Contorsion tensor

The contorsion tensor in differential geometry expresses the difference between a metric-compatible affine connection with Christoffel symbol $\Gamma _{ij}{}^{k}$ and the unique torsion-free Levi-Civita connection for the same metric.

The contortion tensor ${K_{ab}}^{c}$ is defined in terms of the torsion tensor ${T_{ij}}^{k}={\Gamma _{ij}}^{k}-{\Gamma _{ji}}^{k}$ as

$K_{ijk}={\frac {1}{2}}(T_{ijk}-T_{jki}+T_{kij}),$

where the indices are being raised and lowered with respect to the metric:

$T_{ijk}\equiv g_{kl}{T_{ij}}^{l}$.

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

${\Gamma _{kj}}^{i}={\bar {\Gamma }}_{kj}{}^{i}+{K_{kj}}^{i},$

where ${\bar {\Gamma }}_{kj}{}^{i}$ is the torsion-free Levi-Civita connection.