Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

DOI: 10.4236/apm.2013.32038   PDF   HTML   XML   3,400 Downloads   5,716 Views   Citations

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 Rξ=,ξ) ξ is φξξ-parallel and Rξ commute with the shape operator, then M is a Hopf hypersurface. Further, if Rξ is φξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrRξ is constant.

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U. Ki and H. Kurihara, "Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 264-276. doi: 10.4236/apm.2013.32038.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] 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.
[2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 19, 1973, pp. 495-506.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.

  
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