consider a topological real analytic curve in K by (10.4). Such a curve is extendable

˜˜

to a holomorphic curve c by (9.5), hence the composite f —¦ c is holomorphic and its

˜

˜

restriction f —¦ c to R is real analytic.

26.4. De¬nition. (Holomorphic maps on complex vector spaces)

Let K ⊆ E be a convex subset with non-empty interior in a complex convenient

vector space. And map f : E ⊇ K ’ F is called holomorphic i¬ it is real analytic

and the derivative f (x) is C-linear for all x ∈ K o .

26.5. Theorem. Holomorphic maps are germs.

Let K ⊆ E be a convex subset with non-empty interior in a complex convenient

vector space. Then a map f : E ⊇ K ’ F into a complex convenient vector space

F is holomorphic if and only if it extends to a holomorphic map de¬ned on some

neighborhood of K in E.

Proof. Since f : K ’ F is real analytic, it extends by (25.9) to a real analytic map

˜

f : E ⊇ U ’ F , where we may assume that U is connected with K by straight

˜

line segments. We claim that f is in fact holomorphic. For this it is enough to

show that f (x) is C-linear for all x ∈ U . So consider the real analytic mapping

g : U ’ F given by g(x) := if (x)(v) ’ f (x)(iv). Since it is zero on K o it has to

be zero everywhere by the uniqueness theorem.

26.6. Remark. (There is no de¬nition for holomorphy analogous to (25.7))

In order for a map K ’ F to be holomorphic it is not enough to assume that all

composites f —¦ c for holomorphic c : D ’ K are holomorphic, where D is the open

unit disk. Take as K the closed unit disk, then c(D) © ‚K = φ. In fact let z0 ∈ D

then c(z) = (z ’ z0 )n (cn + (z ’ z0 ) k>n ck (z ’ z0 )k’n’1 ) for z close to z0 , which

covers a neighborhood of c(z0 ). So the boundary values of such a map would be

completely arbitrary.

26.7. Lemma. Holomorphy is a bornological concept.

Let T± : E ’ E± be a family of bounded linear maps that generates the bornology

on E. Then a map c : K ’ F is holomorphic if and only if all the composites

T± —¦ c : I ’ F± are holomorphic.

Proof. It follows from (25.6) that f is real analytic. And the C-linearity of f (x)

can certainly be tested by point separating linear functionals.

26.8. Theorem. Exponential law for holomorphic maps.

Let K and L be convex subsets with non-empty interior in complex convenient vector

spaces. Then a map f : K — L ’ F is holomorphic if and only if the associated

map f ∨ : K ’ H(L, F ) is holomorphic.

Proof. This follows immediately from the real analytic result (25.12), since the

C-linearity of the involved derivatives translates to each other, since we obviously

have f (x1 , x2 )(v1 , v2 ) = evx2 ((f ∨ ) (x1 )(v1 )) + (f ∨ (x1 )) (x2 )(v2 ) for x1 ∈ K and

x2 ∈ L.

26.8

263

Chapter VI

In¬nite Dimensional Manifolds

27. Di¬erentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 264

28. Tangent Vectors ........... . . . . . . . . . . . . . 276

29. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 287

30. Spaces of Sections of Vector Bundles .. . . . . . . . . . . . . . 293

31. Product Preserving Functors on Manifolds . . . . . . . . . . . . . 305

This chapter is devoted to the foundations of in¬nite dimensional manifolds. We

treat here only manifolds described by charts onto c∞ -open subsets of convenient

vector spaces.

Note that this limits cartesian closedness of the category of manifolds. For ¬nite

dimensional manifolds M, N, P we will show later that C ∞ (N, P ) is not locally con-

tractible (not even locally pathwise connected) for the compact-open C ∞ -topology

if N is not compact, so one has to use a ¬ner structure to make it a manifold

C∞ (N, P ), see (42.1). But then C ∞ (M, C∞ (N, P )) ∼ C ∞ (M — N, P ) if and only if

=

∞

N is compact see (42.14). Unfortunately, C (N, P ) cannot be generalized to in¬-

nite dimensional N , since this structure becomes discrete. Let us mention, however,

that there exists a theory of manifolds and vector bundles, where the structure of

charts is replaced by the set of smooth curves supplemented by other requirements,

where one gets a cartesian closed category of manifolds and has the compact-open

C ∞ -topology on C ∞ (N, P ) for ¬nite dimensional N , P , see [Seip, 1981], [Kriegl,

1980], [Michor, 1984a].

