Engel structures and weakly hyperbolic flows on four-manifolds

TL;DR

This paper explores the relationship between pairs of Engel structures on four-manifolds and weakly hyperbolic flows, revealing a new correspondence similar to known relations in three-dimensional contact geometry.

Contribution

It establishes a novel correspondence between pairs of Engel structures with shared contact structures and weakly hyperbolic flows on four-manifolds.

Findings

Identifies conditions under which pairs of Engel structures correspond to weakly hyperbolic flows.

Draws an analogy with bi-contact structures and Anosov flows in three dimensions.

Provides a framework for understanding Engel structures via dynamical systems.

Abstract

We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg--Thurston and by Mitsumatsu.