Foliations Associated to Regular Poisson Structures

Letters in Mathematical Physics, 2013

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed.

Journal of Mathematical Physics, 2021

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $\pi$ is defined on the tubular neighbourhood of the singular locus $Z_{\omega}$ of the 2-form $\omega$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_{\omega}$. We compute its smooth Poisson cohomology, which depends on the modular vector field and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure $\pi$ and the overtwisted contact structure associated to a near-symplectic 4-manifold.

Journal of Symplectic Geometry, 2004

A new technique to construct Poisson manifolds inspired both in surgery ideas used to define Poisson structures on 3-manifolds and in Gompf's surgery construction for symplectic manifolds, is presented. As an application of these ideas it is proved that for all n ≥ d ≥ 4, d even, any finitely presentable group is the fundamental group of a n-dimensional orientable closed Poisson manifold of constant rank d. The unimodularity of some of the Poisson structures constructed is studied.

Pacific Journal of Mathematics, 1990

We present examples of foliated compact nilmanifolds, whose foliations are neither simple nor given by suspensions, admitting various foliated geometric structures. For example, we construct foliations which are: 1. transversely symplectic but not transversely Kahler, 2. transversely symplectic but not transversely holomorphic,

In this thesis we consider various structures on foliations. In Chapter I we look at PL and topological foliations and note that not every topological foliation can he made PL. We show that every proper leaf has a microbundle normal to the foliation with holonomy structure group. For transverse foliations the fibres can be chosen not only normal to the leaves of the foliation containing the base leaf, but contained in the leaves of the other foliation. Thus normal microbundles are unique up to isotopy. We also look into the relationship between the holonony group and the foliated neighbourhood of a leaf. In Chapter II we study differentiable structures on foliations, showing that differentiability conditions are meaningful in a topological sense. We do this by constructing an example of a Cʳ foliation which is not homeomorphic to any Cʳ⁺¹ foliation, r>0. (The example is the suspension of a diffeomorphism of a two-manifold.) Using results from Chapter I we also show that the folia...

Arxiv preprint math/0406611, 2004

We give an intrinsic proof that Vorobjev's first approximation of a Poisson manifold near a symplectic leaf is a Poisson manifold. We also show that Conn's linearization results cannot be extended in Vorobjev's setting

2014

Abstract. In this paper we study some algebraic properties of harmonic forms on Poisson manifolds. It is well known that in the classical case (on Riemannian manifolds) the product of harmonic forms is not harmonic. Here we describe the algebraic and analytical mechanisms explaining this fact. We also obtain a condition under which the product of de Rham cohomology classes, which includes harmonic representatives, can be represented by a harmonic form.

Georgian Mathematical Journal, 1995

The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.

2005

We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).

Balkan Journal of Geometry and Its Applications, 2014

The goal of the paper is to present in a unitary way some conditions that a foliation be Riemannian, involving general conditions on higher order normal bundles (jets or accelerations).