Real hypersurfaces in \(\mathbb CP^2\) and \(\mathbb CH^2\) whose structure Jacobi operator is Lie \(\mathbb D\)-parallel (Q2856418)

The classification problem of real hypersurfaces in complex space forms is of great importance in differential geometry. The study of this problem was initiated by \textit{R. Takagi} [Osaka J. Math., 10, 495--506 (1973; Zbl 0274.53062)]. Many differential geometers have studied real hypersurfaces in terms of the structure Jacobi operator. One of the conditions which has been studied extensively is in terms of the Lie derivative of the structure Jacobi operator. In [\textit{T. A. Ivey} and \textit{P. J. Ryan}, Result. Math. 56, No. 1--4, 473--488 (2009; Zbl 1186.53067); \textit{J. de Dios Pérez} and \textit{F. G. Santos}, Publ. Math. 66, No. 3--4, 269--282 (2005; Zbl 1081.53051); \textit{J. de Dios Pérez} et al., Differ. Geom. Appl. 22, No. 2, 181--188 (2005; Zbl 1108.53026); Bull. Belg. Math. Soc. - Simon Stevin 13, No. 3, 459--469 (2006; Zbl 1130.53039)], results concerning the parallelness of the Lie derivative of the structure Jacobi operator of a real hypersurface with respect to \(\xi\) and to any vector field \(X\) were obtained in both complex projective space and complex hyperbolic space. In the present paper, the problem of existence of real hypersurfaces in \(\mathbb CP^2\) and \(\mathbb CH^2\) with Lie \(\mathbb D\)-parallel structure Jacobi operator is discussed. In fact, it is proved that there do not exist real hypersurfaces in \(\mathbb CP^2\) and \(\mathbb CH^2\) equipped with Lie \(\mathbb D\)-parallel structure Jacobi operator.