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The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups

Levin Maier, Francesco Ruscelli

TL;DR

The work extends the Hopf–Rinow theorem to magnetic geodesic flows on infinite-dimensional half-Lie groups with right-invariant metrics and closed 2-forms. It introduces Mañé's critical value on the universal cover and shows that above this energy threshold the lifted flow becomes a Randers-type Finsler geodesic flow, yielding a complete metric structure and global magnetic exponential. The authors prove magnetically geodesic completeness and the existence of energy-preserving, action-minimizing magnetic geodesics between any pair of points, with these results persisting for $G^{\ell}$ and accommodating Sobolev-diffeomorphism groups as key examples. The framework recovers known finite-dimensional results in the $\sigma=0$ case and provides a robust infinite-dimensional analogue via variational methods and a careful treatment of regularity and invariance properties. These results have implications for geometric hydrodynamics and the study of magnetic Euler–Arnold equations on diffeomorphism groups.

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{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.

The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups

TL;DR

The work extends the Hopf–Rinow theorem to magnetic geodesic flows on infinite-dimensional half-Lie groups with right-invariant metrics and closed 2-forms. It introduces Mañé's critical value on the universal cover and shows that above this energy threshold the lifted flow becomes a Randers-type Finsler geodesic flow, yielding a complete metric structure and global magnetic exponential. The authors prove magnetically geodesic completeness and the existence of energy-preserving, action-minimizing magnetic geodesics between any pair of points, with these results persisting for and accommodating Sobolev-diffeomorphism groups as key examples. The framework recovers known finite-dimensional results in the case and provides a robust infinite-dimensional analogue via variational methods and a careful treatment of regularity and invariance properties. These results have implications for geometric hydrodynamics and the study of magnetic Euler–Arnold equations on diffeomorphism groups.

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 or 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.
Paper Structure (16 sections, 13 theorems, 57 equations)

This paper contains 16 sections, 13 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Right-invariant magnetic systems on half-Lie groups.
  3. Mañé's critical value on the universal cover.
  4. Main results.
  5. First illustrations of \ref{['IThm: HopfRinow half lie group magnetic int']}
  6. Outline of the paper.
  7. Preliminaries
  8. Half-Lie groups
  9. Riemannian metrics on half-Lie groups
  10. Infinite dimensional magnetic system
  11. Mañé's critical value on the universal cover
  12. Definition of the Mañé critical value on the universal cover
  13. The magnetic geodesic flow as a Finsler flow
  14. Nondegeneracy of the geodesic distance and no-gain no-loss
  15. The Hopf--Rinow theorem for magnetic geodesics on half Lie groups.
  16. ...and 1 more sections

Key Result

Proposition 1

Let $(G, \mathcal{G}, \sigma)$ be a $G$-right-invariant magnetic system with $\sigma$ weakly exact and assume its pullback $\hat{\sigma}$ to the universal cover $\hat{G}$ admits a $\hat{G}$-right-invariant primitive $\hat{\alpha}$. Then, for every $\kappa > c(G, \mathcal{G}, \sigma)$, we have defines a $\hat{G}$-right-invariant Finsler metric on $\hat{G}$ satisfying the following properties:

Theorems & Definitions (38)

  • Definition 1: =\ref{['Def: int maneuni half']}
  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Theorem 1.1: =\ref{['IThm: HopfRinow half lie group magnetic']}
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 28 more