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Contracting the assumed Riemann curvature tensor gives the Ricci tensorThe contracted Bianchi identity and metric compatibility now implyThus ${\nabla }_{b}R=0$, so the scalar curvature is constant on each connected component of spacetime. The sectional curvature is consequently the constant $\kappa =R/12$. This is the four-dimensional case of the Schur theorem in pseudo-Riemannian geometry.

Contract the isotropic-curvature formula on $a,c$. In dimension $n$ it givesSubstitution into the contracted Bianchi identity givesSince $n>2$, ${\nabla }_{b}K=0$, so $K$ is constant on each connected component. Thus this is constant sectional curvature, withThis argument is Schur theorem in pseudo-Riemannian geometry.