On the existence of
slice theorems
for moduli spaces
on ¯ber bundles
Dissertation zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
an der Formal und Naturwissenschaftlichen FakultÄat
der UniversitÄat Wien,
eingereicht von
Hermann Schichl
Typeset by A
M
ST
E
X
i
Abstract
After a short introduction to the ¯nite dimensional case of orbit spaces and a
summary of the most important results on Hilbert manifolds and smooth in¯nite
dimensional manifolds,we consider three orbit spaces related to the space Conn(E)
of connections on a ¯ber bundle E!M and its gauge group Gau(E).
This investigation is motivated by a decomposition of the space of metrics
Met(E) on the total space of the bundle into three parts on which Gau(E) acts,
one of them being Conn(E).For the orbit spaces related to Met(E) and to
Conn(E) £ Met(V E) the direct sum of the space of connections and the space
of ¯ber metrics on the vertical bundle,respectively,slice theorems will be proven,
which lead to strati¯cations of the orbit spaces.
For the orbit space related to the space of connections on E counterexamples will
show that,except for the trivial cases of zero dimensional ¯ber or zero dimensional
base,no such slice theorem can exist.
Kurzfassung
Nach einer kurzen EinfÄuhrung in OrbitrÄaume im endlichdimensionalen Fall und
einer Zusammenfassung der wichtigsten Ergebnisse Äuber Hilbert Mannigfaltigkeiten
und glatte unendlichdimensionale Mannigfaltigkeiten betrachten wir drei OrbitrÄau
me,die mit den Konnexionen Conn(E) eines FaserbÄundels E!M und dessen
Eichgruppe Gau(E) zusammenhÄangen.
Dies ist motiviert durch die Zerlegung des Raumes der Metriken Met(E) auf
dem Totalraum des BÄundels in drei Teile,auf denen jeweils die Eichgruppe wirkt.
Einer dieser Teile ist der Raum der Konnexionen Conn(E).FÄur den Orbitraum zu
Met(E) und dem Orbitraum zu Conn(E) £ Met(V E),der direkten Summe der
RÄaume der Konnexionen und der Fasermetriken auf dem vertikalen BÄundel,wird
ein Scheibensatz bewiesen,der zu einer Strati¯zierung der OrbitrÄaume fÄuhrt.
FÄur den Orbitraum,der zum Raum der Konnexionen gehÄort,wird mit Hilfe von
Gegenbeispielen gezeigt,da¼ kein Scheibensatz existieren kann,ausgenommen in
den trivialen FÄallen von nulldimensionaler Basis oder nulldimensionaler Faser.
ii
iii
Preface
At ¯rst I would like to thank my supervisor Prof.Peter W.Michor from the
Institute of Mathematics of the University of Vienna for the possibility to write
this thesis,for the problem,the necessary papers,his support,and his patience.
Furthermore,I want to send special thanks to my former teacher Prof.Wolf
gang Herfort from the University for Technology of Vienna for introducing me to
the theory of Hilbert spaces,especially Sobolev spaces,which was an important
background for this work.
Thanks,also,to my colleague Andreas Cap for much help and many fruitful
discussions,whenever questions arose while trying to solve the problem,and to Prof.
Dan Burghelea of Ohio State University for a special discussion and the reference
to the paper of J.Cerf.
Finally,I have to thank the Institute of Mathematics of the University of Vienna,
especially its head Prof.Harald Rindler,and the Erwin SchrÄodinger Institute of
Mathematical Physics,for a position and for providing me with all the means to
¯nish this thesis.
Vienna,November 17th 1996 Hermann Schichl
iv
Vorwort
Zuallererst mÄochte ich meinem Betreuer Prof.Peter W.Michor vom Institut
fÄur Mathematik der UniversitÄat Wien meinen besonderen Dank aussprechen fÄur
die MÄoglichkeit,bei ihm an einer Dissertation zu arbeiten,fÄur die Stellung des
Problems,die Versorgung mit der nÄotigen Literatur,seine UnterstÄutzung und nicht
zuletzt fÄur seine Geduld.
MeinemfrÄuheren Lehrer Prof.Wolfgang Herfort von der Technischen UniversitÄat
Wien mÄochte ich dafÄur danken,da¼ er mich,unter anderem,in die Theorie der
HilbertrÄaume,speziell der SobolevrÄaume,eingefÄuhrt hat,die als Hintergrund fÄur
die vorliegende Arbeit von besonderer Bedeutung war.
Weiters mÄochte ich mich bei meinemKollegen Andreas Cap fÄur seine bereitwillige
Hilfestellung und seine Diskussionsfreudigkeit bei vielen Fragen,die im Zusammen
hang mit dieser Arbeit aufgetreten sind,und bei Prof.Dan Burghelea von der Ohio
State University fÄur seinen Hinweis auf die Arbeit von J.Cerf und eine instruktive
Diskussion bedanken.
Zuletzt gebÄuhren noch demInstitut fÄur Mathematik der UniversitÄat Wien,dabei
ganz besonders dem Institutsvorstand Prof.Harald Rindler,und dem Erwin SchrÄo
dinger Institut fÄur Mathematische Physik Dank fÄur eine Stelle bzw.die zur Ver
fÄugung Stellung von Mitteln,ohne die die Fertigstellung dieser Dissertation nicht
mÄoglich gewesen wÄare.
Wien,17.November 1996 Hermann Schichl
v
Contents
Abstract..............................i
Kurzfassung............................i
Preface..............................iii
Vorwort..............................iv
1 Introduction............................1
2 G{manifolds and Lie group actions..................3
3 Banach and Hilbert manifolds....................9
4 Smooth in¯nite dimensional manifolds................29
5 Slice theorems for Met(E)= Gau(E) and (Conn(E) £Met(V E))= Gau(E)..43
6 Conn(E)= Gau(E) and why no slice theorem exists............63
References.............................69
Lebenslauf.............................71
vi
1
1.Introduction
In modern mathematics and physics actions of Lie groups on manifolds and
the resulting orbit spaces (moduli spaces) are of great interest.For example,the
moduli space of principal connections on a principal ¯ber bundle modulo the group
of principal bundle automorphisms is the proper con¯guration space for Yang{Mills
¯eld theory (as e.g.outlined in [Gribov 1977],[Singer 1978],and [Narasimhan,
Ramadas 1979]).
Usually,when symmetries and invariance groups are considered,a problem re
duces to the corresponding orbit space,and therefore the structure of these spaces
has to be investigated.This structure theory is quite complicated in general,since
these spaces usually are singular spaces and not again manifolds.In fact,only if the
action of the Lie group is free (i.e.all isotropy subgroups of single points are triv
ial),the resulting orbit space bears a manifold structure and forms together with
the manifold and the quotient map a principal ¯ber bundle,whose structure is well
known.More often,the orbit space admits a strati¯cation into smooth manifolds
with an open and dense largest stratum,the set of principal orbits (see section 2).
This strati¯ed space can then be treated almost like a manifold when taking spe
cial care.The existence of such a strati¯cation is usually shown by proving the
existence of slices at every point for the group action.
All these problems arise already,if both the Lie group and the manifold are ¯nite
dimensional.As shown in section 4,both notions can be very widely generalized to
in¯nite dimensions,and again the structure of the,now often in¯nite dimensional,
moduli spaces is interesting.For example the above mentioned con¯guration space
of Yang{Mills theory is constructed from in¯nite dimensional spaces.Again,a slice
theorem for the action is the way to prove existence of a (generalized) strati¯cation
of the moduli space.This slice theorem turns out to be more di±cult than in the
¯nite dimensional case,because of the lack of an inverse function theorem.In order
to reduce this problem to a very special inverse function theorem with very strong
prerequisites,an approach using Hilbert manifolds and Sobolev completions has to
be taken in order to construct the slice via inverse limits of Hilbert manifolds.
In spite of these technical di±culties,these in¯nite dimensional generalizations
are very interesting,since many problems in modern theoretical physics lead to
moduli spaces in the in¯nite dimensional setting.Not only the above mentioned
Yang{Mills theory is considered,but also the space of Riemannian metrics modulo
the group of di®eomorphisms,which appears in General Relativity,(principal) ¯ber
bundle connections modulo the gauge group arise from gauge theories,and also one
approach to quantization,the geometric quantization method,is taken via orbit
spaces,as described in [Kirillov 1982],[Kirillov 1990],and [Vizman 1994].
Also,very recent research in theoretical physics is connected to moduli spaces:
e.g.invariance of Euler numbers of moduli spaces of instantons on 4{manifolds
[Vafa,Witten 1990],moduli spaces of parabolic Higgs bundles,which are connected
to Higgs ¯elds [Maruyama,Yokogawa 1992],[Nakajima 1996].
In algebraic topology moduli spaces play an important role,either,[Maruyama
1996],[Simpson 1994],[Simpson 1995],and also the de¯nition of the famous Don
aldson Polynomials involves moduli spaces ([Donaldson 1990]).
2
In this thesis,I will try to analyze the structure of some moduli spaces,which
are connected with the space of connections on a general ¯ber bundle with compact
¯ber and compact base space.In section 2 some known results about actions of
¯nite dimensional groups on ¯nite dimensional manifolds will be recalled,such that
the notion of slices will become clear.Then in sections 3 and 4 the basic facts
about Hilbert manifolds and about in¯nite dimensional smooth manifolds and Lie
groups will be introduced.Slice theorems for two moduli spaces will be shown in
section 5.In section 6,¯nally,counterexamples will show that there cannot exist
a slice theorem for the moduli space Conn(E)= Gau(E) of connections on a general
¯ber bundle modulo the gauge group.
The result is connected with a slice theorem for Met(M)= Di®(M),the orbit
space of metrics on a manifold with respect to the action of the group of di®eo
morphisms proved by [Ebin 1968],another slice theorem proved by [Kondracki,
Rogulski 1986] for Conn(P)= Gau(P),the space of connections on a principal ¯ber
bundle modulo the gauge group,[Cerf] for F= Di®(R) the space of functions of ¯nite
codimension at critical points on R modulo the di®eomorphismgroup.For a general
survey on slice theorems and slices see [Isenberg,Marsden 1982],where a slice the
orem for the space of solutions of Einstein's equations modulo the di®eomorphism
group is proven.
Finally,the nonexistence of the slice theorem in the case connections on a ¯ber
bundle modulo the gauge group is connected to the fact,that C
1
(S
1
;R)= Di®(S
1
),
where the di®eomorphisms act by composition,admits in general no slices,except
when restricted to the space of functions of ¯nite codimension at critical points.
3
2.G{manifolds and Lie group actions
In this section,I want to introduce the basic facts about ¯nite dimensional Lie
groups acting on ¯nite dimensional manifolds (some of themwithout proofs).These
facts will be taken for motivation of the terms in in¯nite dimensions.Moreover,a
basic example will be discussed,which helps to understand the properties of slices
and sections.
Throughout this chapter all manifolds and groups are supposed to be ¯nite
dimensional,except if stated otherwise explicitly.
