Curvature and Betti Numbers. (AM-32) 9781400882205, 1400882206

Table of contents :
Frontmatter --
Preface --
Contents --
Chapter I. Riemannian Manifold --
Chapter II. Harmonic and Killing Vectors --
Chapter III. Harmonic and Killing Tensors --
Chapter IV. Harmonic and Killing Tensors in Flat Manifolds --
Chapter V. Deviation from Flatness --
Chapter VI. Semi-simple Group Spaces --
Chapter VII. Pseudo-harmonic Tensors and Pseudo-Killing Tensors in Metric Manifolds with Torsion --
Chapter VIII. Kaehler Manifold --
Chapter IX. Supplements --
Bibliography --
Backmatter

Citation preview

Annals o f Mathematics Studies Number 32

ANNALS OF MATHEMATICS STUDIES Edited by Em il Artin and Marston Morse

1. Algebraic Theory of Numbers, by

Herm an n W eyl

3. Consistency of the Continuum Hypothesis, by 6.

The Calculi of Lambda-Conversion, by

7. Finite Dimensional Vector Spaces, by 10. Topics in Topology, by

K u rt G odel

A lon zo C h urch

Paul

R.

H alm os

S o lo m o n L e fsch e tz

11. Introduction to Nonlinear Mechanics, by N. 14. Lectures on Differential Equations, by

K rylo ff

S o lom on

and N.

B o g o l iu b o f f

L e fsc h e t z

15. Topological Methods in the Theory of Functions of a Complex Variable, by

M a r st o n M o r se

16. Transcendental Numbers, by

C a r l L u d w ig S ie g e l

17. Probleme General de la Stabilite du Mouvement, by M. A. 19. Fourier Transforms, by

S.

B ochner

and

K.

L ia p o u n o f f

C h an d r ase k h ar an

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S . L e fsc h e t z

21. Functional Operators, Vol. I, by

J ohn von N e u m an n

22. Functional Operators, Vol. II, by

J ohn

von

Neum an n

23. Existence Theorems in Partial Differential Equations, by

D oroth y

L.

B e r n s t e in

24. Contributions to the Theory of Games, Vol. I, edited by A. W. T u c k e r 25. Contributions to Fourier Analysis, by A. A. P. C a l d e r o n , and S. B o c h n e r 26. A Theory of Cross-Spaces, by

Zygm un d ,

W.

H.

W

K uhn

and

T r a n su e , M . M o r se ,

R obert Sch atten

27. Isoperimetric Inequalities in Mathematical Physics, by G.P o l y a and G. S z e g o 28. Contributions to the Theory of Games, Vol. II, edited by A. W. T u c k e r

H . K uhn

and

29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L e fsch e t z

30. Contributions to the Theory of Riemann Surfaces, edited by

L . A h lfo rs

et al.

31. Order-Preserving Maps and Integration Processes, by 32. Curvature and Betti Numbers, by 33.

K . Y ano

and

E d w ard J. M c Sh an e

S. B och ner

Contributions to the Theory of Partial DifferentialEquations, edited L. B e r s , S. B o c h n e r , and F. J o h n

by

CURVATURE AND BETTI NUMBERS By K. Yano and S. Bochner

Princeton, New Jersey Princeton University Press

1953

Copyright, 1953> "by Princeton University Press London:

Geoffrey Cumberlege, Oxford University Press L. C. Card 5 3 - 6 5 8 3

Printed in the United States of America

PREFACE

This tract gives a first systematic account of a topic in dif­ ferential geometry in the large, that is, of a topic on curvature and Betti numbers, recently inaugurated by Professor S. Bochner. In the hope that the tract might also be of use as a survey of present-day differential geometry in several of its aspects, and also in order to fix our notation, all pre-requisites from differential geometry as such are presented in Chapter 1, virtually independently. The other Chapters contain the recent work of Professor Bochner on differential geometry in the large, and other results closely related to it. These Chapters contain only a part of recent work in this field, but very fortunately for myself and also for the reader, Professor Bochner was kind enough to add a chapter of supplements and from which the reader will learn wider aspects of this very interesting topic. I wish to express here my hearty thanks to Professor 0. Veblen who gave me the opportunity to stay at the Institute for Advanced Study and also to Professor D. Montgomery who gave me a chance to give a lecture in his seminar on which the first draft of the tract was based. Professor Bochner not only added the last Chapter which is the most important and the most interesting part of the book, but also gave me many valuable suggestions on the first eight Chapters. It is ray pleasant duty to express here my sincere gratitude to Professor Bochner without whose kindness this book would not be possible. Kentaro Yano Institute for Advanced Study May 5, 1952

v

CONTENTS

Preface

v

Chapter I.

Riemannian Manifold 1. 2. 3• b.

5• 6.

3

Riemannian Manifold Tensor Algebra Tensor Calculus Curvature Tensors Sectional Curvature Parallel Displacement

3 5 12 16

19 23

Chapter II. Harmonic and Killing Vectors

26

1.

Theorem of E. Hopf

26

2. 3. 5*

Theorem of Green Some Applications of the Theorem of Hopf-Bochner Harmonic Vectors Killing Vectors

33 37 37

6.

Affine Collineations

bo

k.

7.

A Theorem on Harmonic and Killing Vectors 8. Lie Derivatives 9- Lie Derivatives of Harmonic Tensors 10. A Fundamental Formula 11. Some Applications of the Fundamental Formula 12. Conformal Transformations 1 3 . A Necessary and Sufficient Condition that a Vector be a Harmonic Vector lb. A Necessary and Sufficient Condition that a Vector be a Killing Vector 1 5 . Motions and Affine Collineations Chapter III. 1. 2.

31

b2

^3 bi

50 51 53 55 56 57

Harmonic and Killing Tensors

59

Some Applications of the Theorem of Hopf-Bochner Harmonic Tensors

59

vii

6k

CONTENTS

viii 3 . Killing Tensors

65

b.

A Fundamental Formula 5* Some Applications of theFundamental Formulas 6. Conformal Killing Tensor 7- A Necessary and Sufficient Condition that an Anti-symmetric Tensor be a Harmonic Tensor, or a Killing Tensor Chapter IV.

Harmonic and Killing Tensors In FlatManifolds

1. Harmonic and Killing Tensors in a Manifold of Constant Curvature 2. Harmonic Tensors and Killing Tensors in a Conformally Flat Manifold Chapter V.

Deviation from Flatness 1. 2. 3* b.

Chapter VI-

Deviationfrom Deviationfrom Deviationfrom Deviationfrom

Constancy of Curvature Projective Flatness Concircular Flatness Conformal Flatness

Semi-simple Group Spaces

1. Semi-simple Group Spaces 2. A Theorem on Curvature of a Semi-sinrple Group Space 3* Harmonic Tensors in a Semi-simple Group Space b. Deviation from Flatness Chapter VII.

Pseudo-harmonic Tensors and Pseudo-Killing Tensors in Metric Manifolds with Torsion

1. Metric Manifolds with Torsion

67 70 72

7^ 77

77 78 81 81

8^ 86 88 90 90

93 9b

95

97 97

2 . Theorem of Hopf-Bochner and Some Applications

1 01

3 • Pseudo-harmonic Vectors and Tensors Pseudo-Killing Vectors and Tensors 5• Integral Formulas 6. Necessary and Sufficient Condition that a Tensor be a Pseudo-harmonic or a Pseudo-Killing Tensor 7 • A Generalization

1 05

b.

Chapter VIII. Kaehler Manifold

109

11 1

112 11 5 117

1. Kaehler Manifold

117

2 . Curvature in Kaehler Manifold

123

3 • Covariant and Contravariant Analytic Vector Fields

131

CONTENTS

ix

k . Complex Analytic Manifolds Admitting a Transi­

56.

78. 910.

tive Commutative Group of Transformations Self-adjoint Vector Satisfying = 9f/Sza and A f = 0 Analytic Tensors Harmonic Vector Fields Harmonic Tensor Fields Killing Vector Fields Killing Tensor Fields

134 136

139 1^2

1^5 1^9 1 51

1 1 . The Tensor h^j 12. Effective Harmonic Tensors in FlatManifolds

160

13- Deviation from Flatness

166

Chapter IX.

Supplements (written by S. Bochner)

153

170

1 . Symmetric Manifolds

1 70

2 . Convexity

172

3 . Minimal Varieties

173 17^ 177

k.

Complex Imbedding 5- Sufficiently Many Vector or Tensor Fields 6 . Euler-Poincare Characteristic 7* Non-compact Manifolds and BoundaryValues Zero

Bibliography

180 181

187

C H A P T E R

I

RIEMANNIAN MANIFOLD 1 . RIEMANNIAN MANIFOLD We take a Hausdorff space with a given system of neighborhoods {U} , such that each neighborhood U can be put in one-to-one reciprocal continuous correspondence with the interior of a hypersphere n

in an n-dimensional Euclidean space, and such a space we will call an n-dimensional manifold, where, and in the following, Roman indices run over the values i, 2, ..., n. This correspondence between points in a neighborhood of the manifold and points in the inside of a hypersphere is called a coordinate system, and the coordinates (x1 ) of the point In the Euclidean space which corresponds to the point P in the manifold are called coordinates of the point P in this coordinate system. Moreover, a neighborhood endowed with a coordinate system is called a coordinate neighborhood. If, In a neighborhood U , two coordinate systems (x1, x2, ..., xn ) and

(xfl, x*2

••;

are given, then there is a one-to-one reciprocal continuous correspondence between these two coordinate systems which can be expressed by the equations (i.i) or Inversely (i-2 )

3

I.

RIEMANNIAN MANIFOLD

Equations (1 .1 ) or (1 .2 ) define a so-called coordinate transformation. If the functions x^"(x|a) and x !^(xa ) are of class Cr , that is to say, if they admit continuous partial derivatives of the first, the second, ..., the r-th derivatives, and If, when r > 1 , the Jacobians ax1 dx,a

and

dx,a

are different from zero for any coordinate transformation in the manifold, we say that the manifold is of class Cr . It is evident that, in an n-dimensional manifold of class Cr , i a if we have functions f (x ) satisfying the above mentioned conditions, where (xa ) is an original coordinate system in a neighborhood U , then, on putting X'1 = f1(xa ) we can Introduce (x'^) as a new coordinate system In U • We shall call such a coordinate system an allowable coordinate system in U . Now, if the manifold can be covered entirely by a finite number of neighborhoods U1, U2, ..., , then the manifold is said to be compact. As a rule, our manifolds will be compact. We assume sometimes also the orientability of the manifold. If (x^) and (xfi) are two allowable coordinate systems In a coordinate neighborhood U , then the Jacobian

is different from zero throughout the coordinate neighborhood U , and consequently, being a continuous function of the point in U , it has the same sign throughout U . If this sign is positive, we say that these coordinate systems are positively related, and if it is negative, we say that they are negatively related. If there exists a subset of the set of all allowable coordinate neighborhoods such that it covers the whole manifold and any coordinate systems which belong to this subset and which are valid in the same neigh­ borhood are positively related, then we say that the manifold is orientable. We now assume that, with each coordinate neighborhood U in our n-dimensional manifold of class Cr , there is associated a positive def­ inite quadratic differential form in the differentials dx1 , (1-3)

ds2 = gj^dx^dx^

2.

5

TENSOR ALGEBRA

which does not depend on the coordinate system used, where the coefficients gjk(x) are functions of coordinates (x1, x2, ..., x11) of class Cr~1 , and repeated indices represent the summation over their range. Geometrically, we interpret (1 .3 ) as a formula which gives the infinitesimal distance ds between two points (x1 ) and (x1 + dx1 ) , the length of a curve x1 = x1(t) (t1 < t < t2 ) being given by

(i-^)

s = 3

r J t

2

la.

dxJ* dxk

sj sjk

dt

and we call (1 .3 ) the fundamental metric form of the manifold. An n-dimensional manifold of class Cr in which a fundamental metric form (1 .3 ) is given is called an n-dimensional Riemannian manifold of class Cr and the theory'of such manifolds is called Riemannian geometry. (L. P. Eisenhart [1 ]). 2.

TENSOR ALGEBRA

If in a coordinate system (x1 ) we are given the form (1 .3 ) and if in another coordinate system we correspondingly put ds'2 = g* jk-(xl )dxf^dxfk then we must have dsf = ds In general, if an object is represented by f in a coordinate system (x1 ) and by f 1 in any other coordinate system (xfl) , and if we have (1-5)

f• = f

then we call this object a scalar and f respective coordinate systems (x1 ) and ponent of a scalar. Next from (1 .2 ), we have

and f 1 its components in the (x*1 ) . Thus, ds is the com­

dx.i = ^ i dxr dx In general, if an object Is represented by

n

quantities

v1

in

6

I.

RIEMANNIAN MANIFOLD

a coordinate system (x1 ) and by (x*1 ) , and if we have (1.6)

v'1

V 1 =

in any other coordinate system

Vr

dxr then we call this object a contravariant vector and v1 and v'1 its components in respective coordinate systems (x^-) and (x*^*) • Thus, dx1 are components of a contravariant vector in the coordinate system (x1 ) . If an object is defined at every point of a coordinate neighbor­ hood U , then its components are functions of (x1 ) . We call such an object a field. If we denote by f(x) and f T(x!) the components of a scalar field in respective coordinate systems (x1 ) and (x*1 ) , then we have f»(x* ) = f(x) from which, by partial differentiation, d f ’

B xs

b f

Sx1^

Bx

dxs

In general, if an object is represented by n quantities Vj in a coordinate system (x1 ) and by vi in any other coordinate system (x'1 ) , and if we have (1-7)

vi = 2* J

v^ s

then we call this object a covariant vector and v. and v*. its components in respective coordinate systems (xi ) and (x* i ) . If f(x) is a component of a scalar field, then Sf/Sx1 are components of a covariant vector. We call such a special covariant vector the gradient of the scalar field f . From the assumption ds1 = ds , we have Sxs axfc " Sx'J °3t In general, If an object Is represented by

n-P+l^ quantities

2. in a coordinate system

TENSOR ALGEBRA

7

(x1 ) and by ••1p. J1J2 ***Jq

In any other coordinate system

(x'1 ) , and if we have

( 1 -8 )

d x ’

’'Jq. 11

r Sx 1

B x'

1?

dx

dxSq

Jl ax*j2

dx1 ^

S1S2 '

r dx 2

then we call this object a mixed tensor of contravariant valency covariant valency q , and

p

and of

^ 1^"2 ***^"D . and T

••1p, . ^1^2 * *^q

its components in respective coordinate systems (x1 ) and (x,;i) • A tensor having only contravariant valency is called a contravariant tensor and a tensor having only covariant valency a covariant tensor. Thus, gjk (x) are components of a covariant tensor field. Since we have assumed that (1 .3 ) is positive definite, we have

(1.9)

S 11

S 12

"in

g21

822

^2n

^n2

‘‘’ ^nn

and consequently we can define .. (1-10)

7

Thus we have

> 0

g1^ by

(the cofactor of g.. in g) = __________________J________ g

8

(1-n)

I • RIEMAMIAN MANIFOLD for

i = k

for

i + k

S1JSjk = 8k v. 0

and the

5^ here defined is known as Kroneckerfs delta. It will be easily seen that g1^ = g^1 are components of a contravariant tensor, and 5^ are components of a mixed tensor. We call gjk , g1^ and 5^ fundamental covariant, contravariant and mixed tensors respectively. If, for instance, components a tensor satisfy

TV ■ 1

t w

we say that they are symmetric in

TV

j

and

k , and if they satisfy

- - Tikj

we say that they are anti-symmetric in j and k . It Is easily proved that if the components of a tensor are sym­ metric or anti-symmetric in a coordinate system, then they are so in any other coordinate system. If the components of a contravariant or covariant tensor are symmetric (anti-symmetric) in all the indices, then we call the tensor a symmetric (anti-symmetric) tensor. The g and g1^ are both symmetric tensors. We shall next state some algebraic operations which can be applied to tensors. (i) Addition and subtraction. Let, for instance, R1 and 3V be components of two tensors of the same type, then

RV

+ sljk - T±jk

are components of a tensor of the same type which is called the sum of two given tensors. The difference of two tensors is defined in an analogous way. (ii) Multiplication. Let, for instance, R1^. and be components of two tensors of any type, then = TlJkl

2.

