ^ V^«, A ... A o;,^)
-i{Ej)\/E. (a;,, A ... A a;,, A...Aa;,^)
m
= E(-l)'^n,Kx A ... A V^«, A ... A o;,^)
m
= E(-l)'r».('^nA---A(5;;A---Aa;.„)
n
« = 1
n n
= E ^(^.n-K A • • • A a;.„) = 53 r.V(^,)'«'.
the first equality is also true. Note that the above two equalities imply the
conclusion of the lemma immediately, so the proof is complete.
Lemma 1.4.15
R{E,,E,) = -Y,^3ccfii^c.^)^{E^) : A^U) -. A*(C^),
or equivalently
R{E^,Ej) = --2J^«ja^(-£'a^^ - ^a^^)'
Proof First note that both sides of the equality in the lemma are T{U)-
linear and for ili G A*^(M), 1^2 G A*(M),
R{E,,Ej){ni A 1^2) = (Ve, S/e, - VEj Ve. - V[e.,e,]){^i A 1^2)
= ... = {R{E,,Ej)ni) A 1^2 + ^1 A {R{E^, Ej)n2),
and
X!^Ua/3KA)2(^^)(l^l Al^2) = X]^'J«^K^)(W^/^)^l) A^2)
J2 ^30c|i{^oc^)^{E^)nl A 1^2 + ^1 A ^ R^Ja|i{(^a^)^{E^)n2 .

36 PRELIMINARIES IN RIEMANNIAN GEOMETRY
So in order to prove the lemma it is sufficient to check that
R{E,,Ej)Uk = -Y^Rtja^{(^a/\)i{E^)Uk.
By Formula 1.2.1 in §2 we have
R{E,,Ej)uk = {Ve. VEj - VEj VE.VlE,,Ej])^k
I
and
— 2^ R%3a(i{<^a^Y{Ep)uT^ — — 2^ Rijak^^a = 2^ Rtjka^a-
a,^ a a
Therefore the first equality in the lemma is true. The second equality comes
from the following computations
Y,R^Jkl{<^k^)^{El) ^-Y.R,,^i{E^^E-){E:I--K)
k,i
= \j2^MEtE^-E^Ei-)
k,l
AT.^^3ki{-EtEi^E-E^),
4
k,l k,l
k,l
and
Y,R.M-EtET + E-Ei) = Y,R^M-EtEr) -J2R,jiuE^Et
k,l k,l k,l
= -Y.R^JkI{EtE^ + EfEt) = 0.
k,i
So the lemma is proved.
Theorem 1.4.16 (Weitzenbock formula) Given a Riemannian
manifold M, for an orthonormal moving frame {jE7i,---,jE7„}, (d + <5)^ can be
expressed as
{d + 6f = -Ao - i 53 R.,,iE+E+E^Ef + ^R,
t,3,k,l

1 4 Classtcal Geometrtc Operators 37
where R is the scalar curvature defined by
K = — / ^ -thijtj^
which is positive for the spheres (see chapter 1 §6).
Proof First by Lemmas 1.4.3 and 1.4.11 we have
Then we have
(d + <5)2 = ^^-Ve,^7V£;,
hj
= E ErEj^T^, \/E, + E ^r^7 vs. v^,
= Y,e:e;d{e.,e,)
2
= i '£{ErE;D{E„Ej) - e:e;d{Ej,e,)
-2S,jD{Ej.E,)}
= \Y.e;e;r{e,,e,)-i^o,
2
By Lemma 1.4.15 we further have
IJ2e;-e-r{e,,e,) = -1 Y1 R.3a^E;E;{EtE''^-E-E-\
2 2^ « J ~v - ^/ 8
and by Bianchi (I) we have
= E R^a^^iErE-E-E;+ErE^E-E-+ErE-E-E-)

38 PRELIMINARIES IN RIEMANNIAN GEOMETRY
So
2J Rtafij^t E^E^E^ - -2y^R,j,j,
and
{d + 6f = -Ao - i X) R^MKEjEtEt - e:e;e^e^)
= -Ao - i X) R>,^iKE;EtEt -\Y, RiJh
= -^0 - i X) R^JklEtE;E-E^ + \r.
8
The proof is complete.
In other words Weizenbock formula can give a full understanding for the
operators DqDq and Z^JDi, which will be considered later. Now we introduce
another first order differential operator just as we have done for Dq. Let M be
an oriented manifold of dim 2/, then define
r = Vri'^'+')+'* : A^(M) 0 C -. A^'-\M) 0 C.
Lemma 1.4.17 About the .7^(M)-linear map r,
(i) r' = 1;
(ii) Let
A+(M) = {u e A*(M) 0 C\tu = a;},
A_(M) r= {a; G A*(M) 0 C|ra; = -a;},
then there is a direct sum decomposition
A*(M) 0 C = A+(M) + A_(M).
(iii) d + <5 can be restricted to
D+^d-\-S : A+(M) -^ A_(M),
£>_ =d + <5:A_(M) -^A+(M).

1.4 Classical Geometrtc Operators 39
(iv) For any vector field X G T{TM)
Moreover we have
{uJ^)r ^ Ti{Ej), t{ujA) = i{Ej)t.
Proof If a; G A*^(M), then by using Lemma 1.4.5 v\
2 / r^H^-^)-^^ \
r^u = t(v^1 *a;)
^—r<2/-A;)(2/-A;-l)+/+A;(A;-l)+/
— V ~1 * ^^
^,/::f''\-if^^+^u^u,
thus (i) is true, (ii) is a trivial corollary of (i). By the definition of r and
Definition 1.4.6 we can get (d + 6)r = —r(d + 6), hence (iii) is true. It is trivial
that the equality Vx'T" = tVx is equivalent to Vx* = *Va:5 which is (4) in
Lemma 1.4.5 and unfortunately had not been proven there. So we give the
proof now. For any x £ M choose an orthonormal moving frame {jE7i, • • •, E21}
such that
^«jU = 0, V2,j = l,---,2/.
Without loss of generality, let u — fUi^ A • • • A o;,^ then
VEj *(Mi a---Aa;,J|a:
^VEj{ Yj ^(^1 * * * ^*^-^i * * *32i-k)fi^j^ A • • • A a;j2i_ JU
Jlj <j2l-k
XI ^(^1 • •' ^*^-^i • • • :i2i-k){Ejf)uj, A • • • A Uj^,_^ U
Jl, <j2l-k
- *VEj (Ml A---Aa;,J|a:.
Thus the first equality in (iv) is proved. For u G A^{M) by using Lemma 1.4.5
we have

40 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Similarly we also have
i{Ej)ru = r{uj Au).
Therefore the lemma is true.
Definition 1.4.18 Suppose M is an oriented manifold of dim2/, then
the operator
D+=d + 6:A+(M)-^A_(M)
is called a Signature operator.
Exercise 1.4.19 If we define
A%{M) = (A^^^^(M) 0 C) Pi A+(M),
A;(M) = (A^^^(M) 0 C) flA+(M),
A1(M) = (A^^^^(M) 0 C) Pi A_(M),
A^(M) = (A^^^(M) 0 C) fl A_(M),
then there is a direct sum decomposition
A*(M) 0 C = A%{M) + A;(M) + A1(M) + A^_{M),
and d-\- S can be restricted to
d + S:Al{M)-^At^{M).
Towards the end of this section we introduce a notation to express Vx-
Choose an orthonormal moving frame a = {Ei^ • • •, jEJyi}, and let {ui^- -- ,Un}
be its dual coframe. For any a G A^{M) we have an expression
«i< <«fc
For X G r{TM) define
«= Yj -^'1' ''^'^^'i A-'-Ao;,
<«fc
X"" :A^{U)-^A\U)
by
X^a= ^ (X/,„ ,,Ja;,,A...Aa;,,.
«i< •<«fe
Here X^ depends not only on X but on a as well.

1 5 Normal Coordinates 41
Lemma 1.4.20 Given an orthonormal moving frame a = {Ei, • • •, jEJ^}
and a vector field X G T{TM) we have
Vx - X- + Y^Ukj{X){ukA)x{Ej)
= X<^^\Y,^,,{X){E^Et-E;E-)
:A*(M)-^A*(M).
Proof Let
It is easy to see that $ : A*(M) -^ A*(M) is ^(M)-linear, and
$(a A/3) = $(a) A/3 + a A $(/3) Va,/3 G A*(M).
So in order to prove the lemma it is sufficient to check
$(a;,) = 0, or Vx ^, - (^^ + ^a;fc,(X)a;fc2(^,))a;,.
As we did in the proof of Lemma 1.4.14, we now have
s
and
j,k k
SO $ = 0. Thus the proof is complete.
1.5 Normal Coordinates
In this section we introduce a special coordinate system named the
normal coordinate system, which will help local computations m differential
geometry.
Let M be a Riemannian manifold of dimn, and a point p in M, and Y G
TpM. A path
7 : [0,6] -^ M : t •-► j{t)

42
PRELIMINARIES IN RIEMANNIAN GEOMETRY
is called a geodesic with the starting point p and the initial tangent vector Y,
if it satisfies
d „
7(0) = p,
i \di\o) "■^'
where ^ = {l)'^{§i)'> ^^^ ^ is a positive number. We denote 7(t) by (exp^ Y")*.
Choosing a coordinate system {a^i, • • •, ajn} around p, the path ^{t) can be
expressed as
7(t) = (aJl(t),...,aJnW).
Without loss of generality we may assume 7(0) = (0, • • •, 0). Thus
'--(^)=E';<
dt
T(«)
and
v^^ = v^{E^^WazL>
5a;, l7(t)
7(*)
3aj,
7(*)
«J,*;
dxk
7(*)
therefore the equation satisfied by 7 can be expressed as follows
r x:(<)+x;a;:wx;wrf_,(x(<))=o
where Ai = 1, • • •, n, {ai, • • •, 0^} are determined by
^ = E«'
dXi
and rf is determined by
r*=Eff'*(v^#-'^)-
J /^ 'ax, Qx. OXs

1 5 Normal Coordinates
43
It is well known that the above equation has a unique solution (aji(t), • • •, Xn{t))
= (exp y)* for t belonging to a small interval [0, a).
Lemma 1.5.1 For p G M and Y G TpM,
(i) If s, t, st G [0, a), then
(exp„rr' = (exp„.y)'.
(ii) Consider the geodesic j{t) = (exp^Y")*, if we denote the arclength of 7
from 7(0) to 7(T) by 7(0, T), then
7(0,T) = T.|y|.
Proof Let Y, s be fixed, and in the coordinate system {aji, • • •, aj^}, let
(expp Y)'* = ixi{t), ■■■, x„{t)) = x{t), (expp sY)* = {zi{t), • • •, z„{t)) = z^),
then by a computation similar to that for the geodesic 7(t), x(t) and z(t) satisfy
the following equations
«J
d
I 5a
aJA;(a)|a=:0 = ttfe,
^ik(O) = 0
where the variable a is is. The first system of the equations satisfied by x(t)
x{f) can be changed to
xk{o) = 0
»>j
15 , ,,

44 PRELIMINARIES IN RIEMANNIAN GEOMETRY
therefore x(t) ■= z{t), oi
(exp^r)'' = (exppsy)*.
It proves (i). Still in the coordinate system {aji, • • •, a^n},
hence
where
and
It follows that
/ / / ji . O u . Ill Q O ^ \ 1
+a;.x,x„r„.(—, —) + x,XjX„T^j{—, —)|
= 0.
Therefore W{t) = constant, and hence
7(0, t) = / W^(r)c?r = {lim W{r)] ■ t
Jo '■~*°
= |r|.i.
So the proof is complete.

1 5 Normal Coordinates 45
We define
Expp(r) = (exp^r)^
Due to Lemma 1.5.1 and the compactness of the unit sphere there exists an
6 > 0 such that if \Y\ < 6, we can define (exp^ |y|-y)l^l. By Exp^(y) =
{expp Yy = {expp lyr^)'^', it results a map Exp^ : Vp{€) -^ M, which is called
an exponential map, where Vp{€) = {Y £ TpM\ |y| < e}. The map Exp^ is
a diffeomorphism due to the non-singularity of (Exp^)* at 0 G TpM. (It may
be seen on p. 204 in [40].)
Exercise 1.5.2 If M is a Riemannian manifold, p G M, then
(i) For any geodesic j{t) starting at p, there exists Y G TpM such that
Jit) = {expiry.
(ii) For any two non-trivial geodesies
71 (t) = (exp^Fi)*, J2{s) = {exppY2)\
if Yi is parallel to Y2^ then as a point set, 71 coincides with 72, and
j,{i)=j^{s)^=>i^\Yi\ = S'\Y2\.
Definition 1.5.3 Choose an orthonormal frame {Ei{p), • • •, En{p)} in
TpM^ by the equality
q = Exp^{yiEi{p) + • • • + ynEn{p))
we define {yir " iVn) as coordinates of g, in other words, {yi,"*,yn} is a
coordinate system, which is called a normal coordinate system centering
at p with respect to the orthonormal frame {Ei(p),' •• ,En{p)}, where g in a
small neighborhood of p.
Theorem 1.5.4 In a Riemannian manifold, a local coordinate system
{Vi 5 • • • 5 ^n} is a normal coordinate system centering at p if and only if the
following conditions are true
(i) (yi(p),---,yn(p))-(0,---,0),
(ii) for 2 = 1, • • • ,n
Vz ^^^Y^gzjiy) ' %, where g,j{y) = ( —, —).

46
PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proof Consider a path p{t) = Exppt{aiEi{p) -\- --- -\- anEn{p)), where
(tti, • • •, ttyi) is a fixed nonzero point in R". From
p{t) = {exppt{aiEi{p) + '" + ar^Er^{p))y = (expp(ai^i(p) + • • • + a„^„(p)))*
it follows that p{i) is a geodesic. Let {yir " '>yn} be the normal coordinate
system centering at p with respect to {Ei{p),'--^ En{p)}^ then the normal
coordinates of p{t) are (tai, • • • ,tan). In other words, the path p(t) can be
expressed as
yi{i) = iai
[ Vnit) = tCLfi'
The arclength from p(0) = p to p{t) is
. — —
and by Lemma 1.5.1 we also have p(0,t) = t'\Y\=t' ^/a^^+-- + a^. Taking
the derivative operation of ^ on the above two expressions of p(0,t), we have
«j
where g^j{y{t)) — (^, 3^). The equation for the geodesic p{t) is
«J
where
ij
2^ I 3y, ^ 3y, dyi J •
Due to yiii) — ta^ the geodesic equation turns out to be
or
Ef Sfifh 1 ^g^J \

1 5 Normal Coordinates
Now let US examine a function
The above equalities read
47
SO we have
z,k
= ^X]a,ajk—^(tai,---,tan)
«,fe
= |E«'^
1. ggf,fc(tai,'",tan)
- 2y^a»gf»j
= 2t^
3a,
■{-/(<)},
hence
therefore
So we have
o = /,,,l,,, = lM.
</(<) = constant = limtf{t) = (lim<)/(0) = 0.
m = 0,
which implies that the conditions in the theorem are necessary. The converse
IS easy, so its proof is omitted here. Thus we finish the proof of the theorem.
For a fixed point p G M, we define a distance function p : M -^ 'R.hy
p{q) = inf{L(7)|7 is a path joining p to g},
where ^(7) is the arclength of the path 7.

48 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proposition 1.5.5 There exists a neighborhood Up of p, such that
(i) p : Up — {p} —^ R is a C^-function.
(ii) If {t/i, • • •, t/n} is a normal coordinate system centering at p, q £ Up and
{yir " fUn) are the normal coordinates of g, then
Pi^) = V1
Proof It is sufficient to prove (ii). Let Up be the neighborhood of p, on
which the normal coordinate system can be defined. Consider the geodesic
p{t) = Expj,t{yiEi{p) + ... + ynEn{p)y
The path p{i)\o<t<i is a path joining p to g, and its arclength is p(0,1) —
\/yl^"'^yl'^o
p{q) < ^Jyl + "' + yl
If/3(t)|o<*<i is a path joining p to g, i.e.
r m=p
I /3(i) = g,
then its arclength is
Denote the vector field V. "TT^*'^— ^^ ^v ~ ^^^ ^^ ^' ^^^ ^^^
_ d/3(t) dm -^-
Then
= 0.

1 5 Normal Coordinates 49
Let (yi(^), • • • ,yn(^)) be the normal coordinates of/3(t), we have
= (E
|y(t)l
= {|yWl'}',
= ^W^I^(*^I^)'>^
hence
L{^) = 0(0,1) = J^ ^i\y{t)\ydt
= ['\y{t)\'dt
Jo
> I / y{t)'di\
Jo
= l2/(i)l = Vyl + --- + yl
p{q) > \Jy\-^--- + yl,
So
the proof is complete.
In Ur, — {pi, we had defined a vector field S = > ■;—r-;;—> which is de-
noted by — usually, now we define another vector field grad(/9), which is the
gradient vector field of p defined by
(grad(p),X)=Xp, VXGr(TM).
Proposition 1.5.6 For a fixed point p G TpM^p is defined as above.
Then
grad(p) = p.

50 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proof It is sufficient to prove
{p,X)=Xp, \/XeT{TM),
In a normal coordinate system {t/i, • • •, t/n} centering at p,
then
= I^E^./^.-
So the proposition is proved.
Remark 1.5.7 For a geodesic
^
R^
p{t) - expp —{aiEi{p) + • • • + anEn{p)),
we have mi) — ( -tt? * * * ? tt)? so its tangent vector field is
\a\ \a\
1 /i
\a\ ^-^ ' dy.
Recall the well-known Gauss lemraia, it claims that the tangent vector field
along any geodesic p(t), starting at p, is perpendicular to the geodesic spheres
{q G M\p{q) = constant},
that is to say,
p ± {q £ M\p{q) — constant}.

1 5 Normal Coordinates 51
On the other hand, it is well-known that the gradient vector field gra.d{p) is
perpendicular to the hypersurfa^es, on which p keeps the value. So Proposition
1.5.6 implies Gauss lemma immediately.
Next we introduce an orthonormal moving frame in the small neighborhood
Up, which is in harmony with a normal coordinate system. Suppose {t/i, • • •, i/n }
is a normal coordinate system centering at p with respect to an orthonormal
frame {jE7i(p), • • •, jE7n(p)}, it means
Define a frame field {Ei,- •• ,En} in the small neighborhood Up such that it is
parallel along geodesies passing through the point p, in other words, it satisfies
J Vp'(t)^.(p(<)) = 0, Vi,
1 EMO)) = E.{p), V»,
where p{i) is any geodesic described in the following form
lap
p{t) = Expp—{aiEi{p) + • • • + anEnip)),
By using the differential equation theory, the frame field {Ei^-- - ,En} exists
uniquely. Furthermore the computations
^^{E,{p{mEMm = {Vp-it)MpmEApm
+{E.ipit)),Vp'^t)Ejipm=0,
imply that
(E, (Pit)), E, (Pit))) = {E, (p), E, (p)> = 6,j, Vi,:,.
So the frame field {jE7i, • • • ,En} is orthonormal. Such an orthonormal moving
frame {Ei^-- - ^E^} is called a compatible frame or a frame compatible
with the normal coordinate system {t/i, • • •, t/n}-
Proposition 1.5.8 Given a normal coordinate system {t/i, • • •, t/n } and
a compatible frame {jE7i, • • •, jEJyi}, we have

52 PRELIMINARIES IN RIEMANNIAN GEOMETRY
where ^ = p.
Proof For a fixed nonzero point (ai, • • •, a^) G R'*, consider the geodesic
p(t) = Expp —(ai£^i(p) + • • • + an^n(p)).
All the vectors in the set {p (t), jE7i, • • •, £^n} are parallel along the geodesic
p{t), so among them any R-linear relation at the point p(0) can be kept along
the whole path p{t). The linear relation
implies
especially
Owing to
p{p{t))=p{t) Vt>0,
the above equality implies
pipil)) = p'(l) = 53 |^«.^.(p(l)) = (^ ^^y,E,)ipil)).
So the proof is complete.
For a normal coordinate system {yi,---,yn} and its compatible frame
{El,'" ,En}, define {rf^, H^j^Hj^k} by solving the following equations
d
- 7 .H^jE^,
\7^^j = y^^HkijEk.
dyj
It is easy to get
(5^'5^) -^Hk^HkJ, T^j =Y^H'^'Hkmj,
"* k m

1.5 Normal Coordinates
53
where (H^^) is the inverse matrix of (H^j). We have known an explicit formula
to compute F^- from the metric and the Lie algebra structure (see Theorem
1.1.10 (i)), but we don't have explicit ones for H^j^H^jk. What we can do is to
find equations to determine them.
Theorem 1.5.9 Let d = p^, then
a
dEkil — —Hkil — y ^yaRa(3klH^t,
a,/3
where
Proof
Rtjkl = ((V£7» S/Ej - \/Ej V£7» - S/[E^,Ej]) ^k, El)'
By using the formulas
S/^E^=ps/aE, = {},
^
^=E^'5^ = E^'^-
and the convention that the repeated indices implies addition has been
employed, we have
= {\/^d+ [d, —-] , E,)
Gyj dyj
= (S/e_ymEm , E,) + (bm^— , ^ , E,)
^^3 oym oyj
= ^tj + ykHijk — H^j ,
dHk^l =d{\7aEi , Ek)
= iV^V^ El , Eu)
= m ^)Ei , Ek) + (V[3^, ^^E, , E,)
— ydHa,R/)alk — Hk%l •

54
PRELIMINARIES IN RIEMANNIAN GEOMETRY
The proof is complete.
Corollary 1.5.10 Theorem 1.5.9 implies the Taylor expansion of the
functions H^j, H^kj
H,j = ^ij - -^^ VaVfiRatfij (0) + • • • ,
a,(3
Hktl = "^ Z^ VaRatkliO) H .
Proof For a function / G ^(M), the Taylor expansion of / with respect
to a normal coordinate system {yi, • • •, ^n} is
gmf
-)(0)*y«l -'Vim-
We denote the m-th order term in the expansion by f{m), i.e.
mi -=—' 5j/,i • • • dyt.
■)(0)-yn"-2/«m-
Then Theorem 1.5.9 yields
I" {dH,j){m) = {S,j - H,j ■^^ykH^jk){fh),
^ A;
a,/3
Note {df){m) = d{f{fh)) = m{f{m)), so we have
I (m+ l){H^j{m)) =S^jSom +2^yk - H^Jk{m- 1),
A;
{m + l){Hk,i{m)) =- ^ ^Va' Ra^ki{rni) • H^,{m2).
It implies that ^«j(0) = 6,^, irA;,;(0) = 0, and
(H.,(2)) = -EyfcH.,fc(l) = --Y^yaypRa,^^^),
k a,(3

1.5 Normal Coordtnates 55
thus the corollary is true.
Corollary 1.5.11 We have
^yjHtj =yt, Ylyj^j' '='^«'
3 3
Proof The equalities
9yj
J
Y^VjH.j =y,.
imply
And
Haj(3 = {y.Q^E^.Ea) = T—{Ei3,Ea) - {Ef^.V eEa) = -{Ef^.V eEa)
Gyj oyj ^yj ^yj
imply Haj(3 = -H(3ja'
Let f^{y) := ^yJHJ^ - y^, then
3
^My) = X^(^%)^j« + X^%(^-^jO - ^y^
3 3
= My) + X] % i^3^ - ^J« + X] yk^3^k)
3 k
- ^y3ykHjzk = 0.
3,k
So
Uy) := constant = lim Uy) - 0,
y-*o
hence y, = \^yjHji. Again let F = y]yjHaji3, then
3 3
dF = F + ^ yjdHajf) = -^ yjymRmsafiHsj
3
= — 2^ VsymRmsa^ = 0,
and hence F = 0, So the proof is complete.

56 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Proposition 1.5.12 Denote the matrix whose (i,y)-entry is H^j by
{H,j), and
G=(det(H.,)r=det((^,^»,
then
E^p = ^, Aop =-{n-l + d\og \/G),
where Aq is defined in Definition 1.2.5.
Proof First we have
^.p= (grad(^),^,) = {p,E,) = {-J2yJ^J^^^) = ^•
3
Then due to
^' 3 3 ^^ k
we have the following computations
AoP = Y^E,E,p - J2{s7e.E,)p
h3 « «,J,fc
On the other hand, let Htj be the algebraic complementary determinant of the
entry H^j, then
«,J A;

1.5 Normal Coordinates 57
Therefore
Aop= — + - —j^ + n] = -(n-l + dlogVG),
P P \VG J P
and the proof is complete.
Next we will wander over the Taylor expansion in geometry although it will
not be used later. First we prove
Lemma 1.5.13 Let M be a Riemannian manifold, p G M, and {t/i, • • •,
t/n } be a normal coordinate system centering at p, then for a fixed m, in Up — {p}
we have
dpm 2^ ^ pm >dy.^---dy,J
dp"* ^dp' ' dp'^'
•• V '
m times
Proof We use the induction on m to prove the lemma. The lemma is
plainly true for m = 1. Due to
i<l'-'
we have the following computations
dpm + 1 dp dpm
Qp Z^ y pm ^y^^'"^y^^
L. y pm+1 )dy,^...dy,^^/
And the equality \/_e_j- = (} implies
V ^ . S ^ .
m+1 times m times
Therefore by the induction we get the second equality, so the lemma is proved.

58 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Lemma 1.5.14 Given / G ^{M) and p e M, let (t/i, • • • ,yn) be the
normal coordinates of a point g, then the Taylor expansion of / can be expressed
as follows
fiy) = J2 /(^).
m=0
in which
^" V '
m times
where limp(g)_o means that q goes to p along a geodesic starting at p.
Plainly, the limit value lim D(-r-,' --, -r-)f depends on the choice of the
p(q)-.o dp dp
geodesic, but as a function defined on a neighborhood of p,
(lim £>(!-,..., I-)/).p-
p(q)-*0 dp dp^
m times
does not.
Proof By using Lemma 1.5.13 we have
•1? ?»m
•1? ?*m
Vtl'-'Vln
m times
thus Lemma 1.5.14 is true.
Taylor Expansion Theorem 1.5.15 If {t/i, • • •, t/n}, {^i,**,^n}
are a normal coordinate system centering at p and its compatible orthonormal
frame respectively and denote the normal coordinates of g by (t/i, • • •, t/n)? then

1.5 Normal Coordtnates 59
the m-th order term in Taylor series of / can be expressed as
m
«lj j«m
dyz,' ' ^y^^
A^) = ;^ E {^(^nr-,^,.)/}(o)-yn--y..-
ml
«i, ,«
Proof The computations
^^ V '
m times
*1 J j*m
= y (lim£)(^,---,-^)/)
«1, ,«
J j*m
3 3
2: (-(^•••■■a^»««»^-t
«1, ,«
imply the first equality of 1.5.15. Similarly we can get the second equality, and
the proof is complete.
Remark 1.5.16 By Theorem 1.5.15 we can easily get
= Eii^(^'C.)'-.^'(..))/K0),
where tt runs over the permutations of {zi, • • •, Zy„}.
Theorem 1.5.17 Let {yi, • • •, ^n} be a normal coordinate system
centering at p with a compatible orthonormal frame {jE7i, • • •, jEJyi}, Riji^i be the
components of the curvature with respect to {jE7i, • • •, jEJ^}, then
Proof We only prove the case of m = 1 and leave the general cases as
an exercise. First
Rtjkii^) = (lim -g-Rzjki)p,

60 PRELIMINARIES IN RIEMANNIAN GEOMETRY
then
= {S/^R{E,,Ej,Ekl El) + {R{E,, Ej, E^), V^^/)
= {^^R{E,,E,,Ek\Ei)
Gp
= {{s7eR){E.,Ej,Ek),E,)
Gp
= Y^^{iVE.R)iE„E„E,),E,)
s ^
dp
The proof is complete.
and
P
1.6 Computations on Sphere
In this section let us do some exercises on the sphere of dim 2. It can help
us to understand the notions we introduced before, and shows a complicated
Ao even in this very simple case. As you know that the set
S\r) = {{x,y,z) e Ii^\x^ + y^ + z^ = r^}
IS called a sphere with radius r.
(I) Riemann coordinates
Riemann in 1854 gave a metric
,2 \^, 9 9 ., , dul + \-dul
ds = > (-r , )dUtdu>t = T^ r
in his famous Habilitation address, and claimed that the curvature of this metric
is constant. Of course, here (t^i, • • • ,nn) are called Riemann coordinates.
We will show that the following stereographic coordinates on S'^{r) are just
the Riemann coordinates (see also p. 75 in [36]).

1 6 Computations on Sphere
61
On S^{r) the points On = (0, 0, r) and Og = (0, 0, —r) are called the north
pole and south pole respectively, and
7r= {{x,y,z) eK^\z = -r}
is the tangent plane at the south pole Og (see figure below). For any p G
S^(r) — {on} a line joining p and o„ intersects the plane tt at p' = (ni, ^2, — r),
then (^1,^2) are called stereographic coordinates of p.
It is easy to express (ni, ^2) in terms of the Descartes coordinates (aj, y, z)
of p, that is to say,
4r^n2
r{ul + ^2 — 4r2)
^1+^2 + 4r2
Then it follows that
dx =
^;2 ""^l ^^^ "'^2,
ay = 7^—dui H iz;:^ dn2.
•^2
a;2J = —;^—dui H ;^—du2,
where
So we can get
ds^ - dx^ + dy^ + dz^ -
5 = ^1 + ^2 + 4r^.
du\ + du2
(1+4^(^2 + ^1))
2^^2*
Owing to the above equality, (^1,^2) are Riemann coordinates.

62 PRELIMINARIES IN RIEMANNIAN GEOMETRY
(II) Curvature
We use the orthonormal moving frame method to compute the curvature.
It is well-known that for any sufficient small open set U we can always find
an orthonormal moving frame {Ei,-- .En}- A moving frame is orthonormal
if and only if
where {a;i, • • • ,0;^} is dual to {jE7i, • • •, jE7n}. One of the tricks in the
orthonormal moving frame method is that we may not find the frame {Ei,-- - ,En}
clearly, instead we try to get {a;i, • • • ,0;^} by any method such that the
equality ds'^ = ul + (- tt^n is true. e.g. we choose
dui du2
^1 = . ■ 1 / 9 . ox. i^2 =
plainly we have the equality ds'^ = o^f+a;!, which implies that such a {a;i,a;2} is
dual to an orthonormal frame field {jE7i, E2}. From the following computations
(dui,du)2) =
/ \
I U2dui A du2 uidui A du2 \
[2r\l + ^{ul + ul)f ' 2r2(l + ^(«? + «1))2 j
/ "2 . ■"Ian
(f. U2 Ui
0 ^T'^1 ~ oT'^2
and by lemmas 1.3.2 and 1.3.3 we have
U2 Ui
and then
1^12 = dui2 + 2^ '^^i* ^ '*^«2 = ^2^*^12
du2 dui U2 U2 Ui Ui
= -^<^^ + ^<^2 - ^^'^r A 0,2 - ^^0,1 A W2
= (1 + 4;:2-K +^2));:^'^i Aa;2 - -^f;j-^i A a;2
1

1.6 Computations on Sphere 63
Thus due to Lemma 1.3.1
^1212 = —^12(^1,^2) = —2
and the scalar curvature
R = — 2^ Rtjij = —,
which is a positive number plainly.
(Ill) Normal coordinates
Let {t/i, 2/2} be a normal coordinate system centering at the south pole Og
with -^los ^^^ af~l^» being parallel to the aj-axis and the y-axis respectively.
For any p = (x^y^z) G S'^{r)^ let the normal coordinates of p be (2/1,2/2)-
We are going to find the relation between (aj, y, z) and (t/i, t/2) first, and then
the expression of the Riemann metric in terms of the normal coordinates. Let
o^p be the geodesic arc joining Og to p, and o be the center of the sphere.
Denote IpoOg = 0, then
i P= VVi + 2/2 == *h^ length of Ogp = rO,
\Jx^ + 2/2 = r sin ^,
[ 2; = —rcos^.
and (2/1,2/2) = T(a^5 2/)5 which imply
»sin(9 = x/yM^ ^ ;^y^^2 _|_ ^2 ^ ;^^ ^ ;^^^^

64 PRELIMINARIES IN RIEMANNIAN GEOMETRY
therefore
X =
sin^
sin^
0
yi
2/2
[ 2; = —rcos^,
\/y^~+y^
where 6 = — . From the above equalities it follows that
and
, sin ^ , 0 cos ^ — sin ^ . _
dx = —0-dyi + yi ^2 ^^'
, sin ^ , 0 cos ^ — sin ^ . _
dy = —0-^y^ + 2/2 ^2 ^^'
dz = r sin OdO,
ds'' = {dxf + {dyf + {dzf
sin^^^^, .9 / , x9x / 9 n. fOcosO — siiiO\
= -ff^iidyi? + {dy2f) + {yl + yl) (^ j^ j
+2-^ ^^ [yidyi + y2dy2)de
+r^sm^d{dey
(d^r
Therefore
ds' = C^r{Y:idy,f)+(1 - C^?){dp)\
«=1
where ^ = —. In other words,
r
-/A A\
(Check that the above g^j satisfy 2_^y*9^j ^ yj-)

1.6 Computattons on Sphere
(IV) Compatible orthonormal frame
65
Let
sin^ sin Oy^
then it is easy to see that
^l + 0,1 = CJf.)^iJ2idy.)') + (1 - C^rndpf = ds\
« = 1
SO the frame field {Ei^E2} dual to {ct;i,Ct;2} must be an orthonormal moving
frame. Now let us prove that it must be compatible with the normal coordinate
system {2/1,2/2}, i-e. {Ei^E2} satisfies
\fi= 1,2,
^.(0) = ^|o, Vi=l,2.
oy.
By the definition we have \^ —w^ = dp and
{cosB sin^\ ,„ , /, sin0\ dy,
' " \~9 W) ^ ^' ^ ( ~0~) ^'^P
1 19
= (1 — cosO)-dyi A dp = (1 — cosO)-——-a;, A dp
p psmO
yj.
1 9
J
El — cos 9 , .
r-r{yj^z - Vz^j) A Uj
pr sm 9 -^ -^ •'
and by the Cartan lemma 1.3.3 we also have
1 — cos 9 . .
Then
v^^. = Em^)^.
dp'
COS
r2^sin(9
El - cos ^, , d . / 5 xx„
dp>
'dp'

66 PRELIMINARIES IN RIEMANNIAN GEOMETRY
and
J
v^.sin^- ,^ siiiO.y. y^^ ^
J
_d_
dy.
e ' p p
SO {jE7i, £^2} is compatible with the normal coordinate system {2/1,2/2}-
According to the definition, we have
/ ^ \ .sin^ , ,^ sinO.yt . ., d .
> P > P
rsm- rsin-
= ^<5,, + (l ^)^^,
P P P P
1 — COS 6
dyi
r^e.ine^y''''^W?~'''""^'k?^
p
1 — COS -
\yk^i3 - yjStk)'
2/«
Let Xi = — and
P
^y^'
W =
XiXi X1X2
X2X1 X2X2
W has eigenvectors (aji, aj2) and (aj2, -a^i) with eigenvalues 1 and 0 respectively.
det(A/-Vr) = A(A-l),
where / is the unit matrix. Let
sin ^
rsin
// =

1 6 Computations on Sphere 67
we have
VG = det( ^" ^'' ]
= det(/il + (1 - iJ,)W) = detii/j, - 1){-^I - W)}
. P
rsin -
_ r_
P
And due to Proposition 1.5.12
Aop = -{n-l + d\og VG)
= -(n — 1 + dlogsin dlog -)
p r r
= -(n-2+ -cot-).
p r r
Finally it may be check directly that
r sin ^ "^ rsiii^ p p
(V) Computation of Aq
We are going to compute
Ao : A*(M)-^A*(M),
(see 1.4.2 for definition). Using Lemma 1.4.20, we have
Let a = {El, ■■■, E^), X" = X, a,j = {E+E+ - E;Ej), and
r sm ^ p p

68 PRELIMINARIES IN RIEMANNIAN GEOMETRY
then
take the summation convention that the repeated indexes means the
summation. Then by using
Ao = S7E^ V£?, -VvB,^t
-fHsitS/E,
= /l+/2+/3 + /4 + /5 + /6,
where
'5j// '9j/p' dvjdvp ^dyj ' dyp
l2 = H^\^Y\fHa,pcrap
= ^fH''H^,i,<r,i,(^Y+\B''ii-^rfSa.^)crap,
h = -fH''Hc„l}(Tal}i-g—Y = -^f^Hajl}Cra/}i-Q—Y^
11 1
h = -rfHk,i(Tki-7fHa,/}crQii — —f Hk,iHa,i}(Jkicrai3,
h = -HP'H,p.H'»{^Y = -fHs.,H^'{^Y,
Oyj cfyj

1.6 Computations on Sphere
69
First we check that
( dp
oyt
^ _ he _ \
1 f) f)
-Hui = {xkS,i - xi8,k)<Tki = 2(^)+^+ - 2{—)-E- - ix,
k J/. r
then we have
J+ = HI'HP'
32
dyjdyp
= ifs,. + (1 - /)x,x.)(/V + (1 - f)^P^')
52
dyjdyp
dyjdy,
1-/
3 (ff^i-l)d=^^((n-l)/)d=ffi—^d
p2
1 P)
It = -^fH^'Hcpcrai^i-Q—Y = I3
= ••• = 0

70
PRELIMINARIES IN RIEMANNIAN GEOMETRY
h = -fH,„H''{^y = -f{f6j, + (1 - f)xjX,)g{xA, - xAsX^)"
= -fg{n - l){fS,, + (1 - f)x,xs)x,{-^r = -(^ " 1)/^^
= -^f y {^s{n- l){xkSsi - xiSsk))crki = 0.
Therefore
Ao = /i+ + /f + It +l3 + l4 + h
(PS^, + (l-P)x,x,)
dyjdy^
+
p2 p p
+P9
1
(l.)+E+ -(—)-E-
dy,
where (t—)'^ = (V^?/*^*)^ — Tj?/*^*^- ^^ ^^^ ^^^ of this section it would be
notable that Aq is very complicated.
1.7 Connections on Vector Bundles and Principal Bundles
Definition 1.7.1 A vector bundle (jE7,M, tt : E -^ M) consists of the
following iterms
(1) a differential manifold E of dim {n-\- N), called a total space;
(2) a differential manifold M of dim n, called a base space;
(3) a map ir : E -^ M, which is differentiable and is called a projection;
(4) for any x G M, there is a given linear space structure in 7r~-^(aj),

i.7 Connections on Vector Bundles and Principal Bundles 71
such that
(i) for any x G M, there exists a neighborhood U of x and a homeomorphism
^:U xV --^ -JT-^iU), such that
wo^:UxV-^U
is the projection to the first factor space U, where F is a fixed vector
space of dim N;
(ii) for any u £U, the map
^ur) : V -^ 7r-\U)
is an isomorphism between the vector spaces.
The space 'jr~^{x) is called a fiber over aj, and the above vector bundle
is called one of rank N. Sometimes we denote the vector bundle simply by
TT : E -^ M ov E.
Exercise 1.7.2 Let M be a manifold of dim n, TM be the set consisting
of all vectors on M. Define the map tt : TM -^ M : v i-^ x^ where v G T^M.
Then tt : TM -^ M is a vector bundle of rank n, which is called a tangent
bundle of M as usual.
Definition 1.7.3 Given a vector bundle w : E -^ M^ a. map W : M -^
E is called a section, if it satisfies
TToW = id: M -^ M,
where id is the identity map. We denote the set of C"^-sections by T{E).
It is easy to see that in T{E) we can introduce an addition and a
multiplication by functions.
Definition 1.7.4 Given a (real or complex) vector bundle tt : E -^ M,
a connection is a map
D : r(TM) X r{E) -^ r{E) : (X, W) ^ DxW,
satisfying
(i) DxW is .7^(M)-linear with respect to the variable X,
(ii) for W, Wi, W2 G T{E) and X G r(TM), / G ^(M),

72 PRELIMINARIES IN RIEMANNIAN GEOMETRY
Dx{Wi + W2) = DxWi + DxW2,
DxifW) = {Xf)W-^fDxW.
Sometimes a connection on a vector bundle is also called a covariant
derivative. It is by now routine to define the curvature once a connection
is given.
Definition 1.7.5 Suppose we are given a vector bundle E with a
connection D, then define a curvature
R{X,Y):T{E)^T{E)
by
R{X, Y) = DxDy- DyDx - D[x,Y]'
It is also routine to define the derivatives of higher orders as we did in
Definition 1.4.1.
Definition 1.7.6 The second order derivative operator
£)(-, •) : r{TM) X T{TM) x r{E) -^ r{E)
:{X,Y,W)^D{X,Y)W
is defined by
D{X, Y)W = D{X)D{Y)W - D{S7xY)W,
and the higher order derivatives are defined by the same formulas as in
Definition 1.2.7 or Definition 1.4.1. i.e.
D{X^,■■■,X^+^)W=DxMX2,■■■,Xm+l)W
-DiX7XrX2,X3,---,Xm + l)W DiX^,-' ' , Xm,VX,X,„ + l)W.
Definition 1.7.7 Define a second order differential operator
Ao : TiE) -^ T{E)
by letting
Ao = ^D{E„E,),
and call it Laplace-Beltrami operator, where {Ei, -- - ,En} is an orthonor-
mal moving frame on M.
Lemma 1.7.8 For the second order derivative, we have

1 7 Connections on Vector Bundles and Principal Bundles 73
(i) R(X,Y) and D{X^Y) are .7^(M)-linear with respect to the variables X
and Y;
(ii) R{X,Y) = D{X,Y) - D{Y,X).
Definition 1.7.9 Let M be a manifold, G be a group. A Principal
G-bundle over M is a triple {P; tt; m}, where
(1) P is a manifold, called a total space;
(2) TT: P -^ M is a map, called a projection;
(3) rri: P X G -^ P : (p, fif) «—► p • fif is a right action of G on P, i.e.
p ' e = p, e is the unit element
{P'9i) ' 92 =P'{91-92) ypeP; 91,92 eG,
such that
(i) for any x G M, there exists a neighborhood U oi x and a homeomorphism
$ : C/' X G -^ 'K~'^{U), such that
7ro$: U X G-^U
is the projection to the first factor space U\
(ii) for any u £U^ the map
^{u,'):G-^iz-^{U)
satisfies
^{u,9i) '92 = ^{u,gi -92), Vfifi,fif2 G G.
We call 7r~-^(aj) a fiber, and $ a local trivialization of the principal bundle. And
we always assume that all the maps, e.g. tt, m and $, in the definition are
^00 rpj^g above bundle is usually denoted by tt : P -^ M or P.
Exercise 1.7.10 Suppose P is a principal bundle, then
(1) for any x G M, 7r~-^(aj) is homeomorphic to G,
(2) the right action of G on P is free, i.e. if g is not the unit element e, then
the map
m(-,fif) :P-^P:p\^p'g
does not have a fixed point.

