Magnetic flatness and E. Hopf's theorem for magnetic systems

TL;DR

This paper extends Hopf's theorem to magnetic systems using magnetic curvature, showing that magnetic flatness is a rigid condition linked to specific geometric and magnetic properties.

Contribution

It introduces magnetic curvature to generalize Hopf's theorem, establishing conditions for magnetic flatness and its rigidity in magnetic systems.

Findings

Magnetic flow without conjugate points implies non-positive total magnetic curvature.

Magnetic flatness occurs only under specific geometric and magnetic conditions.

Magnetic flatness is characterized by trivial magnetic form with flat metric or Kähler structure with constant negative curvature.

Abstract

Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf's theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the s-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is K\"ahler, the metric has constant negative sectional holomorphic curvature, and s equals the Ma\~n\'e critical value.