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 0 , 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 0 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 0 up to quadrature. The results described for g 0 also hold for any classified metric whose embedding system is hyperbolic.