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, n
o1 (1980), p. 63-86
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MANIFOLDS OF SMOOTH MAPS, II:
THE LIE GROUP OF DIF FEOMORPHISMS OF A NON-COMPACT SMOOTH MANIFOLD *)
by P. MICHOR
CAHIERS DE TOPOLOGIE E T GEOMETRIE DIFFERENTIELLE
Vol. XXI -1
( 1980 )
ABSTRACT.
It is shown that
Diff(X),
the space ofdiffeomorphisms
of a lo-cally
compact smooth manifoldX ,
is a Lie group.This paper is a
sequel
of[8],
where wepresented
a manifoldstructure on the space
C °° ( X, Y)
of smoothmappings
X - Y for(ar- bitrary non-compact)
finite dimensional manifoldsX, Y , using
the no-tion of
differentiability C°° rr
of Keller[4].
The main idea was the intro- duction of a newtopology.
Here we show that
Diff( X ) ,
the space ofdiffeomorphisms
of alocally
compact manifoldX , equipped
with theDoo -topology of[8],
isa Lie group in the same notion of
differentiability C°°rr.
In
Gutknecht [3]
it is shown thatDiff(X)
forcompact X
admitsa Lie group structure in the stronger notion
Coo .
This is doneby
thefunctorial method of
deriving
theadjunction
relationfor compact
Y .
An easycorollary
of this is theCT·-differentiability
ofthe
composition
onDiff(X).
Unit and counit of thisadjunction
are theC°°T-differentiable mappings :
given by
and evaluation
given by
EIf Y is not compact, then the first
mapping
is not even continuous if*) P artially supported by
a research grant of theCity
of Vienna, 1978.C°° ( Y, X x Y )
isequipped
with theWhitney-C0-topology:
a continuouscurve stays constant outside some fixed compact of Y if the parameter stays within some compact set of R . See
[9]
for a more detailed accountof this. So the above
adjunction
does notexist,
if Y is not compact.Therefore we are forced to prove the
C°°rr-differentiability
of the compo- sitionby
direct«onslaught»:
theproof
iscomplicated
andheavy going
and we excuse for the lack of
elegance.
The method ofproof
of[6]
is of nohelp,
since it is wrong.The reader is assumed to be familiar with
[8], especially
withthe sections
dealing
withtopology (Sections 1,2)
and with theQ-Iemma ( 3 .8 ).
The manifold structure will beexplained again ( 6.3 )
in a some-what
simpler
form aspresented
in[9].
Sections will be numbered from 5onwards, following
thoseof [8] (Sections
1 to4),
citations with lead-ing
number less than 5 refer to[8] ( e. g. 3.8 ).
5. SOME TOPOLOGY AGAIN.
Let X
be a smooth finite dimensional manifold. LetJn(X, X)
be the smooth fibre bundle of
n-jets
of smoothmappings
from X to X( see 1.1).
LetJn(X, X)x,y
be the fibre over(X,Y)f X x X , i, e,,
thespace
of n-jets
at x of mapsf E COO (X, X)
withf(x)
= y .Further,
letJn inv (X, X)x,y
be the open subset of invertiblen-jets
from x to y . It is clear thatwhere
7T k,n: Jn(X, X)-> Jk(X, X)
is the canonicalprojection
forn >
k(cf. 1.3)
andGL( X, X)x,y
denotes the open subset of invertible1-j ets
f rom x to y
in 7 (X, X)x,y :
: in a canonical chartJ1 (X, X)x,y
cor-responds
to the space of alldim X x dim X-matrices
andGL(X , X)x,y
corresponds
to the open subset of invertible ones.By
the construction of the canonical chart forJn(X, Y)
it is clear that5.1.
LEMMA.The mapping
given by
, is smooth andis a
smooth fibre bundle homomorphism
overPROOF.
By looking
at a canonical chart we see thatinv (o-) = o--1
cor-responds just
to the inverse power series of thepolynomial mapping
R dim X , 0-> Rdim X, 0 corresponding
to o-, truncated at order n . Since the coefficients of the inverse power series are rational functions of the coefficients of thepolynomial,
the assertion follows. QED 5.2. PROPOSITION.The
setDiff(X) of diffeomorphisms
X, X is anopen subset
of C°°(X , X)
in theg)-topology
and theD°°-topology.
