Isometric Embedding and Darboux Integrability

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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 the isometric embedding M N have particularly nice solvability properties. In this paper we give a classification of all 2-metrics g whose local 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, g₀, 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 g₀ 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 g₀ up to quadrature. The results described for g₀ also hold for any classified metric whose embedding system is hyperbolic.