Tensor Analysis

TENSOR ANALYSIS BY EDWARD NELSON

PRINCETON U N I V E R S I T Y P R E S S

AND THE U N I V E R S I T Y O F TOKYO P R E S S

PRINCETON, NEW J E R S E Y

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

Preface These are the lecture notes for the f i r s t p a r t of a one-term course on d i f f e r e n t i a l geometry given a t Princeton i n the spring of 1967.

They are an expository account of the formal algebraic aspects

of tensor analysis using both modern and c l a s s i c a l notations. I gave the course primarily t o teach myself.

One d i f f i c u l t y

in learning d i f f e r e n t i a l geometry (as well as t h e source of i t s great

beauty) is the interplay

of algebra, geometry, and analysis.

In the

f i r s t p a r t of the course I presented the algebraic aspects of the study of t h e most familiar kinds of structure on a differentiable manifold and in t h e second p a r t of the course (not covered by these notes) discussed some of the geometric and analytic techniques. These notes may be useful t o other beginners i n conjunction with a book on d i f f e r e n t i a l geometry, such as t h a t of Helgason 12, Nomizu

De Rham [7,

Sternberg [9, $81, or Lichnerowicz

These books, together with the beautiful survey by S. S. Chern of the topics of current i n t e r e s t i n d i f f e r e n t i a l geometry

( B u l l . Am. Math. Soc., vol. 72, pp. 167-219, 1966) were the main sources f o r the course. The principal object of i n t e r e s t i n tensor analysis is the module of the algebra of

contravariant vector f i e l d s on a

manifold over

r e a l functions on the manifold, the module being

equipped with the additional structure of the Lie product.

The f a c t

t h a t t h i s module is " t o t a l l y reflexive" (i.e. t h a t multilinear funct i o n a l ~on it and i t s dual can be identified with elements of tensor product modules ) follows-for a f inite-dimensional second-countable

Hausdorff manifold

by the theorem t h a t

a manifold has a

covering by f i n i t e l y many coordinate neighborhoods. Elementary Differential Topology, p

See J. R.

Annals of Mathematics Studies

Press, 1963.

No. 54, Princeton

I wish t o thank the members of the class, particularly for many

and Elizabeth

.

manuscript so beautifully

for

the

iii.

Multilinear algebra

1

1. The

of scalars 2. 3. Tensor products 4. Multilinear 5. Two notions of tensor f i e l d 6. mappings of tensors 7. Contractions 8. The tensor algebra 9. The algebra. 10. Interior 11. Free modules of f i n i t e type Classical Tensor f i e l d s on manifolds tensor notation 14. Tensors mappings Derivations on scalars 2. Lie modules 3. Coordinate Lie modules Vector f i e l d s flows

1. Lie products

Derivations on tensors

................

1. Algebra derivations 2. Module derivations 3. Lie derivatives derivations 5. Derivations on modules which are free of f i n i t e type The exterior derivative

47

1. The exterior derivative i n l o c a l coordinates

2. The exterior derivative considered globally 3. The exterior derivative and i n t e r i o r m u l t i plication The ring differentiation

57

1. Affine connections i n the sense of 2. The derivative 3. Components 4. Classical tensor of affine connections notation f o r the derivative 6. Torsion 5. connections and tensors 7. affine connections and the exterior derivative 8. 9. connections on Lie 10. The Bianchi i d e n t i t i e s i d e n t i t y 12. Twisting and

f i b e r bundles

1.

2. Lie bundles

3. The r e l a t i o n between the two notions of ion

89

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

$8.

structures 1. structures 2. vector f i e l d s and Poisson brackets 3. Symplectic structures in l o c a l coordinates

Complex structures 1. Complexification 2. Almost complex structures 3. Torsion of an Complex structures i n structure coordinates 5. Almost complex connections 6. structures

Multilinear algebra 1.

The algebra of We make the permanent conventions t h a t

i s a f i e l d of charac-

teristic

and t h a t

Elements of

F w i l l be called scalars and elements of

.

F i s a commutative algebra with i d e n t i t y over w i l l be called

constants. The main example we have i n m i n d i s

the f i e l d on a

numbers and F the algebra of all r e a l M

.

In t h i s

the s e t of all

of r e a l manifold

contravariant vector f i e l d s is a

F , with the additional structure t h a t the

module over

vector f i e l d s a c t on the s c a l a r s v i a d i f f e r e n t i a t i o n and on each other v i a the Lie product.

Tensor analysis is the study of t h i s structure.

In t h i s section we w i l l consider only the module structure. 2.

Modules The term "module" w i l l always mean a unitary module

Thus an

F module E i s an Abelian into

mapping of

(indicated by

in E and f , g

X,Y

If

E

E

(written

in

=

X)

with a such t h a t

F .

is an F module, the dual module

.

i s the module of

MULTILINEAR

2.

F-linear mappings of

into

F

If

we denote the value of

X

in

If

A

i s an F-linear mapping of

i s an F-linear mapping of K: E

E

i n t o E its

.

into

A'

defined by

There i s a natural mapping

defined by

E

is b i j e c t i v e .

i s called reflexive i n case

not i n general injective. on a manifold

For example, i f M

and E

The notions of submodule,

K

i s an

F module of a l l continuous

F module homomorphism, and quotient

and K are

If

F modules

F module hiiomorphism then the quotient module

i s canonically isomorphic t o the image of

a

.

X

of an

module E

contravariant vector f i e l d s or vector f i e l d s and t o elements as

a

See

We w i l l frequently r e f e r t o the elements

module

is

.)

=

module are defined i n the obvious way.

a: H

K

i s the algebra of

i s the

contravariant vector f i e l d s then

3.

w

E by any of the symbols

on

and

.

as

of the dual

vector f i e l d s or 1-forms.

Tensor products If

H

(over F ) i s the

and

K

are two

F modules,

tensor product

F module whose Abelian group is the f r e e 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

where

i s in F

f

, and

the action of =

Let

If

E be an F module.

is

or

Notice t h a t

0

= E

F on

i s given by

.

=

We define

we sometimes omit it, and we s e t

,

=

.

We

0

=

.

define

many components of any

where the sums are weak d i r e c t sums (only element

E =

non-zero). Notice t h a t

the tensor product

o* and E

0

E,

as multiplication.

With t h i s identification,

Let

are associative graded

E be an F-module.

We make the identification

i s an associative

We define

F algebras with

F-algebra.

t o be the s e t of

F-multilineax mappings of times, E

6.

ALGEBRA

and extending t o

of -

of

F- linearity.

is

isomorphic t o

ive then each

,

p a r t i c u l a r , t h e dual module

so t h a t i f

E

i s t o t a l l y reflex-

i s reflexive. The mapping

product, and is an

i s well-defined by t h e d e f i n i t i o n of tensor

F module

It i s obviously i n j e c t i v e

.

surjective.

Suppose t h a t

i s t o t a l l y reflexive.

E

A number of s p e c i a l cases

of Theorem 1 come up s u f f i c i e n t l y often t o warrant discussion.

,

and

, symbol A

We i d e n t i f y

and denote t h e p a i r i n g by any of t h e expressions

, as

,

convenient.

