<< . .

. 67
( : 97)

. . >>

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

ω r = κl ’ (Ad —¦ ν).•

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 ).•

= Ad(g ’1 ).ωh .

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 .

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
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,

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.


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-
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).

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

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

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

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),
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) =

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
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

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

where we put P2 (E) := j=2 Lj (E; E) for the space of all polynomial mappings
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
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
In both cases, clearly, for k ≥ l the mapping πl : GLk (E) ’ GLl (E) is a homomor-

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

Lj (E; E) ’ GLk (E) ’ GLl (E) ’ {e}
{e} ’ sym

splits if and only if l = 1 for dim E > 1 or l ¤ 2 for E = R, see [Kol´ˇ, Michor,
Slov´k, 1993, 13.8] for E = R ; only in this case this sequence describes a semidirect

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. 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
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 =
(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

natural, i.e., they commute with the mappings from (3) and (4).
(6) (J k (M, N ), πl , J l (M, N ), i=l+1 Li (E; F )) are ¬ber bundles for all l ¤
k. For ¬nite k the bundle (J (M, N ), πk’1 , J k’1 (M, N ), Lk (E, F )) is an
k k
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 .

(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).

434 Chapter IX. Manifolds of mappings 41.5

It remains to show (6), and here we concentrate on the a¬ne bundle. Let a1 +
a ∈ GL(E) — i=2 Li (F ; F ), σ + σk ∈ Polyk’1 (E, F ) — Lk (E; F ), and b1 +
sym sym
b ∈ GL(E) — i=2 Li (E; E), then the only term of degree k containing σk in
(a1 + a) • (σ + σk ) • (b1 + b) is a1 —¦ σk —¦ bk , which depends linearly on σk . To this the
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.
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).

(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.

(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

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
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.


<< . .

. 67
( : 97)

. . >>