Ordinary differential equations on vector bundles and chronological calculus

ORDINARY DIFFERENTIAL EQUATIONS ON VECTOR BUNDLES AND CHRONOLOGICAL CALCULUS R. V. Gamrelidze, A. A. Agrachev and S. A. Vakhrameev

UDC 514.7;517.977.1

The elements of the calculus of flows defined by vector fields on smooth manifolds and vector bundles are presented. Applications of the calculus to some problems in differential geometry are considered.

In this paper we present the elements of the calculus of flows defined by vector fields on smooth manifolds, possibly equipped with additional structures. The material is the outcome of development of ideas of earlier papers [2], [3], [4], [9]. The principal tools of investigation here are the variation formula and the general existence theorem for flows in finitely generated modules (see Sec. 3, subsection 3.3 and Sec. 3, subsection 3.4). Here, however, we shall develop an extremely convenient functional approach for the description and study of various kinds of geometrical objects on a smooth manifold. A certain nonstandardnotation will be used in this paper - its meaning will be explained in Secs. 1-2. I.

Vector Fields and Diffeomorphisms

An ordinary differential equation on a manifold is defined by the designation of a time-dependent vector field. In this section we shall present a functional description, without the use of local coordinates, of the basic objects relating to an equation: vector fields, differential forms, diffeomorphisms, and also points of a manifold. We begin with the latter. i.i. Homomorphisms of the Algebra C=(M) into R and Points of a Manifold M. Let M be a smooth n-dimensional manifold (the term "smooth" will always mean infinite differentiability; the manifold is assumed to have a countable base). The basic object associated with a manifold M is the R-algebra C~(M) of all smooth real-valued functions on the manifold. This algebra is commutative, associative and contains an identity - the function e(x), x 9 M, identically equal to unity on M. Essentially, C~(M) contains all the information about M. The detailed meaning of this assertion will be the content of Propositions i.i and 2.1. Let x 9 M be some point of M.

It defines a map x:C~(M) + R,

x:a a(x), which is R-linear and multiplicative:

x(kai~b)=kx(a)q-~x(b), (ab) = x (a) x (b) Va, b~C ~ (~I), ~, ~ R .

.....

....

It turns out that the converse is also true: any linear functional on C~(M) other than zero which possesses the multiplicative property is generated by some point of M. To be precise: Proposition i.i. If ~:C~(M) + R is a nonzero homomorphism of the ring C~(M) into the real line R, then $ is generated by a uniquely determined point of M: there exists a unique point x 9 M satisfying the condition

(a)=Jc(a)=a(x)

Va6C~(M).

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 35, pp. 3,107, 1989.

0090-4104/91/5504-1777512.50 9 1991 Plenum Publishing Corporation

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Proof. We first observe that any ring homomorphism C=(M) + R is an R -linear map, that is, a homomorphism of algebras. Therefore the kernel k e r ~ of a (nonzero) homomorphism is a maximal ideal in the ring Cm(M). Indeed, suppose there is an ideal I ~ k e r $ and let a e I, but a # ker $; then $ ( a ) # By the R - l i n e a r i t y of $, we have b = a - $(a) e ker$, and therefore $(a) = a - b e I, is a maximal ideal in the ring C=(M).

0.

Let I x c C~(M) denote the maximal ideal of functions that vanish at x; obviously, I x = kerx. Let C0~(M) be the ideal in C~(M) consisting of all functions with finite support. The proof of the proposition is based on the following assertions (a)-(b). a) If I is a maximal

ideal of Cm(M) not containing

C0~(M),

then I = I x for some point

xeM. To prove this, consider the annihilator of I, that is, the set A n n I = {x e MIa(x ) = 0 Va e i}. Since I is maximal, it follows at once that A n n I is either the singleton of x or the empty set. In the first case, clearly, I c I x and since I is maximal we have I = I x . The second case is impossible, because, as we shall now show, if A n n I = ~ , then I D C0~(M), contrary to assumption. Let Ann I = ~5, then V x e M, there exists a n e i g h b o r h o o d U x of x such that if the support suppa of a function a e C~(M) is a subset of Ux, then a e I. Indeed, take a function a' e I that does not vanish at x, and let U x be a neighborhood of x in which a' ~ 0. If s u p p a c Ux, then the function a" defined by the conditions: a" = a' on U x, a" = 0 off U x is smooth. Consequently, a e I, since a' e I and a = a'a" e I. Now let a 0 e C0~(M). Cover the compact set s u p p a 0 by a finite number of neighborhoods of the above type and let {a I . . . . . as} be a partition of unity on s u p p a 0 subordinate to

the

cover.

Then

a0--~a 0

2 2 ai=

i =1

is

b ) I f ~ : C ~ ( M ) -~ R i s also not zero.

aoaiEI ,

because

a0a i e I,

i = 1,

...,

s.

i =l

a nonzero

ring

homomorphism,

then

the

restriction

of

~ to

C0~(M)

Indeed, suppose the contrary: M is not compact and the nonzero homomorphism ~ vanishes identically on C 0 ~ ( M ) . Consider a function a e C~(M) e a c h o f w h o s e l e v e l s e t s {x e M l a ( x ) < N}, N = 1, 2 . . . . . is compact. In other words, a is greater than any prescribed number outside a suitable compact set. The existence of such a function will be proved later. Then the function a - ~(a) also satisfies the above condition; consequently, we c a n a d d a f u n c t i o n a 0 with compact support such that the function b = a - ~(a) + a 0 is positive everywhere. Therefore ~ ( b ) # 0 , f o r i t f o l l o w s f r o m g # 0 o n C~(M) t h a t i = ~ ( e ) = ~ ( 1 / b - b ) = ~(1/b)-~(b). On t h e o t h e r h a n d , g ( b ) -- ~ ( a - ~ ( a ) + a 0) = $ ( a 0) = 0. This contradiction proves the assertion of part (b). It remains to prove the existence of a function a, all of whose level sets are compact. o0 Consider an open countable locally finite cover {Ul}i= I of M such that all the U i are compact. Let {Yi}i=l = be a cover of M by compact sets with F i c U i. (For example, define F i to be suppbi, where {bi}i=1 ~ is a partition of unity on M subordinate to the cover {Ui}). Let a i be a smooth function The formula

on M with support

a i c Ui,

identically

equal to i on F i.

oo

i=! N

defines

a smooth function which meets

our demands,

since outside

the compact

set

[3 supp a i

it is -> N + i. Assertions (a) and (b) together at once imply the truth of Proposition i.i. Indeed, if ~ is a nonzero homomorphism C~(M) + R , then by (b) it does not vanish on C0~(M); thus the maximal ideal k e r $ does not contain C 0 ~ ( M ) and therefore, by (a), it has the form k e r ~ = I x = kerx. It follows that ~ = x, since V,a e C~(M) the function a - x(a) is a member of kerx, hence also of ker$, and so ~(a - x(a)) = O, or $(a) = x(a). This completes

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the proof of Proposition

i.i.

1.2. Smooth Maps and Diffeomorphisms. The categories Mn~, MnDiff and C=, CDiff ~. Throughout the sequel, "algebra homomorphism" will always mean a ring homomorphism which preserves the identity element. The category of all smooth manifolds will be denoted by Mn~; the morphisms of this category are the smooth maps f Mor (M, N ) = { f IM-+ N}. The subcategory MnDiff c Mn~ contains the same objects as Mn~ but its morphisms are the diffeomorphisms: Mor (M, N ) = D i f f (M, N). Every smooth map f:M ~ N induces an algebra homomorphism f*

C ~ (M) +-C ~ (N), defined by L

f ' b = bofGC~176 (M) VbfCoo(N). If M f N g L, then

(gof ) * =

f*og* :Coo(L)-+ Coo(M)a.d(idM)* -----idcoo(M),

and if f is a diffeomorphism, then f* is an algebra isomorphism and

(f-l)*

=

f*-I.

Thus, the correspondences

,:M~C~(M),

f~f*:C~(N)~COO(M)

d e f i n e a f u n c t o r from Mn~ t o t h e c a t e g o r y C~ whose o b j e c t s a r e t h e a l g e b r a s o f (sm oot h ) f u n c t i o n s on a r b i t r a r y smooth m a n i f o l d s , i t s morphisms a r b i t r a r y a l g e b r a homomorphisms. The s u b c a t e g o r y C D i f f ~ i s d e f i n e d by a n a l o g y w i t h MnDiff: i t s o b j e c t s a r e t h o s e o f C~, i t s morphisms a r b i t r a r y a l g e b r a i s o m o r p h i s m s . The f u n c t o r * i s n a t u r a l l y c a l l e d t h e s u b s t i t u t i o n f u n c t o r , s i n c e t o e v a l u a t e a f u n c t i o n a = f * b e C~(M), b e C~(N), one must s u b s t i t u t e t h e map f i n t o b. Proposition 2.1.

An a r b i t r a r y

a l g e b r a homomorphism

Coo (M) <-C (N) uniquely determines a smooth map

F~=f:M-+N, such that ----f*, Fidc~(M) =idM. ~ is an isomorphism if and only if f is a diffeomorphism, and in that case

~-,= (f-~)*=t*-~. Proof. Let x:C~(M) + R be the homomorphism corresponding to the point x e M (see Sec. i.i). Then the composition x~p:COO(N)-+R is an algebra homomorphism, since it maps the identity function e(y)------l'intothe identity 1 e R. Therefore, by Proposition i.I, there is a uniquely determined point y such that y = ~o~. Obviously, the map x ~ f(x) = y satisfies the condition f* = ~, moreover, it is smooth, Since iVb e C~(M), by assumption, f*b = @(b) e Cm(M). If ~ is an isomorphism, the analogous construction for i~-i determines a map g:N + M for which g* = ~-l , and so

(gof)*

=

f*og* ____~o~-I__=idc~(M), (fog)* idc~(N), =

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or

gof =idM, /og----ldN. Conversely,

if f is a diffeomorphism,

then 9 is an isomorphism.

It follows from this proposition that the correspondences

r : C~(M)~M, ~ r = / , where

C=(M)~-C~176

M---+N,

define a functor from C = to Mn~ or, respectively, F are clearly inverses of one another, since

ro* =

IdMn~ ( = |dMnD,~f),

~----/*

from CDiff = to MnDiff.

*oF-----Idc=( =

The functors * and

Idc~,f),

where Id is the identity functor on the corresponding category. 1.3. Derivations and Vector Fields. We briefly recall some familiar definitions and simple properties related to the tangent bundle of a smooth manifold. A derivation of the algebra C~(M) is an product differentiation rule

R-linear map X:C~(M) + C~(M), satisfying the

X(ab),= (Xa)b+aXb V ~ b~C | (M). The set of all derivations of C~(M) will be denoted by D e r M . called (smooth) vector fields on the smooth manifold M.

Its elements will also be

If multiplication is defined to be commutation of vector fields, D e r M natural structure of a real Lie algebra:

assumes the

[X, Y~=XoY--YoX, 27, Y~DerM, and with the natural operations of addition of vector fields and multiplication fields by smooth functions we a l s o o b t a i n the structure of a C~(M)-module.

of vector

A tangent vector or a derivation at a point x 9 M to a manifold M is an R -linear functional Tx:C=(M) ~ ~ satisfying the "infinitesimal multplicativity" condition at x (Leibniz' rule):

T~(ab)'=(x~a)b(x)nt-a(x)~b

V ~ b~C ~ (M).

The value of a vector field X at a point x is the composition xoX = Xx:C=(M) + Ri,* which, as is easy to see, is a derivation at x. Conversely, suppose that at each point x e M we have a tangent vector ~x' To each function a e C~(M) we can associate the function b(x) = ~x a, x 9 M. If a 9 C~(M) the function b is smooth, then the correspondence X:a ~ b is R -linear and satisfies the product differentiation rule; hence X e D e r M and the value of the vector field X at x is precisely ~x" The set of all tangent vectors at a point x is called the tangent space to M __at x and denoted by TxM. It may be endowed with a natural real vector space structure, which is easily seen to be n-dimensional (where n is the dimension of M). The support of a vector field X 9 D e r M , such that X x ~ 0.

suppX,

is the closure of the set of all x 9 M

The disjoint union of tangent spaces

TM---- [_] TxM xEm *It omit will mean

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Would be more pedantic to the symbol ^ in the notation not cause confusion: writing that it is being treated for

xoX = Xx; from t h i s m o m e n t on, however, we s h a l l for the homomorphism corresponding to a point x; this the point x to the left of a function or an operator will the present as a functional.

with the natural smoothness (see also Sec. 2) is a 2n-dimensional manifold, known as the tangent bundle of M. Its (smooth) sections are the vector fields (see Sec. 2). A morphism F of tangent bundles F

TM-~TN

is any smooth map of the manifolds TM into TN mapping each fiber TxM linearly into a fiber TyN. It is not hard to see (see also Sec. 2) that the correspondence x + y = f(x) thus created is a smooth map of M into N. In this case we speak of a morphism F over f and write

~=O(F). Clearly, the tangent bundles over arbitrary smooth manifolds and all possible morphisms among them (with the natural composition law) form a category. The isomorphisms in this category are the morphisms which are diffeomorphisms. An isomorphism F:TM + TM is called an automorphism of the tangent bundle if

O(F) = i d a . The category of all tangent bundles is denoted by TMn~. The subcategory TMnDiff c TMn~ has the same objects as the whole category; its morphisms are only those morphisms F of TMn~ such that 8(F) is a diffeomorphism. Let f:M + N be an arbitrary smooth map. the linear map

The differential of f at a point x e M is

Txf:TxlW-+Tt(x)N, t defined by the formula

T . f (Vx)--rxof*, The following formulas are obvious:

"rx6TxM.

if M i N ~ L, then

Tx~of)=Ttcx)goTxf, TxidM--=idr~M. The family Txf, x e M, defines a smooth fibered map (linear on fibers) of tangent bundles

T f :TM-+ TN,

T ftrx M.-.-Txf ,

satisfying the conditions

T (go/) = TgoTf,

T idM =idrM.

Thus, the correspondences

ivI~TM,

f,--,.T f

define a covariant functor from Mn= to TMn~, with 8(Tf) = f. This correspondence is also a functor from MnDiff to TMDiff , and the morphism Tf may be regarded as extension of f from M to the entire tangent bundle TM. If f:M ~ N is a diffeomorphism, then p = (f-z)*:C~(M) + C~(N) is an algebra isomorphism and there is a well-defined map A d p : D e r M + Der N,

X~Y=AdpX=poXop-~Der N, XEDerM, carrying the C~(M)-module Der M into the C~(N)-module Der N. conditions

It clearly satisfies the

Ad p (al Xi + a2X2) = ( p a l ) Ad pXl + (pa2) Ad pX2, A d p - I = ( A d p ) -1 Ad (plop2)= Ad pxoAd P2, Ad (idcoo(M))= idD~M .

%The differential of a map f:M + N at a point x is usually denoted by f,,x or Dxf.

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The first two equalities show that A d p is a semi-isomorphism of the C=(M)-module Der M onto the C~(N)-module Der N over the algebra isomorphism p = (f-l), _ see Sec. 2.2. The second two equalities state that p + A d p is a functor, provided that Ad is regarded as an extension of the algebra isomorphis p:C~(M) ~ C=(N) to a semi-isomorphism of the modules D e r M and D e r N over these algebras. By analogy with the formula 8(Tf) = f, we write

0 (Ad p ) - - p = (f-~)*.

|

k In its turn, A d p can be extended from D e r M to any tensor power | M by the formula

= DerM|

(Adp|174174174174174 k

T h i s e x t e n s i o n i s c l e a r l y a s e m i - i s o m o r p h i s m o v e r p o f t h e C~(M)-module | k C~(N)-module | N, and so

M into the

O(Adp|174 The map Ad p = Ad (f-l), may be expressed in terms of the differential Tf as follows. Let X be an arbitrary vector field in DerM, Y = Ad (f-1)*X 9 DerN. Then

T~[ ()Px) = r m ) = / ( x ) oAd ([-') *X. All the concepts and constructions considered in this subsection carry over automatically to arbitrary open sets U c M, considered as smooth manifolds with the smoothness induced by the canonical embedding i:U § M. This remark will be used in the sequel. For example, if Y e Der U, and s u p p Y is compact, then, letting a 9 Ca(M) be a function with support in U and equal to unity on suppY, we can define a vector field X = a Y e DerM, whose values coincide with those of Y on U and vanish off suppY. 1.4. Dual Language - Differential Forms. module D e r M into the algebra Ca(M) by

Denote the C=(M)-homontorphisms of the C~(M) -

Der*M = H o m (Der M, C | (M)) = H o m Der M. The set Der*M is endowed naturally with the structure of a C~(M)-module and called the dual of the module DerM; its elements are called differential forms (of the first degree) or simply forms. The result of applying a form ~ to a vector field X e D e r M is denoted by

<~,X>~C~(M), ~ D e r * M , X~DerM. I t i s e a s y t o s e e t h a t i f a f i e l d X v a n i s h e s a t a p o i n t x, t h e n V~ <m, X>[x = O. We may t h e r e f o r e s p e a k o f t h e v a l u e ~x o f a f o r m ~ a t a p o i n t x , d e f i n i n g i t by t h e f o r m u l a < ~ , X~> = <~,X>I~ VX~DerM. Consequently, the value ~x is an element of the dual T*xM of the tangent space TxM. The value ~x may be expressed as a composition xo~, since

xo~ (X) = x < o , X> =<~, X> I~The support of a form ~ is defined as the closure of the set of all x such that w x # 0. The vector space T*xM is known as the cotangent space to M at x, and its elements are called covectors. The disjoint union

T*M = t_J T~M x6m with the natural smoothness of a 2n-dimensional manifold g e n t b u n d l e o f M. Thus, every form ~ determines a section in T'M,

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(see Sec.

2) i s c a l l e d

the cotan-

Obviously, a section x ~ a ~ x e Tx*M in T*M determines a form ~ e Der*M by the formula

< ~, X > 1~= < ~, X~ > if and only if

'

X e D e r M the map x ~ <~x, Xx> is smooth.

It is also easy to prove that any point x e M can be surrounded by a neighborhood U x so small that DerUx, Der*U x are free C=(Ux)-modules of rank b and if X I . . . . . X n is a basis in Der Ux, then the forms ~i, ..., oh] in Der*U x defined by <~i, Xj>=61j= form a dual basis in Der*U x. TM, T*M are locally trivial.

0, i=/=j, e, i----j,

This assertion is equivalent to the statement that the bundles

The dual module to Der*M is canonically isomorphic to Der M. The canonical injection D e r M c (Der*M)* is obvious, since there exists a vector field X with any prescribed value of Xx, whatever the point x. The proof that this is in fact a surjection is slightly more complicated in this context. However, this isomorphism is a special case of a more general isomorphism between finitely generated projective C=(M)-modules ~ and ~ **, to be established in Sec. 2. The differential da of a function a e C~(M) is the form in Der*M defined by

=Xa

VXODerM.

Obviously, the map

d : C|

a~da,

is R - l i n e a r and satisfies Leibniz' rule

d(ab) = (da)b+adb Va, b6C| The value of the form da at a point x e M was defined previously as the composition xo(da) = (xod)(a), and so

xoda(Xx) =x
X> = x ( X a ) =

(xoX) (a) =Xxa.

-

" ..........

On t h e o t h e r h a n d , t h e f u n c t i o n a i s a s m o o t h map M ~ R , a n d we h a v e d e f i n e d its differential at x by the formula Txa(X x) = Xxoa*. It is easy to see that these definitions are compatible if any number ~ e R is identified with the derivation ~ d/d t on R. The forms

E btdat,

aibt6C~(~t)

l

form a submodule in Der*M, generated by the differentials da, a e C~(M). that this submodule is just Der*M. In other words:

It i s easily seen

Proposition 4.1. Let (x I . . . . , xn+k):M +-R n+k be a smooth embedding of M into R n+k Then the differentials dx I . . . . . dx n+k generate the module Der*M. Consequently, Der*M is a finitely generated C~(M)-module, and every form ~ e Der*M can be expressed as

~---~ aidx ~,

at6C~ (~,:l).

i=I

Suppose now that we are given a smooth map f:M § N and let Tf:TM + TN be the corresponding tangent space map. We define the map adjoint to Tf as the map (TF)* of the C~(N)-module Der*N into the C~(M)-module Der*M

(Tf)* = T * f : Der* N - ~ Der* M defined by

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w~T*f(v)=~fiDer*~l, vEDer*N, < T * f (v), X > Ix= < ~r(x), T x / ( X x ) = <~r(x), T / ( X A >. It is easy to see that T*f is a semihomomorphism over f* from Der*N into Der*M, which we can also write as

which s t a t e s t h a t t h e f o l l o w i n g diagram i s commutative: Txg

rzf

-

rftxlN

The map T*f may also be considered as an extension of the adjoint map f* of algebras to a semihomomorphism of the corresponding modules over f*. Therefore T*f = (Tf)* is also usually denoted by f* and we obtain

[*(db),=d(f*b),

b~C|

which states that the form of the first differential is "invariant under a change of varibles" or, equivalently, that :the differential d commutes with the "substitution operation" f*. Of course, we might also have defined T*f by the formula

T*/: dO~d (f'b), ~ a, (db~)~,~/* (a~)d (f*bO, i

ai, bi, briCk(N). That this definition is legitimate is readily verified, and then our original definition of T*f becomes the following proposition: if v = Zaidbi, then

<

a,ao,, x > i

(/*o,).

(/*aO < a ( / % ) , x > i

i

I f f is a diffeomorphism, so that p = f* is an isomorphism, then the adjoint map (Tf)* = T*f may clearly be defined by

<(Tf)%,X>=p<~,Adp-~X>. Therefore, if f is a diffeomorphism, then (Tf)* may also be regarded as the adjoint of Adp -z relative to the pairing operation <., .> and we can write

<(Adp-l)%,X>=p. The map (Adp-1) * is an extension of the algebra isomorphism p-Z:C~(M) ~ C~(N) to a semi-isomorphism of the C~(N)-modul~ of forms Der*N on N onto the C~(M)-module Der*M of forms on M. By analogy with Adp, the map (Adp)* also extends in a natural way to tensor powers of the appropriate module 2.

Vector Bundles and Finitely Generated Projective Modules

In Sec. 1 we briefly considered the tangent and cotangent bundles TM, T*M of a smooth manifold M and the C~(M)-modules of their sections - the module of vector fields D e r M and the module of first-order differential forms Der*M, together with the corresponding maps. Each of these categories yielded a purely functorial construction of the other three. The theory of differential equations on a smooth manifold is particularly easy to construct using a "functional language" - in terms of the modules of sections of TM and T*M and semihomorphisms between them. However, it then becomes necessary to consider more general modules than D e r M and Der*M. We have therefore devoted this section to a survey cf the necessary generalizations - some information about vector bundles in general, finitely generated projective C~(M)-modules, and the intimate connection between them. 2.1.

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Vector Bundles.

Categories Bd~, BdDiff.

We recall the basic definitions.

A prebundle over M is a family of vector spaces E x, x e M. The disjoint union E = u E x of these spaces is known as the total spac e of the prebundle; the set E x is the fiber

x~M

over x, and the map

pre=-pr:UEx~M,

pr(Ex)=x VxEM

x6M is projection onto the base M.

A section of the prebundle is any map

s : M-+E,

s (x) oEx

Vx~M

or pros = idM. Every family Z of sections generates a certain C~(M)-module of sections, consisting of all the sections

ats~, a~EC=(M), s~6E. i

A prebundle E is called a vector bundle over M if E is endowed with a smoothness (hence also a topology) such that the local triviality condition is fulfilled: V x e M and for some neighborhood U x there exists a diffeomorphism $: L J Ey = pr-1(Ux ) + U x • R k, under Y~Ux

which every fiber Ey is mapped linearly onto the corresponding fiber y • R k of the direct product. The diffeomorphism ~ will be called a trivializing diffeomorphism and the neighborhoods U x trivializing neighborhoods of the bundle E. Clearly, the dimension of a fiber is constant for every connected component of M, and a vector bundle E with n-dimensional base and k-dimensional fibers is a smooth (k + n)dimensional manifold; the corresponding projection pr is a submersion of E onto M (i.e., a smooth map of maximum rank at each point). Let i:M + E denote the smooth regular embedding (canonical embedding) taking a point x e M to the zero of the corresponding fiber Ex; we shall identify M with i(M) c E. A section in a vector bundle E is any section of the prebundle which is at the same time a smooth map. The set oE of all (smooth) sections of a vector bundle has the natural structure of a C=(M)-module. In fact, let sl, s 2 be two smooth sections and al, a 2 e C~(M). We shall prove that any point x has a neighborhood in which the section als I + azs 2 is a smooth map. Let U x be a trivializing neighborhood of x, g the corresponding diffeomorphism. Since ~ is linear on fibers, it follows that on U x

Be ( als t:-[~a2s2) = a l ( Bos l ) + a 2 ( ~os2) , where the right-hand side is a smooth map of U x into U x X R ~. Since $-1 is a diffeomorphism, it follows that als I + a=s 2 = ~-lo(a1$os I + a=~os 2) is a smooth map of U x into E. The module of sections oE uniquely determines the smooth structure C~(E) of the total space, by virtue of the following. Proposition i.I. A function a is a member of the algebra C~(E) if and only if, for any fixed sections sl, ..., s k e oE, where k is the dimension of a fiber, the map ~:M • R k ~ R , defined by

(x, X~. . . . .

