The cogyrolines of Möbius gyrovector spaces are metric but not periodic
type: Item
instance of: scholarly article
title: The cogyrolines of Möbius gyrovector spaces are metric but not periodic
zbMATH Open document ID: 1269.39007
author: Oğuzhan Demirel
author: Emine Soytürk Seyrantepe
published in: Aequationes Mathematicae
publication date: 2013-03-25T00:00:00Z
review text: A real distance space \(\Delta = (S, d)\) is a nonempty set \(S\) together with a mapping \(d: S \times S \to \mathbb{R}\). The subset \(K \subseteq S\) is called a metric line if there exists a bijection \(f : K \to \mathbb{R}\) such that \(d(x, y) = | f(x) - f(y)|\) for all \(x, y \in K\). The subset \(K\) of \(S\) is called a \(\rho\)-periodic line of \(\Delta\) if and only if there exists a bijection \(f : K \to [0, \rho )\) with \[ d(x, y) = \left \{ \begin{matrix}\l & \qquad \l\\ |f(x) - f(y)|& \text{if \,\,} |f(x)-f(y)|\leq {{\rho} \over 2,} \\ \rho- |f(x) - f(y)|& \text{if \,\,} |f(x)-f(y)| > {{\rho} \over 2} \end{matrix} \right. \] \noindent for all \(x, y \in K\). In this paper, the authors establish the following results: (i) The cogyrolines of \(\Delta\) are metric lines of itself. (ii) The metric lines of \(\Delta\) passing through \(0\) are exactly the origin intercept cogyrolines of itself. (iii) For all \(\rho > 0\), \(\rho\)-periodic lines in \(\Delta\) do not exist.
reviewed by: Prasanna Sahoo
Mathematics Subject Classification ID: 39B42
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M25
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 51B10
zbMATH DE Number: 6148627
zbMATH Keywords: metric space
zbMATH Keywords: functional equations of metric and periodic lines and their solutions
zbMATH Keywords: quartic functional equation
zbMATH Keywords: Poincaré ball model of hyperbolic geometry
zbMATH Keywords: Möbius gyrovector space
zbMATH Keywords: distance space
MaRDI profile type: Publication
full work available at URL: s00010-012-0134-1
OpenAlex ID: W2055203992
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DOI: 10.1007/S00010-012-0134-1
description: scientific article; zbMATH DE number 6148627