—

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 )