  We discuss $C^1$ regularity and developability of isometric immersions of
flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s,
\frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on
Sobolev and H\"older regimes. Ingredients of the proof include analysis of the
weak Codazzi-Mainardi equations of the isometric immersions and study of
$W^{2,\frac2s}$ gradient deformations with symmetric Jacobian derivative and
vanishing distributional Jacobian determinant. On the way, we also show that
the distributional Jacobian determinant, conceived as an operator defined on
the Jacobian matrix, behaves like determinant of gradient matrices under
products by scalar functions.
