In fact since ¦(F ) ⊆ f ( 2 ), and f is injective, we have • as mapping, and since f

is initial • is smooth. By using that incl : F ’ E is initial, we could deduce that

• is a local di¬eomorphism. However we only need that • (0) : F ’ 2 is a linear

isomorphism. Since f (0) —¦ • (0) = ¦ (0)|F is a closed embedding, we have that

• (0) is a closed embedding. In order to see that • (0) is onto, pick v ∈ 2 and

27.12

270 Chapter VI. In¬nite dimensional manifolds 27.12

consider the curve t ’ tv. Then w : t ’ ¦’1 (f (tv)) ∈ F is smooth, and

d

|t=0 (f —¦ •)(w(t))

f (0)(• (0)(w (0))) =

dt

d d

= |t=0 ¦(w(t)) = |t=0 f (tv) = f (0)(v)

dt dt

and since f (0) = ι is injective, we have • (0)(w (0)) = v.

uy w Eu

f

2

j

h

t ’ tv h

hh •

RS ∼¦

SS

=

wT S

Fy wE

incl

Now consider the diagram

U

R

h RRR

j

h

R

h

2

»

h

wuy w Eu u

f

2

¦∼

• =

y wE incl

• —¦ • (0)’1 k ¦ —¦ ¦ (0)’1

F

∼ • (0) ¦ (0) ∼

u ι = f (0) u

= =

y wE

2

i.e.

(» —¦ ¦ —¦ ¦ (0)’1 ) —¦ ι —¦ • (0) = » —¦ ¦ —¦ ¦ (0)’1 —¦ f (0) —¦ • (0)

= » —¦ ¦ —¦ ¦ (0)’1 —¦ ¦ (0) —¦ incl

2

= » —¦ ¦ —¦ incl = » —¦ f —¦ • = —¦ •.

By composing with • (0)’1 : 2 ’ F we get an extension q of q := 2

—¦ k to

˜

E, where the locally de¬ned mapping k := • —¦ • (0)’1 : 2 ’ 2 is smooth and

k (0) = Id. Now q (0) : E — E ’ R is an extension of q (0) : 2 — 2 ’ R given by

˜

(v, w) ’ 2 k (0)v, k (0)w . Hence the associated q (0)∨ : E ’ E — ¬ts into

˜

wy wE

k (0) ι

2 2

∼

=

∼ ∨

u u

q (0)

˜

=

u k (0) u

∼

= E—

2 2

—

— ι

2

’ E. This is a contradiction.

and in this way we get a retraction for ι :

27.12

27.14 27. Di¬erentiable manifolds 271

(3) Let us show now the even stronger statement that there is no local regular

G with f ( 2 ) = ρ’1 (0) locally and ker ρ (0) = ι( 2 ). Otherwise

equation ρ : E

we have ρ (0)(v) = 0 and hence there is a µ ∈ G with µ(ρ (0)(v)) = 1. Thus

µ—¦ρ:E R is smooth µ —¦ ρ —¦ f = 0 and (µ —¦ ρ) (0)(v) = 1. Moreover

0 = ( dt )2 |t=0 (µ —¦ ρ —¦ f )(tx)

d

d

= |t=0 (µ —¦ ρ) (f (tx)) · f (tx) · x

dt

= (µ —¦ ρ) (0)(f (0)x, f (0)x) + (µ —¦ ρ) (0) · f (0)(x, x)

2

= (µ —¦ ρ) (0)(ι(x), ι(x)) + 2 x (µ —¦ ρ) (0) · v,

2

hence ’(µ —¦ ρ) (0)/2 is an extension of along ι, which is a contradiction.

27.13. Theorem. Embedding of smooth manifolds. If M is a smooth mani-

fold modeled on a C ∞ -regular convenient vector space E, which is Lindel¨f. Then

o

there exists a smooth embedding onto a splitting submanifold of s — E (N) where s is

the space of rapidly decreasing real sequences.

Proof. We choose a countable atlas (Un , un ) and a subordinated smooth partition

(hn ) of unity which exists by (16.10). Then the embedding is given by

x ’ ((hn (x))n , (hn (x).un (x))n ) ∈ s — E (N) .

