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

<. . 1

i

1

a

.

The c o e f f i c i e n t s i n t h i s expression ( c a l l e d the components of form o r simply t h e components of

so t h a t t h e components of

If

u

v

a r e given by

a as an r-form a r e r! times t h e components of

.

regarded as an element of If

as an r-

n

,

=

has components

v

has components

.

If

then

If u

in

has components

has

where the

1 if the j's

otherwise. If a!

then

are a permutation of the

and is

has components as an element of

given by

where the if the j's

if the j's

are an odd permutation of the

a! is an r-form and

If X

a!

k t

Then

.

,

are an even permutation of the

, and

is

otherwise. If

is an s-form then the

is an r-form, the

be another basis of E

has components

X J

a!

has components

, and define

I

14.

ALGEBRA

t h e components of

If

Proof.

with r e s p e c t t o t h e new b a s i s a r e

The proof i s trivial.

Notice t h a t the primed indices do not take values i n t h e s e t

..

..

but i n a d i s j o i n t s e t

of t h e same c a r d i n a l i t y .

This notation i s very convenient, a s it makes it impossible t o make a mistake i n w r i t i n g t h e transformation laws.

12.

C l a s s i c a l tensor notation of indices, t h e c l a s s i c a l tensor notation i s

Despite t h e

frequently q u i t e useful, e s p e c i a l l y i n computations involving contractions. The vector f i e l d s over a coordinate neighborhood i n a f i n i t e dimensional manifold a r e a f r e e module of f i n i t e type, but t h e module of f i e l d s does not in general have a b a s i s .

vector

it does, t h e manifold i s called

However, it i s possible t o use t h e c l a s s i c a l tensor notat i o n globally, without any choice of

coordinates, i f we make t h e f o l -

lowing conventions. Let

E

be

module, and l e t

u

.

Consider an expression

of t h e form

Instead of

...

are d i s t i n c t indices.

we may use any other

r+s

indices, provided they indices,

The upper indices are

t h e lower indices are

indices.

Next we suppose t h a t the

variant indices are covariant vector f i e l d s vector f i e l d s .

t h e covariant indices

Then we define (11)t o be t h e s c a l a r

MULTILINEAR

.

would perhaps be b e t t e r t o w r i t e

,

but we

Notice t h a t

although the indices axe required t o be d i s t i n c t indices, t h e mathematical objects they denote need not be d i s t i n c t .

(Thus we may have

vector f i e l d s although obviously

as

=

However, f o r an

we make t h e s p e c i a l convention t h a t

r-form

Now suppose t h a t tensor f i e l d s

Instead of

E

reflexive, so t h a t contractions of

is If

meaningful.

u

we define

we may use any other index, provided it i s d i s t i n c t from

the other indices occurring.

An index which occurs p r e c i s e l y twice, once

as an upper index and once as a lower index, i s c a l l e d a dummy index. Notice t h a t there i s no summation sign i n (13). being summed.

(When dealing with components with respect t o a b a s i s of a

f r e e module of f i n i t e type, we summations occur. ) all d i s t i n c t

contractions.

This i s because nothing i s

continue t o

We may have more than one each other and t h e

summation signs when index, provided they a r e indices, t o indicate repeated of

The notation i s unambiguous because, from t h e def

contraction, the order i n which t h e contractions a r e performed i s immaterial.

In all but the

Here axe some examples of the use of t h i s notation. f i r s t example we assume t h a t v

s

t

then

E

is

reflexive.

If

u

and

If

then

and

c

The notation here i s abusive. of two

then

=

The right hand side of (14) i s not the product

but is written instead of

i n t h i s abuse of notation.

We w i l l indulge r e s t r i c t i o n of

.

to

f o r a unique tensor

Since

,

in

be

The tensor

r!

=

then

and i f

perm permanent of a square

where perm denotes the permanent.

no minus

i s F-linear,

be the

explicitly, and one finds

.i . scalars i s

Now l e t

of

i n t h e same w a y as the determinant except t h a t there are

)

Similarly, i f

f o r a unique tensor

in

then

,

and

where d e t denotes t h e determinant.

If

and

then ( r e c a l l

k S

and i f

X

E

then

Tensor f i e l d s on manifolds Let

p

manifold

be a point i n the

M

.

i s an equivalence c l a s s of d i f f e r e n t i a b l e mappings = p

and

, where

x and y

x:

M

with

are equivalent i n case the coordinates of

.

d i f f e r by

A tangent vector at p

v e r i f i e s t h a t t h i s condition i s independ-

e n t of the choice of l o c a l coordinates, and t h a t addition and multiplication by constants

well-defined on tangent vectors.

gent vectors a t

p

at

p

.

forms a r e a l vector space

A cotangent vector a t

of d i f f e r e n t i a b l e mappings

are equivalent i f i n coordinates of

and

p

M

with

, where

=

d i f f e r by l i t t l e

.

c a l l e d t h e tangent space

i s t h e dual notion: an equivalence c l a s s

f: M

and q

p

Thus the s e t of all tan-

o

f

and g

of the difference

Again, t h e condition i s independent of t h e

choice of l o c a l coordinates, and t h e cotangent vectors form a vector space which i s i n a n a t u r a l way t h e dual vector space t o

s t r u c t u r e of They

manifold as

the s e t

* T (M)

M has a

of all

c a l l e d t h e tangent bundle and cotangent bundle.

equipped with n a t u r a l projections onto each vector t h e point bundle i s c a l l e d a

.

of a l l tangent vectors a t a l l points of

The s e t

uectors.

M P

p

M

They a r e

, the projections which assign t o

a t which it l i v e s .

A

s e c t i o n of t h e tangent

vector f i e l d o r vector f i e l d and a

ALGEBRA

1.

vector

P

Pictures of a tangent vector

.

and a cotangent P A tangent vector gives a direction and speed of X

motion, a cotangent vector i s a l i n e a r approximation t o a scalar.

The tangent vector

arrow twice as long,

2X would be indicated by an P would be indicated a relabeling

P of the hyperplanes (twice as dense). and

In the figure

X P look as i f they are i n some sense the same, but

P t h i s has no meaning unless the tangent space i s equipped

with additional structure, such as a pseudo-Riemannian metric or

structure.

section of the cotangent bundle is called a and

They form modules E

vector f i e l d o r 1-form.

over the algebra F of all scalars

real

Therefore we have the notions of tensor f i e l d s on M and

functions on M ) .

.

tensors a t a point

Tensors are of great

i n d i f f e r e n t i a l geometry because

defined geometrical objects (independent of

they are

dinate system) which l i v e a t points. order for an object t o be a tensor. fine a tensor

,

coor-

are necessary in

Both

Suppose f o r example we attempt t o de-

of

2, by requiring, i n l o c a l coordi-

nates, =

is 1 i f

where i s not

i = j

and

w

6

otherwise.

This l i v e s a t points but

defined, since i n new coordinates

x

it would

x

have components

a certain tensor. )

the components of

axe i n each coordinate system,

(On the other hand,

X be a fixed contravariant

As another example, l e t

vector f i e l d other than

and define

is the Lie product of

X

on E by

and Y

($2).

, where

=

This i s invariant- defined a t a point

but it does not l i v e a t points, because in order t o know p we need t o know something about d i f f e r e n t i a t e it.

In f a c t ,

,

=

Y

i n a neighborhood of

is

so t h a t

p

but not F-linear, since i s not a tensor f i e l d .

The condition of

F- linearity is in f a c t the condition t h a t an points.

If f o r example

, since

we

i s a 1-form write

i n order t o

=

X = Y P P with

object l i v e a t then 0

, and

so

The example of the Lie product shows t h a t not metrical objects axe tensors. second-order

Affine connections axe another

objects.

of

Tensor f i e l d s a r e first- order

objects since the notion of tangent vector

14.

Interesting geo-

one

Tensors and mappings algebras

Suppose we have

, and an

dual

homomorphism f o r any

module

and

with dual

, an

.

