The <Levi-Civita connection> is a <torsion-free connection>. Its <torsion form> is
$$
T(u,v)=\nabla_uv-\nabla_vu-[u,v].
$$
Since $T=0$,
$$
\nabla_uv-\nabla_vu=[u,v].
$$
In coordinates this is the symmetry $\Gamma^i_{jk}=\Gamma^i_{kj}$ of the <Christoffel symbols>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-115/4/d/solution