Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

U-Hang Ki, Hiroyuki Kurihara

Hachinohe National College of Technology, Aomori, Japan.
The National Academy of Science, Seoul, Korea.

DOI: 10.4236/apm.2013.32038                

Abstract

Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we prove that if the structure Jacobi operator R_ξ=(·,ξ) ξ is φ▽_ξξ-parallel and R_ξ commute with the shape operator, then M is a Hopf hypersurface. Further, if R_ξ is φ▽_ξξ-parallel and R_ξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrR_ξ is constant.

Keywords

Complex Space Form; Hopf Hypersurface; Structure Jacobi Operator; Shape Operator; Ricci Tensor

Conflicts of Interest

The authors declare no conflicts of interest.

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