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