Abstract
AbstractGeneralizing 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.
Highlights
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, 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
Summary
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, 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 Hamiltonian and admits an equivariant moment. This moment map identifies (unions of) connected components of the symplectization M with certain coadjoint orbits of the contact group
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