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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 classicalRiemannian curvature of the metric ${g}$ together with terms of perturbationdue to the magnetic interaction of ${\sigma}$. From ${M^{\Omega}}$ we derivethe magnetic sectional curvature ${\mathrm{Sec}^{\Omega}}$ and the magneticRicci curvature ${\mathrm{Ric}^{\Omega}}$. On closed manifolds, with aBonnet-Myers argument, we show that if ${\mathrm{Ric}^{\Omega}}$ is positive onan energy level below the critical value then, on that energy level, themagnetic dynamics carries a contractible periodic orbit. In particular, when${\sigma}$ is nowhere vanishing, this implies the existence of contractibleperiodic orbits on every energy level close to zero. Finally, on closedoriented even dimensional manifolds, we discuss about the topologicalrestrictions which appear when one requires ${\mathrm{Sec}^{\Omega}}$ to bepositive.

Magnetic curvature and existence of closed magnetic geodesics on low energy levels