CHARACTERIZATIONS OF VACUUM SOLUTIONS OF f(R)fGRAVITY IN SPACE-TIMES ADMITTING Z TENSOR OF CODAZZI TYPE

In this work, characterizations of vacuum solutions of f ( R )-gravity are established in a space-time whose Z tensor is of Codazzi type. We prove that the associated covector of a (PZS) n space-time is an eigenvector of the Ricci tensor, with an eigenvalue equals zero. Additionally, it satisfies compatibility conditions with both the Riemann and Weyl tensors. It is proved that a (PZS) n space-time satisfying f ( R )-gravity vacuum solutions is a generalized Friedmann-Robertson-Walker space-time. If n = 4, it becomes a Friedmann-Robertson-Walker space-time.