Minimal conditions determining isometries and similarities (Q1074113)

Let us denote by \(E^ n\) and \(\Lambda^ n\) respectively Euclidean space and Lobachevskian space of dimension n, by \({\mathbb{R}}^+\) the set of positive real numbers and by \(\rho\) (X,Y) the distance of the points X and Y in \(E^ n\) or \(\Lambda^ n\). In this work, in order to characterize minimal conditions determining similarities of \(E^ n\) and isometries of \(\Lambda^ n\), the author proves at first the following proposition. Let the set \(\Omega \subset {\mathbb{R}}^+\), whose cardinality is less than the one c of \({\mathbb{R}}^+\), satisfy the following condition: If \(A_ 1,...,A_ n\) are points in \(E^ n\) or \(\Lambda^ n\), such that for each pair (i,j) \((i=1,...,n\); \(j=1,...,n\); \(i\neq j)\) \(\rho (A_ i,A_ j)\in \Omega\), the dimension of the affine hull of the set \(\{A_ 1,...,A_ n\}\) is n-1. Such a set \(\Omega\) has the following property (\(\alpha)\). (\(\alpha)\). Let \(a>0\); if \(f: E^ n\to E^ n\) (respectively, \(f: \Lambda\) \({}^ n\to \Lambda^ n)\) is an injective map such that from the condition \(\rho (X,Y)=a\), where \(X,Y\in E^ n\) (respectively, \(X,Y\in \Lambda^ n)\), it follows that \(\rho\) (f(X),f(Y))\(\in \Omega\), then in the case of \(E^ n\) the map f is a similarity, and in the case of \(\Lambda^ n\) an isometry. Successively, the following theorems are proved. Theorem 1. We fix the space \(E^ n\) (respectively, \(\Lambda^ n)\), \(n\geq 3\). Let the set \({\mathfrak M}\subset {\mathbb{R}}^+\) be dense in \({\mathbb{R}}^+\). There exists a set \(\Omega\) \(\subset {\mathfrak M}\), dense in \({\mathbb{R}}^+\), and having property (\(\alpha)\). Theorem 2. We fix the space \(E^ n\) (respectively, \(\Lambda^ n)\), \(n\geq 3\), and a number \(a_ 0>0\). There exists an \(\epsilon >0\) such that for each set \(\Omega '\subset]a_ 0-\epsilon,a_ 0+\epsilon [\) (where \(]x_ 1,x_ 2[\) is the open interval with ends \(x_ 1,x_ 2)\), for which the cardinality of \(\Omega\) ' is less than c, there exists a dense set \(\Omega ''\subset R^+\), such that the set \(\Omega\) \(=^{def}\Omega '\cup \Omega ''\) has property (\(\alpha)\).