Length Spaces

  • Martin R. Bridson
  • André Haefliger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 319)


In this section we consider metric spaces in which the distance between two points is given by the infimum of the lengths of curves which join them — such a space is called a length space. In this context, it is natural to allow metrics for which the distance between two points may be infinite. A convenient way to describe this is to introduce the notation [0, ∞] for the ordered set obtained by adjoining the symbol ∞ to the set of non-negative reals and decreeing that ∞ > a for all real numbers a. We also make the convention that a + ∞ = ∞ for all a ∈ [0, ∞]. Having made this convention, we can define a (generalized) metric on a set X to be a map d : X × X → [0, ∞] satisfying the axioms stated in (1.1).


Riemannian Manifold Length Space Geodesic Segment Finsler Manifold Finsler Metrics 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin R. Bridson
    • 1
  • André Haefliger
    • 2
  1. 1.Mathematical InstituteUniversity of OxfodOxfordGreat Britain
  2. 2.Section de MathématiquesUniversité de GenèveGenève 24Switzerland

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