where c.e.c’1 is the curve traveling along c(t) for 0 ¤ t ¤ 1, along e(t ’ 1) for

1 ¤ t ¤ 2, and along c(3 ’ t) for 2 ¤ t ¤ 3. This is equivalent to g ∈ Hol(ω, u0 ).

Furthermore, e is null-homotopic if and only if c.e.c’1 is null-homotopic, so we also

have Hol0 (ω, u1 ) = Hol0 (ω, u0 ).

40. Rudiments of Lie Theory for Regular Lie Groups

40.1. From Lie algebras to Lie groups. It is not true in general that every

convenient Lie algebra is the Lie algebra of a convenient Lie group. This is also

wrong for Banach Lie algebras and Banach Lie groups; one of the ¬rst examples is

from [Van Est, Korthagen, 1964], see also [de la Harpe, 1972].

To Lie subalgebras in the Lie algebra of a Lie group, in general, do not correspond

Lie subgroups. We shall give easy examples in (43.6).

In principle, one should be able to tell whether a given convenient Lie algebra is

the Lie algebra of a regular Lie group, but we have no idea how to do that.

40.2. The Cartan developing. Let G be a connected Lie group with Lie algebra

g. For a smooth mapping f : M ’ G we have considered in (38.1) the right

’1

logarithmic derivative δ r f ∈ „¦1 (M, g) which is given by δ r fx := T (µf (x) ) —¦ Tx f :

Tx M ’ Tf (x) G ’ g and which satis¬es the left (from the left action) Maurer-

Cartan equation

1

dδ r f ’ [δ r f, δ r f ]g = 0.

§

2

40.2

40.2 40. Rudiments of Lie theory for regular Lie groups 427

Similarly, the left logarithmic derivative δ l f ∈ „¦1 (M, g) of f ∈ C ∞ (M, G) is given

by δ l fx := T (µf (x)’1 ) —¦ Tx f : Tx M ’ Tf (x) G ’ g and satis¬es the right Maurer

Cartan equation

1

dδ l f + [δ l f, δ l f ]g = 0.

§

2

For regular Lie groups we have the following converse, which for ¬nite dimensional

Lie groups can be found in [Onishchik, 1961, 1964, 1967], or in [Gri¬th, 1974]

(proved with moving frames); see also [Alekseevsky, Michor, 1995b, 5.2].

Theorem. Let G be a connected regular Lie group with Lie algebra g.

If a 1-form • ∈ „¦1 (M, g) satis¬es d• + 2 [•, •]g = 0 then for each simply connected

1

§

subset U ‚ M there exists a smooth mapping f : U ’ G with δ l f = •|U , and f is

uniquely determined up to a left translation in G.

If a 1-form ψ ∈ „¦1 (M, g) satis¬es dψ ’ 1 [ψ, ψ]g = 0 then for each simply connected

§

2

subset U ‚ M there exists a smooth mapping f : U ’ G with δ r f = ψ|U , and f is

uniquely determined up to a right translation in G.

The mapping f is called the left Cartan developing of •, or the right Cartan devel-

oping of ψ, respectively.

Proof. Let us treat the right logarithmic derivative since it leads to a principal

connection for a bundle with right principal action. For the left logarithmic deriva-

tive the proof is similar, with the changes described in the second part of the proof

of (38.1).

We put ourselves into the situation of the proof of (38.1). If we are given a 1-form

• ∈ „¦1 (M, g) with d•’ 1 [•, •]§ = 0 then we consider the 1-form ω r ∈ „¦1 (M —G, g),

2

given by the analogue of (38.1.1) (where ν : G ’ G is the inversion),

ω r = κl ’ (Ad —¦ ν).•

(1)

Then ω r is a principal connection form on M —G, since it reproduces the generators

in g of the fundamental vector ¬elds for the principal right action, i.e., the left

invariant vector ¬elds, and ω r is G-equivariant:

((µg )— ω r )h = ωhg —¦ (Id —T (µg )) = T (µg’1 .h’1 ).T (µg ) ’ Ad(g ’1 .h’1 ).•

r

= Ad(g ’1 ).ωh .

r

The computation in (38.1.3) for • instead of δ r f shows that this connection is ¬‚at.

Since the structure group G is regular, by theorem (39.2) the horizontal bundle is

integrable, and pr1 : M — G ’ M , restricted to each horizontal leaf, is a covering.

Thus, it may be inverted over each simply connected subset U ‚ M , and the inverse

(Id, f ) : U ’ M — G is unique up to the choice of the branch of the covering and

the choice of the leaf, i.e., f is unique up to a right translation by an element of G.

The beginning of the proof of (38.1) then shows that δ r f = •|U .

40.2

428 Chapter VIII. In¬nite dimensional di¬erential geometry 40.3

40.3. Theorem. Let G and H be Lie groups with Lie algebras g and h, respec-

tively. Let f : g ’ h be a bounded homomorphism of Lie algebras. If H is regular

and if G is simply connected then there exists a unique homomorphism F : G ’ H

of Lie groups with Te F = f .

Proof. We consider the 1-form

’1

ψ ∈ „¦1 (G; h), ψ := f —¦ κr , ψg (ξg ) = f (T (µg ).ξg ),

where κr is the right Maurer-Cartan form from (38.1). It satis¬es the left Maurer-

Cartan equation

dψ ’ 1 [ψ, ψ]h = d(f —¦ κr ) ’ 1 [f —¦ κr , f —¦ κr ]h

§ §

2 2

= f —¦ (dκr ’ 1 [κr , κr ]g ) = 0,

§

2

by (38.1).(2™). But then we can use theorem (40.2) to conclude that there exists

a unique smooth mapping F : G ’ H with F (e) = e, whose right logarithmic

derivative satis¬es δ r F = ψ. For g ∈ G we have (µg )— ψ = ψ, and thus

δ r (F —¦ µg ) = δ r F —¦ T (µg ) = (µg )— ψ = ψ.

By uniqueness in theorem (40.2), again, the mappings F —¦ µg and F : G ’ H

di¬er only by right translation in H by the element (F —¦ µg )(e) = F (g), so that

F —¦ µg = µF (g) —¦ F , or F (g.g1 ) = F (g).F (g1 ). This also implies F (g).F (g ’1 ) =

F (g.g ’1 ) = F (e) = e, hence that F is the unique homomorphism of Lie groups we

have been looking for.

40.3

429

Chapter IX

Manifolds of Mappings

41. Jets and Whitney Topologies . . . . . . . . . . . . . . . . . . . 431

42. Manifolds of Mappings .................. . . . 439

43. Di¬eomorphism Groups . . . . . . . . . . . . . . . . . . . . . 454

44. Principal Bundles with Structure Group a Di¬eomorphism Group . . . 474

45. Manifolds of Riemannian Metrics . . . . . . . . . . . . . . . . . 487

46. The Korteweg “ De Vries Equation as a Geodesic Equation .. . . . 498

Complements to Manifolds of Mappings . . . . . . . . . . . . . . . . 510

Manifolds of smooth mappings between ¬nite dimensional manifolds are the fore-

most examples of in¬nite dimensional manifolds, and in particular di¬eomorphism

groups can only be treated in a satisfactory manner at the level of generality de-

veloped in this book: One knows from [Omori, 1978b] that a Banach Lie group

acting e¬ectively on a ¬nite dimensional compact manifold is necessarily ¬nite di-

mensional. So there is no way to model the di¬eomorphism group on Banach

spaces as a manifold.

