The cogyrolines of Möbius gyrovector spaces are metric but not periodic (Q1941994)

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.