F module

E

We s h a l l use the word

the following: p:

an

algebra homomorphism; p:

a group homomorphism (and

where

and p: E

: F

;

for

a r e homomorphisms

the

a b i l i t y condition f (and

; and finally for

for P:

where the

p: F

,

p: E conditions

p:

satisfying

and

,

let

be a

.

mapping of the manifold M i n t o the

Then

F defined by

= f

i s a homomorphism.

tangent vector a t a point

p

what a

I f we

i n M is, we see t h a t

induces a vector

space homomorphism ( l i n e a r transformation)

It is called the d i f f e r e n t i a l of

at

. p .

duality,

I f we define

then

i s a homomorphism.

of the

,

In the same way we obtain a homomorphism

tensor algebras, which preserves the grading

and sends the

algebra

into A*

.

products

However, it is imme-

d i a t e l y clear t o

anyone t h a t we do not in general obtain a homomor-

phism

since

(X ) = (X ) P P do not a r i s e we

p.

of

whenever not get a

i s not necessarily onto, =

,

even i f these d i f f i c u l t i e s

section of

The mapping

not have

induces

(see Exercise maps

on

MULTILINEAR ALGEBRA

but

does not i n general induce a mapping on Suppose now t h a t

.

sections of

i s a diffeomorphism of

M

onto

.

For

in

Then we

obtain a homomorphism ( i n . f a c t , an isomorphism)

* as follows.

and

is as defined above.

we

define

This homomorphism extends i n a n a t u r a l way t o t h e mixed tensor algebras.

In the same way we obtain a homomorphism (22) i f

is an imbedding of

M

in

It i s unfortunate t h a t covariant tensor f i e l d s transform contraunder point mappings of manifolds, but it is too l a t e t o change t h e terminology.

Early geometers were more concerned with coordinate

changes than point mappings,

coordinates a r e s c a l a r s , which transform

t h e same way as covariant tensor f i e l d s .

*

Notice t h a t we have used t h e notation E

f o r covariant tensor

f i e l d s i n keeping with t h e f a c t t h a t they transform the opposite way t o point mappings. algebra

* A

For example, t h e cohomology r i n g is formed from t h e Grassand it i s universally denoted

*.

I n our study of tensor analysis we s h a l l make no use of points except a t one point i n t h e discussion of harmonic forms need t h e

notion.

where we w i l l -

MULTILINEAR Definition.

The

F module

E

i s punctual i f there e x i s t s a sepa-

r a t i n g family of homomorphisms of the form

where F = and E P P The module of

is a f i n i t e dimensional

vector space.

vector f i e l d s on a

at the point p and E

take

t o be P space a t p

P

.

is punctual:

t o be the tangent

References de Book 2,

Paris.

See especially

Chaps. 2 and 3. Differential Geometry

Academic Press, New York,

Symmetric Spaces,

Derivations on scalars 1.

Lie products A derivation of

F

i s an

,

f

$1.1f o r the assumptions on

If

and i f

X

and

i s in

h

F

then

X

D a

If

X

and

.

.) X+Y defined by

defined by

, so

=

then in

F

such t h a t

F i s an

F

.

It i s denoted If

F

Thus the s e t of a l l derivations of

i s also a derivation. module.

and

F

f,g

are derivations then so i s

Y

X: F

mapping

0

F

that

=

.

By

.

Y are i n

D

, we

define t h e i r Lie product

by

This i s again a derivation:

-

= =

+f

=

=

The s e t of a l l associative ring, and

-

.

+

mappings of D

F

i s a subset of it.

define the Lie product of any two elements

X

i n t o i t s e l f forms an

In any associative r i n g we and

Y

t o be

.

26.

DERIVATIONS ON

A simple computation shows t h a t the Jacobi i d e n t i t y

+

=

+

i n any associative ring.

Since

D

is

under the formation of

in it.

Lie products, the Jacobi i d e n t i t y the Lie product as multiplication

with respect t o

i s not i n

D

associative.)

The

Jacobi i d e n t i t y may be rewritten as

Define

on D

.

=

by

eXY

.

=

Then the Jacobi i d e n t i t y (2) i s

.

=

More generally, i f

...

Y

i s an i t e r a t e d Lie product of

p-th

(5)

of which X

X

and

Y

are i n

The Lie product i s D

i s an

satisfying (6) and

D

.

i s replaced by +

=

If

over

i s the sum of

associated i n any way, then by induction

terms, i n the

cation

n

For example,

.

+

then

so t h a t with the Lie product as algebra (not i n general associative). is

a Lie algebra, so t h a t

An

D i s a Lie

. However, the Lie product i s not

In f a c t ,

n

DERIVATIONS ON 2.

Lie modules i s an

A Lie

Definition.

F module

together with an

E

F-linear mapping X

of

E

i n t o derivations of

o f

into

E

such t h a t with respect t o it

algebra over

for all

in

F and X,Y

i s a Lie algebra over

in E . (since

=

E

the more general r e l a t i o n

We have the following. Theorem 1.

f = Xf

i s a Lie

E

and

Notice t h a t from (8) and the f a c t t h a t

holds.

F and a mapping

D

and

D

F

, with

i s a Lie module.

of i n t e r e s t i s the Lie module

of the algebra F of with the s e t of

.

=

The

be the module of all derivations of

E

of all derivations

functions on a manifold, which may be identified

contravariant vector f i e l d s .

module we did not assume t h a t the mapping X of a Lie module are a Lie algebra over s e t of all vector f i e l d s on Lie group over the algebra

manifold F of

I n the definition of Lie i s injective. when under some

F=

Other

, and the action of

invariant scalars.

Roughly speaking, a Lie module i s l i k e a Lie algebra except t h a t the elements of the module a c t on the coefficients by derivations in a natural way.

28.

DERIVATIONS ON SCALARS

Definition. of

f,

df

,

E

X

A Lie module

.

case there e x i s t scalars

are a basis f o r the

,

a scalar.

The differential

E

i s a 1-form

E i s called a coordinate Lie module i n

n .,x

(called coordinates) whose d i f f e r e n t i a l s

of 1-forms.

be a coordinate Lie module with coordinates

and consequently

therefore

f

modules

Definition.

Then

f

i s F-linear, the d i f f e r e n t i a l of

Coordinate

kt

be a Lie module,

i s defined by

Since

3.

Let

,

E =

reflexive (51.5).

1

x

n x

.

and

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

The

.. are a basis of

.

The dual basis, which i s a basis of

E

, is

denoted

so t h a t

t h a t the elements of the basis products are

commute

Lie

DERIVATIONS ON due t o the f a c t t h a t the

, for

symbol

..

x1,

n

are constants ( 1 or

and l e t

, which

i s given ( i n contrast t o 1 x ).

Thus i f

1 2

x ,x

i s simply the

axe coordinates so are x1'

even

=

f

=

.

be a scalar, l e t X be a vector f i e l d with components

Y be a vector f i e l d with components

.

Then df

has com-

ponents

i s t h e scalar

has components

and

If

x

the

example, has meaning only i f the e n t i r e coordinate

e n t i a l of the s c a l a r

Let

Notice

xn

axe also coordinates then the Jacobian matrix

and similarly for i t s inverse, so t h a t we have the familiar formulas

1

,

$2.

30.

DERIVATIONS ON

Vector f i e l d s and flows Let u s discuss h e u r i s t i c a l l y but i n some d e t a i l why vector f i e l d s on a manifold are t h e same as derivations of the algebra of of s c a l a r multiples, sums, and

scalars, and what t h e geometrical Lie products is. X

be a

vector f i e l d on a manifold

.

t h i n g being assumed t o be of c l a s s v e l o c i t i e s a t each point

p

Thus

of t h e manifold.

X

M, every-

is an

of

The

existence

theorem f o r d i f f e r e n t i a l equations shows how t o integrate t o obtain the of a p a r t i c l e s t a r t i n g a t least i f of

M

point.

In t h i s way we obtain ( a t

i s compact) a one-parameter family of

M onto i t s e l f such t h a t

and i n t h i s

.

=

We c a l l t h i s a flow,

we s h a l l ignore t h e f a c t t h a t i n

i s only defined l o c a l l y if

M

position of a p a r t i c l e a t time in

if

given by

f

at

X P t = O

t

i f it s t a r t s a t

. at

Thus

i s a scalar

=

the tangent vector

is not compact.

p

a t time

we have

so t h a t the derivative of X P

This gives t h e action of vector f i e l d s on s c a l a r s . r e c a l l i n g why t h i s i s a derivation:

= p

is a representative of

(see

isknownif

i s the

It i s perhaps worth

$2.