PROOF. These two
topologies
are described in Sections1, 2 respective- ly. Diff(X)
is open in the coarserWhitney-C°°-topology (see [7],
Pro-position 2.5),
so the assertion follows. QED 5.3. THEOREM.The mapping
is continuous in
the T-topology
andthe D°°-topology.
P ROO F. First we show that this is so for the
D-topology,
"We use thebase for it described in
1.5 ( c ) :
letf6 Diff(X)
and letM’(L, U)
bea basic
D-open neighborhood
off’l
inDiff(X), i, e., L =(Ln)
andU =
(Un) ,
where eachL n
is compact in X with(XBLn0) being
a lo-cally
finitefamily,
and eachUn
is open injn (X, X)
foreach n > 0 .
ThenWe want to construct a
D-open neighborhood P
off
such thatf-1
cM’(L , U)
meansLet
L’n
be a sequence of compacts of X such thatand such that
XBL’ n 0
is still alocally
finitefamily.
LetThese two are sequences of compacts and
(XBK’0n), (XBKnO)
are lo-cally
finite families. FurtherK’
CKn0.
Let d be a metric
on X
and let p be astrictly positive
continuous func- tionon X
such that0 max
{p(x) | x
cK n
distance between the compactL’
andthe
disjoint
closed setXBLn0,
for each n E N . Such a function may be found since
(XBL’0n)
islocally
finite
( cf.
Proof of1.4).
Letthen
V n
is openby 5.1,
and let U =(V n).
Consider the basic
D-open
setM’(K’, V) .
We claim that it containsf .
For let n c N and x f
XBK’0n ,
thenbut
by
the choice ofL’n.
Thisimplies
Now
let Vpof(f)
be theD-open neighborhood
( cf. 1.5 : (supp 1/ ’P
of)
islocally finite ),
and letwe have
and
p(f(x))
is less than the distance betweenL’
andT ake n f N and x f
XBLn0 ,
thenx E
XBLnO implies g-1 (X) E XBK’0n
as we sawabove,
sos ince g f
M’( K’, V),
sojn ( g-1 ) (x ) E Un .
This showsg-1 E M’ ( L , U) .
To see that Inv is continuous for the
D°°-topology
too it suffices to notethat Inv is
compatible
with theequivalence
relation from2.1.
QED5.4.
PROPOSITION. LetX, Y,
Z besmooth locally compact manifolds.
Then the canonical identification
is a
homeomorphism for the T- and the D°°-topologies.
REMARK. A direct
proof
of this fact can begiven along
the lines of([2], Chapter II, Proposition 3.6).
Lemma1.9 plays
a vital role in it.The assertion for the
ÐOO-topology
will be a consequence of6.4 below ;
we will not need more than that later on.
6. DIFFERENTIATION.
6.1. The
notiono f dif ferentiation :
We use the notion ofdifferentiability
C°°rr
of Keller[4],
but in theformally
weaker form ofC°°c.
In[4]
it isshown that
C; == C°°c
holds ingeneral.
Let
E,
F belocally
convex linear spaces, letf : E -> F
be amapping. f
is said to be of classCl if,
for all x, y6E
and kER ,
wehave
in
F ,
whereDf (x)
is a linearmapping
E - F for each x6 E ,
and them apping
is
jointly
continuous.This concept is
applicable
with the obviouschanges if f
isonly
defined on some open subset of
E .
f
is said to be of classC2c
ifD f is
of classC1c
as amapping
E X E ->
F ,
and so on for thehigher
derivatives.We refer to
[4]
for more information.The notion
CPc
was introducedby
Ehresmann( Bastiani [1] ),
whoalso showed the
following -
we include aproof
forcompleteness
sake.6.2.
PROPOSITION(partial derivatives).
LetE1, E 2’
F belocally
convex
linear
spaces.Let f: E1 x E2 ->
F be amapping. f
is0 f
classC i f f
themappings x 1 |-> f ( x1 , x2), x2 |-> f ( x1, x2)
areo f class C1c
for each fixed x2, x1 respectively, with
derivativesD1f (x 1 , x 2 ) y1 and D2 f (x 1 , x 2 ) y2 respectively, which
arejointly continuous in all appear- ing variables. The
derivativeof f is
thengiven by
The
same is truei f f is de fined
on an open subseto f E1
XE2 only.
