A dual pair for the contact group

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden, and leads to a geometric description of some coadjoint orbits of the full diffeomorphism group.


Introduction
Every contact manifold gives rise to a symplectic manifold in a canonical way. If the contact structure is described by a 1-form α on P , then this symplectic manifold can be described as P × (R \ 0) with the symplectic form d(tα), where t denotes the projection onto the second factor. Regarding the contact structure as a subbundle of hyperplanes, ξ ⊆ T P , and denoting the corresponding line bundle over P by L := T P/ξ, this symplectization can be described more naturally as M = L * \P , with the symplectic form induced from the canonical symplectic form on T * P via the natural vector bundle inclusion L * ⊆ T * P .
The group of contact diffeomorphisms, Diff(P, ξ), acts on M in a natural way, preserving the symplectic structure. This action is in fact Hamiltonian and admits an equivariant moment map. This moment map identifies (unions of) connected components of the symplectization M with certain coadjoint orbits of the contact group.
1.1. The EPContact dual pair. In this paper we will introduce a natural infinite dimensional generalization M of the symplectization M = L * \ P with similar features. To this end we fix a closed manifold S, we denote by |Λ| S its line bundle of densities, and we consider the space M of line bundle homomorphisms from |Λ| * S → S to L * → P which restrict to a linear isomorphism on each fiber. Every volume density on S provides an identification M ∼ = C ∞ (S, M) and permits to regard elements Φ ∈ M as pairs consisting of a map ϕ : S → P together with a contact form for ξ along this map. This space M can be equipped with the structure of a Fréchet manifold in a natural way, and admits a canonical (weakly non-degenerate) symplectic form. The symplectization M can be recovered by choosing S to be a single point.
The contact group acts on M in a natural way, preserving the symplectic structure. This action is Hamiltonian and admits an equivariant moment map, see Proposition 2.4. Furthermore, the group of reparametrizations, Diff(S), acts on M in a Hamiltonian fashion, also admitting an equivariant moment map. On the non-linear Stiefel manifold of weighted embeddings, E ⊆ M, the latter action is free. We show that the restrictions of these moment maps to E, constitute a symplectic dual pair in the sense of Weinstein [33], see Theorem 2.6. Here X(P, ξ) denotes the Lie algebra of contact vector fields on P , X(S) denotes the Lie algebra of all vector fields on S, and Ω 1 (S, |Λ| S ) denotes the space of smooth 1-form densities on S. The moment maps are given by J E L (Φ), X = S Φ(X • ϕ) for all X ∈ X(P, ξ), and J E R (Φ), Z = S Φ(T ϕ • Z) for all Z ∈ X(S). Actually, we will show a stronger statement: The group Diff(S) acts freely and transitively on the fibers of J E L , and the group Diff(P, ξ) acts locally transitive on the level sets of J E R , see Proposition 4.2 and Theorem 3.5. Moreover, we will see that the level sets of both moment maps are smooth submanifolds of E. The dual pair in (1) will be referred to as the EPContact dual pair, because the left leg provides singular solutions of the EPContact equation, i.e., the Euler-Poincaré equation associated with the group of contact diffeomorphisms.
Recall that the projectivized cotangent bundle of a manifold Q admits a canonical contact structure. The EPContact dual pair corresponding to the projectivized cotangent bundle of Q is closely related to the EPDiff dual pair, due to Holm-Marsden [17], associated to the action of Diff(Q) and Diff(S) on T * Emb(S, Q), the cotangent bundle of embeddings of S into Q, see Section 5. This comparison leads to a geometric interpretation of some coadjoint orbits of Diff(Q), see Corollary 5.5.
1.2. Coadjoint orbits of the contact group. The EPContact dual pair will be used to identify coadjoint orbits of the contact group via symplectic reduction for the reparametrization action, following the general principle: Symplectic reduction on one leg of a dual pair of moment maps leads to coadjoint orbits of the other group. The same principle was used in [13], where symplectic reduction on the right leg of the ideal fluid dual pair due to Marsden and Weinstein [25] led to coadjoint orbits of the Hamiltonian group consisting of weighted isotropic submanifolds of the symplectic manifold [34,20].
To make this more precise, consider the non-linear Grassmannian of weighted submanifolds, G = E/ Diff(S), consisting of pairs (N, γ) where N is a submanifold of type S in P and γ : |Λ| * N → L| * N is an isomorphism of line bundles which may be regarded as being akin to a trivialization of the contact structure along N. This space G is a Fréchet manifold in a natural way and the projection E → G is a smooth principal bundle with structure group Diff(S). The moment map J E L descends to a Diff(P, ξ)-equivariant injective immersion G → X(P, ξ) * , which permits to identify orbits of the contact group in G with coadjoint orbits. Each 1-form density ρ ∈ Ω 1 (S, |Λ| S ) gives rise to a reduced space G ρ ⊆ G given by Reduction works best for the zero level. The corresponding reduced space G 0 coincides with the subset of weighted isotropic submanifolds, G iso ⊆ G. We will see that G iso is a smooth submanifold of G and that the action of the contact group on G iso admits local smooth sections. In particular, this action is locally transitive. Hence, the restriction of the moment map, G iso → X(P, ξ) * , identifies (unions of) connected components of G iso with coadjoint orbits of the contact group. Moreover, this identification intertwines the Kostant-Kirillov-Souriau symplectic form with the reduced symplectic form on G iso . These facts are summarized in Theorem 4.12.
The situation is more delicate with regard to reduction at more general levels. In this case the reduced spaces are more singular subsets of G and it is unclear, if the contact group acts locally transitive on them. If ρ is a contact 1-form density on S, i.e., if ker ρ is a contact structure on S, then the reduced space G ρ consists of certain weighted contact submanifolds of P which are of type (S, ker ρ). This is an open condition on the submanifold in view of Gray's stability theorem. The condition on the weight, however, is rather singular: The space of all admissible (for G ρ ) weights on a fixed contact submanifold may be identified with the Diff(S, ker ρ)-orbit of ρ. The situation is tamer if we specialize to 1-dimensional S, see Example 4. 19. In particular, (unions of) connected components in the spaces of weighted transverse knots of fixed length in a contact 3-manifold, may be identified with coadjoint orbits of the contact group.

Singular solutions of the Euler-Poincaré equation.
Another motivation for studying the EPContact dual pair is the construction of singular solutions of the geodesic equation on the group of contact diffeomorphisms equipped with a right invariant Riemannian metric. This works analogous to the EPDiff equation, where the EPDiff dual pair has been used by Holm and Marsden [17] to construct singular solutions for the geodesic equation on the full diffeomorphism group. Similarly, point vortices in two dimensional ideal fluids, a geodesic equation on the group of volume preserving diffeomorphisms, have been described using a dual pair by Marsden-Weinstein [25], see the appendix. The same kind of argument has been applied for the Vlasov equation in kinetic theory by Holm-Tronci [18] using the ideal fluid dual pair, and for the Euler-Poincaré equations on the group of automorphisms of a principal bundle in [10] using the EPAut dual pair [12].
In all these cases the singular solutions of the system are obtained, via a moment map, from a collective Hamiltonian dynamics on a symplectic manifold, referred to as Clebsch variables. This moment map turns out to be the left leg of a dual pair associated to commuting actions on the manifold of embeddings, while the right leg moment map gives conserved quantities by Noether's theorem. We show that for the group of contact diffeomorphisms the situation is similar.
To describe this in more detail, let us start by briefly reviewing the geodesic equation on a Lie group with respect to a right invariant Riemannian metric. We write the inner product on the Lie algebra g in the form (u, v) = Qu, v , where the inertia operator Q : g → g * is symmetric and strictly positive. Formally, the right trivialized geodesic equation on the Lie algebra g is the Euler-Arnold equation, where the adjoint of the adjoint action can be characterized by (ad ⊤ u v, w) := (v, ad u w) for all u, v, w ∈ g. In other words, ad ⊤ u = Q −1 ad * u Q, where ad * u : g * → g * denotes the coadjoint action characterized by ad * u m, v = m, ad u v for u, v ∈ g and m ∈ g * . Via Legendre transformation, using the momentum m := Qu, the Euler-Arnold equation (2) becomes the Lie-Poisson equation, which is the Hamiltonian equation on the Poisson manifold g * for the Hamiltonian h : g * → R, h(m) := 1 2 m, Q −1 m . Its solutions are confined to coadjoint orbits, the symplectic leaves of g * .
Let us now turn to the group of contact diffeomorphisms on a contact manifold (P, ξ). Recall that its Lie algebra can be canonically identified with the space of contact vector fields, X(P, ξ) = Γ ∞ (L), where L = T P/ξ. For simplicity, we will assume P to be closed. We consider X(P, ξ) * = Γ −∞ (L * ⊗|Λ| P ), the space of distributional sections of L * ⊗|Λ| P , where |Λ| P denotes the bundle of densities on P . We assume that the inertia operator, Q : Γ ∞ (L) → Γ ∞ (L * ⊗ |Λ| P ), is a pseudo-differential operator of real order s which is symmetric, strictly positive, invertible, and its inverse, , is a pseudo-differential operator of order −s. Hence, the corresponding inner product, (u, v) = Qu, v , generates the Sobolev H s/2 topology on Γ(L). Using elliptic theory, such inertia operators can be easily constructed. For instance, we may use Q = φ(1 + ∆) s/2 , where ∆ is a Laplacian acting on Γ(L) which is non-negative and formally self-adjoint with respect to a volume density on P and a fiberwise Euclidean metric on L, and φ : L → L * ⊗ |Λ| P denotes the isomorphism of line bundles provided by these geometric choices.
The Hamiltonian function h(m) = 1 2 m, Q −1 m is well defined on Γ −s/2 (L * ⊗ |Λ| P ), the space of sections which are of Sobolev class −s/2. Note that the Sobolev space Γ −s/2 (L * ⊗ |Λ| P ) is invariant under the coadjoint action of Diff(P, ξ). If k ∈ Γ −∞ (L ⊠ L) denotes the Schwartz kernel of Q −1 , then extends continuously (regularization) to m ∈ Γ −s/2 (L * ⊗ |Λ| P ). Assuming is the push forward of a smooth section on S along a smooth embedding S → P , cf. Remark 2.9. According to a standard property of the trace map on Sobolev spaces, see for instance [30,Proposition 1.6 in Chapter 4], it thus provides a continuous functional on Γ s/2 (L). The map J E L is actually smooth into Γ −s/2 (L * ⊗ |Λ| P ). Hence, the pull back of the Hamiltonian h to E, L , is smooth. Although the symplectic form on E is only weakly non-degenerate, the function H gives rise to a Hamiltonian vector field X H on (and tangential to) E, cf. the discussion in [5,Section 4.2.2]. Indeed, since J E L is a moment map, we formally have where ζ E and ζ L * denote the infinitesimal Diff(P, ξ)-actions on E and L * , respectively, cf. (25) and (20) below. By microlocal regularity, Q −1 J E L (Φ) is smooth along the submanifold N in P determined by Φ, see for instance [31,Corollary 9.4 in Chapter 7] or [15,Proposition 3.11 in Chapter IV §3]. Furthermore, since ζ L * : Γ ∞ (L) → Γ ∞ (T L * ) is essentially given by a first order differential operator, it extends to distributional sections, and ζ L * In particular, the latter is smooth along Φ and thus X H (Φ) is a tangent vector to E at Φ, cf. (5).
Every solution Φ t ∈ E of the Hamilton equation provides a singular (peakon) solution u t : . The support of the distributional momentum m t coincides with the smooth submanifold determined by Φ t , and this also coincides with the singular support of u t . Due to the dual pair property, each solution Φ t of (6) remains in a level of the other moment map, J E R : E → X(S) * , and is thus confined to a Diff(P, ξ) orbit in E. Hence, its momentum m t = J E L (Φ t ) is constrained to a coadjoint orbit.
If S is a single point, then the assumption in (4) implies that the distributional kernel k of Q −1 is continuous. In this case we have E = L * \ P and H is given by the (smooth) restriction of k to the diagonal.
The initial value problem for the EPContact equation has been studied by Ebin and Preston in [5]. They consider inertia operators of the form Q = 1 + ∆, where the Laplacian is with respect to a Riemannian metric which is associated with the contact structure.
It appears to be interesting [4] to replace the class of inertia operators considered above with operators in the Heisenberg calculus [3,29,27], a calculus of pseudo-differential operators which is closely linked to the contact geometry on P . Using the Rockland theorem, one can construct pseudo-differential operators Q : Γ ∞ (L) → Γ ∞ (L * ⊗ |Λ| P ) of Heisenberg order s which are symmetric, strictly positive, invertible, and such that the inverse, Q −1 : Γ ∞ (L * ⊗ |Λ| P ) → Γ ∞ (L), is of Heisenberg order −s. For instance, we may use Q = φ(1 + ∆) s/2 , where ∆ is a subLaplacian. Everything mentioned above remains valid, provided the Sobolev spaces are being replaced with the corresponding spaces in the Heisenberg Sobolev scale and the assumption (4) is replaced by the stronger condition s/2 > dim P − dim S.

