New York Journal of Mathematics
Volume 15 (2009) 423-429


Baokui Li and Yuefei Wang

A new characterization for isometries by triangles

Published: September 18, 2009
Keywords: Isometry, geodesic, affine transformation, triangle preserving, triangle domain preserving
Subject: 30C35, 51F99

Let Rn be an n-dimensional Euclidean space and Dn be an n-dimensional hyperbolic space with the Poincaré metric for n>1. In this paper, we shall prove the following results. (i) A bijection f:DnDn is an isometry (Möbius transformation) if and only if f is triangle preserving. (ii) A bijection f:RnRn is an affine transformation if and only if f is triangle preserving.


The research was supported by NSFC and MSBRD Program of China

Author information

Baokui Li:
Department of Mathematics, Beijing Institute of Technology, 100081, China

Yuefei Wang:
Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100190, China