∞

⊆ Cc (M ← P [g, Ad]),

u± (χ) = su± (χ) = su—¦„ —¦(±,χ) ,

¯

u’1 (s)(z) = ±(z).u’1 („ g (z, s(p(z)))),

±

u’1 (s) = ±.(u’1 .„ g (IdP , s —¦ p)).

±

For the chart change we see that for s ∈ uβ (U± © Uβ ) we have

(u± —¦ u’1 )(s)(p(z)) = q z, u(„ P (±(z), β(z).u’1 („ g (z, s(p(z)))))) .

β

By (30.9.1), the space of smooth curves C ∞ (R, Cc (M ← P [g, Ad])) consists of all

∞

sections c such that c§ : R — M ’ P [g, Ad] is smooth and the following condition

holds:

(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.

Obviously, the chart change respects the set of smooth curves and is smooth. Thus,

the atlas (U± , u± ) describes the structure of a smooth manifold which we denote by

Gau(P ) ∼ C∞ (M ← P [G, conj]), and we also see that the space of smooth curves

=

C ∞ (R, C∞ (M ← P [G, conj])) consists of all sections c such that the associated

42.21

452 Chapter IX. Manifolds of mappings 42.22

mapping c§ : R — M ’ P [G, conj] is smooth and condition (1) holds. Composition

and inversion are smooth on Gau(P ) since these correspond just to push forwards

of sections via the smooth ¬berwise group multiplication and inversion described at

the beginning of the proof. The Lie algebra of C∞ (M ← P [G, conj]) is Cc (M ←

∞

P [g, Ad]).

Let us now suppose that the Lie group G is regular with evolution operator evolG :

C ∞ (R, g) ’ G. Since the smooth group bundle P [G, conj] is described by a cocycle

of transition functions with values in the group of (inner) automorphisms of G and

since by (38.4) we have evolG —¦ •— = • —¦ evolG for any automorphism • of G, there

is an induced ¬berwise evolution operator

evol : P [C ∞ (R, g), Ad— ] ∼ C ∞ (R, p[g, Ad]) ’ P [G, conj],

=

which, by push forward on sections, induces

evolGau(P ) : C ∞ (R, Cc (M ← P [g, Ad])) ’ Cc (M ← P [G, conj]).

∞ ∞

This maps smooth curves to smooth curves and is the evolution operator of Gau(P ).

The remaining assertions are easy to check.

42.22. Manifolds of holomorphic mappings. It is a natural question whether

the methods of this section carry over to spaces of holomorphic mappings between

complex manifolds. The situation is described in the following result.

Lemma.

(1) Each ¬nite dimensional Stein manifold M admits holomorphic local addi-

tions T M ⊃ U ’ M in the sense of (42.4).

(2) Complex projective spaces do not admit holomorphic local additions.

Proof. (1) A Stein manifold M is biholomorphically embedded as a closed complex

submanifold of some Cn (where n = 2 dimC M + 2 su¬ces), see [Gunning, Rossi,

1965, p. 224], and there exists a holomorphic tubular neighborhood p : V ’ M in

Cn , see [Gunning, Rossi, 1965, p. 257], by an application of Cartan™s theorem B that

a coherent sheaf on a Stein manifold is acyclic. The a¬ne addition • : T Cn ’ Cn ,

given by •(z, Z) := z + Z then gives a local addition p —¦ •|T M : T M ⊃ U ’ M

for a suitable open neighborhood U of 0 in T M .

(2) First we show that CP1 does not admit a holomorphic local addition. The usual

a¬ne charts u0 [z0 : z1 ] = z1 and u1 [z0 : z1 ] = z0 have as chart change mapping

z0 z1

z ’ 1/z on C \ {0}. Its tangent mapping is (z, w) ’ ( z , ’ z12 w). A local addition

1

would be given by two holomorphic mappings ±i : T C ⊃ U ’ C on an open

neighborhood U of the zero section {(z, 0) : z ∈ C} with

1

±0 ( z , ’ z12 w) =

1

for z = 0,

±1 (z, w)

‚

‚w |w=0 ±i (z, w)

±i (z, 0) = z, = 0 for all z.

42.22

42.22 42. Manifolds of mappings 453

‚2

‚

of 1 = ±0 ( z , ’ z12 w)±1 (z, w) are in turn

1

‚w |w=0 ‚w2 |w=0

The derivatives and

1 1

0 = z (‚2 ±1 (z, 0) ’ ‚2 ±0 ( z , 0)),

12 12

2 1 1

0 = z ‚2 ±1 (z, 0) ’ z 2 ‚2 ±0 ( z , 0).‚2 ±1 (z, 0) + z 3 ‚2 ±0 ( z , 0)

22 2 2

1 1

z 3 (z ‚2 ±1 (z, 0) ’ z(‚2 ±1 (z, 0)) + ‚2 ±0 ( z , 0)).

=

1

2 2

Hence, limz’0 ‚2 ±0 ( z , 0) = 0, and consequently ‚2 ±0 (z, 0) = 0 for all z since

it is an entire function on C which vanishes at in¬nity. But then we get that

1

z ’ ‚2 ±1 (z, 0) = z (‚2 ±1 (z, 0))2 ’ 0 has a pole at 0, a contradiction.

2

Now we treat CPn . Suppose that a holomorphic local addition ± : T (CPn ) ⊃ U ’

CPn exists. Let us consider CP1 ‚ CPn , given by [z0 : z1 ] ’ [z0 : z1 : 0 : · · · : 0].

Then we have a holomorphic retraction r : V ’ CP1 given by r[z0 : · · · : zn ] = [z0 :

z1 ] for V = {[z0 : · · · : zn ] : (z0 , z1 ) = (0, 0)}. But then r —¦ ±|(U © T (CP1 )) is a

holomorphic local addition on CP1 , a contradiction.

Results. From the argument given in (42.4) follows that a complex manifold ad-

mitting a holomorphic local addition also admits a holomorphic spray and thus a

holomorphic linear connection on T M . Existence of the latter has been investigated

in [Atiyah, 1957]. Let us sketch the relevant results. For a complex manifold M let

T M be the complex tangent bundle, let GL(Cm , T M ) be the linear frame bundle.

Then using the local description from (29.9) and (29.10) we get in turn:

GL(Cm , T M ) (x, s) ∈ U — GL(m, C),

T (GL(Cm , T M )) (x, s, ξ, σ) ∈ U — GL(m, C) — Cm — gl(m, C),

{f ∈ LC (Cm , T (πM )’1 )(ξ) :

T (GL(Cm , T M )) =

ξ∈T M

πT M —¦ f ∈ GL(Cm , T M )},

(x, Id, ξ, A) = (x, s —¦ s’1 , ξ, σ —¦ s’1 ),

T (GL(Cm , T M ))/GL(m, C)

T (GL(Cm , T M ))

{f ∈ LC (Tx M, T (πM )’1 )(ξ) :

=

GL(m, C)

x∈M ξ∈Tx M

πT M —¦ f = IdTx M },

which turns out to be a holomorphic vector bundle over M . Then we have the

following exact sequence of holomorphic vector bundles over M :

vlT M —¦(Id, ) T (πM )

0 ’ L(T M, T M ) ’ ’ ’ ’ ’ T (GL(Cm , T M ))/GL(m, C) ’ ’ ’ M ’ 0.

’ ’ ’ ’’ ’’

T

A holomorphic splitting of this sequence is exactly a holomorphic linear connection

on T M . This sequence de¬nes an extension of the bundle T M by L(T M, T M ), i.e.,

an element b(T M ) in the sheaf cohomology H 1 (M ; T — M — L(T M, T M )). Thus:

[Atiyah, 1957, from theorems 2 and 5]. A complex manifold M admits a holomor-

phic linear connection if and only if b(T M ) vanishes.

Note that via Cartan™s theorem B this again implies that Stein manifolds admit

holomorphic local additions. Moreover, Atiyah proved the following results:

42.22

454 Chapter IX. Manifolds of mappings 43.1

Result. [Atiyah, 1957, theorem 6]. If M is a compact K¨hler manifold, then the

a

k-th Chern class of T M is given by

√

ck (T M ) = (’2π ’1)’k Sk [b(T M )],

where Sk is the k characteristic coe¬cient gl(m, C) ’ C.

Note that this also implies that CPn does not admit local additions.

[Atiyah, 1957, proposition 22]. Even if all characteristic classes of M vanish, M

need not admit a holomorphic connection.

43. Di¬eomorphism Groups

43.1. Theorem. Di¬eomorphism group. For a smooth manifold M the group

Di¬(M ) of all smooth di¬eomorphisms of M is an open submanifold of C∞ (M, M ),

composition and inversion are smooth. It is a regular Lie group in the sense of

(38.4).

The Lie algebra of the smooth in¬nite dimensional Lie group Di¬(M ) is the conve-

∞

nient vector space Cc (M ← T M ) of all smooth vector ¬elds on M with compact

support, equipped with the negative of the usual Lie bracket. The exponential map-

ping exp : Cc (M ← T M ) ’ Di¬ ∞ (M ) is the ¬‚ow mapping to time 1, and it is

∞

smooth.

Proof. We ¬rst show that Di¬(M ) is open in C∞ (M, M ). Let c : R ’ C∞ (M, M )

be a smooth curve such that c(0) is a di¬eomorphism. We have to show that then

c(t) also is a di¬eomorphism for small t. The mapping c(t) stays in the WO1 -open

(and thus open by (42.1)) subset of immersions for t near 0, see (41.10).

