nLab
holonomy

Idea

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

$${\mathrm{hol}}_{\nabla }(\gamma )\in G$$hol_\nabla(\gamma) \in G

in the automorphism group of the fiber $P_{x_0}$ of the bundle. This is the holonomy of $\nabla$ around $\gamma$.

Fixing a connection $\nabla$ and a point $x\in X$ the collection of all elements ${\mathrm{hol}}_{\nabla }(\gamma )$ for all loops $\gamma$ based at $x$ forms a subgroup of $G$, called the holonomy group.

If the Levi-Civita connection on a Riemannian manifold $(X,g)$ has a holonomy group that is a proper subgroup of the special orthogonal group one says that $(X,g)$ 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 $P\to X$ there is a connection $\nabla$ whose holonomy group is $G$ then the structure group can be reduced to $G$.

(…)

The Ambrose-Singer theorem? states that the Lie algebra of the holonomy group of a connection on a bundle $\nabla$ on $X$ at a point $x\in X$ is spanned by the parallel transport ${\mathrm{Ad}}_{{\mathrm{tra}}_{\nabla }(\gamma )}(F_A(v\vee w))$ of the curvature $F_A$ evaluated on any $v\vee w\in {\wedge }^2{T}_{y}X$ at $y\in X$ along any path $\gamma$ from $x\to y$.

We may think of $\mathrm{Id}+{Ad}_{{\mathrm{tra}}_{\nabla }(\gamma )}(F_A(\varphi ))$ as being the holonomy around the loop obtained by

1. going along $\gamma$ from $x$ to $y$

2. going around the infinitesimal parallelogram spanned by $v$ and $w$;

3. coming back to $x$ along the reverse path $\gamma$.

Theorem

(Ambrose-Singer)

(…)

Applications

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Higher holonomy

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

Related concepts

• connection on a bundle, connection on a 2-bundle, connection on an infinity-bundle

• parallel transport, higher parallel transport

• holonomy

  • holonomy group

  • special holonomy

  • fiber integration in ordinary differential cohomology