Parallel real hypersurfaces in $\mathbb{S}^2\times\mathbb{S}^2$ and $\mathbb{H}^2\times\mathbb{H}^2$ with respect to the $k$-generalized Tanaka--Webster connection

In this paper, we first characterize real hypersurfaces of both the K\"ahler surfaces $\mathbb{S}^2\times\mathbb{S}^2$ and $\mathbb{H}^2\times\mathbb{H}^2$ such that their shape operators have identical covariant derivatives with respect to the Levi-Civita connection and the $k$-generalized Tanaka--Webster connection for a nonzero $k\in\mathbb{R}$. Then, amongst others, we classify all real hypersurfaces of both $\mathbb{S}^2\times\mathbb{S}^2$ and $\mathbb{H}^2\times\mathbb{H}^2$ whose shape operators are parallel with respect to the $k$-generalized Tanaka--Webster connection.