space of n-jets at x of maps f E COO (X, X) with f(x) = y . Further ...... KOBAYASHI, S., Transformation groups in differential Geometry, Springer,. 1974, Ergebnisse 70. 6. LESLIE, J.A., On a differentiable structure for the group of diffeomorphisms,.
C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
P. M ICHOR Manifolds of smooth maps, II : the Lie group of diffeomorphisms of a non-compact smooth manifold Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 1 (1980), p. 63-86.
© Andrée C. Ehresmann et les auteurs, 1980, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Vol. XXI -1 ( 1980 )
CAHIERS DE TOPOLOGIE E T GEOMETRIE DIFFERENTIELLE
MANIFOLDS OF SMOOTH MAPS, II: THE LIE GROUP OF DIF FEOMORPHISMS OF A NON-COMPACT SMOOTH MANIFOLD *)
by P.
MICHOR
ABSTRACT. It is shown that
cally
Diff(X),
compact smooth manifold
This paper is
the space of
X,
is
a
diffeomorphisms
of
lo-
a
Lie group.
[8],
where
presented a manifold structure on the space C °° ( X, Y) of smooth mappings X - Y for (arbitrary non-compact) finite dimensional manifolds X, Y , using the notion of differentiability C°° rr of Keller [4]. The main idea was the introduction of
a new
Here
a
structure
functorial method of
notion of
composition
on
Diff(X)
adjunction
corollary
Diff(X).
C°°T-differentiable mappings
the
of
Doo -topology of[8], differentiability C°°rr. for
Coo .
in the stronger notion
deriving
diffeomorphisms
compact X
This is done
a
is
with the
it is shown that
for compact Y . An easy the
we
the space of
X , equipped
same
Gutknecht [3]
Lie group
Diff( X ) ,
show that
Lie group in the In
of
topology.
compact manifold
locally a
we
sequel
a
admits the
by
relation
CT·-differentiability
of this is the
Unit and counit of this
adjunction
are
of
the
: given by
and evaluation
given by E If Y is
not
compact, then the first
*) P artially supported by
a
mapping
research grant of the
63
is
City
not even
of
continuous if
Vienna, 1978.
C°° ( Y, X x Y ) curve
is
outside
constant
stays
stays within
some
compact
of this. So the above
Therefore sition
and of
by
we
no
we
are
direct
excuse
with the
equipped
some
to
a
continuous
fixed compact of Y if the parameter
of R . See
set
adjunction
forced
Whitney-C0-topology:
does
prove the
[9]
not
for
detailed
a more
exist,
if Y is
C°°rr-differentiability
not
account
compact.
of the compo-
the
«onslaught»:
proof is complicated and heavy going elegance. The method of proof of [6] is
for the lack of
since it is wrong.
help,
The reader is assumed
to
be familiar with
[8], especially
with
topology (Sections 1,2) and with the Q-Iemma ( 3 .8 ). The manifold structure will be explained again ( 6.3 ) in a somewhat simpler form as presented in [9]. Sections will be numbered from 5 onwards, following those of [8] (Sections 1 to 4), citations with leading number less than 5 refer to [8] ( e. g. 3.8 ). the sections
dealing
with
5. SOME TOPOLOGY AGAIN.
Let X be
a
smooth finite dimensional manifold. Let
be the smooth fibre bundle of
( see 1.1). Let space
of n-jets
Jn inv (X, X)x,y
Jn(X, X)x,y at x
of maps
n-jets
of smooth
mappings
Jn(X, X)
from X to X
(X,Y)f X x X , i, e,, the f E COO (X, X) with f(x) y . Further, let be the fibre
over
=
be the open subset of invertible
n-jets
from x
to
y . It
is clear that
where
7T k,n: Jn(X, X)-> Jk(X, X)
is the canonical
projection
for n > k
(cf. 1.3) and GL( X,
X)x,y denotes the open subset of invertible 1-j ets f rom x to y in 7 (X, X)x,y : in canonical chart J1 (X, X)x,y the space of all dim X x dim X-matrices and responds GL(X , X)x,y a
:
cor-
to
corresponds
to
the open subset of invertible
of the canonical chart for
is
a
Jn(X, Y)
smooth subbundle of the fibre
ones.
it is clear that
bundle Jn (X, X). 64
By the construction
5.1.
