Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 4 / e / Solution

The Levi-Civita connection is also a metric connection, soand similarly for the two cyclic permutations. Add the identities with leading derivatives $u$ and $v$, subtract the one with leading derivative $w$, and useCancellation and symmetry of $g$ giveThis is the Koszul formula written with the bracket terms on the right.