nLab
holonomy

Contents

Idea

Given connection on a bundle \nabla over a space XX, its parallel transport around some loop γ:[0,1]X\gamma : [0,1] \to X, γ(0)=γ(1)=x 0\gamma(0) = \gamma(1) = x_0 yields an element

hol (γ)G hol_\nabla(\gamma) \in G

in the automorphism group of the fiber P x 0P_{x_0} of the bundle. This is the holonomy of \nabla around γ\gamma.

Fixing a connection \nabla and a point xXx \in X the collection of all elements hol (γ)hol_\nabla(\gamma) for all loops γ\gamma based at xx forms a subgroup of GG, called the holonomy group.

If the Levi-Civita connection on a Riemannian manifold (X,g)(X,g) has a holonomy group that is a proper subgroup of the special orthogonal group one says that (X,g)(X,g) is a manifold with special holonomy. (More precise would be: “with special holonomy group for the Levi-Civita connection”.)

Properties

Proposition. If on a smooth principal bundle PXP\to X there is a connection \nabla whose holonomy group is GG then the structure group can be reduced to GG.

(…)

Theorem

(Ambrose-Singer) Ambrose-Singer theorem: the Lie algebra of the holonomy group of a connection on a bundle \nabla on XX at a point xXx \in X is spanned by the parallel transport Ad tra (γ)(F A(vw))Ad_{tra_\nabla(\gamma)}(F_A(v \vee w)) of the curvature F AF_A evaluated on any vw 2T yXv \vee w \in \wedge^2 T_y X at yXy \in X along any path γ\gamma from xyx \to y.

We may think of Id+Ad tra (γ)(F A(ϕ))Id + \Ad_{tra_\nabla(\gamma)}(F_A(\phi)) as being the holonomy around the loop obtained by

  1. going along γ\gamma from xx to yy

  2. going around the infinitesimal parallelogram spanned by vv and ww;

  3. coming back to xx along the reverse path γ\gamma.

Applications

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Higher holonomy

Also: The higher holonomy of circle n-bundles with connection is given by fiber integration in ordinary differential cohomology.

Revised on October 26, 2012 15:33:23 by Urs Schreiber (80.187.201.44)