On the Geometry of Hypersurfaces in 𝕊^2×𝕊^2

Urbano introduced C function for hypersurfaces in 𝕊^2×𝕊^2 and used it to classify isoparametric hypersurfaces. In this paper, we first show that for a hypersurface in 𝕊^2×𝕊^2 with |C|<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 |c_0|<1 . Note that all examples with constant C shown in Urbano’s paper have C≡ 0 or ± 1 .