An inequality related to M\"{o}bius transformations

TL;DR

This paper establishes a new inequality for M"{o}bius addition in the unit ball, introduces an invariant metric under M"{o}bius transformations, and characterizes the isometry group of this metric space.

Contribution

The paper proves a novel inequality related to M"{o}bius addition, defines a new invariant metric on the unit ball, and characterizes its isometry group.

Findings

Derived a new inequality for M"{o}bius addition in the unit ball.

Introduced a new invariant metric under M"{o}bius transformations.

Computed and parametrized the isometry group of the metric space.

Abstract

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.

Tables

Table 1. Table 1: Properties of the Möbius gyrogroup (cf. [ 10 , 7 ] ).

gyrogroup identity	name/reference
$L_{\ominus 𝐮}=L_{𝐮}^{-1}$	Inverse of gyrotranslation
$\ominus 𝐮{\oplus }_M\left(𝐮{\oplus }_M𝐯\right)=𝐯$	Left cancellation law
$\ominus \left(𝐮{\oplus }_M𝐯\right)=\mathrm{gyr}[𝐮,𝐯](\ominus 𝐯{\oplus }_M\ominus 𝐮)$	cf. ${\left(gh\right)}^{-1}=h^{-1}{g}^{-1}$
$\left(\ominus 𝐮{\oplus }_M𝐯\right){\oplus }_M\mathrm{gyr}[\ominus 𝐮,𝐯]\left(\ominus 𝐯{\oplus }_M𝐰\right)=\ominus 𝐮{\oplus }_M𝐰$	cf. $\left(g^{-1}h\right)\left(h^{-1}k\right)=g^{-1}k$
$\mathrm{gyr}[\ominus 𝐮,\ominus 𝐯]=\mathrm{gyr}[𝐮,𝐯]$	Even property
$\mathrm{gyr}[𝐯,𝐮]={\mathrm{gyr}}^{-1}[𝐮,𝐯]$, the inverse of $\mathrm{gyr}[𝐮,𝐯]$	Inversive symmetry

Equations

$$\mathbf{u}{\oplus }_M\mathbf{v}=\frac{(1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{v}{\Vert }^2)\mathbf{u}+(1-\Vert \mathbf{u}{\Vert }^2)\mathbf{v}}{1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{u}{\Vert }^2\Vert \mathbf{v}{\Vert }^2}$$

$$\frac{\Vert \mathbf{u}\Vert -\Vert \mathbf{v}\Vert }{1+\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }\le \Vert \mathbf{u}{\oplus }_M\mathbf{v}\Vert \le \frac{\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert }{1-\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }$$

$$d_T(\mathbf{u},\mathbf{v})={\tan }^{-1}\Vert -\mathbf{u}{\oplus }_M\mathbf{v}\Vert ,$$

$$\mathbb{B}=\{\mathbf{v}\in {\mathbb{R}}^n:\Vert \mathbf{v}\Vert <1\},$$

$$d_P(\mathbf{x},\mathbf{y})={\cosh }^{-1}\left(1+\frac{2\Vert \mathbf{x}-\mathbf{y}{\Vert }^2}{(1-\Vert \mathbf{x}{\Vert }^2)(1-\Vert \mathbf{y}{\Vert }^2)}\right)$$

$$\mathbf{u}{\oplus }_M\mathbf{v}=\frac{(1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{v}{\Vert }^2)\mathbf{u}+(1-\Vert \mathbf{u}{\Vert }^2)\mathbf{v}}{1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{u}{\Vert }^2\Vert \mathbf{v}{\Vert }^2}.$$

$$L_{\mathbf{u}}(\mathbf{v})=\mathbf{u}{\oplus }_M\mathbf{v}=\frac{(1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{v}{\Vert }^2)\mathbf{u}+(1-\Vert \mathbf{u}{\Vert }^2)\mathbf{v}}{1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{u}{\Vert }^2\Vert \mathbf{v}{\Vert }^2},$$

$$d_P(\mathbf{x},\mathbf{y})=2{\tanh }^{-1}\Vert -\mathbf{x}{\oplus }_M\mathbf{y}\Vert$$

$${\varrho }_M(\mathbf{x},\mathbf{y})=\Vert -\mathbf{x}{\oplus }_M\mathbf{y}\Vert$$

$$d_M(\mathbf{x},\mathbf{y})={\tanh }^{-1}({\varrho }_M(\mathbf{x},\mathbf{y}))$$

$$(-\mathbf{v}){\oplus }_M\mathbf{v}=0=\mathbf{v}{\oplus }_M(-\mathbf{v}).$$

$$\mathbf{u}{\oplus }_M(\mathbf{v}{\oplus }_M\mathbf{w})=(\mathbf{u}{\oplus }_M\mathbf{v}){\oplus }_M\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}$$

$$(\mathbf{u}{\oplus }_M\mathbf{v}){\oplus }_M\mathbf{w}=\mathbf{u}{\oplus }_M(\mathbf{v}{\oplus }_M\mathrm{gyr}[\mathbf{v},\mathbf{u}]\mathbf{w})$$