X

Conversely, i f the

i s a derivation on the

, where

+

31.

DERIVATIONS

l o c a l coordinates a t

p

f

and

i s allowed t o be

, determines

..

x1,

p

i s a point then n

f o r a s e t of

a curve whose equivalence class (the

tangent vector at p ) i s independent of the choice of l o c a l coordinates. Consequently we i d e n t i f y the vector f i e l d s on a

Notice t h a t the assumption t h a t every-

vations of t h e algebra of thing i s of

with the deri-

i s necessary f o r t h i s identification.

Now we s h a l l discuss the meaning of s c a l a r multiplication, addition, and Lie products i n terms of flows. Let

, let

generate t h e flow

X

generate the flow of time scale.

Y

.

Then

Y

h

be a scalar, and l e t

i s the same as

The new velocity i s

, so we

except f o r a change have ( l e t t i n g s

be the

new time

Thus

where

If

then

=

Now l e t If

X

and

= p

,

X generate if

Y

(locall y ) and

.

52

Y generate =

i s given by

The general case i s more complicated, and

Y

, and

then

X+Y generate and

DERIVATIONS ON

$2.

speaking, t h e flow and

Y

.

i s t h e simultaneous action of the flows

To see (13) formally, notice t h a t formally

*=e , since

=

Then the expansion of

i n powers of

t

, except

tends t o

i s the

i n powers of

e

f o r a f r a c t i o n of terms i n each order such t h a t the fraction as

.

n

The product formula (13) i n the very general spaces is due t o Trotter

on

s e t t i n g of Finally, l e t and

as the expansion of

X

generate the flow

generate the flow

.

,

Y

generate the flow Y ,

Then

To see (14) formally, we make a formal computation t o second order i n

I f we replace

.

+

=

2

t

e

by t

t h i s gives, +

;

DERIVATIONS ON SCALARS computation concerns t h e action of t h e flows on s c a l a r s , result i s

t o (14) f o r the point flows.

gives a proof of (14) f o r the case when X

and

96, 97 and

algebra of a Lie group (see pages

the

Helgason e s s e n t i a l l y Y

are i n t h e Lie

of

should

i n t h e general case o f Vector f i e l d s

not be d i f f i c u l t t o e s t a b l i s h on a manifold.

Consider a car.

We conclude with an example.

The configuration

space of a c a r is t h e four dimensional manifold

, where

parameterized by

a r e t h e Cartesian coordinates

of t h e center of the f r o n t axle, t h e angle which t h e

i s headed, and

measures the d i r e c t i o n i n

is t h e angle made by t h e f r o n t wheels

r e a l i s t i c a l l y , t h e configuration space i s the open

t i t h the

.)

submanifold

See Figure 2.

There a r e two distinguished vector f i e l d s , c a l l e d Steer and Drive, on

M

corresponding t o t h e two ways in which we can change t h e

configuration of a car.

Clearly Steer =

since i n t h e corresponding flow and

remain t h e

s t a r t i n g i n t h e configuration tance

h

8

a

changes a t a uniform r a t e while

To compute Drive, suppose t h a t t h e 8)

, moves

an i n f i n i t e s i m a l dis-

i n t h e d i r e c t i o n i n which the f r o n t wheels a r e pointing.

In

the notation of Figure 3,

Let

=

be t h e length of t h e t i e rod ( i f t h a t i s t h e name of t h e

DERIVATIONS ON

Figure 2.

A car

I

A

3.

A

motion

DERIVATIONS ON

thing connecting the front and r e a r axles).

Then

t i e rod does not change length ( i n readily seen t h a t

mechanics),

, and

=

BCD (which i s the increment i n same.

is

a

= h

since

It i s the angle

, while

h sin

remains the

.

us choose units so t h a t Drive =

too since the

=

a

+

+ sin

a

BY

a

[steer, Drive] =

Slide = - sin Rotate =

a

+

cos

a

a

+

a

Then the Lie product of Steer and Drive is equal t o Slide + Rotate on

, and

generates a flow which i s the simultaneous action of sliding motion is

and rotating. spot.

By formula

what i s needed t o get out of a t i g h t t h i s motion may be approximated

closely, even with a r e s t r i c t i o n a r b i t r a r i l y small, i n the following way:

with steer, drive, reverse steer,

reverse drive, s t e e r , drive, reverse s t e e r , . . so laborious is t h e

..

What makes the process

roots i n

us denote the Lie product (17) of Steer and Drive by Wriggle. Then further simple computations show t h a t we have the commutation relations [steer, Drive]

= Wriggle,

= -Drive,

[Wriggle, Drive] = Slide,

DERIVATIONS ON

Steer, Drive, and Wriggle i s zero.

and t h e commutator of Slide

Thus

t h e four vector f i e l d s span a four dimensional solvable Lie algebra over

. To get out of an extremely t i g h t parking spot, Wriggle i s insuff i c i e n t because it may produce too much rotation.

The l a s t commutation

r e l a t i o n shows, however, t h a t one may get out of an parking spot i n t h e following way:

tight

wriggle, drive, reverse wriggle ( t h i s

requires a cool head), reverse drive, wriggle, drive,.

.. .

The example i l l u s t r a t e s a phenomenon of frequent occurrence i n d i f f e r e n t i a l geometry,

holonomy, o r rather the lack of

The vector f i e l d s Steer and Drive, which a t f i r s t sight give the only over the scalars which i s not

possible motions of a car, span a

closed under t h e formation of Lie products.

That i s , the f i e l d of two

dimensional planes i n the tangent bundle is not integrable (not not holonomic) and so i s not the f i e l d of tangent planes t o a family of two dimensional surfaces.

Motions which a t f i r s t sight

0 impossible can i n f a c t be approximated a r b i t r a r i l y closely ( i n the C topology but not t h e

to p ology) by possible motions.

Reference F. Trotter,

On the product of semi-groups of operators,

ceedings of the American Mathematical Society

Pro-

37.

Derivations on tensors Algebra derivations Let ciative.

be an

K

algebra, not necessarily commutative or asso-

A derivation

X

i t s e l f such t h a t

i s an

K

mapping of

u and

f o r all

=

computation i n

X and

of

v

K in

into K

.

neither commutativity nor associativity, so i f

the

.

K so is

Y are derivations of

algebra of endomorphisms of

so the Jacobi i d e n t i t y holds.

The derivations l i e i n

K as an

vector space,

Thus the derivations of

K

form a Lie

.

algebra over

Now suppose t h a t

i s a graded algebra.

K

That i s ,

K

is the

weak d i r e c t sum

.

where each necessarily

.

An

geneous of degree

are usually but not

l i n e a r mapping X

a

of

K

K

into i t s e l f is homo-

, and homogeneous i f

for some

a

.

it i s

The notions of a bi-graded alge-

,

bi-homogeneous mappings of bi-degree

larly. of

with

a i f each

homogeneous of degree bra,

The

An antiderivation of a graded algebra

are defined simi-

i s an

K

mapping

into i t s e l f such t h a t

The

of

X

and

Y

is

+YX

.

A simple calculation

establishes the following theorem. Theorem 1. algebra

K

,

the anticommutator gree

a+b

.

X

Y

be antiderivations on the graded

of add degrees

a

i s a derivation of

b K

, homogeneous of

Then de-

DERIVATIONS ON TENSORS

2.

Module derivations Let

we defined t h e notion of a

and consequently we have the notion of an auto-

homomorphism of

.

of

Formally, l e t

of

be a one-parameter group of

and let

(For example, fold. )

In

E be an F module.

be the derivative of

may be

at

.

t

i s a flow on

where

By t h e product r u l e f o r d i f f e r e n t i a t i o n we obtain, formally,

, This motivates t h e following d e f i n i t i o n . Definition.

Let

i s a derivation such t h a t (1) holds.

E of

A derivation

F and an

A derivation

and an

of

F module.

be an

mapping

)

of

of E

E

is a derivation such t h a t in addition

mapping

, +

=

,

The motivation f o r (2) and (3) is again t h e product r u l e f o r d i f f e r e n t i a t i o n , as it i s of course t h e motivation f o r the d e f i n i t i o n of derivation of an

algebra.

find it convenient t o indicate

We

derivations of a number of d i f f e r e n t

F modules by t h e same symbol

This is legitimate, provided of course they all give t h e of

F

, since we may regard

and t h e various

.

derivation

as defined on t h e d i s j o i n t union of

F modules.

