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, TEERAPONG and DEMİREL, OĞUZHAN
(2019)
"A metric invariant of Möbius transformations,"
Turkish Journal of Mathematics: Vol. 43:
No.
6, Article 18.
https://doi.org/10.3906/mat-1902-13
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss6/18
