Connection (principal Bundle) - Induced Covariant and Exterior Derivatives

Induced Covariant and Exterior Derivatives

For any linear representation W of G there is an associated vector bundle over M, and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of over M is isomorphic to the space of G-equivariant W-valued functions on P. More generally, the space of k-forms with values in is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative dω from -valued k-forms on M to -valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on .