74 PRELIMINARIES IN RIEMANNIAN GEOMETRY
(3) 7r(p.^) = 7r(p), \/peP, geG,
(4) for pi,P2 G i^, if T^iPi) = '^(^2), then there exists a unique g E G such
that pi=P2'g'
Definition 1.7.11 Given a principal G-bundle tt : P -^ M, a C^ map
(J : C/^ -^ P is called a local section, if it satisfies
TT o cr = id : M -^ M,
where U is an open set of M. We denote the set of all local sections by MF(P).
Remark 1.7.12 Usually a principal bundle may not have any global
section. It shows that the notion of the principal bundle is quite different from
that of a vector bundle. So we do not use the notation T{P) to denote the set
of local sections, instead we use MF(P), where "MF" means "moving frame",
which will be explained later.
Exercise 1.7.13 If M is a Riemannian manifold of dim n, let 0{M)
be the set of all orthonormal frames, then
(1) 0{M) is a principal 0(n)-bundle, called a tangent principal bundle;
(2) MF(0(M)) is the set of all local orthonormal frame fields (i.e. all
orthonormal moving frames) of M.
Let us recall the orthonormal moving frame method in §3. Suppose
M is a Riemannian manifold of dim n, for an orthonormal moving frame a =
{El," ' ,En) defined on an open set U, i.e. a G MF(0(M)), we had assigned
a;<y, Q.a to a with
(^(Tg = g~^ '(^(T'g-\-g~^ 'dg \/g :U -^ 0{n)
where Ua^ and Qa^ are nx n antisymmetric matrices with entries being 1-forms
and 2-forms respectively. Uo^ and Ct^- are the fundamental notions of the
orthonormal moving frame method. Now we try to generalize the orthonormal
moving frame method to a general method for the principal G-bundles. The
general method will be called a moving frame method. In the case of the
tangent principal bundle 0(M), Uo- may be viewed as a 1-form with
antisymmetric matrix value, and any antisymmetric matrix is an element of the Lie
algebra of 0(n). So for a general principal G-bundle P, for a G MF(P), which

1 7 ConnecUons on Vector Bundles and Prtnctpal Bundles 75
is defined on an open set U of M, Ua^ should be understood as a 1-form with
^-value, where Q is the Lie algebra of G, in other words,
In order to explain the relations among Uf^^Uf^ g^^o- we need to explain the
quantities g'^fj^ag-, g~^dg^ dua^ and Uo- A Ua- in the general case.
For any g £G^ the map
Rg-iLg : 0{n) -^ 0{n) : h «—► ghg~^
induces a map
Ad{g) = {Rg-.Lg),:g^g,
Let {Xi, • • •, Xm} be a basis of the vector space Q, each X^ can be viewed
as a left-invariant vector field on G. Then the basis {a;J, • • • ,a;^} dual to
{Xi, • • •, Xm} is an .7^(G)-basis of A^(G). It is easy to see that the element
m
does not depend on the choice of bases for ^, which is called the Maurer-
Cartan form of G and is denoted by [G].
Suppose U is an open set of M and g : U -^ G is a. G*^-map, then
(1) define Ad{g-^) : A^{U) ^^G-^ A^{U) 0^ Q by
Ad(^-^)(a; 0 S) - a; 0 (Ad(^-^))(S),
m
(2) let g*{[G\) = X)ff*K°)^. G ^\U) ®^ g,
« = 1
(3) define d: A\U) 0a ^ -^ A'^{U) 0^ Q by
d{u 0Sj = (c2cj)0S,
(4) define a map
[•,•]: iA''{u) ®, g) X (A^iu) o, g) ^ AP+^u) ®, g
by
[a, (g) H,, a^ <8> Hj] = a, A a^ (gi [S,, H^].

76 PRELIMINARIES IN RIEMANNIAN GEOMETRY
U G = 0{n)y let S,j be an antisymmetric n x n-matrix, whose (Ai,/)-entry
{'^tj)ki satisfies
then {S,j|z < ^} is a basis of 0{n). For a tangent principal bundle 0{M) and
a G MF(M), the connection form u^ is an antisymmetric matrix with entries
in A^{U), i.e. it can be expressed as
(^a = Y.^':f-iJ ^ ^^(^) ^«- ^W-
Now we can prove
(1) g-^ 'U^ 'gz=zAd{g-^)u^,
{2)g-'dg = g*[0{n)l
(3) dw^ = d{u^),
(4) u^ Au^ = -^[^<t,(^(t]'
We only check (4) as follows. Let O^j be an n x n-matrix satisfying
Because
we have
2
^ 8 ^ ^'*'*-^ Aa;fe,)[Sij,SH]
t,3,k,l
— {Oki -Oik)' {Oij -Ojt)}

1 7 Connections on Vector Bundles and Principal Bundles 77
—^hOkj + htOij + SijOki — SkjOu}
«j,fe «j k^l
The above discussion introduce us to
Definition 1.7.14 Let tt : P -^ M be a principal G-bundle, a set
{u^\aeMF{P)}
is called a connection on the G-bundle, if it satisfies
u, g = Ad{g-^)u, + ^*([0(n)]), \/a G MF(P), V^ : U, ^ G,
where 11^ is the definition domain of a.
Definition 1.7.15 Let tt : P -^ M be a principal G-bundle, and
u = {u^\aeMF{P)}
be a connection, then define
and call
1^ = {a^\(j eMY{P)]
the curvature of u.
We list a lemma and a theorem without proofs as follows
Lemma 1.7.16 The notations are as above, then we have
Theorem 1.7.17 We have the following Bianchi identity (II)
Remark 1.7.18 It is well-known that if we want to examine a rigid
body moving in a space, we would like to consider a frame attached to the

78 PRELIMINARIES IN RIEMANNIAN GEOMETRY
body. Plainly the frame depends on the time t , so is called a moving frame.
As Prenet considered a curve in the Euclidean space, he used a family of frames
along the curve instead of a fixed frame in the Euclidean space. The frame in
the Frenet family depends on the point in the curve, it looks like the moving
frame depending on the time t, so we call the Prenet family a moving frame or
a Frenet frame. In this sense the orthonormal moving frame, which was used
in §2, may be viewed as a moving frame. This is the reason for it to receive the
name of the orthonormal moving frame. Therefore the orthonormal moving
frame method as we did in §3, which is characterized by the appearance of
Ua and QfT^ is called a moving frame method now. Plainly this method is
just the theory of principal bundles and their connections. The moving
frame method was initiated by E.Cartan, and culminated in the Chern's paper
[11] and the Chern-Weil theory (see Chapter 4).
1.8 General Tensor Calculus
In the last section we explained the connections on the vector bundles and
the principal bundles, but we had not discussed the relations between these two
kinds of connections yet. In fact they are really related to each other. In §1 and
§4 we had introduced connections on r{TM) and A*(M), which are connections
on vector bundles (especially after looking at the following Exercises 1.8.2 and
1.8.4). And a moment's thinking shows that these two connections can be
determined by {ct;<y}, which is a connection of the tangent principal bundle
0{M). Moreover, in the classical tensor calculus the connections (i.e. the
covariant derivatives) on tensor spaces are determined by {ct;^^} too. It tells us
that a tangent principal bundle with a connection plays the most important
rule. In this chapter we will generalize this fact to one for general principal
bundles not only for tangent principal bundles.
Let TT : P -^ M be a principal G-bundle, and F be a vector space with a
left G-linear action p on it, i.e.
GxV -^V :{g,v) ^ p{g)v
satisfies
{pigi ' 92)v = p{gi)p{g2)v, ^9i,92 eG,\/v eV
p{g) keeps the vector space structure of F Vgf G G.
We introduce an equivalence relation ~ in P x F as follows. For (pi, vi), (p2, '^2)
G P X F, (pi, '^i) ~ (P2? "^2) if and only if there exists g £G such that p2 — Vi'Q

1 8 General Tensor Calculus
79
and V2 = p{g )vi. Then define a quotient space
PXpV = {P xV)/ <^ .
And ioi (p^v) E P X V we denote its equivalence class in P XpV by p Xp v,
or simply by (p^v). By using this notation, if (a : U —* P) ^ MF(P) and
f : U -^ V then (cr, /) is a local section of P XpV. Further we define a map
Ir : P Xp V -^ M : (p^v) \-^ 'jr{p).
Now it is easy to prove that 7r:PXpF-^Misa vector bundle.
Definition 1.8.1 Given a principal G-bundle ir : P -^ M and a left
G-linear action p on F, we call the above vector bundle -ir : P XpV -^ M an
associated vector bundle of P.
Exercise 1.8.2 Let M be a manifold of dim n, and let P ^ GL{M)
be the tangent principal GL(n, R)-bundle, which consists of all frames on M,
and let V = R.^. A linear action pi of GL{n, R) on R" is defined by
PiiA)
1''')
\ x„ )
= A-
( ''' \
\ x„ /
/ ^l\
\/AeGL{n,R), V
GR",
and another linear action p2 is defined by
/ XI \
P2iA)
-l\t
= (A-')
\x„ J
f xi\
\x„ J
\x„ /
(^l \
\/AeGL{n,R), V
GR",
\Xn /
where A is a matrix and A* is its transpose. Then {GL{M) Xp^ R") is the
tangent bundle TM and {GL{M) Xp^K^) is the so-called cotangent bundle
T*M.
Definition 1.8.3 Define A^(n) (or A^(n)) to be a real (or complex)
space spanned by the set
{0^, A---A^,J1 < zi < ••• < Zfc <n},
with a left linear action p of 0{n) on it as follows:
p{A)(e„A...Ae,,)^ E E <
ji< <jkh, ,h
h
n
3k
•^?in '"^lu^u^3l ^"' ^^Jk^

80 PRELIMINARIES IN RIEMANNIAN GEOMETRY
where A G 0{n), and 6( ^ ) is 1,-1 or 0 according to whether
ii • • • jk
{hr " Jk) is an even permutation of (ji, • • •, jfc), an odd permutation or
otherwise respectively. And define
Kin) = 0 A* (n), A;(n) = 0 A* (n).
k k
Moreover we can provide A;^(n) or A* (n) with an associative algebra structure
such that
where
€{IJ;L) = €{ ).
^1 • • • ^Jk ^fc+l • • • lk + 8
Such a Aj^(n) (or A* (n)) is called a Grassmann algebra or an exterior
algebra of rank n.
Exercise 1.8.4 Let M be a Riemannian manifold of dim n, and let
0{M) be the tangent principal 0(n)-bundle. Define
A^(T*M) = 0{M) Xp A^(n), A^(T*M 0 C) = 0{M) x^ A^(n).
Then
A^(M) = r(A\T*M)),
A^(M) 0 C = r(A^(T*M (8) C)).
Definition 1.8.5 Let P be a principal G-bundle with a connection
u = {UfT\cr G MF(P)}, and p be a linear action of G on F, and let jE7 = P Xp V
be the associated bundle. Then define a connection
D : T{TM) X T{E) -^ T{E) : {X,W) ^ DxW
on the vector bundle E as follows: For a local section a G MF(P), W can be
expressed as
W\u = (cr, /), for a function f :U —^ V,
then define
DxW=(a,Xf + u4X)f),

1.8 General Tensor Calculus 81
where a;<y(-X') is a map from U to the Lie algebra ^, and the map p : GxV -^ V
induces p^ : Q x V -^ V, which gives p^(a;<y(-X'),/) to explain Uf^{X)f. A
careful check reveals that the above D is indeed a connection, and is called an
associated connection of a;.
Definition 1.8.6 Let G be a Lie group, /9,(z = 1,2,3) be a left linear
action of G on F,. A bilinear map
m:Vi X V2 -^Vs
is called a G-invariant multiplication, if
p3{9)m{vi,V2) - m{pi{g)vi,p2{g)v2), \fv, G V^, \/g G G.
Theorem 1.8.7 Let P be a principal G-bundle with a connection u; =
{u^\a G MF(P)}, then
(1) for any left linear action p of G on F, there is an associated connection,
defined in Definition 1.8.5, on the vector bundle P XpV;
(2) for three left linear actions of G on Vi{t = 1,2,3) and a G-invariant
multiplication m : Vi x F2 -^ V3, we have
Dxm,{Wi,W2) = m,{DxWuW2) + m,{WuDxW2),
VW, G T{P x,^ F,), X G r(TM),
where
m, : T{P Xp, Vi) x r(P x^, V2) ^ r(P x^, F3)
is induced naturally by m, and all Dx are the associated connections.
We will not give the proof of Theorem 1.8.7 here, because the like theorem
could be found in the classical tensor calculus, where the tensor bundles
are viewed as associated vector bundles of a tangent principal GL(n, R)-bundle
GL{M), and the operations on the tensor fields, such as the tensor product, the
exterior product, the contraction and so on, are all expressed as m*, induced by
GL(n, R)-invariant multiplications. So the above fact shows that this section
may be viewed as a first step for the general tensor calculus. Sometimes a
section of an associated vector bundle is called a general tensor (field).
Exercise 1.8.8 Let M be a Riemannian manifold of dim n, and 0{M)
be the tangent principal 0(n)-bundle. Then any connection u of 0{M) induces

82 PRELIMINARIES IN RIEMANNIAN GEOMETRY
an associated connection D on the associated vector bundle 0{M) Xp^ R",
which is TM of course, such that the connection D must satisfy
X{Y, Z) = {DxY, Z) + (y, DxZ) MX, Y,Z e T{TM),
Conversely, if a connection D on the tangent bundle TM satisfying the above
equality, then it must be an associated connection of a connection on 0{M).
Lemma 1.8.9 Let P be a principal G-bundle with a connection u =
{Ufr\cr G MF(P)}, and p be a left linear action of G on F, and let E = P XpV,
and
D : r{TM) X r{E) -^ r{E) : {X,W) ^ DxW
be the associated connection on the vector bundle E. Then
R{X,Y)W = {a, Q4X,Y)f),
where a G MF(P), W = {a, /), X,Y G T{TM). And i2(-, •), ^<r are defined in
Definitions 1.7.5 and 1.7.15, i.e.
R{X, Y) = DxDy- DyDx - D[x,Yh
Remark 1.8.10 We can use Lemma 1.8.9 to give another proof of
Lemma 1.4.15 as follows
R{E.,E,)(a,f) = {<T,n4E.,E,)-f)
= Y,^al3(E„Ej){u,^A)i{E,3){<T,f)
a,l}
a,(3

CHAPTER 2
SCHRODINGER AND HEAT OPERATORS
In the first chapter we introduced the operator d+6, which can be restricted
to different subspaces to give the de Rham-Hodge operator and the signature
operator. Its square operator A = (d -\- S)^ can be expressed as
A = -(Ao + F)
by the Weizenbock formula, where Aq is Laplace-Beltrami operator defined
in Definition 1.4.2 and F is an !F{M)-\ineai operator. Such kind of A does
not contain first order covariant derivatives, so we may give it a special name,
a Schrodinger operator or an operator of Laplacian type will be named
popularly.
It is well-known that either d + 6 or A is an elliptic operator. But in this
book we will not go into the general theory of the elliptic operators. Instead we
confine ourselves to the Schrodinger operators. The most important research
for the Schrodinger operator is the Hodge theorem, which can be used to solve
the following equation
A(j) = 0 or A(j) = /,
where cj) and / are unknown and known functions respectively. Given a
Schrodinger operator A, the operator ^ + A is called a heat operator. The central
point for the heat operator theory is the existence of a fundamental solution and
its applications. In 1951 Milgram-Rosenbloom [24] proposed an approach to
prove the Hodge theorem by using the fundamental solution. Moreover after
the sixties of twentieth century people found that the fundamental solution
can also be used to give a proof of the local index theorem for some geometric
operators. So all attention will be paid to the corresponding heat equation
instead of the Schrodinger operator equation in this book. We will introduce
basic knowledge about the fundamental solution, and use it to prove the Hodge
theorem in this chapter and the local index theorem later. For the simplicity of
formulations in this chapter we temporarily assume that there are many initial
solutions, and leave the proof of this assumption to Chapter 3. As far as the
83

84 SCHRODINGER AND HEAT OPERATORS
approach to the Hodge theorem is concerned, Milgram-Rosenbloom had not
shown the existence and some nice properties of the fundamental solution yet.
Berger and others[5] had proved the existence. This chapter will supply all the
details after consulting [15] and an opinion of Professor Wang Guan-Yin. The
most important detail is the existence and regularity of the solutions to the
Cauchy initial problem of the heat equation, which were of course needed by
the Milgram-Rosenbloom approach.
2.1 Fundamental Solution and Levi Iteration
Let us begin with a simple example. Consider a Cauchy problem of a heat
equation
lim u(t^x) = <^(aj),
where u, /, cf) are real functions. It is well-known that the solution to this
problem is
u{t,x)= / G{t,x,y)(j){y)dy+ dr G{t - T,x,y)f{T,y)dy,
JK Jo JK
where
G{s,x,y} = —==e 4. .
V47rs
In the above expression of the solution, G{s,x,y) is a key function. If we find
it, we can get the solution immediately. It is easy to see that such a G{s, x,y)
satisfies
^2 \
G{t,x,y) = 0, Vt>0,
\dt dxy
lim / G{t,x,y)<l>{y)dy = <l>{x), \l<t>
where (j) is any continuous function. Note that — ^r = — Aq is a Schrodinger
operator, so ^ — -^ is the corresponding heat operator.
The function G{t,x,y) in the above example motivates the following
definition of the fundamental solution.
Definition 2.1.1 Let M be a Riemannian manifold, tt : E —► M be
a vector bundle with a connection D. Let
Ao: r(^) —^ r(^)

2 1 Fundamental Solution and Levi Iteration 85
be Laplace-Beltrami operator (see Definitions 1.4.2 and 1.7.7), which is defined
by using Levi-Civita connection on M and the connection D on the vector
bundle E. Again let F : V{E) —> V{E) be a .F(M)-linear map. Then
A = -(Ao + F)
is a Schrodinger operator. If a family of R-linear maps
G{i,q,v)-Ej,—^Eq,
with parameters t > 0, and g,p G M, satisfies the following conditions (i), (ii),
(iii), the family is called a fundamental solution of the operator ^ + A,
where Ep = '7r~'^(p), and the conditions are
(i) G{t,q,p) : Ep —► Eq is a R-linear map of vector spaces, it is continuous
for all parameters t > 0,g,p.
(ii) For a fixed v £ Ep^ let
e{t,q) = G{t,q,p)v, Vt>0,
then 0 has first and second continuous derivatives at variables i and q
respectively and satisfies the equation
For the simplicity of language we write this fact as
d_
dt
{- + Ag)Git,q,p) = 0, Vt>0,
where Aq a^ts on the variable q.
(iii) If (^ is a continuous section of the vector bundle E, then
lim / G{iA.v)mdv = m^ V(^
where dp is the volume measure given by the Riemann metric.
Remark 2.1.2 G(t, g, p) is a family of linear maps with the parameters
(t, ^, p) G (0, oo) X M X M. This family can be explained as a section of a vector
bundle over the space (0, oo) x M x M, whose fiber is Hom(jE7p, Eq). Hence the
family is a general tensor (see §1.8). The families H{t^q^p)^U{i,q,p),W(i,q,p)

86 SCHRODINGER AND HEAT OPERATORS
appeared in the next section §2.2 can be explained similarly. This kind of
explanation is very important, because for a general tensor the notions of continuity
and differentiability are easy to understand.
Remark 2.1.3 The fundamental solution has many nice properties
such as existence, uniqueness, regularity (C^-differentiability), solvability for
the Cauchy problem and so on, which seem far away from Definition 2.1.1.
So to deduce these nice properties from the definition must be a serious task.
A trick for the deducing is a proper choice of the initial solutions. The nicer
initial solutions (see Definition 2.2.1) we could find, the better properties of the
fundamental solution we would get.
Glancing at the definition, we meet two natural questions first. (1) Can we
find an explicit formula for the fundamental solution? The answer is easy, we
can't in general! there is no formula like
G{s,x,y)- —e 4,
in most cases. (2) Can we prove the existence and uniqueness of the
fundamental solution? It is the question we will deal with in what follows.
Although there is no explicit formula for the fundamental solution, we may
try to find a series convergent to it. For example, if we want to solve an
equation F{x) — 0, we may use the iteration method. More precisely, after
finding an n-th approximate solution x^^ let x^+i — x^ + h^^ where hn is
an approximate solution to the equation F{xn + /i) = 0, then take Xn+i as a
(n + l)-th approximate solution to F{x) = 0. In most cases one can prove that
lim Xn = xo + y^hn
n=0
is just the solution we wanted. In our case we try to find the fundamental
solution by the iteration method. Suppose we find an n-th approximation
Gm{tyQ,p)y which satisfies
liin / Gm{t,q,p)(l>{p)dp= (piq), \f(t>.
Let
Gm+i{-t,q,p) = Gm{t,q,p) + h{t,q,p).

2.1 Fundamental Solution and Levi Iteration 87
Consider the following equation
f (^ + Aq\{Gm{t.q,p) + h{t,q,p)) = 0, Vt>0,
lim / {Gm{t,q,p)-Vh{t,q,p))<t>{p)dp^<t>{q),'i<t>,
lim/ h{t,q,p)(j){p)dp=0, \/(l),
or
lim/i(t,g,p) = 0.
*->o
Recall our original purpose that G{t,q,p) is for solving the Cauchy problem,
so for the above equation the solution h{i^q^p) should be
h{t,q,p)= f ^-^ j G{t-T,q,z)' - (— + AAGm{r,z,p)
dz.
To meet the need of the iteration method, we choose h{i,q,p) approximately
as follows
h{t,q,p)= dr Go{t - T,q,z) - \- I —+ aA Gm{r, z,p)
Thus the iteration equation is
Gm+i{-t,q,p)= Gm{-t,q,p)+ dr Go{t-T,q,z)
Jo Jm
dz.
-(^-■)
Gm{r,z,p)
dz.
Now let us make a slight change of the above formulation. Let
Km{t,q,p) = (-1)™ (^ + A9) Gmit,q,p),

88 SCHRODINGER AND HEAT OPERATORS
then we have the following "formal" computations
+ f dr f Go{t - T,q,z)[{-ir+'Km{r,z,p)]dz }
Jo Jm
= -Km{t,q,p) + [ Go{t-T,q,z)Km{r,z,p)dz]r=t
Jm
+ dr l — + AqjGo{t-T,q,p)Km{r,z,p)dz
= dr Ko{t-T,q,z)Km{r,z,p)dz,
Jo Jm
and
G{t,q,p) = Go{t,q,p) + XI(^-+1 ~ ^-)
m—O
= Go{t,q,p)
+ E / ^^/ Go{t-r,q,z)-[{-ir^'Km{T,z,p)]dz
m=0^^ ^^
= Go{t,q,p)
+
ft f ^
/ dr Go{t-T,q,z)^[y2{-ir+'Km{r,z,p)]dz.
J^ J^ m=0
The above "formal" computations are not rigorous, but the result is of great
value. It motivates the following definition.
Definition 2.1.4 Suppose we are given a Go(^, ^, p), the following com-

2.2 Existence of Fundamental Solution 89
putation from Go{t,q,p) to G{t,q,p) is called Levi algorithm,
Ko{t,q,p)= f—+ AJGo(t,g,p),
Km+i{-t,q,p)= / ^'7- / Ko{t-T,q,z)Km{r,z,p)dz,
Jo Jm
(X)
K(t,q,p)= Y.{-ir+'K,n(t,q,p)
m-0
G{t,q,p) = Go{i,q,p)
+ dr Go{t-T,q,z)K{T,z,p)dz.
Jo Jm
Sometimes we denote G by Levi(Go). The later discussions will show that if
Go is chosen properly then the Levi algorithm is rational, i.e. all the series
and the integrals in the algorithm are convergent and the result G is just the
fundamental solution of the heat operator ^ + A, i.e. it satisfies Definition
2.1.1.
2.2 Existence of Fundamental Solution
In this section we introduce a notion of the initial solutions, which is a
temporary notion only for proving the existence and the regularity of the
fundamental solution. In Chapter 3 we will show that the Minakshisundaram-
Pleyel parametrix can provide a series of the initial solutions.
Definition 2.2.1 Let M, jE7, A be defined as in Definition 2.1.1. If a
family of linear maps
H{t,q,p):Ep—^Eq
with parameters t > 0, and q^p £ M satisfies the following conditions (i), (ii),
(iii), (iv), the family is called a k-ih. order initial solution of the operator
^ + A. The conditions are
(i) H{t^q^p) : Ep —> Eq is a linear map of vector spaces, it is continuous
and infinite differentiable at all parameters t > 0, g,p. We denote this
condition by
H{t,q,p) e C~((0,oo) xM xM).
(ii) (^ + Aq)H(t,q,p) can be extended continuously over [0,oo) x M x M and
CIS a general tensor it is of class C^ on [0, oo) x M x M. We denote this

90 SCHRODINGER AND HEAT OPERATORS
fact by
(- + Ag)H{t,q,p) e C*([0,oo) xMxM).
(lil) Suppose iV is a manifold, /(p, a) is a general tensor field such that
/(P,«) G C^{M X iV), and for any (p,a) e M x N,
f{p,a)eEp.
Let / be defined by
I I H{t,p,z)f{z,a)dz, ift>0,
t,p,a) = < Jm
fit.
then
/(p,a), ift = 0.
/(t,p,a)GC^([0,oo)xMxi\r).
(Iv) Suppose gf(t,p, a) G ^p is a continuous general tensor field over [0, oo) x
M X N^ and
fif(t,p, a) G C'([0, oo)x M X N)
with 2 < Z < Ai. Define a general tensor field W by
W{t,p,a)- I dr H{t-T,p,z)g{T,z,a)dz,
Jo Jm
then
W{i,p,a) e C^^\[0,oo) xM xN).
Moreover, for any 5 < |, and vector fields Xi, • • • ,-X',,
VXi---Vx, Vr(t,p,a)= / c^r / \/x,"'\7x, H{i-T,p,z)g{T,z,a)d
Jo Jm
■§tW{t,p,a) = g{t,p,a)+ dr —H{i-T,p,z)g{r,z,a)dz,
In the above definition [|] means the greatest integer less than or equal to |.
Remark 2.2.2 The condition (iii) in Definition 2.2.1 tells
Jim / H{t,p, z)f{z, a)dz ^ f{p, a),
^-^^ Jm

2.2 Extstence of Fundamental Solution 91
and a moment's thought shows that it is impossible to find an ordinary integral
kernel H{0,p,z) such that
/ H{(},p,z)f{z,a)dz^f{p,a).
Jm
So condition (iii) implies that considering lim H{t,p,z) will give trouble. In
other words, we may think that H{t^q^p) has a singularity at t = 0. Therefore
the integral, which defines W(t,p, a) for t > 0, should be understood as a
singular integral
lim / dr I H{t — T,p,z)g(T,z,a)dz.
Hence it is not trivial to claim that
W{t,p,a) e Ct2]((0,oo) xM xN),
and still less that
in the condition (iv).
W{i,p,a) e d^l([0,oo) xM xN)
Remark 2.2.3 The initial solutions are introduced for "producing"
the fundamental solution, so the fundamental solution itself should be a good
candidate for an initial solution. But comparing Definition 2.1.1 with Definition
2.2.1 you will find that except condition (ii), the conditions for the fundamental
solutions seem weaker than those for the initial solution. But after a hard
work we can prove that the fundamental solution is also an initial solution.
This case shows a balance for defining the fundamental solution. If we add
stronger conditions to Definition 2.1.1, then we will meet more difficulties for
the existence problem. On the contrary, in Definition 2.1.1 one lists as less
as possible conditions such that the uniqueness could be guaranteed. So after
the latter Theorem 2.3.5, Definition 2.1.1 is satisfied. The discussion on the
fundamental solution goes as follows. In this section we use the initial solutions
to prove its existence and regularity, and in the next section we again use the
initial solutions to prove the solvability of the Cauchy problem for the heat
equation, then get the uniqueness of the fundamental solution. This shows
the reason why we introduce the initial solution here, although it may be a
temporary notion.
Remark 2.2.4 W(t,p, a) in Definition 2.2.1 enjoys a certain vogue
under the name of a potential integral, and usually more attention was paid
to such kind of integrals.

92 SCHRODINGER AND HEAT OPERATORS
Lemma 2.2.5 If Go{t,q,p) is a k-th. order initial solution, then in the
Levi algorithm all the series and the integrals are convergent. Moreover
K{t,q,p) e C^([0,oo) X M xM).
Proof The condition (ii) in Definition 2.2.1 reads
Ko= (^+ A J Go G C^([0, oo) X M X M),
so for a fixed T > 0, there exists c > 0, such that
\Ko{t,q.p)\<c, VtG[0,T]; q.pEM.
Therefore
\Ki{t,q,p)\< f dr f \Ko{t-T,q,z)\'\Ko{T,z,p)\dz
Jo Jm
< / c^vdr = c^vt,
Jo
where v is the volume of M. Further,
\K2{t.q.p)\ < [ dr f \Ko{t-T,q,z)\'\Ki{T,z,p)\dz
Jo Jm
-Jo 2
By induction we have
j.Tn
\K,nit,q,p)\<c"'+'v"^ —
ml
Thus
f^\Km{t,q,p)\< f^c
{cvty
m\
= ce"'^*.
m=0 m=0
which is absolutely convergent on [0,T] x M x M. Therefore
K{t,q,p) e C^([0,oo) X M xM).
Denote
{A^B){t,q,p)= dr A{t - T,q,z)B{r,z,p)dz,
Jo Jm

2,2 Existence of Fundamental Solution 93
then
Ko^Km = Ko^Ko^ Km-1 = '" = Km^Ko,
m>0
= -Ko+J2 (-l)™+'^o * ifm-1 = -Ko-Ko*K
m>0
= -Kq + Ko^Ko- Kq^Kq^K = -Kq -{-Kq^Kq- Kq^K ^Kq.
From
Koe C^([0,oo) X M xM)
it follows that
k eC^dO^oo) X M X M).
The proof is complete.
Theorem 2.2.6 (Existence and Regularity of fundamental
solution) If Go is a k-th. order initial solution with k > 4, and G = Levi{Go),
then G is a fundamental solution of the heat operator (^ + A), and
G{t,q,p) e Gt^]((0,oo) X M xM).
Proof Lemma 2.2.5 and Definition 2.2.1(iv) read
^GG^([0,oo) xM X M),
Go * ^ G Gt2]([0, oo) X M X M).
It gives
G = Go + Go * ^ G G^^1((0, oo) X M X M).
By using the formulas in Definition 2.2.1(iv) for v E Ep we have
(^^ + A,yG{t,q,p)v)
= (^^ + A,yGo{t,q,p)v)+(^^ + A,yiGo*K){t,q,p)v)
= {Ko + K + Ko* k)v
(X) (X)
m—Q m=0
= 0,

94 SCHRODINGER AND HEAT OPERATORS
where
^ "
m
Further we have
= I Go{t,q,p)<l>{p)dp+ I {Go^K){t,q,p)<t>{p)dp
Jm Jm
= I Go{t,q,p)<l){p)dp+ dr Go{t - r,q,z)U{z,r)dz,
JM Jo Jm
where
U{z,t) = f k{T,z,p)(t>{p)dp e CP{%oo) X M).
Jm
Definition 2.2.1(iii) reads
lim / Go{t,q,p)<t>{p)dp = (j){q),
*-*^Jm
lim I dr I Go{t — r, g, z)U{z, T)dz = lim / U{t — r, g, T)dT = 0.
lim/ G{t,q,p)(l){p)dp=<t>{q).
*-*^Jm
It follows that
The proof is complete.
2.3 Cauchy Problem of Heat Equation
As we mentioned at the beginning of this chapter, for the heat equation
the most important thing is how to solve the Cauchy problem. And for the
Cauchy problem the existence and the regularity of a solution are the key
points, because the uniqueness can be deduced from them.
Theorem 2.3.1 (Existence and Regularity of a solution) If Go
is a k-th order initial solution of the heat operator (^ + A) with k > A, let
G = Levi(Go),
and
u{t,q) = / G{t,q,p)<t){p)dp
Jm
+ f dr f G{t-T,q,p)f{r,p)dp,
Jo Jm

2,3 Cauchy Problem of Heat Equation 95
If
<f>(p) e c^{M)
fit,p)eC\[0,oo)xM),
then u{t, q) satisfies the Cauchy problem of the heat equation
{- + A,)u{t,q) = fit,q), VOO,
limu{t,q) = <j){q),
t->o
v.^,
and
u{t,q) e C"((0,oo) X M)nC^([0,oo) X M),
where5 = min([|],[|]).
Proof From
G=Go + Go^K
it follows that
i{t,q)= I G{t,q,z)<t>{z)dz^{G^f){t,q)
Jm
= [ {Go + Go^K){t,q,z)<l>{z)dz
Jm
+(Go*/)(t,5) + (Go*^*/)(t,g)
= / Go{t,q,z)<t){z)dz
Jm
+{Go^{P + f + Q)){t,q),
where
P(<,g)= / K{t,q,z)<j>iz)dzeC\[0,oo)xM)
Jm
Qit, q) = ik* f)it, q) e C«=([0, oo) X M).
By (i) and (iv) in Definition 2.2.1 we get
u{t,q)eC'{{0,oo) xM),
and by (iii) we also get
^'i't, q) G C^([0, oo) X M), and lim u{t, q) - <j){q).

96 S CHR ODINGER A ND HE A T OPERA TORS
By using the formulas in (iv), we have
(^ + A5M<,?) = y {^^+ \)Go{t,q,z),j>{z)dz
+iPit,q) + fit,q)+Q{t,q))
+((l+A,)Go)*iP + f + Q){t,q)
= / Koit,q,z)(f){z)dz
Jm
+iPit,q) + f{t,q) + Q{t,q))
+{Ko*{P + f + Q)){t,q)
= f Ko{t,q,z)<j>iz)dz + Pit,q) + iKo*P)it,q)
JM
+/(*, q) + Qit, q) + {Ko * (/ + Q))(<, g)
= [ [Ko + K + Ko* K]{t, q, z)(t>{z)dz
Jm
+ [f + K * f + Ko* f + Ko*K * f]it,q).
Note
Ko + K + Ko*K = (i,
so the above equalities give
i^ + A,)uit,q) = fit,q).
The theorem is true.
Theorem 2.3.2 (Uniqueness for Cauchy problem) If there exists
a k-th. order initial solution with k > A, then for any
uit, q) e C2((0, oo) X M) n C^([0, oo) X M),
which satisfies
(f+ A,)n(t,g) = 0, Vt>0, VgGM,
n(0,g) = 0, Vg,
we have
u{t,q) = 0, \/t >0, \/qeM.

2 S Cauchy Problem of Heat Equation
97
Proof It is sufficient to prove that for any C^ continuously differen-
tiable (^ G r(M),
Jm
By Theorem 2.3.1 there exists ^{t,q) such that
(^ + A,)^(t,g) = 0, Vt>0,
^0,q) = <j>{q).
Then for 0 < r < t, we have
— / {^t-T,q),u{T,q))dq
{-^^t-T,q),u{T,q))dq
mt-r,q),—u{T,q))dq
= / {Ag^t-T,q),u{T,q))dq
Jm
+ / mt-T,q),-Agu{T,q))dq
Jm
= / {Aq^t-T,q),u{T,q))dq
Jm
+ I {Aq^t-T,q),-u{T,q))dq
Jm
= 0,
hence
lim / {^t-T,q),u{T,q))dq=\im {^{t - r,q),u{r,q))dq,
T-**-Jm ''-*^Jm
fM "^-^^Jm
The left-hand side of the above equality is
/ (^(0,g),n(t,g))dg= / {<l>{q),u{t,q))dq,
Jm Jm
while the right-hand side is
/ mt,q),u{0,q))dq= f ($(t, g), 0)dg = 0.
Jm Jm

98 5 CHR ODINGER AND HE A T OPERA TORS
The proof is complete.
Remark 2.3.3 Theorem 2.3.1 also holds if we replace the continuous
condition for <j) by
where T{M) is the completion of T(M) under the inner product. Of course the
condition
\irnu{t,q) = <l){q)
in the Cauchy problem should be understood in jC2-sense.
Remark 2.3.4 Theorem 2.3.2 still holds if the condition is replaced by
uit^q) e c2((o,oo) X M)n£2([o,oo) x m).
Theorem 2.3.5 (Uniqueness of fundamental solution) If there
exists a k-th. order initial solution with k > 4, then the fundamental solution
is unique.
Proof If there is another fundamental solution G{t,q,p), then for any
(j) let
'^(^'?)= / {G{t,q,p)-G{t,q,p))(l){p)dp,
Jm
it must be zero by Theorem 2.3.2. So we have
G{t,q,p)-G{t,q,p) = 0, Vt>0,
which shows the uniqueness.
2.4 Hodge Theorem
Given a vector bundle E over M, let T{E) be the set of all smooth sections
of E. Assume an inner product on E (i.e. a family of inner products on fibers),
then it induces an inner product on T{E) by using a Riemannian metric on M
(see §4 of Chapter 1 for details). If
A : T{E) -^ T{E)
is a Schrodinger operator, we consider the following equation
A*u = /,
where A* is the adjoint operator of A. For what kind of / the above equation
has a solution? And how many solutions does it have? The questions are clearly
answered by the following orthogonal decomposition of T{E), which claims
r(£^) = KerA + ImA*,

2,4 Hodge Theorem 99
where
KerA = {<^Gr(£^)|A<^ = 0}
ImA* =A*{T{E)),
and
dim(KerA) < oo.
The above decomposition is just the well-known Hodge theorem. From the
above decomposition it follows that
there is a solution to A*n = f <=^ / G ImA* <=^ f ± Ker A,
and in case of the existence of solutions the dimension of the solution space
equals
dim Ker A.
So Hodge theorem may be viewed as a theory for solving the Schrodinger
equation, hence it must be very difficult to prove. However, at a first glance
the decomposition seems very trivial, because it may easily be proved that
(ImA*)-^ =KerA,
from which one is tempted to conclude
r(£^) = ImA* + (ImA*)-^,
which is true for finite dimensional case of course. So here we want to stress
that the decomposition
r(£^) = ImA* +(ImA*)-^
is as difficult to prove as the equality
r(£^) = KerA + ImA*.
A thorough explanation of Hodge theorem could be found in [34], [35], and
some interesting historical materials could be found there as well.
In this section we prove Hodge theorem by using a heat equation method
initiated in [24]. Given a continuous section cj) of the vector bundle E^ consider
the following equation
{-^^+Ag)u{t,q) = 0, VOO,
w(0,g) = ^(9), V<^.

