The Geometry of Marked Contact Engel Structures.

A contact twisted cubic structure ( M , C , γ ) is a 5-dimensional manifold M together with a contact distribution C and a bundle of twisted cubics γ ⊂ P ( C ) 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 σ : M → γ , which "marks" a point in each fibre γ x . We study the local geometry of the resulting structures ( M , C , γ , σ ) , 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 γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.