 Solution

The torsion tensor of an affine connection isThe connection is symmetric, or a torsion-free connection, exactly whenIn a coordinate frame, write ${\nabla }_{{\partial }_i}{\partial }_j={\sum }_k{\Gamma }_{ij}^k{\partial }_k$. Since coordinate fields commute, the condition is ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$; this is the reason for the word symmetric.