Ambrose-Singer theorem

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Statement

For connections on vector bundles

Let M be a differential manifold, E a vector bundle over M, and \nabla a connection on E. Let p \in M. Then, the Lie algebra of the restricted holonomy group for \nabla at p is the subalgebra of the Lie algebra of all endomorphisms of E_p, generated by the values R(v,w) where v,w \in T_pM.

For connections on principal bundles

Fill this in later

Importance

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