Investigations of Pseudo-Shouten symmetric spacetimes in f ( R )-gravity vacuum solutions

Pseudo-Schouten symmetric spacetimes are examined under a differential condition on the Schouten tensor, producing explicit relations for the scalar curvature, the Ricci tensor, and the associated covector field. It is shown that the associated covector is a Ricci eigenvector and is irrotational, acceleration-free, and vorticity-free. Corresponding reductions of the Ricci tensor are obtained, including perfect fluid forms under suitable conditions. Ricci recurrent and Ricci semi-symmetric cases are also considered, leading to further curvature restrictions and compatibility properties involving the Riemann and Weyl tensors. In vacuum f(R)-gravity, the Ricci tensor is decomposed into a perfect fluid part and an anisotropic term governed by the shear tensor, equivalently by the electric part of the Weyl tensor. A comoving frame analysis shows that the shear vanishes, so that the associated covector is torse-forming and the spacetime reduces to a generalized Robertson-Walker model, and in four dimensions to a Robertson-Walker spacetime. Moreover, the Weyl tensor is divergence-free and annihilated by the associated covector. These results give a geometric characterization of pseudo-Schouten symmetric spacetimes in both the general setting and the vacuum f(R)-gravity case.