Extensions of distance preserving mappings in euclidean and hyperbolic geometry (Q1879087)

The inner product in a real inner product space \((X, \langle , \rangle)\) induces two metrics on \(X\): the euclidean metric eucl defined by eucl\((x,y)=\sqrt{(x-y)^2}\), and the hyperbolic metric hyp defined by \(\cosh\) hyp\((x,y)=\sqrt{1+x^2} \sqrt{1+y^2} - xy,\) where \(u^2\) stands for \(\langle u,u \rangle\). Suppose that dim(\(X\)) \(\geq 2\), let \(p\) be any one of these two metrics, and let \(S \subset X\). The author proves the elegant theorem that if \(S\) is contained in a finite-dimensional subspace of \(X\), then every \(f : S \longrightarrow X\) for which \(p(f(x),f(y)) = p(x,y)\) for all \(x\) and \(y\) in \(S\) can be extended to an isometry on \(X\). He also gives simple examples showing that the assumption that \(S\) is contained in a finite-dimensional subspace is irredundant for each of the two metrics.