# Ambrose-Singer theorem

## Contents

## Statement

### 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*

## 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*