Abstract
By associating to a strongly pseudoconvex complex Finsler metric F a Hermitian tensor \({T}\) of type (1, 1), we prove that a weakly Kähler Finsler metric is a Kähler Finsler metric if and only if \({T \equiv 0}\), and a weakly complex Berwald metric is a weakly Kähler Finsler metric if and only if it is a Kähler Finsler metric. We introduce a class of explicitly constructed smooth complex Finsler metrics, called the general complex (\({\alpha, \beta)}\) metric, where \({\alpha}\) is a Hermitian metric and \({\beta}\) is a complex differential form of type (1, 0) on a complex manifold. The holomorphic curvature of the general complex \({(\alpha, \beta)}\) metrics is derived, and a necessary and sufficient condition for these metrics to be weakly Kähler Finsler metrics is obtained. Under the assumption that \({\beta}\) is holomorphic or closed, necessary and sufficient conditions for this class of metrics to be weakly complex Berwald metrics, complex Berwald metrics and Kähler Finsler metrics are obtained, respectively.
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27 September 2017
An erratum to this article has been published.
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An erratum to this article is available at https://doi.org/10.1007/s00025-017-0745-x.
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Xia, H., Zhong, C. On a Class of Smooth Complex Finsler Metrics. Results Math 71, 657–686 (2017). https://doi.org/10.1007/s00025-016-0543-x
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DOI: https://doi.org/10.1007/s00025-016-0543-x