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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)

53C55 Global differential geometry of Hermitian and Kählerian manifolds
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