The space of smooth mappings C ∞ (M, N ) carries a natural atlas with charts in-

duced by any exponential mapping on N (42.1), which permits us also to consider

certain in¬nite dimensional manifolds N in (42.4). Unfortunately, for noncom-

pact M , the space C ∞ (M, N ) is not locally contractible in the compact-open C ∞ -

topology, and the natural chart domains are quite small: Namely, the natural model

spaces turn out to be convenient vector spaces of sections with compact support

of vector bundles f — T N , which have been treated in detail in section (30). Thus,

the manifold topology on C ∞ (M, N ) is ¬ner than the Whitney C ∞ -topology, and

we denote by C∞ (M, N ) the resulting smooth manifold (otherwise, e.g. C ∞ (R, R)

would have two meanings).

With a careful description of the space of smooth curves (42.5) we can later often

avoid the explicit use of the atlas, for example when we show that the composition

mapping is smooth in (42.13). Since we insist on charts the exponential law for

manifolds of mappings holds only for a compact source manifold M , (42.14).

If we insist that the exponential law should hold for manifolds of mappings between

all (even only ¬nite dimensional) manifolds, then one is quickly lead to a more

general notion of a manifold, where an atlas of charts is replaced by the system

of all smooth curves. One is lead to further requirements: tangent spaces should

be convenient vector spaces, the tangent bundle should be trivial along smooth

curves via a kind of parallel transport, and a local addition as in (42.4) should

430 Chapter IX. Manifolds of mappings

exist. In this way one obtains a cartesian closed category of smooth manifolds and

smooth mappings between them, where those manifolds with Banach tangent spaces

are exactly the classical smooth manifolds with charts. Theories along these lines

can be found in [Kriegl, 1980], [Michor, 1984a], and [Kriegl, 1984]. Unfortunately

they found no applications, and even the authors were not courageous enough to

pursue them further and to include them in this book. But we still think that

it is a valuable theory, since for instance the di¬eomorphism group Di¬(M ) of a

non-compact ¬nite dimensional smooth manifold M with the compact-open C ∞ -

topology is a Lie group in this sense with the space of all vector ¬elds on M as

Lie algebra. Also, in section (45) results will appear which indicate that ultimately

this is a more natural setting.

Let us return (after discussing non-contents) to describing the contents of this

chapter. For the tangent space we have a natural di¬eomorphism T C∞ (M, N ) ∼ =

C∞ (M, T N ) ‚ C∞ (M, T N ), see (42.17). In the same manner we also treat mani-

c

folds of real analytic mappings from a compact manifold M into N .

In section (43) on di¬eomorphism groups we ¬rst show that the group Di¬(M )

is a regular smooth Lie group (43.1). The proof clearly shows the power of our

calculus: It is quite obvious that the inversion is smooth, whereas more traditional

treatments as in [Leslie, 1967], [Michor, 1980a], and [Michor, 1980c] needed specially

tailored inverse function theorems in in¬nite dimensions. The Lie algebra of the

di¬eomorphism group is the space Xc (M ) of all vector ¬elds with compact support

on M , with the negative of the usual Lie bracket. The exponential mapping exp

is the ¬‚ow mapping to time 1, but it is not surjective on any neighborhood of the

identity (43.2), and Ad —¦ exp : Xc (M ) ’ L(Xc (M ), Xc (M )) is not real analytic,

(43.3). Real analytic di¬eomorphisms on a real analytic compact manifold form a

regular real analytic Lie group (43.4). Also regular Lie groups are the subgroups of

volume preserving (43.7), symplectic (43.12), exact symplectic (43.13), or contact

di¬eomorphisms (43.19).

In section (44) we treat principal bundles with a di¬eomorphism group as structure

group. The ¬rst example is the space of all embeddings between two manifolds

(44.1), a sort of nonlinear Grassmann manifold, in particular if the image space is

an in¬nite dimensional convenient vector space which leads to a smooth manifold

which is a classifying space for the di¬eomorphism group of a compact manifold

(44.24). Another example is the nonlinear frame bundle of a ¬ber bundle with

compact ¬ber (44.5), for which we investigate the action of the gauge group on the

space of generalized connections (44.14) and show that in the smooth case there

never exist slices (44.19), (44.20).

In section (45) we compute explicitly all geodesics for some natural (pseudo) Rie-

mannian metrics on the space of all Riemannian metrics. Section (46) is devoted

to the Korteweg“De Vrieß equation which is shown to be the geodesic equation of

a certain right invariant Riemannian metric on the Virasoro group. Here we also

compute the curvature (46.13) and the Jacobi equation (46.14).

431

41. Jets and Whitney Topologies

Jet spaces or jet bundles consist of the invariant expressions of Taylor developments

up to a certain order of smooth mappings between manifolds. Their invention goes

back to Ehresmann [Ehresmann, 1951.]

41.1. Jets between convenient vector spaces. Let E and F be convenient

vector spaces, and let U ⊆ E and V ⊆ F be c∞ -open subsets. For 0 ¤ k ¤ ∞ the

space of k-jets from U to V is de¬ned by

k

k k

k

Lj (E; F ).

J (U, V ) := U — V — Poly (E, F ), where Poly (E, F ) = sym

j=1

We shall use the source and image projections ± : J k (U, V ) ’ U and β : J k (U, V ) ’

V , and we shall consider J k (U, V ) ’ U —V as a trivial bundle, with ¬bers Jx (U, V )y

k

for (x, y) ∈ U —V . Moreover, we have obvious projections πl : J k (U, V ) ’ J l (U, V )

k

for k > l, given by truncation at order l. For a smooth mapping f : U ’ V the

k-jet extension is de¬ned by

12 1

j k f (x) = jx f := (x, f (x), df (x),

k

d f (x), . . . , dj f (x), . . . ),

2! j!

the Taylor expansion of f at x of order k. If k < ∞ then j k : C ∞ (U, F ) ’ J k (U, F )

is smooth with a smooth right inverse (the polynomial), see (5.17). If k = ∞ then

j k need not be surjective for in¬nite dimensional E, see (15.4). For later use, we

consider now the truncated composition

• : Polyk (F, G) — Polyk (E, F ) ’ Polyk (E, G),

where p•q is the composition p—¦q of the polynomials p, q (formal power series in case

k = ∞) without constant terms, and without all terms of order > k. Obviously, •

is polynomial for ¬nite k and is real analytic for k = ∞ since then each component

is polynomial. Now let U ‚ E, V ‚ F , and W ‚ G be open subsets, and consider

the ¬bered product

J k (U, V ) —U J k (W, U ) = { (σ, „ ) ∈ J k (U, V ) — J k (W, U ) : ±(σ) = β(„ ) }

= U — V — W — Polyk (E, F ) — Polyk (G, E).

Then the mapping

• : J k (U, V ) —U J k (W, U ) ’ J k (W, V ),

σ • „ = (±(σ), β(σ), σ ) • (±(„ ), β(„ ), „ ) := (±(„ ), β(σ), σ • „ ),

¯ ¯ ¯¯

is a real analytic mapping, called the ¬bered composition of jets.

