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 (M¯=I×fM,g¯,ζ¯,λ), where g¯ is a generalized Robertson-Walker spacetime metric and the potential field ζ¯=h∂t+ζ is conformal, (M¯,g¯) 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 (M¯=I×fM,g¯) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.