On the geometry of magnetic flows

Abstract:

To a Riemannian manifold \((M, g)\) endowed with a magnetic form \(\sigma\) and its Lorentz operator \(\Omega\) we associate an operator \(M^\Omega\), called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric \(g\) together with terms of perturbation due to the magnetic interaction of \(\sigma\). From \(M^\Omega\) we derive the magnetic sectional curvature Sec\(^\Omega\) and the magnetic Ricci curvature Ric\(^\Omega\). On closed manifolds, with a Bonnet-Myers argument, we show that if Ric\(^\Omega\) is positive on an energy level below the Mañé critical value then, on that energy level, we can recover the Palais-Smale condition and prove the existence of a contractible periodic orbit. In particular, when \(\sigma\) is nowhere vanishing, this implies the existence of contractible periodic orbits on every energy level close to zero. On closed oriented even dimensional manifolds, we discuss about the topological restrictions which appear when one requires Sec\(^\Omega\) to be positive. Finally, we give a magnetic version of the classical Hopf’s rigidity theorem for magnetic flows without conjugate points on closed oriented surfaces.