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An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric. [ABSTRACT FROM AUTHOR]

The object of this paper is to classify paracontact metric (κ; _≢)-spaces satisfying certain curvature conditions. We show that a paracontact metric (κ; _≢)-space is Ricci semisymmetric if and only if the metric is Einstein, provided κ < -1. Also we prove that a paracontact metric (κ _≢)-space is ϕ-Ricci symmetric if and only if the metric is Einstein, provided κ ≢= 0-1. Moreover, we show that in a paracontact metric (κ _≢)-space with κ < -1, a second order symmetric parallel tensor is a constant multiple of the associated metric tensor. Several consequences of these results are discussed. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study weakly Z symmetric spacetimes (WZS)4. At first we prove that a weakly Z symmetric spacetime is a quasi-Einstein spacetime and hence a perfect fluid spacetime. Next, we consider conformally flat (WZS)4 spacetimes and prove that such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field p. We also study (WZS)4 spacetimes with divergence free conformal curvature tensor. Moreover, we characterize dust fluid and viscous fluid (WZS)4 spacetimes. Finally, we construct an example of a (WZS)4 spacetime. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study N(k)-quasi-Einstein manifolds. study We an N(k)-quasi-Einstein manifold satisfying the curvature conditions R(ξ,X)· Z = 0, Z(X, ·) · R = 0, and P(·,X) · Z = 0, where R, P and Z denote the Riemannian curvature tensor, the projective curvature tensor and Z tensor respectively. Next we prove that the curvature condition C · Z = 0 holds in an N(k)-quasi-Einstein manifold, where C is the conformal curvature tensor. We also study Z-recurrent N(k)-quasi-Einstein manifolds. Finally, we construct an example of an N(k)-quasi-Einstein manifold and mention some physical examples. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study a conformally flat quasi-Einstein space and its hypersurface. [ABSTRACT FROM AUTHOR]