On Real Hypersurfaces in \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2\times {\mathbb {H}}^2\) with Parallel Normal Jacobi Operator

In this paper, we study real hypersurfaces of the two Kähler surfaces \({\mathbb {S}}^2\times {\mathbb {S}}^2\) ${\mathbb{S}}^2\times {\mathbb{S}}^2$ and \({\mathbb {H}}^2\times {\mathbb {H}}^2.\) ${\mathbb{H}}^2\times {\mathbb{H}}^2.$ As the main results, amongst others, we classify all such hypersurfaces whose normal Jacobi operators are parallel with respect to either the Levi-Civita connection or the k-generalized Tanaka–Webster connection.