In this paper, we prove the existence of an isometric embedding near the origin in R3 of a two-dimensional metric with nonpositive Gaussian curvature. The Gaussian curvature can be allowed to be highly degenerate near the origin. Through the Gauss-Codazzi equations, the embedding problem is reduced to a 2 × 2 system of the first order derivaties and is solved via the method of Nash-Moser-Hörmander iterative scheme.
"Local Isometric Embedding of Surfaces with Nonpositive Gaussian Curvature." J. Differential Geom. 63 (3) 475 - 520, March, 2003. https://doi.org/10.4310/jdg/1090426772