= Affine connection decomposition
{title2=$\Gamma=S+C+W$}
Relative to a nondegenerate <metric tensor>, any <affine connection> is uniquely the <Levi-Civita connection> plus a <contorsion tensor> contribution and a <disformation tensor> contribution. In derivative-last notation, with $T_{ab}{}^c=\Gamma^c{}_{ab}-\Gamma^c{}_{ba}$ and $N_{ab c}=\nabla_cg_{ab}$, the lowered difference is $A_{ab c}=\tfrac12(T_{ab c}+T_{bc a}-T_{ca b}+N_{ab c}-N_{bc a}-N_{ca b})$. This follows by solving $T_{ab c}=A_{ab c}-A_{ba c}$ and $N_{ab c}=-A_{ac b}-A_{bc a}$. The <difference of affine connections is a tensor>.
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