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 preserves the Riemannian metric. | |||
==Related notions== | ==Related notions== | ||
Revision as of 14:57, 1 September 2007
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 preserves the Riemannian metric.