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==
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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

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

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