Rigidity and flexibility of isometric extensions

In this paper we consider the rigidity and flexibility of C-1,C-theta isometric extensions. We show that the H & ouml;lder exponent theta( 0) = 1/2 is critical in the following sense: if u is an element of C-1,C-theta is an isometric extension of a smooth isometric embedding of a codimension one submanifold E and Theta > 1/2, then the tangential connection agrees with the Levi-Civita connection along Sigma. On the other hand, for any theta < 1/2 we can construct C1'8 isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for C1'8 isometric embeddings, Theta < 1/2, of compact Riemannian manifolds with C1 metrics and sharper amount of codimension.