  We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric
immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge
\min\{2k, n\}$. Another parallel consequence of our methods is a similar
regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate
Monge-Amp\`ere equations in $n$ dimensions. The analysis also applies to the
situations when the degeneracy is extended to $(k+1)\times (k+1)$ minors of the
Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min\{2k, n\}$.
