Some Geometric Properties of the m-Möbius Transformations

Dorin Ghisa

York University, Toronto, Canada.

DOI: 10.4236/apm.2022.123013                

Abstract

Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f_m mapping onto . Even for the simplest entity, the pre-image by f_m of a unique point, there is no way of visualization. Pre-images by f_m of figures from C are like ghost figures in C^m. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C^m are well known and vector calculus in C^m is familiar, yet the pre-image by f_m of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C^m. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.

Keywords

Möbius Transformation, Conformal Mapping, Symmetry with Respect to a Circle, Symmetry Principle

1. Introduction

The bi-Möbius transformations are functions $f\mathrm{<mo>\; :\; </mo>}\overline{ℂ}\times \overline{ℂ}\to \overline{ℂ}$ of the form:

$\begin{array}{c}{f}_2\left(z_1,z_2\right)=\frac{A{z}_1{z}_2+a\left(1-z_1-z_2\right)}{a\left(z_1{z}_2-z_1-z_2\right)+A}\\ =\frac{\left(A{z}_2-a\right)z_1+a\left(1-z_2\right)}{a\left(z_2-1\right)z_1+\left(A-z_2\right)}\\ =\frac{\left(A{z}_1-a\right)z_2+a\left(1-z_1\right)}{a\left(z_1-1\right)z_2+\left(A-z_1\right)}\end{array}$ (1)

where $a\in ℂ\backslash \left\{\mathrm{0,1}\right\}$ and $A=a^2-a+1$. By denoting $\omega =a+1/a-1$ and $s_1=z_1+z_2$, $s_2=z_1{z}_2$, we have:

$f_2\left(z_1\mathrm{,}{z}_2\right)=\frac{\omega s_2-s_1+1}{s_2-s_1+\omega }$ (2)

Let us notice that if $z_2\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$, then $\left(A{z}_2-a\right)\left(A-a{z}_2\right)+a^2{\left(z_2-1\right)}^2\ne 0$, thus $f_2\left(z_1\mathrm{,}{z}_2\right)$ is a Möbius transformation in $z_1$ and if $z_1\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$, then $\left(A{z}_1-a\right)\left(A-a{z}_1\right)+a^2{\left(z_1-1\right)}^2\ne 0$, thus $f_2\left(z_1\mathrm{,}{z}_2\right)$ is a Möbius transformation in $z_2$, which justifies the name of bi-Möbius transformation we have given to $w=f_2\left(z_1,z_2\right)$.

It is expected that the fixed points of $f_2$ as a Möbius transformation in $z_1$ depend on $z_2$. In reality, solving for $z_1$ the equation $f_2\left(z_1,z_2\right)=z_1$ we obtain $\left(z_2-1\right)\left[z_1^2-\left(\omega +1\right)z_1+1\right]=0$. Since $f_2\left(z_1\mathrm{,1}\right)=z_1$, all the points $z_1$ are fixed points of the Möbius transformation $f_2\left(z_1\mathrm{,1}\right)$. On the other hand, the equation $z_1^2-\left(\omega +1\right)z_1+1=0$ does not depend on $z_2$, therefore its solutions which are the fixed points ${\xi }_1$ and ${\xi }_2$ do not depend on $z_2$ and are such that ${\xi }_1+{\xi }_2=\omega +1$ and ${\xi }_1{\xi }_2=1$. Similarly, the fixed points of $f_2$ as a Möbius transformation in $z_2$ do not depend on $z_1$. It can be easily checked that in fact they are the same, in other words, $f_2\left({\xi }_k,z_2\right)=f_2\left(z_1,{\xi }_k\right)={\xi }_k,k=1,2$ for every $z_1$ and $z_2$ in $\overline{ℂ}$. In particular $f_2\left({\xi }_1,{\xi }_2\right)={\xi }_1={\xi }_2=\xi$, hence $2\xi =\omega +1$ and ${\xi }^2=1$. When $\xi =1$ we have $\omega =1$, hence $a=1$, which has been excluded. When $\xi =-1$, we have $\omega =-3$, thus $f_2$ defined by $\omega =-3$ has the double fixed point −1.

An easy computation shows that:

$f_3\left(z_1,z_2,z_3\right)=f_2\left(z_1,f_2\left(z_2,z_3\right)\right)=f_2\left(f_2\left(z_1,z_2\right),z_3\right)=\frac{\left(1+\omega \right)s_3-s_2+1}{s_3-s_1+\left(1+\omega \right)}$ (3)

where $s_3=z_1{z}_2{z}_3$, $s_2=z_1{z}_2+z_1{z}_3+z_2{z}_3$ and $s_1=z_1+z_2+z_3$. Moreover, $f_3\left(z_1,z_2,z_3\right)$ is a Möbius transformation in each one of the variables, if the other variables do not take the values a or 1/a. Again, solving for $z_1$ the equation $f_3\left(z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\right)=z_1$ we obtain $\left(z_2{z}_3-1\right)\left[z_1^2-\left(1+\omega \right)z_1+1\right]=0$, which indicates that $f_3\left(z_1,z_2,z_3\right)$ has the same fixed points ${\xi }_1$ and ${\xi }_2$ as the Möbius transformation in every variable when the other two $z_j$ and $z_k$ are such that $z_j{z}_k\ne 1$.

By using formula (3) repeatedly, we have computed in [1] several functions $f_m$. We provide here a list of them, which will be needed in Section 3 when dealing with the uniqueness of m-Möbius transformations. There is no harm to write $f_m\left(z\right)$ instead of $f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$, where $z=\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$.

$f_4\left(z\right)=\frac{\left({\omega }^2+\omega -1\right)s_4-\omega s_3+s_2-s_1+\omega }{\omega s_4-s_3+s_2-\omega s_1+\left({\omega }^2+\omega -1\right)}$

$f_5\left(z\right)=\frac{\omega \left(\omega +2\right)s_5-\left(\omega +1\right)s_4+s_3-s_1+\left(\omega +1\right)}{\left(\omega +1\right)s_5-s_4+s_2-\left(\omega +1\right)s_1+\omega \left(\omega +2\right)}$

$f_6\left(z\right)=\frac{\left({\omega }^3+2{\omega }^2-\omega -1\right)s_6-\left({\omega }^2+\omega -1\right)s_5+\omega s_4-s_3+s_2-\omega s_1+\left({\omega }^2+\omega -1\right)}{\left({\omega }^2+\omega -1\right)s_6-\omega s_5+s_4-s_3+\omega s_2-\left({\omega }^2+\omega -1\right)s_1+\left({\omega }^3+2{\omega }^2-\omega -1\right)}$

$f_7\left(z\right)=\frac{\left({\omega }^3+2{\omega }^2+\omega -1\right)s_7-\omega \left(\omega +2\right)s_6+\left(\omega +1\right)s_5-s_4+s_2-\left(\omega +1\right)s_1+\omega \left(\omega +2\right)}{\omega \left(\omega +2\right)s_7-\left(\omega +1\right)s_6+s_5-s_3+\left(\omega +1\right)s_2-\omega \left(\omega +2\right)s_1+\left({\omega }^3+2{\omega }^2-\omega -1\right)}$

$f_8\left(z\right)=\frac{\omega \left({\omega }^3+3{\omega }^2-3\right)s_8-\left({\omega }^3+2{\omega }^2-\omega -1\right)s_7+\left({\omega }^2+\omega -1\right)s_6-\omega s_5+s_4-s_3+\omega s_2-\left({\omega }^2+\omega -1\right)s_1+\left({\omega }^3+2{\omega }^2-\omega -1\right)}{\left({\omega }^3+2{\omega }^2-\omega -1\right)s_8-\left({\omega }^2+\omega -1\right)s_7+\omega s_6-s_5+s_4-\omega s_3+\left({\omega }^2+\omega -1\right)s_2-\left({\omega }^3+2{\omega }^2-\omega -1\right)s_1+\omega \left({\omega }^3+3{\omega }^2-3\right)}$

