nLab
holonomy group

Contents

Idea

For XX a space equipped with a GG-connection on a bundle \nabla (for some Lie group GG) and for xXx \in X any point, the parallel transport of \nabla assigns to each curve Γ:S 1X\Gamma : S^1 \to X in XX starting and ending at xx an element hol (γ)G hol_\nabla(\gamma) \in G: the holonomy of \nabla along that curve.

The holonomy group of \nabla at xx is the subgroup of GG on these elements.

If \nabla is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup HH of the special orthogonal group, one says that (X,g)(X,g) is a manifold of special holonomy .

References

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Created on August 26, 2011 16:42:01 by Urs Schreiber (82.113.99.51)