A sphere theorem for manifolds of positive Ricci curvature
HTML articles powered by AMS MathViewer
- by Katsuhiro Shiohama
- Trans. Amer. Math. Soc. 275 (1983), 811-819
- DOI: https://doi.org/10.1090/S0002-9947-1983-0682734-1
- PDF | Request permission
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 ${S^m}$.References
- M. Berger, Les variétés Riemanniennes $(1/4)$-pincées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 14 (1960), 161–170 (French). MR 140054
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- Marston Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc. 66 (1960), 113–115. MR 117694, DOI 10.1090/S0002-9904-1960-10420-X
- Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
- Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
- Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297. MR 378001, DOI 10.1007/BF01214381
- R. E. Greene and H. Wu, On the subharmonicity and plurisubharmonicity of geodesically convex functions, Indiana Univ. Math. J. 22 (1972/73), 641–653. MR 422686, DOI 10.1512/iumj.1973.22.22052
- R. E. Greene and H. Wu, $C^{\infty }$ convex functions and manifolds of positive curvature, Acta Math. 137 (1976), no. 3-4, 209–245. MR 458336, DOI 10.1007/BF02392418
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
- Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
- Karsten Grove and Katsuhiro Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), no. 2, 201–211. MR 500705, DOI 10.2307/1971164
- Wilhelm Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47–54 (German). MR 139120, DOI 10.1007/BF02567004
- S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. MR 4518
- H. E. Rauch, A contribution to differential geometry in the large, Ann. of Math. (2) 54 (1951), 38–55. MR 42765, DOI 10.2307/1969309
- T. Benny Rushing, Topological embeddings, Pure and Applied Mathematics, Vol. 52, Academic Press, New York-London, 1973. MR 0348752
- Richard Schoen and Shing Tung Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 209–228. MR 645740
- V. A. Toponogov, Riemannian spaces having their curvature bounded below by a positive number, Dokl. Akad. Nauk SSSR 120 (1958), 719–721 (Russian). MR 0099701
- Alan Weinstein, On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel) 18 (1967), 523–524. MR 220311, DOI 10.1007/BF01899493
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- 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