Novel pathways in $k$-contact geometry
Tomasz Sobczak, Tymon Frelik
TL;DR
This work investigates $k$-contact geometry as a natural generalization of contact structures and connects it to Goursat distributions and Lie systems in dimensions $4$–$6$. By employing Kumpera–Ruiz normal forms, it classifies Goursat distributions on $\mathbb{R}^4$, $\mathbb{R}^5$, and $\mathbb{R}^6$ as Lie systems with explicit Vessiot–Guldberg algebras and identifies which classes correspond to $k$-contact structures, including higher classes with parameterized Reeb fields. The authors provide a practical criterion to distinguish $k$-contact from non-$k$-contact Goursat structures and illustrate these ideas with concrete Reeb fields and symmetry data, as well as connections to parabolic Cartan geometries. Applications to trailer-type control problems demonstrate how $k$-contact geometry yields superposition rules and clarifies the high-dimensional geometric structure behind well-known systems like unicycles, cars with trailers, Martinet spheres, and chained forms. Overall, the work broadens the reach of Lie-system methods to higher-dimensional geometric contexts and builds a bridge between $k$-contact geometry and Cartan-geometric constructions.
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.