The cases (6) and (5) Let (M, µ) be a connected real analytic manifold of dimension

m ≥ 2 with a real analytic positive volume density. We can ¬nd a real analytic

Riemannian metric γ on M whose volume density is µ. We also choose a complete

Riemannian metric g.

First we assume that M is orientable. Then the divergence of a vector ¬eld X ∈

X(M ) is div X = —d—X , where X = γ(X) ∈ „¦1 (M ) (here we view γ : T M ’ T — M

and — is the Hodge star operator of γ). We choose real analytic (m ’ 2)-forms

βk for 1 ¤ k ¤ N = nm such that the vector ¬elds Xk = (’1)m+1 γ ’1 — dβk

satisfy conditions (9). Since these conditions describe Whitney C 1 open subsets,

such (m ’ 2)-forms exist by (30.12). The real analytic vector ¬elds Xk are then

divergence free since div Xk = —d — γXk = —ddβk = 0. Now we may ¬nish the proof

as usual.

43.20

474 Chapter IX. Manifolds of mappings 44.1

˜

For non-orientable M , we let π : M ’ M be the real analytic connected oriented

˜ ˜

double cover of M , and let • : M ’ M be the real analytic involutive covering

˜

map. We let π ’1 (xi ) = {x1 , x2 }, and we pull back both metrics to M , so γ := π — γ

˜

i i

˜

and g := π — g. We choose real analytic (m ’ 2)-forms βk ∈ „¦m’2 (M ) for 1 ¤

˜

k ¤ N = nm whose vector ¬elds Xβk = (’1)m+1 γ ’1 — dβk satisfy the following

˜

p

conditions, where we put Yij := Txp π ’1 .Yij for p = 1, 2:

ij

|Xβk (xp ) ’ Yij |g < µ

p

for k = (i ’ 1)m + j, p = 1, 2,

˜

i

p

|Xβk (xi )|g < µ for all k ∈ [(i ’ 1)m + 1, im], p = 1, 2,

/

(10) ˜

˜

|Xβk |g < 2 for all x ∈ M and all k.

˜

Since these conditions describe Whitney C 1 open subsets, such (m ’ 2)-forms exist

by (30.12). Then the vector ¬elds 1 (Xβk + •— Xβk ) still satisfy the conditions (10),

2

are still divergence free and induce divergence free vector ¬elds Zβk ∈ X(M ), so

that LZβk µ is the zero density, which satisfy the conditions (9) on M as in the

oriented case, and we may ¬nish the proof as above.

44. Principal Bundles with Structure

Group a Di¬eomorphism Group

44.1. Theorem. Principal bundle of embeddings. Let M and N be smooth

¬nite dimensional manifolds, connected and second countable without boundary such

that dim M ¤ dim N .

Then the set Emb(M, N ) of all smooth embeddings M ’ N is an open subman-

ifold of C∞ (M, N ). It is the total space of a smooth principal ¬ber bundle with

structure group Di¬(M ), whose smooth base manifold is the space B(M, N ) of all

submanifolds of N of type M .

The open subset Embprop (M, N ) of proper (equivalently closed) embeddings is satu-

rated under the Di¬(M )-action, and is thus the total space of the restriction of the

principal bundle to the open submanifold Bclosed (M, N ) of B(M, N ) consisting of

all closed submanifolds of N of type M .

This result is based on an idea implicitly contained in [Weinstein, 1971], it was

fully proved by [Binz, Fischer, 1981] for compact M and for general M by [Michor,

1980b]. The clearest presentation was in [Michor, 1980c, section 13].

