EVERY SYMPLECTIC MANIFOLD IS A COADJOINT ORBIT

Transcription

PATRICK IGLESIAS-ZEMMOUR
Abstract. Because symplectic structures have no local invariants, they
have a huge group of automorphisms, always big enough to be transitive. In this paper, we show that, for this group, or for the group of
Hamiltonian transformations, a symplectic manifold is always a coadjoint orbit. In other words, coadjoint orbits are the universal model of
symplectic manifolds. To establish that fact, and there is no heuristic
here, the main tool is the Moment Map for diffeological spaces.
Introduction
At the end of the sixties, last century, coming from different points of
view, Kostant, Kirillov and Souriau showed that a symplectic manifold
(M, ω), homogeneous under the action of a Lie group, is isomorphic — up
to a covering — to a coadjoint orbit [Kos70] [Sou70] [Kir74]. The main
tool for such an identification is the moment map that was introduced
at the same period. Now, the group Diff(M, ω), of the automorphisms
of the symplectic manifold (M, ω), is transitive on M, as well as the subgroup Ham(M, ω) of Hamiltonian diffeomorphisms [Boo69]. It is then
tempting to look for an analogous of the Kostant-Kirillov-Souriau (KKS)
theorem, relative to Diff(M, ω) or Ham(M, ω), even if these groups are
not, strictly speaking, Lie groups. Such a question has been already raised,
and solved using functional analysis techniques [Omo86]. Alternatively,
considering the symplectic manifold (M, ω) and the groups Diff(X, ω) or
Ham(M, ω) as diffeological objects, we can use the construction of the
moment map in diffeology [Piz10] to write a new solution to that question. We shall see that we get this way a rigourous equivalent statement —
avoiding the use of functional analysis techniques — that every symplectic
manifold is a coadjoint orbit, linear or affine, of its group of Hamiltonian
transformations. We shall observe also that there is no covering restriction in this case.
Date: May 6, 2015.
1
2
By the way, we shall prove that the characteristics of a closed 2-form ω
defined on a manifold M, homogeneous under the action of the Hamiltonian transformations group Ham(M, ω), are the connected components
of the prevalues1 of the moment map µ¯ω, associated with Ham(M, ω). In
other words µ¯ω integrates the characteristic distribution m 7→ ker(ω m ).
This gives a new interpretation of a symplectic homogeneous 2-form —
opposed to presymplectic — as a 2-form whose levels of the Hamiltonian
moment map are (diffeologically) discrete.
Vocabulary: For the purpose of unification we shall call parasymplectic a
general closed 2-form, without any other condition. It can be defined on
a manifold or on a diffeological space. A space equipped with a parasymplectic form will be called a parasymplectic space.
Note also that a parasymplectic form ω, on a diffeological space X, will
be said to be presymplectic if its pseudogroup of local automorphisms
Diffloc (X, ω) is transitive on X. This is an interpretation of the (presymplectic) Darboux theorem in diffeology, regarded as a definition.
Thanks I am grateful to the Hebrew University of Jerusalem Israel, who
invited me, and where I spent the wonderful time in which I elaborated
the first version of this article.
Review on the Moment Maps of a Parasymplectic Form
Let (X, ω) be a parasymplectic space. Let G be a diffeological group, with
a smooth action2 g 7→ gX on X, preserving ω, that is, gX∗ (ω) = ω for all
g ∈ G. We denote by G ∗ the space of momenta of G, defined as the
left-invariant differential 1-forms on G,
G ∗ = {ε ∈ Ω1 (G) | L(g )∗ (ε) = ε, for all g ∈ G}.
To understand the essential nature of the moment map, which is a map
from X to G ∗ , it is good to consider the simplest case, and use it then as a
guide to extend this simple construction to the general case.
The Simplest Case. Consider the case where X is a manifold, and G is
a Lie group. Let us assume that ω is exact ω = d α, and that α is also
1The prevalues of a map f are the sets f −1 ( f (x)) when x run over the domain of f .
2A smooth action of a diffeological group G on a diffeological space X is a smooth
morhism ρ : G → Diff(X), where Diff(X) is equipped with the functionnal diffeology.
3
invariant by G. Regarding ω, the moment map3 of the action of G on X is
the map
µ : X → G∗
defined by
µ(x) = xˆ ∗ (α),
where xˆ : G → X is the orbit map xˆ(g ) = gX (x).
As we can see, there is no obstacle, in this simple situation, to generalize, mutatis mutandis, the moment map to a diffeological group acting
by automorphisms on a diffeological parasymplectic space. But, as we
know, not all closed 2-forms are exact, and even if they are exact, they
do not necessarily have an invariant primitive. We shall see now, how we
can generally come to a situation, so close to the simple case above, that
modulo some minor subtleties we can build a good moment map in all
cases.
The General Case. We consider a connected parasymplectic diffeological space (X, ω), and a diffeological group G acting on X and preserving
ω. Let K be the Chain-Homotopy Operator, defined in [Piz13]. The
differential 1-form Kω, defined on Paths (X), is related to ω by d [Kω] =
¯ = (1ˆ∗ − ˆ0∗ )(ω) and
(1ˆ∗ − ˆ0∗ )(ω), and Kω is invariant by G. Considering ω
¯ = Kω, we are in the simple case: ω
¯ = dα
¯ and α
¯ invariant by G. We can