TENSOR ALGEBRA

9

are components of a tensor of the type indicated by the position of the indices, and it is called the product of the two given tensors. (iii) Contraction. Let, for instance, T^.,-, be the components of a mixed tensor. The quantities

are components of a tensor having two less indices than the original one, and in this case, we say that we have contracted ^j^i with respect to i and 1 , obtaining . (iv) Raising and lowering of indices. If aA are components of a contravariant vector, then gj^^ are, by (ii), components of a mixed tensor, and consequently g., A.k are, by (iii), components of a covariant vector. We denote ^ k it by = g*vx . Similarly if are components of a covariant J JK ii vector, then g are' ^ (H)> components of a mixed tensor and con­ sequently, g^'Vj are, by (iii), components of a contravariant vector. We denote it by ii1’ = g1^ * • It is evident that if J

then J1

i

=

,i

A,

We will say that aA and A.^ are conjugate to one another, and we are introducing an object which can be represented by A,1 and A,, alternai 1 tively. We call it a vector, and A, its contravariant components, and A,^ its covariant components. The same thing can be stated for the components of a tensor as is shown in the following examples:

T jk

Tijk = sisT jk

Tljk: —

Ti / ‘ T« s 8 3k

In the first example, we say that we lowered the index i , and in the second, we say that we raised the index k , and that we are dealing with components of the same tensor. (v) Symmetrization and anti-symmetrization. Consider, for example, a covariant tensor Tij^ > a-nd form the

I . RIEMANNIAN MANIFOLD sum of all the components obtainable from J taking all the pos­ sible permutations of the indices i , j , and k , and divide it by 3 J (number of the all possible permutations). We denote the resulting object by T (ijk) = 3T'(Tijk + Tjki + Tkij + Tjik + Tkji + Tlkj)

and call it the symmetric part of . It is easily shown that are components of a symmetric covarianttensor, and the operation — >is called symmetrization of T^ . If the original tensor is symmetric, then we have ijlc) = Tijk ' Consider again, for example, a covariant tensor T . , and all J^ components obtainable from by all possible permutations. Next, give a plus sign to a component obtained from by an even permuta­ tion and a minus sign to a component obtained from ^J an 0dd permu­ tation, and form the algebraic sum of these components, and divide it by 3* f • We denote the resulting object by T [ijk] = 3T (Tijk + Tjki + Tkij “ Tjik

" Tkji * Tikj}

and call it the anti-symmetric part of ^ is easily shown that T[ijk] are components of an anti-symmetric covariant tensor, and the opera­ tion Tj_jk — is ca^led- anti-symmetrization of Tj_j^ • If original tensor is anti-symmetric, then we have = ^ijk * Now, the formula (1 .3 ) shows that thelength of the contravariant vector dx1 is ds , and similarly we define the length X of a contra­ variant vector X1 by (1 .1 2 )

(X)2 = gjk* A k

If we denote the covariant components of this vector by X^ , then the above formula may be written in the following various forms:

U)2 =

and

= V-k =

= S^j^k

A vector whose length is unity is called aunit n1 are both unit vectors, then we have gjkA k = 1

and consequently we can prove that

g.k^ M k = 1

vector.

If

x1

2.

11

TENSOR ALGEBRA

(gjk^JV k )2 < i Thus, we define the angle

e between two unit vectors

X1

and

n1 by

j„k cos e = gjkA'V

(1-13) and the angle

between two arbitrary vectors

u1

and

v1

is given by

cos e =

(l .1*0

U

V

Equation (1.14) gives u v cos

g^u-v” = U^V“ = uJ Vj = gjlCuj.vk

and this is called the inner product of two vectors u1 Prom (l.i1*-), we see that two vectors u1 and to each other if

and v1 . v1 are orthogonal

gjkU^ = o Next, from the transformation law of dxS

g^:

bx^

jk = ax'J ^

Sst

we find g* =

bx

dx*

and, on the other hand, the transformation law of n-tuple integral is

•j p dx dx

d x - W ' 2 ... dx’n = |||l| dx1dx2 ... cten Thus, from these two equations, we get

. dx11 in an

12

I.

RIEMANNIAN MANIFOLD

which shows that (1 .1 5 ) is a scalar. he dv .

dv =

dx1dx2 ... dx11

We define the volume element of our Riemannian manifold to

3.

TENSOR CALCULUS

Takea curve x1(t) joining two points P2(x1(t2 )) and introduce its length

P1(x11(11 ))

and

Jt V If another curve x^t) = x^Ct) + eu^(t)

(e: infinitesimal)

passes through P1and P2 (and consequently ui(t1 ) = ui(t2 ) = 0 ) and is infinitesimallyclose to x^(t) , and if we denote by 51 the first variation of the length integral I , then 51 is given by

where we have put F =

and

x

= dx1 w r

We call the curve for which 51 = 0 for any u^ a geodesic in our Riemannian manifold. A geodesic must satisfy the so-called Euler differential equa­ tions

and it will be proved that x^ are covariant components of a vector. Taking the arc length s as parameter along the geodesic, and calculating the contravariant components A.1 of X^ , we find

3-

,,

,i

(l-l6)

x

where

{

TENSOR CALCULUS

d2x1 , , i , dx*1 ' dxk

13 n

s ^ 5 - + {jk} as-as- = 0

are defined by

,i

,, ,,,

(1-17)

I

m

J

Jk

1 „is /

= p- g

j

^gsk

^Sjk ^

c)xJ

dx3

I ---- + -------3-------- I

2

\ c>xk

/

and are called Christoffel symbols. It will be easily verified that the Christoffel symbols satisfy the following Identities:

(l-1 8 )

- ggk

(1.19)

^

(1 .2 0 )

- gjg

+ g8J

Ci)

+ g 13

(?) = J3 ■/? SxJ

til

=

0

=

0

= diogVi SxJ

Now, from the fact that in (1 .1 6) are contravariant com­ ponents of a vector, we can find the following transformation law of the Christoffel symbols under a coordinate transformation:

(1

pi )

^ x 1* dxr r i i1 dx3 dxt fcfJfcc* " ta’1 ^ " Sx'J

(,.22)

(!■) ax3Sxt

dxr

rr , 3t

(1

3t

Bx3

9xt

Jk

If f(x) is the component of a scalar field, then it is evident that df is also the component of a scalar, and that Sf/Sx1 are com­ ponents of a covariant vector. We call df the covariant differential of the scalar f and df/dx^ the covariant derivative of the scalar f and denote them respectively by (i .2 3 )

6f = df

(1 - 2 U )

f

.

'j If

v1 (x)

SxJ

are components of a contravariant vector field, then

I . RIEMANNIAN MANIFOLD dv'*" are not necessarily components of a contravariant vector. But, com­ bining the transformation law of dv1 with that of {j^} , we can prove that (1 .2 5 )

5V1 = dv1 + vJ* {j^.} dxk

are components of a contravariant vector and

0 .2 6 )

v1^

- ^ 4 . v3 (Jy

are components of a mixed tensor. We call Sv1 covariant differential of and covariant derivative of v^ . Similarly, if v.(x) are components of a covariant vector field, J then dvj are not necessarily components of a covariant vector, but we can prove that (1 .2 7 )

6Vj = d.Vj - v± (jy dxk

are components of a covariant vector and dv.

(1,28)

.

vj;k = ^tc - vi 1jk1

are components of a covariant tensor. We call 5Vj covariant differential of v. and v. v covariant derivative of v. . J J J This operation of covariant differentiation may be applied to a general tensor, say, to T1

.(,.29)

- dT1^

* T^C^dx1 -

- T^^ldx1

ST^". (,-30)

TV ; l

■

* TV s l ) - T \ kI ji> - Tljslkl)

We call 5T1 ., , which is a tensor of the same type as T1 ., , the covariant differential of T and T > which is a tensor having one more covariant index than T1 ., , the covariant derivative of i J T jk If we apply this operation of covariant differentiation to the tensors gjk , g1^ , and 5^ , we get

3-

TENSOR CALCULUS

15

Slj;k ■^ *S3J ^ - »

(,-33)

55^ 5jik ■ ^ * s5 ‘ski -

Ijk' - 0

Thus, the tensors g^k , gij* , and 5^ are all constant under covariant differentiation. It will be easily verified that the covariant differentiation obeys the rules of ordinary differentiation:

s(Rlok i SV

■ 6RV

; 531jk

B and

(Rljk

-

^Jk'jl ' Rljk;l 1 3ljkjl

;m • 3kl ♦ Rlj If we are given a covariant vector field form an anti-symmetric tensor dv.

vj(x ) > then we can

dv,

(1 •3k )

V •. - V, . . = —

^

which is independent of the Christoffel symbols. It is called the curl of the covariant vector v. • J Similarly, if we are given an anti-symmetric tensor field |. . . , then we can form an anti-symmetric tensor 1 2 *** td (1 -35 )

dxJ

ax11

ax12

which is independent of the Christoffel symbols. the anti-symmetric covariant tensor

ax1? It is called the curl of

16

I. RIEMANNIAN MANIFOLD

If we are given a contravariant vector field form a scalar (1-36)

v1., -

^

v^(x) , then we can

* vJ I,1,) -

dx1

J1

dx

which depends only on s/g" . This is called the divergence of the contra­ variant vector v1 . The divergence of a covariant vector v. is defined as the J scalar

(1-37)

and the divergence of a covariant tensor

(1-38)

gradient

. . 1 2 *’’ p

as the tensor

slj S u ...i -i -U"2 ^p* J Now, if we are given a scalar field f .. and the square of its length: >J

(1-39)

f(x) , then we can form

its

A f = g^'f .f .

1

i1 jJ

This is called Beltrami's differential parameter of the first kind of the scalar field f(x) . With the gradient f . , we can form its divergence: 31 (1-^0)

A_f = glj‘(f.,)., =

This is called Beltrami's differential parameter of the second scalar field f(x) . It is also called the Laplaceanof f(x) denoted by ( 1 - ^ 1 )

A f

k.

For a scalar by

=

g l j ’f . . .

kind of the and is

. J

CURVATURE TENSORS

f(x) , the covariant derivative of

f(x)

is given

k.

CURVATURE TENSORS

17

and the second covariant derivative is given by

f

d2f

df

;j;k "

,i ,

" Bx1

J'k

Thus, we see that

f;j;k

f;k;j = 0

However, for vectors and tensors, successive covariant differen­ tiations are not commutative in general. Thus, for example, for a contra­ variant vector v1 , we obtain

(l^ 3)

v±;k;l - v±;l;k = ^ j k l

where1

R±jkl ■ ^

^

' IjlHil

are components of a mixed tensor called Riemann-Christoffel curvature tensor,1 and this tensor needs not be zero. Similarly, if we take a covariant vector v. , then we have j

' ’ •‘ S’

Tj ; k ; l - TJ ;l;k - - Ti Rljkl

and if we take a general tensor

example, then we have

klm Formulas (1 .^3 )^ (l-^5)> and (1 .^6 ) are called the Ricci formulas. From the curvature tensor > we get, by contraction,

(1.-7) moreover, from get

Rj k * RSjk= R.^ , by multiplication by

1Some writers denote our

-

by

g^k

•

and by contraction, we

18

I.

RIEMANNIAN MANIFOLD

(1^8)

R = gJ'kRjk

Rjk and R are called "Ricci tensor" and "curvature scalar" respectively. From the definition .bb) of R^j^i > ^ easily seen that R1 .-,-, satisfies the following algebraic identities:

(l-^9)

Rljkl = “ Rljlk

(1*5°)

Rljkl + Rlklj + Rlljk = 0

and consequently, if we put

(1-50

then

Rijkl = gisRSjkl

^^1

satisfies

(1 •5 2 )

Rijkl = ' Rijlk

(1-53)

Rijkl + Rlklj + Riljk = 0

Equations (1 .5 0 ) and (1 -5 3 ) are called the first Bianchi Identities. Moreover, applying the Ricci formula to g^. , we get s

s

0 = sij;k;l " sij;l;k = - SsjR ikl " sisR jkl

from which

(1•54 *

Rijkl “ ' Rjikl Calculating the covariant components

0.55)

Rljkl

jkl

. 1 / &g«lk 2 y dx^dx1

explicitly, we find

^2gjl

^jk

S2g±l

dx dx

dx dx

dx^dxk /

" grs (fjk}{il} _

\

5-

SECTIONAL CURVATURE

19

which shows that ( 1 • 56 )

Rijkl = Rkllj From (1 .5 0 ), on contracting with respect to

i

and

1 , we

obtain (1-57)

Rjk - Rkj = 0

by virtue of (1 .^9 ) and (1 .5 ^), and equation (1 .5 7 ) shows that the Ricci tensor R ^ is a symmetric tensor. It is to be noted that

g

jkpi

jkl

n.jkn.isT? = crjkcriSP = CTiSR s s sjkl s g jslk s si

or (1.58)

83kRljkl ’ Sl3Rsl < ’ A *

For the covariant derivative . R jki > we can prove

(1 -59)

R^-n^i Jkl;m

of the curvature tensor

+ R1^ ! = °

which is called the second Bianchi identity. with respect to i and m , we find

(1,60)

r Sjkl;s

and on multiplying this by 2 , R and consequently K are absolute constants. Thus, if the sectional curvature at every point of the manifold does not depend on the two-dimensional planes passing through the point, then this sectional curvature is an absolute constant in the whole manifold. Such a Riemannian manifoldis said to be of constant curvature. If this constant Is zero, then we have (,.69)

Rijkl ' 0

In this case, the equations S2x ,:L = dx'1 , r , TdxsT > r st dxF dx obtained from (1 .2 2 ) by putting { } = 0 are completely integrable, and ■? X consequently there exists a coordinate system in which {^ } = 0 , and consequently gj^ = const . Thus, every coordinate neighborhood of the manifold can be mapped isometrically on a certain domain in the Euclidean space.

I . RIEMANNIAN MANIFOLD

22

Conversely, if every coordinate neighborhood can be mapped isometrically on a certain domain in the Euclidean space, then it is evident that we have (1 . 6 9 ). Such a Riemannian manifold is said to be locally Euclidean or to be locally flat. Returning to a general Riemannian manifold, we consider n con­ travariant mutually orthogonal unit vectors x.^ at a point

(a, b, c, ... = i, 2 , •••, n)

(x1 ) . We then have gij^a^b " 5ab

and therefore (..7°) Now, the sectional curvature at this point, as determined by a two-dimensional plane spanned by x,^ and xjjj , is given by

Kab = “ ^ijkl^a^b^a^b

and hence n S-|D=1‘ ^ab

“ ^ijkl^a^a1

or C ' 7 ’)

I 'b=i . ,Kab - R 1kxi ^

and (1‘72)

n n X a=i Y ^ b = iKab ab = R

Formula (1 .71 ) shows that, if we take a unit contravariant vector xA and consider n - 1 sectional curvatures determined by n - 1 twodimensional planes spanned by X.1 and n - 1 unit vectors which are orthogonal to A,1 and to each other, then the sum of these n - 1 seci .k and is independent of the choice of tional curvatures is equal to R.lrxJx i k the Ricci curvaother n - 1 orthogonal unit vectors. We call R-.x^A, ^ ture with respect to the unit vector \i

6.

PARALLEL DISPLACEMENT

23

Equation (1 .7 2 ) shows that the sum of n Ricci curvatures with respect to n mutually orthogonal unit vectors is equal to R and is independent of the choice of these n mutually orthogonal unit vectors. Now, consider the Ricci curvature M with respect to a certain contravariant vector aA : M =

The direction which gives the extremum of

(1-73)

M

is given by

(Rjk - Mgjk n k = 0

and in general, there are n such directions which are mutually orthogonal. We call these directions Ricci directions. A manifold for which the Ricci direction is indeterminate is called an Einstein manifold. For such a manifold, we have 0.7*0

R jk = Mgjk

By multiplication by

and contraction, we obtain R = nM

from which

(1 *75 )

R jk =

R gjk

It will be easily seen from (1 . 6 1 ) that

R

is an absolute

constant. 6 . PARALLEL DISPLACEMENT

If v1 is a contravariant vector at a point (x1 ) and v1 + dv1 its value at an infinitesimally nearby point (x1 + dx1 ) , then we know that (1 .7 6 )

bv1 = dv1 + {j^.)vJ*dxk

are components of a contravariant vector. If bv1 = 0 , we say that the vector

v1

at

(x1 ) and the

2b

I.

