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.



Looking for a reference on conformal mapping on $\Bbb R^n$
My two cents: a proof for $n=3$ is given explicitly by Dubrovin, Fomenko and Novikov in [1], §15.2 pp. 138-142. The authors explain also how to extend their proof to the case $n>3$ and leave the details as an exercise.


**An addendum**


The answer by @Piotr Hajlasz triggered my curiosity and pushed me to go a little bit beyond reference [1], which requires a $C^4$ regularity on the conformal map considered ([1], §15.2 p. 138).  

According to Caraman ([2] section 3, chapter 2, p. 358), the proof of Liouville's theorem requiring a minimal regularity on the mapping was given first by Reshetnyak in [3]. Reshetnyak assumes the mapping to be of class $W^1\_n$: while the paper is short, the offered proof is highly non trivial.


**Reference**


[1] Boris A. Dubrovin, Analtoly T. Fomenko and Sergey P. Novikov, *Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, and fields*, translated by Robert G. Burns. 2nd ed. (English) Graduate Texts in Mathematics, 93, Berlin-Heidelberg-New York: Springer-Verlag, pp. xv+468 (1992), [MR1138462](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1138462), [Zbl 0751.53001](https://zbmath.org/?q=an%3A0751.53001)


[2] Petru Caraman, *$n$-Dimensional Quasiconformal (QCF) Mappings*, revised, enlarged and translated from the Romanian by the Author (English), Tunbridge Wells, Kent: Abacus Press, pp. 551 (1974), ISBN 0-85626-005-3, [MR0357782](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0357782), [Zbl 0342.30015](https://zbmath.org/?q=an%3A0342.30015).


[3] Yuriĭ G. Reshetnyak, "[Liouville’s theorem on conformal mappings for minimal regularity assumptions](https://doi.org/10.1007/BF02196483)", (English, translated from the Russian), Siberian Mathematical Journal 8 (1967), pp. 631-634 (1968), [MR0218544](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0218544), [Zbl 0167.36102](https://zbmath.org/?q=an%3A0167.36102).


