Submersions and curves of constant geodesic curvature

TL;DR

This paper investigates conditions under which sub-Riemannian geodesics project to curves with constant geodesic curvature, extending the metric and analyzing geometric properties with illustrative examples.

Contribution

It provides necessary and sufficient conditions for projecting sub-Riemannian geodesics to constant curvature curves and introduces a canonical extension of the metric.

Findings

Conditions for geodesic projection to constant curvature curves

Extension of sub-Riemannian metric to Riemannian manifold

Examples illustrating the theoretical results

Abstract

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.