If

A

1

i s in

f o r t h e F- linear transformation

of

v

>,

we use t h e same E=

into

i t s e l f , so t h a t

Notice t h a t t h e F- linear transformation the dual

of

A

.

If

A

and

product as F- linear mappings of The i d e n t i t y mapping of If

E

B

2

,

that

for their

.

identifies

f o r t h e product i n t h i s sense

or X

identifies

Also,

AB

F algebra (not necessarily associative) on E .

we write

of the two vector f i e l d s

we

i n t o i t s e l f and s i m i l a r l y f o r

E

i s t o t a l l y r e f l e x i v e then

is in

into

B are in

is

into

i n t o i t s e l f i s denoted 1 .

t h e s e t of all s t r u c t u r e s of If

of

and

Y

, so

that

with t h e s e t of a l l

mappings of

Similarly,

identifies

, so t h a t

of

into

7.

Contractions Let

with t h e s e t of a l l F- linear mappings

be an

E

F

.

We define the

v

t o a l l of

module, and l e t

contraction

where t h e circumflex denotes omission, and by extending by

By t h e d e f i n i t i o n of tensor product, t h i s i s well-defined,

and it i s a module homomorphism.

The Encyclopaedia Britannica c a l l s it an

operation of almost magical, efficiency.

the interesting

on

tensor analysis i n the 1 4 t h e d i t i o n . ) If

8.

1 then

A

1

i s denoted

A

, and

c a l l e d the t r a c e of A .

The symmetric tensor algebra Let letters.

E

be

For

in

. Then

Define

and l e t

F

and

a

..

i s a r i g h t representation of

on

E

by

be t h e symmetric group on

in

define

. .. (r)

, on

by

w

i

; that is,

.

makes sense. )

i s a f i e l d of c h a r a c t e r i s t i c zero, t o the tensor

u

tensor algebra i s c a l l e d symmetric i n case

symmetric i f and

,. .

.

of any p a i r of

The s e t of a l l symmetric

= F

Theorem 2.

.

range

Sym

Sym

.

in

is

in

i s denoted

,

i s denoted

so t h a t

. 2

F- linear and i s a

Consequently

by t h e kernel of

u

Thus

A contravariant

i s invariant under the transposition

t h e s e t of all symmetric tensors i n E,

where of course

.

= u

Sym

1

if

by a d d i t i v i t y .

Extend

may be i d e n t i f i e d with The kernel of

=

quotient o f

is a two-sided i d e a l i n

Sym

.

Consequently t h e multiplication =

makes

into Proof.

from (3)

-

associative commutative graded algebra over Sym

i s c l e a r l y F- linear.

i d e n t i f y S,

of

Sym

with the quotient of

By the d e f i n i t i o n s of

where

a

ranges over

and

.

If

over a group representation

is E,

.

That it i s a projection follows

it i s e a s i l y checked t h a t the

i s a projection.

F

by d e f i n i t i o n , so t h a t we by t h e kernel of

Sym

Sym

,

if

u

or

Sym v

Sym and

is

. v

then

t h i s i s clearly

9.

MULTILINEAR ALGEBRA

,

so t h a t t h e kernel of

i s a two-sided i d e a l , and t h e quotient alge-

b r a is an associative

graded

The algebra algebra.

One

F algebra.

i s c a l l e d t h e (contravariant) symmetric tensor

a l s o construct the

The

symmetric tensor algebra

.

algebra The discussion of t h e

module

S

E

, proceeds

define

algebra, given an

along similar l i n e s .

For

in

a

and

F

in

by

where

is 1 f o r

sgn a

mutation.

Then

; is

an even permutation and -1 f o r a r i g h t representation of

(r)

a

an odd per-

.

on

Define

A l t a: =

and extend

by a d d i t i v i t y t o

.

An element

a of

such t h a t

E

i s c a l l e d a l t e r n a t e or antisymmetric and i s a l s o c a l l e d an

=

e x t e r i o r form.

s e t of a l t e r n a t e tensors i n

elements of denoted

Notice t h a t

a r e c a l l e d r-forms.

,

A

0 A = F

X' S

i s denoted

, and

The s e t of all a l t e r n a t e tensors i n E

and

1 A =

. . .,X

A covariant tensor

is

o f rank

changes sign under the transposition

.

Theorem 3. A

E

so t h a t

a l t e r n a t e i f and only i f of

E

Alt

F- linear and i s a projection with range

be i d e n t i f i e d with the quotient of

E

A

.

by t h e kernel

10.

MULTILINEAR

.

of -

i s a two-sided i d e a l i n

The kernel of

and t h e &ti-

E

ion = Alt

makes A*

i n t o an a s s o c i a t i v e graded algebra over

Proof.

The proof i s q u i t e analogous t o t h e proof of Theorem 2.

Instead of ( 5 ) we have, f o r

*

The algebra A

in

i s called the

algebra.

tensor of rank

However it i s customary in

which i s a l t e r n a t e .

t h e l i t e r a t u r e , and we

follow t h e custom because it i s convenient, t o which d i f f e r from conven-

make from time t o time conventions about t i o n s already

These s p e c i a l conventions have t h e pur-

about tensors.

pose of ridding t h e notation of f a c t o r s

or for of

.

.

algebra

As we have defined t h e notion, an r-form i s simply a co-

Warning:

r!

, etc.

a i s an e x t e r i o r form we denote by

with i t s e l f

,

in

and

can a l s o construct the

If

F satisfying

k

times,

.

=

If

a a n e x t e r i o r product of Notice t h a t

=

=

k

the e x t e r i o r product of this i s

for

in

but not f o r general elements f

in

F =

and a

in A

.

A graded algebra whose multiplication s a t i s f i e s (6) i s sometimes

called

but t h i s miserable terminology w i l l not be used here.

ALGEBRA 10.

I n t e r i o r multiplication X be i n the

Let

X J a: =

if

, and

, and

a

general element of

F module E

A

.

we define

and l e t

X J

The mapping

be an r-form.

by a d d i t i v i t y i f X

a is a

i s F-linear from

t h a t it is an antiderivation of

it follows from

We define

to

A* ; t h a t i s ,

Free modules of f i n i t e type

An in

E

,

module E

is f r e e of f i n i t e type i f there e x i s t

called a basis, such t h a t every element

Y

in

E

..

has a unique

expression of the form

indicated otherwise,

always denotes summation over all repeated

indices. ) Theorem Then E

,...

1

where

4.

E

be f r e e of f i n i t e type, with a basis

i s t o t a l l y reflexive.

n

The

.

module has a unique basis

(called the dual basis) such t h a t

is -

a basis of

=

E

so t h a t every

otherwise.

The

has a unique expression of

12. t h e form

The coefficients in t h i s expression ( c a l l e d the components of

with res-

E ) a r e given by

pect t o t h e given basis of

The

a r e a basis of

,

so t h a t every

a

=

has a unique expression of t h e

form i

c



(2) holds, so t h a t

By (a) we do have a unique extension of

By definition of

is a

Then each

the additional terms

on the two sides of (3) cancel. derivation of

and

,

on

1-form since i f we replace

generate

for

algebra.

