Isometric Embedding and Darboux Integrability

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