On the Local Solvability of Darboux’s Equation

We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature. Let (M, g) be a two-dimensional Riemannian manifold. A well-known problem is to ask, when can one realize this locally as a small piece of a surface in R? That is, if the metric g = gijdx dx is given in the neighborhood of a point, say (x, x) = 0, when do there exist functions zα(x , x), α = 1, 2, 3, defined in a possibly smaller domain such that g = dz 1 + dz 2 2 + dz 2 3? This equation may be written in local coordinates as the following determined system