Let U , U ‚ E and V ‚ F be open subsets, and let g : U ’ U be a smooth di¬eo-

morphism. We de¬ne a mapping J k (g, V ) : J k (U, V ) ’ J k (U , V ) by J k (g, V )(σ) =

41.1

432 Chapter IX. Manifolds of mappings 41.3

σ • j k g(g ’1 (x)), which also satis¬es J k (g, V )(j k f (x)) = j k (f —¦ g)(g ’1 (±(σ))). If g :

U ’ U is another di¬eomorphism, then clearly J k (g , V )—¦J k (g, V ) = J k (g—¦g , V ),

and J k ( , V ) is a contravariant functor acting on di¬eomorphisms between open

k

subsets of E. Since the truncated composition σ ’ σ • jg’1 (x) g is linear, the

¯ ¯

mapping Jx (g, F ) := J k (g, F )|Jx (U, F ) : Jx (U, F ) ’ Jg’1 (x) (U , F ) is also linear.

k k k k

Now let U ‚ E, V ‚ F , and W ‚ G be c∞ -open subsets, and let h : V ’ W be a

smooth mapping. Then we de¬ne J k (U, h) : J k (U, V ) ’ J k (U, W ) by J k (U, h)σ =

j k h(β(σ)) • σ, which satis¬es J k (U, h)(j k f (x)) = j k (h —¦ f )(x). Clearly, J k (U, )

is a covariant functor acting on smooth mappings between c∞ -open subsets of

k k k

convenient vector spaces. The mapping Jx (U, h)y : Jx (U, V )y ’ Jx (U, W )h(y) is

linear if and only if h is a¬ne or k = 1 or U = ….

41.2. The di¬erential group GLk (E). For a convenient vector space E, the k-

jets at 0 of germs at 0 of di¬eomorphisms of E which map 0 to 0 form a group under

truncated composition, which will be denoted by GLk (E) and will be called the dif-

ferential group of order k. Clearly, an arbitrary 0-respecting k-jet σ ∈ Polyk (E, E)

is in GLk (E) if and only if its linear part is invertible. Thus

k

GLk (E) = GL(E) — Lj (E; E) =: GL(E) — P2 (E),

k

sym

j=2

k

where we put P2 (E) := j=2 Lj (E; E) for the space of all polynomial mappings

k

sym

of degree ¤ k (formal power series for k = ∞) without constant and linear terms.

If the set GL(E) of all bibounded linear isomorphisms of E is a Lie group contained

in L(E, E) (e.g., for E a Banach space), then since the truncated composition is

real analytic, GLk (E) is also a Lie group. In general, GL(E) may be viewed as a

Fr¨licher space in the sense of (23.1) with the initial smooth structure with respect

o

to (Id, ( )’1 ) : GL(E) ’ L(E, E) — L(E, E), where multiplication and inversion

are now smooth: we call this a smooth group. Then GLk (E) is again a smooth

group.

In both cases, clearly, for k ≥ l the mapping πl : GLk (E) ’ GLl (E) is a homomor-

k

phism of smooth groups, thus its kernel ker(πl ) = Polyk (E, E) := {IdE } — {0} —

k

l

k j

j=l+1 Lsym (E; E) is a closed normal subgroup for all l, which is a Lie group for

l ≥ 1. The exact sequence of groups

k

Lj (E; E) ’ GLk (E) ’ GLl (E) ’ {e}

{e} ’ sym

j=l+1

splits if and only if l = 1 for dim E > 1 or l ¤ 2 for E = R, see [Kol´ˇ, Michor,

ar

m

Slov´k, 1993, 13.8] for E = R ; only in this case this sequence describes a semidirect

a

product.

41.3. Jets between manifolds. Now let M and N be smooth manifolds with

smooth atlas (U± , u± ) and (Vβ , vβ ), modeled on convenient vector spaces E and F ,

41.3

41.3 41. Jets and Whitney topologies 433

respectively. Then we may glue the open subsets J k (u± (U± ), vβ (Vβ )) of convenient

vector spaces via the chart change mappings

’1

J k (u± —¦ u’1 , vβ —¦ vβ ) : J k (u± (U± © U± ), vβ (Vβ © Vβ )) ’

±

’ J k (u± (U± © U± ), vβ (Vβ © Vβ )),

and we obtain a smooth ¬ber bundle J k (M, N ) ’ M — N with standard ¬ber

Polyk (E, F ). With the identi¬cation topology J k (M, N ) is Hausdor¬, since it is

a ¬ber bundle and the usual argument for gluing ¬ber bundles applies which was

given, e.g., in (28.12).

Theorem. If M and N are smooth manifolds, modeled on convenient vector spaces

E and F , respectively. Let 0 ¤ k ¤ ∞. Then the following results hold.

(1) (J k (M, N ), (±, β), M — N, Polyk (E, F )) is a ¬ber bundle with standard ¬ber

Polyk (E, F ), with the smooth group GLk (E) — GLk (F ) as structure group,

where (γ, χ) ∈ GLk (E) — GLk (F ) acts on σ ∈ Polyk (E, F ) by (γ, χ).σ =

χ • σ • γ ’1 .

(2) If f : M ’ N is a smooth mapping then j k f : M ’ J k (M, N ) is also

smooth, called the k-jet extension of f . We have ±—¦j k f = IdM and β—¦j k f =

f.

(3) If g : M ’ M is a di¬eomorphism then also the induced mapping J k (g, N ) :

J k (M, N ) ’ J k (M , N ) is a di¬eomorphism.

(4) If h : N ’ N is a smooth mapping then J k (M, h) : J k (M, N ) ’ J k (M, N )

is also smooth. Thus, J k (M, ) is a covariant functor from the category

of smooth manifolds and smooth mappings into itself which respects each

of the following classes of mappings: initial mappings, embeddings, closed

embeddings, splitting embeddings, ¬ber bundle projections. Furthermore,

J k ( , ) is a contra-covariant bifunctor, where we have to restrict in the

¬rst variable to the category of di¬eomorphisms.

(5) For k ≥ l, the projections πl : J k (M, N ) ’ J l (M, N ) are smooth and

k

natural, i.e., they commute with the mappings from (3) and (4).

k

(6) (J k (M, N ), πl , J l (M, N ), i=l+1 Li (E; F )) are ¬ber bundles for all l ¤

k

sym

k. For ¬nite k the bundle (J (M, N ), πk’1 , J k’1 (M, N ), Lk (E, F )) is an

k k

sym

1

a¬ne bundle. The ¬rst jet space J (M, N ) ’ M — N is a vector bundle.

It is isomorphic to the bundle (L(T M, T N ), (πM , πN ), M — N ), see (29.4)

and (29.5). Moreover, we have J0 (R, N ) = T N and J 1 (M, R)0 = T — M .

1

(7) Truncated composition is a smooth mapping

• : J k (N, P ) —N J k (M, N ) ’ J k (M, P ).

Proof. (1) is already proved. (2), (3), (5), and (7) are obvious from (41.1), mainly

by the functorial properties of J k ( , ).

