Geodesic
Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$
Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1203-1220

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This paper deals with the problem of isometric immersions and embeddings of two-dimensional locally Euclidean metrics in the Euclidean plane. We find explicit formulae for the immersions of metrics defined on a simply connected domain and a number of sufficient conditions for the existence of isometric embeddings. In the case when the domain is multiply connected we find necessary conditions for the existence of isometric immersions and classify the cases when the metric admits no isometric immersion in the Euclidean plane. 
@article{IM2_1999_63_6_a4,
     author = {I. Kh. Sabitov},
     title = {Isometric immersions and embeddings of locally {Euclidean} metrics in~$\mathbb R^2$},
     journal = {Izvestiya. Mathematics },
     pages = {1203--1220},
     publisher = {mathdoc},
     volume = {63},
     number = {6},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/}
}
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I. Kh. Sabitov. Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$. Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1203-1220. http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/