Conformal Geometry: The Full Version

Abstract

In the previous chapter we have described how the full version of projective geometry can be related to a finite Cartan geometry based on the Lie groupoid of fibre morphisms of a bundle of projective spaces, the bundle $\mathrm {P}\mathcal {W}M\rightarrow M$, and how this is related to the infinitesimal geometry on $TM$ studied in Sect. 8.2. There is a similar relationship between infinitesimal conformal geometry on $TM$ (Sect. 8.4) and a finite geometry, but now the fibres of the corresponding bundle will be spheres rather than projective spaces. In this chapter we shall explain how the action of a suitable group gives rise to such a sphere as a homogeneous space, and how the corresponding Cartan geometry can be constructed.