Proof. Let us ¬x an embedding i ∈ Emb(M, N ). Let g be a ¬xed Riemannian

metric on N , and let expN be its exponential mapping. Then let p : N (i) ’ M

be the normal bundle of i, de¬ned in the following way: For x ∈ M let N (i)x :=

(Tx i(Tx M ))⊥ ‚ Ti(x) N be the g-orthogonal complement in Ti(x) N . Then we have

an injective vector bundle homomorphism over i:

w

¯±

N (i) TN

πN

u u

p = pi

w N.

i

M

44.1

44.1 44. Principal bundles with structure group a di¬eomorphism group 475

Now let U i = U be an open neighborhood of the zero section of N (i) which is so

small that (expN —¦¯)|U : U ’ N is a di¬eomorphism onto its image which describes

±

a tubular neighborhood of the submanifold i(M ). Let us consider the mapping

„ = „ i := (expN —¦¯)|U : N (i) ⊃ U ’ N,

±

a di¬eomorphism onto its image, and the open set in Emb(M, N ) which will serve

us as a saturated chart,

U(i) := {j ∈ Emb(M, N ) : j(M ) ⊆ „ i (U i ), j ∼ i},

where j ∼ i means that j = i o¬ some compact set in M . Then by (41.10) the set

U(i) is an open neighborhood of i in Emb(M, N ). For each j ∈ U(i) we de¬ne

•i (j) : M ’ U i ⊆ N (i),

•i (j)(x) := („ i )’1 (j(x)).

Then •i = ((„ i )’1 )— : U(i) ’ C ∞ (M, N (i)) is a smooth mapping which is bijective

onto the open set

V(i) := {h ∈ C∞ (M, N (i)) : h(M ) ⊆ U i , h ∼ 0}

in C ∞ (M, N (i)). Its inverse is given by the smooth mapping „— : h ’ „ i —¦ h.

i

i i

We have „— (h—¦f ) = „— (h)—¦f for those f ∈ Di¬(M ) which are so near to the identity

that h —¦ f ∈ V(i). We consider now the open set

{h —¦ f : h ∈ V(i), f ∈ Di¬(M )} ⊆ C∞ (M, U i ).

∞

Obviously, we have a smooth mapping from this set into Cc (M ← U i ) — Di¬(M )

given by h ’ (h —¦ (p —¦ h)’1 , p —¦ h), where Cc (M ← U i ) is the space of sections

∞

∞

with compact support of U i ’ M . So if we let Q(i) := „— (Cc (M ← U i ) © V(i)) ‚

i

Emb(M, N ) we have

W(i) := U(i) —¦ Di¬(M ) ∼ Q(i) — Di¬(M ) ∼ (Cc (M ← U i ) © V(i)) — Di¬(M ),

=∞

=

since the action of Di¬(M ) on i is free. Furthermore, the restriction π|Q(i) : Q(i) ’

Emb(M, N )/ Di¬(M ) is bijective onto an open set in the quotient.

We now consider •i —¦ (π|Q(i))’1 : π(Q(i)) ’ C ∞ (M ← U i ) as a chart for the

quotient space. In order to investigate the chart change, let j ∈ Emb(M, N ) be

such that π(Q(i)) © π(Q(j)) = …. Then there is an immersion h ∈ W(i) © Q(j)

and hence there exists a unique f0 ∈ Di¬(M ) (given by f0 = p —¦ •i (h)) such that

’1 ’1

h —¦ f0 ∈ Q(i). If we consider j —¦ f0 instead of j and call it again j, we have

Q(i) © Q(j) = …, and consequently U(i) © U(j) = …. Then the chart change is given

as follows:

•i —¦ (π|Q(i))’1 —¦ π —¦ („ j )— : Cc (M ← U j ) ’ Cc (M ← U i )

∞ ∞

s ’ „ j —¦ s ’ •i („ j —¦ s) —¦ (pi —¦ •i („ j —¦ s))’1 .

44.1

476 Chapter IX. Manifolds of mappings 44.3

This is of the form s ’ β —¦ s for a locally de¬ned di¬eomorphism β : N (j) ’ N (i)

which is not ¬ber respecting, followed by h ’ h —¦ (pi —¦ h)’1 . Both composants are

smooth by the general properties of manifolds of mappings. Therefore, the chart

change is smooth.

We show that the quotient space B(M, N ) = Emb(M, N )/ Di¬(M ) is Hausdor¬.

