= Contorsion tensor
{title2=$C=\nabla^{\mathrm{metric}}-\nabla^{\mathrm{LC}}$}
For a metric-compatible <affine connection>, its difference from the <Levi-Civita connection> is the contorsion <tensor>. It is determined by the <torsion tensor>. In derivative-last notation let $T_{ab}{}^c=\Gamma^c{}_{ab}-\Gamma^c{}_{ba}$, the negative of geometric torsion, and lower the last slot with the <metric tensor>. Then $C_{ab c}=\tfrac12(T_{ab c}+T_{bc a}-T_{ca b})$. This is the actual correction added to the <Levi-Civita connection>; a convention writing $\Gamma=S-K$ has $K=-C$.
Back to article page
