Deformations of Nonholonomic Two-plane Fields in Four Dimensions

TL;DR

This paper explores the global deformation space of Engel structures on four-manifolds, linking them to contact three-manifolds and revealing connections to geodesic flows and Zoll metrics.

Contribution

It introduces a method to analyze Engel structures via Cartan's prolongation, establishing a relationship with contact manifolds and geometric flows.

Findings

Engel structures on real projective three-space relate to geodesic flow on the two-sphere.

A subset of Engel deformations corresponds to Zoll metrics on the two-sphere.

The study provides a framework for understanding deformations of nonholonomic structures in four dimensions.

Abstract

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and deprolongation which lets us pass back and forth between certain Engel four-manifolds and contact three-manifolds. Every Engel manifold inherits a natural one-dimensional foliation. Its leaves are the fibers of the map from Engel to contact manifold, when this map exists. The foliation has a transverse contact structure and tangential real projective structure. As an application of our investigations, we show that a canonical Engel structure on real projective three-space times an interval corresponds to geodesic flow on the two-sphere, and that a subspace of its Engel deformations corresponds to the space of Zoll metrics on the two-sphere.