Foliations associated to regular Poisson structures (Q2764517)

This work addresses to the problem of characterizing regular foliations that arise from Poisson structures. Recall that a Poisson structure \(P\) on a manifold \(M\) is called regular if the characteristic foliation \(\mathcal F\) of \(P\) into symplectic leaves (which is a priori a generalized foliation) is a regular foliation, i.e. has constant rank, say, \(2k\). Then, by definition, the leaves of \(\mathcal F\) are symplectic manifolds and the Poisson structure may be viewed as a \textit{tangential symplectic form} on \(\mathcal F\). NEWLINENEWLINENEWLINEThe question, whether a given rank-\(2k\) foliation \(\mathcal F\) on \(M\) comes from a regular Poisson structure is difficult even in the case \(2k=dim(M)\), especially for compact \(M\). The paper presents examples of foliated manifolds with noncompact leaves which do not carry any leafwise symplectic form, although leafwise almost symplectic forms do exist. NEWLINENEWLINENEWLINEThis is in contrast with the result of \textit{M. Gromov} [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 707-734 (1969; Zbl 0197.20404)] which states that on an open manifold \(M\) any almost symplectic form is homotopic among almost symplectic forms to a symplectic form.