Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smooth diameter and eigenvalue rigidity in positive Ricci curvature


Author: Wilderich Tuschmann
Journal: Proc. Amer. Math. Soc. 130 (2002), 303-306
MSC (2000): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-01-06384-5
Published electronically: July 31, 2001
MathSciNet review: 1855649
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given $m$ and $C$ there exists a positive constant $\varepsilon =\varepsilon (m,C)>0$such that any $m$-dimensional complete Riemannian manifold with Ricci curvature $Ricc\ge m-1$, sectional curvature $K\le C$and diameter $\ge \pi -\varepsilon $is Lipschitz close and diffeomorphic to the standard unit $m$-sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.


References [Enhancements On Off] (What's this?)

  • [An1] M. T. Anderson, Metrics of positive Ricci curvature with large diameters, Man. Math. 68 (1990) 405-415 MR 91g:53045
  • [An2] M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Inv. Math. 102 (1990) 429-445 MR 92c:53024
  • [Br] D. Brittain, A Diameter Pinching Sphere Theorem for Positive Ricci Curvature, Preprint (seemingly unpublished, compare however the result's of the proof in [Gr])
  • [Bes] G. Bessa, Differentiable sphere theorems for Ricci curvature, Math. Z. 214 (1993) 245-249 MR 94g:53035
  • [Cai] M. Cai, Rigidity of manifolds with large volume, Math. Z. 213, No.1 (1993) 17-31 MR 94b:53071
  • [Ch] J. Cheeger, Some examples of manifolds of non-negative curvature, J. Diff. Geom. 8 (1973) 623-628 MR 49:6085
  • [Chg] Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975) 289-297 MR 51:14170
  • [Co] T. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 193-214 MR 96k:53068
  • [Cr] C. B. Croke, An eigenvalue pinching theorem, Invent. Math. 68 (1982) 253-256 MR 84a:58084
  • [E] J.-H. Eschenburg, Diameter, volume, and topology for positive Ricci curvature, J. Diff. Geom. 33 (1991) 743-747 MR 92b:53054
  • [Gr] K. Grove, Metric differential geometry, Differential geometry (Lyngby, 1985), 171-227, Lecture Notes in Math., 1263, Springer, Berlin-New York, 1987 MR 88i:53075
  • [GrP] K. Grove and P. Petersen, A pinching theorem for homotopy spheres, J.A.M.S. 3, 3 (1990) 671-677 MR 91e:53040
  • [GrS] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. Math. 106 (1977) 201-211 MR 58:18268
  • [H] H. Hernández-Andrade, A class of compact manifolds with positive Ricci curvature, Differential Geometry, Proc. Symp. Pure Math. 28 (1975) 73-87 MR 52:1565
  • [It] Y. Itokawa, The topology of certain Riemannian manifolds with positive Ricci curvature, J. Diff. Geom. 18 (1983) 151-155 MR 84i:53044
  • [Kat] A. Katsuda, Gromov's convergence theorem and its applications, Nagoya Math. J. 100 (1985) 11-48; Erratum: Nagoya Math. J. 114 (1989) 173-174 MR 87e:53067; MR 90e:53057
  • [Li] A. Lichnerowicz, Géométrie des groupes de transformations, Dunod, Paris (1958) MR 23:A1329
  • [Na] G. Nakamura, Diameter sphere theorems for manifolds of positive Ricci curvature, Dissertation, Nagoya University 1989
  • [Nash] J. C. Nash, Positive Ricci curvature on fiber bundles, J. Diff. Geom. 14 (1979) 241-265 MR 81k:53039
  • [Ob] M. Obata, Certain conditions for a Riemannian manifold to be isometric to a sphere, J. Math. Soc. Japan 14 (1962) 333-340 MR 25:5479
  • [Ot1] Y. Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z. 206 (1991) 255-264 MR 91m:53033
  • [Ot2] Y. Otsu, On manifolds of small excess, Amer. J. Math. 115 (1993) 1229-1280 MR 95i:53046
  • [Pa] S.-H. Paeng, A sphere theorem under a curvature perturbation. I; II, Kyushu J. Math. 50 (1996) 459-470; Kyushu J. Math. 52 (1998) 439-454 MR 97m:53068; MR 99k:53081
  • [Per] G. Perelman, A diameter sphere theorem for manifolds of positive Ricci curvature, Math. Z. 218 (1995) 595-596 MR 96f:53056
  • [Pet] P. Petersen, Small excess and Ricci curvature, J. Geom. Anal. 1,4 (1991) 383-387 MR 93a:53031
  • [Poor] W. A. Poor, Some exotic spheres with positive Ricci curvature, Math. Annalen 216 (1975) 245-252 MR 53:3945
  • [PT] A. Petrunin and W. Tuschmann, Diffeomorphism Finiteness, Positive Pinching, and Second Homotopy, Geom. and Funct. Anal. (GAFA) 9 (1999) 736-774 MR 2000k:53031
  • [PZ] P. Petersen and S. Zhu, An excess sphere theorem, Ann. Sci. Ec. Norm. Super., IV. Ser. 26, No.2 (1993) 175-188 MR 94b:53077
  • [Shi1] K. Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc. 275 (1983) 811-819 MR 84c:53041
  • [Shi2] K. Shiohama, Sphere Theorems, F.J.E. Dillen and L.C.A. Verstraelen (edts.), Handbook of Differential Geometry, Vol. I, Elsevier Science B.V., Amsterdam 2000, 865-903 MR 2001c:53051
  • [Wi] F. Wilhelm, On radius, systole, and positive Ricci curvature, Math. Z. 218 (1995) 597-602 MR 96d:53040
  • [Wr] D. Wraith, Exotic spheres with positive Ricci curvature, J. Diff. Geom. 46 (1997) 638-649 MR 98i:53058
  • [Wu] J.-Y. Wu, A diameter pinching sphere theorem for positive Ricci curvature, Proc. A.M.S. 107,3 (1989) 797-802 MR 90h:53045
  • [Xia] C. Xia, Rigidity and sphere theorem for manifolds with positive Ricci curvature, Manuscr. Math. 85, No.1 (1994) 79-87 MR 95j:53057
  • [Yam] T. Yamaguchi, Lipschitz convergence of manifolds of positive Ricci curvature with large volume, Math. Ann. 284 (1989) 423-436 MR 90c:53114

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C20

Retrieve articles in all journals with MSC (2000): 53C20


Additional Information

Wilderich Tuschmann
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse, D-04103 Leipzig, Germany
Email: tusch@mis.mpg.de

DOI: https://doi.org/10.1090/S0002-9939-01-06384-5
Keywords: Sphere theorems, injectivity radius, exotic spheres, positive Ricci curvature
Received by editor(s): October 26, 2000
Published electronically: July 31, 2001
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society