Manifolds of smooth maps - Numdam

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dle behaves in a nice functional way. We have ...... H. CARTAN, Topologie différentielle, Séminaire E. N. S. Paris ( 1961- 62). 4. ... pology 6 ( 1967), 263-271. 9.
C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

P. M ICHOR Manifolds of smooth maps Cahiers de topologie et géométrie différentielle catégoriques, tome 19, no 1 (1978), p. 47-78.

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Vol. XIX-1 (1978)

CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE

MANIFOLDS OF SMOOTH MAPS

by

X, Y

Let

P. MICHOR

be smooth finite-dimensional

be the space of all smooth maps from X

Coo ( X , Y )

on a

in Sections 1, 2, and

smooth manifold modelled

pact support of

vector

on

use

manifolds, and let Coo ( X, Y )

Y . We introduce the

to

it then

spaces D(F)

to

[7],

of Keller

which

be

can

ture to

a

bundles F over X with the nuclear

applied

rather strong to some

the tangent bundle of

dle behaves in

a

one.

%°°-topology C°° (X , Y)

of smooth sections with

of L. Schwartz. The notion of differentiation which

C’

show that

We obtain

a

we use

is

com-

( LF ) -topology is the concept

weak inversion theorem

stability. We give a manifold strucCoo ( X, Y ) , and we show that the tangent bunnotions of

nice functional way.

only finite dimensional manifolds X , Y , since topology Doo depends heavily on it. It may be possible to extend the We have considered

the

results for

some

infinite dimensional manifolds Y .

We considered

theory

to

the

case

ory coincides with

I

week

to

am

very

help

me

been able

to

only

smooth maps, but it is clear how

to

adapt

C’, supposed existing theories, e. g. Leslie [8]. r

grateful

> 0 . If X is

to

to

E. Binz and A.

be compact, then

Frblicher,

with Lemma 3.8 and without whose

prove it. I thank

J . Eells

vice.

47

who devoted

help

I would

our

a

the the-

whole

never

have

for encouragement and valuable ad-

D-TOPOLOGY

1. THE

ON

X, Y

1.1. NOTATION. Let

Coo ( X, Y ) be the Jn (X , Y ) be the fibre let

C°°(X, Y).

set

be smooth finite dimensional

of smooth

bundle of

the canonical manifold

mappings from X

to

of maps from X

to

n-j ets

manifolds,

Y ; f or n 6 N , let Y , equipped with

makes j n f : X -> ]n (X, Y) f ( Coo ( X, Y ) , where jn f (x) is the n-j et which

structure

smooth section for each

and

into

a

f

at

of

x E X . Let

be the components of the fibre bundle

proj ection,

i. e.

and

if

Q

See

is

a

n-j et

at x

E X of

a

function

[5].

1.2. DEFINITION. Let

K = (Kn),

n = 0,1,...

be

a

fixed sequence of

com-

pact subsets of X such that

Then consider sequences is

such that

nonnegative integer and pair (m , U ) of sequences define mn

The

S-topology

on

basis for its open a

For each such

a

C°° ( X , Y ) sets.

by:

a set

is

It is easy

given by taking to

all

M (m, U) as a actually a base for

sets

check that this is

topology.

1.3. Let

which is compatible Jn ( X, Y), n = 0 , 1, 2, with the topology of the manifold. Consider families 0 = (dn), n = 0, 1, ... of continuous nonnegative functions on X such that the family of the sets (supp On) is locally finite.

dn

be

a

metric

on

... ,

f E Coo (X, Y). Then each farnily d = (4)n) open neighborhood Vd ( f ) of f by

LEMMA. Let

defines

an

48

as

described above

PROOF. We

claim

that each

in such

inductively

a

vd (f)

is open. Determine

R defined

Then consider the continuous maps

and let

Wl

where

17 1,k

k > 1 .

Then it is

(mn)’

way that

be the open

s et Al-1 ((-1, 1)). Set

denotes the canonical

easily

by

checked up

by writing

out

1.4. Consider sequences

of X such that

proj ection

for

the definitions that

compact subsets

is

locally

finite

on X ,

and sequences U

of open subsets : LEMMA.

The

Each pair

( L , U ) of sequences

family {M’ (L ,U)},

as

where L and U

above

are as

defines

a

above, is

set

a

basis

for

the

2J-topolo gy. be the fixed sequence of compact subsets of X of

PROOF

I.2 ; then set as

in

(XBKno )

clearly locally finite. Let M(m, U) 1.2 ; define L = (Ln) by is

define

Then

clearly

49

be

a

basic open

{M’ ( L , U ))

So the system

itself

a

f

we can

M’ ( L , U)

Given of

basis if

and

contains

f E M’ (L , U)

will

we

construct a

neighborhood

Wn = (j n f r1(Un) XBLn° C Wn since in f(XBLno) C Un . For Wn define

with

X and

V0 (f ) C M’(L, U). For each

and it is

is open.

M’ ( L , U)

show that each

d-topology

basis for the

a

n,

V d(f)

is open in

x c

We claim that

subset C of

is bounded below

an

This is

Wn .

seen as

and contained in the open

and

so

For

x E Wn

set

by

choose

a

positive

constant on

follows: j n f (C)

t/ .

