# Connection on vector bundle equals connection on principal GL-bundle

From Diffgeom

Revision as of 00:22, 12 April 2008 by Vipul (talk | contribs) (New page: ==Statement== ===Conceptual statement=== We know that given a differential manifold, the vector bundles of dimension <math>r</math> over that manifold are in one-one corresponden...)

## Contents

## Statement

### Conceptual statement

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

### Ordinary statement

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).

## 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`