A note on Mitsumatsu's construction of a leafwise symplectic foliation

TL;DR

This paper explores the convergence of contact structures to leafwise symplectic foliations, building on Mitsumatsu's construction, and introduces new methods involving confoliations and 2-forms related to twisted Jacobi structures.

Contribution

It improves Mitsumatsu's results by describing convergence via confoliations with specific 2-forms and constructs new leafwise symplectic foliations on product manifolds.

Findings

Convergence of contact structures to leafwise symplectic foliations is characterized using confoliations and 2-forms.

New leafwise symplectic foliations are constructed on the product of the 4-sphere and the circle.

The work relates to twisted Jacobi structures and weak domination concepts.

Abstract

Mitsumatsu constructed leafwise symplectic structures of certain codimension one foliations of the 5-sphere. This inspired the present author to improve his result on convergence of contact structure to foliation. We describe convergence of contact strcture to leafwise symplectic foliation by means of confoliation equipped with a certain 2-form. Such a 2-form appears in the works of Nunes da Costa and Petalidou on twisted Jacobi structures and partially relates to weak domination due to Massot, Niederkruger and Wendl. We also construct leafwise symplectic foliations on the product of the 4-sphere and the circle.