A metric invariant of Möbius transformations

Authors : Teerapong Suksumran, Oğuzhan Demirel

Pages : 2876-2887

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Publication Date : 9999-12-31

Article Type : Makaleler

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