Solder form ($\theta$)

The solder form on a smooth manifold is the identity $TX$-valued one-form $\theta (V)=V$. In a coordinate frame it is $\theta =d{x}^i\otimes {\partial }_i$.

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.