100 SCHRODINGER AND HEAT OPERATORS
Suppose for any k > 0 there is a k-th. initial solution of the heat operator
(^ + A) (this fact will be proved in Chapter 3 §2.), then by Theorems 2.3.1
and 2.3.2 the above equation has a unique solution
u{t,q) e C^([0,oo) X M) nC~((0,oo) X M).
We denote u{t^q) by {Tt<j)){q). For the simplicity of language we confine
ourselves to discussing A in the self-acyolnt case.
Lemma 2.4.1 (Semi-group property) Tt satisfies the following
properties
(i)To = l, Tt,Tt,=Tt,Tt,=Tt,^t,. Vti,t2>0,
(ii) for any t > 0, T* is self-adjoint,
(iii) for any t > 0, T* is a compact operator, it means that Tt maps any
bounded set into a compact set.
Proof The uniqueness for Cauchy problem (Theorem 2.3.2) implies the
property(i). Let us prove (ii) now. The following computations
= {-ATt<j>,Tr^)
= {%<!>,-ATr^)
imply
il-l)iT.<f>,T.i>) = 0.
The transformation of variables

2.4 Hodge Theorem 101
gives
du ~ 2^dt'^ dr^'
d__ Kd___d_
dv~ 2^dt dr^'
So if we let /(t,r) = {Tt<t),Tr'il)), then we get
(|;)mr) = 0.
It follows that f{t,t) - F{u) = F(t + r) = f{T,t), thus
{Tt<l>,Tri>) = {Tr,f>,TtXl>).
In particular, we have
{Tt<l>,i^) = F{t + 0) = {<l>,Tti>),
hence (ii) is proved. Due to
Tt(l>= I G{t,q,z)<t>{z)dz,
JM
and
G{t,q,z) G d^^((0,oo) X M X M),
Tt is a compact operator for t > 0. Therefore the lemma is true.
Theorem 2.4.2 (Fourier Expansion Theorem) Let M be a Rie-
mannian manifold, jE7 be a vector bundle over M with a metric and an
admissible connection D. Suppose A : T{E) —► r(jE7) is a self-adjoint and
non-negative Schrodinger operator, then there exists a set {(^i, <^2, * * *} C ^{E)
such that
Jm
IM
It is reminded that T{E) is a set of C^ sections, hence here (^, are C°
(ii) Denote the completion of the inner product space T{E) by T{E). Then
{<^i5 <^2? • • •} is a complete set of T{E)^ it means that for any (j) G ^{E)
there is an equality
(X)
<t> = Y.'^{<t>.<l>,))<l>^
«=1
in the space V{E).

102 SCHRODINGER AND HEAT OPERATORS
(iii) {<^i, <^2, • • •} satisfies
A(^, = A,(^„ Tt(|>^ =e-'^^<t>,
with a condition
0 < Ai < A2 < • • •, lim A, = 00,
«—>o
which will be written as
0 < Ai < A2 < ^00.
Proof We only need to prove the theorem for a self-adjoint and positive
Schrodinger operator, because for a non-negative operator A and a positive
number 6, the operator A +6 is positive, and the proof for A + 6 can imply the
theorem.
For any fixed t > 0, T^ is a self-adjoint compact operator. By the famous
Hllbert-Schmidt Theorem (see p. 235 in [42]), as an operator on T{E)^Tt
has eigenvalues {//«}, and the nonzero eigenvalues can be arranged such that
\^^m-^ < i/^2(<)r' < i/^swr' < • • • ^ 00.
Moreover all the corresponding eigenspaces
V,{t) = {(t>eV{E)\Tt(t>^^i^{t)(t>}
are finite dimensional and orthogonal to each other. From
Tt.Tt, = Tt.Tt,
it follows that Tf^ and Tt^ share the common eigenspaces. Therefore there exist
a series of finite dimensional spaces Vi, F2, • • •, they are the eigenspaces for all
Tf. Now confine ourselves to discussing the operator Tt on each V^. Because
{Tt\t > 0} is a semigroup, we get
for some real number A,. For t > 0, due to Theorem 2.3.1 (precisely Remark
2.3.3) Tfcj) is C^, hence so is (^ = e^'^Ttcj). From the following computations
0 = (^ + A,)iTt<j>) = (^ + A,)(e-^-V) = (-A.<^ + A$)e-^-',
we obtain

2.4 Hodge Theorem 103
A is positive and self-adjoint, so A, > 0 and F, and Vj are orthogonal to
each other, hence (i) and (iii) are proved. Hilbert-Schmidt theorem also claims
that if t > 0,(^ G r(jE7), then Ttcj) can be expanded with respect to the set
{<^i5 <^2, • • 'l? hence does cf) because of
\imTt(l) = (j).
(ii) is now proved. The theorem is true.
Theorem 2.4.3 (Hodge Theorem) Let M, £", A be defined as in
Theorem 2.4.2, then
(i) "H — {(t> £ r(jE7)|A(^ — 0} is finite dimensional vector space,
(ii) For any (f) G r(jE7), there is a unique decomposition
(p = (pa + <t>(i',
where
<t>ae%, (^^GA(r(^)).
Proof (i) is a direct consequence of Theorem 2.4.2. Due to H L
A(r(jE7)), the decomposition in (ii) is unique. For any (^ G r(jE7), let
<t>a = ^ {{<!>, <t>i))<t>i G W,
<t>(i = <t> — <t>a ^ r(jE7).
Because of Theorem 2.4.2(ii) it is easy to see that
A.
A,>0
Let n(t, q) be the solution to
(— + A)w = <j>^, V* > 0,
.(o..)=5;<w^,,.
A,>0
By Remark 2.3.3
n(t,g) gC^((0,oo) xM),

104 SCHRODINGER AND HEAT OPERATORS
A moment's thought reveals that
A,>0 ^'
is a jC^-solution to the above equation. By the uniqueness (see Remark 2.3.4),
it is immediate that u = u. From ^u = 0 it follows that
r\ Pi
<^ - <^a = (^ + A)'u = —u-V^u = Aix,
and (ii) is proved.
We may say more about the uniqueness of the fundamental solution (recall
Theorem 2.3.5) by proving
Theorem 2.4.4 Let M, jE7,A be defined as in Theorem 2.4.2, and let
la*
G{t,q^p) be the fundamental solution of the heat operator (^ + A), then we
have
(X)
G{t,q,p) = Y,^-^'*MQ){Mp)r),
« = 1
which means that for any v £ Ep, the equality
(X)
« = 1
holds in r{E).
Proof For any fixed t > 0 and v G Ep, expand G{t,q,p)v in terms of
eigenfunctions,
(X)
G{t,q,p)v = Y^f,{t,p,v)(l)^{q),
where
{t,p,v)= / {MQ)^G{t,q,p)v)dq.
Jm

2,4 Hodge Theorem 105
From
Mt,p,v)= {(t>t{q),^G{t,q,p)v)dq
= / {Mq).-^qG{t,q,p)v)dq
Jm
= - {^q<t>t{q)^G{i,q,p)v)dq
Jm
= -\ / {<t>i{q)^G{t,q,p)v)dq
Jm
= -\fi{t,p,v),
it follows that
ft{t,p,v) = K{p,v)e
-Xtt
Note fi{t^p^v) depend on v linearly, we get k^{p^v) = K{p)v, where kt{p) :
Ep -^ "R. is a linear function. And then there exists Ai,(p)^, which does not
depend on v, such that kt(p)v = {K{p)'^,v). Therefore
(X)
G{t,q,p)v = J2^~^''Mq){k,{p)*,^),
«=1
and consequently for any a G r(jE7),
a{q) = lim / G{t,q,p)a{p)dp
*~*^Jm
oo .
,^1 Jm
Comparing this equality with
oo .
<^iq) = y2Mq) {<t>i{v)^0L{v))dp
,^1 Jm
in Theorem 2.4.2 deduces
/ {K{p)*,OL{p))dp^ / {<t>i{p),a{p))dp
Jm Jm
and hence
K{p)* - Mp) ^^
in T{E). The proof is complete.

106 SCHRODINGER AND HEAT OPERATORS
2.5 Applications of Hodge Theorem
In this section we shall examine the de Rham-Hodge operator and the
signature operator by using Hodge theorem. Let M be a Riemannian manifold
of dim n, and
0 —^ A° ^ ■■■ ^ A"-' ^ A" ^ A^^' ^ ■■■ ^ A" —^0
be the de Rham complex (see Chapter 1 §4). Define
Z''{M) = Kei{d : A*^(M) -^ A*+i(M)}
= {ae A*(M)|da = 0}
B''{M) = lm{d : A''-'^{M) -^ A*(M)} = d{A''-^{M)).
Due to d^ = 0 we know 5*(M) C Z''{M), and we can define
hUm) = Z\M)IB\M),
which is called a k-ih. de Rham cohomology group of M. Let
k
The first question about the definition is whether the dimension of the de
Rham cohomology group is finite? In 1928 de Rham published a paper, in
which it was actually claimed that the de Rham cohomology group H2j^{M)
is isomorphic to the cohomology group with real coefficients H* (M, R) in
algebraic topology. As you know that the dimension of the cohomology group
is finite, so de Rham's paper gave a positive answer to the above question.
Moreover it also showed that the group structure of the de Rham cohomology
group is a topological invariant. More information and details may be found in
[10] and [20]. We do not want to dwell on topology any more, and the de Rham
theorem would not be proved here. Instead we will prove the finiteness of de
Rham cohomology group by using Hodge theorem. By definition, an element in
H^j^{M) is an equivalence class of a closed k-foim a G Z^(M), in other words,
the element can be represented by the closed form a. In 1935 Hodge claimed a
theorem which asserted that every element in H^j^{M) can be represented by
a unique harmonic form a. Here a harmonic form a means
da = 0 and Sa = 0.
We denote the set of all Ai-harmonic forms by 7i^{M).

2 5 Applications of Hodge Theorem 107
Lemma 2.5.1 Let M be a Riemannian manifold of dim n, then
rC'iM) = Ker{d + <5 : A^(M) -^ A*(M)}
= KerjA : A^(M) -^ A^(M)},
where A = (d + <5)2.
Proof First of all, by the equality
A = (d + <5)2 = d<5 + 8d
we know
A(A^(M)) C A^(M),
thus there is no trouble for reading the last term in the equalities in the lemma.
It is easy to see that
W^(M) C Ker{d + <5 : A^(M) -^ A*(M)}
C KerjA :A^(M)-^A*(M)}
= KerjA : A^(M) -^ A^(M)},
so in order to prove the lemma it is sufficient to prove
KerjA : A^(M) -^ A^(M)} C W^(M).
If a G KerjA : A*^(M) -^ A*^(M)}, i.e. Aa = 0, then
0 = ((Aa, a)) = (((d + <5)2a, a)) = (((d + <5)a, (d + 8)a))
= {{da, da)) + {{Sa, Sa)) + 2{{da, 8a))
= {{da, da)) + {{6a, Sa)) + 2{{dda, a))
= {{da, da)) + {{Sa, Sa)),
hence
da = 0 and <5a = 0.
It implies a G 7i^{M), and the lemma is proved.
Theorem 2.5.2 Let M be a Riemannian manifold of dim n, then
(1) 7i^{M) is a finite dimensional vector space, where k = 0,1,-" ,n.

108 SCHRODINGER AND HEAT OPERATORS
(2) There is an orthogonal decomposition
A^(M) = rL^{M) -f d(A^-^(M)) -f <5(A^+^(M)).
Proof By the Weitzenbock formula (Theorem 1.4.16)
A:A^(M)-^ A^(M)
is a Schrodinger operator, hence Theorem 2.4.3 (Hodge theorem) is true. Prom
2.4.3(i)
W^(M) = KerjA : A^(M) -^ A^(M)}
is of finite dimension, so (1) is true. And Theorem 2.4.3 (ii) reads
A^(M) = n^{M) + A(A^(M)).
Because
A(A^(M)) C d(A^-^(M)) + <5(A^+i(M)),
we have
A^(M) = n^{M) + d(A^-^(M)) + <5(A^+^(M)).
Note that three spaces n^{M), d{A^-^{M)), S{A^+^{M)) are orthogonal
to each other, so (2) is true. The proof is complete.
Remark 2.5.3 Theorem 2.5.2 Wcis obtained by Hodge in 1935, so it
is the original Hodge theorem. Theorem 2.4.3 was named after Hodge by
successors. A complete proof of Theorem 2.5.2 was given by H.Weyl in 1940.
Corollary 2.5.4 The imbedding i : 7i^{M) -^ Z^{M) induces a vector
space isomorphism
Proof Theorem 2.5.2 (2) implies that
Z\M) = n\M) + d{A^-\M)),
and then
HlniM) = Z\M)/d{A^-\M)) = Ti^M).
Let M be an oriented Riemannian manifold of dim n, and * : A^{M) -^
A"'~^{M) be the star homomorphism defined in Definition 1.4.4. By Lemma
1.4.8(3) we have
* A ^ *(d<5 + Sd) - {dS + <5d)* = A*,

2 5 Applications of Hodge Theorem 109
consequently we have
which is an isomorphism due to Lemma 1.4.5 (2). Then by the original Hodge
theorem (Theorem 2.5.2) we get
Theorem 2.5.5 (Duality Theorem) For an oriented Riemannian
manifold M of dim n, the star isomorphism
induces an isomorphism
or
ir*^(M,R)-ir"-*^(M,R).
Let us examine the restriction of ((•, •)) to the real vector space 7i^{M)
Jm
It is a real inner product of the finite dimensional vector space 7i^{M), and is
non-degenerate of course. By Lemma 1.4.5 (3) we have
Jm Jm Jm
So if we define a bilinear function
:W^(M) xW"-^(M)-^R" :(a,/3)H-. / a A/3,
Jm
then due to the isomorphism * we have
B(a,/3) = (-l)"("-'=)+("-*)((a,*;9)),
hence B is also non-degenerate. In topology there is a cup product U in the
cohomology group H*{M, R), more precisely, it is
U : H^{M, R) X H^'-^iM, R) -^ R, Vik,
which depends on neither the Riemannian structure nor the differential
manifold structure and can be proved to be the bilinear function B. So the cup
product is non-degenerate too, it is another version of the Duality Theorem.

110 SCHRODINGER AND HEAT OPERATORS
When n = 4ik, the bilinear map
U : H^^{M, R) X H^^{M, R) -^ R,
or
is symmetric because of
B{a,p)= [ aA/3= / /3 A a = B(/3,a), Va,/3 G ^^^^(M),
Jm Jm
thus we can define a signature for B and denote it by Sig(B) or Sig(M).
Because B is a non-degenerate bilinear function on the real vector space ?<^*^(M),
there exists a decomposition
n'\M) = Vi^V2,
such that
(1) for ueVuve F2, B{u, v) = 0,
(2) B is strictly positive on Vi, i.e. for 'y / 0 G Vi, B{v^v) > 0,
(3) B is strictly negative on V2, i.e. for 'y / 0 G F2, B{v,v) < 0.
Then
Sig(B) = dima Vi - dim^ F2.
It is easy to see that the above decomposition can be chosen as follows:
Vi = {a en^^{M)\^ a = a}
V2 = {a en^^{M)\^ a = -a}.
Definition 2.5.6 For an oriented Riemannian manifold of dim Ak
define
ni^{M) = {ae W2^(M)| * a = -a},
and call them the positive and negative harmonic spaces of M respectively.
Theorem 2.5.7 (Signature Theorem) Given an oriented
Riemannian manifold M of dim 4Ai, let the signature operator
D^=d-\-S: A+(M) -^ A_(M)

2 5 Applications of Hodge Theorem 111
be defined as that in Definition 1.4.18. Then
Sig(M) = Sig(B) = dime Ker(£>+) - dime Ker(£>;),
where
is the adjoint operator of Z)+, and Hodge Theorem guarantees that each of the
two terms on the right-hand side of the equality is finite.
Proof First we recall some notations in Definition 1.4.18.
A+(M) = {u e A*(M) 0 C\tu = a;},
A_(M) = {u e A*(M) 0 C\tu = -a;},
where
r = v^'^'+')+'% : A*(M) 0 C -. A^^-'{M) 0 C, V^ = 0,1, • • •, 4k,
It is easy to see that A*(M) 0 C + A^*^~*(M) 0 C is a r-invariant space,
consequently 7i^{M) 0 C + W^*^~*(M) 0 C is also a r-invariant space. So we
can define
{n'{M) 0 C + W^^-*(M) 0 C)o
n {a G n'{M) 0 C + n'^^-'iM) 0 C|ra = a}
(W*(M) 0 C + W^^-*(M) 0 C)i
= {a G W'(M) 0 C + W^^-*(M) 0 C|ra = -a}.
then
dime Ker(Z)+) - dime KeviD"^)
2k-l
= Y^ dimo{n'{M) 0 C + n^^-\M) 0 C)o
2Jk-l
- ^ dime(W*(M)0C + W^^-*(M)0C)i
5=0
+ao - ai.
where
ao = dime W^^(M) 0 C = dima V?^{M),
ai = dime ni^{M) 0 C rz dima'H?.^(M).

112 SCHRODINGER AND HEAT OPERATORS
By an easy check, fov s ^ 2k the homomorphisms
^0 : ^'(M) 0 C -^ {n'{M) 0 C + %^^-\M) 0 C)o : X h-. Ux + rX),
and
ifi :W{M)^C-^ (W*(M)0C+W^*^-'(M)0C)i : X y-^ ]-{X -rX)
are isomorphisms of the complex vector spaces, hence
dime Ker(Z)+) — dim© Ker(Z)^) — a^ — ai — Sig(B).
The theorem is proved.
Exercise 2.5.8 Let M be an oriented Riemannian manifold of dim
4Ai + 2, then
dime Ker(Z)+) - dime Kei{D\) = 0.
Exercise 2.5.9 Let M be a Riemannian manifold of dim n, and the de
Rham-Hodge operator
be defined as that in Definition 1.4.12, then
n
dima Ker(£>o) - dim^ Ker(£>S) = X!(-^)'^«(^)'
where
6, rrdima'H'CM).
By Corollary 2.5.4 6, is the well-known z-th Betti number of M, hence by
n
an Euler-Poincare formula in topology, V^( —l)*6j(M) is the Euler number
«=o
X{M),
2.6 Index Problem
In the last section two interesting numbers dima Ker(Z)+) — dim^ Ker(D!j_)
and dima Ker(jDo) — dim^ Ker(D2) emerged, and they turn out to be the
topological invariants Sig(M) and x(^)- This phenomenon shows that for a linear
operator £ : F -^ F, the number
dimKer(£)-dimKer(r)

2.6 Index Problem 113
may be very important, as soon as both dimKer(£) and dimKer(£*) are
finite. This is the case if, for example, dimF is finite or £ is a Fredholm
operator. The number
dimKer(£)-dimKer(r)
displays an interesting property of the operator. In 1960 a Russian
mathematician Walter found that for a new kind of operators, which is the so-called
elliptic operator, it can be provided with the above interesting number. To be
more precise, if E^ F are two complex vector bundles, and
C : T{E) -^ T{F)
is an elliptic operator, then it may be proved that
dimKer(£) < oo and dimKer(£*) < oo.
Note
dimKer(£*) = dimCoker(£) = dim{T{F)/C{T{E))},
then we can define
Ind(£) = dimKer(£) - dimCoker(£),
and call it the index of the operator £.
Gelfand Problem Can we express Ind(£) in terms of "topological
invariants" ?
In the case when C is D^ or Dq, Ind(£) is indeed the topological
invariant Sig(M) or x{^)' I^ is a natural question whether we can find a more
general invariant which contains Sig(M) and x(^) ^s special cases such that
Ind(£) can be expressed in terms of it? In fact this kind of generalized
topological invariants was available during the thirties in the 20-th century, it is the
characteristic number, which will be introduced in Chapter 4. Now we can
formulate the Gelfand problem in another way.
Gelfand Problem* Can we express Ind(£) in terms of "characteristic
numbers" of M, E, F and £?

CHAPTER 3
MP PARAMETRIX AND APPLICATIONS
When we built the theory of the fundamental solution in Chapter 2 §2, we
had assumed the existence of initial solutions. Now the first aim of this chapter
is to prove the existence by using MP Parametrix, which was introduced by
Minakshisundaram-Pleijel in 1949. And we will show that the MP Parametrix
is also important for finding properties of the fundamental solution, which leads
to the asymptotic expansion, and discussions for the index in a local fashion.
3.1 MP Parametrix
Let M be a Riemannian manifold of dimn, E he a vector bundle over M
with an inner product and an admissible connection Z), i.e. it satisfies
X{^,rj) = {Dxi,v) + {i,Dxv), VX G r(TM); ^,7? G T{E).
Occasionally D is denoted by y ^^ well. Let the Laplace-Beltrami operator
Ao-X^z>(^.,^.):r(^) —r(^)
«=1
be.defined by Definition 1.4.2, where {jE7i, • • •, jEJyi} is an orthonormal moving
frame of E, Again let F : T{E) —► r(jE7) be a self-adjoint ^(M)-linear map.
Define
A = -(Ao + F),
which is a self-adjoint Schrodinger operator. We consider the following formal
2
power series with a special transcendental multiplier e~ 4t and with parameters
[^'iViQ) G (0,oo) X M X M
_ ^ (X)
H^{t,q,v) = / " Vft^(')(p,g) : Ej, —> E,,
115

116 MP PARAMETRIX AND APPLICATIONS
where p = p{q,p) is the distance between q and p, Ep = 7r~-^(p) is the fiber of
E over p, and U^^\p,q) : Ep —> Eq is an R-linear map. Now we are going to
find conditions for the following equality
{— + Aq)Hoo{t,q,p)v = {},
In a small neighborhood of p, we choose a normal coordinate system {t/i, • • •,
yn} centering at p. If the coordinates of q is (t/i, • • •, t/n)? then
p{q,p) = yyl-^"' + yl'
Now denote
d d . ^ , .. . ^ ^ d
"'^^W'S^^' ^ = ^^*(^'^)' ^^"^^'%'
then we have
Pi — I 2 "^
.=0 J
_£_ oo 2
e 4t ^-^ p^
Let
_ £_
e 4t
it is not hard to check that
AoH^{t,q,p)v = (Ao$) • {J2*'U^'Hp,Q)v)
«=0
n (X)
a=l «=0
(X)
where {jE7i, • • •, jE7n} is the compatible frame, i.e. it is parallel along geodesies
passing through p and satisfies
E,ip) = ±-y

S 1 MP Parametrtx 117
Now note that
E^^ = ^'{p)EcP,
a
= Y,{Ec{^'ip) ■ E^p) - $'(p). {S7E^E^)P]
a
*"(/.) = *W(^-i).
and Proposition 1.5.12 yields the following equalities
P
J2(^aPr = 1,
a
n — 1 1 -, rzz
Aop= + -.dlogVG,
p p
^ Y^t^ 2V 2t
Therefore the second term on the right-hand side of the equality of Ao^oo(^, ?,
p)v should be
n (X)
n (X)
a=l ^ «=0
(X)
+ dlog\/G^
«=o
consequently
A zr /^ \ ^V^ r /^^ ^ n-H-dlog\/G
4^2 2t 2t
-^^, +Ao }t'U^'\p,q)v.
T p

118 MP PARAMETRIX AND APPLICATIONS
Thus combining with the computation about ^Hoo, the above equaUty yields
(— +A)H^it,q,p)v^i--{Ao + F))H^{t,q,p)v
dG. 1,
4G^-V
= ^{iVi+^)-^U('\p,9)v
It imphes that the equation
i-^ + A)H^{t,q,p)v = 0
is equivalent to
dG
(Vi + i + 77^)U^'Hp, Q)v = (Ao + F)U^'-'\p, q)v
4G
(Vi = 0,l,2,---),
where we assume U^ ^Hp^Q) = ^- The solvability of the equation is established
in the following
Lemma 3.1.1 For fixed p ^ M and v € Ep, there exists a unique
solution w('H-) e r(£;|o(p)), Vz = 0,1,2 • • •, to
(Vi + ^ + §)W^'^(9) = (Ao + F)U^'-'\q)
w(-i)(-) = 0
where -E'|o(p) is a restriction of E over a small neighborhood around p.
Proof Recall d = pf-, then owing to the equalities

3.2 Existence of Imttal Solutions 119
the equation in the lemma reduces to
f vMp'G'^^'^) = P'-'G4(Ao + F)U^'-^^;
W(-i)(-) = 0
[ U^'^Xp) = V.
Hence it is easy to see that the above equation has the unique C°°-solutions
W(°)(g) = G-i(9).//^,
P G^{q) Jo P f P
G^{q) Jo
where i > 1, //^ : Ep -^ Eq is a parallel translation along a geodesic joining p
to q.
Definition 3.1.2 Denote the solution U^'\q) by U^'\p,q)v, which
depends linearly on v. We get
Define
_ £_ (X)
e 4t
(V47rt)« ^
to be an MP Parametrix for the heat operator (^ + A).
Remark 3.1.3 The notion of the parametrix had been introduced in
analysis for a long time, but it was Minakshisundaram-Pleijel, who first
introduced this one in 1949 for the heat operators corresponding to the Schrodinger
operators in geometry, so we call it MP Parametrix.
3.2 Existence of Initial Solutions
We take an iV-truncated MP Parametrix
_el N
(v47r<)" ^
and choose a smooth function
<f>:M xM^R,

120 MP PARAMETRIX AND APPLICATIONS
such that (J)\d{^) — 1? <^|M-£>(e) = 0? where 6 is a sufficient small positive
number, D{€) is an 6-neighborhood of the diagonal in M x M, i.e.
D{e) = {{q,p) e M X M\p{q,p) <e].
Then
<t>{q,p)HM{t,q,p) :Ep-^ Eg
can be defined for all (q^p) £ M x M and t > 0.
Proposition 3.2.1 Let Go{t,q,p) — (l>{q,p)H]sf{t,q,p), then Go{t,q,p)
is a k-th initial solution of the heat operator (^ + A), where Ai = [y — |] is
the greatest integer less than or equal to (y — ^).
Proof We need to check the conditions (i)-(iv) in Definition 2.2.1. (i)
is too trivial to prove. The computations in §1 show
(^ -{Ao+ F))HN{t,q,p)v
e 4t
-t^-t(Ao + F)C^W(p,g)^,
(-v/47r)"
and the expression of U^^\p,q) in the proof of Lemma 3.1.1 shows that
2
U^^\p,q) is C^ with respect to the variables p, q. Note that {t^~^e~ ^t)
belongs to C^^" 4']([0, oo) x M x M), hence
(^ - {Ao + F)){cl^{q,p)Hj,{i,q,p)) G ^^-^^([O, oo) x M x M),
thus (ii) is true. In order to prove (iii), observe that
/ Go{t,q,z)f{z,a)dz
JM
«=o ^^ yv'^'J^t)
where
P = p{q^z)^
e 4t
^ = (^(g,2;)trW(;^,g)/(;^,a)c^;^,
M,C-) = {zeM\piq,z)<'-}.

3.2 Existence of Imttal Solutions 121
On M — Mq{^) the following limit converges uniformly
e 4t
lim ^ , ^ = 0,
consequently,
lim / <e = 0.
im / I
Choose a normal coordinate system centering at g, denote the coordinates of z
by {zi,-- ,Zn). Then
i= '
=^(j){q,z)U^'\z,q)f{z,a)Jdet({^^ dzi^-dzn,
and
lim / ^ = ^(g,0)£^W(o,g)/(0,a)Jdet((^,^))|,=o
= C^^'H9,9)/(9>«)-
It implies that
lim / Go{t,q,z)f{z,a)dz = limf2f{ f ^ + / A
TV
= ^(limf)£^W(g,g)/(g,a)
«=0
After checking on the uniformity of convergence, the conclusion in (iii) is seen
to be true.
The proof of (iv) is rather long, now we prove a lemma first.
Lemma 3.2.2 Given a general tensor
g{i,T,p,z,a) e C'({0 <T<i<oo}xMxMxN),
with g{t^T,p,z,a) G ^p, and the support of g' in a small neighborhood of the
diagonal in M x M for any fixed (t^r^a), then for any vector field X, there
exists a general tensor field
h{t,T,p,z,a) G C'-^({0 <T<t<oo}xMxMxN),

122 MP PARAMETRIX AND APPLICATIONS
with some other properties as those mentioned for g such that
Vx(p) / Q{t-r,p,z)g{i,T,p,z,a)dz= Q{i - T,p,z)h{i,T,p,z,a)dz,
Jm Jm
where
Q{t,p,z) =
e 4t
(\/47rt)"*
Proof Let us look at the equality
Vx(p) / Q{t-r,p,z)g{i,T,p,z,a)dz
Jm
= I Q[{Q)-^XQ]{i-T,p,z)g{t,T,p,z,a)dz
Jm
+ Q{-t- r, p, z) Vx(p) 9{-t, T,p, z, a)dz,
Jm
in which [[Q)~^XQ]{i — r,p^z) has a singularity at r = t. It prevents us from
assuming
h{t, r,p, z, a) = [{Q)-^XQ]{i - r,p, z)g{t, r,p, z, a) + Vx(p)§(^, r,p, z, a)
to get a proof. However this singularity delivered by [{Q)~^XQ]{t — r^p^ z) may
be contained in a divergence term similar to div(/y) in the latter Theorem
3.2.3, hence can be discarded as far as the integral is concerned.
Fixed a point p G M, for any z near to p, there exists a unique geodesic
joining p to z. Let
//;:TpM-.T,M
be the parallel translation from p io z along the geodesic. For X{p) G TpM,
let Y{z) — //^-X'(p), and p{p,z) be the distance between p and z. Denote the
unit tangent vector at q on the geodesic joining p to 2; by ^ P ^. | , then it is
trivial that
dp{p,z)\q dp{z,p)
By Gauss lemma we have
Y{z)p{.,p)={Y{z),
X{p)p{.,z) = (x{p),
^««L)'
dp{z,p)

consequently,
Y{z)pip,-) = (Y{z),
= (lliX{p)Jh
3.2 Existence of Imttal Solutions 123
z)l/
d
dp{p, z)
^dp{p,z)
,)
X{p)r
dp{p, z)
dp{z,p)
= -X{p)p{zr)'
Sometimes the above formula is denoted by
Y{z)p{p,z) = -X{p)p{p,z).
Now consider the integral
/= / Q[{Q)-^XQ]{t-T,p,z)g{t,T,p,z,a)dz
JM
- \ {X{p)Q(i-T,p,z)\g(i,T,p,z,a)dz
JM
= - [y{z)Q{-t - r,p,z)]g{t,T,p,z,a)dz.
JM
For fixed p, choose a base {ei(p), • • •, em{p)} of Ep, Recall g G Ep, so we can
write down
m
g{t,T,p,z,a) = Y^gj{t,T,p,z,a)ej{p),
where gj{t, r^p, z, a) are functions defined on {0 < r < t < oo} x M x M x N.
Thus by the Divergence Theorem (Theorem 3.2.3 below) we have
m
/ = -V / [Y{z)Q{i - T,p,z)]gj{i,T,p,z,a)dz ' ej{p)
j^^Jm
m .
n-V / Y{z)[Q{i - T,p,z)gj{i,T,p,z,a)]dz ' ej{p)
j^^Jm
m ,
+ y^ / 0(^-'7-,P,^)^(^)9^j(^,r,p,2;,a)d2;-ej(p)
3 = 1^^
m
- X^ / <3(^-'^'^'^)[(^ivy)-§j+y9^j](t,r,p,2;,a)d2;-ej(p).
.-1 JM

124 MP PARAMETRIX AND APPLICATIONS
Because
m
J2[{divY)^g,+Y{z)g,]e,{p)
does not depend on the choice of {ei(p), • • •, e„i(p)}, we may denote it by
(divF) -g + Yg,
Let
which obviously belongs to
C'-^({0 < r < t < 00} X M X M X i\r),
and satisfies the equality of the lemma.
Now let us return to the check of Definition 2.2.1(iv). We try to prove
W{i,p,a) e C^i\[0,oo) xMxN),
where
Let
(}<l<k,
W{t,p,a) = lim W^(c,t,p,a),
W{€,t,p,a)= / dr Go{t-T,p,z)g{T,z,a)dz,
Jo JM
g{T,z,a) e C'([0,oo) x M x N).
N
g{i, r, p, z, a) = (^(p, z) Y^{i - r)*U^'\z,p)g{r, z, a),
«=o
then after recalling the definition of Go we have
W{e,i,p,a)- J dr I Q{t - T,p,z)g{t,T,p,z,a)dz,
Jo Jm
Note that
g{t,T,p,z,a) e C'({0 <T<t < 00} xM X M X N),
so Lemma 3.2.2 implies that for any set of vector fields {Xi, • • •, X^} with
s < /, there exists a general tensor
h{t,T,p,z,a) gC'""'({0< T<i<oo}xM xM xN),

3.2 Existence of Imttal Solutions 125
such that
Vxr-Vx.W{€,t,p,a)= / dr Q{t - T,p,z)h{t,T,p,z,a)dz,
Jo Jm
By using a discussion similar to that in the proof of (iii), we have
Vx. • • • Vx. W{t,p,a) e C^([0,oo) x M x N), \/s < L
Note
W{€,t,p,a) = -g^ dr I Go{i-T,p,z)g{T,z,a)dz
= / Go{€,p,z)g{t-€,z,a)dz
Jm
+ dr j [-Q^Go{t - r, p, z)]g{T, z, a)dz
Jo Jm
where
11= Go{€,p,z)g{i-€,z,a)dz,
Jm
III = J drj [{^-Ao-F)Go{t-T,p,z)]g{T,z,a)dz,
IV = dr FGo{t - T,p,z)g{T,z,a)dz,
Jo Jm
F = Ao / dr Go{t - r,p,z)g{r,z,a)dz
Jo Jm
= AoVr(6,t,p,a).
Then we get
g^W{t,p, a) = AoWit,p, z) + g{t, z, a)
'^J^I [(^-^o-J')Go(<-r,p,0)]g(r,0,a)d^
+ dr FGo{t-T,p,z)g{T,z,a)dz.
Jo Jm

126 MP PARAMETRIX AND APPLICATIONS
By the reason in the check of (ii) we have
Li = g{i,z,a)+ I dr I [{—-/^o - F)Go{t - r,v,z)\g{r,z,a)dz eCK
Jo JM ^^
Let
L2= dr FGo{t-T,p,z)g{T,z,a)dz,
Jo Jm
which is a new potential integral similar to
W{t,p,z)= dr Go{t-T,p,z)g{T,z,a)dz,
Jo Jm
i.e.
L2= dr Q{t-T,p,z)Fg{t,T,p,z,a)dz.
Jo Jm
By Lemma 3.2.2 and
—W{t,p,a) = AoW{t,p,z) + Li + L2,
we get
Vxr--Vx. -g^W{t,p,a) =^x,---Vx.{^oW{t,p,z) + Li + L2)
GC°([0,oo)xMx AT) \/s<l-2.
Similarly we can get
Vxr--Vx., {-^y^Wit,p,a) e C«([0,oo) x M x N),
where si + 2s2 < L It means
W{i,p,a) e d2]([0,oo) xM X N).
The left of (iv) is easy to check, so the proof of Proposition 3.2.1 is complete.
Theorem 3.2.3 (Divergence Theorem) Suppose M is a closed
Riemannian manifold, Y" is a vector field on M, and / : M -^ R is a differentiable
function, then
/ {Yf)dv = - f (divY). fdv,
Jm Jm
where dv is the Riemannian measure and divY is the divergence of Y.

3.3 Asymptotic Expansion for Heat Kernel 127
Proof Recall the definition of the divergence in Exercise 1.4.10. Choose
an orthonormal moving frame {jE7i, • • •, jEJyi}, then
«=i
where y is the Levi-Civita connection. By this definition we can easily get
n
diy{fY)=J2{VEXfy),E.)
« = 1
n n
= Y^{Y,E,){E.f) + J2f{VEY,E,)
«=1 «=1
= Yf + f' divy.
On the other hand, there is another definition for div, see [36] for details. An
exercise on p. 199 in [36] shows an important equality for div, which is
/ div(/y). d'y = 0.
Jm
Therefore the above equality and the preceding one give a proof of the theorem
immediately.
3.3 Asymptotic Expansion for Heat Kernel
The second application of MP Parametrix is the following theorem.
Theorem 3.3.1 (Asymptotic Expansion Theorem) Let M be a
Riemannian manifold of dim n, E he a. vector bundle over M with an inner
product and an admissible connection £), and let A = — (Aq + F) he the
Schrodinger operator, G{t,q,p) he the heat kernel (fundamental solution) for
the heat operator ^ + A, and
_£_
e 4t
he the MP Parametrix. Then as t -^ 0,G(t,p,p) has an asymptotic expansion
It is to say that for any N > 0, the following holds
G{t,p,p) --^f2t'-fu('\p,p) = o(<^-t),

128 MP PARAMETRIX AND APPLICATIONS
where o(t^~^) means a quantity, say a, with the property
^. a ^
lim -Tj—ir = 0.
Proof It is sufficient to prove the theorem for any large N. Let Go{t, g, p)
(j){q,p)HN{t,q^p) just as we did in Proposition 3.2.1. Then the conclusion of
the theorem is equivalent to
G(i,p,p)-Go(<,p,p) = o(i^-t).
Due to Proposition 3.2.1 and Theorem 2.2.6 the result G of the Levi iteration
starting from Go is just the fundamental solution. Hence the difference G — Gq
may come from the Levi iteration. Suppose T is a positive number. An equality
in the proof of Proposition 3.2.1
{^-iAo + F))H:,it,q,p)v
_£_
e 4t
t^-t(Ao + F)C^W(p,g>,
shows that there exists a constant A such that for any t G (0, T] it yields
\Ko{t,q,p)\ = \i^ + A)Goit,q,p)\
< constant • l(^ + A)frjv(<,g,p)|
Imitating the proof in Lemma 2.2.5 we also have
\Kx{tA,p)\< [ dr f \Ko{t-T,q,z)Ko{T,z,p)\dz
Jo Jm
< I [A\t - r)^-^(r)^-^ • vo\{M)]dT
Jo
Jo
- -*■" JV-f + l'

3.3 Asymptotic Expansion for Heat Kernel 129
where B = A^ rpN-^ , vol(M). Further,
\K2{t.q.p)\ < / ^^ / \Ko{t-T,q,z)Ki{T,z,p)\dz
Jo Jm
r* r. r^-f+1
2^ ^^ (iV-f+l)(iV-f + 2)-
By induction we finally have
\Km{t,q,p)\<AB^
(i\r-f + i)(i\r-f + 2)...(i\r-f + m)-
Therefore
(X)
m=0
So the Levi iteration gives an equality
\G{t,q,p) -Go{t.q.p)\
< I / dr Go{t-T,q,z)K{T,z,p)dzl
Jo Jm
IM
hence induces
2
|G(t,g,p)-Go(^,g,p)I < constant • / dr / --^L====—r^-t^^;
7o Jm (\/47r(t - r))"
< constant -t^" 2"+^.
The proof is complete.
Remark 3.3.2 For p-^ q^ because of a trivial equality
|Go(^,g,p)| < constant •t*', VAi > 0,
the inequality in the proof of the Asymptotic Expansion Theorem
|G(t,g,p)-Go(t,g,p)| < constant-t^-2+1
implies
\G(i,q,p)\ < constant -t*^, \/k > 0.