2.

E be an

F module and l e t

be a derivation

. (a)

has a unique extension t o a derivation

(F,

.

)

ON we extend

(b) in E

, where

K

to

39.

then

=

is the natural mapping of

E into

(c)

.

has a unique extension which i s a an

(d)

algebra, and

we extend

define

on

as an

algebra.

*

of

Proof.

E,

i s a derivation on

t o be a derivation on by additivity, then in

For all

We define

(3).

by

X

.

i s a derivation of

,

.

For all X

to in E

for

,

in

, so

that

=

=

The uniqueness assertion i n ( c ) i s clear since

as an

(such t h a t

be dropped from the f i r s t two terms of

K

with (3) we see t h a t

t h i s , and by

is a

The uniqueness i s clear.

it i s a derivation of

K

>

(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

<.

pax-

Thus the choice of any affine connec-

establishes a one-to-one correspondence between all affine

connections and

tensor f i e l d s contravariant of rank 1 and

of rank 2. Affine connections always e x i s t on The proof of the theorem shows t h a t i f

and i s a Lie

manifolds.

E is

over F then the s e t of all

reflexive

ine connection on

6.

Torsion Definition.

.

E

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

Let

The t o r s i o n of

i s t h e mapping T of

The t o r s i o n tensor of

i s t h e mapping T

E XE

of

into E

E'XEXE

defined

into

defined by

The t o r s i o n tensor of an a f f i n e connection is a

Theorem tensor. Proof.

Since

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

is F- bilinear i n X and Y

that

we need only show t h a t

by

t h e torsion, T maps

into

2

maps A'

into A An

.

since

is

in X

i n t h e preceding paragraph

i s t o t a l l y reflexive), but

E

need only show

.

But

E is t o t a l l y r e f l e x i v e we i d e n t i f y t h e t o r s i o n tensor and

If

sion

,

, we

so i t s dual =

T maps

.

The t o r -

into

(if

and by (10) it i s c l e a r t h a t

.

connection i s c a l l e d torsion- free i f i t s t o r s i o n is

i s sometimes c a l l e d symmetric, but t h i s term is generally

reserved f o r a torsion- free a f f i n e connection

such t h a t i n addition

where R

=

i s the curvature tensor.)

The most important class

of a f f i n e connections, the Riemannian connections Theorem 5.

,. ..

1

x

n

,

are torsion- free.

Let E be a coordinate Lie module with coordinates

and l e t

be an affine connection

Then the components of i t s torsion tensor

components

ij

T

Proof.

If

E

is merely a f r e e Lie module of f l n i t e type with basis

in F by

n we define the structure

.

=

Then (11) must be modified t o read =

Theorem 6. E

.

Then so

= V

k

.

V be an a f f i n e connection on the Lie module defined by X

and

-

.

The

+

=

,

connection

2 i s torsion- free. of

,

If =

T

.

i s the torsion of

i s the torsion

Every affine connection

on a Lie module E

can be written

uniquely in the form

where V is a torsion-free affine connection and T F-bilinear mapping of torsion

and

into E

EXE

.

be an affine connection with torsion

the definitions of

T

,

and

i s an affine connection.

,

=

The torsion of

1

connection, and i t s torsion is

, which

.

is

written i n the form

i s given by

the torsion of

V

.

plus

1

can be

It remains t o prove the uniqueness

is represented i n the form

a torsion- free affine connection and T EXE

into E

.

=

.

a skew-symmetric

Then the torsion of V is the

(which i s

is indeed the torsion of

7.

By

i s F-bilinear

Thus every affine connection

so suppose

of the decomposition

F-bilinear mapping of

T

.

V i s the torsion- free affine connection

(13) and T is the torsion of

torsion

and since

T

defined by (13) i s an affine

the second paragraph of Theorem 3,

(14) with

is the

t h i s decomposition T

i s given by (13).

Proof.

torsion of

is a skew-symmetric

, so

T

QED.

Torsion-free affine connections and the exterior derivative Theorem 7.

Lie module E on A

.

and l e t

be a torsion- free d

connection on the

be the exterior derivative.

Then d = A l t

Proof.

Define

on A

is t o t a l l y reflexive, so t h a t

E

by

A*

of

.

and

,

A

Suppose

i s generated as an

that algebra

is an

It i s t r i v i a l t o verify t h a t A

we need only verify t h a t

On scalars,

1- form.

.

by d =

d equals

d on scalars and

i s just the d i f f e r e n t i a l .

be a 1-form.

Then

so that,

V

If

is not t o t a l l y

E

rivative d

i s torsion-free,

r e c a l l t h a t the exterior de-

( 3 ) of

i s defined by

One v e r i f i e s t h a t

by d i r e c t computation. This is a useful theorem. t o be A l t

, but

The exterior

was defined

i s not a mapping of tensor f i e l d s into tensor

.

f i e l d s whereas

Curvature Definition. E

.

The

of mappings of

Let of

E

be an

connection on the Lie module

is the function R

into i t s e l f given by

from E X E

i n t o the s e t

The curvature tensor of

the mapping of

XE XE XE

into

F

given by

Theorem

The curvature tensor of an

connection i s a

tensor. Proof.

.

Since

I f we replace

-

by

-

we replace f +

i s in E

=

X

by

=

,

it i s F-linear i n

,

,

so (16) i s F-linear i n

in

Z

.

Similarly, i f

the coefficient is differentiated via

.

so (16) i s F-linear i n

Y

Since

as well, and so i s a tensor i n

If

above identification of the curvature

R

E is

E 2

into

.

reflexive the

with the curvature tensor is

as the identification

F-linear mappings of

3

vector f i e l d s i n the defi-

n i t i o n (16) of the curvature tensor.

of (where

with the s e t of F-linear mappings of s i t i o n of

clearly

the coefficient i s differentiated v i a

Notice the order of the

not the

(16)

1

with the s e t of

i n turn i s identified

E into i t s e l f ) but i s the compo-

with a permutation.

The definition (16) i s the

universally adopted convention. Theorem

..

and l e t

E be a coordinate Lie module with coordinates be an affine connection with components

The components of the curvature tensor

are given by

.

(18) i s not only

and the computation showing t h a t t h i s is the same t r i v i a l but easy. If

. ..

we

,

by

is merely a free Lie module of f i n i t e type

E

modify

by

and

by adding the term

a

where the

by replacing

basis

are the structure scalars defined by (12).

Af f ine connections on Lie algebras Let

E

be a Lie algebra over

paragraph 5), an affine connection

F = F

.

on E

i s the same as an F -bi-

0

linear map

B

of

If

G

i s a Lie group its Lie algebra

A s we have seen ( i n 0

into i t s e l f .

EXE

(over

identified with the s e t of left- invariant vector f i e l d s on connections

B

on

the same as

such

=

for

=

i n t e r e s t , for t h i s means t h a t each

for X,Y

X

in

in

may be G

.

Affine

(which i s

) are o f particular

invariant vector f i e l d X

is

and the geodesics (with respect t o the affine connection) issuing groups.

the i d e n t i t y of

G

are precisely the one-parameter sub-

The cases (the (-) connection),

= = =

1

(the (0) connection), (the (+) connection),

68.

$5.

DIFFERENTIATION

have been studied by E. connections whenever

E

They make sense as affine

and i s a Lie algebra over

F

.

From the definition (8) of torsion, the torsions of the above three affine connections are respectively curvature (15) of the

connection is

identity.

by fhe Jacobi

For the (0) connection the curvature i s given by

This is

again by the Jacobi identity. certain nilpotent Lie algebras. Since

f o r Abelian Lie algebras and

i s a scalar and the

since, by the Jacobi identity, connection i s

of any in E

t h a t is, the torsion is

is

,

is a derivation (cf .$2

We s h a l l not return t o t h i s subject.

The

.

Let us, as an exercise, compute

the Lie algebra case) we have, f o r any W

10.

The

since the connection i t s e l f

and the curvature of the (+) connection i s

is

.

0, and

and

Thus the =

.

See Helgason

identities Following

and torsion.

A s in

we prove some i d e n t i t i e s r e l a t i n g curvature denotes cyclic sums.

Theorem 10. with torsion

be an

T and curvature

R

E

connection on a Lie

.

