On the Local Solvability of Darboux's Equation

TL;DR

This paper links the local nonsolvability of Darboux's equation, related to surface isometric embedding, to a simpler linear equation determined by Gaussian curvature, simplifying the analysis of such problems.

Contribution

It reduces the complex problem of local nonsolvability of Darboux's equation to a more manageable linear equation based on Gaussian curvature.

Findings

Established a reduction from Darboux's equation to a linear equation

Connected local nonsolvability to the type of the linear equation

Provided a clearer criterion based on Gaussian curvature

Abstract

We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature.