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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.00579 |
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| _version_ | 1866911877484773376 |
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| author | Ni, Lei Zheng, Fangyang |
| author_facet | Ni, Lei Zheng, Fangyang |
| contents | We apply the algebraic consideration of holonomy systems to study Hermitian manifolds whose Chern connection is Ambrose-Singer and prove structure theorems for such manifolds. The main result (Theorem 1.2) asserts that the universal cover of such a Hermitian manifold must be the product of a complex Lie group and Hermitian symmetric spaces, which was previously proved up to complex dimension four by the authors. This in some sense is the Hermitian version of Cartan's classification of Hermitian symmetric spaces. We also obtain results on Hermitian manifolds whose Bismut connection is Ambrose-Singer when the complex dimension $n\le 4$. Furthermore we discuss a project of classifying such manifolds via Alekseevskiĭ and Kimel\!\'{}\!fel\!\'{}\!d type theorems, which we establish for the family of the Gauduchon connections on a compact Hermitian manifold except for the Bismut connection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00579 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A classification of locally Chern homogeneous Hermitian manifolds Ni, Lei Zheng, Fangyang Differential Geometry 53C55 We apply the algebraic consideration of holonomy systems to study Hermitian manifolds whose Chern connection is Ambrose-Singer and prove structure theorems for such manifolds. The main result (Theorem 1.2) asserts that the universal cover of such a Hermitian manifold must be the product of a complex Lie group and Hermitian symmetric spaces, which was previously proved up to complex dimension four by the authors. This in some sense is the Hermitian version of Cartan's classification of Hermitian symmetric spaces. We also obtain results on Hermitian manifolds whose Bismut connection is Ambrose-Singer when the complex dimension $n\le 4$. Furthermore we discuss a project of classifying such manifolds via Alekseevskiĭ and Kimel\!\'{}\!fel\!\'{}\!d type theorems, which we establish for the family of the Gauduchon connections on a compact Hermitian manifold except for the Bismut connection. |
| title | A classification of locally Chern homogeneous Hermitian manifolds |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2301.00579 |