2.1.De¯nition.Let G be a Lie group,M a smooth manifold.A smooth action
of G on M is a C
1
mapping l:G£M!M (l(g;x) = l
g
(x) = l
x
(g) = gx),such
that
ex = x 8x 2 M
(g
1
g
2
)x = g
1
(g
2
x) 8g
1
;g
2
2 G;x 2 M:
We say that G acts on M,or M is a G{manifold.Furthermore,a G{action on M
is called
(1) linear,if M is a vector space,and the action is a representation.
(2) a±ne,if M is an a±ne space,and all the l
g
are a±ne transformations.
(3) orthogonal,if (M;°) is a Euclidean space,and the action is a subgroup of
O(M;°).
(4) isometric,if (M;°) is a Riemannian manifold,and every l
g
is an isometry.
(5) symplectic,if (M;!) is a symplectic manifold,and every l
g
is a symplecto
morphism.
2.2.De¯nition.Let M be a G{manifold,x 2 M
The set G¢ x:= fgx j g 2 Gg is called the (G{)orbit of x.
The closed subgroup G
x
:= fg 2 G j gx = xg of G is called the isotropy
subgroup of x.
Then T
e
G =:g,T
0
(G=G
x
)'g= Lie(G
x
),and
G
l
x
M
G=G
x
p
where the mapping G=G
x
!M is an initial immersion with image G¢ x.
2.3.Lemma.
(1) G
gx
= gG
x
g
¡1
(2) G¢ x\G¢ y 6=;=) G¢ x = G¢ y
(3) T
x
(G¢ x) = T
e
(l
x
):g
Proof.Is clear.
42.4.De¯nition.For a G{manifold M,let M=G be the space of all G{orbits
equipped with the quotient topology;¼:M!M=G.Then M=G is called the
orbit space (moduli space) of M with respect to G.
The set of closed subgroups of G bears an equivalence relation H » H
0
:()
H = gH
0
g
¡1
for some g 2 G.The equivalence classes are called conjugacy classes.
The set of conjugacy classes admits a partial order (H) · (H
0
) if H µ gH
0
g
¡1
for some g 2 G.
For an orbit G¢ x,by Lemma 2.3(1),the isotropy subgroups G
gx
form a conju
gacy class (G
x
),which is called the isotropy type of the orbit G¢ x.Two orbits are
said to be of the same type if their isotropy types coincide.
2.5.De¯nition.For two G{manifolds M,N and a smooth mapping f:M!N
we say f is equivariant i® f(gx) = g f(x).
2.6.De¯nition.Let M be a G{manifold.An orbit G¢ x is called a principal orbit,
if there exists an open neighborhood U of x in M with 8y 2 U:9f:G¢ x!G¢ y
G{equivariant,which is equivalent to 8y 2 U:9a 2 G:G
x
µ aG
y
a
¡1
.
x 2 M is called a regular point if G¢ x is a principal orbit,and is called singular
point otherwise.
M
reg
:= fregular pointsg,M
sing
:= fsingular pointsg.
2.7.De¯nition.Let x 2 M,M a G{manifold.A subset S of M is called a slice
at x,if there exists a G{invariant open neighborhood U of the orbit G¢ x and a
smooth equivariant retraction r:U!G¢ x,such that S = r
¡1
(x).
2.8.Proposition.Let M be a G{manifold and S a slice at x.Then
(1) x 2 S and G
x
¢ S µ S
(2) g ¢ S\S 6=;=) g 2 G
x
(3) G¢ S = fgyjg 2 G;y 2 Sg = U with U as in de¯nition 2.7.
Proof.Let r be the (smooth,equivariant) retraction.Then we know that r is a
submersion and S = r
¡1
(x).Thus G
y
µ G
x
8y 2 S.Therefore,rj
G¢y
:G¢ y!
G¢ x is also a submersion 8y,which implies that x is a regular value for r.That
su±ces to show that S = r
¡1
(x) is a submanifold of U and also of M.
Thus we have y 2 S;gy 2 S =) r(gy) = x = gr(x) = gx =) g 2 G
x
,which in
return implies (2).
Then g 2 G
x
;s 2 S =) r(gs) = gr(s) = gx = x =) G
x
¢ S µ S,which shows
(1).(3) can be shown as follows:y 2 U =) 9g 2 G:gr(y) = x =) r(gy) = gr(y) =
x =) gy 2 S.¤
2.9.Corollary.Let S be a slice at x for the G{manifold M.Then
(1) S is a G
x
{manifold
(2) For y 2 S is G
y
µ G
x
.
(3) If G¢ x is a principal orbit and G
x
compact,then G
x
= G
y
8y 2 S,i.e.all
orbits near G¢ x are principal,too.
(4) Two G
x
{orbits G
x
¢ s
1
and G
x
¢ s
2
are of the same type,i® the two G{orbits
G¢ s
1
and G¢ s
2
in M are of the same type.
(5) S=G
x
is isomorphic to G¢ S=G,an open neighborhood of G¢ x in the orbit
space M=G.
5
Proof.
(1) is clear
(2) is clear
(3) y 2 S =) G
y
µ G
x
(compact) =) G
y
is compact.G¢ x is principal =)
for y near x is G
x
conjugate to a subgroup of G
y
.Therefore G
x
= G
y
.
(4) K = G
x
,s 2 S,K acts on S.K
s
= G
s
(see Proposition 2.8(2)).G
x
s
1
=
G
x
s
2
=) K
s
1
is conjugate to K
s
2
in G
x
.Hence G
s
1
is conjugate to G
s
2
.
(5) follows from 2.8(2) and (3).
¤
2.10.Example.An elementary example will be useful to illustrate the properties
of slices a bit.
Take M = R
n
and G = SO(n) acting by rotations about 0 2 R
n
.The orbit
G¢ 0 = f0g,G
0
= G,and the slice S at zero may be chosen to be any open ball
centered at 0.For any other point x 2 R
n
G¢ x is the sphere of radius kxk
2
centered
at 0,G
x
= feg,and any su±ciently short line segment transversal to G¢ x through
x can be chosen as S.
2.11.Proposition.If S is a slice at x in a G{manifold M.Then there exists a G{
equivariant di®eomorphism G[S] = G£
G
x
S!G¢ S,which maps the\zerosection"
G£
G
x
fxg onto G¢ x.
Proof.The map f:G[S]!G¢ S given by f:[(g;s)] 7!g ¢ s is smooth and has the
required properties.¤
2.12.De¯nition.An action l:G£M!M is called proper,if one,hence all,of
the following equivalent conditions is satis¯ed.
(1) (l;Id):G£M!M £M (g;x) 7!(gx;x) is proper
(2) g
n
x
n
!y and x
n
!x imply g
n
has a convergent subsequence.
(3) K;L ½ M compact =) fg 2 GjgK\L 6=;g is compact
M is then called a proper G{manifold.
Proof.(1) =)(2) is clear.Suppose (2) =)(3) does not hold.Then 9(g
n
) without
cluster point,g
n
K\L 6=;.Choose x
n
2 K with g
n
x
n
2 L.Without loss of
generality x
n
!x,g
n
x
n
!y,which is a contradiction.
(3) =)(1):(l;Id)
¡1
(L £K) = f(g;x):x 2 K;gx 2 Lg µ
closed
fg 2 G:gK\L 6=
;g £K which is compact.¤
2.13.Lemma.(M;°) a Riemannian manifold,l:G £ M!M an e®ective,
isometric action,such that
·
l(G) is closed in Isom(M;°).Then l is a proper action.
Now we have collected most of the important results and terms for the ¯rst
important theorem,which was proven by R.Palais in 1961.
2.14.Slice Theorem.Let M be a G{manifold,x 2 M such that G
x
is compact
and for all open neighborhoods U of G
x
in G there is an open neighborhood V of x
in M with fg 2 GjgV\V 6=;g µ U.Then there exists a slice at x.
Idea of proof.Take a Riemannian metric on M,and construct a G
x
{invariant metric
~°.Now construct a so called\almost slice"
~
S:
~
S:= exp
~°
x
(T
x
(G¢ x)
?
\B(")).If
6you take an open neighborhood U of 0 in G=G
x
,such that there exists a section
Â:U!G with Â(0) = e.Then
f:U £
~
S!M (u;s) 7!Â(u) ¢ s is a di®eomorphism (maybe for somewhat smaller
U,
~
S) onto an open neighborhood of x in M.Using the assumption of the theorem,
you get a neighborhood V of x in M,such that S =
~
S\V is the required slice.
The complete proof can be found in [Palais 1961].
In the proof of the in¯nite dimensional slice theorem most of the\easyto
construct"things like the manifold G=G
x
,Â,and T
x
(G¢ x)
?
will make much more
di±culty,and constructing these will be the ¯rst crucial step in obtaining the slice
theorem.2.15.Theorem.In every proper G{manifold M,there exists a slice at every point
x 2 M.
Proof.G
x
is compact,since the action is proper.Now let U be an open neigh
borhood of G
x
in G.Then there exists an open neighborhood V of G
x
with
G
x
¢ V = V.(G
x
¢ G
x
= G
x
;Thus 8(a;b) 2 G
x
£ G
x
exist open neighborhoods
A
a;b
of a in G and B
a;b
of b in G such that A
a;b
¢ B
a;b
µ U.Since G
x
is compact
G
x
½
open
S
ni=1
B
a;b
i
=:B
a
.Let A
a
:=
T
ni=1
A
a;b
i
.Then A
a
¢ B
a
µ U.Further
more G
x
½
open
S
nj=1
A
a
j
and W:=
T
nj=1
B
b
i
is open in G,G
x
µ W,G
x
¢ W µ U,
V = G
x
¢ W.)
Now let U = G
x
U be a neighborhood of cosets.l
x
:G!G¢ x is a closed
mapping.Thus (G n U) ¢ x is closed in G¢ x,therefore in M.This implies the
existence of a neighborhood W of x such that W\(G n U) ¢ x =;.Without
loss of generality let W be compact.Then fg 2 G j gW\W 6=;g is compact
(2.12(3)).K:= fg 2 G n U j gW\W 6=;g is a compact subset of G n U.
k 2 K =) k x 2 (G n U) ¢ x =) k x 2 M n W (which is open).Therefore,
there exist neighborhoods Q
k
of k,V
k
of x such that Q
k
¢ V
k
µ M n W.Some
Q
k
1
:::Q
k
n
cover K,V:=
T
ni=1
V
k
i
is neighborhood of x in M,without loss of
generality V µ W.
Let gV\V 6=;.Then gW\W 6=;,and therefore g 2 U [ K.If g 2 K,then
there exists i such that g 2 Q
k
i
.But then gV µ Q
k
i
V µ Mn W µ Mn V and this
implies gV\V =;which is a contradiction.Therefore g 2 U.¤
2.16.Theorem (Palais 1961).For a proper G{manifold M,x 2 M the following
are equivalent
(1) G
x
is compact and there exists a slice at x.
(2) There exists a neighborhood U of x in M,s.t.fg 2 GjgV\V 6=;g has
compact closure in G.
Proof.in [Palais 1961].
2.17.Corollary.M proper G{manifold =) M=G is completely regular and
locally compact (hence T
2
).
2.18.Lemma.If M is a proper G{manifold,then there exists a principal orbit
type.
7
2.19.Theorem.M a proper G{manifold.Then the set of all regular points in M
is open and dense.
