Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 37D / a

Because the Levi-Civita connection is torsion-free, its Christoffel symbols satisfy ${\Gamma }_{\rho \alpha }^{\mu }={\Gamma }_{\alpha \rho }^{\mu }$. Substitute the stated coordinate expression for each term in the cyclic sumThe partial-derivative terms cancel pairwise by the lower-index symmetry of $\Gamma$, and the quadratic terms cancel after relabelling their dummy indices. Thereforewhich is the first Bianchi identity.