RIEMANNIAN MANIFOLD

vector v1 + dv1 at (x1 + dx1 ) are parallel to each other, orthat the vector v1 + dv1 at (x1 + dx1 ) has been obtained from v1 at(x1 ) by a parallel displacement. This definition is invariant relative to changes of coordinates. A similar definition applies to any tensor. If we compare the equations Cl 77) n -77}

6V 1

_ dv1

f i , j dxk at-

m r - w r + ljk] v

for the parallel displacement of the vector with the differential equations of geodesic

v1(t)

along a curve

x1(t)

d2x1 ^ f i , dxJ* dxk _ n + ljkJ a s - a r - = 0 then we see that the tangent dx:L/ds of a geodesic is displaced parallelly along the geodesic. Since we have = o , it is easily seen that the length of a vector and the angle between two vectors are invariant by parallel dis­ placements of these vectors. If we want to displace parallelly a vector v1 at a point P q (Xq ) to a point P ^ x 1 ) which is at a finite distance from PQ , we must first assign a curve x1(t) joining two points PQ and P1 (and consequently, a curve x1(t) such that xIL(t0) = x1 and x1(t1 ) = x1 ) and integrate the differential equations (1-77) with initial conditions v1(tn) = v1 . If we denote the solution by v1(t) , then v"L(t1 ) is the i i vector which we get when we displace the vector vQ at the point P0(xQ) parallelly along the curve xIL(t) to the point P ^ x 1 ) . Thus, the parallelism depends on the curve which joins the starting point and the finishing point. If the parallelism of a vector does not depend on the curve joining the starting point and the finishing point, then, at every point of the manifold, we have one and only one vector v^(x) which is parallel to *1 *| the given vector vQ at the point PQ(x0) , and the differential equations &V1 _ i dxk _ as- " v ;k at~ ■ 0

should be satisfied for any curve.

Thus we have

v±;k = 0 from which, by virtue of (1 .^3)>

6.

PARALLEL DISPLACEMENT

''•’" V

25

■ 0

Thus, if the parallelism of any vector does not depend on the curve along which the vector is displaced, then, the above equation having to be satisfied for any v1 , we must have

RV

■ °

and consequently, the manifold must be locally Euclidean.

C H A P T E R

II

HARMONIC AND KILLING VECTORS 1• THEOREM OF E. HOPF In an n-dimensional coordinate neighborhood U , we consider a linear partial differential expression of the second order of elliptic type L (.) - gjk-Sf* dx'5dx

* h1 ^ dx

where g^k (x) and h^(x) are continuous functions of point P(x) in U , and the quadratic form g^kZjZk is supposed to be positive definite every­ where in U . We shall prove an important theorem due to E . Hopf [1]: THEOREM 2.1. In a coordinate neighborhood U , if p a< function 4>(P) of class C satisfies the inequality L() > 0 , and if there exists a fixed point PQ in U such that cd(Pq ) everywhere in U , then we must have (P0) > and we draw a contradiction from it. Regarding (x1 ) as coordinates of a point in an n-dimensional Euclidean domain U , we use hereafter the terminologies of Euclidean geometry. As we have assumed that (P) $ M in U , there exists a point C in U such that o . Thus, on

F^

on

P,0

We now take the center of the sphere S as the origin of the orthogonal coordinate system and consider the function

28

where

II. a

HARMONIC AND KILLING VECTORS

is a positive constant and r2 - (x1 )2 + (x2 )2 + ... + (x11)2

and on applying the operator

L

to the function

\|r , we find

L(t) = e”0^ [4a2gJ’kxJ’xk - 2a(hixi + g11)] Since R 1 < R , the origin of the coordinate system, which is the center of the sphere S , is outside of the sphere S1 . Thus, on the surface of S1 and inside of S1 , we have gjkxjxk > o and consequently g^kx^xk > const. > o Consequently, taking (2 .5 )

a

large enough, we may assume that

LU) > 0

in

S1

On the other hand, we have

{

i|r(P) < 0

on

F0

^(Pn ) = 0

Finally, we put ®(P) = 4>(P) + 5*^(P) where

5 is apositive small number chosen in such 0 (P ) < M

and this choice ispossible by virtue of By (2.4) and (2 .6 ), we have $ (P) < M and therefore, on the whole boundary of * (P) < M

a waythat

on F^ the firstequation

on Fq S1 , we have

of (2.^ .

1. THEOREM OF E. HOPF But, by (2 .1 ) and (2 .6 ), at the center

P1

of

29 S 1 , we have

®(P1 ) = M Consequently, the function ®(P) attains the maximum at a point which is inside of S 1 . But this is impossible, since, in consequence of L( ) > 0

in

S1

L(\|r) > 0

in

S1

and

we have (2 .7 )

L(®) > 0

Now at a point where the function to

in

S1

$ attains the maximum,

L($)

reduces

and we must have - d - * v XJ\ k < 0

for any

A,1 , and consequently,

.z^ being positive definite and

4Sx^Sxr A k being negative definite, we must have

L U ) = gJ’k

k < 0

axJaxK ~

which contradicts (2 .7 )* Thus the first part of the theorem is proved. The second will be proved in a similar way. Now, in a compact manifold V , suppose that a function 2 of class C satisfies

part 4>(x)

30

II.

HARMONIC AND KILLING VECTORS

everywhere in Vn . Since the manifold is compact and the function (x) is continuous in this compact manifold, there exists a point PQ at which the function attains the maximum, that is (2.8)

*(P) < *(P0)

everywhere in

Vn . Therefore by Theorem 2 . 1 we have *(P) < (P0) = M

in a certain neighborhood of PQ . But the points where (P) reaches its maximum form a closed set, and thus we attain the following conclusion.

tion

THEOREM 2.2. In a compact space (x) satisfies

L() = gJ*k (x)

- , ■ + h^x) dx^ Sx

Vn , if a func­

> 0 Sx

everywhere in Vn , where g*^k (x) are coefficients of a positive definite quadratic form at any point of VR , then we have = const everywhere in

Vn .

Moreover, since in a compact Riemannian manifold tive definite metric ds2 = gjkdxJdxk , we have

(2-9)

4 . - gJk . . , . k - gJk - i i - *

- S »

Vn

1 11 ) ^

axJax

JK

ax

we can state the so-called Bochnerfs lemma: THEOREM 2 .3 . In a compact Riemannian manifold with positive definite metric, if a function 4>(x) satisfies a 0 everywhere in the manifold, then we have

= const

with posi­

2.

THEOREM OF GREEN A

’

for an arbitrary vector field

xA(x)

To prove this, we remark first that, if a bounded set contained in a coordinate neighborhood, then we have

X1

Suppose now that A is a "rectangle”: a^ < x 1 < b 1 vanishes on the boundary of A . In this case, we have

J a1

3,1

f b*

Ja2

D

is

and that

.....rbn^ ^ = „

^

J&n

and therefore

(2 . 1 2 )

which

X

i

But, since the Integral of

x1 . is zero over any open set on

* v anishes, e q u a t i o n (2.12) shows tha t (2.11) is t r u e if

x

i

II.

32

HARMONIC AND KILLING VECTORS

vanishes outside some "rectangle" A . Now, since the manifold is compact, we cancover it by finite number of neighborhoods U1, U2, ..., , whose closures are contained in "rectangles" A1, A2, ..., A^ respectively. Corresponding to each a , of = 1, 2, ..., M , we can easily find a neighborhood V^ between and and a non-negative scalar function a of class C1 in Aa such that L* 1> 1 in ULZ and LX = o outside V . Completing the function 4>a by values zero outside A^ , we have,throughoutVn ,

Thus, if we put + ..

then, the function i|r is of class C and has the following property: ta vanishes outside the "rectangle" Aa and

Hence, if we put

then the contravariant vector field X1 has the property that it vanishes outside the "rectangle" A^ . Thus we have

But, on the other hand, we have

and consequently M

Integrating this over the whole manifold, we have fx1 J

dv = y M a=i

f x* dv = 0 J a ’x

3*

THE THEOREM OF E. HOPF-BOCHNER

which proves Theorem 2.b. Since the Laplacean as A*

=

A of a scalar field

O1

.

=

33

4>(x)

can be written

• ) ..

9

9

J

9

Theorem 2-k implies as follows. THEOREM 2 .5 . In a compact orientable Riemannian manifold Vn , for any scalar field (x) , we have (2 .1 3 )

f A dv = 0 Vn If we apply theoperator A

to

p ♦ , then

we get

A4>2 = 2*a * + 2g1J* .* • }1 3J and consequently, on applying Theorem 2 . 5 to the scalar field obtain (2.1*0

I

JV

n

p , we

(A + g1^ ..0 . .)dv = 0 ^ -jJ

Now, if we have A> 0 everywhere in Vn ,then, as is seen from (2 .1 3 )* we must have A= 0 everywhere in Vn . Hence, as is seen from (2.14), we must have g^ .n. . = 0 or * . = 0 , or 0 = const. This gives another proof of Theorem 2 . 3 in case the manifold is orientable. 3- SOME APPLICATIONS OF THE THEOREM OF HOPF-BOCHNER

gjk

of

In this section, we assume that the manifold is of class C^ 2 class C We consider a vector field ^(x) of class C2 and we put

(2 .1 5 )

*

and for the Laplacean

= i1t1d ±= g1J-!'] ') of the latter we have

A* = where we have put

+ S^ijbjo)

and

II. HARMONIC AND KILLING VECTORS

S1;j - S1 ,. a .^ 3 Now

is a positive definite form in equations of the form (2-l6>

cl ^ C , and therefore if

(

S ^ i j b j c = Tij*j

and if the quadratic form

satisfies > 0

then we have A4> = 2 (

J

j

+

> 0

Consequently, from Theorem 2*3> we get

+

= 0

or ei;j “ 0 and also T. = 0 , and if the quadratic form T. .|^gc ^ i i ** definite, then we can conclude from T^.S = 0 that

is positive

I1 - 0 Thus we have THEOREM 2.6. In a compact Riemannian manifold Vn , there exists no vector field which satisfies relations

3-

THE THEOREM OP E. HOPF-BOCHNER

35

TijSV* > 0 unless we have

and then automatically = o . Thus, the only exceptions are parallel vector fields, and there are no such vectors other than zero vectors if the quadratic form is positive definite. (Bochner [1 0 ] ).

Vn

We now take an arbitrary vector field and write down the Ricci identity:

class

^b;i;c ” ^b;c;i ” ~ ^aR bic from which we obtain L;bjc ' ^ijb ' ^bji^c “ 6b;c;i ” or, multiplying by

be g

^^bic

and contracting,

bc b c /^ t \ g ^i;b;c ~ g ^i;b “ ^b;i';c ” ^ ;a;i “

Thus, if the vector field

^

ai^

satisfies

*i;b ” ^b;i^;c + ^ ;a;i then it satisfies also 0 unless we have

and then automatically = 0 . Especially, if the manifold has positive definite Ricci curvature throughout, there exists no harmonic vector other than zero vector and consequently, if the manifold is orientable, B 1 = 0 . (Bochner [2 ], Myers [1 ]). 5-

KILLING VECTORS

An infinitesimal point transformation

38

II.

HARMONIC AND KILLING VECTORS

(2-23)

x1 = x1 + ^(xjst

is said to define an infinitesimal motion in V_ if the infinitesimal distance ds between two arbitrary points (x1 ) and (x + dx ) is equal to the infinitesimal distance ds between two corresponding points (x1 ) and (x1 + dx1 ) , except for higher terms in 5t . Now, we have •^

J

•

ds2 = gjk(*)dx^dxk and ds2 = gjk (x)dx^dxk Thus, a necessary and sufficient condition that (2 .2 3 ) be an infinitesimal motion of the manifold is that gjk (x)dxJ*dxk = gjk (x)dxJdxk or that (Sn-v + ^

5t)(dxJ* + dxa

be satisfied for any say, that

(2 -2 * 0

dxb5t)(dxk + dx

dxc6t) = g .,(x )dxJ*dxk dx

dx1 , except for higher terms in

+ is

i&

Sxa

dxJ

g

+

aK

g

Sx

=

5t , that is to

0

ja

This equation is in tensor form:

or 5j;k + *k;j = 0 and is called Killing’s equation. We shall call a vector satisfying Killing’s equation a Killing vector. Now, if the manifold admits an infinitesimal motion (2 .2 3 ), then

5 • KILLING VECTORS

39

the vector satisfies (2.24). If we choose a coordinate system in which the vector | has the components e1 - 5i then, equation (2.24) becomes

ax' which shows that the components g.v of the fundamental tensor do not contain the variable x in this special coordinate system. Thus the manifold admits a one-parameter group of motions x1 = x1 + s|.t which is generated by Now, if I1

I1 . is a Killing vector, then we have

6i;j + and automatically

•Si Thus, it satisfies (2 .1 9 ), and consequently we have (2 .2 0 ). as a special case of Theorem 2 .8 , we have THEOREM 2 .1 0 . In a compact Riemannian manifold Vn there exists no Killing vector field which satisfies RijlV < 0 unless we have

and then automatically = 0 . Especially, if the manifold has negative definite Ricci curvature throughout, there exists no Killing vector field other than zero vector, and consequently there exists no one-parameter group of motions. (Bochner [2 ]).

Thus,

II.

1*0

HARMONIC AND KILLING VECTORS

6. 6 . AFFINE COLLIKEATIONS COLLINEATIONS The geodesics In

(2.26)

a V

Vn

* ri

dsJK where

= {j^}

and

s

are given by the differential equations

(*) S

ds

s

i

. 0

ds

is the arc length.

An infinitesimal point transformation (2 .2 7 )

x 1 = x 1 + |1 (x)Bt

is said to define an infinitesimal affine collineation in

V

, if the

transformation (2.27) carries, infinitesimally, every geodesic of the mani­ fold into a geodesic and if the arc length

s

receives an affine trans­

formation. Now, if the transformation (2.27) is an infinitesimal affine collineation, then it will carry the geodesic (2.26) into the geodesic

(2.28)

0 ds 2

ds

ds

where (2.29) a

s = as + b

and

b

being constants. From (2.28), we have

a2 !1

( rW

dx-j dxk

. / 6i , & 61 s t \

I f “ ) ( * * * ! £ “ ) (5° ^

d 2xa

6t)

0

=

ds

ds

from which, substituting (2.26), we obtain

I

a2 11

1 Sr|k

a jk

axJ

i ^

A

i \ dxj dxk ja / d s ds

=

0

But, since the transformation (2.27) carries every geodesic into a geodesic, we must have

6.

(2

10)

(

3 ’

AFFINE COLLINEATIONS

* i1 orJk _ *8* rS

!

♦

^

r^

sxJ

^

r?- - o

i?

ja

or, in tensor form, (2.30

s1. ^

* nVi!1 - 0

Also, if the manifold admits an infinitesimal affine collineation (2.27) then the vector I1 satisfies (2 .3 0 ). If we choose a coordinate system in which the vector has the components ^ = 6^ , then equation (2 .3 0 ) becomes

^ dx

= 0

which shows that the Christoffel symbols = {j^.} do not depend on the variable x 1 in this special coordinate system. Thus the manifold admits a one-parameter group of affine collineations x1 = x1 + sj.t which is generated by (2 . 2 7 )• Take a vector field g1 by g^k and contract, we obtain

which satisfies (2 . 3 1 )•

If we multiply

id

8

5i;b;c "

Rai5

and thus from Theorem 2 . 6 we obtain THEOREM 2.11. In a compact Riemannian manifold Vn , there exists no one-parameter group of affine collineations whose generating vector satisfies < 0 unless we have = 0 and then automatically = 0 . Especially, if the manifold has negative definite

II.

k2

HARMONIC AND KILLING VECTORS

Ricci curvature throughout, there exists no oneparameter group of affine collineations in the manifold.

7.

