Existence and nonexistence of half-geodesics on 𝑆²

Abstract

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length L / 2 L/2 , where L L is the length of the geodesic. For each nonnegative integer n n , we construct Riemannian manifolds diffeomorphic to S 2 S^2 admitting exactly n n half-geodesics. Additionally, we construct a sequence of Riemannian manifolds, each of which is diffeomorphic to S 2 S^2 and admits no half-geodesics, yet which converge in the Gromov-Hausdorff sense to a limit space with infinitely many half-geodesics.