×

zbMATH — the first resource for mathematics

Metrics with nonnegative isotropic curvature. (English) Zbl 0804.53058
The notion of positive isotropic curvature generalizes classical curvature conditions such as strict pointwise quarter pinching and positive curvature operator. Among the results obtained is the following theorem. If \((M^{2n},g)\) is an even-dimensional closed Riemannian manifold, then (a) \(b_ 2(M)= 0\) if \(g\) has positive isotropic curvature, and (b) if \(g\) is locally irreducible, has nonnegative isotropic curvature, and \(b_ 2(M)\neq 0\), then \(b_ 2(M)= 1\) and \((M,g)\) is Kähler with positive first Chern class. The paper concludes with results on four-manifolds with nonnegative isotropic curvature.

MSC:
53C20 Global Riemannian geometry, including pinching
53C55 Global differential geometry of Hermitian and Kählerian manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Berger, Sur quelques variétés riemanniennes suffisamment pincées , Bull. Soc. Math. France 88 (1960), 57-71. · Zbl 0096.15503 · numdam:BSMF_1960__88__57_0 · eudml:86993
[2] M. Berger, Pincement riemannien et pincement holomorphe , Ann. Scuola Norm. Sup. Pisa (3) 14 (1960), 151-159. · Zbl 0094.34901 · numdam:ASNSP_1960_3_14_2_151_0 · eudml:83243
[3] M. Berger, Sur les variétés à opérateur de courbure positif , C. R. Acad. Sci.Ser. Paris 253 (1961), 2832-2834. · Zbl 0196.54402
[4] M. Berger, Sur quelques variétés d’Einstein compactes , Ann. Mat. Pura Appl. (4) 53 (1961), 89-95. · Zbl 0115.39301 · doi:10.1007/BF02417787
[5] A. Besse, Einstein manifolds , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001
[6] J. P. Bourguignon, Les variétés de dimension \(4\) à signature non nulle dont la courbure est harmonique sont d’Einstein , Invent. Math. 63 (1981), no. 2, 263-286. · Zbl 0456.53033 · doi:10.1007/BF01393878 · eudml:142798
[7] A. Derdzinski, Self-dual Kähler manifolds and Einstein manifolds of dimension four , Compositio Math. 49 (1983), no. 3, 405-433. · Zbl 0527.53030 · numdam:CM_1983__49_3_405_0 · eudml:89617
[8] S. Goldberg and S. Kobayashi, Holomorphic bisectional curvature , J. Differential Geometry 1 (1967), 225-233. · Zbl 0169.53202
[9] M. Gromov and H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature , Ann. of Math. (2) 111 (1980), no. 3, 423-434. JSTOR: · Zbl 0463.53025 · doi:10.2307/1971103 · links.jstor.org
[10] R. Hamilton, Three-manifolds with positive Ricci curvature , J. Differential Geom. 17 (1982), no. 2, 255-306. · Zbl 0504.53034 · www.intlpress.com · euclid:jdg/1214436922
[11] R. Hamilton, Four-manifolds with positive curvature operator , J. Differential Geom. 24 (1986), no. 2, 153-179. · Zbl 0628.53042
[12] S. Helgason, Differential geometry, Lie groups, and symmetric spaces , Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. · Zbl 0451.53038
[13] N. Hitchin, Compact four-dimensional Einstein manifolds , J. Differential Geometry 9 (1974), 435-441. · Zbl 0281.53039
[14] S. Kobayashi, On compact Kähler manifolds with positive definite Ricci tensor , Ann. of Math. (2) 74 (1961), 570-574. JSTOR: · Zbl 0107.16002 · doi:10.2307/1970298 · links.jstor.org
[15] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II , Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0175.48504
[16] H. B. Lawson and M. L. Michelsohn, Spin geometry , Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. · Zbl 0688.57001
[17] C. LeBrun, On the topology of self-dual \(4\)-manifolds , Proc. Amer. Math. Soc. 98 (1986), no. 4, 637-640. · Zbl 0606.53029 · doi:10.2307/2045741
[18] M. Micallef and J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes , Ann. of Math. (2) 127 (1988), no. 1, 199-227. JSTOR: · Zbl 0661.53027 · doi:10.2307/1971420 · links.jstor.org
[19] M. Micallef and J. Wolfson, The second variation of area of minimal surfaces in four-manifolds , Math. Ann. 295 (1993), no. 2, 245-267. · Zbl 0788.58016 · doi:10.1007/BF01444887 · eudml:165042
[20] Min-Oo Maung and E. Ruh, Curvature deformations , Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 180-190. · Zbl 0634.53030
[21] A. Polombo, De nouvelles formules de Weitzenböck pour des endomorphismes harmoniques. Applications géométriques , Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 4, 393-428. · Zbl 0812.53044 · numdam:ASENS_1992_4_25_4_393_0 · eudml:82323
[22] R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature , Manuscripta Math. 28 (1979), no. 1-3, 159-183. · Zbl 0423.53032 · doi:10.1007/BF01647970 · eudml:154634
[23] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math. 92 (1988), no. 1, 47-71. · Zbl 0658.53038 · doi:10.1007/BF01393992 · eudml:143558
[24] W. Seaman, On manifolds with nonnegative curvature on totally isotropic 2-planes , Trans. Amer. Math. Soc. 338 (1993), no. 2, 843-855. · Zbl 0785.53034 · doi:10.2307/2154431
[25] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class , Invent. Math. 101 (1990), no. 1, 101-172. · Zbl 0716.32019 · doi:10.1007/BF01231499 · eudml:143800
[26] J. A. Wolf, Spaces of constant curvature , Publish or Perish Inc., Boston, Mass., 1977. · Zbl 0373.57025
[27] S.-T. Yau, On the curvature of compact Hermitian manifolds , Invent. Math. 25 (1974), 213-239. · Zbl 0299.53039 · doi:10.1007/BF01389728 · eudml:142288
[28] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I , Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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.