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. Nonregular 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 neighborhoodbasis. 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 hX + : 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 4dimensional 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 nondi¬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, uU © 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
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 Evalued 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 Evalued 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 Evalued 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 = gV . 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) ’
sym
∞ k
k=0 Lsym (E, R). Since E is Banach, the space of formal power series is a Fr´chet
e
∞
space and since E is Cb (E, R)regular the last mapping is injective by (27.19).
By E. Borel™s theorem (15.4) every bounded subset of the space of formal power
series is the image of a bounded subset of C ∞ (E, R). Hence this mapping is a
bornological isomorphism and the inductive limit C ∞ (E ⊇ {a}, R) is regular.
27.20
276 Chapter VI. In¬nite dimensional manifolds 28.1
27.21. Lemma. If M is smoothly regular then each germ at a point of a smooth
function has a representative which is de¬ned on the whole of M .
If M is smoothly paracompact then this is true for germs along closed subsets.
For germs of real analytic or holomorphic functions this is not true.
If M is as in the lemma, C ∞ (M ⊇ {x}, R) is the quotient of the algebra C ∞ (M, R)
by the ideal of all smooth functions f : M ’ R which vanish on some neighborhood
(depending on f ) of x.
The assumption in the lemma is not necessary as is shown by the following example:
By (14.9) the Banach space E := C([0, 1], R) is not C ∞ regular, in fact not even
C 1 regular. For h ∈ C ∞ (R, R) the push forward h— : C ∞ (R, R) ’ C ∞ (R, R) is
smooth, thus continuous, so (h— )— : C([0, 1], C ∞ (R, R)) ’ C([0, 1], C ∞ (R, R)) is
continuous. The arguments in the proof of theorem (3.2) show that
C([0, 1], C ∞ (R, R)) ∼ C ∞ (R, C([0, 1], R)),
=
thus h— : E ’ E is smooth. Let h(t) := t for t ¤ 1 and h(t) ¤ 1 for all t ∈ R.
2
1
In particular h— is the identity on {f ∈ E : f ¤ 2 }. Let U be a neighborhood
of 0 in E. Choose µ > 0 such that the closed ball with radius µ > 0 is contained
in U . Then hµ := µ h— 1 : E ’ E has values in U and is the identity near 0.
µ
Thus (hµ ) : C (U, R) ’ C ∞ (E, R) is a bounded algebra homomorphism, which
— ∞
respects the corresponding germs at 0.
28. Tangent Vectors
28.1. The tangent spaces of a convenient vector space E. Let a ∈ E. A
kinematic tangent vector with foot point a is simply a pair (a, X) with X ∈ E.
Let Ta E = E be the space of all kinematic tangent vectors with foot point a. It
consists of all derivatives c (0) at 0 of smooth curves c : R ’ E with c(0) = a,
which explains the choice of the name kinematic.
For each open neighborhood U of a in E (a, X) induces a linear mapping Xa :
C ∞ (U, R) ’ R by Xa (f ) := df (a)(X), which is continuous for the convenient vector
space topology on C ∞ (U, R) and satis¬es Xa (f · g) = Xa (f ) · g(a) + f (a) · Xa (g), so
it is a continuous derivation over eva . The value Xa (f ) depends only on the germ
of f at a.
An operational tangent vector of E with foot point a is a bounded derivation
‚ : C ∞ (E ⊇ {a}, R) ’ R over eva . Let Da E be the vector space of all these
derivations. Any ‚ ∈ Da E induces a bounded derivation C ∞ (U, R) ’ R over eva
for each open neighborhood U of a in E. Moreover any family of bounded deriva
tions ‚U : C ∞ (U, R) ’ R over eva , which is coherent with respect to the restriction
maps, de¬nes an ‚ ∈ Da E. So the vector space Da E is a closed linear subspace of
the convenient vector space U L(C ∞ (U, R), R). We equip Da E with the induced
28.1
28.2 28. Tangent vectors 277
convenient vector space structure. Note that the spaces Da E are isomorphic for all
a ∈ E.