1.4.
Structure of the paper. The remaining part of the paper is organized as follows. In Section 2 we construct the EPContact dual pair. In Section 3 we show that the level sets of the right moment map are submanifolds on which the contact group acts locally transitive. In Section 4 we study the reduced spaces obtained by factoring out the group of reparametrizations. In Section 5 we compare the EPContact dual pair for the projectivized cotangent bundle with the EPDiff dual pair of Holm and Marsden. In the appendix we provide a comparison with a dual pair due to Marsden and Weinstein for the Euler equation of an ideal fluid. 1.5. Acknowledgments. The first author would like to thank the West University of Timişoara for the warm hospitality and Shantanu Dave for a helpful reference. He gratefully acknowledges the support of the Austrian Science Fund (FWF): project numbers P31663-N35 and Y963-N35. The second author was partially supported by CNCS  UEFISCDI, project number PN-III-P4-ID-PCE-2016-0778.

Weighted non-linear Stiefel manifolds
The aim of this section is to construct the EPContact dual pair, see Theorem 2.6.
2.1. Canonical symplectization of contact manifolds. In this section we set up our notation and recall some well known facts about the symplectization of contact manifolds. We emphasize the structure that will be generalized in the subsequent sections. For more details we refer to [1,Appendix 4.E] and [24,Section 12.3].
Consider a contact manifold (P, ξ) where ξ ⊆ T P denotes the contact subbundle. We write L := T P/ξ for the corresponding line bundle. The vector bundle projection of the dual line bundle will be denoted by π L * : L * → P . The canonical projection T P → L permits to regard the dual bundle as a subbundle of the cotangent bundle, L * ⊆ T * P . We denote by θ L * ∈ Ω 1 (L * ) the pull back of the canonical 1-form on T * P . 1 Hence, the defining equation for θ L * is where β ∈ L * x , x ∈ P , and V ∈ T β L * . The pairing in (7) can be viewed either as a pairing between L * x and L x by considering the class of T β π L * · V in L x = T x P/ξ x , or as a pairing between T * P and T P by considering β an element of L * x ⊆ T * x P . It is well known that the closed 2-form restricts to a symplectic form on M := L * \ P , which will be denoted by ω M = dθ M . The symplectic manifold (M, ω M ) is called the symplectization of the contact manifold (P, ξ). Note that both forms are homogeneous of degree one with respect to the fiberwise scalar multiplication δ t : L * → L * , that is δ * t θ L * = tθ L * and δ * t ω L * = tω L * for all t ∈ R. The action by the contact group. Let us write Diff(P, ξ) for the group of contact diffeomorphisms. Since contact diffeomorphisms preserve the contact subbundle ξ, the Diff(P, ξ)-action on P lifts to an action on the total space of L * . For g ∈ Diff(P, ξ), we let Ψ L * g ∈ Diff(L * ) denote the corresponding (fiberwise linear) diffeomorphism on and Ψ L * g 2 g 1 = Ψ L * g 2 Ψ L * g 1 for all g, g 1 , g 2 ∈ Diff(P, ξ) and t ∈ R. Moreover, the contact group action preserves θ L * and ω L * , that is (Ψ L * g ) * θ L * = θ L * and (Ψ L * g ) * ω L * = ω L * for all g ∈ Diff(P, ξ). Noticing that the symplectic piece M ⊆ L * is invariant under the contact group action, we write Ψ M g for the restricted action.
Let X(P, ξ) denote the Lie algebra of contact vector fields. Via the projection T P → L, every (contact) vector field gives rise to a section of L which may in turn be regarded as a fiberwise linear function on the total space of L * . This provides canonical identifications, where h X ∈ C ∞ lin (L * ) is the fiberwise linear function given by h X (β) = β(X x ) for β ∈ L * x and x ∈ P . Clearly, this identification is equivariant, i.e., for all g ∈ Diff(P, ξ) and X ∈ X(P, ξ). For X ∈ X(P, ξ), we denote the corresponding fundamental vector field (infinitesimal action) on the total space of L * by ζ L * X ∈ X(L * ). Clearly, Remark 2.1. A slightly more explicit, yet less natural description is possible if the contact structure is described by a contact form α ∈ Ω 1 (P ), that is, if ξ = ker α. Such a contact form provides a trivialization P × R ∼ = L * ⊆ T * P , (x, t) ↔ tα x . Via this identification we have θ L * = t(π L * ) * α, and the fiberwise linear function h X from (8) becomes h X (x, t) = t(i X α)(x) where x ∈ P and t ∈ R. A diffeomorphism g of P is a contact diffeomorphism iff it preserves the contact form up to a conformal factor, i.e., iff there exists a (nowhere vanishing) function a g on P such that g * α = a g α. Similarly, a vector field X on P is a contact vector field iff it satisfies L X α = λ X α, for a conformal factor λ X ∈ C ∞ (P ). Both, the group action of Diff(P, ξ) and the Lie algebra action of X(P, ξ) on L * , written in the trivialization L * ∼ = P × R, involve the conformal factors. More explicitly, we have Ψ L * g (x, t) = (g(x), ta g (x)) and ζ L * X (x, t) = (X(x), tλ X (x)∂ t ).
Coadjoint orbits. It is well known that each connected component of a symplectic manifold is equivariantly symplectomorphic to a coadjoint orbit of its Hamiltonian group, see for instance [13]. We will now formulate a similar statement for the group Diff c (P, ξ) of compactly supported contact diffeomorphisms which can be considered as a special case of Theorem 4.12 below. For β ∈ M, the isotropy subgroup Diff c (P, ξ; β) is a closed Lie subgroup of Diff c (P, ξ). Moreover, the map provided by the action, Diff c (P, ξ) → M, g → Ψ M g (β), admits a local smooth right inverse defined in a neighborhood of β. In particular, the group Diff c (P, ξ) acts locally and infinitesimally transitive on M, and the Diff c (P, ξ)-orbit through β is open and closed in M. Denoting this orbit by M β , the map Diff c (P, ξ) → M β is a smooth principal bundle with structure group Diff c (P, ξ; β). Hence, may be regarded as a homogeneous space. The moment map (15) induces an equivariant diffeomorphism between M β and the coadjoint orbit of Diff c (P, ξ) through J M (β) ∈ X(P, ξ) * . By infinitesimal equivariance of J M and (13), this diffeomorphism intertwines the Kostant-Kirillov-Souriau symplectic form ω KKS with ω M . Indeed, for β ∈ M and X, Y ∈ X(P, ξ), we get In particular, each connected component of M is equivariantly symplectomorphic to a coadjoint orbit of the identity component in Diff c (P, ξ). If P connected and the contact structure is not coorientable, then M is connected, hence a coadjoint orbit of Diff c (P, ξ).

