For connections on vector bundles
Let be a differential manifold, a vector bundle over , and a connection on . Let . Then, the Lie algebra of the restricted holonomy group for at is the subalgebra of the Lie algebra of all endomorphisms of , generated by the values where .
For connections on principal bundles
Fill this in later
The significance of this is that the Riemann curvature tensor is in some sense, the differential of the holonomy at the point. Fill this in later