The geometry of marked contact Engel structures

TL;DR

This paper investigates the local geometry of marked contact Engel structures, extending classical work on G_2 symmetry, providing invariants, classifying homogeneous models, and establishing a Kerr theorem analogue.

Contribution

It introduces the concept of marked contact Engel structures, derives their local invariants, classifies models with high symmetry, and proves a Kerr theorem analogue.

Findings

Complete set of local invariants for the structures

Classification of homogeneous models with >5-dimensional symmetry

An analogue of the Kerr theorem for these 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.