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Closed manifolds with small excess

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Abstract

In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess. We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is small enough.

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References

  1. Anderson, M.T. Metrics of positive Ricci curvature with large diameter,Manuscripta Math.,68, 405–415, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown, M. A proof of the generalized Schoenflies theorem,Bull. Am. Math. Soc.,66, 74–76, (1960).

    Article  MATH  Google Scholar 

  3. Cheeger, J. Critical points of distance functions and applications to geometry,Lecture Notes in Math., Springer-Verlag,1054,1-38, (1991).

  4. Cheng, S.Y. Eigenvalue comparison theorems and its geometric applications,Math. Z.,143, 289–297, (1975).

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheeger, J. and Ebin, D.Comparison Theorems in Riemannian Geometry, North-Holland, New York, 1975.

    MATH  Google Scholar 

  6. Dai, X. and Wei, G. A comparison-estimate of Toponogov type for Ricci curvature,Math. Ann.,303, 297–306, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  7. Eschenburg, J.H. Diameter, volume and topology for positive Ricci curvature,J. Diff. Geom.,33, 743–747, (1991).

    MathSciNet  MATH  Google Scholar 

  8. Gromov, M. Curvature, diameter and Betti numbers,Comment Math. Helvetici.,56, 179–195, (1981).

    Article  MathSciNet  MATH  Google Scholar 

  9. Grove, K. Critical point theory for distance functions,Proc. Symposia in Pure Math.,54(3), 357–385, (1993).

    MathSciNet  Google Scholar 

  10. Grove, K. and Petersen, P. Bounding homotopy types by geometry,Ann. of Math.,128, 195–206, (1988).

    Article  MathSciNet  Google Scholar 

  11. Grove, K. and Petersen, P. A pinching theorem for homotopy spheres,J. Am. Math. Soc.,3, 671–677, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  12. Grove, K. and Petersen, P. Manifolds near the boundary of existence,J. Diff. Geom.,33, 379–394, (1991).

    MathSciNet  MATH  Google Scholar 

  13. Grove, K. and Shiohama, K. A generalized sphere theorem,Ann. Math.,106, 201–211, (1977).

    Article  MathSciNet  Google Scholar 

  14. Itokawa, Y. The topology of certain manifolds with positive Ricci curvature,J. Diff. Geom.,17, 151–155, (1983).

    MathSciNet  Google Scholar 

  15. Otsu, Y. On manifolds of positive Ricci curvature with large diameters,Math. Z.,206, 255–264, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  16. Perelman, G. A diameter sphere theorem for manifolds of positive Ricci curvature,Math. Z.,218, 595–596, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  17. Petersen, P. Small excess and Ricci curvature,J. Geom. Anal.,1, 383–387, (1991).

    MathSciNet  MATH  Google Scholar 

  18. Shiohama, K. A sphere theorem for positive Ricci curvature,Trans. Am. Math. Soc.,275, 811–819, (1983).

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, J.Y. A diameter pinching sphere theorem for positive Ricci curvature,Proc Am. Math. Soc.,107, 797–802, (1989).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática - IE, Universidade de Brasília, Campus Universitáro, 70910-900, Brasília DF, Brasil

    Keti Tenenblat & Changyu Xia

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  1. Keti Tenenblat
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  2. Changyu Xia
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Correspondence to Keti Tenenblat.

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Tenenblat, K., Xia, C. Closed manifolds with small excess. J Geom Anal 11, 129–134 (2001). https://doi.org/10.1007/BF02921958

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  • DOI: https://doi.org/10.1007/BF02921958

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