Rigidity for periodic magnetic fields

1999/02/02 by Misha Bialy, M. L. Bialy, Bialy, M. L.
Mathematics · Physics and Astronomy · #58F05 #58F07 #58F17 #58F18 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #math-ph #math.DG #math.DS #math.MP #msc:58F05 #msc:58F07 #msc:58F17 #msc:58F18

paper · pdf · doi:10.48550/arxiv.math/9902013

8 pages, no figures

arxiv created 1999/02/02 · openalex publication_date 1999/02/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ``twisted'' version of the recent generalisation of Hopf's rigidity result obtained by Burago and Ivanov.

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