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Existence and nonexistence of half-geodesics on $ S^2$


Author: Ian M. Adelstein
Journal: Proc. Amer. Math. Soc. 144 (2016), 3085-3091
MSC (2010): Primary 53C22
DOI: https://doi.org/10.1090/proc/12918
Published electronically: March 22, 2016
MathSciNet review: 3487238
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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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Additional Information

Ian M. Adelstein
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: iadelstein@gmail.com

DOI: https://doi.org/10.1090/proc/12918
Received by editor(s): September 3, 2014
Received by editor(s) in revised form: May 29, 2015
Published electronically: March 22, 2016
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society

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