The Geometry of Marked Contact Engel Structures

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