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

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(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

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