The mapping c(t) stays injective for t near 0: For |t| ¤ 1 we have c(t)|(M \

K1 ) = c(0)|(M \ K1 ) for a compact subset K1 ⊆ M , by (42.5). Let K2 :=

c(0)’1 (c§ ([’1, 1] — K1 )) ⊃ K1 . If c(t) does not stay injective for t near 0 then

there are tn ’ 0 and xn = yn in M with c(tn )(xn ) = c(tn )(yn ). We claim that

xn , yn ∈ K2 : If xn ∈ K2 then c(tn )(xn ) = c(0)(xn ), so yn ∈ K1 , since other-

/

wise c(tn )(yn ) = c(0)(yn ) = c(0)(xn ); but then c(tn )(yn ) ∈ c§ ([’1, 1] — K1 ) =

c(0)(K2 ) c(0)(xn ). Passing to subsequences we may assume that xn ’ x and

yn ’ y in K2 . By continuity of c§ , we get c(0)(x) = c(0)(y), so x = y. The map-

ping (t, z) ’ (t, c(t)(z)) is a di¬eomorphism near (0, x), since it is an immersion.

But then c(tn )(xn ) = c(tn )(yn ) for large n.

The mapping c(t) stays surjective for t near 0: In the situation of the last paragraph

interior

c(t)(M ) = c(t)(K2 ) ∪ c(0)(M \ K1 ) is closed in M for |t| ¤ 1 and also open for

t near 0, since c(t) is a local di¬eomorphism. It meets each connected component

of M since c(t) is homotopic to c(0). Thus, c(t)(M ) = M .

Therefore, Di¬(M ) is an open submanifold of C∞ (M, M ), so composition is

prop

smooth by (42.13). To show that the inversion inv is smooth, we consider a

43.1

43.1 43. Di¬eomorphism groups 455

smooth curve c : R ’ Di¬(M ) ‚ C∞ (M, M ). Then the mapping c§ : R — M ’

M satis¬es (42.5.1), and (inv —¦c)§ ful¬lls the ¬nite dimensional implicit equation

c§ (t, (inv —¦c)§ (t, m)) = m for all t ∈ R and m ∈ M . By the ¬nite dimensional im-

plicit function theorem, (inv —¦c)§ is smooth in (t, m). Property (42.5.1) is obvious.

Hence, inv maps smooth curves to smooth curves and is thus smooth. (This proof

is by far simpler than the original one, see [Michor, 1980c], and shows the power of

the Fr¨licher-Kriegl calculus.)

o

By the chart structure from (42.1), or directly from theorem (42.17), we see that the

∞

tangent space Te Di¬(M ) equals the space Cc (M ← T M ) of all vector ¬elds with

compact support. Likewise Tf Di¬(M ) = Cc (M ← f — T M ), which we identify

∞

with the space of all vector ¬elds with compact support along the di¬eomorphism

f . Right translation µf is given by µf (g) = f — (g) = g —¦ f , thus T (µf ).X = X —¦ f ,

and for the ¬‚ow FlX of the vector ¬eld with compact support X we have dt FlX = d

t t

FlX

X X

X —¦ Flt = T (µ t ).X. So the one parameter group t ’ Flt ∈ Di¬(M ) is the

integral curve of the right invariant vector ¬eld RX : f ’ T (µf ).X = X —¦ f on

Di¬(M ). Thus, the exponential mapping of the di¬eomorphism group is given by

∞

exp = Fl1 : Cc (M ← T M ) ’ Di¬(M ). To show that is smooth we consider a

∞

smooth curve in Cc (M ← T M ), i.e., a time dependent vector ¬eld with compact

support Xt . We may view it as a complete vector ¬eld (0t , Xt ) on R — M whose

smooth ¬‚ow respects the level surfaces {t} — M and is smooth. Thus, exp —¦X =

(0,X) ∨

(pr2 —¦ Fl1 ) maps smooth curves to smooth curves and is smooth itself. Again

one may compare this simple proof with the original one [Michor, 1983, section 4].

To see that Di¬(M ) is a regular Lie group note that the evolution is given by

integrating time dependent vector ¬elds with compact support,

evol(t ’ Xt ) = •(1, )

‚

‚t •(t, x) = X(t, •(t, x)), •(0, x) = x.

∞

Let us ¬nally compute the Lie bracket on Cc (M ← T M ) viewed as the Lie algebra

∞

of Di¬(M ). For X ∈ Cc (M ← T M ) let LX denote the left invariant vector ¬eld

on Di¬(M ). Its ¬‚ow is given by FlLX (f ) = f —¦ exp(tX) = f —¦ FlX = (FlX )— (f ).

t t t

LX —

d

From (32.15) we get [LX , LY ] = dt |0 (Flt ) LY , so for e = IdM we have

[LX , LY ](e) = ( dt |0 (FlLX )— LY )(e)

d

t

LX LX

d

dt |0 (T (Fl’t ) —¦ LY —¦ Flt )(e)

=

|0 T (FlLX )(LY (e —¦ FlX ))

d

= ’t t

dt

|0 T ((FlX )— )(T (FlX ) —¦ Y )

d

= ’t t

dt

X X

d

dt |0 (T (Flt ) —¦ Y —¦ Fl’t ),

= by (42.18)

X—

d

dt |0 (Fl’t ) Y = ’[X, Y ].

=

Another proof using (36.10) is as follows:

‚

exp(sX) —¦ exp(tY ) —¦ exp(’sX)

Ad(exp(sX))Y = ‚t 0

= T (FlX ) —¦ Y —¦ FlX = (FlX )— Y,

’s ’s

s

43.1

456 Chapter IX. Manifolds of mappings 43.2

thus

(FlX )— Y = ’[X, Y ]

‚ ‚

Ad(exp(tX))Y = ’t

‚t 0 ‚t 0

∞

is the negative of the usual Lie bracket on Cc (M ← T M ).

It is well known that the space Di¬(M ) of all di¬eomorphisms of M is open in

C ∞ (M, M ) even for the Whitney C ∞ -topology, see (41.10); proofs can be found in

[Hirsch, 1976, p. 38] or [Michor, 1980c, section 5].

∞

43.2. Example. The exponential mapping exp : Cc (M ← T M ) ’ Di¬(M ) sat-

is¬es T0 exp = Id, but it is not locally surjective near IdM : This is due to [Freifeld,

1967] and [Koppell, 1970]. The strongest result in this direction is [Grabowski,

1988], where it is shown, that Di¬(M ) contains a smooth curve through IdM con-

tains an arcwise connected free subgroup on 2„µ0 generators which meets the image

of exp only at the identity.

We shall prove only a weak version of this for M = S 1 . For large n ∈ N we consider

the di¬eomorphism

nθ

sin2 (

2π 1

fn (θ) = θ + + ) mod 2π;

2n

n 2

(the subgroup generated by) fn has just one periodic orbit of period n, namely

{ 2πk : k = 0, . . . , n ’ 1}. For even n the di¬eomorphism fn cannot be written as

n

g —¦ g for a di¬eomorphism g (so fn is not contained in a ¬‚ow), by the following

argument: If g has a periodic orbit of odd period, then this is also a periodic orbit

of the same period of g —¦ g, whereas a periodic orbit of g of period 2n splits into

two disjoint orbits of period n each, of g —¦ g. Clearly, a periodic orbit of g —¦ g is a

subset of a periodic orbit of g. So if g —¦ g has only ¬nitely many periodic orbits of

some even order, there must be an even number of them.

Claim. Let f ∈ Di¬(S 1 ) be ¬xed point free and in the image of exp. Then f is

conjugate to some translation Rθ .

We have to construct a di¬eomorphism g : S 1 ’ S 1 such that f = g ’1 —¦ Rθ —¦ g.

Since p : R ’ R/2πZ = S 1 is a covering map it induces an isomorphism Tt p :

R ’ Tp(t) S 1 . In the picture S 1 ⊆ C this isomorphism is given by s ’ s p(t)⊥ ,

where p(t)⊥ is the normal vector obtained from p(t) ∈ S 1 via rotation by π/2.

Thus, the vector ¬elds on S 1 can be identi¬ed with the smooth functions S 1 ’ R

or, by composing with p : R ’ S 1 , with the 2π-periodic functions X : R ’ R.

Let us ¬rst remark that the constant vector ¬eld X θ ∈ X(S 1 ), s ’ θ has the ¬‚ow

θ θ

FlX : (t, •) ’ • + t · θ. Hence, exp(X θ ) = FlX (1, ) = Rθ .

θ

Let f = exp(X) and suppose g —¦ f = Rθ —¦ g. Then g —¦ FlX (t, ) = FlX (t, ) —¦ g

for t = 1. Let us assume that this is true for all t. Then di¬erentiating at t = 0

yields T g(Xx ) = Xg(x) for all x ∈ S 1 . If we consider g as di¬eomorphism R ’ R

θ

this means that g (t) · X(t) = θ for all t ∈ R. Since f was assumed to be ¬xed point

free the vector ¬eld X is nowhere vanishing, otherwise there would be a stationary

tθ

point x ∈ S 1 . So the condition on g is equivalent to g(t) = g(0) + 0 X(s) ds. We

43.2

43.3 43. Di¬eomorphism groups 457

take this as de¬nition of g, where g(0) := 0, and where θ will be chosen such that

t+2π ds

g factors to an (orientation preserving) di¬eomorphism on S 1 , i.e. θ t X(s) =

2π

ds

g(t + 2π) ’ g(t) = 1. Since X is 2π-periodic this is true for θ = 1/ 0 X(s) . Since

the ¬‚ow of a transformed vector ¬eld is nothing else but the transformed ¬‚ow we

θ

obtain that g(FlX (t, x)) = FlX (t, g(x)), and hence g —¦ f = Rθ —¦ g.

