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

TL;DR

This paper introduces magnetic curvature operators on Riemannian manifolds with magnetic forms, generalizes magnetic curvature notions, and proves the existence of contractible periodic orbits under positive Ricci curvature conditions below a critical energy level.

Contribution

It defines magnetic curvature operators and curvatures, extending classical notions, and establishes the existence of periodic orbits under specific curvature conditions.

Findings

Existence of contractible periodic orbits under positive magnetic Ricci curvature.

Generalization of magnetic curvature concepts to higher dimensions.

Topological restrictions for positive magnetic sectional curvature.

Abstract

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 classical Riemannian curvature of the metric $g$ together with terms of perturbation due to the magnetic interaction of ${\sigma}$. From $M^{\Omega}$ we derive the magnetic sectional curvature $Sec^{\Omega}$ and the magnetic Ricci curvature $Ric^{\Omega}$ 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 $Ric^{\Omega}$ being positive on an energy 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 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 $Sec^{\Omega}$ to be positive.