The mapping
LEMMA.
given by
is
a
,
smooth
PROOF.
fibre
bundle
By looking
responds just
to
at a
homomorphism
over
canonical chart
we see
to o-, truncated
corresponding
the coefficients of the inverse power series
polynomial, The
5.2. PROPOSITION. open subset PROOF.
set
of C°°(X , X)
These
two
are
the
coarser
X, X is
1, 2 respectivePro-
QED
T-topology and the D°°-topology. for the D-topology, "We show that this is so
base for it described in 1.5 ( c ) : let
an
D°°-topology.
Whitney-C°°-topology (see [7],
mapping
we
QED
described in Sections
5.3. THEOREM. The
is continuous in the
rational functions of the
g)-topology and
assertion follows.
in the
order n . Since
Diff(X) of diffeomorphisms
in the
topologies
are
at
the assertion follows.
ly. Diff(X) is open position 2.5), so the
P ROO F. First
inv (o-) = o--1 corpolynomial mapping
that
the inverse power series of the
R dim X , 0-> Rdim X, 0 coefficients of the
is smooth and
use
the
f6 Diff(X) and let M’(L, U) be in Diff(X), i, e., L =(Ln) and
D-open neighborhood of f’l loU = (Un) , where each L n is compact in X with (XBLn0) being cally finite family, and each Un is open in jn (X, X) for each n > 0 . a
basic
a
Then
We
want to construct a
D-open neighborhood
65
P of
f
such that
f-1
M’(L , U)
c
Let
L’n
be
sequence of compacts of X such that
a
and such that
These
cally
two
means
XBL’ n 0
are
is still
family.
Let
(XBK’0n), (XBKnO)
are
lo-
K’ C Kn0.
metric on X and let p be
a
finite
locally
sequences of compacts and
finite families. Further
Let d be
a
a
strictly positive continuous func-
tion on X such that 0
max
for each n
E
{p(x) | x c K n
N . Such
a
function
distance between the compact
L’ and the disjoint closed set XBLn0, may be found since (XBL’0n) is locally
finite ( cf. Proof of 1.4). Let
then
Vn
is open
by 5.1,
and let U
D-open x XBK’0n ,
Consider the basic For let n
c
N and
set
f
=
(V n).
M’(K’, V) .
We claim that it contains
then
but
by
the choice of
Now
let Vpof(f)
( cf. 1.5 :
L’n.
This
be the
(supp 1/ ’P f) o
implies
D-open neighborhood is
locally finite ),
We claim that
66
and let
f.
have
we
p(f(x))
and
T ake n
x E
f
N and x
XBLn0 ,
f
M’( K’, V),
as we saw
that Inv is continuous for the
see
that Inv is
with the
compatible
a
homeomorphism for
so
shows
D°°-topology too
equivalence relation
5.4. PROPOSITION. Let X, Y, Z be smooth Then the canonical identification
is
and
above,
jn ( g-1 ) (x ) E Un . This
so
L’
then
XBLnO implies g-1 (X) E XBK’0n
s ince g f To
is less than the distance between
the T- and the
g-1 E M’
( L , U) .
it suffices
from 2.1.
to note
QED
locally compact manifolds.
D°°-topologies.
proof of this fact can be given along the lines ([2], Chapter II, Proposition 3.6). Lemma 1.9 plays a vital role in REMARK. A direct
ÐOO-topology
The assertion for the we
will
not
need
will be
than that later
more
a
consequence of 6.4
of
it.
below ;
on.
6. DIFFERENTIATION.
6.1. The notion
o f dif ferentiation :
We
[4], but in the formally C; == C°°c holds in general.
of Keller
C°°rr
shown that
E, F
be
locally
is said
to
be of class
Let
mapping. f
convex
Cl
use
the notion of
weaker form of
differentiability C°°c. In [4] it is
linear spaces, let
if,
f:
E -> F be
for all x, y6 E and kE R ,
a
we
have
in
F,
where
Df (x)
is
a
linear
mapping E - F 67
for each x
6E,
and the
m apping is
jointly
continuous.