Xk)~a

~s~ (x) i~l

is smooth. Proof.

Let a e C~(E).

Express the map ~ as a composition

(x, X~. . . . . X g , - ~ X~s~(x),-~a t~l

;Usi (x) . l=l

By local triviality, ~ is a smooth map, and so the composition ~ = ao~ is also smooth.

1785

Now let a be a real-valued function on E~such that the map D : aov = ~*a is smooth for any choice of sections Sl, ..., s k e oE. We shall prove that in some neighborhood of an arbitrary point z e E the function a is smooth, so that a ~ C=(E). Let pr(z) = x and let U x be a trivializing neighborhood, $ the corresponding trivializing diffeomorphism. In U x X R ~ take k (smooth) sections ri, .... r k with compact support which are linearly independent at each point of a certain neighborhood V c U x of x. Then E contains k sections s I . . . . , s k such that ~~ = ri" Since r1(y) . . . . rk(Y) are linearly independent, the map

(V, ~1. . . . . X k ) ~ S l ( V ) + . . . + ~ k S ~ ( y )

=~-~oCX~r,CY)+... +~s~(v)),

v~V, (~ ..... X~)~R~

is a diffeomorphism of V x R ~ onto W = pr-l(V) = y~v[1 Ey.

By assumption,

the map (y, X z, ...,

%k ~ ao~(y, A l, ..., %k), y E V, is smooth, and so aov = v*(alw) ~ C~ (V • R~). an isomorphism of C=(W) onto C=(u • ~ ) , of z ~ E); therefore a e C~(E).

But ~* is

and so alw is a smooth function (for any choice

This proves the proposition. Thus

the prebundle E =

[J

E x is converted into a vector bundle by a suitable choice

of a module of C~(M)-sections in it, thus defining a smoothness ology). We shall use this remark in what follows.

in E (together with a top-

A morphism of a vector bundle A = x~]~ A x over M into a vector bundle B = ~

By over N

is defined to be a smooth map F:A + B which maps an arbitrary fiber A x linearly into a fiber Bf(x) (the fibers A x and By may be of different dimensions). The corresponding base map f:M + N is also smooth, sznce it can be expressed as a composition of smooth maps: f = PrBoFoi M. The category of all vector bundles over arbitrary smooth manifolds and will be denoted by Bd=. The isomorphisms of this category are the morphisms morphisms; the corresponding base map i s a l s o a diffeomorphism. Since f is mined by F, we shall write f = ~(F) and speak of a morphism F over f, or of from the base to the bundle. El

their morphisms which are diffeouniquely deteran extension F

P~

If A + B + C are morphisms, then 8(F2oF I) = e(F2)o6(FI), and moreover 0(id A) = id M. An isomorphism F:A + A will be called an autom0rphism if 8(F) = id M. We define the subcategory BdDiff c Bd= as the category whose objects are those of Bd~ and whose morphisms are just the morphisms F ~ Bd= for which 8(F) is a diffeomorphism. Finally, the category Bd M consists of all possible vector bundles over a fixed base M; its morphisms are the morphisms between such bundles. We let Hkm denote the Grassmann manifold - the set of all k-dimensional subspaces of R m with the natural smoothness. A map M + Hk m, x ~ E x c Hk m, x e M, is smooth if and only if, for any given point x e M, there exist maps s I . . . . . s k ~ C (M, Rm), such that sz(y), .... st(Y) ~ Ey V,y e M and the vectors st(x) . . . . . st(x) are linearly independent. Let us assume that in each fiber E x of a vector bundle E = x~M ~ E x, d i m E x = k, we have a subspace of fixed dimension: A x c Ex, d i m A x = I <_ k. Then we can speak of A x as smoothly dependent on x ~ M, because if x, y lie in one trivializing neighborhood of the bundle then, using the appropriate trivializing diffeomorphism, we may assume that Ax, A y e H ~ . Our previous arguments imply that the map x ~ A x, x e M is smooth if and only if, for every point x, there exist (smooth) sections sz(x) . . . . . sz ~ oE, such that sz(y) . . . . . st(y ) ~ AyVy ~ M and the vectors sl(x) . . . . . sl(x) are linearly independent. By declaring the C~(M)-module of smooth sections of the prebundle A = U

A x to be all

sections s E oE satisfying the condition V x e M, we endow A with the smoothness induced on A c E from E. This makes A a vector bundle over M, which we call a subbundle of E.

1786

E - =[...]E x are

zf

two vector

9

x~M

bundles

over M, we can define another

vector

xEM

bundle over M in a natural way:

xG~ the direct sum of E', E", by declaring sum of the modules cE' , cE". The vector with subbundles of E.

-

Conversely,

the C=(M)-module of sections of E to be the direct bundles E' , E" are identified with a natural manner

if E = _ L J; E x is an arbitrary ~t+w

x~E'xcEx,

vector

x~ExCEx,

bundle and we have smooth maps

x6M,

where E x ' n Ex" = {0}, then x ~ E x' | Ex" = A x c Ex, x e M is a smooth map and the subbundle Ex". U Ax c E is naturally isomorphic to the direct sum LJ Ex' | U x@M There exists a universal Proposition

1.2.

construction

of vector

bundles,

described

by the following.

Given a smooth map A'I~

Let A denote the subbundle A =

~

H'~, x ~ A~fiH"fl, x6M.

A x c M x R m described

xGM

duced from M • R m, and the module of sections satisfying the condition s(x) e A x V x 9 M. It turns out that for any vector

above.

The smoothness

on A is in-

oA is the set of all smooth maps s 9 C~(M, Rm),

bundle E =

[J x6M

E x over M such that d i m e x = k, there

exists a smooth map M + Hk k for sufficiently large m, such that the corresponding subbundle, A = LJ A x c M x R m , d i m A x = k, is isomorphic to E and so the modules of sections oE and

x6M oA are isomorphic. Let B x denote the orthogonal determines

a subbundle

B =

complement

to A x in R m.

13 B x c M x R TM, where, x6~

Then the smooth map x § B x 9 Hm-k m

by the above arguments,

A 9 B = M • R m.

Thus, an arbitrary vector bundle E over M can be embedded (isomorphically) summand in the direct product M • R m for sufficiently large m.

as a direct

Proof. We shall assume that M c E, and that x is the zero point of the fiber E x for x 9 M. Let TE be the tangent bundle to the (n + k)-dimensional smooth manifold E, TxE the tangent plane to E at x. Consider

the prebundle

D =

x~M

TxE"

We convert

it into a vector

bundle

by defining

the

module of sections oD as the C~(M)-module generated by the sections s:M ~ D of the form s = XIM , where X is an arbitrary section of the tangent bundle TE, that is to say, a vector field on M. In other words, the sections of the vector bundle D are the restrictions to M of the vector fields on E. Let TxM, TxE x be tangent planes at x e M to the submanifolds M c E, E x c E, respectively. The maps x § TxM c TxE , x + TxE x c TxE , x e M, are obviously smooth and generate subbundles

U Tx3IcD, x6M It is easy to see that

I 1T~Ex~D, x6M

L J TxE x is isomorphic ~6~

D-= LJTxMOh_lTxEx. x~M ~6M

to E =

hJ E x. ~6M

Now let g:E -+ R TM be a smooth map of the (k + n)-dimensional manifold E into R m which is regular in some neighborhood of M; consequently, the rank of the differential Tg is n + k

1787

at every point of M; for sufficiently large m such a map always exists (Whitney's Embedding Theorem). We obtain a smooth map k,

x~A~=T.g(T~Ex)~H~',

x~M,

which is easily seen to meet our demands. Consider a given prebundle over M, A =

~

x~M

A x with constant fiber dimension d i m A x = k.

We define the dual prebundle to A as the prebundle c o A = A* = x6m[J Ax*, where Ax* is the dual of Ax, which is a k-dimensional space. If A is a vector bundle, then A* can be converted in a natural way into a vector bundle, by letting its module of smooth sections oA* be the set of sections $:x + $(x) e Ax* , such that the maps

x ~ < g (x), s (x) > = < g, s > I., x~M, where <., .> is the pairing between A x and Ax*, are smooth functions for any choice of section s e oA. In the category BdDiff the correspondence A ~+ A* = c o A becomes a contravariant functor F co F of the category into itself if, given a morphism A ~ B, we define the map c o A § c o B by the formula = , where N and s are arbitrary sections in coB, A. The linearity of co F on fibers is obvious, and that it is smooth follows from the definition of the module o(coA) and Proposition i.i. There exists a natural map of the bundle A = LJ Ax** xGM '

LJ A x onto the bundle c o o c o A = A** = xEM

t

L_I A r -+ [.5 A~**, ~GM ~GM defined by identifying A x m Ax**" The map I establishes an isomorphism between the modules of sections oA and o(coocoA) and so we obtain a natural functor equivalence cooco m IdBdDiff , which implies,

in particular,

a natural isomorphism of vector bundles A m c o o c o A = A**.

Using the embedding A c M x R m, just described, we can construct an isomorphism A ~ c o A = A* (which cannot be natural), but we shall not do so, since the same isomorphism will follow directly from the module isomorphism (2.11) below. 2.2. Finitely Generated Projective C~(M)-modules. a nonzero algebra homomorphism

The categories Mode, ModDiff.

q~ : C ~ (M)---~C ~ ( N ) , a C~(M)-module ~ and C=(N)-module modules over % if

@.

We call a map r

Given

(2.1) ~-+@

a semihomomorphism of

(a'm + a"u") = ~p(a') ,D (u') + ~ (a") ~ (u") va', a"~C oo(M), vu', tt"fi~. There clearly exists a smooth map f such that f M~-N, The s e t o f a l l s e m i h o m o m o r p h i s m s o v e r 9 f r o m ~ be converted naturally into a C~(N)-module. If

~=f*. to ~ i s d e n o t e d by @ = C~(N) we w r i t e

Homw(g, @).

It

can

Hom~ (~, C a (N)) = Hom~g. I f M = N a n d 9 -- i d , o f homomorphisms o f
1788

The r e s u l t u> ~ C ~ ( N ) .

then the ~ into

of applying

C ~ ( M ) - m o d u l e Homid (~, @) = Hom (~, @) becomes a C~(M)-module @,, a n d H o m i d ~ = H o m g ~ g * is the dual module to ~.

r ~ Hom ~ t o

u e ~ will

be d e n o t e d

independently

of

~ - by

Hem, may be regarded as a contravariant functor from the category of all C~(M)-modules to the category of all C~(N)-modules 6, where the morphisms in both categories are the appropriate module homomorphisms. This functor is defined on objects by the correspondence ~Hom~, and on morphisms x6Hom(~,~2) by the map X ~ Hom ~Z, where

.(Horn,X)O=Oox, OeHom,~.

(2.2)

It is obvious that the diagram

-- Horn ~~ ~r is commutative. The sense of the functor Hom~ can be somewhat generalized by applying it to pairs (3, ~) in accordance with the commutative diagram (~, e) ~Hom~(~, 6)

(~' e'),-.Hom~ (~', e'), where X, 0 are homomorphisms of C~(M) - and C~(N)-modules, respectively, and defined by the formula

Hom,(x, ~

~----Hom~(x, O) @' =Oo~'oX~Hom~ (~, @), @'~Hom~ (~', @').

is

(2.3)

The following natural isomorphisms are obvious:

Hom~ ( ~ ) ~ , @)~Hom~ (~,, @)@Hom~(~, @), Hom~ (~, @t@@2)~Hom,(~, @l)| (~, ~2). ~

(2.4)

Hom~ (~1 @~2) = Hom,~ ~ @ Hom~ ~2-

(2.s)

In particular,

To define the first isomorphism we take r e Hom~(~,@) , i = i, 2. Then the element r is identified naturally with an element of Hom~(~,@~2,~) by the formula

O~@O~(u~u~) =O~u+@2u~, u ~ , ,

~2

i = 1,2.

This identification is a natural isomorphism. ~t, @2

To define the second isomorphism we let prl, pr 2 denote projections from ~ O ~ 2 onto and identify an arbitrary element @~Hom~(~, ~ i @ ~ ) with the element prio@6Hom~(~, @ i ) ~

Hom~(~, ~2). Given an a r b i t r a r y s u b s e t S c s e t S • g i v e n by t h e formula

~ we d e f i n e t h e a n n i h i l a t o r

sx~={@EHom~ ~] < O, u > ~-0 For a r b i t r a r y

S, t h e a n n i h i l a t o r

S•

of S in Hom~8 to be t h e

ruGS}.

(2.6)

i s a submodule o f Hom~ ~ and t h e f o l l o w i n g obvious

formulas are true

S# ~n S#~ c(S, + $2)• J It is readily seen (see below) that for any C=(M)-module ~ ~ 2 morphism

Hom~ ( ~ i ~ ) ~ ~ (

(2.7) there is a natural •

~, (2.8)

where

and we again arrive at the natural isomorphism (2.5).

1789

If

= id we obtain natural isomorphisms

~Hom where we have denoted

~,

(2.10)

~l--~Hom~,

~L=~•

To prove (2.8) we observe that

since vO6Hom~ .1-~0 O=Oopr l+Oopr~, Oopr~6~2 , Oopr26~ir Furthermore,

~2

~-{0},

because (see (2.7))

~ ~ ~ c ( ~ 2 + ~0 -L~=1o}. Consequently, we obtain (2.8), where the isomorphism is defined by the correspondence + r I | Oopr 2. To define the natural isomorphisms

(2.9) we note that if

@ 1 ~ ~,

O2E~I• ,

then the

correspondences ~i ~ ~i[ ~, ~2 ~+ ~21@2 define natural embeddings

~2 cHore, ~1, ~r and it is easy to see, on the basis of (2.5), that these embeddings are surjective. We note, moreover, that for any C~(M)-module a natural homomorphism of C~(N)-modules

X : (Hom~) |

~

and C~(N)-module @

we can write down

@),

defined by

X(~| v) u= (Ou) v, O(~Hom,~. Our assertion follows from the fact that the expression (~u)v is C~(N)-bilinear in r v and and the tensor product { H o m ~ ) | is universal relative to C~(N)-bilinear mass ( H o m ~ ) X @ - + Hom~(~, @). In the general case one cannot say much about the homomorphism • but X is a natural isomorphism in the category of finitely generated projective modules (see below), to the consideration of which we now proceed. Recall that a module (over an arbitrary ring R, and not necessarily finitely generated) is said to be projective if it can be embedded (i.e., mapped homomorphically and injectively) in a free module as a direct summand. The following property characterizes projective modules. If an R-module ~ is isomorphic to a quotient module-~/@ of an R-module ~, by a submodule @, then ~ can be embedded in ~ as a direct complement to ~, i.e., there exists an injective homomorphism X: ~-+~, such that

A free C~(M)-module of rank m will be represented, as a rule, as a module of smooth maps of M into R'~: C~(M, R TM) = {w = (a I . . . . , am)[a i e C~(M)}. Any finitely generated C~(M) module ~ on m generators u i . . . . u m [is the image of C~(M; R m) under the homomorphism] that carries each w i into ui, where w i . . . . . w m is an arbitrary basis in C~(M; Rm). Consequently, is isomorphic to the quotient module C=(M; R ~ ) / ~ , where ~ is the kernel of this homomorphism. Using the projectivity criterion stated above, we see that any projective C~(M)-module on m generators is embeddable in C~(M; R TM) as a direct summand. Therefore, in order to characterize finitely generated projective C~(M)-modules, it will suffice to characterize the submodules of C~(M; R m) which are direct summands. This is accomplished by the following

1790

Proposition 2.1. Let ~ be a submodule of C~(M; R m) and let M be connected. Let ~ denote the subspace ~x = {u(x):u e.~} c Rm, x ~ M. The following conditions (1)-(4) are equivalent. i) C=(M; R m) has a submodule ~, which is a direct complement of 2) dim ~ ators.

3: C~(M; R m ) = ~ .

does not depend on x and the module ~ is finitely generated with ~m gener-

3) dim ~x does not depend on x and ~ is the set of all smooth maps w 9 C~(M; R m) satisfying the condition w(x) e ~ x V x 9 M. 4) The map h~:x ~ F x is a smooth map of M into Hk m, and the module of sections of the vector bundle [ J ~ x c M x R m coincides with 3. Moreover, there exists a smooth map x6s ~ x 9 Hm_km such that

h2:M + Hm_k m, x ~

L1 ~..

C ~ (M; R~)= ~|

i) ~ 2 ) .

Proof.

W = U + V, U 9

By a s s u m p t i o n ,

3, V 9

x6M

a n y map w 9 C~ (M; R m ) i s

~, and therefore w(x) = u(x) + v(x)

(uniquely)

V x 9 M, i.e.,

expressible

~-~x=R

as

m, or

dim ~x + dim @~x -> m. We claim that ~ x ~ ~ x = {0}. Indeed, if z 9 ~ x fl ~ x , there exist u' 9 ~, v' 9 @~, such that u'(x) = v'(x) = z, whence it follows, taking a basis w I . . . . . m

w m in C~(M; I~m), that w' = u' - v' = ~ wi = u i 9 v i ,

ui 9

~, v i 9

t~,so

m.

Express w i as

that

m

r/Z

i=1

By the uniqueness

aiwi, ai(x) = O, i = i . . . . .

i=I

of the representation,

u' = u", and so u' (x) = u"(x) = z = O.

Thus we

have

R~=~@Nx VxEM,dim

~ - - t - d i m ql~=m.

Since the dimensions dim ~=,, dim ~ are lower semicontinuous (a small change in x can only increase them, since the elements of C=(M; R m) are continuous) and dim ~ x + dim ~x = m, it follows that dim ~ and dim ~= are independent of x (for connected M). is finitely generated, we write the basis wl, ..., w m of C=(M; R m ) as

wt=ttt@vi, Then V u e ~

To prove that ~

i=l,...,m.

ut6~, vi6~,

we have m

m

m

i=I

i=I

i~l

n%

a Y)i~E af~i, because

~A~

= {0},

i.e.,

u I . . . . , u m are generators

2) =>;3). If u E ~, then, verse: if u ~ C ~ (M; R~), u(x)

i~l

of

~.

by assumption, u(x) e ~= V x ~ ~ Vx ~ M, then u E~.-.

~ M.

Wesha11

prove

the con-

Let ui, ... u m be a system of generators of 3, {U i} a sufficiently fine locally finite cover of M by open sets with compact closures such that, Ui, the system ul, ... um contains k = dim ~x elements uil . . . . . Uik, such that the vectors ui1(x) , .... Uik(X) are linearly independent V x 9 U i. condition

Hence there exist functions at(i) . . . . .

am(i) on U i satisfying the

m

7=I

Let {a( i ) } be @smooth p a r t i t i o n of unity subordinate to the cover {Ui}., so that the funca(i)aj(i) = bj I can be considered as members of C=(M),

since s u p p bj~(1) c U i.

For g i

179;1

m

a(O (x) u (x) = ~ b(/) (x) tt~ (x) V x6M. Consequently,

J=, Oj (x) uj (x) VxO.e~l,

i=~, ~

bj(i) 9 C=(M) are well defined since the cover {Ui} is locally

where the functions bj = I

finite.

Thus

m

u = ~ b/t16 i~. ./=1

3) =>4).

The map h1:x ~

~ x 9 Hk TM is smooth.

Indeed,

given x, choose maps u i . . . . ,

u k which are linearly independent at x. They are linearly independent in a sufficiently small neighborhood U x of x and therefore the system ui(Y), ..., uk(Y) is a basis for ~ at any point y 9 U x. Thus (see Sec. 2.1) we obtain a vector bundle

I1 h~ (x)=

~M

1__1~ c M • R ~, ~6M

whose module of sections consists of all smooth maps w 9 C~(M; Vx 9 M, and so

~m), for which w(x) 9

~x

~U ~=~. x6m To define h 2 we need only let ~x denote an orthogonal complement to canonical metric of the coordinate space R m) and put h2:x ~+ ~x. 4) =>i).

~x (e.g.,

in the

Obvious.

COROLLARY i. The module of sections of any vector bundle E over M is finitely generated and projective. Indeed, by Proposition 1.2 there exists a smooth map h:M ~ Hkm, such that E is isomorphic to the vector bundle U h(x) c M • R m. Consequently, the C~(M)-module aE is isomorphic to x~M the module of sections of x@M U h(x) which,

as we have just proved,

is finitely generated and

projective. COROLLARY 2. C=(M;

If

~ is a direct summand of C~(M;

R m) contains a submodule ~ , such that C=(M;

Rm), say C~(M;

R ~) = ~ |

R m) =

~@~,

then

and the relations u e ~,

v' e ~' are equivalent to the equality = 0, i.e., (u(x), v'(x)) = 0 x e M, where (-, -) is an arbitrary scalar product, e.g., the canonical one, in Rmi((a I, ..., am), (b I . . . . .

bm)) =

~

a ibi.

i=I

In fact, letting ~'x denote the orthogonal complement to ~ relative to the scalar product (', "), we take ~, to be the module of sections ~ !_j (~'x" x@M Proposition 2.2. Let ~ be a free C=(M)-module of rank I, a free C~(N)-module of rank m, and ~ :C~(M) + C~(N) a nonzero homomorphism. Then Hom ~(~, ~') is a free C~(N) module of rank m, with basis {~ij}, i = i . . . . . I, i = 1 . . . . m defined by the relations

where {wj}, {wj'}, i = 1 . . . . . Consequently, homomorphisms @i,

1792

Hom~@

I, j = i . . . . .

m are bases in ~

is a free C=(N)-module

of rank

and

~',

l, with basis defined by the semi-

9 ~r176176 Proof.

i---#j.

~oi=0,

Vr e Hom~(~, ~') there i s a unique representation i,j

where bij are defined by the relations

r

=~

1=I

b,~%, i= I.....I.

COROLLARY. If ~, @ are finitely generated projective modules, then Hom~(~, @) is also finitely generated and projective.

Indeed, complete ~ and @ by direct summands to free modules finite rank. Then, using the isomorphism (2.4), we obtain

~

and ~ |

of

Hom~ (~@~', @| (~, @)@Hom~(~, @')| | (~', @)*Hom~ (~', @'), where the left-hand side of the isomorphism is a free C~(N)-module of finite type; hence Hom~(~,@) is a finitely generated projective C~(N)-module. Proposition 2.3. morphism

For any finitely-generated projective C~(M)-module there is an iso-

~8*=Hom though this isomorphism is not natural:

~,

(2.11)

it depends on certain non-invariant constructions.

Proof. We can write C~(M; R m) as C~(M; Corollary 2 to Proposition 2.1) that

R TM) =

~@~,

where

~

is so chosen (see

u~,v~c=>-=O, whence it follows that the following injective embeddings (depending on the choice of exist : il • ~c@

~)

@ci~ 9 / -

The direct sum of these embeddings yields an injective homomorphism (see the first formula of (2.10) and Proposition 2.2)

il~i2:C~(M; R m ) = ~ @ @ - - + @ • 1 7 4

(~@@)

= Hem (C ~ (M; R~))~C ~ (M; R~). Consequently, looking at the outermost terms of these relations, we see that i I | i 2 is an isomorphism, so that ii, i 2 are also isomorphisms. This completes the proof, since by the @ • ~. second isomorphism of (2.1), We now prove that if natural homomorphism

is a finitely generated projective C~(M)-module, there is a

~**=HomoHom

~.

(2.12)

We have a n a t u r a l i n j e c t i v e embedding i : ~ c ~ * * , s i n c e V~6~* t h e e l e m e n t u6~ is p a i r e d w i t h ~ by t h e formula = <E, u>. Let @ be t h e complement o f ~ in C~(M; R m). Taking t h e d i r e c t sum o f t h e a p p r o p r i a t e n a t u r a l homomorphisms il : { ~ H o m o H o m ~, i2 : @cHomoH6na @ and taking note of the natural homomorphisms

iI|

(2.5) and Proposition 2.2, we obtain

Rm)=~@@cHomoHom ~@HomoHom @-----

~ H o m (Hem ~@Hom @)~HomoHom (~@@) ~_HomoHom (C ~ (M; Rm))-----C~ (M; R~). C o n s e q u e n t l y , i 1, i 2 a r e isomorphisms.