2.2.
Moment maps on a manifold of weighted maps. In this section we introduce an infinite dimensional generalization L of L * that also carries a canonical 1-form θ L which is invariant under a natural Diff(P, ξ)-action.
To this end, we fix a closed manifold S. We let |Λ| S denote the line bundle of densities [21, Chapter 16] on S, and we write π |Λ| S : |Λ| S → S for the corresponding vector bundle projection. Recall that sections of |Λ| S can be integrated over S in a natural way. Every orientation of S provides an isomorphism of line bundles |Λ| S ∼ = Λ dim(S) T * S. A nowhere vanishing density, i.e., a section in Γ ∞ (|Λ| S \ S), will be referred to as a volume density.
We denote the space of line bundle homomorphisms from |Λ| * S → S to L * → P by There is a canonical map π L : L → C ∞ (S, P ), characterized by for all Φ ∈ L. For the fiber over ϕ ∈ C ∞ (S, P ) we have a canonical identification, The contact group Diff(P, ξ) acts from the left on L, and the reparametrization group Diff(S) acts on L from the right in an obvious way. More explicitly, these actions are given by where Φ ∈ L, g ∈ Diff(P, ξ), f ∈ Diff(S), and ψ |Λ| * S f ∈ Diff(|Λ| * S ) denotes the induced (fiberwise linear) action of Diff(S) on the total space of |Λ| * S . The two actions on L commute, and the map π L intertwines them with the corresponding actions on C ∞ (S, P ) given by where g ∈ Diff(P, ξ), f ∈ Diff(S), and ϕ ∈ C ∞ (S, P ). More explicitly, we have Ψ L Remark 2.2. Let µ ∈ Γ ∞ (|Λ| S \ S) be a volume density on S, i.e., a nowhere vanishing smooth section of |Λ| S . Such a volume density provides an identification whereμ ∈ Γ ∞ (|Λ| * S ) denotes the section dual to µ, that isμ(µ) = 1. In this picture the actions on L take the form where φ ∈ C ∞ (S, L * ), g ∈ Diff(P, ξ) and f ∈ Diff(S).
The space L can be equipped with the structure of a smooth Fréchet manifold such that the identification L ∼ = C ∞ (S, L * ) in Remark 2.2 becomes a diffeomorphism, for each choice of volume density µ. The map π L : L → C ∞ (S, P ) is a smooth vector bundle. The tangent space at Φ ∈ L can be canonically identified as The actions of Diff(P, ξ) and Diff(S) on L are smooth. For X ∈ X(P, ξ) and Z ∈ X(S), the corresponding fundamental vector fields are where Φ ∈ L and ζ |Λ| * S Z ∈ X(|Λ| * S ) denotes the fundamental vector field of the Diff(S)action on the total space of |Λ| * S . Note that Clearly, • π L , and [ζ L X , ζ L Z ] = 0, where g ∈ Diff(P, ξ), X, X 1 , X 2 ∈ X(P, ξ), f ∈ Diff(S), Z, Z 1 , Z 2 ∈ X(S).
The canonical 1-form. Consider the 1-form θ L on L defined by where η ∈ T Φ L and Φ ∈ L. Note here that, because of (19), inserting η into θ L * leads to a fiberwise linear map θ L * (η) : |Λ| * S → R which, when regarded as a section of |Λ| S , may be integrated over S. By invariance of θ L * , the 1-form θ L is invariant under both actions, i.e., we have (Ψ L g ) * θ L = θ L and (ψ L f ) * θ L = θ L for all g ∈ Diff(P, ξ) and f ∈ Diff(S). The corresponding infinitesimal invariance reads L ζ L X θ L = 0 and L ζ L Z θ L = 0, where X ∈ X(P, ξ) and Z ∈ X(S).
Moreover, we introduce the 2-form ω L := dθ L on L. By invariance of θ L , this 2form is invariant under both actions too. More explicitly, we have (Ψ L g ) * ω L = ω L and (ψ L f ) * ω L = ω L for g ∈ Diff(P, ξ) and f ∈ Diff(S), as well as infinitesimal invariance L ζ L X ω L = 0 and L ζ L Z ω L = 0 for X ∈ X(P, ξ) and Z ∈ X(S). Clearly, see [32,9], where η 1 , η 2 ∈ T Φ L and Φ ∈ L. As before, the fiberwise linear function ω L * (η 1 , η 2 ) on |Λ| * S may be regarded as a section of |Λ| S which can be integrated over S. The exact 2-form ω L = dθ L is not (weakly) non-degenerate, because ω L * is not symplectic on all of L * . In the subsequent section, we will restrict to an invariant open subset of L on which ω L is (weakly) symplectic. On this symplectic part, both actions are Hamiltonian with equivariant moment map. This is a well known formal consequence of the fact that these actions preserve the 1-form θ L , see for instance [24,Section 12.3]. The corresponding Hamiltonian functions and moment maps are given by contraction of the fundamental vector fields with the canonical 1-form. However, these geometric objects make sense on all of L. Hence, we will now formulate their fundamental relations on L.
(48) 2 We will denote the restriction to M of any action, function, form, or vector field on L considered above, by replacing the superscript L with M. Because L * \ P is symplectic, the 2-form ω M = dθ M is (weakly) non-degenerate, whence symplectic, cf. (23). The map π M : M → C ∞ (S, P ) is a principal fiber bundle with structure group C ∞ (S, R × ), provided we restrict to the connected components of C ∞ (S, P ) in the image of π M . If ϕ is in one of these components, then the fiber M ϕ := (π M ) −1 (ϕ) may be canonically identified with the space of nowhere vanishing sections of the line bundle |Λ| S ⊗ ϕ * L * , cf. (17). Thus, disregarding the density part, M ϕ may be considered as the space of contact forms for ξ along the map ϕ : S → P .
Clearly, M is invariant under the action of the groups Diff(P, ξ) and Diff(S). Since both actions preserve the 1-form θ M , they are Hamiltonian with equivariant moment maps obtained by contraction of the 1-form with the infinitesimal generators, see for instance [24,Section 12.3]. We summarize these facts in the following proposition.
Remark 2.5. If S is a single point, then we recover the symplectization discussed in Section 2.1. More precisely, in this case the canonical volume density on S provides a canonical isomorphism between the line bundles π L : L → C ∞ (S, P ) and π L * : L * → P . Up to this identification, we have Ψ L g = Ψ L * g , for all g ∈ Diff(P, ξ), θ L = θ L * and ω L = ω L * . Moreover, M = M and J M L = J M . Clearly, the Diff(S)-action is trivial in this case and J M R = 0.

2.4.
A dual pair on the non-linear Stiefel manifold of weighted embeddings. We will now restrict to an open subset of M on which the Diff(S)-action is free. Let denote the open subset of all (fiberwise linear) embeddings in L = C ∞ lin (|Λ| * S , L * ). Elements of E are automatically isomorphisms on fibers, so E ⊆ M. We consider E as a non-linear Stiefel manifold of weighted embeddings. 3 We will denote the restriction to E of any action, function, form, or vector field on L considered above, by replacing the superscript L with E. The map π E : E → Emb(S, P ) is a principal fiber bundle with structure group C ∞ (S, R × ), provided we restrict to the connected components of Emb(S, P ) in the image of π E . Since E is open in M, the symplectic form ω M restricts to a symplectic form ω E on E. Hence, (E, ω E ) is a (weakly) symplectic Fréchet manifold.
Note that E is invariant under the actions of Diff(P, ξ) and Diff(S). In view of Proposition 2.4, the restrictions of J M L and J M R to E provide equivariant moment maps for the actions of Diff(P, ξ) and Diff(S) on E, respectively. A pair of equivariant moment maps for commuting Hamiltonian actions of (infinite dimensional) Lie groups G and H on an (infinite dimensional) symplectic manifold Q, is called a symplectic dual pair [33] if the distributions ker T J L and ker T J R are symplectic orthogonal complements of one another: (ker T J L ) ⊥ = ker T J R and (ker T J R ) ⊥ = ker T J L . Both identities are needed here, due to the weakness of the symplectic form.
i.e., if the G-orbits and H-orbits are symplectic orthogonal complements of one another, then the actions are said to be mutually completely orthogonal [22]. Since ker T J R = h ⊥ Q , the first identity in (53) can be rephrased as the transitivity of the g-action on level sets of the moment map J R , and similarly for the second identity.
Mutually completely orthogonality of the actions implies that J L and J R form a dual pair. The reverse implication is not always true, due to the weakness of the symplectic form [11].
Theorem 2.6. The moment mappings J E L and J E R in (52) form a symplectic dual pair, called the EPContact dual pair. Moreover, the commuting actions of Diff(P, ξ) and Diff(S) on E are mutually completely orthogonal, i.e., for each Φ ∈ E we have as well as follows immediately from (27) and (25). To show the converse inclusion, suppose A, and ω L * . Thus, for all Z ∈ X(S), where the integrands are fiberwise linear functions on the total space of |Λ| * S , which may be regarded as sections of |Λ| S and integrated over S. By Lemma 2.7 below, there exists a fiberwise linear function According to the identification (8), there exists a contact vector field X ∈ X(P, ξ) Since ω L * is non-degenerate over L * \ P , we conclude A = ζ L * X • Φ, and using (20) we get A = ζ E X (Φ), whence (54). It remains to check the other equality (55). The inclusion follows immediately from (35) and (33), or (54). To show the converse inclusion, suppose Hence, for all X ∈ X(P, ξ), We conclude that B is tangential toÑ . Consequently, there exists a vector field W on the total space of |Λ| * S such that B = T Φ • W . Clearly, δ * t W = W , for all t ∈ R. Using Lemma 2.8 below, we conclude that there exists Z ∈ X(S) such that W = ζ In view of (20), we obtain B = ζ E Z (Φ). This completes the proof of (55).
Proof. We fix a volume density µ on S and identify |Λ| * S ∼ = S × R correspondingly. The two canonical projections shall be denoted by p : S × R → S and t : S × R → R, respectively. The radial vector field becomes R = t∂ t ∈ X(S × R). By homogeneity, where L Z µ =: div(Z)µ and p * Z ∈ X(S × R) denotes the vector field which projects to Z on S and 0 on R. Consequently, Using Stokes' theorem, we obtain In view of the assumption (56), we conclude that for all smooth, fiberwise linear functions h on the total space of |Λ| * S . 5 Then W is a fundamental vector field of the natural Diff(S) action on |Λ| * S , i.e., there exists Z ∈ X(S) such that W = ζ |Λ| * S Z . Proof. As in the proof of the preceding lemma we fix a volume density µ on S, we identify |Λ| * S ∼ = S × R correspondingly, and we denote the two canonical projections by p : S × R → S and t : S × R → R. Hence, the vector field W can be written in the form W = p * Z + (p * w)t∂ t where Z ∈ X(S) and w ∈ C ∞ (S). Every functionh ∈ C ∞ (S) gives rise to a fiberwise linear function h := tp * h on the total space of |Λ| * S . Then dh(W ) = tp * (hw + dh(Z)) and Stokes' theorem yields 4 Note that the integrand is a fiberwise linear function on the total space of |Λ| * S , which may be regarded as a section of |Λ| S and integrated over S. 5 Note that the integrand is a fiberwise linear function on the total space of |Λ| * S , which can be regarded as a section of |Λ| S and integrated over S.
Using the assumption (58), we conclude that w = div(Z). Consequently, see (57), we obtain W = p * Z + (p * w)t∂ t = p * Z + (p * div(Z))t∂ t = ζ |Λ| * S Z . Remark 2.9. Let us give a more explicit description of the EPContact dual pair if the contact structure is described by a contact form, ξ = ker α, and a volume density µ on S has been fixed. We have already pointed out before, see footnote 3, that these choices provide an identification of the non-linear Stiefel manifold E with Emb(S, P ) × C ∞ (S, R × ). Via this identification, the actions of Diff(P, ξ) from the left and Diff(S) from the right are Using the identification X(P, ξ) = C ∞ (P ) provided by the contact form α, the EPContact dual pair (52) becomes with moment maps This follows readily from the formulas provided in Remarks 2.2 and 2.3.
In view of Theorem 2.6 one might expect [2,13] that the contact group acts locally transitive on the level sets of J E R . This is indeed the case, see Theorem 3.5 in the subsequent section. Moreover, one might expect that a coadjoint orbit O ⊆ X(S) * gives rise to a reduced symplectic structure on the quotient (J E R ) −1 (O)/ Diff(S) which is equivariantly symplectomorphic to a coadjoint orbit of Diff c (P, ξ) via the symplectomorphism induced by the moment map J E L . Below we will see that this can be made rigorous for coadjoint orbits corresponding to isotropic embeddings, see Theorem 4.12.

