Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A NONFLAT COMPLEX SPACE FORM WHOSE STRUCTURE JACOBI OPERATOR IS ξ-PARALLEL

  • Received : 2009.05.20
  • Accepted : 2009.06.02
  • Published : 2009.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ of a nonflat complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},{\xi}){\xi}$ is ${\xi}$-parallel. In this paper, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex projective space $P_n{\mathbb{C}}$ or a complex hyperbolic space $H_n{\mathbb{C}}$ when $g({\nabla}_{\xi}{\xi},{\nabla}_{\xi}{\xi})$ is constant.

Keywords

References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces, J. Reine Angew. Math. 395 (1989), 132-141.
  2. J. Berndt And H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc. 359 (2007), 3425-3438. https://doi.org/10.1090/S0002-9947-07-04305-X
  3. T. E. Cecil And P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), 481-499.
  4. J. T. Cho and U-H. Ki, Real hypersurfaces of a complex projective space in terms of Jacobi operators, Acta Math. Hungar. 80 (1998), 155-167. https://doi.org/10.1023/A:1006585128386
  5. J. T. Cho and U-H. Ki, Real hypersurfaces in a complex space form with symmetric Jacobi Reeb flow, Canadian Math. Bull. 51 (2008), 359-371. https://doi.org/10.4153/CMB-2008-036-7
  6. U-H. Ki, H. Kurihara and R. Takagi, Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form, to appear in Tsukuba J. Mach.
  7. U-H. Ki and H. Liu, Some characterizations of real hypersurces of type (A) in a nonflat complex space form, Bull. Korean Math. Soc. 44 (2007), 152-172.
  8. U-H. Ki, J. D. Perez, F. G. Santos end Y. J. Suh, Real hypersurfaces in complex space forms with $\xi$-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc. 44 (2007), 307-326. https://doi.org/10.4134/JKMS.2007.44.2.307
  9. U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221.
  10. N.-G. Kim, U-H. Ki and H. Kurihara, Characterizations of real hypersurfaces of type A in complex space form used by the $\xi$-parallel structure Jacobi operator, Honam Math. J. 30 (2008), 535-550. https://doi.org/10.5831/HMJ.2008.30.3.535
  11. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom Dedicate 20 (1986), 245-261. https://doi.org/10.1007/BF00164402
  12. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364. https://doi.org/10.1090/S0002-9947-1975-0377787-X
  13. M. Ortega, J. D. Perez and F. G. Santos, Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. 36 (2006), 1603-1613. https://doi.org/10.1216/rmjm/1181069385
  14. J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective spaces whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 459-469.
  15. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19 (1973), 495-506
  16. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan 15 (1975), 43-53, 507-516.