| Issue |
ESAIM: COCV
Volume 29, 2023
|
|
|---|---|---|
| Article Number | 11 | |
| Number of page(s) | 17 | |
| DOI | https://doi.org/10.1051/cocv/2022086 | |
| Published online | 19 January 2023 | |
Homogeneous geodesics in sub-Riemannian geometry*
A.K. Ailamazyan Program Systems Institute of RAS,
Pereslavl-Zalesskiy,
Russia
** Corresponding author: alex@alex.botik.ru
Received:
15
June
2022
Accepted:
11
December
2022
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.
Mathematics Subject Classification: 53C30 / 53C17 / 35R03
Key words: Homogeneous space / isometry / geodesic / geodesic orbit manifold / integration / weakly symmetric spaces / Riemannian geometry / sub-Riemannian geometry / Carnot group / geometric control theory
© The authors. Published by EDP Sciences, SMAI 2023
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