  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.
