Tsukuba Journal of Mathematics

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

U-Hang Ki, Hiroyuki Kurihara, and Ryoichi Takagi

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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 $M$.

Article information

Source
Tsukuba J. Math., Volume 33, Number 1 (2009), 39-56.

Dates
First available in Project Euclid: 1 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1251833206

Digital Object Identifier
doi:10.21099/tkbjm/1251833206

Mathematical Reviews number (MathSciNet)
MR2553837

Zentralblatt MATH identifier
1180.53021

Subjects
Primary: 53B20: Local Riemannian geometry 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords
complex space form real hypersurface structure Jacobi operator Ricci tensor

Citation

Ki, U-Hang; Kurihara, Hiroyuki; Takagi, Ryoichi. Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form. Tsukuba J. Math. 33 (2009), no. 1, 39--56. doi:10.21099/tkbjm/1251833206. https://projecteuclid.org/euclid.tkbjm/1251833206


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