A study of the characteristic variety of the isometric embedding problem in the determined dimension. We show that, except for metrics whose Riemann curvature tensor lies in a a very small set of normal forms, a 3-manifold can be isometrically embedded into real 6-space. The method is to show that the system can be made suitably hyperbolic so that a version of the Nash-Moser theorem can be made to apply.
Deane Yang and Johnathan Goodman have since generalized the principal analytic result from hyperbolic systems to systems of real principal type and have used this to prove isometric embedding results for 4-manifolds. In higher dimensions, the characteristic variety tends to have singularities that no analytic methods are known to handle (in the smooth category).