PROOF.
Necessity:
So
D1 f
isjointly
continuous in allappearing
variables.Analogously for D2 f .
Sufficiency :
The
joint continuity
ofD f
in allappearing
variables is clear from thesame property of D1f and D2f . QED
6.3.
Themanifold
structure. Wegive
a short review of the manifold struc- ture onC°° (X , Y), X,
Ybeing locally
compact smooth manifolds. "B’tfeuse a
simplified
form of the manifold structure set up in3.3, 3.4
and3.6 ;
this version is described in
( [9] , 8 ).
Let r:
T Y ->
Y be a smoothmapping
such that for each y in Y themapping Ty: T Y ->
Y is adiffeomorphism
onto an openneighbor-
hood of y in
Y ,
andTy(Oy)
= y. Such a map may be constructedby
an
exponential
mapfollowing
anappropriate
fibrerespecting
diffeomor-phism
fromT Y
onto the openneighborhood
of the zerosection,
on whichthe
exponential
map isdiffeomorphic.
Ifrry : T Y ->
Y denotes the cano-nical
projection,
then themapping (r, 1Ty):
T Y - Y X Y is a diffeomor-phism
onto an openneighborhood
of thediagonal
inY X Y . In Seip [11J
these maps
(which
need not be definedglobally there)
arecalled « local additions».
We willadopt
the same name for convenience sake.If
f6 C°° ( X, Y) ,
consider thepullback f * T Y
which is a vectorbundle over
X ,
and the spaceD( f * TY)
of all smooth sections with compact support of thisbundle, equipped
with the1°°-topology ( which
coincides with the
D-topology here).
This is alocally
convexdually
nuclear
(LF)-space, being
thestraightforward generalization
of the spaceÐ
of test functions with compact support in distributiontheory.
See 2.7for further information.
Let be the
mapping
Denote
by U f
theimage
oft/J f’
which is an open subset ofC°° (X, Y) ;
for let be the open
neighborhood
ofthe
graph {(X,f(x)) | x E X} of f
inX X Y = J 0 ( X, Y) (in
fact atubular
neighborhood
with verticalprojection,
cf.3.3-3.5 ).
ThenUf
consists of
all g E C°° ( X , Y)
such that thegraph of g
is contained inZ f
andg - f ( i. e., g
andf
differonly
on arelatively
compact subset of X),
soU f
isD°°-open.
Vif
is continuousby
2.5 and has a continuous inverseas is
easily
checked up.will serve as canonical chart centered at
f.
Now let us check the coordinate
change.
Letf,
g6C°°(X, Y)
with :we have
so the map
is
given by
by pushing
forward sectionsby
a fibrepreserving (locally defined)
dif-feomorphism.
So the coordinatechange
is continuous and of differentia-bility
classC°°rr by
theQ-Lemma
3.8.6.4
PROPOSITION. LetX, Y,
Z besmooth locally compact manifolds.
Then the canonical
identification
is
of class C°°rr. The identification
iscompatible
with the choiceof
ca-nonical
charts.
P ROO F. Let
(f, g) E L"(X, Y) x C°° (X, Z) .
We writeagain (f, g)
forthe
corresponding
element ofC°° ( X, Y x Z ) given by
Let T :
T Y ->
Y be a local addition onY,
p :TZ -> Z
be one onZ ,
thenr x p is a local addition on Y X
Z .
We haveand
U(f,g) = UfxUg
for the canonicalcharts,
and thefollowing diagram
commutes :
6.5.
PROPOSITION. For each n >0,
themapping
is
of class C°°rr .
PROOF. Let
be local additions for Y and
Jk(X, Y) respectively.
Letf E C°°(X , Y) ,
then jk f E Coo ( X, jk(X, Y)) ;
consider the canonical chart(Uf, cp )
centered at
f
ofC°° (X, Y)
and the canonical chart(U jkf, cpjkf)
center-ed at
jk f
ofC°°(X, Jk(X, y)).
We have to check wether themapping
is of class we have
where
is the
(functorially)
inducedmapping. Now j k s
cC°° ( X , Jk( X , Y)) by
definition but in fact it is an element of the closed
subspace DJk( f * T Y)
of smooth sections with compact support of the vector bundle
¡k (f* T Y )
over
X ,
which consists ofk-jets
of sections of the bundlef * T Y (cf.