(4) It is clear from (41.1) that J k (M, h) is a smooth mapping. The rest follows by

looking at special chart representations of h and the induced chart representations

for J k (M, h).

41.3

434 Chapter IX. Manifolds of mappings 41.5

It remains to show (6), and here we concentrate on the a¬ne bundle. Let a1 +

k

a ∈ GL(E) — i=2 Li (F ; F ), σ + σk ∈ Polyk’1 (E, F ) — Lk (E; F ), and b1 +

sym sym

k

b ∈ GL(E) — i=2 Li (E; E), then the only term of degree k containing σk in

sym

(a1 + a) • (σ + σk ) • (b1 + b) is a1 —¦ σk —¦ bk , which depends linearly on σk . To this the

1

degree k-components of compositions of the lower order terms of σ with the higher

order terms of a and b are added, and these may be quite arbitrary. So an a¬ne

bundle results.

We have J 1 (M, N ) = L(T M, T N ) since both bundles have the same transition

functions. Finally,

J 1 (M, R)0 = L(T M, T0 R) = T — M.

1

J0 (R, N ) = L(T0 R, T N ) = T N and

41.4. Jets of sections of ¬ber bundles. If (p : E ’ M, S) is a ¬ber bun-

dle, let (U± , u± ) be a smooth atlas of M such that (U± , ψ± : E|U± ’ U± — S)

is a ¬ber bundle atlas. If we glue the smooth manifolds J k (U± , S) via (σ ’

j k (ψ±β (±(σ), ))) • σ : J k (U± © Uβ , S) ’ J k (U± © Uβ , S), we obtain the smooth

manifold J k (E), which for ¬nite k is the space of all k-jets of local sections of E.

Theorem. In this situation we have:

(1) J k (E) is a splitting closed submanifold of J k (M, E), namely the set of all

σ ∈ Jx (M, E) with J k (M, p)(σ) = j k (IdM )(x).

k

(2) J 1 (E) of sections is an a¬ne subbundle of the vector bundle J 1 (M, E) =

L(T M, T E). In fact, we have

J 1 (E) = { σ ∈ L(T M, T E) : T p —¦ σ = IdT M }.

(3) For k ¬nite (J k (E), πk’1 , J k’1 (E)) is an a¬ne bundle.

k

(4) If p : E ’ M is a vector bundle, then (J k (E), ±, M ) is also a vector bundle.

If φ : E ’ E is a homomorphism of vector bundles covering the identity,

then J k (•) is of the same kind.

Proof. Locally J k (E) in J k (M, E) looks like u± (U± ) — Polyk (FM , FS ) in u± (U± ) —

(u± (U± ) — vβ (Vβ )) — Polyk (FM , FM — FS ), where FM and FS are modeling spaces

of M and S, respectively, and where (Vβ , vβ ) is a smooth atlas for S. The rest is

clear.

41.5. The compact-open topology on spaces of continuous mappings. Let

M and N be Hausdor¬ topological spaces. The best known topology on the space

C(M, N ) of all continuous mappings is the compact-open topology or CO-topology.

A subbasis for this topology consists of all sets of the form {f ∈ C(M, N ) : f (K) ⊆

U }, where K runs through all compact subsets in M and U through all open subsets

of N . This is a Hausdor¬ topology, since it is ¬ner than the topology of pointwise

convergence.

It is easy to see that if M has a countable basis of the compact sets and is compactly

generated ((4.7).(i), i.e., M carries the ¬nal topology with respect to the inclusions

of its compact subsets), and if N is a complete metric space, then there exists a

complete metric on (C(M, N ), CO), so it is a Baire space.

41.5

41.7 41. Jets and Whitney topologies 435

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

addition. Namely, let v : V ’ W ⊆ g be a chart de¬ned on an open neighborhood

V of e with v(e) = 0 ∈ W where W is open in the Lie algebra g. Then put

T G ⊇ U := g∈G T (µg )V ∼ G — V and let ± : U ’ G be given by ±(ξ) :=

=

’1

πG (ξ).v (T (µπ(ξ)’1 ).ξ) be the local addition.

If a manifold N admits a local addition ±, then it admits a ˜spray™, thus a torsionfree

covariant derivative on T N . Recall from [Ambrose, Palais, Singer, 1960] or [Lang,

1972] that a spray is a vector ¬eld S on T M such that πT M —¦S = IdT M , T (πM )—¦S =

IdT M , so that in induced local charts as in (29.9) and (29.10) we have S(x, y) =

(x, y; y, “x (y)), where ¬nally it is also required that y ’ “x (y) is quadratic. In

‚

order to see this, let •(X) := ‚t 0 ±(tX). Then • : T M ’ T M is a vector bundle

automorphism with inverse (in local charts) •’1 (x, y) = ‚t 0 (pr1 , ±)’1 (x, x + ty).

‚

d2 ’1

dt2 |t=0 ±(t•

Then one checks easily that S(X) := (X)) is a spray.

Theorem. Let M be a smooth ¬nite dimensional manifold, and let N be a smooth

manifold, possibly in¬nite dimensional, which admits a smooth local addition ±.

42.4

442 Chapter IX. Manifolds of mappings 42.7

Then the space C∞ (M, N ) of all smooth mappings from M to N is a smooth mani-

fold, modeled on spaces Cc (M ← f — T N ) of smooth sections with compact support

∞

of pullback bundles along f : M ’ N over M .

Let us remark again that for a compact smooth manifold M and a convenient vector

space E the smooth manifold C∞ (M, E) is di¬eomorphic to the convenient vector

space C ∞ (M, E), which is a special case of (30.1) for a trivial bundle with ¬nite

dimensional base.

42.5. Lemma. Smooth curves in C∞ (M, N ). Let M and N be smooth man-

ifolds with M ¬nite dimensional and N admitting a smooth local addition. Then

the smooth curves c in C∞ (M, N ) correspond exactly to the smooth mappings

c§ ∈ C ∞ (R — M, N ) which satisfy the following property:

(1) For each compact interval [a, b] ‚ R there is a compact subset K ‚ M such

that c§ (t, x) is constant in t ∈ [a, b] for all x ∈ M \ K.

In particular, the identity induces a smooth mapping C∞ (M, N ) ’ C ∞ (M, N ) into

the Fr¨licher space C ∞ (M, N ) discussed in (23.2.3), which is a di¬eomorphism if

o

and only if M is compact or N is discrete.

Proof. Since R is locally compact, property (1) is equivalent to

(2) For each t ∈ R there is an open neighborhood U of t in R and a compact

K ‚ M such that the restriction has the property that c§ (t, x) is constant

in t ∈ U for all x ∈ M \ K.

Since this is a local condition on R, and since smooth curves in C∞ (M, N ) locally

take values in charts as in the proof of theorem (42.1), it su¬ces to describe the

∞

smooth curves in the space Cc (M ← E) of sections with compact support of a

vector bundle (p : E ’ M, V ) with ¬nite dimensional base manifold M , with the

structure described in (30.4). This was done in (30.9).

42.6. Theorem. C ω -manifold structure of C ω (M, N ). Let M and N be real

analytic manifolds, let M be compact, and let N be either ¬nite dimensional, or let

us assume that N admits a real analytic local addition in the sense of (42.4).

