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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


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

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) =

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 = ±

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