This concept is defined
E X E ->
is said
F,
and
be of class
to
so on
for the
to
[4]
for
The notion
CPc
was
We refer
also showed the
6.2.
with the obvious
higher
more
following -
C2c
D f is
if
linear
is
only
of class
C1c
as a
mapping
derivatives.
information.
by Ehresmann ( Bastiani [1] ), who include a proof for completeness sake.
introduced we
(partial derivatives). spaces. Let f: E1 x E2 -> F
PROPOSITION
convex
changes if f
open subset of E .
on some
f
applicable
Let be
a
E1, E 2’
F
mapping. f
be is
locally 0 f class
iff mappings x 1 |-> f ( x1 , x2), x2 |-> f ( x1, x2) are o f class C1c for each fixed x2, x1 respectively, with derivatives D1f (x 1 , x 2 ) y1 and D2 f (x 1 , x 2 ) y2 respectively, which are jointly continuous in all appearing variables. The derivative of f is then given by
C
the
The same is
true
i f f is de fined on
PROOF.
Necessity:
D1 f for D2 f .
is
So
jointly
an
open subset
continuous in all
Sufficiency :
68
o f E1 X E2 only.
appearing variables. Analogously
D f in all appearing same property of D1f and D2f . QED The
joint continuity
6.3. The ture
on
use a
of
manifold structure. We give a short review of the manifold strucC°° (X , Y), X, Y being locally compact smooth manifolds. "B’tfe
simplified
form of the manifold
( [9] ,
this version is described in Let the
variables is clear from the
r:
T Y -> Y be
Y,
and
3.3, 3.4
up in
and
3.6 ;
8 ).
mapping such that for each y in Y diffeomorphism onto an open neighbor-
smooth
T Y -> Y is
mapping Ty:
hood of y in
a
structure set
a
Ty(Oy)
=
y. Such
a
map may be constructed
by
appropriate fibre respecting diffeomorphism from T Y onto the open neighborhood of the zero section, on which the exponential map is diffeomorphic. If rry : T Y -> Y denotes the canonical projection, then the mapping (r, 1Ty): T Y - Y X Y is a diffeomorphism onto an open neighborhood of the diagonal in Y X Y . In Seip [11J an
exponential
map
following
(which need
these maps
be defined
not
additions». We will adopt the If bundle
f6 C°° ( X, Y) ,
over
X,
an
same name
consider the
compact support of this bundle,
D-topology
are
called « local
for convenience sake.
pullback f * T Y
D( f * TY)
and the space
coincides with the
globally there)
equipped
which is
a
vector
of all smooth sections with
with the
here). This is
1°°-topology
( which
locally convex dually nuclear (LF)-space, being the straightforward generalization of the space Ð of test functions with compact support in distribution theory. See 2.7 a
for further information.
be the
Let
Denote
by
Uf
the
image
of
t/J f’
which is
for let
an
mapping
be the open
69
C°° (X, Y) ; neighborhood of
open subset of
the
{(X,f(x))| x E X} of f
graph
in X X Y
= J 0 ( X, Y)
(in fact
a
neighborhood with vertical projection, cf. 3.3-3.5 ). Then Uf consists of all g E C°° ( X , Y) such that the graph of g is contained in Z f and g - f ( i. e., g and f differ only on a relatively compact subset of X ), so U f is D°°-open. tubular
Vif
is
as
is continuous
by 2.5
and has
will
serve
check the coordinate
us
as
canonical chart centered
change. we
Let
f,
g6
C°°(X, Y)
at
f.
with :
have
the map
so
is
given by
by pushing
forward sections
by
a
fibre
feomorphism. So the coordinate change bility class C°°rr by the Q-Lemma 3.8. 6.4
PROPOSITION.
