 Associated bundle

For a principal bundle $P\to B$ and a left action of its structure group on a manifold $F$, the associated bundle is $(P\times F)/\sim$, where $(pg,f)\sim (p,gf)$. Local sections identify it with $U\times F$; principal transition functions act on $F$. Linear actions give associated vector bundles.

An associated bundle is a construction from differential geometry and algebraic topology that pertains to the study of fiber bundles. In the context of a fiber bundle, the associated bundle is a way of "associating" a new fiber bundle with a given principal bundle and a representation of its structure group.