Connection on vector bundle equals connection on principal GL-bundle

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Statement

Conceptual statement

We know that given a differential manifold, the vector bundles of dimension r over that manifold are in one-one correspondence with the principal GL(r)-bundles over the manifold.

It turns out under this correspondence, the notion of connection on the vector bundle, corresponds to the notion of connection on the corresponding principal GL(r)-bundle.

Ordinary statement

Let M be a differential manifold and E be a r-dimensional vector bundle over M. Suppose P is the corresponding principal GL(r)-bundle over M. Then, there is a natural bijection between the set of connections on E (viewed as a vector bundle) and the set of connections on P (viewed as a principal GL(r)-bundle).

Related facts

Definitions used

Connection on a vector bundle

Further information: Connection on a vector bundle

Connection on a principal bundle

Further information: Connection on a principal bundle

Proof

From connection on a vector bundle, to connection on principal bundle

Suppose E is a r-dimensionalvector bundle on the differential manifold M, and \nabla is a connection on E. Let \pi:P \to M be the corresponding principal GL(r)-bundle.

Here's how we use \nabla to get a connection on the principal GL(r)-bundle. Observe, first, that giving a point p \in P is equivalent to specifying a point \pi(p) \in M, and a basis for the tangent space at \pi(p). Further, giving a tangent vector v at the point p \in P, then Fill this in later