Then the space C ω (M, N ) of all real analytic mappings from M to N is a real

analytic manifold, modeled on spaces C ω (M ← f — T N ) of real analytic sections of

pullback bundles along f : M ’ N over M .

Proof. The proof is a variant of the proof of (42.4), using a real analytic Riemann-

ian metric if N is ¬nite dimensional, and the given real analytic local addition oth-

erwise. For ¬nite dimensional N a detailed proof can be found in [Kriegl, Michor,

1990].

42.7. Lemma. Let M , N be real analytic ¬nite dimensional manifolds. Then

the space C ω (M, N ) of all real analytic mappings is dense in C ∞ (M, N ), in the

Whitney C ∞ -topology.

This is not true in the manifold topology of C∞ (M, N ) used in (42.1), if M is not

compact, because of the compact support condition used there.

42.7

42.10 42. Manifolds of mappings 443

Proof. By [Grauert, 1958, theorem 3], there is a real analytic embedding i : N ’

Rk on a closed submanifold, for some k. We use the constant standard inner product

on Rk to obtain a real analytic tubular neighborhood U of i(N ) with projection

p : U ’ i(N ). By [Grauert, 1958, proposition 8] applied to each coordinate of

Rk , the space C ω (M, Rk ) of real analytic Rk -valued functions is dense in the space

C ∞ (M, Rk ) of smooth functions, in the Whitney C ∞ -topology. If f : M ’ N is

smooth we may approximate i —¦ f by real analytic mappings g in C ω (M, U ), then

p —¦ g is real analytic M ’ i(N ) and approximates i —¦ f .

42.8. Theorem. C ω -manifold structure on C∞ (M, N ). Let M and N be real

analytic ¬nite dimensional manifolds, with M compact. Then the smooth manifold

C∞ (M, N ) with the structure from (42.1) is even a real analytic manifold.

Proof. For a ¬xed real analytic exponential mapping on N the charts (Uf , uf )

from (42.1) for f ∈ C ω (M, N ) form a smooth atlas for C∞ (M, N ), since C ω (M, N )

is dense in C∞ (M, N ) by (42.7)

’1

The chart changings uf —¦ u’1 = („f —¦ „g )— are real analytic by (30.10).

g

42.9. Corollary. Let Mi and Ni be smooth manifolds with Mi ¬nite dimensional

for i = 1, 2 and Ni admitting smooth local additions. Then we have:

(1) If f : N1 ’ N2 is initial (27.11) then the mapping

C∞ (M, f ) : C∞ (M, N1 ) ’ C∞ (M, N2 )

is initial, too.

(2) If f : M2 ’ M1 is ¬nal (27.15) and proper then the mapping C∞ (f, N ) :

C∞ (M1 , N ) ’ C∞ (M2 , N ) is initial.

Proof. (1) Let c : R ’ C∞ (M, N1 ) be such that f— —¦c : R ’ C∞ (M, N2 ) is smooth.

By (42.5), the associated mapping (f— —¦ c)§ = f —¦ c§ : R — M ’ N2 is smooth and

satis¬es (42.5.1). Since f is initial, c§ is smooth, and since f is injective, c§ satis¬es

(42.5.1), hence c is smooth.

Proof of (2) Since f is ¬nal between ¬nite dimensional manifolds, it is a surjective

submersion, so R — f is also a surjective submersion and thus ¬nal.

Let c : R ’ C∞ (M1 , N ) be such that f — —¦ c : R ’ C∞ (M2 , N ) is smooth. By

(42.5), the associated mapping (f — —¦ c)§ = c§ —¦ (R — f ) : R — M2 ’ N is smooth

and satis¬es (42.5.1). Since R — f is also ¬nal, c§ is smooth. Since f and thus also

R — f is proper, c§ satis¬es (42.5.1), and thus c is smooth.

42.10. Lemma. Let M and N be real analytic ¬nite dimensional manifolds with

M compact. Let (U± , u± ) be a real analytic atlas for M , and let i : N ’ Rn

be a closed real analytic embedding into some Rn . Let M be a possibly in¬nite

dimensional real analytic manifold.

Then f : M ’ C ω (M, N ) is real analytic or smooth if and only if C ω (u’1 , i) —¦ f :

±

ω n

M ’ C (u± (U± ), R ) is real analytic or smooth, respectively.

42.10

444 Chapter IX. Manifolds of mappings 42.13

Furthermore, f : M ’ C∞ (M, N ) is real analytic or smooth if and only if the

mapping C ∞ (u’1 , i) —¦ f : M ’ C ∞ (u± (U± ), Rn ) is real analytic or smooth, respec-

±

tively.

Proof. The statement that i— is initial is obvious. So we just have to treat

C ∞ (u’1 , N ). The corresponding statement for spaces of sections of vector bundles

±

are (30.6) for the real analytic case and (30.1) for the smooth case. So if f takes val-

ues in a chart domain Ug of C ∞ (M, N ) for a real analytic g : M ’ N , the result fol-

lows. Recall from the proof of (42.1) that Ug = {h ∈ C β (M, N ) : (g(x), h(x)) ∈ V }

where V is a ¬xed open neighborhood of the diagonal in N — N , and where β = ∞

or ω. Let f (z0 ) ∈ Ug for z0 ∈ M. Since M is covered by ¬nitely many of its charts

U± , and since by assumption f (z)|U± is near f (z0 )|U± for z near z0 , so f (z) ∈ Ug

for z near z0 in M. So f takes values locally in charts, and the result follows.

42.11. Corollary. Let M and N be ¬nite dimensional real analytic manifolds

with M compact. Then the inclusion C ω (M, N ) ’ C∞ (M, N ) is real analytic.

Proof. Use the chart description and lemma (11.3).

42.12. Lemma. Curves in spaces of mappings. Let M and N be ¬nite

dimensional real analytic manifolds with M compact.

(1) A curve c : R ’ C ω (M, N ) is real analytic if and only if the associated

mapping c§ : R — M ’ N is real analytic.

The curve c : R ’ C ω (M, N ) is smooth if and only if c§ : R — M ’ N

satis¬es the following condition:

For each n there is an open neighborhood Un of R—M in R—MC

and a (unique) C n -extension c : Un ’ NC such that c(t, ) is

˜ ˜

holomorphic for all t ∈ R.

(2) The curve c : R ’ C∞ (M, N ) is real analytic if and only if c§ satis¬es the

following condition:

For each n there is an open neighborhood Un of R — M in C — M

and a (unique) C n -extension c : Un ’ NC such that c( , x) is

˜ ˜

holomorphic for all x ∈ M .

Note that the two conditions are in fact local in R. We need N ¬nite dimensional

since we need an extension NC of N to a complex manifold.

Proof. This follows from the corresponding statement (30.8) for spaces of sections

of vector bundles, and from the chart structure on C ω (M, N ) and C∞ (M, N ).

42.13. Theorem. Smoothness of composition. If M , N are smooth mani-

folds with M ¬nite dimensional and N admitting a smooth local addition, then the

evaluation mapping ev : C∞ (M, N ) — M ’ N is smooth.