130 MP PARAMETRIX AND APPLICATIONS
Therefore
\G{t,q,p) - Go{t,q,p)\ < constant ^t^, Vfe > 0.
This result is trivial enough, but it could be strengthened in [2], in which we
proved that for any 6 > 0, there exists a constant C{M, N, T, e) such that
|G(i,g,p)-C?o0,9,p)| < C(M,JV,T,c)<^-t+V^1^.
So this inequality may be viewed as a generalization of the Asymptotic
Expansion Theorem.
Remark 3.3.3 It is worthy to note that the solution operator Tt and
the fundamental solution G{t,q,p) are defined globally. It means that if U is
an open set in Af, neither Tt\u nor G{i^q^p)\q^p^u can be determined by the
Riemannian structure in U. On the contrary, by the definition MP Parametrix
can be determined by the local data. In this sense the Asymptotic Expansion
Theorem may be viewed as a bridge between the global research and the local
research.
3.4 Local Index for Elliptic Operators
In Chapter 2 §6 we introduced the notion of index for a special kind of
elliptic operators only by using the Hodge theorem 2.4.3(i). If we use the whole
Hodge Theorem 2.4.3 and Theorem 2.4.2 we can get some formal expressions
of the index. Suppose
D : r{E) -^ r{F)
is an operator, such that D* D and DD* are Schrodinger operators, where
D* : T{F) -^ T{E)
is the adjoint operator of D, Let us consider the operators
D*D : T{E) -^ T{E),
and
DD* : T{E) -^ T{E).
They are self-adjoint, hence have non-negative real eigenvalues. Let
T^{E) = {<t>eT{E)\D*D<f, = ^<j>},
and
T^{F) = {<t>eT{F)\DD*<j> = ^<j>},
then we have
Lemma 3.4.1 For the above D,

3.4 Local Index for Elliptic Operators 131
(1)
ToiE) = ker{D : T{E) ^ T{F)},
ToiF) = kerjD* : r{F) -> T{E)}.
(2) For any /x > 0,
dimr^(jE;) = dimT^{E) < oo.
Proof Lemma 2.5.1 implies (1). The map D : T{E) -^ T{F) can be
restricted as D : T^{E) —> r^{F), which has an inverse map
-D* : T^{F) -^ T^{E), V;x > 0.
Thus the proof is complete.
By using Lemma 3.4.1 we can get a new expression of Ind(jD) from the
following computations.
Ind(D) = dimkerD - dimkerD*
= dimro(£^) - dimro(F)
« = !
(X)
- E /(/^0(dim r^, (^) - dim T^, (F))
(X) (X)
1=1 «=1
= tr/(D*D)-tr/(DD*),
where the sequence {//i = 0 < //2 < A*3 < • • • -^ oo} contains all the eigenvalues
of D*D and DD*, and / is a difFerentiable function defined on [0, oo) satisfying
/(O) = 1 and some kind of conditions, which enables the convergence in the
above equalities.
Remark 3.4.2 Now we get the expression for Ind(jD), which depends
on the choice of /. In the above computations, especially the last equality
sign is full of trouble, e.g. f{D*D) must be checked to be a trace operator.
However the above whole computation is true when T{E) and T{F) are of finite
dimension. In the infinite dimensional case,
dimr(jE7) = dimr(F) = oo.

132 MP PARAMETRIX AND APPLICATIONS
so as far as the equality
CX)
^/(//,)(dimr^,(^) -dimr^,(i?^)) = tr/(D*D) - tr/(DD*)
«=i
is concerned, three problems need to be solved.
(i) How to define the operator f{D*D)?
(ii) How to define the trace of f{D*D)? What kind of / ensures that the trace
of/(D*D) exists?
(iii) How to prove the equality or
Ind(D) = tr/(D*D) - tr/(DD*)?
In what follows we will show that for a concrete function f{x) = e~*^, where
t > 0, the above three problems can be solved. First we easily see that /(O) = 1.
And a moment's thought shows that there are three ways to define /(D*D).
The first way to define it is by
oo .
because of
m=:0
^ 1
m=0
The second way uses the expansion theorem (Theorem 2.4.2), i.e.
to define f{D*D) by letting
/(D*D)|r,,(^) = fifi,) : r^.(^) - r^.(^).
The third way defines f{D*D) to be Tf in Lemma 2.4.1 in the case where A
is assumed to be D*D. After some careful thinking we find that the first two
definitions cannot work well, because they had not mentioned what kind of
function spaces the operator f{D*D) acts on. In fact to understand the image
space of an operator needs a regularity discussion, which is usually a central
topic in the theory of PDE. Without the regularity discussion the first two

3.4 Local Index for Elhpttc Operators 133
definitions cannot meet the need for later considerations. So we choose the
third way by defining the operator f{D*D) = e~*^.
Definition 3.4.3 For a Schrodinger operator A, and t > 0, let
e"*^ : T{E) -^ T{E)
be defined by
(e-*^^)(g) = / Git,q,p)<l>ip)dp,
Jm
where G{t,q,p) is the fundamental solution of ^ + A.
Let 0 < Ai < A2 < • • • -^ oo be all the eigenvalues of A, and {<^i, <^2, * • •}
be the corresponding eigenfunction set. Then the trace of the operator e~*^
intuitively is defined by
(X)
tre-*^=^((e-'^^.,^.)).
«=1
(X)
Plainly it is ^e"'^** or ^e"*^ dimr^(jE7). So the above definition of tr is
«=i fi
well-defined if and only if
(X)
J2e-^'*<oo.
t = l
In order to check the above condition we introduce
Definition 3.4.4 Suppose M is a Riemannian manifold, jE7 is a
vector bundle of rank N, and A : T{E) -^ r(jE7) is a non-negative self-adjoint
Schrodinger operator. For any q^p £ M^ choose orthonormal basis {ei(p), • • •,
e;v(p)} and {/i(^), • • •, /jv(?)} on Ep and Eq respectively, let
N
\GtUv)= E ^G(t,q,v)ea[v).fM?^
After checking that the definition of \Gt\{q^p) does not depend on the choice
of {ei(p),--•,ejv(p)} and {fi{q)r " •, fN{q)}, we may call the function |G*|,
defined on M x M, a norm density for e~*^.
Proposition 3.4.5 The notations are defined as above, then
(0 / / \Gt\{q.p)dqdp < 00, \/t> 0.

134 MP PARAMETRIX AND APPLICATIONS
(ii)f]e-^^*< If \GL\{q,p)dqdp <oo, Vt > 0.
«=l -^ -^
Proof For t > 0, G(i, q^p) is continuous and hence satisfies (i). For any
V G r(jE7), Theorem 2.4.4 reads
(X)
«=i
in T{E), hence Parseval equality yields
. oo
The above equality implies
N
Y) I \G{i.q.v)ea{v)?dq
a=:l ^^
N oo
a=l«=:l
oo N
« = 1 a=l
oo
Thus for any m, we have
m m M
< / 5^e-2^-'(^.(p),<^.(p))dp
= I 'ip I ElG(*,g,pK(p)|'dg
./Af ./Af „_i
= dp E {G{t,q,p)eaip),U{q)fdq
= / / |Gt|(9,p)dgdp< oo.
The proof is complete.

3.4 Local Index for Elliptic Operators 135
The above discussions show that the definition of tr is well-defined. So the
problem (ii) is solved in the case of e"*"^*"^, and the problem (iii) is solved as
well. Thus we get
liid{D) = tre-*^*^ - tre-*^^*, \/t > 0.
Proposition 3.4.6 For any t > 0,
tr(e-*^) = / tvG{t,p,p)dp.
Jm
Proof By using Theorem 2.4.4, we have the following computations
N
tiG{t,p,p) = ^{G{t,p,p)ea{p),ea{p))
a = l
N / oo \
a = l \«=1 /
N oo
a=l«=l
(X)
« = 1
and
/ tiG{t,p,p)dp = f y^e-'^^{4>,{p),4>,{p)Ydp
JM Jm ,_i
(X)
= ^e-*^.=tr(e-'^).
« = 1
The proof is complete.
Remark 3.4.7 Recall Tr(t) = tre"*^ = ^e"*^% then Tr(t) is a sum
of a series with positive terms, which converges uniformly as t -^ oo. Therefore
(X)
lim Tr(t) = X) A™ ^~*^* = dimro(jK).
On the other hand, as ^ -^ 0, the equality
G{t,p,p)= —=^ + o(—)
V 47rt 12

136 MP PARAMETRIX AND APPLICATIONS
and Proposition 3.4.6 imply
lim Ti(t) = oo.
As far as Ind(jD) is concerned, we have
Ind(D) = tre-*^*^ - tre"*^^*
= / tiG+{t,p,p)dp- I trG>(t,p,p)dp, Vt>0.
Jm Jm
By Proposition 3.4.6, where G+(t,p,p) and G_(t,p,p) are the fundamental
solutions of ^ + i)*i) and ^ + DD* respectively. At this stage we would like
to find out some special values of t, such that the above equality can be used
to compute Ind(Z>). A moment's thought shows that there is no such kind oft
except t = oo or t = 0. Remark 3.4.7 shows that the case of t = oo is nothing
but the definition of Ind(D). So only the case of t — 0 is left.
lini / (trG+(t,p,p)-trG_(t,p,p))dp
may be good and can provide a proper way for computing Ind(jD), in which
the integrand
(trG+(t,p,p)-trG_(t,p,p))
needs to be examined carefully as t ^ 0.
As t -^ 0 Theorem 3.3.1 reads
oo .
7^0 (v47rt)«
and hence implies
Ind(D) = lim / (trG+(t,p,p) - trG_(t,p,p))dp
oo . .
«=0
From the equalities it follows that
Ind(D), iii= — = integer
0, if Vz < |.

3.4 Local Index for Elliptic Operators 137
Definition 3.4.8 Suppose M,E,F,D : r{E) -^ T{F) are defined as
before, let the local index be defined by
f -^ (tvUi^\p,p) - tvU[^\p,p)) , if n = even,
(Loc.ind(D)) = I (V4x)« V ^ ^
I 0, otherwise.
From the definition it is easy to see
Proposition 3.4.9 Prom the definition of the local index, it follows
Ind(D) = / {LocAiid{D)){p)dp,
Jm
Remark 3.4.10 It is worthy to pay some attention to the above
equality. On the right-hand side the function Loc.ind(jD) is defined on M, its value
at p depends only on the local data around p, while Ind(£)) is a global notion.
This is the reason that we call the function Loc.ind(jD) the local index of D,
Comparing Proposition 3.4.9 with
Ind(D) = liin / (trG+(t,p,p) - tvG-{t,p,p))dp,
we may easily guess an equality
(Loc.ind(D))(p) = lim(trG+(t,p,p) - trG_(t,p,p)).
In fact it is true, but it is by no means trivial, because (trG+(t,p,p)—trG_(t,p,p)
is a global notion different from Loc.ind(jD). And the asymptotic theorem only
tells that the above equality is equivalent to the equalities
^-^ (tiU^\p,p) - tvU^:\p,p)) =0, Vi < |,
or the limit
lim(trG+(t,p,p) - trG_(t,p,p))
exists. Plainly these equivalent conditions are not trivial.
Mckean-Singer problem 3.4.11 For what kind of first order elliptic
operators D : T{E) -^ r(F) the following (i) and (ii) are true?
(i) For i < f
j;^{t.U^\p,p)-tvU^:\p,p))=0.

138 MP PARAMETRIX AND APPLICATIONS
(ii) If n = even, then
is a characteristic density corresponding to the characteristic number in
the Atiyah-Singer index formula for D.
In the problem the density means a function on M, the characteristic density
means a function, whose integral in the Lebesgue sense is a characteristic
number. The characteristic density is different from the characteristic form in the
notions, so attention needs to be paid later. The above problem was proposed
by Mckean-Singer in 1967 (see [23]) after they checked several simple cases.
In 1971 Patodi published two papers [29], [30] to get positive answers to the
problem for the de Rham-Hodge operator and the Riemann-Roch operator.
Researchers said that Patodi proved Mckean-Singer conjecture or local index
theorems for these two operators. In 1974 Gilkey [17] and Atiyah-Bott-Patodi
[3] published a proof for the signature operator. The proof contains a very
elegant application of the invariant theory, but the start point about the
existence of the invariant polynomial needs to be checked by another method.
A complete proof for the signature operator can be found in [39]. Since 1980
many researchers published proofs for the local index theorem for the Dirac
operator, the information could be found in [16]. Among the proofs for the
Dirac operator Yu's [38] is a proof which had not been influenced on by
mathematical physics, although in this book we also introduce a beautiful idea of
mathematical physics into mathematics, it results in a proof by using a Chern
root algorithm in Chapter 4 §3.

CHAPTER 4
CHERN-WEIL THEORY
4.1 Characteristic Forms and Characteristic Classes
In the thirties of the twentieth century a beautiful chapter in topology
emerged, it is the characteristic class theory, which supplied with a series of
new invariants for bundles over a space M, the invariants are elements in the
cohomology group of M, Afterwards Chern and Weil reconstructed the
characteristic classes by using the curvatures in differential geometry, it is to say,
the representative closed forms of the characteristic classes could be a kind of
curvatures. This chapter is devoted to the Chern-Weil theory.
Let M be a manifold, E -^ M he a real vector bundle of rank N with a
connection
D : r{TM) X r{E) -^ r{E) : (X, W) ^ DxW,
As we did in Chapter 1 §7 we define the curvature (operator)
R{X, Y) : T{E) -^ T{E), MX, Y G T{TM)
by
R{X, Y) = DxDy- DyDx - D^x^y]-
It is easy to see that R{X,Y)W is ^(M)-linear with respect to X, Y and
W, and R{X,Y) = -R{Y,X), The first fact enables us to consider R{X,Y)
locally. It means that for any open set U C M, if X,Y,X,Y G r(TM), and
W,W e T{E) with
X\u = X\u, Y\u = Y\u, W\u = W\u,
then
R{X,Y)W\u = R{X,Y)W\u^
So if choose a local basis {VTi, • • •, Wjv} of E on U, then by the following
equality
139

140 CHERN-WEIL THEORY
RiX,Y){Wi,---,WN) = {Wi,---,WN)n{X,Y),
define an N x N matrix Ct{X^Y), Thus Ct{ , ) is a matrix with entries being
2-forms on U, Similarly the connection D is expressed by a;( ) with respect to
{VTi, • • •, Wjv}? which satisfies
Then it is easy to check that
Ct = du -\- u Au.
If we choose another basis {Wi, • • •, Wjv}? then there exists a non-degenerate
matrix g with entries being functions such that
And we can have
u = g-^ 'U -g + g'^ -dg,
n = g-^ -n-g.
We also have the Bianchi (II)
dft = d{du -\- u Au) = (du) Au — uAdij^CtAu—uACt.
By a discussion similar to that in Chapter 1 §7 we have
i.3.k,l
~ 4 XI (^*J ^ ^ki){^i3 ' Sh - Eki' 'Bij)
^,J.k,l
\ J2i^^J^^kl){{0^J-0J,)'{eU-0lk)
-{Oki - Oik) • {Oij - Oji)}
- 22 (^*-? ^ ^ki){SjkOii - SikOji - SjiOik + SiiOjk
—^hOkj + ^kiOij + SijOkt — SkjOu}
4
hJ,k,l
4
hJ.k.l
^{^ik Aukj)0^j - ^{(^tk A ^kj)Ozj
^,J,k tj,k
Ct Au — u ACt^

4^1 Characteristic Forms and Characteristic Classes 141
hence the Bianchi (II) (see Theorem 1.7.17) turns out to be
Let A be a real {N x iV)-matrix, i.e. A G gl(iV, R). The characteristic
polynomial of (—A) is
det(A/ + A) = X^ + ri{A)X^-^ + ... + r^(A),
where / is the identity matrix. It is easy to see that for any non-degenerate
matrix g G GL{N, R) C gl(iV, R), we have
r,{g-''A'g) = r,{A), Vi = 1, • • • ,i\r.
Replacing A by the matrix Ct^ we can also define r,(l^). Due to Q = g~^ -Q - g
we have
r,{n) = r,{g-'^n'g) = r,{n), Vi = 1, • • • ,i\r,
which shows that r,(l^) does not depend on the choice of the local bases, thus
gives a global 2i-form 7«(i^) on M,
Fundamental Lemma 4.1.1 Let jE7 be a real vector bundle over M
of rank N with a connection D, and let the 2i-form n(D) be defined such that
r,(D) = r^{n), then
(i) 7*«(jO) is a closed form, i.e. under any local bcisis rj(l^) satisfies
dr^{n) = 0.
(ii) For any two connections D and J9, there exists a (2z — l)-form w G
A2«-i(M), such that
r,(D)-r,(5) =^dw,
(iii) For any odd number z, there exists a tz7 G A^*~-^(M), such that
r^{D) -dw.
Proof In order to prove (i) let us introduce a notion of
Polarization. For an m-th order homogeneous polynomial <^(aJi, • • •, ajy^), denote X =
(aji, • • •, aj„) G F, where F is a vector space of dim n. Thus cj) may be viewed
as a function on V. Define
^:Fx ...X F-^C,
' r '
m times

142 CHERN-WEIL THEORY
by
1 3"*
and assume the expression of (f) to be written in the following way
where
Xj = {xji,'",Xjn) eV, Vj = 1, •••,m.
Then call cj) the polarization of cj). It is easy to see that <j) satisfies
(1) <j){Xi^ • • •, Xfn) is linear with respect to Xi, • • •, X^.
(2) ^(X^_^^^ = <^(X).
m times
If we take gl(iV, R) to be the above F, we also have
(3) If <I>{A) is an m-th order homogeneous polynomial with variables being the
entries of A G gl(iV, R) and satisfies
<f,ig-'.A-g) = <t>{A), \/g e GL{N,K),
then
^{g-^-A-g,---,9-^-A-g) = ?(£-_^ = <I>{A).
m times m times
For B G gl(iV, R), then e*^ G GL{N, R). By using the above (3) it follows
that
0=^|e=oWe-*-«Ae'^)-<^(A))
m ,
J=l ^^ V '
(j-th place)
= ;^^(^,...,^ [A,B] ,A,---,A),
(j-th place)

4.1 Characteristic Forms and Characteristic Classes 143
where [A, B] = AB - BA,
Now we are going to prove (i) in Lemma 4.1.1. Let (j) — r^^ then by the above
equality and Bianchi (II), (i) follows from the following formal computations
= ^^(1^,...,!^, do. ,1^,...,!^)
•? (j~th place)
(j-th place)
The above computations are said to be "formal", because H^u are not
ordinary matrices, their entries are differential forms. However after a careful
check on the formal computations, they are correct indeed. Moreover we can
generalize the above last equality sign. If a and j3 are matrices with entries
being Ai-forms and Worms respectively, we denote Ai = |a|, / = |/3| and define
[a,/3] = a A/3 - (-1)1^11^1/3 A a,
then we can prove an equality
J=l
Now we are going to prove (ii). The connections D and D are expressed in
terms of a; and u with respect to a same local basis {VTi, • • •, Wn] respectively.
I.e.
i^(^ir-,^iv) = (^ir-,^iv)-a;,
5(l^ir-,^iv) = (^ir-,^iv)-S.
Let
then
After an easy check we have
d
a — u — (jj^
Uf = u -^ ta^
d
at

144 CHERN-WEIL THEORY
and
Note
<l>{n) - <j>{n) = cj^iQi) - <j>{no) = J ^^<l>{nt)dt,
and
(«-th place)
« s ^ /
(«-th place)
* («-th place)
* •^'^* 0-th place) («-th place)
* •?>* («-th place) (j-th place)
(«-th pletce)
* («-th pletce)
* •^'^* 0-th place) («-th place)
* ^^' («-th place) 0-th place)
(«-th place)
* (t-th place)
SO we get
ct>{?l) - <t>{il) = d f {J2 k^t, • • •, a ,..., nt)}dt.
Jo ^'^'^""'^
* («-th place)

4'1 Characteristic Forms and Characteristic Classes 145
In order to prove (ii) it is sufficient to check that such
Jo ^"^"^"'^
* («-th place)
can be patched together to give a differential form w on the whole manifold
M. In fact, if we choose another local basis (Wj , • • •, W)v), then we have a
g e GL{N, R) such that
The quantities corresponding to Uf^ilf^a with respect to the new bcisis are
" n"
II II II
denoted by u^ ^il^^a , then
therefore
^ // // // ^
n^t r"',<^tr"',^t) = n^t,••-,«*,•••,^t),
which guarantees the existence of w. So (ii) is proved. Now let us prove (iii).
Due to (ii) we may assume that the connection D is compatible with an inner
product < , > on the bundle E, i.e. it satisfies
where the inner product is a notation similar to the Riemannian metric. Then
with respect to an orthonormal basis (VTi, • • •, Wjv), we have
U,j =<DW^,WJ >=:d<W^,WJ >-<W^,DWj >=-U;J^^
It is to say that u is antisymmetric. Thus by the equality Ct =: du + u Au^
Ct is antisymmetric too, i.e.
Q^j = —l^jj, or Ct = —1^*,
where Ct* means the transpose of Ct. Consequently,
det(A/ + n) = det(A/ + ^Y = det(A/ + Q*) = det(A - Q),
which implies
A^ + ri{n)X^-^ + r2{n) + ... + VNi^)
= X^ - ri(l^)A^-i + r2{n) + ... + (-l)^r^(l^).

146 CHERN-WEIL THEORY
It follows
r,(l^) = 0, if i is odd,
hence by using (ii), we get (iii).
By the fundamental lemma 4.1.1(i), r,(D) is a closed 2i-form on M, The
element in irJ]j(M), represented by rt{D)^ is denoted by {r,(D)}. Then the
lemma 4.1.11 (ii) and (iii) read
{r,{D)} = {r,{D)}, \/D,D
{r,(D)}=:0, \/i= odd,
which imply that {ri{D)} depends only on the real vector bundle E,
Definition 4.1.2 Let jE7 be a real vector bundle over M of rank N with
a connection D, and the 2z-form r,(D) be defined in Lemma 4.1.1, then define
and
p,iE) = {p,{D)} € Hfj,(M),
and call Pi{E) the z-th Pontryagin class of the bundle E. By the way we call
Pi{D) an z-th Pontryagin form, which depends on E^ and on D as well.
Remark 4.1.3 Pt{D) is a closed form, usually it is not a harmonic one.
In [26] p.308 the Pontryagin forms satisfy
i + Pi(D) + p2(i>) + ... = i + (^)V2(ii:) + (^)V4(Jir) + ---
det
(-H
while in our case
1 +pi{D)+p2{D) + ••• = ! + (^)'r2(0) + i^)^u{^) + • ■
det
(-^")
Although K = —Q^ the Pontryagin forms are the same. By using the de Rham
theorem we know
p.{E)eH^{M,R),

4-1 Characteristic Forms and Characteristic Classes 147
where H'^^M^ R) is the 4z-th cohomology group in topology. And by a tedious
check we even know
and Pi{E) is just the Pontryagin class defined in topology, where Z is the integer
field.
For an oriented real vector bundle E over M of rank 2k^ an Euler class in
H^^{M,Z) may also be defined. Now we express the Euler class in terms of
the curvatures in Riemannian Geometry as follows. First of all we introduce a
function pi (A) instead of the above functions r,(A).
Definition 4.1.4 For a real antisymmetric matrix A of rank 2k, i.e.
Aij = —Aji, define
Pf (^) ^ 2^ Y^ <hJu • • •, ^kJk)A^,J, •. • A,j,,
where 6(zi,;i,---,ijk,ijk) is 1 or -1 or 0 according as (zi,;i, ••• ,U,yjk)
is the even permutation of {1, 2, • • •, 2k} or the odd permutation or otherwise
respectively. And we call pf (A) the Pfaffian of A.
Lemma 4.1.5 If A is an antisymmetric matrix of rank 2k, then
(1) for any basis {Si,-" ,S2k} in R^*^, pf (A) satisfies
l^i^AjS, ASj)^ = pf {A)Si A ... A S2k;
'<]
(2) ii B e GL{2k,R), then
pf (A) ■ det(5) = pi{B*-A--- B),
where B* is the transpose matrix of B;
(3) if 5 € S0{2k), then
piiA) = piiB-'-A---By,
(4) there holds
{pi{A)f=det{A).
Proof (1) follows from the computation
«l,Jl, •,«2fcj2fc

148 CHERN-^WEIL THEORY
Let (<5i,---,<5jk) = (t/i,.--,7/^)5*, then
^A^JS^ ASj =^^A^JB^kBJl'nk A Vi = ^{B* AB)ki'nk At;,,
«,J k,l k,l
hJ
thus
pf {A)6i A • • • A <52jk = pf (5* • A . 5)7/1 A • • • A 7;2Jk.
So (2) is true. For B G 0{2k) we have B-^ = B* and det(5) = 1. Therefore
(2) implies (3). ^ is an antisymmetric matrix, so there exists an orthogonal
matrix B G 0{2k) such that
/ 0 //I
-//I 0
B 'A'B--^ =
\
0 //2
-//2 0
Then
det{B ' A ' B'"-) = //i---//^
0
pi{B'A'B-^) = fjii
fJ'k
0 /
•/^Jk,
therefore by (3) the assertion (4) is true.
Definition 4.1.6 Let tt : jE7 —> M be an oriented vector bundle of rank
2k over M, Let < , > be an inner product on E, that is to say, for any
X G M, < , > is an inner product on the real vector space E\a; = 7r~-^(x). And
let D be a connection of E compatible with the inner product < , >. With
respect to a local orthonormal basis {PFi, • • •, W2k} the connection is expressed
as a; = (i*^«j), and the curvature is Q = du -\- u Au, Note that u; and Cl are
antisymmetric, so we can define a global 2Worm S{D) G A^^{M) such that
with respect to the orthonormal basis
^(^) = pf(^«) = (^)*pf(fi).
After checking that S{D) is closed (see the Exercise 4.1.7 (i) below), we denote
the cohomology class, representative by S{D)^ by S{E) G ^J^(M), and call it
Euler class of E, By the way we call £{D) a. Euler form.

4'1 Characteristic Forms and Characteristic Classes 149
Exercise 4.1.7 The notation is the same as that in Definition 4.1.6,
then
(i) pf (^^) is closed.
(ii) Although S{D) depends on both the inner product and D, but it does not
depend on the choice of the orthonormal bcises.
(iii) S{E) = {6{D)} does not depend on the choice of the inner products and
the compatible connections (see [37]).
Remark 4.1.8 We would like to point out the importance of the factor
^ in the formula pf{j^Cl), It renders
Further the details for S{D) may be found in [37].
Definition 4.1.9 Let M be an oriented Riemannian manifold of dim
2k. Let E = TM^ D be the Levi-Civita connection V? then by Definitions
4.1.2 and 4.1.6 we define p,(v) G A^*(M) and £{\/) G A^^{M), and call them
the z-th Pontryagin form and the Euler form of M respectively.
Now we will carry out the analogous discussions for complex vector bundles.
Let TT : jE7 —> M be a complex vector bundle of rank N with a connection D.
Choose a local basis {PFi, • • •, Wjv}, and define u, Ct by
Q = du -\- u Au.
Then define a 2z-form c^{D) G A^'{M) 0 C such that
or in other words,
k
27r
l + ^c.(i^) = det(/+^l^).
By the discussions as before, we have
(1) c,(jD) is well defined and is closed,
(2) the cohomology class Ci{E) = {c,(jD)} G HJj^{M) 0 C does not depend
on the choice of the connection D.

150 CHERN-WEIL THEORY
Definition 4.1.10 The notations is as above, define c,(jD) and c,(jE7)
to be the z-th Chern form and Chern class respectively.
Remark 4.1.11 By using de Rham isomorphism and a tedious check
c.(^)G^''(M,Z).
Definition 4.1.12 Let M be a manifold. In A*(M) or A*(M) 0 C if
a differential form can be expressed in terms of the Pontryagin forms, Euler
forms, Chern forms, i.e. the sums of their exterior products, then the form is
called a characteristic form. If M is an oriented manifold of dim n, then for
a characteristic form a G K^{M) or G h.^{M) 0 C the integral
JM
a
M
is called a characteristic number, or more exactly, the a-genus.
Remark 4.1.13 To express the Pontryagin class, the Euler class and
the Chern class in terms of the curvatures is the central point in the Chern-Weil
theory. Now we nearly finish the whole topic. This topic is very important. As
you know, every vector bundle is trivial locally, so the characteristic classes,
which can be used for characterizing the vector bundles, must be of global
features. On the contrary, the characteristic forms are determined locally. So
the Chern-Weil theory built a bridge joining the global research and the local
research in geometry.
Exercise 4.1.14 Find examples to show that the Pontryagin form, the
Euler form and the Chern form are not "trivial".
4.2 General Characteristic Forms
Now we are going to introduce the characteristic forms for principal bundles.
First, a moment's thought reveals that what we have done in the last section
is in fact to introduce the characteristic forms for principal GL(N^ R)-bundles
or GL(N^ C)-bundles or 50(2Ai)-bundles. In the process we only need three
things: a;, ^ and (j). The first two things are available nearly in the notion of
connection for a principal G-bundle. That is to say, given a principal G-bundle
P with a connection {a;<y G MF(P)}, then we have
u^ and eta- = du^, + -[o;^, o;^]
with a property Q.(r g = Ad(fif~-^)l^<y. So in order to introduce the characteristic
forms we only need (^:^—>Ror(^:^—>C, which is a polynomial and satisfies
(t>{M{g-^)A) = (^(A), \fgeG,Aeg,

4*2 General CharactertsUc Forms 151
Such kind of <j) is called a G-invariant polynomial. The set of all G-invariant
polynomials is denoted by PolyQ(^,R) or PolyQ(^, C) according as (j) is real
or complex respectively.
Definition 4.2.1 Let tt : P —* M be a principal G-bundle with a
connection D = {c(^<t}^ and (^ be a G-invariant polynomial. Define the curvature
Cta^ as before, then define
<f>{D) = <f>{n,), <t>{P) = {,j>(D)} e H*^RiM) or H*^r{M) ® C,
and call (l>{D) and <I>{P) the (^-form and (^-class respectively. A (p-fovui is
called a general characteristic form.
Remark 4.2.2 In the last section we have introduced the Pontryagin
forms, the Euler forms and the Chern forms for different vector bundles with
curvatures. Here we would like to point out that all the characteristic forms
(see Definition 4.1.12) can be viewed as (^-forms, which are constructed from
principal bundles with connections and invariant polynomials. For example, the
Pontryagin forms are defined for a vector bundle E of rank N with a connection
D. In this case we define a principal GL{N, R)-bundle P to be the set of all
bases in any fiber of E, then naturally we have a moving frame set MF(P).
For each a G MF(P), cr is a local basis field (VTi, • • •, Wjv), and define Ua^ by
Vx{Wir".WN) = {Wir".WN)MX), VX G r(TM).
By an easy check, the set {u^a} is a connection of the principal bundle P, and
if we let (^ = (^)^*72,, then (^-form is just the i-th Pontryagin form, where 72«
is defined in §1. Again, for an oriented vector bundle of rank 2k with an inner
product < , > and a compatible connection D, we define a principal S0{2ky
bundle P to be the set of all oriented orthonormal bases in fibers of E^ whose
orientations are as the bundle's, and define a connection of P as above, then
we can also express the Euler form by the pf-form up to a constant multiplier.
Similar way can be made for the Chern forms.
From now on we restrict ourselves to discussing the general characteristic
forms or the (^-forms, especially to finding the relations among the general
characteristic forms. In order to do so let us take a look at the invariant
polynomials first.
Exercise 4.2.3 Prove the following facts
(1) Let G be the torus group T^ of dimAi, i.e.
T^ = {e<'^^ '^'')|^i,...,^, GR},

152 CHERN-WEIL THEORY
where
Then Q — T'' —R'' with a trivial Lie bracket structure, and then
Polyo(T^ R) = Poly^[i;i, • ■■,Vk],
where Poly^[vi, • • •, Wfc] is a polynomicJ ring over R generated by {vi, • • •, wjt
(2) If G = Gi(JV, R), g = gC{N, R), then
Polyo(g£(JV, R), R) = Poly^^i, • • •, 7iv] ,
where
f,:gCiN,R)^R:A^j,iA)
is defined in §1.
(3) If G = 0(JV), g = 0{N), where
0(JV) = {^ e GL(iV, R)|^-^ = A'}, 0{N) = {Ae gC{N, K)\A* = -A}
then
Polyo((9(JV), R) = Poly^bi, • • •,p™] ,
where m = [y], and
P.=(^f7...
(4) If G = S0{2k), g = SOi2k), where
50(2fe) = {^ G GL(2fc, R)|^-^ = A\ det{A) = 1},
then
Polyo(50(2A^), R) = Poly,[pi,...,p,, ^]/^,
which is the ring generated by {pi, • * *, Pfe, ^} subject only to the relation
Pj^ = S^, where

4'2 General Characteristic Forms
{5)liG = GL{N,C), g = gC{N,C),then
Polyo(gA^,C),C) = PolyJ71,• • • ,7^] ,
which is the polynomial ring over C, generated by {71, • • • ,7jv}
{6)I{G=U{k), g = U{k), where
153
U{k) = {A-\- ^/^B e GL{k, C)\{A + V^B)-^ = A-\- V^B },
then
where
Remark
Polyo(Z^(Aj)) = Poly^[ci,...,Cfc],
c, = (
27r
)'7«.
The difficult part of Exercise 4.2.3 is to prove
Fo\yo{gC{N, R), R) C Poly^[7i, • • • ,7iv] ,
Polyo(0(i\r), R) C Poly^[pi, •••,!>„,],
Polyo(0(2Aj), R) C Poly^[pi, • • • ,Pik, f]/ - ,
Fo\Yo{gC{N, C), C) C Poly^[7i, • • • ,7iv] ,
[ Polyo(Z^(Aj))CPoly^[ci,---,Cfe].
Some of the above formulas will not be needed later, so now it is enough only
to take a look at them. Moreover we discuss briefly the following
Remainder Let G, be a connected compact Lie group, and let t : Gi -^
G2 be a Lie group homomorphism. Then l induces a Lie algebra homomorphism
^* • ^1 -^ ^2, this in turn induces a map
.* : Polyo(e?2) - Polyo(e?i).
If Gi is a maximal torus group of G2, and l is the natural imbedding, then
L* is an injection and the image l*{Fo\yq{Q2^^)) is a subring of PolyQ(^i, R),
which is invariant under the Weyl group action.
For a special a, we can easily compute a* , hence get a good understanding
about the above Remainder. Let Li : T^ -^ ^(^) be the natural imbedding

154 CHERN'-WEIL THEORY
and define a homomorphism b2 : U{k) -^ S0{2k) such that
i2{A-^y/^B)
-('. -/)
Lemma 4.2.4 Let Ai, b2 be mentioned as above. Further let u^ G
Polyo(T*^,C) be defined by
Mx) = —aj,,
^^ = Jl'^^-Qf\e=oeT,{T%
where e is the unit element of T^ and Te(T^) is the tangent space of T^ at e,
which is the Lie algebra T^ of course. Then
Polyo(T^ C) = Poly^K,. • • ,1^^] ,
and
and
^1(^2) =^'a,'Uj,
[ /.J(Cfe) =1X1 •••'Ujk,
{ {i2iiy{vi) = 'i^i + --- + 'i^L
(t2tl)*(p2) ^X^-l^Nj,
i<3
«i< • <«fc
I (t2tl)*(f) =ni--"Ufe.
Proof The first assertion is trivial, so the proof is omitted. Let us
consider

We have
4^2 General Characteristic Forms 155
('•i)*(a5) = (ii)*(X]'^'^l»=o)
«=i
de,'
= Y^X,{l.i)^{ — \0^o)
.=1
k
= E^'^i<'=o(
de,'
.=1
= ,/^
de,'
Xi
\
and
(t2ti)*(aj) =
( "
Xl
\
-Xl
0
0
Xk
\
-Xk
0 )
It follows that
W(l + ci + ...))(x)=(l + ci + ...)(Ui)*(x))
= det(J+-^(n)»(x))
ZTT
= det
2ir
V
nLi(i+".w)
(nf=i(i+«.))w,
27r /

156
and
CHERN-WEIL THEORY
{{titiYil + Pi + • • •))(^) = (1 + Pi + • • •)((^2M)*(a^))
= det{I-\-—{t2tMx))
and
= (nf=i(i+^?)w.
((i2il)*f)(X) = i^fpi{iV2^l).x) = (-i-)*Xa • • • Xfc = («i .
ZTT ZTT
-'Ufe)(aj).
Therefore the lemma is ecisy to prove.
From Lemma 4.2.4 it is easy to see that the following three maps
f v*, : Polyo(W(fc), C) ^ Polyo(r*, C),
{i^c^y : Polyo(5C?(2fc),R) ^ Polyo(T*,C),
[ {t20ir : Polyo(0(2fc),R) ^ Polyo(T^C)
are infective. One may check that the images of the maps relate to the Weyl
groups.
Definition 4.2.5 Let P be a principal G-bundle, and p : G -^ ^ be a
Lie group homomorphism. Define
P = Px H/ ^,
where the equivalence relation ~ is defined by
(aji,/ii) ~ (aj2,^2) <=> 3gf G G, such that xi = X2'g, hi — g~^h2^
Then we denote P by P Xp H, which is a principal iT-bundle, and we call it
the induced principal bundle of P.
Definition 4.2.6 Let P be a principal G-bundle with a connection
D = {(^<t} , and p : G -^ H he a. Lie group homomorphism. Define a connection
D = {u^} of PXpH by
u^ = p^{ua),
where p^ is induced by the Lie algebra homomorphism
p,:g^n.

4^2 General Characteristic Forms 157
After checking that D is well defined, we denote it by p^{D) and call it the
induced connection of D.
Lemma 4.2.7 Let P be a principal G-bundle with a connection D =
{a;<y}, and />, ; G -^ H«, (i = 1,2), be a Lie group homomorphism, and let
Pt^{D) be the induced connection on the principal ir,-bundle P Xp^ H^^ then
for any cp^ G Ployo('W«, C) we have
(1) MP^^{D)) = {p:{<l>^)){D) e A*(M) or A*(M) 0 C,
(2) if G = S0{2k) or U{k), then
Plih) = P2{<l>2) ^=^ (piO*(^i) = Wr(^2),
where l is (^2^1) or ^1 according as G is S0{2k) or U{k) respectively.
Proof (1) is trivial by the definitions of (j){D) and p^{D). Because
(pt)* = t*p*, (2) follows from that the map l* is injective.
Definition 4.2.8 If G = S0{2k) or U{k), P is a principal G-bundle
with a connection D, and if p : G -^ ^ is a group homomorphism, <j) G
Poly0(7^,0), and denote a = (^2^1) or ai, then
(1) 1^1, • • • ,1^^ are called Chern roots, where
u^ G Polyo(T^ C) = Poly^[^xi, • • - ,Ukl
which is described in Lemma 4.2.4.
(2) Note {pby{<t)) G PolyQ['Ui, • • •, ixjk], and we call a = {pLy{<j)) a Chern root
expression of the characteristic form <t){p^D)^ and denote <j){p^D) =
a{D).
At this point, we digress to make a remark concerning the significance of
the Chern root expression. Suppose we are given a principal G-bundle with
a connection £). If we consider all the forms, each of which can be expressed
as (j){p^D) for some (j) and p, then we find that any two such characteristic
forms are equal if their Chern root expressions are equal. So a Chern root
expression may be viewed as a name or a calling number for each of these
characteristic forms. We only have known that 'u,,a G Polyo(T, C), where T
is the Lie algebra of T, and T is a maximal torus in G.
Lemma 4.2.9 Let A'^{2k) be defined in Definition 1.8.3, which is a
R-real vector space of dim c{2k,m) with an 50(2Ai)-linear action p, where
c(2k,m) = I
\ m
\ ^ (2fc)!
J m!(2Jfe-m)!'

