Pseudo-Q-symmetric Riemannian manifolds

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)n and pseudo-concircular symmetric manifolds $({\rm P}\tilde{{\rm C}}{\rm S})-n$ is defined. This is named pseudo-Q-symmetric and denoted with (PQS)n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A 44 (1992) 1-34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski-Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)n scalar field space-time is considered, and interesting properties are pointed out.