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History and Development of Sphere Theorems in Riemannian Geometry  

Cho, Min-Shik (Department of Mathematics Education, Korea National University of Education)
Publication Information
Journal for History of Mathematics / v.24, no.3, 2011 , pp. 23-35 More about this Journal
Abstract
The sphere theorem is one of the main streams in modern Riemannian geometry. In this article, we survey developments of pinching theorems from the classical one to the recent differentiable pinching theorem. Also we include sphere theorems of metric invariants such as diameter and radius with historical view point.
Keywords
Sphere Theorem; Pinching Theorem; Metric Invariant;
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1 V. A. Toponogov, Riemannian spaces with curvature bounded below, Uspekhi. Mat. Nauk 14(1959), 87-130.
2 F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11(2001), 519-560.   DOI   ScienceOn
3 B. Wilking, Nonnegatively and Positively Curved Manifolds, Surveys in differential geometry, Vol. XI: Metric and Comparison Geometry, ed. Grove K. and Cheeger, J. 7, Internat. Press(2007), 25-62.
4 K. Shiohama & T. Yamaguch, Positively curved manifolds with restricted diameters, Perspectives in Math., Vol. 8: Geometry of Manifolds, ed. Shiohama K., Academic Press, Boston(1989), 345-350.
5 M. Sugimoto, K. Shiohama, & H. Karcher, On the differentiable pinching problem, Math. Ann. 195(1971), 1-16.   DOI   ScienceOn
6 O. Durumeric, A generalization of Berger's theorem on almost 1/4-pinched manifolds II, J. Diff. Geom. 26(1987), 101-139.   DOI
7 Y. Suyama, A differentiable sphere theorem by curvature pinching, II, Tohoku Math. J. 47(1995), 15-29.   DOI   ScienceOn
8 P. Petersen & F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv:Math/DG/0805.0812v3.
9 S. Y. Cheng, Eigenvalue comparision theorems and its geometric applications, Math. Z. 143(1975), 289-297.   DOI   ScienceOn
10 T. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124(1996), 193-214.   DOI   ScienceOn
11 G. Perelman, A diameter sphere theorem for manifolds of positive Ricci curvature, Math. Z. 218(1995), 595-596.   DOI   ScienceOn
12 H. Rauch, A contribution to differential geometry in the large, Ann. of Math. 54(1951), 38-55.   DOI   ScienceOn
13 S. B. Myers, Riemannian manifolds with positive mean curvature, Duke. Math. J. 8(1941), 401-404.   DOI
14 Y. Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z. 206(1991), 255-264.   DOI   ScienceOn
15 S. Brendle & R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. of Amer. Math. Soc. 22(2009), 287-307.
16 M. Berger, Sur les varietes riemanniennes pincees juste au-dessous de 1/4, Ann. Inst. Fourier(Grenoble) 33(1983), 135-150.
17 G. Perelman, Alexandrov's spaces with curvature bounded from below II, preprint.
18 P. Petersen, Riemannian Geometry(2nd ed.), Graduate Texts in Mathematics 171, Springer-Verlarg, New York, 2006.
19 W. Klingenberg, Uber Riemannsche Mannigfaltigkeiten mit positiver Krummung, Comment. Math. Helv. 35(1961), 47-54.   DOI   ScienceOn
20 M. J. Micallef & J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. 127(1988), 199-227.   DOI   ScienceOn
21 U. Abresch & W. Meyer, A sphere theorem with a pinching constant below $\frac{1}{4}$, J. Diff. Geom. 44(1996), 214-261.   DOI
22 M. Anderson, Metrics of positive Ricci curvature with large diameter, Manu. Math. 68(1990), 405-415.   DOI   ScienceOn
23 K. Grove & K. Shiohama, A generalized sphere theorem, Ann. of Math. 106(1977) 201-211.   DOI   ScienceOn
24 S. B. Myers, Riemannian manifolds in the large, Duke. Math. J. 1(1935), 39-49.   DOI
25 K. Grove & F. Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine. Angew. Math. 487(1997), 201-217.
26 H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95(1926), 313-339.   DOI   ScienceOn
27 M. Berger, Les varietes Riemanniennes 1/4-pincees, Ann. Scuola Norm. Sup. Pisa 14(1960), 161-170.
28 H. Im Hof & E. Ruh, An equivariant pinching theorem, Comment. Math. Helv. 50(1975), 389-401.   DOI   ScienceOn
29 D. Gromoll, Differenzierbare Strukturen und Metriken positive Krummung auf Spharen, Math. Ann. 164(1966), 353-371.   DOI   ScienceOn
30 D. Gromoll & W. Meyer, An exotic sphere with nonnegatively sectional curvature, Ann. of Math. 100(1974), 401-406.   DOI   ScienceOn
31 K. Grove & P. Petersen, A radius sphere theorem, Invent. Math. 112(1993), 577-583.   DOI   ScienceOn