Note that the formula from (38.2) for the tangent mapping of the exponential of a

Lie group in the case G = Di¬(M ) looks as follows:

1

(FlX )— Y dt —¦ FlX ,

(1) TX exp .Y = ’t 1

0

by the formula for Ad —¦ exp in the proof of (43.1), and by (42.17).

The break-down of the inverse function theorem in this situation is explained by

the following

Claim. [Grabowski, 1993] For each ¬nite dimensional manifold M of dimension

m > 1 and for M = S 1 the mapping TX exp is not injective for some X arbitrarily

near to 0. So GL(Xc (M )) is not open in L(Xc (M ), Xc (M )).

For M = R this seems to be wrong for vector ¬elds with compact support.

1‚

Proof. Let us start with M = S 1 and the vector ¬elds Xn (θ) := n ‚θ and Yn :=

sin(nθ) ‚θ for θ mod 2π in X(S 1 ). Then FlXn (θ) = θ + n t mod 2π, and hence we

‚ 1

t

1

get 0 (FlXn )— Yn dt = 0.

’t

For a manifold M of dimension m > 1 we now take an embedding S 1 — U ’ M for

∞

an open ball U ‚ Rm’1 , functions g, h ∈ Cc (U, R) with g.h = g. Then the vector

˜ ˜

¬elds Xn (θ, x) = h(x)Xn (θ) and Yn (θ, x) = g(x)Yn (θ) in Xc (S 1 — U ) ‚ Xc (M )

˜

1 ˜

satisfy 0 (FlXn )— Yn dt = 0, since h(x) = 1 if g(x) = 0, too.

’t

43.3. Remarks. The mapping

∞ ∞ ∞

Ad —¦ exp : Cc (M ← T M ) ’ Di¬(M ) ’ L(Cc (M ← T M ), Cc (M ← T M ))

is not real analytic since Ad(exp(sX))Y (x) = (FlX )— Y (x) = Tx (FlX )(Y (FlX (x)))

’s ’s

s

is not real analytic in s in general: choose Y constant in a chart and X not real

analytic.

For a real analytic compact manifold M the group Di¬(M ) is an open submanifold

of the real analytic (see (42.8)) manifold C ∞ (M, M ). The composition mapping is,

however, not real analytic by (42.16).

∞

For x ∈ M the mapping evx —¦ exp : Cc (M ← T M ) ’ Di¬(M ) ’ M is not real

analytic since (evx —¦ exp)(tX) = FlX (x), which is not real analytic in t for general

t

smooth X.

In contrast to this, one knows from [Omori, 1978b] that a Banach Lie group acting

e¬ectively on a ¬nite dimensional manifold is necessarily ¬nite dimensional. So

there is no way to model the di¬eomorphism group on Banach spaces as a manifold.

There is, however, the possibility to view Di¬(M ) as an ILH-group (i.e. inverse limit

of Hilbert manifolds), which sometimes permits to use an implicit function theorem.

See [Omori, 1974] for this.

43.3

458 Chapter IX. Manifolds of mappings 43.6

43.4. Theorem (Real analytic di¬eomorphism group). For a compact real

analytic manifold M the group Di¬ ω (M ) of all real analytic di¬eomorphisms of M

is an open submanifold of C ω (M, M ), composition and inversion are real analytic.

Its Lie algebra is the space C ω (M ← T M ) of all real analytic vector ¬elds on M ,

equipped with the negative of the usual Lie bracket. The associated exponential

mapping exp : C ω (M ← T M ) ’ Di¬ ω (M ) is the ¬‚ow mapping to time 1, and it is

real analytic.

The real analytic Lie group Di¬ ω (M ) is regular in the sense of (38.4), evol is even

real analytic.

Proof. Di¬ ω (M ) is open in C ω (M, M ) in the compact-open topology, thus also

in the ¬ner manifold topology. The composition is real analytic by (42.15), so it

remains to show that the inversion ν is real analytic.

Let c : R ’ Di¬ ω (M ) be a C ω -curve. Then the associated mapping c§ : R —

M ’ M is C ω by (42.14), and (ν —¦ c)§ is the solution of the implicit equation

c§ (t, (ν —¦ c)§ (t, x)) = x and therefore real analytic by the ¬nite dimensional implicit

function theorem. Hence, ν —¦ c : R ’ Di¬ ω (M ) is real analytic, again by (42.14).

Let c : R ’ Di¬ ω (M ) be a C ∞ -curve. Then by lemma (42.12) the associated

mapping c§ : R — M ’ M has a unique extension to a C n -mapping R — MC ⊇

J — W ’ MC which is holomorphic on W (has C-linear derivatives), for each

n ≥ 1. The same assertion holds for the curve ν —¦ c by the ¬nite dimensional

implicit function theorem for C n -mappings.

The tangent space at IdM of Di¬ ω (M ) is the space C ω (M ← T M ) of real analytic

vector ¬elds on M . The one parameter subgroup of a tangent vector is the ¬‚ow

t ’ FlX of the corresponding vector ¬eld X ∈ C ω (M ← T M ), so exp(X) = FlX

t 1

which exists since M is compact.

In order to show that exp : C ω (M ← T M ) ’ Di¬ ω (M ) ⊆ C ω (M, M ) is real

analytic, by the exponential law (42.14) it su¬ces to show that the associated

mapping exp§ = Fl1 : C ω (M ← T M ) — M ’ M is real analytic. This follows

from the ¬nite dimensional theory of ordinary real analytic and smooth di¬erential

equations. The same is true for the evolution operator.

43.5. Remark. The exponential mapping exp : C ω (S 1 ← T S 1 ) ’ Di¬ ω (S 1 ) is

not surjective on any neighborhood of the identity.

Proof. The proof of (43.2) for the group of smooth di¬eomorphisms of S 1 can

be adapted to the real analytic case: •n (θ) = θ + 2π + 21 sin2 ( nθ ) mod 2π is

n

n 2

ω 1

Mackey convergent (in UId ) to IdS 1 in Di¬ (S ), but •n is not in the image of the

exponential mapping.

43.6. Example 1. Let g ‚ Xc (R2 ) be the closed Lie subalgebra of all vector ¬elds

‚ ‚

with compact support on R2 of the form X(x, y) = f (x, y) ‚x + g(x, y) ‚y where g

vanishes on the strip 0 ¤ x ¤ 1.

Claim. There is no Lie subgroup G of Di¬(R2 ) corresponding to g.

43.6

43.7 43. Di¬eomorphism groups 459

If G exists there is a smooth curve t ’ ft ∈ G ‚ Di¬ c (R2 ). Then Xt := ( ‚t ft )—¦ft’1

‚

‚

is a smooth curve in g, and we may assume that X0 = f ‚x where f = 1 on a large

ball. But then AdG (ft ) = ft— : g ’ g, a contradiction.

So we see that on any manifold of dimension greater than 2 there are closed Lie

subalgebras of the Lie algebra of vector ¬elds with compact support which do not

admit Lie subgroups.

Example 2. The space XK (M ) of all vector ¬elds with support in some open set

U is an ideal in Xc (M ), the corresponding Lie group is the connected component

Di¬ U (M )0 of the group of all di¬eomorphisms which equal Id o¬ some compact in

U , but this is not a normal subgroup in the connected component Di¬ c (M )0 , since

we may conjugate the support out of U .

Note that this examples do not work for the Lie group of real analytic di¬eomor-

phisms on a compact manifold.

43.7. Theorem. [Ebin, Marsden, 1970] Let M be a compact orientable manifold,

let µ0 be a positive volume form on M with total mass 1. Then the regular Lie

group Di¬ + (M ) of all orientation preserving di¬eomorphisms splits smoothly as

Di¬ + (M ) = Di¬(M, µ0 ) — Vol(M ), where Di¬(M, µ0 ) is the regular Lie group of

all µ0 -preserving di¬eomorphisms, and Vol(M ) is the space of all volume forms of

total mass 1.

If (M, µ0 ) is real analytic, then Di¬ ω (M ) splits real analytically as Di¬ ω (M ) =

+ +

ω ω ω

Di¬ (M, µ0 ) — Vol (M ), where Di¬ (M, µ0 ) is the Lie group of all µ0 -preserving

real analytic di¬eomorphisms, and Volω (M ) is the space of all real analytic volume

forms of total mass 1.

Proof. We show ¬rst that there exists a smooth mapping „ : Vol(M ) ’ Di¬ + (M )

such that „ (µ)— µ0 = µ.

We put µt = µ0 + t(µ ’ µ0 ). We want a smooth curve t ’ ft ∈ Di¬ + (M ) with

ft— µt = µ0 . We have ‚t ft = Xt —¦ ft for a time dependent vector ¬eld Xt on M .

