The Geometry of Marked Contact Engel Structures

A contact twisted cubic structure [Formula: see text] is a 5-dimensional manifold [Formula: see text] together with a contact distribution [Formula: see text] and a bundle of twisted cubics [Formula: see text] compatible with the conformal symplectic form on [Formula: see text] . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group [Formula: see text] . In the present paper we equip the contact Engel structure with a smooth section [Formula: see text] , which “marks” a point in each fibre [Formula: see text] . We study the local geometry of the resulting structures [Formula: see text] , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of [Formula: see text] by curves whose tangent directions are everywhere contained in [Formula: see text] . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension [Formula: see text] up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.