Tensor Analysis 9781400879236

Table of contents :
Preface
Contents
§1. Multilinear algebra
§2. Derivations on scalars
§3. Derivations on tensors
§4. The exterior derivative
§5. Covariant differentiation
§6. Holonomy
§7. Riemannian metrics
§8. Symplectic structures
§9. Complex structures

Citation preview

Preface

Ihese are the lecture notes for the first part of a one-term course on differential geometry given at Princeton in the spring of 1967· ^hey are an expository account of the formal algebraic aspects of tensor analysis using both modem and classical notations. I gave the course primarily to teach myself.

One difficulty

in learning differential geometry (as well as the source of its great beauty) is the Interplay of algebra, geometry, and analysis.

In the

first part of the course I presented the algebraic aspects of the study of the most familiar kinds of structure on a differentiate manifold and in the second part of the course (not covered by these notes) discussed some of the geometric and analytic techniques.

GJhese notes may be useful to other beginners in conjunction with a book on differential geometry, such as that of Helgason [2,§l], Nomizu [5,§5], De Rham [7>§7], Sternberg [9,§8], or Lichnerowicz [ll,§9]·

These books, together with the beautiful survey by S. S.

Chern of the topics of current interest in differential geometry (Bull. Am. Math. Soc., vol. 72, pp· 167-219, 1966) were the main sources for the course. The principal object of interest in tensor analysis is the module of

C™

the algebra of

contravariant vector fields on a

C°°

manifold over

Cw real functions on the manifold, the module being

equipped with the additional structure of the Lie product.

The fact

that this module is "totally reflexive" (i.e. that multilinear functionals on it and its dual can be identified with elements of tensor product modules) follows-for a finite-dimensional second-countable

ii.

Hausdorff manifold - by the theorem that such a manifold has a covering by finitely many coordinate neighborhoods.

See J. R. Munkres,

Elementary Differential Topology, p.l8, Annals of Mathematics Studies No. 54, Erinceton University Press, 1963. I wish to thank the members of the class, particularly Barry Simon, for many improvements, and Elizabeth Epstein for typing the manuscript so beautifully.

CONTENTS Page §1.

Multilinear algebra

1

1. The algebra of scalars 2. Modules 3· Tensor products k. Multilinear functionals 5· Two notions of tensor field 6. F-Iinear mappings of tensors 7. Contractions 8. The symmetric tensor algebra 9· The Grassmann algebra. 10. Interior multiplication 11· Eree modules of finite type 22. Classical tensor notation 13· Tensor fields on manifolds ll·. Tensors and mappings

§2.

Derivations on scalaxs

25

1. Lie products 2. Lie modules 3 . Coor­ dinate Lie modules 4. Vector fields and flows

§3·

Derivations on tensors

37

1. Algebra derivations 2. Module derivations 3· Lie derivatives k. F-Iinear derivations 5· Derivations on modules which are free of finite type

§4.

The exterior derivative

47

1. The exterior derivative in local coordinates 2. The exterior derivative considered globally 3· The exterior derivative and interior multi­ plication 4. The cohomology ring

§5·

Covariant differentiation 1. Affine connections in the sense of Kbszul 2. The covariant derivative 3· Components of affine connections 1+. Classical tensor notation for the covariant derivative 5. Affine connections and tensors 6. Torsion 7· Torsion-free affine connections and the exterior derivative 8. Curvature 9· Affine connections on Lie algebras 10. The Bianchi identities 11, Ricci's identity 12. Twisting and turning

§6.

Holonomy 1. Erineipal fiber bundles 2. Lie bundles 3· The relation between the two notions of connection

57

iv. Page §7.

Riemannian metrics

89

1. Pseudo-Riemannian metrics 2. The Riemannian connection 3 . Raising and lowering indices 4. The Riemann-Christoffel tensor 5. The codifferential 6. Divergences 7 . The Laplace operator 8. The Weitzenbock formula 9. Operators commuting with the Laplacean. 10. Hodge theory

§8.

