41.6. The graph topology. For f ∈ C(M, N ) let graphf : M ’ M — N be

given by graphf (x) = (x, f (x)), the graph mapping of f .

The WO-topology or wholly open topology on C(M, N ) is given by the basis {f ∈

C(M, N ) : f (M ) ‚ U }, where U runs through all open sets in N . It is not

Hausdor¬, since mappings with the same image cannot be separated.

The graph topology or WO0 -topology on C(M, N ) is induced by the mapping

graph : C(M, N ) ’ (C(M, M — N ), WO-topology).

A basis for it is given by all sets of the form {f ∈ C(M, N ) : graphf (M ) ⊆ U },

where U runs through all open sets in M — N . This topology is Hausdor¬ since it is

¬ner than the compact-open topology. Note that a continuous mapping g : N ’ P

induces a continuous mapping g— : C(M, N ) ’ C(M, P ) for the WO0 -topology,

since graphg—¦f = (Id —g) —¦ graphf .

If M is paracompact and (N, d) is a metric space, then for f ∈ C(M, N ) the sets

{g ∈ C(M, N ) : d(g(x), f (x)) < µ(x) for all x ∈ M } form a basis of neighborhoods,

where µ runs through all positive continuous functions on M . This is easily seen.

41.7. Lemma. Let N be metrizable, and let M satisfy one of the following con-

ditions:

(1) M is locally compact with a countable basis of open sets.

(2) M = R(N) .

Then for any sequence (fn ) in C(M, N ) the following holds: (fn ) converges to f

in the WO0 -topology if and only if there exists a compact set K ⊆ M such that fn

equals f o¬ K for all but ¬nitely many n, and fn |K converges to f |K uniformly.

Note that in case (2) we get fn = f for all but ¬nitely many n, since f di¬ers from

fn on a c∞ -open subset.

Proof. Clearly, the condition above implies convergence. Conversely, let (fn ) and

o

f in C(M, N ) be such that the condition does not hold. In case (1) let Kn ‚ Kn+1

be a basis of the compact sets in M , and in case (2) let Kn := {x ∈ Rn ‚ R(N) :

|xi | ¤ n for i ¤ n}. Then either fn does not converge to f in the compact-open

topology, or there exists xn ∈ Kn with d(fn (xn ), f (xn )) =: µn > 0. Then (xn ) is

/

without cluster point in M : This is obvious in case (1), and in case (2) this can be

seen by the following argument: Assume that there exists a cluster point y. Let N

be so large that supp(y) ‚ {0, . . . , N } and |y i | ¤ N ’ 1 for all i. Then we de¬ne

kn ∈ N and δn > 0 by

for n ¤ N or supp(xn ) ⊆ {1, . . . , n}

kn := n, δn := 1

kn := min{i > n : xi = 0}, δn := |xkn | otherwise

n n

Then xn ’ y ∈ U := {z : |z ki | < δi for all i} for n > N , so y cannot be a cluster

/

point.

Then by a paracompactness argument and the second description of the WO0 -

topology the set {(x, y) ∈ M — N : if x = xn then d(f (xn ), y) < µn } is an open

neighborhood of graphf (M ) not containing any graphfn (M ). So fn cannot converge

to f in the WO0 -topology.

41.7

436 Chapter IX. Manifolds of mappings 41.10

41.8. Lemma. Let E be a convenient vector space, and suppose that M satis¬es

the following condition:

(1) Each neighborhood of each point contains a sequence without cluster point

in M .

Then for f ∈ C(M, E) we have tf ’ 0 in the WO0 -topology for t ’ 0 in R if and

only if f = 0.

Moreover, each open subset in an in¬nite dimensional locally convex space has prop-

erty (1).

Proof. The mapping f ’ g —¦ f is continuous in the WO0 -topologies, so by com-

posing with bounded linear functionals on E we may suppose that E = R.

Suppose that f = 0, say f (x) = 2 for some x. Then f (y) > 1 for y in some

neighborhood U of x, which contains a sequence xn without cluster point in M .

Then {(x, y) ∈ M — R : if x = xn then y < 1/n} is an open neighborhood of

graph0 (M ) not containing any graphtf (M ) for t = 0. So tf cannot converge to 0

in the WO0 -topology.

