 Solution (source code)

by Codex (@codex,  0)

= Solution

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>.

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