Holonomy group of Riemannian metric: Difference between revisions
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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 (that is, [[Levi-Civita transport]]) preserves the Riemannian metric. | |||
==Related notions== | ==Related notions== | ||
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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. | |||
Latest revision as of 19:46, 18 May 2008
Template:Riemannian metric-associated group
Definition
Let be a differential manifold and a Riemannian metric on (turning into a Riemannian manifold). The holonomy group of 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 (that is, Levi-Civita transport) preserves the Riemannian metric.
Related notions
- Holonomy group
- 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.