Taylor expansion induces the dashed arrows in the following diagram.
u
C ∞ (E, R) Lk (E, R)
g
e
eg
sym
e
ee
dk 0 prk
ee
u
“ ww
d
(U, R) '
{0} ‘
''
v‘
∞
Lk (E, R)
C
“ pr u
sym
'' “
’’
‘
k=1
’ '(
‘ ’'
“
‘
u u ’
{∞¬‚at} v w C (E ⊇ {a}, R)‘ w L (E, R)
∞
v
∞ k
‘
(
& u
sym
& ‘“
k=1
& “
‘
u&
( ∞
{d¬‚at} C (E ⊇ {a}, R)/{0}
Note that all spaces in the right two columns except the top right corner are
algebras, the ¬nite product with truncated multiplication. The mappings are
algebrahomomorphisms. And the spaces in the left column are the respective
∞
kernels. If E is a Cb (E, R)regular Banach space, then by (27.20) the vertical
dashed arrow is bibounded. Since R is Hausdor¬ every ‚ ∈ Da E factors over
Taya (E, R) := C ∞ (E ⊇ {a}, R)/{0}, so in this case we can view ‚ as derivation on
the algebra of formal power series. Any continuous linear functional on a countable
product is a sum of continuous linear functionals on ¬nitely many factors.
28.2. Degrees of operational tangent vectors. A derivation ‚ is said to have
d
order at most d, if it factors over the space k=0 Lk (E, R) of polynomials of
sym
degree at most d, i.e. it vanishes on all d¬‚at germs. If no such d exists, then it will
be called of in¬nite order; this may happen only if ‚ does not factor over the space
of formal power series, since if it factors to a bounded linear functional on the latter
space it depends only on ¬nitely many factors. If ‚ factors it must vanish ¬rst on
the ideal of all ¬‚at germs, and secondly the resulting linear functional on Taya (E, R)
must extend to a bounded linear functional on the space of formal power series.
For a results and examples in this direction see (28.3), (28.4), and (28.5). An open
question is to ¬nd operational tangent vectors of in¬nite order.
An operational tangent vector is said to be homogeneous of order d if it factors over
Ld (E, R), i.e. it corresponds to a continuous linear functional ∈ Ld (E, R) via
sym sym
(d)
‚(f ) = ( f d!(0) ). In order that such a functional de¬nes a derivation, we need
exactly that
d’1
Lj (E, R) — Ld’j (E, R) = {0},
Sym sym sym
j=1
28.2
278 Chapter VI. In¬nite dimensional manifolds 28.2
i.e. that vanishes on on the subspace
j’1
Li (E; R) ∨ Lj’i (E; R)
sym sym
i=1
of decomposable elements of Lj (E; R). Here Li (E; R) ∨ Lj’i (E; R) denotes
sym sym sym
the linear subspace generated by all symmetric products ¦∨Ψ of the corresponding
elements. Any such de¬nes an operational tangent vector ‚ j a ∈ Da E of order j
by
‚ j a (f ) := ( j! dj f (a)).
1
Since vanishes on decomposable elements we see from the Leibniz rule that ‚ j
is a derivation, and it is obviously of order j. The inverse bijection is given by
‚ ’ (¦ ’ ‚((¦ —¦ diag)( ’a))), since the complete polarization of a homogeneous
1
polynomial p of degree j is given by j! dj p(0)(v1 , . . . , vj ), and since the remainder
of the Taylor expansion is ¬‚at of order j ’ 1 at a.
Obviously every derivation of order at most d is a unique sum of homogeneous
[d]
derivations of order j for 1 ¤ j ¤ d. For d > 0 we denote by Da E the lin
ear subspace of Da E of operational tangent vectors of homogeneous order d and
(d) d [j]
by Da E := j=1 D the subspace of (non homogeneous) operational tangent
vectors of order at most d.