A THEOREM ON HARMONIC AND KILLING VECTORS

We know that, ifis a harmonic vector,vector, then itthen satisfies is a harmonic it satisfies

*i;j = *Jii

“ 0

and

s and if

^

6i;bjc =

is a Killing vector, then it satisfies

’

'Ui

=

and

g

1

T?1

j

;b;c = " R jiJ

If we apply the operator

A

to the inner product of these two

vectors, we obtain

AU.n1) > g^ei.fe.cn1 ♦ but, on the other hand, we have

,b°*. .

S

*

=

P.

Rij*V = 0

Si^Vjbjc = - Ri j s V and consequently A(|/)

Therefore, by Theorem 2-3>

8.

LIE DERIVATIVES

(2 .3 2 )

43

= consta*nt

and consequently THEOREM 2 .1 2 . In a compact Riemannian manifold Vn , the inner product of a harmonic vector and a Killing vector is constant. (Bochner [8]). 8. LIE DERIVATIVES We know that a necessary and sufficient condition that an in­ finitesimal point transformation (2 .3 3 )

x1 = x1 + S^xjst

be an infinitesimal motion is that (2.34)

gjk (x)dxJ*dxk = gjk (x)dxJ*dxk

be satisfied for any dx1 , except for higher terms in St . But, if we regard (2 .3 3 ) as a coordinatetransformation, then, gjk (x)dx^dxk being a scalar, we have (2.35)

gjk (x)dxJdxk = gjk (x)dxJ*dxk

where are components of the fundamental metric tensor in the coordinate system (x1 ) , and consequently are given by -

(- \ _ bx° dxc

/ \

g3k 1 ' a£TasEgbc< ’ Prom (2.34) and (2 .3 5 ), we have (2-36)

Sjk (x) - Sjk (x) = 0

and thus on putting ^ j k = ( ^ j k )5t “ Sjk(5E) - Sjk(5) we have Lglv . ’j'k

dxa

„ dtf + dxJ gak + SJ'a

l*!*

II.

HARMONIC AND KILLING VECTORS

or (2-38)

Lgjk - tj.k , tk .j

We call the Lie derivative of the tensor gjk with respect to the infinitesimal point transformation (2 .3 3 )* or with respect to the vector field I1 . A necessary and sufficient condition that an infinitesimal point transformation (2-33) he a motion of the manifold is that the Lie deriva­ tive of the fundamental metric tensor with respect to (2 .3 3 ) shall be zero. On the other hand, in order to find a necessary and sufficient condition for (2 .3 3 ) to be an affine collineation, we can proceed as follows: The transformation (2 .3 3 ) carries every geodesic

(2 .3 9 )

4

( x ) ^ ^ = ds ds

0

4

0

ds into the geodesic

ds

JK

ds

ds

or (2 .1*0 ) ds

+ r,k (x) J ds

= 0 ds

Since the left hand side of (2 .3 9 ) are components of a vector, if we regard (2*33) as a coordinate transformation, then equation (2.39) may be written as (2 .1*1 )

= 0

+ r.^ (x) dsd

JK

ds

ds

in the coordinate system (x1 ) , where fh. (x) are Christoffel symbols in ^ the coordinate system (xi ) and consequently are given by

Now, comparing (2.4o) and (2 .41 ), we get relations

( rjk (x) _ rjk (x)) a § - a § ~ - 0

8 . LIE DERIVATIVES

which must be satisfied by any (2 .1*3 )

dx1/ds , from which (x) - fjk (x) = o

and for Drjk = (Lrjk)8t = rjt (x) - rjk (x) we obtain (2 i*i*)

Lr1

=

+ f1 Srj k _ A ii r a

J‘k " d J t o F

Bx1”""

+ Alt r 1

Sxa J‘k

BxJ

+ d|a r 1

J'a

or

(2-,‘5>

^

■ ‘Sjjk * ^jki*1

We call Lrjk Lie derivative of the affine connection r with respect to the infinitesimal point transformation (2-33), or with respect to the vector field I1 . We can see that a necessary and sufficient condition that an in­ finitesimal point transformation (2 .3 3 ) be an infinitesimal affine collinea­ tion of the manifold is that the Lie derivative of the Christoffel symbols with respect to (2 .3 3 ) vanish. In general, when a field o(x) of a geometric object is given, we define theLie derivative Ln of n withrespect to by the equation (2 .46)

DO = (LO)Bt = n(x) - o(x)

where n(x) denotes the components of this object in the coordinate system (x1 ) , (2 .3 3 ) being regarded as a coordinate transformation from (x1 ) to (x1 ) . By a straightforward calculation, we can prove the following formulas: For acontravariant vector v1 : (2.1*7) for a covariant vector

Lv1 = ^ V 1 v. : J

j cl

- i1 y va d

II.

k6

HARMONIC AND KILLING VECTORS

for a mixed tensor, say,

(2 .4 9 )

LT

Tljk :

jk .a - I ;aT

+ I .jT ak + 6

Now, for the fundamental metric tensor

ja

g_ . , we have aJ

-^aj = 6 saj;b + 1 ;asbj + 1 ;jSab and hence Lgaj

+ 6j;a

and this gives ^ a j ^ k = 5a;j;k + 5j;a;k ^Lsak^;j = 6a;k;j + *k;a;j (Lgjk);a = “ 5j;k;a " 5kjj;a Adding these three, we find

^Lgaj^;k + ^Lgak^j " ^ j k ^ a = 2|a;j;k + *bR ajk + ^bR jka + *bR kja 2^a;j;k + Rajkl* ^ by virtue of R ajk + R jka + R kaj

0

and Rbkja

Rajkb

Thus we have

2

or

s# ^ Lgaj^;k + ^ a k ^ j " ^ j k ^ a ^

s ;j;k + R 'jkl^

9 - LIE DERIVATIVES OF HARMONIC TENSORS

kj

which shows that a motion in a Riemannian manifold is necessarily an affine collineation. Next, for a contravariant vector field v1(x) , we obtain, by a straightforward calculation,

(2 .5 1 )

L(v±;k) ’ (Lv±);k = Similarly, for a covariant vector field

(2 .5 2 )

L(vj;k> - (Lvj>;k = -

vi(Lrjk)

Finally, for a general tensor, say, (2 .5 3 )

LfT1^ ) - (LT1^ ) ^ - T ^ L r ^ )

vj(x )> we get

T1

, we obtain

- T^fLI^) - T^CLI^)

These equations show that a necessary and sufficient condition that covariant differentiation and Lie derivation be commutative is that the vector field I1 define an affine collineation. Now, from

LrJk ■ sljj;k + " V i 1" we find ^Lrjk^;l =

+ Rljkm;l|m + Rljkm|m;l

and consequently (2-5^)

jk'jl “ (Ll4, '"LJijl).„ ';k = LR ^ jkl

and thus, for a motion, we have

(2-55)

LR1^

9-

LIE DERIVATIVES OF HARMONIC TENSORS

A tensor conditions: (2 .5 6 )

Ii i 12

- 0

.

.

is called harmonic, if it satisfies the

1 2

^

is anti-symmetric in all the indices,

II.

kQ

HARMONIC AND KILLING VECTORS

(2'57)

*[i 1i2 ...ip*-i] ^ = °

or explicitly

(2.58) ^

±

12

p'J

| J 2

±

^

^

p> i

+ ••• M i 3

i ^ 12

p" 2

.-.4 p-i^^p

and furthermore

(2-59)

g1J' 6 1±

i -i = 0

2 ’•^ p ’J

It is well known that in a compact orientable Riemannian manifold, the number of linearly independent (with constant coefficients) harmonic tensors of order p is equal to the p-dimensional Betti number B^ of the manifold, (Hodge [1])Assume now that the manifold Vn admits a one parameter group of motions generated by x1 = x1 + T]i(x)6t and put

dx so that LgJk - 0

and covariant differentiation and Lie derivation are commutative. If we now apply the operator L to a harmonic tensor

^ i V - ’ip then (2 .6 0 )

L|^ . . 1 2‘* p

is anti-symmetric in all the indices,

(2 .6 1 ) (L£. . . ) . = (LI-m ).j_ + (Lg^ .. 1 2 *** ^ 2* ’* jp 1 ^ 3 *‘* P (Lli -? 12“

-T p-1 J

-T).i

. )..

y±2

+ ... +

9-

LIE DERIVATIVES OP HARMONIC TENSORS

(2 .6 2 )

g

(L 61± . ) •= 0 1 2 '‘'1p ’J

and thus the Lie derivative

L

Ei i

12

. . .i

p

is again a harmonic tensor. But, on the other hand, we have by our general definition T

L

t

1_ a t i i = ^ *1 1 i •a ^ -1 6ai 1 2*' p 1 2 " p* "1 2'

a

71 ^ a i 2-..ipji1 i^i^ ...ip;i2

+T'a Ji 1 W

2 ..

t

1 *'* 11 *i *i i p 9 p 1 2

i a p-1

i1ig...i ^ a; i^)

. .lp+ - •

. .ip _ i a

= ^ a^ai 1 ^*1 +^T)a^i ai i ^*i +••*+( 2‘* p 9 1 1 3 * * *^ ? j l 2

±

±

1 2 ’ *' P ” 1

a^*i ,ILP

which shows that the harmonic differential form ii (L r. . . )dx 31iV - * V

a

ip dx

a

... a

^ dx

p

is the exterior derivative of the form ^2

A

____A

rly

^"D

^

2’ and since the harmonic form which is the exterior derivative of another form is identically zero, we obtain THEOREM 2.13- If a compact orientable Riemannian manifold admits a one-parameter group of motions, then the Lie derivative of a harmonic tensor with respect to this group is identically zero. (Yano [33)* Now, if there exist, In the manifold, a harmonic vector a Killing vector tj1 , then, applying Theorem 2.13> we have

and

50

II.

HARMONIC AND KILLING VECTORS L ^i = A i - a + ^;i*a = A i j i

+ A i^ a

« (ta,a );1 = o from which we conclude (2 .6 3 )

= constant

which gives another proof of Theorem 2 . 1 2 for an orientable manifold. 1 0 . A FUNDAMENTAL FORMULA

In a compact orientable Riemannian manifold AT we consider an i arbitrary vector field £ (x) and we form the new vector field

•

v

whose divergence is (2-64)

(i1. 3 J

^

)

1

J J

9 -L

On the other hand, from the Ricci identity: p1

;j;k

p 2-

;k; j

we have, by contracting with respect to ,i

I

= r1 ajk1 i

and

k ,

,i

or

8l;Jii ' 'SijJ * “i / and on substituting this into (2.64), we obtain (2 -6 5 )

(5 1. 3 J

3 d-

3 -* -3 J

X J

i J

3 d-

51

11 • SOME APPLICATIONS OP THE FUNDAMENTAL FORMULA Next we form the vector field

3 d-

whose divergence is

(2.66)

(i1.iiJ*).i = J J

3 d-

3 d ~3

J

3 d-

3 d

and from (2 .6 5 ) - (2 .6 6 ), we obtain

(2-67)

(l1.,^).., - (i1.i|j)., = R, ^ J

3

3 d-

3

J

J

3

J

J - 1-

J - 1-

3

J

Integrating both members of (2 .6 7 ) over the whole manifold, and applying Theorem 2.4, we obtain the formula

(2.68)

f

JV

(R1 .6lgJ' + I1.-!-3'., -

-LJ

n

3J

3-L

3-L 3J

)dv = 0

for which, on putting 6u j

=

, a.

we can also write (2 .6 9 )

f

J V

(R^-l1^

+

n

,)dv = 0

^3

3-L-3

J

(Yano [3 ]) 3 and this formula, which is valid for any vector field will be used extensively in the following discussions. 11. SOME APPLICATIONS OF THE FUNDAMENTAL FORMULA First, if

|^(x)

is a harmonic vector field, then

Si;J ■ !j;i

i1;! - 0

and consequently, the fundamental formula (2 .6 9 ) gives (2-70)

f

t/v

(R. n

1J

,-)dv = 0

-*-* J

^(x) ,

52

II.

HARMONIC AND KILLING VECTORS

But, since = gaCgbd*a;b!c;d and our metric is positive definite, we have i1;3$,., > o J

equality occurring when and only when

|.. . = 0 ; and thus, if J

RijS1^' > o then, from (2 .7 0 ), we conclude

Riji1!*1 '= 0

Moreover, if (2 .7 0 ), we conclude

and

|± . = °

ls a positive definite form, then from

t± = 0 and this gives another proof of Theorem 2 . 9 for an orientable manifold. (Yano [3 1 )• Next, if ^(x) is a Killing vector field, then . = 0

in.. . +

and automatically

and consequently, the fundamental formula (2 .6 9 ) gives

(2-71)

f (Ri , | 1 | j - | 1;j' | , . . J) d v = 0

JV

n

so that

implies

R 1-1-l1 lJ' = 0 -1-J

and

|,. . = o J

. = 0

12.

CONFORMAL TRANSFORMATIONS

Moreover, if (2 .7 1 ), we conclude

53

is a negative definite form, then, from

t± = 0

and this gives another proof of Theorem 2 . 1 0 for an orientable manifold. (Yano [3 1)• 12. 1 2 . CONFORMAL TRANSFORMATIONS An infinitesimal point transformation x1 = x1 + |i(x)6t is said to define angle 9 between angle 0 between neglecting higher Now,

an infinitesimal conformal transformation in Vn if the two directions dx1 and Sx1 at (x1) isequal to the corresponding directions dx1 and bx1 at (x1), terms in 5t .

g.,(x) dxJ,&xk COS

0 =

V

...

y g j k(x)dxJ*dxlcJ g j k ( x ) b x h x k and g.k(x) dxJ*5xk cos 0 = — ■■ J - ■■ ---J gjk(x)dxJdxkJ g . k ( x ) b x h x k and since the angle 0 is a scalar, the first of these formulas can be written also in the form g..(x) dxJ*6xk cos 0 = -.. . J gjk(x)dxJ'dxkJ gjk(x)5xJ'8xk where are components of the fundamental metric tensor in the co­ ordinate system (x^) , and x^" = x1 + |^(x)5t is regarded as a coordinate transformation (x1) --(x^) . Thus, a necessary and sufficient condition for x1 = x1 + ^(xjst to be an infinitesimal conformal transformation is

5b

II.

HARMONIC AND KILLING VECTORS

gjk(x) = (1 + 2 0 Bt)gjk (x) or °Sjk = Sjk(5) - 8jk = 20SjkBt or L g ^ = |jjk + 6k;J = 20 g.k

(2.72)

and if we assume that the vector field conformal transformation, then we have

|^(x)

sjjk + «k;j = 20Sjk

defines an infinitesimal

S1;! = 110

Thus, the fundamental formula (2 .6 9 ) gives

J

[Rij.51|^ + |1 ;J'(20 glj. - 5 i;j) - n2 0 2Jdv = 0 Vn

or (2 .7 3 )

J

[Rlj.|1|J' -

- n(n - 2)02]dv = 0

Vn and consequently, if R^-lV < 0 then we must have, for

n > 2 , = o

Moreover, if (2.7 3 )> we conclude

s1;j = °

0=0

is a negative definite form, then, from

l± = 0 and hence

THEOREM 2.1^.

In a compact orientable Riemannian

1 3 - HARMONIC VECTORS

55

manifold Vn (n > 2) , there exists no vector field defining a conformal transformation which satisfies Rijl1!*5 ’< 0 unless we have

and then automatically = 0 . Especially, if the manifold has negative definite Ricci curvature throughout, there exists no vector field defining an infinitesimal conformal transforma­ tion other than zero vector, and consequently there exists no one-parameter continuous group of conformal transformations. (Bochner [2 ], Yano [3 ])* 13.

A NECESSARY AND SUFFICIENT CONDITION THAT A VECTOR BE A HARMONIC VECTOR

We know already that, if tl.j - Sjji ■ 0

^(x)

is harmonic, that is, if

art

I1^

- 0

then also {2-lb )

g ^ l 1.,-.1, - R1 ,•I J J

and we are going to prove the converse. For an arbitrary vector field

= 0

^(x)

we put

* = and form A* = 2 1

and by Theorem (2 .7 5 )

2

-1-

J J

-b, we find

/ v I < sjl"e±; J ; f c >ei * n

■ 0

56

II.

HARMONIC AND KILLING VECTORS

On the other hand, we know that (2-76)

f

[R.