Symplectie structures

Ill

1. Almost symplectic structures 2. Hamiltonian vector fields and Poisson •brackets 3' Symplectic structures in local coordinates Hamiltonian dynamics §9.

Complex structures 1. Complexification 2. Almost complex structures 3 . Torsion of an almost complex structure Complex structures in local coordinates 5- Almost complex connections 6. jdlhler structures

117

TENSOR ANALYSIS BY EDWARD NELSON

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OP TOKYO PRESS

PRINCETON, NEW JERSEY

1967

Copyright

(§)

1967, by Princeton University Press All Rights Reserved

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

§1.

1.

Multilinear algebra

The algebra of scalars We make the permanent conventions that

teristic

0

and that

Elements of

F

F0

is a field of charac­

is a commutative algebra with identity over

F will be called scalars and elements of

F0

F0 .

will be called

constants. The main example we have in mind is numbers and

F

the field

the algebra of all real C°° functions on a

M . In this example the set of n.n module over

Fp

]R C°°

of real manifold

C°° contravariant vector fields is a

F , with the additional structure that the contravariant

vector fields act on the scalars via differentiation and on each other via the Lie product.

Tensor analysis is the study of this structure.

In this section we will consider only the module structure.

2.

MbdTiles The term "module" will always mean a unitary module (lX = X).

Thus an

F

mapping of

module F>E

E

into

is an Abelian group (written additively) with a E (indicated by juxtaposition) such that f(X+Y) = fX+fY , (f+g)X = fX+gX , (fg)X = f(gX) , OX = X ,

for all

X,Y If

E

in

E

is an

and F

f,g

in

F.

module, the dual module

E1

is the module of all

§1.

2.

F-linear mappings of on

X

If A

in

E

E

MULTILINEAR ALGEBRA

into

F . If

we denote the value of

by any of the symbols

is an F-linear mapping of

is an F-linear mapping of

E

into

into

E

its dual

defined by

. There is a natural mapping

defined by

and

E

is called reflexive in case

not in general injective. functions on a manifold

M

K

is bijective.

For example, if and

E

is the

(The mapping

K

is

F

is the algebra of all

F

module of all continuous

contravariant vector fields then The notions of submodule,

F

module homomorphism, and quotient

module are defined in the obvious way. is an

F

If

H

and

K

are

F

modules and

module homomorphism then the quotient module

is canonically isomorphic to the image of

. See Bourbaki [l].

We will frequently refer to the elements

X

of an

F

module

contravariant vector fields or vector fields and to elements module 3.

E'

E

as

of the dual

as covariant vector fields or 1-forms.

Tensor products If

H

(over F) is the

and F

K

are two

F

modules, their tensor product

module whose Abelian group is the free Abelian group

generated by all pairs

with

X

in

H

group generated by all elements of the form

and

Y

in

K

modulo the sub-"

§1.

MULTILINEAR ALGEBRA

3

(1)

where

f

iB In

Let

If

r

or

E

s

F , and the action of

"be an

is

0

F

module.

F

on

is given by-

We define

we sometimes omit it, and we set

Notice that

. We also define

where the sums are weak direct sums (only finitely many components of any element are non-zero). Notice that

and

are associative graded

F

algebras with

the tensor product as multiplication. We make the identification With this identification, is an associative bi-graded F-algebra. b.

Multilinear functionals Let

E

be an F-module. We define

to be the set of all

F-multilinear mappings of (E1 r times, E s times)

5.

§1.

into

MULTILINEAR ALGEBRA

F . Thus if

is a scalar, and if all arguments but one are held fixed its value depends in an F-linear way on the remaining argument. of addition and scalar multiplication, is

0

u

in

If

r

or

s

Notice that

. We also define the weak direct sums

v

symbol

in

we define

(this is a different use of the

by

Then

and

ciative bi-graded

5•

is an F-module.

we sometimes omit it, and we set and

For

With the obvious definitions

are associative graded F

algebra, all with

F

algebras and

is an asso-

as multiplication.

