The main aim of this little book is to give a treatment of the rarely taught fact that the only conformal mappings in Euclidean space of dimension greater than \(2\) are those generated by similarities and inversions (reflections) in spheres. This result was proved in 1850 in dimension \(3\) by J. Liouville. Here, a conformal mapping is a transformation which preserves the size but not necessarily the sense of angles.
The text is at the advanced undergraduate level and is suitable for a topics course, seminar or independent study. It is not required that the reader is familiar with complex analysis or differential geometry. The material is organized in 7 chapters whose contents we now describe briefly.
Chapter 1. The basic objects of classical inversion theory in the plane are treated. There are given some applications such as Miquel’s theorem and Feuerbach’s theorem.
Chapter 2. The properties of Möbius transformations and the Poincaré models of hyperbolic geometry are discussed. Furthermore, there is included a paper by C. Carathéodory [Bull. Am. Math. Soc. 43, 573-579 (1937; Zbl 0017.22903)] with the result that any circle preserving transformation is necessarily a Möbius transformation in \(z\) or \(\overline{z}\), where not even the continuity of the transformation is assumed.
Chapter 3. At first, a review of advanced calculus in Euclidean \(n\)-space is given. Then conformal mappings are described analytically with the help of inner products.
Chapter 4. The author develops enough complex analysis to prove the abundance of conformal mappings in the plane.
Chapter 5. Here, an elementary proof of the main result in general dimension is given. This is due to R. Nevanlinna [On differential mappings. (Princeton Math. Ser. 24, 3-9, Princeton University Press) (1960; Zbl 0100.35701)] and works for \(C^4\) mappings. Furthermore, a theorem of Möbius on the characterization of continuous sphere preserving transformations in Euclidean \(n\)-space is proved.
Chapter 6. The author gives the standard or classical proof of Liouville’s theorem in dimension \(3\) which works for \(C^3\) mappings. This is not Liouville’s proof and requires some knowledge of differential geometry of surfaces in Euclidean space.
Chapter 7. The following question is posed and answered: Given a smooth closed convex curve in the plane, what is the set of points in the plane as centers of inversion for which the image of the given curve will again be a convex curve? The answer was given by D. E. Blair and J. B. Wilker [Kodai Math. J. 6, 186-192 (1983; Zbl 0524.53002)]. Furthermore, the corresponding result in Euclidean \(3\)-space is stated without proof.