So

Un

for all xEC. Now

an (x)>E

a

we

continuous function

contains

each compact

In ( X, Y)

is compact in

a set

construct a

6x : Wn ->

continuous function:

R such that

d x ( x ) > 0,

0 dx an ; this is possible since an is bounded below on a compact neighborhood of x in W n . Then use a partition of unity on X subordinate and

to

the

cover

and obtain Now

W’ n

bn .

(XBLn° )

of open

is

sets

locally finite, such that

each

XBLno

XBLno C I/( C Wn

is closed. So there is

and

W’ 71

is

a

again locally

family finite.

Choose continuous functions

un : X - [ 0 , 1]

such that

un = 1

on

XBL no , Un = 0

off

W’n

and let

Then dn: x-> [ 0 , oo ) is continuous and d = (dn) is (supp dn)’ C ’ (W’n) is locally finite, so by 1.3 Vd(f) Given

ge V d (f),

th en

f or all n > 0 and for all

x E X,

i.e.

50

a

family

is

3-open.

such that :

so

Since

for all

this

implies j n g (x ) fUn

get

g f M’ ( L , U ) . We have

x f

XIL no by

the definition of

an , so

we

fE f V d (f)C M’ (L,U). Q.E.D.

1.5.

CO ROL L AR Y.

D-topology on a)

Fix

a

Coo

following equivalent descriptions of

the

(X, Y ) :

sequence K

Then the systems

base

We have the

the

=

(Kn) of compact

sets

in X such that

of sets of the form

9-topology

COO (X, Y),

where

through (mn ) all sequences of integers and U =(U n ), U n open in J mn(X, Y ). The D-topology is independent of the choice of the sequence (Kn). J (X, Y), compatible with b) Fix sequence (dn ) o f metrics dn is

a

for

on

m =

runs

n

a

the

is

on

manifold topologies.

a

neighborhood

Then the system

for f c coo (X, Y) cp = (On) runs through base

open sets, where : X -> [0,oo ] such that

independent of the c) The system

(supp d n)

is

of sets of the form

in the

9-topology, consisting of

all sequences

locally finite.

o f continuous The

a

D-topology is

of the metrics dn . of sets of the form

choice

of open sets for the D-topology on Coo (X , Y ), runs through all sequences of compact sets Ln C X such locally finite and Un open in j n ( X , Y ) . is

maps

base

51

where L =

that

(Ln)

(XBLn o ) is

1.6. REMARKS.

a) The °° in

[9],

D-topology

e. g. described very

as

Whitney C’°°-topology, explicitly in [5], II - 3. It is

is finer than the

the

the

topology topology

°° of Morlet [3].

b) If one

,

produced by applying

then the

is the

on

the seminorms of does

Hence the

9-topology

not

have compact support.

The set D (of all functions with compact support in the

name.

Coo ( Rn , R)) is the maximal linear subspace of Coo ( Rn , R ) which is a topological linear space with the topology induced from the S-topology (and from the Whitney C°°-topology too, but 9 with the Whitney topology has no merits from the point of view of functional analysis and this is our reason for introducing the 2-topology). c) C° (X, Y ) with the D-topology is a Baire space. This is proved by Morlet [3 . A quite straightforward proof is possible using description

space

1.5 b, which and are

can

Coo (X , Y) chosen

to

also be used

is

be

complete complete.

1.7. L EMMA. A sequence

to f iff

there exists

a

of the fn ’s equal f off

to

in this

define

pseudo-metrics for uniformity if the metrics dn

(fn) in Coo (X, Y)

compact K

set

uniformity, on In ( X, F. a

converges in the

K C X such that all but

and j 1 fn , j 1 f «uniformly.

on

finitely many K, for all 1 c N .

Sufficiency is obvious by looking at the neighborhood Necessity follows since fn -+ f in the 2-topology implies fn -> f

PROOF.

ney

topology

and there it is well known

1,8. LEMMA. For each

is continuous in the PROOF. For each

is defined

as

k ) 0 the

S-topology

( [5],

II- 3, page 43

base 1.5 b.

in the Whit-

).

map

9-topology.

1 the

follows : if

mapping

a

rep-

52

d, then

resents

It is

a

end

on

fact,

routine task

special embedding ). By

for each

definition

Now

basic open

set

ak,l (a ) is well defined, i. e. the representative f , and that it

show that

choice of

the

an

to

we

we

use

not

dep-

is smooth

(in

have

the basis

be

a

in and

open in Now

does

set

and Set

which is open in a

basic open

which

can

be

set

and denote

is

and

in

seen

Then

by writing

out

the definitions.

Q.E.D.

1.9. If

are

A , B, P

are

topological

continuous maps,

with the

topology

we

Hausdorff spaces and

consider the

set

A

xpB

defined

by :

induced from A X B . Then A x B is the

pullback

of the

uA , 77B in the category of Hausdorff topological spaces. A continuous map is called proper if the inverse image of a compact subset is compact. The subset C° prop (X, Y ) of proper maps of Coo ( X, Y)

mappings

is open in the

9-topology,

since it is open in the

(see [9], 2, Proposition 4 ).

