Solution
= Solution
The <torsion tensor> of an <affine connection> is
$$
T(X,Y)=\nabla_XY-\nabla_YX-[X,Y].
$$
The connection is symmetric, or a <torsion-free connection>, exactly when
$$
\boxed{\nabla_XY-\nabla_YX=[X,Y]\quad\text{for all }X,Y.}
$$
In a coordinate frame, write $\nabla_{\partial_i}\partial_j=\sum_k\Gamma^k_{ij}\partial_k$. Since coordinate fields commute, the condition is $\Gamma^k_{ij}=\Gamma^k_{ji}$; this is the reason for the word symmetric.