Torsion form ($T=d^{\nabla }\theta$)

The torsion form of a connection on $TX$ is the covariant exterior derivative of the solder form. It satisfies

A connection on the tangent bundle is torsion-free when ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$. Its Christoffel symbols are symmetric in their two lower indices.