Curvature of an induced associated connection via a representation

Let $P\to M$ be a principal G-bundle with principal connection $\omega$ and curvature $\Omega\in\Omega^2(P;\mathfrak g)$. Let $E=P\times_G V$ be an associated vector bundle for a representation $\rho:G\to \mathrm{GL}(V)$, with induced covariant derivative $\nabla$ as in induced covariant derivative . Let $\rho_*:\mathfrak g\to \mathrm{End}(V)$ be the derived representation.

Statement. The curvature of $\nabla$ is the $\mathrm{End}(V)$-valued 2-form obtained by applying $\rho_*$ to the principal curvature:

• On $P$, the horizontal, equivariant $\mathrm{End}(V)$-valued 2-form is $\rho_*(\Omega)$.
• On $M$, in a local gauge $s:U\to P$ with $A=s^*\omega$ and $F=s^*\Omega$, the associated curvature is $F_V = \rho_*(F)\in \Omega^2(U;\mathrm{End}(V)).$

Equivalently, for any section $s$ of $E$,

in local form, expressing that the curvature acts through the infinitesimal representation.

Examples

1. For the defining representation of $\mathrm{GL}(n)$, this recovers the usual matrix-valued curvature of a rank-$n$ vector bundle connection.
2. If the principal connection is flat (principal curvature zero), then every induced associated connection is flat.
3. For the adjoint representation on $\mathfrak g$, the induced curvature on $\mathrm{ad}(P)$ is given by the bracket action $\mathrm{ad}_{F}$ in local form.