 Schur's lemma (Riemannian geometry)

On a connected Riemannian manifold of dimension at least three, pointwise independence of sectional curvature from the tangent two-plane forces the curvature to be constant. The contracted Bianchi identity differentiates the relation $\mathrm{Ric}=(n-1)Kg$ and gives $(n-2)dK=0$.