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

We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2 \times \mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1 (r) × \mathbb{S}^2, r \in (0, 1]$, 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.

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Research partially supported by a MINECO-FEDER grant no. MTM2014-52368-P.

Received 24 June 2016

Accepted 4 June 2017

Published 12 December 2019