Characterizations of pseudo Ricci symmetric space-times satisfying the vacuum solutions of $f(R)$-gravity

This paper explores the characteristics of pseudo Ricci symmetric space-times within the vacuum solutions of $f(R)$-gravity. It is proved that a pseudo Ricci symmetric space-time satisfying the vacuum solutions of $f(R)$-gravity can be described as a perfect fluid space-time and its associated vector field ${\lambda }_i$ is torse-forming, Riemann compatible, Weyl compatible, and annihilates the Weyl tensor. Consequently, it is demonstrated that such a space-time can be classified as a generalized Friedman–Robertson–Walker space-time. In the case of four dimensions, it specifically corresponds to a Friedman–Robertson–Walker space-time.