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
From connection on a vector bundle, to connection on principal bundle
Suppose is a -dimensionalvector bundle on the differential manifold , and is a connection on . Let be the corresponding principal -bundle.
Here's how we use to get a connection on the principal -bundle. Observe, first, that giving a point is equivalent to specifying a point , and a basis for the tangent space at . Further, giving a tangent vector at the point , then Fill this in later