Loose Engel structures

TL;DR

This paper introduces loose Engel structures and proves a homotopy equivalence between loose families, establishing a complete h-principle and classifying certain prolongations by formal data.

Contribution

It defines loose Engel structures and demonstrates their homotopy classification aligns with formal data, establishing a complete h-principle for these structures.

Findings

Loose Engel structures are classified up to homotopy by formal data.

A complete h-principle is established for loose Engel structures.

Lorentz and orientable Cartan prolongations are classified by formal data.

Abstract

This article introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete h-principle when some auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.