2.20.Theorem.M a proper G{manifold,x 2 M.Then x has a G{invariant
open neighborhood U,such that U contains only ¯nitely many orbit types.
2.21.Theorem.M a proper G{manifold,then the space M
sing
=G of all singular
G{orbits does not locally dissect M=G.
2.22.Corollary.M connected proper G{manifold.Then
(1) M=G is connected
(2) M has exactly one principal orbit type.
(3) The set of all principal orbits is open and dense.
Most of these facts essentially follow from the existence of slices.In in¯nite
dimensions analogous results can be likewise proven,if the existence of such slices
is ensured.
However,¯rst we will have to de¯ne most of the involved terms like manifold,
Lie group,tangent space,:::in in¯nite dimensions.The next two chapters are
devoted to this.
8
9
3.Banach and Hilbert manifolds
In this chapter I will give a brief excursion on Banach and Hilbert manifolds
which is in part excerpted from [Lang 1995].More detail can (e.g.) be found there.
3.1.De¯nition.A topological vector space is a vector space E over R equipped
with a topology such that the operations +:E £E!E (addition of vectors) and
¢:R£E!E (multiplication with scalars) are continuous.
Throughout our presentation all topological vector spaces will be Hausdor® and
locally convex (i.e.every neighborhood U of 0 2 E contains an open convex neigh
borhood V of 0.)
The set of continuous linear maps':E!F (E and F being two topological
vector spaces) will be denoted by L(E;F),and we set L(E):= L(E;R).
The vector space of n{linear maps Ã:E£¢ ¢ ¢£E!F will be denoted L
n
(E;F),
and as above L
n
(E):= L
n
(E;R).
3.2.De¯nition.A locally convex topological space E will be called a Fr¶echet
space if its topology is metrizable (i.e.there exists a metric d:E £E!R
+0
,such
that every neighborhood U of 0 2 E contains a ball B
"
:= fv 2 Ejd(0;e) <"g.),
and it is complete (i.e.all Cauchy sequences converge).
A Fr¶echet space space E will be called a Banach space if its metric is de¯ned by
a norm k:k:E!R
+0
(i.e.d(v;w) = kv ¡wk).It is well known,that for Banach
spaces E;F the norm kAk:= sup
kxk
E
=1
kAxk
F
for A 2 L(E;F) makes L(E;F)
into a Banach space.
A Banach space is called a Hilbert space if the norm is de¯ned by an inner
product h:;:i:E £E!R (i.e.kvk =
p
hv;vi).
3.3.Proposition.Let E;F be Banach spaces,f 2 L(E;F).Assume G ½ F is an
(algebraic) linear complement to imf and G is closed in F.Then imf is closed in
F and F = imf ©G.
Proof.See [Palais 1965,proof of Theorem 1] ¤
3.4.De¯nition.Let E;F be two Banach spaces,and f:U
½
open
E!F be a
continuous map.We say that f is di®erentiable at a point x
0
2 U if there exists a
¸ 2 L(E;F) such that
lim
y!0
kf(x
0
+y) ¡f(x
0
) ¡¸(y)k
kyk
= 0:
¸ is then uniquely determined,and we set Df(x
0
):= ¸ and call it the derivative
of f at x
0
.If f is di®erentiable at every point x 2 U then we get a map Df:U!
L(E;F),and we say f is di®erentiable.
If Df is again continuous we say that f is of class C
1
.Maps of class C
p
for p ¸ 1
are then de¯ned inductively.The p{th derivative of f will be D
p
f:= D(D
p¡1
f),
a map of U to L(E;L(E;:::;L(E;F):::)),which can be identi¯ed with L
p
(E;F).
A map f is said to be of class C
p
if D
k
f exists for 1 · k · p,and is continuous.
The usual results like chain rule,and linearity hold as in the ¯nite dimensional
case.
103.5.De¯nition.Let E
1
;E
2
;F be Banach spaces,U
i
½
open
E
i
and f:U
1
£U
2
!F
be a continuous map.If (u;v) 2 U£V and we keep v ¯xed,then f(;v):U
2
!F,
and we de¯ne the partial derivative as
@
1
f(u;v):= D(f(;v))(u);
which is a map U £ V!L(E
1
;F).Similarly,we may de¯ne @
2
f.The total
derivative and the partials are related as follows.
Proposition.Let E
i
,F be Banach spaces,U
i
½
open
E
i
,and f:U
1
£¢ ¢ ¢ £U
n
!F
continuous.Then f is of class C
p
if and only if each partial derivative @
i
f:U
1
£
¢ ¢ ¢ £U
n
!L(E
i
;F) exists and is of class C
p¡1
.In that case for v = (v
1
;:::;v
n
) 2
U
1
£¢ ¢ ¢ £U
n
,and (w
1
;:::;w
n
) 2 E
1
£¢ ¢ ¢ £E
n
we have
Df(v) ¢ (w
1
;:::;w
n
) =
n
X
i=1
@
i
f(v) ¢ w
i
:
3.6.The inverse mapping theorem.The inverse mapping theoremis one of the
main reasons for considering Banach and Hilbert spaces (and the yet to be de¯ned
Banach and Hilbert manifolds),since in most of the smooth in¯nite dimensional
spaces of chapter 4 this theorem does not hold.This arises as the main di±culty in
proving theorems for these spaces,and long ways have to be taken to circumvent
this di±culty.Most of these ways lead to Banach and Hilbert completions of the
spaces.
Both the inverse function theorem and the existence theorem for di®erential
equations (which is extremely important also) are based on the following
Shrinking Lemma.Let E be a complete metric space,with distance function d,
and let f:E!E.Assume the existence of a constant 0 < C < 1,such that,for
any v;w 2 E,we have
d(f(v);f(w)) · Cd(v;w):
Then f has a unique ¯xed point x (x = f(x)).Given any point x
0
2 E,then
x = lim
n!1
f
n
(x
0
) with f
n
(v) = f(f
n¡1
(v)).
Inverse mapping theorem.Let E;F be Banach spaces,U
½
open
E,and let f:U!
F be a C
p
{map with p ¸ 1.Assume that for some point x
0
2 U,the derivative
Df(x
0
):E!F is a toplinear isomorphism (the inverse is continuous also).Then
f is a local C
p
{isomorphism at x
0
.(i.e.there exists an open neighborhood V of x
0
such that fj
V
is a C
p
{isomorphism onto an open subset of F.)
Proof.Since a toplinear isomorphism is C
1
,we may assume without loss of gen
erality that E = F,and Df(x
0
) is the identity (Consider Df(x
0
)
¡1
± f instead of
f).Furthermore,we may assume that x
0
= 0 and f(x
0
) = 0.
Set g(x) = x ¡ f(x).Then Dg(x
0
) = 0,and by continuity there exists r > 0
such that,if kxk < 2r,we have kDg(x)k <
1
2
.Then kg(x)k ·
1
2
by the mean value
theorem.Thus g(
B
r
(0)) ½
B
r
2
(0).
Claim:For y 2
B
1
2
(0) there exists x 2
B
r
(0) such that f(x) = y.
Proof:Consider the map
g
y
= y +x ¡f(x):
11
If kyk ·
r
2
and kxk · r,then kg
y
(x)k · r,and hence g
y
may be viewed as a
mapping of the complete metric space
B
r
(0) into itself.g
y
is a contracting map,
since
kg
y
(v) ¡g
y
(w)k = kg(v) ¡g(w)k ·
1
2
kv ¡wk
by the mean value theorem for v;w 2
B
r
(0).By the Shrinking Lemma we ¯nd that
g
y
has a unique ¯xed point x,which is the solution we were looking for.
By this claim we obtain a local inverse f
¡1
.It is continuous,since
kv ¡wk · kf(v) ¡f(w)k +kg(v) ¡g(w)k · 2kf(v) ¡f(w)k:
Furthermore,f
¡1
is di®erentiable in B
r
2
(0) by the following argument:Let v
1
;v
2
2
B
r
(0),w
1
;w
2
2
B
r
2
(0),with f(v
i
) = w
i
.Then
kf
¡1
(w
1
) ¡f
¡1
(w
2
) ¡Df(v
2
)
¡1
(w
1
¡w
2
)k =
= kv
1
¡v
2
¡Df(v
2
)
¡1
(f(v
1
) ¡f(v
2
))k ·
· kDf(v
2
)
¡1
k kDf(v
2
)(v
1
¡v
2
) ¡f(v
1
) +f(v
2
)k =
= o(jv
1
¡v
2
j) = o(jw
1
¡w
2
j);
where the ¯rst o{term is correct,since f is di®erentiable,and the last equality
follows from the already proved continuity of f
¡1
.So the di®erentiability of f
¡1
is proved,and its derivative is D(f
¡1
)(y) = Df(f
¡1
(y))
¡1
for y 2 B
r
2
(0).Since
f
¡1
;Df are continuous and taking inverses is C
1
,D(f
¡1
) is continuous,so f
¡1
is
C
1
.By induction,it follows that f
¡1
is C
p
if f is.¤
3.7.The Implicit Mapping Theorem.Let E;F;G be Banach spaces,U
½
open
E,
V
½
open
F,and let f:U £V!G be a C
p
mapping.Let (u
0
;v
0
) 2 U £V,and assume
that
@
2
f(u
0
;v
0
):F!G
is a toplinear isomorphism.Let f(u
0
;v
0
) = C.Then there exists a continuous
map g:U
0
!V de¯ned on an open neighborhood U
0
of u
0
such that g(u
0
) = v
0
,
and such that
f(x;g(x)) = C
for all x 2 U
0
.If U
0
is taken su±ciently small then g is uniquely determined,and
is also of class C
p
.
Proof.Without loss of generality we may assume that @
2
f(u
0
;v
0
) is the identity
(simply replace f by @
2
f(u
0
;v
0
)
¡1
± f).Consider the map':U £ V!E £ F
'(x;y) = (x;f(x;y)).Then we compute
D'(u
0
;v
0
) =
µ
Id
E
0
@
1
f(u
0
;v
0
) @
2
f(u
0
;v
0
)
¶
=
µ
Id
E
0
@
1
f(u
0
;v
0
) Id
F
¶
;
which obviously is invertible.Hence,'is locally invertible by 3.6.Since we have
'
¡1
(x;z) = (x;h(x;z)) for some mapping h of class C
p
.We set g(x) = h(x;C).
Then g is also of class C
p
and
(x;f(x;g(x))) ='(x;g(x)) ='(x;h(x;C)) ='('
¡1
(x;C)) = (x;C):
12This proves the existence of g.For the uniqueness we suppose that g
0
is a continuous
map de¯ned near u
0
such that g
0
(u
0
) = v
0
and f(x;g
0
(x)) = C for all x near
u
0
.Then g
0
(x) is near v
0
for such x,and hence'(x;g
0
(x)) = (x;C).Since'is
invertible near (u
0
;v
0
),it follows that there is a unique point (x;y) near (u
0
;v
0
)
such that'(x;y) = (x;C).Let U
0
be a small ball on which g is de¯ned.If g
0
is
also de¯ned on U
0
,then the argument above shows that g and g
0
coincide on some
smaller neighborhood of u
0
.Let x 2 U
0
and let u
0
= x ¡ u
0
.Consider the set
ft 2 [0;1]jg(u
0
+tu
0
) = g
0
(u
0
+tu
0
)g.This set is nonempty,so let T be its least
upper bound.By continuity we get g(u
0
+Tu
0
) = g
0
(u
0
+Tu
0
).If T < 1 we can
apply the existence and uniqueness part we have already proved to show that g and
g
0
are equal in a neighborhood of u
0
+Tu
0
.Therefore,T = 1,and the uniqueness
is proved,as well as the theorem.¤
3.8.Banach Manifolds.Let M be a topological Hausdor® space.A chart on M
is a triple (U;u;E),where U
½
open
M and u:U!V
½
open
E is a homeomorphism onto
an open set in a Banach space E.