‚

Then 0 = ‚t ft— µt = ft— LXt µt + ft— ‚t µt = ft— (LXt µt + (µ ’ µ0 )), so LXt µt = µ0 ’ µ

‚ ‚

and LXt µt = diXt µt + iXt 0 = dω for some ω ∈ „¦dim M ’1 (M ). Now we choose ω

such that dω = µ0 ’ µ, and we choose it smoothly and in the real analytic case

even real analytically depending on µ by the theorem of Hodge, as follows: For any

± ∈ „¦(M ) we have ± = H± + dδG± + δGd±, where H is the projection on the space

of harmonic forms, δ = —d— is the codi¬erential, — is the Hodge-star operator, and G

is the Green operator, see [Warner, 1971]. All these are bounded linear operators,

G is even compact. So we may choose ω = δG(µ0 ’ µ). Then the time dependent

vector ¬eld Xt is uniquely determined by iXt µt = ω since µt is nowhere 0. Let ft

’1

be the evolution operator of Xt , and put „ (µ) = f1 .

Now we may prove the theorem itself. We de¬ne a mapping Ψ : Di¬ + (M ) ’

Di¬(M, µ0 ) — Vol(M ) by Ψ(f ) := (f —¦ „ (f — µ0 )’1 , f — µ0 ), which is smooth or real

analytic by (42.15) and (43.4). An easy computation shows that the inverse is given

by the smooth (or real analytic) mapping Ψ’1 (g, µ) = g —¦ „ (µ).

43.7

460 Chapter IX. Manifolds of mappings 43.9

That Di¬(M, µ0 ) is regular follows from (38.7), where we use the mapping p :

Di¬ + (M ) ’ „¦max (M ), given by p(f ) := f — µ0 ’ µ0 .

We next treat the Lie group of symplectic di¬eomorphisms.

43.8. Symplectic manifolds. Let M be a smooth manifold of dimension 2n ≥ 2.

A symplectic form on M is a closed 2-form σ such that σ n = σ § · · · § σ ∈ „¦2n (M )

is nowhere 0. The pair (M, σ) is called a symplectic manifold. See section (48) for

a treatment of in¬nite dimensional symplectic manifolds.

A symplectic form can be put into the following (Darboux) normal form: For each

u

x ∈ M there is a chart M ⊃ U ’ u(U ) ‚ R2n centered at x such that on U the

’

form σ is given by σ|U = u1 dun+1 + u2 dun+2 + · · · + un du2n . This follows from

proposition (43.11) below for N = {x}.

A di¬eomorphism f ∈ Di¬(M ) with f — σ = σ is called a symplectic di¬eomor-

phism; some authors also write symplectomorphism. The group of all symplectic

di¬eomorphisms will be denoted by Di¬(M, σ).

A vector ¬eld X ∈ X(M ) will be called a symplectic vector ¬eld if LX σ = 0; some

authors also write locally Hamiltonian vector ¬eld. The Lie algebra of all symplectic

vector ¬elds will be denoted by X(M, σ).

For a ¬nite dimensional symplectic manifold (M, σ) we have the following exact

sequence of Lie algebras:

gradσ γ

∞

0 ’ H (M ) ’ C (M, R) ’ ’ X(M, σ) ’ H 1 (M ) ’ 0

0

’’ ’

Here H — (M ) is the real De Rham cohomology of M , gradσ f is the Hamiltonian

vector ¬eld for f ∈ C ∞ (M, R) given by i(gradσ f )σ = df , and γ(ξ) = [iξ σ]. The

spaces H 0 (M ) and H 1 (M ) are equipped with the zero bracket, and the space

C ∞ (M, R) is equipped with the Poisson bracket

{f, g} := i(gradσ f )i(gradσ g)σ = σ(gradσ g, gradσ f ) =

= (gradσ f )(g) = dg(gradσ f ).

The image of gradσ is the Lie subalgebra of globally Hamiltonian vector ¬elds. We

shall prove this for in¬nite dimensional manifolds in section (48) below.

A submanifold L of a symplectic manifold (M 2n , σ) is called a Lagrange submanifold

if it is of dimension n and incl— σ = 0 on L.

43.9. Canonical example. Let Q be an n-dimensional manifold. Let us consider

the natural 1-form θQ on the cotangent bundle T — Q which is given by θQ (Ξ) :=

πT — Q (Ξ), T (πQ ).Ξ T Q , where we used the projections πQ : T — Q ’ Q and

— —

—

T (πQ )

πT — Q

— —

T Q ← ’ T (T Q) ’ ’ ’ T Q.

’’ ’’

The canonical symplectic structure on T — Q is then given by σQ = ’dθQ . If q : U ’

Rn is a smooth chart on Q with induced chart T — q = (q, p) : T — U = (πQ )’1 (U ) ’

—

43.9

43.10 43. Di¬eomorphism groups 461

Rn — Rn , we have θQ |T — U = pi dq i and σQ |T — U = dq i § dpi . The canonical

forms θQ and σQ on T — Q have the following universal property and are determined

by it: For any 1-form ± ∈ „¦1 (Q), viewed as a mapping Q ’ T — Q, we have ±— θ = ±

and ±— σ = ’d±. Thus, the image ±(Q) ‚ T — Q is a Lagrange submanifold if and

—

only if d± = 0. Moreover, each ¬ber Tx Q is a Lagrange submanifold.

43.10. Relative Poincar´ Lemma. Let M be a smooth ¬nite dimensional mani-

e

fold, let N ‚ M be a closed submanifold, and let k ≥ 0. Let ω be a closed (k + 1)-

form on M which vanishes when pulled back to N . Then there exists a k-form •

on an open neighborhood U of N in M such that d• = ω|U and • = 0 along N . If

moreover ω = 0 along N , then we may choose • such that the ¬rst derivatives of •

vanish on N .

If all given data are real analytic then • can be chosen real analytic, too.

Proof. By restricting to a tubular neighborhood of N in M , we may assume that

p : M =: E ’ N is a smooth vector bundle and that i : N ’ E is the zero section

of the bundle. We consider µ : R — E ’ E, given by µ(t, x) = µt (x) = tx, then

µ1 = IdE and µ0 = i —¦ p : E ’ N ’ E. Let V ∈ X(E) be the vertical vector ¬eld

V (x) = vl(x, x) = ‚t 0 x + tx, then FlV = µet . So locally for t near (0, 1] we have

‚

t

d— —

= 1 (FlV t )— LV ω by (33.19)

V

d

dt µt ω = dt (Fllog t ) ω log

t

1—

diV ω) = 1 dµ— iV ω.

= t µt (iV dω + t

t

For x ∈ E and X1 , . . . , Xk ∈ Tx E we may compute

( 1 µ— iV ω)x (X1 , . . . , Xk ) = 1 (iV ω)tx (Tx µt .X1 , . . . , Tx µt .Xk )

tt t

= 1 ωtx (V (tx), Tx µt .X1 , . . . , Tx µt .Xk )

t

= ωtx (vl(tx, x), Tx µt .X1 , . . . , Tx µt .Xk ).

So if k ≥ 0, the k-form 1 µ— iV ω is de¬ned and smooth in (t, x) for all t ∈ [0, 1]

tt

and describes a smooth curve in „¦k (E). Note that for x ∈ N = 0E we have

( 1 µ— iV ω)x = 0, and if ω = 0 along N , then also all ¬rst derivatives of 1 µ— iV ω

tt tt

— —— —

vanish along N . Since µ0 ω = p i ω = 0 and µ1 ω = ω, we have

1

µ— ω µ— ω d—

’

ω= = dt µt ωdt

1 0

0

1 1

d( 1 µ— iV 1—

= ω)dt = d t µt iV ωdt =: d•.

tt

0 0

If x ∈ N , we have •x = 0, and also the last claim is obvious from the explicit

form of •. Finally, it is clear that this construction can be done in a real analytic

way.

43.10

462 Chapter IX. Manifolds of mappings 43.12

43.11. Lemma. Let M be a smooth ¬nite dimensional manifold, let N ‚ M be

a closed submanifold, and let σ0 and σ1 be symplectic forms on M which are equal

along N .

Then there exist: A di¬eomorphism f : U ’ V between two open neighborhoods

U and V of N in M which satis¬es f |N = IdN , T f |(T M |N ) = IdT M |N , and

f — σ1 = σ0 .

If all data are real analytic then the di¬eomorphism can be chosen real analytic,

too.

Proof. Let σt = σ0 + t(σ1 ’ σ0 ) for t ∈ [0, 1]. Since the restrictions of σ0 and σ1

to Λ2 T M |N are equal, there is an open neighborhood U1 of N in M such that σt

is a symplectic form on U1 , for all t ∈ [0, 1]. If i : N ’ M is the inclusion, we also

have i— (σ1 ’ σ0 ) = 0, so by lemma (43.10) there is a smaller open neighborhood U2

of N such that σ1 |U2 ’ σ0 |U2 = d• for some • ∈ „¦1 (U2 ) with •x = 0 for x ∈ N ,

such that also all ¬rst derivatives of • vanish along N .

Let us now consider the time dependent vector ¬eld Xt := ’(σt ∨ )’1 —¦ •, which

vanishes together with all ¬rst derivatives along N . Let ft be the curve of local

‚

di¬eomorphisms with ‚t ft = Xt —¦ ft , then ft |N = IdN and T ft |(T M |N ) = Id.

There is a smaller open neighborhood U of N such that ft is de¬ned on U for all

t ∈ [0, 1]. Then we have

—

= ft— LXt σt + ft— ‚t σt = ft— (diXt σt + σ1 ’ σ0 )

‚ ‚

‚t (ft σt )

= ft— (’d• + σ1 ’ σ0 ) = 0,

so ft— σt is constant in t, equals f0 σ0 = σ0 , and ¬nally f1 σ1 = σ0 as required.

