Mappings preserving two hyperbolic distances (Q5944988)

This article is a contribution to the ``characterizations of geometrical mappings under mild hypotheses''. Let \(X\) be a pre-Hilbert space, \(\dim X\geq 2\), with hyperbolic distance \(h(x,y)\geq 0\) of all \(x,y\in X\), where \(\text{cosh} h(x,y): =\sqrt{1+x^2} \sqrt{1+y^2} -xy\). The following theorem is proved: Let \(\rho>0\) be a fixed real number and \(N>1\) be a fixed integer. If \(f:X\to X\) is a mapping satisfying for all \(x,y\in X\) (1) \(h(x,y)= \rho\Rightarrow h(f(x), f(y))\leq \rho\), (2) \(h(x,y)= N\rho \Rightarrow h(f(x), f(y))\geq N\rho\), then \(f\) is an isometry of \(X\). This characterization is particularly interesting for space \(X^\infty\). \(N=1\) is not generally allowed in the theorem.