Codimension one symplectic foliations and regular Poisson structures

Abstract

Dedicated to the memory of Paulette Libermann whose cosymplectic manifoldsplay a fundamental role in this paper.Abstract. In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we will see in [GMP].

Deformations of log-symplectic structures

“…The cosymplectic structure determines a pair (π Z , V ), where π Z is a Poisson structure on Z whose symplectic foliation is given by the kernel of θ endowed with the pullback of η, and V is the vector field such that ι V η = 0 and θ(V ) = 1. Moreover, this gives a one-to-one correspondence between cosymplectic structures and regular corank-1 Poisson structures endowed with a transverse Poisson vector field (see, for example, [6]). In this paper, we show that, up to diffeomorphism, there are only two ways to deform log-symplectic structures on compact manifolds.…”

Integrable systems and group actions

“…We have recently studied the symplectic and Poisson geometry of these manifolds in [24,25] together with Guillemin and Pires.…”

Weakly Hamiltonian actions

“…By our assumptions we can find u ∈ g such that according to (4) for all x ∈ M , ζ u,v,x (s) is a polynomial with linear coefficient H [v,u] (x) − c u,v , which is a non-constant function on x. Hence if we have a periodic orbit of X v , by compactness ζ u,v,x (s) must be constant and therefore must be in the zero subset of H [v,u] (x)−c u,v (and also in the zero set of functions corresponding to the coefficients of higher order in (4)).…”

“…If (M, π) is a Poisson manifold supporting only trivial Casimirs (constants), then the symplectic splitting theorem holds word by word. As examples of Poisson manifolds with trivial Casimirs we may consider, for instance, the Reeb foliation of S 3 with leafwise area form, compact cosymplectic manifolds with noncompact leaves endowed with natural Poisson structures [4], and other Poisson manifolds constructed out of them via products, surgeries, etc.…”