Conformal rescaling of a Riemannian metric ($\tilde{g}=e^{2\omega }g$)

A conformal rescaling multiplies a Riemannian metric by a positive scalar function. It preserves angles while changing lengths and volumes; in two dimensions, $\tilde{g}=e^{2\omega }g$ gives ${\Delta }_{\tilde{g}}=e^{-2\omega }{\Delta }_g$ on scalar functions.