We start by treating the basic concept of manifolds, existence of smooth bump

functions and smooth partitions of unity. Then we investigate tangent vectors

seen as derivations or kinematically (via curves): these concepts di¬er, and we

show in (28.4) that even on Hilbert spaces there exist derivations which are not

tangent to any smooth curve. In particular, we have di¬erent kinds of tangent

bundles, the most important ones are the kinematic and the operational one. We

treat smooth, real analytic, and holomorphic vector bundles and spaces of sections

of vector bundles, we give them structures of convenient vector spaces; they will

become important as modeling spaces for manifolds of mappings in chapter IX.

Finally, we discuss Weil functors (certain product preserving functors of manifolds)

as generalized tangent bundles. This last section is due to [Kriegl, Michor, 1997].

264 Chapter VI. In¬nite dimensional manifolds 27.3

27. Di¬erentiable Manifolds

27.1. Manifolds. A chart (U, u) on a set M is a bijection u : U ’ u(U ) ⊆ EU

from a subset U ⊆ M onto a c∞ -open subset of a convenient vector space EU .

For two charts (U± , u± ) and (Uβ , uβ ) on M the mapping u±β := u± —¦ u’1 : β

uβ (U±β ) ’ u± (U±β ) for ±, β ∈ A is called the chart changing, where U±β :=

U± © Uβ . A family (U± , u± )±∈A of charts on M is called an atlas for M , if the U±

form a cover of M and all chart changings u±β are de¬ned on c∞ -open subsets.

An atlas (U± , u± )±∈A for M is said to be a C ∞ -atlas, if all chart changings u±β :

uβ (U±β ) ’ u± (U±β ) are smooth. Two C ∞ -atlas are called C ∞ -equivalent, if their

union is again a C ∞ -atlas for M . An equivalence class of C ∞ -atlas is sometimes

called a C ∞ -structure on M . The union of all atlas in an equivalence class is again

an atlas, the maximal atlas for this C ∞ -structure. A C ∞ -manifold M is a set

together with a C ∞ -structure on it.

Atlas, structures, and manifolds are called real analytic or holomorphic, if all chart

changings are real analytic or holomorphic, respectively. They are called topologi-

cal, if the chart changings are only continuous in the c∞ -topology.

A holomorphic manifold is real analytic, and a real analytic one is smooth. By a

manifold we will henceforth mean a smooth one.

27.2. A mapping f : M ’ N between manifolds is called smooth if for each x ∈ M

and each chart (V, v) on N with f (x) ∈ V there is a chart (U, u) on M with x ∈ U ,

f (U ) ⊆ V , such that v —¦ f —¦ u’1 is smooth. This is the case if and only if f —¦ c is

smooth for each smooth curve c : R ’ M .

We will denote by C ∞ (M, N ) the space of all C ∞ -mappings from M to N .

Likewise, we have the spaces C ω (M, N ) of real analytic mappings and H(M, N ) of

holomorphic mappings between manifolds of the corresponding type. This can be

also tested by composing with the relevant types of curves.

A smooth mapping f : M ’ N is called a di¬eomorphism if f is bijective and

its inverse is also smooth. Two manifolds are called di¬eomorphic if there exists

a di¬eomorphism between them. Likewise, we have real analytic and holomorphic

di¬eomorphisms. The latter ones are also called biholomorphic mappings.

27.3. Products. Let M and N be smooth manifolds described by smooth atlas

(U± , u± )±∈A and (Vβ , vβ )β∈B , respectively. Then the family (U± — Vβ , u± — vβ :

U± — Vβ ’ E± — Fβ )(±,β)∈A—B is a smooth atlas for the cartesian product M — N .

Beware, however, the manifold topology (27.4) of M — N may be ¬ner than the

product topology, see (4.22). If M and N are metrizable, then it coincides with the

product topology, by (4.19). Clearly, the projections

pr pr

1

’2

M ←’ M — N ’’ N

’

are also smooth. The product (M —N, pr1 , pr2 ) has the following universal property:

27.3

27.4 27. Di¬erentiable manifolds 265

For any smooth manifold P and smooth mappings f : P ’ M and g : P ’ N

the mapping (f, g) : P ’ M — N , (f, g)(x) = (f (x), g(x)), is the unique smooth

mapping with pr1 —¦(f, g) = f , pr2 —¦(f, g) = g.

