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

2025/10/22 by Levin Maier, Maier, Levin, Francesco Ruscelli +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2510.19323

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

In this article, we investigate 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 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.

Citations

• On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups
• On geometric hydrodynamics and infinite dimensional magnetic systems
• On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem
• The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres
• Well-posedness of the EPDiff equation with a pseudo-differential inertia operator
• On Completeness of Groups of Diffeomorphisms
• The Palais-Smale condition for contact type energy levels for convex lagrangian systems
• The Palais-Smale condition on contact type energy levels for convex Lagrangian systems

Cited by

• On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds
• On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups

Related

• The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres
• On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem
• Magnetic Rigidity of Horocycle flows
• On geometric hydrodynamics and infinite dimensional magnetic systems
• Infinitely many noncontractible closed magnetic geodesics on non-compact manifolds