The One-Sided Isometric Extension Problem

Abstract: Let $\Sigma$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $\Sigma$ into $\mathbb R<sup>q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C<sup>1$-extendable to $M$. Even if this condition is not met, "one-sided" isometric $C<sup>1$-extensions may exist and turn out to satisfy a $C<sup>0$-dense parametric $h$-principle in the sense of Gromov.