On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

TL;DR

This paper investigates geodesic flows on manifolds with symmetries, focusing on the existence of closed geodesics related to magnetic geodesics on orbifolds formed by quotienting by a Lie group action.

Contribution

It analyzes the properties of invariant geodesic flows on manifolds with group actions and explores conditions for closed geodesics projecting to magnetic geodesics on orbifolds.

Findings

Existence criteria for closed geodesics under group actions

Relationship between geodesics on manifolds and magnetic geodesics on orbifolds

Insights into geodesic flow dynamics with symmetries

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 $Q/G$.