On ricci symmetric generalized quasi Einstein spacetimes

The object of the present paper is to prove the existence of a generalized quasi Einstein spacetime, briefly G(QE)4, by constructing a non-trivial Lorentzian metric and to study such spacetime. First, we prove that every W2-Ricci pseudosymmetric G(QE)4 is an N(k)-quasi Einstein spacetime which can be considered as a model of perfect fluid, in general relativity. Then, we consider Ricci symmetric G(QE)4 and we prove that in such spacetime satisfying Einstein's field equations, the energy density and the isotropic pressure are constants. As a consequence of this result, the expansion scalar and the acceleration vector vanish and also the possible local cosmological structures of this spacetime obeying Einstein's field equations are of Petrov I, D or O.