  We are concerned with underlying connections between fluids, elasticity,
isometric embedding of Riemannian manifolds, and the existence of wrinkled
solutions of the associated nonlinear partial differential equations. In this
paper, we develop such connections for the case of two spatial dimensions, and
demonstrate that the continuum mechanical equations can be mapped into a
corresponding geometric framework and the inherent direct application of the
theory of isometric embeddings and the Gauss-Codazzi equations through examples
for the Euler equations for fluids and the Euler-Lagrange equations for elastic
solids. These results show that the geometric theory provides an avenue for
addressing the admissibility criteria for nonlinear conservation laws in
continuum mechanics.
