The contorsion tensor in differential geometry expresses the difference between a metric-compatible affine connection with Christoffel symbol and the unique torsion-free Levi-Civita connection for the same metric.
The contortion tensor is defined in terms of the torsion tensor as
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 contortion 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 is the torsion-free Levi-Civita connection.