Torsion tensor

Summary

The torsion tensor of an affine connection measures how far covariant derivatives are from being symmetric in the two vector fields. The Lie bracket is subtracted to make this a tensor. It is a tensor field of type $(1, 2)$, antisymmetric in its two arguments. Connections with zero torsion are called torsion-free or symmetric; the Levi-Civita connection is the unique torsion-free connection compatible with a given Riemannian metric. Geometrically, torsion describes how infinitesimal parallelograms built by parallel transport fail to close.