Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912437025898496 |
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| author | Pavlovic, Milan Sukilovic, Tijana |
| author_facet | Pavlovic, Milan Sukilovic, Tijana |
| contents | In this study, we investigate two distinct classes of normal geodesic flows associated with the left-invariant sub-Riemannian metric on the (2n + 1)-dimensional Heisenberg group. The first class arises from the left-invariant distribution, whereas the second is derived from the right-invariant distribution. It is established that the Hamiltonian system corresponding to the left-left (LL) configuration is completely integrable in the non-commutative sense. We demonstrate that the left-right (LR) systems exhibit non-commutative integrability in dimensions exceeding 5, while in dimensions 3 and 5, integrability is achieved in the commutative sense. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_06586 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group Pavlovic, Milan Sukilovic, Tijana Differential Geometry 53C17, 22E25, 37J35 In this study, we investigate two distinct classes of normal geodesic flows associated with the left-invariant sub-Riemannian metric on the (2n + 1)-dimensional Heisenberg group. The first class arises from the left-invariant distribution, whereas the second is derived from the right-invariant distribution. It is established that the Hamiltonian system corresponding to the left-left (LL) configuration is completely integrable in the non-commutative sense. We demonstrate that the left-right (LR) systems exhibit non-commutative integrability in dimensions exceeding 5, while in dimensions 3 and 5, integrability is achieved in the commutative sense. |
| title | Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group |
| topic | Differential Geometry 53C17, 22E25, 37J35 |
| url | https://arxiv.org/abs/2404.06586 |