On real hypersurfaces of 𝕊²×𝕊²

In this paper, regarding the Riemannian product S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 of two unit 2 2 -spheres as a Kähler surface, we study its real hypersurfaces with typical geometric properties. First, we classify the real hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with isometric Reeb flow and then, by using a Simons’ type inequality, a characterization of these compact real hypersurfaces is provided. Next, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with constant product angle function. Finally, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with parallel Ricci tensor.