$f_9\left(z\right)=\frac{\left({\omega }^4+4{\omega }^3+3{\omega }^2-2\omega -1\right)s_9-\left({\omega }^3+3{\omega }^2+\omega -1\right)s_8-\omega \left(\omega +2\right)s_7-\left(\omega +1\right)s_6+s_5-s_3+\left(\omega +1\right)s_2-\omega \left(\omega +2\right)s_1+\left({\omega }^3+3{\omega }^2+\omega -1\right)}{\left({\omega }^3+3{\omega }^2-\omega -1\right)s_9-\omega \left(\omega +2\right)s_8+\left(\omega +1\right)s_7-s_6+s_4-\left(\omega +1\right)s_3+\omega \left(\omega +2\right)s_2-\left({\omega }^3+2{\omega }^2-\omega -1\right)s_1+\left({\omega }^4+4{\omega }^3+3{\omega }^2-2\omega -1\right)}$

When we checked for the fixed points of $f_4\left(z\right)$ as a Möbius transformation in $z_4$ we got the equation:

$\left[\omega \left(s_3-1\right)+s_1-s_2\right]\left[z_4^2-\left(\omega +1\right)z_4+1\right]=0$

where $s_k$ are the symmetric sums in $z_1\mathrm{,}{z}_2\mathrm{,}{z}_3$, which shows that $f_4$ has the same fixed points ${\xi }_1$ and ${\xi }_2$ as $f_2$ and $f_3$. Due to the symmetry of $f_4$, this is true when it is considered as a Möbius transformation in any one of its variables if the other three are such that $\omega \left(s_3-1\right)+s_1-s_2\ne 0$. For $f_5\left(z\right)$ the equation is:

$\left[\left(\omega +1\right)\left(s_4-1\right)+s_1-s_3\right]\left[z_5^2-\left(\omega +1\right)z_5+1\right]=0$

where $s_k$ are the symmetric sums in $z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\mathrm{,}{z}_4$ and again we find the same fixed points ${\xi }_1$ and ${\xi }_2$ when $f_5$ is treated as a Möbius transformation in any one of its variables and $\left(\omega +1\right)\left(s_4-1\right)+s_1-s_3\ne 0$.

It is expected similar properties to be true for any m-Möbius transformation, yet for higher values of m, the computation becomes too tedious.

We notice that the coefficients of $s_k$, which are polynomials in $\omega$ become more and more complicated as k increases. Yet, an interesting pattern should be noticed, namely that in every $f_m$ the coefficient of $s_k$ at the numerator is the same as the coefficient of $s_{m-k}$ at the denominator. Also, if we compare the coefficients from $f_m$ and from $f_{m+2}$ we find another surprising pattern, which will be studied in detail in Section 3.

More generally, if $f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)=f_2\left(f_{m-1}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{m-1}\right)\mathrm{,}{z}_m\right)$, then:

$f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)=\frac{a_0\left(\omega \right)s_m+a_1\left(\omega \right)s_{m-1}+\cdots +a_m\left(\omega \right)}{a_m\left(\omega \right)s_m+a_{m-1}\left(\omega \right)s_{m-1}+\cdots +a_0\left(\omega \right)}$ (4)

where $s_j$ are symmetric sums of $z_k$ and $a_j\left(\omega \right)$ are polynomials in $\omega$. Moreover, $w=f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ is a Möbius transformation in every $z_k$ if the other variables belong to $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$. This is an m-Möbius transformation, see [1] [2] [3] for more details about these transformations and their applications.

2. Images of Circles

The geometric properties of the m-Möbius transformations concern the way these mappings transform figures from each one of the planes (z_k) into figures situated in the (w)-plane. As Möbius transformations, they are obviously conformal mappings, hence they will preserve the angles of those figures, except at singular points. They will transform circles (including straight lines, which can be considered circles of infinite radius) into circles. However, there are details which need to be examined.

A circle C centered at $w_0$ and of radius r in the (w)-plane has the equation:

$\left|w-w_0\right|=r$, or ${\left|w\right|}^2-2Re\left({\bar{w}}_{0}w\right)+{\left|w_0\right|}^2=r^2$.

It is convenient to write this equation under the form:

$\alpha {\left|w\right|}^2+2Re\left(bw\right)+\beta =0$, (5)

where $a\mathrm{,}\beta \in ℝ$ and $b\in ℂ$. Indeed, we obtain (5) when we replace $w_0$ by $-\bar{b}/\alpha$ and $r^2$ by ${\left|b\right|}^2/{\alpha }^2-\beta /\alpha$ into the equation $\left|w-w_0\right|=r$. Then, when $\alpha =0$, this equation becomes that of a straight line. We will continue to call it circle (of infinite radius, or centered at infinity).

Theorem 1. The circle (5) is the image by $w=f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$, $z_j\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ fixed, for $j\ne k$ of circles:

${\alpha }_k{\left|z_k\right|}^2+2Re\left(b_k{z}_k\right)+{\beta }_k=0$ (6)

from the (z_k)-planes, $k=\mathrm{1,2,}\cdots \mathrm{,}m$. More exactly, for every k, fixing $z_j\mathrm{,}j\ne k$ there is a unique circle (6) which is mapped bijectively by $f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ onto the circle (5) from the (w)-plane.

Proof: Indeed, the coefficients ${\alpha }_k\mathrm{,}{b}_k$ and ${\beta }_k$ can be uniquely determined as functions of $\alpha \mathrm{,}b$ and $\beta$ by using $f_m$ as follows. With fixed $z_j\mathrm{,}j\ne k$, let us denote:

$p_k=A{f}_{m-1}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{k-1}\mathrm{,}{z}_{k+1}\mathrm{,}\cdots \mathrm{,}{z}_m\right)-a$,

$q_k=a\left[f_{m-1}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{k-1}\mathrm{,}{z}_{k+1}\mathrm{,}\cdots \mathrm{,}{z}_m\right)-1\right]$,

$r_k=A-f_{m-1}\left(z_1,z_2,\cdots ,z_{k-1},z_{k+1},\cdots ,z_m\right)$

and:

$w=f_m\left(z_1,z_2,\cdots ,z_m\right)=\frac{p_k{z}_k-q_k}{q_k{z}_k+r_k}$.

Then:

${\left|w\right|}^2=\frac{p_k{z}_k-q_k}{q_k{z}_k+r_k}\frac{{\bar{p}}_k{\bar{z}}_k-{\bar{q}}_k}{{\bar{q}}_k{\bar{z}}_k+{\bar{r}}_k}$

hence, (5) becomes:

$\begin{array}{l}\alpha \left(p_k{z}_k-q_k\right)\left({\bar{p}}_k{\bar{z}}_k-{\bar{q}}_k\right)/\left(q_k{z}_k+r_k\right)\left({\bar{q}}_k{\bar{z}}_k+{\bar{r}}_k\right)\\ +b\left(p_k{z}_k-q_k\right)/\left(q_k{z}_k+r_k\right)+\bar{b}\left({\bar{p}}_k{\bar{z}}_k-{\bar{q}}_k\right)/\left({\bar{q}}_k{\bar{z}}_k+{\bar{r}}_k\right)+\beta =\mathrm{0,}\end{array}$

or:

$\begin{array}{l}\alpha \left[{\left|p_k\right|}^2{\left|z_k\right|}^2-2Re\left(p_k{\bar{r}}_k{z}_k\right)+{\left|q_k\right|}^2\right]\\ +b\left[p_k{\bar{q}}_k{\left|z_k\right|}^2+p_k{\bar{r}}_k{z}_k-{\left|q_k\right|}^2{\bar{z}}_k-q_k{\bar{r}}_k\right]\\ +\bar{b}\left[{\bar{p}}_k{q}_k{\left|z_k\right|}^2+{\bar{p}}_k{r}_k{\bar{z}}_k-{\left|q_k\right|}^2{z}_k-{\bar{q}}_k{r}_k\right]\\ +\beta \left({\left|q_k\right|}^2{\left|z_k\right|}^2+q_k{\bar{r}}_k{z}_k+{\bar{q}}_k{r}_k{\bar{z}}_k+{\left|r_k\right|}^2\right)=\mathrm{0,}\end{array}$

which gives:

$\begin{array}{l}{\alpha }_k=\alpha {\left|p_k\right|}^2+\beta {\left|q_k\right|}^2+2Re\left(b{p}_k{\bar{q}}_k\right)\in ℝ\\ b_k=-\alpha p_k{\bar{r}}_k+b{p}_k{\bar{r}}_k-\bar{b}{\left|q_k\right|}^2+\beta q_k{\bar{r}}_k\in ℂ\\ {\beta }_k=\alpha {\left|q_k\right|}^2+\beta {\left|r_k\right|}^2-2Re\left(b{q}_k{\bar{r}}_k\right)\in ℝ\end{array}$ (7)

Obviously, different values of $z_j\mathrm{,}j\ne k$ will determine different circles (6) in the (z_k)-plane which are mapped bijectively by $f_m$ onto the circle (5). The bijective nature is assured by the fact that the Möbius transformations are bijective and $f_m$ is a Möbius transformation of the (z_k)-plane as long as every $z_j\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ remains fixed. Varying some of $z_j$, the coefficients $p_k\mathrm{,}{q}_k$ and $r_k$ will all change and then ${\alpha }_k\mathrm{,}{\beta }_k$ and $b_k$ will be different, thus all circles (6) will change, despite of the fact that the circle (5) remains the same.

The concept of pre-image by $f_m$ can be useful in order to describe this change. The pre-image by $f_m$ of a point $w\in \overline{ℂ}$ is by definition $\left\{z\in {\overline{ℂ}}^m\mathrm{<mo>\; |\; </mo>}z=\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)\mathrm{,}{f}_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)=w\right\}$. The pre-image by $f_m$ of a figure $\mathfrak{F}\subset \overline{ℂ}$ is $\left\{z\in {\overline{ℂ}}^m\mathrm{<mo>\; |\; </mo>}{f}_m\left(z\right)\in \mathfrak{F}\right\}$. Since $f_m$ depends only on the symmetric sums $s_k$ the pre-image by $f_m$ of any figure is invariant with respect to the permutations of the variables, i.e., it is a symmetric figure in ${\overline{ℂ}}^m$. In particular, the pre-image W of a single point w is a symmetric figure, which means that if $\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ belongs to W, then so does any point obtained by a permutation of these coordinates.

Let us deal for simplicity with the case of $f_2\left(z_1\mathrm{,}{z}_2\right)$ given by (2). We can chose arbitrarily $z_1\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$. Solving for $z_2$ the equation: $w=f_2\left(z_1,z_2\right)=\frac{\omega z_1{z}_2-z_1-z_2+1}{z_1{z}_2-z_1-z_2+\omega }$ we get:

$z_2=\frac{\left(1-w\right)z_1+\omega w-1}{\left(\omega -w\right)z_1+w-1}$ (8)

so, with this value of $z_2$ we have $f_2\left(z_1\mathrm{,}{z}_2\right)=w$, i.e., $\left(z_1\mathrm{,}{z}_2\right)\in W$. We notice that $z_2$, as a function of $z_1$ is a Möbius transformation and therefore a bijective mapping of $\overline{ℂ}$ onto $\overline{ℂ}$, which means that the pre-image by $f_2$ of a single point $w\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ is a subset of ${\overline{ℂ}}^2$ in one-to-one correspondence with $\overline{ℂ}$. Given $z_1$ in the ( $z_1$ )-plane there is a unique $z_2$ in the ( $z_2$ )-plane, namely that given by (8) such that $\left(z_1\mathrm{,}{z}_2\right)\in W$. Due to the symmetry of $f_2$. this happens if and only if $\left(z_2\mathrm{,}{z}_1\right)\in W$, which means that W is a symmetric figure in ${\overline{ℂ}}^2$.

When $w=0$, then:

$z_2=\frac{z_1-1}{\omega z_1-1}$ (8')

hence every point $\left(z_1\mathrm{,}{z}_2\right)\in {\overline{ℂ}}^2$, where $z_2$ is given by (8') is carried by $f_2$ into the origin. Due to the symmetry of $f_2$, the same is true for every point $\left(z_1\mathrm{,}{z}_2\right)\in {\overline{ℂ}}^2$, where $z_2$ is arbitrary and $z_1=\left(z_2-1\right)/\left(\omega z_2-1\right)$.

Now, suppose that w belongs to the circle C of equation (5). Let us denote by $C$ the pre-image by $f_m$ of the circle C, i.e., $C=\left\{z\in {ℂ}^m|f_m\left(z\right)\in C\right\}$. The formula (8) tells us that for every $w\in C$, we can pick up arbitrarily $z_1$ and if $z_2$ is given by (8), then $\left(z_1\mathrm{,}{z}_2\right)\in C$. But, for $z_1$ fixed (8) is a Möbius transformation in w and it maps the circle C onto a circle $C_2\left(z_1\right)$ into the ( $z_2$ )-plane. We will call it the projection onto the ( $z_2$ )-plane of the section of $C$ by $z_1=const$. Analogously, a projection $C_1\left(z_2\right)$ onto the ( $z_1$ )-plane can be defined of the section of $C$ by $z_2=const$. The circles $C_1\left(z_2\right)$ and $C_2\left(z_1\right)$ are mapped bijectively by $f_2\left(z_1\mathrm{,}{z}_2\right)$ onto the circle C when we keep $z_2\mathrm{,}$ respectively $z_1$ fixed in $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$. An easy computation shows that in terms of $p_2\mathrm{,}{q}_2$ and $r_2$ the equation (8) is $z_2=\left(r_{2}w+q_2\right)/\left(-q_{2}w+p_2\right)$, where $p_2=A{z}_1-a$, $q_2=a\left(z_1-1\right)$ and $r_2=A-z_1$. When $z_1$ varies, all these parameters vary and we obtain different formulas (8) and therefore different circles $C_2\left(z_1\right)$. Hence, there are infinitely many circles into the planes ( $z_1$ ) and ( $z_2$ ) which are mapped bijectively by $f_2$ onto the circle C, one for every $z_2=const$, respectively $z_1=const$. They are all projections onto the two planes of sections of $C$ by $z_2=const$, respectively $z_1=const$. However, we can prove:

Theorem 2. There is a unique Möbius transformation of the ( $z_1$ )-plane into the ( $z_2$ )-plane which maps bijectively every circle $C_1\left(z_2\right)$ onto a circle $C_2\left(z_1\right)$.

Proof. For a given $z_2\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ to every circle C in the (w)-plane corresponds uniquely a circle $C_1\left(z_2\right)$ which is the image of C by the Möbius transformation $z_1\left(w\right)=\frac{\left(\omega -z_2\right)w+z_2-1}{\left(1-z_2\right)w+\omega z_2-1}$ of the (w)-plane into the ( $z_1$ )-plane and for every given $z_1\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ a unique circle $C_2\left(z_1\right)$ exists, which is the image of C by the Möbius transformation $z_2\left(w\right)=\frac{\left(\omega -z_1\right)w+z_1-1}{\left(1-z_1\right)w+\omega z_1-1}$ of the

(w)-plane into the ( $z_2$ )-plane. The function $z_2\circ z_1^{-1}$ is a Möbius transformation of the ( $z_1$ )-plane into the ( $z_2$ )-plane which carries the circle $C_1\left(z_2\right)$ into the circle $C_2\left(z_1\right)$. The affirmation of the theorem is not trivial since it asserts the uniqueness of such a Möbius transformation, while it is known that there are infinitely many Möbius transformations which map a given circle onto another given circle. They differ by rotations around the center of any one of these circles. The theorem states that the mapping $z_2\circ z_1^{-1}$ is uniquely determined by the circles $C_1\left(z_2\right)$ and $C_2\left(z_1\right)$. Indeed, the two circles are in turn uniquely determined by the circle C and the two fixed values of $z_1$ and $z_2$, which define uniquely the functions $z_1\left(w\right)$ and $z_2\left(w\right)$.

