Acta Mathematica

Classification of manifolds with weakly 1/4-pinched curvatures

Simon Brendle and Richard M. Schoen

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Abstract

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R2 has non-negative isotropic curvature.

Note

The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.

Article information

Source
Acta Math. Volume 200, Number 1 (2008), 1-13.

Dates
Received: 5 June 2007
Revised: 25 November 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891956

Digital Object Identifier
doi:10.1007/s11511-008-0022-7

Zentralblatt MATH identifier
1157.53020

Rights
2008 © Institut Mittag-Leffler

Citation

Brendle, Simon; Schoen, Richard M. Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200 (2008), no. 1, 1--13. doi:10.1007/s11511-008-0022-7. https://projecteuclid.org/euclid.acta/1485891956


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References

  • B erger, M., Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France, 83 (1955), 279–330.
  • — Les variétés Riemanniennes (1/4)-pincées. Ann. Scuola Norm. Sup. Pisa, 14 (1960), 161–170.
  • — Trois remarques sur les variétés riemanniennes à courbure positive. C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), 76–78.
  • B ony, J. M., Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble), 19:1 (1969), 277–304.
  • B rendle, S. & S choen, R. M., Manifolds with 1/4-pinched curvature are space forms. Preprint, 2007. arXiv:0705.0766.
  • C heeger, J. & G romoll, D., The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry, 6 (1971/72), 119–128.
  • C how, B. & K nopf, D., New Li–Yau–Hamilton inequalities for the Ricci flow via the space-time approach. J. Differential Geom., 60 (2002), 1–54.
  • C how, B. & Y ang, D., Rigidity of nonnegatively curved compact quaternionic-Kähler manifolds. J. Differential Geom., 29 (1989), 361–372.
  • H amilton, R. S., Four-manifolds with positive curvature operator. J. Differential Geom., 24 (1986), 153–179.
  • J oyce, D. D., Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.
  • K lingenberg, W., Über Riemannsche Mannigfaltigkeiten mit nach oben beschränkter Krümmung. Ann. Mat. Pura Appl., 60 (1962), 49–59.
  • Riemannian Geometry. de Gruyter Studies in Mathematics, 1. de Gruyter, Berlin, 1982.
  • K obayashi, S. & N omizu, K., Foundations of Differential Geometry. Vol. I. Wiley Classics Library. Wiley, New York, 1996.
  • Foundations of Differential Geometry. Vol. II. Wiley Classics Library. Wiley, New York, 1996.
  • M ok, N., The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differential Geom., 27 (1988), 179–214.
  • P etersen, P., Riemannian Geometry, second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006.
  • S imons, J., On the transitivity of holonomy systems. Ann. of Math., 76 (1962), 213–234.