On hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$

Abstract

We classify the homogeneous and isoparametric hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$. In the classification, besides the hypersurfaces ${\mathbb{S}}^1(r)\times {\mathbb{S}}^2,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$ with at most two constant principal curvatures and, under certain conditions, with three constant principal curvatures.

32 pages