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