53

coarser

Whitney topology

([9], 2, L emma 1). Let A , B and P be Hausdorff topological spaces. Suppose P is locally compact and paracompact. Let TT A : A , P and rrB : B - P be continuous mappings. Let K C A and L C B be such that tt A/K and rrB/L are proper. Let U be an open neighborhood of KxPxL in A fr B . Then there exist open neighborhoods V of K in A and Wof L in B LEMMA

such that

KxP L C VxP C U .

1.10. PROPOSITION.

1 f X, Y,

given by ( g, f)- go f ,

Z

are

smooth

is continuous in the

manifolds,

then

composition :

2-topology.

(g , f) E Coo ( Y , Z ) x Coo prop (X , Y) and let M’(L,K) be a 2-open ne ighborhood of g o f in Coo ( X , Z) ( c f . 1.5 c ), i. e . L ( L n )

PROOF. Let

basic and U

ite,

=

(Un),

where each

open in

Jn(X, Z)

=

Un

Now consider the

Ln

is compact in X with

(XBLno )

locally

fin-

and

topological pullback

for

each n > 0 ,

as

in I.9 :

The maps

are

and

well

defined, since

they

without

are

smooth, since they

constant

term, followed

are

locally j ust composition

by truncation

54

to

of

polynomials

order n . We have

So

) is proper, since it is

is proper. So all the

there exist

homeomorphism, inverse

is compact in

is proper

since f

a

hypotheses

to

then

of Lemma 1.10

are

fulfilled,

so

neighborhoods and 1

in ,

for each n > 0 such that

Since

family pact

(f

sets

ite. Then

and

locally compact and XBLno is locally finite, the (XBLno)) i s locally finite too. So there exists a sequence of comK in Y such that f (XBLno) C YBK n and (YBKno) is locally fin-

is proper, Y is

f

we

have

is open.

is open in then

we

have

,

and if

and

is open

Moreover

Now let

since

and

then for

since

all n > 0

and x

for all

( xBL n 0

we

have

55

we set

Now if

so

so

hence

g’o f’ E M’ (L, U). Note that

2. THE

2.1.

DEFINITION. Let

relatively

COO ( X, Y )

and the

compact

This is

are

used Y

locally

compact in the

proof.

Doo- TOPOLOGY ON Coo (X, Y).

f , g E Coo (X , Y ) is

we

is

above relation

an

we

call

f equivalent

equivalence

D-topology

to

relation. The

the weakest among all

now

are

be smooth finite dimensional manifolds. If

set

in X ,

clearly

finer than the

X, Y

g

g

).

Doo-topology

topologies

and for which all

(f~ on

on

the

Coo ( X, Y )

equivalence

set

which

classes of the

open.

2.2. REMARK. The

9°°-topology

on

Coo ( X, Y )

is

given by the following topology induced from the

equivalence classes with the 2-topology and take their dis j oint union. It is clear how to translate the different descriptions of the 19-topology given in 1.5 : In 1.5 a and c , j ust take all intersections of basic 2-open sets with equivalence classes. In 1.5 process : take all

b , add

f-g

to

the definition of

2.3. COROLL ARY. A sequence

Td ( f ) . (fn) in Coo (X , Y)

converges in the Doo-

topology iff there exists a compact set K c X such that all but a finite number o f the fn ’s equal f o f f K and j 1 fn -> jl f «uniformly on K>> for all 1. 2.4. COROLLARY. For each

k> 0 the map

56

is continuous in the

topology.

PROOF. j k

respects the

2.5.

X, Y, Z be smooth manifolds. Then the subset of COO prop (X , Y ) is 2’-open in Coo ( X , Y ) and composition

COROLLARY.

proper maps

is continuous in the

equivalence

relation.

Let

Doo-topology.

PROOF. If

where fl , f2

are

proper, then

glo fl - g2 f2 ’ o

2.6. REMARKS.

a) By 2.3 and 1.7 convergence of sequences in

C’(X, Y)

is

equival-

Whitney C°-topology, the S-topology and the Doo -topology. Therethe Thom Transversality-Theorem and the Multijet-Transversality-Theo( cf. [3L II, Theorems 4.9 and 4.13 ) hold for the 2-topology and the

for the

ent

fore rem

%°-topology

too, since in the

proofs

of

these, convergent

sequences

are

constructed.

b) reason

C°°(X, Y)

with the

g-topology

is in

general

no

Baire space. The

for this will become clear in Section 3. However, it is paracompact

and normal

(cf. 3.9 ).

2.7. PROPOSITION. Let

E - X be

rr :

a

smooth

finite

dimensional

vector

Let D (E ) denote the space of all smooth sections with compact support o f this bundle. Then D(E), with the topology induced from the Dootopology on Coo (X, E ), is a locally convex topological linear space, in fact a dually nuclear (LF)-space. Furthermore it is a Lindelöf sp ace, hence bundle.

paracompact and normal.

