Magnetic Flatness and E. Hopf’s Theorem for Magnetic Systems

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 Kähler, the metric has constant negative sectional holomorphic curvature, and s equals the Mañé critical value.