In this lecture we show how one can combine:

• A prinicpal $G$-bundle $\pi \colon P \to M$,
• A smooth action $\sigma$ of $G$ on a manifold $L$,

to produce a new $(G,\sigma)$-fibre bundle $\pi_L \colon P \times_G L \to M$ with fibre $L$. This is called an associated bundle of $P$. If $L=V$ is a vector space and $\sigma$ is linear then $P \times_G V$ is a vector bundle.

If $E$ is a vector bundle with fibre $V$ then we show that

$$E \cong \operatorname{Fr}(E) \times_{\operatorname{GL}(V)} V,$$

that is, the operation of taking the associated bundle of the frame bundle coming from the canonical representation of $\operatorname{GL}(V)$ on $V$ recovers the original vector bundle.

This in itself is not too thrilling, but by pushing this argument a bit further we obtain a proof of the following metatheorem:

Anything you can do with vector spaces, you can do with vector bundles.

We will see more applications of this next lecture.