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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - S. N. Mikhalev
AU  - I. Kh. Sabitov
TI  - Isometric Embeddings of Locally Euclidean Metrics in~$\mathbb R^3$ as Conical Surfaces
JO  - Matematičeskie zametki
PY  - 2014
SP  - 234
EP  - 247
VL  - 95
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a6/
LA  - ru
ID  - MZM_2014_95_2_a6
ER  - 
%0 Journal Article
%A S. N. Mikhalev
%A I. Kh. Sabitov
%T Isometric Embeddings of Locally Euclidean Metrics in~$\mathbb R^3$ as Conical Surfaces
%J Matematičeskie zametki
%D 2014
%P 234-247
%V 95
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a6/
%G ru
%F MZM_2014_95_2_a6
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/

[1] I. Kh. Sabitov, “Izometricheskoe pogruzhenie lokalno-evklidovykh metrik v $\mathbb R^3$”, Sib. matem. zhurn., 26:3 (1985), 156–167 | MR | Zbl

[2] I. N. Vekua, Obobschennye analiticheskie funktsii, Fizmatgiz, M., 1959 | MR | Zbl