An obstruction for existence of codimension two contact embeddings in a Darboux chart

TL;DR

This paper establishes that the first Chern class acts as an obstruction to embedding certain contact manifolds into Darboux charts, providing conditions under which such embeddings are possible.

Contribution

It proves the vanishing of the first Chern class as an obstruction and characterizes when a contact 3-manifold can embed into 5-dimensional Euclidean space.

Findings

First Chern class vanishes for certain contact submanifolds.

Vanishing Chern class is necessary for codimension 2 contact embeddings.

Contact 3-manifolds with zero first Chern class can embed into 5D Euclidean space.

Abstract

We prove the vanishing of the first Chern class of a codimension 2 closed contact submanifold of a cooriented contact manifold with trivial integral 2-dimensional cohomology group. Hence the first Chern class is an obstruction for the existence of codimension 2 contact embeddings in a Darboux chart. For the existence of such an embedding, we prove that a closed cooriented contact 3-manifold can be a contact submanifold of the 5-dimensional Euclidean space for a certain contact structure, if its first Chern class vanishes.