Then, for

,

vector f i e l d s

the following i d e n t i t i e s hold:

, ,

= =

.

+

In particular, i f

,

+

=

is torsion-free

o, = Proof. (20).

We have already noted the trivia3 i d e n t i t i e s (19) and

To prove (21) and (22) we need

The relation (25) is an immediate consequence of the f a c t t h a t a derivation of the mixed tensor algebra. $3.2 as well (notice t h a t

is

For (26) we need Theorem 3,

i s not a Lie product of vector

f i e l d but a commutator of operators on vector f i e l d s ) . To prove

we use the definition (8) of torsion and (19) t o

find

W e use

t o re-express the f i r s t two terms on the r i g h t of (27) and

make the discovery t h a t

by the Jacobi i d e n t i t y and t h e d e f i n i t i o n

(15) of curvature.

Thus (21)

holds. use t h e d e f i n i t i o n (8) of t o r s i o n and

To prove

to

find =

-

+

=

.

Sum c y c l i c a l l y and use (26) t o obtain

By t h e d e f i n i t i o n (15) of curvature

t h e Jacobi identity,

-

The r e l a t i o n (24) (and sometimes (23)) f o r a torsion- free a f f i n e connection i s called

identity.

a f f i n e connection with curvature tensor

If

R then

i s a torsion- free

since these are merely

and (20) i n a d i f f e r e n t notation.

identity The i d e n t i t i e s we prove next the

of

of an

given i n $3.4, where A

module

Lie module

into

E

and

E

,

and

be used

If

i s in

we

i s an

i s an a f f i n e

on a

use the notations

(x)

it being understood t h a t X and X,Y

contravariant arguments i n again tensors i n E Theorem with torsion

and

curvature

mixed tensors v v u X Y

In

if

is

in Proof.

u

,

Y

xu

.

be an affine connection on the Lie module

Let T

are the f i r s t

Thus

s

and

E

mapping

R

-

.

Then f o r

vector

=

i s torsion- free then

. For given

X,Y

in

E

, let

[v

$3.2 t h i s is the derivation on the mixed tensor the derivation

it only on

-

on E

, so t o prove

induced by (31) we need

(where both sides axe 0 ) and vector f i e l d s (where

it i s t h e d e f i n i t i o n of curvature).

DIFFERENTIATION To prow

let

be in

in

.

Then

so t h a t , by the d e f i n i t i o n (8) of torsion,

If

This i s of writing

is torsion- free we may w r i t e (33) as

identity.

We emphasize again t h a t we follow t h e custom

turning

12.

Suppose we have an affine connection on the manifold

t

be a parameterized

C

and l e t

Y

Let

M

, where k y

Y

.

i n M with tangent vector

be a vector f i e l d on

ponents

M

.

I n l o c a l coordinates X

k

corn-

.

i s the k-th coordinate of

.

at

,

X

Then

at

has corn-

ponent s

That i s , we

C

need t o know Y off

points i n the direction of

C

k y (o!

I f we a t the

point

.

Y

since X

. as the nomponents of a tangent vector

p =

of

C

a well-posed i n i t i a l value problem. vectors

t o compute

and s e t (35) equal t o

, we

When we integrate we obtain

along C

=

and

Y

P

at

be the f i n a l point of the curve segment C

.

Then we obtain a

by

.

P

obtain a

i n t h i s way. along C

A

This mapping is

and invertible, and we

The mapping

is called

. case of ( 3 5 ) i s

along C

, which has

compo-

nents

This is called the acceleration vector.

Notice t h a t although the velocity

vector of a p a r t i c l e moving on a differentiable manifold makes sense,

$5. t h e r e i s no meaning t o the notion of acceleration vector unless we have additional structure such a s an a f f i n e connection.

is

Notice t h a t

connections having the same torsion- free p a r t (see

the same f o r 6), since

i s symmetric i n

we s t a r t a p a r t i c l e a t

p

a t time

i

j

.

with prescribed i n i t i a l velocity

, we

and required t h e acceleration (36) t o be

have a well-posed

i n i t i a l value problem f o r a second-order d i f f e r e n t i a l equation.

This

second-order d i f f e r e n t i a l equation has t h e s p e c i a l property t h a t i f we X by a constant k t h e p a r t i c l e t r a v e l s i n the same P jectory with velocity k times t h e previous velocity, and i s c a l l e d a

multiply

spray.

I n f a c t , t h i s is t h e geodesic spray of t h e a f f i n e connection

and t h e t r a j e c t o r i e s a r e called the geodesics of the a f f i n e connection. Affine connections with t h e same torsion- free p a r t give r i s e t o t h e same geodesic spray and t h e same geodesics.

Any spray i s t h e geodesic

spray of a unique torsion- free a f f i n e connection.

a manifold with law

an a f f i n e connection (or with a i s meaningful.

(The proper s e t t i n g f o r

= ma

dynamics i s a

p l e c t i c manifold ($8).) Suppose we have a torsion- free a f f i n e connection p

be a point i n M and l e t

on M

.

be two tangent vectors a t p P P Choose curves a t p with tangent vectors X Y and l e t and P be the points on these curves corresponding t o t h e parameter value (see Figure 4).

X

Parallel translate

Y to P t o obtain

parallel translate X to r P and q with these tangent vectors. and

are

at

p

,

is

t o obtain

,

Since at

end points (corres p ondin g t o t h e parameter

and

and choose curves a t i s torsion- free and

p

.

Therefore ($2.4) t h e t ) of t h e curves

.

DIFFERENTIATION

Figure

and Y

representing X

A parallelogram

are equal t o second order i n t

.

is

f a l s e i f t h e a f f i n e connection has torsion. Given a spray or a f f i n e connection on M

, let

be a tangent P vector a t p l e t a p a r t i c l e s t a r t a t p with velocity Y and P t r a v e l f o r u n i t time with zero acceleration. Its position a t u n i t time Y

so t h a t (the exponential mapping) i s a mapping i s denoted exp Y P ' of the tangent-space M i n t o the manifold M It i s always P defined l o c a l l y (and l o c a l l y i s a diffeomorphism) and i s sometimes well-

.

defined globally ( i n which case the spray o r a f f i n e connection i s called

kt

vectors

C

be a curve segment i n M s t a r t i n g a t

, and

be P the t r a n s l a t e of Y i n M along C t o . For YP P sufficiently the curves exp Y ( t ) sweep out a tubular P If i s torsion- free then the tangent vectors these hood of C X

.

given an a f f i n e connection

p with tangent

on M l e t

Y (t)

curves are, t o f i r s t order, obtained by in Figure

translation of

Now consider the affine connection with the same torsion-

f r e e p a r t but with torsion T

Since

,

=

k

1

.

Then i n (35) we add the term

i s a l i n e a r transformation acting

t o the direction of the curves around C

X

.

, and

exp

it twists them

This is torsion. consider curvature.

Now l e t

A Frame a t a point

p

in the

. The s e t of P p is the principal homogeneous space of the general

manifold M is simply a basis of the tangent space all frames

l i n e a r group

) )

element of

.

M

I f any frame i s singled out, there i s a unique

taking it into any given frame.

frames a t all points i s a principal f i b e r bundle over

.

group

The projection maps each

The s e t of M with s t r u c t u r a l

t o the point p

in

M

a t which it l i v e s . be an

Let M

.

Then

point p C

to q

i n t o frames a t the f i n a l point

, picks

station mobile. C

,

C

a curve segment on

t r a n s l a t i o n along C takes frames a t the initial q

.

The frame t r a v e l s along

up a tensor and brings it back t o p

or other purposes.

around

connection on M

It i s similar t o the repair truck which leaves the and i n f a c t the notion i s called the

tow back a If

C

is a closed loop a t p p a r a l l e l t r a n s l a t i o n

gives an automorphism of a group called the

Recall from

for d i f f e r e n t i a t i o n

M

P

, and

the s e t of such

group.

the geometrical meaning of the Lie product of

two vector f i e l d s X and Y

, which

be expressed by saying t h a t

Figure 5 .

A loop

Figure 5 i s a closed loop t o second order i n late a

.

trans-

I f we

around t h i s loop we w i l l find in general t h a t it has turned.

In fact, p a r a l l e l translation of a tangent vector

around the loop

gives t o second order i n =

.

