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Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

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Abstract

Let M n be a closed Riemannian manifold homotopy equivalent to the product of S 2 and an arbitrary (n–2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M n there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M n.

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References

  1. Balacheff F., Croke C., Katz M. (2009) A Zoll counterexample to a geodesic length conjecture. GAFA, Geom. funct. anal. 19:(1)1–10

    Article  MathSciNet  MATH  Google Scholar 

  2. Burago Ju., Zalgaller V. (1988) Geometric Inequalities. Springer-Verlag, Berlin

    MATH  Google Scholar 

  3. Croke C.B. (1988) Area and the length of the shortest closed geodesic. J. Diff. Geom. 27: 1–21

    MathSciNet  MATH  Google Scholar 

  4. C.B. Croke, M. Katz, Universal volume bounds in Riemannian manifolds, Surveys in Differential Geometry VIII (Boston, MA 2002), Surv. Diff. Geom. VIII, Int. Press. Somerville, MA (2003), 109–137.

  5. Y. Felix, S. Halperin, J.C. Thomas, Rational Homotopy Theory, Springer, 2001.

  6. Frankel S., Katz M. (1993) The Morse landscape of a Riemannian disk. Ann. Inst. Fourier (Grenoble) 43: 503–507

    MathSciNet  MATH  Google Scholar 

  7. Gromov M. (1983) Filling Riemannian manifolds. J. Diff. Geom. 18: 1–147

    MathSciNet  MATH  Google Scholar 

  8. W. Klingenberg, Lectures on Closed Geodesics, Springer, 1978.

  9. Nabutovsky A., Rotman R. (2007) Lengths of geodesics between two points on a Riemannian manifold. Electron. Res. Announc. (ERA) Amer. Math. Soc. 13: 13–20

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Nabutovsky, R. Rotman, The length of the second shortest geodesic, Comment. Math. Helv., to appear.

  11. Nabutovsky A., Rotman R. (2009) Lengths of geodesics on a two-sphere. American J. Math. 131: 545–569

    Article  MathSciNet  MATH  Google Scholar 

  12. Rotman R. (2008) The length of a shortest geodesic loop at a point. J. Differential Geom. 78:(3)497–519

    MathSciNet  MATH  Google Scholar 

  13. Sabourau S. (2004) Global and local volume bounds and the shortest geodesic loops. Commun. Anal. Geom. 12: 1039–1053

    MathSciNet  MATH  Google Scholar 

  14. Schwarz A. (1958) Geodesic arcs on Riemannian manifolds. Uspekhi Math. Nauk (translated as “Russian Math. Surveys”) 13: 181–183

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Alexander Nabutovsky & Regina Rotman

  2. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alexander Nabutovsky & Regina Rotman

Authors
  1. Alexander Nabutovsky
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  2. Regina Rotman
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Correspondence to Alexander Nabutovsky.

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Nabutovsky, A., Rotman, R. Short Geodesic Segments between Two Points on a Closed Riemannian Manifold. Geom. Funct. Anal. 19, 498–519 (2009). https://doi.org/10.1007/s00039-009-0004-8

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  • DOI: https://doi.org/10.1007/s00039-009-0004-8

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