Article Title

A metric invariant of Möbius transformations

Authors

TEERAPONG SUKSUMRAN
OĞUZHAN DEMİREL

DOI

10.3906/mat-1902-13

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 $(\mathbb{D}, d_T)$.

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