Search

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 × f M , g ̄ , ξ ̄ , λ) , where g ̄ is a generalized Robertson–Walker spacetime metric and the potential field ξ ̄ = h ∂ t + ξ 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 × f M , g ̄ ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study spacetimes admitting pseudo-projective curvature tensor. At first we prove that a pseudo-projectively flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein's field equation with cosmological constant is covariant constant. Next, we prove that if the perfect fluid spacetime with vanishing pseudoprojective curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field U. Moreover, it is shown that a pseudo-projectively flat spacetime satisfying Einstein's equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space. We also prove that under certain conditions a perfect fluid spacetime with divergence-free pseudo-projective curvature is a Robertson-Walker spacetime and the possible local cosmological structure of such a spacetime is of type I, D or O. We also study dust-like fluid spacetime with vanishing pseudo-projective curvature tensor. [ABSTRACT FROM AUTHOR]

A perfect-fluid space-time of dimension n = 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an Einstein fiber. Condition (1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson- Walker space-time with null divergence of the Weyl tensor is a perfect-fluid spacetime. [ABSTRACT FROM AUTHOR]