On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

Abstract

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.

Closed geodesics on compact orbifolds and on noncompact manifolds

“…Most notably, Guruprasad and Haefliger [GH06] showed that every nondevelopable compact Riemannian orbifold, i.e. one which cannot be written as a global quotient of a Riemannian manifold M by a proper, cocompact, isometric action of a discrete group G, has a closed geodesic (see also [AS18] for an alternative proof). On the other hand, the assumption that there are no closed geodesics on a developable compact Riemannian orbifold M/G imposes strong restrictions on M , G and its action on M .…”