Associated bundle

Summary

Let $P \to B$ be a principal bundle with group $G$, and let $G$ act on a space $F$. The associated bundle $P \times_G F$ is a fiber bundle over $B$ with fiber $F$. It is built by gluing copies of $F$ along $B$ with the same twisting as $P$. In this way one principal bundle produces many bundles: vector bundles, sphere bundles, bundles of frames and so on. For example, the tangent bundle of a manifold is associated to its frame bundle through the action of $GL_n(\R)$ on $\R^n$.