— —

43.12. Theorem. Let (M, σ) be a ¬nite dimensional symplectic manifold. Then

the group Di¬(M, σ) of symplectic di¬eomorphisms is a smooth regular Lie group

and a closed submanifold of Di¬(M ). The Lie algebra of Di¬(M, σ) agrees with

Xc (M, σ).

If moreover (M, σ) is a compact real analytic symplectic manifold, then the group

Di¬ ω (M, σ) of real analytic symplectic di¬eomorphisms is a real analytic regular

Lie group and a closed submanifold of Di¬ ω (M ).

Proof. The smooth and the real analytic cases will be proved simultaneously; only

once we will need an extra argument for the latter.

Consider a local addition ± : T M ’ M in the sense of (42.4), so that (πM , ±) :

T M ’ M —M is a di¬eomorphism onto an open neighborhood of the diagonal, and

±(0x ) = x. Let us compose ± from the right with a ¬ber respecting di¬eomorphism

T M — ’ T M (coming from the symplectic structure or from a Riemannian metric)

and call the result again ± : T — M ’ M . Then (πM , ±) : T — M ’ M — M also is a

di¬eomorphism onto an open neighborhood of the diagonal, and ±(0x ) = x.

We consider now two symplectic structures on T — M , namely the canonical sym-

plectic structure σ0 = σM , and σ1 := (πM , ±)— (pr— σ ’ pr— σ). Both have vanishing

1 2

—

pullbacks on the zero section 0M ‚ T M .

43.12

43.12 43. Di¬eomorphism groups 463

Claim. In this situation, there exists a di¬eomorphism • : V0 ’ V1 between two

open neighborhoods V0 and V1 of the zero section in T — M which is the identity on

the zero section and satis¬es •— σ1 = σ0 .

First we solve the problem along the zero section, i.e., in T (T — M )|0M . There is

a vector bundle isomorphism γ : T (T — M )|0M ’ T (T — M )|0M over the identity

on 0M , which is the identity on T (0M ) and maps the symplectic structure σ0 on

each ¬ber to σ1 . In the smooth case, by using a partition of unity it su¬ces

to construct γ locally. But locally σi can be described by choosing a Lagrange

subbundle Li ‚ T (T — M )|0M which is a complement to T 0M . Then σi is completely

determined by the duality between T 0M and Wi induced by it, and a smooth γ is

then given by the resulting isomorphism W0 ’ W1 .

In the real analytic case, in order to get a real analytic γ, we consider the principal

¬ber bundle P ’ 0M consisting of all γx ∈ GL(T0x (T — M )) with γx |T0x (0M ) = Id

—

and γx σ1 = (σ0 )0x . The proof above shows that we may ¬nd a smooth section of

P . By lemma (30.12), there also exist real analytic sections.

Next we choose a di¬eomorphism h : V0 ’ V1 between open neighborhoods of

0M in T — M such that T h|0M = γ, which can be constructed as follows: Let u :

N (0M ) ’ V0 be a tubular neighborhood of the zero section, where N (0M ) =

(T (T — M )|0M )/T (0M ) is the normal bundle of the zero section. Clearly, γ induces

a vector bundle automorphism of this normal bundle, and h = u —¦ γ —¦ u’1 satis¬es

all requirements.

Now σ0 and h— σ1 agree along the zero section 0M , so we may apply lemma (43.11),

which implies the claim with possibly smaller Vi .

We consider the di¬eomorphism ρ := (πM , ±) —¦ • : T — M ⊃ V0 ’ V2 ‚ M — M

from an open neighborhood of the zero section to an open neighborhood of the

diagonal, and we let U ⊆ Di¬(M ) be the open neighborhood of IdM consisting of

all f ∈ Di¬(M ) with compact support such that (IdM , f )(M ) ‚ V2 , i.e. the graph

{(x, f (x)) : x ∈ M } of f is contained in V2 , and πM : ρ’1 ({(x, f (x)) : x ∈ M }) ’

M is still a di¬eomorphism.

For f ∈ U the mapping (IdM , f ) : M ’ graph(f ) ‚ M — M is the natural

di¬eomorphism onto the graph of f , and the latter is a Lagrangian submanifold if

and only if

0 = (IdM , f )— (pr— σ ’ pr— σ) = Id— σ ’ f — σ.

1 2 M

Therefore, f ∈ Di¬(M, σ) if and only if the graph of f is a Lagrangian submanifold

of (M — M, pr— σ ’ pr— σ). Since ρ— (pr— σ ’ pr— σ) = σ0 this is the case if and only

1 2 1 2

if {ρ (x, f (x)) : x ∈ M } is a Lagrange submanifold of (T — M, σ0 ).

’1

We consider now the following smooth chart of Di¬(M ) which is centered at the

identity:

u

Di¬(M ) ⊃ U ’ u(U ) ‚ Cc (M ← T — M ) = „¦1 (M ),

∞

’ c

u(f ) := ρ’1 —¦ (IdM , f ) —¦ (πM —¦ ρ’1 —¦ (IdM , f ))’1 : M ’ T — M.

43.12

464 Chapter IX. Manifolds of mappings 43.13

Then f ∈ U © Di¬(M, σ) if and only if u(f ) is a closed form, since u(f )(M ) =

{ρ’1 (x, f (x)) : x ∈ M } is a Lagrange submanifold if and only if f is symplec-

tic. Thus, (U, u) is a smooth chart of Di¬(M ) which is a submanifold chart for

Di¬(M, σ). For arbitrary g ∈ Di¬(M, σ) we consider the smooth submanifold

chart

ug

Di¬(M ) ⊃ Ug := {f : f —¦ g ’1 ∈ U } ’ ug (Ug ) ‚ Cc (M ← T — M ) = „¦1 (M ),

∞

’’ c

ug (f ) := u(f —¦ g ’1 ).

Hence, Di¬(M, σ) is a closed smooth submanifold of Di¬(M ) and a smooth Lie

group, since composition and inversion are smooth by restriction. If M is compact

then the space of closed 1-forms is a direct summand in „¦1 (M ) by Hodge theory,

as in the proof of (43.7), so in this case Di¬(M, σ) is even a splitting submanifold

of Di¬(M ). The embedding Di¬(M, σ) ’ Di¬(M ) is smooth, thus it induces a

bounded injective homomorphism of Lie algebras which is an embedding onto a

closed Lie subalgebra, which we shall soon identify with Xc (M, σ).

Suppose that X : R ’ Xc (M, σ) is a smooth curve, and consider the evolution curve

f (t) = Evolr ‚

Di¬(M ) (X)(t), which is the solution of the di¬erential equation ‚t f (t) =

X(t) —¦ f (t) on M . Then f : R ’ Di¬(M ) actually has values in Di¬(M, σ), since

ft σ = ft— LXt σ = 0. So the restriction of evolr

‚—

Di¬(M ) to Xc (M, σ) is smooth into

‚t

Di¬(M, σ) and thus gives evolr Di¬(M,σ) . We take now the right logarithmic derivative

of f (t) in Di¬(M, σ) and get a smooth curve in the Lie algebra of Di¬(M, σ) which

maps to X(t). Thus, the Lie algebra of Di¬(M, σ) is canonically identi¬ed with

Xc (M, σ).

Note that this proof of regularity is an application of the method from (38.7), where

p(f ) := f — σ ’ σ, p : Di¬(M ) ’ „¦2 (M ).

43.13. The regular Lie group of exact symplectic di¬eomorphisms. Let us

assume that (M, σ) is a connected ¬nite dimensional separable symplectic manifold

1

such that the space of exact 1-forms with compact support Bc (M ) on M is a

1

convenient direct summand in the space Zc (M ) of all closed forms. This is true if

M is compact, by Hodge theory, as in (43.7).

In the setting of the last theorem (43.12) we consider the universal covering group

Di¬(M, σ) ’ Di¬(M, σ), which we view as the space of all smooth curves c : [0, 1] =

I ’ Di¬(M, σ) such that c(0) = IdM modulo smooth homotopies ¬xing endpoints.

For each such curve the right logarithmic derivative (38.1) δ r c(‚t ) : I ’ Xc (M, σ)

is given by δ r c(‚t |t ) = ‚t 0 c(t) —¦ c(t)’1 . Then i(δ r c(‚t ))σ is a curve of closed 1-

‚

forms with compact supports since di(δ r c(‚t |t ))σ = Lδr c(‚t |t ) σ = 0. For a smooth

homotopy (s, t) ’ h(s, t) with h(0, t) = c(t) we have by the left Maurer Cartan

1

equation dδ r h ’ 2 [δ r h, δ r h] = 0 in lemma (38.1)

‚s δ r h(‚t ) = ‚t δ r h(‚s ) + d(δ r h)(‚s , ‚t ) + δ r h([‚s , ‚t ])

= ‚t δ r h(‚s ) + [δ r h(‚s ), δ r h(‚t )]Xc (M,σ) + 0.

