  Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$,
$n\geqslant 2$. We study the isometric extension problem for isometric
immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the
Euclidean standard metric. We prove a general curvature obstruction to the
existence of merely differentiable extensions and an obstruction to the
existence of Lipschitz extensions of $f$ using a length comparison argument.
Using a weak form of convex integration, we then construct one-sided isometric
Lipschitz extensions of which we compute the Hausdorff dimension of the
singular set and obtain an accompanying density result. As an application we
obtain the existence of infinitely many Lipschitz isometries collapsing the
standard two-sphere to the closed standard unit $2$-disk mapping a great-circle
to the boundary of the disk.
