= Torsion tensor
{title2=$\mathcal T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]$}
The torsion of an <affine connection> is the displayed antisymmetric <tensor>. It measures the failure of the antisymmetrized <covariant derivative> to agree with the <Lie bracket of vector fields>. With derivative-last coefficients $\nabla_\mu Y^\rho=\partial_\mu Y^\rho+\Gamma^\rho{}_{\nu\mu}Y^\nu$, its components are $\mathcal T^\rho{}_{\mu\nu}=\Gamma^\rho{}_{\nu\mu}-\Gamma^\rho{}_{\mu\nu}$. Some conventions use the negative component tensor. Geodesic equations depend on the symmetric connection coefficients, so they alone cannot detect torsion.
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