In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by $$p=\frac{3-n}{n-1} \sigma $$p = 3 - n n - 1 σ , or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that spacetimes are Ricci flat in trivial, , and subspaces, whereas perfect fluid in subspaces , , and , and have zero scalar curvature in subspace . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

The main object of this paper is to investigate spacetimes admitting concircular curvature tensor in f(R) gravity theory. At first, concircularly flat and concircularly flat perfect fluid spacetimes in fR gravity are studied. In this case, the forms of the isotropic pressure p and the energy density σ are obtained. Next, some energy conditions are considered. Finally, perfect fluid spacetimes with divergence free concircular curvature tensor in f(R) gravity are studied; amongst many results, it is proved that if the energy-momentum tensor of such spacetimes is recurrent or bi-recurrent, then the Ricci tensor is semi-symmetric and hence these spacetimes either represent inflation or their isotropic pressure and energy density are constants.

The main aim of this study is to investigate the effects of the curvature flatness, divergence-free characteristic, and symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the curvature flat warped manifold are Einstein manifold. Besides that, the forms of the curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, symmetric warped product manifold is considered.

In this article, we characterize a spacetime endowed with Q-curvature tensor. We acquire that a Q-flat perfect fluid spacetime is either, vacuum or, a de-Sitter spacetime. Moreover, it is established that a Q-flat perfect fluid spacetime becomes a Robertson-Walker spacetime. Besides, it is proved that a Q-recurrent spacetime is a generalized Ricci recurrent spacetime. Finally, we examine the impact of Q-flat perfect fluid spacetime solutions in f(R,T,RijTij)-gravity.