# Difference between revisions of "Connection on vector bundle equals connection on principal GL-bundle"

From Diffgeom

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

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+ | ==Related facts== | ||

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+ | * [[Metric connection on metric bundle equals connection on principal O-bundle]] | ||

==Definitions used== | ==Definitions used== |

## Revision as of 00:23, 12 April 2008

## 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).

## 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`