A new characterization for isometries by triangles

摘要

Let Rn be an n-dimensional Euclidean space and Dn be ann-dimensional hyperbolic space with thePoincaré metric for n1. In thispaper, we shall prove the following results. (i) A bijectionf:Dn→Dn is an isometry (Möbius transformation) if andonly if f is triangle preserving. (ii) A bijection f:Rn→Rnis an affine transformation if and only if f is trianglepreserving.