Coo (E), the space Coo ( X , E ), and 19 ( E )

is

clearly 2-closed

PROOF.

of all

space

is closed and open in

sections,

bundle 17 I : F - X such that the Whitney space X X Rn . Then Coo ( E O F ) = c:o (X,

a vector

thus on

a

it is

exactly

the

topology

induced from the

57

Coo ( E ).

sum

Rn)

in the

There exists

EO F -> X is trivial, and the Doo -topology

gm-topology on C°°(X, E (9 F ).

Then l%(E ) and the

is

a

topology b with

paring 1.5

in fact,

topological subspace,

is

exactly

one

topology

the

of L.

direct summand of

a

Schwartz,

as

is

of the well known systems of seminorms

seen

by

com-

for D (X x R

(compare [6] page 170). Therefore it is a (LF)-space, nuclear and dually nuclear, and locally convex inductive limit of countably separable Frechet spaces

(which

where K

=

of these is

its closed

ular,

can

(Kl) a

be identified with

is

a

sequence of compact

Lindel8f space

subspace D ( E )

a

of X

as

in 1.2. Each of

metrizable ), so D (X x R)n Lindel6f space, and clearly completely

( separable

is

sets

and

and reg-

hence paracompact and normal.

Q.

3. THE MANIFOLD STRUCTURE ON

Coo(X, Y)

EQUIPPED WITH

E. D.

THE

Doo-TOPOLOGY. 3.1.

DEFINITION.

tubular a

neighborhood

submersion

a)

Let X be

rr :

b) the

rr :

submanifold of the smooth manifold Y . A

of X in Y is

an

open subset Z of Y

together

with

manifolds,

let

Z -> X such that :

Z -> X is

a vector

embedding X->

3.2. PROPOSITION. Let

f E Coo (X, Y )

a

Z is the

X,

and denote the

Then there exists

bundle ; zero

section of this bundle.

Y be smooth

finite

dimensional

graph

Z f of X f in X X Y with vertical proj ection, i. the submersion Zf -31 X f is j ust the restriction to Zf o f the mapping (x , y ) -> ( x , f (x ) ) from X x Y onto X f. PROOF. We have Xf C X X Y , so TX (X f) is subbundle of TX f (X X Y). We claim that for each (x, f (x)) E X f the space T (x,f (x))(Xf) is transverby

X f.

a

tubular

neighborhood

e.

7T:

a

sal

to

58

in

:

is

where

unit

for

vector

in ’

smooth

a

any

.

Such

a

is of the form

path

with

path iff

so

if

has

is the

and

path with

smooth

a

has the form

vector

iff

then

non-zero

first coordinate and

Thus

and

is

transversality follows since

a vector

dle

u

XxY

is

a

bundle

over

Xf

x}x Y) T(x,y)({ x,y

since it

over

X

X

is j ust

exponential maps, defined tions respectively. Then

an

on

to

So

the

Y via the

realization of the normal bundle

be

is

dim X f = dim X .

pullback embedding

neighborhoods

a

vector

X

Y,

bun-

and it

Y ).

Now let :

U and V of the

zero sec-

in

diffeomorphism

59

X f -> X

TXf (Xf) TX (X

exponential for X X Y , given explicitly by

gives

of the

of

X

onto our

open

neighborhood

of

Zf

if vx E V . We denote it by exp . Clearly there is a diffeomorphism of the

section of E

zero

onto

Xf

cp

in X

from the

proj ection.

It is

easily

commutes, where p : Z -

So

have

we

making

p:

a

fibre

Zf -> X f

Xf

neighborhood

tt o d = tt/W,

checked that the

is the restriction

into

where

preserving 77;

E +

Xf

Z,

thus

diagram

to

preserving diffeomorphism a vector

which is fibre

E,

the whole of

(cf. [5], II, Lemma 4.7 ), i. e. such that is the

Y , given by

X

Z of the map

r

=

exp o

d-1:E->

bundle .

Q. E. D . 3.3. For further reference

Coo (X, Y ) ,

an

open

ection

and

a

we

have

a vector

neighborhood

p f:

fibre

commutes.

we

Zf

of

repeat the situation in detail : For each

bundle

Xf

?7f: E (f) ->

in X

X

Y

Xf

together

f

given by

with the vertical

proj -

Zf 4 X f, preserving diffeomorphism t f: E (f ) -i’

For

f, g (Coo (X, Y)

we

have the

and

60

in

Zf,

i.

e.

the

diffeomorphisms

diagram

They satisfy

3.4. Since the main idea of the construction of the manifold will be blurred

give an outline of it now, disregarding ologies, continuity and differentiability. For f E Coo (X, Y ) we have the setting of 3.3. Let

up

by technicalities later

and denote Then

Zf .

by

Coo (Zf)

on,

we

the space of smooth sections of the

define df: Uf -> Coo (Z f)

vector

top-

bundle

by

and for is the canonical

where 77,

uses

We

now

ping.

declare that

is

the vertical

a

chart for

We will check the coordinate

cond map such that with

VI

V f n Vg # 0 .