α
apply the construction above and define then the Moment Map of Paths by
Ψ : Paths (X) → G ∗
with Ψ (γ ) = γˆ ∗ (Kω),
and γˆ : G → Paths (X) is the orbit map γˆ (g ) = gX ◦ γ of a path γ . The
moment of paths is additive with respect to the concatenation,
Ψ (γ ∨ γ 0 ) = Ψ (γ ) + Ψ (γ 0 ).
This paths moment map Ψ is equivariant by G, acting by composition on
Paths (X), and by coadjoint action on G ∗ . Next, defining the Holonomy
of the action of G on X by
Γ = {Ψ (`) | ` ∈ Loops (X)} ⊂ G ∗ ,
the Two-Points Moment Map is defined by pushing Ψ forward on X × X,
ψ(x, x 0 ) = class (Ψ (γ )) ∈ G ∗/Γ ,
where γ is a path connecting x to x 0 , and where class denotes the projection from G ∗ onto its quotient G ∗/Γ . The holonomy Γ is the obstruction
3Precisely, one moment map, since they are defined up to a constant.
4
for the action of G to be Hamiltonian. The additivity of Ψ becomes the
Chasles’ cocycle condition
ψ(x, x 0 ) + ψ(x 0 , x 00 ) = ψ(x, x 00 ).
The group Γ is invariant by the coadjoint action. Thus, the coadjoint
action passes to the quotient G ∗/Γ , and ψ is equivariant. Because X is
connected, there exists always a map
µ : X → G ∗/Γ
such that
ψ(x, x 0 ) = µ(x 0 ) − µ(x).
(1)
The solutions of this equation are given by µ(x) = ψ(x0 , x)+c, where x0 is
a chosen point in X and c is a constant. But this map is a priori no longer
equivariant. Its variance introduce a 1-cocycle θ of G with values in G ∗/Γ
such that
µ(g (x)) = Ad∗ (g )(µ(x)) + θ(g ),
with θ(g ) = ψ(x0 , g (x0 )) + ∆c(g ), where ∆c is the coboundary due to the
constant c in the choice of µ. This construction extends to the category
{Diffeology}, the moment map for manifolds introduced by Souriau in
[Sou70]. The remarkable point is that none of the constructions brought
up above involves differential equations, and there is no need of considering a putative Lie algebra either. That is a very important point. The
momenta appear as invariant 1-forms on the group, naturally, without
intermediary, and the moment map as a map in the space of momenta.
Note that the group of automorphisms Gω = Diff(X, ω) is a legitimate diffeological group. The above constructions apply and give rise to universal
objects: universal momenta Gω∗ , universal path moment map Ψω, universal holonomy Γω, universal two-points moment map ψω, universal moment
maps µω, universal Souriau’s cocycles θω.
A parasymplectic action of a diffeological group G is a smooth morphism
h : G → Gω, and the objects, associated with G, introduced by the above
moment maps constructions, are naturally subordinate to their universal counterparts. That applies in particular to the group of Hamiltonian
diffeomorphisms, denoted by Hω or by Hamω(M, ω), as it has been defined in [Piz10] and by Hω∗ the space of its momenta. The moment maps
¯ω, ψ
¯ ω, µ¯ω, ¯θω and σ¯ω.
objects associated to Ham(M, ω) will be denoted by Ψ
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The universal moment maps of a symplectic manifold
1. Value of the moment maps for manifolds Let M be a connected manifold equipped with a closed 2-form ω. The value of the paths moment
map Ψω at the point p ∈ Paths (M) = C∞ (R, M), evaluated on the n-plot
F : U → Diff(M, ω) is given by
Z1
Ψω( p)(F) r (δ r ) =
ω p(t ) ( ˙p (t ), δ p(t )) d t
(♦)
0
where r ∈ U and δ r ∈ Rn , δ p denotes the lifting in the tangent space TM
of the path p, defined by
δ p(t ) = [D(F(r ))( p(t ))]−1
∂ F(r )( p(t ))
(δ r ) for all
∂r
t ∈ R.
(♥)
Note 1 — Let us remind that if a differential 1-form is defined by its values
on all the plots, it is however characterized by the values it takes on the 1plots. Moreover, any momentum of a diffeological group is characterized
by its values on the 1-plots plointed at the identity. Thus, in order to
characterize Ψ ( p), it is sufficient, in the formula above, to consider F as a
1-plot pointed at the identity, F(0) = 1M , to choose r = 0 and δ r = 1.
Note 2 — The same formula (♦) gives the paths moment map associated
with the Hamiltonian group. For any plot F of the group Ham(X, ω) ⊂
Diff(X, ω) and any path p of M we have
¯ ω( p)(F) r (δ r ) = Ψω( p)(F) r (δ r ).
Ψ
Now, since by construction the holonomy of Ham(M, ω) is trivial, this
expression gives also the values of the 2-points moment map and we have,
for any pair m, m 0 ∈ M
¯ ω(m, m 0 )(F) = Ψ
¯ ω( p)(F),
ψ
where p is a path of M such that m = p(0) and m 0 = p(1). And, we get
also the values of the moment maps
¯ ω(m0 , m) + c,
¯ω : m 7→ ψ
µ
where m0 is any base point of M and some c ∈ Hω∗ .
Proof. By definition, Ψω( p)(F) = ˆp ∗ (Kω)(F) = Kω( ˆp ◦ F), where ˆp is the
orbit map ϕ 7→ ϕ ◦ p, from Ham(X, ω) to Paths (M). The expression of
6
the Chain-Homotopy operator K , given in [Piz10], applied to the plot
ˆp ◦ F : r 7→ F(r ) ◦ p of Paths (M) gives
Z 1 s
1
0
(Kω)( ˆp ◦ F) r (δ r ) =
ω
7→ ( ˆp ◦ F)(u)(s + t )
d t.
u
0 δr
s=0
0
( u=r
)
But ( ˆp ◦ F)(u)(s + t ) = F(u)( p(s + t )), let us denote temporarily by Φt the
plot (s, u) 7→ F(u)( p(s + t )), then F(u)( p(s + t )) writes Φt (s, u). Thus, by
definition of differential forms, the integrand
s
1
0
(I ) = ω
7→ Φt (s, r )
r
0 δr
( 0r )
of the right term of this expression writes:
1
0
∗
(I ) = Φt (ω)( 0 )
r
0 δr
1
0
, D(Φt )( 0 )
= ωΦ ( 0 ) D(Φt )( 0 )
t r
r
r
0
δr
§
ª
§
ª
∂
∂
= ωF(r )( p(t ))
F(r )( p(s + t ))
,
F(r )( p(t )) (δ r ) .
∂s
s=0 ∂ r
But,

