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A note on holomorphic sectional curvature of a hermitian manifold

Part of: Global differential geometry Local differential geometry Complex manifolds

Published online by Cambridge University Press:  04 March 2022

Hongjun Li Open the ORCID record for Hongjun Li [Opens in a new window]  and
Chunhui Qiu
Hongjun Li
Affiliation:
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China Email: lihj@vip.henu.edu.cn
Chunhui Qiu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China Email: chqiu@xmu.edu.cn
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Abstract

As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.

Keywords

holomorphic sectional curvaturesectional curvatureKahler manifold

MSC classification

Primary: 53C55: Hermitian and Kählerian manifolds
Secondary: 53B35: Hermitian and Kählerian structures 32Q15: Kähler manifolds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 739 - 745
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

*

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12001165, 11971401), Postdoctoral Research Foundation of China (Grant No. 2019M652513), Postdoctoral Research Foundation of Henan Province (Grant No. 19030050).

References

Lu, Q.(K. Look) and Xu, Y.(I. Hsü), A note on transitive domains, Acta Math. Sin. 11(1) (1961), 11–23 (Chinese).Google Scholar
Moroianu, A., Lectures on Kähler geometry (Cambridge University Press, Cambridge, 2007).CrossRefGoogle Scholar
Siu, Y., The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. Math. 112(1) (1980), 73–111.CrossRefGoogle Scholar
Zheng, F., Complex differential geometry (American Mathematical Society, International Press, Boston, 2000).Google Scholar