Two notions of tensor field The preceding paragraphs suggest two different notions of a tensor

field:

an element of

or an element of

. Happily, the two notions

coincide for finite dimensional differentiable manifolds (assumed to be paracompact). F

The second notion is of greater importance, so that if

module we will refer to elements of

E

is an

as tensor fields or tensors.

20.

§1.

A tensor in rank

s ,

MULTILINEAR ALCEBRA

is said to be contr arariant of rank

r

is the contravariant tensor algebra,

algebra, and

and covariant of the covariant tensor

the mixed tensor algebra.

There is a natural

F

algebra homomorphism

preserving the bi-grading, defined by setting

and extending to all of product, K

K

by

is well-defined.

F

linearity.

This agrees with our previous definition of

as a mapping of

. We call the

totally reflexive in case ule of all

By the definition of tensor

K

is bijective.

F

module

E

As mentioned before, the mod-

contravariant vector fields on a finite dimensional para-

compact manifold is totally reflexive.

6.

F-linear mappings of tensors Theorem 1.

isomorphic to the

Let F

E

(2)

F

module.

Then

module

of all F-linear mappings of is defined by setting

be an

into

The isomorphism

is canonically

6.

§1-

MULTILINEAR ALGEBRA

and extending to all of of

by F-linearity.

is canonically isomorphic to

ive then each

In particular, the dual module

, so that if

E

is totally reflex-

is reflexive.

Proof.

The mapping

product, and is an surjective.

F

i

is well-defined by the definition of tensor

module homomorphlsm.

It is obviously infective and

QED.

Suppose that

E

is totally reflexive.

A number of special cases

of Theorem 1 come up sufficiently often to warrant discussion. and

, and denote the pairing by any of the expressions , as convenient.

symbol A

We identify

If

A

is in

we use the same

for the F-linear transformation

into

itself, so that

Notice that the F-linear transformation the dual

of A .

If

A

and

product as F-linear mappings of The identity mapping of If

E

E

B

is in

into

F

AB

for their

X

identifies

identifies

with

algebra (not necessarily associative) on E .

we write

, so that

we write

is

into itself and similarly for

is totally reflexive then

of the two vector fields

Also,

E

are in

into

into itself is denoted 1 .

the set of all structures of If

B

of

for the product in this sense and

Y , so that

with the set of all F-linear mappings of

8. §1.

Similarly,

t(o,l,l,o)

of

into

7-

Contractions Let

MULTILINEAR ALGEBRA

identifies

with the set of all F-linear mappings

, so that

E

be an

F

module, and let

We define the

contraction

by

where the circumflex denotes omission, and by extending by F-linearity.

to all of

By the definition of tensor product, this is well-defined,

and it is a module homomorphism.

The Encyclopaedia Britannica calls it an

operation of almost magical efficiency.

(See the interesting article on

tensor analysis in the 1^-th edition.) If

8.

letters.

Then

(3)

is denoted

tr A , and called the trace of A

The symmetric tensor algebra Let

r

then

Define

E

be an

For

u

F in

module and let and

is a right representation of

Sym

on

by

in

be the symmetric group on define

on

; that is,

by

8.

§1.

(Since Sym

MULTILINEAR ALGEBRA

is a field of characteristic zero,

to the contra-variant tensor algebra

tensor

u

by addltivity.

is called symmetric in case

symmetric if and only if of any pair of

makes sense.) Extend

. Thus

u

A contravariant in

is

i is invariant under the transposition

. The set of all symmetric tenors in

and the set of all symmetric tensors in

Er

is denoted

is denoted

, so that

where of course Theorem 2. range

Sym is F-linear and is a projection

. Consequently

by the kernel of

with

may be identified with the quotient of

Sym . The kernel of

Sym

is a two-sided ideal in

Consequently the multiplication

00 makes

into an associative commutative graded algebra over Proof.