Clearly, we can form products of ¬nitely many manifolds. The reader may now

wonder why we do not consider in¬nite products of manifolds. These have charts

which are open for the so called ˜box topology™. But then we get ˜box products™

without the universal property of products. The ˜box products™, however, have the

universal product property for families of mappings such that locally almost all

members are constant.

27.4. The topology of a manifold. The natural topology on a manifold M

is the identi¬cation topology with respect to some (smooth) atlas (u± : M ⊇

U± ’ u± (U± ) ⊆ E± ), where a subset W ⊆ M is open if and only if u± (U± © W )

is c∞ -open in E± for all ±. This topology depends only on the structure, since

di¬eomorphisms are homeomorphisms for the c∞ -topologies. It is also the ¬nal

topology with respect to all inverses of chart mappings in one atlas. It is also

the ¬nal topology with respect to all smooth curves. For a (smooth) manifold

we will require certain properties for the natural topology, which will be speci¬ed

when needed, like smoothly regular (14.1), smoothly normal (16.1), or smoothly

paracompact (16.1).

Let us now discuss the relevant notions of Hausdor¬.

(1) M is (topologically) Hausdor¬, equivalently the diagonal is closed in the

product topology on M — M .

(2) The diagonal is closed in the manifold M — M .

(3) The smooth functions in C ∞ (M, R) separate points in M . Let us call M

smoothly Hausdor¬ if this property holds.

We have the obvious implications (3)’(1)’(2). We have no counterexamples for

the converse implications.

The three separation conditions just discussed do not depend on properties of

the modeling convenient vector spaces, whereas properties like smoothly regu-

lar, smoothly normal, or smoothly paracompact do. Smoothly Hausdor¬ is the

strongest of the three. But it is not so clear which separation property should be

required for a manifold. In order to make some decision, from now on we re-

quire that manifolds are smoothly Hausdor¬. Each convenient vector space

has this property. But we will have di¬culties with permanence of the property

˜smoothly Hausdor¬™, in particular with quotient manifolds, see for example the

discussion (27.14) on covering spaces below. For important examples (manifolds of

mappings, di¬eomorphism groups, etc.) we will prove that they are even smoothly

paracompact.

The isomorphism type of the modeling convenient vector spaces E± is constant

on the connected components of the manifold M , since the derivatives of the chart

changings are linear isomorphisms. A manifold M is called pure if this isomorphism

type is constant on the whole of M .

27.4

266 Chapter VI. In¬nite dimensional manifolds 27.6

Corollary. If a smooth manifold (which is smoothly Hausdor¬ ) is Lindel¨f, ando

if all modeling vector spaces are smoothly regular, then it is smoothly paracompact.

If a smooth manifold is metrizable and smoothly normal then it is smoothly para-

compact.

Proof. See (16.10) for the ¬rst statement and (16.15) for the second one.

27.5. Lemma. Let M be a smoothly regular manifold. Then for any manifold N

a mapping f : N ’ M is smooth if and only if g —¦ f : N ’ R is smooth for all

g ∈ C ∞ (M, R). This means that (M, C ∞ (R, M ), C ∞ (M, R)) is a Fr¨licher space,

o

see (23.1).

Proof. Let x ∈ N and let (U, u : U ’ E) be a chart of M with f (x) ∈ U .

Choose some smooth bump function g : M ’ R with supp(g) ‚ U and g = 1 in a

neighborhood V of f (x). Then f ’1 (carr(g)) = carr(g —¦ f ) is an open neighborhood

of x in N . Thus f is continuous, so f ’1 (V ) is open. Moreover, (g.( —¦ u)) —¦ f

is smooth for all ∈ E and on f ’1 (V ) this equals —¦ u —¦ (f |f ’1 (V )). Thus

u —¦ (f |f ’1 (V )) is smooth since E is convenient, by (2.14.4), so f is smooth near

x.

27.6. Non-regular manifold. [Margalef, Outerelo, 1982] Let 0 = » ∈ ( 2 )— ,

let X be {x ∈ 2 : »(x) ≥ 0} with the Moore topology, i.e. for x ∈ X we take

{y ∈ 2 \ ker » : y ’ x < µ} ∪ {x} for µ > 0 as neighborhood-basis. We set

X + := {x ∈ 2 : »(x) > 0} ⊆ 2 .

Then obviously X is Hausdor¬ (its topology is ¬ner than that of 2 ) but not regular:

In fact the closed subspace ker » \ {0} cannot be separated by open sets from {0}.

It remains to show that X is a C ∞ -manifold. We use the following di¬eomorphisms

(1) S := {x ∈ 2 : x = 1} ∼C ∞ ker ».

