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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 6, Pages 147–166 (Mi izv270)  

This article is cited in 5 scientific papers (total in 5 papers)

Isometric immersions and embeddings of locally Euclidean metrics in $\mathbb R^2$

I. Kh. Sabitov

M. V. Lomonosov Moscow State University

Abstract: 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.

DOI: https://doi.org/10.4213/im270

Full text: PDF file (1629 kB)
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English version:
Izvestiya: Mathematics, 1999, 63:6, 1203–1220

Bibliographic databases:

MSC: 53A05, 53B20
Received: 20.10.1998

Citation: I. Kh. Sabitov, “Isometric immersions and embeddings of locally Euclidean metrics in $\mathbb R^2$”, Izv. RAN. Ser. Mat., 63:6 (1999), 147–166; Izv. Math., 63:6 (1999), 1203–1220

Citation in format AMSBIB
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\paper Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$
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\pages 147--166
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\pages 1203--1220
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  • https://doi.org/10.4213/im270
  • http://mi.mathnet.ru/eng/izv/v63/i6/p147

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Borisenko, “Isometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant curvature”, Russian Math. Surveys, 56:3 (2001), 425–497  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. I. Kh. Sabitov, “Locally Euclidean Metrics with a Given Geodesic Curvature of the Boundary”, Proc. Steklov Inst. Math., 266 (2009), 210–218  mathnet  crossref  mathscinet  zmath  isi  elib
    3. I. Kh. Sabitov, “On the Extrinsic Curvature and the Extrinsic Structure of Normal Developable $C^1$ Surfaces”, Math. Notes, 87:6 (2010), 874–879  mathnet  crossref  crossref  mathscinet  isi  elib
    4. S. N. Mikhalev, I. Kh. Sabitov, “Isometric Embeddings in $\mathbb{R}^3$ of an Annulus with a Locally Euclidean Metric which Are Multivalued of Cylindrical Type”, Math. Notes, 98:3 (2015), 441–447  mathnet  crossref  crossref  mathscinet  isi  elib
    5. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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