158 CHERN-WEIL THEORY
And let GL{A'^{2k)) be the group consisting of all non-degenerate R-linear
maps on A^(2Ai), which is isomorphic to GL(c(2Ai,m),R) of course. Then we
have
r, = r^{m) G Fo\yo{gC{A'^{2k)), R), Vz = 1, • • •, c{2k,m),
and
p : S0{2k) -^ GL{A^{2k)),
which is given in Definition 1.8.3. Further let
t = A2ti : T^ -^ S0{2k)
be given in Lemma 4.2.4, then {puy {r^{m)) satisfies
5](M*7.(m))A<2*,™)-. ^ -Q (F(A,p,g)U=2,„)«(''-'),
«>0
p.q
where (F(A,p, g')), <^(p,^) will be defined in the course of the proof.
Proof Let us recall the action
p : S0{2k) X A™(2fc) ^ A™(2fc) : {g,u;,, A •• • Ao;.^) ^ p{g){u>,^ A ••• Aa;.„),
where
Mfl')(wnA---Aw,„)= ]^ ^ e(
h
L
}1< ■ <]mll,- ,/„
31 ••• 3n
)
•9h.r--9l„,.,n'^ji A---Awj
If ffo = <.2ii(e^/^ (*'••■ •*")), then
go = 1-2
\
/ 9
where
On =
ayf=^Q^^
COS ^, — sin
11
^ifcjfc /
in^. \
sin ^, cos i
The above p naturally induces an action
/>o = pHix ■■ T* X A™(2fe) ^ A™(2fe): (e^=^(''" •"'•), w) h^ ^^0)^.

4 2 General CharactertsUc Forms 159
This action is ecisy to describe, because there is a tensor decomposition
Al{2k) = A;(a;i,a;2) 0 • • • 0 Al{u2k-u(^2k),
with actions
p^Q^ : T^ X A;(a;2,-i,a;20 ^ A;(a;2,-i,a;2,) : (e'^^^^u) ^ p{e,,)u,
such that
where A* (ct;2«-i,Ct;2«) is the complex linear space spanned by {l,Ct;2«-i,Ct;2«,Ct;2«-i
U2t}. It is easy to check that
p^o\e'^^^^){l,U2z-UU2i,i^2z-l ^i^2z)
= p{0^^){l,U2^-l,U2^,U;2^-l^(^2^)
/ 1 0 0 0 \
,^ X 0 COS Ot — sin 9i 0
= (l,a;2,-i,a;2,,^2«-iAa;20 ^ . . . ^
I 0 sin^, cos^, 0
\ 0 0 0 1 J
The left action Pq of T^ on A* (a;2«-i, tA^2«) can be expressed as a group homo-
morphism
p'o':e^''^
/ 1 0 0 0 \
0 COS Oi — sin Oi 0
0 sin^, cos^, 0
\ 0 0 0 1 /
0,,.
Therefore
iPo^)*i^^^O=o) = X,■^\0=,oO^^ =
'dO
dO,'
/ 0 0 0 0 \
0 0 -X, 0
0 X, 0 0
\ 0 0 0 0 /
and
Po
^,^Heu '^^)) = pW(e^^^O^--^P^^(e^'^)-^0--0^

160 CHERN^WEIL THEORY
/ 1 0 0 0 \
0 cos^i — sin^i 0
0 sin^i cos^i 0
\ 0 0 0 1 /
/ 1 0 0 0 \
0 cos^jk — sin^jk 0
0 sin^jk cos^jk 0
\ 0 0 0 1
/
d
/ 0 0 0 0 \
2j^4t-4
«=1
^hk-
4t5
0 0 -X, 0
0 X, 0 0
\ 0 0 0 0 /
where Ig is the unit matrix of rank s.
If we choose another basis {ej^ , - " •,ey} for A* (a;2t-i, '*^2t)5 where
J0_
»
1,
e^'^ =a;2t-i + \/^a;2t,
e^'^ =a;2t-i - A/=T'*^2t,
p(0
'*^2t-i Aa;2,.
Then with respect to the new basis the corresponding (Po )*(^«^l^=o) is
proved to be
/ 0 0 0 0 \
0 -^/-Tx, 0 0
0 0 V^x, 0
\ 0 0 0 0 /
aW,
consequently
{po)*{x) = ^ J4 0 ' " 0 i"4^0^^'^ 0 /4 '
^h = A.
t=i
(t-l) times (*^-«) times
It is easy to see that the set
{e« = eW0...0eW|i<ai,...,ajk<4}
is a bcisis for A* (a;i, • • • ,a;2Jk)5 and also a set of the eigenvectors of A. The
eigenvalue of Ca is
2k
X{ea) = V^ Y^{S-2,a^ + h,a^)x^.
t=l

4'2 General Charadertsttc Forms 161
Let
Then
2k
rjsiea) = Xl^*'«»' Vs= l,---,4.
«=i
{ea = e^J-^ 0 • • • 0 e^J^^l r}2{ea) + mM + 27;4(ea) = m}
is a basis for A2*(tA^i, • • •, (^2k) and
{A(ea)| r;2(ea) + mi^a) + 27;4(ea) = m}
is the set of eigenvalues. For any two fixed integers p, g, if there is an e^ with
r r;2(ea) = p
\ V3{ea) = q
[ mi^a) + r73(ea) + 27;4(ea) = m,
then m — p — q =: even and the number of the set {ea|A(ea) = A(ea)} is
k —p — q \
^o{p,q) =
m—p—q
2
Let <^(p, g) be $,q{p^ q) or 0 according as m—p—g is an even number or otherwise
respectively, hence the eigenvalues of the eigenvectors in
{cal mi^oc) + m{ea) + 27;4(ea) - m, r}2{eoc) - p, rj3{ea) = q}
constitute ^{p,q) copies of the set
where
f^P,Q
fr . . . J *i < ••• < V; 3i<"' <3q \
S 1*15 • * • 5 V' ^«' * * * ' ^d r. • T ^ r • • T I
t I {Hr"^^p}(^{jzr",3q} = 0 J
And the characteristic polynomial of —A is
det(AI + A) = n(^(^'P'?))^^''''^
P,Q
where
F{X,p,q) =
[«1,- ,^p^,j^, ,Jq] e flp,q
^jJ)-

162 CHERN-WEIL THEORY
So
J2{{pcrj,(m))ix)X<''''"'^-' = det(AJ + ^) = H (F(A,p,«))«("•«),
«>o p,q
and
t>0 p,q
It is notable that F{X^p^q)\x=-2Tru turns out to be symmetric with respect to
the variables nj, • • •, n^.
Finally we follow [21] to give
Definition 4.2.10 Let M be an oriented Riemannian manifold of dim
2k. We define
In "- ]
I -y- tanh Us J
and call L{\/) an ^-characteristic form of M. The number
A
M
and the function
*1
are called the X-genus and an L-density of M respectively, where L^^^\\/)
is the 2Ai-form component in L{\/) and *1 is the oriented volume element.
It is ecisy to see that X is a symmetric polynomial of the variables n^, • • •, n^,
while Pi, • • • ,pjk are the elementary symmetric polynomials. So L can be
expressed in terms of pi, • • • ,pjk, and the characteristic form L{\y) can be
expressed in terms of pi(v), • • • ,pjk(v) ^^ well.
4.3 Chern Root Algorithm
In the last section we have introduced the Chern root expression for
characteristic forms. In fact such kind of expression was first shown in [9] and [8],
although that was only in the cohomology level, where two characteristic classes
of equal Chern root expression are equal to each other. Now in what follows
we will make a "wild" generalization of Definition 4.2.8. We will introduce the
new Chern root expression for the curvature forms. Let us recall the
discussions in §4.1. If jE7 be a real vector bundle of rank 2k over M with an

4^3 Chern Root Algorithm 163
inner product (,) and a compatible connection D, then the Pontryagin forms
satisfy
1 +pi(D) +P2(I>) + ••• +Pik(I>) = det (l + ^QJ .
By Lemma 4.2.4 we have
k
t=i
where u^ G Poly(T*^, C). So by Definition 4.2.8 (nf=i(l + '^?)) is the Chern
root expression of 1 + pi(jD) + 'P2{D) + f- pjk(i^), and
k
t=i
At this time we are apt to assume there exist n,(jD) such that
k k
«=i
.=1
Here w,(jD) are called the new Chern roots. Similarly, for a complex vector
bundle of rank k the new Chern roots u,{D) can also be used to deduce
k
l + ci(I>) + ... = ]][(l + «'(^))-
«=i
Let
/ 0 Mi(I>)
-ui{D) 0
U
0 Uk{D)
-Uk{D) 0 /
/ -^/=Tul(I>)
Uo
-^-luk{D)

164 CHERN-WEIL THEORY
then we have
det {l+^^ = l+pi(l>)+ ... = nLiCl + KW)') = det(I+ CT),
pf(i-n) = e{D) = u^{D)---u^{D) = pf(tr),
/—r- h k
det(I + ^Q') = 1 + y]c.(I>) = TT(1 + «.(£))) = det(J + v^ £^0).
«=1 t=l
Now we are again apt to assume the following equalities
New Chern root expression 4.3.1 Let jE7 be a real (or complex)
vector bundle over M of rank 2k (or k) with an inner (or Hermitian) product
(,) and a compatible connection D, and let Q (or Q^) be the curvature form
respect to a moving frame (j, then assume
Q = 2'kU or Q^ = 27rC^o
according cis jE7 is a real vector bundle or a complex one respectively, where U
and Uq are defined by using the new Chern roots as above. And call U and
Uo the new Chern root expression of the curvature form — and —
27r 27r
respectively.
In 4.3.1 we encounter problems. First people may ask what is the meaning
of the new Chern roots n,(D)? Where are the new Chern roots u^{D) located?
To tell the truth, we do not have a satisfying answer yet. Second, we do not
know what is the meaning of the equality Q = 2'kU either. Moreover, from
the equality Q — 2'kU it follows that each n,(i)) is a 2-form, and can join
in computations just like a number. If we accept the new Chern roots and
4.3.1, and do computations by using the new Chern roots, then we say we
play a Chern root algorithm. Plainly such an algorithm may be incredible
in mathematics, however it sometimes can lead us to correct and marvelous
results, and hence shows a magic power. This case will be shown in the next
section, where the Chern root algorithm is used to prove the local index theorem
for the signature operator.
For a fixed JD, sometimes we denote the new Chern root u^{D) by n,. So
later when we encounter a notation n,, we must point out whether n, is the

4-4 Formal Approach to Local Index of Signature Operator 165
Chern root in Poly(T*^, C) or the new Chern root n,(D). Let O^j be an 2k x 2k-
matrix satisfying {0^j)aji — ^la^j^-, then
k
U = 2^(^2«-l,2« — ^2«,2«-l)l^«,
8 = 1
where u^ are new Chern roots. So in the real bundle case, by 4.3.1 we get
k
Qtj = 27r2_^(^2«-l,2« — ^2«,2«-l)«j'i^«
8=1
k
= 27r2_J(<52«-l,«<52«j -S2s,tS28-l,j)y'8,
8=1
and by Lemma 1.3.1 we get
Tricky formula 4.3.2 The new version of the new Chern root
expression 4.3.1 is
k
Rtja/3 = -^tj{^a^^(3) = -27r ^J "^^^ (^a, ^^)(<52«-l,«<52« j - h8,t^28-l,j)'
8 = 1
4.4 Formal Approach to Local Index of Signature Operator
Now we consider the signature operator
D+=d + S:A+{M)-^A. (M),
which is defined in Definition 1.4.18 under the cissumption that M is an oriented
Riemannian manifold of dim 2n. Let
D.=d + 6:A.{M)-^ A+(M),
and
n+ = D-D+: A+{M) -^ A+(M),
n_ = D+D_ : A_(M) -^ A_(M).
Then by Weitzenbock formula (Theorem 1.4.16) we have
n+ or n_ = -Ao - ^ X) KkiE+E+E^E,- + ^R,

166 CHERN-WEIL THEORY
where Aq is defined in Definition 1.4.2. Choosing an orthonormal moving frame
a = {^1, • • •, ^2n}? by using Lemma 1.4.20 we have
= {E, + \r\,iE+E+ - E^Er))iE, + \Tt{E+E+ - E'E^))
-r^E, + \r\,{EtE^ - E-ED),
where the repeated indices imply addition has been employed. When we choose
{El,- •• ,E2n} to be the frame compatible with a normal coordinate system
{yi,--- ,y2n}, then
and putting the equalities of Corollary 1.5.10, i.e.
H^j = S^j - -yay(3Rat(3j{^) H ,
Hkil — --VaRatkli^) H 5
into the above computations we have
Ao = {^ - ly,R..ikmE+E+ - E'E-)}
1 /^ 1
+2^«.j.(o)y«5r- - j^Ra.j,mRi}jikmyay0{EtE+ - E^ED
+ •••
64
+ •••
R,M0)R>jp4(i)y.yj{E+E+ - E-Er){E+E+ - E-E-)

4-4 Formal Approach to Local Index of Signature Operator 167
consequently
+^R„ik{(i)Rsji}aiO)y.yjiE+E+ - E^Er){E+E+ - E^E^)
-\R,,k,{Q)EtE^E;Er + \R
+ •••
In the above computations the notation "• • •" means the remainder terms. The
remainder terms are usually chosen such that they have no effect on the results.
It is the case that the remainder terms listed above can be discarded for getting
the local index. The proof of this will be discussed in Chapter 6.
Let us recall the Tricky formula 4.3.2, it reads
«=i
We are going to apply the Tricky formula to the above expression of Dj.. First
we denote
^*J ~ 9 zJ ^'J^(^(^)^a ^^ '
2 ^
and
a, = J2^M^a,Ef))E-^^ and &, = ^t-i^l
a,(3

168 CHERN-WEIL THEORY
then
n
a,(3 8 = 1
n
8 = 1
1 1
1 '^
8 = 1
1 **
~ 7 X^^«(<^2«-l,«<52«j -S28,tS28-l,j)E^E'j'
8 = 1
*=i «=i
Therefore
2' ^2 1 ' 1 '
« = 1 * 8 = 1 8 = 1
Note that a^ in the equality is defined by using the new Chern root n^, so the
right-hand side of the equality may be viewed cis the Chern root expression of
the operator n^.
Let us recall what we have done about the index of □+ . We had introduced
a series of notions, they are the heat operator ^ + n±, the solving operator
e~*°=t^ the fundamental solution G±(t,^,p), and the local index
Loc.ind(n+)(p) = _l^(tr(tr(.")) - tr(£ri"))).
V47r
And had proved the asymptotic expansion
v47r ™>o

/(
JM
4-4 Formal Approach to Local Index of Signature Operator 169
and had got a following computation
Ind(n+) =tre-*°+ - tre-*°-
(trG+(t,p,p) - tiG-{t,p,p))dp
r
= lini / (trG+(t,p,p)-trG_(t,p,p))
= f (Loc.ind(n^.))(p)dp.
JM
For simplicity of notations, in the above equalities we denote the terms
tre-*°+ - tre-*°-, trG+(t,p,p) - trG>(t,p,p)
by
tre-*°, ti G{t,p,p)
respectively. It should be stressed that the existence of the limit
A
lim tr G{t,p,p)
is still not known yet, hence we have no idea about the following equality
lim t^'r G{t,p,p) = (Loc.ind(n^.))(p),
although it sounds true in the above computation. However at this stage we
take it for granted and try to use the Chern root algorithm to compute
A
lim tr G(t,p,p).
In other words, try to find its Chern root expression. To this end, let us find a
Chern root expression for G(t,p,p) first, which should satisfy the heat equation
« = 1 * S = l «=1
\imy/A^'^''G{t,{},{}) = l.
In the above equation we omit the remainder terms in the Chern root expression
of □. This omission of the remainders is tricky, it with the Chern root algorithm
will let us to get the correct result later.

170 CHERN-WEIL THEORY
Lemma 4.4.1 The following equation
lim \/47rt H(t, 0) = 1
has a solution
la
^STTsinh^^^^^
-exp{-
y/^ a \/^ at
8
-(coth
)y'],
where a is a real number.
Proof Suppose the solution can be expressed as
then
H{t,y)=—exp{y^c{t)},
dy^
= ^.{2c(t) + Vc'(<)}
Thus the equation turns out to be
s{t)
= 2c{t)
16
c\t) = Ac^{t)-h
lim —-—- = 1.
«->o 8{t)
The above second equation is equivalent to
dt = ^-^ dlog ^
which has solutions
,{t) =
8 l-AeV^«*'
where A is a constant. And the first equation is equivalent to
-dlogsit) = d (^ at - ilog(l - Ae^^"')^ ,

4 4 Formal Approach to Local Index of Signature Operator 171
which has solutions
8{t) =z ;/ . (e-^^ «* - Ae^^ «*)^,
where // is a constant. By Using
8{t) =//•(!-A ^V^at + o(t))^
and
we get
Therefore
lim —--- = 1,
«->o 8{t)
A= 1
2_ W^
/^
a
e(,) = _^^coth^^"'
s{t) =
8 2
The proof is complete.
Lemma 4.4.2 The following equation
s = l
limV47rt^"F(t,0,---,0)= 1
has a solution
-A
5=1 I
/-Ta,
STTsinh^^^l^
•exp
(-^(coth ^)(.L., + vl) - ^) } ,
where a, and hg are numbers.
Proof It is a trivial corollary of Lemma 4.4.1.

172 CHERN-WEIL THEORY
Because of Lemma 4.4.2, we would like to say that
'<'.«.•••.»)^^{J;;^■-(-^)}
is the Chern root expression of G(t,p,p), where a, and bg are not numbers,
they are
a, = J2''u,{Ea,E^)E-E- : A*(M) 0 C\p -^ A*(M) 0 C\p
and
bs = El.^El : A*(M) 0 C|p -. A*(M) 0 C|p.
Then
Q = lim tr^+(t, 0, • • •, 0) - lim tr^_ (t, 0, • • •, 0) = lim tr ^(t, 0, • • •, 0)
is the Chern root expression of
A
lim(trG+(t,p,p) - trG_(t,p,p)) = lim tr G(t,p,p).
In order to compute Q we need
Lemma 4.4.3 If 1 < ii < • • • < Zp < 2n; 1 < ji < • • • < jg < 2n, then
^ ( (-4^^)^*, if p = 0 and g = 2n,
tr(^+...^+^-...^-)=<^
( 0, otherwise.
Proof By using a method similar to that in Lemma 4.2.9 we can prove
the lemma. The details are omitted here. Or, we may consult Proposition
5.3.3, which is just the lemma here.
Let us now examine the value of
^ A
P =tv {a^^ . . . tttpKi&ji) • • • {O'jM).
From Lemma 4.4.3 it follows that
(1) if p+ g < n, or ii, • • •, jg do not appear in pairs, then
P = 0,

4.4 Formal Approach to Local Index of Signature Operator 173
(2) if p-^ q = n, and ji, • * * ? jq appear in pairs, then
^ A .— .—
P =tr {a^, . "a^^{aj,V-l) • • • (a^^V-1)),
(3) we have the following computation
tr (a,, .. .a, J = TT** ^n,,(^ai, ^^J • •-^^t.C^a., ^^ J
A
• tr {Ea^Efi^ '"Ea^,Ei3^)
= 'K-J2^,,{E^,,E^,)^^^u,^{Ea^,E^J{-Ay/=Tr
..(_4^^/ZT)n(2-n,, A...An,J(^l,...,^2n)
in which we have used an equality similar to
If /(ai, • • •, cin) is a power series with respect to the variables ai, • • •, a^,
we denote the sum of the m-th order terms in the series by (/(ai, • • •, an)){m).
Then by the above result for computing P, we have
Qo = lim tr H{t, 0, • • •, 0)
^limt^r|n( "^^ .expf-^))|
*-o U=i STTsinh^^^ "^V 2 ;^j
1 ^
= lim 7-—T" tr
n<^-(-^)>l<^
=^M&5^-(-t^)>l<"'}

174 CHERN-WEIL THEORY
'^<^''{[n'
a^
sinh
• exp
+
- V'-lfii:::^!.")}
(4^)n 1^ [^11 ^^j^j^ ,^=Ta^
Sttv^)"
L,^i tanh
(47r)» 1^
So the Chern root algorithm gives
2
A^«.
(£^l,---,£^2n) ?
lim t^r G{t,p,p) = lim t^r H{t, 0, • • •, 0)
=1
n
s = l
tanh Us
(£^l,---,£^2n),
where Ug are the new Chern roots Us{D).
By using the following equality
n "-(^) =(17 "- )(£>)
we get a formal proof of the following local index theorem for the signature
operator.
Theorem 4.4.4 Let M be an oriented Riemannian manifold of dim
2n, and let
DA+(M)-^A_(M)
be defined in Definition 1.4.18. Then for p G M,
lim t^ Git,p,p) = |n I^} (^- • • > ^3„)|p,
where the right-hand side of the equality is the L-density understood by using
Definition 4.2.10.
Theorem 4.4.4 gives a positive answer to the Mckean-Singer problem 3.4.11
and
^oc.india^) = {l[-^m,...,E,„)
j:(^'')(V)
»=i
tanh Ug'
*1
By Proposition 3.4.9 we also know that Ind(n^.) is the i-genus of M.

CHAPTER 5
CLIFFORD ALGEBRA AND SUPER ALGEBRA
Prom now on we are interested in finding formulas for local indexes. By
Definition 3.4.8 the local index is
j^{tvUi^\p,p)-ivu[^\p,p)),
SO we need to understand Lemma 3.1.1 in order to get the information for U^^^
and the local index. A, rough thinking of Lemma 3.1.1 shows that the local
index is a universal polynomial because it can be computed with any coordinate
system or any moving frame. But as far as the variables of the polynomial are
concerned, things seem complicated. In fact, the set of variables should contain
p, the orientations of the manifold and coordinate systems, the expressions of
the Riemann metric, the curvatures and its derivatives with respect to the
coordinate system, furthermore it should contain some traces for basic linear
maps. Without a full understanding of such kind of traces we can't get a
universal polynomial only containing p, the expressions of the Riemann metric,
the curvatures and its derivatives as the variables. So first thing is to consider
these traces. Note that in the expression of the local index, the traces appear
in pairs, counting the pairs together induces a notion of supertrace. We had
A
met the supertrace in Chapter 4 §4 already, where it was denoted by tr. This
chapter is for computing the supertraces, as a particular example we prove the
formula in Lemma 4.4.3. The other purpose is to give a preparation for defining
the Dirac operator later.
5.1 Clifford Algebra
We follow references [25] and [33] to define the Clifford algebra by using
generators and relations. Such a definition is not as modern as others, but it
could provide a clear picture for understanding the Clifford algebra.
Definition 5.1.1 For each integer n, let the Clifford algebra Cn( —1)
175

176 CLIFFORD ALGEBRA AND SUPER ALGEBRA
be the associative algebra with unit 1 over R which is generated by elements
ei, • • • 5 ^n subject only to the relations
ef = -l, Vi,
Example 5.1.2 C2( —1) is isomorphic to the quaternion space. The
isomorphism is induced by a 1-1 correspondence
€i *-> i,
[ 6162 *-> k.
As a real vector space, Cn(—1) is to have a basis consisting of the monomials
{e,i ••-6,^11 < ii < ^2 <•••< **fe < ^; 0 <k <n},
in which the monomial corresponding to Ai = 0 is defined to be 1. Clearly
dimH.Cn(—1) = 2^*. Three structures can be introduced in Cn(—1):
(1) a natural inner product (•,•} such that the monomials defined as above
are orthonormal to each other;
(2) a canonical anti-automorphism of algebra
(.)':C„(-1)-C„(-1)
such that
(3) a Z2-graded decomposition
C„(-l) = C+(-l) + C-(-l)
such that
C+(-l) • C+(-l) C C+(-l), C-(-l) • C-(-l) C C+(-l),
C+(-l) • C-(-l) C C-(-l), C-(-l) • C+(-l) C C-(-l),
where C^(—1) and C~(—1) are the real vector spaces spanned by the sets
{e,j • • • e,^ |1 < z'l < ^2 < • • • < ife < ^, k = even},
{^ti • • • e,^ |1 < ii < 22 < • • • < ijk < n, Ai = odd}
respectively.

5.1 Clifford Algebra 177
Definition 5.1.3 Four subsets in Cn(—1) are defined as follows
(i) R" is a real vector space spanned by the set {ei, • • •, Cn}.
(ii) S''-^ = {aeir\{a,a) = 1}.
(iii) Pin{n) = {a = ui" 'U]e\ui, " ' •fUje G 5""-^}.
(iv) Spin{n) — {a = ^i • •-nfelni, • • •, njb G 5""-^, k = even}.
It is obvious that Spin(n) C C^( —1) C Cn(—1), Pin{n) and Spin{n) are
groups under the multiplication in Cn{ — ^)-
There are two important topics for Clifford algebra, one is about the inner
automorphism, and the other is about the irreducible algebra representation.
We shall discuss them in turn.
Let u G Cn( —1) be a unit, i.e. there exists n"-^, then define
U^ ' Cn(—1) —> Cn(—1) : V I—► UVU~^.
It is easy to see that u^ is an algebra isomorphism, which is called an inner
automorphism.
Proposition 5.1.4 The inner automorphism u^ satisfies
(i) If u G 5""-^, then n^ can be restricted to R", i.e. it can induce
gn-l X R" -^ R" : (u,v) ^ U^{v),
And the map {—y'*{v)\-£in) is a mirror reflection with respect to u^, where
u-^ is the hyperplane perpendicular to u,
(ii) If n G Spin{n), then n^ can be restricted to R", and the restriction u^ |j^^
is a rotation in R". We denote this rotation by p{u) G SO{n).
(iii) The map
p : Spin{n) -^ SO{n)
is a group epimorphism and a 2-fold covering.
Proof (i) For u^v £ R", by Definition 5.1.1 it is easy to see that
U' V -^ V ' u = —2(n, v).

178 CLIFFORD ALGEBRA AND SUPER ALGEBRA
n n
In fact if u = 2_] ^t^t ^^^ '" = Tj A*«6«) then
t=i
n
= -2{u,v).
Thus for ueS""-^, v eTC",
—u^{v) = —u ' V ' u ^ — w V ' u — {—V ' u — 2(n, v))i
= v- 2{u,v)u e R'*,
which is a reflection in R" by seeing the figure
(ii) For any u G Spin{n), it can be expressed as
U = Ui'U2' -U2k,
hence
U^ = (Ul)^ ' (^2)* • • • {U2k)* = [-{y'l)*] ' [-('^2)*] • • • [-('^2fe)*].
Consequently n* is a composition of even number times of reflections by (i),
and is plainly a rotation in R". Therefore we may define
p : Spin(n) —> SO{n) : a 1—► p(a),
(aela"^ • • •, acntt"^) = (ei, • • •, e„) • p{a).
by

5.1 Clifford Algebra 179
(iii) The following computation
(ei,---,en) ' p{ci' b) = {ab€i{ab)~'^,'" .abcniaby^)
= {ab€ib~^a~^, • • •, ab€nb~^a~^)
= a{b€ib~^^ • • •, benb~^)a~^
= a[{eir",en)p{b)]a-^
= [a{€i,'",€n)a-^]p{b)
= {€ir-,en)p{a)p{b)
implies
p{a'b) = p{a)'p{b),
i.e. p is a group homomorphism. Since every rotation can be expressed as a
composition of even number times of reflections, the map p is onto. If a G
Spin{n) with p{a) = 1, then
(aela"^•••,aena"^) = (ei,--•,€„),
e~^ae, = a, Vi.
or
It implies
If
then
« = Xl XI '^Jl J2S^1"'^2S,
S>0 Ji< <j2s
^ = Yj Y1 ^^^ j2s^i'"^2s = J2 J2 '^Ji •J2.e/eji---ej,,e,, Vz.
»>0 3i< <328 s>0 3i<' '<32s
A direct computation shows
^« ^Jl * * *^J28^«
I 1,
if s > Oandz€ {3i,---,J2s}
if s > Oandi^ {ji,---,j2,}
if s = 0.

180 CLIFFORD ALGEBRA AND SUPER ALGEBRA
Applying this result to the above equality implies that a must be a real number
A. Note
a* = X* = A, a* = {ui • --^2^)* = U2k -"Ui = a~^ = A""^,
so we have a = ±1. The proof is complete.
Now we are going to discuss the representation problem. A representation
of Cn(—1) is an algebra homomorphism
^:Cn(-l)^Enda(Vi),
or
^:Cn(-l)0C^Endc(F2),
where Vi, V2 are real and complex vector spaces respectively, and EndR(Vi), End
are algebras consisting of all linear maps.
Let A^(n), A^(n) be the real and complex vector spaces spanned by
{^,, A...^,Jl<zi<...<Zfe}
respectively (see Defintion 1.8.3). Let
K{n) = ^Aiin),
k
k
and define the following linear maps (see Exercise 1.4.10)
6, : A;(n) -^ A;(n) : e^, A ---AO^^ ^ 0^ Ae^, A ---AO^^
A, :A;(n)^A;(n):^,, A...A^,,
J](-l)*-i<5,,,,^,^A...A^;, A...A^,,
8=1
Lemma 5.1.5 We have
(6, + L,){€j - Lj) + {€j - Lj){€, + L,) = 0,
(6, - L,){€j - Lj) + {€j - Lj){€, - L,) = -26,j.

5.1 Clifford Algebra 181
Proof The proof is straightforward, so it is omitted here.
Definition 5.1.6 Let Cn(l,—1) be the associative algebra with unit 1
over R which is generated by elements ef, • • •, ej,e][, • • •, e~ subject only to
the relations
e," ^7 + ej e - = - 26, j, Vi, j.
Proposition 5.1.7 Let
M:Cn(l,-l)^Enda(A;(n))
be an algebra homomorphism satisfying
^/'l{et) = e, + A,,
then Afi is an isomorphism.
Proof By Lemma 5.1.5 the map Afi is well-defined. If ai,---,a7n G
EndR(A;^(n)), we denote the subalgebra generated by ai, • • •, am by A{ai^ • • •,
ojm)- Then it is ecisy to see
A{^rliet),■■■,^fliet),^flie^),■^■,^rlie-))
—- •^V^l 5 * * * 5 ^n 5 ^15 * * * 5 ^n j
= Enda(A;(n)).
Therefore Afi is onto. As real vector spaces, both Cn(l, — 1) and EndR(A;^(n))
are of dim 2^^*, hence Afi is an isomorphism.
Theorem 5.1.8 (Fundamental Theorem of Clifford Algebra) There
exists a complex vector space V of dim 2", such that C2n(—1)<S>C and Endc(V)
are algebra isomorphic. Moreover, the isomorphism induces a representation
of C'2n(—1) on y, which is the unique irreducible complex representation of
C2„(-l).
Proof We may determine an algebra homomorphism
^2 : C2„(-l) ® C ^ Endc(A;(n))

182 CLIFFORD ALGEBRA AND SUPER ALGEBRA
by the conditions
Af2{e2k-i) = ^i(efe ) \/l<k<n
^f2{e2k) = V^Afiiet) \fl<k<
n.
Proposition 5.1.7 implies that A/2 is an algebra isomorphism. So letting V ~
A* (n) gives the first part of the theorem. Recall for any F, Endc(F) is a full
matrix algebra. By an important theorem about the full matrix algebra it has
just one irreducible representation. The proof is complete.
Example 5.1.9 Let us look at the Ccise of n = 1. The isomorphism A/2
in Proposition 5.1.8 satisfies
^2(ei) = (61 - Ai),
^2(^2) = \/^(ei + ti).
Thus
A/-2(ei)(l,^i) = (^1,-1) = (1,^1) f° -^ j,
^2(e2)(l,^i) = (V^^i,V^) = (l,^i)f ^ ^j,
and then A/'2(ei), A/2(e2) and ^2(^162) correspond to
/ 0 -1 W 0 V^\ ( -V^
[1 0 )'\V^ 0 j'V 0
0
which are the well-known Pauli matrices.
The action of C2n( —1) on -^c(^) induces an action of Sptn{2n). Denote
S{2n) = Al(n),
5+(2n) = A-«"(n) = Ar"(n) ® C,
5-(2n) = AZ^in) = A°J'^{n) ® C,
then we have
Theorem 5.1.10 There exists a 2"-diniensional complex vector space
5(2n) with a decomposition
5(2n) = 5+(2n) + 5-(2n),

5,2 Super Algebra 183
such that 5(2n) is an irreducible C2n(—l)-niodule, and 5'^(2n), 5~(2n) are two
irreducible 5pm( 2n)-modules with dim2'*~-^. Moreover the decomposition is
characterized by
5+(2n) = {ve S{2n)\Tv = v},
5-(2n) = {ve S{2n)\Tv = -v},
where
T=(V^)"ei...e2nGC2n0C.
Proof We only prove that the decomposition of 5(2n) can be
characterized by T. The isomorphism ^2 in Theorem 5.1.8 maps T to
(VZi)«.(Vrr)nere+...e-e+=e+er...e+e-.
It may be checked in a straightforward manner that
A|-"(n) = {ve AUn)\ete:---ete-v = v},
Af^in) = {ve A;(n)|e+er • • -e+e^^ = -v},
hence we prove what we wanted.
Remark 5.1.11 Theorem 5.1.10 is a complement of Theorem 5.1.8.
The actions of Spin{2n) on 5^(2n) may be viewed as double value
representations of 50(2n), and are called the Spin representations. The elements in
5(2n) are termed Spinors, and the space 5(2n) is termed Spinor space.
Exercise 5.1.12 Check
C2+(-1) •5+(2n) C 5+(2n), C^" (-1) • 5-(2n) C 5+(2n),
C2+J-I) • 5-(2n) C 5-(2n), C2-J-I) • 5+(2n) C 5-(2n).
5.2 Super Algebra
The notions of super vector space and super algebra were introduced in
geometry in connection with the effort to understand the Atiyah-Singer index
theory by the idea of super symmetry in physics. Some mathematicians felt that
the symmetry is nothing but a Z2 graded structure, which is also named super
structure in line with fashion. The Z2 graded structure appeared in
mathematics for many years. The Z2-decompositions for C2n(—1), A* (n), 5(2n) and
EndR(A* (n)) are just the super structures in those spaces. The reason for
mathematicians not to advertise the notion is that they had not obtained
satisfying results from it yet. However, now things may be improved, physicists

184 CLIFFORD ALGEBRA AND SUPER ALGEBRA
have done a lot of computations on it. This chapter will explain how to use
the super structure to compute the supertraces.
Definition 5.2.1 Let F be a vector space, a super structure of it is
a decomposition of the vector space
an element in Vq or Vi is called an even or odd element respectively. A
vector space with a fixed super structure is called a super vector space or a
super space.
If F is a super vector space, then we define a linear map 6 : F —> F such
that
€\vo =: 1 : Fo ^ Fo, 6|v, = -1 : Fi ^ Fi.
It is easy to see that 6^ = 1. Conversely if we are given a linear map e : V ^^ V
with the property 6^ = 1, then we can define a super structure of F such that
Vo = {xe V\€x = x},
Vi = {x e V\€x = -x}.
So such kind of e is called a super structure too.
The classical tensor spaces are constructed from a single vector space (a
tangent space as usual) by using two operations, first is the tensor product 0,
and the second is Hom(-, •). So it is natural to cisk how to introduce the super
structure into the tensor spaces.
Definition 5.2.2 If F, U are two super spaces, then there is a natural
super structure on the tensor product space F (S> C/^, such that
{V^U)o = Vo^Uo'\-Vi^Uu
{V^U)i = Vo^Ui'^Vi^Uo.
(Here we only need to check an equality
V^U = {V^U)o'\-{V^U)i.)
With this super structure the space V ^U is called a super tensor product
of the super spaces F and U, and is denoted by the same notation V ^U,
Definition 5.2.3 If F, U are two super spaces, then there is a natural
super structure on
Hom(F, U) = {(f) :V ^ U\(p is a linear map}.

5.2 Super Algebra 185
such that
Hom(F, U)o = {<f>:V^ U\<f>{Vo) C C^o, ^(Vi) C C^i},
Hom(F, U)i = {<t>:V^ U\<t>{Vo) C U^ <t>{Vi) C Uo},
The above notion of super structure for vector spaces is as plain as the nose
on one's face, but the super structure for algebras introduced below seems to
be different, it causes a little surprise.
Definition 5.2.4 For an algebra U^ a super structure of it is a super
structure of the vector space underlying the algebra U
with an additional condition:
U,Uj CZ^t+j, Vz,jf G Z2.
An algebra with a super structure is called a super algebra.
After seeing Definition 5.2.4 two questions arise immediately.
Question 1 If F is a super vector space, the space
End(F) = Hom(F,y)
is an algebra. Does the super structure of the vector space End(V'), defined by
Definition 5.2.3, result in a super structure of the algebra?
Question 2 IfV^U are two super algebras, by Definition 5.2.2 V ^U
is a super vector space. Can we introduce a multiplication inV^U such that
it becomes a super algebra?
The answer to question 1 is affirmative, the check is straightforward, so is
not mentioned here. The answer to question 2 looks a little strange. InV^U
we can introduce two multiplications, both of them can make super algebras
from it. The two multiplications are denoted by • and A, which are defined
below. For ai,6i G V, ^2,^2 G U let
(ai 0 02) • {h 0 ^2) = (ai^i 0 0262),
(ai 0 02) A {h 0 &2) = (-l)'^^'*'^^'(ai6i 0 a2&2),
where the second formula is only for the case that each of 02, &i is even element
or odd element. In this Ccise
is an even element,
{0, if a2 is
1, ifa2is
l«2| ,
"* ^ is an odd element.

