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Scalar curvatures in almost Hermitian geometry and some applications

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Abstract

On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the Kähler metric, the balanced metric, the locally conformal Kähler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier (2020) and a conjecture by Angella et al. (2018).

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 10831008, 11025103 and 11501505). The second author thanks Professor Jiagui Peng and Professor Xiaoxiang Jiao for their helpful suggestions and encouragement. Part of this work was done while the second author was visiting Laboratory of Mathematics for Nonlinear Science, Fudan University. The second author thanks it for the warm hospitality and support.

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Authors and Affiliations

  1. Institute of Mathematics, Fudan University, Shanghai, 200433, China

    Jixiang Fu

  2. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China

    Xianchao Zhou

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  1. Jixiang Fu
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  2. Xianchao Zhou
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Correspondence to Xianchao Zhou.

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Fu, J., Zhou, X. Scalar curvatures in almost Hermitian geometry and some applications. Sci. China Math. 65, 2583–2600 (2022). https://doi.org/10.1007/s11425-021-1943-8

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  • DOI: https://doi.org/10.1007/s11425-021-1943-8

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