Level sets of the right moment map
In this section we will show that each level set of the right moment map J E R : E → Ω 1 (S, |Λ| S ) ⊆ X(S) * is a smooth splitting Fréchet submanifold in E. Furthermore, we will see that the contact group acts locally transitive on each level set. More precisely, we will show that this action admits local smooth sections. Hence, (unions of) connected components of these level sets may be regarded as homogeneous spaces of the contact group. These results are summarized in Theorem 3.5 below.
A similar transitivity statement has been established in [9, Proposition 5.5] using methods quite different from the approach presented here.
Let π J 1 L : J 1 L → P denote the 1-jet bundle of sections of L. Recall that each section h ∈ Γ ∞ (L) gives rise to a section j 1 h ∈ Γ ∞ (J 1 L). We equip the total space of J 1 L with the contact structure uniquely characterized by the following property: A section s ∈ Γ ∞ (J 1 L) has isotropic image iff there exists h ∈ Γ ∞ (L) such that s = j 1 h. 6 In this case h = π J 1 L L • s, where π J 1 L L : J 1 L → L denotes the natural projection. 6 If L ∼ = P ×R is a trivialization of L, then J 1 L ∼ = T * P ×R, and the contact structure can be described by the contact form p * θ − dt, where θ denotes the canonical 1-form on T * P , while p : T * P × R → T * P and t : T * P × R → R denote the canonical projections.
Consider the line bundle p : hom(p * 1 L, p * 2 L) → P × P where p 1 , p 2 : P × P → P denote the two canonical projections. We let P := isom(p * 1 L, p * 2 L) denote the open subset of fiberwise invertible maps. We equip the total space of P with the contact structure where a ∈ P. 7 Note that a diffeomorphism g ∈ Diff(P ) is contact if and only if there exists a smooth map a : P → P with isotropic image satisfying p 1 • p • a = id and ), for all x ∈ P . Here Ψ L g,x denotes the restriction of Ψ L g to the fiber L x . It is well known [23, Theorem 1] that there exists a contact diffeomorphism from an open neighborhood V of the zero section P ⊆ J 1 L onto an open neighborhood U of the diagonal P ⊆ P intertwining the contact structure obtained by restriction from J 1 L with the contact structure obtained by restriction from P. Moreover, for all x ∈ P , we have Ξ(0 x ) = id Lx .
(63) It is also well known, see [19,Theorem 43.19] for the coorientable case, that the map provides a chart for the Lie group Diff c (P, ξ) at the identity. Here W is a C ∞ -open neighborhood of zero such that, for each h ∈ W, the image of j 1 h is contained in V and Clearly, F (0) = id P , see (63). Moreover, for h ∈ W and x ∈ P , we have in hom(L x , L F (h)(x) ). In particular, Proof. Put ϕ = π E (Φ) ∈ Emb(S, P ) and N := ϕ(S). For the chart F in (64) we obtain (66) and (18). Since N is a closed submanifold in P , the linear space on the right hand side admits a linear complement in Γ ∞ c (L). To construct such a complement, let π W : W → N denote the normal bundle of N, where W = T P | N /T N; fix a tubular neighborhood W ⊆ P of N such that N corresponds to the zero section in W ; and choose an isomorphism of line bundles L| W ∼ = (π W ) * L| N . This provides a linear map by regarding sections of L| N as π W -fiberwise constant sections of L| W , and by regarding sections of L| N ⊗ W * as π W -fiberwise linear sections of L| W . Let χ ∈ C ∞ c (W, R) be 7 If ξ = ker α, and P ∼ = P × P × (R × ) denotes the corresponding trivialization, then the contact structure can be described by the contact form tp * 1 α − p * 2 α on P × P × (R × ).
a compactly supported bump function such that χ ≡ 1 in a neighborhood of the zero section. Multiplication with χ and extension by zero provides a linear map Γ ∞ (L| W ) → Γ ∞ c (L). Composing this with (67), we obtain a linear map we will denoted by The image of χ provides a linear complement of h ∈ Γ ∞ c (L) ∀x ∈ N : . For x ∈ S consider the restrictions to the fibers, Φ 1,x : |Λ| * S,x → L * ϕ 1 (x) and Φ 2,x : |Λ| * S,x → L * ϕ 2 (x) , and define a smooth map G(Φ 1 , Φ 2 ) : S → P by for x ∈ S. Clearly, S,x , and write a := G(Φ 1 , Φ 2 )(x). Then: The lemma follows at once.
Proof. Fix Φ 1 ∈ E ρ , put ϕ 1 := π E (Φ 1 ) ∈ Emb(S, P ), and consider the submanifold N := ϕ 1 (S) of P . Let π W : W → N denote its normal bundle, W := T P | N /T N. Choose a tubular neighborhood W ⊆ P of N in P such that the zero section in W corresponds to N. As in the proof of Lemma 3.1, we fix an isomorphism of line bundles, and a compactly supported bump function χ ∈ C ∞ c (W, R) such that χ ≡ 1 on an open neighborhood X of the zero section in W . The corresponding map (68) extends uniquely to a linear mapχ such that the following diagram commutes: The line bundle isomorphism in (71) also provides an isomorphism Using this isomorphism to replace the lower right corner in the diagram (72), we obtain linear maps γ and Γ ∞ (J 1 L)| N → Γ ∞ c (J 1 L), s →s, such that the following diagram commutes: For every ν ∈ Γ ∞ (W ) with ν(N) ⊆ X we obtain a linear isomorphism Moreover,ν and its inverseν −1 are given by first order differential operators depending smoothly on ν.
We will next show that the following map is a diffeomorphism from V onto the C ∞ -open subset U in G ∈ C ∞ (S, P) : p 1 • p • G = ϕ 1 consisting of all G ∈ C ∞ (S, P) with the following five properties: . To see that (78) is a diffeomorphism, let s ∈ V and observe that (77) and (78) yield as well as ψ Gs = ψ s and ν Gs = ν s . Hence, We conclude that G s ∈ U and s Gs = s, for all s ∈ V, see (j). This shows that the map U → V, G → s G , is left inverse to the map (78). To show that it is right inverse too, consider G ∈ U and note that (75) and (j) yields Composing with p 1 • p and using (b), (f) we obtain, Combining the latter two equations, we get In other words, G s G = G, for all G ∈ U, cf. (78). This shows that (78) is indeed a diffeomorphism. Using (76), (79), and the fact that Ξ is a contact diffeomorphism we find G s has isotropic image in P ⇔ s ∈ img(γ).
(81) The construction in (69), cf. also (70), provides a diffeomorphism Combining this with the diffeomorphism in (78), we see that the map This shows that (82) is a submanifold chart for E ρ in E, centered a Φ 1 .
Combining Lemmas 3.1, 3.3, and 3.4 we obtain the following result: Theorem 3.5. Suppose ρ ∈ Ω 1 (S, |Λ| S ). Then the level set E ρ is a smooth splitting Fréchet submanifold of E. For Φ ∈ E ρ , the isotropy subgroup Diff c (P, ξ; Φ) is a closed Lie subgroup of Diff c (P, ξ). Moreover, the map provided by the action, Diff c (P, ξ) → E ρ , g → Ψ E g (Φ), admits a local smooth right inverse defined in a neighborhood of Φ in E ρ . In particular, the group Diff c (P, ξ) acts locally and infinitesimally transitive on E ρ , and the Diff c (P, ξ)-orbit of Φ is open and closed in E ρ . Denoting this orbit by E ρ Φ , the map Diff c (P, ξ) → E ρ Φ is a smooth principal bundle with structure group Diff c (P, ξ; Φ). Hence, E ρ Φ = Diff c (P, ξ)/ Diff c (P, ξ; Φ) may be regarded as a homogeneous space.

Weighted non-linear Grassmannians
We continue to consider a manifold P endowed with a contact structure ξ, and a closed manifold S. Recall that the Diff(S) action is free on the non-linear Stiefel manifold E of weighted embeddings. We will now factor out this action and consider the corresponding space G = E/ Diff(S) of unparametrized weighted submanifolds of P .

