  We study a class of design problems in solid mechanics, leading to a
variation on the classical question of equi-dimensional embeddability of
Riemannian manifolds. In this general new context, we derive a necessary and
sufficient existence condition, given through a system of total differential
equations, and discuss its integrability. In the classical context, the same
approach yields conditions of immersibility of a given metric in terms of the
Riemann curvature tensor. In the present situation, the equations do not close
in a straightforward manner, and successive differentiation of the
compatibility conditions leads to a new algebraic description of integrability.
We also recast the problem in a variational setting and analyze the infimum of
the appropriate incompatibility energy, resembling the "non-Euclidean
elasticity." We then derive a $\Gamma$-convergence result for the dimension
reduction from $3$d to $2$d in the Kirchhoff energy scaling regime.