Xh C Zf n Zg . Let

us

proj ection.

proj ection

f,

and that

change That

check the

now.

means

Then

of ’

Vif

=d-1 f

since

and

CPt is the coordinate map-

Let

that

g E Coo (X, Y) be a sethere is h E Coo (X, Y)

map df o wg /dg(Uf n Ug). If

then

So Of 0 09 (s) structure

=

s

oaf

in the notation of 3.3. But, of course, the

changes .

61

linear

REMARK. If

3.5. duce

topology

a

in 3.4 should be

Uf

Coo ( X, Y)

on

it is clear that such

topology

a

such that must

f,

chart for

a

then

we

have

to

intro-

is open. From the form of

Uf

be finer than the

Whitney

Uf

C 0-topology.

topology (and the VJhitney CO -topology too) the space Coo(Zf) is not a topological vector space, only a topological module over the topological ring Coo (Xf, R). One could choose this solution and use differential calculus on such modules, as sketched by F. Berquier [1]. The other But with this

possible solution, presented here, of Coo

(Zf )

logy ;

this is the

which is

is

space D(Zf)

look

at

the maximal linear

linear space with the

topological

a

to

Whitney

Whitney C°°-topology on it has no merits from the point functional Analysis (cf. the topological conclusions of 2.7), so to introduce the g-topology. The equivalence relation 2.1 is we want

how

to

to

modify

tioned, and

model the manifold the construction

most

of the

to

topological

on

obtain

proofs which

we

3.6. THEOREM. Let X, Y be smooth

Doo-topology is

a

smooth

of view of

So

Uf

is

set

compact support of

Let

will

give

choose

necessary,

spaces. It is clear

of the other models

just

men-

remain valid.

manifolds.

discussion of

up in 3.3. For

2’-open.

one

vector

we

Then C’

(X, Y) with the

manifold.

PROOF. We postpone the

the notation

CO -topo-

of smooth sections with compact support.

But the

if

subspace

f E Coo (X , Y)

D(E (f))

rr f: E( f ) ->

differentiability we

to

3.7, and

define the chart

we use

Uf

by:

be the space of smooth sections with

X . We

define 95f: U f -* D(E ( f ))

and

Then

and

62

by:

both maps

so

verse to

If

on

Now

are

Doo-continuous by 2.5. We claim

each other. Note that rrY

the other

we

study

hand g c U f ,

open

to

=

s E D (E ( f )).

g . Let

change. Let g f Coo ( X , Y) h f Coo (X, Y ) with

check the map

in D(E( f )) . Let

are

in-

Then

then

the coordinate

be non-void. So there is

We have

oj0g

that d f and qjf

such that

UfnUg

d f o wg/d g(Ug n Uf). Clearly dg (Ug n Uf)

is

s E dg(Ug n Uf) C D(E(g)).

So

or

Coo (dfg*, E(g )): pullback

So

we

s ->

s o df isust carrying

(dfg ) * E ( g ) ,

have j ust

to

a

vector

check the

bundle

over

over

Xf,

differentiability

63

sections of

and

so

E(g)

is linear :

of the map

to

the

t-1f

given by s->

o

r

o

s , and this is assured

Lemma 3.8.

by

Q. E. D. 3.7. We a

have

now

to

fix the notion of

rather strong concept, the notion

locally convex spaces expansion to hold.

and strong

Let E and F be

continuous we

enough

locally

f

convex

is of class

of Keller

[7],

It is

valid for

arbitrary the Taylor

for the chain rule and

f : E - F be a x, y E E and X 6 R

linear spaces, let

C1c,

if for all

x , and the

at

F.

to

so on.

Keller If

f

mapping E , F for each x E E , the derivative of mapping (x, y) -> D f (x )y is continuous as a mapping from

is

Df (x)

E X E

linear

a

is of class

f [7]

is of class

fulfills the

following

f

(HL)

C1tt , then

it is

actually

Coott , then

even

For each seminorm p

3.8. LEMMA. Let X be

bundles

and

differentiable in

an

apparent-

Keller [7], 1.2.8 ) :

on

F there is

over

a

following

on

F’ there is

smooth

X. Let U be

smooth section with compact support

seminorm q

on

E such that

a

seminorm q on E

Taylor

with

series.

E -> X and p : F -> X be open neighborhood of the image of a

manifold, an

a

stronger condition holds :

the

Similar conditions hold for the remainders in

vector

C1c,

is of class

C; == Coott .

condition (see

For each seminorm p

is of class

(x,y) -> Df (x)y

if

the remainder

sense :

(HL’)

Cc

has shown that

ly stronger

If

Then

Coott

we want to use.

have

where

f

mapping.

differentiability

s

64

of E

let

n :

and let

a :

U , F be

a

smooth

fibre preserving is

of class

Coott,

map. Then the map

where

in D (E). Furthermore, derivative o f a ,

D D (a)=D (df a),

is op en

PROOF. If

we can

holds in the

since the

so D (a ) D (a) , so So

we

continuity

condition

C1tt,

it is of class

("

holds in the

converges

for s

is of class

to

to

show

and

D D (a ) = D(dfa )

again,

and

on

of

d fa

And for this it suffices

form

same

use

of

Corollary 2.3.