$$\mathrm{gyr}[\mathbf{u}{\oplus }_M\mathbf{v},\mathbf{v}]=\mathrm{gyr}[\mathbf{u},\mathbf{v}]\text{and}\mathrm{gyr}[\mathbf{u},\mathbf{v}{\oplus }_M\mathbf{u}]=\mathrm{gyr}[\mathbf{u},\mathbf{v}].$$

$$\mathbf{u}{\oplus }_M\mathbf{v}=\mathrm{gyr}[\mathbf{u},\mathbf{v}](\mathbf{v}{\oplus }_M\mathbf{u}).$$

$$\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=-(\mathbf{u}{\oplus }_M\mathbf{v}){\oplus }_M(\mathbf{u}{\oplus }_M(\mathbf{v}{\oplus }_M\mathbf{w}))$$

$$S_{\mathbf{x}}^2=(L_{\mathbf{x}}\circ \iota \circ L_{\mathbf{x}}^{-1})\circ (L_{\mathbf{x}}\circ \iota \circ L_{\mathbf{x}}^{-1})=L_{\mathbf{x}}\circ {\iota }^2\circ L_{\mathbf{x}}^{-1}=L_{\mathbf{x}}\circ L_{\mathbf{x}}^{-1}=I.$$

$$B(\mathbf{u},\mathbf{v})=-⟨\mathbf{u},\mathbf{v}⟩,\mathbf{u},\mathbf{v}\in {\mathbb{R}}^n.$$

$$\{e_I:I=\varnothing \text{ or }I=\{1\le i_1<i_2<\cdots <i_k\le n\}\},$$

$${\mathbf{e}}_i^2=-1\text{and}{\mathbf{e}}_i{\mathbf{e}}_j=-{\mathbf{e}}_j{\mathbf{e}}_i$$

$${\Gamma }_n=\{g\in \mathrm{C}{\ell }_n:g\text{ is invertible and }\hat{g}\mathbf{v}{g}^{-1}\in {\mathbb{R}}^n\text{ for all }\mathbf{v}\in {\mathbb{R}}^n\}.$$

$$\eta (a)=a\bar{a},a\in \mathrm{C}{\ell }_n.$$

$$\hat{\mathbf{v}}\mathbf{w}{\mathbf{v}}^{-1}=\frac{1}{\Vert \mathbf{v}{\Vert }^2}\mathbf{vwv}$$

$$\Vert q\mathbf{w}{q}^{-1}\Vert =\sqrt{\eta (q\mathbf{w}{q}^{-1})}=\sqrt{\eta (\mathbf{w})}=\Vert \mathbf{w}\Vert .$$

$$\mathbf{u}{\oplus }_M\mathbf{v}=(\mathbf{u}+\mathbf{v})(1-\mathbf{uv}{)}^{-1}$$

$$q=\frac{1-\mathbf{uv}}{|1-\mathbf{uv}|},$$

$$\frac{\Vert \mathbf{u}\Vert -\Vert \mathbf{v}\Vert }{1+\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }\le \Vert \mathbf{u}{\oplus }_M\mathbf{v}\Vert \le \frac{\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert }{1-\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }$$

$$-\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert \le ⟨\mathbf{u},\mathbf{v}⟩\le \Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert$$

$$\eta (\mathbf{u}+\mathbf{v})=\Vert \mathbf{u}{\Vert }^2-(\mathbf{uv}+\mathbf{vu})+\Vert \mathbf{v}{\Vert }^2=\Vert \mathbf{u}{\Vert }^2+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{v}{\Vert }^2\le (\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert {)}^2$$

$$\eta (1-\mathbf{uv})=1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{u}{\Vert }^2\Vert \mathbf{v}{\Vert }^2\le (1+\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert {)}^2.$$

Full text

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{B}$. In this article, we prove the inequality

[TABLE]

in $\mathbb{B}$. This leads to a new metric on $\mathbb{B}$ defined by

[TABLE]

which turns out to be an invariant of Möbius 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.

Keywords. Möbius transformation, Poincaré metric, Euclidean norm inequality, isometry group, gyrogroup.

2010 MSC. Primary 51B10; Secondary 46T99, 15A66, 51F15, 20N05.

1 The unit ball of $\boldsymbol{n}$-dimensional Euclidean space $\boldsymbol{\mathbb{R}^{n}}$

Let $\mathbb{B}$ denote the open unit ball of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, that is,

[TABLE]

where $\|\cdot\|$ denotes the usual Euclidean norm on $\mathbb{R}^{n}$. It is known in the literature that $\mathbb{B}$ forms a bounded symmetric domain, naturally associated with the Poincaré and Beltrami–Klein models of $n$-dimensional hyperbolic geometry. In fact, the Poincaré metric $d_{P}$ corresponding to a curvature of $-1$ is given by

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$ [4, p. 1232]. Further, the Cayley–Klein metric associated with the Beltrami–Klein model is defined via cross-ratios; see, for instance, [4, p. 1233].