,]dv = 0

n and thus, we find

f

Jy

K g ^ i 1 ., - . 1 , - R1 ,-^’) ^ + >J>K

n

J

.„• - 1 ,-^) +

-*•yJ

-L

Jdv = 0

>J- jJ

which may be also written in the form

2 '77)

f v l o 12* "" ”

unless we have

^1112** and automatically

Fl'i)v V ° Especially, if the form

is positive definite, then there exists no anti­ symmetric tensor field other than zero which satisfies (3 •*0 • Dually to (3*3) we also have

kl

1 . THE THEOREM OF E. HOPF-BOCHNER

63

-pg

^ i1l3**’ip^a^i2

^2**'^p-l^l

. Ra i1 *'-1s-ials+i *'‘“ ^p *s

S

+

. '3 < t

' #1S-ialS+l'*+1**#ip

Thus if the anti-symmetric tensor

(3 -7 ) g ^ P S ^ . . . ^

+ 8^...^.^

^ 12* ■-1p ;a;11

. Rab 1Sit

£. . . l 2’’"Tp

satisfies

+ ••• + eill2...lp _1j;lp )jk

1 i1i3‘'*1p;a;12

S x2 ***^p-l 11;a; ^ ^

then it satisfies also

(3 .8 )

±

,

1 2

V

i

.,.k +

P ^3

1

al

. / ,

3-13+1

T>

3

and consequently we obtain the relation

with the minus sign, where the symbol and hence we have

Ftl^ “1 2

. } is defined as before,

THEOREM 3*3* In a compact Riemannian manifold Vn , there exists no nc anti-symmetric tensor field which satisfies (3•7) and

6k

III.

HARMONIC AND KILLING TENSORS Fit,

. , }< o 1 2 ‘ "- i p

unless we have

and then automatically F U ±1i2 ...i) -^p = 0 Especially, if the form PIS

12 is negative definite, then there exists no anti­ symmetric tensor field other than zero tensor field which satisfies (3-7). 2.

HARMONIC TENSORS

Now, if ^i1i2‘*’*p is a harmonic tensor field of order p , then it satisfies (2 .5 8 ) and (2 '59), and consequently it satisfies (3-^)- Thus from Theorem 3-2, we have THEOREM 3-k. In a compact Riemannian manifold Vn , there exists no harmonic tensor field of order p which satisfies

V unless we have I and then automatically Fit*

4

1

1 1 1 2 ‘*mXp

®

0

3Especially,

KILLING TENSORS

If the f o r m

is p o s i t i v e d e f i n i t e , field of order field,

p

then there exists no harmonic

other th a n the zero tensor

and consequently,

we have

B

[1], M o g i

65

i f t h e m a n i f o l d is o r i e n t a b l e ,

= 0 (p = 1, 2, [1], T o m o n a g a

3.3

••., n - l ) .

[1], Y a n o

(Lichnerowicz

[h]).

KILLING TENSORS

For a Killing vector

^

and a geodesic

x 1 (s)

of the manifold,

we have

6 , ds

dx1 , _ 1 / u i 3 s“ > - 2 u i; j +

along thegeodesic,and thus the Killingvector

onthe

Conversely,

len g t h of

^ dx1 dxJ _ Q 3 s ~ 3 s- ' 0 the orthogonalprojection

t a n g e n t o f a g e o d e s i c is c o n s t a n t a l o n g

of

a

thegeodesic.

if the leng t h of the orthogonal p r o j e c t i o n of a

v e c t o r f i e l d o n t h e t a n g e n t o f a n y g e o d e s i c is c o n s t a n t a l o n g t h i s g e o d e s i c , then

1

iy , as-1

+ p

(p

2 u i;j +

^ dx

dxJ

as- ■ 0

implies

5i ; j + 6j ; i Thus,

0

a necessary and sufficient condition that a vector field

b e a K i l l i n g v e c t o r is t h a t t h e l e n g t h o f t h e o r t h o g o n a l p r o j e c t i o n o f the vector on the tangent of any geodesic b e constant along the geodesic. Next,

for an anti-symmetric tensor field

5.

quantity

1 2 dx1

is p a r a l l e l a l o n g a n y g e o d e s i c

.

x^(s)

, if and only if

. ^

the

66

III.

HARMONIC AND KILLING TENSORS

that is, if and only if

*il2. . . V

J ^ 0 i 2...V

l - °

and such an anti-symmetric tensor field

^iii2‘’#ip we will call a Killing tensor. derivative

Equation (3-9) shows that the covariant

6i1i2-•-ipSJ is not only anti-symmetric in i1, ig, ..., ip i1 , and j . Thus we can see that

but alsoanti-symmetric in

'*ip ;

is anti-symmetric in all its indices, and consequently, equation (3-9) is equivalent to I

(3‘ 10)

Sli 12” - V j = l [ 1 112’ - - V j]

or explicitly (3-11) pe± ±

12

i

-1 + i n .. i -i + ••• + J 2 p i

1 ...1 i-1 1 2 p-1J,T)

=

0

If 5ii±2 ..-ip is a Killing tensor, then it is evident from (3 •10) that satisfies ( 3 . 12)

i

.

1 2 *

'

.

D

y

= 0

2 ""’ p but (3-11 ) and (3*12) imply (3*7), and consequently (3-8), and thus, as a special case of Theorem 3 .3 , we have

b.

67

A FUNDAMENTAL FORMULA

THEOREM 3 .5 . In a compact Riemannian manifold Vn , there exists exist no Killing tensor field of order p which satisfies Fit* 4 1

* )< 0

2 * * *

jp

unless we have

V

2" - V J = 0

and then automatically FU± 1 1 ) - 0 1 2'*,Ap Especially, if the form F{| '12 is negative definite, then there exists no Killing tensor of order p other than zero tensor. (Mogi [1], Yano [k ]). ^ . A FUNDAMENTAL FORMULA In a compact orientable Riemannian manifold symmetric tensor

± ..

we form

iV

and the divergence ii„- •

From the Ricci identity:

Vn , with an anti-

68

III.

n 2...ip

HARMONIC AMD KILLING TENSORS

iig .-.lp ;j;k

=

|a l 2 “ '

5

;kjj

V

•

+

^

-

R ajk + 1

"

V

K

2

+

+

61 1 2 , , ' V i a R 1P

ajk + ••• + 1

we have, by contracting with respect to

i

and

R

ajk

k ,

...... ^ - v . y » aji

and consequently, on substituting this into (3 *1 3 ), we obtain

(iLX2'--1V

j

)

=

,J’

i2 ,--1p ; 1

;ljj

iki •..i + (P - i)Rljkli

i2 *,-1p

jl 5

+R

112'--1p J lj

ii ...l

i3...ip + 5

1 2 ," ip

j Jj6 i2 ...lpjl

by virtue of Rijkl = Rlkji But, according to the identity: Rijkl + Rlklj + Riljk “ 0 the term iki ...i jl (p - l )R±jkl6 I

appearing in the right-hand member of the above equation can also be written as iki0...i jl ^ , iji0...i kl (p-i)RnviS 3 = L R,,_ie 3 'ijkl5 5 i3...lp - E-2“ "ijkls -8! i3---lp and thus we have

4.

(3-l*0

(I

A FUNDAMENTAL FORMULA

69

ii ...L j iip•••ip j X-;! ± L ).± = I -i-i6 i .. i 2*** p * 2 p + R ^ + Rij5

‘" V

+ Erl R g1,5^ ' ■'1p.kl 5 i2...lp + 2 Rijkl5 5 V " 1*

+ ili2' " ^

-5j.

Next, we consider

sl l 2" ' V ;i. J 12*''1p . and the divergence ii_... i_

j

U p •••in

(3‘15) J V ” V

iip-•-in

- 8

J

;l‘ V - .ly j) * '- °

where

■

1

;a®

70

III.

HARMONIC AND KILLING TENSORS

Now, the tensor 11....Ip 1V I Is anti-symmetric in all the indices, and hence li2...ip;j

_ 1 el1l2‘''-4)’

J'

*j^2‘

P

^1^2‘‘'^p’*^

“

p+1 P 4

‘'ip;^ ^i^^.-ipij]

where

il,!«•••: denotes the anti-symmetric part of the tensor

^11i2‘‘'^p’j and, on introducing this into (3-17) we obtain the relation:

( 3 .- 8 )

f "V

n

12

p _ 2HM

P

lx2

^

J

[i,i0...ip)j] -1-2 U" l 12 - * - V j] iip-.-ln -

which will be of equal importance. 5If

I

j -i*

i

1

• i>d v

=

0

(Yano [^]).

SOME APPLICATIONS OF THE FUNDAMENTAL FORMULAS . . (x) 1 2 " ’1?

is harmonic, then the substitution of

I[l1l2” - V j] = ° and |a . . = o 12‘",:Lp ;a

5-

SOME APPLICATIONS OF THE FUNDAMENTAL FORMULAS

71

in (3 •1 8 ) gives

(3 -1 9 )

f

(FU±

JvT>

± j} + 1 1 2 * “" Jr p p

I

1 2

± * 1112 * V

-)dv = 0 J

and thus F{£. .

, }> 0

*'T) implies .

3iii2 -.-ipj j

= o

and

and Ff^ , , }> 0 12*** implies

This gives another proof of Theorem 3 •^ for an orientable mani­ fold • (Yano [h ]). Similarly for a Killing tensor, (see (3-10) and (3.12)), we obtain n

(3.20)

I JV

(FU± n

U p ' - U j t } - | h i i .,-)dv = 0 1 2 *** p 1 2*“ T),J ^ * ±

and thus

implies

' 0 and

72

III.

HARMONIC AND KILLING TENSORS

P[*i 1i2 . . .“T) J =0 and

X1i2 ...i ’Lp}< 0 implies

= 0

fold .

This gives another proof of Theorem 3.5 for an orientable mani­ (Yano [k ] ).

6.

CONFORMAL KILLING TENSOR

If a vector field defines a one-parameter continuous group of conformal transformations, then f±;J ♦ !j;i ’ a Ll1^

and

R = g1^

. > nL > 0

and if we fix a point in the manifold and take a coordinate system in which gij = 8 ^. atthefixed point, so that contravariant and covariant compo­ nents of a tensor have the same values at this point, then, for n > 2 p , we have n(t

1 ^ n-2p T h x2 ‘••1 p. , n(p-l ) i ^ g - •-ip - n - 2 L? ^ i g . ..ip (n-0 (n-2 ) 5

=

^ig.-.ip

a=E L 6l l l 2 ‘ ' , l p 6 n-i L|

' W " 1*

Thus the quadratic form F(ii i . ) l 1 l2 - " lp is positive definite, and this is so also in all coordinate systems, and by Theorem 3 .k , we have B_ = o for p = 1 , 2 , ..., [n /2 ] . If we now apply ir Poincare’s duality theorem for Betti numbers we obtain THEOREM k .1 . In a conformally flat compact orientable Riemannian manifold Vn , if the Ricci

80

IV.

FLAT MANIFOLDS

quadratic form is positive definite, then we have Bp = 0 , (p = l, 2, ..., n - i). (Bochner [5 ], Lichnerowicz [1 ]). Next, if we assume that the Ricci quadratic form negative definite and denote by the matrix

the biggest (negative) eigenvalue of

||Rj_j II > then we have R1 .|1IJ < - Ml1^

and for

- M

g^. =

and

we obtain for

Fl5i.i„...ip > i - l£iE M ‘1'v “1 2

R =

< - nM < 0

n > 2p

" ipV

2 ...ip

n(p-i )

w.1!12 ’""1p,

(n-1;(n-2) M5 = _ £=£ Mi1112" ' 113?. .

n-1

and hence we have THEOREM b.2. In a conformally flat compact orientable Riemannian manifold Vn , if the Ricci quadratic form is negative definite, then there exists no (conformal) Killing tensor field other than zero for

is

p = 1 , 2, ..., [n/2] .

C H A P T E R

V

DEVIATION FROM FLATNESS

1 •

DEVIATION FROM CONSTANCY OF CURVATURE

If

(5-1 )

R1(jkl = K(gJ.kg11 - g ^ g ^ ) >

then, for an anti-symmetric tensor

K > 0

, the quantity

------------ = 2 K

I

**

and is a positive constant. Assume now more generally that we have

Riikl|lt]'|kl

(5 -2 )

0 < A < --- --------- < B

for every anti-symmetric tensor

, A

and

B

being constants.

If we put = p-^qJ - p^q1

for two unit vectors get, from (5 -2 ),

p1

and

^ A < -

where

which are mutually orthogonal, then we

to each other, we have 81

) > '**’Q-(n-i

)

82

v.

DEVIATION PROM FLATNESS

" RijklplpJpkpl = 0

i A < - RijklP1(l?a)Pkcl(a) < i B

(a = 1’ 2’

and from this and n-1 p ip l

• gjl

we obtain

(5 •3 )

\ (n - 1 )A < RjjfP^P^ < \ (n - 1 )B

Prom (5.3 )> we have

R1j!iSJ" > \ (n - 1 )A|1i1

for any vector

(5-4)

l1

and consequently

Rlj|la5Ja - r (n ' 1>A *lj'5lj

for any anti-symmetric tensor

(5-5)

t^

and, from (5-2), we have

R 1 J k l ! 1 J lk l >

-

Thus, from (5 -b) and (5*5)> we find

Rij-Ila|,5 *a +

\ t(n - 1 )A - (p - 1 JB]!1* ^

or i,i~- ••: p{si 1

* } > i- [(n - l )A - (p - 1 )B] | 1 2

1 2 ‘"‘T>

and for

this Is positive. But, since we have

£>£{ -

.

1

1112“ *;T)

1.

DEVIATION FROM CONSTANCY OF CURVATURE

J > n=T

for

p = 1' 2' •••’ [n/25

we can say that, for ^ - 1

or

A - 1 B

the form F(S

12 is positive definite for Theorem 3 •^ > we have

p = i, 2, ..., [n/2] . Thus, applying

THEOREM 5 .1 . In a compact orientable Riemannian manifold Vn , if the curvature tnesor satisfies

1

0 < 1 B < -

(5-7)

Riikl5lJ|kl

-----< B 6 J6• • 5 5iJ

for any anti-symmetric tensor , B being a constant, then all the Betti numbers Bp vanish, (p = 1 , 2, ..., n - 1 ) . (Bochner and Yano [1 ]). This result may be compared with a recent result of H. E. Rauch [1]. Next, if we assume that --------- < - B < 0 - A < ---BiikiS1J«kl

(5 .8 )

for any anti-symmetric tensor

I1 *1 , then

.

F( I, , , } < 1 [(p 12'** p ~ 2

1

i i ...L %

)A - (n - l )B] I

and for
we have THEOREM 5.2. In a compact Riemannian manifold, if the curvature tensor satisfies

< -

(5 -9 )

A < 0

for any anti-symmetric tensor I1*' , A being a constant, then there exists no Killing tensor of order p where p = 1, 2, ..., [n/2] . 2.

DEVIATION FROM PROJECTIVE FLATNESS

Consider an n-dimensional Riemannian manifold. If there exists, for any coordinate neighborhood of the manifold, a one-to-one correspondence between this neighborhood and a domain in Euclidean space such that any geodesic of the Riemannian manifold corresponds to a straight line in the Euclidean space, then we say that the Riemannian manifold is locally pro­ jectively flat. For n > 3 > a necessary and sufficient condition that the mani­ fold be locally projectively flat is that the so-called Weyl projective curvature tensor vanish, where (5. 10)

W ijkl " Rijkl ~ rdr ^jk^il “R jl8ik^ Now

(5. 1 1 )

W ijkl = Rijkl " n=T

R jlgik^ " 0

implies R jksil " R jlgik + Riksjl ~ Ril8jk " 0 and hence

and substituting this into (5*11 (5-12)

), we

find

Rijkl = n(n-i ) ^jk^il ”^jlsik^

2.

