Iso-contact embeddings of manifolds in co-dimension $2$

TL;DR

This paper investigates conditions under which closed contact manifolds can be iso-contact embedded in higher-dimensional contact manifolds, providing new embedding criteria and applications to 3- and 5-dimensional cases.

Contribution

It establishes a criterion for iso-contact embeddings based on contact embeddings with trivial normal bundle and homotopic induced structures, and applies this to classify embeddings of certain 3- and 5-manifolds.

Findings

A closed contact manifold embeds in a higher-dimensional contact manifold if it contact embeds with trivial normal bundle and homotopic almost-contact structures.

A closed contact 3-manifold with no 2-torsion in its cohomology embeds in the standard 5-sphere iff its first Chern class is zero.

Discussion of iso-contact embeddings of simply connected contact 5-manifolds.

Abstract

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$ provided $M$ contact embeds in $(N, \xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $\xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.