Steady gradient Ricci soliton scalar identity
= Steady gradient Ricci soliton scalar identity
{title2=$R+|\nabla f|^2=\mathrm{constant}$}
For $\operatorname{Ric}+\nabla^2 f=0$, the <contracted Bianchi identity> and the commutation identity $\Delta\nabla f=\nabla\Delta f+\operatorname{Ric}(\nabla f)$ imply $\nabla R=2\operatorname{Ric}(\nabla f)$. Meanwhile $\nabla|\nabla f|^2=-2\operatorname{Ric}(\nabla f)$, so their sum is constant on every <connected component>.