An atlas of class C
p
(0 · p · 1) is a set A = f(U
i
;u
i
;E
i
)ji 2 Ig of charts
satisfying the following conditions:
(1) fU
i
ji 2 Ig is a covering of M
(2) For each pair of indices (i;j) 2 I £I the map
u
j
± u
¡1
i
:u
i
(U
i
\U
j
)!u
j
(U
i
\U
j
)
is a C
p
di®eomorphism.
If p ¸ 1 we see by (2) that if U
i
\U
j
is nonempty E
i
'E
j
.Therefore,on each
connected component of M we can assume that all E
i
are equal,say E.In the
following we will assume that all E
i
are isomorphic,which is not a big assumption,
since we could prove all theorems for each connected component separately.
We say that two atlases A
1
and A
2
are C
p
equivalent if the atlas A
1
[ A
2
is a
C
p
atlas.An equivalence class of C
p
atlases is said to de¯ne the structure of a C
p
Banach manifold on M,and if all E
i
are isomorphic to the space E we say M is
modeled on E.If the modeling space E is a Hilbert space,we call M a Hilbert
manifold.
Let M and N be two Banach manifolds,and let f:M!N be a continuous
map.We shall say f 2 C
p
(M;N) if for all x 2 M,there exist charts (U;u) of M
with x 2 U and (V;v) of N with f(x) 2 N such that v ± f ± u
¡1
:u(U)!v(V )
is C
p
as a mapping of Banach spaces.A bijective mapping is said to be a C
p
di®eomorphism if f and f
¡1
are both C
p
.
Manifolds,mappings,:::of class C
1
will also be called smooth in the sequel.
3.9.Submanifolds,Immersions,Submersions.Let M be a C
p
Banach mani
fold.A subset N ½ M is called submanifold of M if at each point x 2 N there exists
a chart (U;u) of M such that there are two Banach spaces E
1
;E
2
with E
1
£E
2
'E,
and u(U) = V
1
£V
2
with V
i
½
open
E
i
and u(N\U) = V
1
£f0g.
Then the collection of pairs (U\N;pr
1
±(uj
U\N
)) is an atlas of class C
p
for N.
This structure satis¯es a universal mapping property,which characterizes it:
Given any map f:Z!M from a manifold Z into M such that f(Z) is contained
in N.Let f
N
:Z!N be the induced map.Then f is C
p
if and only if f
N
is C
p
.
13
A submanifold N is always locally closed in M (i.e.every point x 2 N has an
open neighborhood U in M such that N\U is closed in U).We say that N is a
closed submanifold of M if N is a closed topological subspace of M.
Let f:Z!M be a C
p
map,and let z 2 Z.We say that f is an immersion at z
if there exists an open neighborhood V of z such that fj
U
induces an isomorphism
of V onto a submanifold of M.We call f an immersion if it is an immersion at
every point z 2 Z.An immersion f is called a (closed) embedding if it gives an
isomorphism onto a (closed) submanifold of M.
A mapping f:M!Z is called a submersion at a point x 2 M if there exists
a chart (U;u) at x and a chart (V;v) at f(x) such that u gives an isomorphism of
U on a product U
1
£U
2
(U
1
;U
2
open in some Banach spaces),and such that the
map vfu
¡1
:U
1
£U
2
!V is a projection.We say that f is a submersion if it is a
submersion at every point.Submersions are open mappings.
Acriterion for immersions and submersions will be given in 3.10 using the tangent
space:On every point x 2 M we consider triples (U;u;X) where (U;u) is a chart at x
and X 2 E.We call two such triples (U;u;X),(V;v;Y ) equivalent if and only
if D(vu
¡1
)(u(x)):X = Y.An equivalence class of such triples is called a tangent
vector of M at x.The set T
x
M of such tangent vectors is called the tangent space
of M at x.Each chart (U;u) determines a bijection of T
x
M to the modeling space
E of M by which T
x
M gets the structure of a Banach space.Using the tangent
spaces,we can interpret the derivative of a C
p
mapping f:M!N by means of
charts as a continuous linear mapping T
x
f:T
x
M!T
f(x)
N (essentially as in the
¯nite dimensional case).
Proposition.Let M and N be Banach manifolds of class C
p
(p ¸ 1),and let
f:M!N be a C
p
mapping.Take x 2 M.Then
(1) f is an immersion at x if and only if the map T
x
f is injective and splits
(i.e.it exists a toplinear isomorphism ®:T
f(x)
N!F
1
£ F
2
such that
® ± T
x
f induces a toplinear isomorphism of T
x
M onto F
1
£f0g.)
(2) f is a submersion at x if and only if the map T
x
f is surjective and its kernel
splits (i.e.is closed and has closed complement).
Proof.This is an immediate consequence of the inverse mapping theorem.¤
3.10.Partitions of unity.Unlike for ¯nite dimensional manifolds the existence
of partitions of unity is even in the Banach case not always satis¯ed.The problem
for constructing di®erentiable partitions of unity is the existence of a di®erentiable
norm.However,if we put strong restrictions on the topology of M,we will get the
existence:Theorem.Let M be a Banach manifold modeled on E which is locally compact,
and whose topology has a countable base.Then M admits partitions of unity:i.e.
for every open covering fV
i
g there exists a subordinate open covering fU
i
g of M
and a family of functions f
i
:M!R satisfying the following conditions
(1) For all x 2 M we have f
i
(x) ¸ 0.
(2) The support of f
i
is contained in U
i
.
(3) The covering is locally ¯nite (i.e.every x 2 M has a neighborhood which
intersects only ¯nitely many U
i
).
14
(4) For each x 2 M we have
P
i
f
i
(x) = 1.
Proof.The proof will be left out,but can be found in [Lang 1995,II x3].¤
However,since the only locally compact Banach spaces are ¯nite dimensional we
have not won too much.The proof can be carried over to the situation where M
is paracompact,and there is a di®erentiable norm on E which is equivalent to the
original one,but it is di±cult to check the existence of such a norm.However,in
one case we can always construct di®erentiable partitions of unity:
3.11.Theorem.Let M be a paracompact manifold of class C
p
modeled on a
separable Hilbert space H.Then M admits partitions of unity of class C
p
.
Proof.For this proof we will need a few de¯nitions and some lemmas.
De¯nition.A subset V of a metric space (X;d) is called scalloped,if there exist
open balls B
r
i
(x
i
) in X such that
V = B
r
0
(x
0
)\C(
B
r
1
)\¢ ¢ ¢\C(
B
r
n
):
where C(A) shall denote the set theoretical complement of A.
Lemma 1.Let (X;d) be a metric space and fB
r
i
(x
i
)g (i 2 N) a countable covering
of a subset W by open balls.Then there exists a locally ¯nite open covering fV
i
g
(i 2 N) of W such that V
i
½ B
r
i
(x
i
) for all i,and such that V
i
is scalloped for all
i.
Proof.De¯ne V
i
inductively by the following construction.Let V
1
:= B
r
1
(x
1
).
Then set r
0
ji
:= r
j
¡
1
i
,and let
V
i
:= B
r
i
(x
i
)\
i¡1
\
j=1
C(
B
r
0
ji
(x
j
));
replacing all balls of negative radius by the empty set.By construction,each V
i
is
scalloped and is contained in B
r
i
(x
i
).Take x 2 W,and let k be the smallest index
such that x 2 B
r
k
(x
k
).Then x 2 V
k
,because otherwise,x would be in C(V
k
).But
C(V
k
) = C(B
r
k
) [
k¡1
[
j=1
B
r
0
kj
(x
j
);
and thus x lies in some B
r
j
(x
j
) with j · k ¡1 which is a contradiction.
For proving the locally ¯niteness,take again x 2 W.Then x 2 B
r
k
(x
k
) for some
k.Let"> 0 be so small that B
"
(x) ½ B
r
k
(x
k
).For all su±ciently large i the
ball B
"
2
(x) ½
B
r
0
ki
(x
k
),and therefore by construction B
"
2
(x)\V
i
=;.Thus B
"
2
(x)
meets only ¯nitely many V
i
.¤
15
Lemma 2.Let U be an open ball in the Hilbert space H,and let V be a scalloped
open subset.Then there exists a C
1
function Ã:H!R such that Ãj
C(V )
´ 0 and
Ãj
V
> 0.
Proof.Since V is scalloped,we have V = B
r
0
(x
0
)\
T
ni=1
C(
B
r
i
(x
i
)).For i =
1;:::;n choose a function'
i
:H!R such that
0 <'
i
(x) · 1 if x 2 C(
B
r
i
(x
i
))
'
i
(x) = 0 if x 2
B
r
i
(x
i
):
Let'
0
:H!R be a function such that'(x) > 0 on U and'(x) = 0 outside U.
Set
Ã(x):=
n
Y
i=0
'
i
(x):
Then Ã satis¯es the requirement.¤
Proposition.Let A
1
,A
2
be two nonempty,closed,disjoint subsets of a separable
Hilbert space H.Then there exists a smooth function Ã:H!R such that Ãj
A
1
´ 0
and Ãj
A
2
´ 1,and 0 · Ã(x) · 1 for all x (i.e.H is smoothly normal).
Proof.By the theorem of LindelÄof,we can ¯nd countably many open balls B
r
i
(x
i
)
(i 2 N) covering A
2
such that each B
r
i
(x
i
) is contained in C(A
1
).Let W =
S
i2N
B
r
i
(x
i
).By Lemma 1 we can ¯nd a locally ¯nite re¯nement fV
i
g of scalloped
open sets.By Lemma 2,we ¯nd functions'
i
positive on V
i
and zero outside of V
i
.
Let'=
P
i2N
'
i
(the sum is ¯nite at each point of W,since the V
i
are a locally
¯nite covering).Then'is positive on A
2
,and'j
A
1
´ 0.