Sym

is clearly F-linear.

F .

That it is a projection follows

from ( 3 ) - it is easily checked that the average over a group representation is a projection. identify

The range of

Sym

with the quotient of

By the definitions of

and

is

by definition, so that we may by the kernel of

Sym , if

Sym . and

then

(5)

where

ranges over

. If

Sym u

or

Sym v

is

0

this is clearly

9.

§1.

0 , so that the kernel of

MULTILINEAR ALGEBRA

Sym

is a two-sided ideal, and the quotient alge-

bra is an associative commutative graded The algebra algebra.

9.

F

algebra.

QED.

is called the (contravariant) symmetric tensor

One may also construct the covariant symmetric tensor algebra

The Grassmann algebra The discussion of the (covariant) Grassmann algebra, given an

module

E , proceeds along similar lines.

define

where

in

and

in

by

sgn

mutation. Alt

For

F

is 1 for Then

on

an even permutation and -1

for

is a right representation of

an odd per-

on

. Define

by

and extend Alt

Alt

by additivity to

. An element

of

such that

is called alternate or antisymmetric and is also called an

exterior form. elements of is denoted

The set of alternate tensors in axe called r-forms.

is denoted

, and

The set of all alternate tensors in

, so that

Notice that

. A covariant tensor

alternate if and only if

of rank

r

is

changes sign under the transposition

of anjr two Theorem 3. Consequently

Alt

is F-linear and is a projection with range

may be identified with the quotient of

by the kernel

§1.

10.

of

MULTILINEAR ALGEBRA

Alt . The kernel of Alt

is a two-sided ideal in

and the multi-

plication

makes

into an associative graded algebra over

F

satisfying

(6) Proof.

The proof is quite analogous to the proof of Theorem 2.

Instead of (5) we have, for

in

and

,

(7)

QED. The algebra

is called the (covariant) Gr as smarm algebra.

One

can also construct the contravariant Grassmann algebra Warning;

As we have defined the notion, an r-form is simply a co-

variant tensor of rank

r

which is alternate.

However it is customary in

the literature, and we will follow the custom because it is convenient, to make from time to time conventions about r-forms which differ from conventions already made about tensors.

These special conventions have the pur-

pose of ridding the notation of factors If with itself or for of

r! , etc.

is an exterior form we denote by k

times,

the exterior product of

If

this is

0

for

in

an exterior product of 1-forms, but not for general elements

. Notice that

for

f

in

and

a

in

A graded algebra whose multiplication satisfies (6) is sometimes called "commutative," but this miserable terminology will not be used here.

12.

§1.

10.

MULTILINEAR ALGEBRA

Interior multiplication Let

X

be in the

F

module

E

and let

he an r-form. We define

by

(8) , and we define general element of

by additivity if

. The mapping

is a

is F-linear from

, and it follows from (7) that it is an antiderivation of

to

; that is,

(9)

11.

Free modules of finite type An

in

F

module

E

is free of finite type if there exist

E , called a basis, such that every element

Y

in

E

has a unique

expression of the form

(Unless indicated otherwise,

always denotes summation over all repeated

indices.) Theorem k. Then

E

Let

E

be free of finite type, with a basis

is totally reflexive.

The dual module has a unique basis

(called the dual basis) such that

where

is 1 if

are a basis of

and

0

, so that every

otherwise.

u

In

The

has a unique expression of

§1.

12.

MULTILINEAR ALGEBRA

the form

The coefficients In this expression (called the components of pect to the given basis of

u

with res-

E ) are given by

The

(10) are a basis of

, so that every r-form

has a unique expression of the

form

The coefficients in this expression (called the components of form or simply the components of

so that the components of a

regarded as an element of

are given by

as an r-form are . If

If

If

r! times the components of then

.

has components

and

as an r-

then

has components

13.