43.13

43.13 43. Di¬eomorphism groups 465

Then we get

1 1 1 1

iδr h(‚t |(1,t) ) σ dt ’ iδr h(‚t |(0,t) ) σ dt = i‚s δr h(‚t ) σ ds dt

0 0 0 0

1 1

= i‚t δr h(‚s )+[δr h(‚s ),δr h(‚t )] σ ds dt

0 0

1 1 1 1

= ‚t iδr h(‚s ) σ ds dt + i[δr h(‚s ),δr h(‚t )] σ ds dt

0 0 0 0

1 1 1 1

iδr h(‚s |(s,1) ) σ ds ’ Lδr h(‚s ) , iδr h(‚t ) σ ds dt

= iδr h(‚s |(s,0) ) σ ds +

0 0 0 0

1 1

=0’0+ Lδr h(‚s ) iδr h(‚t ) σ ds dt ’ 0

0 0

1 1 1 1

iδr h(‚s ) Lδr h(‚t ) σ ds dt

= d iδr h(‚s ) iδr h(‚t ) σ ds dt +

0 0 0 0

1 1

σ (δ r h(‚t ), δ r h(‚s )) ds dt.

=d

0 0

Thus, we get a well de¬ned smooth mapping into the de Rham cohomology with

compact supports

1

“ : Di¬(M, σ) ’ Hc (M ),

1

i(δ r c(‚t ))σdt ,

“([c]) :=

0

which is a homomorphism of regular Lie groups: the multiplication in Di¬(M, σ)

is induced by pointwise multiplication of curves. But note that t ’ c1 (t) —¦ c2 (t) is

homotopic to the curve which follows ¬rst c2 and then c1 ( ) —¦ c2 (1). The right log-

arithmic derivative does not feel the right translation, thus the integral “([c1 ].[c2 ])

equals “([c1 ]) + “([c2 ]).

Note that, under the assumption on M made above, “ admits a global smooth

section s as follows:

wC

Ψ ∞

1

Zc (M ) (I, Di¬(M, σ))

u

u

w Di¬(M, σ),

s

1

Hc (M )

’1

where Ψ(ω) = (Flσ ω )0¤t¤1 is smooth. Since the canonical quotient mapping

t

1 1

Zc (M ) ’ Hc (M ) admits a section Ψ induces a section of “.

Claim. The closed subgroup ker “ ‚ Di¬(M, σ) is simply connected.

First note that Di¬(M, σ) is also a topological group in the the topology described

in (42.2), thus a fortiori its universal covering Di¬(M, σ) is also a topological

43.13

466 Chapter IX. Manifolds of mappings 43.13

1 1

group. Hc (M ) is a direct summand in Zc (M ), which is smoothly paracompact

as a closed linear subspace of „¦1 (M ) by (30.4). Since it admits a continuous sec-

c

1

tion, “ : Di¬(M, σ) ’ Hc (M ) is a ¬bration with contractible basis. The long exact

homotopy sequence then implies that ker(“) is simply connected, too.

Theorem. The subgroup ker “ ‚ Di¬(M, σ) is a splitting regular Lie subgroup with

∞ 0

Lie algebra Cc (M, R)/Hc (M ).

Proof. Recall from the proof of (43.12) the chart (U, u) of Di¬(M, σ) near the

identity, which we consider also as a chart on the universal covering Di¬(M, σ). It

ρ

is induced by a di¬eomorphism T — M ⊃ V0 ’ V2 ‚ M — M satisfying ρ— (pr— σ ’

’ 1

—

pr2 σ) = σM in the following way. To a symplectic di¬eomorphism f near IdM

we ¬rst associate its graph, a Lagrange submanifold in (V2 , pr— σ ’ pr— σ), then its

1 2

inverse image L under ρ, a Lagrange submanifold in V0 , and ¬nally a closed 1-form

u(f ) = ω ∈ „¦1 (M ) with ω(M ) = L. The form ω is exact if and only if the pullback

c

of the natural 1-form θM ∈ „¦1 (T — M ) (see (43.9)) on L is exact. Equivalently,

the form θ1 = (ρ’1 )— θM on V2 pulls back to an exact form on the graph of f ,

or (Id, f )— θ1 is exact on M . Let ft ∈ U for t ∈ [0, 1] with f0 = IdM , and let

Xt = dt ft —¦ ft’1 . Then

d

1

“(f ) = “([ft ]) = i(Xt )σ dt ,

0

—

= (IdM , ft )— L0—Xt θ1

d

dt (IdM , ft ) θ1 compare (33.19)

= (IdM , ft )— di0—Xt θ1 + (IdM , ft )— i0—Xt dθ1

= d(IdM , ft )— i0—Xt θ1 + ft— iXt σ,

since ’dθ1 = pr— σ ’ pr— σ. Thus, iXt σ is exact for all t if and only if (IdM , ft )— θ1

1 2

is exact for all t. If ω is exact let ft := u’1 (tω), and it follows that “(f ) = 0.

If conversely f ∈ U © ker “ ‚ Di¬(M, σ), there exists a smooth curve t ’ ht in

1

U ‚ Di¬(M, σ) from IdM to f . Then “(ht ) is a closed smooth curve in Hc (M ),

’1

which we may lift smoothly to gt ∈ Di¬(M, σ). Then gt —¦ ht lies in ker(“) for

all t. Thus, for f near IdM in ker(“) we may ¬nd a smooth curve t ’ ft ∈ U

1 t

which lies in ker(“). Then 0 i(Xts )σ ds = t 0 i(Xt )σ dt is exact, so i(Xt )σ is

1

exact, and ¬nally u(ft ) is exact in Zc (M ). Hence, for some smaller U we have

1

f ∈ U © ker “ ‚ Di¬(M, σ) if and only if ω = u(f ) ∈ u(U ) © Bc (M ), and ker(“) is

a smooth splitting submanifold.

The Lie algebra of ker(“) consists of all globally Hamiltonian vector ¬elds: for a

smooth curve ft in ker(“) we consider Xt = dt ft —¦ ft’1 ; from above we see that

d

i(Xt )σ = dht for some ht ∈ Cc (M, R) and then gradσ (ht ) = Xt . Conversely,

∞

1

gradσ (h)

i(gradσ (h))σ dt = [dh] = 0 ∈ Hc (M ).

1

“([Flt ]) =

0

That ker(“) is regular follows from (38.7), using p = “.

43.13

43.15 43. Di¬eomorphism groups 467

43.14. Remark. In the situation of (43.13) above, the fundamental group π1 =

π1 (Di¬(M, σ)) is a discrete subgroup of the universal covering Di¬(M, σ). Then

1

“(π1 ) ⊆ Hc (M ) is a subgroup, and we have an induced homomorphism “ of groups:

w H (M )

“ 1

Di¬(M, σ) c

u

u

w

1

Di¬(M, σ) Hc (M )

“

Di¬(M, σ)

π1 “(π1 )

1 1

Note that Hc (M )/“(π1 ) is Hausdor¬ if “(π1 ) is a ˜discrete™ subgroup of Hc (M ) in

the sense of (38.5), and then “ is a smooth homomorphism of regular Lie groups.

In any case, the group π1 © ker(“) is a ˜discrete™ (in a sense analogous to (38.5))

central subgroup of ker(“), thus ker(“) is a regular Lie subgroup of Di¬(M, σ) with

universal cover ker(“) and with Lie algebra the space of globally Hamiltonian vector

¬elds.

1

It is known that “(π1 ) is ˜discrete™ in Hc (M ) if M is compact and either dim(M ) = 2

or M is a K¨hler manifold or σ has integral periods on M . There seems to be no

a

known example where “(π) is not discrete, see [Banyaga, 1978] and [Banyaga, 1980].

The next topic is the Lie group of contact di¬eomorphisms.

43.15. Contact manifolds. Let M be a smooth manifold of dimension 2n+1 ≥ 3.

A contact form on M is a 1-form ± ∈ „¦1 (M ) such that ± § (d±)n ∈ „¦2n+1 (M ) is

nowhere zero. This is sometimes called an exact contact structure. The pair (M, ±)

is called a contact manifold.

A contact form can be put into the following normal form: For each x ∈ M there

u

is a chart M ⊃ U ’ u(U ) ‚ R2n+1 centered at x such that ±|U = u1 dun+1 +

’

u2 dun+2 + · · · + un du2n + du2n+1 . This follows from proposition (43.18) below, for

a simple direct proof see [Libermann, Marle, 1987].

The vector subbundle ker(±) ‚ T M is called the contact distribution. It is as

non-involutive as possible, since d± is even non-degenerate on each ¬ber ker(±)x =

ker(±x ) ‚ Tx M . The characteristic vector ¬eld X± ∈ X(M ) is the unique vector

¬eld satisfying iX± ± = 1 and iX± d± = 0.

Note that X ’ (iX d±, iX ±) is isomorphic T M ’ {• ∈ T — M : iX± • = 0} — R, but

we shall use the isomorphism of vector bundles

T M ’ T — M, X ’ iX d± + ±(X).±,

(1)

A di¬eomorphism f ∈ Di¬(M ) with f — ± = »f .± for a nowhere vanishing function

»f ∈ C ∞ (M, R \ 0) is called a contact di¬eomorphism. The group of all contact

di¬eomorphisms will be denoted by Di¬(M, ±).

A vector ¬eld X ∈ X(M ) is called a contact vector ¬eld if LX ± = µX .± for a

smooth function µX ∈ C ∞ (M, R). The linear space of all contact vector ¬elds will

43.15

468 Chapter IX. Manifolds of mappings 43.18

be denoted by X(M, ±); it is clearly a Lie algebra. Contraction with ± is a linear

mapping also denoted by ± : X(M, ±) ’ C ∞ (M, R). It is bijective since we may

apply iX± to µX .± = LX ± = iX d± + d(±(X)) to get µX = 0 + X± (±(X)), and

since by using (1) we may reconstruct X from ±(X) as

iX d± + ±(X).± = µX .± ’ d(±(X)) + ±(X).±

= X± (±(X)).± ’ d(±(X)) + ±(X).±.

