A new characterization for isometries by triangles (Q1043778)

Let \(\mathbb{R}^n\) be the usual \(n\)-dimensional Euclidean space, and let \(\mathbb{D}^n\) be the open unit sphere in \(\mathbb{R}^n\) with the Poincaré metric, \(n> 1\). In the literature, there are several conditions which are necessary or sufficient for maps \(f: \mathbb{R}^n\to\mathbb{R}^n\) or for maps \(f: \mathbb{D}^n\to\mathbb{D}^n\) to be an isometry or an affine transformation. These conditions involve geometric properties of \(f\) like angle preserving, triangle preserving, or geodesic preserving. In this paper, the authors show that a bijection \(f: \mathbb{R}^n\to\mathbb{R}^n\) is an affine transformation if and only if \(f\) is triangle preserving. Further, they also show that \(f: \mathbb{D}^n\to\mathbb{D}^n\) is an isometry if and only if \(f\) is triangle preserving.