4.1.
Principal bundles over non-linear Grassmannians. Let Gr S (P ) denote the non-linear Grassmannian of all smooth submanifolds of P which are diffeomorphic to S. It is well know that Gr S (P ) can be equipped with the structure of a Fréchet manifold such that the canonical map Emb(S, P ) → Gr S (P ) becomes a principal bundle with structure group Diff(S).
The Diff(P, ξ)-actions on P and on L * induce a left action on G. For g ∈ Diff(P, ξ) we let Ψ G g denote the corresponding action on G, that is, Ψ G g (N, γ) = (g(N), g * γ).
by identifying (N, ν) with (N, ν ⊗ α| N ) ∈ G. The weighted Grassmannian can be equipped with a smooth structure such that the canonical forgetful map Gr wt S (P ) → Gr S (P ) is a smooth fiber bundle. Indeed, it can be canonically identified with the bundle associated to the principal fiber bundle Emb(S, P ) → Gr S (P ) via the Diff(S)-action on the space Γ ∞ (|Λ| S \ S) of volume densities on S. Note that the induced smooth structure on G does not depend on the contact form α for ξ. Via the identification (85), the Diff(P, ξ)-action becomes where g ∈ Diff(P, ξ) and (N, ν) ∈ Gr wt S (P ). Indeed, g * (ν ⊗ α| N ) = g * α α g(N ) g * ν ⊗ α| g (N ) . The space G in (84) can be equipped with the structure of a smooth manifold such that the canonical forgetful map π G : G → Gr S (P ) becomes a smooth fiber bundle with typical fiber Γ ∞ (|Λ| S \ S). Indeed, if (N, γ) ∈ G, then locally around N, the contact structure on P is coorientable and can be described by a contact form. We can therefore use Remark 4.1 to equip G with a smooth structure. In view of (86) the Diff c (P, ξ)-action on G is smooth.
To an element Φ ∈ E = Emb lin (|Λ| * S , L * ) over the embedding ϕ = π E (Φ) ∈ Emb(S, P ) we associate a pair (N, γ) ∈ G in the following way: N = ϕ(S) and γ is the composition of Φ (corestricted to L * | N ) with the isomorphism |Λ| * ϕ : |Λ| * N → |Λ| * S induced by the diffeomorphism ϕ : S → N. It is easy to see that the map q : E → G, given by q(Φ) = (N, γ), is a smooth principal bundle with structure group Diff(S). We summarize this in the following Diff(P, ξ)-equivariant commutative diagram: By Diff(S) invariance, see Proposition 2.4(a), the moment map J E L descends to a smooth map (88) In view of (49) we have the explicit formula where (N, γ) ∈ G and X ∈ X(P, ξ). On the right hand side X is regarded as a section of L, see (8), restricted to N and contracted with γ to produce a density on N which can be integrated. 8 (a) The map J G L : G → X(P, ξ) * is a Diff(P, ξ)-equivariant injective immersion. (b) We have Diff(P, ξ; (N, γ)) = Diff(P, ξ; J G L (N, γ)), where the left hand side denotes the isotropy group of (N, γ) ∈ G and the right hand side denotes the isotropy group of J G L (N, γ) ∈ X(P, ξ) * for the coadjoint action. (c) The group Diff(S) acts freely and transitively on level sets of J E L : E → X(P, ξ) * . Proof. In view of Proposition 2.4(a), the smooth map J G L is Diff(P, ξ)-equivariant. It follows from the dual pair symplectic orthogonality condition (55) that J G L is immersive. To check injectivity, suppose (N 1 , γ 1 ) and (N 2 , γ 2 ) are two elements in G such that J G L (N 1 , γ 1 ) = J G L (N 2 , γ 2 ). Since γ i is nowhere vanishing, we have supp(J G L (N i , γ i )) = N i , see (89), whence N 1 = N 2 . Assume, for the sake of contradiction, γ 1 = γ 2 . Then there existsX ∈ Γ ∞ (L| N ) such that γ 1 ,X = γ 2 ,X with respect to the canonical pairing between Γ ∞ (|Λ| N ⊗ L| * N ) and Γ ∞ (L| N ). ExtendingX to a global section X ∈ Γ ∞ (L), we obtain J G L (N 1 , γ 1 ), X = J G L (N 2 , γ 2 ), X using (89). Since this contradicts our assumption J G L (N 1 , γ 1 ) = J G L (N 2 , γ 2 ), we must have γ 1 = γ 2 . This shows that J G L is injective.
The assertion about the isotropy groups in (b) follows readily from the injectivity and equivariance of J G L . The assertion in (c) also follows from the injectivity statement in (a), since the Diff(S)-action on the fibers of q : E → G is free and transitive.

4.2.
Right leg symplectic reduction. In this section we study the spaces obtained by symplectic reduction for the right moment map J E R : E → Ω 1 (S, |Λ| S ) ⊆ X(S) * . For a 1-form density ρ ∈ Ω 1 (S, |Λ| S ) we put where E ρ = (J E R ) −1 (ρ). By Diff(S)-equivariance of J E R , and since Diff(S) acts transitively on the fibers of q : E → G, the definition of G ρ may be rephrased equivalently as Here ρ · Diff(S) ⊆ Ω 1 (S, |Λ| S ) ⊆ X(S) * denotes the coadjoint orbit through ρ. Note that q induces a bijection where Diff(S, ρ) = {f ∈ Diff(S) : f * ρ = ρ} denotes the isotropy group of ρ. Thus, G ρ is the underlying set of the symplectically reduced space at ρ.
We have the following more explicit description of G ρ : Here ι N : N → P denotes the inclusion and the pull back ι * N γ ∈ Ω 1 (N, Proof. Consider Φ ∈ E over ϕ := π E (Φ) ∈ Emb(S, P ) and put (N, γ) := q(Φ). By definition of q, we have ϕ(S) = N and the "triangle" on the top of the following diagram commutes: The left rectangle in this diagram commutes in view of the formula for J E R in (50); the right rectangle commutes in view of the definition of ι * N γ; and the "triangle" at the bottom commutes trivially. We conclude that (N, ). Using the description (90) of G ρ we obtain the lemma.
Remark 4.4. We have seen in Remark 4.1 that the choice of a contact form α on P permits to identify G with a weighted Grassmannian. Under this identification, the reduced space becomes Remark 4.5. A general fiber of the forgetful map π G : G → Gr S (P ) will intersect several of the spaces G ρ , for many different ρ. A notable exception are fibers over isotropic submanifolds, cf. (97) in Section 4.3 below.
Since we do not expect G ρ to be a submanifold in G for general ρ, we will consider G ρ as a Frölicher space with the smooth structure induced from the ambient Fréchet manifold G.
Recall that a Frölicher space, see [19,Section 23] and [6,7,8], is a set X together with a set C X of curves into X and a set of functions F X on X with the following two properties: A map g : X → Y between Frölicher spaces is called smooth if g • c ∈ C Y for all c ∈ C X . Equivalently, smoothness of g can be characterized by f • g ∈ F X for all f ∈ F Y . Note that C X coincides with the set of smooth curves into X, and F X coincides with the set of smooth functions on X, provided R is equipped with the standard Frölicher structure C R = C ∞ (R, R) = F R . Frölicher spaces and smooth maps between them form a category which is complete, cocomplete and Cartesian closed, see [19,Theorem 23.2].
Any subset A of a Frölicher space X admits a unique Frölicher structure such that the inclusion A ⊆ X is initial, i.e., a curve into A is smooth iff it is smooth into X. The c ∞ -topology on a Frölicher space X is the strongest topology such that all smooth curves into X are continuous. If U is a cover of X by c ∞ -open subsets, then a function f on X is smooth iff the restriction f | U is smooth (with respect to the induced Frölicher structure) for all U ∈ U.
Any Fréchet manifold, together with the usual smooth curves into and smooth functions on it, constitutes a Frölicher space. More generally, manifolds modeled on c ∞ -open subsets of convenient vector spaces [19,Section 27] are Frölicher spaces. For Fréchet manifolds the c ∞ -topology coincides with the Fréchet topology, see [19,Theorem 4.11(1)].
We consider G ρ as a Frölicher space with the smooth structure induced from G. Hence, a curve in G ρ is smooth iff it is smooth into G; and a function on G ρ is smooth iff it is smooth along smooth curves. Moreover, we equip E ρ / Diff(S, ρ) with the induced Frölicher structure. Hence, a function on E ρ / Diff(S, ρ) is smooth iff the corresponding (fiberwise constant) function on E ρ is smooth, with respect to the Frölicher structure on E ρ considered before; and a curve in E ρ / Diff(S, ρ) is smooth iff it is smooth along smooth functions. One readily checks that the maps in the commutative diagram   (N, γ).
Clearly, this is a Diff(S, ρ)-equivariant smooth bijection. To see that these are actually diffeomorphisms of Frölicher spaces, we use the fact that E → G is a smooth principal bundle. This implies that the mapδ : E × G E → Diff(S), implicitly characterized by is smooth. Restricting δ, we obtain a smooth map δ : E ρ × G ρ E ρ → Diff(S, ρ), which can be used to express the inverse of (94): E ρ U → U × Diff(S, ρ), Φ → (q(Φ), δ(σ(q(Φ)), Φ)). This shows that the trivialization (94) is a diffeomorphism, whence E ρ → G ρ is a locally trivial smooth principal fiber bundle. The remaining assertions in (a) are now obvious.
Using local sections of E ρ → G ρ and the fact that the Diff c (P, ξ)-action on E ρ admits local smooth sections, see Theorem 3.5, we readily see that the Diff c (P, ξ)-action on G ρ admits local smooth sections. The remaining assertions in (b) are then obvious.
Part (c) follows at once, see Proposition 4.2(a).
Note that the assumption in Proposition 4.6 is trivially satisfied for ρ = 0. This isotropic case will be discussed in Section 4.3; and we will obtain a more precise conclusion than formulated in Proposition 4.6 above, see Theorem 4.12. In particular, we will show that in this case G 0 is a smooth submanifold of G which inherits a reduced symplectic form from E 0 . Moreover, the map in (93) is a symplectomorphism onto the coadjoint orbit equipped with the Kostant-Kirillov-Souriau form.
For more general ρ (e.g. ρ of contact type) the situation is more delicate. If we equip G ρ with the trace topology induced from G, then q restricts to principal fiber bundle (J E R ) −1 (ρ · Diff(S))/ Diff(S) → G ρ with structure group Diff(S), see (91). However, with respect to this topology, the action of Diff(P, ξ) on G ρ will in general not admit local sections, see Proposition 4.20 below.
The next lemma provides a criterion for the premise in Proposition 4.6 above.  We do not know if these (equivalent) properties hold true for all contact 1-form densities.

