The open unit ball $\mathbb{ B} = \{ \mathbf{ v}\in\mathbb{ R}^n\colon\|\mathbf{ v}\|<1\}$ is endowed with M\"{ o}bius addition $\oplus_M$ defined by $$\mathbf{ u}\oplus_M\mathbf{ v} = \dfrac{ (1 + 2\langle\mathbf{ u},\mathbf{ v}\rangle + \|\mathbf{ v}\|^2)\mathbf{ u} + (1 - \|\mathbf{ u}^2)\mathbf{ v}}{ 1 + \langle\mathbf{ u},\mathbf{ v}\rangle + \|\mathbf{ u}\|^2\|\mathbf{ v}\|^2\|}$$ for all $\mathbf{ u},\mathbf{ v}\in \mathbf{ B}$. In this article, we prove the inequality $$ \dfrac{ \|\mathbf{ u}\|-\|\mathbf{ v}\|}{ 1+\|\mathbf{ u}\|\|\mathbf{ v}\|}\leq \|\mathbf{ u}\oplus_M \mathbf{ v}\| \leq \dfrac{ \|\mathbf{ u}\|+\|\mathbf{ v}\|}{ 1-\|\mathbf{ u}\|\|\mathbf{ v}\|} $$ in $\mathbb{ B}$. This leads to a new metric on $\mathbb{ B}$ defined by $$d_T(\mathbf{ u},\mathbf{ v}) = \tan^{ -1}{ \|-\mathbf{ u}\oplus_M\mathbf{ v}\|},$$ which turns out to be an invariant of M\"{ o}bius transformations on $\mathbb{ R}^n$ carrying $\mathbb{ B}$ onto itself. We also compute the isometry group of $(\mathbb{ B}, d_T)$ and give a parametrization of the isometry group by vectors and rotations.
Categories: math.MG