Holonomy group of Riemannian metric - Revision history

Vipul: 4 revisions

2008-05-18T19:46:56Z

4 revisions

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Vipul at 02:34, 2 September 2007

2007-09-02T02:34:08Z

← Older revision		Revision as of 02:34, 2 September 2007
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	* [[Holonomy group]]		* [[Holonomy group]]
	* [[Restricted holonomy group of Riemannian metric]]		* [[Restricted holonomy group of Riemannian metric]]
			* [[Riemannian metric with special holonomy]]

			==Facts==

			===Orientedness===

			If the Riemannian manifold is oriented, the holonomy group is a subgroup of the [[special orthogonal group]].

			===Generic and special holonomy===

			For a generic Riemannian metric on an oriented manifold, the holonomy group is the whole [[special orthogonal group]]. A Riemannian metric where the holonomy group is a proper subgroup of the special orthogonal group is termed a [[Riemannian metric with special holonomy]].

			===Flat manifolds===

			For a [[flat metric]], the [[restricted holonomy group]] is trivial, or in other words, the holonomy of any loop homotopic to the identity, is trivial. Thus, the holonomy group is a homomorphic image of the [[fundamental group]], and thus gives rise to a [[linear representation]] of the fundamental group.

Vipul: /* Definition */

2007-09-02T02:28:00Z

Definition

← Older revision		Revision as of 02:28, 2 September 2007
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	Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math> (turning <math>(M,g)</math> into a [[Riemannian manifold]]). The '''holonomy group''' of <math>g</math> is defined as the [[holonomy group]] for the [[Levi-Civita connection]] on the [[tangent bundle]].		Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math> (turning <math>(M,g)</math> into a [[Riemannian manifold]]). The '''holonomy group''' of <math>g</math> is defined as the [[holonomy group]] for the [[Levi-Civita connection]] on the [[tangent bundle]].

	The holonomy group is a subgroup of the [[orthogonal group]]. This is because [[transport along a curve]] using the Levi-Civita connection preserves the Riemannian metric.		The holonomy group is a subgroup of the [[orthogonal group]]. This is because [[transport along a curve]] using the Levi-Civita connection (that is, [[Levi-Civita transport]]) preserves the Riemannian metric.

	==Related notions==		==Related notions==

Vipul at 14:57, 1 September 2007

2007-09-01T14:57:11Z

← Older revision		Revision as of 14:57, 1 September 2007
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	Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math> (turning <math>(M,g)</math> into a [[Riemannian manifold]]). The '''holonomy group''' of <math>g</math> is defined as the [[holonomy group]] for the [[Levi-Civita connection]] on the [[tangent bundle]].		Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math> (turning <math>(M,g)</math> into a [[Riemannian manifold]]). The '''holonomy group''' of <math>g</math> is defined as the [[holonomy group]] for the [[Levi-Civita connection]] on the [[tangent bundle]].

			The holonomy group is a subgroup of the [[orthogonal group]]. This is because [[transport along a curve]] using the Levi-Civita connection preserves the Riemannian metric.

	==Related notions==		==Related notions==

Vipul at 14:55, 1 September 2007

2007-09-01T14:55:29Z

New page

{{Riemannian metric-associated group}}

==Definition==

Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math> (turning <math>(M,g)</math> into a [[Riemannian manifold]]). The '''holonomy group''' of <math>g</math> is defined as the [[holonomy group]] for the [[Levi-Civita connection]] on the [[tangent bundle]].

==Related notions==

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