Today we introduce another special class of fibre bundle, called a principal bundle. Prinicipal bundles are equally (if not more so) fundamental as vector bundles. Roughly speaking, a principal bundle is a fibre bundle where the fibre is a Lie group acting freely on the total space, and all associated data has to be equivariant with respect to this action.

Two key examples of principal bundles are:

• If $M$ is a homogeneous space $M \cong G/H$ then $G \to M$ is a principal $H$-bundle. (See Lecture 12 if you forgot the definition of a homogeneous space.)
• If $E$ is any vector bundle of rank $k$ then its frame bundle $\operatorname{Fr}(E)$ is a principal $\operatorname{GL}(k)$-bundle.

Next semester it will be convenient to be able to switch back and forth between vector bundles and principal bundles (and indeed, which you prefer is essentially a matter of taste).