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Characterizations of Complex Finsler Metrics

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Abstract

Munteanu (Complex spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Academic Publishers, Dordrecht, 2004) defined the canonical connection associated to a strongly pseudoconvex complex Finsler manifold (M, F). We first prove that the holomorphic sectional curvature tensors of the canonical connection coincide with those of the Chern–Finsler connection associated to F if and only if F is a Kähler-Finsler metric. We also investigate the relationship of the Ricci curvatures (resp. scalar curvatures) of these two connections when M is compact. As an application, two characterizations of balanced complex Finsler metrics are given. Next, we obtain a sufficient and necessary condition for a balanced complex Finsler metric to be Kähler-Finsler. Finally, we investigate conformal transformations of a balanced complex Finsler metric.

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Acknowledgements

The authors would like to thank Prof. Chunhui Qiu and Chunping Zhong for giving us useful discussions. The authors are very grateful to the referees for providing many valuable suggestions. This research is supported by the National Natural Science Foundation of China (Grant Nos. 12001165, 12071386, 11701494), the Nanhu Scholars Program for Young Scholars of Xinyang Normal University and the Key Research Project of Henan Higher Education Institutions(China) (No. 22A110021).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Hongjun Li

  2. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Hongchuan Xia

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  1. Hongjun Li
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  2. Hongchuan Xia
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Correspondence to Hongchuan Xia.

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Dedicated to Professor Yichao Xu on the occasion of his 90th birthday.

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Li, H., Xia, H. Characterizations of Complex Finsler Metrics. J Geom Anal 33, 208 (2023). https://doi.org/10.1007/s12220-023-01272-3

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