On the Geometry of Magnetic Flows

A magnetic system is the toy model for the motion of a charged particle moving on a Riemannian manifold endowed with a magnetic force. To a magnetic flow we associate an operator, called the magnetic curvature operator. Such an operator encodes together the geometrical properties of the Riemannanian structure together with terms of perturbation due to magnetic interaction, and it carries crucial informations of the magnetic dynamics. For instance, in this talk, we see how a level of the energy positively curved, in this new magnetic sense, carries a periodic orbit. We also generalize to the magnetic case the classical Hopf's rigity and we introduce the notion of magnetic flatness for closed surfaces