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 \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2\times {\mathbb {H}}^2.\) 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.