Associated vector bundle ($E=P{\times }_{\rho }V$)

A linear representation $\rho :G\to GL(V)$ of a principal bundle's structure group defines the quotient of $P\times V$ by $(p,v)\sim (pg,\rho (g{)}^{-1}v)$. It is a vector bundle over the same base. A principal connection induces parallel transport and a local covariant derivative $d+{\rho }_{\ast }(A)$ on its sections.

The adjoint bundle is the associated vector bundle for the Adjoint representation of a Lie group on its Lie algebra. The gauge curvature of a principal connection is globally a two-form with values in this bundle. Differences of principal connections and their infinitesimal variations are one-forms with values in the same bundle.