186 CLIFFORD ALGEBRA AND SUPER ALGEBRA
It is ecisy to check that V ^ U is a. super algebra under either of the two
multiplications. It is worthy to note that the multiplication A depends on the
super structure, while "•" does not.
Definition 5.2.5 Suppose V^U are two super algebras, we denote the
super algebra V^U, which is defined by using the multiplication "•", by V (S> Z^,
or simply by V ^U.
Definition 5.2.6 Suppose V^U are two super algebras, we denote the
A
super algebra V ^U, which is defined by using the multiplication A, by V 0 U.
Exercise 5.2.7 Let us introduce the super structures in Aj^(n) and
Cn(—1) as follows
(A;(n))o = A--(n), (A;(n)), = A°<i<i(n),
(C„(-l))o = C+i-1), (C„(-l))i = C-(-l).
It may be checked that they are super algebras.
Exercise 5.2.8 Prove that
(i) there are two following natural super algebra isomorphisms
Cm{-1) ^ Cn{-1) = Cm+n{-iy
(ii) There exist two super algebras V and U such that as algebras, V 0 V
A
and V ^U are not isomorphic, especially as super algebras they are not
isomorphic either. (Hint: V =U = Ci(—1).)
Suppose F, U are two super vector spaces. According to the answer to
question 1 all End(F), End(C/') and End(F 0 U) are super algebras.
Proposition 5.2.9 Suppose F, U are two super vector spaces, then
there are two super algebra isomorphisms
A/*: End(F) 0 End(C^) -^ End(F 0 U),
AT: End(F) 0 End(C^) -^ End(F 0 U),
A
where A/,A/* are defined as follows. For <t> G End(F), ip G End(C/'), v e V,
ueu,
{Af{(t> 0 i^)){v 0 n) = (f){v) 0 xl){u),

5.2 Super Algebra 187
{AT {<t> 0 '^)){v (8) n) = (-l)l^l'IH^(^) 0 ^(u),
A
Proof First check that N and ^ are super algebra homomorphisms,
then both are epimorphisms. Consequently they are isomorphisms by
computing the dimensions.
A
Remark 5.2.10 Sometimes we denote the elements Af {(f>^ip),A/{(f>^if)
A
in End(F ^ U) by<^(8)'0,<^(8>'0 respectively in order to avoid using the worse
A . . .
notations ^, Af^ One is inclined to think that Proposition 5.2.9 is trivial, but
in fact it is not. The following examples will show that. First, Proposition
5.2.9 can induce an isomorphism
Af-^ AT: End(y) 0 End(C^) -^ End(F) 0 End(C^),
which does not hold in general cfise (see Exercise 5.2.8(ii)). Another corollary
is as follows. Regard A* (n) as a super vector space, there is a super vector
space isomorphism
A;(n)-A;(1)0...0A;(1)
' y '
n times
(see Exercise 5.2.8). Then Proposition 5.2.9 induces a super algebra
isomorphism
Endc(A*(n))=Ende((A*(l))®")
^(Endc(A;(l)))®".
As soon as we check Endc(A* (1)) and C2(—1)(S>C are super algebra isomorphic,
it immediately follows that
Endc(A*(n))=(Endc(A*(l)))^-
= (C2(-1)(8)C)^"
= C2n(-l)0C.
It is just Proposition 5.1.8.
Definition 5.2.11 Suppose F is a super vector space, and e is its super
structure. If <^ G End(F), let ti<j) be the trace of (f). Then define
A
tr= tr(6 o (p).

188 CLIFFORD ALGEBRA AND SUPER ALGEBRA
A
Here tr and tr are called the trace and the super trace of (^ respectively.
Proposition 5.2.12 Suppose F, U are two super vector spaces, (p G
End(y) and ip G End(C/'). By using the notations in Remark 5.2.10, we have
<p^if,(t>^if e End(F 0 U). Then
tr((^(8)'0) = (tr (^) • (tr ^),
tr ((^ 0 V) = (tr <^) • (tr ^).
Proof The first equality is well-known, so the proof is omitted. To
prove the second one, consider the following decompositions of vector spaces
F 0 C^ = Vb 0 C^o + Vb 0 C^i + Fi 0 C^o + Vi 0 C^i,
we get three projections to the factor subspaces
irY :V^V,C F,
7r,j : F 0 C^ -► V; 0 C/j C F 0 C^,
where i, j G Z2. If let
,j = irr<f>^; :V^V,
V'.j = <V'<:C^-C^,

5.j3 Super Algebra
189
then
A A A r A 1
tr {(p 0 if) =tv <Af {(f>^if)o (tToo + TTii + TToi + TTio) >
= tl<At {(f)^i))o (tToo + TTii - TToi - TTio) >
= tr {M'{(p 0 ('000 + '001 + '010 + '01l)) O (^00 - -TToi)
+ Af{(f) 0 ('000 - '001 - '010 + '01l)) O (^11 - -TTio)}
= tr {Af {(poo 0 Voo - <^oo 0 '011 + <^ii 0 '011 - <^ii 0 '0oo)}
= (tr<^oo)(tr'0oo) - (tr<^oo)(tr'0ii) + (tr(^ii)(tr'0ii)
-(tr^ii)(tr'0oo)
= (tr^oo - tr(^ii)(tr'0oo - tr'0ii)
= (t''r^)(t''rV),
A
in which the second equality sign turns tr into the ordinary tra^e, because of
^ f tr onEnd(Vb0C^o) + End(Vi0C^i),
tr=
( -tr onEnd(Vb0C^i)-tEnd(Vi0C^o).
A
The third equality sign turns Af into Af because of
( Af{(t>^ip) on Vb0C^o + Vb0C^i
M {4>^i))- I M{4> 0 '0) on Fi 0 C^o + Vi <S) Ui and V G (EndC/')o
[ -M{4> 0 '0) on Fi 0 C^o + Vi 0 Ui and V G (EndC/')i,
and
From
'000,'011 G (EndC^)o, '001,'010 G (EndC^)i.
tr(jy/'((^aia2 0 '0^i^2) o 7r,j) 7^ 0
it is ecisy to see that
«i = «2 = i G Z2, and I3i = 132 = j e Z2,

190 CLIFFORD ALGEBRA AND SUPER ALGEBRA
hence the fourth equality sign holds. The last three equality signs are trivial,
so the proposition is true.
5.3 Computations on Supertraces
Suppose F is a super vector space, then by Definition 5.2.11 there is a
A A
function tr: Enda(F) —> R or tr: Endc(F) —> C according cis F is a real
space or a complex one. This kind of functions appears both in mathematics
and in mathematical physics. For example it appears in the definition of the
local index (see Chapter 4 §4). This section aims to discuss further those
functions. We will deal with some special cases, where V is any of the vector
spaces Aj^(2n), 5(2n) and A* (2n) with the following proper super structures.
For Aj^(2n) and 5(2n), the super structures are defined by
(A;(2n))o = A--(2n), (A;(2n))i = A^/^{2n),
(5(2n))o = 5+(2n), {S{2n))i = 5-(2n)
(see Exercise 5.2.7, Theorem 5.1.10), and for A* (2n) the super structure Tn :
A* (2n) -^ AU^n) is defined by
Tn = (v^)^(^-^)+"* : A^(2n) ^ A2--^(2n),
where * is the Hodge homomorphism (see Chapter 1 §4), and the check for the
equality (r^)^ = 1 may be seen in Lemma 1.4.17. We denote the vector space
A* (2n) with the super structure Tn by Ac(2n). In other words
(Ac(2n))o ^{ue Al{2n)\TnU = u},
(Ao(2n))i = {ue Al{2n)\TnU = -u}.
A
In order to understand the function tr we need to identify the elements in
End(F) first. For V = A;^(2n) or A* (2n), by using the algebra isomorphisms
M:Cn(l,-l)-.End,(A;(n)),
Afi : Cn(l, -1) 0 C ^ Endc(A;(n)),
in Proposition 5.1.7. we choose
as an algebraic basis for EndR(A;^(n)) or Endc(A* (n)). And for V = S{2n) by
using the algebra isomorphism
^2 : C2n(-1) 0 C -^ Endc(A;(n)) = Endc(5(2n))

5,3 Computations on Supertraces 191
in Theorem 5.1.8. We choose
{ei,---,e2n}
as an algebraic basis for Endc(5(2n)).
Proposition 5.3.1 The super tra^e (function)
tr:Endc(5(2n)) -^ C
is a complex linear map satisfying
^ [ 0, ifg<2n;
where 1 < ii < " - < iq < 2n.
Proof Before the proof it is worthy to note that in Proposition 5.3.1 the
A
supertrace tr is a function defined on the set Endc(5(2n)). It can be defined,
because we know the super vector space structure in 5(2n).
By Definition 5.2.3 Endc(5(2n)) is a super algebra. Recall the definition
of 5(2n) and Remark 5.2.10 we have super vector space isomorphisms
5(2n) = A;(n) = A;(1) 0 ... 0 A;(1) = (A;(1))®^
^^ V '
n times
and a super algebra isomorphism
C2n(-1) 0 C = Endc(A;(l)) 0 ... 0 Endc(A;(l)).
n times
The above algebra isomorphism shows that for any (f) G C2n(—1) (S> C =
Endc(5(2n)) it can be expressed as
EA A
where <^, is in
Endc(A;(l)) = Endc(A;(».)),
which is the Grcissmann algebra generated by ^,. Then by Proposition 5.2.12
we have

192 CLIFFORD ALGEBRA AND SUPER ALGEBRA
An easy computation shows
tr {<!>,) = I
[ 0, if.^, = l,e, ore+,
consequently
A A A f (T^r, ii<^>^ = v^e;et^i;
tr ((^1 0 • • • 0 (^n) = <
( 0, if 3i, <^, = l,e~ or e,"*".
By a direct check we have
Therefore the proof is complete.
Proposition 5.3.2 Suppose the super structure of Aj^(2n) is defined
by
(A;(2n))o = A--(2n),
(A;(2n))i = A°dd(2n).
If 1 <ii <••• <ip <2n; 1 < ji < • • • < jq < 2n, then
A ^ ^ f (-4)", ifp=9 = 2n;
^ 0, if Pt^ 2n or g 7^ 2n.
Proof By using the super algebra isomorphism
A;(2n) = A;(l)®---®A;(l),
2n times
A
and the fact that tr: A;^(l) —> A;^(l) is determined by
A f -4, if^ = ei"e;[;
tr (^) = I
(^ 0, if <^ = l,e][" or e^ ,
we can get the proof easily.

5.3 Computations on Supertraces 193
A ^
Proposition 5.3.3 The complex linear map tr: Endc(Ac(2n)) —> C
satisfies
^ r (-4V^)'*, ifp=0andg = 2n,
tr (el*" ' "ct e~ ' "e~) ■= <
\ 0, otherwise,
where Ac(2n) is the super algebra, whose super structure is Tn^ and 1 < ii <
• • • < Zp < 2n, 1 < jf'i < • • • < jf^ < 2n.
Proof First by a direct check the proposition is true forn = 1. Secondly
the canonical isomorphism between the vector spaces
A* (2m) 0 A;^ (2n) = A^ (2m + 2n)
induces a super vector space isomorphism
Ac (2m) 0 Ac (2n) = Ac (2m + 2n).
Of course we need to prove it. It is equivalent to prove that: if a G A* (2m),
p e A*(2n) then
'Tm-\-n
Without loss of generality we may assume
aG A^^(2m), ^G A^^(2n).
Denote the Hodge star homomorphism of A* (2m) by *y„, then
Tm-^-n {a^/3)= Tm^n (« A /3)
= (yZT)(fei + fe2)(fei+fe2-l)+m+n ^^^^ (^^^)
y * m-\-n •'■ y
•(*m«) A {*nl3)
= (yZT)(fei+fe2)(fei+fe2-l)+m+n . (_l)feife2(^^^) ^ (^^^)
Thus we prove the super vector space isomorphism. Consequently we have a
super vector space isomorphism
Ac(2m + 2n) = Ac(2) 0 • • • 0 Ac(2),
"^ "^^ '
m-\-n times

194 CLIFFORD ALGEBRA AND SUPER ALGEBRA
and a super algebra isomorphism
End(Ac(2m + 2n)) = End(Ac(2)) 0 • • • 0 End(Ac(2))
^^ V ^
m+n times
by Proposition 5.2.9. The rest of the proof is similar to that of Proposition
5.3.1, and is omitted here.

CHAPTER 6
DIRAC OPERATOR
In this chapter we will discuss a generalized Dirac operator, which wfis
introduced by Atiyah-Singer in the sixties of the 20-th century by imitating the
original Dirac operator in physics. This new operator influenced the
development of mathematics and mathematical physics in the 20-th century greatly.
It may be called an Atiyah-Singer operator, but inventors called it the Dirac
operator modestly.
6.1 Spin Structure
Definition 6.1.1 Let M be an oriented Riemannian manifold of dim
2n, and SO{M) be the set which consists of all oriented orthonormal frames
with the same orientation as that of M, A principal 5pm(2n)-bundle P =
Spin{M) is called a spin structure of M, if
PxpS0{2n) = S0{M),
where p : Spin{2n) —> 50(2n) is the standard 2-covering (see Proposition
5.1.4).
In topology we can find a condition for an oriented manifold to have a
spin structure, however we would not describe it here. Instead we pay all the
attention to the analysis of a given spin structure. As we know there is a Levi-
Civita connection v on M, by Exercise 1.8.8 it is also a connection £ on the
tangent principal 50(2n)-bundle SO{M)^ that is to say,
a; = {£-1? is a local section of SO{M)},
which satisfies
where A is a map from an open set of M to 50(2n), and if ? = {^i, * * *, ^2n},
then
195

196 DIRA C OPERATOR
a;ii ••• c*^i,2n
\
hJ «<J
in which
and Otj is an n X n matrix, whose {k^l)-th. entry is <5,fe<5j;, and S,j = Otj — Ojt.
Now we are going to introduce a connection
u; = {u^a\o' is a local section of Spin{M)}
on Spin{M), Note u^ is a 1-form with value in the Lie algebra Spm(2n), and
by the following natural homomorphisms
5pm(2n) = Ti(5pin(2n)) ^ Ti(C2n(-l)) = C2n(-1)
it is a 1-form with value in C2n(—!)• It is remained that by Proposition 5.1.4(iii)
Spin(2n) is a manifold, so Ti(Spin(2n)) can be defined. And it is easy to see
that under the above natural homomorphisms, Spin{2n) is an R-linear space
containing a subspace Spi'nP{2n) spanned by
{etej eC2n{-l)\i<j}'
Also the standard covering p : Spin{2n) —> S0{2n) induces an isomorphism
p^ : Spin(2n) = TiSpin{2n) ~> TiS0{2n) = 50(2n),
where 1 and / are the unit elements.
Lemma 6.1.2 For i ^ jf,
e,ej G SpivP{2n) C Spin{2n) and /9*(e,ej) = —2S,j.
Sometimes the above equality is denoted simply by
Further, the above equality implies
SpivP(2n) = Spin{2n).
Proof Consider a path
7 : [0,6) -^ Spin{2n) : t h-^ j{t),

6.1 Spin Structure 197
where
j(t) = —e,(cos t • e, + sint - Cj) = cost — sint • e,Cj.
Then by the equality
(ei, • • •, e2n) • P7{t) = (tW • ei • (7W)~S • • •, tW • e2n ' (tW)"')
we have
/ 1 \
ptW
1
cos2t
sin2t
— sin 2t
cos2t
i-th row
jf-th row
1/
Hence
therefore
And consequently
we also have
p*(-e,ej) = 2S,j.
p^{SpinP{2n)) = Span^{S,j} = 50(2n).
SpivP{2n) = Spin{2n).
The lemma is proved.
Lemma 6.1.2 implies that the unknown a;<y can be expressed as
t<j

198 DIRAC OPERATOR
Comparing with
2
we have to choose a^j = — |a;,j, it is to say
Lemma 6.1.3 Let M be an oriented Riemannian manifold of dim 2n,
and SO{M) be the tangent principal 50(2n)-bundle. Suppose there is a
principal 5pm(2n)-bundle Spin{M) which is a spin structure of M, For a local
section a of Spin{M)^ let p*(cr) = {^i, • • •, ^2n}, and the Levi-Civita
connection 5- = (i^tj) be defined cis above, and define
"iSl^'J^'^J'
4
where
p^ : Spin{M) = Spin{M) x,d Spin{2n) -^ Spin{M) Xp 50(2n) = SO{M)
is induced by p : Spzn{2n) —> 50(2n). Then
is a connection of Spin{M), and the Levi-Civita connection lj is the induced
connection of a; (see Definition 4.2.6).
Proof We need to prove that for any map gf : C/" —> 5pin(2n), where U
is the definition domain of (j, we have
u^ g = Ad{g-^)u^ + g*{ [Spin{2n)] )
(see Definition 1.7.14), or
a;, g{X) = Ad(ff-i)«,(X) + g*( [Spin{2n)] )(X),
for all vectors X on M. The above equality is the one in the Lie algebra
Spin(2n), By the natural homomorphism it is an equality in C2n(—1). It is
easy to see that in C2n(—1) we have
Ad{g-')u;,{X)^g-''U;,{X)'g

6.1 Spin Structure 199
and
g*{[Spin{2n)]){X)=g-'-{Xg).
So we are going to prove
(^a-giX) - g~^ • (^a{X) -g-^g'^ ' Xg.
By the definition
hence
(f 1,..., E2n) = p4a'g) = {Eir",E2n)' p{g)^
If we let {(j^tj) and (i^tj) be the Levi-Civita connection matrices with respect to
the moving frames (-E7i, • • •, E2n) and (-^i, • • •, E2n) respectively, and we have
known
i<^ll ••• ^Mn \ / i*'!! ••• i^l,2n
A-^A-^dA,
\ ^2n,l ' ' ' i^2n,2n
= A-^
^2n,l ' • • ^2n,2n
i.e.
k,l k
where p{g) = A = (Atj) G 50(2n). This time we have
1 1
w„g{X) = --^£.j(X)e,ej = --Y^{J2AktWi:i{X)Aij+Y^Ak,{XAkj)}e,t
*,} «.j fc.'
1
"4^ J2 ^kl{X){Ak^e,){AlJeJ)+Y,{A-^■XA),,e^eJ}
= -\{Y^'^kiiX)ig-'eug){g-'etg)+'£{A-'-XA).,e,e,}
= g-'-u;4Xyg -i^(^-i.X^).,e.e,.
So the lemma is true if we can prove the equality
4
',3

200 DIRA C OPERATOR
which is the conclusion of Lemma 6.1.4.
Lemma 6.1.4 If U is an open set in M, g : U ^^ Spin(2n) is a map,
then for any tangent vector X G Tp{U) ,
(9(p))-i ■Xg = -\ X]((^(P))"' • XA)„e,e„
where A = p{g)'
Proof Let us choose a path
such that
where ■^\t=o(t>{t) is understood to be <^*(^|«=o) in the standard way. Then
in order to prove the lemma it is sufficient to check the equality
(9 o <l>m-\j^9 o <l>{t))t=o = -\J2(^{0)-' ■ ^^\,^oA{t))„e,e„
where A{t) = po g o <f>{t).
Consider a path starting at 1 G Spin{2n)
7 : [0,6) ^ Spin{2n) : t ^ {g{p))-^ - {g o <t>){t).
Plainly we have
(g o '^(0))"'(^9 ° <l>it))t=o = J^\t=oJ(t) e Ti{Spin{2n)).
Thus we may cissume
Further by Lemma 6.1.2 we have
^(0)-^ • ^\t=oAit) = ^|*=o(^(0)-^ • A{t))
= ft\t=oiip°9ip))-'-po9o<t>it))
= ft\t=opi'yit))
= P*ift\t=of{t))
= P*(Y^IJ'kieket) = -2^/ijfc,S)fcj,
fc<J ifc</

6.2 Sptnor Bundle 201
hence
-iZ)(^W • '^\t=oA{t)\,e,e, = --J2(-2J2m^kl)^Je^eJ
So the lemma is true.
Lemma 6.1.5 The principal bundle Spin{M) and the connection a; =
{ci^a} ^^^ given as above, then the curvature form Ct^^, defined by
satisfies
il(r = —- 2^ "tj ^« ^j 5
where
Proof Because
4
= (—2<5aje,e^ + 2<5j^e,ecr — 2<5,ae^ej + 2<5,^eaej),
we get
The proof is complete.
4
6.2 Spinor Bundle
Let us recall Theorem 5.1.10. Spin{2n) C C2n(—1), and there is a canonical
representation of the Clifford algebra C2n(—1) on the spinor space S = 5(2n)
Af:C2n{-l)xS^S
with properties as follows

202 DIRAC OPERATOR
(1) 5 is a complex vector space of dim 2",
(2) as an algebra representation, Af is irreducible,
(3) as a representation of the group Spin{2n), Jsf is reducible, more exactly,
there exists a decomposition
such that
and
s^s-^ -vs-,
M : 5pm(2n) x S^^ ^ S^^
Af : Spin{2n) x S- ^ S-
are two irreducible representations. The space 5"^ and S are
characterized by
Ta = a, \/ae 5+,
where
T = {^/^re^ - - e2n G C2n(-1) 0 C,
and {ei, • • •, e2n} are the standard generators of C2n(—1).
Definition 6.2.1 The vector bundles
E = Spin{M) X AT 5, E^ = Spin{M) Xj^S"^
are called spinor bundles.
Definition 6.2.2 The covariant derivative
V : T{TM) X T{E) -^ T{E)
is defined by
where W = ((j, /) G r(^), / : C^^ -. 5, and X G T{TM).
Definition 6.2.3 The second covariant derivative S7{X,Y), the
curvature operator R{X,Y) and the Laplace-Beltrami operator Aq are
defined by

6.3 Dtrac Operator 203
R(X,Y) = ^{X,Y)-^{Y,X),
where {^i, • • •, E2n} is an orthonormal moving frame.
Lemma 6.2.4 For an orthonormal moving frame {-E7i, • • •, E2n}^
R{X,Y):T{E)^T{E)
where {a;i, • • •,a;2n} is the coframe dual to {Ei,-" ,E2n}, R%jki = {Ri^t, Ej)Eje
El) and ((j, a) is a "local" map from r(^) to r(^) defined by
((J, a) ((J,/) = ((j,a-/),
in which a G C2n(—l^and ((J,/) is a local section of r(^),
Proof Foi W = ((J, /), by Lemma 1.8.9 we have
R{X,Y)W = {cr,n,{X,Y)-f) = -^J2^,,,{X,Y){cr,{e^ei)-f)
k,l
= -i E MX)u;j{Y)iiui{E„Ej){a,{e,,e,)-f)
= i E '^.(^H(i^)i2.jH(or,(efce,)-/)
Hence the proof is complete.
6.3 Dirac Operator
Definition 6.3.1 Define Dirac operator
D : T{E) -> T{E)
by
t
= Y^ia,e,iEJ + u;4E,)f)),

204 DIRAC OPERATOR
where (j is a local section of Spin(M)^ {^i, • • •, E2n} = P*{<^)^ ^^^ W = ((J, /).
Theorem 6.3.2 (Weitzenbock Formula)
where R is the scalar curvature mentioned in Theorem 1.4.16.
Proof First we make a remark on the notation ((J, a), which appeared
in Lemma 6.2.4. It is a "local" linear map similar to the notations a;,A , i{E^) .
As we did in the proof of Lemma 1.4.14, now we have
V£7,(<^,ej)((j,/) = VE^cr^ejf) = {G,E,{ejf)-^ uj^{E^)ejf)
= {(T.ejEJ)- -^Y^{(T,u;i,i{E^)eT,eiejf)
= (<T,ej£^J) - -^^{<j,ujj,i{E,){ejej,ei + 2e,<5fej - 2ej,6ij)f)
= (or, ej) Ve, (or, /) + Y^UJi,j{E^){(T, Cfe/).
h
Comparing with
VE,(o^,ej)((j,/) = (VE.(or,ej))((j,/) + ((j,ej) Ve, (o^,/)
we get
VE,(or,ej) = 5]a;fej(^,)(^.efe) = X!^«V^'^fe)-
(The above equality can be explained by using Theorem 1.8.7(2).) Now we
have
D^ = X^(<^. e,) Ve, {cr, ej)\/E,
= X](^' eO[VE,(or, ej)] Ve, + X!(^' ^0(<^^ ej) Ve, Ve,
= 5^((j,e,ej)v(^t,^j)
l''
-Y,6,jS7{Ej,E,)
1 '''
2

6.3 Dtrac Operator 205
and by Lemma 6.2.4
The Bianchi (I) reads
Rtjkl + Rtljk + Rtklj = 0.
So we have
0 = ^J (^jfe' + ^«'jfe + Rtklj)Gtejekei
= 2^ Rtjui^tejejcei + e,efee;ej + e,e;ejefe)
— 3 2;_^ RijklGt€j€je€l + 2 2_^(i2jufe - 2i2j,fe, + Rjktt)Gjek
= 3 2_^ RtjklGt€j€je€l + 67^ Rtjtj^
thus
2J RtjklGtejejcei = -22j^tjtj = 2i2.
Therefore we get the Weitzenbock formula.
It is easy to check that there is a Hermitian product ( , ) on 5 such that
(e,n,e,'y) = (n,'y), \/u^v £ S ; z=l,---,2n,
which is called a spinor product. In fact, the model space of 5 can be A^(n),
and the canonical Hermitian product in A* (n) is easily checked to be a spinor
product. For a spinor product we have
(an, av) = (n, v)^ Vn, V £ S \ a£ 5^'*~-^(unit sphere),
which can be checked by the computation
[au^av) — / ^ \t\j{etU^ejv)
= — 2j'^tAj(n, e,ej'y) = —{u^aav) = (u^v)^
hj
where a = \jA,e,, and A, is a real number. So the action of Spin{2n) on
t
S keeps the spinor product, thus it induces a Hermitian product ( , ) in the
spinor bundle E such that
{Wi, W2) = (A, /2), V P^i = (a, A), W2 = {<T, A).

206 DIRA C OPERA TOR
And hence induces an inner product (( , )) in T(E)
{{WuW2))= I {WuW2)dv,
JM
where dv is the measure given by the Riemann metric.
Proposition 6.3.3 The Dirac operator
D : V{E) -^ V{E)
is self-adjoint, i.e.
{{DWuW2)) = {{WuDW2)), V WuW2 e T{E),
Proof Suppose Wa = (<t, /«), and we denote e, = (<t, e,), then
{DWuW2) = X)<^~' V£?, Wi, W2)
= Y^{{VE,e~,Wi,W2) - {{VeAWi,W2)}
t
= Y,{M^^Wl.W2) - {e^Wu\7E.W2) -Y,^i{^~jW^^W2)}
= Y.{Wue. S7E. W2)+Yli^^i^^f^^f^) - El^n(^./l./2)}.
t t J
It is easy to see that
(i) T = 2j(^«-^i'-^2)^t is a vector field on the whole M,
(ii) div T = Y^iVEX E.) = E{^«(^' E') - {T, VE.E,)}
t t
= X;{^.(e./i,/2)-X;rfi(e,/i,/2)}
(see Exercise 1.4.10).
Then
{DWi, W2) = {Wu DW2) + div T,
and
{{DWi,W2)) = {{Wi,DW2)).
The proposition is proved.

6.4 Index of Dtrac Operator 207
Corollary 6.3.4 Let the restrictions of D
and
be called the Dirac operators too, then D~ is the adjoint operator of jD+.
6.4 Index of Dirac Operator
Let
n+ = D-D+ : r(£^+) -^ r(£^+),
n_ = D^D- : T{E-) -^ T{E-).
Weitzenbock formula claims that □+ and □_ are Schrodinger operators, so by
Theorem 2.4.3
Ker Di < oo.
Lemma 6.4.1 An argument similar to that in Lemma 2.5.1 yields
Ker {D"^ : r(^+) -^ T{E-)} = Ker {n± : r(^+) -^ r(^+)},
which with Corollary 6.3.4 implies
Ind £)+ = dime Ker n+ - dime Ker n_.
Let e~*°=*= be the solving operators of the heat operators ^ -f D-j. and
G±{t,q^p) the fundamental solutions, then By Proposition 3.4.6
lndD+ = tre-*°+ - tre-*°-
= / tiG^{t,p,p)dp- I tiG-{t,p,p)dp, Vt > 0
Jm Jm
= I ti G{t,p,p)dp, Vt > 0
JM
= lim / ti G{t,p,p)dp,
where G(t, q^p) is the fundamental solution for the heat operator ^ + D, and

208 DIRA C OPERA TOR
In what follows we will use the Chern root algorithm to compute
A
lim tr G{t^p,p),
By Definition 2.1.1 G{t^q^p) satisfies
lim / G{t,q,p)(f>{p)dp-(f){q), ycf)^
where □ = Z)^ acts on the variable g, <^ is a continuous section of the vector
bundle E^ and dp is the measure given by the Riemann metric. We try to solve
the above equation in a neighborhood U of p, although it is impossible to get
G{t^q^p) (see [2]). Let a he a. local section of Spin{M) defined on U^ denote
D = (or, D^), i.e.
then by Theorem 6.3.2
According to our experience in Chapter 1 §6, Aq is too complicated to be
written down, however we would like to get its Taylor expansion at p. Choose
(J such that pc = {Ei^ • • •, E2n} is an orthonormal moving frame parallel along
geodesies passing though the point p, and denote the normal coordinate system
centering at p and compatible with {Ei^ • • •, E2n} by {t/i, • • •, y2n}. Then
(Ao)<r = Y,{E^ - -^Y^ujj,i{E,)ej,ei){E, - - Y,^a^{E^)eae^)
I k,l a,(3
= J2{E.E, - i^rf,^,e,e, - jE^^rr^^a
ei3
k,l a,^
a,i3 k,l
+ T^ E "^^l^ffi^keieaC^}-^'
k,l,a,^

6.4 Index of Dvrac Operator 209
By using Corollary 1.5.10 we have
E, rf^ = J2 ^" g-H""Hcm/} = 0 + • • •,
l,m ^^
t,k,l s,i,k,l
and
t,k,l,a,l3 j,m,«,fc,J,o,/3
Therefore
D,, = -(Ao)<r + -"
= -2:ey2-i2^RM0)ysQ^^e,e.
t J,t,K,l
■^64 X^ Rjtki{0)Rima(3{0)yjymekei€aef) + • • •
^ ~ XI 5^ "^ 54 XI Rjthi{^)Rtma(3{^)yjymekeiec,€f) + • • •.
Let {ni, • • •, ^n} be the new Chern roots, and recall Trick formula 4.3.2
n
Rtjki = -27r2j'i^«(^fe?^0(<^2«-i,t<52«,j -hs^ths-ij)^
s = l
and denote
. 2n 2n
then
n
s = l
and
g2 2^
°<r = -Xl^+64 XI Rjthi{0)Rtma^{0)yjymekeiec€^ + • • •

210 DIRA C OPERATOR
By using Lemma 4.4.1 the equation
limV47rt^'*ir(t,0,---,0) = 1
hfis a solution
H{t,yir-^y2n) = n 1
V^Ws
87rsinh>^^^*
•exp
(-
T; )(2/2.~l + 2/2J
8
-(coth ■
Consequently,
lim tr G(t,p,p) = lim tr H(t, 0, • • •, 0) = lim tr TT(
1 ^,
«=1
Airt sinh ^.
where
^. = ^^Ws = ^^7r^n,(^40),^^(0))e«e^.
a,^
Proposition 5.3.1 reads
^^{^ti"'^tj= {
0, if m < 2n;
[ (V^)n^(^ir"^^2n), ifm = 2n.
Thus if Ai ^7^ n, then
^ . 1
Denote
|1?S*' (f-^^- •••^•*) = Jli^*"^ ((^f**""»^n •••W^.J = 0.
a,/}

6.4 Index of Dvrac Operator 211
where {a;i, • • • ,cc^2n} is the coframe dual to {^i, • • •, E2n}^ then
t'r((^rW^M---W^.n) = W.A---AT^.J(^i(0),...,£2„(0))
and
Similarly we can get
limt'r(^/(ei,...,e„)) = (/(P^i,---,W^n))(^i(0),---,^2n(0)),
where / is a power series. Note Ug is thought of as a 2-form, so
Therefore if we let
. sinh^.'
then we have
Let
limt'rG(<,p,p)=^limt'r(lM.,...,«.„))
= 7^/(2^wi, • • •, 27r«„)(^i(0), • • •, E^nm
= (4^(4vrr/(y, • • •, y)(^i(0), • • •,-E;2„(0))
= (n-|Ml)(^i(0)'-'-^2'.(0))-
« = 1 2
J-'; sinh ^
«=1 -^
Just like L defined in Definition 4.2.10, A is a symmetric polynomial of the
variables n^, • • •, n^, hence it can be expressed in terms of pi,p25 * * *• As usual
we call A(v) an A-characteristic form, and call

212 DIRA C OPERATOR
the A-genus and the A-density of M respectively. Then we can write down
the well-known Atiyah-Singer index theorem for Dirac operator as follows.
Theorem 6.4.2 For the Dirac operator
we have
and consequently
t—*o *i
Ind {D'^) = A-genus of M.
At this stage we would like to think what we have done in this section is just
to give a formal proof for the local index theorem and the Atiyah-Singer index
theorem for Dirac operator, because we have used the Chern root algorithm.

CHAPTER 7
LOCAL INDEX THEOREMS
In Chapter 4 §4 and Chapter 6 §4 we proved the Atiyah-Singer index
theorem and the local index theorem for signature operator and Dirac operator by
using Chern root algorithm. In the process of the proof we took the equality
lim tr G{t,p,p) = lim tr H{t, 0)
for granted, but which is by no means trivial because neither the equality
G{t,q,p) = H{t,y)
nor the equality
G{t,p,p) = H{t,0)
is true, where y is the coordinates of q and G{t,q,p) is the fundamental solution
for the heat operator (^ -f- (c2 + <5)^) or (^ -f- D) . For the operators over a
manifold of dim 2n, the cisymptotic theorem reads
^ ^ t=0
so we have
lim t^r Git,p,p) = lim t^ ^ g\.-"£.(.)(p,p)+ t', _l_tr(")(p,p).
Hence to prove
lim tr G(t,p,p) = lim tr H(t, 0)
is equivalent to prove
A FACT : For the concerning operator
D : T{E) -^ T{E)
213

214 LOCAL INDEX THEOREMS
over a Riemannian manifold M of dim 2n, the following (i) and (ii) are true
(i) For t < n
(ii)
where the right-hand side of the equality is the result computed by using
Chern root algorithm.
The fact plainly implies that the Mckean-Singer problem 3.4.11 hcis an
affirmative answer. This chapter is devoted to proving the fa^t in several cases of
operators.
7.1 Local Index Theorem for Dirac Operator
Let M be an oriented Riemannian manifold of dim 2n with a Spin(2ny
structure Spin{M), and D^ ; r(^+) —> r(^") be the Dirac operator. And let
G{t,q,p) and
_p1 oo
ff ■ ^ ^ "
^(''^'^)^(4;^g''^^'^(^'«)
be the fundamental solution and the MP Parametrix for the heat operator
^ -^ n respectively, where q is near to p and U^^\p,q) : Ep ^^ Eg is a. linear
map determined by Lemma 3.1.1, i.e.
A
(V<i + ^ + ^W^'Hq-, «) = (Ao + F)U('-'\q; v)
where v £ Ep and F = — |-R due to Weitzenbock formula 6.3.2. When fix
the point p we can denote
where U<^'\p,q) : Ep ^ Eg and U^'\q) : Eg ^ Eg are linear maps, //^ is the
parallel transformation along geodesies from p to q. Now it is easy to see that
t^'r {U^'\p,p) : Ep ^ Ep} ^tv U^^^p).

7.1 Local Index Theorem for Dtrac Operator 215
And we will try to compute tr U^^\p) later.
As we did in Chapter 6 §4, choose a such that pa = {Ei, • • •, E2n} is an
orthonormal moving frame parallel along geodesies parsing though the point p,
and denote the normal coordinate system centering at p and compatible with
{^1, • • •, E2n} by {t/i, • • •, y2n}' Let Mo- be the definition domain of (j, then
define maps
C^W-M, ^C2n(-l)0C, Vz,
such that
It is reminded that as define Hom(^g,^g), we need the following action
Spin(2n) x C2n(-1) -^ C2n(-1) : {g,a) ^ gag~^.
Lemma 7.1.1 Let o", U^^\p,q), U^^\q), ui^\q) be defined as above,
then
Ct U^'\p,p) = tT Ui'\q),
and
{U^'\p,q)}Hp),vo) = (a(g), Ui'\q) ■ vo).
Proof The first cissertion is trivial, let us prove the second. We have
A
d
and
A 1 V—\ A 1 ,^-^
tj,m
SO
Then
ma{plvo) = {a{q),vo).
{U^'\p.q)}v = U('\q){l/lv) = U(^\q){{cr{qlvo))
= {^{q),ui'\q)){a{q),vo) = {a{q)M'\q)'Vo)^
where v = {o'{p),vo). Therefore the lemma is proved.

216 LOCAL INDEX THEOREMS
For a differential operator
which can be expressed as a composition of covariant derivatives and T{My
maps plainly, there exists a unique differential operator
U:C°°{M^,C2„i-l)®C)^C°°(M^,C2„(-l)®C)
such that
C{a,f- vo) = {<T, (C.f) ■ vo), V/ e C°°(M,,C2„(-1) ® C), ^vo e 5,
where C°°(M<r,C2n(-l) ® C) is the set of all C°°-maps from M„ to
C2„(-1)®C . Thus for
D = -(Ao) + ^R : r(^) ^ T{E)
we have D^ = — (Ao)<r + ^R, which is a differential operator on
C°°(M<„C2„(-1)®C).
Lemma 7.1.2 Let
£^W:M, ^C2„(-1)(8>C
be defined as above, then the equation in Lemma 3.1.1 turns out to be
W+i + f)
ui-^\-) = 0
c/-f \o) = 1.
id+i + ^)ui^' = -oM'-'^
Proof Because
A A
iVi + i+ 3^)U^'\<1\ v) = (Vj + i + ^)K Ui'^. vo)
AG'
AG'
A

7.1 Local Index Theorem for Dtrac Operator 217
A
and
(Ao + F)U^'-'\q; v) = (Aq + F){a, U*^'^^ ■ vo)
by the equation in Lemma 3.1.1 we have
A
So the lemma is true.
The expression of B^ is too complicated to be written in Lemma 7.1.2, so a
good thing we can do is to list the main terms of O^, The following Definition
7.1.3 can be used to rank terms both for D^ and for Us^K
Definition 7.1.^ Let Ma- be the definition domain of a.
(i) If a differential operator
a : C~(M^,C2n(-l) 0 C) ^ C~(M^, C2n(-1) 0 C)
can be expressed as
then we denote
X(a) < m + 5 - z/(y?,, ... y?,^^ J,
where (pi^ are functions, and i^{(pii '"^im+i) ^^ the order of the zero of
i^ii • • • ^im+i) with respect to the variables y,.
(ii) If an operator
13 : C~(M,, C2n(-1) 0 C) ^ C~(M,, C2n(-1) 0 C)
N
can be expressed as 13 = TJ a, with
t=i
X(at) < m, Vi,
then we denote
X{I3) < m.

