Ribaucour pairs corresponding to dual pairs of conformally flat hypersurfaces

Abstract

We discuss generic conformally flat hypersurfaces of dimension 3 in Euclidean space and, in particular, the duality of such hypersurfaces. There exists a one-to-one correspondence between dual pairs of conformally flat hypersurfaces and Ribaucour pairs of their canonical Guichard nets. These Ribaucour pairs encode important geometric information for dual pairs of hypersurfaces: the Ribaucour pairs determine the geometry of dual pairs of hypersurfaces intrinsically and also extrinsically. In this paper, a method to construct Ribaucour pairs of canonical Guichard nets directly from one Guichard net is proposed.