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) 11-dimensional supergravity. 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.
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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.
Metric geometry
In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol
The contorsion tensor
where the indices are being raised and lowered with respect to the metric:
The reason for the non-obvious sum in the definition is that the contorsion tensor, being the difference between two metric-compatible Christoffel symbols, must be antisymmetric in the last two indices, whilst the torsion tensor itself is antisymmetric in its first two indices.
The connection can now be written as
where
It can be shown that the geodesics for both connections are the same: that is, the any geodesic of
Affine geometry
In affine geometry, one does not have a metric nor a metric connection, and so one is not free raise and lower indices on demand. One can still achieve a similar effect by making use of the solder form, allowing the bundle to be related to what is happening on its base space. This is an explicitly geometric viewpoint, with tensors now being geometric objects in the vertical and horizontal bundles of a fiber bundle, instead of being indexed algebraic objects defined only on the base space. In this case, one may construct a contorsion tensor, living as a one-form on the tangent bundle.
Recall that the torsion of a connection
where
That is, one may construct a tensor
Here
The quantity
The vanishing of the torsion is then equivalent to having
or
This can be viewed as a field equation relating the dynamics of the connection to that of the contorsion tensor.
Relationship to teleparallelism
In the theory of teleparallelism, one encounters a connection, the Weitzenböck connection, which is flat (vanishing Riemann curvature) but has a non-vanishing torsion. The flatness is exactly what allows parallel frame fields to be constructed. These notions can be extended to supermanifolds.