4.3.
Weighted isotropic non-linear Grassmannians. We will now specialize to the isotropic case, ρ = 0. Let us introduce the notation where Emb iso (S, P ) denotes the space of isotropic embeddings, cf. (41), (47), or (60). This can equivalently be characterized as the elements in E = Emb lin (|Λ| * S , L * ) which restrict to isotropic embeddings |Λ| * S \ S → L * \ P = M. 10 Let Gr iso S (P ) denote the space of isotropic submanifolds of type S and consider the space of all weighted isotropic submanifolds of type S, In view of (87) and (95) 0). Hence, G iso coincides with the reduced space G ρ for ρ = 0, i.e., Remark 4.9. If α is a contact form for ξ, then isotropic submanifolds N are characterized by ι * N α = 0 and the identification in Remark 4.4 becomes Lemma 4.10. The subset Gr iso S (P ) is a smooth splitting submanifold of Gr S (P ). Proof. This follows from the tubular neighborhood theorem for contact structures near isotropic submanifolds, see [14,Theorem 2.5.8] or [23, Theorem 1]. Since we were not able to locate this statement in the literature, we will sketch a proof in the subsequent paragraph.
Suppose S ∼ = N ⊆ P is an isotropic submanifold, and let E := T N ⊥ /T N denote its conformal symplectic normal bundle, see [14,Definition 2.5.3]. Using the relative Poincaré lemma, one easily constructs a 1-form ε on the total space of E such that (1) ε vanishes along the zero section; (2) i X dε = 0 for every vector X tangent to the zero section; and (3) such that (dε)| N represents the conformal symplectic structure on each fiber of E, cf. the proof of [23, Proposition in Section 4]. Hence α := p * 1 ε + p * 2 θ + dt is a contact form in a neighborhood of the zero section of E ⊕ T * N × R, where p 1 , p 2 , t denote the canonical projections onto the three summands, and θ denotes the canonical 1-form on T * N. Assuming, for simplicity, that the contact structure on P is coorientable near N, the tubular neighborhood theorem for isotropic submanifolds asserts that there exists a contact diffeomorphism ψ between an open neighborhood of the zero section in E ⊕ T * N × R and an open neighborhood of N in P which restricts to the identity along N. Using this diffeomorphism, we obtain a manifold chart for Gr S (P ) centered at N by assigning to a smooth section σ of E ⊕T * N ×R, which is sufficiently C 1 -close to the zero section, the submanifold ψ(σ(N)) in P . As ψ is contact, the part of Gr iso S (P ) covered by this chart corresponds to sections (N) and writing σ = (s, β, f ) accordingly, the latter condition is equivalent to s * ǫ + β + df = 0. Hence, Gr iso S (P ) corresponds to the part of the chart domain contained in the splitting linear subspace 10 Using a volume density µ on S to identify L ∼ = C ∞ (S, L * ) as in Remark 2.2, the subset E iso corresponds to C ∞ (S, M ) ∩ (π L ) −1 (Emb iso (S, P )). If moreover ξ = ker α, then the corresponding diffeomorphism L ∼ = C ∞ (S, P ) × C ∞ (S) provides an identification E iso ∼ = Emb iso (S, P ) × C ∞ (S, R × ).
This shows that Gr iso S (P ) is a splitting smooth submanifold of Gr S (P ). Remark 4.11. Lemma 4.10 implies that Emb iso (S, P ) is a smooth splitting submanifold of Emb(S, P ), because the natural map Emb(S, P ) → Gr S (P ) is a (locally trivial) smooth principal bundle with typical fiber Diff(S). Since π E : E → Emb(S, P ) is a (locally trivial) smooth fiber bundle, this also implies that E iso is a smooth submanifold of E, see (95). Using the isotropic isotopy extension theorem for contact manifolds, see [14,Theorem 2.6.2] for instance, one can show that the group Diff c (P, ξ) acts locally and infinitesimally transitive on E iso . Hence, for ρ = 0, Theorem 3.5 is essentially known.
As mentioned before, one expects that connected components of G iso , endowed with a reduced symplectic form, are symplectomorphic to coadjoint orbits of Diff c (P, ξ) via the restriction of J G L : G → X(P, ξ) * . The following theorem makes this precise. Theorem 4.12. (a) The subset G iso is a smooth splitting submanifold of G. Moreover, the map provided by the action, Diff c (P, ξ) → G iso , g → Ψ G g (N, γ), admits a local smooth right inverse defined in a neighborhood of (N, γ) in G iso . In particular, the group Diff c (P, ξ) acts locally and infinitesimally transitive on G iso , and the Diff c (P, ξ)orbit of (N, γ) is open and closed in G iso . Denoting this orbit by G iso (N,γ) , the map Diff c (P, ξ) → G iso (N,γ) is a smooth principal bundle with structure group Diff c (P, ξ; (N, γ)) in the sense of Frölicher spaces. Hence, G iso (N,γ) = Diff c (P, ξ)/ Diff c (P, ξ; (N, γ)) may be regarded as a homogeneous space in the sense of Frölicher spaces.
(b) The projection q restricts to a smooth principal bundle q iso : E iso → G iso with structure group Diff(S). The restriction of the symplectic form ω E to E iso descends to a (reduced) symplectic form ω G iso on G iso . The Diff(P, ξ)-equivariant injective immersion provided by restriction of J G L from (89), identifies G iso (N,γ) with the coadjoint orbit through J G L (N, γ) of the contact group Diff c (P, ξ), such that where ω KKS denotes the Kostant-Kirillov-Souriau symplectic form on the coadjoint orbit through J G L (N, γ), cf. Remark 4.13 below. Remark 4.13. To avoid discussing differential forms on coadjoint orbits, we consider the Kostant-Kirillov-Souriau form on the coadjoint orbit through J G L (N, γ) as a formal object only. We actually work with its pull back along J G iso L , that is, the well defined smooth 2-form on G iso characterized by where X, Y ∈ X(P, ξ) and (N, γ) ∈ G iso . To motivate this definition, recall that for a Lie algebra g the Kostant-Kirillov-Souriau symplectic form on the coadjoint orbit through λ ∈ g * is (formally) given by , where X, Y ∈ g and ζ g * X denotes the infinitesimal coadjoint action. Since J G iso L is equivariant, we are being lead to (100).
Proof of Theorem 4.12. We have already observed that Gr iso S (P ) is a smooth submanifold of Gr S (P ), see Lemma 4.10. Since the forgetful map π G : G → Gr S (P ) is a smooth fiber bundle, we conclude that G iso is a smooth submanifold of G, see (96). In particular, the map provided by the action Diff c (P, ξ) → G iso , g → Ψ G g (N, γ), is smooth. The remaining assertions in (a) thus follow from Proposition 4.6(b). Note that in the isotropic case the assumption in the latter proposition is trivially satisfied.
In view of E iso = q −1 (G iso ), the smooth principal bundle q : E → G restricts to a smooth principal bundle q iso : E iso → G iso with structure group Diff(S). By Proposition 4.2 the map J G iso L is a Diff(P, ξ)-equivariant injective immersion. In view of (the trivial inclusion in) Equation (55), we have ω E (ζ E X , ζ E Z ) = 0 for all X ∈ X(P, ξ) and Z ∈ X(S). Since Diff c (P, ξ) acts infinitesimally transitive on E iso , the 1-form ω E (−, ζ E Z ), thus, vanishes when pulled back to E iso . Hence, the restriction of ω E to E iso is vertical. We conclude that there exists a unique 2-form ω G iso on G iso such that (q iso ) * ω G iso coincides with the pull back of ω E to E iso . Clearly, ω G iso is closed. The 2-form ω G iso is (weakly) non-degenerate in view of (the non-trivial inclusion in) Equation (55). From (100), (29), (26) and the equivariance of q we immediately obtain (q iso ) * (J G iso L ) * ω KKS = (q iso ) * ω G iso , whence (99). The remaining assertions are now obvious.
Remark 4.14. We expect that the isotropy group Diff c (P, ξ; (N, γ)) in Theorem 4.12(a) is a closed Lie subgroup in Diff c (P, ξ). If this is the case then G iso (N,γ) may be regarded as a homogeneous space in the category of smooth manifolds.
Example 4.15. If S is the circle S 1 and P is a 3-dimensional contact manifold, then the weighted non-linear Grassmannian G becomes the manifold of weighted (unparametrized) knots in P , and G iso is the (symplectic) manifold of weighted Legendrian knots in P . By Theorem 4.12, its connected components can be identified with coadjoint orbits of the identity component of the contact group.

