Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 35D / b

At any chosen point take normal coordinates for $g$, so $\Gamma =0$ there. Substitute $\tilde{\Gamma }=\Gamma +T$ into the stated Riemann curvature tensor formula and subtract the original curvature. At that point the mixed products with $\Gamma$ vanish and ordinary derivatives of $T$ equal its covariant derivatives. The result isIn unit-weight antisymmetrization notation this is $\boxed{2T^a{}_{b[d;c]}-2T^a{}_{e[d}{T}^e{}_{c]b}}$. Every term is a tensor, so the identity established in normal coordinates holds in every coordinate system.