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 Engel structures are locally diffeomorphic. We investigate the deformation space of Engel structures obtained by deforming certain canonical Engel structures on four-manifolds with boundary. When the manifold is RP 3 × I where I is a closed interval, we show that this deformation space contains a subspace corresponding to Zoll metrics on the two-sphere (metrics all of whose geodesics are closed) modulo ‘projective’ equivalence. The main tool is a construction of an Engel manifold from a three-dimensional contact manifold and a method of reversing this construction. These are special instances of Cartan’s method of prolongation and deprolongation. The double prolongation of a surface X is an Engel manifold of the form SX×S where SX denotes the unit tangent bundle to X. The RP 3 × I example occurs in this way, since the unit tangent bundle of the two-sphere is RP . Besides proving these new results, the article has the flavour of a review.