A regularity gain for some classes of Sobolev isometric immersions

I will first discuss and present a proof of the following statement. Let $m>1$. Any $W^{2,m}$ isometric immersion from a domain $R^m$ into $R^{m+1}$ is $C1$ regular inside the domain. I will proceed to dicuss some open problems. In particular strategies and obstacles in generalizing this result to isometric immersions of any codimension will be discussed.