Geodesic
Isometric Embeddings of Locally Euclidean Metrics in~$\mathbb R^3$ as Conical Surfaces
Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 234-247

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It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane $\mathbb R^2$ with the standard metric, then it can be isometrically embedded in $\mathbb R^3$ as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.
Keywords: locally Euclidean metric, isometric embedding, isometric immersion, conical surface, planar graph. 
@article{MZM_2014_95_2_a6,
     author = {S. N. Mikhalev and I. Kh. Sabitov},
     title = {Isometric {Embeddings} of {Locally} {Euclidean} {Metrics} in~$\mathbb R^3$ as {Conical} {Surfaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {234--247},
     publisher = {mathdoc},
     volume = {95},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a6/}
}
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S. N. Mikhalev; I. Kh. Sabitov. Isometric Embeddings of Locally Euclidean Metrics in~$\mathbb R^3$ as Conical Surfaces. Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 234-247. http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a6/