Research on conformally flat hypersurfaces

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We had a great progress in the study on generic conformally flat hypersurfaces of dimension 4. The most important result is that we fund several kinds of transformations between these hypersurfaces. This means that our study on these hypersurfaces made progress to the theory of integral systems for higher dimensional hypersurfaces. Existence of transformations is useful for the construction of new hypersurfaces and for the study toward complete classification of these hypersurfaces: A hypersurface has a one parameter associated family of hypersurfaces. Each hypersurface has a dual hypersurface. We gave an integral formula for the dual. Generally, the members of dual pair are not conformally equivalent, and both duals of a hypersurface and its change by a conformal transformation are also not. There is a one to one correspondence between dual pairs of hypersurfaces and Ribaucour pairs of Guichard nets. We gave an algebraic method for a Guichard net to construct its Ribaucour partner.