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arXiv · 2510.19323

The Hopf--Rinow Theorem and Ma\~n\'e's Critical Value for Magnetic Geodesics on Half Lie-Groups

Levin Maier·Francesco Ruscelli
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Abstract

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{Ma\~n\'e'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 Ma\~n\'e'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.

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Levin Maier, Francesco Ruscelli. 2025-10-22. The Hopf--Rinow Theorem and Ma\~n\'e's Critical Value for Magnetic Geodesics on Half Lie-Groups. https://arxiv.org/abs/2510.19323

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