Magnetic Curvature and Existence of a Closed Magnetic Geodesic on Low Energy Levels.

The article focuses on the introduction and study of the magnetic curvature operator \( M^{\Omega}_k \) associated with a magnetic system \((M,g,\sigma)\), where \((M,g)\) is a closed Riemannian manifold and \(\sigma\) a closed 2-form called the magnetic form. This operator generalizes classical Riemannian curvature by incorporating magnetic perturbations via the Lorentz operator \(\Omega\). From \(M^{\Omega}_k\), the authors define magnetic sectional curvature \(\mathrm{Sec}^\Omega_k\) and magnetic Ricci curvature \(\mathrm{Ric}^\Omega_k\), extending previous notions from surfaces to arbitrary dimensions. The main results establish existence of contractible closed magnetic geodesics on energy levels below the Mañé critical value \(c\) under positivity assumptions on \(\mathrm{Ric}^\Omega_k\), including a new theorem guaranteeing such geodesics for all small energies when \(\sigma\) is nowhere vanishing. Additionally, topological restrictions on the manifold arise from positivity of \(\mathrm{Sec}^\Omega_k\), generalizing classical results like Synge’s theorem to the magnetic setting. The work employs variational methods on the magnetic action form \(\eta_k\), index estimates, and a magnetic Bonnet–Myers type argument to prove compactness and existence results, contributing a comprehensive framework for magnetic curvature and its dynamical implications.