ON CONFORMALLY FLAT HYPERSURFACES, CURVED FLATS AND CYCLIC SYSTEMS

It will be shown that suitable ''Gauss maps'' associated to a conforma lly flat hypersurface in S-n+1 (n greater than or equal to 3) yield no rmal congruences of circles having a whole 1-parameter family of confo rmally flat orthogonal hypersurfaces. However such a ''cyclic system'' is not uniquely associated to a conformally flat hypersurface. The ke y idea is to show that these Gauss maps are ''curved flats'' in a pseu do Riemannian symmetric space. Additionally, in this context some char acterizations of 3-dimensional conformally flat hypersurfaces arise wi th a new flavour. The curved flat approach allows us to handle conform ally flat hypersurfaces in the context of integrable system theory.