=

(2) • : 2 \ {0} ∼C ∞ ker » — R+ .

=

+∼ ∞

(3) S © X =C ker ».

(4) ψ : X + ’ ker » — R+ .

(1) This is due to [Bessaga, 1966].

(2) Let f : S ’ ker » be the di¬eomorphism of (1) and de¬ne the required di¬eo-

morphism to be •(x) := (f (x/ x ), x ) with inverse •’1 (y, t) := t f ’1 (y).

(3) Take an a ∈ (ker »)⊥ with »(a) = 1. Then the orthogonal projection 2 ’ ker »

is given by x ’ x ’ »(x)a. This is a di¬eomorphism of S © X + ’ {x ∈ ker » :

x < 1}, which in turn is di¬eomorphic to ker ».

(4) Let g : S © X + ’ ker » be the di¬eomorphism of (3) then the desired di¬eo-

morphism is ψ : x ’ (g(x/ x ), x ).

We now show that there is a homeomorphism of h : X + ∪ {0} ’ 2

, such that

h(0) = 0 and h|X + : X + ’ 2 \ {0} is a di¬eomorphism. We take

(•’1 —¦ ψ)(x) for x ∈ X +

h(x) := .

0 for x = 0

27.6

27.10 27. Di¬erentiable manifolds 267

zu u

ψ ’1 •

z

+

ker » — R+ 2

\ {0}

X ∼ ∼

= =

u u

u u

u

h

+

X ∪ {0} E

y y

u | {0}

{0}

Now we use translates of h as charts 2 ’ X + ∪ {x}. The chart changes are

then di¬eomorphisms of 2 \ {0} and we thus obtained a smooth atlas for X :=

+

x∈ker » (X ∪ {x}). The topology described by this atlas is obviously the Moore

topology.

If we use instead of X the union x∈D (X + ∪ {x}), where D ⊆ ker » is dense and

countable. Then the same results are valid, but X is now even second countable.

Note however that a regular space which is locally metrizable is completely regular.

27.7. Proposition. Let M be a manifold modeled on smoothly regular convenient

vector spaces. Then M admits an atlas of charts de¬ned globally on convenient

vector spaces.

Proof. That a convenient vector space is smoothly regular means that the c∞ -

topology has a base of carrier sets of smooth functions, see (14.1). These functions

satisfy the assumptions of theorem (16.21), and hence the stars of these sets with

respect to arbitrary points in the sets are di¬eomorphic to the whole vector space

and still form a base of the c∞ -topology.

27.8. Lemma. A manifold M is metrizable if and only if it is paracompact and

modeled on Fr´chet spaces.

e

Proof. A topological space is metrizable if and only if it is paracompact and locally

metrizable. c∞ -open subsets of the modeling vector spaces are metrizable if and

only if the spaces are Fr´chet, by (4.19).

e

27.9. Lemma. Let M and N be smoothly paracompact metrizable manifolds.

Then M — N is smoothly paracompact.

Proof. By (16.15) there are embeddings into c0 (“) and c0 (Λ) for some sets “ and

Λ which pull back the coordinate projections to smooth functions. Then M — N

embeds into c0 (“) — c0 (Λ) ∼ c0 (“ Λ) in the same way and hence again by (16.15)

=

the manifold M — N is smoothly paracompact.

27.10. Facts on ¬nite dimensional manifolds. A manifold M is called ¬nite

dimensional if it has ¬nite dimensional modeling vector spaces. By (4.19), this is

the case if and only if M is locally compact. Then the dimensions of the modeling

spaces give a locally constant function on M .

27.10

268 Chapter VI. In¬nite dimensional manifolds 27.11

If the manifold M is ¬nite dimensional, then Hausdor¬ implies smoothly regular.

We require then that the natural topology is in addition to Hausdor¬ also paracom-

pact. It is then smoothly paracompact by (27.7), since all connected components

are Lindel¨f if M is paracompact.

o

Let us ¬nally add some remarks on ¬nite dimensional separable topological man-

ifolds M : From di¬erential topology we know that if M has a C 1 -structure, then

it also has a C 1 -equivalent C ∞ -structure and even a C 1 -equivalent C ω -structure.

But there are manifolds which do not admit di¬erentiable structures. For example,

every 4-dimensional manifold is smooth o¬ some point, but there are some which

are not smooth, see [Quinn, 1982], [Freedman, 1982]. Note, ¬nally, that any such

manifold M admits a ¬nite atlas consisting of dim M +1 (not necessarily connected)

charts. This is a consequence of topological dimension theory, a proof may be found

in [Greub, Halperin, Vanstone, 1972].