42.13

42.14 42. Manifolds of mappings 445

If P is another smooth ¬nite dimensional manifold, then the composition mapping

comp : C∞ (M, N ) — C∞ (P, M ) ’ C∞ (P, N )

prop

is smooth, where C∞ (P, M ) denotes the space of all proper smooth mappings

prop

P ’ M (i.e. compact sets have compact inverse images). This space is open

in C∞ (P, M ).

In particular, f— : C∞ (M, N ) ’ C∞ (M, N ) and g — : C∞ (M, N ) ’ C∞ (P, N ) are

smooth for f ∈ C ∞ (N , N ) and g ∈ C∞ (P, M ).

prop

The corresponding statement for real analytic mappings can be shown along similar

lines, using (42.12). But we will give another proof in (42.15) below.

Proof. Using the description of smooth curves in C∞ (M, N ) given in (42.5), we

immediately see that (ev —¦(c1 , c2 ))(t) = c§ (t, c2 (t)) is smooth for each smooth

1

∞

(c1 , c2 ) : R ’ C (M, N ) — M , so ev is smooth as claimed.

The space of proper mappings C∞ (P, M ) is open in the manifold C∞ (P, M ) since

prop

changing a mapping only on a compact set does not change its property of being

proper. Let (c1 , c2 ) : R ’ C∞ (M, N ) — C∞ (P, M ) be a smooth curve. Then we

prop

§ §

have (comp —¦(c1 , c2 ))(t)(p) = c1 (t, c2 (t, p)), and one may check that this has again

property (44.5.1), so it is a smooth curve in C∞ (P, N ). Thus, comp is smooth.

42.14. Theorem. Exponential law. Let M, M , and N be smooth manifolds

with M ¬nite dimensional and N admitting a smooth local addition.

Then we have a canonical injection

C ∞ (M, C∞ (M, N )) ⊆ C ∞ (M — M, N ),

where the image in the right hand side consists of all smooth mappings f : M—M ’

N which satisfy the following property

(1) If M is locally metrizable then for each point x ∈ M there is an open

neighborhood U and a compact set K ‚ M such that f (x, y) is constant in

x ∈ U for all y ∈ M \ K.

(2) For general M: For each c ∈ C ∞ (R, M) and each t ∈ R there exists

a neighborhood U of t and a compact set K ‚ M such that f (c(s), y) is

constant in s ∈ U for each y ∈ M \ K.

Under the assumption that N admits smooth functions which separate points, we

have equality if and only if M is compact, or N is discrete, or each f ∈ C ∞ (M, R)

is constant along all smooth curves into M.

If M and N are real analytic manifolds with M compact we have

C ω (M, C ω (M, N )) = C ω (M — M, N )

for each real analytic (possibly in¬nite dimensional) manifold M.

Proof. The smooth case is simple: The equivalence for general M follows directly

from the description of all smooth curves in C∞ (M, N ) given in the proof of (42.5).

42.14

446 Chapter IX. Manifolds of mappings 42.15

It remains to show that for locally metrizable M a smooth mapping f : M ’

C∞ (M, N ) satis¬es condition (1). Since f is smooth, locally it has values in a

chart, so we may assume that M is open in a Fr´chet space by (4.19), and that f

e

∞

has values in Cc (M ← E) for some vector bundle p : E ’ M .

∞

We claim that f locally factors into some CKn (E) where (Kn ) is an exhaustion

of M by compact subsets such that Kn is contained in the interior of Kn+1 . If

not there exist a (fast) converging sequence (yn ) in M and xn ∈ Kn such that

/

f (yn )(xn ) = 0. One may ¬nd a proper smooth curve e : R ’ M with e(n) = xn

and a smooth curve g : R ’ M with g(1/n) = yn . Then by (30.4), Pt(e, )— —¦ f —¦ g

∞

is a smooth curve into Cc (R, Ee(0) ). Since the latter space is a strict inductive

limit of spaces CI (R, Ee(0) ) for compact intervals I, the curve Pt(e, )— —¦ f —¦ g

∞

locally factors into some CI (R, Ee(0) ), but (e— —¦ f —¦ g)(1/n)(n) = f (yn )(xn ) = 0, a

∞

contradiction.

We check now the statement on equality: if M is compact, or if N is discrete then

(2) is automatically satis¬ed. If each f ∈ C ∞ (M, R) is constant along all smooth

curves into M, we may check global constancy in (2) by composing with smooth

functions on N which separate points there.

For the converse, we may assume that there are a function f ∈ C ∞ (M, R), a curve

c ∈ C ∞ (R, M) such that f —¦c is not constant, and an injective smooth curve e : R ’

N . Then M — M (x, y) ’ e(f (x)) is in C ∞ (M — M, N ) \ C ∞ (M, C∞ (M, N ))

since condition (2) is violated for the curve c.

Now we treat the real analytic case. Let f § ∈ C ω (M—M, N ) ‚ C ∞ (M—M, N ) =

C ∞ (M, C∞ (M, N )). So we may restrict f to a neighborhood U in M, where it takes

values in a chart Ug of C ∞ (M, N ) for g ∈ C ω (M, N ). Then f (U ) ‚ Ug ©C ω (M, N ),

one of the canonical charts of C ω (M, N ). So we may assume that f : U ’ C ω (M ←

g — T N ). For a real analytic vector bundle atlas (U± , ψ± ) of g — T N the composites

U ’ C ω (M ← g — T N ) ’ C ω (U± , Rn ) are real analytic by applying cartesian

closedness (11.18) to the mapping (x, y) ’ ψ± (πN , exp)’1 (g(y), f § (x, y)). By the

description (30.6) of the structure on C ω (M ← g — T N ), the chart representation of

f is real analytic, so f is it also.

Let conversely f : M ’ C ω (M, N ) be real analytic. Then its chart representation is

real analytic and we may use cartesian closedness in the other direction to conclude

that f § is real analytic.

42.15. Corollary. If M and N are real analytic manifolds with M compact

and N admitting a real analytic local addition, then the evaluation mapping ev :

C ω (M, N ) — M ’ N is real analytic.

If P is another compact real analytic manifold, then the composition mapping

comp : C ω (M, N ) — C ω (P, M ) ’ C ω (P, N ) is real analytic.

In particular, f— : C ω (M, N ) ’ C ω (M, N ) and g — : C ω (M, N ) ’ C ω (P, N ) are

real analytic for real analytic f : N ’ N and g ∈ C ω (P, M ).

Proof. The mapping ev∨ = IdC ω (M,N ) is real analytic, so ev too, by (42.14).

42.15

42.17 42. Manifolds of mappings 447

The mapping comp§ = ev —¦(IdC ω (M,N ) — ev) : C ω (M, N ) — C ω (P, M ) — P ’

C ω (M, N ) — M ’ N is real analytic, thus comp too.

42.16. Lemma. Let Mi and Ni be ¬nite dimensional real analytic manifolds with

Mi compact. Then for f ∈ C ∞ (N1 , N2 ) the push forward f— : C∞ (M, N1 ) ’

C∞ (M, N2 ) is real analytic if and only if f is real analytic. For f ∈ C ∞ (M2 , M1 )

the pullback f — : C∞ (M1 , N ) ’ C∞ (M2 , N ) is, however, always real analytic.

