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

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{{further|[[Connection on a principal bundle]]}} | {{further|[[Connection on a principal bundle]]}} | ||

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+ | ==Proof== | ||

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+ | ===From connection on a vector bundle, to connection on principal bundle=== | ||

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+ | Suppose <math>E</math> is a <math>r</math>-dimensional[[vector bundle]] on the differential manifold <math>M</math>, and <math>\nabla</math> is a connection on <math>E</math>. Let <math>\pi:P \to M</math> be the corresponding principal <math>GL(r)</math>-bundle. | ||

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+ | Here's how we use <math>\nabla</math> to get a connection on the principal <math>GL(r)</math>-bundle. Observe, first, that giving a point <math>p \in P</math> is equivalent to specifying a point <math>\pi(p) \in M</math>, and a basis for the tangent space at <math>\pi(p)</math>. Further, giving a tangent vector <math>v</math> at the point <math>p \in P</math>, then {{fillin}} |

## Latest revision as of 19:36, 18 May 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`

## Proof

### 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*