Is there any theorem which states any general linear connection can be decomposed into another linear connection plus the contortion tensor ? i didn't find any References
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Check out Kobayashi and Nomizu's Foundations of Differential Geometry, Volume 1. On page 159, when proving the existence of the LeviCivita connection (Theorem IV.2.2), they pick an arbitrary metric connection and add the contorsion tensor to it and show that it is a metric connection with vanishing torsion. Hence any metric connection can be written as the difference of the LeviCivita connection and its contorsion tensor. Another reference is Section 7.2.6 of Nakahara's Geometry, Topology, and Physics. See equations (7.30)(7.35) for Nakahara's derivation. 

