Generalized quasi-Einstein metrics and applications on generalized Robertson–Walker spacetimes

Abstract

In this paper, we study generalized quasi-Einstein manifolds ( Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on ( Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton [Formula: see text], where [Formula: see text] is a generalized Robertson–Walker spacetime metric and the potential field [Formula: see text] is conformal, [Formula: see text] can be considered as the model of perfect fluids in general relativity. Moreover, the fiber ( M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of [Formula: see text] is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.