The general case can be treated similarly. We choose $\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{k-1}\mathrm{,}{z}_{k+1}\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ arbitrarily in ${\overline{ℂ}}^{m-1}$ and let:

$z_k\left(w\right)=\frac{\left(1-w\right)f_{m-1}\left(z_1,z_2,\cdots ,z_{k-1},z_{k+1},\cdots ,z_m\right)+\omega w-1}{\left(\omega -w\right)f_{m-1}\left(z_1,z_2,\cdots ,z_{k-1},z_{k+1},\cdots ,z_m\right)+w-1}$ (9)

such that $f_m\left(z_1,z_2,\cdots ,z_m\right)=w$. Due to the symmetry of $f_m$, this happens if and only if $f_m\left({z^{\prime }}_1,{z^{\prime }}_2,\cdots ,{z^{\prime }}_m\right)=w$ where $\left({z^{\prime }}_1,{z^{\prime }}_2,\cdots ,{z^{\prime }}_m\right)$ is an arbitrary permutation of $\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$. The formula (9) represents a Möbius transformation in w which maps a circle C of equation (5) onto a circle $C_k$ into the ( $z_k$ )-plane when all the other variables are kept constant in $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$. The theorem 2 in the general form states that for every k and j, $j\ne k$ there is a unique Möbius transformation of the ( $z_k$ )-plane into the ( $z_j$ )-plane which carries every circle $C_k$ into a circle $C_j$.

Let us deal now with the symmetry with respect to a circle (see [4], page 80) of Equation (5). As we have seen, that equation represents a proper circle when $\alpha \ne 0$ or a line when $\alpha =0$. When that line is the real axis, we say that the points w and $\bar{w}$ are symmetric with respect to it. Yet, for any line it is known what symmetric points with respect to that line mean, namely z and $z^{\ast }$ are symmetric points with respect to the line L if and only if L is the bisecting normal of the segment determined by z and $z^{\ast }$. This concept can be extended to the case when $\alpha \ne 0$. In that case the Equation (5) is $\left|w-w_0\right|=r$, where $w_0$ is the center of the circle and r is its radius. As shown in [4], page 81 by using the tool of cross ratios, w and $w^{\ast }$ are symmetric with respect to this circle if and only if:

$w^{\ast }=r^2/\left(\bar{w}-{\bar{w}}_0\right)+w_0$. (10)

The symmetry principle states that if a Möbius transformation carries a circle $C_1$ into a circle $C_2$, then it transforms any pair of symmetric points with respect to $C_1$ into a pair of symmetric points with respect to $C_2$. Here circle means proper circle or line. This principle can be extended to m-Möbius transformations in the following way.

Theorem 3 (The Main Theorem). Let w and $w^{\ast }$ be symmetric points with respect to the circle (5) and let W and $W^{\ast }$ be the pre-images by $f_m$ of w and respectively $w^{\ast }$. Then the projection onto the ( $z_k$ )-plane of any section of W and $W^{\ast }$ obtained by keeping $z_j\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ fixed, $j\ne k$ are points symmetric with respect to the circle (6) corresponding to that section.

Proof: By the Theorem 1, the circle (5) is the image by Möbius transformations of every circle (6) from the ( $z_k$ )-plane when $z_j\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\},j\ne k$ are kept fixed. Then the symmetry principle applied to these circles is exactly this theorem.

This theorem describes in fact a phenomenon happening in ${\overline{ℂ}}^m$ related to m-Möbius transformations. If we consider $C$ as a generalized circle in ${\overline{ℂ}}^m$ and W as the equivalent of a point from $\overline{ℂ}$, then it makes sense to say that $W^{\ast }$ is the symmetric of W with respect to $C$, since this is true for the projections on every ( $z_k$ )-plane of any section of them obtained by keeping $z_j\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ fixed, $j\ne k$. The projection of the respective section of $C$ is $C_k$ and that of the sections of W and $W^{\ast }$ is $z_k$ and $z_k^{\ast }$. The symmetric of $z_k$ with respect to $C_k$. The theorem states that a m-Möbius transformation carries symmetric points with respect to $C$ into symmetric points with respect to C.

There is a one-to-one mapping of W onto $W^{\ast }$ assigning to every $z=\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)\in W$ the point $z^{\ast }=\left(z_1^{\ast }\mathrm{,}{z}_2^{\ast }\mathrm{,}\cdots \mathrm{,}{z}_m^{\ast }\right)$, where $z_k^{\ast }$ is the symmetric of $z_k$ with respect to $C_k$. It can be called reflection in $C$ , extending to ${\overline{ℂ}}^m$ a concept pertinent to $\overline{ℂ}$ (see [4], page 81). The reflection in a circle is an involution and the composition of two reflections with respect to different circles result in a Möbius transformation. More generally, the composition of an odd numbers of reflections with respect to different circles is again a reflection and the composition of an even number of reflections is a Möbius transformation. Moreover, as shown in [4], page 172, every Möbius transformation can be obtained as the composition of two or four reflections in circles.

The symmetry principle states that if a Möbius transformation $w=M\left(z\right)$ carries the circle $C_1$ into the circle $C_2$, then M composed with reflection in $C_1$ is reflection in $C_2$. In other words ${\left[M\left(z\right)\right]}^{\ast }=M\left(z^{\ast }\right)$, where $\ast$ on the left hand side means reflection in $C_2$ and on the right hand side reflection in $C_1$.

We notice that (10) does not represent a Möbius transformation since it is anticonformal, while Möbius transformations are conformal mappings. Yet, in terms of mapping properties there are some similarities between the reflections in circles and Möbius transformations. One of them is that of preservation of circles ( [5], page 126). Even more can be said: if a line L does not pass through the center $w_0$ of the circle C, then its reflection in C is a circle $C_1$ passing through $w_0$ and vice-versa, if a circle $C_1$ passes through $w_0$ then its reflection in C is a line L not passing through $w_0$. If the proper circle $C_1$ does not pass through $w_0$ then its reflection in C is another proper circle not passing through $w_0$.

Next we will investigate some similar properties of the reflections with respect to $C$ and m-Möbius transformations of $C$.

Theorem 4. Let $C$ be the pre-image by a m-Möbius transformation $w=f_m\left(z\right)$ of the circle $C\mathrm{<mo>\; :\; </mo>}\left|w-w_0\right|=r$. Then $f_m$ composed with reflection in $C$ is reflection in C. If L is a line not passing through $w_0$, then $L\cap w_0=\varnothing$, where $L$ and $w_0$ are the pre-images of L and $w_0$. Moreover, if $L^{\ast }$ is the image of $L$ by reflection in $C$, then $w_0\subset L^{\ast }$. Reciprocally, if $C_1$ is a proper circle which passes through $w_0$ then $w_0\subset C_1$ and $C_1^{\ast }\cap w_0=\varnothing$.

Proof: By the formula (9) and the symmetry principle we have $z_k\left(w^{\ast }\right)={\left[z_k\left(w\right)\right]}^{\ast }$ for every $k=\mathrm{1,2,}\cdots \mathrm{,}m$. Then: $w^{\ast }=f_m\left(z_1\left(w^{\ast }\right)\mathrm{,}{z}_2\left(w^{\ast }\right)\mathrm{,}\cdots \mathrm{,}{z}_m\left(w^{\ast }\right)\right)=f_m\left({\left[z_1\left(w\right)\right]}^{\ast }\mathrm{,}{\left[z_2\left(w\right)\right]}^{\ast }\mathrm{,}\cdots \mathrm{,}{\left[z_m\left(w\right)\right]}^{\ast }\right)$

and the first term shows reflection in C, while the last one is $f_m$ composed with reflection in $C$. The second affirmation is true since the pre-image of the intersection of two sets is equal to the intersection of the pre-images of those sets. Next, $f_m$ moves $L^{\ast }$ into the reflection of L in C which passes through $w_0$, hence the pre-image of $w_0$ is included in $L^{\ast }$. Finally, if a proper circle $C_1$ passes through $w_0$, then $C_1^{\ast }$ should pass through $\infty$, which is the reflection in C of $w_0$. Thus $C_1^{\ast }$ is a line not passing through $w_0$ and then $C_1^{\ast }\cap w_0=\varnothing$.

