Metric transformations of \({\mathbb{R}}\) into \(H^ n\) (Q1068362)

[This paper has been already published in Zbl 0544.51013 but with a wrong referee report which actually belongs to the paper above.] A metric transformation between two metric spaces \(M_ 1\) and \(M_ 2\) is defined to be a function f such that for some function \(\rho: {\mathbb{R}}^+\to {\mathbb{R}}^+,\) called the scale function associated with f, \(\rho (d_ 1(x,y))=d_ 2(f(x),f(y))\) for all \(x,y\in M_ 1\). Here neither f nor \(\rho\) are assumed to be continuous. In Rocky Mt. J. Math. 15, 199-206 (1985; see the review above) the author characterized all metric transformations from \({\mathbb{R}}\) into \(E^ n\). The major result of this paper (Theorem 2) is a characterization of all metric transformations of \({\mathbb{R}}\) into the n-dimensional hyperbolic space \(H^ n\).