Then the canonical
is
continuous inverse
checked up.
easily
Now let
a
of
class
C°°rr.
Let
preserving (locally defined)
dif-
is continuous and of differentia-
X, Y, Z be smooth locally compact manifolds.
identification
The
identification
is
compatible
with the choice
of
ca-
nonical charts. P ROO F. Let
the
(f, g) E L"(X, Y) x C°° (X, Z) .
corresponding
element of
We write
C°° ( X, Y x Z ) given by
70
again (f, g)
for
Let r x
T Y -> Y be a local addition on
T :
p is
and
local addition
a
U(f,g) = UfxUg
on
Y, p : TZ -> Z
be one on
Z,
then
Y X Z . We have
for the canonical
charts,
and the
following diagram
commutes :
6.5.
is
PROPOSITION.
For each
n >
0, the mapping
of class C°°rr .
PROOF. Let
be local additions for Y and
Jk(X, Y)
respectively.
Let
f E C°°(X , Y) ,
then jk f E centered ed
at
Coo ( X, jk(X, Y)) ; consider the canonical chart (Uf, cp ) at f of C°° (X, Y) and the canonical chart (U jkf, cpjkf) center-
jk f
of
C°°(X,
Jk(X,
is of class
y)).
We have
we
to
check wether the
mapping
have
where
Now j k s C°° ( X , Jk( X , Y)) by element of the closed subspace DJk( f * T Y) definition but in fact it is bundle ¡k (f* T Y ) of smooth sections with compact support of the is the
(functorially)
induced
mapping.
c
an
vector
71
X , which consists of k-jets of sections [10]). We will write jk s if we consider it to be
over
The
f * T Y (cf. jk (f* T Y) .
of the bundle a
section of
mapping
is linear and continuous space of
tors on a
test
of class
77 Therefore
and it is
easily
(being
a
complex
functions - in we
a
of
differential opera-
partial
canonical chart )
so
it is
trivially
have
that
seen
I.-i
is
fibre
a
it is of
preserving smooth mapping, so pushing forward sections by class q by the Q-Lemma 3.8. The assertion follows by the
chain rule.
6.6.
QED
REMARK. The
converging
sequence
fixed compact
differs from
Tf
from
on
for
shown that the
where
and
in
infinitely no
at some x E
many n
Tx X ,
T fn
so
chance for T
even
to
a
some
cannot
X,
then
converge
T fn to
Tf
be differentiable. But it
differs in gecan
be
( compare [7], 2, Proposition 6),
differentiable
may write
comp:
T X x X ,J1 (X, Y) -> T Y
just composition of matrices smooth. We will use this technique in a
locally, which is complicated situation
vectors
much
be
mapping
is continuous and we
f
the whole fiber
neral. Thus there is
since
to
neither in the
in X for all but finitely many n (2.3 or [9] ). But if
set
f
D°°-topology : let fn Coo ( X, Y). Then fn equals f off
:D-topology,
continuous in the
fn
mapping T : C°°(X, Y) -> C°°( T X, T Y) is not even
more
6.7. THEOREM. Let X be
a
later
is
on.
locally compact
72
smooth
manifold. Then
the
Diff(X) of top olo gy.
space
Diff(X) is
topological
a
X is
a
C°°(X , Y) in the Sby 2.5 and inversion is
is open in
In the latter it is
that it is
dif feomorphisms of
is continuous
composition
Diff(X)
all
an
group in the
and the
open submanifold of
C°°( X , Y) ,
the
and
structure.
D°°-
D°°- topology,
continuous
D-topology and in
Lie group in this induced manifold
a
Lie group in the
by 5.3,
so
D°°-topology. we
will show
The
proof
of
this will occupy Sections 7 and 8.
7. THE COMPOSITION IS DIFFERENTIABLE.
7.1. Before p: E -> X is
we
can
a vector
with the
begin
we
need
some
preparation.If
by V ( E) ker T (p) the TE - E . If E, = p.1 (x) is the fibre
bundle then let
vertical subbundle of the bundle over x c
proof, us
denote
=
X and
T v( Ex)
ix : Ex -> E is the embedding, then we may identify Ex itself for v E Ex via the affine structure of Ex and
with
define
V = VE : E ® E -> V ( E )
It is clear that
dles
over
V(v, w )
X.
is
an
isomorphism
will be called the vertical
lift of
w
vector
bun-
over v .
