The One-Sided Isometric Extension Problem

Let Sigma be a codimension one submanifold of an n-dimensional Riemannian manifold M, n >= 2. We give a necessary condition for an isometric immersion of Sigma into R-q equipped with the standard Euclidean metric, q >= 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.