In more detail any operational tangent vector ‚ ∈ Da E has a decomposition
k’1
‚ [i] + ‚ [k,∞] ,
‚=
i=1
which we obtain by applying ‚ to the Taylor formula with remainder of order k,
see (5.12),
k’1 1
(1 ’ t)k’1 k
1i
d f (a)y i + d f (a + ty)y k dt.
f (a + y) =
(k ’ 1)!
i! 0
i=0
Thus, we have
1i
‚ [i] (f ) := ‚ x ’ d f (a)(x ’ a)i ,
i!
1
(1 ’ t)(k’1) k
[k,∞]
d f (a + t(x ’ a))(x ’ a)k dt .
(f ) := ‚ x ’
‚
k ’ 1!
0
A simple computation shows that all ‚ [i] are derivations. In fact
k
1 k
‚ [k] (f · g) = ‚ (f (j) (0) —¦ ∆) · (g (k’j) (0) —¦ ∆)
k! j
j=0
k
f (j) (0) —¦ ∆ g (k’j) (0)(0(k’j) )
·
= ‚
(k ’ j)!
j!
j=0
k
f (j) (0)(0j ) g (k’j) (0) —¦ ∆
·‚
+
(k ’ j)!
j!
j=0
= g(0) · ‚ [k] (f ) + 0 + · · · + 0 + f (0) · ‚ [k] (g).
28.2
28.4 28. Tangent vectors 279
Hence also ‚ [k,∞] is a derivation. Obviously, ‚ [i] is of order i, and hence we get a
decomposition
d
[j] [d+1,∞]
Da • Da
Da E = ,
j=1
[d+1,∞]
where Da denotes the linear subspace of derivations which vanish on polyno
mials of degree at most d.
28.3. Examples. Queer operational tangent vectors. Let Y ∈ E be an
element in the bidual of E. Then for each a ∈ E we have an operational tangent
vector Ya ∈ Da E, given by Ya (f ) := Y (df (a)). So we have a canonical injection
E ’ Da E.
Let : L2 (E; R) ’ R be a bounded linear functional which vanishes on the subset
[2]
E — E . Then for each a ∈ E we have an operational tangent vector ‚ a ∈ Da E
[2]
given by ‚ a (f ) := (d2 f (a)), since
(d2 (f g)(a)) = (d2 f (a)g(a) + df (a) — dg(a) + dg(a) — df (a) + f (a)d2 g(a))
= (d2 f (a))g(a) + 0 + f (a) (d2 g(a)).
Let E = ( 2 )N be a countable product of copies of an in¬nite dimensional Hilbert
space. A smooth function on E depends locally only on ¬nitely many Hilbert space
[kn ]
variables. Thus, f ’ n ‚Xn (f —¦ injn ) is a well de¬ned operational tangent vector
in D0 E for arbitrary operational tangent vectors Xn of order kn . If (kn ) is an
unbounded sequence and if Xn = 0 for all n it is not of ¬nite order. But only for
k = 1, 2, 3 we know that nonzero tangent vectors of order k exist, see (28.4) below.
28.4. Lemma. If E is an in¬nite dimensional Hilbert space, there exist nonzero
operational tangent vectors of order 2, 3.
Proof. We may assume that E = 2 . For k = 2 one knows that the closure of
L( 2 , R) ∨ L( 2 , R) in L2 ( 2 , R) consists of all symmetric compact operators, and
sym
the identity is not compact.
For k = 3 we show that for any A in the closure of L( 2 , R) ∨ L2 ( 2 , R) the
sym
following condition holds:
A(ei , ej , ek ) ’ 0 i, j, k ’ ∞.
(1) for
Since this condition is invariant under symmetrization it su¬ces to consider A ∈
2
—L( 2 , 2 ), which we may view as a ¬nite dimensional and thus compact operator
2
’ L( 2 , 2 ). Then A(ei ) ’ 0 for i ’ ∞, since this holds for each continuous
linear functional on 2 . The trilinear form A(x, y, z) := i xi yi zi is in L3 ( 2 , R)
sym
and obviously does not satisfy (1).
28.4
280 Chapter VI. In¬nite dimensional manifolds 28.7
28.5. Proposition. Let E be a convenient vector space with the following two
properties:
(1) The closure of 0 in C ∞ (E ⊇ {0}, R) consists of all ¬‚at germs.
