Left invariant complex Finsler metrics on a complex Lie group
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917165519601664 |
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| author | Xu, Xiyun Xu, Ming |
| author_facet | Xu, Xiyun Xu, Ming |
| contents | In this paper, we consider a left invariant complex Finsler metric $F$ on a complex Lie group. Using the technique of invariant frames, we prove the following properties for $(G,F)$. First, the metric $F$ must be a complex Berwald metric. Second, its complex spray $χ=w^iδ_{z^i}$ on $T^{1,0}G\backslash0$ can be extended to a holomorphic tangent field on $T^{1,0}G$. If we view $χ$ as a real tangent field on $TG$, it coincides with the canonical bi-invariant spray structure on $G$. Third, we prove that the strongly Kähler, Kähler, and weakly Kähler properties for $F$ are equivalent. More over, $F$ is Kähler if and only if $G$ has an Abelian Lie algebra. Finally, we prove that the holomorphic sectional curvature vanishes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Left invariant complex Finsler metrics on a complex Lie group Xu, Xiyun Xu, Ming Differential Geometry In this paper, we consider a left invariant complex Finsler metric $F$ on a complex Lie group. Using the technique of invariant frames, we prove the following properties for $(G,F)$. First, the metric $F$ must be a complex Berwald metric. Second, its complex spray $χ=w^iδ_{z^i}$ on $T^{1,0}G\backslash0$ can be extended to a holomorphic tangent field on $T^{1,0}G$. If we view $χ$ as a real tangent field on $TG$, it coincides with the canonical bi-invariant spray structure on $G$. Third, we prove that the strongly Kähler, Kähler, and weakly Kähler properties for $F$ are equivalent. More over, $F$ is Kähler if and only if $G$ has an Abelian Lie algebra. Finally, we prove that the holomorphic sectional curvature vanishes. |
| title | Left invariant complex Finsler metrics on a complex Lie group |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.19353 |