For the last assertion we have to show that the unit ball of each seminorm p in

an in¬nite dimensional locally convex vector space M contains a sequence without

cluster point. If the seminorm has non-trivial kernel p’1 (0) then (n.x)n for 0 =

x ∈ p’1 (0) has this property. If p has trivial kernel, it is a norm, and the unit

ball in the normed space (M, p) contains a sequence without cluster point, since

otherwise the unit ball would be compact, and (M, p) would be ¬nite dimensional.

This sequence has also no cluster point in M , since M has a ¬ner topology.

41.9. The COk -topology on spaces of smooth mappings. Let M and N

be smooth manifolds, possibly in¬nite dimensional. For 0 ¤ k ¤ ∞ the compact-

open C k -topology or COk -topology on the space C ∞ (M, N ) of all smooth mappings

M ’ N is induced by the k-jet extension (41.3) from the CO-topology

j k : C ∞ (M, N ) ’ (C(M, J k (M, N )), CO).

We conclude with some remarks. If M is in¬nite dimensional it would be more

natural to replace the system of compact sets in M by the system of all subsets

on which each smooth real valued function is bounded. Since these topologies will

play only minor roles in this book we do not develop them here.

41.10. Whitney C k -topology. Let M and N be smooth manifolds, possibly

in¬nite dimensional. For 0 ¤ k ¤ ∞ the Whitney C k -topology or WOk -topology

on the space C ∞ (M, N ) of all smooth mappings M ’ N is induced by the k-jet

extension (41.3) from the WO-topology

j k : C ∞ (M, N ) ’ (C(M, J k (M, N )), WO).

A basis for the open sets is given by all sets of the form {f ∈ C ∞ (M, N ) : j k f (M ) ‚

U }, where U runs through all open sets in the smooth manifold J k (M, N ). A

41.10

41.11 41. Jets and Whitney topologies 437

smooth mapping g : N ’ P induces a smooth mapping J k (M, g) : J k (M, N ) ’

J k (M, P ) by (41.3.4), and thus in turn a continuous mapping g— : C ∞ (M, N ) ’

C ∞ (M, P ) for the WOk -topologies for each k.

For a convenient vector space E and for a manifold M modeled on in¬nite di-

mensional Fr´chet spaces (so that there the c∞ -topology coincides with the locally

e

convex one) we see from (41.8) that for f ∈ C ∞ (M, E) we have t.f ’ 0 for t ’ 0

in the WOk -topology if and only if f = 0. So (C ∞ (M, E), WOk ) does not contain

a non-trivial topological vector space if M is in¬nite dimensional.

If M is compact, then the WOk -topology and the COk -topology coincide on the

space C ∞ (M, N ) for all k.

41.11. Lemma. Let M , N be smooth manifolds, where M is ¬nite dimensional

and second countable, and where N is metrizable. Then J ∞ (M, N ) is also a metriz-

able manifold. If, moreover, N is second countable then also J ∞ (M, N ) is also

second countable.

o

Let Kn ‚ Kn+1 ‚ Kn+1 be a compact exhaustion of M . Then the following is a

basis of open sets for the Whitney C ∞ -topology:

M (U, m) := {f ∈ C ∞ (M, N ) : j mn f (M \ Kn ) ‚ Un },

o

where (mn ) is any sequence in N and where Un ‚ J mn (M, N ) is an open subset.

Proof. Looking at (41.3) we see that J ∞ (M, N ) is a bundle over M — N with

Fr´chet spaces as ¬bers, so it is metrizable. We can also write

e

M (U, m) := {f ∈ C ∞ (M, N ) : j ∞ f (M \ Kn ) ‚ (πmn )’1 Un }.

