Let $M$ be a smooth manifold, $G$ a Lie group and $P\rightarrow M$ a smooth principal $G$-bundle. Let $\Omega^1_{eq} (P;\mathfrak{g})$ denote the space of $G$-equivariant $\mathfrak{g}$-valued 1-forms in $P$. Let $\Omega^2_{heq} (P;\mathfrak{g})$ denote the space of horizontal and $G$-equivariant $\mathfrak{g}$-valued 2-forms in $P$. Let us say that a 2-form $F\in \Omega^2_{heq} (P;\mathfrak{g})$ is covariant if it is the exterior covariant derivative of someone. More precisely, if there is a connection 1-form $D$ in $P$ and a 1-form $\alpha\in \Omega^1_{eq} (P;\mathfrak{g})$ such that $F=d_D\alpha = d\alpha + \frac{1}{2}\alpha [\wedge]D$.
My question is:
- under which conditions on $P$, $G$ and $M$ every 2-form in $\Omega^2_{heq} (P;\mathfrak{g})$ is covariant?
Thanks a lot.