§
ª

∂
∂ p(s + t ) F(r )( p(s + t ))
= D(F(r ))( p(t ))
∂s
∂s
s =0
s =0
= D(F(r ))( p(t ))( ˙p (t )).
Then, using this expression and the fact that, for all r in U, F(r )∗ (ω) = ω,
we have:

∂ F(r )( p(t ))
(I ) = ωF(r )( p(t )) D(F(r ))( p(t ))( ˙p (t )),
(δ r )
∂r

−1 ∂ F(r )( p(t ))
˙
= ω p(t ) p (t ), [D(F(r ))( p(t ))]
(δ r )
∂r
= ω p(t ) ( ˙p (t ), δ p(t )).
Therefore,
Ψω( p)(F) r (δ r ) := Kω( ˆp ◦ F) r (δ r ) =
Z
0
1
ω p(t ) ( ˙p (t ), δ p(t )) d t ,
with δ p given by (♥), is the expression announced above.

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2. The paths moment maps for symplectic manifolds Let (M, ω) be
a Hausdorff symplectic manifold. Let m0 and m1 be two points of M
connected by a path p, m0 = p(0) and m1 = p(1). Let f ∈ C∞ (M, R)
with compact support. Let
F : t 7→ e t gradω ( f )
be the exponential of the symplectic gradient of the f . Then, F is a 1¯ω,
parameter group of Hamω(M, ω) and the Hamiltonian moment map Ψ
computed at the path p, evaluated to the 1-plot F, is the constant 1-form
of R,
¯ ω( p)(F) = [ f (m1 ) − f (m0 )] × d t ,
Ψ
where d t is the standard 1-form of R.
Proof. Let us remark that, in our case, the lift δ p defined by (♥) of (art.
1) writes simply, with ξ = gradω( f ),
δ p(t ) = [D(e r ξ )( p(t ))]−1
∂ e r ξ ( p(t ))
(δ r ) = ξ( p(t )) × δ r,
∂r
where r and δ r are reals. Then, the expression (♦) of (art. 1) becomes
Z1
ω p(t ) ( ˙p (t ), ξ( p(t )) d t × δ r
Ψω( p)(F) r (δ r ) =
0
=
Z
1
ω p(t ) ( ˙p (t ), gradω( f )( p(t )) d t × δ r
0
1

d p(t )
df
=
d t × δr
dt
0
= [ f ( p(1)) − f ( p(0))] × δ r
Z

We remind that, by definition, gradω( f ) = −ω−1 (d f ). Now, it is clear
that for all loop ` of M, Ψω(`)(F) = 0, thus, F is a plot of Ham(M, ω).
¯ ω( p)(F) = Ψω( p)(F) = [ f (m1 ) − f (m0 )] × d t .
And therefore, Ψ

