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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.14979 |
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| _version_ | 1866916013021331456 |
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| author | González, Jorge Alcázar |
| author_facet | González, Jorge Alcázar |
| contents | The main purpose of the present paper is to investigate the symmetry properties of a Kähler manifold involving the Ricci tensor. In this context, the most symmetric manifolds are Kähler-Einstein spaces, and their natural generalizations are Ricci-parallel Kähler manifolds, Ricci-semisymmetric Kähler manifolds and holomorphically Ricci-pseudosymmetric Kähler manifolds. Unlike their Riemannian counterparts, we prove that all these conditions also admit a characterization solely in terms of holomorphic planes, analogously to the symmetries related to the Riemannian curvature tensor in Kähler manifolds. A key finding is that the concept of holomorphic Ricci pseudosymmetry is distinct from the classical Ricci-pseudosymmetric condition introduced by Deszcz. By carefully analyzing the interplay between these definitions, we clarify the precise geometric role of the so-called Ricci curvature of Deszcz. Additionally, we also present a geometric interpretation of the complex Tachibana-Ricci tensor and we establish a new criterion for a Kähler manifold to be Einstein based on holomorphic planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14979 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Ricci symmetries of a Kähler manifold González, Jorge Alcázar Differential Geometry The main purpose of the present paper is to investigate the symmetry properties of a Kähler manifold involving the Ricci tensor. In this context, the most symmetric manifolds are Kähler-Einstein spaces, and their natural generalizations are Ricci-parallel Kähler manifolds, Ricci-semisymmetric Kähler manifolds and holomorphically Ricci-pseudosymmetric Kähler manifolds. Unlike their Riemannian counterparts, we prove that all these conditions also admit a characterization solely in terms of holomorphic planes, analogously to the symmetries related to the Riemannian curvature tensor in Kähler manifolds. A key finding is that the concept of holomorphic Ricci pseudosymmetry is distinct from the classical Ricci-pseudosymmetric condition introduced by Deszcz. By carefully analyzing the interplay between these definitions, we clarify the precise geometric role of the so-called Ricci curvature of Deszcz. Additionally, we also present a geometric interpretation of the complex Tachibana-Ricci tensor and we establish a new criterion for a Kähler manifold to be Einstein based on holomorphic planes. |
| title | On the Ricci symmetries of a Kähler manifold |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.14979 |