§1.

If

and

where the otherwise.

MULTILINEAR ALGEBRA

then

is 1 if the If

has components

are a permutation of the

then Alt

i's

and is

0

has components as an element of

given by

where the if the

is 1 if the

are an even permutation of the

are an odd permutation of the

is an r-form and

If

and

, and is

0

is an s-form then the

is an r-form, the

, is -1

otherwise.

has components

-form

has components

by

Then

Let

be another basis of

If

E , and define

and

§1.

14.

If

MULTILINEAR ALGEBRA

the components of

Proof.

n

with respect to the new basis are

The proof is trivial.

QED.

Notice that the primed indices do not take values in the set but in a disjoint set

of the same cardinality.

This notation is very convenient, as it makes it impossible to make a mistake in writing the transformation laws. 12.

Classical tensor notation Despite the profusion of indices, the classical tensor notation is

frequently quite useful, especially in computations involving contractions. The vector fields over a coordinate neighborhood in a finite dimensional manifold are a free module of finite type, but the module of all vector

(11)

fields does not in general have a basis, Instead of parallelizable.)

(if it does, the manifold is called

use any other indices, provided they However, we it may is possible to •use r+s the classical tensor nota-

are are called contravariant tiondistinct globally,indices. without The any upper choice indices of local coordinates, if we make indices, the folthe lower indices are covariant indices. lowing conventions.

Next we suppose that the contra-

variant Let indices are an eovariant vector and the covariant indices are E be F module, andfields let Consider an expression contravariant vector fields. of the form

Then we define (ll) to be the scalar

§1.

MULTILINEAR ALGEBRA

(it would perhaps be better to write

15-

, but we don't.)

Notice that

although the indices are required to be distinct indices, the mathematical objects they denote need not be distinct.

(Thus we may have

covariant vector fields although obviously r-form

as However, for an

we make the special convention that

(12) Now suppose that

E

is totally reflexive, so that contractions of

tensor fields are meaningful.

If

we define

(13)

Instead, of

we may use any other index, provided it is distinct from

the other indices occurring.

An index which occurs precisely twice, once

as an upper index and once as a lower index, is called a dummy index. Notice that there is no summation sign in (13). being summed.

This is because nothing is

(When dealing with components with respect to a basis of a

free module of finite type, we will continue to write summation signs when summations occur.) We may have more than one dummy index, provided they are all distinct from each other and the remaining indices, to indicate repeated contractions.

The notation is unambiguous because, from the definition of

contraction, the order in which the contractions are performed is Immaterial. Here are some examples of the use of this notation. first example we assume that then

E

is totally reflexive.

If

In all but the and

16.

§1.

MUI/rmUEAR ALGEBRA

If

If

then

(14)

The notation here is abusive.

The right hand side of (l4) is not the product

of two scalers hut is written instead of

We will indulge freely in this abuse of notation. restriction of

Sym

to

. Since

for a unique tensor

The tensor

Sym

Now let

be the

is F-linear,

and if

then

may be computed explicitly, and one finds

where perm denotes the permanent.

(The permanent of a square array of

scalers is defined in the same way as the determinant except that there are no minus signs.) Similarly, if

for a unique tensor

in

, and

then

17. §1.

MULTILINEAR ALGEBRA

where det denotes the determinant.