The
of
mapping:
is called the vertical L EMMA. 1°
20 Let
dles
as
projection.
V(E), VE, a :
ÇE
commute
E , E’ be
a
with
smooth
pullbacks o f vector fibre mapping between
bundles. vector
bun-
given by the diagram
then the
fibre
derivative
of
a ,
the
mapping
73
dfa : E ® E - E’
over
(3
is
given by
f:
P ROOF. If
given
as
the
In view of
where
f*E REMARK.
X
VE 0X’ .
we
a
f*E
is
have :
zero
If f : X->
then
dition,
we
is the =
bundle
mapping, then the pullback following categorical pullback:
this,
0x
and V
X’-> X is
Y is
section of
X,
both
as a
manifold and
a
assertion 2 of the lemma is clear.
a
smooth
w ill denote by
mapping
Tf: f* T Y -
and
T:
T Y -> Y is
Y x X the
mapping, Q ED
a
local ad-
diffeomorphism
into
given by
Clearly w ith
have
we
Tf (x)
if
we
identify
(f*TY)x
Y.
X, Y, Z be locally compact smooth manifolds. Then the composition mapping 7.2.
THEOREM.
Let
given by Comp (g, f) Here of Coo
=
g o
C°°prop (X, Y)
f,
is
o f class C’
denotes the open subset of proper
mappings
(X, Y) (cf. 1.9, 2.5 ).
P ROO F. Let
We w ill show that
is differentiable in the canonical charts centered
pectively,
as
described in 6.3. Let
74
at
g,
Comp f and g o f res-
be local
the canonical charts.
additions, inducing
We will suppose that
Uf
and
Ug
are
small
enough
such that
precise we should restrict to open subsets of Uf and Ug respectively, using continuity of the composition ( 2.5 ), but we will not specify this to save notation. So by some abuse of language we consider the mapping ( 1 ) :
-
to
be
We have
to
show that
c
C’-
The
given
mapping
There is
guish
some
bundle
rs~f
will do this
for
,
along f
f *T Y , i, e.,
and
f
by showing
have identified
we
is proper,
too:
we
by
did
not
distin-
with compact support from sections of the
morphic space D( f* T Y ), to ferentiability of the mapping
Since
we
abuse of notation in this formula
fields
vector
vector
is
and
using 6.2.
that it is of class c
class -
is of
save
Ts
Df(X, T Y)
notation. Let
is proper too,
so
us
with the iso-
first look
the
at
the dif-
mapping
is continuous and linear. Then
we
consider the fibre
feomorphism
over
respecting (but
IdX :
75
not
everywhere defined) dif-
It is clear that
By the chain rule and
C°°rr
the Q-lemma the
and its derivative is
Here
we
again
(g r s ) * TZ
mapping ( 3 )
given by
considered tr s,
is therefore of class
,
t’r s both
mappings X -> TZ .
as
sections of the bundle
heavily Lemma 7.1. The last line of formula ( 7 ) shows that D 1 c ( t, s ) t’ is jointly continuous in s, t, t’ (use 6.4, the continuity of the composition and the fact that
the
is
and
as
We used
mapping
continuous).
