Contorsion

Every covariant derivative operator that one can define in a manifold is related in some sense to the others. See \cite{malament} page 51:

(\nabla_{m}^{'} - \nabla_{m}) α_{b_{1} \dots b_{s}}^{a_{1} \dots a_{r}} = α_{n b_{2} \dots b_{s}}^{a_{1} \dots a_{r}} C_{m b_{1}}^{n} + \dots

In particular, for a vector field we would have:

(\nabla_{m}^{'} - \nabla_{m}) ξ^{a} = - ξ^{n} C_{m n}^{a}

This let us show (see \cite{malament} page 57) that if two derivative operators define the same set of geodesics then they are equals.
Moreover, the expression $\nabla_{m}^{'} - \nabla_{m}$ defines a tensor called difference tensor or contorsion.