Maps which almost preserve angles
Let \({\mathbb{R}}^ n\) be equipped with the usual euclidean structure: \(x\cdot y\)- the inner product, \(\| x\| =\sqrt{x\cdot x}\) the corresponding norm and \[ \sphericalangle y_ 1xy_ 2:=\arccos (y_ 1- x)\cdot (y_ 2-x)/\| y_ 1-x\| \| y_ 2-x\|,\quad y_ 1\neq x\neq y_ 2, \] the measure of the angle of rays \(xy_ 1\) and \(xy_ 2.\) The main result of this paper is the following: Let \({\mathcal V}\subset {\mathbb{R}}^ n\), \(n\geq 2\), an open set, f: \({\mathcal V}\to {\mathbb{R}}^ m\), \(m\leq 2n\), a mapping such that for each \(x\in {\mathcal V}\) there exists \(\alpha_ x\in]0,\pi /2[\) and a most countable set \(\Omega_ x\subset [0,\pi [\setminus \{\pi /2\}\) such that \(\sphericalangle y_ 1xy_ 2=\alpha_ x(y_ 1,y_ 2\in {\mathcal V}\setminus \{x\})\) implies \(f(y_ 1)\neq f(x)\neq f(y_ 2)\) and \(\sphericalangle f(y_ 1)f(x)f(y_ 2)\in \Omega_ x\). Then f is a similarity i.e. there exists \(k>0\) such that \(\| f(x)-f(y)\| =k\| x-y\|\), for every \(x,y\in {\mathcal V}\).
scientific article; zbMATH DE number 4094272