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)

 

 

Classification of almost quarter-pinched manifolds


Authors: Peter Petersen and Terence Tao
Journal: Proc. Amer. Math. Soc. 137 (2009), 2437-2440
MSC (2000): Primary 53C21
Published electronically: January 30, 2009
MathSciNet review: 2495279
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if a simply connected manifold is almost quarter-pinched, then it is diffeomorphic to a CROSS or a sphere.


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

  • 1. Uwe Abresch and Wolfgang T. Meyer, Pinching below \frac14, injectivity radius, and conjugate radius, J. Differential Geom. 40 (1994), no. 3, 643–691. MR 1305984
  • 2. Uwe Abresch and Wolfgang T. Meyer, A sphere theorem with a pinching constant below 1\over4, J. Differential Geom. 44 (1996), no. 2, 214–261. MR 1425576
  • 3. Marcel Berger, Sur les variétés riemanniennes pincées juste au-dessous de 1/4, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 135–150 (loose errata) (French). MR 699491
  • 4. C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167 (2008), 1079-1097.
  • 5. S. Brendle and R. Schoen, Manifolds with $ 1/4$-pinched curvature are space forms. J. Amer. Math. Soc. 22 (2009), 287-307.
  • 6. Simon Brendle and Richard M. Schoen, Classification of manifolds with weakly 1/4-pinched curvatures, Acta Math. 200 (2008), no. 1, 1–13. MR 2386107, 10.1007/s11511-008-0022-7
  • 7. Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. MR 2061425
  • 8. Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006. MR 2274812
  • 9. O. Durumeric, A generalization of Berger’s theorem on almost \frac14-pinched manifolds. II, J. Differential Geom. 26 (1987), no. 1, 101–139. MR 892033
  • 10. Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255–306. MR 664497
  • 11. Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
  • 12. Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
  • 13. Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  • 14. P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv:math.DG/0805.0812
  • 15. Xiaochun Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. (2) 143 (1996), no. 2, 397–411. MR 1381991, 10.2307/2118648

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C21

Retrieve articles in all journals with MSC (2000): 53C21


Additional Information

Peter Petersen
Affiliation: Department of Mathematics, University of California, 520 Portola Plaza, Los Angeles, California 90095
Email: petersen@math.ucla.edu

Terence Tao
Affiliation: Department of Mathematics, University of California, 520 Portola Plaza, Los Angeles, California 90095
Email: tao@math.ucla.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09802-5
Received by editor(s): July 11, 2008
Received by editor(s) in revised form: October 16, 2008
Published electronically: January 30, 2009
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2009 American Mathematical Society