Isotropic Ricci curvature (${\mathrm{Ric}}_p=c_p{g}_p$)

At a point of a Riemannian manifold, Ricci curvature is isotropic when $\mathrm{Ric}(v,v)/g(v,v)$ is independent of nonzero tangent direction. Polarization makes this equivalent to the displayed tensor identity. In dimension three, sectional curvatures from Ricci curvature in dimension three gives every sectional curvature at that point the value $c_p/2$. In higher dimensions isotropic Ricci curvature does not determine every two-plane curvature.