From an algebraic point of view, the unit ball has a group-like structure when it is endowed with Möbius addition $\oplus_{M}$ defined by

[TABLE]

Möbius addition governs the unit ball in the same way that ordinary vector addition governs the Euclidean space; see, for instance, [11, 3, 6]. Further, Möbius addition induces the well-known Möbius transformation of $\mathbb{B}$ of the form

[TABLE]

called the hyperbolic translation by $\mathbf{u}$, for all $\mathbf{u}\in\mathbb{B}$ [6, p. 124]. A remarkable result of Kim and Lawson shows strong connections between the geometric and algebraic structures of the unit ball. In fact, they relate the Poincaré metric with Möbius addition:

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$; see Theorem 3.7 of [4]. Equation (1.5) includes what Ungar refers to as a gyrometric [10, Definition 6.8]. More precisely, the (Möbius) gyrometric and the rapidity metric of $(\mathbb{B},\oplus_{M})$ are defined by

[TABLE]

and by

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$, respectively.

1.1 A nonassociative structure of the unit ball

The space $(\mathbb{B},\oplus_{M})$ shares many properties with abelian groups, called by some a gyrocommutative gyrogroup and by others a Bruck loop or a K-loop. Henceforth, $(\mathbb{B},\oplus_{M})$ is referred to as the Möbius gyrogroup.

The group-like axioms satisfied by the Möbius gyrogroup are as follows.

1. (I)

(identity) The zero vector $\mathbf{0}$ satisfies $\mathbf{0}\oplus_{M}\mathbf{v}=\mathbf{v}=\mathbf{v}\oplus_{M}\mathbf{0}$ for all $\mathbf{v}\in\mathbb{B}$. 2. (II)

(inverse) For each $\mathbf{v}\in\mathbb{B}$, the negative vector $-\mathbf{v}$ belongs to $\mathbb{B}$ and satisfies

[TABLE] 3. (III)

(the gyroassociative law) For all $\mathbf{u},\mathbf{v}\in\mathbb{B}$, there are automorphisms ${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$ and ${\mathrm{gyr}[{\mathbf{v},\mathbf{u}}]}{}$ in $\mathrm{Aut}\,{(\mathbb{B},\oplus_{M})}$ such that

[TABLE]

and

[TABLE]

for all $\mathbf{w}\in\mathbb{B}$. 4. (IV)

(the loop property) For all $\mathbf{u},\mathbf{v}\in\mathbb{B}$,

[TABLE] 5. (V)

(the gyrocommutative law) For all $\mathbf{u},\mathbf{v}\in\mathbb{B}$,

[TABLE]

The automorphism ${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$ mentioned in Item (III) is called the gyroautomorphism generated by $\mathbf{u}$ and $\mathbf{v}$. It is uniquely determined by its generators via the gyrator identity described by the formula

[TABLE]

for all $\mathbf{w}\in\mathbb{B}$. Sometimes it is convenient to denote $-\mathbf{v}$ by $\ominus\mathbf{v}$, the (unique) inverse of $\mathbf{v}$ with respect to Möbius addition. Some elementary properties of the Möbius gyrogroup are collected in Table 1.

1.2 Isometries of the unit ball

It is known in the literature that the transformation $L_{\mathbf{u}}\colon\mathbf{v}\mapsto\mathbf{u}\oplus_{M}\mathbf{v}$ preserves the gyrometric $\varrho_{M}$; see, for instance, [4, Lemma 3.2 (v)]. Thus, $L_{\mathbf{u}}$ preserves the rapidity metric $d_{M}$. In fact, every isometry of $(\mathbb{B},d_{M})$ must be of the form $L_{\mathbf{u}}\circ\tau$, where $\tau$ is the restriction of an orthogonal transformation on $\mathbb{R}^{n}$ to the unit ball, due to the fact that any Möbius transformation that fixes $\mathbf{0}$ is orthogonal. The following theorem shows that the metric geometry of $\mathbb{B}$ with respect to $d_{M}$ is homogeneous.

Theorem 1.1** (Homogeneity).**

For each pair of points $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{B}$, there is an isometry $T$ of $(\mathbb{B},d_{M})$ such that $T(\mathbf{x})=\mathbf{y}$. In particular, $\mathbb{B}$ is homogeneous.

Proof.

Let $\mathbf{x},\mathbf{y}\in\mathbb{B}$. Define $T=L_{\mathbf{y}}\circ L_{\ominus\mathbf{x}}$. Then $T$ is an isometry of $\mathbb{B}$, being the composite of isometries of $\mathbb{B}$. Further, $T(\mathbf{x})=\mathbf{y}\oplus_{M}(\ominus\mathbf{x}\oplus_{M}\mathbf{x})=\mathbf{y}$. ∎

By using the gyrogroup formalism, a point-reflection symmetry of $\mathbb{B}$ is easy to construct, as shown in the following theorem.

Theorem 1.2** (Symmetry).**

For each point $\mathbf{x}\in\mathbb{B}$, there is a symmetry $S_{\mathbf{x}}$ of $\mathbb{B}$; that is, $S_{\mathbf{x}}$ is an isometry of $(\mathbb{B},d_{M})$ such that $S_{\mathbf{x}}^{2}$ is the identity transformation $I$ of $\mathbb{B}$ and $\mathbf{x}$ is the unique fixed point of $S_{\mathbf{x}}$.

Proof.

Let $\iota$ be the inversion map of $\mathbb{B}$, that is, $\iota(\mathbf{v})=\ominus\mathbf{v}$ for all $\mathbf{v}\in\mathbb{B}$. Since $\ominus\mathbf{v}=-\mathbf{v}$ for all $\mathbf{v}\in\mathbb{B}$, $\iota$ is simply the negative map: $\mathbf{v}\mapsto-\mathbf{v}$. Note that $\iota$ is an isometry of $(\mathbb{B},d_{M})$ for $\iota$ is linear and preserves the Euclidean norm. Furthermore, $\iota(\mathbf{v})=\mathbf{v}$ if and only if $\mathbf{v}=\mathbf{0}$.