1793

We now introduce the category of modules Mod= and its subcategories ModDiff, Mod M, corresponding to the categories of vector bundles Bd=, BdDiff, Bd M and equivalent to them respectively (see Secs. 2.3-2.4). The objects of Mod~ are all the finitely generated projective C=(M)-modules, where M ranges over all smooth manifolds. The morphisms of Mod= are arbitrary semihomomorphisms r 9 Hom,(~,~), where ~,~ range over all objects in Mod~ and ~ are arbitrary (nonzero) homomorphisms of the corresponding algebras. The isomorphisms of this category are semi-isomorphisms over ~, i.e., one-to-one semihomomorphic maps ~-+$. over ~. It is readily seen that in that case ~ is an algebra isomorphism. The subcategory ModDiff c Mod~ consists of the same objects as Mod=, but its morphisms are only the semihomomorphisms P over algebra isomorphisms p:C=(M) + C=(N). Finally, the objects of Mod M c Mod= are all the Ca-modules for a fixed M, and its morphisms arbitrary semihomomorphisms over endomorphisms ~:C=(M) § Ca(M). The same map r ~-+~ may correspond to several semihomorphisms P over algebra isomorphisms p:C=(M) +Ca(N); an example is the zero semihomomorphism, which maps the entire module ~ into zero. Nevertheless, there is a large class of semihomomorphisms r 9 Hom for which 9 is uniquely determined, and one can then write

~=0(~).

(2.13)

For example, this is the case for the class of all semi-isomorphisms - see below. In that case, since a composition of semi-isomorphisms is again a semi-isomorphism, any two algebra isomorphisms 8(P~), 8(P=) determine an isomorphism O(P~oP~) and

0 (PloP2) = 0 (Pl)oO (P~, C l e a r l y , an a u t o m o r p h i s m o f a C ~ ( M ) - m o d u l e for which 8(P) = idca(M ).

~

0 (p-m) = 0 (P)-L

(2.14)

is a semiisomorphism of the indicated

type,

I t f o l l o w s f r o m ( 2 . 1 4 ) t h a t i f s e m i - i s o m o r p h i s m s P1, P2 a r e s u c h t h a t 8(P 1) = O ( P 2 ) , then PloP2 -z and Pz-~oP2 are automorphisms. In particular, together with P, the inverse p-Z is also an automorphism. The following proposition furnishes a sufficient condition for the validity of (2.13), i.e., for a given semihomomorphism r 9 Hom $(~,~) to uniquely determine an algebra homomorphism~ :Ca(M) + Ca(N). Proposition 2.4. A semihomomorphism of modules ~ 9 Hom~(~,~) , where ~ is a C=(M) module and ~ a C=(N)-module, uniquely determines the homomorphism % if V y e N there exist u 9 3, ~ 9 ~*, such that

I,,# O. In particular, this condition is valid for semi-isomorphisms. Proof. Supposing the contrary, let ~I,~2: Ca(M) + C ~ ( N ) b e different algebra homomorphisms such that r (au),= ( ~ l a ) O u = ( ~ a ) ~ u

Va~C | (M), V u ~ . Then t h e r e e x i s t a p o i n t b2, and c o n s e q u e n t l y

y e N and a f u n c t i o n

a e C~(M), s u c h t h a t

b 1 = ~la(y)

r ~2a(y) =

(bl--b2):<~, Cu>[u=0=er< ~, ~u>[u=O V ~ @ * , u ~ , contrary to assumption. To conclude this subsection we prove a natural isomorphism Hom~ (~, ~ ) - - ( H o m ~ ) | in Mod=.

1794

(2.15)

It was mentioned previously that the map

(Hom~|

(~, @),

(2.16)

d e f i n e d by (X(O|

(Ou)v, u ~ , v~@, O~Hom,~,

is a natural homomorphism. We first prove that X is an isomorphism if ~ and @ are free modules of finite type. Let {Uk} be a basis in ~, {vl} a basis in ~, {#i} a basis in H o m ~ ~iUk = 6ik. Then the system {X(#i | vj)} satiss the relations

Z~O,| vj) u,= (O,u,) v~=vj, X (~, | v~) ul= (@~u0 vj =0, i ~ l, t h a t i s , by P r o p o s i t i o n 2.2, i t is a b a s i s of the f r e e module Hom,(3.@) ; hence X is indeed an isomorphism. In the general case, let ~=~i, @=@l in suitable free modules of finite type.

and let ~2, @2 be direct complements of ~t, @t Further, let

(Hom~i)| be the corresponding natural homomorphisms following chain of natural homomorphisms: i, ~

(Homq, (~, | {~=))|

XU

Hom~ (~i, @:)

(2.16) and EXij their direct sum. :XX I / ~

We have the

it

~9@,,)----~ (Homw{~i)@@j:~_. Z Homq, (~i, @j)----Home ({~l@~=, @,@@=). z,i

Here the composition i2oExijoil

i,j

is the natural isomorphism (2.16) of the outermost terms,

because ~i~2, @i@@2 are free modules of finite type, and il, i 2 are also natural isomorphisms, see (2.5). Consequently, Y Xij is an isomorphism and therefore all the direct summands Xij are isomorphisms. 2.3.

Equivalence of the Categories Bd~, Mode.

C~(M)-module

~

If a finitely generated projective

is embedded by an injective homomorphism X into C~(M: R rn) as a direct summand,

C~ (M; RTM) =X (~) 6D@, then it may be associated with a vector bundle over M, by using the smooth map (see Proposition 2.1)

x"+Z(~)x6

k, xEM,

k--dimz(~)x,

which introduces a vector bundle

U z ( ~ t c : 4 • Rm,

x6M

(3. i)

(cf. Sec. 2.1). This procedure, however, is not satisfactory, as it depends on the embedding X and is not defined in terms of functors. In fact, it is not even clear that the dimension of the fiber dimx (3)x is independent of X. However, there is an invariant procedure which, given an arbitrary finitely generated projective C~(M)-module ~ , produces a vector bundle ~ over M, known as passage to the bundle of 0-jets of ~ , which we now proceed to describe. We first observe that, given a representation C~(M; R m) = @ ~ @ , if there is a point x e M at which an element s e ~ (a smooth map M + R m) v,,~ishes, s(x) = 0, then there exist smooth functions al, ..., a m e Ca(M) and elements sl, ..., sm e , such that

s = ~ aist,

ai (x)=O,

i = l . . . . . m.

i=1

1795

To prove this, we write s as s = ~

aiw i, where w I . . . . , w m is a basis in C=(M; Rm).

i=!

Since

ai(x)wi(x) = 0, it follows that at(x) = ... = am(x) = 0.

Expression w i as w i =

i=l

s i + vi, s i 9 we obtain

@,, v i e

~, i = 1 . . . . .

m, and remembering that

s:~aisi+2a~vi-~-~aisi, i=I

i:1

~

and @

are direct sum~ands,

a~ (x)=O.

i=l

Suppose now that ~ is an arbitrary finitely generated projective module over C~(M). Let I x denote the maximal ideal in C=(M) of all functions that vanish at x (see Sec. i). There is a natural isomorphism C~(M)/Ix m R. Consider the submodule

(3.2) and construct the quotient module

~l&~ = {ul&~ l u6~} ,

(3.3)

w h i c h may b e c o n s i d e r e d in a natural w a y a s a r i n g C ~ ( M ) / I x ~ R. Consequently, this quotient m o d u l e i s a r e a l v e c t o r s p a c e o f d i m e n s i o n ~m, w h e r e m i s t h e n u m b e r o f g e n e r a t o r s u~, ..., u m i n ~ , s i n c e t h e m e l e m e n t s u i / I x ~, i = 1, . . . . m, a r e o b v i o u s l y generators of ~/G~. ment

The vector space ~/~ will be called the (abstract) fiber of U/Ix~E~/l~ the value of u at x. We shall denote it by

~

over x, and the ele-

Ux=U/fx~.

(3.4)

We define a family of ~ -linear surjective maps Jx ~

J~:~'-->'~/Ix~, u~J~u=Ux,

x 9 M, by the formula

u6~.

(3.5)

The family of fibers

~/I~=J~=={u~lufi~ }, xfiM, forms a prebundle

~ = N J ~0 ,

(3.6)

x6M

which we shall now make into a vector bundle by using only the module structure of We first define an injective homomorphism j0 of of the prebundle ~ , by the formula 0 0 u~Yu:x~YxU=Ux,

The C~(M)-linearity of the map j0 is obvious. J~

= 0, then Jx~

= 0 forVx

~

~..

into the C~(M)-module of all sections

x~M.

(3.7)

We shall prove that it is injective.

e M, that is, u 9 I x ~ V x

If

9 M, or (3.8)

i

But we know t h a t aix(ui),

there

exists

an injective

h o m o m o r p h i s m X: ~

C~(M;

Rm), a n d s o X ( u ) =

ai(x) = 0 or X(u)(x) = 0 Vx e M~=~u = O.

i

"Thus the image j0 ~, which is a certain submodule of the C~(M)-module of all sections of ~ , is isomorphic to ~. We now prove that the dimension of a fiber J x ~ is independent of x. Since %(~) is a direct summand in C~(M; a smooth map

1796

Rm), it follows by Proposition 2.1 that we have

We claim that

dim J ~ = k Define a linear map of the fiber J ~

Vx6M.

onto the subspace X(~)x c ~m by the formula

~.

J~u=u~z(u)(x), The map is well defined, because if Jx~

= 0, then u = ~

aiui, ai(x) = 0 (see (3.8)). i

Consequently, z

(~)(x)= ~

a~ (x)z

(u~)(x)=o.

I

In addition, this map is surjective, since any point of the subspace X(3)x can be expressed as X(U)(X), u 9 8. We shall show that it is also injective. If Jx~

~ X(U)(X) = O, then, by the assertion proved above, there exist functions

ai 9 C~(M) and elements ui 9 ~, such that X(u) = ~i

aix(ui)=

ai(x)=O=~u6fx~=>-ux=O.

is injective, we have u = ~ a t ~ ,

X (~=~i),

ai(x) = O. Since x

Thus the dimension of the fiber

i

jo~

is indeed independent of x.

Considering the prebundle ~ , we declare 7~ to be the module of all smooth sections. It is easy to see that this makes 63 a vector bundle over M (see Proposition 2.2) with module of sections naturally isomorphic to 3,

o' ( ~ ) : jo.~__ ~ .

( 3.9

)

j'o

We claim that the isomorphism ~_~j0~ is natural in the category Mode. Indeed, j0 becomes a covariant functor from Mod~ into itself, which is naturally equivalent to the identical functor IdMod~ , if it is defined on objects by the correspondence

(3.10)

~jo~, a n d V ~ 9 Hom~(~,~), where ~ is a C~(M)-module, (nonzero) algebra homomorphism. We define

~ a C~(N)-module and ~:C~(M) ~ C~(N) a

jof~: jo~.+ jo@,

jo~ (jou) Clearly,

J~

9

( j 0 ~, j o , ~ ) ,

=

(3. ii)

jo (f~u), u~.

and t h e diagram ]o

~_.j0~ ~-+ JOi8 i s commutative. bundle o f ( 3 . 1 ) ,

I n d e p e n d e n t l y o f t h e embedding X, t h e bundle ~ where t h e isomorphism i s g i v e n by t h e map

is isomorphic to a sub-

J~u=ux~z(u)(x), u6~, xEM. The vector bundle 63

is called the bundle of.0"jets of the module

I t i s e a s y t o s e e t h a t t h e d u a l bundle t o ~N i s 8(Hom co (63) ~

3.

3),

(Horn 3).

(3.12)

Indeed, letting $ denote an arbitrary element of Hom 9, we have

I~(Horn ~) = LA j o (Hom ~) = LA {gx I gEHom ~}. ,~GM x6m

1797

The vector spaces {gxlg e Hem defined by

~} and {UxlU e ~ }

are related by a natural duality, as

<~,u.>=<~,u>l., x~M. Consequently,

j o (Hom ~)~(Jx) o 9 , and therefore

~(Hom ~ ) = U J~(Hom ~ ) ~ d Horn (J~)----co(~). x6M *6a In addition, it is clear that the modules of sections of co(~ ~) and ~(Hom ~) are naturally isomorphic. The above constructions can be summarized as a proposition: Proposition 3.1. prebundle of 0-jets:

Every finitely generated projective C=(M)-module ~

determines its

(3.13)

x6m

whose fibers are of constant (i.e., independent of x) dimension and may be represented as

~l Ix~--=-~/ Ix~=Ux= J~u; u6~}, where u x : Jx~

(3.14)

is the value of u at x.

The map j0 of ~ by the formula

into the C~(M)-module of all sections of the prebundle ~ ,

u~(J~

defined

J~u = ux, x6M),

(3.15)

is an injective homomorphism. By designating the image j0~ as the module of all smooth sections, we convert ~ a vector bundle (over M) - the bundle of 0-jets of 3:

U j o g = [_] {uxlu6.~}. *Era .~M

~= The module of sections of }~

into

(3.16)

is given by the formula (3.17)

ja

and the map ~_~j0~

is a natural isomorphism

~j0~,

(3.18)

since the correspondences (3.10)-(3.11) make j0 a covariant functor which is naturally equivalent to the identity functor. We shall now prove that Bd~ is equivalent to Mode. two covariant functors

To be precise:

we shall define

E.:Bdo~--+Mode, h : Modoo->Bd~ and prove that the covariant functors AoE and EoA are naturally equivalent to the identity functors IdBd ~ and IdMod . Let A, B be arbitrary vector bundles over manifolds M, N, respectively, and F a morphism between them: F

A~B,

F6MorI(A,B),

where

M-~' N, 1798

C~(M)~__COO(N)

are a smooth map of M into n and the corresponding algebra homomorphism. Consider the functor E defined on objects by the formula

(3.19)

A ~ EA =Hom (oA), and on morphisms by the formulas Horn (~A) +-Horn (aB),

< (Zf)n, s > I x = < ~t(x), f s ( x ) > V~Hom (~B), VsGoA.

(3.20)

A direct check shows that these maps are well defined and constitute a functor. is also easy to see that ZF6Homr. (Horn (aB), Horn (oA)). Similarly, let homomorphism

It

(3.2i)

~, O E Mod~ be C~(M) - and C~(N)-modules, respectively, and ~ a semi~-O,

OGHom~(O, ~),

where ~:C~(N) ~ C~(M) is an algebra homomorphism. Let f:M + N denote the smooth map corresponding to ~ :~ = f*. Then the functor A is defined on objects by the formula

A~- ~ (Hom ~), AJ = ~ (Horn gO)

(3.22)

13(Horn ~)-+ 13(Hom @), < (aO) G, va~) > = < ~, Ov > t., V~EHom ~, Vv60.

(3.23)

and on morphisms by the relations

Clearly,

AOfiMor/(13 (Horn ~), 13(Horn O)). There obviously exists a natural isomorphism A ~ ~(oA); moreover, we have a natural isomorphism Hom(Hom ~)--~, (see subsection 2.2). Consequently (see also (3.17)-(3.18)),

AoEA~ 13(Horn (2A)) := [~(Horn [Horn (~A)]) ~=13(HomoHom (~A))~13 (~A)~A, XcA~= I (13(Horn ~)) = Horn (~ [~ (Horn ~)i) ~Hom (yo (Horn ~))~_H0m~H0m ~ . Thus,

(3.25) The equivalence thus proved of the categories Bd~ and Mod= establishes a one-to-one correspondence between invariant constructions in vector bundles over smooth manifolds and in finitely generated projective modules over algebras of smooth functions. For example, many constructions over vector bundles are more conveniently done first in their modules of sections; then, after completing the construction in that context, one returns to bundles (cf. the next subsection). 2.4.

The Categories BdDiff, ModDiff. f=~*

Adjoint Morphisms. ~

The maps

f*=~

A4--+N, C (M)+--C~(N) are "adjoints" of one another relative to the pairing

<x,a>=a(x), =b(y), x~M, y6N; a~C~(M), b6C| that is, we have identities

= <x, [%> = <x, r 1799

One frequently encounters similar situations of mutually adjoint maps R, S of pairings, where the latter are R -linear in both arguments, the maps themselves are linear and their ranges are either R or the algebra of smooth functions. In all such cases the result of switching from R to S and back is naturally denoted by an asterisk

S=R*, R=S*, and moreover the operator * in such contexts is functorial and can be viewed, as in the case f + f*, as a "substitution functor." We shall now consider mutually adjoint morphisms of the categories BdDiff , ModDiff , that is to say, mutually adjoint maps of bundles over diffeomorphisms of the appropriate base spaces or semihomomorphisms over appropriate algebra isomorphisms. We shall show, moreover, that in these categories the correspondences 13 and o introduced above can be made into "mutually inverse" functors (see below) representing the equivalence of the categories BdDiff and ModDiff. Given dif feomorphisms f

M_~N r-,

(4.1)

let us assume that

A = [_JAx, x6M are vector bundles over M and N.

B = I IBy ~6~

The dual bundles will be denoted by

c o A = I lAx, x6M

c o B = L_IBy, p6N

(see subsection 2.1). Sections in A and B will be denoted by r and s, sections in c o A and coB by D, v:r e oA, s e o B , V e coA, v e coB. Let F be an arbitrary morphism over f: F

A~B,

F~Mor/(A, B)

(4.2)

F*6Mort-, (co co AB,)

(4.3)

and define the adjoint morphism F*

co A + c o B , by t h e formula

=

(4.4)

which is obviously equivalent to the formula <w(f(x), Fr(x)>=
r(x)>.

(4.5)

In these formulas the symbol <-, .> denotes the pairing operator in the corresponding dual fibers of A, c o A and B, coB. Clearly, formula (4.4) uniquely determines F* given F, and formula (4.5), conversely, determines F given F*. Moreover,

F=F**,

(4.6)

(of course, provided that co(coA) is naturally identified with A). Denote r(f-1(y)) = z e A; then the formula (4.4) for F* becomes t

=p*~ (y) (z) --v(U) (F(z)) =~ (y)or(Z),

(4.7)

that is, to evaluate the linear functional F*~(y) in the appropriate fiber it is sufficient to "substltute F into v(y). Suppose now that we are given two mutually inverse isomorphisms

c ~ (m) ~_ c ~(N) p--I and let ~, @ be finitely generated Ca(M) - and C~(N)-modules, respectively, and H o m ~ , M o m @ the dual modules. The elements of these modules will be denoted by

1800

u~, we have n a t u r a l

rE@; ~6Hom~5,

~6HomO.

pairings

~O*(M), <~,v>~C| Let P be an a r b i t r a r y

morphism o v e r p, i . e . , P

~-+@,

a semihomomorphism o v e r p

PEMorp(~, @)=Homp(~, @)

(4.9)

and d e f i n e t h e a d j o i n t morphism as t h e semihomomorphism P* o v e r p-1 p*

(4.10)

Hem ~ ~-Hom @, P*EHomp-, (Hem @, Hem ~) defined by the formula (

P*tl, tt ) =p-~ ( ~, Ptt >,

(4.11)

which is equivalent to the formula

<~, Pu>=p.

(4.12)

Formula (4.11) determines the semihomomorphism P* given P, formula (4.12) determines P given P*, and (in view of the isomorphism ~ m HomoHom 3),

P=P**.

(4.13)

Here again it is easy to verify that all the above notions are well defined. If the pairing is written as a function u ~ q(u) = , formula (4.11) can be rewritten as

P*~ (u) =p-i (~ (Pu) ) =p-~ (~oP (u) )

(4.14)

that is, to evaluate P*q at u ~ ~ one evaluates the result of the substitution n ~ qop at u and transfers the resulting function, which is an element of C~(N), to C~(M) by applying p - i We now define a covariant functor

O : BdDtfr-+MOdDttt

(4.15)

as follows. To an arbitrary object of BdDiff, i.e., a vector bundle A over M, we associate the C~(M)-module of sections oA (see subsection 2.1); to a morphism (4.2) over f we associate a semihomomorphism over f-1*" oF

etA-+ etB,

oF~Homr_~,(etA, ~B),

(4.16)

where oF is defined by the formula

(gF) r [ y = F (rof-1 (y))EBy,

y~N,

(4.17)

which can be rewritten as

( aF ) r

=

Fore f - l = s~et B.

The legitimacy of this definition is obvious; F

O

it is also clearly functorial, 0*F

(4.18) since if

OO

A-+ B -+ C , a A-+ etB -+ crC , then

e (QoF) = ~OoetF, Thus, the correspondence

(r ida = i d , a 9

(4.15) is a covariant functor.

Similarly, we define a covariant functor

[~:Modmff -+ Bdmff,

1801

which, to an arbitrary object of ModDiff, i.e., a finitely generated projective C=(M)-module , associates a vector bundle 8~ by formula (3.16):

~3=

u J~=

~l {u~lu~}.

In order to define $ for a morphism (4.9), we let f denote a diffeomorphism (4.1) satisfying the condition

P

(f-9*,

=

(4.20)

and define a morphism gP over f [~P

[33-~[3@,

~PGMorz(~3, 1~@)

(4.21)

by the formula

(f~P) u,

=

Pu Ir(~).

(4.22)

It is readily verified that the morphism is well defined:

(~P) (auA --- P (au) )(~) = pa Ir(~)Ptt It(~) = f * op* a [xPu [W:) =

=a(x) (13P)u~. We have the following obvious (and easy to verify formally) natural equivalences between contravariant functors Homo~ ~ ooco,1 coo~ ~_ fioHom. J (4.23) We now write the adjoint map (oF)* of the map oF in (4.16), using the definition (4.11): HomoaA~ (~F)*=H~176 .HomocrB,

(crF)*GHomr, (Homo~B, Homo~A).

(4.24)

Similarly, we write the adjoint map (~P)* of the map ~P in (4.21) and obtain

co (~3)~-(~P)* co (~@), (~P)*EMorr-, (co (~@), co (~3)). Using t h e isomorphisms ( 4 . 2 3 ) ,

we can w r i t e f o r m u l a s ( 4 . 2 4 ) - ( 4 . 2 5 )

OoCOA+-(oF)*-0"oCOB, [M-Iom 3+- (~P)*~oHom ~,

(4.25) in an e q u i v a l e n t

form:

(eF)*EHomr, (~oco B, c~ocoA),

([3P)*~Morf-~ @Horn ~, i~oHom 3).

(4.26)

Comparing (4.26), (4.11) and (4.26), (4.3) defining (oF)* and (~P)* with the formulas defining EF, AF (see (3.20), (3.23)), we obtain natural equivalences

Y, _--__Homoc~~_ o'~co; A _--_--i~Hom ~ COol~,

(4.27)

which again leads to the equivalences (3.25) for the more restricted categories BdDiff, ModDiff: , A0E ~ i~HomoHomo~ ------~o~ ~ IdBdmfP /

EoA ------o'ocoocoo[~~ ao~ -----IdModDm. j

(4.28)

These formulas impart a rigorous meaning to the statement made at the beginning of this subsection that the functors G and ~ are inverses of one another. To end this subsection, we consider the functors representing passage to tangent and cotangent bundles or, respectively, vector fields and differential forms, from the standpoint of the constructions introduced in this subsection. We confine ourselves to the category of smooth manifolds and diffeomorphisms between them. Let f:M ~ N be a diffeomorphism. Denote the tangent bundles to M and N by TM, TN, the differential of f by Tf; in other words, T is a covariant functor

Tf : TM-v~ZN, TfOMori (TM, TN).

1802

Then o ( s e e

(4.16)-(4.17))

defines

correspondences

: TM where X is a vector field on M.

D e r M , o(T[) =ooTf:X,-+Ad(f-I)*X

Thus,

o o T ~ D e r , o ( T ~ ) = A d ( / - l ) *. Next,

aoco-----Homoo :. TM-+Der*M, or

oocooT~HomoooT~Der*, since Homoo(TM) = Hom(Der(M))

is the C=(M)-module of differential forms on M.

