Deformations of Nonholonomic Two-plane Fields in Four Dimensions

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.