Integrability conditions for the Grushin and Martinet distributions

TL;DR

This paper derives integrability conditions for Grushin and Martinet distributions by realizing them as symmetry reductions of well-known distributions, enabling the construction of differential operator complexes.

Contribution

It introduces a novel approach of using symmetry reductions of Heisenberg and Engel distributions to analyze Grushin and Martinet distributions.

Findings

Derived integrability conditions for Grushin and Martinet distributions.

Constructed well-behaved complexes of differential operators.

Showed these complexes resolve non-regular distributions.

Abstract

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive the integrability conditions for the Grushin and Martinet distributions and build certain complexes of differential operators. These complexes are well-behaved despite the distributions they resolve being non-regular.