Novel pathways in $k$-contact geometry

TL;DR

This paper introduces new types of $k$-contact distributions derived from Goursat distributions, explores their applications to Lie systems, and develops geometric methods for superposition rules and structural characterization.

Contribution

It characterizes $k$-contact distributions within Goursat structures and applies these to control systems, offering new geometric insights and methods.

Findings

Goursat distributions lead to new $k$-contact structures

Characterization of $k$-contact distributions on $\

$k$-contact geometry relates to parabolic Cartan geometries

Abstract

Our study of Goursat distributions originates new types of $k$-contact distributions and Lie systems with applications. In particular, families of generators for Goursat distributions on $\mathbb{R}^4, \mathbb{R}^5$ and $\mathbb{R}^6$ give rise to Lie systems and we characterise Goursat structures that are $k$-contact distributions. Our results are used to study the zero-trailer and other systems via Lie systems and $k$-contact manifolds. New ideas for the development of superposition rules via geometric structures and the characterisation of $k$-contact distributions are given and applied. Some relations of $k$-contact geometry with parabolic Cartan geometries are inspected.