  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.