Given $\mathbf{x}\in\mathbb{B}$, define $S_{\mathbf{x}}=L_{\mathbf{x}}\circ\iota\circ L_{\ominus\mathbf{x}}$. Then $S_{\mathbf{x}}=L_{\mathbf{x}}\circ\iota\circ L_{\mathbf{x}}^{-1}$ and so

[TABLE]

Note that $S_{\mathbf{x}}\neq I$; otherwise, we would have $L_{\mathbf{x}}\circ\iota\circ L_{\mathbf{x}}^{-1}=I$ and would have $\iota=I$, a contradiction. It is clear that $S_{\mathbf{x}}$ is an isometry of $\mathbb{B}$. By construction, $\mathbf{x}$ is a fixed point of $S_{\mathbf{x}}$. Suppose that $\mathbf{y}$ is a fixed point of $S_{\mathbf{x}}$, that is, $S_{\mathbf{x}}(\mathbf{y})=\mathbf{y}$. It follows that $\mathbf{x}\oplus_{M}\iota(\ominus\mathbf{x}\oplus_{M}\mathbf{y})=\mathbf{y}$ and hence $\iota(\ominus\mathbf{x}\oplus_{M}\mathbf{y})=\ominus\mathbf{x}\oplus_{M}\mathbf{y}$. As mentioned previously, $\mathbf{0}$ is the unique fixed point of $\iota$ and so $\ominus\mathbf{x}\oplus_{M}\mathbf{y}=\mathbf{0}$. This implies that $\mathbf{x}=\mathbf{y}$. ∎

We close this section with the following theorem whose proof is straight-forward (and so is omitted).

Theorem 1.3**.**

If $\tau\in\mathrm{Aut}\,{(\mathbb{B},\oplus_{M})}$ and $\|\tau(\mathbf{v})\|=\|\mathbf{v}\|$ for all $\mathbf{v}\in\mathbb{B}$, then $\tau$ is an isometry of $\mathbb{B}$ with respect to $d_{M}$. In particular, the gyroautomorphisms of $(\mathbb{B},\oplus_{M})$ are isometries.

2 The negative Euclidean space and its Clifford algebra

It seems that the formalism of Clifford algebras is a suitable tool for the study of the Möbius gyrogroup [2, 5]. Let us begin with the definition of an underlying vector space that will be used to built a unital associative algebra in which Möbius addition has a compact formula. The negative Euclidean space has $\mathbb{R}^{n}$ as the underlying vector space, but its inner product is a variant of the Euclidean inner product defined by

[TABLE]

Note that (2.1) defines a nondegenerate symmetric bilinear form on $\mathbb{R}^{n}$. Also, the associated quadratic form is given by $Q(\mathbf{v})=-\|\mathbf{v}\|^{2}$ for all $\mathbf{v}\in\mathbb{R}^{n}$.

The negative Euclidean space induces a real unital associative algebra, which is unique up to isomorphism, called the Clifford algebra of $(\mathbb{R}^{n},B)$ denoted by $\mathrm{C}\ell_{n}$ [5]. To describe the structure of $\mathrm{C}\ell_{n}$, let $\{\mathbf{e}_{1},\mathbf{e}_{2},\ldots,\mathbf{e}_{n}\}$ be the standard basis of $\mathbb{R}^{n}$. Then $\mathrm{C}\ell_{n}$ has a basis of the form

[TABLE]

where $e_{I}=\mathbf{e}_{i_{1}}\mathbf{e}_{i_{2}}\cdots\mathbf{e}_{i_{k}}$ for $I=\{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n\}$ and $e_{\emptyset}=1$, the multiplicative identity of $\mathrm{C}\ell_{n}$. Hence, a typical element of $\mathrm{C}\ell_{n}$ is of the form $\displaystyle\sum_{I}\lambda_{I}e_{I}$ with $\lambda_{I}$ in $\mathbb{R}$. The binary operations of vector addition and scalar multiplication in $\mathrm{C}\ell_{n}$ are defined pointwise. The product of two elements in $\mathrm{C}\ell_{n}$ is obtained by using the distributive law (but not assuming that algebra multiplication is commutative) subject to the defining relations

[TABLE]

for all $i,j\in\{1,2,\ldots,n\}$ with $i\neq j$. The base field $\mathbb{R}$ is embedded into $\mathrm{C}\ell_{n}$ by the map $\lambda\mapsto\lambda 1$, and the original space $\mathbb{R}^{n}$ is embedded into $\mathrm{C}\ell_{n}$ by the inclusion map [7, Section 3].