Proof. If f is real analytic and if g ∈ C ω (M, N1 ), then the mapping uf —¦g —¦f— —¦u’1 : g

∞ — ∞ —

C (M ← g T N1 ) ’ C (M ← (f —¦ g) T N2 ) is a push forward by the real analytic

mapping (pr1 , (π, expN2 )’1 —¦ (f —¦ g —¦ pr1 , f —¦ expN1 —¦ pr2 )) : g — T N1 ’ (f —¦ g)— T N2 ,

which is real analytic by (30.10).

The canonical mapping evx : C∞ (M, N2 ) ’ N2 is real analytic since evx |Ug =

expN2 —¦ evx —¦ug : Ug ’ C ∞ (M ← g — T N2 ) ’ Tg(x) N2 ’ N2 , where the second evx

is linear and bounded. Furthermore, const : N1 ’ C∞ (M, N1 ) is real analytic since

the mapping ug —¦ const : y ’ (x ’ (πN1 , expN1 )’1 (g(x), y)) is locally real analytic

into C ω (M ← g — T N1 ) and hence into C ∞ (M ← g — T N1 ).

If f— is real analytic, also f = evx —¦f— —¦ const is.

For the second statement choose real analytic atlas (U± , ui ) of Mi such that

i

±

f (U± ) ⊆ U± and a closed real analytic embedding j : N ’ Rn . Then the dia-

2 1

gram

w

f—

∞

C∞ (M2 , N )

C (M1 , N )

C∞ ((u1 )’1 , j) C∞ ((u2 )’1 , j)

u u

± ±

wC

(u2 —¦ f —¦ (u1 )’1 )—

± ±

C∞ (u1 (U± ), Rn ) ∞

1

(u2 (U± ), Rn )

2

± ±

commutes, the bottom arrow is a continuous and linear mapping, so it is real

analytic. Thus, by (42.10), the mapping f — is real analytic.

42.17. Theorem. Let M and N be smooth manifolds with M compact and N

admitting a local addition. Then the in¬nite dimensional smooth vector bundles

T C∞ (M, N ) and C∞ (M, T N ) ‚ C∞ (M, T N ) over C∞ (M, N ) are canonically iso-

c

morphic. The same assertion is true for C ω (M, N ) if M is compact.

Here by C∞ (M, T N ) we denote the space of all smooth mappings f : M ’ T N

c

such that f (x) = 0πM f (x) for x ∈ Kf , a suitable compact subset of M (equivalently,

/

such that the associated section of the pull back bundle (πM —¦ f )— T N ’ M has

compact support).

One can check directly that the atlas from (42.1) for C∞ (M, N ) induces an atlas

for T C∞ (M, N ), which is equivalent to that for C∞ (M, T N ) via some natural iden-

ti¬cations in T T N . This is carried out in great detail in [Michor, 1980c, 10.13].

We shall give here a simpler proof, using (42.5).

42.17

448 Chapter IX. Manifolds of mappings 42.18

Proof. Recall from (28.13) the diagram

u eC

C ∞ (R, C∞ (M, N ))/ ∼ ∞

(R, C∞ (M, N ))

δ ee

ee

∼δ

u h

e u

ev0

=

wC

T C∞ (M, N ) ∞

(M, N ),

πC∞ (M,N )

From (42.5) we see that C ∞ (R, C∞ (M, N )) corresponds to the space Clc (R—M, N )

∞

of all mappings g § ∈ C ∞ (R — M, N ) satisfying

(1) For each compact interval [a, b] ‚ R there is a compact subset K ‚ M such

that g § (t, x) is constant in t ∈ [a, b] for all x ∈ M \ K.

Now we consider the diagram

wC y wC

∼

=

C ∞ (R, C∞ (M, N )) ∞ ∞

— M, N ) (R — M, N )

lc (R

•

u

δ

‚ ‚

C ∞ (R, C∞ (M, N ))/ ∼ ‚t 0 ‚t 0

∼δ

u u u

=

wC y wC

¦

T C∞ (M, N ) ∞ ∞

c (M, T N ) (M, T N ).

∼

=

‚

The vertical mappings on the right hand side are ‚t 0 f = T f —¦ (‚t — 0M )|(0 — M ).

‚

The middle one is surjective since f (x) = ‚t 0 exp(h(t).f (x)) for suitable h, and h

can be chosen uniformly for f in a piece of a smooth curve into C∞ (M, T N ). By

construction the top isomorphism factors to a bijection ¦.

The mapping ¦ is smooth by (28.13) since ¦—¦δ factors over •, which maps the space

C ∞ (R2 , C∞ (M, N )) to Clc,c (R — M, T N ) ∼ C ∞ (R, C∞ (M, T N )). The inverse of

∞

= c

¦ is smooth by a similar argument, using again (28.13).

42.18. Corollary. Some tangent mappings. For f ∈ C ∞ (M1 , M2 ) and g ∈

C ∞ (N1 , N2 ) we have

T C∞ (M2 , N ) ∼ C∞ (M2 , T N ) T C∞ (M, N1 ) ∼ C∞ (M, T N1 )

=c =c

T C∞ (f, N ) C∞ (f, T N ) T C∞ (M, g) C∞ (M, T g)

u u u u

T C (M1 , N ) ∼ T C (M, N2 ) ∼

∞

C∞ (M1 , T N ) ∞

C∞ (M, T N2 ).

= =

c c

The tangent mapping of the composition

comp : C∞ (M, N ) — C∞ (P, M ) ’ C∞ (P, N )

prop

at (f, g) in direction of (X, Y ) ∈ Cc (M ← f — T N ) — Cc (P ← g — T M ) is given by

∞ ∞

T(f,g) comp .(X, Y ) = T f —¦ Y + X —¦ g ∈ Cc (P ← (f —¦ g)— T N ).

∞

42.18

42.20 42. Manifolds of mappings 449

The tangent mapping of the evaluation ev : C∞ (M, N ) — M ’ N at (f, x) in

direction of (X, ξ) ∈ Cc (M ← f — T N )—Tx M is given by T(f,x) ev .(X, ξ) = Tx f.ξ +

∞

X(x) ∈ Tf (x) N .

Proof. By (42.17), we may take a tangent vector X ∈ Tf (0, ) C∞ (M, N1 ) of the

form X = ‚t 0 f (t, ) ∈ Cc (M ← f — T N1 ), where f ∈ Clc (R — M, N1 ). Then we

∞ ∞

‚

‚ ‚

have (Tf (g— ).X)(x) = ‚t 0 g(f (t, x)) = T g. ‚t 0 f (t, x) = T g.X(x).

T (g — ) = g — is similar but easier, and the tangent mappings of the composition and

the evaluation can be computed either from the partial derivatives, or directly by

a variational computation as above.

42.19. The tangent mapping T : C∞ (M, N ) ’ C∞ (T M, T N ) is not smooth,

since the condition (42.5.1) is not preserved. But it is smooth as a mapping

T : C∞ (M, N ) ’ C∞ (M, L(T M, T N )), and its tangent mapping is given by

w

∼

=

T C∞ (M, N ) C∞ (M, T N )

c

u u

(κN )— —¦ T

T (T )

T C∞ (M, L(T M, T N )) ⊆ C ∞ (T M, T 2 N ),

where κN : T 2 N ’ T 2 N is the canonical ¬‚ip mapping, compare with (29.10).

