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

TL;DR

This paper classifies various types of homogeneous and isoparametric hypersurfaces in the product space , identifying specific families with constant principal curvatures and their geometric properties.

Contribution

It provides a comprehensive classification of homogeneous and isoparametric hypersurfaces in , including new families with three constant principal curvatures.

Findings

Identified hypersurfaces (r) with r in (0,1]

Discovered a family with three different constant principal curvatures and zero Gauss-Kronecker curvature

Classified hypersurfaces with at most two constant principal curvatures

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.