Pseudo generalized quasi-Einstein manifolds with applications to general relativity

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Quasi-Einstein manifold and generalized quasi-Einstein manifold are the generalization of Einstein manifold. In the present paper we discuss about a set of some geometric properties of pseudo generalized quasi-Einstein manifold and we give three and four examples (both Riemannian and Lorentzian) of pseudo generalized quasi-Einstein manifold to show the existence of such manifold. We also discuss PG(QE)4 spacetime with space-matter tensor and some properties related to it. Lastly we prove the existence of a pseudo generalized quasi-Einstein spacetime by constructing a non-trivial example.