(2) The quotient Tay0 (E, R) = C ∞ (E ⊇ {0}, R)/{0} with the bornological
topology embeds as topological linear subspace into the space k Lk (E; R)
sym
of formal power series.
Then each operational tangent vector on E is of ¬nite order.
∞
Any Cb regular Banach space, in particular any Hilbert space has these properties.
Proof. Let ‚ ∈ D0 E be an operational tangent vector. By property (1) it factors
to a bounded linear mapping on Tay0 (E, R), it is continuous in the bornological
topology, and by property (2) and the theorem of HahnBanach ‚ extends to a
continuous linear functional on the space of all formal power series and thus depends
only on ¬nitely many factors.
∞
A Cb regular Banach space E has property (1) by (27.19), and it has property (2)
∞
by E. Borel™s theorem (15.4). Hilbert spaces are Cb regular by (15.5).
28.6. De¬nition. A convenient vector space is said to have the (bornological)
approximation property if E — E is dense in L(E, E) in the bornological locally
convex topology.
For a list of spaces which have the bornological approximation property see (6.6)“
(6.14).
28.7. Theorem. Let E be a convenient vector space which has the approxima
tion property. Then we have Da E ∼ E . So if E is in addition re¬‚exive, each
=
operational tangent vector is a kinematic one.
Proof. We may suppose that a = 0. Let ‚ : C ∞ (E ⊇ {0}, R) ’ R be a derivation
at 0, so it is bounded linear and satis¬es ‚(f · g) = ‚(f ) · g(0) + f (0) · ‚(g). Then
we have ‚(1) = ‚(1 · 1) = 2‚(1), so ‚ is zero on constant functions.
Since E = L(E, R) is continuously embedded into C ∞ (E, R), ‚E is an element
of the bidual E . Obviously, ‚ ’ (‚E )0 is a derivation which vanishes on a¬ne
functions. We have to show that it is zero. We call this di¬erence again ‚. For
f ∈ C ∞ (U, R) where U is some radial open neighborhood of 0 we have
1
f (x) = f (0) + df (tx)(x)dt,
0
1
thus ‚(f ) = ‚(g), where g(x) := 0 df (tx)(x)dt. By assumption, there is a net
± ∈ E — E ‚ L(E, E) of bounded linear operators with ¬nite dimensional image,
which converges to IdE in the bornological topology of L(E, E). We consider g± ∈
1
C ∞ (U, R), given by g± (x) := 0 df (tx)( ± x)dt.
Claim. g± ’ g in C ∞ (U, R).
1
We have g(x) = h(x, x) where h ∈ C ∞ (U —E, R) is just h(x, y) = 0 df (tx)(y)dt. By
cartesian closedness, the associated mapping h∨ : U ’ E ‚ C ∞ (E, R)) is smooth.
28.7
28.9 28. Tangent vectors 281
Since : L(E, E) ’ L(E , E ) is bounded linear, the net ± converges to IdE in
L(E , E ). The mapping (h∨ )— : L(E , E ) ‚ C ∞ (E , E ) ’ C ∞ (U, E ) is bounded
linear, thus (h∨ )— ( ± ) converges to h∨ in C ∞ (U, E ). By cartesian closedness, the
net ((h∨ )— ( ± ))§ converges to h in C ∞ (U — E, R). Since the diagonal mapping
δ : U ’ U — E is smooth, the mapping δ — : C ∞ (U — E, R) ’ C ∞ (U, R) is
continuous and linear, so ¬nally g± = δ — (((h∨ )— ( ± ))§ ) converges to δ — (h) = g.
Claim. ‚(g± ) = 0 for all ±. This ¬nishes the proof.
n
•i — xi ∈ E — E ‚ L(E, E). We have
Let =
± i=1
1
g± (x) = df (tx) •i (x)xi dt
0 i
1
= •i (x) df (tx)(xi )dt =: •i (x)hi (x),
0
i i
•i · hi =
‚(g± ) = ‚ ‚(•i )hi (0) + •i (0)‚(hi ) = 0.
i i
28.8. Remark. There are no nonzero operational tangent vectors of order 2 on
E if and only if E ∨ E ‚ L2 (E; R) is dense in the bornological topology. This
sym
seems to be rather near the bornological approximation property, and one may
suspect that theorem (28.7) remains true under this weaker assumption.
