Holonomy group - Revision history

Vipul: /* Reduction of structure group to holonomy group */

2011-07-24T22:09:07Z

Reduction of structure group to holonomy group

← Older revision		Revision as of 22:09, 24 July 2011
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	* In particular, any two bases at <math>m</math> differ by the [[holonomy of a loop]], which lies in <math>H</math>. Thus <math>H</math> acts freely and transitively on the fibre at each point.		* In particular, any two bases at <math>m</math> differ by the [[holonomy of a loop]], which lies in <math>H</math>. Thus <math>H</math> acts freely and transitively on the fibre at each point.

	Moreover the holonomy group is the smallest group to which ~~thestructure~~ group can be reduced (for the same reasons).		Moreover the holonomy group is the smallest group to which the structure group can be reduced (for the same reasons).

Vipul: 4 revisions

2008-05-18T19:46:55Z

4 revisions

← Older revision	Revision as of 19:46, 18 May 2008
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Vipul: /* Reduction of structure group to holonomy group */

2007-09-02T03:34:03Z

Reduction of structure group to holonomy group

← Older revision		Revision as of 03:34, 2 September 2007
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	===Reduction of structure group to holonomy group===		===Reduction of structure group to holonomy group===

	Any path-connected [[differential manifold]] can be treated as a [[principal bundle]] with structure group]] being the holonomy group. The description is as follows:		Any path-connected [[differential manifold]] can be treated as a [[principal bundle]] with structure group being the holonomy group. The description is as follows:

	* Pick (arbitrarily) a basis at a particular point <math>p</math>		* Pick (arbitrarily) a basis at a particular point <math>p</math>

Vipul at 03:33, 2 September 2007

2007-09-02T03:33:37Z

← Older revision		Revision as of 03:33, 2 September 2007
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	* [[Restricted holonomy group]]		* [[Restricted holonomy group]]
	* [[Holonomy group of Riemannian metric]]		* [[Holonomy group of Riemannian metric]]

			==Facts==

			===Reduction of structure group to holonomy group===

			Any path-connected [[differential manifold]] can be treated as a [[principal bundle]] with structure group]] being the holonomy group. The description is as follows:

			* Pick (arbitrarily) a basis at a particular point <math>p</math>
			* Now, for each point <math>m</math>, the fibre at that point is the set of all bases at <math>m</math> that can arise from the basis at <math>p</math> by means of [[transport along a curve]] using the connection <math>\nabla</math>.
			* In particular, any two bases at <math>m</math> differ by the [[holonomy of a loop]], which lies in <math>H</math>. Thus <math>H</math> acts freely and transitively on the fibre at each point.

			Moreover the holonomy group is the smallest group to which thestructure group can be reduced (for the same reasons).

Vipul at 14:50, 1 September 2007

2007-09-01T14:50:54Z

← Older revision		Revision as of 14:50, 1 September 2007
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	==Definition==		==Definition==

	Let <math>M/math> be a [[differential manifold]], <math>E</math> a [[vector bundle]] over <math>M</math> and <math>\nabla</math> a [[connection]] for <math>E</math>. For a point <math>m \in M</math> the holonomy group at <math>m</math> is the subgroup of <math>GL(E_p)</math> comprising those linear transformations that arise as the [[holonomy of a loop]] at <math>m</math>.		Let <math>M</math> be a [[differential manifold]], <math>E</math> a [[vector bundle]] over <math>M</math> and <math>\nabla</math> a [[connection]] for <math>E</math>. For a point <math>m \in M</math> the holonomy group at <math>m</math> is the subgroup of <math>GL(E_p)</math> comprising those linear transformations that arise as the [[holonomy of a loop]] at <math>m</math>.

	If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of <math>GL(E_p)</math> so we can talk of '''the holonomy group'''.		If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of <math>GL(E_p)</math> so we can talk of '''the holonomy group'''.

Vipul at 14:50, 1 September 2007

2007-09-01T14:50:34Z

New page

==Definition==

Let <math>M/math> be a [[differential manifold]], <math>E</math> a [[vector bundle]] over <math>M</math> and <math>\nabla</math> a [[connection]] for <math>E</math>. For a point <math>m \in M</math> the holonomy group at <math>m</math> is the subgroup of <math>GL(E_p)</math> comprising those linear transformations that arise as the [[holonomy of a loop]] at <math>m</math>.

If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of <math>GL(E_p)</math> so we can talk of '''the holonomy group'''.

==Related notions==

* [[Restricted holonomy group]]
* [[Holonomy group of Riemannian metric]]