Existence of Engel structures

TL;DR

This paper introduces a method to construct Engel structures on 4-manifolds using handle decompositions, proving their existence on all parallelizable 4-manifolds and exploring connected sums of Engel manifolds.

Contribution

It provides a new construction technique for Engel structures on 4-manifolds and demonstrates their existence on all parallelizable cases, expanding understanding of Engel geometry.

Findings

All parallelizable 4-manifolds admit Engel structures.

A new construction method based on handle decompositions.

Existence of Engel structures on connected sums of certain Engel manifolds.

Abstract

We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, there is an Engel structure on M_1#M_2#(S^2xS^2) which is closely related to the original Engel structures.