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Recall that a smooth Riemannian metric on a simply connected domain canbe realized as the pull-back metric of an orientation preserving deformation ifand only if the associated Riemann curvature tensor vanishes identically.When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem byintroducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the properscaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W2,2 isometric immersion of a given 2d metricinto $\mathbb R^3$
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