Abstract
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 C3-isometric embedding into E3. 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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E. V. Shikin, “On the regularity of horocyclic coordinates,” Matem. Zametki,17, No. 3, 477–486 (1975).
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E. V. Shikin, “On the regular embedding in the large of metrics of class C4 of negative curvature into R3,” Matem. Zametki,14, No. 2, 261–266 (1973).
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Translated from Matematicheskie Zametki, Vol. 17, No. 5, pp. 765–781, May, 1975.
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Shikin, E.V. On the existence of solutions of the system of Peterson-codazzi and gauss equations. Mathematical Notes of the Academy of Sciences of the USSR 17, 455–466 (1975). https://doi.org/10.1007/BF01155803
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DOI: https://doi.org/10.1007/BF01155803