DEVIATION PROM PROJECTIVE FLATNESS

85

and this shows that the manifold is also of constant curvature* Conversely, if the manifold is of constant curvature, then its Riemannian-Christoffel curvature tensor has the form (5-12) and its Ricci tensor has the form R^j =(R/n)gij , and consequently, it will be easily verified that = 0 , that is to say, that the manifold is locally projectively flat. If we substitute Rijkl = ^ijkl +

^Rjkgil “ Rjlgik^

into (3*6), we find

FI!w

V

- i ? r Ri / v

' ' V l2...lp + P_1 W

1j13

^

kl

and in order to measure deviation from projective flatness, we Introduce the quantity

(5-1^)

by L have

W = sup

Iw . tiJ t^lI — ------ , ! Jtlj

U 1J’ = - 5j:l)

Now, if we assume that is positive definite and denote the smallest (positive) eigenvalue of the matrix ||Rj_jl I > then we

R^-i1^* > Li1^ and thus, for

and

R = g1JR1j > nL > 0

g . . = 5.. , we have J J

Rij* J

^

i2 ...i Xp >

1 2 ...ip 1l12''•1p,

Consequently, we have, from (5•i3 )>

86

V.

DEVIATION FROM FLATNESS

and we obtain the following conclusion: THEOREM 5.3 • In a compact orientable Riemannian manifold VR which has a positive Ricci curvature, if (5 -1 5 )

L > Erl w then there exists no harmonic tensor of order p other than the zero tensor, and consequently Bp = (p = 1, 2, ..., n - i) . (Bochner [5 ]> Yano [k]).

- M

Similarly if is negative definite and if we denote by the biggest (negative; eigenvalue of the matrix ||Rj_j| I then

*i,i2 ...V

£. 2p ,

^I^p M __ P - 1 ___ R > Pzl c n- 2 (n-1 )(n-2 ) 2 then there is no (conformal) Killing tensor of order p other than zero, p = 1 , 2 , ..., [n/2 ] . (Bochner [5 ]* Mogi [1 ], Yano !>]).

C H A P T E R

VI

SEMI-SIMPLE GROUP SPACES 1.

SEMI-SIMPLE GROUP SPACES

Take a compact semi-simple group space with Maurer-Cartan equations dh^ (6 .1 )

where

h^

° bx1

Shf — 2 . = ch _V*

0 bx1

.

.

.

.

(a, b, c, ... = 1 , 2 , ..., n)

bc a

are the constants of structure, (Eisenhart [2]). If we put She = " chefccfS

(6 -2 )

then, for a semi-simple group the rank of the matrix | |g^c lI is n and, since the space is compact, the quadratic form g, nz°zcis positive defab inite. Thus, denoting by ||g || the inverse of thematrix ||guc l I > V we can use g and g^Q to raise and lower the indices a, b, c, ..., f Thus multiplying the Jacobi identity: q

j

cabeccef + cbce°aef + “ca®0^ by

and contracting over a

which shows that C^C(^ d . If we put (6 .3 )

and

f

= 0

, we get

anti-symmetric In all the indices

g1J - b & ’s’*” 90

b , c , and

1. and denote by

SEMI-SIMPLE GROUP SPACES

||g | | the Inverse matrix of j-*

91

|Ig1 "1 1 | , then we have

gjk = hjh^gbc

(6.k)

where /ZT r-N (6-5)

JO hj =

g

JO On, gjkhc

and the quadratic differential form (6 .6 )

ds2 = gjkdxJ*dxk

is positive definite. We give this metric to our semi-simple group space. As we have h^h^ = 5^ , we have, from (6 . 1 ), a J J

.

J -

"jk1 ’

i

from which ■ tjV * "jk1 The curvature tensor formed with the affine connection being zero, we have _ RV i

+ ^ k 1;! " nji^k + ^jk^si1

E^.

S^ 1 jl nsk

from which, by virtue of (6 .1 2 ) and of the Jacobi identity, we find _ s i K jkl “ kl sj or (6,13)

1

Rijkl " " °ijsQklS

Multiplying this equation by , we find

(6 .1 * 0 by virtue of

g

Rjk “ T gjk

and contracting over

i

and

2.

(6 .1 5 )

SEMI-SIMPLE GROUP SPACES

- Q jsnkr

njs °kr

93

T &jk

Thus, our space is an Einstein space with positive scalar curva­ ture. 2. A THEOREM ON CURVATURE OP A SEMI-SIMPLE GROUP SPACE Now, we shall prove the following THEOREM 6 . 1 . In a semi-simple group space with the metric tensor (6 .^), we have 0 > R

—

for any

i * ijkl*s1^* 1 > — -T

= - I"*1 .

To prove this we fix a point in the space and take a coordinate _n which g ^ = system in at this point, and write all indices as subscripts. We have, from (6 .1 5 ),

X.

. J

^°ijsnijt

5st

or 1 ...n

I

k

’ 6st

Consequently, 2 (1< i; s = i, 2, ..., n) represent n J unit vectors orthogonal to each other in (i/2)n(n - 1 )-dimensional Euclidean space. Thus, if we denote by < A = n + 1, ...,(i/2)n(n - 1)) the (i/2)n(n - i )-n unit vectors orthogonal to each other and also to 2\[2 Q . . , then we have -LJ3

^ s=1(2'/i”n ijs)(2‘^ n k1s ) + I J =n+, nijAnklA = 5(ij )(kl) (i < j; k < 1) from which

VI.

SEMI-SIMPLE GROUP SPACES

and consequently

Thus, we have proved the tensor inequality

at a fixed point in a special coordinate system, and therefore this holds in general. (Yano [k ]). 3- HARMONIC TENSORS IN A SEMI-SIMPLE GROUP SPACE We now assume that there exists a harmonic tensor field

in our semi-simple group space, and then formula (3 *1 8 ) gives

For

p = 1 , we have

which shows that = 0 , and hence we have For p = 2 , we have

but we have, by Theorem 6 .1 ,

B1 = 0 .

b.

1

. 1 p

T 5

i i12

DEVIATION FROM FLATNESS

t i j e k 1 ■*» 1

2

e "Ll "*"2 e

^

l 1l 2l 3 t

i

J-i i 2

Thus we must have £j_j = 0 > For p = 3 9 we have

f ,1 J (T

95

2 T 5

hence we also

have

1_1. - h 1

^

5ii12

B2 = 0 .

R slj;L3.kl + 1 1 jkl 3 3

J*. )d j-T 1 2 1 3 ; j*

= 0

But, if we fix a point in the space and choose a coordinate system in w hi c h g^. = at this point, then we have, b y Theorem 6 .1 , i ,1 i1 2;l3 e T

6

r

1 ^'2±3

,1 3 1 3 £kl ^

> 1 £ll l 2 1 3 E

i E1 i1 2;L3 t

13 ~ T

£. . . . . = o

and thus, we must have

~

W

M a ^

.

1 2 39

THEOREM 6 .2 . In a compact semi-simple group space, we have = B 2 = 0 as well known. Also a harmonic tensor of the third order must have vanishing covariant derivative. But the tensor n ijk is such a tensor whi c h is not identically zero and hence we have B^ > 1 . 4. k.

DEVIATION FFRO ROMM FLATNESS

Now, in In our semi-simple group space, we have

W ijkl = Z ijkl = Cijkl = R ijkl “ b ( n - i )

jkg il " s jls lk^

and consequently i.r ei j ekl _ W ijkl5 1 -

rj

tij,kl _ n «ljtkl _ D ti j ekl , 1 ei j e ijkl^ 1 " Cijkll 1 - i jkl^ 1 + 2Th~ lT 1 Sij

and. h e n c e , by Theo r em 6 .1 ,

VI.

96

SEMI-SIMPLE GROUP SPACES

THEOREM 6 .3 . In a semi-simple group space, we have

1 ^ 2(n-i ) -

ijkl5

5

ij

_

ijkl5 5 .ij ij

_

ijkl5

xj 5

1

n-3 - " 4(n-i )

^

ij

and consequently we have

2(n-i )

(n < 5 )

n-3 Mn-1 )

(n > 5 )

W = Z = c =

(Bochner [5 ], Yano [k]).

C H A P T E R

VII

PSEUDO-HARMONIC TENSORS AND PSEUDO-KILLING TENSORS IN METRIC MANIFOLDS WITH TORSION 1 . METRIC MANIFOLDS WITH TORSION We consider an n-dimensional compact manifold is given a positive definite metric

(7-i)

on which there

ds2 = gjkdxj'±jck

and a metric connection

(7 *2 )

Vn

E^- > so that

s jk|l H

®sk^jl - & js®kl = 0

where the solidus denotes covariant differentiation with respect to i i i The connection Ejk needs not be symmetric, Ejk ^ E^j , and the entity (7 -3 )

sjkX " h ^E jk "

is is i will be called the torsion tensor. We define g by g g . = 5 . , and 3j J we will use this to pull indices up and down, so that for instance (7-10

S1^

=

and we note that, in virtue of (7-2), the pulling up and down of indices is commutative with covariant differentiation. From (7*2), we have dgs A +4^ - StjE sk - gstE jk = 0 97

98

VII.

METRIC MANIFOLDS WITH TORSION

“ f1 "

‘ gstEkj " 0

-^

* Sjt 4 » - -

and on multiplying the sum of these equations by (1 / 2 )gis over s we find

( 7 -5 )

Ejk = {jk } + Sjk

and contracting

S j k - S kj

by virtue of (7-k ). Prom (7 •5 )> we have

? « jk * Ski> ‘ ‘jk> - s ljk - 3±:kj so that, the symmetric part of E^v does not necessarily coincide with the i ^ Christoffel symbols { .^} . In order that such be the case, we must have

S jk + S kj or Sijk + Sikj = 0 Thus, the covariant torsion tensor , which is by definition anti-symmetric in i and j , must be anti-symmetric in all indices. The converse being evident, we have THEOREM 7*1 • A necessary and sufficient condi­ tion that the symmetric part of e V coincide with the Christoffel symbols { is that the covariant components tiie torsion tensor be anti-symmetric in all indices. In the case of semi-simple group space explained in Section 1, of Chapter VI, we have °jki = sjki and so since^3 anti-symmetric in all indices, we obtain

1 . METRIC MANIFOLDS WITH TORSION (7.6)

1 (E Jk ♦ Bj^) - (|k )

Now, taking a general tensor, say, " 1 1 P jk|l|m |-i i™ i„ - P jk|m|l Ji,i„n find, the Ricci formula: ji,n then we find

),

(7'7)

99

PV l l | l - pljk|»|l ’ PV

1slm -

^

we calculate

- pljsE3klm - 2pijk13^1mS

where dE*,,.

dE*-,

( 7-8)

E1^

=+E ^ E ^ - E ^ k

is the curvature tensor of the metric connection • Applying the Ricci formula to , we find 0 " gij|k|l " sij|l|k ““ssjE ikl _ gisE jkl and on putting Eijkl = si3ESjkl we obtain (7-9)

Eijkl " " E jikl

^

Eijkl " " Eijlk

It will be easily verified that the components of the curvature tensor satisfy, instead of the usual ones, the following Bianchi identities: (7.10)

E1^

*■ E1^

♦ B1^

-

J - v

- !i % .

(Ejk - Ekj ) = Sjk ,t

and thus the tensor we have

(7.21)

E^k

is not symmetric in general.

But, from (7*19),

Ej k ^ k = Rj k ^ k “ (SjrS|J')(Skrs|k)

and hence we have THEOREM 7-2. In a metric manifold with anti­ symmetric torsion tensor, if E-v + Ev . = 0 , then ik J ^ R.kld| is non-negative. THEOREM 7-3- In a metric manifold with antisymmetric torsion tensor, if RjklJ£ is non-positive ik then Ejk£J£ is also non-positive. 2 . THEOREM OF HOPF-BOCHNER AND SOME APPLICATIONS Now, in a compact manifold with positive definite metric ds2 = gjkdx^dxk and with linear connection Ejk , we have, for a scalar function (x) >

IJ 5^ ljlk " and consequently

do yl ' ax1 J'k

1 02

VII.

METRIC MANIFOLDS W I T H TORSION

" 8

IJ'k

8

8

Jk i ?

and hence, applying Theorem 2.2, we obtain T HEOREM 7 -k. In a compact manifold w it h positive definite metric, if, for a scalar (x) , we have

A * 3 gJ'k *l j|k - 0 then we have a «jj

= o

As an application of this theorem, we have THEOREM 7 •5 • In a compact metric m a nifold wi t h torsion, if a vector satisfies the relation

(7 .2 2 )

8j k ! i u . | k - V

5 * 2Vl r 3 ! r l s

t hen we cannot have

unless equality holds. More generally, if a tensor

ffjk f . .

^1 ^ 2 * ' ’

I j ^k

= u

£. . . 1 2 * ’* p

.

.

^ 1 ^ 2 ‘ ‘ '^ p * j 1 ^ 2 ‘ ' ^"p

satisfies

‘‘ ^p

r ir 2'--rp |3 + 2V. . , „ ^ „ 0| P 1 2 **‘ Y l r 2 ’’' p then we cannot have A

a TJ.......................s J'l JV

• -j p

^1^2"' ’ ^p^l*^' ' ‘ Jp

+ 2V

| 1 l 1 2 - - - i p 5I *1I12 - - - r p l S i ll 2 - - - l p I,1r 2 - - - r p 3 P 1r 2 ‘ ‘ ' P p l 3 t l t 2 - - , t p |U

+

sr 1t 1sr 2t 2 ‘

unless equality holds.

5

—0

2.

THEOREM OP HOPF-BOCHNER AND SOME APPLICATIONS

For the proof we note that if 1 2

a

^ ^

^i|j|k

=

> then we have

„ ,r|s*t|u ^rt^su^ ®

where

i1 *'3= t v and thus if

*3

satisfies (7*22), then we have 1 A4> = A

and hence the conclusion by Theorem 7 •^ • The extension to tensors is by analogy. THEOREM 7-6. In a compact metric manifold with torsion, there exists no vector field which satisfies

(7 -23 )

gJk(5i|j “ ^ j l i V + £ |j |i = ° and

V 1*3 - * W 1‘r | s * e rtW r|,’«t | u 2:° unless equality holds. In fact, we have the general identity:

(7.24)

gjkl1(j.|k - g Jk(51 |j - fijiPik -

and thus if

= Eai|a " 2Sirs?1

satisfies (7-23) then it also satisfies

(7’25)

- BaiSa - 23lr3SI'IS

and now apply Theorem 7*5* Similarly we obtain THEOREM 7-7- In a compact metric manifold with torsion, there exists no vector field which satisfies

1 ou

VII. METRIC MANIFOLDS WITH TORSION

(7.26)

♦ tj|±)|k - ,JU|± - 0 and EllSV

-

2SlP3tV

| a - S pt8sutr|V

u we have

(7*28)' A. . *ijkl5 > S ^ - ' V5 1 . •••ip • v

61J13 " ' V ■ 11 ljrst5 5 3. 13-#‘1p rsiQ...i |t uv |w + ft £ p e rstuvwb s i^...ip

where

Kijkl = 2" ^ik®lj ~ ^jk^li “ Eil%j + Ejl%i^ “ ^ 2

^iklj ~ Ejkli

- En k j + Ejiki> ’ Sijrst = \ ^Sirtsjs “ Sjrtsis “ Sistsjr + S,jstsir^ ’ ^rstuvw

(§ru^sv

^rv^su^tw 9

and if, on the other hand, the tensor satisfies (7.29)

S ^ P S ^ . . . ^

.

0

unless equality holds. Also, if, it satisfies (7-29), then we cannot have

k (p) . ^ V ' - v 1 Kijkl! ! 13- " 1p

, 3 (P ) . i 3i 3 - - - v s 1 iJrst* ! 13- " 1p rsi^..-ip|t uv

|w

unless equality holds. 3-

PSEUDO-HARMONIC VECTORS AND TENSORS

We shall call a vector pseudo-harmonic, if

(7-31)

SjL|j = Sj(1

and

5k |k = 0

Such a vector satisfies evidently (7 *2 3 ) and consequently (7 and for

= ^^i > we have

VII.

