Homogeneous geodesics in sub-Riemannian geometry

TL;DR

This paper investigates the properties of homogeneous geodesics in sub-Riemannian manifolds, providing criteria for their homogeneity, characterizing geodesic orbit spaces, and exploring specific examples such as Carnot groups.

Contribution

It introduces a criterion for homogeneous geodesics based on initial momentum and characterizes geodesic orbit spaces, especially in the context of Carnot groups.

Findings

Weakly commutative sub-Riemannian homogeneous spaces are geodesic orbit.

Geodesic orbit Carnot groups are only of step 1 and 2.

Conditions for the existence of at least one homogeneous geodesic.

Abstract

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step $1$ and $2$. Finally, we get a broad condition for existence of at least one homogeneous geodesic.