A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e., if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then

$$\cos (Tx(t_0),Ty(t_0)= \cos (x(t_0),y(t_0)= \frac{x'(t_0)\cdot y'(t_0)}{|x'(t_0)|| y'(t_0)|}.$$

A typical example of conformal mapping is the inversion $I:\Bbb R^n \to \Bbb R^n$ $I(x)= \frac{x}{|x|^2}$, with the convention that $I(0)=\infty $ and $I(\infty)=0$.

Trivial examples are rigid motions i.e., a combination of orthogonal group, Scallings or homothety and/or translations.

I am barely looking for a proof or a reference for the following Theorem:

Theorem: Every conformal mapping is the composite of finely many rigid motions and the Inversion mapping.

There are several book complex analysis dealing with the case $n=2$ on the complex plane.

But I haven't seen any for the higher dimensional situation.

A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e., if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then

$$\cos (Tx(t_0),Ty(t_0))= \cos (x(t_0),y(t_0))= \frac{x'(t_0)\cdot y'(t_0)}{|x'(t_0)|| y'(t_0)|}.$$

A typical example of conformal mapping is the inversion $I:\Bbb R^n \to \Bbb R^n$ $I(x)= \frac{x}{|x|^2}$, with the convention that $I(0)=\infty $ and $I(\infty)=0$.

Trivial examples are rigid motions i.e., a combination of orthogonal group, Scallings or homothety and/or translations.

I am barely looking for a proof or a reference for the following Theorem:

Theorem: Every conformal mapping is the composite of finely many rigid motions and the Inversion mapping.

There are several book complex analysis dealing with the case $n=2$ on the complex plane.

But I haven't seen any for the higher dimensional situation.