I am quite sure that this is all standard, but I couldn't find any reference:
Let $(M,g)$ be a compact, oriented $d$-dimensional Riemannian manifold with boundary. Let $f:M\to R^{d+1}$ be a $W^{2,2}$-map such that $\det df>0$ a.e. Can $f$ be approximated in $W^{2,2}$ by smooth immersions? (Note that the only role of the metric $g$ is to provide a norm on differential forms and a volume form for integration.)