There is a unique involutive algebra anti-automorphism of $\mathrm{C}\ell_{n}$ that extends the identity automorphism $I$ of $\mathbb{R}^{n}$, called the reversion, denoted by $a\mapsto\tilde{a}$. Further, the grade involution denoted by $a\mapsto\hat{a}$ is a unique involutive automorphism of $\mathrm{C}\ell_{n}$ that extends $-I$, whereas the (Clifford) conjugation denoted by $a\mapsto\bar{a}$ is a unique involutive anti-automorphism of $\mathrm{C}\ell_{n}$ that extends $-I$. The grade involution is used to define a Clifford group (also called a Lipschitz group), which is a group under multiplication of $\mathrm{C}\ell_{n}$ defined by

[TABLE]

The conjugation of $\mathrm{C}\ell_{n}$ gives rise to a group homomorphism of $\Gamma_{n}$. In fact, define a map $\eta$ by

[TABLE]

Then the restriction of $\eta$ to $\Gamma_{n}$ is a homomorphism from $\Gamma_{n}$ to the multiplicative group of nonzero numbers, denoted by ${\mathbb{R}}^{\times}$ [8, Proposition 2]. If an element $a$ in $\mathrm{C}\ell_{n}$ has the property that $\eta(a)\in\mathbb{R}$ and $\eta(a)\geq 0$, we define $|a|=\sqrt{\eta(a)}$. It is not difficult to see that $|\mathbf{v}|=\|\mathbf{v}\|$ for all $\mathbf{v}\in\mathbb{R}^{n}$.

The following theorem summarizes basic properties of $\mathrm{C}\ell_{n}$ that will be used in Section 3, especially the proof of Theorem 3.1.

Theorem 2.1** (Proposition 5, [8]).**

The following properties hold in the Clifford algebra $\mathrm{C}\ell_{n}$.

1. (1)

$\mathbf{u}\mathbf{v}+\mathbf{v}\mathbf{u}=-2\langle\mathbf{u},\mathbf{v}\rangle$* for all $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$.* 2. (2)

$\mathbf{v}^{2}=-\|\mathbf{v}\|^{2}$* for all $\mathbf{v}\in\mathbb{R}^{n}$.* 3. (3)

$1-\mathbf{u}\mathbf{v}\in\Gamma_{n}$* and $(1-\mathbf{u}\mathbf{v})^{-1}=\dfrac{1-\mathbf{v}\mathbf{u}}{\eta(1-\mathbf{u}\mathbf{v})}$ for all $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$ with $\|\mathbf{u}\|\|\mathbf{v}\|\neq 1$.* 4. (4)

$\eta(\mathbf{w}(1-\mathbf{u}\mathbf{v})^{-1})=\dfrac{\eta(\mathbf{w})}{\eta(1-\mathbf{u}\mathbf{v})}$* for all $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{R}^{n}$ with $\|\mathbf{u}\|\|\mathbf{v}\|\neq 1$.*

In view of Theorem 2.1 (2), if $\mathbf{v}\neq\mathbf{0}$, then $\mathbf{v}$ is invertible with respect to multiplication of $\mathrm{C}\ell_{n}$ and $\mathbf{v}^{-1}=-\dfrac{1}{\|\mathbf{v}\|^{2}}\mathbf{v}$. Furthermore, by Lemma 1 of [8],

[TABLE]

belongs to $\mathbb{R}^{n}$ for all nonzero vectors $\mathbf{v}\in\mathbb{R}^{n}$ and all $\mathbf{w}\in\mathbb{R}^{n}$. This implies that $\mathbb{R}^{n}\setminus\{\mathbf{0}\}\subseteq\Gamma_{n}$ and we obtain the following theorem.

Theorem 2.2**.**

Every transformation of the form $\mathbf{w}\mapsto q\mathbf{w}q^{-1}$, where $\mathbf{w}\in\mathbb{R}^{n}$ and $q\in\Gamma_{n}$, defines an orthogonal transformation on $\mathbb{R}^{n}$.

Proof.

Let $\mathbf{w}\in\mathbb{R}^{n}$ and let $q\in\Gamma_{n}$. Clearly, $\|q\mathbf{0}q^{-1}\|=0=\|\mathbf{0}\|$. Therefore, we may assume that $\mathbf{w}\neq\mathbf{0}$ and hence $\mathbf{w}\in\Gamma_{n}$. Since $\eta$ is a homomorphism from $\Gamma_{n}$ to ${\mathbb{R}}^{\times}$, it follows that $\eta(q\mathbf{w}q^{-1})=\eta(q)\eta(\mathbf{w})\eta(q)^{-1}=\eta(\mathbf{w})$ and so

[TABLE]

It is clear that the map $\mathbf{w}\mapsto q\mathbf{w}q^{-1}$ is linear and bijective for $\mathbf{w}\mapsto q^{-1}\mathbf{w}q$ defines its inverse with respect to composition of maps. ∎

Using the Clifford algebra formalism, one gains a compact formula for Möbius addition, as shown in the following theorem.

Theorem 2.3** (Theorem 5.2, [5]).**

In $\mathrm{C}\ell_{n}$, Möbius addition can be expressed as

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{B}$. The gyroautomorphisms are given by ${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{\mathbf{w}}=q\mathbf{w}q^{-1}$, where

[TABLE]

for all $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{B}$.

3 Metrics on the Möbius gyrogroup and their isometry groups

In this section, we prove a useful inequality involving Möbius addition and the Euclidean norm as an application of the Cauchy–Schwarz inequality, using the Clifford algebra formalism. This enables us to define a variant of norm metric on the Möbius gyrogroup. This metric turns out to be a characteristic property of Möbius transformations on $\hat{\mathbb{R}}^{n}$ carrying $\mathbb{B}$ onto itself, where $\hat{\mathbb{R}}^{n}$ is the one-point compactification of $\mathbb{R}^{n}$. We then give a complete description of the corresponding isometry group via a gyrogroup approach.