For x

as

to

c

expressions give

zero.

to x

converge

uniformly

to

So

only «partial dewe

those of

E supp s all partial neighborhood of x, . So we can

show that for each xo

uniformly

on

locally trivializing chart

X,

to

the compact support of the section 9 all

rivatives >> of ( 2 ) with respect

a

is of the

so on.

will make

we

by the definition

vatives » converge

fibre

that, for y E RB{ 0 }:

3T.topology ;

show that

is the

E V, s E D(E), then D(a ) is of class Coott , for D(d fa ) is automatically fulfilled by 2.5,

Outside of the compact support of 9 both have

d fa

show that

Doo -topology

have

where

some

about xo

65

and have

now

the situation

( 3 ). deritake

In this

set

up

we

have and

for x

c

where t,t: W -> Rr

W,

Step

uniformly

the

we use

on some

R r, Rs .

smooth and ax :

1. We will show that for h - 0 the

converges

For that

are

neighborhood

expression

of xo

to

[4]

value theorem in the form of Dieudonn6

mean

( 8.

5 .4 ) : Let E and F be

into F of If

f

We

In

is

set

two

neighborhood differentiable on S , a

f - f ’(d )

our case

for

f

f be a continuous mapping S j oining two points a, b of E .

Banach spaces, let of

a

segment

then

and get

this looks like

sup

and

Step

2. "We

uniformly on this by reducing

verges do

and

now

( 5 ).

But

by

the uniform

show that the differential of some

neighborhood of xo

to

continuity

of the

(4) with respect

mapping

to x

con-

the differential of (5 ). We

Step 1. So we compute first the differentials of (4 ) first some preliminaries : Let p : Rn x Rs - Rs be the canoto

66

let j x ; Rr -> Rn+r be have ax p a o j x , so

the

partial derivative.

is

nical

and for x E X

jx(

Then

proj ection, y ) = (x , y).

where

to

it

d2 designs

corresponds

given by

we

the second

=

d2 a (x, y, z)= d2 a ( x, y ) z .

designs evaluation,

i.

e.

e(f, y)

=

d2 a

If

f (y ),

then

we

have

since

So

we can

since

Now

e

compute the derivative of

is

we are

bilinear,

ready

to

such that

o

mapping

a

inj ection

d2 a :

so

compute the derivative of

67

(5 ) :

a

mapping:

Now

we set

then p* is

again

linear and

so

for any function

f

into L

(Rr , Rn x RS ),

have

So:

since

( 5 ) is the following expression

by (7).

So the derivative of

Now

compute the derivative with respect

we

68

to x

of

(4):

we

Now

the 1

we

put

together pieces

expression

we

can

). First

we

in the square bracket is of the

conclude that this converges

for h - 0

to

Then

consider

we

from ( 11

consider

same

uniformly

form

on a

again of the same form as ( 2 ), so by Step uniformly on a neighborhood of xo for y -> 0 to

This is

69

( 2 ),

by Step neighborhood of xo as

1 this

so

again

converges

Then

is

half of (10). The

one

converges

for h - 0

Step

uniformly 3: The

higher to x

(11),

by uniform continuity remaining member of (10).

of xo

neighborhood

on a

the

derivatives. In Step 2

we

of

dfa

have shown that the der-

of the convergence situation

convergence situations

two

member of

to df a (x, t (x)) d t ( x) y ,

ivative with respect of

remaining

(4) -> ( 5 ) is the

sum

(4)-> (5) and something which converges

clearly uniformly in all derivatives. So for the apply Step 2 to the two parts, and we continue in

second derivative that way for the

we j ust higher der-

ivatives.

Q. E. D. 3.9. REMARKS.

(a) In the proof of Lemma 3.8, considered have compact support ; 2.1

so

brought advantage. (b) By 2.7 each chart

C°° (X , Y)

pact.

with the

But D (Zf)

is

no

longer

be

an

absolute

lopped

a

is

so

used

of : Uf -> D(Zf )

2’-topology

not a

heavily that all sections we introducing the equivalence relation

we

is

of

locally

Baire space, if X is

Coo (X , Y)

paracompact, thus paracomnot

neighborhood

compact,

so

Coo ( X , Y )

Baire space. One would hope that retract, but the

is paracompact,

theory

is

not

Coo ( X , Y )

turns out to

very much deve-

for non-metrizable spaces.

4. MISCELLANY.

4.1. LEMMA. Let

serving at

rr :

smooth map.

E - X be

a

derivative

I f the s0 E D (E) is surj ective,

then

bundle and

E - E be

fibre preD D (a) (s0) : D (E)-> 2(E) of D (a ) there is an open neighborhood U of the

vector

70

a :

a

image of

and

s,

U -> V is

a :

a

of the image of a

V

neighborhood diffeomorphism. an

open

o

so , such

that

PROOF.

by 3.8. Fix x E X . Let ex Then by hypothesis there is

a s

s E D (E)

and let

Ex

E3(E)

such that

s (x ) = ex

such that

i.e.

