Existence and Non-existence of Half-Geodesics on $S^2$

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