218 LOCAL INDEX THEOREMS
(iii) Let x{0) < ^ mean x{0) <rn—l. Then denote the set
{^\xm < a}
by (x < ^)^ ^^^ 1^^ ^he notation
A = ^92 + (X < «)
mean that if there exists a ^ G (x < ^) such that /3i = 02-^ Po-
(iv) For U e C~(M^,C2n(-l) O C), if it can be expressed as C^ = ^(1) for
some
^ : C°°(M^,C2„(-1) ® C) ^ C°°(M,,C2„(-1) ® C)
with x(/?) < "^) then we denote xC^) < ^t where 1 means the map
l:M^^C2„(-l)(8>C:gK^l.
Proposition 7.1.4 Let □ be the square of the Dirac operator, then
D, = -(Ao). '^^R=-J2dy2-4J2 ^jMO)yj g-ekej
■^64 XI ^jtfe/(0)i2«ma^(0)yjymefee;eae^ + (X < 2).
Proof By using the definition of Aq and Corollary 1.5.10, the
proposition can be proved by a direct computation.
Lemma 7.1.5 liU e C^(M^,C2n(-l) O C) is defined as above with
X(C^) < 2n, then
ii (U{id)) = 0.
Proof Without loss of generality we may assume U = ck(1)5 where a
was defined in Definition 7.1.3 (i). From x(<^) < 2n it follows either m —
^{^h '"^im+i) < 0 or s < 2n. If m < ^{(pi^ "'(pim+i)^ then U = a(l) = 0,
A
hence tr {U{0)) = 0 . If s < 2n, then Proposition 5.3.1 implies the conclusion.
For a map / : Ma^ —> C2n(—1) <S> C we expand / into Taylor series
oo
m=0

7.1 Local Index Theorem for Dtrac Operator 219
where /(m) is the m-th order homogeneous polynomial in the variables yi, • • •,
y2n' Sometimes we denote /(m) by f{fh).
Lemma 7.1.6 For m, i > 0, fis elements in C°°(M<r, C2n(—1) 0 C) we
have
u^'\m)= y:
""'•••""■ t^W(m + ai + --- + a.) + (x<2i),
ai,«««,a,
r(ai,...,a,;m)
r(o)
where C/<^ Ms a function with value in R, (x < 2i) is understood by Definition
7.1.3 (iv) and
«-i
r(ai,---,a,;m) = JJ(m + i-j + ai H hCKj),
the summation in the above equality runs over ai, • • •, a, such that all the
t-i
factors in TT(m + z ^ jf + ai H H ckj) are positive, and
2n g2
a, = <
if s = 2,
if 5 = 0,
~M. X^ ^jtfe/(0)i2tma^(0)yjymefee;eae^ ifs=-2,
1 fi
j,m,«,fe,J,a,^
otherwise.
Proof Owing to
rC)
d try'(m) = m£ry^(m)
('■)/
and
dG ^
4G
=(l h = y. rnh{m),
m = l
where h = log G 4, and comparing the Taylor series on both sides of the equality
in Lemma 7.1.2, we have
(m + i)ui'\m) + Yl mih{mi)ui'\m2) = -(n,C^^'-i))(m).
mi-\-m2=m , mi>0

220 LOCAL INDEX THEOREMS
Proposition 7.1.4 implies that
-{oM'~^^){m) = a2Ui'~'^\m+2)+aoUi'~^\m)+a_2U^'-^\m - 2)
where Cj G (x < 2). So we have a formula Ft,m-
Ui'\m) = ^
m + z
E
mi+m2=ni , mi>0
« J J
where £j G (x < 2). Note that
X(fc(m)) < -m ; x(«a) < 2, Va = 2,0, -2.
And applying the formula i^,,m2 ^^ ^^e formula Ft^rn several times we get
a J
where jCj G (x < 2). This equality leads to the first conclusion of the lemma
immediately. Solving the equation in Lemma 7.1.2
£^i°^(0) = 1,
we get a function (p with value in R and (p{0) = 1 such that
So the proof is complete.
Theorem 7.1.7 For i <n
t'^r U^'\p,p) = 0.
Proof Prom the equality Ui ' = <p in 7.1.6 it follows that
Ui°\m) = ip{m).

7.1 Local Index Theorem for Dtrac Operator 221
By using Lemma 7.1.6 we have
Owing to the equality
X{aa, ''' aa,(p{ai + • • • + a,)) < 2i < 2n
and Lemmas 7.1.1 and 7.1.5, we have
t'i: U^'\p,p) =tv Ui'\0) =tv Ui'\0) = 0.
The proof is complete.
Lemma 7.1.8
(ai,...,an)
where the summation runs over the set
{(ai,---,an)|«t = -2 or 2, and ai H f-CKn = 0}.
Proof By using Lemma 7.1.6 we have
nr. ... nr V"l' * * * '"n? ";
If ai H h CKn > 0, then
X(^(«i+ ••• + ««)) <0,
and if ai + \- an < 0, then
^(ai + httn) = 0.
So in the above two cases
V r(ai,...,an;0) /
Therefore we have

222 LOCAL INDEX THEOREMS
Now it is plainly that if we can prove the following lemma then the theorem is
true.
Lemma 7.1.9 If one among a^i, • * * ? «an is ao, then
Ki •••«««!) = 0 + (x <2n).
Proof If Ga^ = tto, then (a^i • • • a^^l) = 0, hence the lemma is true. If
ttan ¥^ ^0, we try to move ao to the place occupied by a^^. Hence we need to
consider the commutators. First we have
1 /^ /^
[ao,a2] = -7:y]Rstki{0)^—^-ekei = (}'^{x < 4),
[ao,a_2] = -— ^i25ja6(0)i2jtfe/(0)i2tma/3(0)y5ymeae6efee;eae^ + (X < 4).
Then because
[ao,a_2] = -Y^^i25ja6(0)i2jtfe/(0)i2tma^(0)y,ymeae6efee,eae^ + (x < 4)
e^ + (X < 4)
~ 128 X ^«o6(0)i2tjfe/(0)i2j5a^(0)ymy5eae^efee;eae6 + (x < 4)
= -[ao,a_2] + (x <4),
we further have
[ao,a-2] = 0 + (x<4).
Without loss of generality we may assume aa^ = oq. Then
^ai •••^an — ^O^aa * * * ^a^
= (bo, Oa Jaa3 • • • aa„) + (^a J^q, tta^] ' ' • ^a^) + • • •
+ (aa2«a3 • • • [«0, ttan]) + ^aa * * * ^an^O
= aa2---aa„ao + (x < 2n),
and hence
Ki •••«««!) = 0 + (x <2n).
So the proof is complete.

7.1 Local Index Theorem for Dtrac Operator 223
As we did in Chapter 6 §4, we define
2 ^
and let A be the matrix whose (i, jf)-entry is Aty Then the (i, jf)-entry of ^*^ is
A^ - S^ A A ...4
Let
In general we have
xK)<2fc, xiA''iy))<2k-2.
Note that A is antisymmetric, so A^fe+i jg ^jg^ antisymmetric up to adding
terms in (x < 4Ai + 2). It immediately implies
X(tr A^^-^^) < 4ik + 2, x{A^^'^\y)) < 4k.
A
In order to compute a term tr {{a^^ • • • aa^l){0)} in Lemma 7.1.8, where a^, =
a2 or a_2, we try to move a2 to the place occupied by a^^, and hence
need to consider the commutators, or more exactly, to consider the Lie algebra
generated by {a2,a_2}. Recall the notations in Lemma 7.1.6, we have
Let
then
a2 -
a-2
»2Jk _
Zn
-T.
t = l
= —
•.J
32
16^
>
(y)-
d
[a2,A^\y)] = 2 • tr^^' + 2-B^' + {x< 41),
[B^\A^\y)] = 2 • ^2fc+2J(y) + (^ < 4fc + 4i _ 2),
[a2,tr^2'] = 0,
[B^'',tiA^'] = 0 + (x < 4fc + 4/).

224 LOCAL INDEX THEOREMS
If we let A^^^^ be the set of polynomials generated by
{A^''iy),txA^'\k,l^l,2,---},
then by using the above commutators we can move the differential operators
a2 or B^^ acting on A^^^^ towards the right. If we start at aa^ • • • a^^ , by
using these moves, a^i • • -CLan ^^ changed into a sum of several terms up to
adding terms in (x < 2n), some of these several terms are in A^^^^ and the
others are terminated by 03 or B^^. It shows that
hence
Ki---«anl)(0)
is a polynomial of the variables trA^, trA^, • • • with adding a term in (x < 2n).
Thus we come to
Lemma 7.1.10 We had shown a way to determine a polynomial
such that
C^i")(0) = F(trA2,..., trA2% ...) + (X < 2n),
where //(/3i, • • • ,/3„i) G R.
Lemma 7.1.11 Let R[xi,---,Xn] be the polynomial ring generated
by xi,---,Xn , and let
be the solution to the equation
(a 2n ^2 1 "
,=1 "'^' " j=i
r(-i)(-) = 0
[ K(°)(0) = 1.
Then
F(")(0) = F{-2{xl + ... + xl), •••, {-iy2ix{ + ... + x'J,. •.),

7.1 Local Index Theorem for Dtrac Operator
where F{zi^ • • •, 2;,, • • •) is determined in Lemma 7.1.10.
225
Proof Let
X =
/ 0 xi
-xi 0
\
\
0 Xn
-Xn 0 /
and let
A = X
hj
Then
A\y) = -J2''i(y2i-i'^y2i)'
1=1
So the equation in Lemma 7.1.11 is
«=i ^'
f(-i)(-) = 0
[ FW(0) = 1.
By using the same reasoning as that in Lemmas 7.1.6 and 7.1.8 we get
W- 2. r(ai,...,a„;0)'
where the summation runs over the set
{(ai,---,an)|«t = -2 or 2, and ai H [-«« = 0},
and
16

226 LOCAL INDEX THEOREMS
Comparing the above expression of V^"'\(}) with that of Ua^\(}) in Lemma
7.1.8, we find that they are similar. So we will deal with the expression of
V^^\0) by the same way for getting lemma 7.1.10. Let
then it is easy to check that
[a2,A^'iy)] = 2-tvA^' + 2-B^',
\a2,tvA^'] = 0,
[52*,trl2'] = 0.
Thus by the same procedure for getting F{zi,- • •) in Lemma 7.1.10 we get the
same F{zi,- • •) and the equality
y(")(0) = F(trP, • • •, trP% • • •).
Owing to
tiA^' ^{-iy2-{xl' + --- + xl')
we have
The proof is complete.
Lemma 7.1.12 Let
FW:M,^R[xi,...,Xn]
be the solution of the equation in Lemma 7.1.11, then
F(")(0) = /(n),
where

7.1 Local Index Theorem for Dtrac Operator 227
Proof Note that {F^'^l i = 0,1, •• •} satisfies the equation in Lemma
7.1.11. So if we let
_kii oo
then ir(t, t/i, • • •, y2n) satisfies the equation
( f) 2n ^2 1 ^
limV47r<^"ff(t,0,---,0)= 1.
t->0
By using Lemma 4.4.2 the above equation has a solution
Mt,m,-,.3„) = n{^;^;^^
•exp
—-I (coth W2S-1 + 2/2.)
))
Because
e-^r- 5=1 smh ^ ^ ^
_ (47rt)"
•exp
{|:(5('-^co.h£i^)„i._,+„i,))}
can be expanded into a power series with respect to the variable t, the set of
the coefficients { — Z{i)} satisfies the equation in Lemma 7.1.11, and hence
by the uniqueness of solutions we have
1 ^,
especially
F(")(0) = ^Z{n)\y=^ = /(n).
The proof is complete.
Lemma 7.1.13 The polynomial
2{(3i + 2/32+'"+m(3m)=n

228 LOCAL INDEX THEOREMS
determined in Lemma 7.1.10, satisfies
Fi-2{xl + ... + xl),..., (-I)^2(x2' + •.. + xl%•..)
2
Moreover, the above equality determines F uniquely.
Proof Lemmcis 7.1.11 and 7.1.12 imply the first part of the lemma.
Note that /(n) is an Tvth order symmetric polynomial with respect to the
variables xi, • • •, Xy^, so by using the fundamental theorem for the symmetric
polynomials we get the second part of the lemma.
Theorem 7.1.14 For the Dirac operator over an oriented Riemannian
manifold of dim 2n, we have
where /(ni, • • •, Un){n) means the n-th order homogeneous polynomial in
Taylor series of /(ni, • • •, n^), and {t^i, • • •, t^n} are Chern roots defined in
Definition 4.2.8(1), and the right-hand side of the equality is understood by 4.2.8(2).
Proof By using Lemmas 7.1.1, 7.1.5 and 7.1.10 we have
ti U^''\p,p) ^t'i: Ui^'^O) =tv F{tvA\ • • •, tvA^'r ' •)•
Now we examine trA^* and tr F{tvA^, • • •, trA^*, • • •). By using the definition
of Atj in Chapter 6 §4, we have
a,/3 a,(3
If we define
then
a,/3
A^J = -lT.^^JiE-^^
a,(3
A --ii

7.1 Local Index Theorem for Dtrac Operator 229
Due to the equality
A 2^*
tr (eti • • • e,,J = /-y (^ti A • • • A ic^t2n)(^i, * * *, ^2n)
in Proposition 5.3.1, we have
tiF{tiA^---,tiA^\---) = -^=-F{tTA\---,tiA'',---){Ei,---,E2n)
(v-l)"
(V-l)"
Define .^, G Polyo(50(2fe),R) by
where X is an antisymmetric matrix of rank 2n, then tr 1^^* = (f)s{fl). By
using the notations in Lemma 4.2.4, we have
t*{<f>,){x) = M^*{^)) = tr((i*(x)2^) = (-1)^2 • (xf + • • • + xl')
= i-iy2 . ((27rMa(x))2« + • • • + i2^u„ix))l')
= {-iy2-i2nr-{ur+...+ui'){x),
hence
i*(^.) = (-ir2.(27r)2'.« + ... + «2«).
By Definition 4.2.8(2), the Chern root expression of <t>s{Ct) is
(-iy2.(2-Kf'.(ul' + ... + ul'),
where n, G Poly(T",R), thus the Chern root expression of
F{---,tTn'\.--) = F{---,M^)r--)
is
F{...,{-iy2.{2-Ky'.{ul' + ... + ul'),.--).
So
F(---,tr02*,---) = -F'("-.(-l)'2-(27r)2'.(«f+ ... + «2^),...)(V).
If we denote the above equality simply by
F(---, tiil'',-••) = F{---, i-iy2 ■ (27r)2* • (ul' + • • • + u^*), • • •),

230 LOCAL INDEX THEOREMS
then
2"
-F(.--,tr02«,...)
on / " 3«i \
= ;— (27rA/^)" TT ^-f^ (n)
= (47r)
Therefore we have
(n «jL \
(47r)" ^^'^-^ (4-nr)" "^ "" '
(4^)
^-t'rF(trA2,...,trA2%...)
^ ^" -F(tr02,---,tr02',---)(£;i,---,^2n)
(47r)» (v'^T)"'
= {(ns;^)'""'}*^--'^'"'
The proof is complete.
In the above proof we have used the Chern root expression but had not used
the Chern root algorithm. The Chern root expression is a good mathematical
notion described in Chapter 4 §2 while the Chern root algorithm described in
Chapter 4 §3 contains some mysteries. Due to the equality
{(nslf)r«)}(^..--.^..)={(n;^)}(^..--.^.).
Theorem 7.1.14 with Theorem 7.1.7 is the following
Theorem 7.1.15 Let M be an oriented Riemannian manifold of dim 2n
with a 5pin(2n)-structure, and D+ : T{E'^) -^ T{E~) be the Dirac operator.
And let G{t,q^p) be the fundamental solution for the heat operator ^ + D,
then

7.2 Local Index Theorem for Signature Operator 231
A
(i) The limit lim tr G{t,p,p) exists,
(ii)
lim t; Git,p,p) = I (n J^) } (E.,...,E.„).
Usually Theorem 7.1.14 with Theorem 7.1.7 is called the local index
theorem for Dirac operator, in which 74^ tr U^'^\p^p) is called the local index
by Definition 3.4.8. Due to the asymptotic theorem 3.3.1, Theorem 7.1.14 with
Theorem 7.1.7 is equivalent to Theorem 7.1.15, so we would like to call The-
A
orem 7.1.15 and lim tr G{t^p,p) the local index theorem and the local
index for Dirac operator respectively.
7.2 Local Index Theorem for Signature Operator
Let M be an oriented Riemannian manifold of dim 2n,
£>+ = d + 6 : A+(M) -^ A_ (M)
be the signature operator, which is defined in Definition 1.4.18 and first
discussed in Chapter 2 §5. In Chapter 4 §4 we had used the Chern root algorithm
to prove the Atiyah-Singer index theorem for it (Theorem 4.4.4). Now we are
going to prove the index theorem by the traditional reasoning. Let us recall
the super structure
A*(M) 0 C = A+(M) + A_(M),
and
D+ = (1-^6: A+{M) -^ A_(M),
in Lemma 1.4.17, and define
Then Weitzenbock formula (Theorem 1.4.16) and Proposition 5.3.3 read

232 LOCAL INDEX THEOREMS
and
^ f (-4x/=T)'*, ifp=0andg = 2n,
( 0, otherwise,
where 1 < ii < • • • < ip < 2n, 1 < jfi < • • • < jfg < 2n. They are the two basic
formulcis which are the points we start at.
First of all let us explain A±{M) as T{E±) just as we did in Exercise
1.8.4. Let SO{M) be the tangent 50(2n)-principal bundle, and A* (2n) be the
Grassmann algebra of rank 2n with an 50(2n)-action p on it, the action is
defined in Definition 1.8.3. Then if let E = SO{M) x^ A* (2n), we have
A*(M)(8)C = r(£^).
Further, we had defined * in Definition 1.4.4 and r in Lemma 1.4.17, now define
*:A;(2n)^A;(2n)
and
r:A;(2n)^A;(2n)
in a similar way. (Please give full details of them.) Then we define
A±(2n) = {ee A;(2n)|re = ±6),
and
and we have
E± =SO{M) XpA±(2n),
A±(M) = r(^±).
For p £ M, Choose an orthonormal moving frame (j = {jE7i, • • •, E2n} in
MF(50(M)), which is parallel along geodesies passing though the point p,
and denote the normal coordinate system centering at p and compatible with
{jE7i, • • •, E2n} by {t/i, • • •, y2n}' Let Mo- be the definition domain of (J, define
maps
irW:M^^Endc(A;(2n)), Vi,
as we did in §1, then
U^'\p,q)icr,Vo)) = {CT, Ui'\q) ■ Vo), V ^o G Ki^n),
where U^^\p,q) are given by MP parametrix in Theorem 3.3.1. And we also
have the following 7.2.1-7.2.9 similar to Lemma 7.1.1-7.1.9.

7.2 Local Index Theorem for Signature Operator 233
Lemma 7.2.1 Let (j, U^^\p,q), TJa be defined as above, then
tWW(p,p)=tWW(0),
and
{Cr(')(p, g)}(or(p), «o) = {cr{q), Ui'\q) ■ «o).
Lemma 7.2.2 The map
C^i'):M^^Endc(A*(2n))
satisfies
ui-'\-) = 0
ui°\o) = 1.
Definition 7.2.3 If a differential operator
a : C°°(M<„A* (2n)) ^ C°°(M^, A;(2n))
can be expressed as
then we denote
And the notations
xiP) <m, (x < m), ^1 = ^2 + (x < rn)
are defined in the same way as in Definition 7.1.3. Moreover, for U G C^{Ma, A*
we can also define x(^) < ^•
Proposition 7.2.4
"^64 J2 ^jthiWRtma(3{(^)yjymej,ej e^e^
j,m,t,k,l,a,(3
-g E ^^M^X^t^^^r + (x < 2).
7 Jb /

234 LOCAL INDEX THEOREMS
Let C2n(—1) is a Clifford algebra generated by {ej , • • •, €2^}, C'2n(+1) =
C2n(-1) and let C2„(-l) be a C2n(-l)-algebra generated by {e^", • • •, ej„},
and Cf'yf (+1) be a C2n(—l)-niodule (or C2n( — 1)-vector space), spanned by
{e+ • • • e+ I s > 1 and 1 < H < " - < is < 2n}.
Then as C2n(—l)-modules,
C2*„(+1) = C'°„(+1) + C2>°(+1).
Let
^0 : C2*„(+l) ^ C°„(+l)
be the projection, which is a C2n(—l)-niodule homomorphism of course.
Lemma 7.2.5 For U G C~(M^, A;^(2n)) with x{U) < 2n, then
tv (U) = 0.
Lemma 7.2.6 For m,i > 0, we have
ui'\m)= y:
Cl(Xi ' ' ' ^n
T{ai,---,a,;m)
^^—nr{m + ai + • • • + a.) + (X < 2i),
ai,-..,a,
where C/"^^) is a function with value in C,
t-i
r(ai,---,a,;m) = ]][(m + i-jf+ ai + f-CKj),
the summation in the above equality runs over such ai, • • •, a, that all the
t-i
factors in TT(m + z — jf + ai + H cKj) are positive, and
( 2n ;^2
1 _ /)
as = <
ifs = 2,
+ ^ E ^.jfe/(0)e+e;e-ef if . = 0,
«,j,fe,'
~64 X^ ^jtfe/(0)i2«ma^(0)yjymefee, e^e^ ifs=-2.
j,m,t,fe,/,a,^
I 0
otherwise.

1.2 Local Index Theorem for Signature Operator 235
Theorem 7.2.7 For i < n
Lemma 7.2.8
where 62 = ^2, 6_2 = a_2 and
^0=8^ ^tjfe/(0)e+e+e;^e,-.
Lemma 7.2.8 is a corollary of Lemma 7.2.9.
Lemma 7.2.9
{aa,'"aaj) - (6ai • • • 6or, 1) + (X < 2n).
Lemma 7.2.10 There is a way, similar to that in Lemma 7.1.10, to
determine a polynomial
2(^i+2^2+-+m^„,)=n
such that
£^W(o) = F{ixA\---,ixA^',---) + {x < 2n) + C2>„°(+l),
where A is a matrix whose (i, j)-entry is
proof We can prove that the element
oc,^ oil,' •,a2m
can be expressed in terms of trA^, • • •, trA^"*° plus an element in (x < 4mo),
where mo is the largest integer less than or equal to y. By using this fact with
the reasoning in the proof of Lemma 7.1.10 we have Lemma 7.2.10.

236 LOCAL INDEX THEOREMS
Lemma 7.2.11 Let R[aji, • • •, ajn, ^i, • * *, ^n] be the polynomial ring
generated by a^i, • * *, 2J„, ^i, • • •, ^n , and let
be the solution to the equation
[ f(°)(o) = i.
Then
5ro{F(")(0)} = Fi-2{xl + • • • + xl), • • •, i-iy2(x{ + • • • + <), • • •),
where F{zi, • • •, 2;,, • • •) is determined in Lemma 7.2.10, and ttq = 7r2 o tti,
which does not contain any ^^,
Proof If we define
^'J ^ n^^jM^)^a^^^
a,(3
and
^-^ = -T^T.y^y^K = -T^^'iy)^
U'^"'"'' 'J 16
1 *'■' _ _ 1
then the equation in Lemma 7.2.1 turns out to be
(d +i + ^)Ui'^ = (62 + 60 + 6_2 + • • -Wi'-'^
ui''\o) = 1.

7.2 Local Index Theorem for Signature Operator 237
Now we define A — X, where X is a matrix defined in the proof of Lemma
7.1.11, and define
b-2 =
1 r-
16^
^'(2/)=TfiS)'''(^2'-i + ^2')'
16
j=i
*•« = IE) ^'j^t^t = ^J2 ""'^^'
'J
»=i
where ^^ = e2g_ie2g- Then we have an equation similar to that in Lemma
7.2.2, which is
{d +i)Fi') = (62 + &o + 6-2)Fi*"^)
V^-^\-) = 0
Vi°^(0) = 1.
The procedure for getting
from the equation satisfied by U^' can also induce a similar equality
«.4^a r(ai,...,a„;0)
from the equation satisfied by V^ - Applying some commutators
simultaneously on the expressions
{hg^ --hg^l)
and yj
(^ai '-Kn 1)
a. ...a r(ai,.--,an;0) f^ r(ai, •..,«„; 0)'
they are changed into the right-hand sides of the following equalities
respectively
tri")(0) = Y^iY^ Aa^ete+rF,n{irA\- ■ ■, trA^™") + (x < 2n),
Vi"^(0) = X)(E ^^e+e+)™F™(trl2,. • •, trP™o)
= 5];(46o)'" J-^CtrP, • • •, trP™o).

238 LOCAL INDEX THEOREMS
Further, the right-hand sides can be written as
F(trA^ • • •) + (X < 2n) and F(trP,...).
Similarly, as far as V^"^^ (0) is concerned, we also have
2
It is easy to check that
s a,(3 s
SO
m s
= F(trl2^...)-
Due to
trP"* = (-1)"*2 • (2^5" + • • • + C)
the proof is complete.
Lemma 7.2.12 Let
be defined in Lemma 7.2.11, then
io(F(»)(0)) = /(n),
where
n >/—1 or.
Proof By using the consideration in Lemma 7.1.12 we solve a heat
equation corresponding to the equation in Lemma 7.2.11 and get

7,2 Local Index Theorem for Signature Operator 239
where
{'T^ / 1 ., y/^Xgt . ^/-A Xgt^. 2 2 X 1 > A 1
^ Vit ^ 2 2 ^^^2.-^ "^ ^2 J + 2 ^*^* V I '
The terms in Zo{t), which contain odd orders of ^,, will be discarded in
i^(F(")(0)) , so
^iWie=-i = TT (^^^—■<^08h{^Q^}\,
the lemma is proved.
Lemma 7.2.13 The polynomial
2((3i + 2/32+ +m/3m)^n
determined in Lemma 7.2.10, satisfies
F{-2{xl + ... + xl),..., (-iy2(xr +... + xl'),...)
where
Therefore
Due to
where /(jci, • • •, Xn){n) means the n-th order homogeneous polynomial in
Taylor series of /(xi, • • •, x^)- Moreover, the above equality determines F uniquely.

240 LOCAL INDEX THEOREMS
Theorem 7.2.14 For the signature operator over an oriented Rieman-
nian manifold of dim 2n, we have
ii^''"""'("•'') = {(r[5:^)(8))(^..-.^-).
where {^i, • • •, n„} are Chern roots, and the right-hand side of the equality is
understood by Definition 4.2.8 (2).
Proof By Lemmas 7.2.5 and 7.2.10 we have
If we define
then Proposition 5.3.3 is
t^r (e- •• -e-J = (-4A/=I)"(a;.. A--■ Au;„J{Ei,--■ ,E2n),
hence
t'rF(tr^2^---,tr^2.^...)^(_4^/n)nj?(trl2,---,trl2^,.-.)(^i,---,^2«)
= {-4^/^rF{tI n^--- ,ti n^\---)(Ei,--- ,E2n).
It is easy to see that the Chern root expression of F{- • •, trfl^*, • • •) is
F{...,i-iy2.{2irf'-iul' + .-- + ul'),---).
Therefore
^ t'r £^(")(p,p) = ^ t'r FitrA\...,trA'\...)
7r» yj-J-^ tanh ^/^ iru, J ^ '
'^" ^ ' \^^^^A tanhu.y ^
= (n ^) (^)-

1,3 Local Index Theorem for de Rham-Hodge Operator 241
Thus the proof is complete.
Theorem 7.2.15 Let M be an oriented Riemannian manifold of dim
2n , and let D^ = d-^S : A+(M) -^ A_(M) be the signature operator. And let
D = (d + (5)^, and G(t, q^p) be the fundamental solution for the heat operator
^ + D, then
A
(i) the limit lim tr G{t,p,p) exists,
(ii)
The above theorem is called the local index theorem for signature
operator.
7.3 Local Index Theorem for de Rham-Hodge Operator
Let M be an oriented Riemannian manifold of dim 2n,
Do = d-^-S: A^^^^(M) -^ A^^^(M)
be the de Rham-Hodge operator, which is defined in Definition 1.4.12. Let us
recall the super structure
and we define
and
A*(M) = A^^^(M) + A^^^(M),
Di = d'Jt-6: A^^^(M) -^ A^^^(M),
n = (d-^(5)2:A*(M)-^A*(M),
D+ = DiDo, □- = DqDi,
Let SO{M) be the tangent 50(2n)-principal bundle, and Aj^(2n) be the Grass-
mann algebra of rank 2n with an 50(2n)-action p on it, and let
E:=zSO{M) XpA;(2n),
^+ = 50(M)XpA^^^^(2n),
E_ =SO{M) XpA^^^(2n),

242 LOCAL INDEX THEOREMS
then we have
A*{M) = T{E),
A^^^^(M) = T{E+),
A°^^{M) = T{E_).
It shows that the super structure
A;(2n) = Ar"(2n) + A°'id(2n)
induces the super structure
r(^) = r(^+) + r(^-).
For p G M, in MF(50(M)) choose an orthonormal moving frame a = {^i, • • •,
E2n}y which is parallel along geodesies passing though the point p, and denote
the normal coordinate system centering at pand compatible with {jE7i, • • •, E2n}
by {yi, • • •, y2n}' Let Ma^ be the definition domain of cr, define maps
Ui'^ : M, ^ Enda(A;(2n)), Vz,
as before, then
U^'Hp,q)ia{p),vo) = i(T,Ui'H<l)' «o), V t>o G A;(2n).
The Weitzenbock formula (Theorem 1.4.16) and Proposition 5.3.2 are
= -Ao - J X) RnkiKK^kEr + \r,
and
8
^ [0, if p / 2n or g / 2n;
i (-4)", ifp=g = 2n,
where 1 < zi < • • • < i^ < 2n and 1 < ji < • • • < Jg < 2n.
We follow the way in §2 to deal with the local index theorem for de Rham-
Hodge operator. We omit 7.3.1-7.3.2 which are corresponding to 7.2.1-7.2.2,
and we also omit 7.3.6-7.3.7, 7.3.9-7.3.13, because they can be proved in the
same way as the corresponding theorems in §2.
Definition 7.3.3 If a differential operator
a : C°°(M,, A;(2n)) ^ C°°{M,,h^{2n))

7 3 Local Index Theorem for de Rham,-Hodge Operator 243
can be expressed as
"="'' h:"'- • • • 5^^'-^- • • • '-'^" • • • '^"''
then we denote
X{a) <m- u{(pi^ ... y?,^^ J + a + 6.
Proposition 7.3.4 There holds
hJ,Jc^l
Lemma 7.3.5 If U is defined as above with x(^) < 4n, then
tv {U{0)) = 0.
Lemma 7.3.8
^^^0) = ^il Yl R^M^^We^e^err + (X < 4m).
hj,k,i
Theorem 7.3.14
1 tWW(p,p) = (ni...nn)(^l,--,^2n),
(47r)"
where ni, • • •, n^ are Chern roots, {ui -- -Un) is a Chern root expression of a
characteristic form, {jE7i, • • •, E2n} is an oriented orthonormal moving frame.
Proof First we have
(47r)" ''^''^' (47r)
^ ^ hJ,k,l
~ (47r)"8"n! ^ •^l«2jlj2(P) * * * •^2n-l«2nj2n-lj2n
•6(zi, 22, • • • 5 *2n-l*2n)^0l, J2, * * * , J2n-lj2n)
— /^\ngn^] / ^^«l«2(-^Jl?-^j2) * * * ^«2n-l«2n (^J2n-1 ' ^J2n )
•6(zi, 22, • • • 5 i2n-l*2n)^(jl, J2, * * * , J2n-lj2n)
~ (7r)«8"n! ^ ^'I'^C^Jl ^^J^)'" ^«2n-l«2n (^J2n-1 ' ^J2n )
•^(n, ^2, • • • , i2n-li2n)(t*^ji A 0;^, A • • • A t*^j2,_it*^j2 J(^l, • • • , ^2n)

244 LOCAL INDEX THEOREMS
^ (x)"4"n! ^^^'^' *^' ■ ■ ■' *2n-i*2«)
•fill,J A--- Afi,j„_i,2„(^l,---,^2n),
then by Definitions 4.1.4 and 4.1.6 we have
(^)n4n^| Z) ^(*1' *2' • • • ' »2«-l*2n)0,„, A • • • A «,,„_,.,„
1
= (^^PfW = ^W-
Furthermore, by Lemma 4.2.4, the Chern root expression of £^ is ni • • -Un- So
the proof is complete.
By using the above discussions we come to a theorem similar to Theorems
7.1.15 and 7.2.15.
Theorem 7.3.15 Let M be an oriented Riemannian manifold of dim
2n, and let
A;(2n) = A^^^^(2n) + Af^{2n)
be a super structure. Then we can construct A* (M) and
n = (d + (5)2:A*(M)-^A*(M),
and we have a fundamental solution G{t,q,p) for the heat operator ~ + D.
Then
A
(i) The limit lim tr G{t,p,p) exists,
(ii)
A
limtr G(t,p,p) = {ui-"Un){Ei,'--,E2n)'
Theorem 7.3.15 is called the local index theorem for de Rham-Hodge
operator.
If one meets so-called twisted (Dirac, signature, de Rham-Hodge)
operators, he/she may use the method in this chapter to prove the local index
theorems. For the de Rham-Hodge-signature operator described in Exercise
1.4.19 see [32].

CHAPTER 8
RIEMANN-ROCH THEOREM
Let M be a compact manifold of dim 2n. Suppose M has a i7(n)-structure,
i.e. there exist a principal {7(n)-bundle U{M) and an identification
U{M) XpR2" = TM,
then we call M a Hermitian manifold, where R^" is a real vector space
spanned by {5i, • • • ,^2n},
p : U{n) X R2" ^ R2" : {A + ^/^5,■y) h^ />( (^ + V^5) , v)
is a left action in which p{-, •) is defined by
p{{A^yf^B), {6^,■■■,82n)) = iSl,■■■,62n)p{A + ^/^B),
and
p{A + y/^B)
<'.-:)
Sometimes we denote p{ (A + y/^B) , v) by p{ A-^- y/^B )v.
Note that U{M) is not only a principal C/'(n)-bundle, but also satisfies
U{M) XpR2" =TM,
so we call U{M) a tangent principal C/^(n)-bundle of M.
8.1 Hermitian Metric
Suppose we are given a tangent principal C/'(n)-bundle of M, we try to
develop a theory from this bundle. It is to say, we will introduce a Hermitian
metric on M, and explain that its real part is a Riemann metric while
the image part is a Kahler form on M, And we will introduce an almost
complex structure as well.
245

246 RIEMANN'-ROCH THEOREM
By the identification
U{n) XpR2" =TM
we define E^ — (cr, <5,), thus we assign a local frame field (in other words, a
moving frame) {jE7i, • • •, E2n} to cr. In R^" the standard inner product
2n 2n
(.,.) : R'" X r2" ^ R : (5]; A,6„ 5];//,6,) H-5]; A,//.
«=i j=i I
is invariant under the C/'(n)-action, then it induces a Riemann metric on M
such that
{{(T,vi),{(T,V2)) = ('yi,'y2).
It is easy to see that {^i, • • •, ^2n} is orthonormal under this Riemann metric,
and it determines a conversely. Similarly let C" be a complex vector space
spanned by {(SJ, • • •, <5^} with a standard C/'(n)-action on it, the action
Pe : U{n) X C" -► C" : (A + y/^B, v) h-. p^( A + yf^B , v)
is complex linear and satisfies
We denote the induced bundle U{M) Xp^ C"' by TcM , which is an tv
dimensional complex vector bundle. We call it a tangent complex bundle
of M, The standard Hermitian (inner) product
(., .)^ : C" X C" - C : (^ A.^^f.f^/.^^;) - E A^ ^
is invariant under the C/'(n)-action, i.e. for any g G U{n)
{g^t^g^jV = Yligkz ^l^gij ^Y = YlgkzgkJ
k,l k
= ^gki{g~^)jk = <5tj = {^t^^jV,
k
where g^j is the (z,j)-th entry of the complex matrix gf, thus the standard
Hermitian product induces a Hermitian metric on TcM,
Let E^ = {a,6^), then {^f,---,^^} is a unitary frame (field) on M,
which can be checked to satisfy

8.1 Hermthan Metric 247
Define a real linear homomorphism
such that
<f>{6^) = 6,, <t>iV^6^) = 8„+,, Vi = 1, • • •, n.
For fif = ^ + y/^B € U{n),
<l>Pc{g) {^l---,K^ ^/^*^ • • •, V^K)
= {^(^^•••,6^,^/=T6^.•.,v^6^)}
B
-:)
= (^l> • • •,^n,^n+l, •••,^2n)
A -B
B A }
= p(ff)^ (6f, . . • , 6^, V^5f, • • • , V^5^) ,
SO <^ is C/^(n)-invariant, and hence it induces a real vector bundle isomorphism
^^ : TcM -^ TM,
which may be called a realization map. For a^^bt^Ct^di G R, where t =
1,- •• ,n, let
then
t=i
n
«=1
n
1=1

248 RIEMANN-ROCH THEOREM
n
1 = 1
n
(^1,^2)^ - E«=i+^/^^o(c. - v^d,)
«=i
n n
«=i «=i
n
((^('yi), (^('^2)) = E(a«c, + 6,d,),
«=i
so {(I>{vi),<j){v2)) is the real part of ('yi,'y2)^- Now we identify C* with R^'*,
TcM with TM by (^, (^* respectively, then (•, •) is the real part of (•, •)^, and the
Riemann metric is the real part of the Hermitian metric as well. Now define
the image part of the Hermitian metric to be a Kahler form and is denoted
by $(•, •). More precisely for X,Y e T{TM),
$(x,y) = im((x,yn
"^^((x,yr-(x;ir)
2
2
It is easy to see that
Thus $ is R-linear and antisymmetric, and hence is a 2-form.
An almost complex structure
J :TM ^TM
is the composition of the following maps
TM = TaM y TaM =: TM.
According to the common definition, M is a Kahler manifold if and only if
the almost complex structure J is integrable and the Kahler form $ is closed.
By a well-known Newlander-Nirenberg theorem ([28]) it is equivalent to
( [JX, JY] - [X, Y] - J[X, JY] - J[JX, y] = 0 VX, y G V{TM)
\ (d$)(x,y,z) = o vx,y,^Gr(TM).

8 2 HermtUan Connection 249
8.2 Hermitian Connection
It is well-known that there is a very natural connection on the tangent
principal 50(2n)-bundle SO{M), which is the Levi-Civita connection
{u^\a = fXT, aeMF{U{M))},
where a bundle map
p : U{M) = U{M) x,d U{n) -^ U{M) Xp 50(2n) = SO{M)
is induced by the group homomorphism p : U{n) —> 50(2n) appeared in §8.1.
Let us recall the definition of the Levi-Civita connection as follows. For a local
section cr G MF(C/'(M)), ? = per is a local section of SO{M), We have known
that the local sections and the moving frames are one-to-one corresponding to
each other, that is to say, pa corresponds one-to-one to {Ei,-- -, E2n}, and
similarly a corresponds one-to-one to the moving frame {El^ • • •, jE7^}. And
If we identify
and let {a;i, • • • ,a;2n} be the coframe dual to {jE7i, • • •, E2n}^ then by the Cartan
equation (Lemma 1.3.2)
we get a unique solution
{ua(3\ oc,f3= l,---,2n}.
Then the Levi-Civita connection form is
which is the matrix with entries being 1-forms. In what follows we are going
to make a connection on U{M) from the Levi-Civita connection {a;-} with the
help from Lemmas 8.2.1- 8.2.4.
Lemma 8.2.1 A -f- y/^B G U{n) if and only if
p{A + ^/^B)= ( ^ ~f ) eS0{2n),

250 RIEMANN-ROCH THEOREM
Where A and B are real n x n matrices.
Proof It is ecisy to see that
A + ^/^B e U{n) ^=> (A + y/^By{A + y/^B) = I
{A* - ^f^ B'\A + V^^) = I
{ A^A^-B^B^I
\ A^B-B^A^^,
and
SO the lemma is true.
Lemma 8.2.2 The element of Lie algebra V(n) can be expressed as
X + v^—ly, where X and Y are antisymmetric and symmetric real n x n
matrices respectively. In other words, V(n) is a real vector spa^e spanned by
{H^j., V^'S'rnimall <^<3<'^\ 1 < TTli < 7712 < n},
where Hf^ and Smima ^^^ matrices of rank n, whose (Ai,/)-entries are
^\j)u — ^tk^jl — Stidjje
Proof It is a straightforward check, so is omitted. However if we use a
tricky notation g~^dg, for g G U{n), then the proof could be easy to see. In
the equalities
g-^dg = g-^dg = g' • d{g-^)' = -g* • {g-' ■ dg • g-^ - -{g-'dg)',
we explain g~^dg as X + y/^Y , then
X + V^y = -{X + V^yY,
this equality gives the lemma immediately. Here g~^dg is actually the
Maurer-Cartan form [C/'(n)] explained in §1.7.

