We introduce the new notion of pseudo-$\mathbb D $-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.
This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m(4\epsilon )$. In the second, we give a complete classification of real hypersurfaces in $M_m(4\epsilon )$ which satisfy the above geometric facts.
In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form $M_m(c)$, $c\ne 0$ as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].