Three-dimensional curvature from the Ricci tensor
= Three-dimensional curvature from the Ricci tensor
{title2=$\operatorname{Ric}=0\ \Longrightarrow\ \operatorname{Riem}=0\quad(n=3)$}
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.