=

. F, E, and

To prove existence, we need only show

93.

H

that i f

and

K

are

F modules,

i s well-defined on

free

a derivation on

and

(agreein g on F ) , then

a derivation on

.

DERIVATIONS ON TENSORS

and extends by a d d i t i v i t y t o a derivation of

To see t h i s , notice t h a t (5) i s obviously well-defined on the group used i n t h e d e f i n i t i o n of tensor product (91.3) and

that

sends

i n t o t h e subgroup generated by t h e re-

.

l a t i o n s imposed i n $1.3, and so i s well-defined on clear t h a t

.

i s a derivation on

is the dual of

( a ) and ( c ) and the f a c t ($1.6) t h a t has

.

unique extension as a derivation of

by a d d i t i v i t y t o

.

Let

It is then

u

We extend

v

E

.

Then,

so t h a t +

That i s ,

in

, so

=

algebra.

+

that

The f i n a l formula

*

i s a derivation of is

(3) for

u

in

as and

S

Notice t h a t by ( b ) i f and

,

the

derivations given by Theorem 2

E

i s t o t a l l y reflexive, so t h a t we m a y

d e f i n i t i o n s of

agree.

The various

be called the derivations induced

$3. DERIVATIONS ON by the derivation Theorem 3.

E

.

of

be an

F module and l e t

The commutators of t h e derivations induced by

.

a r e t h e derivations induced by Proof.

be deri-

the commutator

.

derivation of and

.

of

We have =

, so

and s i m i l a r l y f o r

and we know t h a t

that

is a derivation of

F

.

The l a s t statement

of t h e theorem is an immediate consequence of t h e uniqueness assertions

i n Theorem 2 . Theorem

and $1.6, i f

i s a derivation of

, and

have the following

I n general it does not Theorem 4. and -

,

and E be an

=

F

.

=

(

-( a derivation of

not 0

. ,

DERIVATIONS ON TENSORS

Roof.

the definition ($1.6) of

.

then

=

, if

,

y

,

Therefore, by

so t h a t (6) holds.

3.

Lie derivatives Suppose t h a t

Lie module, i f

X

E is a Lie module ($2.2). E and we l e t

.

i s a derivation of

then

By the definition of

The induced derivation

the mixed tensor algebra is called the Lie derivative. defined on 1-forms

X

of rank

and

of rank s

is a vector f i e l d on a manifold, generating the

then f o r any tensor f i e l d (see

is

which gives

by

and f o r tensors

If

Thus

on

u

,

i s the derivative a t

of

DERIVATIONS ON

4.

F-linear derivations Let

t i o n of

E be an F module and l e t A be an F-linear transforma-

E into i t s e l f .

i s reflexive the s e t of F-linear

E

transformations of

into i t s e l f can be identified

Define

by

on

on F is of t h i s type.

which i s tions are also denoted

is the

of

We

A

.

By

By

(4) we have f o r

in

mapping

into the

of the mixed tensor algebra B

, where

,

F module.

, o

each

.

i n t o i t s e l f such t h a t

. into i t s e l f

i s in

There

F module of

c =

each A

is

on

E be a t o t a l l y reflexive

is a unique

F-linear

.

The induced deriva-

have occasion l a t e r ($7) t o use a related notion.

Theorem

itself,

, and every derivation of

is clearly a derivation of

Then

, by $1.6).

each

ON TENSORS

- Since which sends

E i s t o t a l l y reflexive,

is F-bilinear

to

of

1

so

for

on F

B

,

1

in

sends each

the

and

f i c e s t o consider t h e

.

A =

r

where we have made use of the f a c t t h a t

, we find

basis

u

..

the

be an

E

kt

.

i n t o i t s e l f and is

To prove (11) it suf-

By

is

a If we use

.

of f i n i t e type F module, f r e e of f i n i t e type with

be a derivation of

of

c

(11)f o r the case A =

on modules which are

6.

The map-

to

...

.

5.

.

properties.

The notation in (11) is t h a t of $1.12.

(12) t o compute

1

the definition

has a unique F-linear extension

Since

If

=

and define

DERIVATIONS ON TENSORS

.. .

i s another basis

and i f

are defined

i s defined by

then

Let

E

be a coordinate Lie module with coordinates

and l e t the vector f i e l d X sponding t o the

If

X

.

have components

..

x1,

Then the

n

corre-

derivative

has components

with respect t o new

x

,..

then

where we use the notation Proof.

a2

t o mean

a a

ax

.

The proof i s t r i v i a l .

Formula (13) shows the basic f a c t that the p a r t i a l derivatives of the components of a tensor do not in general form the components of

n

46.

DERIVATIONS ON

a tensor.

This was what l e d

differentiation

t o t h e notion of

($5).

Reference

A d e f i n i t i o n of the e x t e r i o r derivative

R. S. of Lie derivatives,

Proceedings of the American Mathematical Society

In the d e f i n i t i o n on with

terms

r a t h e r than with

itself.

should be i d e n t i f i e d

$4. 1.

The exterior derivative

The exterior derivative i n l o c a l coordinates Let

E

be an F

A*

the

an exterior derivative we mean an

algebra mapping d of

*

A

into

i t s e l f such t h a t

Theorem 1. Let

E

be a coordinate Lie module.

a unique exterior derivative d such t h a t f o r

. 1 ,..

the d i f f e r e n t i a l of Proof.

Let

n be coordinates.

Then there is

scalars

Then each

, in

is uniquely of t h e form i

a

.

=

i

1

If

and

hold

That is, the

we

have

of

This proves uniqueness. To prove existence, choose coordinates and define extending t o

To prove

of

* A

by additivity.

Then d

relation

since the

let

.

is

the e x p l i c i t

d by and [ D l ]

in $1.11

for the components of

and by

holds.

The proof shows, of course, t h a t ( 1 ) holds for any choice of coordinates.

i s c e r t a i n l y the quickest approach t o the exterior

derivative on a manifold, f o r once t o define it globally.

d

i s known l o c a l l y it i s t r i v i a l

However, a coordinate-free treatment of the

e x t e r i o r derivative i s worthwhile f o r several reasons.

For one, it

applies t o Lie modules which do not have coordinates (even locally, such as a Lie

over

The invariant expressions f o r

Finally, it deepen% one's understanding of the exterior

useful.

derivative and shows it t o be the natural dual object duct

.

2.

The exterior derivative considered globally Theorem 2.

E

unique e x t e r i o r derivative the d i f f e r e n t i a l of and

Y

d

f

t h e Lie pro-

reflexive Lie module there is a

is a

d such t h a t f o r all scalars

and f o r all 1-forms

f

,

df

and vector f i e l d s X

,

If

E

is any Lie module and we define

and by extending derivative.

d

t o all of

*

A

. d

*

A

by additivity, then

d

i s an exterior

DERIVATIVE

Proof. a s an

For E

t o t a l l y reflexive,

generate A

algebra, so the uniqueness assertion is clear.

the requirements t h a t

r=

special cases that

and

be the d i f f e r e n t i a l of of (3). )

and r =

and (2) are the

f

Therefore we need only prove

defined by (3) i s an exterior derivative.

d

denote the s e t of a l l of

F multilinear)

. E

If

, and

(not necessarily

E ( r times) into

E

i s in

Z

, we

i s in

F

.

