Engel Manifolds and Contact 3-Orbifolds

TL;DR

This paper extends the theory of Engel manifolds by introducing Cartan prolongation and development maps for contact 3-orbifolds, providing conditions for when these prolongations are manifolds and relating Engel manifolds to contact orbifolds.

Contribution

It generalizes the concept of Cartan prolongation to contact 3-orbifolds and establishes criteria for the prolongation to be a manifold, linking Engel manifolds to contact orbifolds.

Findings

Characterization of when Cartan prolongation is a manifold

Extension of development map to orbifolds and Lie groupoids

All Engel manifolds from Cartan prolongation originate from contact 3-orbifolds

Abstract

In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper, we introduce the Cartan prolongation of a contact 3-orbifold and the development map associated to a more general Engel manifold, and we give necessary and sufficient condition for the Cartan prolongation to be a manifold. Moreover, we explain the Cartan prolongation of a 3-dimensional contact \'etale Lie groupoid and the development map associated to any Engel manifold, and we proof that all Engel manifolds obtained as the Cartan prolongation of a "space" with contact structure are obtained from a contact 3-orbifold.