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Singularities of Surfaces with Constant Negative Gauss Curvature

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1000)

Abstract

In this chapter we shall be concerned with (open) surfaces and their imbeddings in E3. The definition of an open surface is identical with definition II, 1.1 except that condition 1) that S be compact is no longer true. We will show that a surface with constant negative Gauss curvature cannot be imbedded as a general (open) surface in E3 without singularities (in a sense to be defined below). The first proof of this was given by Hilbert (~ 1900) for analytic surfaces. Our proof works for C3 surfaces and the theorem is still true for C2 surfaces. However, Kuiper has given a C1 isometric imbedding of the hyperbolic plane in E3 without singularities. For details see N.H. Kuiper, on C1-isometric Imbeddings I and II; Indagationes Mathematicae, Vol. 17 (1955) pp 545–556 and pp 683–689.

Keywords

  • Singular Point
  • Fundamental Form
  • Finite Length
  • Hyperbolic Plane
  • Asymptotic Line

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1983 Springer-Verlag Berlin Heidelberg

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Hopf, H. (1983). Singularities of Surfaces with Constant Negative Gauss Curvature. In: Differential Geometry in the Large. Lecture Notes in Mathematics, vol 1000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21563-0_14

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  • DOI: https://doi.org/10.1007/978-3-662-21563-0_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12004-9

  • Online ISBN: 978-3-662-21563-0

  • eBook Packages: Springer Book Archive