3. Symplectic manifolds A closed 2-form ω on a (connected) Hausdorff
manifold M is symplectic if and only if:
1. The manifold M is homogeneous under the action of the group of
automorphisms Diff(M, ω).
2. The universal moment map µω : M → Gω∗ /Γω is injective.
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We can replace the group of automorphisms Diff(M, ω) by the group
Ham(M, ω) of Hamiltonian automorphisms, and the the universal moment map µω by the universal Hamiltonian moment map µ¯ω : M → Hω∗ .
Hence, the moment map identifies M with a (Γω, θω)-coadjoint orbit Oω
of Diff(M, ω),
µω(M) = Oω ⊂ Gω∗ /Γω,
Also, the Hamiltonian moment map µ¯ω identifies M with a ¯θω-coadjoint
orbit O¯ of Ham(M, ω), µ¯ (M) = O¯ ⊂ H ∗ . This is what we summarize
ω
ω
ω
ω
by the sentence: Every symplectic manifold is a coadjoint orbit.
Diff(M, ω)
πM
M
πO
µω
Oω
On this diagram: on the left M ' Diff(M, ω)/St(x0 ), where x0 is any
point in M, and πM : ϕ 7→ ϕ(x0 ) is a principal fibration4 with group the
stabilizer St(x0 ) ⊂ Diff(M, ω). On the right, Oω ' Diff(M, ω)/St(µω(x0 )),
where St(µω(x0 )) is the stabilizer for the affine coadjoint action on Gω∗ /Γω,
with respect to the universal cocycle θω. The Moment Map µω being then
a diffeomorphism.
Proof. A) Let us assume that ω is symplectic, that is, nondegenerate.
Then, the group Diff(M, ω) is transitive on M [Boo69]. Moreover, for
ˆ : ϕ 7→ ϕ(m) is a subduction [Don84].
every m ∈ M, the orbit map m
Thus, the image of moment moment map µω is one orbit Oω of the affine
coadjoint action of Gω on Gω∗ /Γω, associated with the cocycle θω. Hence,
for the orbit Oω, equipped with the quotient diffeology of Gω, the moment map µω is a subduction.
Now, let m0 and m1 two points of M such that µω(m0 ) = µω(m1 ), that
is, ψω(m0 , m1 ) = µω(m1 ) − µω(m0 ) = 0. Since M is connected, let p ∈
Paths (M) such that p(0) = m0 and p(1) = m1 . Thus, ψω(m0 , m1 ) = 0 is
equivalent to Ψω( p) = Ψω(`), where ` is some loop of M, we can choose
`(0) = `(1) = m0 . Now, let us assume that m0 6= m1 . Since M is Hausdorff
there exists a smooth real function f ∈ C∞ (M, R), with compact support,
such that f (m0 ) = 0 and f (m1 ) = 1. Let us denote by ξ the symplectic
4In the category {Diffeology}.
9
gradient field associated to f and by F the exponential of ξ. Thanks to
(art. 2), on one hand we have Ψω( p)(F) = [ f (m1 ) − f (m0 )]d t = d t , and
on the other hand Ψω(`)(F) = [ f (m0 ) − f (m0 )]d t = 0. But d t 6= 0, thus
ψω(m0 , m1 ) 6= 0, and the moment map µω is injective. Therefore, µω is an
injective subduction on Oω, that is, a diffeomorphism.
The proof for the group of Hamiltonian diffeomorphisms is similar.
A’) Let us assume that ω is symplectic. We know that the group of
ˆ restricted
Hamiltonian diffeomorphisms is transitive. The orbit map m
to the group Ham(M, ω) is still a subduction [Don84]. Thus, M is homogeneous under the action of Ham(M, ω). Now let m0 and m1 be
two different points of M. Let p be a path connecting m0 to m1 , thus
¯ ω( p). Since M is Hausdorff there exists a smooth real
¯ω(m1 )− µ¯ω(m0 ) = Ψ
µ
∞
