Three-dimensional curvature from the Ricci tensor ($\mathrm{Ric}=0⟹\mathrm{Riem}=0(n=3)$)

In dimension three the Riemann curvature tensor is $R_{ijpq}=g_{ip}{S}_{jq}-g_{iq}{S}_{jp}-g_{jp}{S}_{iq}+g_{jq}{S}_{ip}$, where $S$ is the Schouten tensor. Consequently every three-dimensional Ricci-flat Riemannian manifold is a flat Riemannian manifold. The implication fails in higher dimensions.