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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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Table of 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.