Let U be the open neighborhood of A
2
on which'> 0.Then A
2
and C(U) are
disjoint closed sets.We then apply the construction above to get another function
¾:H!R which is positive on C(U) and is identically zero on A
2
.By setting
Ã:=
'
'+¾
we get the required function.¤
Proof of Theorem 3.11.Let B
r
(x) be an open ball in H.By
y 7!
y
p
r
2
¡hy;yi
B
r
(x) is di®eomorphic to H.Take any point x 2 M,and a neighborhood V of x.We
can ¯nd a chart (U;u) of M at x such that u(U) = H,and U 2 V.Given an open
covering of M we,therefore,can ¯nd an atlas f(U
®
;u
®
)g such that u
®
(U
®
) = H for
all ®,and the U
®
are subordinate to the given covering.By paracompactness,we
can ¯nd a re¯nement f
~
U
i
g of the covering fU
®
g which is locally ¯nite.Each
~
U
i
is
contained in some U
®(i)
.Let ~u
i
= u
i
j
~
U
i
.Again by paracompactness we ¯nd open
re¯nements fV
i
g and fW
i
g such that
W
i
½ V
i
½
V
i
½
~
U
i
:
16By construction ~u
i
(
W
i
) and ~u
i
(
V
i
) are closed in H.By the proposition,we can
¯nd functions'
i
on H with'
i
j
~u
i
(
W
i
)
= 1 and'
i
j
H¡~u
i
(V
i
)
= 0,being between 0
and 1,otherwise.Set Ã
i
='
i
± u
i
.Then Ã
i
is 0 on M ¡ V
i
and 1 on
W
i
.Set
Ã =
P
i
Ã
i
,and f
i
=
Ã
i
Ã
.Then the ff
i
g are the desired partition of unity.¤
Since partitions of unity are the only known means of gluing together local
mappings,this theorem gives a hint on the importance of Hilbert manifolds (i.e.
manifolds modeled on Hilbert spaces).A very important class of Hilbert spaces
will be considered in paragraph 3.21.
3.12.Vector bundles.The partitions of unity discussed above are an essential
tool when considering vector bundles.
Let M be a C
p
Banach manifold modeled on a Banach space B,let E be another
Banach manifold,and ¼:E!M be a C
p
map.Let F be a Banach space.Let
fU
i
g be an open covering of M,and for each i suppose that we have a mapping
¿
i
:¼
¡1
(U
i
)!U
i
£F satisfying the following conditions:
(1) The map ¿
i
is a C
p
di®eomorphism such that the following diagram com
mutes:
¼
¡1
(U
i
)
¿
i
¼
U
i
£F
pr
1
U
i
In particular,we obtain an isomorphism on each ¯ber
¿
ix
:¼
¡1
(x)!F:
(2) For each pair of open sets U
i
,U
j
the map
¿
jx
¿
¡1
ix
:F!F
is a toplinear isomorphism.
(3) If U
i
and U
j
are two members of the covering,then the map of U
i
\U
j
into
L(F;F) given by
x 7!(¿
j
¿
¡1
i
)
x
is a C
p
mapping.
Then we call (U
i
;¿
i
) a trivializing covering for ¼ or E,and that f¿
i
g are its trivial
izing maps.If x 2 U
i
,we say that (U
i
;¿
i
) trivializes at x.Two trivializing coverings
are called to be equivalent if their union is a trivializing covering.An equivalence
class of such trivializing coverings is said to give the quadruple (E;¼;M;F) the
structure of a vector bundle.M is called the base (space),E the total space,¼ the
bundle (or footpoint) projection.The Banach space F is called the standard ¯ber.
The space ¼
¡1
(x) is called the ¯ber over x.
Note the di®erence to the ¯nite dimensional case:(3) is implied by (2) there.In
the in¯nite dimensional case it has to be stated explicitly.
The maps ¿
ijx
= ¿
ix
± ¿
¡1
jx
,are called the transition functions associated with
the covering.They satisfy the so called cocycle condition
¿
kjx
± ¿
jix
= ¿
kix
;(i.p.¿
ijx
= ¿
¡1
jix
):
17
As in the ¯nite dimensional case,the cocycle of transition functions characterizes
the vector bundle.
A vector bundle (E;¼;M;F) is called trivializable if it is isomorphic to (M £
F;pr
1
;M;F).
Let (E;¼;M;F) and (E
0
;¼
0
;M
0
;F
0
) be two vector bundles.A C
p
vector bundle
morphism f between these bundles consists of a pair of C
p
mappings f
0
:M!M
0
and f:E!E
0
satisfying the following conditions:
(1) The diagram
E
f
¡¡¡¡!E
0
¼
??y
??y¼
0
M ¡¡¡¡!
f
0
M
0
is commutative for each x 2 M f
x
:E
x
!E
0
f(x)
,and the induced map is a
continuous linear map.
(2) For each x
0
2 M there exist trivializing maps
¿:¼
¡1
(U)!U £F
¿
0
:¼
0
¡1
(U
0
)!U
0
£F
0
at x
0
and f(x
0
) respectively,such that f
0
(U) is contained in U
0
,and such
that the map
U!L(F;F
0
)
x 7!¿
0
f
0
(x)
± f
x
± ¿
¡1
x
is C
p
.
We will usually write f:E!E
0
to denote a vector bundle morphism.
Let (E;¼;M;F) be a vector bundle,and f:N!M a C
p
map.Then
(f
¤
(E);f
¤
(¼);N;F) is a vector bundle called the pull back of E along f,and the
pair (f;¼
¤
(f)) is a vector bundle morphism.
f
¤
(E)
¼
¤
(f)
¡¡¡¡!E
f
¤
(¼)
??y
??y
¼
N ¡¡¡¡!
f
M
An important vector bundle is the tangent bundle of a manifold.Let M be a
manifold of class C
p
with p ¸ 1.Let TM be the disjoint union of the vector spaces
T
x
M from 3.9.Let ¼:TM!M map T
x
M to x,and set F = B the modeling
space of M.Take an atlas (U
i
;u
i
) of M.From the de¯nition of tangent vectors as
triples (U
i
;u
i
;X
i
) we immediately get a bijection
¿
i
:¼
¡1
(U
i
) = TU
i
!U
i
£F
18which commutes with the projection on U
i
,that is such that
¼
¡1
(U
i
)
¿
i
¼
U
i
£F
pr
1
U
i
is commutative.Furthermore,if we set for any two charts (U
i
;u
i
) and (U
j
;u
j
)
u
ij
= u
j
u
¡1
i
,then we obtain transition mappings
¿
ji
= ¿
j
¿
¡1
i
:u
i
(U
i
\U
j
) £F!u
j
(U
i
\U
j
) £F
by the formula
¿
ji
(x;X) = (u
ji
(x);Du
ji
(x) ¢ X)
for x 2 U
i
\U
j
and X 2 F.Since the derivative Du
ji
is of class C
p¡1
and is an
isomorphism at x,we ¯nd all conditions for a vector bundle satis¯ed.Therefore,
TM is a vector bundle of class C
p¡1
.
Given a C
p
map f:M!N,we can de¯ne Tf:TM!TN to be simply
T
x
f on each ¯ber T
x
M.It is easy to check that Tf is a vector bundle morphism
TM!TN of class C
p¡1
called the tangent map of f.Locally,the map is given as
Tf(x;X) = (f(x);Df(x) ¢ X).
Another useful de¯nition follows:A mapping f:E!E
0
between vector bundles
(E;¼;M;F) and (E
0
;¼
0
;M;F
0
) is called ¯ber preserving,if f(¼
¡1
(x)) ½ ¼
0
¡1
(x)
for all x 2 M.
3.13.Sections of bundles,vector ¯elds.Let M be a C
p
manifold,and take a
C
q
vector bundle (M;E;¼;F) over M (q · p).A C
r
section of E (r · q) is a C
r
map »:M!E with ¼ ± » = Id
M
.The set of all such sections will be denoted by
C
r
(E).
If E = TM such a section » of class C
p¡1
will be called a (timeindependent)
vector ¯eld on M.The set of all vector ¯elds on M will be denoted by X(M).
Like in the ¯nite dimensional case,some constructions can be applied to vector
bundles,but more care has to be taken with the topology:V ©W,V W,V
¤
=
L(V;R),and ¤
n
V can e.g.be constructed.Sections of tensor products of TE and
TE
¤
are also called tensor ¯elds.
3.14.The existence theoremfor di®erential equations.There is an existence
theorem for the °ow of vector ¯elds similar to the ¯nite dimensional case.Since we
will only need the existence of local °ows,only that result will be mentioned.
Let f:J £U!E be a C
p
mapping (p ¸ 0),0 2 J an open interval in R and
U ½ E open (i.e.the local representation of a timedependent vector ¯eld).
For a point x
0
2 U,an integral curve for f with initial condition x
0
is a mapping
of class C
r
(r ¸ 1)
®:J
0
!U
®(0) = x
0
®
0
(t) = f(t;®(t));
19
where 0 2 J
0
is an open subinterval of J.
A local °ow for f at x
0
is a mapping
®:J
0
£U
0
!U
®
x
(t) = ®(t;x);
where 0 2 J
0
is an open subinterval of J,and x
0
2 U
0
½ U is an open subset,and
®
x
is an integral curve for f with initial condition x.
Having these de¯nitions in mind,we ¯nd the following results,similar to the
¯nite dimensional case.
Proposition.In the situation above,let 0 < a < 1 be a real number such that
B
3a
(x
0
) lies in U.Assume that f is a continuous map bounded by a constant
L ¸ 1 on J £U and satis¯es a Lipschitz condition on U,uniformly with respect to
J,with constant K ¸ 1.If b <
a
LK
,then for each x 2
B
a
(x
0
) there exists a unique
local °ow
®:(¡b;b) £B
a
(x
0
)!U:
If f is of class C
p
,so is each integral curve ®
x
.
The local °ow ® is continuous,and the map x 7!®
x
of
B
a
(x
0
) into the space of
curves satis¯es a Lipschitz condition.
If we take f C
p
with p ¸ 1 then we get stronger results.
Theorem.Let f be a (local) vector ¯eld on U of class C
p
(p ¸ 1),and let x
0
2 U.
Then there exists a unique local °ow for f at x
0
.We can select a (maximal)
open subinterval J
0
of J containing 0 and an open subset U
0
of U containing x
0
,
such that the unique local °ow
®:J
0
£U
0
!U
is of class C
p
,and such that @
2
® satis¯es the di®erential equation
@
1
@
2
®(t;x) = @
2
f(t;®(t;x))@
2
®(t;x)
on J
0
£U
0
with initial condition @
2
®(0;x) = Id.Usually,® will then be denoted by
Fl
f
.
Proof.The proof can e.g.be found in [Lang 1995,IV.x1].It depends heavily on
the Shrinking Lemma,and on the Banach space norm.
3.15.Corollary.Let U,V be open sets in Banach spaces E,F respectively.Let
J be an open interval of R containing 0,and let g:J £U £V!F be a C
r
map
(r ¸ 1).Let (u
0
;v
0
) be a point in U £V.Then there exist open balls J
0
,U
0
,V
0
centered at 0,u
0
,v
0
respectively,and a unique map of class C
r
h:J
0
£U
0
£V
0
!V
such that h(0;u;v) = v and
@
1
h(t;u;v) = g(t;u;h(t;u;v))
for all (t;u;v) 2 J
0
£U
0
£V
0
.
Proof.This follows fromthe existence and uniqueness of the local °ow of the vector
¯eld on U £V G:J £U £V!E £F given by G(t;u;v) = (0;g(t;u;v)).Then
h(t;u;v) = pr
2
±Fl
G
(t;u;v).¤
203.16.Corollary.The function h from Corollary 3.15 satis¯es the equation
@
1
@
2
h(t;u;v) ¢ x = @
2
g(t;u;h(t;u;v)) ¢ x +@
3
g(t;u;h(t;u;v)) ¢ @
2
h(t;u;v) ¢ x
for all x 2 E.
