General properties of $\mathbf{f}(\mathbf{R})$ gravity vacuum solutions

General properties of vacuum solutions of $f(R)$ gravity are obtained by the condition that the divergence of the Weyl tensor is zero and $f^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$. Specifically, a theorem states that the gradient of the curvature scalar, $\mathrm{\nabla }R$, is an eigenvector of the Ricci tensor and, if it is time-like, the space-time is a Generalized Friedman-Robertson-Walker metric; in dimension four, it is Friedman-Robertson-Walker.