A pinching theorem for homotopy spheres

Grove, Karsten; Petersen, Peter

doi:10.1090/S0894-0347-1990-1049696-6

A pinching theorem for homotopy spheres
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by Karsten Grove and Peter Petersen

J. Amer. Math. Soc. 3 (1990), 671-677

DOI: https://doi.org/10.1090/S0894-0347-1990-1049696-6

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References

Uwe Abresch and Detlef Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355–374. MR 1030656, DOI 10.1090/S0894-0347-1990-1030656-6
Michael T. Anderson, Metrics of positive Ricci curvature with large diameter, Manuscripta Math. 68 (1990), no. 4, 405–415. MR 1068264, DOI 10.1007/BF02568774
Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
J.-H. Eschenburg, Diameter, volume, and topology for positive Ricci curvature, J. Differential Geom. 33 (1991), no. 3, 743–747. MR 1100210
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
Karsten Grove, Metric differential geometry, Differential geometry (Lyngby, 1985) Lecture Notes in Math., vol. 1263, Springer, Berlin, 1987, pp. 171–227. MR 905882, DOI 10.1007/BFb0078613
Karsten Grove and Peter Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), no. 1, 195–206. MR 951512, DOI 10.2307/1971439
Karsten Grove and Peter Petersen, Manifolds near the boundary of existence, J. Differential Geom. 33 (1991), no. 2, 379–394. MR 1094462

Excess and rigidity of inner metric spaces

K. Grove, P. Petersen V, and J. Y. Wu, Controlled topology in geometry, Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 2, 181–183. MR 974425, DOI 10.1090/S0273-0979-1989-15755-8
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
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Yoe Itokawa, The topology of certain Riemannian manifolds with positive Ricci curvature, J. Differential Geometry 18 (1983), no. 1, 151–155. MR 697986
Yukio Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z. 206 (1991), no. 2, 255–264. MR 1091941, DOI 10.1007/BF02571341
Peter Petersen V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), no. 2, 387–395. MR 1037407

On the upper estimate of the Betti numbers of long Riemannian manifolds

Katsuhiro Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc. 275 (1983), no. 2, 811–819. MR 682734, DOI 10.1090/S0002-9947-1983-0682734-1

Generalized Poincaré’s conjecture in dimensions greater than four

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Jyh Yang Wu, A diameter pinching sphere theorem for positive Ricci curvature, Proc. Amer. Math. Soc. 107 (1989), no. 3, 797–802. MR 984822, DOI 10.1090/S0002-9939-1989-0984822-X

Collapsing and pinching in lower curvature bound