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 = ±