Following Klein's Erlanger program [160], a geometry can be described in terms of a transformation group acting on an underlying space (see also [254]): Two geometric configurations are considered equivalent if one can be mapped onto the other by an element of the transformation group under consideration. Thus, we describe Möbius geometry as the geometry given by the Möbius group Möb(n) acting on the conformal n-sphere Sn. The goal of this chapter is to describe Möbius geometry as a subgeometry of projective geometry, as the classical geometers did; see [239], [86], and [88], or [161], where interesting historical remarks also are to be found. This description provides a linearization of Möbius geometry: Spheres will be described by certain linear subspaces and Möbius transformations by linear transformations.

After the description of this model of Möbius geometry, culminating in a “translation table” for the occurring geometric objects and relations (see Figure T.1), we will discuss the metric geometries (hyperbolic, Euclidean, and spherical geometry) as subgeometries of Möbius geometry — and give the announced proof of Liouville's theorem by making use of the subgeometry description of hyperbolic (resp. Euclidean) geometry.

Turning to differential geometry, we will introduce the concept of (hyper)sphere congruences and their envelopes, which is central to many constructions in Möbius differential geometry (cf., Figure T.2).