[10]). We
willwrite jk
s if we consider it to be a section ofjk (f* T Y) .
The
mapping
is linear and continuous
(being
acomplex
ofpartial
differential opera-tors on a space of test functions - in a
canonical chart )
so it istrivially
of class
77
Therefore we haveand it is
easily
seen thatI.-i
is a fibre
preserving
smoothmapping,
sopushing
forward sectionsby
it is of class
q by
the Q-Lemma 3.8. The assertion followsby
thechain rule. QED
6.6.
REMARK. Themapping T : C°°(X, Y) -> C°°( T X, T Y) is not even
continuous in the
:D-topology,
neither in theD°°-topology :
letfn
be asequence
converging
tof
inCoo ( X, Y).
Thenfn equals f
off somefixed compact set
in X
for all butfinitely
many n(2.3 or [9] ).
But iffn
differs fromf
forinfinitely
many n at some x EX,
thenT fn
differsfrom
T f
on the whole fiberTx X ,
soT fn
cannot converge toT f
in ge-neral. Thus there is no chance for T to be differentiable. But it can be shown that the
mapping
is continuous and even differentiable
( compare [7], 2, Proposition 6),
since we may write
where comp:
T X x X ,J1 (X, Y) -> T Y
isjust composition
of matrices and vectorslocally,
which is smooth. We will use thistechnique
in aspace
Diff(X) of all dif feomorphisms of
X is a Lie group in theD°°-
top olo gy.
Diff(X)
is open inC°°(X , Y)
in theS-
and theD°°- topology,
composition
is continuousby 2.5
and inversion is continuousby 5.3,
soDiff(X)
is atopological
group in theD-topology
and in theD°°-topology.
In the latter it is an open submanifold of
C°°( X , Y) ,
and we will showthat it is a Lie group in this induced manifold structure. The
proof
ofthis will occupy Sections 7 and 8.
7. THE COMPOSITION IS DIFFERENTIABLE.
7.1.
Before we canbegin
with theproof,
we need somepreparation.If
p: E -> X is a vector bundle then let us denote
by V ( E)
=ker T (p)
thevertical subbundle of the bundle TE -
E .
IfE, = p.1 (x)
is the fibreover x c X and
ix : Ex -> E
is theembedding,
then we mayidentify
T v( Ex)
withEx
itself forv E Ex
via the affine structure ofEx
anddefine
It is clear that
V = VE : E ® E -> V ( E )
is anisomorphism
of vector bun-dles over
X . V(v, w )
will be called thevertical lift of
w over v . Themapping:
is called the vertical
projection.
L EMMA. 1°
V(E), VE, ÇE
commutewith pullbacks o f
vectorbundles.
20 Let a : E ,
E’
be asmooth fibre mapping between
vectorbun-
dles
asgiven by
thediagram
then the fibre derivative of
a ,the mapping dfa :
E ®E - E’
over(3
isgiven by
P ROOF. If
f : X’-> X
is amapping,
then thepullback
bundlef*E
isgiven
as thefollowing categorical pullback:
In view of
this,
we have :where
0x
is the zero section ofX ,
both as a manifold and amapping,
and V
f *
E=
VE
X0X’ .
assertion 2 of the lemma is clear. Q EDREMARK.
If f : X->
Y is a smoothmapping
and T: T Y -> Y is a local ad-dition,
then we w ill denoteby Tf: f* T Y -
Y x X thediffeomorphism
intogiven by
Clearly
we have if weidentify (f*TY)x
w ith
Tf (x) Y.
7.2. THEOREM. Let
X, Y,
Z belocally compact
smoothmanifolds.
Then the composition mapping
given by Comp (g, f)
= g of,
iso f
classC’
Here
C°°prop (X, Y)
denotes the open subset of propermappings
of
Coo (X, Y) (cf. 1.9, 2.5 ).
