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 λi 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.