function f ∈ C (M, R) with compact support such that f (m0 ) = 0 and
f (m1 ) = 1. Let us denote by ξ the symplectic gradient field associated to
¯ ω( p)(F) = d t by (art. 2). Hence,
f and by F the exponential of ξ. Thus, Ψ
(¯µω(m1 ) − µ¯ω(m0 ))(F) = d t 6= 0 and µ¯ω(m0 ) 6= µ¯ω(m0 ). Therefore µω is injective, that is, an injective subduction on Oω, and thus a diffeomorphism.
The converse is the same proof for the only symplectic and the Hamiltonian cases
B) – B’) Let us assume that M is an homogeneous space of Diff(M, ω) and
µω is injective. Let us notice first that, since Diff(M, ω) is transitive, the
rank of ω is constant. In other words, dim(ker(ω m )) = const. Now, let
us assume that ω is degenerate, that is, dim(ker(ω m )) > 0. Since m 7→
ker(ω m ) is a smooth foliation, for any point m of M there exists a smooth
path p of M such that p(0) = m and for t belonging to a small interval
around 0 ∈ R, ˙p (t ) 6= 0 and ˙p (t ) ∈ ker(ω p(t ) ) for all t in this interval.
Then, we can re-parametrize the path p and assume now that p is defined
on the whole R and satisfies p(0) = m, p(1) = m 0 with m 6= m 0 , and
˙p (t ) ∈ ker(ω p(t ) ) for all t . Now, since ˙p (t ) ∈ ker(ω p(t ) ) for all t , using the
expression (♦) given in (art. 1), we get Ψω( p) = 0Gω∗ and thus µω(m) =
µω(m 0 ). But m 6= m 0 and we have assumed that µω is injective. Hense, the
kernel of ω is reduced to {0}, ω is nondegenerate, that is, symplectic.
4. The homogeneous case Let (M, ω) be a connected symplectic manifold. Assume that M is homogeneous under a subgroup G ⊂ Ham(M, ω).
Then, the moment map µ associated with G, as defined in the first section,
is a covering onto its image.
10
For G a Lie group, this is the Souriau’s theorem [Sou70] on homogeneous
symplectic manifolds, but proved the diffeology way. It is illustrated by
the example of (art. 7).
Proof. Let p be a path of M such that µ ◦ p = const. Then, Ψ ( p) = 0G ∗ ,
where Ψ is the paths moment map of G. Thus, for any integer n and
any n-plot F in G, we have Ψ ( p)(F) r (δ r ) = 0, for all r ∈ dom(F) and
all δ r ∈ Rn . Using the expression of Ψ given in (art. 3) part B, we get
R1
ω p(t ) ( ˙p (t ), δ p(t )) d t = 0. Considering the 1-parameter family of paths
0
p s : t 7→ p(s t ), the derivative of Ψ ( p s )(F) r (δ r ) = 0, with respect to s at
s = s0 , gives ω x (u, δ x) = 0, with x = p(s0 ), u = ˙p (s0 ) ∈ T x M and
∂ F(r )(x)
(δ r ) ∈ T x M.
∂r
Now, let v ∈ T x M be any vector, and let γ be a path in M such that
γ (0) = x and v = γ˙ (0). Since M is assumed to be homogeneous under
G, there exists a smooth path r 7→ F(r ) in G such that F(r )(x) = γ (r ),
with F(0) = 1G . Thus, for this F and for r = 0, δ x = v. Therefore, for
all v ∈ T x M, ω x (u, v) = 0, that is, u ∈ ker(ω x ). But ω is symplectic, then
u = 0. Hence, ˙p (s0 ) = 0, for all s0 . Therefore, the path p is constant.
p(t ) = x for all t . Thus, the preimages of the values of the moment map
µ are (diffeologically) discrete. Thanks to the double homogeneity: G
over M, and by equivariance, G over the (possibly affine) coadjoint orbit
O = µ(M), µ is a covering onto its image.