P ROO F. Let We w ill show that
Comp
be local
additions, inducing
the canonical charts.We will suppose that
U f
andUg
are smallenough
such that- to be
precise
we should restrict to open subsets ofUf
andUg
respec-tively, using continuity
of thecomposition ( 2.5 ),
but we will not spe-cify
this to save notation. Soby
some abuse oflanguage
we considerthe
mapping ( 1 ) :
We have to show that c is of
class -
and we will do thisby showing
that it is of class
C’- using
6.2.The
mapping
c isgiven
for ,by
There is some abuse of notation in this formula too: we did not distin-
guish
vector fieldsalong f
with compact support from sections of thevector bundle
f *T Y , i, e.,
we have identifiedDf(X, T Y)
with the iso-morphic
spaceD( f* T Y ),
to save notation. Let us first look at the dif-ferentiability
of themapping
Since
r s ~ f
andf
is proper, Ts is proper too, so themapping
is continuous and linear.
Then we consider the fibre
respecting (but
noteverywhere defined)
dif-feomorphism
overIdX :
It is clear that
By
the chain rule and the Q-lemma themapping ( 3 )
is therefore of classC°°rr
and its derivative isgiven by
,Here we
again
considered tr s,t’r s
both as sections of the bundle(g r s ) * TZ
and asmappings
X ->TZ .
We usedheavily
Lemma7.1.
Thelast line of formula
( 7 )
shows thatD 1 c ( t, s ) t’
isjointly
continuousin
s, t, t’ (use 6.4,
thecontinuity
of thecomposition
and the fact that themapping
is
continuous).
Now we
investigate
thedifferentiability
of themapping
for fixed t For fixed t we define the
mapping
where
a(t)
is a fibremapping
overIdX .
Soby
theQ-Lemma
andby
the chain rule the
mapping ( 8 )
is of class77
and the derivative isgiv-
en
by:
But the
m appin g t |-> a (t)
is no t continuous( there
is a non proper openembedding
on theright of t )
neitheris t 1, dfa ( t).
So we have to re-arrange the
expression ( 11 )
in such a way as to see thejoint continuity in t,
s,s’. We
compute asfollows, again using
Lemma7.1 :
So it remains to show that the
mapping
is continuous. For that we use the manifold
which is a submanifold of the
product,
and thefollowing composition
evaluation
mapping
11: M ->T Z given by :
Since p
islocally just multiplication
ofmatrices,
it isC°° .
Then we h ave :This
expression
isjointly
continuousin t,
s,s’ by 5.4, 6.4, 6.5
andby
the fact that
s |-> T s
is continuousIn view of
6.2
we have shown that c is of classCIc
and that(17) The higher derivatives :
If we want to show that c is of classC2c
we have to check that D c is of classC1c .
In order toapply 6.2
a-gain
we have to compute allpartial
derivatives of D c and have to show thatthey
arejointly
continuous in allappearing
variables. Now( 7 )
and( 15 )
arecomposita
ofexpressions
that look like( 2 ) again
andby 6.5
jl (t)
is of classC°°rr
in t . So we mayapply
what we havealready
prov- ed and getC2c. By
induction we getC°°c= C°°rr .
Q ED7.3. COROLLARY. Let
X,
Y belocally compact
smoothmanifolds. Then
the
evaluation mapping Ev: X X C°° (X, Y)-> Y is o f class C°°rr (and
consequently D°°-continouous).
P ROO F. First we show that
diffeomorphically,
where * denotes theone-point
manifold.L et r :
T X - X
be some local addition. Then the canonical chart(Ufl Of)
centered at
f : * -> X corresponds
to the chart(Im T f(* ), T f (*)-1)
center-ed at
f (*) of X.
Now the
following diagram
commutes and so the assertion follows from thedifferentiability
of thecomposition :
7.4.
COROLLARY. LetX, Y,
Z belocally compact smooth manifolds.
Then the canonical mapping
notion of
differentiability
inC°°(X , C °° ( Y, Z )) ,
since all reasonable notions coincide( s ee [4] ); C°°( Y, Z)
isequipped
with its canonicalC;
-manifold structure.° This canonical
mapping
is notsurjective
onC°°( Y x X, Z ) by topological reasons,
as wealready
mentioned in the introduction. It issurjective, however,
if Y is compact. That has been shownby
Gutknecht[3]
for the stronger notion ofdifferentiability CF ;
in order to be com-plete
we will prove this fact in oursetting
too(7.5 ).
PROOF OF 7.4. Let
f E C°°(X, COO(Y,Z)).
Then the canonical map-ping
associates tof
themapping f : Y x X ->
Zgiven by
7.5. THEOREM. Let
X, Y,
Z besmooth manifolds, X,
Zlocally
com-pact
and Ycompact. Then
via the canonical
identification.
