Ricci soliton ($\mathrm{Ric}+\frac{1}{2}{\mathcal{L}}_{X}g=\lambda g$)

A Riemannian metric satisfying the displayed equation for a vector field $X$ and constant $\lambda$. A gradient Ricci soliton has $X=\nabla f$; it is steady when $\lambda =0$.

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