Note that the inverse f ’ grad± (f ) of ± : X(M, ±) ’ C ∞ (M, R) is a linear

di¬erential operator of order 1.

A smooth mapping f : L ’ M is called a Legendre mapping if f — ± = 0. If f is also

an embedding and dim M = 2 dim L + 1, then the image f (L) is called a Legendre

submanifold of M .

43.16. Lemma. Let Xt be a time dependent vector ¬eld on M , and let ft be

the local curve of local di¬eomorphisms with ‚t ft —¦ ft’1 = Xt and f0 = Id. Then

‚

LXt ± = µt ± if and only if ft— ± = »t .±, where »t and µt are related by ‚»»t = ft— µt .

t

t

Proof. The two following equations are equivalent:

1—

±= f ±,

»t t

‚

‚t »t —

1— 1— 1

ft LXt ± = ft— (’µt .± + LXt ±).

‚

=’

0 = ‚t f± f± +

»t t »2 t »t »t

t

43.17. Canonical example. Let N be an n-dimensional manifold, let θ ∈

„¦1 (T — N ) be the canonical 1-form, which is given by θ(ξ) = πT — N (ξ), T (π — ).ξ T N ,

and which has the following universal property: For any 1-form ω ∈ „¦1 (N ), viewed

as a section of T — N ’ N , we have ω — θ = θ.

Then the 1-form θ ’ dt = pr— θ ’ pr— dt ∈ „¦1 (T — N — R) is a contact form. Note that

1 2

T N —R = J (N, R), the space of 1-jets of functions on N . A section s of T — N —R =

— 1

J 1 (N, R) ’ N is of the form s = (ω, f ) for ω ∈ „¦1 (N ) and f ∈ C ∞ (N, R). Thus,

s is a Legendre mapping if and only if 0 = s— (θ ’ dt) = ω — θ ’ f — dt = θ ’ df or

s = j1f .

43.18. Proposition. ([Lychagin, 1977]) If L is a Legendre submanifold of a (¬nite

dimensional) contact manifold (M, ±), then there exist:

(1) an open neighborhood U of L in M ,

(2) an open neighborhood V of the zero section 0L in T — L — R,

(3) a di¬eomorphism • : U ’ V with •|L = IdL and •— (θL ’ dt) = ±.

If all data is real analytic then • may be chosen real analytic, too.

Proof. By (41.14), there exists a tubular neighborhood N (L) = (T M |L)/T L ⊃

•

˜’

U ’ U ‚ M of L in M .

43.18

43.18 43. Di¬eomorphism groups 469

Note that ker(d±) is a trivial line bundle, framed by the characteristic vector ¬eld

X± , and that T M = ker(±) • ker(d±). Thus, for the normal bundle we have the

following chain of natural isomorphisms of vector bundles:

ker(±) d±•Id

• ker(d±) ’ ’ ’ T — (L) — R.

N (L) = (T M |L)/T (L) = ’∼’

T (L) =

Therefore, we may assume that the tubular neighborhood is given by T — L — R ⊃

•

V ’ U ‚ M , where now V is an open neighborhood of the 0-section 0L (which we

’

identify with L) in T — L — R.

Next we consider the contact structure ± := •— ± ∈ „¦1 (V ). Note that on the

˜

subbundle T L = T (0L ) ‚ T V both contact structures ± and ±0 := θ ’ dt vanish.

˜

We will ¬rst arrange that both contact structures agree on T V |0L : We claim that

there exists a vector bundle isomorphism γ : T V |0L ’ T V |0L which satis¬es

γ — d˜ = d±0 , γ — ± = ±0 , and such that γ|T (0L ) = Id. Note that we have two

± ˜

symplectic vector subbundles (ker ±, d˜ ) and (ker ±0 , d±0 ). We ¬rst choose a vector

˜±

bundle isomorphism γ : ker ±0 ’ ker ± with γ — d˜ = d±0 and γ |T (0L ) = Id, as in

˜ ˜ ˜± ˜

the proof of the claim in (43.12), and then we complete γ to γ in such a way that

˜

for the characteristic vector ¬elds we have γ(X±0 ) = X± .˜

There exists a di¬eomorphism ψ : V ’ V between two open neighborhoods V

and V of 0L in V such that T ψ|0L = γ, and since γ|T (0L ) = Id we even have

ψ|0L = Id. We put ±1 := ψ — ± = ψ — •— ±. Then ±0 |0L = ±1 |0L , and we put

˜

±t := (1 ’ t)±0 + t±1 ,

and since ±t |T (0L ) = ±0 |T (0L ) = ±1 |T (0L ) the 1-form ±t is a contact structure on

an (again smaller) neighborhood V of 0L in T — M — R.

Let us now suppose that ft is a curve of di¬eomorphisms near 0L which satis¬es

T (ft )|T (0L ) = IdT V |0L , with time dependent vector ¬eld Xt = ( ‚t ft ) —¦ ft’1 . Then

‚

we have

‚— ‚— —‚

‚t ft ±t = ‚t ft ±s s=t + fs ‚t ±t s=t

ft— LXt ±t + ft— (±1 ’ ±0 )

=

ft— (iXt d±t + d iXt ±t + ±1 ’ ±0 ) .

(4) =

We want a time dependent vector ¬eld Xt with iXt d±t + d iXt ±t + ±1 ’ ±0 = 0

near 0L and we ¬rst look for a time dependent function ht de¬ned near 0L such

that dht (X±t ) = iX±t (±1 ’ ±0 ). Since ±1 = ±0 along 0L and vanishes on T (0L ),

the vector ¬eld X±t equals X±0 along 0L and is not tangent to 0L . So its ¬‚ow lines

leave 0L and there is a submanifold S of codimension 1 in T M containing 0L which

is transversal to the ¬‚ow of X±t for all t ∈ [0, 1], and we may take ht as

s

X X

ht (Fls ±t (z)) (±1 ’ ±0 )(X±t )(Flr ±t (z)) dr for z ∈ S.

=

0

43.18

470 Chapter IX. Manifolds of mappings 43.19

Now we use (43.15.1) and choose the unique time dependent vector ¬eld Xt which

satis¬es

iXt d±t + ±t (Xt ).±t = ±0 ’ ±1 ’ d ht + ht .±t .

Then for the curve of di¬eomorphisms ft which is determined by the ordinary

di¬erential equation ‚t ft = Xt —¦ft’1 with initial condition f0 = Id we have ‚t ft— ±t =

‚ ‚

0 by (4), so ft— ±t = f0 ±0 = ±0 is constant in t. Since (±1 ’ ±0 )|T (0L ) = 0, also

—

ht |0L = 0, and dht |T (0L ) = 0, the vector ¬eld Xt vanishes along 0L , and thus the

curve of di¬eomorphisms ft exists for all t near [0, 1] in a neighborhood of 0L in

T — L — R. Then f1 ±1 = ±0 and f1 —¦ ψ —¦ • is the looked for di¬eomorphism.

—

43.19. Theorem. Let (M, ±) be a ¬nite dimensional contact manifold. Then

the group Di¬(M, ±) of contact di¬eomorphisms is a smooth regular Lie group.

The injection i : Di¬(M, ±) ’ Di¬(M ) is smooth, TId i maps the Lie algebra of

Di¬(M, ±) isomorphically onto Xc (M, ±) with the negative of the usual Lie bracket,

and locally there exist smooth retractions to i, so i is an initial mapping, see (27.11).

If (M, ±) is in addition a real analytic and compact contact manifold then all as-

sertions hold in the real analytic sense.

Proof. For a contact manifold (M, ±) let M = M — M — (R \ 0), with the con-

tact structure ± = t. pr— ± ’ pr— ±, where t = pr3 : M — M — (R \ 0) ’ R. Let

ˆ 1 2

f ∈ Di¬(M, ±) be a contact di¬eomorphism with f — ± = »f .±. Inserting the char-

acteristic vector ¬eld X± into this last equation we get

»f = iX± »f ± = iX± (f — ±) = f — (if— X± ±).

(1)

Thus, f determines »f , and for an arbitrary di¬eomorphism f ∈ Di¬(M ) we may

de¬ne a smooth function »f by (1). Then »f ∈ C ∞ (M, R \ 0) if f is near a contact

di¬eomorphism in the Whitney C 0 -topology. We consider its contact graph “f :

M ’ M , given by “f (x) := (x, f (x), »f (x)), a section of the surjective submersion

pr1 : M ’ M . Note that “f is a Legendre mapping if and only if f is a contact

di¬eomorphism, f ∈ Di¬(M, ±), since “— ± = »f .± ’ f — ±.

fˆ

Let us now ¬x a contact di¬eomorphism f ∈ Di¬(M, ±) with f — ± = »f .±. By

proposition (43.18), and also using the di¬eomorphism “f : M ’ “f (M ) there are:

an open neighborhood Uf of “f (M ) ‚ M , an open neighborhood Vf of the zero

•f

section 0M in T — M — R, and a di¬eomorphism M ⊃ Uf ’’ Vf ‚ T — M — R, such

that the restriction •f |“f (M ) equals the inverse of “f : 0M ∼ M ’ “f (M ), and

=

•— (θM ’ dt) = ±.

