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To a Riemannian manifold $(M, g)$ endowed with a magnetic form ${\sigma}$ andits Lorentz operator ${\Omega}$ we associate an operator $M^{\Omega}$, calledthe magnetic curvature operator. Such an operator encloses the classicalRiemannian curvature of the metric $g$ together with terms of perturbation dueto the magnetic interaction of ${\sigma}$. From $M^{\Omega}$ we derive themagnetic sectional curvature $Sec^{\Omega}$ and the magnetic Ricci curvature$Ric^{\Omega}$ which generalize in arbitrary dimension the already known notionof magnetic curvature previously considered by several authors on surfaces. Onclosed manifolds, under the assumption of $Ric^{\Omega}$ being positive on anenergy 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 acontractible periodic orbit on every energy level close to zero. Finally, onclosed oriented even dimensional manifolds, we discuss about the topologicalrestrictions which appear when one requires $Sec^{\Omega}$ to be positive.

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