Real hypersurfaces of S 2 × S 2 $\mathbb {S}^2\times \mathbb {S}^2$ and H 2 × H 2 $\mathbb {H}^2\times \mathbb {H}^2$ with parallel operators

On real hypersurafces of the Kähler surface $S^2\times S^2$ and $H^2\times H^2$, there are three typical operators: the shape operator, the structure Lie operator, and the contact Lie operator. In this paper, we study real hypersurfaces in $S^2\times S^2$ and $H^2\times H^2$ related to these operators. As the main results, we classify real hypersurfaces of both $S^2\times S^2$ and $H^2\times H^2$ for which one of the preceding three operators is either parallel or $\eta$-parallel.