In this paper, we study real hypersurfaces of the product manifold \(M_{\kappa _1}^2\times M_{\kappa _2}^2,\) where \(M_{\kappa _1}^2\) and \(M_{\kappa _2}^2\) are 2-dimensional real space forms with constant curvature \(\kappa _1, \kappa _2\in \{-1,0,1\}\) and \(\kappa _1\ne \kappa _2.\) We classify real hypersurfaces whose normal Jacobi operators with respect to the Levi-Civita connection or the k-generalized Tanaka–Webster connection satisfy one of the four conditions: (1) parallel, (2) recurrent, (3) Codazzi type, (4) Killing type. We also classify Hopf real hypersurfaces with constant product angle function as well as with constant Reeb function.
