Möbius transformations preserving the unit ball

In the Euclidean space R^n, composition of a finite number of reflections in hyperplanes and spheres is called a Möbius transformation. These transformations are conformal (angle preserving) and when n ≥ 3 there are no conformal transformations other than the Möbius transformations. In this thesis we first study the properties of reflections in hyperplanes and spheres. We show that these reflections are conformal and preserve symmetry. We show that Euclidean isometries and, more generally, the similarities can be written as compositions of reflections. The main purpose of this thesis is characterizing the Möbius transformations that preserve the unit ball. For this we first determine the reflections in hyperplanes and spheres that preserve the unit ball, then define the canonical Möbius transformations and study their properties. As the main result of the thesis we show that every Möbius transformation that preserve the unit ball can be written as a composition of a canonical transformation and an orthogonal transformation. Finally, on the unit ball we define the pseudo-hyperbolic metric and show that this metric is Möbius invariant.