Weakly Z symmetric manifolds

TL;DR

This paper introduces weakly Z symmetric manifolds, a new class of Riemannian manifolds generalizing several symmetric types, and explores their geometric properties and conditions for special vectors and tensors.

Contribution

It defines weakly Z symmetric manifolds, generalizes existing symmetric structures, and analyzes their geometric properties and conditions for special vectors and tensors.

Findings

Conditions for the existence of proper concircular vectors.

Characterization of Ricci tensor in conformally harmonic case.

Local form of the metric tensor for conformally flat cases.

Abstract

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective- Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named "weakly Z symmetric" and denoted by (WZS)_n. If the Z tensor is singular we give conditions for the existence of a proper concircular vector. For non singular Z tensor, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally harmonic case, and the general form of the Ricci tensor. For conformally flat (WZS)_n manifolds, we derive the local form of the metric tensor.