Proof.This is just calculation,using the result above.¤
3.17.Corollary.Let J be an open interval of R containing 0 and take U
½
open
E.
Let f:J £ U!E be a continuous map,which is Lipschitz on U uniformly for
every compact subinterval of J.Let t
0
2 J and let'
1
,'
2
be two C
1
maps such
that'
1
(t
0
) ='
2
(t
0
) and satisfying the relation
'
0i
(t) = f(t;'
i
(t))
for all t 2 J.Then'
1
(t) ='
2
(t).
Proof.This follows directly fromthe existence and uniqueness result for di®erential
equations.¤
3.18.Integrable subbundles.Let V be a tangent subbundle over M.We say
V is integrable at a point x
0
if there exists a submanifold N of M containing x
0
such that the tangent map of the inclusion i:N!M induces a vector bundle
isomorphism of TN with the subbundle V restricted to N.Equivalent is,for each
y 2 N the tangent map T
y
j:T
y
N!T
y
M induces a toplinear isomorphism of
T
y
N onto V
y
.
We say that V is integrable if it is integrable at every point.
3.19.Frobenius'Theorem.Let M be a Banach manifold of class C
p
for p ¸ 2
and let S be a subbundle of TM.Then S is integrable if and only if S is involutive
(i.e.for each point z 2 M and vector ¯elds X,Y de¯ned on an open neighborhood
of z which lie in V,the bracket [X;Y ] also lies in S).
Proof.The part integrable =)involutive follows just by the functoriality of vector
¯elds and their relations under tangent maps.The converse is the di±cult direction.
The proof will be carried out locally.We ¯rst try to ¯nd a suitable description
of the bundle S in local terms.
Take z 2 M.We can then ¯nd a product decomposition of an open neighborhood
W of z,say U £ V,open subsets of Banach spaces E and F,respectively,such
that the point has coordinates (u
0
;v
0
) and such that Sj
W
can be written as the
image of an injective vector bundle map f:U £V £E,!U £V £E £F with
f(u
0
;v
0
):E,!E £F is the canonical embedding E,!E £f0g.Without loss of
generality we may assume that pr
1
±f(u;v) = Id
E
for all (u;v) 2 U £V.Thus we
may describe f by a C
p¡1
mapping (also called) f:U £V!L(E;F).
Further note that a subbundle S of TM is integrable at a point z 2 M if and only
if there exists an open neighborhood W of z and a di®eomorphism':U £V!W
of a product of open subsets of Banach spaces onto W such that the composition
T
1
(U£V ),!T(U£V )
T'
!TW is a bundle isomorphismonto Sj
W
,where T
1
(U£V )
is the subbundle of T(U£V ) whose ¯bers are T
x
U£0 µ T
x
U£T
y
V = T
(x;y)
(U£V ),
x 2 U,y 2 V.
21
Now take the local representations of a vector ¯eld X over W = U £V.Then
X 2 Sj
W
if and only if X
2
(u;v) = f(u;v) ¢ X
1
(u;v),where X
1
and X
2
are the
projections of X to E and F respectively.In other words,i® X is of the form
X(u;v) = (X
1
(u;v);f(u;v):X
2
(u;v)),for some C
p¡1
map X
1
:U £V!E.If X,
Y are vector ¯elds of this type,then [X;Y ] 2 Sj
W
if and only if
Df ¢ X ¢ Y
1
= Df ¢ Y ¢ X
1
;
which can be calculated from the local representation of [;].Having expressed
all data locally,the following result remains to be shown.
Proposition.Let U,V be open subsets of Banach spaces E,F respectively.Let
f:U £V!L(E;F) be a C
r
map (r ¸ 1).Assume that if X
1
;Y
1
:U £V!E
are two C
r
maps and that
Df ¢ (X
1
;f ¢ X
1
) ¢ Y
1
= Df ¢ (Y
1
;f ¢ Y
1
) ¢ X
1
:
Let (u
0
;v
0
) 2 U £V.Then there exist open neighborhoods U
0
½ U,V
0
½ V of u
0
,
v
0
respectively,and a unique C
r
map ®:U
0
£V
0
!V such that
@
1
®(u;v) = f(u;®(u;v));
and ®(u
0
;v) = v for all (u;v) 2 U
0
£V
0
.
Proof.By acting by a translation we can without loss of generality assume that
(u
0
;v
0
) = (0;0) 2 E £ F.Now set g(t;u;v) = f(tu;v) ¢ u.u 2 B
"
(0) ½ U a
small ball in E.Then by Corollary 3.15 we obtain h:J
0
£ E
0
£ V
0
!V with
initial condition h(0;u;v) = v for all u 2 E
0
,satisfying the di®erential equation
@
1
h(t;u;v) = f(tu;h(t;u;v)) ¢ u.Changing variables by t = at
0
and u = a
¡1
u
0
for
a small a > 0,we can assume that 1 2 J
0
,provided E
0
is small enough.
Set ®(u;v) = h(1;u;v).Then we have to calculate @
2
h(t;u;v).From Corollary
3.16 we obtain for any vector x 2 E,
@
1
@
2
h(t;u;v) ¢ x = t@
1
f(tu;h(t;u;v)) ¢ x ¢ u+
+@
2
f(tu;h(t;u;v)) ¢ @
2
h(t;u;v) ¢ x ¢ u +f(tu;h(t;u;v)) ¢ x:
Now let k(t) = @
2
h(t;u;v) ¢ x ¡tf(tu;h(t;u;v)) ¢ x.Then k(0) = 0,and using the
local version of the integrability for the ¯elds u and x,we get
Dk(t) = @
2
f(tu;h(t;u;v)) ¢ k(t) ¢ u:
By Corollary 3.17 we know that k(t) ´ 0 is the unique solution.Thus
@
1
h(t;u;v) = tf(tu;h(t;u;v));
and hence
@
1
®(u;v) = f(u;®(u;v)):
¤
22
Having this result,we can set':U
0
£ V
0
!U £ V as'(u;v) = (u;®(u;v)).
Then
D'(u
0
;v
0
) =
µ
Id 0
f(u
0
;v
0
) Id
¶
;
which,obviously,is a toplinear isomorphism.Thus by the inverse mapping theorem
3.6'is a local di®eomorphism at (u
0
;v
0
).Furthermore,for (x;y) 2 E£F we have
@
1
'(u;v) ¢ (x;y) = (x;@
1
®(u;v) ¢ x) = (x;f(u;®(u;v)) ¢ x);
which shows that the bundle is integrable.¤
3.20.Corollary.Let M be a Banach manifold,S an involutive subbundle of TX.
Then for any x 2 M,there is a neighborhood W of x and a di®eomorphism':
U £ V!W (U,V open neighborhood in Banach spaces) such that'(0;0) = x,
and the composition T
1
(U £V ),!T(U £V )
T'
!TW is a bundle isomorphism onto
Sj
W
,where T
1
(U £V ) is the subbundle of T(U £V ) whose ¯bers are T
x
U £0 µ
T
x
U £T
y
V = T
(x;y)
(U £V ),x 2 U,y 2 V.
Proof.This is just a reformulation of the local version of V being integrable.
3.21.Sobolev spaces.These important spaces have been developed in [Sobolev
1936].Lets start with the space L
2
(R
m
;C
n
) of all Lebesgue square integrable
functions.This space is,as is well known,a Hilbert space with the inner product
hf;gi
0
:=
Z
hf(x);g(x)i dx:
De¯nition.The space of rapidly decreasing functions S(R
m
;C
n
) is the vector
space of all C
1
{functions f:R
m
!C
n
,which satisfy that for every multiindex ®
and every p 2 N
0
exists a c
®p
¸ 0 such that for all x 2 R
n
kxk
p
kD
®
f(x)k · c
®p
:
Obviously,S(R
m
;C
n
) ½ L
2
(R
m
;R
n
).
For all f 2 S(R
m
;C
n
) we de¯ne the Fourier transformation F
0
as
(F
0
f)(x):= (2¼)
¡
m
2
Z
e
¡ihx;yi
f(y) dy:
Having this,we get the well known result
Proposition.F
0
S(R
m
;C
n
) ½ S(R
m
;C
n
),and for every f 2 S(R
m
;C
n
) and every
multiindex ®
D
®
F
0
f = (¡1)
j®j
F
0
M
®
f;M
®
F
0
f = F
0
D
®
f;
where (M
®
f)(x) = x
®
f(x) componentwise.
F
0
is a bijective linear mapping of S(R
m
;C
n
) onto itself and
(F
¡1
0
g)(x) = (2¼)
¡
m
2
Z
e
ihx;yi
g(y) dy:
Furthermore,(F
0
f)(x) = (F
¡1
0
f)(¡x) for all f 2 S(R
m
;C
n
) and F
4
0
= Id.
The following describes the extension of F
0
to L
2
(R
m
;C
n
),which is well known,
also.
23
Theorem.F
0
and F
¡1
0
preserve the L
2
{norm,and there exist unique extensions
F,
~
F of F
0
and F
¡1
0
,respectively,as bounded unitary operators on L
2
(R
m
;C
n
),
and
~
F = F
¤
= F
¡1
,F
4
= Id.The operator F is called Fourier transformation on
L
2
(R
m
;C
n
).
Furthermore,the following are equivalent
(1) Ff ¢ Fg 2 L
2
(R
m
;C
n
),
(2) F
¡1
f ¢ F
¡1
g 2 L
2
(R
m
;C
n
),
(3) f ¤ g 2 L
2
(R
m
;C
n
),
and in that case f ¤ g = F
¡1
(Ff ¢ Fg).
De¯nition.In the following set for x 2 R
m
k
s
(x):= (1 +kxk
2
)
s
2
and
L
2s
(R
m
;C
n
):= ff 2 L
2
(R
m
;C
n
)jk
s
f 2 L
2
(R
m
;C
n
)g:
L
2s
(R
m
;C
n
) is a dense subspace of L
2
(R
m
;C
n
).
hf;gi
(s)
:=
Z
hf(x);g(x)ik
s
(x)
2
dx
for f;g 2 L
2s
(R
m
;C
n
) de¯nes an inner product on L
2s
(R
m
;C
n
).This inner product
makes L
2s
(R
m
;C
n
) to a separable Hilbert space,isomorphic to L
2
(R
m
;C
n
) by U
s
:
f 7!k
s
f:L
2s
(R
m
;C
m
)!L
2
(R
m
;C
n
).
The Sobolev space of order s is de¯ned by
H
s
(R
m
;C
n
):= ff 2 L
2
(R
m
;C
n
)jFf 2 L
2s
(R
m
;C
n
)g = F
¡1
L
2s
(R
m
;C
n
):
H
s
(R
m
;C
n
) is a dense subspace of L
2
(R
m
;C
n
).It is a separable Hilbert space by
de¯ning for f;g 2 H
s
(R
m
;C
n
)
hf;gi
s
:= hFf;Fgi
(s)
kfk
s
:=
p
hf;fi
s
:
Obviously,H
0
(R
m
;C
n
) = L
2
(R
m
;C
n
).
