# 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 $v,w \in T_pM$.

### 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