Magnetic curvature and existence of a closed magnetic geodesic on low energy levels

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 $Se{c}^{\Omega }$ and the magnetic Ricci curvature $Ri{c}^{\Omega }$ which generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $Ri{c}^{\Omega }$ being positive on an energy level below the Mañé critical value, with a Bonnet-Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $\sigma$ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions which appear when one requires $Se{c}^{\Omega }$ to be positive.