Local smooth retracts to this embedding are given by ((tn ), (xn )) ’ u’1 ( t1 xk )

k k

de¬ned for tk = 0.

27.14. Coverings. A surjective smooth mapping p : N ’ M between smooth

manifolds is called a covering if it is the projection of a ¬ber bundle (see (37.1))

with discrete ¬ber. Note that on a product of a discrete space with a manifold the

product topology equals the manifold topology. A product of two coverings is again

a covering.

A smooth manifold M is locally contractible since we may choose charts with star-

shaped images, and since the c∞ -topology on a product with R is the product of

˜

the c∞ -topologies. Hence the universal covering space M of a connected smooth

manifold M exists as a topological space. By pulling up charts it turns out to be a

˜ ˜˜

smooth manifold also, whose topology is the one of M . Since M — M is the universal

˜

covering of M — M , the manifold M is Hausdor¬ even in the sense of (27.4.2). If

˜

M is smoothly regular then M is also smoothly regular, thus smoothly Hausdor¬.

As usual, the fundamental group π(M, x0 ) acts free and strictly discontinuously

˜ ˜

on M in the sense that each x ∈ M admits an open neighborhood U such that

g.U © U = … for all g = e in π(M, x0 ).

˜

Note that the universal covering space M of a connected smooth manifold M

can be viewed as the Fr¨licher space (see (23.1), (24.10)) C ∞ ((I, 0), (M, x0 )) of

o

all smooth curves c : [0, 1] = I ’ M , such that c(0) = x0 for a base point

x0 ∈ M modulo smooth homotopies ¬xing endpoints. This can be shown by

27.14

272 Chapter VI. In¬nite dimensional manifolds 27.16

the usual topological proof, where one uses only smooth curves and homotopies,

and smoothes by reparameterization those which are pieced together. Note that

ev1 : C ∞ ((I, 0), (M, x0 )) ’ M is a ¬nal (27.15) smooth mapping since we may

construct local smooth sections near any point in M : choose a chart u : U ’ u(U )

on M with u(U ) a radial open set in the modeling space of M . Then let •(x) be

the smooth curve which follows a smooth curve from x0 to u’1 (0) during the time

from 0 to 2 and stops in¬nitely ¬‚at at 1 , so the curve t ’ u’1 (ψ(t).u(x)) where

1

2

ψ : [ 2 , 1] ’ [0, 1] is smooth, ¬‚at at 2 , ψ( 1 ) = 0, and with ψ(1) = 1. These local

1 1

2

˜

smooth sections lift to smooth sections of C ∞ ((I, 0), (M, x0 )) ’ M , thus the ¬nal

˜

smooth structure on M coincides with that induced from the manifold structure.

If conversely a group G acts strictly discontinuously on a smooth manifold M , then

the orbit space M/G turns out to be a smooth manifold (with G.U ™s as above as

charts), but it might be not Hausdor¬, as the following example shows: M = R2 \0,

G = Z acting by FlX where X = x‚x ’ y‚y .

z

The orbit space is Hausdor¬ if and only if R := {(g.x, x) : g ∈ G, x ∈ M } is

closed in M — M with the product topology: M ’ M/G is an open mapping, thus

the product M — M ’ M/G — M/G is also open for the product topologies, and

(M — M ) \ R is mapped onto the complement of the diagonal in M/G — M/G.

The orbit space has property (27.4.2) if and only if R := {(g.x, x) : g ∈ G, x ∈ M }

is closed in M — M with the manifold topology: the same proof as above works,

where M — M ’ M/G — M/G is also open for the manifold topologies since we

may lift smooth curves.

We were unable to ¬nd a condition on the action which would ensure that M/G is

smoothly Hausdor¬ or inherits a stronger separation property from M . Classical

results always use locally compact M .

27.15. Final smooth mappings. A mapping f : M ’ N between smooth

manifolds is called ¬nal if:

A mapping g : N ’ P into a manifold P is smooth if and only if

g —¦ f : M ’ P is smooth.