This i s curvature.

K. Nomizu,

Lie Groups and Differential Geometry,

the Mathematical Society of Japan 2 , (1956).

of

Holonomy 1.

Principal f i b e r bundles The

a principal f i b e r bundle we have a

f i b e r bundle.

n:

of the bundle

B

is an example of a principal

bundle discussed i n

manifolds.

P

onto the base

Each f i b e r

space of a Lie group

G

p

, the

each f i b e r i s the same as

G

G acts

the whole bundle

P

, and

,

, where

P and

is a

B are

homogeneous

structure group of the bundle.

That is,

except t h a t it has forgotten which ele-

ment i s the i d e n t i t y ( c f . frames structure group

B

B

mapping

the general l i n e a r group).

The

translations on each f i b e r and therefore on t h i s action is

.

One can define the notion of connection in t h i s s e t t i n g and discuss

and t h e r e l a t i o n of connections t o reduc-

t i o n s of the structure group (see

The discussion is

simpler i f we confine ourselves e n t i r e l y t o bundle

P

.

The invariant scalars on P are isomorphic t o the algebra

scalars on B

of

objects on the

,

the invariant vector f i e l d s on P form a Lie

module over them. Each invariant vector f i e l d on P projects onto a vector f i e l d on B

, and

onto

.

of them (those which l i e along the f i b e r s ) project

We shall discuss t h i s s i t u a t i o n algebraically.

our r e s t r i c t i o n t o

Because of

objects, the terminology i s a b i t d i f f e r -

ent from the standard terminology

The usual treatment

com-

plicated by t h e notion of the connection form, a notion which we avoid. A t each point i n a principal f i b e r bundle we have a

linear

of the tangent space, called the v e r t i c a l space, con-

s i s t i n g of those tangent vectors tangent t o the f i b e r .

A connection is

Figure 6.

A horizontal curve

a Ginvariant choice of a

linear

a t each point,

The f i e l d of horizontal subspaces is not

called t h e horizontal space.

in general integrable, and t h i s is where t h e notion of

(see

A

l i e s in

in P

describes the

horizontal i f i t s tangent vector

is

horizontal space.

f i b e r in many points (see

enters

curve may cut a given

A

and the

of the connection

directions of motion which can be approximated

by horizontal

2.

Lie bundles collection of a l l Lie modules over

category i f we define a F-module homomorphism p: Q such t h a t

fixed F

a

of two Lie modules Q and P t o be

P

., an

of

Q

P

,

=

, A morphism i n t h i s category w i l l be called a Lie homomorphism.

An ideal V

i n a Lie module

is a

P

such t h a t XEP,

Z E V ,

and such t h a t =

If

P

ZEV,

is a Lie module then

P:

=

f o r all f

F)

is

and every ideal is contained in it.

clearly an If

(X

V

is an ideal i n the Lie module P then the quotient module

is i n the obvious way a Lie module, and the projection is a Lie homomorphism. kernel

V

i s an ideal, the image

of

induced mapping of

of a Lie module P

module and the inclusion

. We

For

R

X

let and

the curvature

H

Y

H

Then

K

i s a Lie

is a Lie homomorphism. H

a submodule (not necessarily a

be the projection of

in

.

P onto the module

define

the curvature of the submodule.

Theorem 1.

only i f

P

P be a Lie module and

Lie submodule),

the

i s called a Lie submodule

and Y a r e in

whenever X

in case

i s a Lie module,

is an isomorphism.

onto

A

kt

is any Lie homomorphism then the

p: P

If

P

H

be a

R

is a Lie submodule.

and

of the Lie and is

. and

The curvature i s F-linear in X

Proof.

Since

is

R

since

it i s also F-linear i n

Y

.

The l a s t

statement i n the theorem is obvious. Definition. modules.

The kernel V

ments of it

P

L:

a Lie bundle.

is an

of Lie

i s called the v e r t i c a l module

ele-

A reduction of a Lie bundle

P

i s a monomorphism of Lie modules and a: Q

B

A Lie bundle

t i o n with Q = B and a P

B

sum of

H

B

P

the identity.

is a submodule H

of

and the v e r t i c a l module

B

is trivial i f there is a A connection i n a Lie bundle

P such t h a t

V

.

is

P

is the module d i r e c t

The module H

in a connection

the horizontal module and elements of it are

i s also

The projections of respectively. into

B

diagram

where

a:

of

vertical.

is a

n: P

A Lie bundle

P

onto

are denoted

V

curvature of a connection is the mapping R

V given by

A connection is

.

in case i t s curvature is

L of a connection i s the smallest

of

The

P such t h a t

v

h of

HXH

and

Notice that we

identify the module

with t h e previous one.

the definition of curvature wish extend R t o a

This has the

B

of s e t t i n g

is

t o the kernel V of H

of

.

=

module V and l e t

.

is.

B

or

, since X

is called the

of

into

n: P

i f we

i s vertical.

it is then

is in B the element X

.

W e also def

by s e t t i n g

B be a

the

.

=

with v e r t i c a l

Suppose there is a connection with horizontal module H

L be

Then

We

that

i n a connection of the Lie bundle

B

curvature as a mapping of Theorem

, so

However, i f the connection is not

is not a Lie module whereas H such t h a t

if

=

as a module t o

n

V

PXP i n t o V by

of

The horizontal module H

n: P

with

,

module. L is contained in V

,

modules

L

where

is the

map and a

reduction of t h e Lie bundle

P

the modules

L

H+L

Lie

in

is the r e s t r i c t i o n of B

.

n,

is a connection in

B and i t s

H+L

.

module

A Lie bundle is

a f l a t connection.

if and only i f

Proof.

V s a t i s f i e s (1) and

By definition of the curvature,

since V is an i d e a l (being the kernel of a ) it s a t i s f i e s (2). is the smallest module satisfying (1) and

V

.

i s spanned a s a module by the s e t of

module

The

so

But L

all vector f i e l d s of the form

z

,

=

since each

is an F-linear

Let

of elements of the form (3).

Z be given by

be of the

and l e t

By the Jacobi identity,

as an induction hypothesis on m t h a t f o r all W of the form (4) and all Z of the form ( 3 ) for any value of i s in

.

The r e l a t i o n ( 5 )

is

in

, the

t h a t if

hypothesis is t r u e f o r m i f it is t r u e f o r show t h a t

n

Lie the induction

.

L we need only show t h a t

to

is in L

.

By t h e definition of

R

,

=

.

+

horizontal, so t h a t fore we need only

.

Since

.

Therefore

L is contained in V

of modules.

, the

shows t h a t it

in H+L

argument shows t h a t

L

, and

.

+

(6) applied t o

The

H+L is a Lie

and

Z in

f

i f we l e t

H+L then a: H+L

and the

, gives

I : H+L

that

is a connection in a: H+L

P

a

L is contained

since

be the r e s t r i c t i o n of

B is a Lie bundle and, together with the in-

clusion

L

is in L

Therefore

i n the i d e a l V ).

is again

Then

L is an ideal i n H+L ( i f we notice i n addition

for

=

.

module H+L is a d i r e c t

L

L is a Lie module,

H

Z

L i s a Lie module.

.

and

H

is in L by the definition of

to

,

H,

hand side is in L

that

L

There-

But by the Jacobi identity

and the

Since

.

that

, is i n L

i s in L

a reduction of

P

B

B and t h a t i t s

.

It i s module

. i f the connection is f l a t then

s a t i s f i e s (1)

(2).

Then H

isomorphism of Lie modules, and

is a Lie module, o

B

a: H is a

L

since B is

of

Lie modules,

that

i s t r i v i a l we

take as our connection H the image of

the

3.

B i s trivial. Conversely, i f n: P

P

and H

i s a f l a t connection.

B

B

i n P in

QED.

The r e l a t i o n between the two notions of connection On a manifold we have the notion of an affine connection in the

($5.1) and the notion of a connection in the

sense of

bundle, and the notions have a close a f f i n i t y . correspondence between

There i s a one-to-one

vector f i e l d s on the frame

bundle and derivations ($3.2) of the module of vector fields, and t h i s motivates the following construction. Theorem 3. and l e t

P

define

B be the Lie module of

in B

t o be the r e s t r i c t i o n of

a: P

V

i s the s e t of

derivations of

Let

be a connection i n a: P

into H which takes X

( i n the sense of

B i s a Lie bundle.