Let i, j ∈ Emb(M, N ) with π(i) = π(j). Then i(M ) = j(M ) in N for otherwise

put i(M ) = j(M ) =: L, a submanifold of N; the mapping i’1 —¦ j : M ’ L ’ M

is then a di¬eomorphism of M and j = i —¦ (i’1 —¦ j) ∈ i —¦ Di¬(M ), so π(i) = π(j),

contrary to the assumption.

Now we distinguish two cases.

Case 1. We may ¬nd a point y0 ∈ i(M ) \ j(M ), say, which is not a cluster point of

j(M ). We choose an open neighborhood V of y0 in N and an open neighborhood

W of j(M ) in N such that V © W = …. Let V := {k ∈ Emb(M, N ) : k(M ) ‚ V }

W := {k ∈ Emb(M, N ) : k(M ) ‚ W }. Then V is obviously open in Emb(M, N ),

and V is even open in the coarser compact-open topology. Both V and W are

Di¬(M ) saturated, i ∈ W, j ∈ V, and V © W = …. So π(V) and π(W) separate π(i)

and π(j) in B(M, N ).

Case 2. Let i(M ) ‚ j(M ) and j(M ) ‚ i(M ). Let y ∈ i(X), say. Let (V, v) be a

chart of N centered at y which maps i(M )©V into a linear subspace, v(i(M )©V ) ⊆

Rm © v(V ) ‚ Rn , where m = dim M , n = dim N . Since j(M ) ⊆ i(M ) we conclude

that we also have v((i(M ) ∪ j(M )) © V ) ⊆ Rm © v(V ). So we see that L :=

i(M ) ∪ j(M ) is a submanifold of N of the same dimension as N . Let (WL , pL , L)

be a tubular neighborhood of L. Then WL |i(M ) is a tubular neighborhood of i(M )

and WL |j(M ) is one of j(M ).

44.2. Result. [Cervera, Mascaro, Michor, 1991]. Let M and N be smooth mani-

folds. Then the di¬eomorphism group Di¬(M ) acts smoothly from the right on the

manifold Immprop (M, N ) of all smooth proper immersions M ’ N , which is an

open subset of C∞ (M, N ).

Then the space of orbits Immprop (M, N )/ Di¬(M ) is Hausdor¬ in the quotient

topology.

Let Immfree, prop (M, N ) be set of all proper immersions, on which Di¬(M ) acts

freely. Then this is open in C∞ (M, N ) and it is the total space of a smooth principal

¬ber bundle

Immfree,prop (M, N ) ’ Immfree,prop (M, N )/ Di¬(M ).

44.3. Theorem (Principal bundle of real analytic embeddings). [Kriegl,

Michor, 1990, section 6]. Let M and N be real analytic ¬nite dimensional manifolds,

connected and second countable without boundary such that dim M ¤ dim N , with

M compact. Then the set Embω (M, N ) of all real analytic embeddings M ’ N is an

open submanifold of C ω (M, N ). It is the total space of a real analytic principal ¬ber

bundle with structure group Di¬ ω (M ), whose real analytic base manifold B ω (M, N )

is the space of all real analytic submanifolds of N of type M .

44.3

44.5 44. Principal bundles with structure group a di¬eomorphism group 477

Proof. The proof of (44.1) is valid with the obvious changes. One starts with

a real analytic Riemannian metric and uses its exponential mapping. The space

of embeddings is open, since embeddings are open in C ∞ (M, N ), which induces a

coarser topology.

44.4. The nonlinear frame bundle of a ¬ber bundle. [Michor, 1988], [Michor,

1991]. Let now (p : E ’ M, S) be a ¬ber bundle, and let us ¬x a ¬ber bundle atlas

(U± ) with transition functions ψ±β : U±β — S ’ S. By (42.14) we have

C ∞ (U±β , C∞ (S, S)) ⊆ C ∞ (U±β — S, S)

with equality if and only if S is compact. Let us therefore assume from now on

that S is compact. Then we assume that the transition functions ψ±β : U±β ’

Di¬(S, S).