∞

o

By pulling up to higher jet bundles, we may assume that mn is strictly increasing. If

we put Vn = (πmn )’1 Un , we may then replace Vn by V0 © · · · © Vn without changing

∞

o o

M (U, m). But then we may replace M \ Kn by Kn+1 \ Kn without changing the

set. Using compactness of j ∞ f (Kn+1 \ Kn ) and that J ∞ (M, N ) carries the initial

o

topology with respect to all projections πl : J ∞ (M, N ) ’ J l (M, N ) by (41.3.6),

∞

we get an equivalent basis of open sets given by

M (U ) := {f ∈ C ∞ (M, N ) : j ∞ f (Kn+1 \ Kn ) ‚ Un },

o

where now Un ‚ J ∞ (M, N ) is a sequence of open sets. It is obvious that this

basis generates a topology which is ¬ner than the WO∞ -topology. To show the

converse let f ∈ M (U ). Let d be a compatible metric on the metrizable manifold

J ∞ (M, N ), and let 0 < µn be smaller than the distance between the compact set

j ∞ f (Kn+1 \ Kn ) and the complement of its open neighborhood Un . Let µ be a

o

o

positive continuous function on M such that 0 < µ(x) < µn for x ∈ Kn+1 \ Kn ,

and consider the open set W := {σ ∈ J ∞ (M, N ) : d(σ, j ∞ f (±(σ))) < µ(±(σ))} in

J ∞ (M, N ). Then f ∈ {g ∈ C ∞ (M, N ) : j ∞ g(M ) ‚ W } ⊆ M (U ).

41.11

438 Chapter IX. Manifolds of mappings 41.14

41.12. Corollary. Let M , N be smooth manifolds, where M is ¬nite dimen-

sional and second countable, and where N is metrizable. Then the COk -topology is

metrizable. If N is also second countable then so is the COk -topology.

Proof. Use (41.11) and [Bourbaki, 1966, X, 3.3].

41.13. Comparison of topologies on C ∞ (M, E). Let p : E ’ M be a smooth

¬nite dimensional vector bundle over a ¬nite dimensional second countable base

∞

manifold M . We consider the space Cc (M ← E) of all smooth sections of E with

compact support, equipped with the bornological locally convex topology from

(30.4),

∞ ∞

Cc (M ← E) = lim CK (M ← E),

’’

K

∞

where K runs through all compact sets in M and each of the spaces CK (M ←

f — T N ) is equipped with the topology of uniform convergence (on K) in all deriva-

tives separately, as in (30.4), reformulated for the bornological topologies. Consider

also the space C ∞ (M, E) of all smooth mappings M ’ E, equipped with the Whit-

ney C ∞ -topology, and the subspace C ∞ (M ← E) of all smooth sections, with the

induced topology.

Lemma. Then the canonical injection

Cc (M ← E) ’ C ∞ (M, E)

∞

is a topological embedding. The subspace C ∞ (M ← E) is a vector space, but scalar

multiplication is jointly continuous in the induced topology on it if and only if M

is compact or the ¬ber is 0. The maximal topological vector space contained in

C ∞ (M ← E) is just Cc (M ← E).

∞

Proof. That the injection is an embedding is clear by contemplating the descrip-

tion of the Whitney C ∞ -topology given in lemma (41.11), which obviously is the

lim ∞

inductive limit topology ’ CKn (E). The rest follows from (41.7) since t.f ’ 0 for

’

t ’ 0 in in C (M, E) for WO∞ if and only if t.j ∞ f ’ 0 in C ∞ (M, J ∞ (E)) for

∞

the WO0 -topology.

41.14. Tubular neighborhoods. Let M be an (embedded) submanifold of a

smooth ¬nite dimensional manifold N . Then the normal bundle of M in N is the

π

vector bundle N (M ) := (T N |M )/T M ’ M with ¬ber Tx N/Tx M over a point

’

x ∈ M . A tubular neighborhood of M in N consists of:

˜

(1) A ¬berwise radial open neighborhood U ‚ N (M ) of the 0-section in the

normal bundle

˜

(2) A di¬eomorphism • : U ’ U ‚ N onto an open neighborhood U of M

in N , which on the 0-section coincides with the projection of the normal

bundle.

It is well known that tubular neighborhoods exist.

41.14

439

42. Manifolds of Mappings

42.1. Theorem. Manifold structure of C∞ (M, N ). Let M and N be smooth

¬nite dimensional manifolds. Then the space C∞ (M, N ) of all smooth mappings

from M to N is a smooth manifold, modeled on spaces Cc (M ← f — T N ) of smooth

∞

sections with compact support of pullback bundles along f : M ’ N over M .

