Magnetic curvature and existence of a closed magnetic geodesic on low energy levels - Assenza, Valerio (International mathematics research notices, 2024)
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 $Ric^{Ω }$ 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 $Ric^{Ω }$ 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 that appear when one requires $Sec^{Ω }$ to be positive.