Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form II
Bull. Korean Math. Soc. 2011 Vol. 48, No. 6, 1315-1327
https://doi.org/10.4134/BKMS.2011.48.6.1315
Published online November 1, 2011
U-Hang Ki and Hiroyuki Kurihara
Kyungpook National University, Hachinohe National College of Technology
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 $R_{\xi}\phi S=R_{\xi} S\phi$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.
Keywords : complex space form, real hypersurface, structure Jacobi operator, Ricci tensor
MSC numbers : Primary 53B20, 53C15, 53C25
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