28.9. Let U ⊆ E be an open subset of a convenient vector space E. The operational
tangent bundle DU of U is simply the disjoint union a∈U Da E. Then DU is in
bijection to the open subset U —D0 E of E—D0 (E) via ‚a ’ (a, ‚—¦( ’a)— ). We use
this bijection to put a smooth structure on DU . Let now g : E ⊃ U ’ V ‚ F be
a smooth mapping, then g — : C ∞ (W, R) ’ C ∞ (g ’1 (W ), R) is bounded and linear
for all open W ‚ V . The adjoints of these mappings uniquely de¬ne a mapping
Dg : DU ’ DV by (Dg.‚)(f ) := ‚(f —¦ g).
Lemma. Dg : DU ’ DV is smooth.
Proof. Via the canonical bijections DU ∼ U — D0 E and DV ∼ V — D0 F the
= =
mapping Dg corresponds to
U — D0 E ’ V — D0 F
+a)— —¦ g — —¦ ( ’g(a))—
(a, ‚) ’ g(a), ‚ —¦ (
+a) ’ g(a))— .
= g(a), ‚ —¦ (g(
In order to show that this is smooth, its enough to consider the second component
and we compose it with the embedding D0 F ’ W 0 C ∞ (W, R) . The associated
mapping U — D0 E — C ∞ (W, R) ’ R is given by
(a, ‚, f ) ’ ‚ f —¦ (g( +a) ’ g(a)) ,
(1)
28.9
282 Chapter VI. In¬nite dimensional manifolds 28.11
where f —¦ (g( +a) ’ g(a)) is smooth on the open 0neighborhood Wa := {y ∈ E :
g(y + a) ’ g(a) ∈ W } = g ’1 (g(a) + W ) ’ a in E. Now let a : R ’ U be a smooth
curve and I a bounded interval in R. Then there exists an open neighborhood UI,W
of 0 in E such that UI,W ⊆ Wa(t) for all t ∈ I. Then the mapping (1), composed
with a : I ’ U , factors as
I — D0 E — C ∞ (W, R) ’ C ∞ (UI,W , R) — C ∞ (UI,W , R) ’ R,
given by
(t, ‚, f ) ’ ‚UI,W , f —¦ g( ’ ‚UI,W (f —¦(g(
+a(t))’g(a(t)) +a(t))’g(a(t))))),
which is smooth by cartesian closedness.
(k)
28.10. Let E be a convenient vector space. Recall from (28.2) that Da E is the
space of all operational tangent vectors of order ¤ k. For an open subset U in a
convenient vector space E and k > 0 we consider the disjoint union
Da E ∼ U — D0 E ⊆ E — D0 E.
(k) (k)
D(k) U := (k)
=
a∈U
Lemma. For a smooth mapping f : E ⊃ U ’ V ‚ F the smooth mapping Df :
DU ’ DV from (28.9) induces smooth mappings D(k) f : D(k) U ’ D(k) V .
(k) (k)
Proof. We only have to show that Da f maps Da E into Df (a) F , because smooth
ness follows then by restriction.
The pullback f — : C ∞ (V, R) ’ C ∞ (U, R) maps functions which are ¬‚at of order k
at f (a) to functions which are ¬‚at of the same order at a. Thus, Da f maps the
(k) (k)
corresponding annihilator Da U into the annihilator Df (a) V .
28.11. Lemma.
(1) The chain rule holds in general: D(f —¦ g) = Df —¦ Dg and D(k) (f —¦ g) =
D(k) f —¦ D(k) g.
(2) If g : E ’ F is a bounded a¬ne mapping then Dx g commutes with the
restriction and the projection to the subspaces of derivations which are ho
mogeneous of degree k > 1.