So the linear map

The

Jacobi

df a (x, so (x)): Ex -+ Ex

differential matrix of

a

(in

a

is

surj ective,

thus invertible.

local trivialization of

E )

looks

like

so

it is invertible

Theorem there is

neighborhood Vx phism. If we take

then

a :

U -> v is

have

to

force

Let that

be

a

r(x)

is

a

sets

of

a

open a o

so

local

diffeomorphism

E , i. e. positive definite, symmetric

metric

of the

on

the bundle

where W is open in X and

we we

and

c >

of

So

we

each fibre

a

section

r :

X -

S2 ( E *)

such

bilinear form for all x E X .We

Ex = Rn

r ( tt (p ) ) ( p - S0 tt (p ), p - S0 tt (p ) ) E} ,

0 . We

neighborhoods consisting choose a locally trivializing equip

trivially surj ective.

form :

following

M ( W , E ) = I p E E | tt (p ) E W of

and is

inj ectivity.

r

consider

an

image of so , and by the classical inversion neighborhood Ux of so (x ) in E and an open (x ) in E such that a : Ux ->Vx is a diffeomor-

the

too on

sets

assert

that

each so (x )

has

chart S about xo ,

with the inner

71

a

basis

M (W, f ). To prove that, so E/ S S X R n and Let product r (x). k (x ) be

of the form

=

the linear

automorphism

which carries

r(x)

into the standard inner

product

of Rn .

is

homeomorphism (in

a

is then

WC

homeomorphism S exactly on an open

these So

a

are

obviously

we assume

We

fact

now assert

diffeomorphism).

which carries each

The map

set

of the form

M (W,E)

with

set

base of

a

without that

a

a :

neighborhoods. loss of generality that each Ux U - V is inj ective. If

is of the form

M(W,E).

then

If

p :;-p

Then

in U and

7T (p )

=

tt (p) = tt (p),

tt (p) E

then

W n W and if e.g.

c >

i,

then

so

since

a/ M (W, ()

is

a

diffeomorphism. Q. E.D.

for special mappings). Let tt: E , X be a vector bundle and a : E , E be a fibre preserving smooth map. I f the derivative D D(a) (s0): D(E)-> D(E) is surjective, then there are open neighborhoods U of so in D (E ) and V of a s, in D ( E) such that D (a): U -> V is a dif feomorphism. 4.2. COROLLARY

(Inversion

Theorem

o

PROOF.

a so

(X)

By 4.1, there

are

in E such that

a :

neighborhoods Uo of so (X) and Vo Uo - Vo is a diffeomorphism. If open

72

of

and

then 3(a): U- V

has the smooth inverse

Q. 4.3. If fields

f f Coo ( X, Y ) , let us denote by Df along f >> with compact support, i. e,

(X , TY )

the space of

E. D.

rector-

the space of smooth maps

such that

relatively We have

that D f (X , T Y) = D ( f*T Y),

compact in X . the space of smooth sections with

compact support of the

pullback f*T Y. Comparing

D(f*T Y) = D(E( f))

via the linear map

s - s

with 3.3

o af .

Thus

we

we

see

that

have:

PROPOSITION.

of all smooth mappings a and I x E X I a (x) # 0} is relatively compact in X . proj ection, then the space

such that

If

a :

X -> T Y

77 y: T Y -> Y is the

is the

proj ection of

in the obvious

T Coo

and T Coo (X,

c

show that the tangent bundle

properties with the only exception that may only be applied to proper mappings.

note tor

a

vector

bundle

and

torial

PROOF.

becomes

a -+ a 0 g.

REMARK. b and

tor

Y)

sense.

given by a -> (T f)o a ,

given by

(X, Y)

a

is clear from the discussion

T Coo (X , Y)

has nice func-

the contravariant

preceding

the

partial

proposition.

-We

func-

only

that the tangent space to so E D(E) is again D (E) where E is a vecbundle over X . We will show in 4.4 that the definition using smooth

paths

is

equivalent.

73

b ) By the usual definition of the tangent bundle

give D (X , T Y )

We an

open subset of

ection,

a vector

on

Y . We

is

a

is

a

for

structure

Zf

f E Coo (X, Y)

bundle. Let

again

c .

Now

and let

is the

we

be the

operate with the data from 3.3. It is

easily

be

to

chart for

so

small that for each x E X the

f (x) E Y ,

by being canonical proj -

prove that

Oy

is chosen

trivializing

which it inherits

COO (X, T Y ). Clearly D (X , ttY)

and that it is smooth follows from

is

and if

the differentiable

zero seen

vectorfield that

set

then

neighborhood with vertical proj ection E (OY f ) we can choose the Whitney sum tubular

of

X 0 Y of C X X T Y

and

o

and then

and the latter space is

D(X , tt Y)

linearly homeomorphic

becomes the

proj ection

under these identifications. If

f E Ug

( i. e.