Theorem 5. Given a circle $C_k\mathrm{<mo>\; :\; </mo>}{\alpha }_k{\left|z_k\right|}^2+2Re\left(b_k{z}_k\right)+{\beta }_k=0$ in the ( $z_k$ )-plane, for every $j\ne k$ there are infinitely many circles $C_j$ in every ( $z_j$ )-plane such that $C_k$ and $C_j$ have the same image by $f_m$ when all the other variables are kept fixed.

Proof: Suppose that a circle C of Equation (5) is given in the (w)-plane. We are looking for a circle (6) which is mapped bijectively by $f_m$ onto the circle C when $z_j\mathrm{,}j\ne k$ are kept all fixed. With the notations of Theorem 1, we have that if $z_k$

is on $C_k$ and w is on C where $w=f_m\left(z_1,z_2,\cdots ,z_m\right)$, then $z_k=\frac{r_{k}w+q_k}{-q_{k}w+p_k}$, as in Theorem 1, thus ${\bar{z}}_k=\frac{{\bar{r}}_k\bar{w}+{\bar{q}}_k}{-{\bar{q}}_k\bar{w}+{\bar{p}}_k}$, and the equation of: $C_k\mathrm{<mo>\; :\; </mo>}{\alpha }_k{z}_k{\bar{z}}_k+2Re\left(b_k{z}_k\right)+{\beta }_k$ becomes:

${\alpha }_k\frac{r_{k}w+q_k}{-q_{k}w+p_k}\frac{{\bar{r}}_k\bar{w}+{\bar{q}}_k}{-{\bar{q}}_k\bar{w}+{\bar{r}}_k}+b_k\frac{r_{k}w+q_k}{-q_{k}w+r_k}+{\bar{b}}_k\frac{{\bar{r}}_k\bar{w}+{\bar{q}}_k}{-{\bar{q}}_k\bar{w}+{\bar{r}}_k}+{\beta }_k=0$, or

$\begin{array}{l}{\alpha }_k\left(r_{k}w+q_k\right)\left({\bar{r}}_k\bar{w}+{\bar{q}}_k\right)+b_k\left(r_{k}w+q_k\right)\left(-{\bar{q}}_k\bar{w}+{\bar{r}}_k\right)\\ +{\bar{b}}_k\left({\bar{r}}_k\bar{w}+{\bar{q}}_k\right)\left(-q_{k}w+r_k\right)+{\beta }_k\left(q_{k}w-r_k\right)\left({\bar{q}}_k\bar{w}-{\bar{r}}_k\right)\mathrm{,}\end{array}$

which is:

$C\mathrm{<mo>\; :\; </mo>}\alpha w\bar{w}+2Re\left(bw\right)+\beta$,

where:

$\begin{array}{l}\alpha ={\alpha }_k{\left|r_k\right|}^2+{\beta }_k{\left|q_k\right|}^2-2Re\left(b_k{r}_k{\bar{q}}_k\right)\\ \beta ={\alpha }_k{\left|q_k\right|}^2+{\beta }_k{\left|r_k\right|}^2+2Re\left(b_k{q}_k{\bar{r}}_k\right)\\ b={\alpha }_k{\bar{q}}_k{r}_k-{\beta }_k{q}_k{\bar{r}}_k-{\bar{b}}_k{\left|q_k\right|}^2+b_k{\left|r_k\right|}^2\end{array}$ (11)

Hence the circle $C_k$ is mapped bijectively by $f_m$ onto the circle C when $p_k\mathrm{,}{q}_k$ and $r_k$ are kept fixed. On the other hand, the projection onto the ( $z_j$ )-plane of any section of the pre-image $C$ of C by $z_l=const,l\ne j$ is a circle $C_j$ which is mapped by $f_m$ bijectively onto C when all the variables except $z_j$ are kept fixed in $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$, therefore there are infinitely many circles $C_j$ in every ( $z_j$ )-plane such that $C_k$ and $C_j$ have the same image by $f_m$.

The theory of Apollonius circles and of Steiner nets ( [4], page 85) can be extended word by word to m-Möbius transformations. Let us deal first with the case $m=2$. With the notation $k\left(z_2\right)=\left(A{z}_2-a\right)/a\left(z_2-1\right)$, $\phi \left(z_2\right)=a\left(z_2-1\right)/\left(A{z}_2-a\right)$ and $\psi \left(z_2\right)=\left(z_2-A\right)/a\left(z_2-1\right)$ the formula (1) becomes $w=k\left(z_2\right)\left[z_1-\phi \left(z_2\right)\right]/\left[z_1-\psi \left(z_2\right)\right]$, which shows that $z_1=\phi \left(z_2\right)$ corresponds to $w=0$ and $z_1=\psi \left(z_2\right)$ corresponds to $w=\infty$, thus straight lines through the origin of the (w)-plane are images by $f_2\left(z_1\mathrm{,}{z}_2\right)$ of circles $C_l$ through the limit points $\phi \left(z_2\right)$ and $\psi \left(z_2\right)$ from the ( $z_1$ )-plane for every $z_2\in \overline{ℂ}$. On the other hand, the concentric circles about the origin $\left|w\right|=\rho$, are the images by $w=f_2\left(z_1\mathrm{,}{z}_2\right)$ of circles with equation $\left|\left[z_1-\phi \left(z_2\right)\right]/\left[z_1-\psi \left(z_2\right)\right]\right|=\rho /\left|k\left(z_2\right)\right|$ for every $z_2\in \overline{ℂ}$. These are the Apollonius circles $C_a$ with the limit poins $\phi \left(z_2\right)$ and $\psi \left(z_2\right)$. Together with the $C_a$ circles they form the Steiner net. Every $C_l$ circle is orthogonal to all $C_a$ circles and every $C_a$ circle is orthogonal to all $C_l$ circles.There is exactly one $C_a$ circle and one $C_l$ circle from the Seiner net defined by $z_2$ passing through every point of the ( $z_1$ )-plane except the points $\phi \left(z_2\right)$ and $\psi \left(z_2\right)$. Reflection in a $C_a$ circle switch $\phi \left(z_2\right)$ and $\psi \left(z_2\right)$ and transforms every $C_l$ circle into itself and every $C_a$ circle into another $C_a$ circle.

The pre-image by $f_2$ of any Steiner net from the (w)-plane is an object in ${\overline{ℂ}}^2$ whose sections by $z_2=const$ and $z_1=const$ are Steiner nets in the ( $z_1$ )-plane, respectively the ( $z_2$ )-plane. Indeed, this is true due to the formula (8) and the preservation of circles by Möbius transformations. The Theorem 2 implies the following:

Corollary 1. There is a unique Möbius transformation of the ( $z_1$ )-plane into the ( $z_2$ )-plane which carries the Steiner net determined by $z_2=const$ into the Steiner net determined by $z_1=const$.

Proof: Indeed, by the Theorem 2 there is a unique Möbius transformation $z_1=M\left(z_2\right)$ which carries an Apollonius circle $C_a\left(z_2\right)$ into another Apollonius circle $C_a\left(z_1\right)$. Then, by the symmetry principle every $C_l\left(z_2\right)$ circle is transformed into a $C_l\left(z_1\right)$ circle. Yet the family of these circles determines uniquely the family of the $C_a\left(z_1\right)$ circles, which are all the orthogonal circles to them. Finally, the whole Steiner net from the ( $z_2$ )-plane is mapped by $z_1=M\left(z_2\right)$ into the Steiner net determined by $z_2=const$.

The pre-image by $f_m$ of a Steiner net from the (w)-plane is an object in ${\overline{ℂ}}^m$ whose sections obtained by keeping all $z_j$ fixed for $j\ne k$ is a Steiner net in the ( $z_k$ )-plane, due to the formula (9) and the preservation of circles by Möbius transformations. Every point of ${\overline{ℂ}}^m$, except the pre-image of the limit points of the net, belongs to the pre-image of both families of circles belonging to the Steiner net from the (w)-plane.

