Closed geodesics on orbifolds

Abstract

In this paper, we try to generalize to the case of compact Riemannian orbifolds $Q$ some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds $M$. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics. In the classical case the solution of those problems involve the consideration of the homotopy groups of $M$ and the homology properties of the free loop space on $M$(Morse theory). Those notions have their analogue in the case of orbifolds (see [7]). The main part of this paper will be to recall those notions and to show how the classical techniques can be adapted to the case of orbifolds.