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

To a Riemannian manifold $(M,g)$ endowed with a magnetic form σ and its Lorentz operator Ω we associate an operator MΩ, 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 σ. From MΩ we derive the magnetic sectional curvature SecΩ and the magnetic Ricci curvature RicΩ 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 RicΩ 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 σ 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 SecΩ to be positive.