Symmetries of sub-Riemannian surfaces

TL;DR

This paper investigates the symmetries of sub-Riemannian surfaces, especially contact types, using invariants to determine when symmetries exist and how to find them, with explicit conditions for contact cases.

Contribution

It provides new criteria based on invariants for the existence of symmetries in sub-Riemannian surfaces, including non-contact cases, and offers methods to explicitly identify these symmetries.

Findings

Conditions for non-existence of symmetries in contact sub-Riemannian surfaces.

Explicit methods to find symmetries when invariants satisfy certain conditions.

Analysis of invariants on non-contact regions of sub-Riemannian surfaces.

Abstract

We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.