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.


4 Answers 4


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.


[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, Zbl 0751.53001

[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, Zbl 0342.30015.

[3] Yuriĭ G. Reshetnyak, "Liouville’s theorem on conformal mappings for minimal regularity assumptions", (English, translated from the Russian), Siberian Mathematical Journal 8 (1967), pp. 631-634 (1968), MR0218544, Zbl 0167.36102.


See the following Wikipedia page: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)


Theorem (Liouville). If $\Omega\subset\mathbb{R}^n$, $n\geq 3$ is open and $f:\Omega\to\mathbb{R}^n$ is conformal, then $f$ is a Mobius transformation.

While the theorem is true for $f\in C^1$, there is no easy proof in that case. Standard proofs assume that $f\in C^3$ or even $f\in C^4$. The classical and well know proof due to Nevanlinna can be found here (see page 265):



Have you tried the first chapter of Riccardo Benedetti Carlo Petronio, Lectures on Hyperbolic Geometry? It contains a proof of Liouville's theorem from which you can easily deduce the result you are looking for.


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