Finally,

HomoaoTf=Hom(ooT[) =Hom(Adf-1)*=T*[, (see Sec. I). Thus all the functors considered in Sec. 1 can be expressed in terms of the functor T and the universal functors Hem, co, o (as well as 8) applied to the categories Bd=, Mod~ or BdDiff, ModDiff. The functor T is specific to the category of tangent bundles and extends maps f from manifolds to the appropriate tangent bundles. 2.5. Induced Modules - the Substitution Functor * . a vector bundle over N,

Given a smooth map f:M + N and

B=ly ~ WI B y C N X R m. They define a prebundle over M by the formula

A= U Ax= U Bt(.)cM X R m. xGM .tim The map x ~ B f ( x ) e Hkm, k = d i m B f ( x ) i s s m o o t h , s o t h a t A b e c o m e s a v e c t o r b u n d l e and it is easy to show that, independently of the embedding B c M • R ~ all bundles A thus formed are isomorphic. Therefore, A is called the vector bundle induced by the bundle B under the map f. We are going to present a "functional" description of this construction in the language of modules of sections; this in turn will lead to a general concept of a substitution functor * generalizing the functor

/~f*:C~(N)-+C~(M),

f*O-~-bof, bECk(N)

and enabling us to regard f* as an operator "substituting" f not only into elements of the ring C~(N), but also into elements of an arbitrary C=(N)-module ~ e Mode: [*v =

We shall consider natural isomorphism

Vof,

v~q~.

as a module of sections; this is always possible thanks to the

_~_JO@, J%:y~J~,

yGN.

The expression f*v = vof will be treated as a section

f*v:x~vr(x),

x6M.

The C~(M)-module generated by all sections of this type for arbitrary v e the maps

(5.1) consists of

i

The resulting C~(M)-module will be denoted by

1803

..,,:{%

"m,

(5.2)

and we shall call it the module induced by the C~(N)-module ~ under the substitution f*. The elements of the module f * @ are naturally identified with semihomomorphisms in Homf, @*, since the action of an element f*v on G ~ ~* may be defined by

< S*~,~ ) =S* ( ~ , n > , and then any element

~

aif*v i ~ f* ~

will act according to the formula

i

The legitimacy of the definition is easily verified. The following inclusions are obvious:

{f*vl v(7t!l}c"f*l!t~ Hornt~I!i*,

(5.3)

{/*
(5.4)

and if f*(C=(M)) = C=(M), then

since in this case

Z a,i*~', -- Z (i-b,) i * v , : Z i * @m,): i " i

i

i

To each homomorphism of C~(N)-modules X: phism of C~(M)-modules

f.~u f*x:f*~'-,'-~ ,

~,_+~r~ it

Z bin,: S"~. i

is natural to associate the homomor-

f ' ~ a i j"*v ~'~] = ~ l'W a IJr #~ i . f*%tz..~

(5 5)

It is easy to see that in the diagram

(5.6) f*e,Z~, f*qj,,t__~f*~ ," we have

f* (%%X')-~f*%%f*x', f* id@=idff~,

(5.7)

whence it follows that the relation ~ ~ f* ~ is functorial. In particular, the fact that f* is functorial implies the following statement about isomorphisms:

~' ~r If

f*~' ~ f*~".

(5.8)

~, 0" are C~(N)-modules, there is a natural isomorphism

f* (~0~)~/*~|

(5.9)

which is uniquely determined by the relations

f* (Xep)~f*z~f*p, xfiHom(~, ~'), p6Hom(0, ~,)._

(5.10)

It iis easily proved that the homomorphism

f* (~@~)~ f*@| f*~, defined by the first relation of (5.10) is well defined, injective and surjective. also obvious that the diagram

1804

It is

f*(XE)p) ~

'~ f*X~f*O

(5.11)

f* is commutative.

We now prove the fundamental

isomorphism

f*~Homr.@*, as stated in the following Proposition 5.1.

Let

be a finitely generated projective C~(N)-module.

The embedding

f*~cHomr,~*,

(5.12)

=/*, v6@, ~6@*,

(5.13)

uniquely determined by the formula

i s an isomorphism, and i t i s n a t u r a l

thanks t o t h e c o m m u t a t i v i t y o f t h e diagram

/*@cHomr.@* :*z.t .l.rton~,z~ ~crtomt@ c~,~

(5.14)

/*~'cHomf,@" Thus, we have two equivalent isomorphisms:

f *@~Homr,(~*.~,-f*@*_~Homf,@,

(5.15)

since ~__--~**. Since the module Homfr is finitely generated and projective (see subsection 2.2), the same is true of f* ~, and so f* is a functor from the category Mod~ to the category Mode. Proof. We first prove the isomorphism (5.12) for a free C~(N)-module of finite type. Let v I . . . . , v m be a basis in @, qz . . . . . qm the dual basis in ~*. Then f*v z . . . . . fr m is a basis in the free C~(M)-module Homf, 6* of rank m, since

<

> =s* <

>

o

i=/=j.

Consequently, f* @ = Homf, ~, because for all the basis elements we have f~Tsi~f*@. That the diagram is commutative follows directly from the definition of Homf*x* in subsection 2.2. In the general case, let 52 be a direct complement to @ finite type. Consider the diagram

f~@|

-~Homr~(~* |

f * ((~|

in a free module @ Q @

52*

~Homr, (52|

whore T1, 12 a r e t h e n a t u r a l isomorphisms ( 5 . 9 ) , ( 2 . 5 ) , id i s t h e i d e n t i t y whose e x i s t e n c e has j u s t bean proved f o r f r e e modules, and f i n a l l y

il:f*~ are the natural idol 2 = 1 2 ,

embeddings ( 5 . 1 2 ) .

of

isomorphism,

}tatar. @*, i2: f * ~ - ~ Homr. 0" This diagram i s o b v i o u s l y commutative, and so i o I 1 =

that is, i = i I | i 2 = 12oI~ -z is a natural isomorphism.

Hence

ii:/~@-~ Hom~.@ is the natural isomorphism. As a direct corollary of this proposition we obtain a natural isomorphism

/*oHom ~--~Homo/*@.

(5.16)

Indeed, we have a natural embedding f*oHom ~ c Hom=f* ~, since every element of the form f'q, D e ~*,. acts naturally as a homomorphism on elements fev e f*:~ by the formula

1805

( f*% .f*v ) = f * < ~, v

).

(5.17)

0 n t h e other hand, if ~ e Hem| ~. then $ may be identified naturally with elements of Homf, @, since by putting $(v) = <~, f'v> (the easy proof that this definition is legitimate is omitted) we obtain

~(bv) =<~,[*bv>=<~,(i*b)f*v> = (f*b)<~, ]*v> = ( f * b ) ~ ( o ) .

Consequently, by t h e second of t h e n a t u r a l isomorphisms ( 5 . 1 5 ) , we may assume t h a t ~ e f*oHom ~, which implies the reverse inclusion f*oHom @ c Homer* @. One more useful corollary of the natural isomorphisms (5.15), (5.16) is the natural isomorphism

~•

•

Indeed (see subsection 2.2), we have

(f*@)x={~:fittomo/*@l < ~, f*v

> = 0 VvE~} -----{f2EHomr,~ < ~, v ) = 0 vv(~@}=~•

An important and very useful isomorphism is the following natural isomorphism for tensor products, which is easily proved: 9

f * (v|

f * v e f*w,

(5.18)

1

rE@, wE~.~

If X: @-+@', 9:~-+~' are C~(N)-module homomorphisms, we have a commutative diagram

f* (v|

,.

Sr*(x|

f*(zv|

f*

I t*xer*~ (xv)| f* (pw).

Using this isomorphism, one easily establishes the following generalization of the natural isomorphism (5.16). For arbitrary finitely generated projective C~(N)-modules morphism

@,

we have a natural is|

!

f*oHom (@, ~) =Hem(f*~, /*~), where I is defined by identifying the element f*~, with a homomorphism of f*@ the formula

1 ff*~) (ay*v) =ai* (~v), a~C~ (M), v~@. To 9

(5.19) into f*@

by

(5.20)

this w e use the natural isomorphism (see (2.15))

Hem (@,@)~ (Hem@)| and the isomorphisms (5.16),

The correspondence f * ~

(5.21)

(5.18); this gives

f*oHom (@, ~) ~ f * (Hem @)| -- f*oHom @| (Homof*@)| ~ Horn (f*@, f*O). I ( f * ~ ) f o l l o w s from ( 5 . 2 1 ) , (5.18) and ( 5 . 1 6 ) .

The fact that the isomorphism I is natural is readily demonstrated with the aid of the following commutative diagram. Suppose we are given C=(N)-module homomorphisms X @~-~, . ~P. ~ t.

Then we define a functorial homomorphism

Hem (~, ~)

Hom(X,p)

~Hom (~', ~')

by the formula .e~Hom (X, p) fl = pofloZ. With this notation, the diagram in question is

1806

f*oHom (0, .9)/-~Hom (f*@, f*.9) f*oHom(Z,9) 4'

I

4" Hom(#*Z,f*O)

f * o H o m (~', Sy)--->Hom (f*~3', f*~') Its commutativity is easily proved. Proposition 5.2.

Given smooth maps

M4LAN and a finitely generated projective C=(N)-module

@.

There is a natural isomorphism

f*og*@~--_-(gof)*@, /*og* (~)~z (~)=(go/)* (~),

(5.22)

which consequently acts as described by the formula

I : ~ a, ( / * o g * ~ 3 ~ a, (go/)*v,, a,6C" (M), v,6r t

(5.23)

i

In view of the isomorphism (5.15), the isomorphism (5.22) can be written as Homf, oHom~ -----Hem(got),.

(5.24)

If X e Hom( @,-9 ), then the following diagram is commutative:

f *og*@ t*~ (gof)*g3 tg=f) z-~(gof)-O. Proof.

To a v o i d c o n f u s i o n o f n o t a t i o n

of the natural

isomorphisms already

we d e n o t e f * [ C ~ ( L ) = ~, g*[C~(N) = X.

established,

By virtue

we h a v e

f*og*@ _~ Homr, oHomog*@ ~ Homr,og* (Horn @). Consequently, any element

tt = 2 aif*~ can be applied to the element

g'q, q e Hem

f * ~ *~

@, according to the formula (see (5.23))

< u, g*n > = < ~ a,/*og*~,, g*n > = ~ a,~ < g*~, g*~ > = i

=2a,r

< v~, n > = 2 a ~ ( g o f ) * < %, ~ > = <2 a ~ ( g o f ) * v , ~ > = < Iu, ~ >.

i

i

i

It follows at once from this formula that I n d e e d , Vq e Hem @ we h a v e < I u , q> = =
t h e homomorphism I i s w e l l d e f i n e d and i n j e c t i v e . g*~> = 0, so t h e homomorphism i s w e l l d e f i n e d . g ' n > = 0 f o r V q e Hem @, so u = 0, s i n c e { g ' q ; g*oHom @ - i n j e c t i v i t y I

i s o b v i o u s , a s t h e r i g h t - h a n d s i d e o f ( 5 . 2 3 ) e x p r e s s e s an It is also easy to verify that the diagram is commutative.

I f f:M + N i s a d i f f e o m o r p h i s m , r

t h e n t h e semihomomorphism o v e r ~ = f *

~-~ f*@,

~I~v=

D e f i n e a semihomomorphism o v e r

~-1 by

f*v

is a semi-isomorphism. Proof.

I~-, : / * ~ - , - (f-')*o/**, i

7

This map is inverse to I m, because

1807

I~_,oI~o@ ~__f - l * o f * @ ~ (/of-1)*@ ~_ @,

(/*e)

Go4-,/*e 3.

(/-'o f)* /*e

f*e.

Flows on Modules and Vector Bundles

In this section we shall study differential equations in the context of the "functional language" developed above, which enables us to treat nonlinear objects - ordinary differential equations on smooth manifolds - as linear operator equations in an appropriate function space. The construction of a meaningful calculus of the flows generated by these equations necessitates consideration of more general operator equations, on finitelygenerated modules. The main results of this section are the variation formula (see (3.1)) and the general existence theorem for flows on finitely generated projective modules (subsection 3.4). 3.1. Derivations and Connections in the Category Mod M. will prove useful later.

We begin with a remark that

If ~ is a finitely generated projective C=(M)-module other than {0}, it is exact (or strict), in the sense that au = 0 Vu 9 ~ implies a = 0. Indeed, let X be an embedding of ~ in C~(M; R m) as a direct summand: ~ X ( ~ ) 9 ~ = C=(M; R m) for ~ c C=(M; Rm). Suppose that the above assertion is false, so there exists a point x 9 M such that a(x) ~ 0. Then

{0} =a(x)z(~)~={a(x)u(x)I u~X(9) } = {u(x)] u~x(~)} =X(9) ~. By P r o p o s i t i o n 2.1 o f S e c . 2, d i m • x = dim• and t h e r e f o r e whence i t f o l l o w s t h a t {0} = X ( 9 ) , c o n t r a r y t o a s s u m p t i o n . L e t ~ be a f i n i t e l y g e n e r a t e d p r o j e c t i v e m o d u l e . We d e f i n e t o be a n y R - l i n e a r map D: 9-+~ s u c h t h a t f o r Va 9 C~(M), u e

X(~)~ = {0} f o r Yy 9 M, a derivation

of this

module

D (au) = (Xa) u-{-aDu, where X is a vector field on M. This vector field X is uniquely defined. Indeed, if D(au)= Xl(a)u + aDu = X2(a)u + aDu for any a 9 C=(M), u 9 with XI, X 2 9 DerM, then X l = X 2 since (X1(a) - X2(a))u = O, whence, since the module 9, is exact, X1(a) = X2(a) V a 9 C~(M), i.e., X I = X 2. Consequently, we have a well-defined map 8 which, with any derivation D of the module , associates a vector field X:

O(D)=X. Let D e r 9 denote the set of all derivations of ~; this set has a natural C~(M)-module structure, the map @:Der $ + D e r M is a homomorphism of the C~(M)-modules Der 9, DerM, and its kernel ker @ is the set End 9 of endomorphisms (i.e., C~(M)-linear maps of $ into itself). Any vector field X e D e r M definition, so that

is a derivation of the C~(M)-module C=(M) in the sense of this

Der M c D e r C ~ (M) and this inclusion is proper, since any endomorphism of the C~(M)-module C=(M) is also a derivation, but the operator of multiplication by an arbitrary function a0, a ~ a0.a Va 9 C~(M), is an endomorphism. Besides its module structure, Der ~ has the natural structure of a real Lie algebra, in which the multiplication is commutation of derivations DI, D 2 9 Der 9:

[Dl, In this context, 8:Der

+ DerM

D2}=DIoD2--D2oD,

is also a homomorphism of the Lie algebras Der 9 and DerM:

0 ([Db D2] ) = [0 (D0, 0(D2) ]i VD~, DfiDer ~. Indeed, this follows from the following formal chain of equalities, which is true for Va e C~(M), u 9 ~ :

1808

[D1, D2I (au) = (D1oD2-- D2oD1) (art) = D, (0 (D2) (a) u + aD2u) --- D2 (0 (DI) (a) tz + aDlu) = 0 (D:)o0 (Dz) (a) u + 0 (D2) (~z) D1 u + + 0 (DO (a) D2tt+aDloD2tz-- 0 (D2)oO (D1) (a)"u-- 0 (D1) (a) D2u--- 0 (D2) (a) Dxtt-- aD2oDttz = [0 (D1), 0 (D2)] (a) u + a [Dx, D2] u.

8:Der

Proposition i.i. Let ~ be a finitely generated projective C~(M)'module. + D e r M has the following properties: i) kerO = End

The map

~.

2) There exists a module homomorphism 7:Der M + Der ~ such that

0oV~---~idDerM.

3) Der ~ = E n d ~ @ v D e r M ~ E n d g@Der ~f. Proof. Part 1 has already been proved, so we proceed to Part 2. It will clearly suffice to consider the case in which ~ is a direct summand in C=(M; Rm): there exists a submodule ~ c C~(M; R m) such that C=(M; R m ) = ~ @ ~ . Let pr~ be the endomorphism of the free module C=(M; Rm), that carries an element w = u + v(u e ~, v 9 @) to the element u 9 ~. For any vector field X 9 DerM, define a derivation X

of C=(M; R m) by the formula

~ (~ .....

w ~ ) = ( X w ~. . . . . Xw~),

Now put

w = ( w ' . . . . . w~)6C~(M; R~). -->

Vx = pr3oX I~, thus defining a map V:DerM-+Der

~,

X~XTx, X~DetM.

Then 7 is a homomorphism of the modules Der ~ and Der M, since Va e C=(M), u e

V x (au) = pr~oX (a~ = pr~ ((Ya) u + aXu) = ~Xa) u + a p rsoXu = (Xa) u + a V x u

and V a x (u)----pr~ (aXu) Moreover,

by the very definition,

=

aVx~.

O(7x) = X, so that

OoV~--~-idDorM. Part 3 of the proposition

is a direct Corollary of Parts i and 2.

This completes the proof. The map V figuring in the statement of Proposition i.i is known as a connection on the module If V is a connection on 8(D) = X can be expressed as

~, then 7x e Der ~ and any derivation D 9 Der ~ such that

D=Vx+H, where H is an endomorphism of

~.

The operator 7X is called covariant differentiation field X.

in the direction of the vector

We shall come back to connections again in this section (subsection 3.6), and also in Sec. 4. To end this subsection, we note that if ~ is a finitely generated projective module, then the set Der ~ of its derivations is also a finitely generated projective module, so that for any integer k the iterates

1809

Der (~Li!(Der ~)...) k tad form a series of finitely generated projective modules. Indeed, Der ~ m End ~ 9 DerM. Since ~ is finitely generated and projective, it follows that End ~ is a finitely generated projective module (see See. 2, Corollary to Proposition 2.2). That the module D e r M is projective follows from Proposition 2.1 of Sec. 2 (see Corollary 1 to the latter). 3.2. Nonstationary Vector Fields, Derivations and Flows on Manifolds and Projective Modules. From this point on we shall assume that the algebra C~(M) is endowed with the standard topology that makes it a Frechet space (complete metrizable locally convex space). This topology may be defined in various ways, e.g., through a family of seminorms

11a II~,~,•

sup _ _ I X , o . . . oXja (x) l + Pa (x) l ,

a~C ~ ( M ) ,

~E~ L T ,

where K ranges over all compact subsets of M, s e N, X = (XI, ..., Xs), X" e DerM. If the manifold M is regularly embedded in R d, then it is readily seen that t~is topology coincides with the topology of C~(M) described in [2], [4]. Let ~ ( M ) denote the associative algebra of all R-linear continuous maps of C~(M) into itself with the topology of simple (pointwise) convergence: a sequence of operators {Ti} c ~ ( M ) converges to the zero operator in ~ ( M ) if and only if Tj § 0, j + ~ for V a e C~(~). It follows at once from the definition of the topology in C~(M) that D e r M c ~ ( M ) . In addition, IsoM c ~'(M), where IsoM is the set of all automorphisms of the algebra C~(M); the elements of the latter will sometimes also be called diffeomorphisms (see Proposition 2.1 of Sec. i, which justifies this terminology). Let ~ ( M ; R) denote the set of all continuous linear functionals on Ca(M); the points x of M (considered as linear functionals, which have the multiplicativity property - see Sec. i, Proposition i.i) are obviously elements of ~(]W; R). The free module C~(M; R m) is canonically equipped with the topology of a Frechet space - the topology of the direct product of m copies of C=(M). If 8 is a finitely generated projective C~(M)-module, one can endow ~ with the topology of a Frechet space, defining it as the projective (or initial) topology generated by the dual module Hom ~ : the weakest topology relative to which all the maps

C (M), u~+,

z6Hom

are continuous. It follows from the definition that any homomorphism X e Hom (~,~) of modules ~,~, is continuous; this is true, in particular, of the endomorphisms of 8. It also follows that if ~ is a submodule of the free module C~(M; R~), then the topology induced on it from C=(M; R TM) is precisely that defined above. Even more: it can be proved that the topology of ~ is the projective topology generated by the family {Homp~; p e IsoM}, in particular, this implies that any module semi-isomorphism P e Homp(~,~) over a diffeomorphism p is continuous. We prove that the set ~(~) of all R-linear continuous maps of a finitely generated projective module ~ into itself contains the set Der(~) ~ of derivations of the module. Indeed, let ~ be a direct summand in C~(M; Rm). Then (see Proposition i.i of this section) any derivation D 9 Der ~ with e(D) = X can be expressed as D = VX + H, where H is an endomorphism of ~, and VX is the canonical corvariant differentiation, VX = pr ~oXI~,"X~ a derivation of the free module C~(M; Rm), (~, ..., ~)=

(x~' .....

x~)

v (~ .....

~ ) ~ c ~ (M, R~).

Since H 9 End ~ , pr~ 9 End C~(M; R m) are continuous, it follows that to prove D continuous it will be enough to show that the derivation X of the free module C=(M; R m) is continuous; this in turn follows from the continuity of the field X 9 Der M and the definition of the topology in C~(M; Rm).

1810

Using the topologies of C~(M) and the finitely generated projective C~(M)-module 9, one can transfer all the fundamental constructions of analysis to one-parameter families a t , t e R, and ut, t e R, of elements of C~(M) and ~, respectively. The continuity and differentiability of these families with respect to the parameter t e R require no special explanations, since C~(M), ~ are linear topological spaces. A family at, t e R, of elements of C~(M) is said to be measurable if Vx e M the scalar function x ~ a(x); is measurable; similarly, a family u t, t e R , of elements of ~ is said to be measurable if VX 9 H o m ~ the family a t = <X, ut>, t e R, of elements of C=(M) is measurable. A measurable family at, t 9 R, of elements of C=(M) is said to be locally integrable if the scalar function t ~ llatils,k, X is locally integrable for any seminorm Jl"lls,K, . The integral of a locally integrable family at, t e R, from t I to t 2 is defined to be the funct ion tz

x ~ j" at(x)clt. t,

It can be proved (see, e.g., [3]) that this function is indeed a member of C~(M) and moreover for any seminorm I["[Is,K, and t 2 ~ t I tj ta

5",K.X

tx

Accordingly, a family u t, t e l~, of elements of ~ is said to be locally integrable if V X 9 Hem 3 the family a t = <X, ut>, t 9 R, of elements of C~(M) is locally integrable. In t,

that case there is a uniquely determined element

~ utdt of

3, called the integral

of

the

t,

(locally integrable) family from t I to ti; it is characterized by the relation (subject to the natural identification of HomoHom 3 with 3 : t'2

fj

< z, S.,at > = j tt

< z, u, > at. ft

A family at, t e l{, of elements of C~(M) is said to be absolutely continuous if there exists a locally integrable family bt, t e R, such that for any to, t e R, t

at = a t e +

J' ~

b~dT:.

to

I n t h a t c a s e a t , t 9 R, i s d i f f e r e n t i a b l e f o r a l m o s t a l l tER, a n d d / d t a t = b t ( f o r a p r o o f see, e.g., [1]). Similarly, a family u t, t e R, of elements of 3 is said to be absolutely continuous if there exists a locally integrable family v t, t 9 R, of elements of 3 such that Vt, t o ~ R , t

ut = Uto+ j" ~,d*; to

here, again, for almost all t e R we have dut/dt = v i. We can now endow the algebra ~ ( ~ ) , like ~(M), with the topology of simple (pointwise) convergence. Then measurability, local integrability and differentiability of a family Tt, t e R, of operators in ~(~) (or in ~ ( M ) ) are defined by stipulating that for any u e ~ (or any a e C~(M)) the family v t = Ttu, t 9 R (b t = Tta, t 9 R) have the appropriate property. The derivative of a family Tt, t e R, of operators in ~ ) ferentiable at a point t o is defined as the linear operator

T ;.

(or in 5f(M)) which is dif-

{r , . + o , -

That the derivative is continuous follows from the Banach-Steinhaus Theorem. The integral of a locally integrable family of operators Tt, t c, in 2 ( 3 ) (or in ~ (M)) is the operator

1811

T~clr tt.-=~T4tdT, ttE~

(uEC~(M)).

i,

The proof that it is continuous follows the same lines as in [3]. A family Pt, t e R of operators in ~ ( ~ ) ( o r in ~ ( M ) ) is said to be absolutely integrable if there exists a locally integrable family Qt, t e R, such that Vt, t0ER, t

Pt=P,o+ S Q,dr. to

In that case for almost all t e R we have dPt/dt = Qt Note that if families of operators Tt', Tt" , t e R, in ~ ( ~ ) o r in ~ (M)) are absolutely continuous, then the family T t = T'oTt", t e R, is also absolutely continuous and Leibniz' formula holds:

a-T Tt~ ~---~7-~ + Tt~ at " The above notions carry over quite naturally to one-parameter families of functionals St, t e R, in ~ ( M , R ) . A nonstationary (or time-dependent) vector field X t, t ~ R, or simply a field, on a manifold M is an arbitrary locally integrable family Xt, t e R, of elements of D e r M c ~ (M). If a field Xt, t e R, is given, we can consider an ordinary differential equation

Otd__At = A~oX,

(2. i )

with initial condition

A'l'=~

(2.2)

where id = idc~(M ) is the identity map of C=(M) and the unknown is a family At, t e R of elements of ~ ( M ) ,

and the analogous equatibn d B t-'--'--Xt"oBt

(2.3)

B,It =o= id,

(2.4)

with initial condition

known as the equation adjoint to (2.1). We emphasize that these equations are linear. A solution of equation (2.1) with condition (2.2) is, b y definition, an absolutely continuous family At, t e R, of elements of c~(M) satisfying equation (2.1) for almost all t e R and the initial condition (2.2). Absolute continuity guarantees that (2.1), (2.2) together are equivalent to the integral equation t

At = At + .[A~oX~d~.