Clearly, a ¬nal mapping f : M ’ N is smooth, and surjective if N is connected.

Coverings (27.14) are always ¬nal, as are projections of ¬ber bundles (37.1). Be-

tween ¬nite dimensional separable manifolds without isolated points the ¬nal map-

pings are exactly the surjective submersions. We will use the notion submersion in

¬nite dimensions only.

27.16. Foliations. Let F be a c∞ -closed linear subspace of a convenient vector

space E. Let EF be the smooth manifold modeled on F , which is the disjoint union

of all a¬ne subspaces of E which are translates of F . A di¬eomorphism f : U ’ V

between c∞ -open subsets of E is called F -foliated if it is also a homeomorphism

(equivalently di¬eomorphism) between the open subsets U and V of EF .

Let M be a smooth manifold modeled on the convenient vector space E. A foliation

on M is then given by a c∞ -closed linear subspace F in E and a smooth (maximal)

27.16

27.17 27. Di¬erentiable manifolds 273

atlas of M such that all chart changings are F -foliated. Each chart of this maximal

atlas is called a distinguished chart. A connected component of the inverse image

under a distinguished chart of an a¬ne translate of F is called a plaque.

A foliation on M induces on the set M another structure of a smooth manifold,

sometimes denoted by MF , modeled on F , where we take as charts the restrictions

of distinguished charts to plaques (with the image translated into F ). The identity

on M induces a smooth bijective mapping MF ’ M . Clearly, MF is smoothly

Hausdor¬ (if M is it). A leaf of the foliation is then a connected component of MF .

The notion of foliation will be used in (39.2) below.

27.17. Lemma. For a convenient vector space E and any smooth manifold M

the set C ∞ (M, E) of smooth E-valued functions on M is also a convenient vector

space in any of the following isomorphic descriptions, and it satis¬es the uniform

boundedness principle for the point evaluations.

(1) The initial structure with respect to the cone

c—

C (M, E) ’ C ∞ (R, E)

∞

’

for all c ∈ C ∞ (R, M ).

(2) The initial structure with respect to the cone

(u’1 )—

C (M, E) ’ ’ ’ C ∞ (u± (U± ), E),

∞ ±

’’

where (U± , u± ) is a smooth atlas with u± (U± ) ‚ E± .

Moreover, with this structure, for two manifolds M , N , the exponential law holds:

C ∞ (M, C ∞ (N, E)) ∼ C ∞ (M — N, E).

=

For a real analytic manifold M the set C ω (M, E) of real analytic E-valued functions

on M is also a convenient vector space in any of the following isomorphic descrip-

tions, and it satis¬es the uniform boundedness principle for the point evaluations.

(1) The initial structure with respect to the cone

c—

C (M, E) ’ C ∞ (R, E) for all c ∈ C ∞ (R, M )

ω

’

c—

C ω (M, E) ’ C ω (R, E) for all c ∈ C ω (R, M ).

’

(2) The initial structure with respect to the cone

(u’1 )—

C (M, E) ’ ’ ’ C ω (u± (U± ), E),

ω ±

’’

where (U± , u± ) is a real analytic atlas with u± (U± ) ‚ E± .

27.17

274 Chapter VI. In¬nite dimensional manifolds 27.18

Moreover, with this structure, for two real analytic manifolds M , N , the exponential

law holds:

C ω (M, C ω (N, E)) ∼ C ω (M — N, E).

=

For a complex convenient vector space E and any complex holomorphic manifold

M the set H(M, E) of holomorphic E-valued functions on M is also a convenient

vector space in any of the following isomorphic descriptions, and it satis¬es the

uniform boundedness principle for the point evaluations.

(1) The initial structure with respect to the cone

c—

H(M, E) ’ H(D, E)

’

for all c ∈ H(D, M ).

(2) The initial structure with respect to the cone

(u’1 )—

±

H(M, E) ’ ’ ’ H(u± (U± ), E),

’’

where (U± , u± ) is a holomorphic atlas with u± (U± ) ‚ E± .

