On some hypersurfaces of S2×S2and H2×H2

We first classify Hopf hypersurfaces of both S2×S2and H2×H2which satisfy one of the three conditions: (1) constant mean curvature, (2) constant scalar curvature, (3) constant squared norm of the shape operator. It follows that these three conditions are equivalent for a Hopf hypersurface of both S2×S2and H2×H2. Then, we classify hypersurfaces of both S2×S2and H2×H2whose structure Jacobi operator is of Codazzi type. As its direct consequence, we obtain the classification of hypersurfaces in both S2×S2and H2×H2for which the structure Jacobi operator satisfies one of the six conditions: (1) vanishing, (2) parallel, (3) recurrent, (4) semi-parallel, (5) Lie parallel, (6) Killing type.