The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) eleven-dimensional supergravity.[1] That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role.
The elimination of torsion in a connection is referred to as the absorption of torsion, and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures.