  In this paper we consider the rigidity and flexibility of $C^{1, \theta}$
isometric extensions and we show that the H\"older exponent $\theta_0=\frac12$
is critical in the following sense: if $u\in C^{1,\theta}$ is an isometric
extension of a smooth isometric embedding of a codimension one submanifold
$\Sigma$ and $\theta> \frac12$, then the tangential connection agrees with the
Levi-Civita connection along $\Sigma$. On the other hand, for any
$\theta<\frac12$ we can construct $C^{1,\theta}$ isometric extensions via
convex integration which violate such property. As a byproduct we get moreover
an existence theorem for $C^{1, \theta}$ isometric embeddings,
$\theta<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and
sharper amount of codimension.