Given a Steiner net in the (w)-plane, its pre-image by $f_m\left(z\right)$ into ${\overline{ℂ}}^m$ projects into Steiner nets in every ( $z_k$ )-plane, the image of which by $f_m$ is the original Steiner net from the (w)-plane. By Theorem 2, there is a unique Möbius transformation of any ( $z_j$ )-plane into the ( $z_k$ )-plane carying one such Steiner net into the other.

The question arises: what is the pre-image by $f_m$ of a triangle, rectangle, or in general of an arbitrary polygon? We cannot describe these pre-images, yet we can imagine their sections when keeping constant all the variables except one. Since such a section of the pre-image of a line segment is an arc of a circle or a half line, when one of the ends is sent to infinity, the answer to that question is the following. The projection onto the ( $z_k$ )-plane of the sections as previously defined of the pre-image by $f_m$ of a triangle is a curvilinear triangle having the same angles as the original one. Some of these triangles can be infinite, in the sense that one side is an arc of a circle and the other two are half-lines. An analogous situation appears for the pre-image of an arbitrary polygon.

3. Uniqueness of m-Möbius Transformations

It is known that there is a unique Möbius transformation in the plane moving three distinct points into other three distinct points. In what follows, we will study similar properties of m-Möbius transformations.

Theorem 6. There is a unique m-Möbius transformation $f_m$ moving a given point $z\in {\overline{ℂ}}^m\mathrm{,}m=\mathrm{2,3}$ into a given point $w\in \overline{ℂ}$.

Proof: Indeed, such transformations have the form see ( [1] ) $f_2\left(z_1\mathrm{,}{z}_2\right)=\frac{\omega s_2-s_1+1}{s_2-s_1+\omega }$, respectively $f_3\left(z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\right)=\frac{\left(1+\omega \right)s_3-s_2+1}{s_3-s_1+\left(1+\omega \right)}$. Then, in the first case, $w=f_2\left(z\right)$ implies $\omega s_2-s_1+1=w\left(s_2-s_1+\omega \right)$, hence $\omega$ is uniquely determined by $\omega =\frac{w\left(s_2-s_1\right)+s_1-1}{s_2-w}$. Analogously, in the second case, $w=f_3\left(z\right)$ implies $\left(1+\omega \right)s_3-s_2+1=w\left(s_3-s_1+1+\omega \right)$, thus $\omega$ is uniquely determined by $\omega =\frac{\left(s_3+1\right)\left(w-1\right)+s_2-w{s}_1-1}{s_3-w}$.

This theorem is a particular case of the following:

Theorem 7. For $m=2k$ and $m=2k+1$, $k=1,2,3,\cdots$ the equation $f_m\left(z\right)=w$ has the degree k in $\omega$ and therefore, with the exception of multiple roots, it has k solutions, which means that with those exceptions there are k different m-Möbius transformations $f_m\left(z\right)$ moving a given point $z\in {\overline{ℂ}}^m$ into a given point $w\in \overline{ℂ}$.

Proof: We need a pretty elaborate induction argument. It can be made more obvious if we write the m-Möbius transformations as matrices $\left[\begin{array}{c}{r}_{m\mathrm{,1}}\\ r_{m\mathrm{,2}}\end{array}\right]$ whose

entries are the polynomials in $\omega$ appearing at the numerator and at the denominator of $f_m\left(z\right)$ and arrange the coefficients of these polynomials also as matrices. The examples of $f_m$ which follow can be found in [1].

For $f_2\left(z\right)$ we have the matrix expression:

$\left[\begin{array}{c}{r}_{\mathrm{2,1}}\\ r_{\mathrm{2,2}}\end{array}\right]=\left[\begin{array}{ccc}\omega & -1& 1\\ 1& -1& \omega \end{array}\right]$

For $f_3\left(z\right)$ the matrix is:

$\left[\begin{array}{c}{r}_{\mathrm{3,1}}\\ r_{\mathrm{3,2}}\end{array}\right]=\left[\begin{array}{cccc}1+\omega & -1& 0& 1\\ 1& 0& -1& 1+\omega \end{array}\right]$

We notice that starting with $m=4$ every matrix $\left[\begin{array}{c}{r}_{m\mathrm{,1}}\\ r_{m\mathrm{,2}}\end{array}\right]$ is built around the matrix $\left[\begin{array}{c}-r_{m-\mathrm{2,1}}\\ -r_{m-\mathrm{2,2}}\end{array}\right]$ by adding a first and a last column as follows:

$\left[\begin{array}{c}{r}_{4,1}\\ r_{4,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^2+\omega -1& -r_{2,1}& \omega \\ \omega & -r_{2,2}& {\omega }^2+\omega -1\end{array}\right]$

$\left[\begin{array}{c}{r}_{5,1}\\ r_{5,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^2+2\omega & -r_{3,1}& \omega +1\\ \omega +1& -r_{3,2}& {\omega }^2+2\omega \end{array}\right]$

$\left[\begin{array}{c}{r}_{6,1}\\ r_{6,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^3+2{\omega }^2-\omega -1& -r_{4,1}& {\omega }^2+\omega -1\\ {\omega }^2+\omega -1& -r_{4,2}& {\omega }^3+2{\omega }^2-\omega -1\end{array}\right]$

$\left[\begin{array}{c}{r}_{7,1}\\ r_{7,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^3+3{\omega }^2+\omega -1& -r_{5,1}& {\omega }^2+2\omega \\ {\omega }^2+2\omega & -r_{5,2}& {\omega }^3+3{\omega }^2+\omega -1\end{array}\right]$

$\left[\begin{array}{c}{r}_{8,1}\\ r_{8,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^4+3{\omega }^3-3\omega & -r_{6,1}& {\omega }^3+2{\omega }^2-\omega -1\\ {\omega }^3+2{\omega }^2-\omega -1& -r_{6,2}& {\omega }^4+3{\omega }^3-3\omega \end{array}\right]$

$\left[\begin{array}{c}{r}_{9,1}\\ r_{9,2}\end{array}\right]=\left[\begin{array}{ccc}{\omega }^4+4{\omega }^3+3{\omega }^2-2\omega -1& -r_{7,1}& {\omega }^3+3{\omega }^2+\omega -1\\ {\omega }^3+3{\omega }^2+\omega -1& -r_{7,2}& {\omega }^4+4{\omega }^3+3{\omega }^2-2\omega -1\end{array}\right]$

Let us notice that a simplification with $\omega -1$ occurs in every $f_{2k+1}\left(z\right)$ such that $f_{2k}\left(z\right)$ and $f_{2k+1}\left(z\right)$ have the same degree as rational functions of $\omega$. We need to prove this affirmation thoroughly by induction. It is clear that the induction hypothesis should be:

$\left[\begin{array}{c}{r}_{2k\mathrm{,1}}\\ r_{2k\mathrm{,2}}\end{array}\right]=\left[\begin{array}{ccc}{p}_k\left(\omega \right)& -r_{2k-\mathrm{2,1}}& p_{k-1}\left(\omega \right)\\ p_{k-1}\left(\omega \right)& -r_{2k-\mathrm{2,2}}& p_k\left(\omega \right)\end{array}\right]$

$\left[\begin{array}{c}{r}_{2k+\mathrm{1,1}}\\ r_{2k+\mathrm{1,2}}\end{array}\right]=\left[\begin{array}{ccc}{q}_k\left(\omega \right)& -r_{2k-\mathrm{1,1}}& q_{k-1}\left(\omega \right)\\ q_{k-1}\left(\omega \right)& -r_{2k-\mathrm{1,2}}& q_k\left(\omega \right)\end{array}\right]$

where $p_k\left(\omega \right)$ and $q_k\left(\omega \right)$ are polynomials of degree k and $p_{k-1}\left(\omega \right)$ and $q_{k-1}\left(\omega \right)$ are polynomials of degree $k-1$. Moreover, it can be easily checked that for every $k=1,2,3,4$ we have $q_k\left(\omega \right)-q_{k-1}\left(\omega \right)=p_k\left(\omega \right)$, $p_k\left(\omega \right)-p_{k-1}\left(\omega \right)=\left(\omega -1\right)q_k\left(\omega \right)$, $\omega p_k\left(\omega \right)-p_{k-1}\left(\omega \right)=\left(\omega -1\right)q_k\left(\omega \right)$ and $\omega q_k\left(\omega \right)-q_{k-1}\left(\omega \right)=p_{k+1}\left(\omega \right)$ and these equalities should be a part of the induction hypothesis.

