The structure Jacobi operator and the shape operator of real hypersurfaces in CP^{2} and CH^{2}

TL;DR

This paper investigates the properties of real hypersurfaces in complex projective and hyperbolic spaces, proving the non-existence of hypersurfaces with certain structure Jacobi and shape operator conditions.

Contribution

It establishes the non-existence of real hypersurfaces in CP^{2} and CH^{2} satisfying specific Lie derivative conditions on their structure Jacobi and shape operators.

Findings

Real hypersurfaces with structure Jacobi operator satisfying the condition do not exist.

Real hypersurfaces with shape operator satisfying the condition do not exist.

Abstract

This paper presents two results conserning real hypersurfaces in CP^{2} and CH^{2}. More precisely, it is proved that real hypersurfaces equipped with structure Jacobi operator satisfying condition $\mathcal{L}_{X}l=\nabla_{X}l$, where \emph{X} is a vector field orthogonal to structure vector field $\xi$, do not exist. Additional real hypersurfaces equipped with shape operator \emph{A} satisfying relation $\mathcal{L}_{X}A=\nabla_{X}A$, where \emph{X} is a vector field orthogonal to $\xi$, do not exist.