Thus

( r times),

i s in

as an abbrevi-

use the notation

ation for

in

For

we define

=

and

if

r = O

general i n

if

alternating elements in we define

For that

in

is in

Then

.

i s in

,

, and

.

be the s e t of

.

so t h a t

t h i s definition of

d:

Let

and i s not in

For

in

; t h a t is,

=

We claim t h a t so is

.

d:

To see t h i s , l e t

It i s trivial

agrees with (3).

simple computation shows t h a t

; t h a t is, i f

,

f

F , and l e t

is

DERIVATIVE

7

We must show t h a t

.

which proves the claim. only t o show t h a t

It

.

=

A simple computation

,

a in

shows

2

us therefore compute

f o r X and

,

Y

in E

and

in E

that

E

.

= =

(r times).

-1

,

Let

=

THE

Since

-

we

that

where we have used t h e Jacobi i d e n t i t y t o cancel the f i r s t , second, and fourth terms o f t h e second l i n e and the t o r e - m i t e t h e t h i r d term.

Consequently (4) i s equal t o

in t h e i r natural order,

if

changes sign under distinct.

By the

of the Lie product

and

=

if

identity again, the l a s t sum is

a r e not

, so t h a t

THE

If

E

Lie module we

c a l l t h e operator

defined

by (3) t h e exterior derivative. Theorem 3.

E

exterior derivative on A

.

be a

reflexive

Define

X

F module,

d

a vector f i e l d and

a scalar by

for

and define

Y vector f i e l d s by

X

Then with

a

,

t o these reflexive

E

WE

E ' .

is a Lie module.

for

F module there is a one-to-one correspondence be-

tween structures of Lie module on E

and e x t e r i o r derivatives on the

algebra. Proof. Since

+ t o show t h a t in

Recall the definition of Lie module i n is a 1-form,

,

each

X

i s F- linear.

is a derivation of

F

.

is i n E ; t h a t is, t h a t

Since Next we need is

.

and by d e f i n i t i o n of t h e exterior product,

+

.

=

=

+

EXTERIOR DERIVATIVE

, we

=

Since

.

=

have

Since

is

a 2-form,

so t h a t

.

+

=

i n X and Y

, so

so t h a t

and t h a t

is

reflexive,

generated

A*

=

a

.

an

Therefore we L e t us use

If we l e t

=

Since E

algebra by

i s totally

.

and

use (3) t o compute

=

t o denote cyclic

where K i s any is a 2-form, (3)

into

maps

did not use the Jacobi

an

so remains valid under our present assumptions.

for

equals d on scalars

proof given in Theorem 2 t h a t

and

is

it remains only t o prove the Jacobi identity.

by formula

Define

It is clear t h a t

E X E XE

t o an additive group.

If

be written

, where

is a 1-form, we obtain

Since t h i s is true f o r all 1-forms

, the Jacobi

identity holds.

Con-

E is an a r b i t r a r y F

More generally, the proof shows t h a t i f module, i f for X

in

and

x

for

, and we define

is an exterior derivative on A

d

and

holds, then

in

and

Y in

i s a Lie module.

Lie products and exterior

derivatives are dual notions, and the Jacobi i d e n t i t y corresponds t o the t h a t an exterior derivative has square

3.

.

The exterior derivative and i n t e r i o r multiplication Theorem

be a Lie module,

E

in -

X

E

.

Then the anti-

commutator of the exterior derivative and i n t e r i o r multiplication by X i s the Lie derivative

If

on exterior forms.

is exact.

i s a closed exterior form, Proof.

That i s ,

We give the proof first under the assumption t h a t E

is t o t a l l y reflexive, since t h i s i s the case of i n t e r e s t in d i f f e r e n t i a l

geometry and the proof is l e s s computational. derivation of

by X degree -1

*

A

which is homogeneous of degree

is an

of

A*

1-form,

and i n t e r i o r multi-

which is homogeneous of

A

(Theorem 1, $3.1).

is generated as an

we need only scalar,

is an anti-

t h e i r anticommutator is a derivation of

is homogeneous of degree

reflexive,

Since d

for

says t h a t

=

If

df (x)

, which

t h a t f o r all vector f i e l d s

is true. Y

,

which

E is t o t a l l y 1 and A

, so

If

=

is a

If

=

is a

algebra

a scalar or 1-form.

*

A

$4.

THE

which i s the same as (6).

DERIVATIVE

Thus

i s true i f

E

i s t o t a l l y reflexive.

The proof f o r the general, case i s similar: one v e r i f i e s r-form by the

(3) for

d

,

for

the definition (formula

a

an

$1.10)

of i n t e r i o r multiplication, and the formula ($3.3) f o r Lie derivatives. The computation is omitted. The l a s t assertion i n the theorem i s an immediate consequence

The cohomology r i n g Let of an

d

be an exterior derivative on the Grassmann algebra

F module E

exact i f

An exterior form

5.

on the Grassmann algebra graded -

i s called closed i f

E A

be an

.

F module,

d

an exterior derivative

The s e t of closed exterior forms is a

algebra i n which the s e t of exact exterior forms is a homo-

geneous ideal, so t h a t t h e quotient i s a graded Proof.

algebra.

The proof i s t r i v i a l .

The quotient algebra i s denoted

.

=

.

for some exterior form

=

Theorem

.

A

It i s called the cohomology ring.

vector space i s denoted

H

,

with homogeneous subspaces

The dimension of

and called the r - t h

number.

theorem a s s e r t s t h a t t h e cohomology ring formed from the forms on a

as an

exterior

manifold is a topological invariant, the cohomology r i n g

of the manifold with r e a l coefficients.

differentiation 1. Affine connections i n t h e sense of Koszul

On a d i f f e r e n t i a b l e manifold there i s no i n t r i n s i c

of d i f -

f e r e n t i a t i n g tensor f i e l d s t o obtain tensor f i e l d s , covariant of one higher, and t o obtain such a

we must impose

In Riemannian geometry t h e r e i s a n a t u r a l notion

additional

of covariant d i f f e r e n t i a t i o n ($7) discovered by later

Many

discovered the geometrical

of covariant d i f -

f e r e n t i a t i o n by i n t e g r a t i n g t o obtain p a r a l l e l t r a n s l a t i o n along curves. A number of people, e s p e c i a l l y E l i e

studied non-Riemannian

connections," and t h e notion was way by Koszul,

follows.

Definition. on

i n a convenient

Let

E i s a function

E be a Lie module.

An a f f i n e connection

from E t o the s e t o f mappings of

X

E

into i t s e l f satisfying

all vector f i e l d s

and s c a l a r s

f

and

g

. of

Thus an a f f i n e connection i s an F- linear mapping X E

i n t o derivations on

(see $3.2).

such t h a t f o r

scalars

The derivation of t h e mixed tensor algebra

($3.2) i s i n t h e d i r e c t i o n of

denoted X

f E,

and i s c a l l e d t h e

,

Xf = induced by

derivative

(with respect t o t h e given a f f i n e

In p a r t i c u l a r we have

YEE.