Proof. Choose a smooth Riemannian metric on N . Let exp : T N ⊇ U ’ N be

the smooth exponential mapping of this Riemannian metric, de¬ned on a suitable

open neighborhood of the zero section. We may assume that U is chosen in such

a way that (πN , exp) : U ’ N — N is a smooth di¬eomorphism onto an open

neighborhood V of the diagonal.

For f ∈ C ∞ (M, N ) we consider the pullback vector bundle

w

—

πN f

—

M —N T N f TN TN

πN

f — πN

u u

w N.

f

M

For f , g ∈ C ∞ (M, N ) we write f ∼ g if f and g agree o¬ some compact subset in

M . Then Cc (M ← f — T N ) is canonically isomorphic to the space

∞

Cc (M, T N )f := {h ∈ C ∞ (M, T N ) : πN —¦ h = f, h ∼ 0 —¦ f }

∞

via s ’ (πN f ) —¦ s and (IdM , h) ← h. We consider the space Cc (M ← f — T N )

— ∞

of all smooth sections with compact support and equip it with the inductive limit

topology

Cc (M ← f — T N ) = inj lim CK (M ← f — T N ),

∞ ∞

K

∞

where K runs through all compact sets in M and each of the spaces CK (M ←

f — T N ) is equipped with the topology of uniform convergence (on K) in all deriva-

tives separately, as in (30.4), reformulated for the bornological topology; see also

(6.1). Now let

Uf := {g ∈ C ∞ (M, N ) : (f (x), g(x)) ∈ V for all x ∈ M, g ∼ f },

uf : Uf ’ Cc (M ← f — T N ),

∞

uf (g)(x) = (x, exp’1 (g(x))) = (x, ((πN , exp)’1 —¦ (f, g))(x)).

f (x)

Then uf is a bijective mapping from Uf onto the set {s ∈ Cc (M ← f — T N ) :

∞

s(M ) ⊆ f — U = (πN f )’1 (U )}, whose inverse is given by u’1 (s) = exp —¦(πN f ) —¦ s,

— —

f

where we view U ’ N as ¬ber bundle. The set uf (Uf ) is open in Cc (M ← f — T N )

∞

for the topology described above, see (30.10).

Now we consider the atlas (Uf , uf )f ∈C∞ (M,N ) for C∞ (M, N ). Its chart change

mappings are given for s ∈ ug (Uf © Ug ) ⊆ Cc (M ← g — T N ) by

∞

(uf —¦ u’1 )(s) = (IdM , (πN , exp)’1 —¦ (f, exp —¦(πN g) —¦ s))

—

g

’1

= („f —¦ „g )— (s),

42.1

440 Chapter IX. Manifolds of mappings 42.3

where „g (x, Yg(x) ) := (x, expg(x) (Yg(x) )) is a smooth di¬eomorphism „g : g — T N ⊇

g — U ’ (g — IdN )’1 (V ) ⊆ M — N which is ¬ber respecting over M .

Smooth curves in Cc (M ← f — T N ) are smooth sections of the bundle pr— f — T N ’

∞

2

R—M , which have compact support in M locally in R. The chart change uf —¦u’1 = g

’1

(„f —¦ „g )— is de¬ned on an open subset and it is also smooth by (30.10).

Finally, following (27.1), the natural topology on C∞ (M, N ) is the identi¬cation

topology from this atlas (with the c∞ -topologies on the modeling spaces), which is

obviously ¬ner than the compact-open topology and thus Hausdor¬.

’1

The equation uf —¦ u’1 = („f —¦ „g )— shows that the smooth structure does not

g

depend on the choice of the smooth Riemannian metric on N .

42.2. Remarks. We denote the manifold of all smooth mappings from M to N by

C∞ (M, N ) because otherwise the set C ∞ (M, Rn ) would appear with two di¬erent

convenient structures, see (6.1) or (30.1), where the other one was treated. From

the last sentence of the proof above it follows that for a compact smooth M the

manifold C∞ (M, Rn ) is di¬eomorphic to the convenient vector space C ∞ (M, R)n .

