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

1778

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

'

X e D e r M the map x ~ 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 =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 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

1783

w~T*f(v)=~fiDer*~l, vEDer*N, < T * f (v), X > Ix= < ~r(x), T x / ( X x ) = . 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


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

=p. 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

=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.

1784

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 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 '~/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

=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)>=.

(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), ~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

=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)

{/*=/*, 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) = = 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 belt

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, Ot"

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).