Scalar curvature under conformal rescaling ($\tilde{R}=e^{-2\omega }[R-2(N-1){\Delta }_g\omega -(N-1)(N-2)|\nabla \omega {|}_g^2]$)

In dimension $N$, the conformal rescaling of a Riemannian metric $\tilde{g}=e^{2\omega }g$ changes the Ricci scalar by the displayed formula. In two dimensions the gradient-square term vanishes; for a flat base metric, $\tilde{R}=-2e^{-2\omega }\Delta \omega$.