2.

The

derivative derivatives have an

property which is not

enjoyed by Lie derivatives: Definition. E , and l e t

be an affine connection on the Lie module

Let

.

u be i n

Theorem 1.

Proof.

derivative

by

be an affine connection on the Lie module

u be in

and l e t

We define the

.

Since

F- linearity of (2) in

is in

.

i s a tensor the only point a t issue i s t h e

, and

t h i s is an immediate consequence of

(which remains t r u e f o r the induced derivation of the mixed tensor

3.

Components of affine connections Theorem 2 .

basis

If

..

i s in

E

, and l e t

be a f r e e Lie module of f i n i t e type, with be an a f f i n e connection on E

the components of

.

Define

If t h e -

a r e defined i n t h e same way with respect t o an-

.

other b a s i s

and

a r e defined as i n

then

. Proof.

J

a

=

b

This follows e a s i l y from a r e an

Notice t h a t if the t h e r e is a unique basis.

The

.

+

s e t of

connection on E

n3 s c a l a r s

s a t i s f y i n g ( 3 ) f o r t h e given

called t h e components of t h e a f f i n e connection

(with respect t o t h e given b a s i s ) .

The components may be

with re-

spect t o one basis b u t not with respect t o another. If

,. ..,xn

1

x

E

i s a coordinate Lie module (52.3) with coordinates

then (5) takes t h e more

form

For some applications i n d i f f e r e n t i a l geometry, a coordinate system does not give t h e most convenient l o c a l b a s i s f o r t h e vector f i e l d s . For example, on a Lie group it i s usually convenient t o choose a b a s i s of l e f t - i n v a r i a n t vector f i e l d s .

These do not i n

come from a

coordinate system since they do not i n general commute.

It i s customary t o denote t h e l e f t hand s i d e of (4) by

The notations

.

i

and

a r e a l s o i n use t o mean t h e same thing.

u

.. . However, it i s convenient in

7) t o have

connection with the exterior derivative (see new

index be the f i r s t covariant index.

Classical tensor notation f o r the

derivative.

be an a f f i n e connection on t h e Lie module E and

Let

the global meaning of the c l a s s i c a l tensor notation

Following

convention, we write

Notice therefore t h a t

On the other hand, i f u X Y

s

i s again in

tensor i n

and

X

Y

are vector f i e l d s and u

,

i s in

and i s in general quite d i f f e r e n t from the

obtained by substituting X

.

contravariant arguments of

and

Y

i n the first two

See paragraph

5 . Affine connections and tensors Let

E

t h a t tensors into E

be a

reflexive Lie module.

in

may be regarded as F-bilinear

B

, and we write

kt

or

f o r the

be the s e t of all

pings of i s an

E

.

vector

i s an

Thus

of

i s , i n case element of

over

E XE vector f i e l d .

(not necessarily F - bilinear) map-

.

vector space,

other examples of elements of

are affine connections and the Lie product. called a Lie

We have seen ($1.6)

The Lie module E

F i n case the Lie product i s i n

i s F-bilinear i n X

and Y

.

2

When we

a tensor we mean t h a t it l i e s in the vector

is that that

Theorem 3.

be a t o t a l l y reflexive Lie mod

E

not a Lie algebra over

F

.

which

Then no affine connection i s a tensor.

two affine connections i s a tensor, and i f

difference of

is a tensor i n

affine connection and B

is

i s an

connection. s e t of affine connections and l e t

be a be scalars.

Then a

an affine connection if

Proof.

i

f o r all vector f i e l d s

=

i s a tensor i f and

X

and

and only i f E is a Lie algebra over F

if

and

i s in

is

, so it

then

affine connection.

Y

and

f ;

.

be affine connections. and l e t

.

F-linear i n Y B

.

an affine connection

By

only i f

=

a

is a tensor.

in X If

and by

it i s

is an affine connection

satisfies

and

and so i s an

The second paragraph of the theorem i s obvious.

the theorem, the s e t of all affine connections on a t o t a l l y reflexive Lie module E t o but d i s j o i n t t i o n on E

is e i t h e r empty or is an affine

m

Since

e

i > m or

. =

are

by what we have

this

i,j

basis of

V

,

Let

the definition of

are in V for

By the

is in

m of them

>,

.

.

.

seen t h i s

($3.4) we have

that

.

with

Notice that a mapping L with d or

.

form

into its component i n

nection

For example, l e t

but If

g is

Lie

Then

in V

be a basis of the

be the dual basis.

we have

pings

for each A

in V

.

j

4

that

=

e

commutes with each

and Y in E , we have

each

symmetry and

in

spanned by

L of

is

is

A*

L be the mapping which sends each exterior

.

d on

and

, we

Ld

f o r any affine con-

is on

i n t o i t s e l f which have an

algebra.

on

.

t o t a l l y reflexive map-

the s e t of

denote by

defined as follows. is given by

Then

pseudo-Riemannian E

need not commute with

=

.

derivative

The algebra If

L

possesses a then

L

in

*

and

i s extended t o A,

by additivity.

Then

*

maps

into i t s e l f ,

and

*,

.

=

8 each L in

By

maps the harmonic forms into themselves,

so t h a t by r e s t r i c t i o n we obtain a *-representation of the bi-graded with

10.

.

on the graded vector space

Hodge theory If

M

is a differentiable manifold

dimensional) then we

construct a Riemannian metric

i s trivial t o do p a r t i t i o n of unity.

and f i n i t e

and then we construct

g

g on it.

This

globally using a

Since a convex combination of Riemannian

i s again a Riemannian metric, there is no d i f f i c u l t y .

A manifold M metric

g

M

is

0

or not, with a pseudo-Riemannian

has a distinguished

which we denote

If

, orientable

.

element ( a measure, not an n-form)

In l o c a l coordinates,

i s compact then the i n t e g r a l of any scalar which i s a divergence

. For the r e s t of t h i s discussion, l e t

manifold. define the

*

Then the exterior forms A

M be a compact Riemannian

form a pre-Hilbert space i f we

inner product of two forms

and

be

=

Theorem

5,

,

=

Therefore, i f

is

are positive and

Since the

they are

quently, on a compact Riemannian manifold a

manifold.)

This

why t h e

was introduced in

and

space

integrals by

A

.

is a symmetric, positive operator on t h e

A

As a p a r t i a l d i f f e r e n t i a l operator it is e l l i p t i c ,

for by the

formula i n l o c a l coordinates

plus lower order terms, and the matrix

denote

A was introduced.

by Bidal and The operator

type.

Conse-

pseudo-Riemannian

operator theory of

.

form i s closed and

converse is t r i v i a l l y t r u e on

co-closed.

0

ij

g

A

is

i s of s t r i c t l y positive

It follows t h a t the closure of the operator

A

(which we again

A ) i s a self- adjoint, positive operator on the completion of

*,

the pre- Hilbert space

A

A has d i s c r e t e spectrum;

and define

G =

by the

H =

for

operators on

.