Now we de¬ne the nonlinear frame bundle of (p : E ’ M, S) as follows. We consider

the set Di¬{S, E} := x∈M Di¬(S, Ex ) and equip it with the in¬nite dimensional

di¬erentiable structure which one gets by applying the functor Di¬(S, ) to the

cocycle of transition functions (ψ±β ). Then the resulting cocycle of transition func-

tions for Di¬{S, E} induces the structure of a smooth principal bundle over M with

structure group Di¬(M ). The principal action is just composition from the right.

We can now consider the smooth action ev : Di¬(S) — S ’ S and the associated

bundle Di¬{S, E}[S, ev] = Di¬{S,E}—S . The mapping ev : Di¬{S, E} — S ’ E

Di¬(S)

is invariant under the Di¬(S)-action and factors therefore to a smooth mapping

Di¬{S, E}[S, ev] ’ E as in the following diagram:

w Di¬{S, E} — S

pr

Di¬{S, E} — S

Di¬(S)

ev

u

u

E Di¬{S, E}[S, ev].

The bottom mapping is easily seen to be a di¬eomorphism. Thus, the bundle

Di¬{S, E} may in full right be called the (nonlinear) frame bundle of E.

44.5. Let now ¦ ∈ „¦1 (E; T E) be a connection on E, see (37.2). We want to lift

¦ to a principal connection on Di¬{S, E}, and for this we need a good description

of the tangent space T Di¬{S, E}. With the method of (42.17) one can easily show

that

{f ∈ C ∞ (S, T E|Ex ) : T p —¦ f = one point

T Di¬{S, E} =

x∈M

in Tx M and πE —¦ f ∈ Di¬(S, Ex )}.

Starting from the connection ¦ we can then consider ω(f ) := T (πE —¦ f )’1 —¦ ¦ —¦ f :

S ’ T E ’ V E ’ T S for f ∈ T Di¬{S, E}. Then ω(f ) is a vector ¬eld on S, and

we have:

44.5

478 Chapter IX. Manifolds of mappings 44.10

Lemma. ω ∈ „¦1 (Di¬{S, E}; X(S)) is a principal connection, and the induced con-

nection on E = Di¬{S, E}[S, ev] coincides with ¦.

Proof. The fundamental vector ¬eld ζX on Di¬{S, E} for X ∈ X(S) is given by

ζX (g) = T g —¦ X. Then ω(ζX (g)) = T g ’1 —¦ ¦ —¦ T g —¦ X = X since T g —¦ X has vertical

values. Hence, ω reproduces fundamental vector ¬elds.

Now let h ∈ Di¬(S), and denote by rh the principal right action. Then we have

((rh )— ω)(f ) = ω(T (rh )f ) = ω(f —¦ h) = T (πE —¦ f —¦ h)’1 —¦ ¦ —¦ f —¦ h

= T h’1 —¦ ω(f ) —¦ h = AdDi¬(S) (h’1 )ω(f ).

44.6. Theorem. Let (p : E ’ M, S) be a ¬ber bundle with compact standard ¬ber

S. Then connections on E and principal connections on Di¬{S, E} correspond to

each other bijectively, and their curvatures are related as in (37.24). Each principal

connection on Di¬{S, E} admits a global parallel transport. The holonomy groups

and the restricted holonomy groups are equal as subgroups of Di¬(S).

Proof. This follows directly from (37.24) and (37.25). Each connection on E is

complete since S is compact, and the lift to Di¬{S, E} of its parallel transport is

the global parallel transport of the lift of the connection, so the two last assertions

follow.

44.7. Remark on the holonomy Lie algebra. Let M be connected, let ρ =

’dω ’ 1 [ω, ω]X(S) be the usual X(S)-valued curvature of the lifted connection ω on

2

Di¬{S, E}. Then we consider the R-linear span of all elements ρ(ξf , ·f ) in X(S),

where ξf , ·f ∈ Tf Di¬{S, E} are arbitrary (horizontal) tangent vectors, and we call

this span hol(ω). Then by the Di¬(S)-equivariance of ρ the vector space hol(ω) is

an ideal in the Lie algebra X(S).

