Connection on vector bundle equals connection on principal GL-bundle

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

• Metric connection on metric bundle equals connection on principal O-bundle

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