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 $σ$ and its Lorentz operator $Ω$ we associate an operator $M^Ω$, 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 $M^Ω$ we derive the magnetic sectional curvature $Sec^Ω$ and the magnetic Ricci curvature $Ric^Ω$ 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 $Ric^Ω$ 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 which appear when one requires $Sec^Ω$ to be positive.