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

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.