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The Existence of Three Short Closed Geodesics

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Differential Geometry and Complex Analysis

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Abstract

A famous theorem of Lusternik and Schnirelmann [LS] states that, for a Riemannian manifold M given by an arbitrary Riemannian metric on the differentiate 2-sphere, there are at least three closed geodesics without self-intersections. See [Ly] for a more complete proof.

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References

  1. Abraham, R.: Bumpy metrics. In: Global Analysis, Proc. Symp. Pure Math. Vol. XIV, Amer. Math. Soc, Providence, R.I. 1970

    Google Scholar 

  2. Anosov, D.: Some homology classes in the space of closed curves in the n-dimensional sphere. Izv. Akad. Nauk SSSR Ser. Mat. 45 No. 2 (1981) (Russian) — Mat. USSR Izv. 18 (4) 3–422 (1982)

    Google Scholar 

  3. Bott, R.: Non-degenerate critical manifolds. Ann. of Math. 60 (1954), 248–261

    Article  MathSciNet  MATH  Google Scholar 

  4. Chern, S. S.: On the multiplication of the characteristic ring of a sphere bundle. Ann. of Math. 49 (1948), 362–372

    Article  MathSciNet  MATH  Google Scholar 

  5. Klingenberg, W.: The theorem of the three closed geodesics. Bull. Amer. Math. Soc. 71 (1965), 601–605

    Article  MathSciNet  MATH  Google Scholar 

  6. Klingenberg, W.: Closed geodesics. Ann. of Math. 89 (1969), 68–91

    Article  MathSciNet  MATH  Google Scholar 

  7. Klingenberg, W.: Lectures on Closed Geodesies. Springer, Berlin Heidelberg New York, 1978

    Book  Google Scholar 

  8. Klingenberg, W.: Über die Existenz unendlich vieler geschlossener Geodätischer. Abh. Math. Nat. Klasse, Akademie Wiss. Lit. Mainz, Jg. 1981, Nr. 1

    Google Scholar 

  9. Klingenberg, W.: Riemannian Geometry. Studies in Mathematics Vol. 1. de Gruyter, Berlin New York, 1982

    Google Scholar 

  10. Klingenberg, W.: Closed Geodesies on Riemannian manifolds. CBMS Regional Conference Ser. no. 53. Amer. Math. Soc: Providence, R.I. 1983

    Google Scholar 

  11. Klingenberg, W.; Takens, F.: Generic properties of geodesic flows. Math. Ann. 197 (1972), 323–334

    Article  MathSciNet  MATH  Google Scholar 

  12. Lyusternik, L.; Schnirelmann, L.: Sur le problème de trois geodésiques fermées sur les surfaces de genre O. C.R. Acad. Sci. Paris 189 (1929), 269–271

    Google Scholar 

  13. Lyusternik, L.: The topology of function spaces and the calculus of variations in the large. Trudy Mat. Inst. Steklov 19 (1947), (Russian); Translations of Math. Monographs Vol. 16, Amer. Math. Soc, Providence, R.I., 1966

    Google Scholar 

  14. Sullivan, D.: Singularities in spaces. Lecture Notes in Mathematics, Vol. 209 (1971), 196–206. Springer, Berlin Heidelberg New York, 1971

    Google Scholar 

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Authors
  1. Wilhelm Klingenberg
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Editors and Affiliations

  1. The City College of the City University of New York, 138th Street & Convent Avenue, New York, NY, 10031, USA

    Isaac Chavel

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel

    Hershel M. Farkas

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© 1985 Springer-Verlag Berlin, Heidelberg

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Klingenberg, W. (1985). The Existence of Three Short Closed Geodesics. In: Chavel, I., Farkas, H.M. (eds) Differential Geometry and Complex Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69828-6_12

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  • DOI: https://doi.org/10.1007/978-3-642-69828-6_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-69830-9

  • Online ISBN: 978-3-642-69828-6

  • eBook Packages: Springer Book Archive

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