Now
we
investigate
the
differentiability
of the
mapping
for fixed t For fixed t
Then
we
we
define the
mapping
have
76
a(t)
where
is
a
the chain rule the en
mapping over IdX mapping ( 8 ) is of class fibre
. So by the Q-Lemma and by 77 and the derivative is giv-
by:
m appin g t |-> a (t) is no t continuous ( there is a non proper open embedding on the right of t ) neither is t 1, dfa ( t). So we have to rearrange the expression ( 11 ) in such a way as to see the joint continuity in t, s, s’. We compute as follows, again using Lemma 7.1 :
But the
So it remains
to
show that the
is continuous. For that
which is
evaluation
a
mapping
we use
the manifold
product, and M -> T Z given by :
submanifold of the
mapping
11:
the
following composition
Since p is locally just multiplication of matrices, it
is
C°° . Then
we
h ave :
This
expression
is
jointly
continuous in t, s, s’
77
by 5.4, 6.4, 6.5
and
by
s |-> T s
the fact that In view of 6.2
we
is continuous
have shown that
c
(17) The higher derivatives : If
C2c
have
we
gain
we
to
have
to
is of class
CIc
we want to
show that
check that D c is of class
compute all
partial
and that
C1c . In order
to
c
is of class
apply 6.2
derivatives of D c and have
to
a-
show
jointly continuous in all appearing variables. Now ( 7 ) and ( 15 ) are composita of expressions that look like ( 2 ) again and by 6.5 jl (t) is of class C°°rr in t . So we may apply what we have already provthat
they
are
ed and get
C2c. By induction
we
get
C°°c= C°°rr .
Q ED
X, Y be locally compact smooth manifolds. Then the evaluation mapping Ev: X X C°° (X, Y)-> Y is o f class C°°rr (and 7.3. COROLLARY. Let
consequently D°°-continouous). P ROO F. First
we
diffeomorphically, L et
r :
T X - X be
centered ed at
at
f (*)
show that
where some
denotes the
*
local addition. Then the canonical chart
f : * -> X corresponds
to
the chart
following diagram commutes and differentiability of the composition :
7.4.
(Ufl Of)
(Im T f(* ), T f (*)-1)
center-
of X.
Now the
the
one-point manifold.
COROLLARY.
Let X,
so
the assertion follows from
Y, Z be locally compact smooth manifolds.
Then the canonical mapping
takes its values in Coo ( Y REMARKS.
10 Since X
X
X, Z ).
is finite dimensional
78
we
need
not
specify
the
notion of
C°°(X , C °° ( Y, Z )) , since all reasonable ( s ee [4] ); C°°( Y, Z) is equipped with its canonical
differentiability
notions coincide
C; -manifold
in
structure.
mapping is not surjective on C°°( Y x X, Z ) by topological reasons, as we already mentioned in the introduction. It is surjective, however, if Y is compact. That has been shown by Gutknecht [3] for the stronger notion of differentiability CF ; in order to be complete we will prove this fact in our setting too (7.5 ). ° This canonical
f E C°°(X, COO(Y,Z)). Then the f the mapping f : Y x X -> Z given by
PROOF OF 7.4. Let
ping
associates
7.5.
THEOREM.
to
canonical map-
X, Y, Z be smooth manifolds, X, Z locally
Let
com-
pact and Y compact. Then
via the canonical
identification. 7.4 it remains
P ROOF. In view of
cation
mapping
suffices
to
is
onto
show that the
to
C°° ( Y x X, Z) , mapping
is of class
is of class of class
terms
C°°rr
to
by
abstract
non-sense
it
too; this latter
f (77
by 7.2, so f*o n : X -> Coo ( Y, Z) is mapping is easily seen to be the canonical
the Q-Lemma
or
is the so-called unit of the
xo c X
and let
r :
adjunction,
in
categorical
T X -> X and p : T Y - Y be local additions.
Then
a
by
).
Now fix
is
and
then
C°°rr
associate
show that the canonical identifi-
local addition; let
79
C°°( Y, Y x X ) , centered at 77(x0), which 77(x0) is given by y |-> (y, Xo) we see that
be the canonical chart of
comes from p X r . Since
bundle
as
Y,
over
so
In view of this identification
So
cp77(x0) o 71
which is
is
we
given by the
clearly differentiable
have for x
c
X
near xo
and
sequence
in any sense, where V is
a
suitable
borhood of xo in X and where B is the continuous linear maps each
of
point
T x0 X
into the
constant
7.6. PROPOSITION. The tangent
neigh-
mapping which
function Y -
well defined and continuous iff Y is compact.
is
y6Y:
7" X
. B is
QED
mapping o f the composition
given by
PROOF. Since
we
pute the tangent g and
f
know
already
that
Comp
mapping by considering
with «tangent vectors » t and
position pointwise, i, e.,
for fixed x
c
is differentiable we may
one s
parameter variations
and
X , using
differentiating
com-
through
their
com-
Lemma 4.4.