By the

theorem f o r e l l i p t i c operators,

into i t s e l f .

A*

so t h a t

fact,

Ha = a

H

and 6

with

is the

AG =

A

G

.

,

functions

G

From the definition of G , onto the orthogonal

Therefore we have the

, =

+

+

i s closed, the second term t o its

form i s

then i t s harmonic part is have the Hodge theorem:

form into its

. is

=

past

, since

Ha

.

If

=

, so t h a t

a closed

a i s exact, =

.

Therefore we

On a compact Riemannian manifold every

form is closed and eo-closed, and t h e

onto

H and

projection

a If

G

Space into A

, and

and

H

complement of t h e harmonic

a in

and

a is harmonic.

commute with A

and 6

A ,

H

maps the e n t i r e

i f and only i f

Since of

They are bounded

is the orthogonal projection onto the null-space of

operators, and H A

space.

which sends a

class i s a vector space isomorphism of

* H . The Hodge theorem i s of great importance i n Riemannian

It can

groups but i t s chief im-

be used t o compute

portance l i e s in t h e p o s s i b i l i t y it affords of deducing global r e s t r i c t i o n s on the topology of a manifold in order t h a t it admit certain types of differentid- geometric structures. By Theorem

8 the algebra

i t s e l f with Hedge theorem, on

derivative

*.

H

of F- linear mappings of a c t s on

*

*

A

and therefore, by the

This places r e s t r i c t i o n s on the topology of a

$7.

METRICS

n-manifold which admits a Riemannian metric

) as holonomy g r o w .

group of

Theorem 8 is due t o Suppose

and a number of examples a r e well-known.

i s an orientable compact Riemannian n-manifold.

M

is an F- linear operator with A of

onto

mapping

d u a l i t y (weak be-

c a t i o n of Hodge theory i s t o t h e topology of

is the

with

A = L

with

and

induces an isomorphism

with r e a l c o e f f i c i e n t s ) .

/-

Then t h e r e

onto

. By t h e Hodge theorem, it . This is a weak form of

cause we have

and

a given sub-

The p r i n c i p a l applimanifolds.

derivative

then

and so induce operators on

la

* H .

If =

The

existence of these operators places strong r e s t r i c t i o n s on t h e cohomology of a

manifold. The application of Hodge theory requires t h e geometrical

s t r u c t u r e under study t o be r e l a t e d t o a Riemannian metric.

One of the

main open f i e l d s of research in d i f f e r e n t i a l geometry i s t h e problem of f i n d i n g global r e s t r i c t i o n s on a manifold i n order f o r it t o admit a more general geometrical s t r u c t u r e , such as a f o l i a t i o n o r complex struct u r e , s a t i s f y i n g i n t e g r a b i l i t y conditions.

References A. Weil,

No. Noordhoff, Groningen,

R.

differentiables Hermann, Paris,

- Formes,

$8. 1.

structures

Almost symplectic structures A 2-form

i s i n particular a

so determines a mapping

tensor of rank 2 and

E

by

,

=

and i s called non-degenerate i f the mapping is b i j e c t i v e ($7.1). Definition.

An almost symplectic structure on an

module E

is a non-degenerate 2-form. Thus the definition i s the same as t h a t of a pseudo-Riemannian metric except f o r a minus sign. basis

n

n

is f r e e of f i n i t e type with

E

is an almost symplectic structure with

and

then c l e a r l y

nents Also,

If

det

ij

$ 0 ,

be even since det

=

Definition.

det

=

det

=

. is a

A symplectic structure on a Lie module E

closed almost symplectic structure. That is, a non-degenerate 2-form is

a symplectic structure.

satisfies

=

As i n $7, we begin by studying

connections which preserve t h e structure. be an almost symplectic structure on the Lie module E

.

V i s an a f f i n e connection such t h a t =

In particular, if there i s a torsion-Free that

=

Proof.

is a symplectic =

we have

. connection

=

by the definition of torsion (05.6).

When we take cyclic sums the f i r s t

l a s t terms of the r i g h t hand side cancel, s o

the l e f t hand side of t h i s

As we remarked before (formula

.

is

o r the remark that

2.

in the theorem

The l a s t

for V

=

from t h i s

torsion- free.

Hamiltonian vector f i e l d s and Poisson brackets be a symplectic structure on the Lie module E

Let

then

is a

.

If h

is a vector field, and a vector f i e l d of t h i s

That is,

form is

is a Hamiltonian vector f i e l d

X

field X called locally

such t h a t

lemma a closed

by

on a manifold is l o c a l l y exact). the 1-forms as follows: i f

We and

is closed is

transport the Lie product t o 1-forms t h e i r Poisson

bracket is 2 If

f

and

g

Theorem 2. E

.

Then:

.

we define t h e i r Poisson bracket t o be

be a symplectic structure on the Lie module

(a)

X

A vector f i e l d

i f and

i s locally

X i s closed then

=

n u 1 =

=

.

=

Proof.

By Theorem

of $4.3,

is closed,

proves

is locally

J

If

.

+ XJ

Since

is closed then

so by ( a )

proves (b).

prove (c), notice

.

By Theorem

t

it is

By the definition of

.

.

of

so t h i s is also again,

=

=

Therefore, by (b) and

3.

in As we saw

coordinates

in paragraph 1, a coordinate Lie module must have an

even number of coordinates t o admit a symplectic structure. Theorem 3. 1

.

n

E

be a coordinate Lie

with coordinates

...+

= 2

is a symplectic structure.

If

h

is a

I t s components are given by the matrix

then

We have

and

g

If

Proof.

Everything e l s e is a

consequence of

which

is a t r i v i a l consequence of the definition of the wedge product.

The theorem of Darboux says t h a t on a manifold with a symplectic structure

choose l o c a l coordinates so t h a t

one

given locally by (1).

Thus all symplectic manifolds of a given

sion are l o c a l l y the same. variety of

is

This is i n strong contrast t o the great

non-isometric Riemannian manifolds.

Theorem

if

h

i s any

on a symplectic manifold preserves the

then the flow generated by the vector f i e l d s p l e c t i c structure.

Again t h i s i s i n strong contrast t o t h e

case, where there seldom e x i s t s a flow of are

more r i g i d than symplectic

Riemannian metrics

a distinguished affine connection, and the connection is

preserving an f i n i t e l y many

of

a Lie group (parameterized by structures do not

,

a distin-

guished a f f i n e connection, and t h e l o c a l

the

symplectic structure form a pseudogroup

by a

dynamics Let of

T ( M)

M be a manifold,

i t s cotangent bundle.

a cotangent vector

is

,...,qn

q1

that i f

T ( M)

a t some point

l o c a l coordinates

i

are

and

q of

M

,

q then

.

=

Now the

An element

and are i n f a c t a

on T ( M )

coordinate system, so t h a t i

i s a 1-form on

( M)

.

The 1-form

is well-defined globally ( i t

it has t h e in-

does not depend on the choice of l o c a l coordinates) variant description =

) T ( M)

is a tangent vector on M

at the point

is t h e projection sending each

the induced mapping of t h e tangent

Then

to =

and

(and

is a

, and

structure on T

n

1 q

l o c a l coordinates

is given by (1) with respect t o the

.