(2.5)

0

Similarly one defines a solution of equation (2.3) with condition (2.4) and the equivalence of (2.3), (2.4) to the integral equation

Bt= Bo-- 5~X~~ ~.

(2.6)

0

An absolutely continuous family T t, t e R, of elements o f ~ ( M ) is said to be invertible if every operator T t (t e R) is invertible and the inverse family Tt -I, t e R, is absolutely continuous. A flow on a manifold M is an arbitrary absolutely continuous family Pt, t e R, of diffeomorphisms (i.e., an absolutely continuous family of automorphisms of the algebra C=(M)) satisfying the condition P0 = id.

1812

If Pt, t 9 R, is a flow on M, then the family ft, ft* = Pt of diffeomorphisms of M is absolutely continuous and is also called a flow. It follows from the Inverse Function Theorem that the family ft -I, t e R, is absolutely continuous, and hence so is the family pt ~I = (ft-l) *. Thus every flow is invertible. Proposition 2.1. If either of equations (2.1), (2.2) and (2.3), (2.4) has an invertible solution, then the other equation also has an invertible solution. These solutions are unique and are mutually inverse flows. Proof. We first observe that any solution At, t 9 R, of equation (2.1) with initial condition (2.2) is a left inverse for any solution Bt, t 9 R, of equation (2.3) with initial condition (2.4), since by Leibniz' formula d n n dAt d---f~toDt=--g~Bt+Ato

dd~

,, --~.AtoAtoBt--AtoXtoBt-~-O,

and since the family AtoB t is absolutely continuous we have AtoBt~---id. Continuing: if Tt, t 9 R, is an invertible solution of one of equations (2.1) or (2.3), then Tt -I, t 9 R, is an invertible solution of equation (2.3) or (2.1), respectively. Indeed, since id=TtoTt -I, it follows by Leibniz' formula that 0

dTtoT-1-dT-t ="-d~ t -T-Tt ~ dt '

whence we have ~ r 7 ~ __ dt

--~lt

~--1

dTt o-~oT

--I t .

dT~ 1

Consequently, if, say, Tt, t e R, satisfies equation (2.3), then

at

T?l~ ~

* TFI~176176 I : T - lto .At. Therefore the solutions are unique. Indeed, if Tt, At, t e R, are solutions of (2.1)-(2.2) and A t is invertible, then by what we have proved B t = At -I is a solution of (2.3) and so TtoBt----id, whence it follows that T t = Bt -I = A t . Similarly, if Tt, Bt, t e R, are solutions of (2.3) and B t is invertible, then A t = Bt -I is a solution of (2.1) and again AtoT t = id, so that T t = At -I = B t. Since the solutions of equations (2.1), (2.2) and (2.3), (2.4) are invertible and unique, it follows (for the full details see [3]) that At, B t are mutually inverse flows. This completes the proof of Proposition 2.1. Let Pt, t e R, be a flow satisfying equation ~2.1). Then for any point x e M the family xopt, t e R, is an absolutely continuous family of multiplicative functionals in ~ ( M : R), i.e., an absolutely continuous curve on the manifold M satisfying the ordinary differential equation dxt

d

t6R.

dt = Y ~ 1 7 6 1 7 6 1 7 6

Conversely, if ft(x), t e R, is an absolutely continuous family of diffeomorphisms of M satisfying the equation d d--~ L (X) = Jet (X)~ / o (X) = X, then Pt = ft ~, t e R, is a flow satisfying (2.1), since the family P t t

e R, is invertible

and ft(x) = x~ so that d/dt ft(x) = d/dt xopt = xod/dt Pt = x~176 since x is an arbitrary point, that

whence it follows,

dpt dt = Pt~

Consequently, the question of existence of flows reduces to verification of existence conditions for a flow defined by an ordinary differential equation ex, = x,oXt,

dt

Xo = x.

(2.7)

1813

If one considers this as an equation for an absolutely continuous family xt, t e R, of (not necessarily multiplicative) functionals in ~ ( M ; R) then under the assumptions of Proposition 2.1 this equation cannot have more than one solution. Indeed, if Bt, t 9 R, is a flow satisfying (2.3), then, "multiplying" (2.7) by Bt, we obtain

d---Xto~t-~-xtoXtoBt=--xtodBt dt dr' whence

xtoBt = X, and so x t = xoA t, t 9 R.

Since At, t 9

is a flow, it follows that x t = xoA t is a multi-

plicative functional ~ and therefore equation (2.7) cannot have solutions which are not absolutely continuous curves in M. Thus the only possible invertible solutions of equation (2.1) is the flow defined by the ordinary differential equation (2.7). It follows from the existence theorem for solutions of ordinary differential equations on a manifold that the solution of equation (2.7) exists locally with respect to time t and locally with respect to x. A flow will exist (and then, by Proposition 2.1, it is unique) if, for example, the supports of the fields X t, t 9 R, are contained in some fixed compact subset K c M:{suppXt; t 9 R} c K. A nonstationary vector field Xt, t 9 R, is said to be complete if there exists a flow Pt, t 9 R, satisfying equation (2.1). t

Let exp

t

X~d~ and exp

-XTd~

denote the (unique) solutions of equations (2.1),

0

(2.2) and (2.3), (2.4); if they exist then, by Proposition 2.1, they are mutually inverse flows, as expressed by the formula

exp

X~d'~

= exp

- - X~d'r

(2.8)

0 -..+ /~

We call

exp J

.+-_

X~dT and

0

t

exp~

XTdT the right and left chronological exponentials,

0

respectively, generated by the (nonstationary) vector field Xt, t 9 R, on M. The arrow in the notation for exp indicates the direction in which the field X t is "taken out" when the exponential is differentiated with respect to t: f

t

t

d -4 -4 g-F exp S X , d , = exp j' X , d z o X t , 0

0

d . . . . ~ exp ; X , d * = 0

t

Xtoexp S X~d~'. 0

We now consider the analogous notions for operator equations on a finitely generated projective C~(M)-module {L A flow on ~ is an arbitrary absolutely continuous family of semi-isomorphisms Pt, t 9 R, of the module such that P0 = id. If Pt, t 9 R, is a flow on a C~(M)-module ~, it uniquely determines (see Sec. 2) a a family Pt 0 (lOt), t6R, =

of automorphisms of C=(M) or, equivalently, an absolutely continuous family of diffeomorphisms Pt*, t e R, of M. Since the family Pt, t e R, is absolutely continuous, it follows that the family Pt = e(Pt), t 9 R, is also absolutely continuous and, since P0 = id ~, Pt, t 9 R, is a flow on

M.

*If x is multiplicative.

1814

A flow Pt, t e R, on :~ (if it exists) is naturally considered as the mesult of extendding a flow Pt, t e R, on the base (i.e., the manifold M) to ~a flow on the mo~. ~ ~and a correspond:ing vector ~bundle. The existence of this extension will be proved bel
the inv,erse family Pt -I,

OyPZ~ = ~ ~(P~y~. L e t P t , t 9 R, be a f l o w on a f i n i t e l y g e n e r a t e d p r o j e c t i v e family of operators -I d d -t

C~(M)-module

3.

Define the

D t = P t O~T Pt, At=2-f Pt ~

Then Dt, A t , t e R, are locally integrable families of derivations of ~

O(Dt)!=Xt,

and we have

At= --Dr,

where X t, t 9 R, is the vector field corresponding to the flow t

Pt=O(Pt)=exp f X~dn o

Indeed, considering,

s a y , Dt ,

t e R, we i n f e r

from L e i b n i z '

f o r m u l a t h a t Va ~ C~(M),

u ~

D t (au) =Pt -i ~ d Pt (au)=Pt-1 o~a p # . P,u= =PT'

dPt- u) = P71 (Pt(X,a)Ptu + (pta ) aP,._~u] <-'~--t/dPt= o u ".-F(Pta)---~ -~ (X,a)u+ aP-;t ~

u=

(X,a)u + aDtu.

Denote t

t

t

0

Formally, these right and left chronological exponentials of the families D t and -D t, t 9 R, of derivations of ~ are solutions of the operator differential equations

apt "-~-'----"P ot D ,t

Po ---id,

(2.9)

dQt -~F=--DtoQt,

Q0=id,

(2.1o)

and

which are analogs of the pair of adjoint equations (2.1), (2.2) and (2,3), (2.4). Solutions of these equations are defined to be any absolutely continuous families Pt, Qt, t 9 R, of elements of ~(~),satisfying the equations for almost all t e R and satisfying the appropriate initial conditions. We can prove the following analog of Proposition 2.1. proposition 2.2. Let Dt, t E R, be a locally integrable family of derivations of a finitely generated projective module ~. If either of equations (2.9), (2.10) has an invertible solution, then the same is true of the other equation; both solutions are unique and are mutually inverse flows i

D~d*

exp

= exp

-- D,d*----- PT t.

0

Under these assumptions t

t

Pt = 8 e-xp ! D~dx = exp. OD~dz,

1815

t

p7 l=Oexp

t

--D~d,c=exp

--OD,dT 0

are flows on M generated by the nonstationary vector field 8D t, t E R. The proof follows the same lines as that of Proposition 2.1. In fact, any solution Pt, t e R, of equation (2.9) is a left inverse for any solution Qt, t s R, of equation (2.10), because d dPt dQt dr- Pt~ ~ "~-~ t 3c pfo..~- ~ PtoDtoQt__ PtoDtoQt __~O. On the other hand, if Pt, t ~ R, is an invertible solution of one of equations (2.9), (2.10), then the inverse family Qt = Pt -I, t e R, is a solution of the other and therefore the solutions

I

t

exp

I

D~d~ or exp

-D~d~ (if it exists) is a solution of equation (2.9) or (2.10); 0

since the flow is invertible, there are no other solutions (thanks to uniqueness). This completes the proof. COROLLARY. exists a flow

Let ~

be a finitely generated projective C=(M)-module and suppose there

exp D~d~,

tGR.

Then there exists a unique solution of the differential equation

(2.11)

e

d t Ut = D tut with initial condition

Ut It=0= ~6~.

(2.12)

(By a solution of this equation we mean any absolutely continuous family u t, t eR,, of elements of ~, which satisfies the initial condition (2.12) and equation (2.11) for almost all t e R.) Proof.

Existence is obvious:

the solution is the absolutely continuous family u t =

t

exp 1.

DTd~u0, since t

c/t = • e x p

t

D:d'~Uo==Dtoex-p D:d'~uo~ Dtut. 0

0 ,

By Proposition 2.2, there exists a flow exp ~

-D~d~ =

i

0 U t ,

t

DTd~

Therefore,

if

0

R, is an arbitrary solution of the equation, then t

t

t

whence it follows that 1

t

0

0

i.e., the solution is unique. 3.3.

Basic Formulas of Chronological Calculus.

In this subsection we shall establish t

a number of formulas for the chronological exponentials exp

t

D~d~ and exp 0

D~d~ gener0

ated by a locally integrable family of derivations of a finitely generated projective C=(M) -

1816

module ~

and corresponding formulas for the chronological expoenentials t

t

0

t

0

t

0

0

Uniqueness arguments (see Proposition 2.2) at once imply the formulas t

t

exp ~ Ot_r = ~_xpexp~ O~d ~, 0

0

ex? ~ D d v = exp ~ D~d,o~ O cl~ V t,t,fi R, 0

tt

0 t

T

0

(for a strictly

0

monotone a b s o l u t e l y continuous f u n c t i o n O, 0(0) = O, O(t) = T).

The most important formula is the variation formula, which expresses a flow perturbed

by a l o c a l l y i n t e g r a b l e family Dt", t ~ R, of d e r i v a t i o n s , t

;-x-bj" (ol + o;)e 0

in terms of the unperturbed flow f

exp -"~D'dr. 0

Before formulating the proposition, we observe that if ~ is a finitely generated projective C~(M)-module, then the set Der ~ of its derivations is a finitely generated projective C~(M) module (see Sec. 3, subsection 3.1). Any derivation D ~ Der ~ of ~ generates a derivation a d D of the module Der ~ in a natural way:

ad ~ D ' = [D, D'] VD'~Der~. Indeed, the fact that a d D is a derivation is proved formally: V a 9 C~(M), D' ~ Der adD(aD') = [D, aD'] = D(aD') - aD'oD = X(a)DoD' + aDoD t - aD'oD = (Xa)D' + adD(D'). Moreover, 8(ad D) = 8(D). Consequently, we can speak of the chronological exponential (flow) of the family adDt, t ~ R, on the finitely generated projective module Der ~, where Dt, t 9 R, is a given family of derivations of a finitely generated projective module As before (see Sec. i), given a semi-isomorphism P : ~ - + ~ operator

of ~ , we let A d P denote the

D Ad PD:Der~-+Der ~, Ad PD=PoDoP-1.

Proposition 3.1. Let Dt', Dt", t 9 R, be locally integrable families of derivations of a finitely generated projective C~(M)-module ~. If there exists a flow t

0

then the flows t

ex? ;

t

(O'~.-[-D;)d~, exp ~ exp ~ adDodOD"d~

0

0

0

either both exist or both do not exist, and if they do they satisfy the variation formula ---~ t

0

t

0

T

t

0

1817

t

t

=exp S Ad

t

exp S O'odOO;d~:oexpS O',d~.

0

0

(3.1)

0

In particular, t

t

Ad exp d D,d'r = exp 5 ad D'~dT.

(3.2)

0

By the definition of Ad, for VD e Der ~

Proof.

t

f

-Y

d

t

Ad exp -- D',dT. - s D'~d,D----exp D~d,oDoexp ' 0

0

Differentiating with respect to t, we obtain d

t

---~

t

t

t

---~

~f Adexp S D',d,:=ffxp ~ D'~dzo(DjoD--DoD;)ogxp5 --D~cl* =Adexp S D',dzoadO;D. 0

0

0

0

t

Thus, AdexpS. D.c'd'r is a solution of the equation 0

a_ dt pt=PtoadD; ' with initial condition P0 = id, in the module Der ~, and consequently, by Proposition 2.2, t

This proves formula (3.2).

To prove (3.1), we write t

t

ex? I (D~--}-D~) d~=Ctoexp I D~dT. 0

(3.3)

0

Then, differentiating this relation, we see that the required flow C t (if it exists) satisfies the equation

-~t (Ct~

t

t

t

0

0

0

ac, -~ I D;d, +CtoeX*plD'cd~D;' = C,oexpI (D~ q-D:) d~,

'

or d

t

-'-*-

-d-fCt---CtoAdexp ; D~d~D;, 0

so that by Proposition 2.2 the flow C t is given by the formula t

Ct=exp S Adexp S D'odOD~dx. 0

Consequently,

0

in view of (3.2), (3.3), we obtain (3.1).

This completes the proof. A similar formula can be established for the left exponential. implies the formula t

t

t

~

Note that (3.1) also

t

t

exp ~ (Diq- D:)dx=ffxp "D~d~offxpS Adexp S D;dOD~d* --~exp f D:droffxp S exp ~ adD;clOD:d,. 0

1818

0

t

0

0

t

(3.4)

In particular,

if ~

f i e l d s on M, i . e . ,

= C~(M) and D t' = X t, Dt" = Yt, where Xt, Yt are (nonstationary)

vector

elements of DerM, we obtain the following formulas: t

t

"c

t

exp ] (X, + Y,)dz = exp S exp j' ad X odOYTd~o 0

0 t

X,d,----

0

'~

t

Adexp ] Xodor,d~oexp ~ X,d'~,

=exp

0

(3.5)

0

a?:ad X~d'~-~-Ad e-xp S X~d'r, t

t

(3.6)

0 t

t

t

%

ex*O] (X.~+ Y~)dz-----exp S X:lzoexp ~ ex~p] adXedOrtd'~= 0

0

0

t

1

t

-----e~xp~ A%dxoexp S Adexp S XodOY~d~, 0

0

(3.7)

t

and corresponding formulas for the l e f t e x p o n e n t i a l s . Using the v a r i a t i o n formula ( 3 . i ) one can prove an important formula for the d e r i v a t i v e with respect to a parameter e of the chronological

exponential

exp,[ D, (s) d~, 0

where Dt(c), t 9 R, is a locally integrable family of derivations of a finitely generated projective C~(M)-module ~, on the assumption that the dependence on the parameter e 9 R is smooth:

o-~exp yD,(8)d,= 0

exp adDo(8)d0 0

dxoexp D,(~)a,.

(3.10)

0

Indeed,

exp ,D~ (e q- 6s) dx-- exp D,(s)d'~ =

O'-~exp D, (a) d~ = lim ~ 0

6~--~-0

0

= 68-~qlim~ lexp ~ D~ (s)+68 ----lira 1

--~----t o~ ~e,ld~-- exl~.[6D~(8)d'~ =

exPSexp [adOo(e)d0

6e.-,-O

0

+0,(8) dl:o

0

oexp S D~ (e) d'r-- exp y D~(e)d'rj = 0

=lim-~

0

exp D~(8)d~+

6e~O

~

exp adD0(8)d0 ---~--+o~(8) dxo 0

0

oexp DP)d~+...-t - --exp

D(e~

=

0

0

.

0

Similar arguments yield the formula:

Oexp Os

adD,(s)d~:D'=

0

FD' 9 D e r ~

e~pyadDe(8) dO o

dx, expSadX,(e)dxD'

(3.11)

o

Another important formula i s a m u l t i p l i c a t i v e analog of the formula for i n t e g r a t i o n by p a r t s , e s t a b l i s h e d by A. V. Sarychev [6]: I f Dr, t 9 R, is an a b s o l u t e l y continuous

1819

family of derivations of a finitely generated projective C~(M)-module able and bounded function, then t

t

I

0

0

0

and u is a measur-

(3.12) t

where v(t) = ~

u(~)d~, D~ = d/dv D~

0

!

e~

!

] Dd,=exp

~ D d x VDfiDer ~.

0

0

t

Let Pt = exp S D%d% be a flow on a finitely generated projective module

~,~ generated

by a locally integrable family of derivations Dt, t ~ R. Then (see Sec. 2) we can consider the adjoint flow Pt ~'c in the module Der ~. By definition (see Sec. 2),

< P*,X,# > =p71 < X, P,u ), Pt=O (P,) V~3, Now for V X e Hom

x 6 H o m 3.

3, u ~ d

,

d

1

d-T < P,x, u > = ~f P7 ( X, P,u ) = -- Xept -~ ( X, Ptu ) +

(3.13)

+ p-;lo < %, PtoDtu > , and so, letting Dt~'~ denote the operator D ; : ~*-+ 3",

DtX ---- - - Xtox + x o D .

(3.14)

we see that Dt*, t 9 R, is a locally integrable family of derivations of the dual module Hom 3 - - - - ~ * , since Va ~ C~(M), X e Hom D~ (aX) = -- Xtoaxt + xoDt--= -- ( X t ~ X+ Dtz,

and moreover, O(Dt*) = -X t.

Therefore (3.13) yields

0

In particular, applied to the flow t

0

on the module D e r M (X t, t 9 R, being a nonstationary vector field on M), this construction yields the important formula ad

'

*

(

'"

~'

'~-'

(3.1s)

where Lxt~ is the Lie derivative of the i-form ~ along the vector field Xt:

Lx,o ----Xto-- o0Sd Xt = -- (ad X,)*.

(3.16)

t

The chronological exponential exp ]. D~d~ uniquely determines the locally integrable 0

family Dt, Ini:

1820

t 9 ~, o f d e r i v a t i o n s

of ~

that

generates

it.

H e n c e we h a v e a w e l l - d e f i n e d

map

t

expS ' D~d~.

which we call the chronological logarithm of the flow

0

It follows from the variation formula that the chronological logarithm of a product of t

-t

flowsixp!iD.r'd'ro

e x p ~ D,r"d'r i s e q u a l t o 0

11

] O~a,~xp [ D;a, ----] Adeip o

/

~

D;aoD, + D:

d'r

o

(3.17)

3.4. Existence Theorem for Flows. We are now going to prove the fundamental existence theorem for flows in arbitrary finitely generated projective modules ~ over C~(M). It will follow as an easy corollary from a general theorem concerning the "stability of a finitely generated C~(M)-module '' - Theorem 4.2 below. m.

3.4.1. Existence Theorem for Free Modules. COnsider the free module C~(M; ~m) of rank On this module we have a well-defined canonical connection V: D e r M + Der C~(M; ~m),

X + Vx = X, where ~X(u 1 , ..., u m) = (Xu I, ..., Xu m) u = (u I . . . . , u m) e C~(M; ~m) (see subsection 3.1 of this section). If D x is an arbitrary derivation of C~(M; ~m) and 8(Dx) = X, then by Proposition i.i of Sec. 3, Dx=Vx+H,

where H is some endomorphism of C~(M:

R~), which

whose elements h8 ~ are elements of C~(M). D x (u I . . . . .

I f P i s an a r b i t r a r y may be r e p r e s e n t e d as

may be thought of as an m • m matrix (h8 a)

Thus,

u = (u I, .... u m) e C~(M; R ~

u ~ = ( X u 1, . . . , X a m) +

....

h~

.

s e m i - i s o m o r p h i s m o f C~(M; R m.) o v e r an a u t o m o r p h i s m p e I s o M ,

it

P=poA, where p is the canonical semi-isomorphism of C=(M; R'n), generated by p: h ( g l , . . . . /~m)= (p/~l . . . . .

pu m)

V~ ~-

(U1. . . . .

/~m)~C ~~( m ; Rm),

and A i s an a u t o m o r p h i s m o f C~(M; Rm), t h a t i s , an m • m m a t r i x A = ( a ~ a ) , whose e l e m e n t s a~ a a r e e l e m e n t s o f C~(M) s u c h t h a t d e t ( a B a ) ( x ) ) ~ O,

LEMMA 4.1.

Let p e IsoM, H e End C~(M;

Rm). Then

(4.1)

Ad >H---->H= (pk~). Proof.

By direct calculation:

V u = (u I . . . . , u m) e C~(M; R m)

=i,

--_ \~=i p ( h i p - ~ u = ) . . . . .

LEMMA 4.2.

there exists

~p(h2p-'u ~=I

....

~) =

o,, . ) -

~%~ip h ~ u ~,, . . . , ~" y1 p k~u~,

----( p H ) (u ~, . . ., u"9.

Let X t, t e R, be a locally integrable family of elements of D e r M such that

a flow

t

&p I::,a, 0

1821

Then the family of derivations VXt, t e R, where V is the standard connection on C=(M; R~), is locally integrable and there exists a flow

0

0

Proof.

A

A _--_

X , d x u 1. . . . , exp

exp

X,dx

t -~I Therefore, by the uniqueness of the flow exp

dT

Xtu' . . . . . X , u ~ .

X~d~ ( P r o p o s i t i o n 2 . 2 ) ,

t

t

exp S X,d*=exp S Vx,d*. 0

0

This proves the len~na. LEMI~ 4.3. Let Nt, t e R, be an a r b i t r a r y l o c a l l y i n t e g r a b l e family of endomorphisms of the free module C~(M; Rm). Then there exists a flow t

exp

H,d~.

t

Proof.

By d e f i n i t i o n , the flow e-xp~

H~d~ ( i f i t e x i s t s ) is a solution of the equation

0

dPt.~PfHt,

Po=ld.

~t Let x ~ M be some point of M. sides of (4.3) we obtain

(4.3)

Applying the multiplicative functional x:C=(M) + R d

to both

d

xoTf P, = xop,oH, = ~-[ xP~ = xPt" xHt,

(.)), that is,

a xPt = xPt. xHt. a--/-

(4.4)

Equation (4.4) is an ordinary matrix differential equation (depending on a parameter x ~ M). Consequently, it has a unique solution which is a smooth function of x e M and may be expressed as a uniformly and absolutely convergent matrix series

xH,dv---I+~,~Sdx... S d~k(xH,,)...(xH~)=I-rl-xo 2

xPt=exp 0

k=lO

0

k~lO

d*,... 5 d*k(H,,o .... H,,). 0

At the same time, t

exp ~ xH,dx6Aut C~ (71,/; Rm). 0

Consequently, equation (4.3) has a solution (which is unique, by Proposition 2.2 of this section), and the solution satisfies the formula t

xo 0

1822

t

H~dx = ex

xHxdx.

