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

The symmetrized covariant derivative equation is ${\nabla }_a{\xi }_b+{\nabla }_b{\xi }_a=\psi g_{ab}$. Lower ${\xi }_b=g_{bc}{\xi }^c$ and expand both derivatives. Using the Christoffel symbols of the metric and their symmetry, the coefficient of ${\xi }^c$ simplifies byHenceThis is the coordinate formula for the Lie derivative ${\mathcal{L}}_{\xi }g=\psi g$, proving equivalence with the Conformal Killing vector field condition.