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