Turkish Journal of Mathematics
Abstract
The complex unit disk $\mathbb{D} = \{z\in\mathbb{C}\colon z < 1\}$ is endowed with Möbius addition $\oplus_M$ defined by $$ w\oplus_M z = \dfrac{w+z}{1+\overline{w}z}. $$ We prove that the metric $d_T$ defined on $\mathbb{D}$ by $d_T(w, z) = \tan^{-1} -w\oplus_M z $ is an invariant of Möbius transformations carrying $\mathbb{D}$ onto itself. We also prove that $(\mathbb{D}, d_T)$ and $(\mathbb{D}, d_P)$, where $d_P$ denotes the Poincaré metric, have the same isometry group and then classify the isometries of $(\mathbb{D}, d_T)$.
DOI
10.3906/mat-1902-13
Keywords
Möbius transformation, Poincaré metric, transformation invariant, isometry group, gyrogroup
First Page
2876
Last Page
2887
Recommended Citation
SUKSUMRAN, T, & DEMİREL, O (2019). A metric invariant of Möbius transformations. Turkish Journal of Mathematics 43 (6): 2876-2887. https://doi.org/10.3906/mat-1902-13
