18. Associated Bundles
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.
Comments and questions?