A note on global smoothability of small $L^p$-connection with prescribed curvature

We show that if $Omega$ is a connection $1$-form on a vector bundle $eta$ over a closed $n$-dimensional Riemannian manifold $mathcal{M}$ with $L^p$-regularity ($p>n$) and smooth curvature $2$-form $mathscr{F}$, then it can be approximated in the $L^p$-norm by smooth connections of the same curvature, provided that $|Omega|_{L^p(mathcal{M})}$ is smaller than a uniform constant depending only on $p$ and $mathcal{M}$.