If

1

and

then (recall

(12))

and if

13.

then

Tensor fields on manifolds Let

p

he a point in the

manifold

M . A tangent vector at p

is an equivalence class of differentiable mappings , where and

x

and

y

differ by

with

are equivalent in case the coordinates of x(t)

. One verifies that this condition is independ-

ent of the choice of local coordinates, and that addition and multiplication by constants are well-defined on tangent vectors. gent vectors at at

p

forms a real vector space

p . A cotangent vector at

p

g(q)

in coordinates of

q

and

, called the tangent space

is the dual notion: an equivalence class

of differentiate mappings are equivalent if

Thus the set of all tan-

with

and

g(q)

differ by little

, where o

f

and

g

of the difference

p . Again, the condition is independent of the

choice of local coordinates, and the cotangent vectors form a vector space which is in a natural way the dual vector space to The set

T(M)

of all tangent vectors at all points of manifold as 4 0 e s the set

natural structure of

M

has a

of all cotangent

rectors. They are called the tangent bundle and cotangent bundle. equipped with natural projections onto each vector the point

p

They are

M , the projections which assign to

at which it lives.

section of the tangent

bundle is called a contravariant vector field or vector field and a

§1.

MULTILINEAR ALGEBRA

,/ u>„

Xp

Eigure 1. Pictures of a tangent vector vector

ω

.

and a cotangent

A tangent vector gives a direction and speed of

motion, a cotangent vector is a linear approximation to a scalar.

The tangent vector

arrow twice as long,

2ω^

2X^

would he indicated by an

would he indicated a relabeling

of the hyperplanes (twice as dense). and (Op

In the figure

look as if they are in some sense the same, hut

this has no meaning unless the tangent space is equipped with additional structure, such as a pseudo-Riemannian metric or sympleetic structure.

19. §1.

MULTILINEAR ALGEBRA

section of the cotangent bundle is called a covariant vector field or 1-form. They form modules

E

and

E'

over the algebra

F

of all scalers

real

functions on M).

Therefore we have the notions of tensor fields on

M

and

tensors at a point p . Tensors are of great inqportance in differential geometry because they are invariantly defined geometrical objects (independent of any coordinate system) which live at points. order for an object to be a tensor. fine a tensor

Both characteristic's are necessary in Suppose for example we attempt to de-

u , contravariant of rank 2, by requiring, in local coordi-

nates,

where

is 1 if

and

0

otherwise.

This lives at points but

is not invariantly defined, since in new coordinates

it would

have components

(On the other hand,

are in each coordinate system,

a certain tensor.) As another example, let vector field other than

0

and define

is the Lie product of

X

X

on

and

the components of

be a fixed contravariant E

by

, where

This is invariantly defined

but it does not live at points, because in order to know p

we need to know something about

differentiate it.

Ia fact,

Y

1 at a point

in a neighborhood of

p

in order to

is IR-linear but not F-linear, since

, so that

is not a tensor field.

The condition of

F-linearity is in fact the condition that an 3R-multilinear object live at points.

If for example

is a 1-form

, since we may write

and with

then , and so

20.

§1.

MULTILINEAR ALCEBRA

The example of the Lie product shows that not all Interesting geometrical objects are tensors.

Affine connections are another example of

second-order geometrical objects.

Tensor fields are first-order geometrical

objects since the notion of tangent vector involves one derivative.

14.

Tensors and mappings Suppose we have two

W±th dual

, and an

algebras

module

F

and

with dual

, an

F

module

E

. We shall use the word

homomorphism for any of the following:

(15) an

algebra homomorphism;

(16) a group homomorphism (and similarly for

I;

(IT) where

are homomorphisms satisfying the compat-

ability condition

(18) (and Similarly for

; and finally for

(19) where the compatability conditions (l8) and (2)

are homomorphisBfa satisfying

21. §1.

MULTILINEAR ALGEBRA

(21) Now let

be a

mapping of the manifold

defined by

M

into the manifold

is a homomorphism.

tangent vector at a point

p

in

M

. Then

If we recall what a

is, we see that

induces a vector

space homomorphism (linear transformation)

It is called the differential of

at

p .

By duality,

If we define

then is a homomorphism.

In the same way we obtain a homomorphism

phism p.83 of diately do ,the not and ofcovariant arise Helgason clear sendswe to the may tensor almost [ 2Grassmarm whenever ] not ) . algebras, anyone The get since mapping aalgebra that which is section we not do preserves induces necessarily ,into not of andin even general theif maps onto, (see However, grading these obtain Exercise we difficulties may a&d ita is products homomornot A.k immehave on

22.

but

§1.

