Apply the definition to the coordinate basis, for which $[\partial_\mu,\partial_\nu]=0$ and $\nabla_\mu\partial_\tau=\Gamma^\rho{}_{\tau\mu}\partial_\rho$. Comparing coefficients gives
$$
\boxed{R^\sigma{}_{\tau\mu\nu}
=\partial_\mu\Gamma^\sigma{}_{\tau\nu}
-\partial_\nu\Gamma^\sigma{}_{\tau\mu}
+\Gamma^\sigma{}_{\rho\mu}\Gamma^\rho{}_{\tau\nu}
-\Gamma^\sigma{}_{\rho\nu}\Gamma^\rho{}_{\tau\mu}},
$$
equivalent to the paper's ordering after commuting scalar factors. Although the individual <Christoffel symbols> are not tensors, the preceding intrinsic definition proves that this complete combination is tensorial.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 1 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-309/1/a/ii/solution