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

TL;DR

This paper investigates the existence and non-existence of special closed geodesics called half-geodesics on spheres, constructing examples with any finite number and sequences with none converging to a space with infinitely many.

Contribution

It provides explicit constructions of Riemannian spheres with prescribed numbers of half-geodesics and explores their convergence properties in the Gromov-Hausdorff sense.

Findings

Constructed spheres with exactly n half-geodesics for each nonnegative integer n.

Created sequences of spheres with no half-geodesics converging to a space with infinitely many.

Established existence and non-existence results for half-geodesics on $S^2$.

Abstract

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/2$. 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.