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A sphere theorem for manifolds of positive Ricci curvature


Author: Katsuhiro Shiohama
Journal: Trans. Amer. Math. Soc. 275 (1983), 811-819
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-1983-0682734-1
MathSciNet review: 682734
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Abstract: Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that $ {\text{Ri}}{{\text{c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}$ and the volume of $ M \geqslant $ the volume of the $ (\pi - \varepsilon )$-ball on the unit $ m$-sphere, $ m = {\text{dim }}M$. Then for a suitable choice of $ \varepsilon = \varepsilon (m,k)$ every $ M$ belonging to this class is homeomorphic to $ {S^m}$.


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  • [1] M. Berger, Les variétés riemanniennes $ (1/4)$-pincées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1960), 161-170. MR 0140054 (25:3478)
  • [2] R. Bishop and R. L. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
  • [3] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 0117695 (22:8470b)
  • [4] J. Cheeger, Finiteness theorems for riemannian manifolds, Amer. J. Math. 92 (1970), 61-74. MR 0263092 (41:7697)
  • [5] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam and New York, 1975. MR 0458335 (56:16538)
  • [6] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971), 119-129. MR 0303460 (46:2597)
  • [7] S. Y. Cheng, Eigenvalue comparison theorem and its geometric applications, Math. Z. 143 (1975), 289-297. MR 0378001 (51:14170)
  • [8] R. Greene and H. Wu, On the subharmonicity and plurisubharmonicily of geodesically convex functions, Indiana Univ. Math. J. 22 (1973), 641-653. MR 0422686 (54:10672)
  • [9] -, $ {C^\infty }$ convex functions and manifolds of positive curvature, Acta Math. 137 (1976), 209-245. MR 0458336 (56:16539)
  • [10] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Grossen, Springer-Verlag, Berlin, Heidelberg and New York, 1966. MR 0229177 (37:4751)
  • [11] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), 179-195. MR 630949 (82k:53062)
  • [12] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201-211. MR 0500705 (58:18268)
  • [13] W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47-54. MR 0139120 (25:2559)
  • [14] S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404. MR 0004518 (3:18f)
  • [15] H. Rauch, A contribution to differential geometry in the large, Ann. of Math. (2) 54 (1951), 38-55. MR 0042765 (13:159b)
  • [16] T. Rushing, Topological embeddings, Academic Press, New York and London, 1973. MR 0348752 (50:1247)
  • [17] B. Schoen and S. T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. Math. Studies, no. 102, Univ. of Princeton, Princeton, N.J., 1982, pp. 209-228. MR 645740 (83k:53060)
  • [18] V. A. Toponogov, Riemannian spaces having their curvature bounded below by a positive number, Amer. Math. Soc. Transl. (2) 37 (1964), 291-336; translated from Uspehi Mat. Nauk 14 (1959). MR 0099701 (20:6139)
  • [19] A. Weinstein, On the homotopy of positively pinched manifolds, Arch. Math. (Basel) 18 (1967), 523-524. MR 0220311 (36:3376)

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DOI: https://doi.org/10.1090/S0002-9947-1983-0682734-1
Article copyright: © Copyright 1983 American Mathematical Society

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