Holonomy group: Difference between revisions
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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'''. | ||
Revision as of 14:50, 1 September 2007
Definition
Let be a differential manifold, a vector bundle over and a connection for . For a point the holonomy group at is the subgroup of comprising those linear transformations that arise as the holonomy of a loop at .
If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of so we can talk of the holonomy group.