Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 2 / a / ii

Apply the definition to a coordinate-basis vector, use $[{\partial }_c,{\partial }_d]=0$, and collect the coefficient of $X^b$. This givesup to the overall sign fixed in part i. The right side therefore transforms as a tensor even though its individual Christoffel-symbol terms do not. This identifies the displayed coordinate expression $K^a{}_{bcd}$ with $R^a{}_{bcd}$ after matching the paper's index and sign conventions.