The Chern Sectional Curvature of a Hermitian Manifold
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929281690501120 |
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| author | Cao, Pandeng Li, Hongjun |
| author_facet | Cao, Pandeng Li, Hongjun |
| contents | On a Hermitian manifold, the Chern connection can induce a metric connection on the background Riemannian manifold. We call the sectional curvature of the metric connection induced by the Chern connection the Chern sectional curvature of this Hermitian manifold. First, we derive expression of the Chern sectional curvature in local complex coordinates. As an application, we find that a Hermitian metric is Kähler if the Riemann sectional curvature and the Chern sectional curvature coincide. As subsequent results, Ricci curvature and scalar curvature of the metric connection induced by the Chern connection are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01533 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Chern Sectional Curvature of a Hermitian Manifold Cao, Pandeng Li, Hongjun Differential Geometry 53C55, 53B35 On a Hermitian manifold, the Chern connection can induce a metric connection on the background Riemannian manifold. We call the sectional curvature of the metric connection induced by the Chern connection the Chern sectional curvature of this Hermitian manifold. First, we derive expression of the Chern sectional curvature in local complex coordinates. As an application, we find that a Hermitian metric is Kähler if the Riemann sectional curvature and the Chern sectional curvature coincide. As subsequent results, Ricci curvature and scalar curvature of the metric connection induced by the Chern connection are obtained. |
| title | The Chern Sectional Curvature of a Hermitian Manifold |
| topic | Differential Geometry 53C55, 53B35 |
| url | https://arxiv.org/abs/2211.01533 |