On some hypersurfaces of S2×S2 and H2×H2.

We first classify Hopf hypersurfaces of both S 2 × S 2 and H 2 × H 2 which 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 S 2 × S 2 and H 2 × H 2 . Then, we classify hypersurfaces of both S 2 × S 2 and H 2 × H 2 whose structure Jacobi operator is of Codazzi type. As its direct consequence, we obtain the classification of hypersurfaces in both S 2 × S 2 and H 2 × H 2 for 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. [ABSTRACT FROM AUTHOR]