Hypersurfaces of \(\mathbb{S}^2\times\mathbb{S}^2\) with constant sectional curvature (Q6568732)

This paper classifies hypersurfaces in \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) with constant sectional curvature. Recall that there is a natural product structure on \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) given by \(P(X_{1},X_{2})=(X_{1}, -X_{2})\) and the geometry of hypersurfaces in \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) is related to the product angle function \(C=g(PN,N)\), where \(g\) is the standard metric on \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) and \(N\) is a unit normal vector field on the hypersurface.\N\N\textit{F. Urbano} [Commun. Anal. Geom. 27, No. 6, 1381--1416 (2019; Zbl 1429.53078)] found remarkable classes of hypersurfaces in \(\mathbb{S}^{2}\times \mathbb{S}^{2}\), all with product angle function equal to \(0\). Let \(\langle \cdot, \cdot \rangle\) denote the standard metric on \(\mathbb{S}^{2}\), then\N\begin{itemize}\N\item \(M_{t}=\{(p,q)\in \mathbb{S}^{2}\times \mathbb{S}^{2} \ | \ \langle p,q\rangle =t\}\) is a homogeneous isoparametic hypersurface with three constant principal curvatures (hence with constant mean curvature), among which, only \(M_{0}\) is minimal.\N\item \(M_{a,b}=\{(p,q)\in \mathbb{S}^{2}\times \mathbb{S}^{2} \ | \ \langle p,a\rangle +\langle q,b \rangle =0\}\) is minimal with non-constant scalar curvature;\N\item \(\widehat{M}_{a,b}=\{(p,q)\in \mathbb{S}^{2}\times \mathbb{S}^{2} \ | \ \langle p,a\rangle^2 +\langle q,b \rangle^2 =1\}\) has constant sectional curvature \(1/2\) with non-constant mean curvature.\N\end{itemize}\N\NThese surfaces play a fundamental role in the authors' classification. Indeed they prove that if \(M\) is a hypersurface of \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) with constant sectional curvature \(\kappa\), then \(\kappa=1/2\), the product angle function \(C = 0\) and \(M\) is congruent to an open part of a parallel hypersurface of a minimal hypersurface in \(\mathbb{S}^{2}\times \mathbb{S}^{2}\) with \(C = 0\).