Notes on pseudo symmetric and pseudo Ricci symmetric generalized Robertson–Walker space-times

We establish two key results regarding pseudo symmetric and pseudo Ricci symmetric space-times. Firstly, we demonstrate that in pseudo symmetric generalized Robertson-Walker space-times either the scalar curvature remains constant or the associated vector field \(B_{i}\) is irrotational. Secondly, in pseudo Ricci symmetric generalized Robertson-Walker space-times, we establish that either the scalar curvature is zero or the associated vector field \(A_{i}\) is irrotational. We identify the conditions to ensure both \(B_{i}\) and \(A_{i}\) of these manifolds are acceleration-free and vorticity-free. We provide evidence that a pseudo symmetric and pseudo Ricci symmetric GRW space-time can be described as a perfect fluid. In a pseudo symmetric space-time, the state equation is given by \(p=\frac{4-n}{ 2n-2}\mu \), whereas in a pseudo Ricci symmetric space-time, the state equation takes the form \(p=\frac{3-n}{n-1}\mu \), where p and \(\mu \) are the isotropic pressure and the energy density. It is noteworthy that if \(n=4\) , a pseudo symmetric space-time corresponds to the dust matter era, while a pseudo Ricci symmetric space-time corresponds to the phantom era.