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.
We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians
${\rm SU}_{2,m}/S(U_2{\cdot}U_m)$ with commuting conditions between the restricted normal Jacobi operator $\overline{R}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$)., Doo Hyun Hwang, Eunmi Pak, Changhwa Woo., and Obsahuje bibliografii
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, first we introduce a new notion of commuting condition that $\phi \phi _{1} A = A \phi _{1} \phi $ between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for real hypersurfaces in $G_2({\mathbb C}^{m+2})$. Suprisingly, real hypersurfaces of type $(A)$, that is, a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ satisfying the commuting condition. Finally we get a characterization of Type $(A)$ in terms of such commuting condition $\phi \phi _{1} A = A \phi _{1} \phi $.
Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type ${\rm (A)}$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for $M$ in $G_2({\mathbb C}^{m+2})$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi \phi _{1}$ induced by two structure tensors $\phi $ and $\phi _{1}$. That is, this commuting shape operator is given by $\phi \phi _{1} A = A \phi \phi _{1}$. Using this condition, we prove that $M$ is locally congruent to a tube of radius $r$ over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
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].