Zf

small

to

we

enough ),

choose another

then

74

trivializing

chart

Ug with

so D(f* T Y)

is

to D(g* T Y).

linearly homeomorphic

c ) Let h f Coo ( X, Y ) and consider again the data from 3.3: behave to

check that

is smooth. For s

so

the

(Compare the

we

have

mapping is j ust composition

followed by

tion

E D(E(h))

pulling

back

to

another

with

a

fibre

preserving

smooth function

bundle, and so is smooth by 3.8. proof of 3.6 .) Under the identifica-

vector

with the last argument in the

or D(E(h)) with Dh (X , T Y) and of D (E ( f o h )) with D fo h ( X , T Y’) mapping T Coo ( X, f) just coincides with D(X , T f ) which is seen by

writing

out

the definitions. Likewise

is j ust pulling

back

sections,

[(dh )-1 od ah 0 g] * E (h). easily

seen

to

coincide

one can

check that

linear, if

thus

E (h o g )

is chosen

to

be

Coo(g, Y) is smooth too and T Coo(g, Y) is with D ( g, T Y) under the appropriate identificaSo

tions.

Q. E. D. 4.4. Given U open in as

the space of all

d (0)

=

a

space D (E),

equivalence

then

we can

define

classes of smooth

s, , where

75

T So

paths 0

U for so E U : R -> U with

(L)

Via 0 -+ (d d)0 cides

and

s -

(t ->

so

it is

+ts)

easily

seen

that

Ts0 U coin-

with D(E ). Furthermore

LE MMA.

If d :

we

have :

COO (X,

R ,

Y)

is

a

smooth

path

then

iff for all x E X , where ex : C°° ( X , Y ) -> Y is evaluation at x. That is, two paths through f E Coo (X , Y) are equivalent iff they are pointwise equivalent at f ( x ) in Y for all x c X . P ROO F.

locally. Q. E. D.

sketchy development of an application of the inversion to stability. The result is probably well known to specialists. Let us denote by Diff (X ) the open subset of diffeomorphisms of Coo (X , X). It is a group and there is a right action of it on Coo ( X, Y ) . A mapping f E Coo (X , Y) is called source-g’ -stable if the orbit of f under

4.5. We

give principle 4.1

Diff (X)

is

is smooth

now a

Doo-open

Coo (X , Y).

by 4.3, and

infinitesimally source-S’ -stable if surj ective. Following the lines of [5], V- 5 - 6 it is possible if f is source-Doo -stable, then it is infinitesimally source-

is its derivative

Df *(IdX) to

in

is

show that

at

IdX . f

is called

s-stable. Now let

95f : Uf-> D (Zf)

dIdv: UId X -> D (ZIdX) be

a

chart of

f

in

be

a

chart of

COO (X, Y) .

76

Then

IdX

in

Diff(X),

and

is

given by s -> (I dX xf) o s ,

as

is

followed

linear

isomorphism (pullback) proof f infinitesimally source-Doo-stable, then f * is in a neighborhood of the image of IdX ( = X ) given by composition with the fibre preserving map IdX x f , and the derivative is surj ective. As in 4.1 we conclude that the fibre maps are locally subseen

in the

of 4.3. If

by

a

is

(IdX x f )/ (ZIdX)*

mersions, i. e. f So

we

is

a

submersion,

and submersions

are

source-2c*- stable.

obtain the result :

The source-Doo-stable

in Coo

mappings

(X, Y)

are

exactly

the sub-

mersions.

4.6. In

pings

analogous way one can characterize the image-2’-stable mapCoo prop (X, Y ) as the proper immersions, but one has to replace

an

in

the argument of 4.5

consider blem in on

vector a

later

by

one

using

the

adj oint

bundle valued distributions.

article,

as we

of

D ( f *)(IdY),

Maybe

we

will do with the canonical

Diff(X).

77

and has

to

will tackle this pro-

Lie-group

structure

REFERENCES. 1. F.

BERQUIER, Calcul différentiel dans les modules quasi-topologiques. Variétés Esquisses Mathématiques 24, Amiens ( 1976).

différentiables.

2. C. BESSAGA and A.

ology,

PELCZYNSKI, Selected topics Publishers, 1975.

Topologie différentielle,

3. H. CARTAN,

4. J. DIEUDONNE, Foundations

infinite

dimensional top-

Séminaire E. N. S. Paris ( 1961- 62).

of modern Analysis I,

5. M. GOLUBITSKY and V. GUILLEMIN, Stable Springer G.T.M. 14, 1973.

6. J. HORVATH, 7. H. H.

in

Polish Scientific

Topological

vector

Academic Press, 1969.

mappings and

spaces and distributions

KELLER, Differential calculus in locally Springer (1974).

convex

their

singularities,

I, Addison-Wesley.

spaces, Lecture Notes in

Math. 417, 8.

J. A. LESLIE, On a differentiable 6 ( 1967), 263-271.

structure

for the group of

diffeomorphisms,

To-

pology

9. J. MATHER, ility, Annals

Stability of C~-mappings : II, of Math. 89 ( 1969), 254- 291.

Mathematisches Institut der Universitat

Strudlhofgasse

4

A-1090 WIEN AUTRICHE

78

Infinitesimal

stability implies

stab-