# 41. Holonomy and principal bundles

In this lecture we define the holonomy of a connection $ \varpi$ on a principal $G$-bundle $ \pi \colon P \to M$. The construction is slightly more elegant than the corresponding treatment for vector bundles (see Lecture 32) as we can view the holonomy group $H_{ \varpi}(p)$ 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 32.

This concludes the first half of Differential Geometry II.

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