On the Geometry of Hypersurfaces in S-2 x S-2

Urbano introduced C function for hypersurfaces in S-2 x S-2 and used it to classify isoparametric hypersurfaces. In this paper, we first show that for a hypersurface in S-2 x S-2 with vertical bar C vertical bar < 1, the first fundamental form and Urbano's C function determines its second fundamental form. Moreover, we derive examples of hypersurfaces with C c(0) for any vertical bar c(0)vertical bar < 1. Note that all examples with constant C shown in Urbano's paper have C 0 or +/- 1.