Isometric embedding and Darboux integrability

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold (M, g) and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold (N, h) , one can ask under what circumstances does the exterior differential system I for an isometric embedding M↪ N have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics g whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds (N, h) is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, g, showing how to use its Darboux integrability in order to construct all its embeddings in finite terms of arbitrary functions. Additionally, the geometric Cauchy problem for the embedding of g is shown to be reducible to a system of two first-order ODEs for two unknown functions—or equivalently, to a single second-order scalar ODE. For a large class of initial data, this reduction permits explicit solvability of the geometric Cauchy problem for g up to quadrature. The results described for g also hold for any classified metric whose embedding system is hyperbolic.