The Geometry of Marked Contact Engel Structures.

Abstract

A contact twisted cubic structure $(M,C,\gamma )$ is a 5-dimensional manifold $M$ together with a contact distribution $C$ and a bundle of twisted cubics $\gamma \subset P\left(C\right)$ compatible with the conformal symplectic form on $C$ . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $G_2$ . In the present paper we equip the contact Engel structure with a smooth section $\sigma :M\to \gamma$ , which "marks" a point in each fibre ${\gamma }_x$ . We study the local geometry of the resulting structures $(M,C,\gamma ,\sigma )$ , 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 $\gamma$ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $\ge 6$ up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.