# Equivalence between principal $O(n)$-bundles and vector bundles

There's a well-known result (for example, Th. 14.2.7 in tom Dieck's book) that the category of principal $\operatorname{GL}_n(\mathbb{R})$-bundles and bundle maps is equivalent to the category of $n$-plane bundles and bundle maps by sending a principal $\operatorname{GL}_n(\mathbb{R})$-bundle to to its associated fiber bundle.

A number of people have mentioned to me that a similar result holds between the categories of $n$-plane bundles over paracompact spaces, and principal $O(n)$-bundles over paracompact spaces. However, I can't seem to find a reference for this. I'm wondering if anyone knows of a reference for this (of if it's actually not true).

• You don't get an equivalence of categories because e.g. the automorphism groups of the trivial bundle don't match up. Loosely, you get an equivalence of topologically enriched categories up to homotopy, coming from the fact that $GL_n(\mathbb{R})$ deformation retracts onto $O(n)$. – Qiaochu Yuan Nov 5 '15 at 18:30

First, notice that the claimed equivalence between principal $\operatorname{GL}_n(\mathbb{R})$-bundles and $n$-plane bundles holds only if you consider only those vector bundle maps which are invertible, i.e. they are linear isomorphisms on fibers.
Your claim about principal $O(n)$-bundles is not true. The category of principal $O(n)$-bundles is equivalent to the category of $n$-plane bundles endowed with an Euclidean metric, and having as morphisms bundle maps which are linear isometries fiberwise.
What is true is that on a $n$-plane bundle on a paracompact space you can always put such an Euclidean metric using a partition of unit, but then an arbitrary vector bundle map is not necessarily an isometry, and in this case it does not induce a map of principal $O(n)$-bundles.