Moreover, with this structure, for two manifolds M , N , the exponential law holds:

H(M, H(N, E)) ∼ H(M — N, E).

=

Proof. For all descriptions the initial locally convex topology is convenient, since

the spaces are closed linear subspaces in the relevant products of the right hand

sides. Thus, the uniform boundedness principle for the point evaluations holds for

all structures since it holds for all right hand sides. So the identity is bibounded

between all respective structures.

The exponential laws now follow from the corresponding ones: use (3.12) for c∞ -

open subsets of convenient vector spaces and description (2), for the real analytic

case use (11.18), and for the holomorphic case use (7.22).

27.18. Germs. Let M and N be manifolds, and let A ‚ M be a closed subset. We

consider all smooth mappings f : Uf ’ N , where Uf is some open neighborhood

of A in M , and we put f ∼ g if there is some open neighborhood V of A with

A

f |V = g|V . This is an equivalence relation on the set of functions considered.

The equivalence class of a function f is called the germ of f along A, sometimes

denoted by germA f . As in (8.3) we will denote the space of all these germs by

C ∞ (M ⊃ A, N ).

If we consider functions on M , i.e. if N = R, we may add and multiply germs, so

we get the real commutative algebra of germs of smooth functions. If A = {x},

this algebra C ∞ (M ⊃ {x}, R) is sometimes also denoted by Cx (M, R). We may

∞

consider the inductive locally convex vector space topology with respect to the cone

C ∞ (M ⊇ {x}, R) ← C ∞ (U, R),

27.18

27.20 27. Di¬erentiable manifolds 275

where U runs through some neighborhood basis of x consisting of charts, so that

each C ∞ (U, R) carries a convenient vector space topology by (2.15).

This inductive topology is Hausdor¬ only if x is isolated in M , since the restriction

to some one dimensional linear subspace of a modeling space is a projection on a

direct summand which is not Hausdor¬, by (27.19). Nevertheless, multiplication is

a bounded bilinear operation on C ∞ (M ⊇ {x}, R), so the closure of 0 is an ideal.

The quotient by this ideal is thus an algebra with bounded multiplication, denoted

by Tayx (M, R).

27.19. Lemma. Let M be a smooth manifold modeled on Banach spaces which

admit bump functions of class Cb (see (15.1)). Then the closure of 0 in C ∞ (M ⊇

∞

{x}, R) is the ideal of all germs which are ¬‚at at x of in¬nite order.

Proof. This is a local question, so let x = 0 in a modeling Banach space E. Let f

be a representative in some open neighborhood U of 0 of a ¬‚at germ. This means

∞

that all iterated derivatives of f at 0 vanish. Let ρ ∈ Cb (E, [0, 1]) be 0 on a

neighborhood of 0 and ρ(x) = 1 for x > 1. For fn (x) := f (x)ρ(n.x) we have

germ0 (fn ) = 0, and it remains to show that n(f ’ fn ) is bounded in C ∞ (U, R). For

1

this we ¬x a derivative dk and choose N such that dk+1 f (x) ¤ 1 for x ¤ N .

Then for n ≥ N we have the following estimate:

k

k

ndk (f ’ fn )(x) ¤ n dk’l f (x) nl dl (1 ’ ρ)(nx)

l

l=0

k 1

(1 ’ t)l+1 k+1

k l+1 l

n dl (1 ’ ρ)(nx)

¤ n d f (tx) dt x

l (l + 1)!

0

l=0

0 for nx > 1

¤ k k1

dl (1 ’ ρ) for nx ¤ 1.

∞

l=0 l l!

∞

27.20. Corollary. For any Cb -regular Banach space E and a ∈ E the canonical

mapping

∞

Lk (E, R)

Taya (E, R) ’ sym

k=0

is a bornological isomorphism.

Proof. For every open neighborhood U of a in E we have a continuous linear

∞

mapping C ∞ (U, R) ’ k

k=0 Lsym (E, R) into the space of formal power series,

∞

hence also C ∞ (E ⊇ {a}, R) ’ k=0 Lk (E, R), and ¬nally from Taya (E, R) ’