In this lecture we show how one can combine:

• A prinicpal $G$-bundle $\pi \colon P \to M$,
• An effective action $\mu$ of $G$ on a manifold $F$,

to produce a new fibre bundle $\wp \colon P \times_G F \to M$ with fibre $F$ and structure group $G$. This is called an associated bundle of $P$. If $F$ is a vector space and $\mu$ is linear then $P \times_G F$ is a vector bundle.

We explain how many of our earlier vector bundle constructions can be rephrased in this language: for example, if $E$ is a vector bundle over $M$ with typical fibre $V$, then if $\mathrm{Fr}(E)$ denotes the frame bundle one has:

$$\mathrm{Fr}(E) \times_{ \mathrm{GL}(V)} V^* = E^*,$$

$$\mathrm{Fr}(E) \times_{ \mathrm{GL}(V)} (V \otimes V) = E \otimes E,$$

$$\mathrm{Fr}(E) \times_{ \mathrm{GL}(V)} \bigwedge(V) = \bigwedge(E),$$

and so on.