Geodesic
On the existence of solutions of the system of Peterson--Codazzi and gauss equations
Matematičeskie zametki, Tome 17 (1975) no. 5, pp. 765-781.

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This paper is concerned with isometric embeddings of complete two-dimensional metrics, defined on the plane, whose curvature is bounded by negative constants (metrics of type L). It is proved that under certain conditions any horocycle in a metric of type L (an analog of a horocycle in the Lobachevskii plane) admits a $C^3$-isometric embedding into $E^3$. The proof is based on the construction of a smooth solution of the system of Peterson–Codazzi and Gauss equations in an infinite domain. 
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     author = {E. V. Shikin},
     title = {On the existence of solutions of the system of {Peterson--Codazzi} and gauss equations},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {5},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_5_a10/}
}
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E. V. Shikin. On the existence of solutions of the system of Peterson--Codazzi and gauss equations. Matematičeskie zametki, Tome 17 (1975) no. 5, pp. 765-781. http://geodesic.mathdoc.fr/item/MZM_1975_17_5_a10/