Associated principal bundle of the universal vector bundle

Remark 9.2.13. Under the bijection of Proposition 9.2.7, the principal bundle \(V_k(\C ^{\infty }) \to \Gr _k(\C ^{\infty })\) is in fact the principal bundle associated to the universal rank \(k\) vector bundle \(E_{\univ }\) from Definition 9.2.2.