The One-Sided Isometric Extension Problem

TL;DR

This paper investigates conditions under which a submanifold's isometric immersion can be extended to the ambient manifold, revealing necessary conditions and the existence of dense extension phenomena in the $C^1$ topology.

Contribution

It establishes a necessary condition for local $C^1$ isometric extension and demonstrates the generic existence of one-sided extensions satisfying a $C^0$-dense parametric $h$-principle.

Findings

Necessary condition for $C^1$ isometric extension of submanifolds.

Existence of one-sided $C^1$-extensions satisfying a $C^0$-dense $h$-principle.

Extension phenomena occur even when the necessary condition is not met.

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 $\mathbb R^q$ 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.