Ambrose-Singer theorem

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 $v,w\in T_{p}M$.

For connections on principal bundles

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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