Flat Hermitian Lie algebras are Kähler
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914396719022080 |
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| author | Zhang, Dongmei Zheng, Fangyang |
| author_facet | Zhang, Dongmei Zheng, Fangyang |
| contents | In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flat Hermitian Lie algebras are Kähler Zhang, Dongmei Zheng, Fangyang Differential Geometry 53C55 In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author. |
| title | Flat Hermitian Lie algebras are Kähler |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2504.08512 |