(3) If g : E ’ F is a bounded a¬ne mapping with linear part = g ’ g(0) :
[k] [k]
E ’ F then Dx g : Dx E ’ Dg(x) F is induced by the linear mappings
(Lk ( ; R))— : Lk (E, R)— ’ Lk (F, R)— .
sym sym sym
[1]
(4) If g : E ’ R is bounded linear we have Dg.Xx = D(1) g.Xx .
Remark that if g is not a¬ne then in general Dg does not respect the subspaces of
derivations which are homogeneous of degree k > 1:
[k]
In fact let g : E ’ R be a homogeneous polynomial of degree k on which ‚ ∈ D0 E
does not vanish. Then by (4) we have that 0 = ‚(g) = Dg(‚) ∈ R ∼ D0 R = D0 R.
[1]
=
28.11
28.12 28. Tangent vectors 283
Proof. (1) is obvious.
For (2) let Xx ∈ Dx E and f ∈ C ∞ (F, R). Then we have
(Dg.Xx )[k] (f ) = (Dg.Xx )( k! dk f (g(x))( ’g(x))k )
1
k
) ’ g(x))k )
1
= k! Xx (d f (g(x))(g(
[k] [k]
Xx (f —¦ g)
(Dg.Xx )(f ) =
Xx ( k! dk (f —¦ g)(x)( ’x)k )
1
=
k
’x))k ).
1
= k! Xx (d f (g(x))( (
These expressions are equal.
ww y wD E
[k]
Dx E Dx E x
u
u u
Dg Dg
Dg
ww y wD F
[k]
Dg(x) F Dg(x) F x
For (3) we take • ∈ Lk (E; R) which vanishes on all decomposable forms, and let
sym
[k]
k
Xx = ‚• x ∈ Dx E be the corresponding homogeneous derivation. Then
k k
(Dg.‚• x )(f ) = ‚• x (f —¦ g)
= •( k! dk (f —¦ g)(x))
1
= •( k! dk f (g(x)) —¦ k
1
)
= (Lk ( ; R)— •)( k! dk f (g(x)))
1
sym
[k]

= ‚Lk (f ).
;R)— • g(x)
sym (
y wL
[k] k
Dx E Sym (E, R)
Lk ( , R)—
u
D[k] g
u Sym
y wL
[k] k
Dg(x) F Sym (F, R)
(4) is a special case of (2).
28.12. The operational and the kinematic tangent bundles. Let M be a
u±
manifold with a smooth atlas (M ⊃ U± ’’ E± )±∈A . We consider the following
’
equivalence relation on the disjoint union
D(u± (U± )) — {±},
D(u± (U± )) :=
±∈A ±∈A
(‚, ±) ∼ (‚ , β) ⇐’ D(u±β )‚ = ‚.
We denote the quotient set by DM and call it the operational tangent bundle
of M . Let πM : DM ’ M be the obvious foot point projection, let DU± =
28.12
284 Chapter VI. In¬nite dimensional manifolds 28.12
’1
πM (U± ) ‚ DM , and let Du± : DU± ’ D(u± (U± )) be given by Du± ([‚, ±]) = ‚.
So Du± ([‚ , β]) = D(u±β )‚ .
The charts (DU± , Du± ) form a smooth atlas for DM , since the chart changings are
given by
Du± —¦ (Duβ )’1 = D(u±β ) : D(uβ (U±β )) ’ D(u± (U±β )).
This chart changing formula also implies that the smooth structure on DM depends
only on the equivalence class of the smooth atlas for M .
The mapping πM : DM ’ M is obviously smooth. The natural topology is
automatically Hausdor¬: X, Y ∈ DM can be separated by open sets of the form
’1
πM (V ) for V ‚ M , if πM (X) = πM (Y ), since M is Hausdor¬, and by open subsets
of the form (T u± )’1 (E± — W ) for W open in E± , if πM (X) = πM (Y ) ∈ U± .
