Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form

TL;DR

This paper investigates real hypersurfaces in complex space forms with a focus on the structure Jacobi operator being b1$-parallel, characterizing certain homogeneous hypersurfaces under specific conditions.

Contribution

It establishes a characterization of homogeneous real hypersurfaces of type A in complex space forms via the b1$-parallelism of the structure Jacobi operator and a commutation condition.

Findings

The structure Jacobi operator is b1$-parallel on certain hypersurfaces.

Homogeneous real hypersurfaces of type A are characterized by b1$-parallelism and commutation conditions.

The results apply to hypersurfaces in both complex projective and hyperbolic spaces.

Abstract

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(\phi, \xi, \eta, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_\xi=R(\cdot,\xi)\xi$ is $\xi$-parallel. In particular, we prove that the condition $\nabla_{\xi} R_{\xi}=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_{\xi} \phi S=S \phi R_{\xi}$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.