Curvature of left-invariant complex Finsler metric on Lie groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915701266055168 |
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| author | Luo, Kuankuan Xiao, Wei Zhong, Chunping |
| author_facet | Luo, Kuankuan Xiao, Wei Zhong, Chunping |
| contents | Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a Kähler-Finsler metric and a weakly Kähler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a Kähler-Berwald metric iff $G$ is an Abelian complex Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curvature of left-invariant complex Finsler metric on Lie groups Luo, Kuankuan Xiao, Wei Zhong, Chunping Differential Geometry 53C30, 53C60 Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a Kähler-Finsler metric and a weakly Kähler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a Kähler-Berwald metric iff $G$ is an Abelian complex Lie group. |
| title | Curvature of left-invariant complex Finsler metric on Lie groups |
| topic | Differential Geometry 53C30, 53C60 |
| url | https://arxiv.org/abs/2512.24791 |