‚

For the tangent mapping of the tangent mapping we consider ξx = ‚s |0 c(s) ∈ Tx M ,

and X ∈ Tf (0, ) C∞ (M, N ) of the form X = ‚t 0 f (t, ) ∈ Cc (M ← f — T N ) as

∞

‚

in the beginning of the proof. Then we have

‚

‚s |0 f (t, c(s))

T (f (t, )).ξx =

‚ ‚

(Tf (0, ) (T ).X)(ξx ) = ‚t 0 T (f (t, )) (ξx ) = ‚t 0 T (f (t, )).ξx

‚ ‚ ‚ ‚

‚t 0 ‚s |0 f (t, c(s)) = κN ‚s |0 ‚t 0 f (t, c(s))

=

‚

κN ‚s |0 X(c(s)) = κN .T X.ξx .

=

42.20. Theorem. Let M and N be smooth ¬nite dimensional manifolds, and

let q : N ’ M be smooth. Then the set C∞ (q) of all smooth sections of q

is a splitting smooth submanifold of C∞ (M, N ), whose tangent space is given by

T C∞ (q) = C∞ (M, ker(T q)) ‚ C∞ (M, T N ). If q : E ’ M is a ¬nite dimensional

c c

∞

vector bundle, the convenient vector space Cc (M ← E) is a splitting smooth sub-

manifold of C∞ (M, E).

Let now M and N be real analytic ¬nite dimensional manifolds with M compact,

and let q : N ’ M be real analytic. Then the set C ω (q) of all real analytic sec-

tions of q is a splitting real analytic submanifold of C ω (M, N ), and also C∞ (q)

is a is a splitting real analytic submanifold of C∞ (M, N ). If q : E ’ M is a

real analytic ¬nite dimensional vector bundle with M compact, the convenient vec-

tor space C ω (M ← E) is a splitting real analytic submanifold of C ω (M, E), and

C ∞ (M ← E) is a splitting real analytic submanifold of C∞ (M, E).

It is possible to extend this result at least to the case of a ¬ber bundle p : E ’ M

with in¬nite dimensional standard ¬ber by requiring certain properties. We do not

42.20

450 Chapter IX. Manifolds of mappings 42.21

present it here since in the only possible application (42.21) we have a simpler direct

proof.

Proof. If a smooth section s : M ’ N of q exists, then q, restricted to an open

neighborhood of s(M ), is a surjective submersion. Thus, there exists an open

neighborhood Ws of s(M ) in N such that ps := s —¦ q|Ws : Ws ’ s(M ) is a

surjective submersion, and we may assume that Ws is a tubular neighborhood, so

that ps : Ws ’ s(M ) is a vector bundle. Since C∞ (M, Ws ) is open in C∞ (M, N ),

we may replace N by Ws or assume that q : N ’ M is a vector bundle, and that

s is the zero section.

Claim. There exists a local addition ± : T N ’ N such that

(1) ± restricts to a local addition T 0(M ) ’ 0(M ) on the zero section.

(2) On each ¬ber Nx the local addition ± restricts to the addition T Nx ∼=

Nx — Nx ’ Nx .

In fact, choose a second vector bundle E ’ M such that N • E = M — Rk is

trivial, choose a local addition ±M on M , and let ±k be the addition on Rk . Then

±M — ±k restricts to a local addition on N with the required properties.

Now we consider the atlas for C∞ (M, N ) induced by ±, as in (42.4), i.e., we use the

formulas of (42.1) with exp replaced by ±. In particular, for the zero section s = 0

and for g ∈ U0 ‚ C∞ (M, N ) we have

u0 (g) = (IdM , (πN , ±)’1 —¦ (0, g)) ∈ Cc (M ← 0— T N ) ∼

∞

=

∼ C ∞ (M ← T M • N ) ∼ C ∞ (M ← T M ) — C ∞ (M ← N ),

=c =c c

∞

so that u0 (g) ∈ 0 — Cc (M ← N ) if and only if g is a section of the vector bundle.

∞

Moreover, Cc (M ← N ) ‚ U0 , so the second statement follows.

The statement about T C∞ (q) follows from (42.17) by noting that the derivative

of smooth curves in C∞ (q) are precisely sections s : M ’ ker(T q) such that s =

0 —¦ πE —¦ s o¬ some compact set in M .

This proof also works in the real analytic cases.

42.21. Theorem. Let (p : P ’ M, G) be a principal ¬ber bundle with ¬nite

dimensional base manifold M and a possibly in¬nite dimensional Lie group G as

structure group.

Then the gauge group Gau(P ) = C∞ (M ← P [G, conj]) from (37.17) carries the

∞

structure of a smooth Lie group modeled on Cc (P [g, Ad]).

If G is a regular Lie group then Gau(P ) is regular, too. If G admits an exponen-

tial mapping then Gau(P ) also admits an exponential mapping. If G is compact

then Gau(P ) is di¬eomorphic to the splitting submanifold C∞ (P, G)G of all G-

equivariant smooth mappings in C∞ (P, G).

If, moreover, M is compact and (p : P ’ M, G) is a real analytic principal bundle

with real analytic Lie group G, possibly in¬nite dimensional, then Gauω (P ) :=

42.21

42.21 42. Manifolds of mappings 451

C ω (M ← P [G, conj]) is a real analytic Lie group with the corresponding properties

as above.

Proof. The associated bundle P [G, conj] = P —(G,conj) G is a group bundle over

M with typical ¬ber G. It admits transition functions with values in Aut(G).

Therefore, the multiplication in G induces a smooth ¬berwise group multipli-

cation µ : P [G, conj] —M P [G, conj] ’ P [G, conj], also the ¬berwise inversion

ν : P [G, conj] ’ P [G, conj] is smooth.

The associated bundle P [g, Ad] = P —(G,Ad) g is a bundle of Lie algebras with

the same cocycle of transition functions. Thus, the bracket in g induces a smooth

∞

¬berwise bilinear mapping [ , ] : P [g, Ad]—M P [g, Ad] ’ P [g, Ad] and Cc (M ←

P [gAd]) is a convenient Lie algebra.

We shall use the canonical mappings q : P — g ’ P —G g = p[g, Ad] from (37.12.1),

„ G : P —M P ’ G from (37.8), and „ g : P —M P [g, Ad] ’ g from (37.12). We

also recall the the bijection C ∞ (P, g)G ∼ C ∞ (M ← P [g]) from (37.16), denoted by

=

f ’ sf and given by sf (p(u)) = q(u, f (u)), with inverse s ’ fs = „ g —¦ (IdP , s —¦ p).

Let u : U ’ V ⊆ g be a chart of G centered at e. Note that any model space of a

Lie group is isomorphic to the Lie algebra.

U± := {χ ∈ Gau(P ) : „ (±(z), χ(z)) ∈ U for all z ∈ P

and p({x : ±(x) = χ(x)}) has compact closure in M },

u± : U± ’ C ∞ (P, g)G ,

¯

u± (χ) = u —¦ „ —¦ (±, χ),

¯

˜ ∞