If there is a C 1 -di¬eomorphism between M and N , then there is also a C ∞ -

di¬eomorphism. There are manifolds which are homeomorphic but not di¬eomor-

phic: on R4 there are uncountably many pairwise non-di¬eomorphic di¬erentiable

structures; on every other Rn the di¬erentiable structure is unique. There are

¬nitely many di¬erent di¬erentiable structures on the spheres S n for n ≥ 7. See

[Kervaire, Milnor, 1963].

27.11. Submanifolds. A subset N of a manifold M is called a submanifold, if for

each x ∈ N there is a chart (U, u) of M such that u(U © N ) = u(U ) © FU , where

FU is a closed linear subspace of the convenient model space EU . Then clearly N

is itself a manifold with (U © N, u|U © N ) as charts, where (U, u) runs through all

these submanifold charts from above.

A submanifold N of M is called a splitting submanifold if there is a cover of N by

submanifold charts (U, u) as above such that the FU ‚ EU are complemented (i.e.

splitting) linear subspaces. Then every submanifold chart is splitting.

Note that a closed submanifold of a smoothly paracompact manifold is again

smoothly paracompact. Namely, the trace topology is the intrinsic topology on

the submanifold since this is true for closed linear subspaces of convenient vector

spaces, (4.28).

A mapping f : N ’ M between manifolds is called initial if it has the following

property:

A mapping g : P ’ N from a manifold P (R su¬ces) into N is smooth

if and only if f —¦ g : P ’ M is smooth.

Clearly, an initial mapping is smooth and injective. The embedding of a submani-

fold is always initial. The notion of initial smooth mappings will play an important

role in this book whereas that of immersions will be used in ¬nite dimensions only.

In a similar way we shall use the (now obvious) notion of initial real analytic map-

pings between real analytic manifolds and also initial holomorphic mappings be-

tween complex manifolds.

27.11

27.12 27. Di¬erentiable manifolds 269

If h : R ’ R is a function such that hp and hq are smooth for some p, q which are

relatively prime in N, then h itself turns out to be smooth, see [Joris, 1982.] So the

mapping f : t ’ (tp , tq ), R ’ R2 , is initial, but f is not an immersion at 0.

Smooth mappings f : N ’ M which admit local smooth retracts are initial. By

this we mean that for each x ∈ N there are an open neighborhood U of f (x) in M

and a smooth mapping rx : U ’ N such that r —¦ f |(f ’1 (U )) = Idf ’1 U . We shall

meet this class of initial mappings in (43.19).

27.12. Example. We now give an example of a smooth mapping f with the

following properties:

(1) f is a topological embedding and the derivative at each point is an embed-

ding of a closed linear subspace.

(2) The image of f is not a submanifold.

(3) The image of f cannot be described locally by a regular smooth equation.

This shows that the notion of an embedding is quite subtle in in¬nite dimensions.

ι

Proof. For this let 2 ’ E ’ 2 be a short exact sequence, which does not split,

’

see (13.18.6) Then the square of the norm on 2 does not extend to a smooth

function on E by (21.11).

Choose a 0 = » ∈ E — with » —¦ ι = 0 and choose a v with »(v) = 1. Now consider

f : 2 ’ E given by x ’ ι(x) + x 2 v.

(1) Since f is polynomial it is smooth. We have (» —¦ f )(x) = x 2 , hence g —¦ f = ι,

where g : E ’ E is given by g(y) := y ’ »(y) v. Note however that g is no

di¬eomorphism, hence we don™t have automatically a submanifold. Thus f is a

topological embedding and also the di¬erential at every point. Moreover the image

is closed, since f (xn ) ’ y implies ι(xn ) = g(f (xn )) ’ g(y), hence xn ’ x∞ for

some x∞ and thus f (xn ) ’ f (x∞ ) = y. Finally f is initial. Namely, let h : G ’ 2

be such that f —¦ h is smooth, then g —¦ f —¦ h = ι —¦ h is smooth. As a closed linear

2

embedding ι is initial, so h is smooth. Note that » is an extension of along

f : 2 ’ E.

(2) Suppose there were a local di¬eomorphism ¦ around f (0) = 0 and a closed

subspace F < E such that locally ¦ maps F onto f ( 2 ). Then ¦ factors as follows

uy w Eu

f

2

∼¦

• =

y wE

incl

F