Thus the

bundle of an

a r b i t r a r y manifold admits a natural symplectic structure.

In

mechanics the configuration space of a mechanical

system is a manifold M and T (M) is the momentum phase space. If 1 q, are l o c a l coordinates on M are the conju-

...

...,

gate momenta.

The

structure

nates and t h e i r conjugate momenta.

knits together the coordi-

In

all transformations which preserve

mechanics one even i f the distinction between

coordinates and momenta is l o s t . system is a scalar

The energy of a c l a s s i c a l the momentum phase space. of the

.

Hamilton's equations

on

t h a t the

n by the flow with generator

system i s

By ( 3 ) t h i s means t h a t i n l o c a l coordinates

Abraham gave a course a t Princeton on t h i s subject l a s t year and I

no more.

and Sternberg's

is minus Abraham's

account of Poisson brackets

is twice Abraham's

Note:

.

Therefore the different

s l i g h t l y but all are such t h a t one local

obtains the

References R.

Hall

and J.

Foundations of Mechanics,

Sternberg,

Lectures on D i f f e r e n t i a Geometry, N . J .),

Mathematical Foundations of Quantum Mechanics, Benjamin

1963.

§9. Complex structures Complexification

1.

On an a r b i t r a r y differentiable manifold it makes sense t o con-

sider complex-valued scalars and tensors, covariant derivatives i n the direction of a oomplex-valued vector f i e l d , etc.

Algebraically, we do

t h i s as follows. be the algebra obtained by joining an element

Let 2 i = -1

with

no square root of

. (Thus - 1.) If

is a f i e l d i f and only i f is any

V

= V+iV

by X+iY = X-iY into

.

If u is an

; if

mapping of

X..

.

X

,

. i s again a commutative algebra with u n i t

E is an F module the

is a Lie module so is

Thus

extension, again denoted u

into

1

In particular,

; if

V

is an

module; i f

i s an affine connection on E

extension V is an affine connection on

2.

.

be

We define complex oonjugation on

it has a unique

V

mapping

over

.

i(X+iY)= -Y+iX

and

contains

vector space we l e t

module obtained by extending coefficients t o

the

to

i

E

its

.

Almost complex structures Definition.

An almost complex structure on an

i s an F-linear transformation J

of

E

i n t o i t s e l f such t h a t

E is f r e e of f i n i t e type with basis XI,.

If

an almost complex structure then the components of

and conversely such no scalar

f

F module E

J

..

= -1

and J

.

is

satisfy

give an almost complex structure.

in F such t h a t

= -1

If there is

then n must be even, since

COMPLEX STRUCTURES

(det

= det(-1) =

= det

,

It must be emphasized t h a t an almost complex s t r u c t u r e J very d i f f e r e n t from i . whereas

maps

i

extension

.

E

.

JZ = -iZ E

0,1

We define

(0,1)

Theorem 1. module

The transformation

0,1 E

The projections onto

Proof. i f and only i f (0,1)

1,0

. Let

module d i r e c t sum of

and

E

l,o

=

Z is of type

and

= Z

and let

.

1,0

= 0

, and t h a t

=

+

C

C

(F ,E )

=

.

PZ = Z Z is

i f and only i f

(F,E)

.

E 0,1

QED.

.

be an almost complex structure on the

be a derivation of

=

QED•

J

F

and -

E

The l a s t statement in t h e theorem i s obvious.

Let

in

are given by

= 1,

(1,0)

F

, those

J be an almost complex structure on the

It is c l e a r t h a t

Proof. for

(l,0)

to 1,0 -

.

.

Z in

Complex conjugation is b i j e c t i v e from E

Theorem 2. E

iJ = Ji

such t h a t

are s a i d t o be of type

Recall ($3.2) the notion of a derivation of

module

has a unique

J

Z in

t o be the s e t of

is the

respectively.

into i t s e l f

E

t o be the s e t of vector f i e l d s

1,0

.

E

of type

E

Elements of

of type

maps

be an almost complex structure on the

Let J

JZ = iZ and E

such t h a t

J

as i n the preceding paragraph, and

Definition. module

.

E into

to

J

The transformation

is

Then

, and

F

and similarly

COMPLEX STRUCTURES

Torsion of an almost complex structure

3.

Definition. module

.

E

Let

be an almost complex structure on the Lie

J

The torsion

T of

i s defined by

J

and the torsion tensor by

=

A complex structure

is an almost complex structure whose torsion is

on the Lie module E

We may rewrite (2) as Theorem 3. module

Let J be an almost complex structure on the Lie

with torsion

E

The module

-

+

E

T

.

Then T is antisymmetric and

J

i s a Lie module i f and only i f

1,0

is a complex

structure. For all X

Proof. so t h a t T

T

and

Y

By Theorem 2,

i s too.(and

consequently t h e ' t o r s i o n tensor i s a tensor).

and

,

so t h a t

1,0

T

is

X

by observing t h a t f o r 0

.

J

i s a complex structure then

, so

E

is a Lie module then so i s

E

verifying it f o r

give

If

are each always If

are

and

is c l e a r l y antisymmetric.

Lie modules.

,

in

and X

and E

are

E

.

= sinde 0,1 The formula (3) is most e a s i l y proved by

Y

ir.

and

and f o r

X

and

Y

in

E

and 0,1 Y of different types both sides of (3) E

1,0

QED.

Theorem

4. Let

E

be a coordinate Lie module with coordinates

x1,..., xn and l e t J be an almost complex structure on E with components

e.

Then the components of the torsion tensor T are given by

C0MPLEX STRUCTURES

Proof.

4.

QED.

The proof is t r i v i a l .

Complex structures i n l o c a l coordinates Theorem 5 . E be a coordinate Lie module with coordinates n n 1 .,x ,y and let J i n have components given by the matrix

,..

1 1

x ,y

Then J Then E

1,0 is

,..

1

=

xn+iyn

.

..., z , and

1 a coordinate Lie module with coordinates z ,

the basis dual t o

Proof.

x1

L e t

i s a complex structure.

...,dzn

i s given by

It is clear t h a t

J

i s an almost complex structure.

The elements (5) a r e simply P applied t o clear t h a t they are a basis of Lie module and J

E

1,0

'

...

, and

Since they commute,

E

0 (This also

is a

1,

is a complex structure by Theorem 3.

follows from Theorem 4, since the components of

it is

J

are constants. )

QED.

The Newlander-Nirenberg theorem a s s e r t s t h a t on a diiferentiable manifold with an almost complex structure J whose torsion i s

0

(i.e.,

a complex structure as we have defined i t ) one can choose l o c a l coordinates i n the neighborhood of any point so t h a t J has the above form.

This theorem is the j u s t i f i c a t i o n f o r calling an almost complex struct u r e with torsion theorem

0 a complex structure.

k contrast t o the Darboux

the Newlander-Nirenberg theorem i s quite d i f f i c u l t and

the r e s u l t was unknown f o r a long time.