’1
For x ∈ M the set Dx M := πM (x) is called the operational tangent space at x or
the ¬ber over x of the operational tangent bundle. It carries a canonical convenient
vector space structure induced by Dx (u± ) := Du± Dx M : Du± (x) E± ∼ D0 (E± ) for
=
some (equivalently any) ± with x ∈ U± .
Let us construct now the kinematic tangent bundle. We consider the following
equivalence relation on the disjoint union
U± — E± — {±},
±∈A
(x, v, ±) ∼ (y, w, β) ⇐’ x = y and d(u±β )(uβ (x))w = v
and denote the quotient set by T M , the kinematic tangent bundle of M . Let
’1
πM : T M ’ M be given by πM ([x, v, ±]) = x, let T U± = πM (U± ) ‚ T M ,
and let T u± : T U± ’ u± (U± ) — E± be given by T u± ([x, v, ±]) = (u± (x), v). So
T u± ([x, w, β]) = (u± (x), d(u±β )(uβ (x))w).
The charts (T U± , T u± ) form a smooth atlas for T M , since the chart changings are
given by
T u± —¦ (T uβ )’1 : uβ (U±β ) — Eβ ’ u± (U±β ) — E± ,
(x, v) ’ (u±β (x), d(u±β )(x)v).
This chart changing formula also implies that the smooth structure on T M depends
only on the equivalence class of the smooth atlas for M .
The mapping πM : T M ’ M is obviously smooth. It is called the (foot point)
projection of M . The natural topology is automatically Hausdor¬; this follows
from the bundle property and the proof is the same as for DM above.
’1
For x ∈ M the set Tx M := πM (x) is called the kinematic tangent space at x or
the ¬ber over x of the tangent bundle. It carries a canonical convenient vector
space structure induced by Tx (u± ) := T u± Tx M : Tx M ’ {x} — E± ∼ E± for some
=
(equivalently any) ± with x ∈ U± .
Note that the kinematic tangent bundle T M embeds as a subbundle into DM ; also
for each k ∈ N the same construction as above gives us tangent bundles D(k) M
which are subbundles of DM .
28.12
28.15 28. Tangent vectors 285
28.13. Let us now give an obvious description of T M as the space of all velocity
vectors of curves, which explains the name ˜kinematic tangent bundle™: We put
on C ∞ (R, M ) the equivalence relation : c ∼ e if and only if c(0) = e(0) and in
d
one (equivalently each) chart (U, u) with c(0) = e(0) ∈ U we have dt 0 (u —¦ c)(t) =
d
dt 0 (u —¦ e)(t). We have the following diagram
u e C (R, M )
C ∞ (R, M )/ ∼ ∞
δ ee
ee
∼δ
uh
e u
ev
= 0
wM
TM π M
where to c ∈ C ∞ (R, M ) we associate the tangent vector δ(c) := [c(0), ‚t 0 (u± —¦
‚
c)(t), ±]. It factors to a bijection C ∞ (R, M )/ ∼’ T M , whose inverse associates to
[x, v, ±] the equivalence class of t ’ u’1 (u± (x) + h(t)v) for h a small function with
±
h(t) = t near 0.
Since the c∞ topology on R — E± is the product topology by corollary (4.15), we
can choose h uniformly for (x, v) in a piece of a smooth curve. Thus, a mapping g :
T M ’ N into another manifold is smooth if and only if g—¦δ : C ∞ (R, M ) ’ N maps
˜smooth curves™ to smooth curves, by which we mean C ∞ (R2 , M ) to C ∞ (R, N ).
28.14. Lemma. If a smooth manifold M and the squares of its model spaces
are smoothly paracompact, then also the kinematic tangent bundle T M is smoothly
paracompact.
If a smooth manifold M and V — D0 V for any of its model spaces V are smoothly
paracompact, then also the operational tangent bundle DM is smoothly paracom
pact.
Proof. This is a particular case of (29.7) below.