δ x = [D(F(r ))(x)]−1
5. Presymplectic spaces and the Nœther–Souriau theorem The Darboux theorem applied to a presymplectic manifold (M, ω) tells, in particular, that the local automorphisms are transitive. Indeed, M is locally
diffeomorphic, at each point, to (R2k × R` , ωst ), where ωst is the standard
symplectic form on the factor R2k and vanishes on the factor R` .
1. — That is the reason why we shall say that a parasymplectic form
ω, on a diffeological space X, is presymplectic if its pseudogroup of local
automorphisms Diffloc (X, ω) is transitive.
2. — Back to the case of a manifold M, the Nœther–Souriau theorem,
applied to the whole group Ham(M, ω) (which is not a Lie group stricto
sensu), states that the universal Hamiltonian moment map µ¯ω is constant
on the characteristic of ω.
Note that the second proposition applies also for the universal moment
map µω, the proof is identical.
11
Proof. Let us recall that the characteristics of ω are the integral manifolds
of the distribution x 7→ ker(ω x ). Then, the second proposition is an immediate consequence of the explicit formula of (art. 1). If p connects m
to m 0 and ˙p ∈ ker(ω p(t ) ), then for every n-plot F of Ham(M, ω), for every
r ∈ dom(F), for every δ r ∈ Rn , we have
Z1
¯ ω( p)(F) r (δ r ) =
Ψ
ω( ˙p (t ), δ p(t )) d t = 0.
0
¯ ω( p) = ψ
¯ ω(m, m 0 ) = µ¯ω(m 0 )−¯µω(m), and therefore µ¯ω(m) =
Thus, 0H ∗ = Ψ
¯ω(m 0 ).
µ

6. Moment maps and characteristics Let M be a manifold and ω be
parasymplectic. If the group Ham(M, ω) is transitive, which implies in
particular that (M, ω) is presymplectic, then the characteristics of ω coincide with the connected components of the µ¯ω prevalues. In other words,
the Hamiltonian moment map µ¯ω integrates the characteristics of ω.
Proof. Nœther’s theorem states that If m and m 0 are on the same characteristic then µ¯ω(m) = µ¯ω(m 0 ), see (art. 5). Conversely, let us assume
that m and m 0 are connected by a path p such that µ¯ω( p(t )) = µ¯ω(m)
for all t . Then, let s 7→ p s be defined by p s (t ) = p(s t ), for all s and
¯ ω( p s ) = 0, for all s . But, after a
t . Thus µ¯ω( p s (1)) = µ¯ω( p s (0)), that is, Ψ
change of variable,Rand noting that δ p s (t ) = δ p(s t ), we get for all s , 0 =
¯ ω( p s )(F) r (δ r ) = s ω p(t ) ( ˙p (t ), δ p(t )) d t , that is, ω p(t ) ( ˙p (t ), δ p(t )) = 0,
Ψ
0
for all t , and δ p is given in (art. 1) (♥). Now, let v ∈ T p(t ) M, there exists
a path c of M such that c(0) = p(t ) and c˙(0) = v. But we have assumed
that M is homogeneous under the action of Ham(M, ω), that is for any
ˆ : ϕ 7→ ϕ(m), from Ham(M, ω) to M, is a subducm ∈ M the orbit map m
tion. Hence, by choosing m = p(t ), there exists a smooth lifting s 7→ F(s)
defined on a small open superset of 0 ∈ R with values in Ham(M, ω), such
that F(s)( p(t )) = c(s), we can even choose F(0) = 1M . Thus, for s = 0 and
δ s = 1, by (art. 1) (♥), we have
d F(s)( p(t )) δ p(t ) =
= c˙(0) = v.
ds
s=0
Hence, for every v ∈ T p(t ) M, ω( ˙p (t ), v) = 0, e.g. ˙p (t ) ∈ ker(ω p(t ) ) for all
t . Therefore, if two points are connected by a path taking its values in a
connected component of a µ¯ω prevalue, then it is contained in a characteristic of ω.

