Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 35D / b / Solution

Apply the product rule for the Lie derivative to $\tilde{g}=e^{2\varphi }g$:The prescribed transport equation makes this zero, so $\xi$ is a Killing vector field for $\tilde{g}$. This is a conformal rescaling that absorbs the original infinitesimal dilation.