44.8. Lemma. Let f : S ’ Ex0 be a di¬eomorphism in Di¬{S, E}x0 . Then

f— : X(S) ’ X(Ex0 ) induces an isomorphism between hol(ω) and the R-linear span

of all g — R(CX, CY ), X, Y ∈ Tx M , and g : Ex0 ’ Ex any di¬eomorphism.

The proof is obvious.

44.9. Gauge theory for ¬ber bundles. We consider the bundle Di¬{E, E} :=

x∈M Di¬(Ex , Ex ) which bears the smooth structure described by the cocycle of

’1

transition functions Di¬(ψ±β , ψ±β ) = (ψ±β )— (ψβ± )— , where (ψ±β ) is a cocycle of

transition functions for the ¬ber bundle (p : E ’ M, S).

44.10. Lemma. The associated bundle Di¬{S, E}[Di¬(S), conj] is isomorphic to

the ¬ber bundle Di¬{E, E}.

Proof. The mapping A : Di¬{S, E} — Di¬(S) ’ Di¬{E, E}, given by A(f, g) :=

f —¦g—¦f ’1 : Ex ’ S ’ S ’ Ex for f ∈ Di¬(S, Ex ), is Di¬(S)-invariant, so it factors

to a smooth mapping Di¬{S, E}[Di¬(S)] ’ Di¬{E, E}. It is bijective and admits

locally over M smooth inverses, so it is a ¬ber respecting di¬eomorphism.

44.10

44.16 44. Principal bundles with structure group a di¬eomorphism group 479

44.11. The gauge group Gau(E) of the ¬nite dimensional ¬ber bundle (p : E ’

M, S) with compact standard ¬ber S is, by de¬nition, the group of all principal

bundle automorphisms of the Di¬(S)-bundle (Di¬{S, E} which cover the identity

of M . The usual reasoning (37.17) gives that Gau(E) equals the space of all smooth

sections of the associated bundle Di¬{S, E}[Di¬(S), conj] which by (44.10) equals

the space of sections of the bundle Di¬{E, E} ’ M . We equip it with the topology

and di¬erentiable structure described in (42.21).

44.12. Theorem. The gauge group Gau(E) = C∞ (M ← Di¬{E, E}) is a regular

Lie group. Its exponential mapping is not surjective on any neighborhood of the

identity. Its Lie algebra consists of all vertical vector ¬elds with compact support

on E (or M ) with the negative of the usual Lie bracket. The obvious embedding

Gau(E) ’ Di¬(E) is a smooth homomorphism of regular Lie groups.

Proof. The ¬rst statement has already been shown before the theorem. A curve

through the identity of principal bundle automorphisms of Di¬{S, E} ’ M is a

smooth curve through the identity in Di¬(E) consisting of ¬ber respecting map-

pings. The derivative of such a curve is thus an arbitrary vertical vector ¬eld with

compact support. The space of all these is therefore the Lie algebra of the gauge

group, with the negative of the usual Lie bracket.

The exponential mapping is given by the ¬‚ow operator of such vector ¬elds. Since

on each ¬ber it is just conjugate to the exponential mapping of Di¬(S), it has all

the properties of the latter. Gau(E) ’ Di¬(E) is a smooth homomorphism since

by (40.3) its prolongation to the universal cover of Gau(E) is smooth.

44.13. Remark. If S is not compact we may circumvent the nonlinear frame

bundle, and we may de¬ne the gauge group Gau(E) directly as the splitting closed

subgroup of Di¬(E) which consists of all ¬ber respecting di¬eomorphisms which

cover the identity of M . The Lie algebra of Gau(E) consists then of all vertical

vector ¬elds on E with compact support on E. We do not work out the details of

this approach.

44.14. The space of connections. Let J 1 (E) ’ E be the a¬ne bundle of

1-jets of sections of E ’ M . We have J 1 (E) = { ∈ L(Tx M, Tu E) : T p —¦ =

IdTx M , u ∈ E, p(u) = x}. Then a section of J 1 (E) ’ E is just a horizontal lift

mapping T M —M E ’ T E which is ¬ber linear over E, so it describes a connection

as treated in (37.2), and we may view the space of sections C ∞ (E ← J 1 (E)) as the

space of all connections.

