On the Geometry of Magnetic Flows

To a Rimeannian manifold endowed with a magnetic form we associate an operator called the magnetic curvature operator. Such an operator encodes the geometrical properties of the Riemannian structure together with the terms of perturbation due to the magnetic interaction and carries significant informations about the magnetic dynamics. For instance, in the first part of the talk we show that a level of the energy positively curved in this new magnetic sense carries a contractible periodic orbit. The second part is devoted to the generalization of the classical Hopf's rigidity to the magnetic case and the notion of magnetic flatness for closed surfaces.