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˜
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

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