ˆ

f

˜

Now let Uf be the open set of all di¬eomorphisms g ∈ Di¬(M ) such that g equals

f o¬ some compact subset of M , “g (M ) ‚ Uf ‚ M , and π —¦ •f —¦ “g : M ’ M is

a di¬eomorphism, where π : T — M — R ’ M is the vector bundle projection. For

˜

g ∈ Uf and

sf (g) := (•f —¦ “g ) —¦ (π —¦ •f —¦ “g )’1 ∈ Cc (M ← T — M — R)

∞

∞

=: (σf (g), uf (g)) ∈ „¦1 (M ) — Cc (M, R)

c

43.19

43.19 43. Di¬eomorphism groups 471

the following conditions are equivalent:

(2) g is a contact di¬eomorphism.

(3) “g (M ) is a Legendre submanifold of (M , ±).

ˆ

•f (“g (M )) is a Legendre submanifold of (T — M — R, θM ’ dt).

(4)

The section sf (g) satis¬es sf (g)— (θM ’ dt) = 0, equivalently (by (43.17))

(5)

σf (g) = d(uf (g)).

Let us now consider the following diagram:

u {u wuy wC

sf

u u

˜ ˜ ∞

← T — M — R)

Di¬(M ) Uf Vf c (M

y y y y

j linear, splitting

j

u {U wV y w

uf ∞

Di¬(M, ±) Cc (M, R).

∼ f f

=

In this diagram we put j(h) := (dh, h), a bounded linear splitting embedding. We

˜

let Vf ‚ Cc (M ← T — M — R) be the open set of all (ω, h) ∈ „¦1 (M ) — Cc (M, R)

∞ ∞

c

with (ω, h)(M ) ‚ Vf and such that pr1 —¦•’1 —¦ (ω, h) : M ’ M is a di¬eomorphism.

f

We also consider the smooth mapping

˜

wf : Vf ’ Di¬(M )

wf (ω, h) := pr2 —¦•’1 —¦ (ω, h) —¦ (pr1 —¦•’1 —¦ (ω, h))’1 : M ’ M,

f f

˜

and let Vf = (wf —¦j)’1 Uf . Then wf —¦sf = Id, and so we may use as chart mappings

for Di¬(M, ±):

˜ ˜

uf : Uf := Uf © Di¬(M, ±) ’ Vf := (wf —¦ j)’1 (Uf ) ‚ Cc (M, R),

∞

uf (g) := pr2 —¦(•f —¦ “g ) —¦ (π —¦ •f —¦ “g )’1 ∈ C ∞ (M, R),

u’1 (h) = (wf —¦ j)(h) = wf (dh, h).

f

The chart change mapping uk —¦ u’1 is de¬ned on an open subset and is smooth,

f

’1

because uk —¦uf = pr2 —¦sk —¦wf —¦j, and sk and wf are smooth by (42.13), (43.1), and

by (42.20). Thus, the resulting atlas (Uf , uf )f ∈Di¬(M,±) is smooth, and Di¬(M, ±)

is a smooth manifold in such a way that the injection i : Di¬(M, ±) ’ Di¬(M ) is

smooth.

Note that sf —¦ wf = Id, so we cannot construct (splitting) submanifold charts in

this way.

But there exist local smooth retracts u’1 —¦ pr2 —¦sf : (pr2 —¦sf )’1 (Vf ) ’ Uf . There-

f

fore, the injection i has the property that a mapping into Di¬(M, ±) is smooth if

and only if its prolongation via i into Di¬(M ) is smooth. Thus, Di¬(M, ±) is a Lie

group, and from (38.7) we may conclude that it is a regular Lie group.

A direct proof of regularity goes as follows: From lemma (43.16) and (36.6) we see

that TId i maps the Lie algebra of Di¬(M, ±) isomorphically onto the Lie algebra

Xc (M, ±) of all contact vector ¬elds with compact support. It also follows from

lemma (43.16) that we have for the evolution operator

∞

Evolr r

Di¬(M ) |C (R, Xc (M, ±)) = EvolDi¬(M,±)

so that Di¬(M, ±) is a regular Lie group.

43.19

472 Chapter IX. Manifolds of mappings 43.20

43.20. n-Transitivity. Let M be a connected smooth manifold with dim M ≥ 2.

We say that a subgroup G of the group Di¬(M ) of all smooth di¬eomorphisms acts

n-transitively on M , if for any two ordered sets of n di¬erent points (x1 , . . . , xn )

and (y1 , . . . , yn ) in M there is a smooth di¬eomorphism f ∈ G such that f (xi ) = yi

for each i.

Theorem. Let M be a connected smooth (or real analytic) manifold of dimension

dim M ≥ 2. Then the following subgroups of the group Di¬(M ) of all smooth

di¬eomorphisms act n-transitively on M , for every ¬nite n:

(1) The group Di¬ c (M ) of all smooth di¬eomorphisms with compact support.

(2) The group Di¬ ω (M ) of all real analytic di¬eomorphisms.

(3) If (M, σ) is a symplectic manifold, the group Di¬ c (M, σ) of all symplectic

di¬eomorphisms with compact support, and even the subgroup of all globally

Hamiltonian symplectic di¬eomorphisms.

(4) If (M, σ) is a real analytic symplectic manifold, the group Di¬ ω (M, σ) of

all real analytic symplectic di¬eomorphisms, and even the subgroup of all

globally Hamiltonian real analytic symplectic di¬eomorphisms.

(5) If (M, µ) is a manifold with a smooth volume density, the group Di¬ c (M, µ)

of all volume preserving di¬eomorphisms with compact support.

(6) If (M, µ) is a manifold with a real analytic volume density, then the group

Di¬ ω (M, µ) of all real analytic volume preserving di¬eomorphisms.

(7) If (M, ±) is a contact manifold, the group Di¬ c (M, ±) of all contact di¬eo-

morphisms with compact support.

(8) If (M, ±) is a real analytic contact manifold, the group Di¬ ω (M, ±) of all

real analytic contact di¬eomorphisms.

Result (1) is folklore, the ¬rst trace is in [Milnor, 1965]. The results (3), (5), and

(7) are due to [Hatakeyama, 1966] for 1-transitivity, and to [Boothby, 1969] in the

general case. The results about real analytic di¬eomorphisms and the proof given

here is from [Michor, Vizman, 1994].

Proof. Let us ¬x a ¬nite n ∈ N. Let M (n) denote the open submanifold of all

n-tuples (x1 , . . . , xn ) ∈ M n of pairwise distinct points. Since M is connected and

of dimension ≥ 2, each M (n) is connected. The group Di¬(M ) acts on M (n) by

the diagonal action, and we have to show that any of the subgroups G described

above acts transitively. We shall show below that for each G the G-orbit through

any n-tuple (x1 , . . . , xn ) ∈ M (n) contains an open neighborhood of (x1 , . . . , xn ) in

M (n) , thus any orbit is open. Since M (n) is connected, there can be only one orbit.

The cases (2) and (1). We choose a complete Riemannian metric g on M , and we

let (Yij )m be an orthonormal basis of Txi M with respect to g, for all i. Then we

j=1

choose real analytic vector ¬elds Xk for 1 ¤ k ¤ N = nm which satisfy:

|Xk (xi ) ’ Yij |g < µ for k = (i ’ 1)m + j,

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

(9) /

|Xk (x)|g < 2 for all x ∈ M and all k.

43.20

43.20 43. Di¬eomorphism groups 473

Since these conditions describe a Whitney C 0 open set, such vector ¬elds exist by

(30.12). The ¬elds are bounded with respect to a complete Riemannian metric, so

they have complete real analytic ¬‚ows FlXk , see e.g. [Hirsch, 1976.] We consider

the real analytic mapping

f : RN ’ M (n) ,

X1 XN

«

(Flt1 —¦ . . . —¦ FltN )(x1 )

f (t1 , . . . , tN ) := ...

(FlX1 —¦ . . . —¦ FlXN )(xn ),

t1 tN

which has values in the Di¬ ω (M )-orbit through (x1 , . . . , xn ). To get the tangent

mapping at 0 of f we consider the partial derivatives

‚

‚tk |0 f (0, . . . , 0, tk , 0, . . . , 0) = (Xk (x1 ), . . . , Xk (xn )).

If µ > 0 is small enough, this is near an orthonormal basis of T(x1 ,...,xn ) M (n) with

respect to the product metric g — . . . — g. So T0 f is invertible, and the image of f

thus contains an open subset.

In case (1), we can choose smooth vector ¬elds Xk with compact support which

satisfy conditions (9).

For the remaining cases we just indicate the changes which are necessary in this

proof.

The cases (4) and (3) Let (M, σ) be a connected real analytic symplectic smooth

manifold of dimension m ≥ 2. We choose real analytic functions fk for 1 ¤ k ¤

N = nm whose Hamiltonian vector ¬elds Xk = gradσ (fk ) satisfy conditions (9).

Since these conditions describe Whitney C 1 open subsets, such functions exist by

[Grauert, 1958, Proposition 8]. Now we may ¬nish the proof as above.

The cases (8) and (7) Let (M, ±) be a connected real analytic contact manifold of

dimension m ≥ 3. We choose real analytic functions fk for 1 ¤ k ¤ N = nm such

that their contact vector ¬elds Xk = grad± (fk ) satisfy conditions (9). Since these

conditions describe Whitney C 1 open subsets, such functions exist. Now we may

¬nish the proof as above.

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,