H

and

i s the l i f t of

With respect

B

X X

.

The v e r t i c a l module

.

B

F

Then the mapping of

is an affine connection

i n t o i t s lift

of

,

which are

Conversely, if V

the s e t

.

an affine connection

is a connection in

curvature R

P

as a mapping B X B

is i d e n t i c a l with the curvature of the a f f i n e connection Proof.

the s e t

P

of

Theorem 1, $2 derivations of

B

F

in P

-

i s a Lie module and

H

F

=

P

V

.

be the module of all derivations of

t o the operations

B

derivations of

i s indeed a Lie module. i s an

. $3.2

Lie algebra and it

i s readily seen t o be a Lie module, and

B

is

a Lie

bundle, with v e r t i c a l module as described. be a connection, and define

X

.

Then X

F-linear and each

is an

so

t o be the l i f t

connection.

nection the s e t

H

of

is a derivation of

Conversely, i f is a

is

the sum of

element of

in V

then is a connection.

is the l i f t of

.

.

, but

X

+

of

is

i f we l e t

V

Equally

By definition

X

V

con-

, since

of

We need t o show t h a t any derivation

is

of

X = =

, so =X

is horizontal and since

,

we may identify the two notions of con-

nection. The

a mapping of

BXB =

Since of

=

.

,

into V

i s by definition

-

the horizontal p a r t of

. is the

That i s ,

QED.

$7. 1.

Riemannian metrics

Pseudo-Riemannian metrics Let

E

be

F module and l e t

If

X

is

vector f i e l d we l e t

rank 2.

be a

tensor of be the 1-form def ined

.

=

The tensor

is called non-degenerate in case the mapping u: E

is bijective.

If

of

is non-degenerate

modules, so t h a t E i s reflexive.

also a mapping of

The inverse mapping of

u: E

, so t h a t

E

contra- iant tensor Definition.

.

enjoy

only i f

The inverse =

F module

,

Riemannian

g

f o r all vector f i e l d s

points

p

such t h a t

#

X

.

n

.

Riemannian metrics

t h e same.

and g

nents by g

E is

of f i n i t e type

is a pseudo-Riemannian metric we denote i t s those of

E is

(and consequently

Riemannian metrics, but the formal algebraic properties, which we

F

.

is a pseudo-Riemannian

geometrical and analytic properties not shared by

If the

.

tensor of rank

>

consider now,

=

determines t h e

A pseudo-Riemannian metric on an

M

is

non-degenerate is denoted

of rank 2 given by

On a manifold

at

if

for

a symmetric non-degenerate

metric such t h a t

Therefore the dual

is

and

E to

is an

u: E

by

, so

that

RIEMANNIAN

the

a r e given e x p l i c i t l y i n

but

2.

the

complicated

by the

f o r the inverse of a matrix.

The Riemannian connection Theorem 1.

module E

.

such t h a t

Let

g be a pseudo-Riemannian metric on a Lie

Then there is a unique torsion- free

.

=

F i r s t we prove uniqueness. fields.

Since

connection V

Since

we have

=

Let

, so

=

X,Y

,

and

be vector

that

is torsion-free,

where (2) and (3) a r e obtained from (1) by cyclic permutation. t r a c t (2) from (1)plus (3).

We obtain

The r i g h t hand side of (4) does not involve for

z

on X

.

Since

Now sub-

g

, so we have

i s non-degenerate

X

a

is

is uniquely determined, so V i s unique. To prove existence, consider the

is t i a t e d via

in

X

, for

i f we replace

X

hand side of

by

then

f

is differen-

side of (4) i s a 1-form

Therefore i f we fix Y and Z the r i g h t

w and we Z

, for

define

t o be

i f we replace

Z by

i f w e replace

Y

by

1

.

then f

is differentiated via

then f

Then

is

in

is differentiated via

so that +

Thus

is an

Let side of (1).

Therefore (5)

.

connection. be the

hand side of (1)

W e have seen that (4) is equivalent t o

and by cyclic

and

=

.

That is,

.

T be the torsion of

We also have = By

and

+

+

the r i g h t hand

METRICS

We need t o show that both sides of the common value.

The

u

.

are

is symmetric in the f

the l e f t hand side of ( 9 ) ) and

u =

.

and since g

=

Thus

The connection o r the

of Theorem connection.

Theorem 2.

Therefore

is non-degenerate,

T=O

.

It is due t o Christoffel.

g be a pseudo-Riemannian metric on a

Riemannian connection

xn

.

Then the components of the

are given by

(4) t o

x

=

Y =

.

a we obtain (10).

we multiply by E

two variables

is called the Riemannian

1 Lie module E with coordinates x

If

denote

in the f i r s t and

variables (by t h e r i g h t hand side of (9)).

and

Let

is merely f r e e of

type we obtain three additional

terms involving the structure scalars due t o t h e f a c t t h a t the vector not commute.

f i e l d s of the

The discovery of t h e Christoffel symbols (10) and of differentiation

a

discovery.

Despite the f a c t t h a t tensor

analysis has been studied so intensively t h a t it has become easy it remains an amazing

87. 3.

METRICS

Raising and lowering indices g be a pseudo-Riemannian metric on the

kt

let

tensor of rank r .

u be a

a tensor

v

, contravariant

module E

.

kt

of rank 1 and

we define

of rank

by

In the c l a s s i c a l tensor notation (81.12) t h i s is indicated by

(where i =

and i = X

passable c l a r i t y .

for

p).

This is a notation of

Notice t h a t it i s important t o leave a space below

the raised index t o indicate precisely which tensor in sponding t o the given tensor in

i s meant.

corre-

Notice t h a t we

extend the notation t o allow several raised indices. when working with a fixed pseudo-Riemannian metric

It is g t o write

tensors with the upper and lower indices ordered r e l a t i v e t o each other the

t o indicate the r e l a t i o n and lowering indices.

If

E

tensors formed by r a i s i n g

is

reflexive, so t h a t contractions

of tensors make sense, we continue t o indicate contractions by the use of

indices as before.

4.

The

tensor kt

g be a pseudo-Riemannian

the Riemannian connection and R

,

of rank

called the

on the Lie module E

i t s curvature.

,

We define a tensor tensor by

Thus

where R

is the curvature tensor.

In the c l a s s i c a l tensor notation we denote the values of the curvature tensor by

Thus the

tensor i s obtained by lowering t h e contra-

variant index

i

, and i t s

values are denoted

t h a t no t i l d e is necessary.) tensor see pp. Theorem 3. module E

.

of

be a pseudo-Riemannian metric on

Then the

Proof. tensor of

Therefore

g

For t h e geometric meaning of the

Lie

tensor s a t i s f i e s

We have already seen ($5.10) t h a t i f

R

is the

connection then

hold.

The equation (13) may be re-written as

RIEMANNIAN

95

is a skew-symmetric operator

(The equation (15) says t h a t each with respect t o

at

t=

g

, which means,

formally, t h a t it is the derivative

)

of a one-parameter group of

-

=

we

, obtaining

to

-

(The l e f t hand side is

by

To prove

+

+

since

i s a scalar.)

f i r s t term on the r i g h t hand side is

since

=

.

The

Therefore (13)

holds.

and

The equation (14) is a consequence of ( l l ) , follows

.

By (12

,

since each l i n e is

.

By (13) the top raw

so t h a t

s i x terms are symmetric about the main

.

+ By (12) the f i r s t two terms of

add up t o

+

with the and (13) the remaining

diagonal of the remaining nine terms.

Thus

as

J

so t h a t

= 0 .

holds. Let us suppose t h a t

may be contracted.

By

E

R

is t o t a l l y reflexive, so t h a t tensors i

=

.

This property i s not shared

RIEMANNIAN by

affine connections : it

is

t h a t the trace of each

.

Consider however

.

=

This is a symmetric (by

and

the Ricci

in

The corresponding tensor

is called the Ricci tensor.

1

danger of

is called the s c a l a r curvature.

I t s trace

wish t o

Since the Ricci curvature is

be written as

metric, the Ricci tensor fusion.

tensor of rank 2,

indices t h e Ricci tensor (17)

For those who

be

as

a notation of almost magical inefficiency.

