The geometry of marked contact Engel structures

Abstract

A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of the most symmetric contact twisted cubic structure; we thus refer to this one as the contact Engel structure. In the present paper we equip the contact Engel structure with a smooth section s: M-> S that `marks' a point in each twisted cubic. We study the local geometry of the resulting structures (M,C,S,s), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of M by curves whose tangent directions are everywhere contained in S. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension greater than 5, and we prove an analogue of the classical Kerr theorem from relativity.