Advertisement

Journal of Geometry

, Volume 70, Issue 1–2, pp 8–16 | Cite as

Mappings preserving two hyperbolic distances

  • Walter Benz Hamburg
  • 33 Downloads

Abstract.

Suppose that X is the set of points of a hyperbolic geometry of finite or infinite dimension \( \geq2 \), and that \( \varrho>0 \) is a fixed real number and N>1 a fixed integer. Let \( f:X\to X \) be a mapping such that for every \( x,y\in X \) if h(x, y)=\( \varrho \), then h(f,(x),f,(y)) \( \leq\varrho \), and if h(x,y) = N\( \varrho \), then h(f,(x),f,(y)) \( \geq N\varrho \), where h,(p,q) designates the hyperbolic distance of p,q \( \in X \). Then f is an isometry of X. Note that there is no regularity assumption on f, like continuity or even differentiability. Moreover, we present an example showing that the assumption that one fixed distance > 0 is preserved does not characterize hyperbolic isometries.

Keywords

Real Number Fixed Distance Regularity Assumption Hyperbolic Geometry Infinite Dimension 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag Basel, 2001

Authors and Affiliations

  • Walter Benz Hamburg
    • 1
  1. 1.Department of Mathematics, University of Hamburg, Bundesstrasse 55, D--20146 Hamburg, Germany, e-mail: benz@math.uni-hamburg.deDE

Personalised recommendations