On the existence of closed magnetic geodesics via symplectic reduction

Abstract

Let $(M,g)$ be a closed Riemannian manifold and $\sigma$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair $(g,\sigma )$ can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle $T^{\ast }E$ of a suitable $S^1$-bundle $E$ over $M$ or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of $E$. We then study the relation between the stability property of energy hypersurfaces in $(T^{\ast }M,dp\wedge dq+{\pi }^{\ast }\sigma )$ and of the corresponding codimension 2 coisotropic submanifolds in $(T^{\ast }E,dp\wedge dq)$ arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.

arXiv commentIn this new version, besides some minor modifications, the study of the relation between stability/contact property of energy hypersurfaces and of the corresponding coisotropic submanifolds via symplectic reduction has been added