P ROOF. In view of 7.4 it remains to show that the canonical identifi- cation
mapping
is ontoC°° ( Y x X, Z) ,
andby
abstract non-sense itsuffices to show that the
mapping
is of class then
is of class
C°°rr by
theQ-Lemma
orby 7.2,
sof*o n : X -> Coo ( Y, Z)
isof class
C°°rr
too; this lattermapping
iseasily
seen to be the canonical associate tof (77
is the so-called unit of theadjunction,
incategorical
terms
).
Now fix
xo c X
and let r : T X -> X and p :T Y -
Y be local additions.Then
is a local
addition;
letbe the canonical chart of
C°°( Y, Y x X ) ,
centered at77(x0),
whichcomes from p X r . Since
77(x0)
isgiven by y |-> (y, Xo)
we see thatas bundle over
Y ,
soIn view of this identification we have for x c X near xo and
y 6 Y :
So
cp77(x0)
o 71 isgiven by
the sequencewhich is
clearly
differentiable in any sense, where V is a suitableneigh-
borhood of xo
in X
and where B is the continuous linearmapping
whichmaps each
point
ofT x0 X
into the constant function Y -7" X . B
iswell defined and continuous iff Y is compact. QED
7.6.
PROPOSITION.The tangent mapping o f
thecomposition
is
given by
PROOF. Since we know
already
thatComp
is differentiable we may com- pute the tangentmapping by considering
one parameter variationsthrough
g and
f
with «tangent vectors » t and s anddifferentiating
their com-position pointwise, i, e.,
for fixed x cX , using
Lemma4.4.
8. THE INVERSION IS DIFFERENTIABLE.
v ersion I nv :
Diff(X) -> Diff(X)
iso f
classC°°rr.
PROOF.
(1)
It suffices to show that Inv:Diff(X) -> Diff(X)
is of class in aneighborhood
U of theidentity I dX
ofX .
For letf E Diff(X) ,
then
is a
neighborhood
off by 2.5.
For anyg E V f
we havethus
J
Since
(f -1)*
and(r-1)*
are of classC°°rr ( by
7.1 orby
the A- and n- Lemmarespectively)
the chain ruleimplies
thatIn v | Vf
is of class( 2 )
Now let T : T X -> X be a local addition and let canonical chart centered atI d E Diff(X) ,
and letand
be the chart maps. We have to show that the
mapping
is of class
C°°rr.
There isagain
some abuse of notation involved: to beprecise
we have to restrict the domain of themapping
Inv0 f/J
to an open subset ofD(TX)
so that itsimage
is contained in the domain llof cp.
This is
possible by 5.3,
and we willsilently
assume this in the follow-ing
to save notation. Consider themapping
of 7.2
( 1 ) (here
too we have some silent restrictionsinvolved).
Thenfor any
s E D( T X)
we have:Experience
with finite dimensional Lie groups or the formula of VerEecke about the derivative of
implicitly given
functions suggests to try thefollowing
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
granted
and we willinvestigate
this formula. Recall from 7.2
( 11 )
thatwhere
df
denotes the fibre derivative and whereis a fibre
preserving
smoothdiffeomorphism, given by
7.2(9).
Nowso
a (i ( s ))
is invertible and(6 )
is invertible too, and we haveSo
by
theimplicit
function theorem in finite dimensions we havePutting
this back into(5 )
we getFrom this formula and from 7.2
( 7 )
and( 15 )
it is clear thatD i (s ) s’
isjointly
continuous in s ands’ ,
so i is of classC1c. This implies
in turnthat D i is of class
C1c again, applying
the chain rule and 7.2( 17 )
tothe
right-hand
side of(7 ),
so i is of classC2c .
Now astraightforward
induction shows that i is of class
C°°c = C°°rr .
QEDP ROO F. Let T be near
enough
0 in R . Thenby 7.1
themapping
is
differentiable,
is of classC°°
in the classical sense. So the usual rules of Calculushold,
inparticular
the relations betweenintegrals
andderivatives,
theintegrals being
Riemannianintegrals
with values in thecomplete locally
convex spaceD( T X) .
