Every covariant derivative operator that one can define in a manifold is related in some sense to the others. See \cite{malament} page 51:
$$ \left( \nabla _ { m } ^ { \prime } - \nabla _ { m } \right) \alpha _ { b _ { 1 } \ldots b _ { s } } ^ { a _ { 1 } \dots a _ { r } } = \alpha _ { n b _ { 2 } \dots b _ { s } } ^ { a _ { 1 } \dots a _ { r } } C _ { m b _ { 1 } } ^ { n } + \ldots $$In particular, for a vector field we would have:
$$ \left( \nabla _ { m } ^ { \prime } - \nabla _ { m } \right) \xi^{a} =-\xi^n C _ { m n } ^ { a } $$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 $\nabla _ { m } ^ { \prime } - \nabla _ { m }$ defines a tensor called difference tensor or contorsion.
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Author of the notes: Antonio J. Pan-Collantes
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