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

{title2=$T=d^\nabla\theta$}

The torsion form of a connection on $TX$ is the covariant exterior derivative of the <solder form>. It satisfies
$$
T(U,V)=\nabla_UV-\nabla_VU-[U,V].
$$

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