44.15. Theorem. The action of the gauge group Gau(E) on the space of connec-

tions C ∞ (E ← J 1 (E)) is smooth.

Proof. This follows from (42.13)

44.16. We will now give a di¬erent description of the action. We view a connection

¦ again as a linear ¬ber wise projection T E ’ V E, so the space of connections

44.16

480 Chapter IX. Manifolds of mappings 44.18

is now Conn(E) := {¦ ∈ „¦1 (E; T E) : ¦ —¦ ¦ = ¦, ¦(T E) = V E}. Since S

is compact the canonical isomorphism Conn(E) ’ C ∞ (E ← J 1 (E)) is even a

di¬eomorphism. Then the action of f ∈ Gau(E) ‚ Di¬(E) on ¦ ∈ Conn(E) is

given by f— ¦ = (f ’1 )— ¦ = T f —¦ ¦ —¦ T f ’1 . Now it is very easy to describe the

in¬nitesimal action. Let X be a vertical vector ¬eld with compact support on E

and consider its global ¬‚ow FlX .

t

Then we have dt |0 (FlX )— ¦ = LX ¦ = [X, ¦], the Fr¨licher Nijenhuis bracket, by

d

o

t

(35.14.5). The tangent space of Conn(E) at ¦ is the space T¦ Conn(E) = {Ψ ∈

„¦1 (E; T E) : Ψ|V E = 0}. The ”in¬nitesimal orbit” at ¦ in T¦ Conn(E) is {[X, ¦] :

∞

X ∈ Cc (E ← V E)}.

The isotropy subgroup of a connection ¦ is {f ∈ Gau(E) : f — ¦ = ¦}. Clearly, this

is just the group of all those f which respect the horizontal bundle HE = ker ¦.

The most interesting object is of course the orbit space Conn(E)/ Gau(E).

44.17. Slices. [Palais, Terng, 1988] Let M be a smooth manifold, G a Lie group,

G — M ’ M a smooth action, x ∈ M, and let Gx = {g ∈ G : g.x = x} denote

the isotropy group at x. A contractible subset S ⊆ M is called a slice at x, if it

contains x and satis¬es

(1) If g ∈ Gx then g.S = S.

(2) If g ∈ G with g.S © S = … then g ∈ Gx .

(3) There exists a local continuous section χ : G/Gx ’ G de¬ned on a neigh-

borhood V of the identity coset such that the mapping F : V — S ’ M,

de¬ned by F (v, s) := χ(v).s is a homeomorphism onto a neighborhood of x.

This is a local version of the usual de¬nition in ¬nite dimensions, which is too

narrow for the in¬nite dimensional situation. However, in ¬nite dimensions the

de¬nition above is equivalent to the usual one where a subset S ⊆ M is called a

slice at x, if there is a G-invariant open neighborhood U of the orbit G.x and a

smooth equivariant retraction r : U ’ G.x such that S = r’1 (x). In the general

case we have the following properties:

(4) For y ∈ F (V — S) © S we get Gy ‚ Gx , by (2).

(5) For y ∈ F (V — S) the isotropy group Gy is conjugate to a subgroup of Gx ,

by (3) and (4).

44.18. Counter-example. [Cerf, 1970], [Michor, Schichl, 1997]. The right action

of Di¬(S 1 ) on C ∞ (S 1 , R) does not admit slices.

Let h(t) : S 1 = (R mod 1) ’ R be a smooth bump function with h(t) = 0 for

t ∈ [0, 1 ] and h(t) > 0 for t ∈ (0, 1 ). Then put hn (t) = 41 h(4n (t ’ (1 ’ 41 )/3))

/ n n

4 4

which is is nonzero in the interval (1 ’ 41 )/3, (1 ’ 4n+1 )/3 , and consider

1

n