5 . The codifferential Let V be the Riemannian connection with respect t o a pseudoRiemannian metric

g on a t o t a l l y reflexive Lie module E

.

If

a!

an r-form we define

and

if

=

that for

a!

a!

in

The (r-1)-form called co-closed if

is a 0-form.

the convention

of $1.12,

,

i s called the codifferential of =

, co-exact

i f f o r some

, and

a!

we have

is = a!

.

$7. The operator

6

97.

is called the codifferential.

by

It was introduced in

t o whom the following theorem and proof are due. Theorem 4.

6 be the codifferential, with respect t o a

pseudo-Riemannian metric

g on a t o t a l l y reflexive

Lie

E

.

Then

. identity

By

Now r a i s e the indices The

and

and contract with a and b 2 hand side of (19) gives 6 , the next term gives

2 -6 (1 due t o the

of

ba axe a reason.

a i n b and

.

tensors, the next two terms give

Since f o r the

Therefore

Cyclic permutations of

by

k

and

and a

so t h a t (20) i s

0

leave

.

unchanged, but

and

Modern treatments (see

give

t h i s theorem by use of the operator

.

much shorter proof of

However, the introduction of

leads t o needless complications r e l a t e d t o orientability. the above definition of

and proof t h a t

2

Furthermore,

generalize t o i n f i n i t e

dimensional pseudo-Riemannian manifolds (and Could conceivably be of i n t e r e s t ) , provided one pays attention t o convergence problems when taking contractions.

6.

There i s no

on an

manifold.

Divergences g be a pseudo-Riemannian

Let

F module E

.

W e define

for

on a t o t a l l y reflexive u and

in

s

by

(where we have not indicated r e l a t i v e order of contravariant and variant indices). that

A s with an r-form a

with raised indices i s

r!

i t s e l f , we make the convention

times what it would be i f regarded

simply as a tensor, and we make the convention t h a t f o r

and

in

,

If

u

define

where

and

and we extend

by a d d i t i v i t y i n each variable t o all of the

mixed tensor algebra.

Similarly

define

f o r a r b i t r a r y ex-

.

terior If

is the Riemannian connection with respect t o a

Riemannian metric on a t o t a l l y reflexive Lie

and

is a

METRICS

we

1

,

the divergence of

=

99 scalar of t h i s form

is a divergence.

Theorem 5 .

be the codifferential, with respect t o a

pseudo-Riemannian metric

reflexive Lie module

on a

Then

f o r all exterior forms

is a divergence. Proof.

.

It suffices t o prove t h i s f o r

us write

f

g in case

f-g

i n A'

is a divergence.

and

in

Recall t h a t

since the Riemannian connection i s torsion-free, we have ($5.7)

Therefore

This theorem Notice t h a t quite apart

that

6

is the

d

.

any question of notational conventions, it

is necessary t o define the g-inner product on of

operator t o

by weighting forms

degrees d i f f e r e n t l y i n order f o r t h i s t o be the case.

7.

The

operator If

f

.

is a scalar so is

on

operator, and the same terminology

scalars i s called the be used f o r the operator for

The operator

on

tensors.

=

.

Notice that

a

f

=

It is natural t o study the operator

forms, and t h i s

f i r s t done by

Unfortunately, the wrong

sign convention has been adopted i n tegrals and one r e f e r s t o The

on exterior

recent accounts of harmonic in-

as the Laplacean!

operator i n i t s various manifestations is the most

beautiful and central object in

of mathematics.

Probability theory,

mathematical physics, Fourier analysis, p a r t i a l d i f f e r e n t i a l equations, differential

the theory of Lie

all revolve around

t h i s sun, and its l i g h t even penetrates such theory and algebraic geometry.

regions as number

with pain do I adopt the sign con-

vention which is standard in the theory of Definition.

kt

reflexive Lie module.

forms.

g be a pseudo-Riemannian metric on a

The

operator

A

*

on A

is

fined by

An exterior form Notice t h a t

is harmonic in case A maps each

.

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

i f and only i f each of i t s homogeneous components i s . set

of

forms is a graded

geneous subspaces are denoted

r

.

vector space, and i t s

is The

8.

The

formula Theorem 6.

A

be the

operator, with respect

t o a pseudo-Riemannian metric on a t o t a l l y reflexive Lie module. r-forms

For

,

+ V=l

a

r

V

the formulas (18) and (21) f o r d and 6

,

and

so t h a t

I f we use

identity

and the f a c t t h a t

we see t h a t t h i s i s equal t o

=

a

.

v b a V

i s alternating,

$7.

RIEMANNIAN

c l a s s i c a l tensor notation i s a convenient and f l e x i b l e computational device, but it i s not immediately clear what the meaning of the additional terms i n the formula, l e t

formula is. 2

be the tensor i n

given by r a i s i n g the second co-

variant index i n the curvature tensor ; t h a t is, operator

To reformulate the

R

(so t h a t

has values and r e c a l l the

=

of Theorem 7.

A

be the

with respect

t o a pseudo-Riemannian metric on a t o t a l l y reflexive Lie module. all exterior forms

a!

,

Compare

.

+

=

Proof.

For

of $3.4 with (22) and r e c a l l t h a t

Operators commuting with the Laplacean

+

From the d e f i n i t i o n A = and the f a c t t h a t with

.

Let

where

A

,

it i s c l e a r t h a t

operator d

and 6 commute

Here we s h a l l investigate F-linear mappings of

which commute with

ping of

2 2 d = 6 =

of the

*

E

A

.

be a t o t a l l y reflexive

*.

into A

Let

L

i s the component in A

* A

F-linear mapping of there is a tensor

L

into A

A

in

a! S

into A

of

.

L an F- linear map-

F module,

be the component i n a!

.

Then

Now l e t

such t h a t

L

L= L

of i s a l s o an BY

,

.j .i

.i Since

for a l l

La!

a!

,

in

.

t h e tensor

L

i s alternating i n

i t s covariant indices, and t h e r e i s no l o s s of g e n e r a l i t y i n assuming

it t o be a l t e r n a t i n g i n i t s F- linear mappings of

.

algebra A,

A'

into

for

=

d i r e c t sum.)

i n t h e contravariant

i d e n t i f y F- linear mappings of

*.

with the elements of

If

is a Lie module,

E

L an F- linear mapping of

t o mean t h a t f o r each tensor

L

, into i t s e l f

in

into i t s e l f

A

an a f f i n e con-

A

we define

we have

Recall t h e d e f i n i t i o n a t t h e end of tual

.

s u f f i c i e n t l y large, t h i s i s t h e same a s t h e weak

Then we

nection, and

s

be t h e strong d i r e c t sum of t h e

A,

Let

t h e s e t of

may be i d e n t i f i e d with

is t h e elements of rank

where

if

indices.

.

of the notion of a punc-

F module. Theorem 8.

A be the

operator, with respect

t o a pseudo-Riemannian metric on a t o t a l l y r e f l e x i v e punctual Lie module E

.

L

be an F- linear mapping of the e x t e r i o r forms

with Then

derivative L commutes with

and

X

=

for

A

(with respect t o t h e Riemannian connection).

.

It suffices t o prove t h i s f o r

Proof.

[V

A

A*

L: A vector f i e l d s

.

for X

.

L= L

By

Therefore

of $3.2,

If

#

for

s

and

a

.

both

with

R

Consequently,

.

operator L

is also in A"'

then

,

so t h a t

L

with the

Theorem 7, we need only show t h a t

. is torsion-free, we have by

Since the Riemannian connection $5.11 t h a t

= vX vY

YX

of $3.2 again,

so t h a t by Theorem

with each

Suppose there i s an

is not

0

.

Then there i s a

such t h a t the scalar ive.

Since

where W

E

7

such t h a t

7

in

i s not

( i n fact, we

, since

E

homomorphism

is t o t a l l y reflex-

p

, such

induces a homomorphism

*

mixed tensor algebra =

in

i s punctual, there is a homomorphism

i s a finite-dimensional vector space over

.

take

into the mixed tensor algebra W,

.

.

Therefore,

of the

, and

Consequently we need only

prove t h a t =

p

that

.

$7. V be the vector

Let

Since

RIEMANNIAN

of

,

commutes with each

commutes with

and so For each

, so

is i n V

..,en

vector

such t h a t the f i r s t

and l e t

..

< fi @

is

if

j>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