On the Ricci symmetries of a K\"ahler manifold

TL;DR

This paper investigates Ricci symmetry properties of Kähler manifolds, characterizing various Ricci-related conditions through holomorphic planes and clarifying distinctions between different pseudosymmetries.

Contribution

It provides new characterizations of Ricci symmetries in Kähler manifolds using holomorphic planes and clarifies the difference between Ricci pseudosymmetry types.

Findings

All Ricci symmetry conditions admit characterization via holomorphic planes.

Holomorphic Ricci pseudosymmetry is distinct from classical Ricci pseudosymmetry.

A new criterion for a Kähler manifold to be Einstein based on holomorphic planes.

Abstract

The main purpose of the present paper is to investigate the symmetry properties of a K\"ahler manifold involving the Ricci tensor. In this context, the most symmetric manifolds are K\"ahler-Einstein spaces, and their natural generalizations are Ricci-parallel K\"ahler manifolds, Ricci-semisymmetric K\"ahler manifolds and holomorphically Ricci-pseudosymmetric K\"ahler 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 K\"ahler 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 K\"ahler manifold to be Einstein based on holomorphic planes.