(4.5)

This completes the proof. A simple check shows that if D 9 Der ~, H 9 End THEOREM 4.1.

~, then [D, H] ~ End

I~

Let C~(M; R m) be the free module of rank m, DXt , t e 'R,a locally inte-

grable family of derivations of this module, 8(Dxt) = Xt, such that there exists a flow t

0 t

Then there exists a flow exp

DX dT.

If DXt = VXt + Ht, H t 9 EndC~(M;

Rm), then

0 t

t

0

Proof.

Z

0

t

0

(4,6)

0

Since the family DXt, t e R, can be represented as DXt = VXt + Ht, where VXt is

the canonical covariant differentiation of the free module in the direction of Xt, and Ht, t e ~, a locally integrable family of endomorphisms of C~(M; Rm), we can use the variation formula (3.1) to prove the existence of the flow

Dx~d~= exp ~ (~TxT+ HO d~. 0

0

In fact, formally,

exp j'Dxr

and the flows e x p ]

j'Vx~dr

= exp ~ exp t ad VxodOHr

0

0

DxTd~ and exp~

0

0

e4 3

0

0

ad VXodOHTdT, either both exist or both do not

0

t

exist, thanks to the existence

(by Lemma 4.2) of the f l o w exp j'

VxTd~.

We shall prove the

0

existence of the flow t

exp e•

ad Vx~dOH,d'r

Indeed, by (3.2),

0

0

0

0

and the flows on the left and right of this equality either both exist or both do not exist. But the flow r

exp VxodO exists, and hence so does the flow

Since

1823

e-xp ~ V x, d'r-=

p X,d'r ,

0

it follows by Lemma 4.1 that

,

t A

0

0

This means that there exists a locally integrable family of endomorphisms t

-+

Ade-xp ~

t

m_~

Vx,d~Ht----exp~ adVx,dvHt

0

of C~(M; Rm).

0

By Lemma 4.3, this implies the existence of t

'g

0

0

Consequently, by Proposition 3.1, there exists a flow t

.[ 0

f

t

= Cx?S cw, +-oe

t

t

=ox? .f a; S WoaO-,e oox?

0

0

0

0

3.4.2. Stability Theorem for Finitely Generated Modules. Let ~ be a finitely generated submodule of the free module C~(M; Rm),and D a derivation of C~(M; Rm). We shall say that D preserves ~, if

Du~

Vu~.

Now let Dt, t 9 R, be a locally integrable family of derivations of C~(M; Rm). It is natural to say that this family preserves 9, if Dtu 9 ~ V u e ~ and for almost all t e R. However, it need not follow from this that for every system of generators u I, ... us of the module there exist locally integrable families b~,tS, t 9 R, such that I

D,u~= Y, b~d:.=, ~-----1..... l.

(4.7)

We shall therefore say that a_family of derivations Dr, t 9 R, preserves the module ~, if, for every system of generators ul, ..., us of ~ , there exist locally integrable families b ,t~, t 9 R of elements of C~(M), such that (4 9 7) holds 9 It is clearly sufficient to verify this condition for only one system of generators uz, ..., uz. THEOREM 4.2. Let Dt, t 9 R, be a locally integrable family of derivations of the free module C~(M; R m) such that 8(Dt) = Xt, and there exists a flow t

exp .f X ~ d x . 0

Assume that the family Dt, t 9 R, preserves a finitely generated submodule Then there exist flows t

0

which preserve

~, i.e., for Vu 9

~ c C~(M; R m).

i

0

~, t 9 R t

t

0

In particular, if X is a complete vector field on M and the derivation D X of ~ with 8(DX) = X preserves a submodule ~:: Dxu 9 ~ ~'u e ~, then the flow

1824

preserves the same submodule:

Vte

t

t

o

o

R, u ~ :~

etOu6~. This is known as the Stability Theorem for Finitely Generated Modules. Proof.

We shall prove that the equation

dut dt where u 0 9 ~ ,

=

u,l,-0=%

--Dtu,,

has a solution u t e ~ V t e

R.

Indeed,

(4.s)

for any a = (a I, 9 . . ,

as

e Cm(M, R O

define the family l

. u,= ~ (P,a)~,

where Pt is the flow in the free module C~(M;

dPt

dt

R0, satisfying

the equation

=-(vx +B,)oP,,P, f,=o=id,

V is the standard connection on C~(M; R0, Ul, ..., ufi is a system of generators of ~ and B t is the matrix B t = (bs,t ~) corresponding to ul, .... us in the definition of a family Dt, t 9 R, preserving a module ~ ( s e e (4.7)): l

O,u==

Z

k tttk. ha,

Then a direct check verifies that dut/dt = -Dtu t. Since aJ are arbitrary and u I . . . . , us is a family of generators of ~, it follows that a3 may be so chosen that !

u, l,=o=, , =~! a~'tt~,=tto ~.

Vu 0 9

By t h e c o r o l l a r y t o P r o p o s i t i o n 2 . 2 i n Sec. 3 , e q u a t i o n b u t t h i s e q u a t i o n i s s a t i s f i e d by t h e f l o w

(4.8) has a unique solution;

t

--D~d~%,

vt = exp 0

and so vt----ut e ~ V t

e

!R.

We now prove that for Vt e R, u 0 t

0

To that end it will suffice to prove the following assertion: For any u 0 e 3, s e R, there exists an element u e:~ such that exp j'

--D~d~--~.

0

Indeed,

if this is true, then since l

0

it follows that us = ,exp y

DTd~u 0 = fie ~ , by uniqueness.

0

1825

To prove the assertion, we define a = ps

c, c 9 C~(M,

Rt),. where Pt is the flow in

!

C~(M; RI), defined above.

Then the family w t =

~

( P t a ) ~ = satisfies the equation

a=l

d~r t

l

~vt It=0----~ a=u=,

at = - - D t ~ " and at t = { l

1

l

(PrP;- ' c)

*; = E

u,~ = ~ '

c"~=.

Consequently, for u 0 = Ec~u~ and any { 9 R, ws = u 0. 3.4.3. Existence Theorem for a Finitely Generated Projective Module. Let ~ be a finitely generated projective C~(M)-module, embedded as a direct sun,hand in the free module C~(M; Rm): there exists a submodule @ c C~(M; R m) such that C~(M; R m) = ~ @ @ Let Dr, t 9 R, be a locally integrable family of derivations of ~, 8(D t) = X t. This family is extended to a family Dt, t 9 ~, of derivations of C~(M;

R m) by the formula

D , = D t e p r ~ V x t, w h e r e p r @: nection

C~(M;

on C~(M;

R m) --+@ i s t h e

projection

o f C~(M; k s) o n t o

O, V i s t h e c a n o n i c a l

con-

~m).

It is easy to see that the family Dt, t e R, preserves ~ and so, by Theorem 4.2, we immediately obtain the following existence theorem for flows on an arbitrary finitely generated projective module 3: THEOREM 4.3. Let Dt, t 9 R, be a locally integrable family of derivations of a finitely generated projective C~(M)-module ~ with 8(D t) = Xt, such that there exists a flow t

X~dT.

exp ~ 0

Then there exists a flow t

0

on

~.

T h u s , we s e e t h a t

an a r b i t r a r y

f l o w exp

. XTd~ on a m a n i f o l d M c a n be l i f t e d

to a flow

0

on an a r b i t r a r y finitely generated responding vector bundle.

projective

C~(M)-module ~ , h e n c e a l s o t o a f l o w on t h e c o r -

3.5. Some A p p l i c a t i o n s - t h e S t e f a n - S u s s m a n n O r b i t Theorem and I n t e g r a b i l i t y of Distributions. I n t h i s s u b s e c t i o n we s h a l l a p p l y t h e t e c h n i q u e d e v e l o p e d a b o v e t o p r o v e a t h e o r e m o f S t e f a n and Sussmann [ 1 1 ] , [ 1 2 ] , and t h e n show how t h e l a t t e r can be u s e d t o e s t a b l i s h c o n d i tions for smooth distributions on a m a n i f o l d M t o be c o m p l e t e l y i n t e g r a b l e . R e c a l l ( s e e S e c . 1) t h a t a d i f f e o m o r p h i s m f:M ~ M g e n e r a t e s an a u t o m o r p h i s m p = f * : C~(M) ~ C~(M) o f t h e a l g e b r a C~(M) by t h e f o r m u l a f * b = b o f u e C~(M), I f x 9 M, t h e n x c a n be i d e n t i f i e d with a multiplicative f u n c t i o n a l on C~(M), and u n d e r t h i s i d e n t i f i c a t i o n xop = f ( x ) ; i n t h i s n o t a t i o n t h e d i f f e r e n t i a l Tf:~1,1~ TM o f f a c t s on t a n g e n t v e c t o r s i n accordance with the formula

T.~[(X~) =[ (x) oAd [*-lX=xopoAd p-iX

VXEDer M.

Let N c D e r M be an arbitrary family of complete vector fields on M. group of the automorphism group I s o M of C=(M) generated by the elements

p--_e,, x ......et~Xk, tj6R, X j6~, kGN,

1826

Let ~

be the sub-

t

t

Xd~ = exp

where e tX = exp

{Ad#X; ~p e

Xdx.

Also, let .~ b e the family of vector f~elds ~

=

~ , X e ~} a n d IIx = s p a n { x o , ~ ; X e @}. Ox of the :f'~9.miiI.I~r ,~ ~thrQ~gh ,-a pg~int x to be the fam~:i~y

W e d~fiDe the orbit

~x as the final ~ l ~ ! o g Y ,

We define ~lle ttQpol,.o~gyron

(tl,

G),-., xoe,,X,~

Xi6~, that is to say, the topology of cent inuous.

t = (ii,

generate4 [by ,the family of maps

~',~xk.,o~ ~ D

.... t#~:~,

7~:fN,

~x is the stronge.s~ t o , p o l o g y

LD ,which :all such maps are

~ . 5.1. Let ~ be a family of complete vector fields on a manifold M. Then fer a n ~ ~o ~ .M ~lae ~or!~i~ ~ ~f ~, through x o is an (immersed) submanifold of M. Moreovea~,~ ~he tangent space t,o

~|

Oxo is precisely Hx:

at a ?l~o~,t x e

T.~Dxo = IIx VxfiDxo. Proof.

We first prove that the pla~e IIx as of constant dimension on any orbit

Indeed, let x, y e Dx0; we claim that dim~ x = dimly. p e ,~ such that y = x=p = f(x). Consequently, q e ~, X e

Dxo.

Since x, y c Dx0, there exists

yo Ad q X ~ xopoAd q X = xopoAd p-%Ad poAd qX---= xopoAd p-~ Ad (pq) X = / (x)oAd (f<)* Ad p q X

= Tx/(Ad pqX)6TxfII;, whence it follows that Ny c TxfH x.

Similarly, H x c Tyf-ZHy.

Since Txf , Tyf -I are isomor-

phisms, it follows that dimH x = dimHy. It is therefore legitimate to speak of the dimension of the orbit

Dx,

equal by defini-

tion to the dimension of the plane ~x0. Let k be the dimension of

Ox, and x e

Ox0 an arbitrary point.

Then we can find k

vector fields

Ad PtXt . . . . .

Ad p~X~,

such that the vectors

xoAd Pl XI . . . . .

xoAd pkXk

are linearly independent and therefore span the plane Hx"

~x:(tl . . . . .

Define a map

tk)~xoAdplet~X . . . . . . AdpketkX~:Rk-+Dx,

The rank of the differential of this map at the point t = (tl,

T ~

d~' [o=0-~:i 0 q~x(O. . . . .

O)= xoAd p j X j,

..., t k) = 0 is k, because

y = 1. . . . .

k.

Consequently, by the Inverse Function Theorem, there exists a neighborhood V x of the origin in ~k, such that S X = ~ x ( V x ) i s a k-dimensional submanifold of M and ~ x a diffeomorphism of V x onto Sx, so that, in particular, the rank of ~x is equal to k everywhere in V x. We note that S x c

Ox,.

It is natural to define any pair (Sx,

can prove that the family of charts {(Sx, the maps

~ i ]Sx) ; x e

~ I Isx) as a chart on

Dx~

If we

Dx.}, is compatibl e , i.e., that

(5.1) are differentiable,

then

~x. will be a smooth submanifold of M. 1827

7'Isy)

Let us call charts (Sx, 9~'Is x) and (Sy, incident if their centers x and y are such that x e Sy and y e S x. It will suffice to prove that the maps (5.1) are differentiable for any pair of incident charts. Let the charts (Sx, ..., tk ~

= xoAdple t1~

~ i [ S ~) and (S , q~l [Sy) be incident. Then t o 9 Vx: y : ~x(ti ~ ..... A9a2p k e tk~ y. Consider the map Fx:Vy x V x + M, defined by

Fs(sl,...,s~, tl . . . . . t~)---= x o A d p l e , , X . . . . . . Adp~e'~X~oe-~r~Adq~ 1. . . . . e-*,r, Adq~ ~,

where Yj, q j , j = 1, . . . ,

k, a r e d e f i n i n g

elements of th~ chart

(Sy,

~l[Sy):

~ ( s l . . . . . s~)=yo A d q i e , , Z t o . . . o A d e ' ~ . Since Fx(0, t o ) = y and the ra,g ~' of the map t ~ F(s, t) at the point (0, t o ) is k (because k is the rank at t o of the map t +

f,,v:V, is also k at (0, to).

~x(t)

= Fx(0, t)), it follows that the rank of the map

XVx-+Vy,

fx~(s,t)=~ioF,(s,t)

Since fx,y(0, t e) = 0, it follows by the Implicit Function Theorem

that there are neighborhoods of zero in R k and t o ~ Vx, and also a smooth function t = t(s), such that

/.y (s, t (s))=O in the neighborhood of zero. Consequently, s + t(s) = ~iO~y(S) lar argument proves that the map t + ~io~x[(t) is smooth. ~Xo

is a smooth map.

A simi-

To complete the proof of the theorem it remains to observe that the tangent space to at x e Ox0 is the plane Hx" Indeed, let y 9 ~x, be an arbitrary point of Dx0 and

(S x, ~x[S x) a chart such that Y=~.

(6 . . . . .

tD.

Then for t = (tl, ..., t k) we have

O )t~ = to O-ds 0 x oAciple " t,x . . . . . . T , % /t-a-~j

-----xoAd Ple t'x . . . . . . Ad p / j X j X p A d

Adpke'P:k=

pmets+,xm . . . . . Ad pxetixlk -~

-~ xoAd P, e', x . . . . . . Ad p / k X k o e -'Prk Ad p~-i. . . . . e-ts+,xm Ad PT-~, X , Ad p /+let i+,x i+,o . . . oAd peet kXk = xoAd plet 'X'o . . . oAd p~e t ~ . Ad (e -t i+,xs+, Ad pT_~,o . . . . Ad p ; ' ) = % (tl . . . . . where

q = e-ts+*xJ +, Ad PT~l . . . . . e -t~'xlAd p;i.

tk)oAd q X = roAd qXEIIy,

Consequently, Ty~xi-~'IIy VyEOx,.

This completes the proof. Remark. If M is a real analytic manifold and ~ a family of real analytic vector fields on M, then the orbit ~x0 through an arbitrary point x 0 e M is a real analytic submanifold. We define the reduced orbit or t-orbit ~ x ( t ) of a family x to be the set

~ c DerM, through a point

~x(t)={xoet'X'o . . . . eqXk; /sER, X t j = t , XSE~, kEN}. Let

~,= {e"X'o ....e'Mk; XsEs

tsER, Xts----t,kEN),

II'x={xoY; yEe,}, where

1828

~,~ = {Adp(X - Y); p e ~r, X, Y 9 ~} c DerM.

A slight change in the arguments used to prove Theorem 5.1 yields the following Proposition 5.1.

Any t-orbit

Dx,(t) of a family

on a manifold M is an immersed submanifold of M. Hyt.

In addition,

Dx0(t) c

For any t', t" 9 R the orbits

~ c Der M of complete vector fields

The tangent space Ty :Ox0(t) tO

Dx, and the codimension of

Dxo(t') and

~Dx,(t) in

Ox * is

~Dx0 is zero or one.

Dx0(t") are diffeomorphic.

Theorem 5.1 is frequently used to study questions of integrability of smooth distributions on a manifold M. Recall that a distribution 4 on a manifold M is a map that associates to each point x 9 M a linear subspace 4(x) c TxM. Any family ~ of vector fields on M naturally generates a distribution 4 on M - the distribution defined by

A (~) = span{xoX; XO$} By no means all distributions are generated in this way. For example, in the case M = R, the distribution A, A(x) = R for x = 0 and 4(x) = 0 for x # 0 is not generated by any family of smooth vector fields. Those distributions generated by some family ~ of vector fields are called smooth distributions, and the family ~ itself is called a family of generators for the distribution in question. Any given smooth distribution 4 on M has a maximal set Jr(4) of generators. This set J/(4) consists of all vector fields X 9 D e r M such that xoX 9 A(x) x 9 X (in that case we shall say that the vector field X belongs to the distribution 4, writing X 9 4). Clearly, J/(4) is a C~(M)-submodule of the C~(M)-module Der M, which is in turn uniquely determined by 4. On this basis, the smooth distributions 4 are often identified with Js that is to say, a smooth distribution on a manifold M is defined to be a C~(M)-submodule ~ c DerM, such that if xoX 9 ~x = {xoY; Y 9 I}, then X 9 i~. This last condition is also known as the completeness condition for ~. Throughout the sequel we shall consider only smooth distributions. A distribution A(x), x 9 M, is said to be involutive if X, Y e 4 implies

[x, Y] 9 4.

A distribution A(x), x e M, is said to be regular if

dim A (x) -----const. Regular distributions are identified in a natural way with subbundles of the tangent bundle TM of M. A connected smooth submanifold N c M is called an integral manifold of a distribution A(x), x e M, if x 9 N TxN = A(x). A distribution A on M is said to be completely integrable if it has an integral manifold through each point x 9 M. An obvious necessary condition for a distribution A to be integrable is that it be involutive. However, not every involutive smooth distribution A on M is integrable. Nevertheless, one can prove Proposition 5.2. the module

If A(x), x 9 M is a smooth distribution on M which is involutive and

J / ( A ) = {X~Der MlxoX~A(x) Vx~M} is finitely generated,

then A(x), x e M, is completely integrable.

Proof. It follows from the maximality of JK(A) and the fact that A is involutive that the C ~ ( M ) - m o d u l e ~ ( A ) is involutiuve, i.e.,

X, Y~,/I (A):=~[X, Y]~dt' (A). Therefore, for any X edK(A) the operator a d X preserves the finitely generated module d/(A). Consequently, by Theorem 4.2 (stability of finitely generated modules),

eta~ Y~J[ (A) Vt~R, X, rodt" (A). In particular,

TdA (x)~=A ([ (x)) Vx6M, t~ R, rze [* = etX.

1829

Hence it follows that for any X, X I, ..., X k e J ( A ) ,

tj e R,

e''adx'o... oetkadXkx6df(A). Let ~X be the orbit of the family J ( ~ ) to ~ at a point y e ~x, is

through x e M.

By Theorem 5.1, the tangent space

Ty~x= span {yoAd pX; XG./Ar (A), p~}----IIy, where ~)=: {et'X'o . . . . etkXk; Xi6~}.

By what we have a l r e d y p r o v e d ,

Vp----et'X'o . . . . . etkXkE~)

Ad p X = e" ~dx'o . . . . e' ka~Xkx = Ad (e' 'x'o ... oe'kxk) X6/g (A). Therefore n y c a (g)={yoX; XGX (a)}, so that ny = Ay. Therefore, since an orbit is a smooth submanifold of M, we see that ~ is completely integrable and its integral manifolds are the orbits of the family J ( & ) of vector fields. This completes the proof. Remark. The conclusion of Proposition 5.2 is also true when ~f(A) is a locally finite generated module, i.e., the C~(M)-module X (a) Jv= {X lu;

X 6 X ta)},

where U is a neighborhood of an arbitrary point x e M, is finitely generated. As direct corollaries of Proposition 5.2 we obtain the following classical results: Proposition 5.3 (Frobenius' Theorem; see [8]). If & is a distribution on a real analytic manifold M, generated by a family of real analytic vector fields, then it is completely integrable if and only if it is involutive. Its maximal integral manifolds are real analytic submanifolds of M. 3.6. Further Applications - Lifts of Base Diffeomorphisms, Flat Connections and Parallel Translation. In this subsection we consider a few more applications of the main results of this section - the variation formula and the existence theorem. 3.6.1. Lifts of Base Diffeomorphisms. Let ~ be a finitely generated projective C~(M) moule; thanks to the natural isomorphism ~70~ (see Sec. 2), we may always assume that is actually the C~(M)-module of smooth sections of a suitable vector bundle. If N c M is a smooth submanifold of M, iN:N + M an embedding, then any flow t

0

determines a semi-isomorphism over Pt of the C~(N)ymodules ftoiN:

~a~,i~

i* N

and i* t ~, where it =

~p,ac, i*.vPtU., ~ 6 C ~176 (N),

t

where Pt = exp

Vx~d~, V a connection on

3.

0

By a simple generalization of this procedure, one can also prove that if f, g: N + M are homotopic maps, then the C=(M)-modules f*~ and g*~ are isomorphic. Indeed, let F: [0, i] • M + M be a smooth homotopy, F(0, x) = f(x), F(I, x) = g(x), x e N. To avoid having to deal with manifolds with boundary, we shall assume that F is extended in some way to the whole manifold R • N. Define embeddings i~:N ~ R x N, t e R, by i~(x) = (T, x). It is easy to see that io*oF*~[*~,i,*oF*~N.g*~.,

1830

At the same time, the flow 0t:(~, x) ~ (~ + t, x) in R x

M determines

isomorphisms

it~*F*~-,i~,*F*~, Vll, t~R. Hence it follows,

inter alia, that any vector bundle over a contractible m a n i f o l d i s

3.6.2. Flat Connections. projective C~(M)-manifold 3. modules D e r M and Der ~:

Let V:DerM ~ Der ~ be a connection on a finitely generated Recall that V is an injective homomorphism of the C=(M) -

V=x = a V x ,

~x,+x, = ~Tx, + ~Tx,

X, Xt, X26.DerMand0"V-----i..

VaGC ~ (M), However,

trivial.

it is only a module homomorpism;

in general,

[Vx, Vy] ~

Vlx.~l.

The difference between V and a Lie algebra homomorphism of D e r M and Der ~ is characterized by the curvature R V of V - the bilinear map Rv:DerM • D e r M ~ End ~, defined by the formula

R v ( X , Y)=[Vx, Vr]--~7[x,rl v X, YEDer M.

(6.1)

A connection V is said to be flat if it has zero curvature:

Rv=0. Proposition 6.1. The map

Let V be a connection on a finitely generated projective C~(M)-module f

t

I:ex% I X~dz~ex~p SoVx~dz,

t6R,

is a homomorphism of the group of flows on M into the group of flows on ~ V is flat. Proof. interest.

if and only if

The proof is based on the following assertion, which is also of independent

LEMMA 6.1.

The relations

~-]t

exp adVx~d~vv=v

,

-'! a d v x ~ d ~ V r = V ,

(6.2)

t

exp

expJa~x~ezr 0

are true if and only if V is flat. Proof.

Suppose that V is flat.

Then

[XYxt, Vy] = Vtxt,rl and so d

w~,

= v_,,

eXPo~adXg~'~Y t

Y]= v~e p]'adXgdT 'o [v:~,, w,l;

[x,,

exp~[adXgd-~

t

hT exp 0

0

t

whence it follows that the flows exp

ad~Tx~dr, ~7~t 0

satisfy the same equation.

But

exp~adXTdz 0

at t = 0 we have

1831

t

exp ~ ad Vx~d~Vy----V and hence,

by uniqueness

(see Proposition

,

=Vr,

2.2),

t

exp I ad v x ~ d ~Y ----V

__>t

0

Conversely, if this last relation This proves the lemma.

0.

exp~adX,cd,cF 0

is true then, retracing our steps, we see that R v =

Now, using the lemma and assuming that R v = 0, we have

I exp. X,d~ oI exp r~d~ = exp f Vx~dxoexp f V r,d~ = 0

0

0

L

= ~p i;xp j - .d w0e0 (vx, + v ~,)a~= 0

0

---exp.V+_t (I

exp

~adFodO(X~+r,c)d'~

=I

p

X~d~oexp Y~d~ .

