# 42. The Ambrose–Singer Holonomy Theorem

In this lecture we define the holonomy of a connection $\omega$ on a principal $G$-bundle $ \pi \colon P \to M$. The construction is slightly more elegant than the corresponding treatment for vector bundles as we can view the holonomy group $H^{ \omega}(p)$ as a subgroup of $G$ itself.

We then state and prove the principal bundle version of the *Ambrose-Singer Holonomy Theorem*, thus completing the unfinished business from Lecture 35.

This concludes the first part of Differential Geometry II. Next lecture we begin Riemannian Geometry.

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