28.15. Tangent mappings. Let f : M ’ N be a smooth mapping between
manifolds. Then f induces a linear mapping Dx f : Dx M ’ Df (x) N for each
x ∈ M by (Dx f.‚x )(h) = ‚x (h —¦ f ) for h ∈ C ∞ (N ⊇ {f (x)}, R). These give a
mapping Df : DM ’ DN . If (U, u) is a chart around x and (V, v) is one around
f (x), then Dv —¦ Df —¦ (Du)’1 = D(v —¦ f —¦ u’1 ) is smooth by lemma (28.9). So
Df : DM ’ DN is smooth.
By lemma (28.10), Df restricts to smooth mappings D(k) f : D(k) M ’ D(k) N
and to T f : T M ’ T N . We check the last statement for open subsets M and
N of convenient vector spaces. (Df.Xa )(g) = Xa (g —¦ f ) = d(g —¦ f )(a)(X) =
dg(f (a))df (a)X = (df (a)X)f (a) (g).
If f ∈ C ∞ (M, E) for a convenient vector space E, then Df : DM ’ DE =
E — D0 E. We then de¬ne the di¬erential of f by df := pr2 —¦ Df : DM ’ D0 E. It
(k)
restricts to smooth ¬berwise linear mappings D(k) M ’ D0 E and df : T M ’ E.
If f ∈ C ∞ (M, R), then df : DM ’ R. Let Id denote the identity function on R,
then (T f.‚x )(Id) = ‚x (Id —¦f ) = ‚x (f ), so we have df (‚x ) = ‚x (f ).
28.15
286 Chapter VI. In¬nite dimensional manifolds 28.16
The mapping f ’ df is bounded linear C ∞ (M, R) ’ C ∞ (DM, R). That it is linear
and has values in this space is obvious. So by the smooth uniform boundedness
principle (5.26) it is enough to show that f ’ df.Xx = Xx (f ) is bounded for all
Xx ∈ DM , which is true by de¬nition of DM .
28.16. Remark. From the construction of the tangent bundle in (28.12) it is
immediately clear that
T (pr ) T (pr )
T M ’ ’ 1 T (M — N ) ’ ’ 2 T N
’’ ’ ’’ ’
is also a product, so that T (M — N ) = T M — T N in a canonical way.
We investigate D0 (E — F ) for convenient vector spaces. Since D0 is a functor for 0
preserving maps, we obtain linear sections D0 (injk ) : D0 (Ek ) ’ D0 (E1 — E2 ) and
hence a section D0 (inj1 )+D0 (inj2 ) : D0 (E1 )•D0 (E2 ) ’ D0 (E1 •E2 ). The comple
ment of the image is given by the kernel of the linear mapping (D0 (pr1 ), D0 (pr2 )) :
D0 (E1 • E2 ) ’ D0 (E1 ) • D0 (E2 ).
w D (E ) )
Id
D0 (E1 )&
&
x )
x (pr
x
0 1
&prD (E
&D0 (injE
x
injD0 (E1 ) )
)
&
(
&
x
x
0 1
1
D E1 )
0
w D (E w D (E ) • D (E )
• E2 )&
D0 (E0 ) • D0 (E1 )
)
&
x
x (inj
x
0 1 0 0 0 1
& &D0 (prE
xpr
)
injD (E ) &
7 (
&
x x
2
D0 (E2 )
D E2 )
0
0 2
w D (E )
D0 (E2 ) 0 2
2
Lemma. In the case E1 = = E2 this mapping is not injective.
Proof. The space L2 (E1 — E2 , E1 — E2 ; R) can be viewed as L2 (E1 , E1 ; R) —
L2 (E1 , E2 ; R) — L2 (E2 , E1 ; R) — L2 (E2 , E2 ; R) and the subspace formed by those
forms whose (2,1) and (1,2) components with respect to this decomposition are
compact considered as operators in L( 2 , 2 ) ∼ L2 ( 2 , 2 ; R) is a closed subspace.
=
So, by HahnBanach, there is a nontrivial continuous linear functional : L2 ( 2 —
22
, — 2 ; R) ’ R vanishing on this subspace. We claim that the linear mapping
‚ : C ∞ ( 2 — 2 , R) f ’ (f (0, 0)) ∈ R is an operational tangent vector of 2 — 2