Weakly Z-symmetric manifolds

Abstract

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 (W ZS)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 (W ZS)n manifolds, we derive the local form of the metric tensor.

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

“…The results were extended by Mantica et al to pseudo Z-symmetric spaces [24] and to weakly Z-symmetric spaces [25].…”

𝒵 Tensor on N(k)-Quasi-Einstein Manifolds

“…In Refs. [23], [26] and [24] various properties of the Z tensor were pointed out. The derivation conditions R(ξ, X) · R = 0 and R(ξ, X) · S = 0 have been studied in [36], where R and S denote the curvature and Ricci tensor respectively.…”

On weakly cyclic Z symmetric spacetimes

“…We may say that the Ricci symmetric condition is only a special case of weakly Ricci symmetric manifold. Recently, Mantica and Molinari [14] introduced weakly Z symmetric manifolds which generalize the notion of weakly Ricci symmetric manifolds. Also De, Mantica and Suh [11] introduced the notion of weakly cyclic Z symmetric manifolds (W CZS) n .…”

“…In a recent paper [14] the authors introduced weakly Z symmetric manifolds which is denoted by (W ZS) n . A Riemannian or a semi-Riemannian manifold is said to be weakly Z symmetric, denoted by (W ZS) n , if the generalized Z tensor satisfies the condition…”