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.