0

Again, r e t r a c i n g our s t e p s and using Lemma 6.1, we see t h a t RV = 0 i f I is a homomorphism, proving the p r o p o s i t i o n . Let V be a flat connection and Pt(S), t e R, a family of flows on M depending on a parameter s e R:

smoothly

t 0

0

Assume that at t = i the flow Pt(S) is independent t

. . Os exp i'

- -

--

0

.

.

D~(s)d~

.

t

t

:=expj'D~(s)d~o.f

0

T

(s)

exp ~ a d O o ( s ) d O - - - 5 7 - - a ' c 0

0

OO,

of s e R: .

,

0--s P1(S) = 0.

Then,

since

we have

t

.

1

.

.

1

.

ox~ (s)

p, (S)= exp j'X~(s)d'~o yexp f a d X o ( s ) d o ~ d %

0 =0

0

0

1

I

whence it follows,

canceling out the flow exp

X~(s)d~,

that

0

Consequently,

1

t

0

I

by Lemma 6. i, 1

0

1

--'*"

I

-'-+

Os exp S Vx~(s)dr =exp f 0

0

0 1

1

_..+ 1

=oxp S v~,,~,e,ov --> ,

1832

0 expfadX~(s)d~--~-Xl(s)d,r

=oxp S w ~ e , Vo=O, 0

1

and this proves Proposition module ~, and

1

t

1

0

L

Vx~(~)clrofex? fad Vx~(s)d'co Vxgs)dt-.~

6.2.

If V is a flat connectiion on a finitely generated projective

C~(M) -

I

3"

o

then 1

;

=o.

0

3.6.3. Parallel Translation. As before, let ~ be a finitely generated projective C~(M)-module, realized as a C~(M)-module of sections: the elements of ~ are smooth sections u:x ~ u x e Jx ~ ~. In this case, by analogy with the notation used above for the algebra C~(M), we shall write xou for the "value" of an element u at a point x:

XoU=Ux==ullx~. In just the same way we shall write the point to the left of operators ~. For example, if D is a derivation of 3, then (xoD)f = (Df) x and xo(D(af)) = xo(aDf + (Xa))f) = a(x)o(Df) x + (Xxa)f = a(x)o(Df) x + (xoX)(a)f = a(x)(Df) x + (Xa)(x)f = a(x)(xoDf) + (xoX)(a)f, in agreement with our previously adopted notation (see Sec. i).

"S exp

t

Let Pt =

t

xopt

=

vtER

xoqt

Yxd~ be flows on M such that for some point x e M,

X~d~, qt = exp

0

0

It turns out that then, for any connection V on ~ , t

t

xoexp ] Vx~dzu = xoexp y Vr~d~u VuE~. 0

(6.3)

0

The proof of this statement is rather cumbersome and will not be presented here. mention that the proof makes use of the following composition formula: ~t(x)

We only

t

0

(6:4)

0

__~ ~t (x) where ~

~t(x) = A i ~xoexp

O(D~)d~),

D t e Der

~, A i ~ C~(M).

This formula, which is

proved using the variation formula, is a natural generalization of the formula for changing the variable in a chronological exponential (see subsection 3.3 in this section) to the case of a substitution that depends smoothly on a point of M. Thanks to formula (6.3) and the availability of a connection V, one has a well-defined notion of the parallel translation of a family ut, t e R, of elements of ~ along a curve t ---~ j'

xt

=

xo,exp

XTd~, t e R, on M.

Indeed, we shall say that ut, t e R, is parallel along the

0

curve x t, t eR, on M if

_-+

t

where Pt = exp j" Vxxd~. 0

4.

Bundles and Modules With Additional Structures

In this section we shall consider families of derivations of finitely generated projective modules, including connections of a special type. Descriptions will be given of the

1833

Lie algebras they generate and the corresponding groups of flows. We shall also point out the relationship between these notions and the traditional geometrical objects: reductions of principal bundles, holonomy groups, etc. 4.1. Lie Groups. This subsection collects the Lie group prerequisites needed for the sequel, The basic information about Lie theory is presented in a rather more general form than usual in the textbook literature, but everything follows easily from the variation formula and the existence theorem for flows; proofs will be omitted. Recall that a Lie group is a group G with the structure of a smooth finite-dimensional manifold defined on G, in such a way that the group operation (x, y) ~ (xy) and inverse x ~ x -I are smooth maps of G • G into G and of G into itself, respectively. Let G be a Lie group; for every x 9 G we let %(x):G ~ G denote the left translation %(x):y + xy, y e G. It is easy to see that %(x) is a diffeomorphism of the manifold G; accordingly, %(x)* is an automorphism of the algebra C~(G). In addition %(xy)* = %(x)*%(y)*, and the correspondence x ~ %(x)* is an isomorphism of G onto a subgroup %(G)* of the automorphism group of C~(G). Now consider an arbitrary smooth manifold M. group of C~(M).

Recall that IsoM denotes the automorphism

IsoM is a subset of the Frechet space ~ (M) of all continuous linear operators in C=(M). Lie subgroups of IsoM are defined as subgroups that have the structure of (finite-dimensional) submanifolds of ~ (M), with the group operation continuous in their topologies. The left translations described above yield a canonical representation of an arbitrary Lie group as a Lie subgroup of a suitable automorphism group. Let D e r M c ~

be an arbitrary set of vector fields on M.

X~d~ I X~E~,

Exp ~ = exp

Define

t, ~ 1 t c I s o M.

I t i s e a s y t o s e e t h t Exp ~ i s a l w a y s an a r c w i s e c o n n e c t e d s u b g r o u p o f I s o M ( b u t n o t n e c e s sarily a Lie subgroup). On t h e o t h e r h a n d , l e t I s o M c ~ be an a r b i t r a r y subset of lsoM. Define en ~={P71o-~i

P, Jpt--aflow, P,6~ vt6~}cDer ~4.

THEOREM i.i. i) Let G c IsoM be a Lie subgroup, ~ = LnG. Then g is a Lie subalgebra of DerM, all of whose elements are complete vector fields, dim ~ = dimG . . . . . 2) Let g be a finite-dimensional Lie subalgebra of DerM, all of whose elements are complete vector fields. Then Exp ~ is a connected Lie subgroup of IsoM. 3) The operations Exp and Ln are inverses of one another, where the former is defined on the set of finite-dimensional Lie subalgebras in DerM, all of whose elements are complete vector fields, and the latter on the set of connected Lie subgroups of IsoM. 4.2.

tions

Holonomy Group and Principal Bundles.

of a finitely

generated

projective

Let ~ c D e r ~

C~(M)-module

ft.

be an arbitrary set of deriva-

D e f i n e Exp

~ ~---

.f D~d'~ ID~E~, 0

t, ~ER/.

It

is easy to see that

Exp ~) i s a s u b g r o u p o f t h e g r o u p I s o ~ o f a l l

semi-isomor-

phisms of @ onto itself. Denote ~=Exp~. It follows from Theorem 5.1 of Sec. 3 (the Stefan-Sussman Theorem) that the orbits of the group 8(9) c IsoM in M are smooth manifolds o f m. H o w e v e r , a s y e t we know n o t h i n g a b o u t t h e " v e r t i c a l part" of the group ~, i n p a r t i c u lar, about the structure o f t h e g r o u p Ker 8 Cl $ c A u t ~ . L e t M ~ N b e a s u b m a n i f o l d , iN: ~ M an e m b e d d i n g , iN* ~ t h e m o d u l e o v e r C~(N) i n d u c e d by t h e e m b e d d i n g ( c o r r e s p o n d i n g t o " r e s t r i c t i o n " of the bundle ~ t o N; s e e S e c . 2 ) . Let P t e I s o ~ b e s u c h t h a t 8 ( P ) maps t h e s u b m a n i f o l d N i n t o i t s e l f : x o S ( P ) e N x 9 N. Then

1834

we have a well-defined semi-morphism iN*P e Iso iN @*. tor field 8 ( D ) i s

Similarly,

tangent to N (i.e., xoD e TxN c T x M V X

if D e Der @, and the vec-

e N), we have a well-defined deri-

vation iN*D e Der iN* @. We now turn to the group ~ = Exp ~. Let M c N be an orbit of the group 8 (~). We wish to study iN* ~; in order to avoid cumbrous notation, we shall assume henceforth that N = M (this clearly involves no loss of generality). We are thus assuming that 8(~) acts transitively on M. For any x e M, we denote ~x = {P e ~ [xoS(P) = x} - the stable subgroup of the point x under 8 (~), and H x = i{x } ~X" Let ~ @ = I l E x = E. Then H x is a subgroup of GL(Ex) , x e M . xEM Proposition 2.1. similar. Proof.

For any x i, x 2 e M, the subgroups Hxi c GL(Exi ) and Hx2 c GL(Ex2 ) are

Let P e ~, where x2oS(P) = x i.

Then i{Xi}*(AdP):Exi

§ Ex2 maps the subgroup

Exi c GL(Exi ) onto the subgroup Hx2 c GL(Ex2 ). Remark. Let A be an arbitrary group and B e A a subgroup. Just as a group is usually defined only up to isomorphism, a subgroup is defined to within similarity, i.e., isomorphism of the pair (A, B). c

Definition. c Der @.

The subgroup H x c GL(Ex) is known as the holonomy group of the family

The rest of our account revolves around the concept of the principal bundle of frames associated with ~. Let dime x = n, x e M. Consider the module @ n = ~ O . . . ~ and let E n = 8~ n denote the corresponding vector bundle, Exn = E x | ... 9 Ex, x e M. There is a natural right action of the group GL(n) by automorphisms of the module @ n (this action of course preserves the fibers of the right action of GL(n) in En). This action is defined as row-multiplication of a vector by a matrix:

(e~. . . . . G) A =

eia~, . . . . .

eia~ ,

\i =I

i=i i=1

D e n o t e R(E x) = { ( e l ( x ) ,

...,

en(x))

....

,

AEGL(n),

e~Ee,

/ n.

e Exn[ei(x ) .....

en(x) are linearly

independent vectors

in Exn},

R(E)= xstJ

R ( E x ) c E~"

It is easy to see that R(E) is mapped into itself under the action of the group GL(n). In addition, on each fiber R(E x) the action of GL(n) is free and transitive. The manifold R(E) with the above-specified right action of GL(n) is called the principal bundle of frames (associated with @) and denoted by R(~). Let pair (~, the set smoothly

G be a Lie subgroup of GL(n) and ~ a smooth connected submanifold of R(E). The G) is called a reduction of the frame bundle to the subgroup G if, for any x e M, S x = R(Ex) 0 $ is an orbit of the action of G on R(Ex), and in addition $ x depends on x.

In this context, "smoothness" is defined as folows: every point of M has a neighborhood U such that some smooth section of the bundle ~iu* @ n takes values in $. To each semi-isomorphism P of @ we associate, in an obvious way, a semi-isomorphism p 9 ... 9 p of ~ n and hence also a diffeomorphism of the total space E n which, as is easily seen, maps R(E) into itself. If the corresponding diffeomorphism maps $ into itself~ then P is called a diffeomorphism of the reduction ($ , G). The set of all such diffeomorphisms is denoted by Diff( $; G). Clearly, Diff( $ ; G) is a subgroup of ~ Iso 6, consisting of the diffeomorphisms of E that map fibers linearly onto fibers. A similar definition yields the subgroup A u t ( ~ ; G) of the group 8Aut q, consisting of the diffeomorphisms of the manifold E that map each fiber linearly into itself. Thus, 9 In a more general way, if 8(P) maps a submanifold N i onto a submanifold N2, we have a semi-isomorphism iNi*P:(iNl* ~) ~ (iN2* ~) . 1835

Aut(~; G) =Diff(~; G)N~ Aut ~. Henceforth, given an arbitrary topological group F, we shall denote its arcwise connected component containing the identity F ~ The subset r ~ is obviously a subgroup (in fact, a normal divisor) of r. For example, for the group D i f f M of diffeomorphisms of a manifold M, the subgroup (Diff M) ~ is precisely the set of all diffeomorphisms that can be included in a flow. Proposition 2.2.

I) The group D i f f ( ~ ; G) ~ acts transitively in ~

and

8 (Diff (~; G) ~ = (Diff M) o. 2) Any orbit of the group Dill( ~ ; G) ~ in R(E) has the form ~ A for some A 9 GL(n). In particular, such an orbit determines a reduction of the frame bundle to the subgroup A-IGA. We shall state a few more assertions, senting their proofs.

only then, at the end of the subsection, pre-

We return to the group ~ = Exp ~. Any semi-automorphism of the module ~ defines a diffeomorphism of the manifold R(E) in a standard way, with the composition of semi-automorphisms corresponding to composition of diffeomorphisms. In particular, we obtain an action of ~ on R(E). Since no other actions of ~ on R(E) will be considered here, we shall not use any special symbols to denote the standard action. THEOREM 2.1.

i) The holonomy group H x is a Lie subgroup of GL(Ex).

2) Any orbit of ~ in R(E) is a reduction of the frame bundle to a subgroup similar to H x. 3) The Lie algebra h x of H x is described by:

h~=span{xoAd PD I Pe~, D6~}Ngl (E~). COROLLARY. Let ~ be an orbit of ~ in R(E) and H a subgroup of GL(n) similar to the holonomy subgroup, so that ( 9 , H) is a reduction of the frame bundle to H. The orbits of the (right) action of H ~ on $ determine an equivalence relation on $; the quotient space modulo this relation is a cover of M. It is easy to see that this cover is connected, since , as an orbit of the connected group ~ , is connected. Consequently, the connected components of the holonomy group H x are in one-to-one correspondence with the cosets of the fundamental group ~I(M) modulo some subgroup. COROLLARY. Let L(~) be the Lie subalgebra of Der 6, generated by ~. module generated by L ( ~ ) has finitely many generators, then h x = x o L ( ~ ) particular, this is true if ~ is a set of analytic fields.

If the C~(M) N gl (Ex). In

Our last goal in this section is to give an infinitesimal description of reductions of the bundle of frames R ( 6 ) ; this will be utilized to generalize the ordinary Lie theory to a certain class of subgroups of Iso ~ and Lie subalgebras of Der 6 Definition. A Lie subalgebra ~ c Der ~ is said to be principal if ~ is a projective submodule of Der ~ and 8 ( ~ ) = DerM. Proposition 2.3. For any principal Lie subalgebra ~ of Der @ , the ideal ~ N E n d ~ = ~ N ker 8 in ~ has the following property: the Lie subalgebras i { x } * ( ~ 0 ker 8) c GL(Ex) are similar to one another for all x 9 M. For an arbitrary arcwise connected subgroup

~ c Diff ~, define

L n ~ = { P ? l o ~ P , IP~E~ V~ER, P 0 = i d } c D e r ~. Let G c GL(n) be a L i e subgroup and

g c gl(n)

i t s Lie subalgebra.

THEOREM 2 . 2 . 1) Let ~ = D i f f ( ~; G) ~ f o r some r e d u c t i o n ( ~, G) of t h e frame bundle t o Then Ln ~ i s a p r i n c i p a l L i e s u b a l g e b r a of Der ~, and t h e Lie s u b a l g e b r a i { x } * ( k e r 8 N Ln ~:) c gl(Ex) is similar to ~ Vx e M.

G.

2) Conversely, let ~ be a principal Lie subalgebra of Der i{x}*(ker 8 0 ~ ) c gl(Ex) are similar to ~ ~ x e M.

1836

@, such that the subalgebras

Then Exp ~ = Diff( ~0; G0) ~ for some reduction ,(fro, Go) of the frame bundle to a subgroup Go, such that the components of the identity in G O and G coincide. 3) The operations Exp and Ln are inverses of one another on the set of groups Diff( $; G) ~ of diffeomorphisms of reductions of the principal frame bundle and the set of principal Lie subalgebras in Der ~. COROLLARY. If ~ = DerM, called G-structures on M.

reductions of the principal frame bundle to a subgroup G are

(For example, a Riemannian metric is an O(n)-structure; a distribution of k-dimensional planes is a G-structure, where G is the subgroup of GL(n) preserving a fixed k-dimensional plane.) It follows from Theorem 2.2 and Proposition 2.1 that there is a natural one-to-one correspondence between the principal Lie subalgebras of Der D e r M and the G-structures on M. The proof of most of the above statements of reductions of frame bundles is based on the existence of certain special connections in ~, which are strongly related to reductions. Definition.

A connection 7 on a module

@ is called a connection on a reduction ($ , G)

t

of the frame bundle if exp~

Vx~dT 9 Diff( ~; G) ~ for any X T 9 DerM,

T 9

R.

0

Proposition 2.4.

On any reduction of the principal frame bundle there is a connection.

Proof. Let (~, G) be a reduction to a subgroup G and g the Lie algebra of G. Assume first that ( ~, G) has a section r:M ~ ~, r e @~. Clearly, any connection in ~ is uniquely determined on r, and 7xr = rA(X), where X + A(X) is an arbitrary homomorphism of D e r M into the module C~(m, gl(n)). In addition, for any martix B e gl(n), 7x(rB) = (Vxr)B. Taking into account that ~ = U r(x)G and recalling the usual properties of connections, we obtain xEM the following description of all connections on the reduction" ($ , G). LEMMA 2.1. A connection 7 in ~ is a connection on ( f; G) if and only if 7xr = rA(X), and then X ~ A(X) is a homomorphism of D e r M into C~(M, @ c C~(M, gl(n)). An arbitrary reduction of a frame bundle need not have global sections, but it always has local sections. A standard application of partitions of unity enables one to glue local sections together to get a global section. Proof of proposition 2.2. i) The equality D i f f M ~ = O(Diffi(~; G) ~ follows immediately from the existence of a connection on (~, G). Next, as the manifold ~ is connected, transitivity of the action of D i f f ( ~ ; G) ~ in ~ will be proved if we show that, whatever the point P0 e ~, the orbit of the group Diff(~; G) ~ through P0, contains a neighborhood of Po in ~. Let P0 e R(Ex0 ) and let r be some local section of the bundle

(~ , G) passing through

P0, r(x0) = P0" In addition, let a e Cm(M) be a function equal to unity near x 0 and vanishing off a sufficiently small neighborhood of x 0. Then for any B e g c gl(n), we have a welldefined automorphism (reaBr -1) e Aut( ~; G); for each x e M, r(x)ea(x)Br-i(x) is a transformation of the space E x represented in the basis r(x) = (ei(x) . . . . . en(x)) by the matrix by the matrix ea(x)B; or (which is the same in other words) a transformation which maps the frame r(x) into the frame r(x)e a(x)B. Let 7 be a connection on ($ , G). eVX0(reaBr -i) 9 Diff( ~; G) ~

For

In addition,

arbitrary X 9 DerM, B 9 ig , we have the map

(X, B) ~ eVXo(re~Br-9 JOo is clearly of full rank at the point (0, 0). this map covers a neighborhood of P0 in ~.

Consequently,

by the implicit Function Theorem,

2) Follows easily from part I of the proposition and the obvious identities

P(pA)=P(p)A,

vP~Iso~, A6~GL(n), p6R(E).

1837

Proof of Theorem 2.1. Let P 9 leo ~ A 9 GL(n), p 9 R(E). It follows from the formula P(p) A = P(pA) that A carries orbits of the group H to orbits. Define G = {A e GL(n)[ A = def

}; G is a subgroup of GL(n). Clearly, $ x = R(Ex) D ~ is an orbit of G on R(Ex) for any x 9 M. Now it follows from the definition of the monodromy group G x that it acts transitively (and, of course, freely) on ~ . L e t p = (e I . . . . , en) 9 ~x and let ~:E x + E x be an element of Hx; then ~p = pA for some uniquely determined A 9 G c GL(n). We thus obtain a one-toone correspondence ~ ~-+ A between the groups H x and G. This correspondence is an isomorphism: l (# (~) ----A (pB) = (A (@) B----p A B = A B (p). Since this isomorphism extends in an obvious way to an isomorphism of GL(Ex) and GL(n), the groups H x and G are similar (el. the proof of Proposition 2.1). The proof of smoothness and the description of the tangent space are based on the same arguments as the proof of the Stefan-Sussmann Theorem (Theorem 5.1 in See. 3). Proof of Proposition 2.3.

Let D< 9

~, ~ 9 [0, t], then the isomorphism

t

eXp~

ad D~d~ of the Lie algebra Der ~ maps the subalgebra

~

into itself;

if x2oexp]

0

0

6(D~)d~ = x 1, t h e n i { x l } *

exp

subalgebra

~}.

i{x2}*(ker 8 A

adD~d~

maps t h e s u b a l g e b r a

i{x~}*(ker

e n

~) o n t o t h e

P r o o f o f Theorem 2 . 2 . 1) The e q u a l i t y O ( l n ~) = DerM f o l l o w s f r o m t h e e x i s t e n c e o f a c o n n e c t i o n on (~ , G). L e t U c M be a n e i g h b o r h o o d s u c h t h a t t h e r e e x i s t s a s e c t i o n r : U ~ N. Then ~ , and h e n c e a l s o iu*End ~ , a r e f r e e C ~ ( U ) - m o d u l e s . In this situation ( k e r ~ N Ln ~) c IU* End ~ i s t h e s e t o f a l l e l e m e n t s

x~r(x)B(x)r-~(x),

B(x)Eg, VxEU.

It follows at once from this local representation that ker 8 N Ln ~ is a projective module and i{x}*(ker 8 0 Ln ~) is a Lie subalgebra similar to ~ in gl(n). It remains to prove that Ln ~ is a Lie subalgebra of Der ~. We again use the local representation. Let V be some connection on ($ , U). Then it follows from the previous arguments that iu*(In ~) is the set of elements

V x + r B ( . ) r -1, Recall that Vxf = rA(X), where A(X)(x) e

XGDerU, g, Vx e U.

B(-):U~g. The fact that [VXz + rB1(.)r -l, VX2 +

rB2(.)r -z] e iu*Ln~0 can now be verified directly. 2) By Theorem 2.1, any orbit $ of Exp ~B in R(E) is a reduction of the frame bundle to some subgroup Go, and g is the Lie algebra of G o . Let V:DerM + ~ be a connection on ~, taking values in ~D. Then obviously V is a connection on the reduction ($ , G) of the frame bundle.

Let Pt e Diff(

if; G), P t = exp

So ,

C.rd~ and Xt = 8 ( C t ) .

Then Ptoexp

0

VX d'r 9 Aut (~" ; G). Thus i t w i l l s u f f i c e t o p r o v e t h a t f o r any f l o w Qt 9 Aut(l~ ; G), Ln~Qt) e Der 8 Cl ~. But t h i s f o l l o w s f r o m t h e o b v i o u s i n c l u s i o n s ( i { x } * L n Qt) e k e r O Cl and t h e f a c t t h a t t h e module k e r 0 A N i s p r o j e c t i v e . Part

3 follows

from t h e arguments in p a r t s

1 and 2.

4.3. Gauge T r a n s f o r m a t i o n s . On any f i n i t e l y g e n e r a t e d p r o j e c t i v e C~(M)-module @ t h e r e e x i s t s a c o n n e c t i o n V. At t h e same t i m e , t h i s by no means i m p l i e s t h a t t h e c o n n e c t i o n is uniquely determined. We s h a l l now t r y t o d e s c r i b e a l l t h e p o s s i b l e c o n n e c t i o n s on ~ . First, !et us consider the projective module Hom(DerM, must be members of this module.

1838

Der

~).

All the connections

If V I, V 2 are connections on ~, then obviously for any X E DerM, Consequently, (V ~ . E7~)EHom (Der 3~, End ~)c Horn (Der M, Der ~). On the other hand, for any H e Hom (DerM, End tion.

(VX I - VX 2) e End

~.

~) the homomorphism V1 + H is again a connec-

Finally, we obtain: Let V be a fixed connection on ~, then all other connections are of the form V + H, where H E Hom(DerM, End ~) is arbitrary. Thus the set of all connections is an "affine plane" in the module Hom(DerM, Der which is the "translate by V" of the submodule Hom(DerM, End ~).

~),

The group Iso 6 acts in a natural way on Der 6 and D e r M by inner automorphisms. Namely: if P E Diff 6 and 8(P) = p e DiffM, then the inner automorphisms operate in accordance with the rule

D ~ A d PD=PoDoP-I, X ~ A d pX=poXop -~, YP~Der~, X~DerM. These representations yield the natural action of the same group Iso Hom(DerM, Der 6).

in the module

This action, denoted by P,, is defined by the formula

P.A =Ad PoAoAd p-', VAEHom(DerM, Der ~), VP~Diff6, 0 ( P ) = p . It is easy to see that P, is a semi-isomorphism of Hom(DerM, Der ~), and moreover e(P,) = 9(P) = p. In addition, it can be verified directly that P, takes connections into connections; such transformations of connections are usually called gauge transformations. If V' = P,V for some P e Diff ~ we say that V and V' are gauge-equivalent. In other words, the gauge-equivalence classes are the orbits of the natural action of the group Iso 6 on the connection. Let R V be the curvature tensor of a connection V, R V ~ Hom(DerM A DerM, End ~ ).

