Published Paper
Inserted: 18 sep 2018
Last Updated: 1 dec 2020
Journal: Adv. Math.
Volume: 363
Year: 2020
Abstract:
We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"older space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $\alpha < \frac{1}{2}$ we construct $C^{1,\alpha}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.
Download: