Closed Magnetic geodesics on Heisenberg nilmanifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916029334028288 |
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| author | Ovando, Gabriela P. Subils, Mauro |
| author_facet | Ovando, Gabriela P. Subils, Mauro |
| contents | In this work we study the existence of closed magnetic geodesics on three-dimensional Heisenberg nilmanifolds for every left-invariant Lorentz force. Our first objective is to establish the existence of closed contractible magnetic geodesics on $H_3$. Once the invariant magnetic field is induced to a compact quotient $M=Λ\backslash H_3$, we study magnetic geodesics on $M$. Firstly, we determine conditions on a lattice $Λ\subset H_3$ to ensure that a given magnetic geodesic projects to a closed curve on $M$. In particular, we prove that for {\it any} energy level below the Mañé critical value there always exists a contractible closed magnetic geodesic on the compact manifold $M$. On the other hand, we show that closed magnetic geodesics do not necessarily exist in every homotopy class. Finally, we present examples of compact quotients $Γ_k\backslash H_3$ that admit infinitely many closed magnetic trajectories, as well as examples for which no closed non-contractible magnetic trajectories exist for a given left-invariant Lorentz force. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05515 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed Magnetic geodesics on Heisenberg nilmanifolds Ovando, Gabriela P. Subils, Mauro Differential Geometry 53C99, 70G65, 22E25 In this work we study the existence of closed magnetic geodesics on three-dimensional Heisenberg nilmanifolds for every left-invariant Lorentz force. Our first objective is to establish the existence of closed contractible magnetic geodesics on $H_3$. Once the invariant magnetic field is induced to a compact quotient $M=Λ\backslash H_3$, we study magnetic geodesics on $M$. Firstly, we determine conditions on a lattice $Λ\subset H_3$ to ensure that a given magnetic geodesic projects to a closed curve on $M$. In particular, we prove that for {\it any} energy level below the Mañé critical value there always exists a contractible closed magnetic geodesic on the compact manifold $M$. On the other hand, we show that closed magnetic geodesics do not necessarily exist in every homotopy class. Finally, we present examples of compact quotients $Γ_k\backslash H_3$ that admit infinitely many closed magnetic trajectories, as well as examples for which no closed non-contractible magnetic trajectories exist for a given left-invariant Lorentz force. |
| title | Closed Magnetic geodesics on Heisenberg nilmanifolds |
| topic | Differential Geometry 53C99, 70G65, 22E25 |
| url | https://arxiv.org/abs/2407.05515 |