4.4.
Weighted contact non-linear Grassmannians. Let us now consider a 1-form density ρ ∈ Ω 1 (S, |Λ| S ) of contact type, i.e., ker ρ ⊆ T S is assumed to be a contact hyperplane distribution. Then the reduced space G ρ , see (91), consists of weighted contact submanifolds. More precisely, according to Lemma 4.3 we have where Gr contact (S,ker ρ) (P, ξ) ⊆ Gr S (P ) denotes the subset of contact submanifolds which are of type (S, ker ρ). In contrast to the isotropic case, see (97), the inclusion (101) is strict.
The maps in (87) restrict to a Diff(P, ξ)-equivariant commutative diagram where Emb contact (S,ker ρ) (P, ξ) ⊆ Emb(S, P ) denotes the subset of contact embeddings inducing the contact structure ker ρ on S. Since ρ is nowhere vanishing, the map in (d) is a bijection. This map is smooth because the inclusion Emb contact (S,ker ρ) (P, ξ) ⊆ Emb(S, P ) is initial. To see that its inverse is smooth too, we fix a vector bundle homomorphism σ : T S/ ker ρ → T S splitting the canonical projection T S → T S/ ker ρ. Let W denote the set of embeddings ϕ ∈ Emb(S, P ) for which the composition is an isomorphism of line bundles over S. Clearly, W is an open neighborhood of Emb contact (S,ker ρ) (P, ξ) in Emb(S, P ). We obtain a smooth map s : W × Γ ∞ (T S/ ker ρ) * ⊗ |Λ| S → L, characterized by π L (s(ϕ, β)) = ϕ and J L R (s(ϕ, β)) • σ = β, for all ϕ ∈ W and β ∈ Γ ∞ (T S/ ker ρ) * ⊗ |Λ| S . Its restriction provides the smooth inverse for the map in (d).
Remark 4.17. If α is a contact form for ξ, then contact submanifolds N are characterized by the fact that ι * N α is a contact form on N, and the identification in Remark 4.4 becomes G ρ = (N, ν) : N ∈ Gr contact S (P ), ν ∈ Γ ∞ (|Λ| N \ N), and (N, ι * N α ⊗ ν) ∼ = (S, ρ) . If (N, ν) ∈ G ρ then any other weight on N allowed in G ρ must be of the form for a contact diffeomorphism f ∈ Diff(N, ker ι * N α). Thus, unlike the isotropic case (98), in the contact case not all weights on a contact submanifold N ∈ Gr contact S (P ) are allowed in G ρ , i.e., the inclusion in (101) is strict.
Remark 4.18. Let ρ ∈ Ω 1 (S, |Λ| S ) be a contact 1-form density. Since G ρ may not be a manifold, we refrain from considering the Kostant-Kirillov-Souriau form on G ρ . However, formally pulling back the Kostant-Kirillov-Souriau form along J E ρ L : E ρ → X(P, ξ) * , we obtain a well defined smooth 2-form ( , where Φ ∈ E ρ and X, Y ∈ X(P, ξ), cf. Remark 4.13 and Theorem 3.5. Proceeding exactly as in the proof of Theorem 4.12, we see that this coincides with ω E ρ , the pull back of the symplectic form ω E to E ρ , i.e., The discussion in the next example shows that the situation is as nice as one could wish for 1-dimensional S. Subsequently, we will see that the situation is considerably more delicate in general, see Proposition 4.20.
Example 4.19. Let us specialize to the circle, S = S 1 . In this case, any contact 1form density ρ ∈ Ω 1 (S, |Λ| S ) gives rise to an orientation and a Riemannian metric on S. We write |ρ| for the induced volume density on S, and denote the total volume by vol(ρ) := S |ρ|. Using parametrization by arc length it is easy to see that two contact 1-form densities lie in the same Diff(S)-orbit iff they have the same total volume. In particular, the Diff(S)-orbits through contact 1-form densities are closed submanifolds in Ω 1 (S, |Λ| S ). Moreover, parametrization by arc length provides local smooth sections for the Diff(S)-action on these orbits. In particular, the assumption in Lemma 4.7 is satisfied in this case.
Suppose (P, ξ) is a contact manifold and let ρ ∈ Ω 1 (S, |Λ| S ) be a contact 1-form density on S = S 1 . Using (103) we conclude that G ρ is a closed submanifold of G. Parametrization by arc length provides local smooth sections of E ρ → G ρ and the latter is a locally trivial smooth principal bundle. Note that the structure group Diff(S, ρ) ∼ = SO(1) is a closed Lie subgroup of Diff(S). By Proposition 4.6, the Diff c (P, ξ)-action on G ρ admits local smooth sections. Moreover, its orbits are open and closed subsets in G ρ which may be identified with coadjoint orbits of the contact group via the restriction of J G L . The symplectic form on E gives rise to a reduced symplectic form on G ρ which coincides with the pull back of the Kostant-Kirillov-Souriau symplectic form via J G L as in Theorem 4.12(b). If P is 3-dimensional, then G ρ is a (symplectic) manifold of weighted transverse knots.
A slightly more explicit description can be given if the contact structure is admits a contact form, ξ = ker α. Then, via the identification in Remark 4.4, we have , for every contact 1-form density ρ.
The following result shows that the trace topology on G ρ induced from G is not the appropriate topology for general contact ρ.
Proposition 4.20. There exist a compact contact manifold (P, ξ), a compact manifold S, and a contact 1-form density ρ ∈ Ω 1 (S, |Λ| S ) such that the continuous bijection induced by the natural inclusion is not a homeomorphism with respect to the quotient topologies. In particular, the continuous bijection E ρ / Diff(S, ρ) → G ρ induced by q is not a homeomorphism where the right hand side is equipped with the trace topology induced from G. Moreover, for (N, γ) ∈ G ρ the map provided by the action, Diff(P, ξ) → G ρ , g → Ψ G g (N, γ), does not admit a continuous local (with respect to the trace topology induced from G) right inverse defined in a neighborhood of (N, γ).
The following lemma will be crucial in the proof of Proposition 4.20.
Lemma 4.21. There exists a compact contact manifold (S, α) and a sequence of diffeomorphisms f n ∈ Diff(S) with the following properties: (a) f * n α → α with respect to the C ∞ -topology. (b) There does not exist a sequence of diffeomorphisms g n ∈ Diff(S) such that g * n α = f * n α for all n and g n → id S with respect to the C 0 -topology.
Proof. Let (M, ω) be a connected compact symplectic manifold with integral symplectic form. Choose a sequence of non-empty open subsets U, U 1 , U 2 , U 3 , . . . of M such that their closures are mutually disjoint. Choose points x ∈ U and x n ∈ U n . Assume that the sequences of closuresŪ n only accumulates at a single point. Choose Hamiltonian diffeomorphisms h n ∈ Ham(M, ω) such that for each n we have (i) h n (y) = y for all y ∈ i =nŪ i , and (ii) h n (x) = x n . Shrinking U n , we may moreover assume (iii) h −1 n (U n ) ⊆ U. For each n let λ n be a compactly supported smooth function on U n such that λ n is constant and strictly positive in a neighborhood of x n . Let λ : M → R denote the function which coincides with λ n on U n and vanishes outside n U n . Multiplying λ n with a sufficiently fast decreasing sequence of constants, we may assume that the following hold true: (iv) λ is smooth on M, and (v) h * n λ → λ with respect to the C ∞ topology on M.
By construction, we have: (vi) λ is constant and strictly positive on a neighborhood of x n , for each n, and (vii) λ vanishes on U.
Let p : S → M be the circle bundle with Chern class [ω] and letα ∈ Ω 1 (S) be a principal connection 1-form with curvature ω. Hence,α is a contact form on S. It is well known that Hamiltonian diffeomorphisms on M can be lifted to strict contact diffeomorphisms on S. Hence, there exist diffeomorphisms f n ∈ Diff(S) such that f * nα = α and p • f n = h n • p. We consider the contact form α := e −p * λα on S. From (v) we immediately obtain f * n α → α, whence (a). To see (b), let E denote the Reeb vector field of α. From (vii) we see that α coincides with the principal connectionα on p −1 (U). Over p −1 (U), the Reeb vector field E thus coincides with the fundamental vector field of the principal circle action. For each y ∈ p −1 (U) we thus have Fl E t (y) = y ⇔ t ∈ 2πZ. Hence, if g ∈ Diff(S) is sufficiently close to the identity with respect to the C 0 -topology, then Note that g * E is the Reeb vector field of g * α, and f * n E is the Reeb vector field of f * n α. For each n there exists a constant 0 < c n < 1 such that f * n α coincides with c n α on a neighborhood of p −1 (x), see (ii) and (vi). Hence, f * n E coincides with c −1 n E on a neighborhood of p −1 (x). In particular, Comparing (105) and (106) and using c n = 1, we conclude g * E = f * n E and thus g * α = f * n α. This shows (b). Proof of Proposition 4.20. We consider a closed manifold S of dimension 2k+1, a contact form α on S, and diffeomorphisms f n ∈ Diff(S) as in Lemma 4.21. Using Gray's stability result [14, Theorem 2.2.2], we may w.l.o.g. assume that each f n is a contact diffeomorphism. Hence, there exist smooth functions λ n on S such that f * n α = λ n α. Since f * n α → α, we have λ n → 1, as n → ∞. In particular, we may assume λ n > 0. We let µ := |α ∧ (dα) k | denote the volume density associated with the volume form α ∧ (dα) k . Note that f * n µ = λ k+1 n µ. Moreover, we put ρ := α ⊗ µ ∈ Ω 1 (S, |Λ| S ). We consider the manifold P := S equipped with the contact structure ξ := ker(α). Using the volume density µ on S and the contact form α on P , we may identify E = Emb(S, P ) × C ∞ (S, R × ), see Remark 2.2. Using this identification we define a sequence Φ n ∈ E by Φ n := (id S , λ k+2 n ). Clearly, Φ n converges to Φ := (id S , 1) ∈ E. Using (47) we In particular, we have Φ ∈ E ρ , Φ n ∈ (J E R ) −1 (ρ · Diff(S)) and Φ n → Φ, as n → ∞. We will now show that the corresponding sequence in E ρ / Diff(S, ρ) does not converge, cf. (104). Suppose, by contradiction, there exists a sequence of diffeomorphisms g n ∈ Diff(S) such that ψ E gn (Φ n ) ∈ E ρ and ψ E gn (Φ n ) converges in E ρ / Diff(S, ρ). W.l.o.g. we may moreover assume that ψ E gn (Φ n ) converges in E ρ . In particular, g n converges to a diffeomorphism g ∈ Diff(S). Using the Diff(S) equivariance of J E R , the relation ρ = In particular, letting n → ∞, we obtain g ∈ Diff(S, ρ). Replacing g n with g n • g −1 we may w.l.o.g. assume that g n → id S . Combining (107) and (108) we see that g n is a contact diffeomorphism. Hence, there exist smooth functionsλ n with g * n α =λ n α. We obtain λ k+2 n ρ = g * n ρ =λ k+2 n ρ and thusλ k+2 n = λ k+2 n . Since g n converges to the identity, we may assumeλ n > 0. Hence,λ n = λ n and thus g * n α = f * n α. This contradicts the choice of f n , see Lemma 4.21(b). Hence, the sequence in E ρ / Diff(S, ρ) corresponding to Φ n does not converge.
This shows that the continuous bijection (104) is not a homeomorphism. The remaining statements follow immediately from the fact that the projection q : E → G admits local (smooth) sections.

Comparison with the EPDiff dual pair
A pair of moment maps has been introduced by D. D. Holm and J. E. Marsden [17] in relation to the EPDiff equations describing geodesics on the group of all diffeomorphisms. The left moment map provides singular solutions of these equations, whereas the right moment map provides a constant of motion for the collective dynamics of these singular solutions. It has been shown in [9] that the pair of moment maps, when restricted to an appropriate open subset, do indeed form a symplectic dual pair. In this section we relate the EPDiff dual pair of a manifold with the EPContact dual pair of its projectivized cotangent bundle.
5.1. The dual pair for the EPDiff equation. The (regular) cotangent bundle to the space of smooth maps from a closed manifold S into a manifold Q can be equipped with the canonical symplectic structure. Recall that the tangent space at η ∈ C ∞ (S, Q) is T η C ∞ (S, Q) = Γ ∞ (η * T Q). Using the canonical pairing, we regard the space of 1-form densities along η, as the regular cotangent space at η. In this way we identify the space of smooth fiberwise linear maps from |Λ| * S to T * Q with the regular cotangent bundle: Via this identification, the canonical 1-form on T * C ∞ (S, Q) reg can be written in the form where A is a tangent vector at Φ ∈ T * C ∞ (S, Q) reg , i.e., and θ T * Q denotes the canonical 1-form on T * Q. As before, the integrand θ T * Q (A) is a fiberwise linear function on the total space of |Λ| * S , which may be regarded as a section on |Λ| S and integrated over S. The differential dθ T * C ∞ (S,Q)reg is the canonical (weakly non-degenerate) symplectic form on T * C ∞ (S, Q) reg .
The cotangent lifted actions of the groups Diff(Q) and Diff(S) on the manifold C ∞ (S, Q) preserve the canonical 1-form θ T * C ∞ (S,Q)reg . In particular, these actions are Hamiltonian with equivariant moment maps J Sing : T * C ∞ (S, Q) reg → X(Q) * , and J S : T * C ∞ (S, Q) reg → X(S) * , respectively, where Φ ∈ T * C ∞ (S, Q) reg . Here ζ denote the fundamental vector fields on T * C ∞ (S, Q) reg corresponding to the (infinitesimal) action of Y ∈ X(Q) and Z ∈ X(S), respectively. More explicitly, using the identification (109), these cotangent moment maps are and where η ∈ C ∞ (S, Q) and Φ ∈ T * η C ∞ (S, Q) reg = Γ ∞ (|Λ| S ⊗ η * T * Q). In particular, the second formula shows that J S takes values in Ω 1 (S, |Λ| S ) ⊆ X(S) * . More precisely, J S (Φ) is the 1-form density on S corresponding to the 1-homogeneous vertical 1-form Φ * θ T * Q on the total space of |Λ| * S where we regard Φ : |Λ| * S → T * Q, cf. (110). We denote by T * C ∞ (S, Q) × reg the open subset of (110) that corresponds to the space C ∞ lin, inj (|Λ| * S , T * Q) of smooth maps that are linear and injective on fibers. Restricting further the actions and moment maps to the open subset T * Emb(S, Q) × reg , we obtain the EPDiff symplectic dual pair [9]: The left moment map J Sing provides the formula for singular solutions of the EPDiff equations, whereas the right moment map J S provides a Noether conserved quantity for the (collective) Hamiltonian dynamics of these singular solutions in terms of the canonical variable Φ ∈ T * Emb(S, Q) × reg , see [17]. Remark 5.1. Fixing a volume density µ on S, we obtain identifications T * C ∞ (S, Q) reg ∼ = C ∞ (S, T * Q) and T * C ∞ (S, Q) × reg ∼ = C ∞ (S, T * Q \ Q), cf. (110), as well as Ω 1 (S, |Λ| S ) ∼ = Ω 1 (S). Using these identifications, the moment maps may be written in the form where φ ∈ C ∞ (S, T * Q) and Y ∈ X(Q), cf. [9, Section 5].

