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The Total Curvature (Curvatura Integra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements

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

Abstract

A line element on a surface S is determined by a non-zero tangent vector to the surface. The same line element is determined by all non-zero multiples of the vector. Hence there is no distinguished direction on a line element. Strictly speaking, a line element is a one dimensional linear subspace of the tangent vector space.

Keywords

  • Line Element
  • Euler Number
  • Total Curvature
  • Spherical Image
  • Barycentric Subdivision

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). The Total Curvature (Curvatura Integra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements. 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_8

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  • DOI: https://doi.org/10.1007/978-3-662-21563-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12004-9

  • Online ISBN: 978-3-662-21563-0

  • eBook Packages: Springer Book Archive