The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups
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| Format: | Preprint |
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2025
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| _version_ | 1866908605924507648 |
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| author | Maier, Levin Ruscelli, Francesco |
| author_facet | Maier, Levin Ruscelli, Francesco |
| contents | In this article, we investigate \emph{right-invariant magnetic systems} on half-Lie groups, which consist of a strong right-invariant Riemannian metric and a right-invariant closed two-form. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. In this setting, we define \emph{Mañé's critical value} on the universal cover for weakly exact right-invariant magnetic fields. First, we prove that the lift of the magnetic flow to the universal cover coincides with a Finsler geodesic flow for energies above this threshold. Finally, we show that for energies above Mañé's critical value, the full Hopf--Rinow theorem holds for such magnetic systems, thereby generalizing the work of Contreras and Merry from closed finite-dimensional manifolds to this infinite-dimensional context. Our work extends the recent results of Bauer, Harms, and Michor from geodesic flows to magnetic geodesic flows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups Maier, Levin Ruscelli, Francesco Symplectic Geometry Differential Geometry Dynamical Systems In this article, we investigate \emph{right-invariant magnetic systems} on half-Lie groups, which consist of a strong right-invariant Riemannian metric and a right-invariant closed two-form. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. In this setting, we define \emph{Mañé's critical value} on the universal cover for weakly exact right-invariant magnetic fields. First, we prove that the lift of the magnetic flow to the universal cover coincides with a Finsler geodesic flow for energies above this threshold. Finally, we show that for energies above Mañé's critical value, the full Hopf--Rinow theorem holds for such magnetic systems, thereby generalizing the work of Contreras and Merry from closed finite-dimensional manifolds to this infinite-dimensional context. Our work extends the recent results of Bauer, Harms, and Michor from geodesic flows to magnetic geodesic flows. |
| title | The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2510.19323 |