History and Development of Sphere Theorems in Riemannian Geometry

리만기하학에서 구면정리의 발전과 역사

  • Cho, Min-Shik (Department of Mathematics Education, Korea National University of Education)
  • 조민식 (한국교원대학교 수학교육과)
  • Received : 2011.06.27
  • Accepted : 2011.08.22
  • Published : 2011.08.31


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.

본 논문에서는 어떤 기하학적 양이 핀치되어 있으면 위상적 또는 미분위상적인 구면이 된다는 구면정리의 발전과 역사를 다루었다. 단면곡률의 핀칭과 관련하여, 고전적 핀칭 구면 정리에서 최근에 증명된 기념비적인 미분 핀칭 구면정리로 발전하는 과정의 역사를 기술하였다. 또 직경, 반경, 부피 등과 관련하여 계량불변량 구면정리와 미분 계량불변량 구면정리의 발전의 과정을 소개하였고, 구면정리와 관련된 미해결문제에 대한 역사를 기술하였다.



Supported by : 한국교원대학교


  1. U. Abresch & W. Meyer, A sphere theorem with a pinching constant below $\frac{1}{4}$, J. Diff. Geom. 44(1996), 214-261. https://doi.org/10.4310/jdg/1214458972
  2. M. Anderson, Metrics of positive Ricci curvature with large diameter, Manu. Math. 68(1990), 405-415. https://doi.org/10.1007/BF02568774
  3. M. Berger, Les varietes Riemanniennes 1/4-pincees, Ann. Scuola Norm. Sup. Pisa 14(1960), 161-170.
  4. M. Berger, Sur les varietes riemanniennes pincees juste au-dessous de 1/4, Ann. Inst. Fourier(Grenoble) 33(1983), 135-150.
  5. S. Brendle & R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. of Amer. Math. Soc. 22(2009), 287-307.
  6. S. Y. Cheng, Eigenvalue comparision theorems and its geometric applications, Math. Z. 143(1975), 289-297. https://doi.org/10.1007/BF01214381
  7. T. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124(1996), 193-214. https://doi.org/10.1007/s002220050050
  8. O. Durumeric, A generalization of Berger's theorem on almost 1/4-pinched manifolds II, J. Diff. Geom. 26(1987), 101-139. https://doi.org/10.4310/jdg/1214441178
  9. D. Gromoll, Differenzierbare Strukturen und Metriken positive Krummung auf Spharen, Math. Ann. 164(1966), 353-371. https://doi.org/10.1007/BF01350046
  10. D. Gromoll & W. Meyer, An exotic sphere with nonnegatively sectional curvature, Ann. of Math. 100(1974), 401-406. https://doi.org/10.2307/1971078
  11. K. Grove & K. Shiohama, A generalized sphere theorem, Ann. of Math. 106(1977) 201-211. https://doi.org/10.2307/1971164
  12. K. Grove & P. Petersen, A radius sphere theorem, Invent. Math. 112(1993), 577-583. https://doi.org/10.1007/BF01232447
  13. K. Grove & F. Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine. Angew. Math. 487(1997), 201-217.
  14. H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95(1926), 313-339. https://doi.org/10.1007/BF01206614
  15. H. Im Hof & E. Ruh, An equivariant pinching theorem, Comment. Math. Helv. 50(1975), 389-401. https://doi.org/10.1007/BF02565758
  16. W. Klingenberg, Uber Riemannsche Mannigfaltigkeiten mit positiver Krummung, Comment. Math. Helv. 35(1961), 47-54. https://doi.org/10.1007/BF02567004
  17. 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. https://doi.org/10.2307/1971420
  18. S. B. Myers, Riemannian manifolds in the large, Duke. Math. J. 1(1935), 39-49. https://doi.org/10.1215/S0012-7094-35-00105-3
  19. S. B. Myers, Riemannian manifolds with positive mean curvature, Duke. Math. J. 8(1941), 401-404. https://doi.org/10.1215/S0012-7094-41-00832-3
  20. Y. Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z. 206(1991), 255-264. https://doi.org/10.1007/BF02571341
  21. G. Perelman, A diameter sphere theorem for manifolds of positive Ricci curvature, Math. Z. 218(1995), 595-596. https://doi.org/10.1007/BF02571925
  22. G. Perelman, Alexandrov's spaces with curvature bounded from below II, preprint.
  23. P. Petersen, Riemannian Geometry(2nd ed.), Graduate Texts in Mathematics 171, Springer-Verlarg, New York, 2006.
  24. P. Petersen & F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv:Math/DG/0805.0812v3.
  25. H. Rauch, A contribution to differential geometry in the large, Ann. of Math. 54(1951), 38-55. https://doi.org/10.2307/1969309
  26. 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.
  27. M. Sugimoto, K. Shiohama, & H. Karcher, On the differentiable pinching problem, Math. Ann. 195(1971), 1-16. https://doi.org/10.1007/BF02059412
  28. Y. Suyama, A differentiable sphere theorem by curvature pinching, II, Tohoku Math. J. 47(1995), 15-29. https://doi.org/10.2748/tmj/1178225633
  29. V. A. Toponogov, Riemannian spaces with curvature bounded below, Uspekhi. Mat. Nauk 14(1959), 87-130.
  30. F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11(2001), 519-560. https://doi.org/10.1007/BF02922018
  31. 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.