Apply the definition to a coordinate-basis vector, use $[\partial_c,\partial_d]=0$, and collect the coefficient of $X^b$. This gives
$$
R^a{}_{bcd}
=\partial_c\Gamma^a{}_{db}-\partial_d\Gamma^a{}_{cb}
+\Gamma^a{}_{ce}\Gamma^e{}_{db}
-\Gamma^a{}_{de}\Gamma^e{}_{cb},
$$
up 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.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-309/2/a/ii/solution