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.
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Hungerbühler, N., Wasem, M. The One-Sided Isometric Extension Problem. Results Math 71, 749–781 (2017). https://doi.org/10.1007/s00025-016-0593-0
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DOI: https://doi.org/10.1007/s00025-016-0593-0