Isometric embedding and Darboux integrability

Abstract

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.