Killing vector preserves scalar curvature
= Killing vector preserves scalar curvature
Every <Killing vector field> $K$ preserves the <scalar curvature>: $\mathcal L_KR=K^\alpha\nabla_\alpha R=0$. One index proof rewrites this derivative as a doubly contracted curvature commutator acting on the antisymmetric tensor $\nabla_{[\mu}K_{\lambda]}$; both terms vanish by symmetry of the <Ricci tensor>.