Theorem 3.1**.**

The inequality

[TABLE]

holds in the Möbius gyrogroup.

Proof.

Using the Cauchy–Schwarz inequality, we have

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$. This implies that

[TABLE]

and that $\eta(\mathbf{u}+\mathbf{v})\geq(\|\mathbf{u}\|-\|\mathbf{v}\|)^{2}$ for all $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$. Let $\mathbf{u},\mathbf{v}\in\mathbb{B}$. As in the proof of Proposition 5 (4) of [8], we have $\eta(1-\mathbf{u}\mathbf{v})\geq(1-\|\mathbf{u}\|\|\mathbf{v}\|)^{2}$ and

[TABLE]

Hence, by Theorem 2.1 (4),

[TABLE]

and similarly

[TABLE]

as required. ∎

In view of (3.1) and the well known trigonometric identity, the tangent function is needed in order to obtain a bounded metric on the unit ball of $\mathbb{R}^{n}$. In fact, define a function $\|\cdot\|_{T}$ by

[TABLE]

for all $\mathbf{v}\in\mathbb{B}$. Here, $T$ stands for “$\tan^{-1}$”.

Theorem 3.2**.**

$\|\cdot\|_{T}$* satisfies the following properties:*

1. (1)

$\|\mathbf{x}\|_{T}\geq 0$* and $\|\mathbf{x}\|_{T}=0$ if and only if $\mathbf{x}=\mathbf{0}$;* 2. (2)

$\|\ominus\mathbf{x}\|_{T}=\|\mathbf{x}\|_{T}$; 3. (3)

$\|\mathbf{x}\|_{T}-\|\mathbf{y}\|_{T}\leq\|\mathbf{x}\oplus_{M}\mathbf{y}\|_{T}\leq\|\mathbf{x}\|_{T}+\|\mathbf{y}\|_{T}$; 4. (4)

$\|{\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{\mathbf{x}}\|_{T}=\|\mathbf{x}\|_{T}$**

for all $\mathbf{u},\mathbf{v},\mathbf{x},\mathbf{y}\in\mathbb{B}$.

Proof.

Item (1) follows from the fact that $\tan^{-1}$ is a strictly increasing injective function on $(-\infty,\infty)$. Item (2) follows from the fact that $\|-\mathbf{x}\|=\|\mathbf{x}\|$.

To prove (3), set $x=\tan^{-1}{\|\mathbf{x}\|}$ and $y=\tan^{-1}{\|\mathbf{y}\|}$. By Theorem 3.1,

[TABLE]

and so $\tan{(x-y)}\leq\|\mathbf{x}\oplus_{M}\mathbf{y}\|\leq\tan{(x+y)}$. Since $\tan^{-1}$ is an increasing function, it follows that $x-y\leq\tan^{-1}{\|\mathbf{x}\oplus_{M}\mathbf{y}\|}\leq x+y$, as claimed. By Theorem 2.3, there is an element $q\in\Gamma_{n}$ for which ${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{\mathbf{x}}=q\mathbf{x}q^{-1}$. It follows from Theorem 2.2 that

[TABLE]

which proves (4). ∎

As a consequence of Theorem 3.2, we obtain a new metric on the Möbius gyrogroup. Unlike the Poincaré metric, this metric is bounded as shown in the following theorem.

Theorem 3.3**.**

Define $d_{T}$ by

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$. Then $d_{T}$ is a bounded metric on $\mathbb{B}$.

Proof.

By Theorem 3.2 (1), $d_{T}(\mathbf{x},\mathbf{y})\geq 0$ for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$ and $d_{T}(\mathbf{x},\mathbf{y})=0$ if and only if $\mathbf{x}=\mathbf{y}$. Let $\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{B}$. Using appropriate properties of the Möbius gyrogroup in Table 1, together with Theorem 3.2, we obtain

[TABLE]

and so $d_{T}(\mathbf{y},\mathbf{x})=d_{T}(\mathbf{x},\mathbf{y})$. Furthermore, we obtain

[TABLE]

This proves that $d_{T}$ satisfies the defining properties of a metric.

Note that $d_{T}(\mathbf{0},\mathbf{v})=\|\mathbf{v}\|_{T}=\tan^{-1}{\|\mathbf{v}\|}<\tan^{-1}{1}=\dfrac{\pi}{4}$ for all $\mathbf{v}\in\mathbb{B}$. Hence,

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$. ∎

Although $d_{T}$ is quite different from the Poincaré metric, both generate the same topology on the unit ball. It is clear that the Poincaré metric and the rapidity metric of the Möbius gyrogroup generate the same topology since the former is twice the latter.

Theorem 3.4**.**

The topologies induced by $d_{T}$ and $d_{M}$ are equivalent.

Proof.

