Research ArticleOpen AccessGoogle Scholar indexed
Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms
The National Academy of Science, Seoul, Korea
Hachinohe National College of Technology, Aomori, Japan
- 1 The National Academy of Science, Seoul, Korea
- 2 Hachinohe National College of Technology, Aomori, Japan
Advances in Pure Mathematics·Volume 03 (2013)·Pages 264–276·Published 5 March 2013·DOI10.4236/apm.2013.32038
Copy link · social · email
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 ▽ ξ ξ -parallel and R ξ commute with the shape operator, then M is a Hopf hypersurface. Further, if R ξ is φ ▽ ξ ξ -parallel and R ξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that Tr R ξ is constant.
KeywordsComplex Space FormHopf HypersurfaceStructure Jacobi OperatorShape OperatorRicci Tensor
- T. E. Cecil and P. J. Ryan, “Focal Sets and Real Hypersurfaces in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 269, No. 2, 1982, pp. 481-499.
- R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 19, 1973, pp. 495-506.
- R. Takagi, “Real Hypersurfaces in a Complex Projective Space with Constant Principal Curvatures I, II,” Journal of the Mathematical Society of Japan, Vol. 15, No. 43-53, 1975, pp. 507-516.
- M. Kimura, “Real Hypersurfaces and Complex Submanifolds in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 296, No. 1, 1986, pp. 137-149.
- J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperblic Spaces,” Journal für die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132-141.
- J. T. Cho and U-H. Ki, “Jacobi Operators on Real Hypersurfaces of a Complex Projective Space,” Tsukuba Journal of Mathematics, Vol. 22, 1988, pp. 145-156.
- J. T. Cho and U-H. Ki, “Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Jacobi Operator,” Canadian Mathematical Bulletin, Vol. 51, No. 3, 2008, pp. 359-371.
- U-H. Ki, H. Kurihara, S. Nagai and R. Takagi, “Characterizations of Real Hypersurfaces of Type A in a Complex Space Form in Terms of the Structure Jacobi Opera- tor,” Mathematics Journal of Toyama University, Vol. 32, 2009, pp. 5-23.
- J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Nonflat Complex Space Forms with Commuting Structure Jacobi Operator,” Houston Journal of Mathematics, Vol. 33, 2007, pp. 1005-1009.
- M. Ortega, J. D. Pérez and F. G. Santos, “Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms,” Rocky Mountain Journal of Mathematics, Vol. 36, No. 5, 2006, pp. 1603-1613.
- J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Spaces Whose Structure Jacobi Operator is D-Parallel,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 13, No. 3, 2006, pp. 459-469.
- U.-H. Ki, H. Kurihara and R. Takagi, “Jacobi Operators along the Structure Flow on Real Hypersurfaces in a Nonflat Complex Space Form,” Mathematics Journal of Toyama University, Vol. 33, 2009, pp. 39-56.
- U.-H. Ki and H. Kurihara, “Real Hypersurfaces and ξ-Parallel Structure Jacobi Operators in Complex Space Forms,” Journal of Korean Academy Sciences, Sciences Series, Vol. 48, 2009, pp. 53-78.