Connection on vector bundle equals connection on principal GL-bundle
Let be a differential manifold and be a -dimensional vector bundle over . Suppose is the corresponding principal -bundle over . Then, there is a natural bijection between the set of connections on (viewed as a vector bundle) and the set of connections on (viewed as a principal -bundle).
Connection on a vector bundle
Further information: Connection on a vector bundle
Connection on a principal bundle
Further information: Connection on a principal bundle