Note that $d_{T}(\mathbf{u},\mathbf{v})\leq d_{M}(\mathbf{u},\mathbf{v})$ for all $\mathbf{u},\mathbf{v}\in\mathbb{B}$ since

[TABLE]

defines a strictly increasing function on the open interval $(0,1)$. This implies that the topology generated by $d_{M}$ is finer than the topology generated by $d_{T}$. Next, we prove that the topology generated by $d_{T}$ is finer than the topology generated by $d_{M}$. Let $\mathbf{u}\in\mathbb{B}$ and let $\epsilon>0$. Choose $\delta=\tan^{-1}{(\tanh{\epsilon})}$. Let $\mathbf{v}\in B_{d_{T}}(\mathbf{u},\delta)$. Then $d_{T}(\mathbf{u},\mathbf{v})<\delta$, that is, $\|\ominus\mathbf{u}\oplus_{M}\mathbf{v}\|_{T}<\tan^{-1}{(\tanh{\epsilon})}$. It follows that

[TABLE]

for $\tan$ and $\tanh^{-1}$ are strictly increasing functions. Hence, $\mathbf{v}\in B_{d_{M}}(\mathbf{u},\epsilon)$. This proves $B_{d_{T}}(\mathbf{u},\delta)\subseteq B_{d_{M}}(\mathbf{u},\epsilon)$. ∎

Let $\mathrm{O}\,({\mathbb{R}^{n}})$ be the orthogonal group of $\mathbb{R}^{n}$, that is,

[TABLE]

Set

[TABLE]

where ${\tau}\hskip-3.0pt\mid_{\mathbb{B}}$ is the restriction of $\tau$ to $\mathbb{B}$. It is clear that $\mathrm{O}\,({\mathbb{B}})$ forms a group under composition of maps since $\mathbb{B}$ is preserved under orthogonal transformations on $\mathbb{R}^{n}$. Given $\mathbf{u},\mathbf{v}\in\mathbb{B}$, note that ${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$ satisfies the following properties:

1. (1)

${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{\mathbf{0}}=\mathbf{0}$; 2. (2)

${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$ is an automorphism of $(\mathbb{B},\oplus_{M})$; 3. (3)

${\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$ preserves the Möbius gyrometric.

Hence, by Theorem 3.2 of [1], there is a bijective orthogonal transformation on $\mathbb{R}^{n}$, denoted by ${\mathrm{Gyr}[{\mathbf{u},\mathbf{v}}]}{}$, for which ${{\mathrm{Gyr}[{\mathbf{u},\mathbf{v}}]}{}}\hskip-3.0pt\mid_{\mathbb{B}}={\mathrm{gyr}[{\mathbf{u},\mathbf{v}}]}{}$. This proves the following inclusion:

[TABLE]

Next, we compute the isometry group of $(\mathbb{B},d_{T})$.

Lemma 3.5**.**

The left gyrotranslation $L_{\mathbf{u}}\colon\mathbf{v}\mapsto\mathbf{u}\oplus_{M}\mathbf{v}$ defines an isometry of $(\mathbb{B},d_{T})$ for all $\mathbf{u}\in\mathbb{B}$.

Proof.

By Theorem 10 (1) of [9], $L_{\mathbf{u}}$ is a bijective self-map of $\mathbb{B}$. Using appropriate properties of the Möbius gyrogroup in Table 1, we obtain

[TABLE]

It follows that

[TABLE]

Theorem 3.6**.**

The isometry group of $(\mathbb{B},d_{T})$ is given by

[TABLE]

Proof.

For convenience, if $\rho\in\mathrm{O}\,({\mathbb{R}^{n}})$, then the restriction of $\rho$ to $\mathbb{B}$ is simply denoted by $\rho$. By Lemma 3.5, $L_{\mathbf{u}}$ is an isometry of $\mathbb{B}$ with respect to $d_{T}$. Let $\rho\in\mathrm{O}\,({\mathbb{R}^{n}})$. Using (1.3), we have $\rho(\mathbf{x})\oplus_{M}\rho(\mathbf{y})=\rho(\mathbf{x}\oplus_{M}\mathbf{y})$ for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$ since $\rho$ is linear and preserves the Euclidean inner product. Hence, the restriction of $\rho$ to $\mathbb{B}$ is indeed an automorphism of $(\mathbb{B},\oplus_{M})$ since $\rho(\mathbb{B})\subseteq\mathbb{B}$ and $\rho^{-1}\in\mathrm{O}\,({\mathbb{R}^{n}})$. It follows that

[TABLE]

Thus, $\rho$ is an isometry of $\mathbb{B}$ and so $\{{L_{\mathbf{u}}\circ\tau}\colon{\mathbf{u}\in\mathbb{B},\tau\in\mathrm{O}\,({\mathbb{B}})}\}\subseteq\mathrm{Iso}\,{(\mathbb{B},d_{T})}$.

