Magnetic Curvature and Existence of a Closed Magnetic Geodesic on Low Energy Levels

Abstract

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 $\textrm{Sec}^{\Omega }$ and the magnetic Ricci curvature $\textrm{Ric}^{\Omega }$ that 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 $\textrm{Ric}^{\Omega }$ 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 that appear when one requires $\textrm{Sec}^{\Omega }$ to be positive.