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Geometry IV

Non-regular Riemannian Geometry

  • Book
  • © 1993

Overview

Editors:
  1. Yu. G. Reshetnyak

    1. Institute of Mathematics, Siberian Branch of the Russian, Academy of Sciences, Novosibirsk, Russia

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Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 70)

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Table of contents (2 chapters)

  1. Front Matter

    Pages i-2
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  2. Two-Dimensional Manifolds of Bounded Curvature

    • Yu. G. Reshetnyak
    Pages 3-163

  3. Multidimensional Generalized Riemannian Spaces

    • V. N. Berestovskij, I. G. Nikolaev
    Pages 165-243

  4. Back Matter

    Pages 245-252
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Keywords

About this book

The book contains a survey of research on non-regular Riemannian geome­ try, carried out mainly by Soviet authors. The beginning of this direction oc­ curred in the works of A. D. Aleksandrov on the intrinsic geometry of convex surfaces. For an arbitrary surface F, as is known, all those concepts that can be defined and facts that can be established by measuring the lengths of curves on the surface relate to intrinsic geometry. In the case considered in differential is defined by specifying its first geometry the intrinsic geometry of a surface fundamental form. If the surface F is non-regular, then instead of this form it is convenient to use the metric PF' defined as follows. For arbitrary points X, Y E F, PF(X, Y) is the greatest lower bound of the lengths of curves on the surface F joining the points X and Y. Specification of the metric PF uniquely determines the lengths of curves on the surface, and hence its intrinsic geometry. According to what we have said, the main object of research then appears as a metric space such that any two points of it can be joined by a curve of finite length, and the distance between them is equal to the greatest lower bound of the lengths of such curves. Spaces satisfying this condition are called spaces with intrinsic metric. Next we introduce metric spaces with intrinsic metric satisfying in one form or another the condition that the curvature is bounded.

Editors and Affiliations

  • Institute of Mathematics, Siberian Branch of the Russian, Academy of Sciences, Novosibirsk, Russia

    Yu. G. Reshetnyak

Bibliographic Information

  • Book Title: Geometry IV

  • Book Subtitle: Non-regular Riemannian Geometry

  • Editors: Yu. G. Reshetnyak

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-662-02897-1

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1993

  • Hardcover ISBN: 978-3-540-54701-3Published: 21 October 1993

  • Softcover ISBN: 978-3-642-08125-5Published: 01 December 2010

  • eBook ISBN: 978-3-662-02897-1Published: 14 March 2013

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: VII, 252

  • Additional Information: Originally published by VINITI, Moscow, 1989

  • Topics: Differential Geometry

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