5.2.
The projectivized cotangent bundle. We will compare the EPDiff dual pair described in the preceding paragraph with the EPContact dual pair associated with the projectivized cotangent bundle. Recall that the projectivized cotangent bundle, admits a canonical contact structure [1, Appendix 4] given by where ℓ ∈ P and β ∈ T * Q is any non-zero element of ℓ. As the natural action of Diff(Q) on P preserves the contact structure ξ, we obtain an injective group homomorphism Diff(Q) → Diff(P, ξ).
The line bundle L * , see Section 2.1, associated with the projectivized cotangent bundle is naturally isomorphic to the canonical line bundle over P : Indeed, the vector bundle homomorphism χ : γ → T * P over the identity on P , given by χ(ℓ, β) := β • T ℓ p, induces an isomorphism of line bundles, χ : γ → L * . Furthermore, where pr 2 : γ → T * Q denotes the canonical projection, i.e., the blow-up of the zero section in T * Q. We consider the map κ : L * → T * Q, κ := pr 2 •χ −1 . One readily checks: The map κ is a vector bundle homomorphism over the bundle projection p, / / Q which has the following properties: (a) κ is equivariant over the homomorphism Diff(Q) → Diff(P, ξ).
Composition with κ provides a map, cf. (110), reg which fits into the following diagram: Here i * denotes the dual of the Lie algebra homomorphism i : X(Q) → X(P, ξ) corresponding to the homomorphism of groups Diff(Q) → Diff(P, ξ). Clearly, i * is equivariant over the homomorphism Diff(Q) → Diff(P, ξ). Note that via (8) and κ, the Lie algebra X(P, ξ) = C ∞ lin (L * ) may be regarded as the space of homogeneous functions on T * Q \ Q, while the image of i consists of those which extend to fiberwise linear functions on T * Q.
Proposition 5.3. The diagram (117) commutes. The map κ * is equivariant over the homomorphism Diff(Q) → Diff(P, ξ) and also Diff(S)-equivariant. It restricts to a symplectic diffeomorphism from M ⊆ L onto T * C ∞ (S, Q) × reg . Proof. The map κ * is equivariant over the homomorphism Diff(Q) → Diff(P, ξ) since κ has the same property, see Lemma 5.2(a). Clearly, κ * is Diff(S)-equivariant too. Hence, the fundamental vector fields are κ * -related, that is, for Y ∈ X(Q) and Z ∈ X(S). Using Lemma 5.2(c), (22), and (111), we obtain Combining the latter with the first equation in (118), we see that the square on the left hand side in (117) commutes, cf. (112) and (29). Combining (119) with the second equation in (118), we see that the square on the right hand side in (117) commutes, cf.
(113) and (37). As κ restricts to a diffeomorphism from L * \ P onto T * Q \ Q, the map κ * restricts to a diffeomorphism from M onto T * C ∞ (S, Q) × reg which is symplectic in view of (119). 5.3. Coadjoint orbits of the diffeomorphism group. The first line in (117) becomes a dual pair when restricted to E = Emb lin (|Λ| * S , L * ). The second line has to be restricted to T * Emb(S, Q) × reg to become a dual pair. The latter is a proper open subset of the image κ * (E). To make this more precise, note that is a Diff(S) invariant open subset of E. Since p : P → Q is Diff(Q) equivariant, E Q is invariant under Diff(Q) too. According to Proposition 5.3, the map κ * restricts to a Diff(Q) and Diff(S) equivariant symplectomorphism which makes the following diagram commute: Here J In view of (89) we have the explicit formula where (N, γ) ∈ G Q and Y ∈ X(Q). On the right hand side i(Y ) is regarded as a section of L, see (8), restricted to N and contracted with γ ∈ Γ ∞ (|Λ| N ⊗ L| * N ) to produce a density on N. Recall from Theorem 4.12 that G iso is a closed submanifold of G. Hence, G iso Q := G Q ∩ G iso is a closed splitting submanifold of G Q . Consequently, E iso Q := E Q ∩ E iso = q −1 (G iso Q ) is a closed splitting submanifold of E Q . The projection q Q : E Q → G Q restricts to a smooth principal bundle q iso Q : E iso Q → G iso Q with structure group Diff(S). Via κ * the manifold G iso Q = E iso Q / Diff(S) = {(N, γ) ∈ G : N ∈ Gr iso S (P ), p| N ∈ Emb(N, Q)} identifies with J −1 S (0)/ Diff(S), the reduced space at zero of the right Diff(S)-action on Emb(S, Q) × reg . According to [9,Proposition 5.5], the group Diff c (Q) acts locally transitive on E iso Q and, thus, on G iso Q too. In particular, the Diff c (Q)-orbit (G iso Q ) (N,γ) through (N, γ) ∈ G iso Q is open and closed in G iso Q . From Theorem 4.12 we thus obtain: Corollary 5.5. The projection q restricts to a smooth principal bundle q iso Q : E iso Q → G iso Q with structure group Diff(S). The restriction of the symplectic form ω E to E iso Q descends to a (reduced) symplectic form ω G iso Q on G iso Q . The Diff(Q)-equivariant injective immersion Here (N, γ) ∈ G iso Q and ω KKS denotes the Kostant-Kirillov-Souriau symplectic form on the coadjoint orbit through J G iso Q L (N, γ) ∈ X(Q) * . Remark 5.6. In the Legendrian case one has a description of the coadjoint orbit that does not use contact geometry. A transverse Legendrian submanifold N ⊆ PT * Q projects to a codimension one submanifold N 0 = p(N) ⊆ Q, while N 0 has a unique Legendrian lift to the projectivized cotangent bundle N 0 ∋ x → ann(T x N 0 ) ∈ PT * x Q. Moreover, the line bundle L = T (PT * Q)/ξ restricted to N is canonically isomorphic to the pull back of the normal line bundle, p| * N T N ⊥ 0 , since the contact hyperplane at y ∈ N is ξ y = (T y p) −1 (T x N 0 ) for x = p(y) ∈ N 0 . Hence, the coadjoint orbit of Diff c (Q) described above can be seen as (N 0 , γ 0 ) N 0 ⊆ Q has codimension one and γ 0 ∈ Γ ∞ (|Λ| N 0 ⊗ (T N ⊥ 0 ) * ) is nowhere vanishing , embedded into X(Q) * via Y → N 0 γ 0 (Y | N 0 mod T N 0 ). Note that we have a canonical identification |Λ| N 0 ⊗ (T N ⊥ 0 ) * = |Λ| Q ⊗ O ⊥ N 0 , where O ⊥ N 0 denotes the orientation bundle of the normal bundle T N ⊥ 0 . Hence, disregarding the latter orientation bundle, we may regard points in this coadjoint orbit as codimension one submanifolds N 0 in Q, weighted by a volume density of the ambient space Q along N 0 .
Appendix A. Comparison with the dual pair for the Euler equation A dual pair of moment maps associated to the Euler equations of an ideal fluid has been described by J. E. Marsden and A. Weinstein [25]; it justifies the existence of Clebsch canonical variables for ideal fluid motion and also explains the Hamiltonian structure of point vortex solutions in a geometric way. It has been shown in [9] that the pair of moment maps restricted to the open subset of embeddings does indeed form a symplectic dual pair. In this section we relate this dual pair to the EPContact dual pair, see (52).
A.1. The dual pair for the Euler equation. The space of smooth maps from a closed manifold S into a symplectic manifold (M, ω) can be equipped with a symplectic structure once a volume density µ ∈ Γ ∞ (|Λ| S \ S) has been fixed. Recall that the space of maps C ∞ (S, M) is a Fréchet manifold in a natural way. The symplectic form on C ∞ (S, M) can be described by Here the first arrow is given by pull back of θ; the second identification is via the volume density µ; the third is the inclusion of smooth sections into distributional sections of T * S ⊗ |Λ| S ; and the fourth map is the dual of the canonical inclusion X(S, µ) ⊆ X(S). We will write this as where φ ∈ C ∞ (S, M) and X ∈ X(S, µ). where φ ∈ C ∞ (S, M) and h ∈ C ∞ (M). This moment map is in fact equivariant with respect to the natural action of the full symplectic group, Diff(M, ω). 11 Restricting the actions and moment maps to the open subset Emb(S, M) ⊆ C ∞ (S, M) of embeddings, we obtain a symplectic dual pair, see [9] and [11,Section 4.2]: A.2. Comparison with the EPContact dual pair. We will now specialize to the symplectization of a contact manifold (P, ξ), that is, we consider M = L * \ P equipped with the symplectic form ω M obtained by restricting the canonical 2-form ω L * on the total space of L * , cf. Section 2.1. We will relate the dual pair for the Euler equation whereμ ∈ Γ ∞ (|Λ| * S ) denotes the section dual to µ. Let j : X(P, ξ) → C ∞ (M), j(X) := h M X , denote the Lie algebra homomorphism provided by (8), see also (14). In view of (9), j is equivariant over the homomorphism Diff(P, ξ) → Diff(M, ω M ). Note that the composition of j with the action C ∞ (M) → X ham (M, ω M ) yields a Lie algebra homomorphism X(P, ξ) → X ham (M, ω M ) ⊆ X(M, ω M ) corresponding to the homomorphism of groups Diff(P, ξ) → Diff(M, ω M ), see (12). Finally, let i : X(S, µ) → X(S) denote the natural inclusion. Clearly, i is equivariant over the inclusion Diff(S, µ) ⊆ Diff(S).
These maps give rise to the following diagram: Here i * and j * denote the (equivariant) maps dual to the homomorphisms i and j, respectively. where π M : M → P denotes the restriction of the canonical projection π L * : L * → P .
Proof. Clearly, ι µ is an equivariant diffeomorphism, see Remark 2.2. It is symplectic in view of (43) and (124). The right hand side of the diagram commutes in view of (47) and (125). The left hand side of the diagram commutes in view of (46) and (126).
The first line in (129) becomes a dual pair only when restricted to E, while the second line needs to be restricted to Emb(S, M) to become a dual pair. Note that the image ι µ (E) is an open subset (strict, in general) of Emb(S, M).