Spherical maps of euclidean spaces (Q1290399)

The authors classify continuous maps \(f : {\mathbb R}^m \to {\mathbb R}^n\) that satisfy \(| f(a)-f(b)| = | f(b)-f(c)| \) whenever \(| a-b| =| b-c| \). Such maps are called \textit{spherical}, because they map spheres to spheres. For \(m=1,n=2\) such maps have been classified by M. Mc. Kemie and \textit{J. D. Vaaler} [Ann. Acad. Sci. Fenn., Ser. A I, 12, 163-167 (1987; Zbl 0632.30026)]. Starting with a preparatory section on spherical maps \(f : {\mathbb Z} \to {\mathbb R}^n\) defined on the integers the authors first tackle the case \(m=1\). They prove that a map \(f : {\mathbb R} \to {\mathbb R}^n\) is spherical if and only if it is constant, a similarity, a \(k\)-winding map or a \(k\)-helical map. A \(k\)-winding map \(f : L \to {\mathbb R}^n\), \(n\geq 2k\), is defined on some line \(L\) of \({\mathbb R}^m\) and has the form \(f : x \mapsto \alpha( r_1e^{i\theta_1x},\dots,r_ke^{i\theta_kx})\), where \(\alpha : {\mathbb R}^{2k} \to {\mathbb R}^n\) is a similarity, \(r_j, \theta_j > 0\), and the numbers \(\theta_j\) are pairwise distinct. A \(k\)-helical map \(f : L \to {\mathbb R}^n\), \(n\geq 3\), is defined also on some line \(L\) of \({\mathbb R}^m\) and has the form \(f : x \mapsto \alpha( r_1e^{i\theta_1x},\dots,r_ke^{i\theta_kx},ax)\), where \(\alpha : {\mathbb R}^{2k+1} \to {\mathbb R}^n\) is an similarity, \(r_j, \theta_j > 0,a>0\), and the numbers \(\theta_j\) are pairwise distinct. Such maps are obviously spherical. It turns out that for \(m\geq 2\) spherical maps are either constant or a similarity. This nicely written paper is self-contained and the proofs mostly use elementary geometric facts.