Consequently terms such as

pseudocomplex structure and integrable almost complex structure are used i n many places f o r an almost complex structure with torsion

0

.

Once a manifold has a complex structure the theory of functions of several complex variables may be applied. When using c l a s s i c a l tensor notation when an almost complex structure is given, the convention is made t h a t Greek covariant indices represent vector f i e l d s of type

,

(1,0)

h a t Greek covariant indices

with a bar over them represent vector f i e l d s of type of a bar, a dot or a star i s sometimes used. tensor of the almost complex structure,

Thus i f =

(

0 T

.

Instead

is the torsion

.

5 . Almost complex connections If J

is an almost complex structure we l e t J be the tensor

.

in

given by

VJ =

i s called an almost complex connection.

requiring t h a t Theorem 6.

=

X ,J]

=

Let

affine connection on E E

.

If

Then

0 or E

An affine connection

such t h a t

This i s the same as

X ,P] = 0 f o r all vector f i e l d s X

.

be a Lie module such t h a t there e x i s t s an

, and l e t

J

be an almost complex structure on

is an affine connection on E l e t

is an almost complex connection and

i s an almost complex connection.

Let

=

i f and only i f

be i t s torsion and l e t

COMPLEX STRUCTURE

*

Then

i s an almost complex connection, and if

then the torsion of structure.

*

i s the torsion T of the almost complex

is a complex structure i f

J

f r e e almost complex connection. with torsion

is torsion-free

i s any almost complex connection

If

then the torsion

and only i f there is a torsion-

of the almost complex structure

T

i s given bv

+

Proof.

and by Theorem 2 t h i s is

*

so t h a t

is an affine connection.

connection.

They a r e almost complex connections since

clearly commute with P connection too,

If

are

torsion- free and l e t

T

, so

T

Now suppose t h a t =W

.

.

= Z

that

=

.

Suppose t h a t

.

For

Z

for

Z

and W of type

Similarly f o r

Z

and W of type

This term is annihilated by P have

and

is an almost complex connection

be the torsion of

, so

i s also an

is an almost complex

then

=

conversely if

and

then

.

Therefore

and W are of different types, say

Then =

-

-

-

+

so t h a t T

=

-

-

and W

of

we

. = Z

and

*

Therefore T = T if

is torsion-free. Since E has an affine

connection, it has a torsion-free affine connection has an almost complex connection

if

(95.6) and so

whose torsion is T

.

Therefore

is a complex structure there is a torsion-free almost complex

connection. Conversely, if

is a torsion-free almost complex con-

nection then V

into itself (this is clearly true for

almost complex connection) so that if Z

-

=

, and by Theorem 3,

also follows from the last

, the

-

so is

is a complex structure. This

right hand side of (6) is

of type

W

,

=

types then both sides of (6) are

6.

are in E

of the theorem, which we now prove.

For Z and W of type

and similarly for Z

W

. If .

Z and W are of

structures We have discussed pseudo-Riemannian

structures, and

almost symplectic

complex structures. We conclude our study of

tensor analysis by discussing

the interrelationships among these

three types of structure.

A pseudo-Riemannian metric is a bijective symmetric mapping

,

g: E

an almost symplectic structure is a bijective

metric mapping 9: E jective mapping J: E

F-linear).

An almost

and an almost complex structure is a bi-

E such that

=

(all

structure on a Lie module E is a

pseudo-Riemannian metric g and an almost complex structure that

=

is

being

such that

such

COMPLEX

Then

is an almost symplectic structure, since

n-'

jective with

.

=

Since

=

is clearly bi-

-1 , the relation(7) is

equivalent to

a manifold, the term almost Hermitian structure is usually reserved for the case that g is a Riemannian metric. Perhaps we should use the term almost pseudo-Hermitian structure, but we won't.)

We

also give

structure by means of a pseudo-Riemannian metric g

an almost

and an almost symplectic structure

is an almost

such that J =

complex structure, or by an almost symplectic structure complex structure J such that g =

=

and an almost

- 4 J is symmetric and con-

and indi-

sequently a pseudo-Riemannian metric. We

be

cate an almost Hermitian structure by

where g is

Riemannian, J is almost complex, is an almost

If

structure then

g,J,

is almost symplectic, and

=

structure and J is a complex

is called a

structure. There is no

name for an almost Hermitian structure in which

is a symplectic

structure. An

such that J is a

complex structure and

Hermitian structure

is a symplectic structure is called a

structure. Theorem 7.

be an

Hermitian structure,

the Riemannian connection, T the torsion of J

.

Z

J.

STRUCTURES

in

of the same type then

If Z

W

of opposite types then

Proof.

Let

Z

so t h a t (10) holds in t h i s case. holds in

Since

Since

Next observe

,

=

=

.

and W be of type

J

preserves types,

is torsion-free,

since

t h i s implies t h a t

-

.

=

That i s ,

Since

= Alt

is

, so

of t h i s is of type

.

.

, so

that the l e f t hand side

z

that

complex conjugates we see t h a t

By

and (12) also hold f o r

=

Z

and W

of type

.

and

w

126.

STRUCTURES

Now l e t = W

.

and W be of opposite types, say PZ = Z and

Z

Then

is as i n Theorem

where

6 and we have used (9).

it i s clear t h a t each

of Now

d i f f e r s from

that

=

C

F

commutes with

we need only show t h a t

are g-antisymmetric.

We

structure =

=

8.

structure.

,

so

=

show

(which are

=

=

= W ,

.

and

there i s an a f f i n e connection =

If

and

=

=

is a

Conversely, i f

Since

=

=

.

be an almost Hermitian structure,

the Riemannian connection, then

then

, to

and (10).

and

and

Let

the Riemannian connection,

Proof.

Z

= 0.

t h a t the l a s t p a r t of the proof shows t h a t i f

i s an

a

=

that

Since

and so

=

for

=

, so

J

and

But, by

.

and similarly f o r

such t h a t

.

+

by

the d e f i n i t i o n

.

and

=

=

=

structure,

, if

VJ =

. or

= 0

The Riemannian connection i s torsion- free,

i s a complex structure by Theorem 6 and

is a

COMPLEX STRUCTURES

structure by or

Therefore

is a

.

=

Conversely, l e t torsion

structure i f

T

of

be a

is

J

and

=

structure, so t h a t the

.

By Theorem 7,

and so

=

also. Complex projective space has a

metric.

t i v e algebraic v a r i e t i e s without

Complex projec-

are complex analytic

manifolds of complex projective space and so have an induced metric. =

Hodge theory was developed the operators

L and

mute with the

A

A

Since

and

com-

given by

operator.

and

f o r t h i s situation. A =

By the theorems of Hodge and

a c t on the r e a l cohomology

and place strong

r e s t r i c t i o n s on the r e a l cohomology of a non-singular complex projective algebraic variety (see

Also, i f we define

C

on

by

then

on

=

and

C

follows t h a t odd-dimensional

commutes with A since numbers of compact

=

.

manifolds

are even.

References Lichnerowicz, d'holonomie, Edizioni

Nazionale

Ricerche, Monografie Matematiche

Rome, Introduction

Paris,

des connexions e t des groupes

des