8. THE INVERSION IS DIFFERENTIABLE.
8.1.
THEOREM.
Diff(X)
Let X be
be the open subset
locally compact smooth manifold and let of diffeomorphisms of C°°(X , X J. Then ina
80
Diff(X) -> Diff(X)
v ersion I nv : PROOF.
in
a
(1)
It suffices
to
neighborhood U
o f class C°°rr.
is
show that Inv:
Diff(X) -> Diff(X) is of class identity I dX of X . For let f E Diff(X) ,
of the
then
is
a
of
neighborhood
f by
2.5. For any
gEVf
we
have
thus J
Since
(f -1)*
(r-1)*
and
respectively)
Lemma
(2)
( by 7.1 or by the A- and nimplies that In v| Vf is of class
of class
are
the chain rule
Now let T : T X -> X be
canonical chart centered
IdE
at
a
C°°rr
local addition and let
Diff(X) ,
and let
and
be the chart maps. We have
is of class
precise
we
subset of
This is
ing
for any
have
There is to
D(TX)
again
so
that its
and
image we
will
too we
s E D( T X)
we
have
mapping
abuse of notation involved:
some
notation. Consider the
( 1 ) (here
Experience
show that the
restrict the domain of the
possible by 5.3,
to save
of 7.2
C°°rr.
to
mapping Inv 0 f/J
to an
is contained in the domain ll
silently
assume
to
be
open
of cp.
this in the follow-
mapping
some
silent restrictions involved). Then
have:
with finite dimensional Lie groups
81
or
the formula of Ver
Eecke about the derivative of the
following
implicitly given
functions suggests
to
try
ansatz :
That this is indeed the derivative of i will be shown in Lemma 8.2 be-
low. For the
moment we
will take it for
this formula. Recall from 7.2
where
is
a
df
fibre
preserving
a (i ( s ))
So
by
smooth
implicit
the
have
function theorem in finite dimensions
we
have
is of class
right-hand
(5 )
we
get
( 7 ) and ( 15 ) it
s’ ,
C1c
side of
L EMMA.
C1c. This
the chain rule and 7.2
again, applying (7 ), so i is of class
C2c . Now
C°°c = C°°rr .
QED
With the notation
D i (s ) s’ is implies in turn
is clear that
so i is of class
induction shows that i is of class
8.2.
Now
we
continuous in s and
that D i
investigate
that
(6 ) is invertible too, and
From this formula and from 7.2
jointly
will
we
diffeomorphism, given by 7.2 (9).
is invertible and
this back into
Putting
and
denotes the fibre derivative and where
so
the
( 11 )
granted
of 8.1
we
82
have
a
( 17 )
to
straightforward
P ROO F. Let T be
is
differentiable,
near
enough
is of class
0 in R . Then
by 7.1
C°° in the classical
the
mapping
sense.
So the usual
hold, in particular the relations between integrals and integrals being Riemannian integrals with values in the
rules of Calculus
derivatives,
the
complete locally convex space D( T X) . These relations can be derived very simply from the one-dimensional case by using the Hahn-Banach Theorem. Our first aim is
to
show that: stays bounded in (
By 8.1 (4)
So
we
we
know that
have
For h - 0 the first summand converges
Therefore
we
To show that
that there is
a
to
conclude that
( 1 ) stays bounded in
T( T X)
compact set K C X such that
83
for k -> 0
we
have
to
show
and that is
«uniformly
bounded’) for
(x, À)
in
K x [- 1 , 1]B { 0 } ) for any k. ( 4 ) is easily checked:
is
a
some
continuous compact
compact K
this K satisfies To prove
(5)
K there is stays
a
(compare
so
it
only inside introduction);
can move
the fact mentioned in the
that it suffices
neighborhood Ux0 =
D( T X) ,
( 4 ).
we note
«uniformly
We choose U
C X
in
path
to
be
so
show that for each k and xo in
of xo in X such that the 0
bounded» for
Ux
to
(x, k) c Ux 0
x ([ -1, 1]B{0}).
small that T X
0
expression ( 5 )
|U =U X Rn .
