Title: Homogeneous geodesics in sub-Riemannian geometry

Authors: A.V. Podobryaev

Journal title: ESAIM: Control, Optimisation and Calculus of Variations

Year: 2023

Issue: 11

Volume: 29

Pages: 17 p.

Citation:

A.V. Podobryaev. Homogeneous geodesics in sub-Riemannian geometry // ESAIM: Control, Optimisation and Calculus of Variations. 29, 11 (2023)

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.