Fixing a connection and a point the collection of all elements for all loops based at forms a subgroup of , called the holonomy group.
If the Levi-Civita connection on a Riemannian manifold has a holonomy group that is a proper subgroup of the special orthogonal group one says that is a manifold with special holonomy. (More precise would be: “with special holonomy group for the Levi-Civita connection”.)
Proposition. If on a smooth principal bundle there is a connection whose holonomy group is then the structure group can be reduced to .
(Ambrose-Singer) Ambrose-Singer theorem: the Lie algebra of the holonomy group of a connection on a bundle on at a point is spanned by the parallel transport of the curvature evaluated on any at along any path from .
We may think of as being the holonomy around the loop obtained by
going along from to
going around the infinitesimal parallelogram spanned by and ;
coming back to along the reverse path .
The higher holonomy (see there) of circle n-bundles with connection is given by fiber integration in ordinary differential cohomology.