Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudo Z symmetric manifold and denoted by (PZS) n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS) n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS) n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc. 47(3) (1983) 1526; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A 44 (1992) 134]). Finally, we study some properties of (PZS) 4 spacetime manifolds.