The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it.
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.
In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol
as (up to a sign, see below) where the indices are being raised and lowered with respect to the metric: The reason for the non-obvious sum in the definition of the contorsion tensor is due to the sum-sum difference that enforces metric compatibility.
The full metric compatible affine connection can be written as: where
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.
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.
Convention for derivation (Choose to define connection coefficients this way.
The motivation is that of connection-one forms in gauge theory): We begin with the Metric Compatible condition: Now we use sum-sum difference (Cycle the indices on the condition): We now use the below torsion tensor definition (for a holonomic frame) to rewrite the connection: Note that this definition of torsion has the opposite sign as the usual definition when using the above convention
for the lower index ordering of the connection coefficients, i.e. it has the opposite sign as the coordinate-free definition
Rectifying this inconsistency (which seems to be common in the literature) would result in a contorsion tensor with the opposite sign.
Substitute the torsion tensor definition into what we have: Clean it up and combine like terms The torsion terms combine to make an object that transforms tensorially.
We will define it here with the motivation that it match the indices of the left hand side of the equation above.
Cleaning by using the anti-symmetry of the torsion tensor yields what we will define to be the contorsion tensor: Subbing this back into our expression, we have: Now isolate the connection coefficients, and group the torsion terms together: Recall that the first term with the partial derivatives is the Levi-Civita connection expression used often by relativists.
The flatness is exactly what allows parallel frame fields to be constructed.