The proof consists in showing that the formula:

$f_{m+1}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{m+1}\right)=\frac{\left(\omega z_{m+1}-1\right)f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)+1-z_{m+1}}{\left(z_{m+1}-1\right)f_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)+\omega -z_{m+1}}$

produces matrices obtained from the last two by replacing k with $k+1$, i.e.:

$\left[\begin{array}{c}{r}_{2k+2,1}\\ r_{2k+2,2}\end{array}\right]=\left[\begin{array}{ccc}{p}_{k+1}\left(\omega \right)& -r_{2k,1}& p_k\left(\omega \right)\\ p_k\left(\omega \right)& -r_{2k,2}& p_{k+1}\left(\omega \right)\end{array}\right]$

$\left[\begin{array}{c}{r}_{2k+3,1}\\ r_{2k+3,2}\end{array}\right]=\left[\begin{array}{ccc}{p}_{k+2}\left(\omega \right)& -r_{2k+1,1}& p_{k+1}\left(\omega \right)\\ p_{k+1}\left(\omega \right)& -r_{2k+1,2}& p_{k+2}\left(\omega \right)\end{array}\right]$

We have:

$\begin{array}{c}{f}_{2k+2}\left(z_1,z_2,\cdots ,z_{2k+2}\right)=\frac{\left(\omega z_{2k+2}-1\right)\frac{q_k\left(\omega \right)s_{2k+1}+\cdots +q_{k-1}\left(\omega \right)}{q_{k-1}\left(\omega \right)s_{2k+1}+\cdots +q_k\left(\omega \right)}+1-z_{2k+2}}{\left(z_{2k+2}-1\right)\frac{q_k\left(\omega \right)s_{2k+1}+\cdots +q_{k-1}\left(\omega \right)}{q_{k-1}\left(\omega \right)s_{2k+1}+\cdots +q_k\left(\omega \right)}+\omega -z_{2k+2}}\\ =\frac{\left[\omega q_k\left(\omega \right)-q_{k-1}\left(\omega \right)\right]z_{2k+2}{s}_{2k+1}+\cdots +q_k\left(\omega \right)-q_{k-1}\left(\omega \right)}{\left[q_k\left(\omega \right)-q_{k-1}\left(\omega \right)\right]z_{2k+2}{s}_{2k+1}+\cdots +\omega q_k\left(\omega \right)-q_{k-1}\left(\omega \right)}\\ =\frac{p_{k+1}\left(\omega \right)s_{2k+2}+\cdots +p_k\left(\omega \right)}{p_k\left(\omega \right)s_{2k+2}+\cdots +p_{k+1}\left(\omega \right)},\end{array}$

which shows that indeed, the matrix corresponding to $f_{2k+2}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{2k+2}\right)$ is:

$\left[\begin{array}{c}{r}_{2k+\mathrm{2,1}}\\ r_{2k+\mathrm{2,2}}\end{array}\right]=\left[\begin{array}{ccc}{p}_{k+1}\left(\omega \right)& -r_{2k\mathrm{,1}}& p_k\left(\omega \right)\\ p_k\left(\omega \right)& -r_{2k\mathrm{,2}}& p_{k+1}\left(\omega \right)\end{array}\right]$

The computation for the second matrix is similar. This proof provides more information than that about the degree of the equation $f_m\left(z\right)=w$, namely it shows the structure of $f_m\left(z\right)$.

4. Vectors in ${ℂ}^m$

The orthogonality of two vectors in ${ℂ}^m$ is expressed by the cancellation of their inner product (see [5], page 151). With the help of m-Möbius transformations we can say more, namely an angle of two arbitrary two vectors in ${ℂ}^m$ can be defined, such that the respective angle is $\pi /2\left(\mathrm{mod}2\pi \right)$ when their inner product is zero. For simplicity, we deal first with the case $m=2$. Let $z,\zeta \in {ℂ}^2$ be arbitrary points. Following a tradition (see [4], page 12), we will keep the same notation for their position vectors, i.e., vectors pointing from the origin to those points. The inner product of the vectors $z$ and $\zeta$ is by definition $\langle z,\zeta \rangle =z_1{\bar{\zeta }}_1+z_2{\bar{\zeta }}_2$. We say that $z$ and $\zeta$ are orthogonal if and only if $\langle z,\zeta \rangle =0$. Suppose that $z$ and $\zeta$ are not those given by (8'), i.e., their images by $f_2$ is not zero. We denote by $z_0$ a point which is mapped by $f_2$ into zero, i.e., $z_0=\left(z_1\mathrm{,}{z}_2\right)$, for an arbitrary $z_1$ and $z_2=\left(z_1-1\right)/\left(\omega z_1-1\right)$. Then $z-z_0$ and $\zeta -z_0$ are vectors with the initial point $z_0$ and the final points respectively $z$ and $\zeta$. Their images by $f_2$ are position vectors u and v in the (w)-plane. They make an angle $\alpha =\sphericalangle \left(u\mathrm{,}v\right)$ which remains invariant to a conformal mapping.

Theorem 8. Let $f_2$ be a bi-Möbius transformation of parameter $\omega$ moving the points $z-z_0$ and $\zeta -z_0$ into u and v. Then the projection of the section of the pre-image of u and v by $z_2=1$ does not depend on $\omega$.

Proof: Indeed, let us deal with the mapping $z_1\left(w\right)=\frac{\left(\omega -z_2\right)w+z_2-1}{\left(1-z_2\right)w+\omega z_2-1}$ obtained by solving for $z_1$ the equation $w=f_2\left(z_1\mathrm{,}{z}_2\right)$. For $z_2$ fixed this is a

conformal mapping of the (w)-plane onto the ( $z_1$ )-plane. For $z_2=1$ we have $z_1\left(w\right)=w$, which implies that $\sphericalangle \left(z_1\left(u\right)\mathrm{,}{z}_1\left(v\right)\right)=\sphericalangle \left(u\mathrm{,}v\right)$. Due to the symmetry of $f_2$ we obtain a similar result for the projection onto the ( $z_2$ )-plane.

If u and v are orthogonal, so are $z_1\left(u\right)$ and $z_1\left(v\right)$, respectively $z_2\left(u\right)$ and $z_2\left(v\right)$. Then $z_1\left(u\right)\overline{z_1\left(v\right)}=0$ and $z_2\left(u\right)\overline{z_2\left(v\right)}=0$, thus $z_1\left(u\right)\overline{z_1\left(v\right)}+z_2\left(u\right)\overline{z_2\left(v\right)}=0$, which means that $z$ and $\zeta$ are orthogonal. We can put by definition $\sphericalangle \left(z,\zeta \right)=\sphericalangle \left(u\mathrm{,}v\right)$, which agrees with the the definition of orthogonality. These concepts generalize trivially to ${ℂ}^m$.

5. The Cross-Ratio

The cross-ratio of four points used by Desargues in his studies of projective geometry (see [5], page 154), reappears in complex analysis as a means of dealing with Möbius transformations. It is known that there is a unique Möbius transformation which carries three arbitrary distinct points $q\mathrm{,}r\mathrm{,}s\in \overline{ℂ}$ into

$\mathrm{0,1,}\infty$. This is $\left[z\mathrm{,}q\mathrm{,}r\mathrm{,}s\right]=\frac{\left(z-q\right)\left(r-s\right)}{\left(z-s\right)\left(r-q\right)}$ and it is called the cross-ratio of the

four points $z\mathrm{,}q\mathrm{,}r\mathrm{,}s$. In other words, the cross-ratio $\left[z\mathrm{,}q\mathrm{,}r\mathrm{,}s\right]$ is the image of z by the Möbius transformation which carries $q\mathrm{,}r\mathrm{,}s$ into $\mathrm{0,1,}\infty$. It is also known (see [4], page 79) that for any Möbius transformation $w=M\left(z\right)$ and for any four distinct points $z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\mathrm{,}{z}_4\in \overline{ℂ}$ we have $\left[w_1\mathrm{,}{w}_2\mathrm{,}{w}_3\mathrm{,}{w}_4\right]=\left[z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\mathrm{,}{z}_4\right]$, where $w_j=M\left(z_j\right)$, $j=1,2,3,4$, hence the cross-ratio of four points is an invariant with respect to Möbius transformations.