MUIfflLINEAR ALGEBRA

does not in general induce a mapping on Suppose now that

sections of

is a diffeomorphism of

M

T(m) .

onto

. Then we

For

in

obtain (22) a homomorphism (in fact, an isomorphism) as follows.

On

and

is as defined above.

we

define

This homomorphism extends in a natural way to the mixed tensor algebras. In the same way we obtain a homomorphism (22) if

is an imbedding of

M

in It is unfortunate that covariaat tensor fields transform contravariantly under point mappings of manifolds, but it is too late to change the terminology.

Early geometers were more concerned with coordinate

changes than point mappings, and coordinates are scalars, which transform the same way as covariant tensor fields. Notice that we have used the notation

for covariant tensor

fields in keeping with the fact that they transform the opposite way to point mappings. mann algebra

For example, the cohomology ring is formed from the Grassand it is universally denoted

In our study of tensor analysis we shall make no use of points except at one point in the discussion of harmonic forms (§7), where we will need the following notion.

§1.

23.

Definition.

The

MULTILINEAR ALGEBRA

F

module

E

is punctual if there exists a sepa-

rating family of homomorphisms of the form

where

and

is a finite dimensional

-vector space.

The module of contravariant vector fields on a manifold is punctual: take

to be evaluation at the point

space at

p .

p

and

to be the tangent

References [1]

N. Bourbaki,

Elements de mathematique, Hermann, Paris.

See especially

Book 2, Algebre, Chaps. 2 and 3 . [2]

Sigur''

u

and

v

K

into

in K . The

used neither commutativity nor associativity, so if

are derivations of

K

so is

. The derivations lie in

the associative algebra of endomorphisms of BO the Jacobi identity holds.

K

as an

Thus the derivations of

vector space, K

form a Lie

algebra over Now suppose that

K

is a graded algebra.

That is,

K

is the

weak direct stem

where each necessarily

. The An

geneous of degree

a

with

linear mapping if each

homogeneous of degree

a

are usually but not X

of

K

into itself is homo-

, and homogeneous if it is

for some

a . The notions of a bi-graded alge-

bra, and bi-homogeneous mappings of bi-degree

(a,b) , are defined simi-

larly.

K

of

An antiderivation of a graded algebra

K

is an F°-linear mapping

into itself such that

The anticommutator of X

and

Y

is

. A

simple calculation

establishes the following theorem. Theorem 1. algebra

X

and

Y

be antiderivations on the graded

K , homogeneous of odd degrees

the anticommutator gree

Let

a+b .

a

and

1b a derivation of

b

respectively. Then

K , homogeneous of de-

38.

2.

§3-

DERIVATIONS ON TEilBORS

Module derivations Let

E

be an

homomorphism of

P

(F,E)

module.

In §1.

and consequently we have the notion of an auto­

morphism of (F,E) . Formally, let automorphisms of (For example, fold.)

(F,E)

p(t)

we defined the notion of a

and let

may be

p(t)

φ

be a one-parameter group of

be the derivative of

®(t)* where

®(t)

ρ

at

t =0 .

is a flow on a mani­

By the product rule for differentiation we obtain, formally, φ(ίΧ) = ίφ(χ) + [12]). Also, if we define

C on

by

then

and C

commutes with

since

It

follows that odd-dimensional Betti numbers of compact K&hler manifolds are even.

References [ll]

Andre Lichnerowicz, Theorie globale des connexions et des groupes

d'holonomie, Consiglio Hazionale delle Ricerche, Monografie Matematiche 2, Edizioni Cremonese, Rome, 1955[32]

Andre Weil,

Introduction a 1'etude des varietes kaehleriennes,

Hermann, Paris, 1958.