Non-existence of certain singularities in Legendrian foliations

TL;DR

This paper proves that the singular set of Legendrian foliations is a compact submanifold with components of codimension at most two, leading to uniqueness results for contact structures near certain submanifolds.

Contribution

It establishes the non-existence of higher codimension singularities in Legendrian foliations and shows contact structures are uniquely determined by characteristic foliations near coisotropic submanifolds.

Findings

Singular locus of Legendrian foliation is a compact submanifold.

Connected components of the singular locus have codimension at most two.

Contact structure near a coisotropic submanifold is uniquely determined by its Legendrian foliation.

Abstract

In this paper we show that the singular locus of a Legendrian foliation as defined in [Hua13] is a compact submanifold whose connected components are of codimension at most two. As a consequence, given any closed $(n+1)$-dimensional coisotropic submanifold $W$ in a contact $(2n+1)$-manifold, the contact structure in a sufficiently small neighborhood of $W$ is uniquely determined by the characteristic (Legendrian) foliation.