These relations can be derivedvery
simply
from the one-dimensional caseby using
the Hahn-Banach Theorem.Our first aim is to show that:
stays bounded in
( By
8.1(4)
we know thatSo we have
For h - 0 the first summand converges to
Therefore we conclude that
To show that
( 1 )
stays bounded inT( T X)
for k -> 0 we have to show that there is a compact set K C X such thatand that
is
«uniformly
bounded’) for(x, À)
inK x [- 1 , 1]B { 0 } ) for any k.
( 4 )
iseasily
checked:is a continuous compact
path
inD( T X) ,
so it can moveonly
insidesome
compact K
C X(compare
the fact mentioned in theintroduction);
this K satisfies
( 4 ).
To prove
( 5 )
we note that it suffices to show that for each k and xo in K there is aneighborhood Ux0
0of xo
in X
such that theexpression ( 5 )
stays
«uniformly
bounded» for(x, k) c Ux 0 x ([ -1, 1]B{0}).
We choose U =
Ux
0 to be so small that T X|U =U
XRn .
For x c U wemay write
Then (3)
looks likeuniformly
for xe v and for each derivative with respect to x . Let us write( 6 )
for short in the formso that
B( x, À)
is the localrepresentative
over U of( 1 ).
Then letG : L ( Rn )-> R+
be the continuousmapping
Since
A ( x, À, 11): Rn -> Rn
is invertible(cf. 8.1 )
and continuous in x,and
by (7) B (x, X) I
has to be bounded for k ->0 .
Let
dx
denote the derivative with respect to x . Then we haveThe first summand is
already
bounded and for the second we may repeat the aboveargument. A simple
induction then shows thatdkx B(x, k)
isbounded for k -> 0 for any
k ,
so( 5 )
isproved.
Now we
proceed
to prove the Lemma. Sinceis bounded and
Ð( T X)
is a Monte 1space, M
is precompact, so thereare cluster
points
of M for k ,0 .
Let t be such a clusterpoint,
thenthere is a net
such
that ta -> t
inD( T X ) . By
thejoint continuity
ofD2 c
in all va-riables we conclude that
since ta -> t
andka -> 0 . By ( 3 ) again
we conclude thatSince
D 2 c ( s , i ( s ))
is invertible we getThis holds for any cluster
point of M for k -> 0 ,
so the lemma isproved.
QED
8.3. PROPOSITION.
The tangent mapping o f the
inversioni s
given by
P ROO F. As in 7.6 one may compute the tangent
mapping by
differentiat-ing
the inverse of a one parameter variationthrough f
with « tangent vec-tor» s. The
computation
is a little more difficult than in 7.6.REFERENCES.
1. E HRESM ANN
(BASTIANI),
A.,Applications
différentiables et variétés de dimensioninfinie,
J.d’Analyse
Math.13, Jérusalem ( 1964),
1- 114.2. GOLUBITSKY, M. & GUILLEMIN, V., Stable mappings and their
singulari-
ties,
Springer, 1973,
G.T.M. 14.3. GUTKNECHT, J., Die
C~0393-Struktur auf
derDiffeomorphismen
gruppe einerkompakten Mannifaltigkeit,
Dissertation ETH 5879,Zürich,
1977.4. KELLER, H.H., Differential Calculus in locally convex spaces, Lecture Notes in Math.
417,
Springer ( 1974).5. KOBAYASHI, S.,
Transformation
groups indifferential
Geometry,Springer, 1974, Ergebnisse
70.6. LESLIE, J.A., On a differentiable structure for the group of
diffeomorphisms, Topology
6 (1967),
263- 271.7. MATHER,
J. N.,
Stability ofC~-mappings.
II : Infinitesimal stability implies stability, Annalsof Math.
89 (1969),
254- 291.8. MICIIOR, P., Manifolds of smooth maps, Cahiers
Topo.
et Géo.Diff.
XIX- 1( 1978 ),
47- 78.9. MICHOR, P., Manifolds of differentiable maps, Proc. CIME
Conf.
ondifferen-
tial
Topology (foliations),
Varenna, 1976.10. P ALAIS, R., Seminar on the
Atiyah- Singer
Index Theorem, Ann. Math. Stu- dies 57 ( 1965).11. SEIP, U.,
Kompakt
erzeugteAbbildungsmannigfaltigkeiten, Preprint,
T. H.Darmstadt, 1974
Institut fur Mathematik Universitat Wien