For x
c
U
we
may write
Then (3) looks like
uniformly write ( 6 )
for
xe
v and for each derivative with respect
x .
Let
us
for short in the form
B( x, À) is the local representative G : L ( Rn )-> R+ be the continuous mapping so
to
that
over
U of ( 1 ). Then let
A ( x, À, 11): Rn -> Rn is invertible (cf. 8.1 ) and continuous in x, À, IL, we conclude that G(A(x, k, 03BC)) > E for all relevant x, À, /l.
Since
Therefore
84
and
by (7) B (x, X) I
Let
dx
has
denote the derivative with respect
The first summand is the above
argument. A simple
Now
we
proceed
is bounded and
there is
such
we
induction then shows that
so
have
may repeat
we
dkx B(x, k)
is
( 5 ) is proved.
prove the Lemma. Since
is
a
Monte1
space, M
is precompact,
for k , 0 . Let t be such
a
cluster
so
point,
there
then
in
D( T X ) . By
the
joint continuity
of
D2 c
in all
va-
conclude that
since ta -> t Since
we
a net
that ta -> t
riables
to
Ð( T X)
points of M
cluster
k,
Then
to x .
bounded and for the second
already
bounded for k -> 0 for any
are
be bounded for k -> 0 .
to
and
ka -> 0 .
D 2 c ( s , i ( s ))
By ( 3 ) again
is invertible
This holds for any cluster
we
we
conclude that
get
point of M for k -> 0 ,
so
the lemma is
proved. QED
8.3. PROPOSITION. The tangent
is
mapping o f the
given by
85
inversion
P ROO F. As in
ing
7.6
the inverse of
tor» s.
The
one
mapping by differentiatthrough f with « tangent vec-
may compute the tangent
a one
parameter variation
computation
is
a
little
more
difficult than in 7.6.
REFERENCES.
(BASTIANI), A., Applications différentiables infinie, J. d’Analyse Math. 13, Jérusalem ( 1964),
1. E HRESM ANN dimension
variétés de
et
1- 114.
2. GOLUBITSKY, M. & GUILLEMIN, V., Stable mappings and their
singulari-
ties, Springer, 1973, G.T.M. 14.
C~0393-Struktur
3. GUTKNECHT, J., Die
kompakten Mannifaltigkeit,
auf
der
Diffeomorphismen
Dissertation ETH
5879, Zürich,
4. KELLER, H.H., Differential Calculus in locally Notes in Math. 417, Springer ( 1974).
Transformation
5. KOBAYASHI, S.,
1974, Ergebnisse
groups in
gruppe einer
1977. spaces, Lecture
convex
differential Geometry, Springer,
70.
6. LESLIE, J.A., On a differentiable Topology 6 ( 1967), 263- 271.
structure
for the group of
diffeomorphisms,
7.
MATHER, J. N., Stability of C~-mappings. II : Infinitesimal stability implies stability, Annals of Math. 89 ( 1969), 254- 291.
8.
MICIIOR, P., Manifolds
( 1978 ),
of smooth maps, Cahiers
Topo.
et
Géo. Diff. XIX- 1
47- 78.
9. MICHOR, P., Manifolds of differentiable maps, Proc. CIME tial Topology (foliations), Varenna, 1976. 10. P ALAIS, R., Seminar dies 57 ( 1965).
on
the
Atiyah- Singer
Index
Conf.
on
differen-
Theorem, Ann. Math. Stu-
Kompakt erzeugte Abbildungsmannigfaltigkeiten, Preprint, Darmstadt, 1974
11. SEIP, U.,
Institut fur Mathematik
Universitat Wien
Strudlhofgasse 4 1090 WIEN. AUTRICHE
86
T. H.