12
Examples
7. The cylinder and SL(2, R) This is a classical example for which the moment maps of a transitive Hamiltonian action of a Lie group is a nontrivial
covering. I use this example here to show how the algorithm of the moment map in diffeology works in a concrete case. Let us consider the real
space R2 equipped with the standard symplectic form surf = d x∧d y, with
(x, y) ∈ R2 . The special linear group SL(2, R) preserves the standard form
ω. Its action on R2 is effective and has two orbits, the origin 0 ∈ R2 and
the «cylinder» M = R2 −{0}. The restriction ω = surf M is still symplectic and invariant by SL(2, R). Since R2 is simply connected the holonomy
of SL(2, R) is trivial, so its action is Hamiltonian. And since 0 is a fixed
point, the 2-points moment map ψ is exact. Then, there exists an equivariant moment map µ : R2 → sl(2, R)∗ such that ψ(z, z 0 ) = µ(z 0 ) − µ(z),
for all z, z 0 ∈ R2 [Piz10]. Moreover, we know an explicit expression for µ.
For every z ∈ R2 , let p z = [t 7→ t z] ∈ Paths (R2 ) connecting 0 to z. The
general expression given in (art. 1) (♦) and (♥) gives, in the particular
case of p = p z and Fσ = [s 7→ e s σ ], with5 σ ∈ sl(2, R), the following:
1
2
µ(z)(Fσ ) = surf(z, σ z) × d t .
By choosing various σ in sl(2, R), we can check that µ(z) = µ(z 0 ) if and
only if z 0 = ±z. Restricting this construction to M, which is an orbit
of SL(2, R), and thanks to the functoriality of the moment maps [Piz10],
the moment map µM = µ M of SL(2, R) on M is a non trivial double
sheets covering onto its image O = µ(M). It is possible to complicate this
˜ of M, equipped with
example by considering the universal covering M
˜ → M. Then, the action of
˜ of ω by the projection π : M
the pullback ω
f R) on M
˜ is still effective homogeneous and
the universal covering SL(2,
Hamiltonian, and the moment map µ˜ factorizes through π and has the
same image O .
8. The linear cylinder The example of the cylinder is interesting because
it shows simply and explicitly what happens when a symplectic form is
exact but not its primitive. So let M = R × S1 equipped with the 2-form
ω = d α, and α = r × d z/i z, where (r, z) ∈ R × S1 and S1 is identified with
the complex numbers of modulus 1. The manifold M is also a group G,
5 sl(2, R) denotes the Lie algebra of SL(2, R), that is, the vector space of real 2 × 2
traceless matrices.
13
acting by gM (r, z) = (r + ρ, ζ z), with g = (ρ, ζ). Now, for all g ∈ G,
gM∗ (α) = α + β(g ) with β(g ) = ρ
dz
,
iz
β ∈ C∞ (G, Z1DR(M)).
The form β(g ) is closed for every g ∈ G as it must be. The holonomy
group Γ is the subgroup of all Ψ (`) = `ˆ∗ (Kω), where ` runs over the
loops of M (notations [Piz10]). We have,
ˆ
`ˆ∗ (Kω) = `ˆ∗ (K d α) = `ˆ∗ (1ˆ∗α − ˆ0∗α − d [K α]) = d [K α ◦ `],
ˆ ) = g ◦ `, thus K α ◦ `(g
ˆ ) = K α(g ◦ `), and then
but `(g
Z
Z

