Real hypersurfaces of S2×S2$\mathbb {S}^2\times \mathbb {S}^2$ and H2×H2$\mathbb {H}^2\times \mathbb {H}^2$ with parallel operators

On real hypersurafces of the Kähler surface S2×S2$\mathbb {S}^2\times \mathbb {S}^2$ and H2×H2$\mathbb {H}^2\times \mathbb {H}^2$, 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 S2×S2$\mathbb {S}^2\times \mathbb {S}^2$ and H2×H2$\mathbb {H}^2\times \mathbb {H}^2$ related to these operators. As the main results, we classify real hypersurfaces of both S2×S2$\mathbb {S}^2\times \mathbb {S}^2$ and H2×H2$\mathbb {H}^2\times \mathbb {H}^2$ for which one of the preceding three operators is either parallel or η$\eta$‐parallel.