The functions in H
s
(R
m
;C
n
) are in a weak sense di®erentiable:
Theorem.
(1) Let s ¸ 1,w
j
= (±
j1
;:::;±
jm
),(M
j
g)(x) = x
j
g(x) and f
j;"
(x) = f(x+"w
j
)
for j = 1;2;:::;m,and ±
ij
denotes the Kronecker ± symbol.Then for all
f 2 H
s
(R
m
;C
n
) and j = 1;2;:::;m
lim
"!0
1
i"
(f
j;"
¡f) = F
¡1
M
j
Ff:
in the L
2
{sense.Is f 2 S(R
m
;C
n
),then the limit coincides with D
®
f
with ® = (±
j1
;:::;±
jm
).We write D
®
f for this limit in any case,even if
f =2 S(R
m
;C
n
).
24
(2) If ® is a multiindex with j®j · s,then the derivative D
®
f can be calculated
iteratively.The order of di®erentiation can be exchanged.
(3) If s 2 N
0
,then kfk
s
is equivalent to the norms
kfk
s;0
=
s
X
j®j·s
kD
®
fk
2
kfk
s;1
=
s
kfk
22
+
X
j®j=s
kD
®
fk
2
:
(4) For all g 2 H
j®j
(R
m
;C
n
) is hD
®
f;gi
0
= hf;D
®
gi
0
.
(5) The function D
®
f 2 L
2
(R
m
;C
n
) is uniquely determined by
hD
®
f;gi
0
= hf;D
®
gi
0
for all g 2 C
1
0
(R
m
;C
n
),the space of all functions R
m
!C
n
of compact
support.
(6) For s ¸ 0 are C
1
0
(R
m
;C
n
) ½ S(R
m
;C
n
) and C
1
0
(R
m
;R
n
) ½ H
s
(R
m
;C
n
)
dense subspaces with respect to the norm k k
s
.
De¯nition.The spaces L
2
(R
m
;R
n
),L
2s
(R
m
;R
n
),and H
s
(R
m
;R
n
) shall be just
the subspaces of almost everywhere real valued functions of the spaces L
2
(R
m
;C
n
),
L
2s
(R
m
;C
n
),and H
s
(R
m
;C
n
),respectively.The results above are true for the
spaces of real valued functions,also.
The result,which is one of the reasons for the importance of Sobolev spaces is
the3.22.Sobolev Lemma.Let s > k +
m
2
,then the inclusions H
s
(R
m
;C
n
) ½
C
k
(R
m
;C
n
) and H
s
(R
m
;R
n
) ½ C
k
(R
m
;R
n
) are continuous linear maps.
Proof.in [Sobolev 1938].
Analogously,we de¯ne the spaces H
s
(U;R
n
) for open subsets U of R
m
.
Next we will prove some useful results,which we will need in the next sections.
3.23.Lemma.Let D
n
denote the open unit ball in R
n
,and let f:D
n
!D
n
and
g:D
n
!R
k
be H
s
{maps (s >
n
2
+1) such that Df has everywhere maximal rank.
Then g ± f 2 H
s
(D
n
;R
k
),and the map (f;g)!g ± f is jointly continuous near
(f;g),as a map ±:H
s
(D
n
;D
n
) £H
s
(D
n
;R
k
)!H
s
(D
n
;R
k
).
Proof.Recall that by the Sobolev Lemma 3.22 H
s
(D
n
;D
n
) ½ C
1
(D
n
;D
n
) is
continuous.By induction,we will prove that ±:H
s
(D
n
;D
n
) £ H
r
(D
n
;R
k
)!
H
r
(D
n
;R
k
) is continuous.
For k = 0 we check
R
D
n
kg ± fk
2
.But
R
D
n
kg ± fk
2
=
R
f(D
n
)
kgk
2
1=j det(J(f))j
with J(f) the Jacobian of f.Since D
n
is compact,J(f) is bounded away fromzero.
Therefore,±(H
s
(D
n
;D
n
) £H
0
(D
n
;R
k
)) ½ H
0
(D
n
;R
k
).Take"> 0,f
0
such that
R
D
n
kf ¡f
0
k <"=(4 max
x2D
n
kD
x
gk
2
) (this is possible since the L
1
norm is weaker
25
than the H
s
norm) and (
R
D
n
kg ¡g
0
k)
2
<"=(4 max
x2D
n
f1=j det(J(f
0
))(x)jg).Fur
ther choose ± such that max
x2D
n
f1=j det(J(f))(x)j;1=j det(J(f
0
))(x)jg <"=4±,and
pick g
1
2 C
1
(D
n
;R
k
) so that
R
D
n
kg
1
¡gk < ±.Then we compute
Z
D
n
kg ± f ¡g
0
± f
0
k
2
·
Z
D
n
kg ± f ¡g
1
± fk
2
+
Z
D
n
kg
1
± f ¡g
1
± f
0
k
2
+
+
Z
D
n
kg
1
± f
0
¡g ± f
0
k
2
+
Z
D
n
kg ± f
0
¡g
0
± f
0
k
2
·
Z
D
n
kg
1
¡gk
2
1
j det(J(f))j
+ max
x2D
n
kD
x
gk
2
Z
D
n
kf ¡f
0
k+
+
Z
D
n
kg
1
¡gk
2
1
j det(J(f
0
))j
+
Z
D
n
kg ¡g
0
k
1
j det(J(f
0
))j
<
"
4
+
"
4
+
"
4
+
"
4
<";
which proves the continuity of ±:H
s
(D
n
;D
n
) £H
0
(D
n
;R
k
)!H
0
(D
n
;R
k
).
For the inductive step we will need the following
3.24.Lemma.Let l >
n
2
,k · l.Let B be any bilinear map B:R
p
£ R
q
!
R
r
.Then
~
B:H
l
(B
n
;R
p
) £ H
k
(B
n
;R
q
)!H
k
(D
n
;R
r
) de¯ned by
~
B(f;g)(x) =
B(f(x);g(x)) is a continuous bilinear map.
Proof.In [Palais 1968,9.13] ¤
Next we assume the lemma for H
s
£H
r
!H
r
for r < s.We then prove it for
H
s
£H
r+1
!H
r+1
.Since g ± f 2 H
r+1
(D
n
;R
k
) if D(g ± f) 2 H
r
(D
n
;L(R
n
;R
k
))
we compute D(g ± f) = (Dg ± f) ¢ Df.But Dg 2 H
r
(D
n
;L(R
n
;R
k
)),thus
Dg ± f 2 H
r
(D
n
;L(R
n
;R
k
)) by the induction assumption.Furthermore,Df 2
H
s¡1
(D
n
;L(R
n
;R
n
)),and s¡1 ¸ r,so we ¯nd that by Lemma 3.24,(Dg±f)¢Df 2
H
r
(D
n
;L(R
n
;R
k
)).
Take (f
0
;g
0
) near (f;g) in H
s
(D
n
;D
n
) £H
r+1
(D
n
;R
k
).Then (f
0
;Dg
0
) is near
(f;Dg) in H
s
(D
n
;D
n
) £ H
r
(D
n
;L(R
n
;R
k
)),so Dg ± f is near Dg
0
± f
0
by the
induction assumption.Also Df
0
is near Df in H
s¡1
(D
n
;L(R
n
;R
n
)).Since we
have r · s¡1,s¡1 >
n
2
,(Dg
0
±f
0
)¢ Df
0
is near (Dg±f)¢ Df in H
r
(D
n
;L(R
n
;R
k
))
by Lemma 3.24.¤
3.25.Sobolev completions of spaces of vector bundle sections.To over
come certain di±culties which arise while working with manifolds modeled on
Fr¶echet spaces (or on general convenient spaces (see section 4)),we construct Hilbert
manifold completions of these spaces in the following way.
For any vector bundle V over M construct the sth jet bundle J
s
(V ),and endow
J
s
(V ) with an inner product h;i
s
.Taking any volume form dvol on M,we get
an inner product (;)
s
on C
1
(J
s
(V )),the space of smooth J
s
(V ){sections by
(a;b)
s
:=
Z
M
ha;bi
s
dvol:
Since there exists the natural map j
s
:C
1
(V )!C
1
(J
s
(V )),(;)
s
de¯nes an
inner product on C
1
(V ),also.
26
De¯ne H
s
(V ) as the Hilbert space completion of C
1
(V ) with respect to (;)
s
.
Taking other choices for dvol and h;i
s
changes the inner product,but any two
such constructed inner products are equivalent.
3.26.Lemma (Sobolev Lemma).Let s > k +
1
2
dim(M),then the inclusion
H
s
(V ) ½ C
k
(V ) is a continuous linear map.
Proof.See [Palais 1965],but follows essentially from the Sobolev Lemma 3.22.
3.27.Theorem.If V and W are vector bundles over M and f:V!W is
a smooth ¯ber preserving map,then for s >
1
2
dim(M),the map
~
f:H
s
(M;V )!
H
s
(M;W) de¯ned by
~
f(®) = f ±® is smooth,and its derivatives satisfy the formula
D
k
®
~
f(x
1
;:::;x
k
)(p) = D
k
®(p)
f(x
1
(p);:::;x
k
(p)),with p 2 M.
Proof.In [Palais 1968,11.3].¤
3.28.Di®erential operators.Let V and W be vector bundles over M,and let
D:C
1
(V )!C
1
(W) be a kth order di®erential operator.Let V and W have
smooth inner products h;i
V
and h;i
W
,respectively,and let vol be a smooth
volume form on M so that H
0
(V ) and H
0
(W) have explicit inner products.
A kth order di®erential operator D
¤
:C
1
(W)!C
1
(V ) is called an adjoint
of D if for all v 2 C
1
(V ),w 2 C
1
(W),
R
M
hDv;wi
W
dvol =
R
M
hv;D
¤
wi
V
dvol.
Every operator D has a unique adjoint.
For any x 2 M,and » 2 T
¤
x
,the symbol of D at »,¾
»
(D) is a linear map
V
x
!W
x
.It can be shown that
¾
»
(D
1
± D
2
) = ¾
»
(D
1
) ± ¾
»
(D
2
);and ¾
»
(D
¤
) = ¾
»
(D)
¤
;
where ¾
»
(D)
¤
:W
x
!V
x
is the adjoint of ¾
»
(D) (with respect to the given inner
products on V
x
and W
x
).
We say that D has injective symbol if ¾
»
(D) is an injective map for all » 6= 0,
and we call D elliptic,if ¾
»
(D) is an isomorphism for » 6= 0.
The equations above show that D
¤
± D is elliptic if and only if D has injective
symbol.
Any kth order operator D extends uniquely from a map C
1
(V )!C
1
(W) to
a continuous linear map D
s
:H
s
(V )!H
s¡k
(W).
3.29.Proposition.If D is a kth order elliptic operator from V to W.Then
(1) ker D = ker D
s
is a ¯nite dimensional subspace of C
1
(V ),and similarly
ker D
¤
= ker D
¤
s
is a ¯nite dimensional subspace of C
1
(W).