∗
∗
ˆ
Ψ (`) = ` (Kω) = d g 7→
α = d g 7→
g (α)
`
g ◦`

= d g 7→
Z
α+
Z
`
Z

β(g ) = d g 7→ β(g )
`
`
Z

dz
= d g 7→ ρ
= d [g 7→ 2π k ρ]
` iz
= 2π k × d ρ,

where k ∈ Z is the class of the loop `. Hence, the form a = d ρ is a good
closed (even exact) invariant 1-form of G, that is a momenta of G. And,
Γ = {2π k × a | k ∈ Z}
with a = d ρ.
Now, the space G ∗ of momenta of the Lie group G is generated by a = d ρ
and b = d ζ/i ζ, the quotient G ∗/Γ is thus equal to [Ra/2π Za]×Rb which
is equivalent to S1 × R.
9. The holonomy of the torus We shall compute the holonomy group Γω
for the 2-torus T2 = R2 /Z2 , equipped with ω = class ∗ (d x ∧d y), the canonical volume form on T2 . We denoted by class : R2 → T2 , the canonical
projection.
We know that Γω is a homomorphic image of the first homotopy group
of T2 , that is, π1 (T2 ) = Z2 . We choose then a canonical representant of
every homotopy class:
`n,m = [t 7→ class (nt , mt )],
with n, m ∈ Z.
We will show now that the map j : (n, m) 7→ Ψω(`n,m ) is injective. Since
Ψω(`) is a closed 1-form on the group Diff(T2 , ω) for any loop ` [Piz10],
14
it is sufficient, if (n, m) 6= (0, 0), to find a loop γ in Diff(T2 , ω) such that
R
Ψ (` ) 6= 0. We have
γ ω n,m
Z
Z1
Z 1 Z 1
˙
Ψ (`) =
Ψω(`)(γ ) s (1) d s =
ω`(t ) (`(t ))(δ`(s, t )) d t d s ,
γ
0
0
0
with
∂ γ (s)(`(t ))
∂s
Consider now two integers j , k ∈ Z, we check immediately that
x
x+sj
γ (s) = class
7→ class
y
y + sk
δ`(s, t ) = [D(γ (s))(`(t ))]−1
is a loop in Diff(T2 , ω) based at the identity. For that γ , and for ` = `n,m ,
we have:
n
j
`˙ n,m (t ) = class ∗
and δ`(s , t ) = class ∗
.
m
k
Then,
Thus,
n j
˙
ω`(t ) (`(t ))(δ`(s, t )) = det
= nk − m j .
m k
Z
γ
Ψω(`n,m ) = nk − m j .
Hence, j (`n,m ) = 0 only for n = m = 0. Therefore, j is injective and
Γω ' Z2 .
References
[Boo69]
William M. Boothby. Transitivity of the automorphisms of certain geometric
structures. Trans. Amer. Math. Soc. vol. 137, pp. 93–100, 1969.
[Don84] Paul Donato Revêtement et groupe fondamental des espaces différentiels homogènes. Thèse de doctorat d’état, Université de Provence, Marseille, 1984.
[Kos70] Bertram Kostant. Orbits and quantization theory. In Actes, Congrès intern.
Math., 1970. vol. 2, pp. 395–400.
[Kir74] Alexandre A. Kirillov. Elements de la théorie des représentations. Ed. MIR,
Moscou, 1974.
[Piz13] Patrick Iglesias-Zemmour. Variations of integrals in diffeology. Canad. J. Math.
Vol. 65 (6), 2013 pp. 1255–1286.
[Piz10] Patrick Iglesias-Zemmour The moment maps in diffeology. Memoirs of the
American Mathematical Society, vol. 207, RI 2010.
[Omo86] Stephen Malvern Omohundro. Geometric Perturbation Theory in Physics.
World Scientific, 1986.
[Sou70]
15
Jean-Marie Souriau. Structure des systèmes dynamiques. Dunod Ed., Paris,
1970.
Institut de Mathématique de Marseille, CNRS, 39, rue F. Joliot-Curie, 13453
Marseille Cedex 13