Lie Sphere Transformations

In this chapter, we study the sphere geometries of Lie, Möbius and Laguerre from the point of view of Klein’s Erlangen Program. In each case, we determine the group of transformations which preserve the fundamental geometric properties of the space. All of these groups are quotient groups or subgroups of some orthogonal group determined by an indefinite scalar product on a real vector space. As a result, the theorem of Cartan and Dieudonné, proven in Section 3.2, implies that each of these groups is generated by inversions. In Section 3.3, we give a geometric description of Möbius inversions. This is followed by a treatment of Laguerre geometry in Section 3.4. Finally, in Section 3.5, we show that the Lie sphere group is generated by the union of the groups of Möbius and Laguerre. There we also describe the place of Euclidean, spherical and hyperbolic metric geometries within the context of these more general geometries.