Homogeneous geodesics in sub-Riemannian geometry
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910320159621120 |
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| author | Podobryaev, A. V. |
| author_facet | Podobryaev, A. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09085 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homogeneous geodesics in sub-Riemannian geometry Podobryaev, A. V. Differential Geometry Optimization and Control 53C30, 53C17, 35R03 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. |
| title | Homogeneous geodesics in sub-Riemannian geometry |
| topic | Differential Geometry Optimization and Control 53C30, 53C17, 35R03 |
| url | https://arxiv.org/abs/2202.09085 |