Associated bundles

Let $\pi :P\to M$ be a smooth right principal $H$-bundle, meaning that the topological principal charts of Principal g bundle and associated fiber bundle are diffeomorphisms, and let $\rho :H\to GL(V)$ be a smooth finite-dimensional real representation. Define a right action on $P\times V$ by

$(p,v)\cdot h=(ph,\rho (h{)}^{-1}v)\mathrm{.}$

The vector bundle associated to $P$ and $\rho$ is the quotient set with quotient topology

$P{\times }_{H}V=(P\times V)\mathrm{/}H\mathrm{.}$

Write $[p,v]$ for the orbit of $(p,v)$. Equivalently, the generating relation is

$[ph,v]=[p,\rho (h)v]\mathrm{.}$

The projection is

$r:P{\times }_{H}V\to M,r([p,v])=\pi (p)\mathrm{.}$

It is well defined because $\pi (ph)=\pi (p)$ and continuous by the quotient universal property For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map. The inverse in the diagonal action is essential: it makes the displayed relation and the right action law agree. The quotient construction itself uses no choices. The homogeneous principal bundle of G to G/H is a smooth principal H-bundle is one instance, not a hypothesis of the general definition. The next theorem supplies the smooth vector-bundle atlas in the sense of Vector bundle charts and transition functions.