$C^{1,\alpha}$ Isometric Embeddings of Polar Caps

TL;DR

This paper investigates the regularity threshold for isometric embeddings of 2D manifolds, showing that $C^{1,1/2}$ regularity is critical for the Levi-Civita connection to be Euclidean, with constructions below this threshold.

Contribution

It establishes the criticality of the $C^{1,1/2}$ regularity for isometric embeddings and constructs examples below this threshold where the Levi-Civita connection differs.

Findings

For $eta > 1/2$, Levi-Civita connection is Euclidean.

For $eta < 1/2$, constructed embeddings where Levi-Civita connection is not Euclidean.

$C^{1,1/2}$ is the critical regularity for the property.

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.