×

zbMATH — the first resource for mathematics

Kähler manifolds and 1/4-pinching. (English) Zbl 0725.53068
The complex sectional curvature was introduced by Y. T. Siu [Ann. Math., II. Ser. 112, 73-111 (1980; Zbl 0517.53058)]. Later, J. H. Sampson has used this curvature in the study of harmonic maps from Kähler to Riemannian manifolds. With these results as starting point the author studies the restrictions coming from the existence of a metric with negative complex sectional curvature.
The main results are the following two theorems: Theorem 1. Let M be a compact manifold of dimension greater than 2. If M admits a metric with negative complex sectional curvature at every point then M cannot admit a Kähler metric. Theorem 2. Let (M,g) be a compact Kähler manifold, and let h be another metric for M. Assume that (M,h) has negative sectional curvature with pointwise pinching at least 1/4. Then (M,h) is a complex hyperbolic spaceform \(\pm biholomorphic\) to (M,g).
Reviewer: N.Bokan (Beograd)

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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
[2] 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
[3] J. Carlson and D. Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces , Inst. Hautes Études Sci. Publ. Math. (1989), no. 69, 173-201. · Zbl 0695.58010 · doi:10.1007/BF02698844 · numdam:PMIHES_1989__69__173_0 · eudml:104050
[4] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds , Amer. J. Math. 86 (1964), 109-160. JSTOR: · Zbl 0122.40102 · doi:10.2307/2373037 · links.jstor.org
[5] M. Gromov, Synthetic geometry in Riemannian manifolds , Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 415-419. · Zbl 0427.53018
[6] S. Marchiafava, Variétés riemanniennes dont le tenseur de courbure est celui d’un espace symétrique de rang un , C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 7, 463-466. · Zbl 0512.53024
[7] 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
[8] J. H. Sampson, Applications of harmonic maps to Kähler geometry , Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984), Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 125-134. · Zbl 0605.58019 · doi:10.1090/conm/049/833809
[9] Y. T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds , Ann. of Math. (2) 112 (1980), no. 1, 73-111. JSTOR: · Zbl 0517.53058 · doi:10.2307/1971321 · links.jstor.org
[10] R. J. Spatzier and R. J. Zimmer, Fundamental groups of negatively curved manifolds and actions of semisimple groups , · Zbl 0744.57022 · doi:10.1016/0040-9383(91)90041-2
[11] M. Ville, On \(1\over 4\)-pinched \(4\)-dimensional Riemannian manifolds of negative curvature , Ann. Global Anal. Geom. 3 (1985), no. 3, 329-336. · Zbl 0575.53020 · doi:10.1007/BF00130484
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.