(2) H
s¡k
(W) = imD
s
©ker D
¤
,in particular imD
s
is closed in H
s¡k
(W).
Proof.The proof can be found in [Palais 1965,pp.178{179].¤
3.30.Proposition.If D has injective symbol then
(1) H
s¡k
(V ) = im(D
¤
± D)
s+k
+ker D
¤
± D.
(2) ker D
¤
± D = ker D and im(D
¤
± D)
s+k
= imD
¤
s
.
(3) imD
s+k
is closed in H
s
(W),and H
s
(W) = imD
s+k
©ker D
¤
s
.
27
Proof.(1) follows from 3.29(2) and the fact that D
¤
± D is selfadjoint.
(2):Let h;i
V;0
be the inner product on H
0
(V ) and h;i
W;0
be the one on
H
0
(W).Of course,ker(D
¤
± D) ¾ ker D.Also if D
¤
Dv = 0,0 = hD
¤
Dv;vi
V;0
=
hDv;Dvi
W;0
,so Dv = 0.Therefore ker D
¤
± D = ker D.Furthermore,im(D
¤
±
D)
s+k
½ imD
¤
s
,since (D
¤
± D)
s+k
= D
¤
s
± D
s+k
.By (1) it has only to be shown
that imD
¤
s
\ker D
¤
±D = f0g or that imD
¤
s
\ker D = f0g.If Dv = 0 and v = D
¤
w
then D±D
¤
w = 0,so 0 = hD±D
¤
w;wi
W;0
= hD
¤
w;D
¤
wi
V;0
,and so v = D
¤
w = 0,
which in turn implies (2).
(3):Since ker D
¤
s
is closed by Lemma 3.3 we need only show that H
s
(W) =
imD
s+k
©ker D
¤
s
is true in the algebraic sense.
Let w 2 H
s
(W) such that D
¤
s
w = 0 and w = D
s+k
v.Then 0 = hD
¤
s
±
D
s+k
v;vi
E;0
= hD
s+k
v;D
s+k
i
F;0
,so w = 0.Therefore,imD
s+k
\ker D
¤
s
= f0g.
D
¤
s
(H
s
(W)) = D
¤
s
± D
s+k
(H
s+k
(V )) by (2),H
s
(W) = D
¤
s
¡1
(D
¤
s
(H
s
(W))),and
thus
H
s
(W) = D
¤
s
¡1
(D
¤
s
± D
s+k
(H
s+k
(V ))) = ker D
¤
s
+imD
s+k
:
This implies (3).¤
28
29
4.Smooth infinite dimensional manifolds
Throughout this and the following chapters,I will use the term smooth manifold
in the sense of [Kriegl,Michor 1997].I will use the notion of FrÄolicher{Kriegl
calculus and of convenient vector spaces.
I will not de¯ne more than the most basic facts of the FrÄolicher{Kriegl calculus,
since that would exceed the goal of this thesis.However,the most important results
can be found in [FrÄolicher,Kriegl 1988] or [Kriegl,Michor 1997].Most of the results,
which are presented in this chapter are taken from[Kol¶a·r et al.1993],[Michor 1988],
[Michor 1991],and [Kriegl,Michor 1997].
First we will de¯ne a suitable generalization of Fr¶echet spaces which will ¯t
extraodinary well in a category theoretical way to in¯nite dimensional calculus.
These spaces will later serve for the local models of manifolds.
4.1.De¯nition.Let E be a locally convex vector space.A curve c:R!E will
be called di®erentiable if the derivative c
0
(t) = lim
h!0
1
h
(c(t +h) ¡c(t)) at t exists
for all t.A curve c is called smooth (or C
1
) if all iterated derivatives exist.The
set of all smooth curves will be denoted by C
1
(R;E).It will be equipped with
the bornologi¯cation of the topology of uniform convergence on compact sets,in all
derivatives separately.(Note:Smoothness is not primarily a topological concept,
but rather a concept of bounded sets,hence a bornological concept.Therefore,the
bornologi¯cation.)
A sequence fx
n
g in E is called Mackey{convergent to x if there exists a positive
sequence f¹
n
g in R with ¹
n
!0 and
1
¹
n
(x
n
¡x) is bounded.
The c
1
{topology on a locally convex vector space is the ¯nal topology with re
spect to all smooth curves R!E.(Note:The c
1
{topology is in general not a
vector space topology on E.However,the ¯nest locally convex topology coarser
than the c
1
{topology is the bornologi¯cation of the original locally convex topol
ogy.)
A locally convex vector space E is called convenient (or c
1
{complete) if one
of the following equivalent conditions is satis¯ed.(There are more equivalent con
ditions than these,which can be found together with the proof of equivalence in
[Kriegl,Michor 1997,Theorem 1.22].)
(1) Any MackeyCauchy sequence converges (i.e.E is Mackey{complete).
(2) E is c
1
{closed in any locally convex space.
(3) For any smooth curve c
1
2 C
1
(R;E) there exists a curve c
2
2 C
1
(R;E)
with c
02
= c
1
(i.e.the existence of an antiderivative).
(4) If c:R!E is a curve such that`± c:R!R is smooth for all`2 E
¤
,
then c is smooth.
4.2.Theorem.The following constructions preserve c
1
{completeness:
(1) limits,
(2) direct sums,
(3) strict inductive limits of sequences of closed embeddings,
(4) formation of`
1
(X;),where X is a set together with a family Bof subsets
of X containing the ¯nite ones,which are called bounded,and`
1
(X;F)
30
denotes the space of all functions f:X!F bounded on all B 2 B,
supplied with the topology of uniform convergence on the sets in B.
Proof.See [Kriegl,Michor 1997,Theorem 1.23].
4.3.De¯nition.A mapping f:E ¶ U!F between convenient spaces,de¯ned
on a c
1
{open subset U of E is called smooth (C
1
) if it maps smooth curves in U
to smooth curves in F.
By C
1
(U;F) we will denote the space of all smooth maps U!F.This space
is locally convex,with pointwise linear structure and the bornologi¯cation of the
initial topology with respect to all mappings c
¤
:C
1
(U;F)!C
1
(R;F) for c 2
C
1
(R;U).Then C
1
(U;F) is a convenient vector space.
4.4.Proposition.A (multi)linear map f:E
1
£¢ ¢ ¢ £E
n
!F,where E
i
,F are
convenient spaces,is smooth if and only if f is bounded.
We equip the space L(E
1
;:::;E
n
;F) of all such maps with the topology of uni
form convergence on bounded sets.Then L(E
1
;:::;E
n
;F) is a closed linear sub
space of C
1
(E
1
£¢ ¢ ¢ £E
n
;F),hence convenient.
There are natural bornological isomorphisms
L(E
1
;:::;E
n
;F)'L(E
1
;:::;E
l
;L(E
l+1
;:::;E
n
;F)):
This is called the exponential law for the linear maps.
Proof.See [Kriegl,Michor 1997,Propostion 3.2]
4.5.Theorem(Cartesian closedness).The category of convenient vector spaces
and smooth mappings is cartesian closed.So there is natural bijection
C
1
(E £F;G)'C
1
(E;C
1
(F;G)):
Furthermore,the following canonical mappings are smooth.
ev:C
1
(E;F) £E!F;ev(f;x) = f(x)
ins:E!C
1
(F;E £F);ins(x)(y) = (x;y)
( )
^
:C
1
(E;C
1
(F;G))!C
1
(E £F;G)
( )
_
:C
1
(E £F;G)!C
1
(E;C
1
(F;G))
comp:C
1
(F;G) £C
1
(E;F)!C
1
(E;G);comp(f;g)(x) = f(g(x))
C
1
(;):C
1
(F;F
0
) £C
1
(E
0
;E)!C
1
(C
1
(E;F);C
1
(E
0
;F
0
))
(f;g) 7!(h 7!f ± h ± g)
Y
:
Y
C
1
(E
i
;F
i
)!C
1
(
Y
E
i
;
Y
F
i
)
Proof.See [Kriegl,Michor 1997,1.36]
The following lemma provides strong means for proving many results in the
FrÄolicher{Kriegl calculus.
31
4.6.Lemma.Uniform boundedness principle.Let E be a locally convex
vector space and let S be a point seperating set of bounded linear mappings with
common domain E.Then the following conditions are equivalent.
(1) If F is a c
1
{complete locally convex vector space and f:F!E is linear
and ¸ ± f is bounded for all ¸ 2 S,then f is bounded.
(2) If fb
n
g is an unbounded sequence in E with ¸(b
n
) bounded for all ¸ 2 S,
then there is some ft
n
g 2`
1
such that
P
t
n
b
n
does not converge in E for
the initial locally convex topology induced by S.
We then say that E satis¯es the uniform Sboundedness principle if these conditions
are satis¯ed.
A convenient vector space E satis¯es the uniform Sboundedness principle for
each point separating set S of bounded linear mappings on E if and only if there
exists no strictly weaker ultrabornological topology than the bornological topology of
E.
The space C
1
(U;E) satis¯es the uniform boundedness principle for the set S:=
fev
x
:x 2 Ug.
Proof.See [Kriegl,Michor 1997,3.21{3.25]
The importance of the FrÄolicher{Kriegl calculus is also due to the fact that the
function spaces in ¯nite dimensions are all convenient vector spaces.
4.7.Proposition.Let M be a smooth ¯nitedimensional paracompact manifold.
Then the space C
1
(M;R) of all smooth functions on M is a convenient vector
space and satis¯es the uniform boundedness principle for the point evaluations.
The structure is e.g.given by the following description:The initial structure with
respect to the cone
C
1
(M;R)
c
¤
¡!C
1
(R;R)
for all c
¤
2 C
1
(R;M).
Proof.Other equivalent descriptions and the proof can be found in [Kriegl,Michor
1997,3.31].
By considering smooth spaces,which are the category theoretical basis of the
smooth calculus,many results can be proved by using the Cartesian closedness.
However,one cannot di®erentiate in smooth spaces,so I will not show the devel
opment of the theory here,and rather give a cite:[Kriegl,Michor 1997,section
4].
A very important notion,as always in the theory of manifolds,is the existence
of smooth partitions of unity,since they are the only known means of gluing local
results together to yield global results.
4.8.De¯nition.A convenient vector space is said to be smoothly normal if for any
two closed disjoint subsets A
1
;A
2
½ X there is a smooth function f with fj
A
1
´ 0
and fj
A
2
´ 1.
It is called smoothly paracompact if it is paracompact and smoothly normal.
Then the vector space admits smooth bump functions (i.e.for any neighborhood
U of x there exists a smooth function f such that f(x) = 1 and the carrier carr(f) ½
U).
32
Note:A nuclear convenient space admits smooth bump functions.
4.9.De¯nition.A chart (U;u) on a set M is a bijection u:U!u(U) ½ E
U
from
a subset U ½ M onto a c
1
{open subset of a convenient vector space E
U
.For two
charts (U
®
;u
®
) and (U
¯
;u
¯
) on M the mapping u
®¯
:= u
®
± u
¯
:u
¯
(U
®
\U
¯
)!
u
®
(U
®
\U
¯
) is called the chart changing.
A family (U
®
;u
®
)
®2A
of charts is called an atlas of M,if fU
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