Download Manifolds of smooth maps

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
&#x3E; 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 -&#x3E; ]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 &#x3E; 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)&#x3E;E
a
we
continuous function
contains
each compact
In ( X, Y)
is compact in
a set
construct a
6x : Wn -&#x3E;
continuous function:
R such that
d x ( x ) &#x3E; 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-&#x3E; [ 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 &#x3E; 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 -&#x3E; [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 -&#x3E; 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 &#x3E; 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 &#x3E; 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 &#x3E; 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 -&#x3E; jl f «uniformly on K&#x3E;&#x3E; for all 1.
2.4. COROLLARY. For each
k&#x3E; 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 -&#x3E; 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 -&#x3E; X such that :
Z -&#x3E; X is
a vector
embedding X-&#x3E;
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 ) -&#x3E; ( 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 -&#x3E; 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 -&#x3E; 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-&#x3E;
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) -&#x3E;
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 -&#x3E; 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 ) -&#x3E;
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 -&#x3E;
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-&#x3E;
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) -&#x3E; 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 -&#x3E; X and p : F -&#x3E; 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) -&#x3E; 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 &#x3E;&#x3E; 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 -&#x3E; 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 -&#x3E; 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 -&#x3E; 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) -&#x3E; ( 5 ) is the
sum
(4)-&#x3E; (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 -&#x3E; 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)-&#x3E; 2(E) of D (a )
there is an open neighborhood U of the
vector
70
a :
a
image of
and
s,
U -&#x3E; 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 -&#x3E; 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 &#x3E;
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 -&#x3E;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 &#x3E;
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)-&#x3E; 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 -&#x3E; 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 &#x3E;&#x3E; 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 -&#x3E; T Y
77 y: T Y -&#x3E; 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 -&#x3E; (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 -&#x3E; U with
(L)
Via 0 -+ (d d)0
cides
and
s -
(t -&#x3E;
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 ) -&#x3E; 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-&#x3E; D (Zf)
dIdv: UId X -&#x3E; 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 -&#x3E; (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-