Compact Hermitian surfaces and isotropic curvature. (English) Zbl 0957.53035

Apostolov, Vestislav; Davidov, Johann

Ill. J. Math. 44, No. 2, 438-451 (2000).

In the present paper, the authors consider the question: which compact Hermitian surfaces have non-negative sectional curvature for all 2-planes which are totally isotropic with respect to the complexified metric (or shortly non-negative isotropic curvature)? M. Micaleff and J. Moore proved that every compact simply connected Riemannian manifold of dimension \(\geq 4\) with positive isotropic curvature is homeomorphic to the unit sphere [Ann. Math 127, 199-227 (1988; Zbl 0661.53024)]. Recently R. Hamilton has proved that, in case of dimension four, the manifold is actually diffeomorphic to the unit sphere [Commun. Anal. Geom. 5, 1-92 (1997; Zbl 0892.53018)]. He also has provided a diffeomorphic classification of four manifolds with positive isotropic curvature and fundamental group \(\mathbb{Z}_2\) or \(\mathbb{Z}\). The authors prove that a compact Hermitian surface of non-negative isotropic curvature is either biholomorphically isometric to one of: the flat Kählerian torus, the flat Kählerian hyper-elliptic surface, the product \(\mathbb{C} P^1\times \mathbb{C} P^1\) equipped with a product metric with non-negative sum of Gauss curvatures of the factors, a unitary flat \(\mathbb{C} P^1\)-bundle over Riemann surface \(\Gamma\) of genus \(\geq 1\) with a metric which is locally a product of metrics on \(\Gamma\) and \(\mathbb{C} P^1\) with the sum of the Gauss curvatures of the factors non-negative; or is biholomorphic to \(\mathbb{C} P^2\) or a Hopf surface. The proof uses a detailed analysis of the properties of the zero-order term in a Weitzenböck formula and the Kodaira’s classification of compact complex surfaces.

Reviewer: Gueo Grantcharov (Storrs)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Kodaira classification; compact Hermitian surface; isotropic curvature; Gauss curvature; Weitzenböck formula

Citations:

Zbl 0661.53024; Zbl 0892.53018