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The length of a shortest geodesic loop at a point

Regina Rotman University of Toronto

Differential Geometry mathscidoc:1609.10059

Journal of Differential Geometry, 78, (3), 497-519, 2008
In this paper we prove that given a point p ∈ Mn, where Mn is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(Mn) at this point is bounded above by 2nd, where d is the diameter of Mn. Moreover, we show that on a closed simply connected Riemannian manifold Mn with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of Mn, or the length of a shortest closed geodesic on Mn is bounded from above by 4d.
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@inproceedings{regina2008the,
  title={THE LENGTH OF A SHORTEST GEODESIC LOOP AT A POINT},
  author={Regina Rotman},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909105721646048716},
  booktitle={Journal of Differential Geometry},
  volume={78},
  number={3},
  pages={497-519},
  year={2008},
}
Regina Rotman. THE LENGTH OF A SHORTEST GEODESIC LOOP AT A POINT. 2008. Vol. 78. In Journal of Differential Geometry. pp.497-519. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909105721646048716.
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