On some hypersurfaces of S2 x S2 and H2 x H2

We first classify Hopf hypersurfaces of both S-2 x S(2 )and H(2 )x( )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 x S(2 )and H(2 )x( )H(2). Then, we classify hypersurfaces of both S-2 x S(2 )and H(2 )x( )H(2 )whose structure Jacobi operator is of Codazzi type. As its direct consequence, we obtain the classification of hypersurfaces in both S-2 x S(2 )and H(2 )x( )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.