We

have Rp, V = P,Rv, where P,R V is the result of the natural action of P on RV, i.e., (P,R V) (X, Y) = A d P R v ( A d p -I X, Adp-iY). Suppose now that we have a flow t

j D~d~ B D i f f ~. 0

It is not hard to find an infinitesimal generator of the flow Pt," X t . Then

In fact, let 0(P t) =

Pt.A = Ad Pto(ad DtoA-- Aoad Xt)oAd p71 : ----Pt. (ad DtoA-- Aoad Xt). Let p(D t) denote the derivation of the module Hom(DerM, Der 6), and also of the modules of homomorphisms from "tensor modules" over M to "tensor modules" over 6, defined by the rule: P(Dt)A = adDtoA - AoadD t. Then

exp S D,dz~ ----exp~ p (DO d't. 0

7.

0

Note that

Otp(D))=O(D). It is easy to show that p(D) maps the submodule Hom(DerM, Der ~6) into itself, and also maps an arbitrary connection into the same submodule. We verify this, e.g., for a connection:

1839

0 (p (D) Vx)---- 0 ([D, Vx]) - - 0 ( V lo(o),xl) = = [0 (D), X] - - [0 (D), X] = 0. If D = 7X, then p(D) = 0(V X) is called the covariant differentiation of the field X (determined by the connection V). In particular, we have

in the direction

(9(Vx)V)(Y)=[Vx, Vr]--Vlx,~'l=Rv(X, Y). Thus, curvature is the covariant derivative of the connection determined by itself. For gauge transformations we obtain

(Pt,v)~'---- exp 9 (Vx~)d~v

=

Vr-]- ~(P~,op(Vx~)V)(Y)dT=

Y

0

f

= S AdP~Rv (X~, Adp~:Y)d'~+ V~'. 0

The identity t

f

( P , , v ) v - - V ~ ' = ] Ad P~Rv (X~, AdPTW ) dx ----] (P**Rv) (Ad p,X,, Y) dT

(3.1)

0

0

establishes a relation between gauge transformations and curvature. In particular, if R7--0 then P~,7 = 7. The full import of this relationship is described by the Holonomy Theorem. Let im7 = {7xIX e DerM} c Der ~ ; this is a C~(M)-submodule. Let Hx(V) denote the holonomy group of the submodule imV at the point x and hx(V) the Lie algebra of the group. THEOREM 3.1

(Holonomy Theorem).

For any connection V on ~ and any x e M:

hx (V) = span {xo(P,Rv) (X, Y) ]PE Exp (ira V), X, YGDer M}. COROLLARY.

If V is an analytic connection,

then

hx (V) = span {xo9 (Vx~) . . . . . 9 (Vx,) Vxo [ Xi6 Der M, k > 1}. Proof of the Holonomy Theorem.

Denote

~

= Exp(im7).

It follows from Theorem 2.1

that

hx (V)-- span {xoAd PVx] P6~, XE Der 34} N gl (EJ. On the other hand, the following identity is obvious:

span {Ad P Vxl PE~, X6 Der M}-- span {(P,V)y ]PE~, Ys Der M}. Consequently,

hz (V) = span {(P,V)r -- VY [ P ~ , YG Der M} --s~an{(P.V)v--(Q.V)y ]P, QE~, YE Der M}.

The assertion of the theorem now follows from (3.1). 4.4. Linear Connections. fields. Let

A linear connection is a connection on a module of vector

V :Der 7V/~ Der (Der M ) be a connection on DerM. For any submanifold N of M, we can define a connection UN*V:DerN + Der (iN* DerM) (where iN:N ~ M is an embedding). Note that the submodule D e r N is a direct summand of iN* DerM. The submodule N is said to be completely geodesic for V if, for any X e DerN, the derivation (iN*V) X of iN*Der M leaves the submodule Der N invariant. One-dimensional completely geodesic submanifolds are called nonparametrized geodesics. A vector field X e Der N (where N is not necessarily completely geodesic) is called a geodesic field for V if 7xX = 0. A smooth curve 7('): R ~ M is called a (parametrized)

18~0

d? geodesic if its velocity dy/dt is a geodesic vector field on y(R) c M, i.e.,~Tdv-~-=O

.

dt

It can be shown that a submanifold N of positive dimension is completely geodesic if and only if, for any geodesic curve y(.) and any t e R, the condition dy/dt e Ty(t)N implies y(~) e N for all 9 close to t. integral curves are geodesics.

A vector field X is geodesic if and only if all its

We now consider a nonstationary vector field X t on N, depending smoothly on t e R. the integral curves of this field are geodesics if and only if

All

~TXt+VxtXt~O. Nonstationary fields satisfying this equality are called geodesics. The sets of geodesic curves for two connections V and V' are the same if and only if S(X, Y) = V x Y - Vx'Y is a skew-symmetric C~(M)-bilinear map of Der M into itself. For any connection V on DerM, def

Tv (X, Y)~

----

~xY-- V~X--[X, YI

is known as the torsion tensor of V. It is clear that TV(. , ") is a skew-symmetric Ca(M) bilinear map of D e r M into itself. It follows from the aforesaid that the connection Y is uniquely determined by the set of its geodesic curves together with its torsion tensor. In this context any element of Hom(DerM h DerM, DerM) m a y b e taken as torsion, maintaining the same set of geodesics. 4.5. Left(right)-Invariant Connections. Let G be a Lie group. We may assume without loss of generality that G is a subgroup of the automorphism group of the algebra Ca(M) for some smooth manifold M: G c Iso M (see subsection 4.1). Let fl denote the Lie algebra of G. Thus, G is a finite-dimensional submanifold of the space of continuous linear operators on Ca(M) and fl the tangent space to G at the identitv. An arbitrary tangent vector to G at a point p e G has the form pox = yop, where X, Y e g, Y = AdpX. Arbitrary smooth sections of the tangent bundle are vector fields on G. A vector field is left(right)-invariant if it has the form p ~ poX(P ~ yop), where X (resp., Y) is independent of p e G. Thus, the map that carries each left(right)-invariant field onto its value at the identity is a canonical isomorphism of the space of these fields and g. A linear connection on G is said to be left(right)-invariant if the covariant derivative of a left(right)-invariant field in the direction of a left(right)-invariant field is again a left(right)-invariant field. Let (X, Y) ~ VxY be an arbitrary bilinear map of the Lie algebra g into itself. Defining the covariant derivative of the field poyp in the direction of the field poXp, p e G, by the formula

po(VxpYp J- (TpY) Xp),

(5.1 s )

we obtain a left-invariant connection on G. Similarly, defining the covariant derivative of ypop in the direction of Xpop, p e G, by the formula

(5.1 r)

(-- VxpYp + (TpY) Xp)op, we obtain a right-invariant

connection.

All left(right)-invariant connections are described in this way, since they are completely determined by their values on left(right)-invariant fields. Set Rv(X, Y) = [YX, Vy] - V[X,y], Tv(X, Y) = VxY - YyX - [X, Y], ture and torsion tensors for the connection (5.1,s are

X, Y e g.

The curva-

(poXp, poYp)~poRv (Xp, Yp) and

1841

9(poXp, PoYp)~poTv (Xp, Yp), and t h e a n a l o g o u s t e n s o r s f o r t h e c o n n e c t i o n (5.1 r ) a r e

(X pop, y pop)~ Rv (X p, Yp)op and

(Xpop, Ypop)~Tv (Xp, y,,)op, pGO. Let Pt be an absolutely continuous curve in G.

The element X t = pt-lod/dt Pt =

d/dt In Pt ~ g is called the left angular velocity of Pt at time t, and Yt = d/dt ptopt -l = d/dt in Pt e g the right angular velocity. This terminology emphasizes the analogy with the traditional concepts, which have to do with the group SO(3). A parallel translation along the curve p,=p0oexp ~ XodO=exp 0

SYodeopo, O<*-.
with respect to the left-invariant connection (5.1s

transforms a trangent vector p 0 o z e T p 0 G

t

into the vector pto,~pj '. - V x d ~ .

A parallel translation along the same curve with respect

~0

to t h e r i g h t

invariant

c o n n e c t i o n (5.1 r ) t r a n s f o r m s a t a n g e n t v e c t o r zop0 e TpoG i n t o t h e

t

vector exp!'

Vy d~opt.

A curve Pt is a geodesic for the connection (5.1 ~) (for the connection (5.1r)) if and only if it is smooth and

ot Xt~-VxtXt~O 5.

(~--~-Yt~vYtYt)

Perturbations of Flows and Chronological Connection

Let M be a smooth manifold, ~ a finitely generated projective C~(M)-module, Dt(e) an infinitely differentiable family, depending on e e R, of complete nonstationary derivations of ~ and V(E) an analogous family of stationary derivations. Denote t

0

0

this is a family, depending smoothly on the parameter g, of flows on ~. To every semiisomorphism P of ~ there is a canonically corresponding diffeomorphism of the bundle of frames R(E) (see subsection 4.2), which we denote by r(P):R(E) + R(E). Let t e R and let p e R(E) be a frame. One of the results of our subsequent calculations will be an invariant expression for the tangent vector at the "point" R(Qt(0))(p) to the curve e ~ R(Qt(~))(p) in R(E), valid even when there are singularities of arbitrarily high order. Using formula (3.10) of Sec. 3 for the differential of the chronological exponential, we obtain

Qt (~)=zl D (~)~ (~), t

AdQ~(E) 8/8~ D~(e)d~.

where Zt(1)(e) = V(E) + O

1842

(1)

Denote

t

Z~O (~)=--~_, 0t-* A (e)+ I Ad Q~ (s) ~O~ D~ (~)dr, i - - 1 , 2 . . . . . 0

In that case (see (3.11) in Sec. 3) t

O f+'

o

_

,

o

t

=Z}'+')

0~

t

+ I (ad Z(~') (e))0 Z(,).~z 7('+') 0

where "*" is the operation of chronoloKical multiplication, which associates a pair of nons t a t i o n a r y d e r i v a t i o n s A t , Bt w h i c h a r e a b s o l u t e l y c o n t i n u o u s f u n c t i o n s o f t w i t h t h e d e r i v a tion

(A'B) =

A~, ~

B~

dx.

Note that the operation "*" is not associative, but the

0 following weaker relationship is valid:

A,(B,C)--B,(A,C)=(A,B--B,A),C VA, B, C. Thus

~176176176 Z~~ (0)= ~~

i----1, 2, t

a (0)+ I Ad Q~ (0)-~- D~ (0)dr. 08i

(2)

0

The correspondence e ~ Qt(g) defines a smooth curve in the space of nonstationary derivations of ~. By equation (i), this curve has the sense of a "right angular velocity" for the curve g ~ Qt(e) in the group of flows on ~.

Let Zt(1) (e) ~ ~ i=i

gi-1~ti be the Taylor expansion.

System (2) yields recurrent formu-

las for successive evaluation of the nonstationary derivations ~t i, which use only the flow Qt(O) and t h e h i g h e s t - o r d e r d e r i v a t i v e s w i t h r e s p e c t t o e o f t h e d e r i v a t i o n s b(E) and D t ( ~ ) at zero. Denote

A(0-- Os---O~? A(O), In particular, we have

D~0 ~ 0-a~7 Dt (0).

t o

[

~o

j

o

In order to obtain a compact notation for ~t i for arbitrary i, we shall introduce a special sequence of polynomials in nonassociative variables. The definitions are as follows. Consider a free nonassociative algebra with countably many generators %1, %2, ---; denote it by ~ (%1, %2, -..). The quotient algebra~(X1, %=, ...) by the ideal generated by the elements a(bc) - b(ac) - (ab - ha)c, a, b, c e ~ (%1 . . . . ) is called the free ch r algebra on generators X l, %2, .-.; denote it by A. We make A into a graded algebra by defining deg X i = i, i = i, 2 . . . . (for details about chronological algebras, including in particular the construction of a basis for the free chronological algebra, see [4]). We define a derivation ~ of degree I in A in terms of its action on the generators, by the formula ~X i = XIX i + Xi+ I. The polynomials in which we are interested have the form

1843

In particular, v1(lx) = Ii, ~2(XI, 12) = ~k I = I X I + %=, ~a(Xl, l=, l S) = ~(ll~ I + l=) = (XIXI)X 2 + XI(XIXI) + 2XIX 2 + X2X I + I s. The correspondence

~t~ AU-O+ AdQ,(O) D$~ uniquely determines a homomorphism of the free chronological algebra A into the algebra of nonstationary derivations on M with the multiplication operation "*", We have

;7-- ~, A(~ +

Ad Qo (0) D(o')dO, '"

., A("-) + 5 Ad Qo (0) Dg')ao 0

(3) t"

where the polynomial Vn is considered in terms of the multiplication "*". Fix some time T e R, and a frame 00 e R(Ex0 ) c R(E), and denote pT(e) = R(QT(e))(p0). Then ~ ~ 0T(e) is a smooth curve in R(E). Our calculations enable us to write down an explicit expression for the tangent to this curve at the point PT(0) e R(E), even if PT(0) is a singular point of the curve, of arbitrarily high order. To each derivation D of @ we associate a vector field r(D) on R(E), defined by

Let k be the least index such that ~T k does not vanish at the point x 0 ~ M. It follows without difficulty from equation (1) that 8k

pr (e) = pr (0) + ~- R (Ad Q~' (0) ;~) (Or (0)) + O (e~'). It is not hard to derive recurrent formulas that yield hdQt-l(O) r (k) directly.

(4) Denote

yjk) (8)= Ad QT' (0) Z~k) (e), Y~')(e)= '~ #-'n~, i=!

where

Using equation (2), we obtain a~ -6~

0

O

'[.'.

-~'i ~t

J T-a-[ ~t

,

-

0

I~at+ ad D, (0)) Y~) (0) ----D~k), yo(k)(0) ----A(k-~ As we see, the equation for Yt (k) differs from the equation for zt(k) only in that a/at is replaced by the operator O

a~+ ad Dt

(0).

The coefficients of the expansion of Yt (k) in powers of ~ are obtained from the equations

'0b) + ad Dt (0)) ri~ = O~), r(ol)= AO),

(aO~+adD,(O))n~=[nLD~"lq-DT',

n~,=W')

and so on. Consequently t

q~ ----exp 5 0

1844

i

ad D~ (0)dTA(~

t

S exp ~ 0

'v

-- ad Do (O) dOD~')d'r;

t

f

t

0

0

'~

and so on. We now devote some more attention to the expression obtained above for ~k, expressing ~k as a polynomial with respect to chronological multiplication "*"; we shall assume that A is independent of E, so that ~0 k = 0 for k e 2. Integrating by parts shows that, for arbitrary At, Bt,

(A.B),--(B,A), = [A, B , ] - [Ao, Bol. In particular, if B 0 = 0, then A , B - B,A = [A, B]. Let x 0 E M. In view of (4), it is particularly important for us to know how to evaluate St k at the point x0, on condition that ST i vanishes at this point for i < k. It is clear that the value remains unchanged if the derivation Ek$tk is changed by adding an arbitrary commutator polynomial of aistm , i < k. In this sense the above sequence of chronological polynomials is not the only one possible. Sequences other than (3) were described in [3], [4] (see also [2]). The lemma formulated below provides an overview of all the possibilities. As before, let A denote the free chronological algebra on generators Xl, 12, .... degl i = i. The commutation operation (a, b) ~ (ab - ha), where a, b e A, defines the structure of a graded Lie algebra on A -- the Jacobi identity is readily derived from the relation a(bc) b(ac) = (ab - ba)c. We denote this Lie algebra by [A], writing, as usual, (ada)b = ab ba, V a , b e A. LEMMA. Let @ i = 1, 2 . . . . , and

be the Lie subalgebra of [A] generated by the elements wi(X I . . . . , li) , ~k the set of all homogeneous elements of degree k in 9. Then

~2=Span{~,l~,[-{-~,2}, .~k+r=~k+~(ad ~,l)~h, k~2. Proof. Clearly, ~ is the minimal Lie subalgebra of [A] containing ~i and preserving the derivation ~. We can endow the space [A] 9 R~ with the structure of a graded Lie algebra by putting def

[~,a]=~a,

Va~A, deg~=l

(the Jacobi identity follows from Leibniz' rule for the derivation 11). Then the Lie subalgebra of [A] * R ~ generated by the elements Ii, ~ is precisely ~ R 11, and the homogeneous component of this subalgebra of degree k is ~A for k _> 2. The statement of the lemma now follows from standard properties of Lie algebras (see, e.g., [7], Chap. 4, Sec. 8). We give separate attention to the case in which the family of derivations Dt(E ) is an affine function of E (but A, as before, is independent of E), in other words, Dt(E) = A t + EB t. In this situation Dt(i) = zt(i)(c)--0 for i -> 2, and we may legitimately write Zt (I) = Z t.

The basic equations

(i), (2) become 0

~8 Qt = Z p Q .

t

os Z, ----j'

t

[Z~, 2~1 dT, Z, (0) = A + j' Ad Q~ (0) B~d'~.

0

0

Thus, the curve E ~ Qt(E) is a geodesic for some right-invariant connection q on the group of flows, and for any absolutely continuous nonstationary field Z~, dependent on ~, we have t

(Vz~Z~),= ~ [z+, 2,1 d~, 0

Remark. Strictly speaking, right-invariant connections and their geodesics were defined above for finite-dimensional Lie groups only. Nevertheless, the formal relations involved carry over without change to the group of flows. The curve E ~ Qt(E) and its expansion are obtained by a transparent geometrical interpretation. The connection we are looking for is already defined "on the diagonal," so that in order to define it uniquely everywhere we need only specify the torsion. If we demand

1845

that T V = 0, then for any nonstationary absolutely continuous fields AT, B0, t

(VA, B,)t-----~ [A.~, B~I d'c-q- 1 [Ao, Bol.

(5)

0

The curvature in this case is Rv(AT, B~) = 1/4 ad [B0, A0]. We now consider in more detail the geodesic flow defined by a connection Y on the group of flows. Proposition. Let A t be a nonstationary derivation on ~, which is absolutely continuous with respect to t, Qt ~ a flow on ~. Then there exists a unique geodesic e ~ Qt(g) of V which satisfies the conditions Qt(0) = Qt ~ 3/3~ Qt(0) = AtoQt~ This geodesic is defined by the formula t

The fact that the above expression indeed defines a geodesic of V follows from our previous

arguments.

Uniqueness w i l l be proved presently.

Let at(s, A) denote the right angular velocity of the geodesic t

Qt (0=e~A'oexp ~ ~A~d~, & Qt (~)=~t (8, A)oQt(0. 0

It follows from (i) that I

f~t (e, A) = Ao+ SoAd Q, (8) ATdz. A direct calculation using the variation formula yields the identity

f~t(el:~e2, A) =fl(et, fl(e2, A)).

(6)

It turns out that an important property is preserved if the scalar is replaced by a scalar function of t. To be precise: let u: ~-+R be an absolutely continuous function. Define t

Qt (u) = e'(~

t

j" u (~) fi~az, f~t (u, A) = uoAoq- SAd Q, (u) A~a~. 0

.s

Then

~tCu+~, A)=~t(u, ~(~, A)), Vu, ~, A.

(7)

This identity is also a direct consequence of the variation formula. It expresses the fact that the correspodnence u ~ a(u, .) defines a (nonlinear) action of the additive group {u} = R 9 L=(R) in the space of nonstationary vector fields which are absolutely continuous with respect to t. We shall use (6) to prove the uniqueness of a geodesic with given initial data. Strictly speaking, we must prove that the Cauchy problem for the following equation has a unique solution I

~sV, (e)----~ [V~ (e), V~(8)] dz,

Vt (0)----At.

(8)

0

Let Vt(s) be a solution; it will suffice to show that 8/8s ~t (--g, V(g))--0. V0(E)--A0, it will suffice to prove that 82/BESt ~t (-g, V(E))--0. t

0

1846

Denote

Since clearly

and i

o, (s)= e-~da.oexp f -- s ad V, (s) d~V, (s). 0

Differentiation of the chronological exponentials yields the identities

o-~o

=

~--

8

[co, ~], ~ (0

=

A0

and

o-~ o = [~, o ] - -

~ + 8 [~, o] d*, o --[A0, o]. 0

Consequently,

- •[ 0 = 0 . 8s

The relations established in this section find applications in the theory of smooth control systems. In that context one actually uses a slightly simplified variant of the theory developed here, in which families of derivations Dt(e) , A(E) of the module @ , are replaced by families of vector fields on M. We end this section with an account of invariant commutator expressions for the highest-order variations of a control system. Consider the system

p=poft(u),

uER~, pCIsoM.

(9)

Here f t ( u ) , where u e I{~, i s an i n f i n i t e l y d i f f e r e n t i a b l e f a m i l y o f n o n s t a t i o n a r y v e c t o r f i e l d s on M. A d m i s s i b l e c o n t r o l s a r e d e f i n e d as e l e m e n t s o f t h e space L = r [ o , T] o f e s s e n t i a l l y bounded m e a s u r a b l e r - d i m e n s i o n a l v e c t o r - v a l u e d f u n c t i o n s on [0, T]. I t i s assumed t h a t a l l n o n s t a t i o n a r y f i e l d s of t h e form f t ( u ( t ) ) , u ( - ) e L ~ r [ o , T] a r e complete. F i x some a d m i s s i b l e c o n t r o l u ( ' ) -+~

e L ~ r [ o , T] and d e n o t e k tad

Ok

t 0

Perturbation of the initial conditions by a vector field eA e D e r M and perturbation o f t h e c o n t r o l u ( t ) by a c o n t r o l f u n c t i o n e v ( t ) l e a d t o t h e f a m i l y t 0

Using the above calculations, we obtain 0

0---8Qt = Z ( 1 ) ( ~ ) ~

Qt (O)-~->t"

As t o t h e c o e f f i c i e n t s

of the Taylor expansion of the angular

o f e, Zt (1) (e) ~ ~

e k - l ~ t k , these are (see (3)):

velocity Zt(1) (E)

in powers

k=t

....

This expression is homogeneous of degree k in the perturbation (A, v(.)).

(A,~('))~k

A+

The map

Adpovk(O)fodO

Adp0v(0)]0d0 . . . . . 0

1847

is known as the k-th variation of the control system (9) "along u(t)." Suitable expressions can of course be developed for qt k = AdPt-1~t k. LITERATURE CITED i. 2.

3. 4. 5. 6. 7. 8. 9.

i0. ii. 12.

1848

A. A. Agrachev and S. A. Vakhrameev, "Chronological series and the Cauchy-Kovalevskaya Theorem," Itogi Nauki i Tekh., VINITI, Probl. Geometrii, 12, 165-189 (1981). A. A. Agrachev, S. A. Vakhrameev and R. V. Gamkrelidze, "Differential-geometric and group-theoretic methods in optimal control theory," Itogi Nauki i Tekh., VINITI, Probl. Geometrii, 14, 3-56 (1983). A. A. Agrachev and R. V. Gamkrelidze, "Exponential representation of flows and chronological calculus," Mat. Sb., 107, No. 4, 467-532 (1978). A. A. Agrachev and R. V. Gamkrelidze, "Chronological algebras and nonstationary vector fields," Itogi Nauki i Tekh., VINITI, Probl. Geometrii, ii, 135-176 (1980). S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, II, Interscience, New York (1969). A. V. Sarychev, "Integral representation of trajectories of a control system with generalized right-hand side," Differents. Uravn., 24, No. 9, 1551-1564 (1988). J. -P. Serre, Lie Algebras and Lie Groups, Benjamin, New York (1965). F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foreman and Co., London (1971). R. Gamkrelidze, "Exponential representation of solutions of ordinary differential equations," in: Equadiff. IV (Proc. Czechoslovak Conf. Diff. Equations and Their Applications, Prague 1977), Springer, Berlin (1979), pp. 118-129. T. Nagano, "Linear differential systems with singularities and applications to transitive Lie algebras," J. Math. Soc. Jpn., 18, No. 4, 398-404 (1966). P. Stefan, "Accessibility, orbits and foliations with singularities," Proc. London Math. Soc., 29, No. 4, 699-713 (1974). H. J. Sussmann, "Orbits of families of vector fields and integrability of distributions," Trans. Am. Math. Sot., 180, 171-188 (1973).