Isometric embedding and Darboux integrability

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold \((M, \varvec{g})\) and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold \((N, \varvec{h})\), one can ask under what circumstances does the exterior differential system \(\mathcal {I}\) for an isometric embedding \(M\hookrightarrow N\) have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics \(\varvec{g}\) whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds \((N, \varvec{h})\) is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, \(\varvec{g}_0\), 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 \(\varvec{g}_0\) 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 \(\varvec{g}_0\) up to quadrature. The results described for \(\varvec{g}_0\) also hold for any classified metric whose embedding system is hyperbolic.