Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 3 / a / i / Solution

Write $h_{ab}=\delta g_{ab}$ and $C^a{}_{bc}=\delta {\Gamma }^a{}_{bc}$. The difference between two affine connections is a tensor, so its infinitesimal change $C$ is a type $(1,2)$ tensor field. Both affine connections are torsion-free, giving $C^a{}_{bc}=C^a{}_{cb}$. Varying metric compatibility givesLower the first index of $C$. Add the versions with derivatives $b,c$ and subtract the one with derivative $d$; the lower-slot symmetry cancels the unwanted terms, leavingThereforeAll covariant derivatives use the original Levi-Civita connection. The PDF has ${\nabla }_b{h}_{dc}$ as its second term; the converted TeX's ${\nabla }_d{h}_{dc}$ is a transcription error.