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 $Sec^{\Omega}$ and the magnetic Ricci curvature $Ric^{\Omega}$ 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 $Ric^{\Omega}$ being positive on an energy level below the Ma\~n\'e 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 $Sec^{\Omega}$ to be positive.