As a first application of the just-elaborated classical model of Möbius geometry we want to study conformally flat hypersurfaces in the conformal n-sphere. Here, we will consider the (regular) hypersurface as the primary object of interest and an enveloped (1-parameter) family of spheres as its “Gauss map,” in contrast to the way we considered channel hypersurfaces in most of Section 1.8.

Conformally flat hypersurfaces have been a topic of interest for some time. In [61], É. Cartan gave a complete local classification of conformally flat hypersurfaces) in Sn when n ≥ 5, that is, in the case where conformal flatness is detected by the vanishing of the Weyl tensor (cf., §P.5.1). Cartan's result was reproven several times; we will present a proof close to Cartan's original proof to show that any conformally flat hypersurface in dimension n ≥ 5 is (a piece of) a branched channel hypersurface, using the model for Möbius geometry developed in the previous chapter. Later, the global structure of compact conformally flat (submanifolds and) hypersurfaces was analyzed, showing that the conformal structure has, in general, no global flat representative (see [199]) deriving restrictions on the intrinsic structure (cf., [198], [222], [194]), and giving a complete geometric description of their shape (see [102]; cf., [173]).