Nonpositively curved almost Hermitian metrics on product of compact almost complex manifolds. (English) Zbl 1326.53106

F. Zheng [Ann. Math. (2) 137, No. 3, 671–673 (1993; Zbl 0779.53045)] gave a classification of Kähler metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds. The author of the paper under review extended this classification to Hermitian metrics [Proc. Am. Math. Soc. 139, No. 4, 1469–1472 (2011; Zbl 1216.53050)]. In the present paper, he obtains a classification of almost Hermitian metrics with the above properties. The main difficulty encountered in this generalization was the lack of a curvature formula for almost Hermitian metrics. The author introduces a new holomorphic frame and obtains such a formula with respect to this frame. The new frames might have a role in studying some other problems from geometry of complex manifolds.


53C55 Global differential geometry of Hermitian and Kählerian manifolds
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