A theorem of Hadamard-Cartan type for Kahler magnetic fields

We study the global behavior of trajectories for Kahler magnetic fields on a connected complete Kahler manifold M of negative curvature. Concerning these trajectories we show that theorems of Hadamard-Cartan type and of Hopf-Rinow type hold: If sectional curvatures of M are not greater than c (< 0) and the strength of a Kahler magnetic field is not greater than $\sqrt{|c|}$, then every magnetic exponential map is a covering map. Hence arbitrary distinct points on M can be joined by a minimizing trajectory for this magnetic field.