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arXiv · 2202.09085

Homogeneous geodesics in sub-Riemannian geometry

A. V. Podobryaev
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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.

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A. V. Podobryaev. 2022-02-18. Homogeneous geodesics in sub-Riemannian geometry. https://doi.org/10.1051/cocv%2F2022086

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