What can be said if $w=M\left(z\right)$ is a m-Möbius transformation? Obviously, the cross-ratio cannot be defined in ${\overline{ℂ}}^m$. Yet, we can use the pre-image by M of the four points $w_j\in \overline{ℂ}$ and the fact that the projections of the sections obtained by keeping $z_l$ fixed, $l\ne k$ of this pre-image onto any ( $z_k$ )-plane are complex numbers. More exactly, keeping fixed all $z_l$, $l\ne k$, the section of the pre-image of $w_j$ is a unique point. Let us denote by $z_k^{\left(j\right)}$ the projection onto the ( $z_k$ )-plane of the section of the pre-image of $w_j$ obtained by keeping $z_l$ fixed, $l\ne k$. We have:

$w_j=\frac{p_k{z}_k^{\left(j\right)}-q_k}{q_k{z}_k^{\left(j\right)}+r_k}$ (12)

as in Section 2. Thus, we can state:

Theorem 9. For every m-Möbius transformation $w=M\left(z\right)$ the cross-ratio of four points from every ( $z_k$ )-plane is preserved.

Proof: Let us take $z_k^{\left(j\right)},j=1,2,3,4$ four distinct points in the ( $z_k$ )-plane and for $z_l^{\left(j\right)}$ arbitrary in the ( $z_l$ )-planes, $l\ne k$ denote

$w_j=M\left(z_1^{\left(j\right)},z_2^{\left(j\right)},\cdots ,z_m^{\left(j\right)}\right)=\frac{p_k{z}_k^{\left(j\right)}-q_k}{q_k{z}_k^{\left(j\right)}+r_k}$, $j=1,2,3,4$. The projections onto the

( $z_k$ )-plane of the sections of the pre-image by M of $w_j$ obtained when we keep $z_l=z_l^{\left(j\right)},l\ne k$ fixed are exactly the points $z_k^{\left(j\right)}$. Since (9) is a Möbius transformation, we have $\left[z_1^{\left(k\right)}\mathrm{,}{z}_2^{\left(k\right)}\mathrm{,}{z}_3^{\left(k\right)}\mathrm{,}{z}_4^{\left(k\right)}\right]=\left[w_1\mathrm{,}{w}_2\mathrm{,}{w}_3\mathrm{,}{w}_4\right]$, which proves the theorem.

Theorem 10. The cross-ratio $\left[z_1^{\left(k\right)}\mathrm{,}{z}_2^{\left(k\right)}\mathrm{,}{z}_3^{\left(k\right)}\mathrm{,}{z}_4^{\left(k\right)}\right]$ of four points in any ( $z_k$ )-plane is real if and only if for arbitrary $z_l\mathrm{,}l\ne k$, the points $w_j=M\left(z_j\right)$, where $z_j=\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{k-1}\mathrm{,}{z}_j^{\left(k\right)}\mathrm{,}{z}_{k+1}\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ lie on a circle in the (w)-plane.

Proof: It is known that cross-ratio of four points in the complex plane is real if and only if the four points lie on a circle in that plane (see [4], page 79). Let the plane be a ( $z_k$ )-plane and let $z_j^{\left(k\right)},j=1,2,3,4$ lie on a circle. Then $\left[z_1^{\left(k\right)}\mathrm{,}{z}_2^{\left(k\right)}\mathrm{,}{z}_3^{\left(k\right)}\mathrm{,}{z}_4^{\left(k\right)}\right]$ is real and $\left[z_1^{\left(k\right)}\mathrm{,}{z}_2^{\left(k\right)}\mathrm{,}{z}_3^{\left(k\right)}\mathrm{,}{z}_4^{\left(k\right)}\right]=\left[w_1\mathrm{,}{w}_2\mathrm{,}{w}_3\mathrm{,}{w}_4\right]$. The images of that circle by $M\left(z\right)$ for different choices of $z_l\mathrm{,}l\ne k$ are circles in the (w)-plane which contain the points $w_j$. Reciprocally, if $w_j$ lie on a circle in the (w)-plane, let us take the pre-image by $M\left(z\right)$ of that circle. Keeping $z_l\mathrm{,}l\ne k$ constant we obtain a section of that pre-image whose projection onto the ( $z_k$ )-plane is a circle containing the points $z_j^{\left(k\right)}$, therefore, the cross-ratio of these points must be real.

6. Conclusion

The purpose of this work was to extend to ${\overline{ℂ}}^m$ the geometric concepts pertinent to Möbius transformations in the plane. The introduction of a function $f_m\mathrm{<mo>\; :\; </mo>}{\overline{ℂ}}^m\to \overline{ℂ}$, which is a Möbius transformation in each one of the variables, when keeping the others constant allowed us to perform this task. The most remarkable achievement was the extension to ${ℂ}^m$ of the symmetry principle. The concept of the angle of two vectors in ${ℂ}^m$ has been also dealt with. However, just a few properties have been visited, so the potential for other developments is obvious. In particular, visualization in the style done in [6] might be possible having in view the fact that $f_m$ is a classic Möbius transformation in every one of its variables when others are kept fixed. For the same reason, characterizations as those made by Haruki and Rassias in [7] [8] [9] [10] are expected.

Acknowledgements

I thank Aneta Costin for her support with technical matters.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Ghisa, D. (2021) A Note on m-Möbius Transformations. Advances in Pure Mathematics, 11, 883-890. https://doi.org/10.4236/apm.2021.1111057
[2]	Cao-Huu, T. and Ghisa, D. (2021) Lie Groups Actions on Non Orientable n-Dimensional Complex Manifolds. Advances in Pure Mathematics, 11, 604-610. https://doi.org/10.4236/apm.2021.116039
[3]	Barza, I. and Ghisa, D. (2020) Lie Groups Actions on Non Orientable Klein Surfaces. In: Dobrev, V., Ed., Lie Theory and Its Applications in Physics, Springer, Singapore, Vol. 335, 421-428. https://doi.org/10.1007/978-981-15-7775-8_33
[4]	Ahlfors, L.V. (1979) Complex Analysis. McGraw-Hill, New York.
[5]	Needham, T. (1997) Visual Complex Analysis. Clarendon Press, Oxford.
[6]	Arnold, D.N. and Rogness, J. (2008) Möbius Transformations Revealed. Notices of the American Mathematical Society, 55, 1226-1231.
[7]	Haruki, H. and Rassias, T.M. (1994) A New Invariant Characteristic Property of Möbius Transformations from the Standpoint of Conformal Mapping. Journal of Mathematical Analysis and Applications, 181, 320-327. https://doi.org/10.1006/jmaa.1994.1024
[8]	Haruki, H. and Rassias, T.M. (1996) A New Characterization of Möbius Transformations by Use of Apollonius Points of Triangles. Journal of Mathematical Analysis and Applications, 197, 14-22. https://doi.org/10.1006/jmaa.1996.0002
[9]	Haruki, H. and Rassiss, T.M. (1998) A New Characterization of Möbius Transformations by Use of Apollonius Quadrilaterals. Proceedings of the American Mathematical Society, 126, 2857-2861. https://doi.org/10.1090/S0002-9939-98-04736-4
[10]	Haruki, H. and Rassias, T.M. (2000) A New Characterization of Möbius Transformations by Use of Apollonius Hexagons. Proceedings of the American Mathematical Society, 128, 2105-2109. https://doi.org/10.1090/S0002-9939-00-05246-1