Existence and nonexistence of half-geodesics on 𝑆²

Adelstein, Ian

doi:10.1090/proc/12918

Existence and nonexistence of half-geodesics on $S^2$
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by Ian M. Adelstein PDF

Proc. Amer. Math. Soc. 144 (2016), 3085-3091 Request permission

Abstract:

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length $L/2$, where $L$ is the length of the geodesic. 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.

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