We describe now another topology on C∞ (M, N ): Consider ¬rst the WO∞ -topology

on C ∞ (M, N ) from (41.10) and re¬ne it such that each equivalence class (of smooth

mappings di¬ering only on compact subsets) from the beginning of the proof above

becomes open. For this topology all chart mappings are homeomorphisms into open

subsets of Cc (M ← f — T N ) with the bornological topology, and the chart changes

∞

are also homeomorphisms, by (41.10) and (41.13). With this topology C ∞ (M, N )

is also a topological manifold, modeled on locally convex spaces Cc (M ← f — T N ),

∞

which, however, do not carry the c∞ -topologies. It is even a smooth manifold

in a stronger sense (all derivatives of chart changes are continuous), and this is

the structure used in [Michor, 1980c]. This smooth structure and the natural

one described above in (42.1) have the same smooth curves (use (30.9) and (42.5)

below). The natural topology is the ¬nal topology with respect to all these smooth

curves. It is strictly ¬ner if M is not compact.

42.3. Proposition. For ¬nite dimensional second countable manifolds M , N the

smooth manifold C∞ (M, N ) has separable connected components and is smoothly

paracompact and Lindel¨f. If M is compact, it is metrizable.

o

Proof. Each connected component of a mapping f is contained in the open equiv-

alence class {g : g ∼ f } of f consisting of those smooth mappings which di¬er

from f only on compact subsets. This equivalence class is the countable induc-

tive limit in the category of topological spaces of the sets {g : g = f o¬ K} of

all mappings which di¬er from f only on members Kn of a countable exhaus-

tion of M with compact sets, see (30.9), since a smooth curve locally has values

in these steps {g : g = f o¬ Kn }. By (41.12) the steps are metrizable and sec-

ond countable. Thus, {g : g ∼ f } is Lindel¨f and separable. Since its model

o

spaces Cc (M ← h— T N ) are smoothly paracompact by (30.4), by (16.10) the space

∞

{g : g ∼ f } is smoothly paracompact, and since C∞ (M, N ) is the disjoint union of

such open sets, it is smoothly paracompact, too.

42.3

42.4 42. Manifolds of mappings 441

42.4. Manifolds of mappings with an in¬nite dimensional range space.

The method of proof of theorem (42.1) carries over to spaces C ∞ (M, N ), where

M is a ¬nite dimensional smooth manifold, and where N is a possibly in¬nite

dimensional manifold which is required to admit an analogue of the exponential

mapping used above, i.e., a smooth mapping ± : T N ⊃ U ’ N , de¬ned on an

open neighborhood of the zero section in T N , which satis¬es

(1) (πN , ±) : T N ⊃ U ’ N — N is a di¬eomorphism onto a c∞ -open neighbor-

hood of the diagonal.

(2) ±(0x ) = x for all x ∈ N .

A smooth mapping ± with these properties is called a local addition on N .

Each ¬nite dimensional manifold M admits globally de¬ned local additions. To

see this, let exp : T M ⊃ U ’ M be the exponential mapping with respect to

a Riemannian metric g, where U is an open neighborhood of the 0-section, such

that (πM , exp) : U ’ M — M is a di¬eomorphism onto an open neighborhood of

the diagonal. Thus, exp is a local addition. One can do better. We construct a

¬ber respecting di¬eomorphism h : T M ’ U with h|0M = IdM as follows. Let

µ : M ’ (0, ∞) be a smooth positive function such that U := {X ∈ T M :

g(X, X) < µ(πM (X))} ‚ U . Let h : T M ’ U be given by

µ(πM (X)) 1

h’1 (Y ) :=

h(X) := X, Y.

))2 ’ g(Y, Y )

1 + g(X, X) µ(πM (Y

Then ± = exp —¦h : T M ’ M is a local addition.

If M is a real analytic ¬nite dimensional manifold, then there exists a real analytic

globally de¬ned local addition T M ’ M constructed as above with a real analytic

Riemannian metric g and real analytic µ; these exist by [Grauert, 1958, Prop. 8],

see also (42.7) below.

The a¬ne structure on each convenient vector space is a local addition, too.

Let G be a possibly in¬nite dimensional Lie group (36.1). Then G admits a local