8.2 HermtUan Connection 251
Similarly, the element of Lie algebra 50(2n) can be expressed as W, which
is an antisymmetric 2nx 2n matrix, hence 50(2n) is a real vector space spanned
by
{S,j|l<z< j<2n},
where H,j is the antisymmetric matrix of rank 2n whose {k^ /)-entry is {d^kdji —
Lemma 8.2.3 p induces a monomorphism p* : U{n) —> 50(2n) such
that
In other words.
Proof Prom
X + ^f^Y = (yl + ^f-VB)-'^ ■ d(A + ^fAB)
= (A* - ^/^B') • d{A + x/=TB)
= {A*dA + B*dB) + yfA{A*dB - B*dA),
p,{X + ^/=Tr) = {p{A + V^B)-i • dp{A + x/^B),
A -B \~^ / A -B \ ( A* B* \ ( A -B
A j \B A ) \-B* A* )
B*dA \
rdB )
B A J \ B A J \ -B' A' I \ B A
A*dA + B*dB -A*dB + B*
-B*dA + A*dB A*dA + B*
it follows
p,{X + V^Y)
■(; -;)
So the lemma is true.
Lemma 8.2.4 There is a well-known inner product in 50(2n) such
that
{Wi,W2) = -tv{WiW^), \/Wi,W2 e 50(2n),

252
RIEMANN-ROCH THEOREM
and let ttq : 50(2n) —> p^(U{n)) C 50(2n) be the orthogonal projection under
this inner product, then for any
W={f^ f' e 50(2n),
\ ^21 ^22 /
(1)
(2)
(-X"ll -X"l2 \ _ I 2 (-^11 ■''■^22) 2(-^12 —-X^2l) \
X21 X22 / \ 2(-^21 —-X^12) 2(-^" +-^22) /
= p. (i(Xii + X22) - ^(Xa2 - X21)) ,
ipig))-' ■ MW) ■ Pig) = MiPig))-' ■ W ■ Pig)), V g G Uin).
Proof We can prove (1) by a direct computation, so the detail is
omitted. In order to prove (2), let
[J]
'{•• -:)•
then pig)[J] = [J]pig) and MW) = Iw - \[J]W[J], so
1.
1,
ipig))-' ■ MW) ■ Pig) = ipig))-' ■ i-W - -[J]W[J]) ■ pig)
= \ipig))-' • W ■ Pig) - \[J]ipig))-' • W . pig)[J]
= Mipi9))-'-W-pig)).
The proof is complete.
Let
a;ii
a;(l)
;(2) =
i^ln \
(^nn I
^n-\-\,2n
^2n,2n /

8.2 Hermtttan Connectton 253
^l,n+l ••• i^l,2n \
a;(3) =
^n,n+l ••• ^n,2n )
then
uy- —
a;(l) a;(3) \
-w(3)' w(2) y
and therefore
TTqW-
^( |Hl)+a,(2)) |(a;(3) + a;(3)*) \
V-|(a;(3)+a;(3)') i(a;(l)+ 0,(2)) j
= p, (|(a;(l) + a;(2)) - >^(a;(3) + a;(3)')) .
Now define
< - i(a;(l) + a;(2)) - ^(a;(3) + 0,(3)')
Lemma 8.2.5 The above
{a;^|c7GMF(C^(M))}
is a connection of C/'(M).
Proof \i g — A-^ yJ—\B : Ma- —> U{n) then we want to prove
^ag^g'^'^-g-^g'^dg,
which is changed into
p4< g) = (pig))-' • P*iO ■ Pig) + ipig))-'dpig)
by a monomorphism p^. By the definition we have
and ^
Hence in order to prove the lemma it is sufficient to show
M'^p(g)) = ipig))''^ ■ M<^) • pig) + ipig))~^dp(g).

254 RIEMANN-ROCH THEOREM
By Lemma 8.2.4(2) it is equivalent to
The equality
^ P(9) ^ (^(^))'^ * ^'^ ' ^(^) "^ iP(9))~^M9)
is just the formula for 50(2n)-connection on SO{M), which can be proved by
using Cartan lemma 1.3.3. Therefore the proof is complete.
Definition 8.2.6 Suppose we are given a tangent principal U{n)-
bundle U{M)^ we call the above connection
{ul\aeMF{U{M))}
a Hermitian connection on U{M),
Definition 8.2.7 Choose an orthonormal moving frame pa = {Ei,- • •,
E2n}, define
and
/ 6i ••• 6n \ / r/ii
K]^
Then we have
and
\ U
Snn /
\ Vnl
Vln \
rjnn J
^«J ~ ^J«' ^«J ~ ~^Jt5
i^p<T — T^Qf^pa + 2
1/ K]
Proof The proof is a direct check, so is omitted.
Lemma 8.2.8 For a fixed a G MF{U{M)), let
^^7 - "oC'^^^n+J +'*^J,n+t) ,

\ Onl ' " 0.
nn
(^Jl •
m = : ■
\Kx •
^f. = ^;..
•• o\„
•
•• Kn
8,2 Hermitian Connection 255
and
I Oil ' " Oin
[9] =
then we have
where {ct;^^} is the solution to the Cartan equation.
Proof The conclusion is just the definition of o;^ in Definition 8.2.6.
Theorem 8.2.9 Suppose we are given a tangent principal C/'(n)-bundle
U{M), Let
{u;l\(T eMF{U{M))]
be the Hermitian connection defined above, then it induces a set
which can be checked to be a connection on SO{M), We call this new
connection an induced connection. Then the following three conditions are
equivalent to each other.
(1) The induced connection is the Levi-Civita connection.
(2) V cr G MF(C/'(M)), let {it'tj}i<t,j<2n be the solution to the Cartan equation
(Lemma 1.3.3), then
/
^11 ••• ^l,2n
\ i^2n,l '" ^2n,2n
(3) M is a Kahler manifold with the Hermitian product, the almost complex
structure and the Kahler form given in §8.1. ,
Before we prove the Lenrnia let us examine the conditions for Kahler
manifold first, i.e. M is a Kahler manifold if and only if
i^{x, y) = 0 vx, y G r(TM)
(d^)(x, y, ^) = 0 vx, y, ^ G v(tm)

256 RIEMANN'ROCH THEOREM
where
N{X,Y) = [JX.JY] - [X,Y] - J[X,JY] - J[JX,Yl
which is called a torsion of the almost complex structure J. It is easy to
see that
(i) N{X,Y) is ^(M)-linear with respect to X and Y", where ^{M) is the real
function ring on M,
(ii) N{JX,Y) = -JN{X,Y), N{X,Y) = -i\r(y,X).
Lemma 8.2.10 For the simplicity of notations we denote
where i, jf = 1, • • •, n, and a = 1, • • •, 2n. Then
(1) N{',') is determined by the set
{{NiE^Ej) , Ek) , {N(E„E,) , En+k) \l<i,j,k<n}
and
{N{E„Ej) , Ek)^^k,{j)-ikj{i)-Vk,{n + j) + Vkjin+i)
{N{E„Ej) , En+k)=^k,in+j)-^kji^+i)+r,k,{j)-Vk3ii)-
(2) d$ is uniquely determined by {d^){E„Ej,Ek) , {d^)(E,,Ej,En+k) ,
{d^)(En+r,En+j,Ek) and {d^){En+„E„+j,E„+k). And
{d^){E, ,Ej,Ek) = v,k (i) + Vk, (i) + v.] (fe)
{d^){E„Ej,E„+k) = v,j{n+k)-ik,{j)+^kjii)
(d$)(£!„+., En+j,Ek) = -n,j(k) - Ck,(n + j) + 6j(n + i)
{ {d^){E„+t,E„+j,E„+k) = -Vjkin + i) - r]k,{n + j) - r],j{n + k).
Proof N{X,Y) is ^(Af)-linear, so N{-,-) is determined by the set
{{NiEc,Ei,),Ey)\a,0,f = l,-..,2n}.
Owing to the equahties
{NiJX,Y),Z) = {-JN{X,Y),Z) = {N{X,Y),JZ)
{N{JX,JY),Z) = -{N{X,Y),Z)

8.2 Hermtttan Connection 257
the first half of (1) is true. Note that
n
a=l
where \/ is the Levi-Civita connection, so we have
{N{E,,E,),Ek)
= {[En+t,En+j] - [^t,^j] - J[^t,^n-]-j] - J[En+t,Ej] , Ek)
— {rk rk I r"+*^ _l r"+*^ ^
— /T*^ _ r*^ -4-r^+*^ -4-r"+*^ ^
-i*'fe,n+t (n 4- jf) + i^k,t (jf) - i^n-^k,t {n + jf) - i*'n+fe,n+t (jf)
= 6t (jf) - 6j (^) - Vkt {n 4- jf) + rjkj (n 4- i) .
Similarly we also have
{N{E,,Ej) , ^n+fe) = 6t(n -h i) - 6j(n + i) + r;fe,(i) - r/^j (i).
Therefore (1) is proved. From the definition of the Kahler form $ we have
^{E^,E^)= {
0 if (a,^) = (i,i),
6^J if (a,/3) = (n4-i,jf),
-<5,j if {a, 13) = (i,n4-jf),
0 if (a,^) = (n4-i,n4-jf),
and then by a formula in §1.3 we also have
{d^){Ec„E^,E^) = -^{[Ec„E^] ,E^) + ^{[Ec„E^] ,Ep) - ^{[E^,E^] ,E^).

258 RIEMANN-ROCH THEOREM
Therefore
= -^[E„E,] ,E,) + mE„E,] ,Ej)-^{[Ej,E„] ,E,)
= -{t:;" - r;+*) + (r."+^ - r^+^) - (r;+' - r^/')
— ^t,n+k ^J,n+fe ^t,n+j ^ ^fe,n+j ^ ^j,n+t ^ fe,n+t
= ^jfeW+^fet(jf)+r7tj(Aj),
and similar computations yield (2). The proof is complete.
Proof of Theorem 8.2.9 It is easy to see that (2) holds if and only if
6j(^)=6j(n+A^) = 0,
r}tj{k) = 'rJ^J{n^^k) = 0, Vi,jf,Ai = l,---,n.
So (2) implies that M is a Kahler manifold owing to Lemma 8.2.10. Conversely,
if M is a Kahler manifold, then
{N{E,,Ej) , E^) = 6t(i) - 6j(*) - r7fe.(n +j) + r7fej(n + z) = 0
{d^){E,,Ej,En+i.) = r7,j(n+ A^) - 6t(i) +6j(^) = 0
(d$)(^n+t,^n+j,^n+fe) = -r7jfe(n + i) - 7;fe,(n + jf) - T/ij(n + Aj) = 0,
thus adding the above first equality to the second and using the third, we have
= -^fet(n + jf) - r;jfe(n-|-i) -7;,j(n +Ai) + 27;,j(n +Ai)
= 2r}^j{n'\-k),
Similarly we can also get rjtj{k) = 0. These equalities with Lemma 8.2.10 imply
CkziJ) =Ckj{i), 6t(n + jf) =^fej(n + z).
Note ^,j = —^j, and by using the trick in Theorem 1.1.3 we further have
6.(i) = 6j(i) = -CAi) = -^Ak) = ^v(^) = ^.fc(j) = -6.(j),

8.3 Rtemann-Roch Operator 259
hence Cki{j) = 0. Similarly we also have ^fet(^ + jf) = 0. Therefore the proof is
complete.
8.3 Riemann-Roch Operator
Suppose we are given a tangent principal G-bundle P of M with a
connection {u^a^}, then for a representation
X:G-^ End(F)
we can define a vector bundle
E = PxxV
and the negative Laplace-Beltrami operator
-Ao: r(^) ^ r(^),
which is the simplest Schrodinger operator of course. A general Schrodinger
operator can be expressed as
where F is a global section of the bundle
Px3^End(F),
where the action
A : G X End(F) -► End(F)
is induced by A, that is to say, for gf G G, 'y G F and C G End(F), the equality
Xig,C)v = X{g)-C-X{g-'){v)
determines A. Sometimes we denote the above equality simply by
{gC)v = gCg'^v,
For our purpose we are only interested in the case in which F is a super vector
space, then we may consider the interesting function
A
lim tr Git.v.v)
by using the heat equation method just as we did in Chapter 7. Nevertheless,
the Schrodinger operators with square roots are very common both in
mathematics and mathematical physics. A square root of a Schrodinger operator is

260 RIEMANN-ROCH THEOREM
called a Dirac operator. In what follows we try to introduce a Dirac operator
for a tangent principal U{n)-h\ind\e and a special A.
Let A* (n) be defined in Definition 1.8.3, which is a complex space spanned
by the set
{0,^A'"A0,J1 < ii < ••• <ife <n}.
Now define a C-linear action A of U{n) on A* (n) as follows:
where g G U{n)^ and gf,j is the (i, jf)-entry of the n x n complex matrix g.
Define a complex vector bundle
E=U{M)x^Al{n),
and an induced covariant derivatives
V : T{TM) X T{E) -> T(E) : {X, W) >-^ VxW
by
Vxi<T,f) = i<T, X/ + «J(X)./),
where cr is a local section of U{M)^ / is a map from the definition domain Ua
of cr to A* (n), and
K = '^o<^\<^ e MF{U{M))}
is the Hermitian connection given in §8.2.
Lemma 8.3.1 The above definition for Vx is well defined. That is to
say, for any g \Ua ^ U{n), we have (cr, /) = (ag , g~^f), then
VxK/) = Vx(cT^,rV).
Proof It is equivalent to check the equality
Xf + u>l{X) ■f = g{ Xig-'f) + «^,(X) • (9-V) ).
The check goes cis follows
S(X(ff-V)+<W-(9-V))
= g {-g-\Xg)g-'f + g-^Xf + ig-'u;^,iX)g + g-^dgiX)) ■ (g-'f)}
= g{g-^Xf + g-^u,%{X)f} = Xf + u>%{X)-f,

8.3 Rtemann-Roch Operator 261
SO the proof is complete.
Now we describe a technique, by using which we can construct a Dirac
operator from a C/'(n)-invariant algebra homomorphism
C2n(-l)^Endc(A;(n)).
Here the actions of U{n) on C2n( —1) and Endc(A* (n)) are given by
U{n) X C2n(-1) ^C2n(-1) : {A + y/=AB , (ei,...,e2n))
(ei,--^e2n)
B A )
and A respectively. For a G MF{U{M)), let pa = {^i, • • •, E2n}^ where
^a = (^, 6a) e T{U{M) Xp R2") = r(TM), V a = 1,..., 2n,
and define
2n
a=l
Lemma 8.3.2 There exists an operator
D : T{E) -^ T{E)
such that for any a eMF{U{n))
D = D{a),
Proof For any g : U^ ^^ U{n)^ we want to prove D{a) = D{ag). We
denote p{crg) = {^i, • • •, E2n}, then
2n 2n
^a = {(^g , <5a) = (O- , gf- 6a) = {O- ,^{p(T)^a6^) = ^{pa)f)aEf).
(3=1 ^=1
Thus
2n 2n
t=i «=i
2n 2n
«,J,fe=l J,fe = l
= i?(a).

262 RIEMANN-ROCH THEOREM
The lemma is proved.
Now we try to find a C/'(n)-invariant homomorphism
C2n(-l)^Endc(A;(n)).
First let us recall an algebra homomorphism in Theorem 5.1.8, it is
^2 : C2n(-1) 0 C ^ Endc(A;(n))
satisfying
^2(e2fe-i) = {€k -t^k) \/ l<k<n ,
^f2{e2k) = V^ (^ik + ^fe) \fl<k<n,
where 6^, a^ G Endc(A* (n)) so that
m
For fif = ^ + -v/^B G U{n),
i9€k)(e.,A---Ae,J = gekg-\e,,A---Ae,J
= 9(k J2 (ff"')j..«.---(r')j„,.„^j. A-.-A^j^
= 9 J2 i9~')juH---i9~')j,n,,r.0kA0j,A---A9j^
Jl}"'}Jm
= 22 XI (9~%i,ti"'{9~^)jm,tm9skgi^,jr"9lm,Jm
Jly'-'iJm S,li, ",lm
■9,A9i^A---A Oi^
= Y^ g,kO, A0„A---A9,^=J2 9ske,ie„ A • • • A e,J,
s s
thus
{9€k) = 9€kg~^ = J^Osk^s G Endc(A;^(n)).
s
Similarly we can get

8.3 Rtemann-Roch Operator 263
therefore
9 • (^1, ••• ,^n,^ir* • ,^n) — (^1, •• ^^n,^!,- * * , ^n) I
-(61,...,6„,M,...,.„)! ^ A-V^5
:)
)
Let
where // is a constant determined later, then
g[ei, • • •, e2nj — —(61, • • •, 6yi, Ai, • • •, A„j I A—r„
Denote
[-v^ -1 j
= (ei,--,e2n)l ^
A/'2(odd) = (^2(ei),^2(e3)---,A/'2(e2„-l)),
A/2 (even) = (A/'2(e2), A/2(e4) • • • , A/2(e2„)),
then
(A/'2 (odd), M (even))

264 RIEMANN-ROCH THEOREM
so we have
iel,■■■,e2n) = H^/'2{odd),^f2{even))l^ j [_/Zl _1 J
^ "*" ^(A/'2(odd),M(even))
(-.;)
, ,..„ ™ ,,• •)
/xv5 V ^v ;> V //^ \^ -1 1 y
2/i
-(A/'2(odd),A/'2(even))
Note
and {^2(ei),-")-^2(e2n)} is a standard set of generators of C2n(—1), and we
choose
1 + V^
then it follows that {e'l, • • • ,?2n} is also a standard set of the generators,
i.e the set satisfies
€i€j + €j€j = —20,j.
Therefore the setting
€a = ^ay V a = 1, • • • , 2n,
gives the desired algebra homomorphism from C2n(—1) to Endc(A* (n)).
Definition 8.3.3 Suppose we are given a tangent principal U{n)-h\iiid\e
U{M) of a 27vdimensional manifold M, let A,e'a, and the C/'(n)-connection be
defined as above, and
then by Lemma 8.3.2 define
D : T{E) -^ T{E),

8.3 Rtemann-Roch Operator 265
and call D a pre-Dirac operator. For a pre-Dirac operator, if it is self-adjoint
and its square is a Schrodinger operator, then we call it a true Dirac operator
or a Dirac operator.
Note that the action
X:U{n)xAl{n)^Al{n)
induces
which can be checked to satisfy
X,{X + V^Y , ^., A • • • A ^.J = 5](X + V^Y\^,€,Lj{e,, A • • • A ^. J ,
and then implies the following lemma immediately.
Lemma 8.3.4 Let u;^ = O-^y/^O^ be the Hermitian connection defined
in 8.2.5, then we have
and this equality may be put into the induced covariant derivative, i.e.
Vx(<T,/) = (a, Xf + Y,iKjW + V^OtJX))e,c,f).
Define a Hermitian product on A* (n)
(., •)^:A*(n)xA*(n)^C
such that
{Ot^ A • • • A ^, J 1 < zi < • • • < ijk < n}
is a unitary bfisis. After checking that this Hermitian product is invariant under
the action of U{n), it induces a Hermitian metric on the vector bundle
E = U{M) Xx A* (n). As we did in §1.4, now we introduce a Hermitian
product on the oo-dimensional complex vector space T{E) by
{{WuW2)y= I {WuW2Ydv, \/WuW2ev{Ey
Jm
Lemma 8.3.5
(1) E,{ Wi,W2 Y = { ^E.Wi,W2 r + { Wi,Ve.W2 r.

266 RIEMANN-ROCH THEOREM
(2) {{<T,ea)Wi,{<T,ea)W2Y = {Wi,W2Y, Va = l,.--,2n.
2n
(3) Vs„(^,e^) = YlfX^('^'^-r)> where T^^ = ip,{c,'^)\p{E^).
7 = 1
Proof Fixing cr G MF(J7(M)), we may assume Wi — {a, /i) and W2 =
(<T,/2), then
^a( W^, W2 Y = E^( fl,f2Y={ Eafl,f2 Y + ( fu E^f2 Y
^ { {Eafl + CvUEa) ■ /l),/2 Y - { <iEa) • fl, /2 Y
+ { fl, {Eaf2 + <^%{Ea) ■ A) Y - { fl,<^%{Ea) ■ A )^
From
{9fi,gf2Y = {fiJ2Y, ^geuin)
it follows
{z-fi,f2Y + {fi,z-f2Y = (i, ^zeu(n)_,
therefore
Ea{Wi,W2Y
= {(E^fi + u>%{Ea) ■ A), A )^ + (A, (^aA + <(^a) • A) )<=
= {{a,Eafi + w%{Ea)-fi),{<T, f2)Y
+( (a , A), {<T , Eaf2 + <^%{Ec) ■ A) )^
= ( Vs^m, W^2 )<= + ( m, Vs„W^2 Y-
(1) is proved. Further it is easy to see that
(c.A,e.A Y + (^.A,^.A Y = (A, A )^ (e.A.^.A Y = o,
then we have
((e.±..)A,(c.±i.)A)'= = (A,Ar,
thus
( Ar2(e„)A,A/'2(e„)A Y = ( A, A r. Va = l,---,2n.
Because the action of an element in 0(2n) on {^2(^1), • • • ,^2(e2n)} results
{e'l, • • • ,e'2n}, the above equality induces

8.3 Riemann-Roch Operator 267
Therefore (2) is true. Due to
\B A )
we have
(3) is true, thus the proof is complete.
For a C-linear operator
C : V{E) -^ V{E),
its adjoint operator
C* : T{E) -^ T{E)
is C-linear and satisfies
(( CWi,W2 )r = (( Wi,C*W2 )r, ^Wi,W2 G T{E).
Lemma 8.3.6 Let D be the pre-Dirax: operator defined in Definition
8.3.3, then
1 ""
1 ""
"2 X^(^J«(^ + 0+^J«W)(^'^«+j)-
Proof By Lemma 8.3.5 and Theorem 1.8.7 we have
2n 2n
DWi,W2 r = ( Y^ia,e^)VE^Wi , T^2 T = X)^«( (^,ea)m,W^2 V
a=l a=l
2n 2n
a=l a=l
2n 2n
= div{^((a,e„)m,P^2ri;a}+ X) ^L{i<^,e^)Wi,W2r
a=l a,13=1
2n 2n
- E( i^EA<^,e„))Wi,W2 r + J2i Wt,{a,e„)VE^W2 Y
a=l

268 RIEMANN'ROCH THEOREM
2n
= div{5]( {a,ea)WuW2 yEa} + ( WuDW2 Y
a=l
2n
a,/3=l
2n
= div{5]( {a,ec,)WuW2 )^^a} + ( WuDW2 Y
2n
E(rL-rL)(w^i,Ke/3)w^2r
a,/3 = l
Therefore
2n
a,(3 = l
By a direct computation we have
2n
- E (rL-rL)(^,e^) = -^ E(^^'« + '?^'(" + ^))('^'^7)
a,(3 = 1 t,i = l
1 '^
SO the proof is complete.
2
8.4 Weitzenbock Formula
Theorem 8.4.1 Let
2n
a = l
be the pre-Dirac operator defined in Definition 8.3.3, then
. 2n
£>2 = _Ao + - ^ (o-, eae^)R{Ea, Ef))
a,(3 = l
^ 2n n
a = l t,j = l
^ 2n n
2
a = l «,J = 1

8 4 Wettzenbock Formula 269
where Aq is the Laplace-Beltrami operator and R is the curvature operator of
V.
Proof First we have
2n
/•I- JO jV7t-i i/t iO/ijV7i^,
2n 2n
2n
= J2 {(T,ea){(T,ef3)V{Ea,Ef3)
2n
where W{Ea,E^) is the second covariant derivative. Then by a direct
computation for (r^^ — r^^) we can prove the theorem.
Corollary 8.4.2 If M is a Kahler manifold, then the pre-Dirac operator
is a true Dirac operator, i.e. D = D"^ and D^ is a Schrodinger operator.
Proof It is the corollary of Theorem 8.2.9, Lemma 8.3.6 and Theorem
8.4.1.
Definition 8.4.3 If M is a Kahler manifold, we call the restriction of
pre-Dirac operator
D : T{E^''^'^) -^ r(£^^^^)
a Riemann-Roch operator, where
^even ^ ^^^^ ^^ A^^^^(n), E""^^ = U{M) Xa K^\n).
Problem 8.4.4 For a Hermitian manifold M, is there an element
FGEndc(£^)
such that D -^ F is 3, true Dirac operator?
From now on we assume that M is a Kahler manifold, i.e. there is
a tangent principal C/'(n)-bundle U{M)^ such that the conditions described in
Theorem 8.2.9 hold. Then for a G MF{U{M))
P*K)
'^p<r5

270 RIEMANN-ROCH THEOREM
where u^ is the natural Hermitian connection expressed as a complex n x tv
matrix while cjpa^ is the Levi-Civita connection expressed as a real 2n x 27V
matrix. As in Lemma 1.3.2, the curvature forms are
ni=du;t+u,lAu,l
and
Then from p*(a;J) = ijjpa it follows that
Recall
' X -Y
p* : U{n) -^ S0{2n) : X + y/^Y y-^
Y X
^^pcr —
0^ =
/ fin'
/ 011
\0n,l
fil,
2n
ei,„
/ e
+ v^
\Ql,l
and fla^, ©tj 5 0^j are real 2-forms satisfying
so the equality p*(l^^) = Qpa^ means exactly that
f\c _ o — O
Let US examine the term
2n
X) K eae^)R{Ea. Ef,) : T{E) ^ r(^)
in Theorem 8.4.1, where
R{E^, Ef,) = ni{E^, Ef,) : r(^) ^ r(^).

8.4 Wettzenbock Formula 271
I.e.
From
R{Ea,Ef3) = J2 {K{Ea.E^)),j{cT,e,Lj) : r(^) ^ r(^).
«,J=1
n
it follows that
1
6. =
-(e, - \/^e„+,)
""^/2(l-^/^)'
Let
2(1-^31)2- 4'
and consider the map
Y,{K{Ea.E^)\,e,Lj:Kl{n)^h.l{ny
«j=i
We have
n
«,j=i
— Z-^ (^^(^a, E^))tjK,(€i€j + en^tCn^j - y/^en^tCj + \/-le,en+j)
«,J=1
= /c 2J (0(^a, Ef))\j{etej + en+,en+j + \/^ejen+, + \/^e,en+j)
«,j = i
+\/^/c 2J (0''(^a, ^^))tj(e,ej + en+,en+j + \/^ejen+, + \/^e,e„+j)
«,j=i
= /C ^ (0(^a, Ef)))^J{e^eJ + Cn+tCn+j)
«,J = 1
+ V^/c5^(0''(^a, ^^))««(e,e, + Cn+tCn+O
« = 1

272 RIEMANN-ROCH THEOREM
n
n
+^/^/c^(0^(^a,^^))..(-2)
n
Cn+t — ^t,n+j(^a5 ^^)v^e,e„+j)
2n n
71,72 = 1 « = 1
By using
tr 1^^ = tr 0 + x/^ tr 0^ = V^ tr 0%
we have
«,j = i
2n
2n
= -/C 2J ^ai37i72^7i^72 " ^/^(tr l^^)(^a, ^^)
71,72 = 1
1 ^'^ 1
= 7 X/ ^«i37i72^7i^72 + 9 (tr ^^)(^a, ^^),
4
71,72 = 1
and thus
2n 2n
^ (cT,eae^)5(£^a,^^)= X! K^a?^)K Yl {Ki^a, E^))t3etij)
f,^ = l a,^=l «,j = l
^ 2n
a,(3,71,72 = 1
^ 2n
+ - ^(trl^^J(^a,^^)(cT,?«?^).
Theorem 8.4.5 (Weitzenbock Formula) For a Kahler manifold M,
let
2n
D=J2i<',e^WE^:T{E)^T{E)
a = l

8.5 Index Theorem 273
be the pre-Dirac operator defined in Definition 8.3.3, then
. r—:r 2n
1 _ V —Itt
Proof Theorem 8.4.1 yields
^ 2n
2
SO
2n
D'^ = -Ao + - 2J Ra(3jij2 ' (^ 5 eae^e^iC^J
8
Of,^,71,72 = 1
2n
+ - f] (tr Ql){Ea, E^) • (c7 , e^e^y
4
By using the reasoning in Theorem 6.3.2, we have
2n
yj i2a^7i72 • (^ 5 eae^ey^eyj = 2i2,
a,^,7i,72 = l
where i2 is the scalar curvature. Further by Definition 4.1.9
then the theorem is ecisy to see.
8.5 Index Theorem
Let us begin with an approximate formula for — Aq. The computation in
§6.4 shows
^^ d^ I ^r^ ^ ,,, d ^ ^
-Ao = - 2^ ^ - - 2^ Ra^s.sMVa^es.es,
a,^,7,«l,«2,«8,«4

274 RIEMANN'ROCH THEOREM
Denote '
a,/3
and
2
2J Ci{Ea,Ef)) 'Caef).
Recall Trick Formula 4.3.2
n
« = 1
where {t^i, • • •, t^n} are Chern roots. Then
For the heat operator approximate to -;— -^ D^
at
the equation
, \im{4-Kt)"H{t,yi,---,y2n) = l
\ r—►00
has a solution
Hit,y^,---,y2n) = exp{ct}-n:=i{ ^^^..
l^ Stt sinh J'^—^ '
■ ^^^ V 8 ^ 2 )(^2.-i + ^2.) 11,
hence
i.,,0....,0) = expp<).n{j;^^}.
2
Recall that the Chern root algorithm in §6.4 had shown an equality
A ^ f 1 yELl^ 1 i «x

8.5 Index Theorem 275
in the case
e, = e,, Vi = 1, •• -,271.
The right-hand side of the above equality may be viewed as the left-hand side
after replacing '^~ ^ '* by ^ and discarding the factor ^ and with some
properly changes of marks. We detail the replacing as follows
y/^ Wst _ yf-i-Kt
2 " 2 ^
X;^.(^a(0),^^(0))
thus we are apt to assume a similar replacing
^ \/—Ttt^ V—V ._ _ . _ _ c\
ct= —-— 2^ ci{Ea,E^)'eaei3^ —,
a,(3=1
So we would like to believe that the Chern root algorithm can induce an equality
^ "^ ( I V^w^t ^
lim tr exp(c't) • TT < 1—„, >
=/x |exp(|) • n-jj^ I (i;i(0), • • •, ^2„(0)),
and
Loc.index(D) = lim tr H{t, 0, • • •, 0)
"I
«^P(f )• n ^±^ iE^iO),..-,E2nm,
s = l
where // is a constant, which comes from the difference between {e,} and {?,}
and is determined by
tr(e,, •••?,,) = //• tr(e,, •••e,J, Vs < 2n,
(see the following Lemma 8.5.2 for detail). After we determine the constant
// and prove the above equality by using the method in §7.2 we get the local
index formula for Riemann-Roch operator.
Theorem 8.5.1 For a Kahler manifold M of complex dimn, let
be the Riemann-Roch operator (see Definition 8.4.3), then
Loc.index(D) = // • f (^i, • • •, ^2n-i, ^2, •'', ^2n),

276 RIEMANN-ROCH THEOREM
where
The following lemma is used to determine the constant //.
Lemma 8.5.2 Define a super structure
AJ,(n) = A«-"(n)0AS'^V),
then for zi <
tr(e,j
Proof
•••<**,
•••?.J = {
Due to
(?!,••• ,e2ri
(-
0,
) =
-1)^
■ {J^2{o
li 2"
(V^)"
^(n,
even))
if s = 2n,
otherwise.
I V2 ^^ j
\ "75 71 /
we have
ei • • • e2n = ^2(ei) • • • A/'2(e2n-1)^2(62) • • • ^2(e2n)
= (-1)''^^^2(61)^^2(62) • • • ^f2{e2n^
By Theorem 5.1.10 and Proposition 5.3.1 we get the proof of the lemma.
Remark 8.5.3 The constant // = (—1) 2 wfis explained on p. 125
in [21]. And Theorem 8.5.1 turns out to be the local version of Theorem 20.2.2
in [21], as soon as we explain that the Riemann-Roch operator in this section
is just the traditional one. We will give the explanation in §8.6.
8.6 Riemann-Roch Operator in Complex Analysis
Let us restrict ourselves to the space C". Suppose {zi^-- - ^Zn] and {lyi, • • •,
Wn] are two complex analytic coordinate systems, if 2;, = x, -f- \/^y,, and
w^ - n, -l-V^'y,, then {xi,-• • ,Xn,yi,-* * ,yn} and {ui,-- ,Un,vi," ^ ,Vn] are
two real coordinate systems. Let
A^V^B^ 5(^l,--,^n)
d{wi,''',Wn)
I dzj^ _ , dzi
' dwi * * * dwn
dZn ... dZn
dw 1 dw„

8 6 Rtemann-Roch Operator in Complex Analysts
277
then it is easy to see that
A =
and then
B =
il
dui
dui
dUn
dUn
dUn
dyn
dUn
I a-.
dvx
dyn
dvi
( ^]b.
' dui
dyn
dui
dyi
dVn
dyn
dVn
dyi
dUn
dyn
dUn
■) = (-
d
)-{A + V^B),
(
d
' dwi' ' dw„ dzi' ' dz„
{dwi,--- ,dwn) - (dzi,--- ,dz„)- (L + V^M),
d d d ^ , d d d 5 ,
^■)=(,^
I A -B
\ B A
L M \
-ml)
dui' ' dun' dvi' ' dvn dxi' ' dxn dyi' ' dyn
{dui,'",dun,dvi,'",dvn) = {dxi,'",dxn,dyi,"',dyn)' I
{dw^, • • •, dm;;;) ■= (dz^, • • •, d^) • {L - v^M),
where
{L + V^M) = {A* + x/ri5*)-i.
From the above five equalities we can introduce five vector bundles in complex
analysis, they are TcM, T*M, TM, T*M, and T^M respectively, where
M = C*. As an example let us describe how to introduce a complex vector
bundle from the first equality and explain why this bundle is TcM in
complex analysis. For any complex coordinate system z = {zi^ • • •, 2;^}, denote its
definition domain by Uz, then define a vector bundle E by
^ = (J(£r, X C")/~,
z
where ^ is an equivalence relation defined by
(p€ U,,{vi,---,v„)) ~ {pe i7„,(ui, ••-,«„))
/ «i \
= iA+V^B)\,
\ «« /
\ Un J

278 RIEMANN-ROCH THEOREM
in which (p G Uzi (^i, * * *, '^n)) means an element (p, (ui, • • •, '^n)) in ^-^ x C'*.
From the above equality, which explains the equivalence re),ation, it follows that
t = l
Thus it induces a vector bundle isomorphism from E to T^M, So the first
equality corresponds to TcM, whose elements are complex linear combinations
of 37~?"*5af~* ^^ other words, [A-^ y/^-lB) corresponds to T^M.
Similarly, the second, third, fourth and fifth equalities or the four matrices in these
equalities
r— ( A -B\ ( L m\ .—
''^^''^'[s A )\-M ,)'''-^'»
correspond to T*M, TM, T*M, and T^M respectively.
If {A + ^/-IB) e U{n), then L - y/-[M = A + V^B, hence {A + y/^B)
can correspond to T^M also. It is to say, by using the notations in §3 we have
UiM)xxAl{n) = T^M,
Note that an element in T^M can be expressed by a complex linear
combinations of {dzi^ • • •, dz^}, it follows that
T{U{M) xx Ai{n)) = A^^^^\M),
And we also have
r(C^(M)xAA;(n)) = A(^'*)(M),
where A^^'*^ was defined in [21]. More careful checks show that the Riemann-
Roch operator D in the Kahler case is just the operator
V2(a + a^): a(^'*)(m) ^ a(^'*)(m).
So we finish the explanation mentioned in Remark 8.5.3.

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A-characteristic form
A-density
A-genus
admissible connection
almost complex structure
associated connection
associated vector bundle
Asymptotic Expansion
Theorem
127,
Atiyah-Singer index theorem
174, 212, 231, 241,
Betti number
Bianchi identities
calling number
Cartan Lemma
Cauchy problem
characteristic clciss
Chern class
Euler clciss
Pontryagin class
characteristic form
A-characteristic form
Chern form
244,
5, (
INDEX
211
212
212
115
245,
248
81
79
130
275
112
5,11
157
18, 20
94
150
149
146
151
211
150
Euler form
^-characteristic form
<^-form
Pontryagin form
characteristic number
Betti number
Chern class
Chern form
Chern roots
(new) Chern roots
Chern root algorithm
Chern root expression
146,
113,
(new) Chern root expression
Chern root expression of the
vature form
Chern-Weil theory
Fundamental Lemma
Clifford algebra
Fundamental Theorem
coframe
compatible frame
connection
admissible connection
induced connection
coordinates
Riemann coordinates
<
157,
149
162
151
, 149
, 150
112
150
150
157
163
164
157
164
cur-
164
139
141
175
181
17
51
}, 71
115
255
60
283

284
Index
stereographic coordinates
cotangent bundle
curvature
curvature operator
density
^-density
L-density
de Rham complex
60
79
4, 72
202
212
162
24
Euler number
Euler class
Euler form
even element
exponential map
exterior algebra
exterior differential
fiber
112
148
149
184
45
80
15
71
de Rham-Hodge operator 33
de Rham cohomology group 106
derivative
covariant derivative 72
covariant derivative 202
induced covariant derivatives
260
higher order derivatives 23, 72
second order derivative 23, 72,
202
m-th derivative 14, 23
differential form 14
Dirac operator 203, 207, 260
pre-Dirac operator 265
true Dirac operator 265, 269
distance function 47
divergence Theorem 126
duality Theorem 109
Egregium theorem 1
formal proof 174, 212
Fourier Expansion Theorem 101
Frenet frame 78
fundamental solution 85
Gauss lemma 50
Gelfand Problem 113
genus
A-genus 212
a-genus 150
L-genus 162
geodesic 42
G-invariant multiplication 81
G-invariant polynomial 151
gradient vector field 49
Grassmann algebra 80
Habilitation address 1
harmonic form 106
Hermitian connection 254

Inde
285
Hermitian manifold 245
Hermitian metric 245, 246, 265
Hermitian product 265
Hilbert-Schmidt theorem 102
Hodge Theorem 98, 103
index
local index
initial solution
induced connection
induced principal bundk
inner automorphism
inner product
invariant point of view
Jacobian identity
Kahler form
Kahler manifold
113,
89,
157,
245,
137
137
120
255
156
177
26
7
2
248
248
Levi-Civita connection 3
Fundamental Theorem 3
Laplace-Beltrami operator 24,
70, 72, 202
Levi algorithm 89
^-characteristic form 162
L-density 162
L-genus 162
local basis 7
local index 137, 231
local index theorem
for Dirac operator 212, 231
for Signature operator 174, 241
for de Rham-Hodge operator244
for Riemann-Roch operator 275
local section 74
Maurer-Cartan form 75, 250
Mckean-Singer problem 137
metric 2
moving frame method 14, 74, 78
moving frame 78
MP Parametrix 89, 119
natural frame 10
normal coordinate system 41, 45
norm density 133
odd element 184
orthonormal moving frame 11
orthonormal moving frame
method 14
171
oscillator
parallel
Pauli matrices
Pfaffian
<^-form
polarization
51
182
147
151
142

286
Index
Pontryagin clciss 146
Pontryagin form 146, 149
positive or negative harmonic
spaces 110
potential integral 91
Principal G-bundle 73
induced principal bundle 156
tangent principal C/^(n)-bundle
245
quaternion
realization map
Remainder
Riemann coordinates
Riemannian manifold
176
247
153
60
3
Riemann metric 2, 245, 246
Riemann-Roch operator 269
spin-structure
spinor
spinor bundles
spinor product
spinor space
195
183
202
205
183
standard set of the generators 264
star homomorphism 26
super algebra 185
super space 184
super structure 183
Z2-decompositions 183
Z2-graded decomposition 176
Z2 graded structure 183
super symmetry 183
super tensor product 184
super trace 188
super vector spa^e 184
scalar curvature
section
Schrodinger operator
simplest Schrodinger
semi-group property
Signature
Signature operator
Signature Theorem
spin representations
37
71
83
operator
259
100
110
40
110
183
tangent bundle 71
tangent complex bundle 246
tangent principal bundle 74
Taylor expansion 57
tensor
classical tensor calculus 81
general tensor calculus 81
general tensor (field) 81
torsion of
the almost complex structure
256

trace
Trick 4.3.2
twisted
unitary frame
133
165
244
246
Index 287
vector bundle 70
Weitzenbock Formula 36, 204,
272
d
6
div
_d_
dp
15
28, 31
56
30
33
181
49
grad(^)
H,j, H,jk
R{E„E,)
T
P
MF(P)
[G\
49
52, 53
35, 203, 270
38
49
74
75