The structure Jacobi operator and the shape operator of real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$

Abstract

In this paper two new results concerning real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$ are presented. The first result is the non-existence of real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$, whose structure Jacobi operator satisfies the relation $\mathcal {L}_{X}l=\nabla _{X}l$, for any vector field $X$ in the holomorphic distribution $\mathbb {D}$, i.e. $X$ is orthogonal to $\xi$. The second result concerns the non-existence of real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$, whose shape operator satisfies the relation $\mathcal {L}_{X}A=\nabla _{X}A$, for any vector field $X$ in the holomorphic distribution $\mathbb {D}$, i.e. $X$ is orthogonal to $\xi$.