  We are concerned with two interrelated problems: smoothability of connection
1-forms with low regularity on bundles with prescribed smooth curvature
2-forms, and existence of isometric immersions with low regularity. We first
show that if $\Omega$ is an $L^p$-connection $1$-form on a vector bundle over a
closed Riemannian $n$-manifold $\mathcal{M}$ with small $L^p$-norm ($p>n$) and
smooth curvature $2$-form $\mathscr{F}$, then $\Omega$ can be approximated in
the $L^p_{\rm loc}$-topology by smooth connections of the same curvature (not
necessarily gauge equivalent). Our proof, adapted from S. Mardare's work on the
fundamental theory of surfaces with $L^p$-second fundamental form, is
elementary in nature and uses only Hodge decomposition and fixed point
theorems. This result is then applied to the study of isometric immersions of
Riemannian manifolds with low regularity. We revisit the proof for the
existence of $W^{2,p}$-isometric immersion $\mathcal{M}^n \to \mathbf{R}^{n+k}$
with arbitrary $n$ and $k$ given weak solutions to the Gauss--Codazzi--Ricci
equations, aiming at elucidating some global vs. local issues, and also we
provide a characterisation for metrics on $\mathcal{M}$ that admit $W^{2,p}$-
but no $C^\infty$-isometric immersions.
