Principal bundle - Revision history

Vipul: 4 revisions

2008-05-18T19:51:03Z

4 revisions

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Vipul: /* Facts */

2007-09-02T07:31:56Z

Facts

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	The process of finding a subbundle that is principal for a subgroup is termed ''reduction'' of the structure group.		The process of finding a subbundle that is principal for a subgroup is termed ''reduction'' of the structure group.

			===Lifting of structure group===

			{{further\|[[Lifting of structure group]]}}

			Let <math>G</math> be a [[Lie group]], <math>N</math> a [[normal subgroup]] that is also a Lie subgroup, and <math>G/N</math> the [[quotient group]]. Then any principal <math>G</math>-bundle gives rise to a <math>G/N</math>-bundle as follows: if <math>\pi:P \to M</math> is the <math>G</math>-bundle, then the <math>G/N</math>-bundle is simply <math>P</math> modulo the action of <math>N</math>. The fibre at each point is the quotient space by <math>N</math>.

			The question of ''lifting'' the structure group is the other way around: given a principal bundle over <math>H = G/N</math>, find a principal <math>G</math>-bundle that would give rise to this on quotienting by <math>N</math>.
	===Vector bundles as principal bundles===		===Vector bundles as principal bundles===

Vipul: /* Facts */

2007-09-02T03:35:00Z

Facts

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	{{further\|[[Reduction of structure group]]}}		{{further\|[[Reduction of structure group]]}}

	If <math>H</math> is a [[subgroup]] of <math>G</math> and <math>M</math> hs a principal <math>G</math>-bundle <math>P</math>, we may be able to find a subbundle <math>P'</math> that is a principal <math>H</math>-bundle. Intuitively, what we are doing is that at each <math>m \in M<math>, we are choosing a subset of the copy of <math>G</math> that is the orbit of a single point under the <math>H</math>-action. The tricky part is to make a choice that varies smoothly with the point.		If <math>H</math> is a [[subgroup]] of <math>G</math> and <math>M</math> hs a principal <math>G</math>-bundle <math>P</math>, we may be able to find a subbundle <math>P'</math> that is a principal <math>H</math>-bundle. Intuitively, what we are doing is that at each <math>m \in M</math>, we are choosing a subset of the copy of <math>G</math> that is the orbit of a single point under the <math>H</math>-action. The tricky part is to make a choice that varies smoothly with the point.

	The process of finding a subbundle that is principal for a subgroup is termed ''reduction'' of the structure group.		The process of finding a subbundle that is principal for a subgroup is termed ''reduction'' of the structure group.

Vipul: /* Definition with symbols */

2007-09-02T03:34:24Z

Definition with symbols

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	===Definition with symbols===		===Definition with symbols===

	Let <math>M<math> be a [[differential manifold]] and <math>G</math> a [[Lie group]]. A princial bundle on <math>M</math> with structure group <math>G</math> is defined as the following data:		Let <math>M</math> be a [[differential manifold]] and <math>G</math> a [[Lie group]]. A princial bundle on <math>M</math> with structure group <math>G</math> is defined as the following data:

	* A [[differential manifold]] <math>P</math> with a differentiable map <math>\pi:P \to M</math>		* A [[differential manifold]] <math>P</math> with a differentiable map <math>\pi:P \to M</math>

Vipul at 12:55, 1 September 2007

2007-09-01T12:55:08Z

New page

==Definition==

===Definition with symbols===

Let <math>M<math> be a [[differential manifold]] and <math>G</math> a [[Lie group]]. A princial bundle on <math>M</math> with structure group <math>G</math> is defined as the following data:

* A [[differential manifold]] <math>P</math> with a differentiable map <math>\pi:P \to M</math>
* A [[Lie group action]] of <math>G</math> on <math>M</math> such that the fibres of <math>\pi</math> are precisely the orbits of <math>G</math>, and also the group action is free (that is no two group elements have the same effect on any point)

satisfying the following condition called ''local triviality'':

Given any point <math>m \in M</math> there exists an open neighbourhood <math>U</math> of <math>M</math> such that the map <math>\pi:\pi^{-1}(U) \to U</math> is the same as the projection map <math>U \times G \to U</math>.

Intuitively a principal <math>G</math>-bundle means a copy of <math>G</math> at each point, varying smoothly with the point. However, the copy of <math>G</math> at each point, does not have any natural "origin" or identity element.

==Facts==

===Reduction of structure group===

{{further|[[Reduction of structure group]]}}

If <math>H</math> is a [[subgroup]] of <math>G</math> and <math>M</math> hs a principal <math>G</math>-bundle <math>P</math>, we may be able to find a subbundle <math>P'</math> that is a principal <math>H</math>-bundle. Intuitively, what we are doing is that at each <math>m \in M<math>, we are choosing a subset of the copy of <math>G</math> that is the orbit of a single point under the <math>H</math>-action. The tricky part is to make a choice that varies smoothly with the point.

The process of finding a subbundle that is principal for a subgroup is termed ''reduction'' of the structure group.

===Vector bundles as principal bundles===

A [[vector bundle]] on a [[differential manifold]] can be viewed as a principal bundle with structure group <math>GL(n,\R)</math>. If <math>M</math> is the differential manifold, the fibre at <math>m \in M</math> is the set of all ordered bases for <math>T_pM</math>.

===Connections on principal bundles===

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

Typically the term [[connection]] is used for a [[vector bundle]] on a [[differential manifold]], and it is intended to be a rule that allows us to differentiate sections of the vector bundle alogn [[vector field]]s (viz, sections of the tangent bundle).

We can generalize this to define a notion of connection on a principal bundle. {{fillin}}