  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 $\mathbb R^q$ equipped with the standard
Euclidean metric, $q\geqslant n+1$, to be locally isometrically
$C^1$-extendable to $M$. Even if this condition is not met, "one-sided"
isometric $C^1$-extensions may exist and turn out to satisfy a $C^0$-dense
parametric $h$-principle in the sense of Gromov.
