No abstract

Let (M, g) be a closed Riemannian manifold and σ 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, σ) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle T * 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 * M, dp ∧ dq + π * σ) and of the corresponding codimension 2 coisotropic submanifolds in (T * E, dp∧dq) arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.

Let Q be a closed manifold admitting a locally-free action of a compact Lie group G. In this paper we study the properties of geodesic flows on Q given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group G, since they project to closed magnetic geodesics on the quotient orbifold Q/G.