On pseudosymmetric manifolds

TL;DR

This paper investigates the relationships between different notions of pseudosymmetry in manifolds, establishing necessary and sufficient conditions for various types of pseudosymmetric and weakly symmetric manifolds, with examples illustrating their independent existence.

Contribution

It provides a comprehensive analysis of conditions linking Chaki and Deszcz pseudosymmetry, as well as weak symmetry, and introduces simplified forms and examples of these manifolds.

Findings

Conditions for Chaki and Deszcz pseudosymmetry are established.

Necessary and sufficient conditions for weakly symmetric manifolds are derived.

Examples demonstrate the existence of manifolds with various pseudosymmetry properties.

Abstract

In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], and there are many papers related to these notions. The object of the present paper is to deduce necessary and sufficient conditions for a Chaki pseudosymmetric [7] (resp. pseudo Ricci symmetric [8]) manifold to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also study the necessary and sufficient conditions for a weakly symmetric [58] (resp. weakly Ricci symmetric [59]) manifold by Tam\'assy and Binh to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also obtain the reduced form of the defining condition of weakly Ricci symmetric manifolds by Tam\'assy and Binh [59]. Finally we give some examples to show the independent existence of such types of pseudosymmetry which also ensure the existence of Roter type and generalized Roter type manifolds and the manifolds with recurrent curvature 2-form ([2], [29]) associated to various curvature tensors.