METRIC MANIFOLDS WITH TORSION

thus we obtain THEOREM 7 •9 • In a compact metric manifold with torsion, if the symmetric matrix

Ejk +

' (Sirs + ^Lsr*

M ^irs + ^isr^

®rtgsu + ®ru®st

defines a non-negative quadratic form in the variables i rs sr £ and I = £ , then every pseudo-harmonic vector must satisfy

(Ejk + Ekj)|j|k - 2(Slr3 +

+ (grtg8U + grugst)|1’|s|t|u = 0

If the matrix M defines a positive definite form, then there exists no pseudo-harmonic vector other than zero. Now, if we have Ejk +

= 0

ax>&

then, for a pseudo-harmonic vector

Sirs + Sisr = 0

6^ , we have

A * = 25j|k|jik = 0 from which = 0 follows. Thus, there exists in this case at most n linearly independent (with constant coefficients) pseudo-harmonic vectors. Moreover, if such a pseudo-harmonic vector exists, it must satisfy

■ «j;k from which *j;k + *k;j = 0

’ 0

3 • PSEUDO-HARMONIC VECTORS AND TENSORS The last equation shows that and thus we have

I1

Is an ordinary Killing vector,

THEOREM 7 -1 0 . In a compact metric manifold with torsion satisfying Ejk + Ekj = 0 and S ^ + Sisp = 0 , a pseudo-harmonic vector must have vanishing covariant derivative with respect to the connection of the mani­ fold, and consequently the number of linearly independ­ ent (with constant coefficients) pseudo-harmonic vectors is at most n . Moreover, if such a pseudo-harmonic vector exists, it is then an ordinary Killing vector. Now, if Ejk + Ekj = o and Sips + Sisp = o , then by Theorem 7.2, is non-negative. Hence by Theorem 2-9, an ordinary harmonic vector must have vanishing covariant derivative with respect to the Christoffel symbols and satisfy

Rj k ^ k = Sjrs3/ 3^

= 0

Thus, if the rank of the matrix HSjrsSkrS|| conclude that there exists no ordinary harmonic vector.

is n 9 we can Thus, we have

THEOREM 7*11* In a compact metric manifold with torsion satisfying Ejk + E^- = 0 and S ^ + S±3p = o the ordinary harmonic vector must have vanishing covariant derivative with respect to the Christoffel symbols. Moreover if the rank of the matrix ||SjrsSkrSN is harmonic vector.

n 9 then there exists no ordinary

A compact semi-slmple group space falls under Theorem 7*10 and 7*11 • On the other hand, in such a space, a pseudo-harmonic vector can be written as = f a^x ^h j and by Theorem 7 -1 0 , it must have a vanishing covariant derivative with respect to the connection of the manifold; consequently, the covariant derivatives of h^ being zero, fQ(x) must be constants. Thus we have J d THEOREM 7-12. space, there exist

In a compact semi-simple group n linearly independent pseudo­

VII.

METRIC MANIFOLDS WITH TORSION

harmonic vectors, and any pseudo-harmonic vector is a linear combination with constant coefficients of these vectors. Moreover, from >a ,a a0 i 0 ■ hoik • hjjk - V j k we have ,a i a _ jjk + k;j = 0 and thus hj are all ordinary Killing vectors and the manifold admits a simply transitive motions as well known. Now, we shall call an anti-symmetric tensor

i2‘*#ip pseudo-harmonic if it satisfies the conditions:

^7‘33^

',:Lplp] = °

or explicitly

+ V v - ' V ^

+

+ l i i 1 2 - - - 1p - i r | i P

and ^rig.-.lpli, + ^ r ^ .. .ip|i2

*'ip-i:r,lip^ and automatically

(7 -39)

S^rV-.ipIs = 0

Such an anti-symmetric tensor evidently satisfies

gjk(p5i1i2.•-ipl j + lji2...ip |i1 + ••• + fta

_ ta

^2 ’ ‘ *

Ia l-^1

i

_ ta t

• • •i p la li 2

) = o

i 2 " * ’^p-i

I

^p

and consequently, for

•••ip we get

A4> = - —

P

K N (P)

5 ijkl5

5

,• 4 V " 1?

- 22 3D.

.1 3 i 3 .| ijrst5

' " i P .|r S

is---1?

rsi3-•-ipIt uv * i,...ip ■? - pGrstuvw5 5 3 and thus we have THEOREM 7-15- The second half of Theorem 7-8 applies in particular to pseudo-Killing tensors.

5•

INTEGRAL FORMULAS

5-

INTEGRAL FORMULAS

11 1

In this section, we shall consider a compact orientable metric manifold with torsion, and suppose that the torsion tensor satisfies the condition Sjj1

(7.^0)

= 0

and this condition is satisfied automatically if the covariant torsion tensor is anti-symmetric in all indices. First, for any vector v1 , we have

11 - • ; i

’

’S

i 1 ‘ ’ S i

* ^ S i 1

from which

(7.4.)

1

s/g dx

by virtue of the assumption (7*^0), where g is the determinant formed with g ^ . Thus, for any vector field v1(x) , we have (7*^2)

Iv ,.dv = 0

the integral being taken over the whole manifold, where dv is the volume element. Applying first (7-^2) to the vector > we find

/(l^l-Mlldv - /(e1,j|±eJ ♦ !1|j!J'|1)av ■ 0 or

(7-*3)

+ Ejk8jtk - 2|i lsSJ.i35j + 11 ]j5^'|i )dv = 0

by virtue of Ricci identity:

el|j|k - sl|k|j * Applying it next to the vector

- 2* V j k 3 > we have

112

VII. METRIC MANIFOLDS WITH TORSION

(7.-M

- 0

and hence

f(Ejki h k

(7.45)

* i 1 | j S J , 1 - e1 | l 5 J | jM '' - 0

-

If the vector

is pseudo-harmonic, then equation (7*^5)

becomes

f i a Jk * Ekj)!jlk - 2(3lrs * 3lsl. ) t V |8 • (grtg3u * g pug s t ) t r | S l t | u J a v - 0

and this gives another proof of Theorem 7 . 9 for a compact orientable metric manifold with torsion satisfying = 0 . If the vector is pseudo-Killing, then equation (7*^5) becomes

/'« j k * V * jSk - 2 (Sir‘s - ■ W

- lr| V

l V

|u]dv -

and this gives another proof of Theorem 7-1^- for a compact orientable metric manifold with torsion satisfying ^>j±L = 0 • The generalization of formula (7-^5) to the case of anti-symmetric tensor is 11 ...1 I[E. -I r

(7 .^6 ) v

'

J

iJ

lrs£

P

iji . . . 1

j

.

. + (p - 1 )En .n .6

1 2 **‘1p

1 2 - " 1p

t1 ^

Ijki 5

P

•- - i p I j ]

kl 5

I3 ...

1 !1 2 ' " 1 plj

li1

and Theorems 7-13 and 7-15 can be again reproved for

= 0 •

6.

NECESSARY AND SUFFICIENT CONDITION THAT A TENSOR BE A PSEUDO-HARMONIC OR PSEUDO-KILLING TENSOR Under the same assumption as in Section 5 , we have

6.

PSEUDO-HARMONIC AND PSEUDO-KILLING TENSORS

/ s jk(^i>|j|kdV = 0 for any vector field

, or

(7^7)

+ 6l8jksi|j|k)dv = °

and hence

(7.W)

/ [ i V X u i k

- SsA ♦ aW*'*)

* ? + 5±|iEJ|j]jkl

the Ricci tensor (8.14)

Rjk " gllRijkl

and the scalar curvature (8.15)

R = SJ'kRjk

are all self-adjoint. Here, and always, a scalar is self-adjoint, if it is real valued. Thus, if we denote the covariant differentiation with respect to Tjk by a semi-colon: e1 5 *

_ ^ + tJ'r1 8? 1 Jk

then, we can see that the self-adjointness is preserved by a covariant dif­ ferentiation. Assume now that, in our complex analytic manifold, there is given a positive definite quadratic differential form (8 - 1 6 )

ds2 = gj^.dz^dzk

where the symmetric tensor

gjk

is self-adjoint and satisfies

122

VIII. KAEHLER MANIFOLD

From the complete separation of the components gjk into four blocks , g^g , g-p , g-g , it is evident that conditions (8 .1 7 ) are preserved by any coordinate transformation of the form (Q.k). Also, by virtue of condition (8 .1 7 ), the metric form (8 .1 6 ) can be written in the form (8 .1 8 )

ds2 = 2ga^dzadzp

where (8*19)

Sap = Spa =

= ^5

and a metric (8 .1 8 ) satisfying (8 .1 9 ) is called a Hermitian metric. Taking account of gaB ^ = g5S p - 0 ,

we obtain for the Christoffel symbols

a

1

r P7 " 2 g

are

r t h e

/ Sg«p ^ 3gir \ \ dz7

1 Qf€ ( % 1

Of

\ ar

* ■

relations

dz^

}

dgPr aze

\

/

«

and the values of other components are given by symmetry and self-adjoint­ ness. From the law of transformation pti = dzfl / dzq dzr •>k ' dzP \ Bz-J

„p ^

d2zp \ az*jSz,k j

we get ptor _ dz|Qf dz^ dzv px_ " Szx Sz'p Sz'7 tiV

2.

CURVATURE IN KAEHLER MANIFOLD

123

and thus the condition (8 .2 0 )

■?5-«

is invariant under a coordinate transformation of the form (8.4). equivalent to

This Is

( 8.21 ) dz'

dz1

or to

*S5p = aSgy

(8 .22 )

bz7

bz^

or, further to

(8.23)

g - = SaP bzabzP The self-adjointness of

gag

demands that

$

be a real valued

function. The condition (8 .2 0 ) or (8 .2 1 ) or (8 .2 2 ) or (8 .2 3 ) is called Kaehler fs condition, and a metric satisfying (8 .1 9 ) and (8 .2 1 ) will be called a Kaehler metric. Thus, in a Kaehler metric, we have

ae

(8.24)

bz7

'

and the covariant derivative of a vector

= Alf + ra

Pa

s il-

|

i5

’

8

Si’'

, say, is given by

;7

bz7

(8.2 5 ) a

5. >7

dg'a bZ7 2.

_ = Ilf. + r|-i^

i7

dg7

P7S

CURVATURE IN KAEHLER MANIFOLD

Prom the definition of the curvature tensor

VIII. KAEHLER MANIFOLD Ri = arjk _ arjl Jk l 5^ ^

s i _ rs i J k 31 J 1 sk

we obtain - i OC

(8 .2 6 )

and

Pkl

Rl f3kl

and for g

Rijkl = sisR jkl we have apkl = o

(8.27)

and

Rapkl = 0

Also, Rijkl = Rklij implies (8 .2 8 )

R. .R = o 1J/5

and

R.1J7& = 0

From (8.27) and (8 .2 8 ), we can see that only the components of the form

can be different from zero, and consequently that only the components of the form -pQ? n P7 & ,

'R^' ^

6 '

K P75 '

"R^ K £76

can be different from zero, and also we obtain dr

(8 •29)

a

Ra~ E = —

This equation shows that if the components analytic functions of za , then all components R^^l tensor vanish.

are complex the curvature

2.

CURVATURE IN KAEHLER MANIFOLD

125

From the Bianchi identity:

Rljkl + Rlklj + Riljk - 0 we have

R“£76r + Ra

+ R“?f t = 0 0^7

But, the last term of the left-hand member being zero, we have

r

n £ 67

- -Ra _

7 5(3

*

(8 .3 0 )

and this can also be obtained directly from (8 .2 9 ). Next, from the definition of > ve have drIfi &ae ^ 7

? p7&

arP75

’ae az7

and hence

aP78

az7a55

azr

az5

and also

/Q 0 0 )

R

_

_

4 ____^ $ dz^zP&z^z8

_

____________

a^ 6

— .

p-e T

o o d$ d$ SzGazaaz7 azTaz^i£ _________________

and the latter implies

(8-33>

RaPr5

R7Pa8 ~ Ror6rP

For the Ricci tensor

R . . , we have

Rr6ap

126

VIII.

KAEHLER MANIFOLD

and consequently (8.35)

Rg-

= 0

and

hra Rapa p y = R“Qr~ P7Qf = - Ra po>.y - - ^ -7 But, since blogJg

rca pa we have b2lo&Jg

(8 .3 6 )

R„- ----- 5----

where

g = IS±JI = lgapl2 We now introduce a sectional curvature linearly independent vectors u1 and v^ :

(8.37)

K

determined by two

Ri iklulv,3*ukvl K = ------ ---------- 1 U 1 (gjkgil - gjiSuc^ v^u v

If this sectional curvature is the same for all possible 2 -dimen­ sional section, then the curvature tensor must have the form (6 -38)

Rijkl = ^ S j k 8!! " sjlslk^

but in the present case this reduces to

and on substituting this into

2.

CURVATURE IN KAEHLER MANIFOLD

127

RorP76 we find

If we multiply this by

ft y

g 7g

and contract, we obtain

n2K = nK and hence the following conclusion. THEOREM 8 . 1 . For n > 1 , if, at every point of a Kaehler manifold, the sectional curvature is the same for all possible 2 -dimensional sections, then the curvature tensor is identically zero. Now, if the two vectors zn „ v

(8 .3 9 )

V

= 1U

a

,

u^~

.a

and 5

v

v^

satisfy the conditions: .a

= - iu

then the section is called a holomorphic section. tion, we have Rijklu l v W

For a holomorphic sec­

= -

(SjkSil ' and consequently (8 A 0 )

K =

Thus, if we assume that at all points of the manifold, the holo­ morphic sectional curvature are all the same, then we must have [R,[a fi7S ~ f (SapSr5 + Sa5S7p )]u%Pu7u& = 0

128

for any

VIII.

KAEHLER MANIFOLD

ua , from which

On the other hand, from the Bianchi identity: Rijkl;m + Rijlm;k + Rijmk;l " 0 we obtain Rap s - fT 7 &;e

+

Rap5e;7 ^ r + Rape q 7 ;5 . c

=

0

or Rap75;e

(8 -h 2 )

Rape&;7

Substituting (8.4i ) into (8.42), we find K;e(SapSr5 + Sa§Srg> = K;7(SapS€5 + ScrfM* and contracting with

gapgrB , we obtain n(n + 1 )K._ = (n + 1 )K.. 3

hence, for

9

n > 1 , K; e = 0

and K.? - 0 Hence THEOREM 8.2. If, at all points of a Kaehler mani­ fold, the holomorphic sectional curvature K is the same, then the curvature tensor has the form (8.41 ) and K is an absolute constant. A.

We shall call such a manifold, manifold of constant holomorphic curvature.

2.

CURVATURE IN KAEHLER MANIFOLD

129

THEOREM 8 .3 . In a manifold of constant holomorphic curvature k , for the general sectional curvature K , we have (8 A 3 )

0 < tJ- k < K


0

k < K < •£- k
0 , a self-adjoint covariant vector field /y

3-

MALYTIC VECTOR FIELDS

whose components are analytic functions of coordinates must have vanishing covariant derivative, and if Rag£a£^ is positive definite, no such covariant vector field exists other than zero. (Bochner [2 ]). For a contravariant

with complex analytic components, we

have (8-51 )

)I

= 0

and the Ricci identity:

6

_ _ !« _ = |PR« _ 6 ;&;7 5 n PrS

implies

and hence (8.52)

gr&|a .7.g + RQpiP = 0 We now have

A. - V

8 "(gap- i V ) ;r;&-

= ^ g apSr5^

p.8 + gag(s75ia .r.5)ip]

and on substituting from (8 .5 2 ), we find

(8.53)

A. - ■.[g^g’V . / . g

- RajsV)

and hence the following conclusion: THEOREM 8 .6 . In a compact Kaehler manifold with Rap£a£^ < 0 7 a self-adjoint contravariant vector field whose components are analytic functions of co­ ordinates must have vanishing covariant derivative, and if Rap£a!^ is negative definite, no such contra­ variant vector field exists other than zero. (Bochner [2]).

133

VI I I . KAEHLER MANIFOLD h.

