Isotopy Classification of Engel Structures on Circle Bundles

TL;DR

This paper introduces an invariant for classifying Engel structures on circle bundles over three-manifolds, focusing on those with characteristic line fields tangent to the fibers, and provides an isotopy classification.

Contribution

It defines a new isotopy invariant for Engel structures on circle bundles and applies it to classify structures with tangent characteristic line fields.

Findings

Introduces an isotopy invariant for Engel structures.

Provides a classification for structures with tangent characteristic line fields.

Enhances understanding of Engel structures on circle bundles.

Abstract

We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classification of Engel structures on circle bundles with characteristic line field tangent to the fibers.