Let $T\in\mathrm{Iso}\,{(\mathbb{B},d_{T})}$. By definition, $T$ is a bijective self-map of $\mathbb{B}$. By Theorem 11 of [9], $T=L_{T(\mathbf{0})}\circ\rho$, where $\rho$ is a bijective self-map of $\mathbb{B}$ fixing $\mathbf{0}$. As in the proof of Theorem 18 (2) of [7], $L_{T(\mathbf{0})}^{-1}=L_{\ominus T(\mathbf{0})}$ and so $\rho=L_{\ominus T(\mathbf{0})}\circ T$. Therefore, $\rho$ is an isometry of $(\mathbb{B},d_{T})$. Since $d_{T}(\rho(\mathbf{x}),\rho(\mathbf{y}))=d_{T}(\mathbf{x},\mathbf{y})$ and $\tan^{-1}$ is injective, it follows that

[TABLE]

for all $\mathbf{x},\mathbf{y}\in\mathbb{B}$. Thus, $\rho$ preserves the Möbius gyrometric. By Theorem 3.2 of [1], $\rho={\tau}\hskip-3.0pt\mid_{\mathbb{B}}$, where $\tau$ is a bijective orthogonal transformation on $\mathbb{R}^{n}$. This proves that

[TABLE]

By Theorem 3.6, every isometry of $\mathbb{B}$ with respect to $d_{T}$ can be expressed as the composite of a left gyrotranslation with an orthogonal transformation restricted to $\mathbb{B}$. This expression is unique in the sense that if $L_{\mathbf{u}}\circ\alpha=L_{\mathbf{v}}\circ\beta$ with $\mathbf{u},\mathbf{v}$ in $\mathbb{B}$ and $\alpha,\beta$ in $\mathrm{O}\,({\mathbb{B}})$, then $\mathbf{u}=\mathbf{v}$ and $\alpha=\beta$. Furthermore, we have the following composition law of isometries of $(\mathbb{B},d_{T})$:

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{B}$ and $\alpha,\beta\in\mathrm{O}\,({\mathbb{B}})$, a formula comparable to the composition law of Euclidean isometries.

Since $\mathbf{v}\mapsto L_{\mathbf{v}}$ defines a one-to-one correspondence from $\mathbb{B}$ to the set of left gyrotranslations of $\mathbb{B}$, we have

[TABLE]

Here, $\mathbb{B}\rtimes_{\rm gyr}\mathrm{O}\,({\mathbb{B}})$ is the semidirect-product-like group whose underlying set is

[TABLE]

with group law

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{B}$ and $\alpha,\beta\in\mathrm{O}\,({\mathbb{B}})$. This is a result analogous to the fact that the isometry group of the Euclidean space is the semidirect product of $\mathbb{R}^{n}$ and $\mathrm{O}\,({\mathbb{R}^{n}})$:

[TABLE]

where the group law is given by

[TABLE]

for all $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$ and $\alpha,\beta\in\mathrm{O}\,({\mathbb{R}^{n}})$. The group $\mathbb{B}\rtimes_{\rm gyr}\mathrm{O}\,({\mathbb{B}})$ is known as the gyrosemidirect of $\mathbb{B}$ and $\mathrm{O}\,({\mathbb{B}})$ [10, Section 2.6].

Theorem 3.7**.**

Let $T$ be a self-map of $\mathbb{B}$. The following are equivalent:

1. (1)

$T$* preserves the Poincaré metric $d_{P}$;* 2. (2)

$T$* preserves the rapidity metric $d_{M}$;* 3. (3)

$T$* preserves the Möbius gyrometric $\varrho_{M}$;* 4. (4)

$T$* preserves the metric $d_{T}$ generated by $\|\cdot\|_{T}$.*

Proof.

The theorem follows directly from the fact that $d_{P}(\mathbf{x},\mathbf{y})=2d_{M}(\mathbf{x},\mathbf{y})$ and that $\tanh^{-1}$ and $\tan^{-1}$ are injective. ∎

Corollary 3.8**.**

$\mathrm{Iso}\,{(\mathbb{B},d_{P})}=\mathrm{Iso}\,{(\mathbb{B},d_{M})}=\mathrm{Iso}\,{(\mathbb{B},\varrho_{M})}=\mathrm{Iso}\,{(\mathbb{B},d_{T})}$.

Recall that a Möbius transformation of $\hat{\mathbb{R}}^{n}$ that leaves $\mathbb{B}$ invariant is called a Möbius transformation of $\mathbb{B}$ [6, p. 120]. It is known that the isometry group of the Poincaré ball model $(\mathbb{B},d_{P})$, also called the conformal ball model, can be identified with the group of Möbius transformations of $\mathbb{B}$; see, for instance, [6, Corollary 1 on p. 125]. By Corollary 3.8, Equation (3.10) provides a parametric realization of the Möbius transformation group of $\mathbb{B}$ in terms of vectors and rotations. Further, $d_{T}$ is an invariant of Möbius transformations of $\mathbb{B}$ in the sense of the following theorem.

Theorem 3.9**.**

Every Möbius transformation of $\mathbb{B}$ restricts to an isometry of $(\mathbb{B},d_{T})$, and every isometry of $(\mathbb{B},d_{T})$ extends to a unique Möbius transformation of $\mathbb{B}$.

Proof.

Let $\phi$ be a Möbius transformation of $\mathbb{B}$. By Theorem 4.5.2 of [6], $\phi$ restricts to an isometry of $(\mathbb{B},d_{P})$. By Corollary 3.8, ${\phi}\hskip-3.0pt\mid_{\mathbb{B}}$ is an isometry of $(\mathbb{B},d_{T})$. Let $\sigma$ be an isometry of $(\mathbb{B},d_{T})$. By the same corollary, $\sigma$ is an isometry of $(\mathbb{B},d_{P})$ and hence extends to a unique Möbius transformation of $\mathbb{B}$ by the same theorem. ∎