COMPLEX ANALYTIC MANIFOLDS ADMITTING A TRANSITIVE COMMUTATIVE GROUP OF TRANSFORMATIONS We consider a complex analytic manifold of real dimension 2n which admits a transitive commutative group of transformations whose in­ finitesimal operators are

where (p, q, ••• = 1, 2, ..., r) are r holomorphic contravariant vector fields. By the transivity of the group, we have r > n and the rank of the matrix (it*) is n . By the commutativity of the group, we Jr have (R

I

cb

p;5;r ' V

PrS

the left side vanishes identically, and hence also the right side. has everywhere maximal rank, so that

But

R°f£76 t r = 0 and thus our manifold is a flat Kaehler manifold. Therefore, in the neigh­ borhood of every point we can allowably normalize the metric tensor to

136

VIII. KAEHLER MANIFOLD This normalization shows that the covariant vector fields

P _ are likewise holotnorphic and parallel, so that in particular we have

dug

dug

&zp ' aza = Therefore the

r

Abelian integrals

may be introduced, and if, for instance, the first n among them are linearly independent at the point zQ , then they will be so everywhere and they map the manifold holomorphically and locally one-to-one into the Euclidean manifold. THEOREM 8 .7 . Let V2n , n > 1 , be a complex ana­ lytic manifold of real dimension 2n . If, for r > n , there are on it r holomorphic contravariant vector fields rip such that its rank has everywhere its maxi­ mal value n and that the bracket expressions (8-55) all vanish identically, then there are on the manifold n simple Abelian integrals of the first kind by which it is mapped holomorphically and locally one-to-one into the Euclidean manifold. In particular, if V2n is compact, it is a complex multi-torus. (Bochner [ 9] )•

5• SELF-ADJOINT VECTOR SATISFYING = df/ dza AND Af = 0 We now consider a (self-adjoint) vector field which in the neigh­ borhood of every point can be represented in the form

It is not a local gradient field in the proper sense unless

5.

A CLASS OP SELF-ADJOINT VECTORS

f = f , that is, if f is real valued; and In our application it will definitely not be so. We introduce the associate vector

/O CO \

_ df

_ df

and these two vectors have the following properties: ^cc;& "

~

^or;p = % ; a ,

y

^5;P =

'

” ^(3;or ' Now, for

0 = 2gQfP|a £p , we obtain

A = A + B + C where A

g Pgp (Sa .plp;5 + ia .jig.p)

B = 8e;Q?Psg:pae -p _ l c c ; p ; a l fi

C^ = SQ*aPp;P^t t_ S §o:§P;p;a If we substitute

— £ R^ 5 X,

— + £

po:cr

—

5 p;a;o:

we obtain

B = Rap- i V + ^

and if we put

^

\

i5);atp

137

138

VIII • KAEHLER MANIFOLD

6g;p;S

’•pjg;?

’’pj S;? + (,)p;P;a

T'P;5;p)

’’p; o;& we obtain

C - saP'(sp% p;5).p -ia

Finally we introduce the assumption

(8-59)

gp5ip.- = 0

that is (8.60)

Af = 0 This will also imply

gB\ . ; and if we interchange the variables ing theorem:

■ 0 (za ) and

(za ) , we obtain the follow­

THEOREM 8.8. If on a compact manifold with posi­ tive Ricci curvature a (self-adjoint) vector field has the property that in the neighborhood of every point the components g- can be expressed in the form (8. 61 )

with a

then

Af = 0

Bza

g - = 0 , that is,

f is complex analytic. If the Ricci curvature is only non negative, then |-.i = o , that is, the derivatives df/dza are not necessarily zero but have covariant derivative zero. (Bochner [2]).

6.

ANALYTIC TENSORS

6.

139

ANALYTIC TENSORS

If the components OlOfg.-.Otp 5

PiV-.eq

of a self-adjoint tensor of mixed type are complex analytic functions of the coordinates (za ) , then we again have or orp . . .or

(8'62)

!

V 1Bo...S r ' ° 2 q;'

and from the Ricci identity: a^ctg...ap

_ Xcc2 ''‘“p - 1

_ _ P1P2 -•-Pq;r;5

e ^ . . . P qR

_ “l“2 " ,0tp

«ia2---ap P1P2-•

5

X7S

n

^2---Pq P17" ‘ by using (8 .6 2 ) and contracting with

«,

+ «l“2-*-VlXr“p

aia2 - " ap

"

X?8

n

Pl^-'-Pq-^

g7& , we obtain

V"

140

VIII.

KAEHLER MANIFOLD

then we have P.6 .P 5 "

=

-

V

ctt a ...a

' " g

ps 3.5,

+ K S o:1 7 1

K

8

3 5

- K 1 1

y ^ - - - 7

1

V - v *

V y

ax a, . . .a

K ^q q«8 I 1

P

-

v

;

... y

-I 1

P-

- 1

3 1 ...3 q ;a;x^ V

and substituting from (8 .6 3 ), we have B 6B £

A = 2 [g

17

1

... g - S sV p

ax a

7

. . .a

...S s

£ 8

£ 1 1

P

. . . 7^

t 1 Pi-*-Pq;a

+ G{|}] where Xa ...a (8.6*)

G(S)

- -

Pi-'-Pn

a

1

1

.

R“p , |!. " - eq S, ■ ■ ■ \ “ A.'-'-p

* ... . S“ '"'aP

R“ | p. ' " V i “ p9 a r " “p

Hence the following conclusion. THEOREM 8 .9 . In a compact Kaehler manifold, if the complex analytic components “1*2-*•“!) * 8 1 8„...8 2 pq of a self-adjoint tensor of mixed type satisfy the inequality: G(£]

>

0

P- _ V - - V T

"

q

6. then we must have

14i

ANALYTIC TENSORS G U ) = 0 and

o

Also this assertion applies not only to tensor fields satisfying

o

but also to those satisfying or, ...or P

o

(Bochner [1 1 ]). Now, if, at every point of the manifold, we denote by M and the algebraically largest and smallest eigenvalues of the matrix R^g respectively, then we have

G{|) > (qm - pM)!^1 ‘“ °'p

and we obtain the following conclusion: THEOREM 8 .1 0 . If just stated, and if

M

and

m

have the meaning

qm - pM > 0 then every complex analytic tensor field of mixed type

0, - --P.

must satisfy o

m

VIII. KAEHLER MANIFOLD If qm - pM > 0

everywhere and

qm - pM > o

somewhere,

then there exists no complex analytic tensor field of mixed type “ia2---ap 6 PlP2 -..Pq other than zero.

(Bochner [11] ).

As a corollary to this Theorem, we can state THEOREM 8.11 . If a compact Kaehler manifold is an Einstein manifold RaP = Xgap for X > 0 , there exists no analytic tensor field of the type cr . . .a

S

and for

PB . . .pq f (*>r>

\ < o , none of the type

a . . .or p

^

Hi

.. .p

Hq

(o.


plaRV

' pajRlakl

and sVjkl ■ ^/akl and multiplying (8 .1 1 0 ) by

QJh

and contracting, we find

or (8 .1 1 2 )

plapjbRa\ l = 0 Similarly we can prove

(8-113)

^

5/ \ l

Next, multiplying (8 .1 1 0 ) by p^- pj r b or

= 0 PJt> and contracting, we find

_ p& -pi jkl ' ^ bK akl

157

158

V III. KAEHLER MANIFOLD

(8.114)

P V b Rabkl = QJaRi\ l

and similarly 0)

1 2 . EFFECTIVE HARMONIC TENSORS IN FLAT MANIFOLDS

ii . . . 1 j R .£ Pf Klj6 ? i2 ''

= 2 (R - E p

a7i . . . i 5

Pe

^

ap

a7i ...i

-3

Pp

p

rl3 " ,;LP api,...i

(8‘12^

+ R _t

161

= (n+l)k(|

“P7l3*’' V

api • . . i P ^ p i 3...ip + ^ P W i 3...ip )

On the other hand, we have iji ...i Rijkl5

kl 1

orpi .. .i ±3 •••±p

Rap7&5

75 1

I3 •••ip

and consequently ( 8. 1 25 )

i j i , . . . i kl R,,m S 6 P5 *

*

=

orpi . . . i - 2k| 3

P l „ 5n-

by virtue of

%j3 Altogether we have api ...i Flti , v

V

■ o , we have B21 = 1

B21+1 = 0

. )

(0 < 21 , 21+1 < n)

We will now envisage as formal analogues to the Weyl projective and conformal curvature tensors, the following tensors:

162

VIII.

KAEHLER MANIFOLD

= Ftap7g - 2 (n+1 ) ^sapRr6 + sa6R 7 P + g75 RaP + g7 pRaS^

(8.127) and (

8 . 128)

Ka g y g

=

Ra p 7 g

-

(S a p R 7 6

+

S q ;5R 7 p

+

S r sRap

+

SygR ag)

+ 2 (n+1 }(n+2 ) ^agS7 § + ga5g7 ^ all of which were introduced in Bochner [6]. These tensors satisfy (8 .,29)

g°XW

' 0

«“5k« M - - 0

and since, for a manifold of constant holomorphic curvature, we have

RaP75 = 2n(n+i) (gapS75 + gaBg 7 P^ R - = —

£ -

it is evident that, for such a manifold, we have GaP7§ = 0

HaP75 = 0

Conversely, if we assume that GorP7& = 0 then (8,13°)

RcfP7& = n+T ^gaPR 7& + gct5R7P^

and on substituting this into ^(*£76

^ 7 Por6

we find g « p R75 + g a 6 R7 p = g r PRa 6

Multiplying this by

gaP

+ g r 6 RaP

and contracting, we find

1 2 . EFFECTIVE HARMONIC TENSORS IN FLAT MANIFOLDS " V

+ V

163

= Rr5 + SyS T

or R 76- = — s; 2 n **7 6 and, by (8 .1 3 0 ), we obtain RaP76 = 2 n(n+l ) ^8 ccpS78 + sa88 7 P^ the conclusion then being that the manifold is of constant holomorphic curvature. Similarly for ^ar£7& ” 0 we obtain (8 . 1 3 1 )

Ragr§ = 2 (n+1 ) ^sagR75 + sa8R 7 P + s7 8RaP + srPRor8 ^

and by contraction with

g

, R- = — gnP7 2n SP7

and then from (8 . 1 3 1 ), Rap76 = 2 n(n+i) ^sagg 78 + ga5 g7 ^ and thus the manifold is again of constant holomorphic curvature. For (8.,32)

KoM5 - 0

that is, Ra?y& = n+2" ^gQrpRr6 + 3aSRyP + g7SRaP + s?pRaS ^

2 (n+1 )(n+2 ) ^SQfpS 76 + SQf6 g 7 P^

we do not claim this conclusion, but the effect of (8 .1 3 2 ) on Betti numbers will be the same as of constant holomorphic curvature.

l64

VIII.

KAEHLER MANIFOLD

In fact, for an effective tensor, we have ii ...i "ii1

j

a7i ...i

p

ocyl ...i

i i2 - i p - !|W

5

PS ? 1 3

and iji,...i

kl

api,...i

76

••-ip

n+2

ap

3

P5 r 7l3.--Xp

2R “P13 * - - V "(n+i )(n+2 ) 5

‘

and consequently

_ 2(P-,1.) 1R .|Qfrl3,-'1p |P_. *ap5 ^ J '!1 ri3...ip n+2r 1 ]|fi

+ 2

_Ez]____ QfPl^ *••Ip + (n+1 )(n+2 ) R| 6o tp i3

. . . i p

and hence the following conclusion: THEOREM 8.25- In a compact Kaehler manifold in which Kag7g = 0 and Raplal^ is positive definite, we have F U i 1i2 ...i ] > 0 for

i < p < n/2 + 2 , and therefore

B21 = 1

B21 +1 = °

(0 < 21 , 21 +1 < S + 2 )

(Bochner [6]). In order to obtain a result for all tensor

p , we introduce here a

12. (8 ‘1

165

EFFECTIVE HARMONIC TENSORS IN FLAT MANIFOLDS

^

Sa|3 = RorP “ 2n Sap

and the quantity (8.135)

3 - sup. I

|Sag!a5P l ^ f

a

which measures deviation from being an Einstein manifold. Substituting Rorf3 = Sag + 2n in (8.133), we get o r .

,

F{|i1i2 .•-ip5

_fa7l3‘■ ' V P RaP5 1

=

2 R

I

n :p + 2

.

n(n+l )

R c a g l 3 "

' l p t

-

lapi3...ip

2 (n-2p+^ ) „ _.ar:L3 " •1P,P_ n+2 orp5 5 yi^...ip But, for

2

we have 2 W and for

•

^api^.-ip

api-. ..L ?5 3 ^ ij.-.ip]

n/2 + 3 < p < n , this is

[

or/i-... i

Ra S !

i

p

Ps r i 3 . . . l p

m k

* ‘nCSTTT 11 -

and hence the conclusion in case (8 .1 ^3 ).

0fj3iQ...i 3 - (n-,)K>*

3

P W i 3.--ip]

C H A P T E R

IX

SUPPLEMENTS S. Bochner 1.

SYMMETRIC MANIFOLDS

On a Riemannian manifold we take an arbitrary tensor but we im­ mediately denote it by Rjjkl >because it will very soon bethe curvature tensor itself. For thesquare length = r^J^Ir * * ijkl .

we form the LaplaceanAo , and the quantity

^ A4>

isthen the sum

of

is arbitrary continuous, we can take for D€ the entire manifold Vn itself and on the (empty) set Vn - D€ we then have | | < e , as required. 8.

ALMOST -AUTOMORPHIC VECTOR AND TENSOR FIELDS

A less trivial generalization arises in the following manner. Assume Vn compact, but introduce its universal covering space Vn , even if it is non-compact, and make no restriction on the nature of the funda­ mental group r = Vn/Vn . If the elements of r are 70> 7V

72> •••

then each 7 p defines a homeomorphism of we associate the sequence of points (9-33)

?r = 7r (?)

VR . With each point

P

of

Vn

r = 0, 1, 2 , •••

and any two points of this sequence are "equivalent.” The original mani­ fold Vn can be identified with the space of sequences (9.34)

P = {r0(P)> 7}(?)> 7 2 ( P ) ,

••• )

in a suitable manner, and on the other hand there is a compact subset R in Vn such that to any point in Vn there is a point equivalent to it. For any given sequence (9- 3^ ) we may say that each Pp "covers" P or "lies over" P , and conversely that P is a "projection" of Pp . If we are given any structure on Vn , differentiable or analytic, then there is a structure on VR of which the given one is a projection, and this structure of Vn is "periodic" (or "automorphic") in the sense that if U is a coordinate neighborhood of Vn and 7p is an element of

8. r

then

ALMOST -AUTOMORPHIC VECTOR AM) TENSOR FIELDS

183

7 p (U)

is again such a coordinate neighborhood. We say that a function on VR , scalar or tensor, i3 periodic (or automorphic) if we have (7 rP) = ♦(P) for all r , and ^ ( 7 pP) = |^(P) for a vector, and in the same manner for any tensor. Any periodic function on Vn gives rise to a function on Vn itself (its "projection") by put­ ting (P) = *>(P) , and conversely any function (P) • In particular if we aregiven a metric tensor g^. on Vn then it has a periodic exten­ sion ontoVR , and we will denote it again by g ^ . Now, we call a continuous function (P) on Vn "almost peri­ odic" (or "almost automorphic," in either case "relative to the given group r ") if every infinite sequence of elements contains an infinite subsequence such that the sequence of functions {o(7 rP)}

r = 1 , 2 , -- -

is convergent uniformly on the entire space Vn . The definition also applies to vectors and tensors, the uniformity of convergence being rela­ tive to the uniform structure of the space, and the best way of expressing this uniformity is to utilize the tensor g^j , assumed periodic, in the following manner: given, say, a vector !^(P) then the sequence of "translated" vectors 5 . ( 7 P) converges uniformly towards a limiting vector (^(P) if the square length g l j ( P ) [ 6 i ( 7 r P) -

6 i ( ? ) ] t 8 j ( 7 r P)

-

lj(P)]

converges to 0 ,as r --- ► , uniformly in Vn , and similarly for tensors. Now, almost periodic functions and tensors have the following properties. First of all, due to the compactness of the set R previously introduced, any continuous periodic function is almost-periodic, and any almost periodic function is bounded. A constant function is of course almost periodic. The sum and the product of two almost periodic functions, scalar or tensor, is almost periodic, and the contractions of an almost periodic tensor is again almost periodic, and finally there is the follow­ ing property which will be all-decisive in our argument. If a function $(P) , scalar or tensor, is almost periodic, and if for a sequence of ele­ ments {7 ) the sequence #(7 ^?) converges uniformly, and if we denote the limit function by