Hypersurfaces of $$\mathbb {S}^2\times \mathbb {S}^2$$ with constant sectional curvature

Abstract

In this paper, we classify the hypersurfaces of \(\mathbb {S}^2\times \mathbb {S}^2\) with constant sectional curvature. We prove that the constant sectional curvature can only be \(\frac{1}{2}\). We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in \(\mathbb {S}^2\times \mathbb {S}^2\), and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous “sinh-Gordon equation” \( \left( \frac{\partial ^2}{\partial u^2}+\frac{\partial ^2}{\partial v^2}\right) h =-\tfrac{1}{\sqrt{2}}\sinh \left( \sqrt{2}h\right) \).