×

zbMATH — the first resource for mathematics

Classification of manifolds with weakly 1/4-pinched curvatures. (English) Zbl 1157.53020
In the present paper the authors classify up to diffeomeorphism Riemannian manifolds whose sectional curvature is weakly \(1/4\)-pinched, i.e., \(0\leq K(\pi_1)\leq 4K(\pi_2)\) for all two planes \(\pi_1, \pi_2\) in the tangent space at any point. Namely, they show that if \(M\) is a compact Riemannian manifold of dimension greater than four with weakly \(1/4\)-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, the authors classify all compact, locally irreducible Riemannian manifolds \(M\) with the property that \(M \times \mathbb{R}^2\) has non-negative isotropic curvature.

MSC:
53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berger, M., Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France, 83 (1955), 279–330. · Zbl 0068.36002
[2] – Les variétés Riemanniennes (1/4)-pincées. Ann. Scuola Norm. Sup. Pisa, 14 (1960), 161–170.
[3] – Trois remarques sur les variétés riemanniennes à courbure positive. C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), 76–78. · Zbl 0143.45001
[4] Bony, 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. · Zbl 0176.09703
[5] Brendle, S. & Schoen, R. M., Manifolds with 1/4-pinched curvature are space forms. Preprint, 2007. arXiv:0705.0766. · Zbl 1251.53021
[6] Cheeger, J. & Gromoll, D., The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry, 6 (1971/72), 119–128. · Zbl 0223.53033
[7] Chow, B. & Knopf, D., New Li–Yau–Hamilton inequalities for the Ricci flow via the space-time approach. J. Differential Geom., 60 (2002), 1–54. · Zbl 1048.53026
[8] Chow, B. & Yang, D., Rigidity of nonnegatively curved compact quaternionic-Kähler manifolds. J. Differential Geom., 29 (1989), 361–372. · Zbl 0668.53041
[9] Hamilton, R. S., Four-manifolds with positive curvature operator. J. Differential Geom., 24 (1986), 153–179. · Zbl 0628.53042
[10] Joyce, D. D., Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000. · Zbl 1027.53052
[11] Klingenberg, W., Über Riemannsche Mannigfaltigkeiten mit nach oben beschränkter Krümmung. Ann. Mat. Pura Appl., 60 (1962), 49–59. · Zbl 0112.13603 · doi:10.1007/BF02412764
[12] – Riemannian Geometry. de Gruyter Studies in Mathematics, 1. de Gruyter, Berlin, 1982.
[13] Kobayashi, S. & Nomizu, K., Foundations of Differential Geometry. Vol. I. Wiley Classics Library. Wiley, New York, 1996. · Zbl 0119.37502
[14] – Foundations of Differential Geometry. Vol. II. Wiley Classics Library. Wiley, New York, 1996. · Zbl 0866.53007
[15] Mok, N., The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differential Geom., 27 (1988), 179–214. · Zbl 0642.53071
[16] Petersen, P., Riemannian Geometry, second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006. · Zbl 1220.53002
[17] Simons, J., On the transitivity of holonomy systems. Ann. of Math., 76 (1962), 213–234. · Zbl 0106.15201 · doi:10.2307/1970273
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.