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Remarks on the fundamental group of positively curved manifolds

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1255)

Abstract

Let M be a complete Riemannian manifold with sectional curvature KM>-H>0, Bonnet's theorem tells us that the fundamental group π1(M) of M is finite. In this note, we'll determine π1(M) under some conditions on the closed geodesics in M.

Keywords

  • Riemannian Manifold
  • Fundamental Group
  • Sectional Curvature
  • Closed Geodesic
  • Complete Riemannian Manifold

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References

  1. W. Ballmann, G. Thorbergsson & W. Ziller, Some existence theorems for closed geodesics, Comm Math. Helv 58(1983)416–432.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. J. Cheeger & D. Ebin, Comparison Theorems in Riemannian Geometry. North-Holland. Publishing Company, Amsterdam 1975.

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  3. J. Cheeger & D. Gromoll, On the lower bound for the injectivity radius of 1/4-pinched manifolds, J.Diff Geom 15 (1980) 437–442.

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  4. K. Orove & K. Shiohama, A generalized sphere theorem, Ann. Math. 106 (1977), 201–211.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. W. Klingenberg, Riemannian Geometry. Walter de Gruyter. Berlin New York. 1982.

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© 1987 Springer-Verlag

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Xia, C.Y. (1987). Remarks on the fundamental group of positively curved manifolds. In: Gu, C., Berger, M., Bryant, R.L. (eds) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077690

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  • DOI: https://doi.org/10.1007/BFb0077690

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17849-1

  • Online ISBN: 978-3-540-47883-6

  • eBook Packages: Springer Book Archive