On Legendrian foliations in contact manifold I: Singularities and neighborhood theorems

TL;DR

This paper investigates the structure of singularities in Legendrian foliations within contact manifolds and establishes local existence and uniqueness results for contact structures near such foliations, extending classical 3D contact geometry to higher dimensions.

Contribution

It provides a structure theorem for singularities of characteristic foliations and proves existence and uniqueness of germs of contact structures near Legendrian foliations in higher dimensions.

Findings

Structure theorem for singularities of characteristic foliations.

Existence of germs of contact structures near Legendrian foliations.

Uniqueness of these contact structure germs.

Abstract

In this note we study several aspects of coisotropic submanifolds of a contact manifold. In particular we give a structure theorem for the singularity of the characteristic foliation of a coisotropic submanifold. Moreover we establish the existence and uniqueness results of germs of contact structures near Legendrian foliations, which is a special case of coisotropic submanifold. This note can be thought of as an attempt to generalize the study of surfaces in three-dimensional contact geometry to higher dimensions.