Torsion tensor of an affine connection

For an affine connection, define its torsion on local vector fields by $T(X,Y)={\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]\mathrm{.}$ The connection is as in Affine connection on a smooth manifold, and the bracket is the smooth vector field given by Coordinate formula for the Lie bracket. The next lemma proves this operation is a smooth alternating tangent-valued two-tensor. It is called torsion free when $T(X,Y)=0$ for all local fields.

This construction uses the tangent bundle: general bundle sections have no Lie bracket with which to form this difference. In dimension zero all terms vanish. Smooth fields on a manifold with boundary need not be boundary-tangent; bracket and covariant derivative use smooth coordinate derivatives there as well.