Curvature Form - Bianchi Identities

Bianchi Identities

If is the canonical vector-valued 1-form on the frame bundle, the torsion of the connection form is the vector-valued 2-form defined by the structure equation

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

The second Bianchi identity takes the form

and is valid more generally for any connection in a principal bundle.