  We derive a continuum model for incompatible elasticity as a variational
limit of a family of discrete nearest-neighbor elastic models. The discrete
models are based on discretizations of a smooth Riemannian manifold
$(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The
metric $\mathfrak{g}$ determines local equilibrium distances between
neighboring points; the connection $\nabla$ induces a lattice structure shared
by all the discrete models. The limit model satisfies a fundamental rigidity
property: there are no stress-free configurations, unless $\mathfrak{g}$ is
flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional
systems, however, all our results readily generalize to higher dimensions.
