Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE STRUCTURE JACOBI OPERATOR ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM

  • KI, U-HANG (DEPARTMENT OF MATHEMATICS, KYUNGPOOK NATIONAL UNIVERSITY) ;
  • KIM, SOO-JIN (DEPARTMENT OF MATHEMATICS, CHOSUN UNIVERSITY) ;
  • LEE, SEONG-BAEK (DEPARTMENT OF MATHEMATICS, CHOSUN UNIVERSITY)
  • Published : 2005.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let M be a real hypersurface with almost contact metric structure $(\phi,\;\xi,\;\eta,\;g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor S, then M is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of M is constant or $g(S\xi,\;\xi)$ is constant.

Keywords

References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in a complex hyperbolic space, J. Reine Agnew. Math. 395 (1989), 132-141
  2. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), 481-499 https://doi.org/10.2307/1998460
  3. J. T. Cho and U.-H. Ki, Real hypersurfaces of a complex projective space in terms of the Jacobi operators, Acta Math. Hungar. 80 (1998), 155-167 https://doi.org/10.1023/A:1006585128386
  4. J. T. Cho, Jacobi operators on real hypersurfaces of a complex projective space, Tsukuba J. Math. 22 (1998), 145-156 https://doi.org/10.21099/tkbjm/1496163476
  5. U.-H. Ki, Cyclic-parallel real hypersurfaces of a complex space form, Tsukuba J. Math. 12 (1988), 259-268 https://doi.org/10.21099/tkbjm/1496160647
  6. U.-H. Ki, H.-J. Kim, and A.-A. Lee, The Jacobi operator of real hypersurfaces in a complex space form, Comm. Korean Math. Soc. 13 (1998), 545-560
  7. U.-H. Ki, S.-J. Kim, and S.-B. Lee, Some characterizations of a real hypersurfaces of type A, Kyungpook Math. J. 31 (1991), 73-82
  8. U.-H. Ki, A. A. Lee, and S.-B. Lee, On real hypersurfaces of a complex space form in terms of Jacobi operators, Comm. Korean Math. Soc. 13 (1998), 317-336
  9. U.-H. Ki and H. Song, Jacobi operators on a semi-invariant submanifold of codimension 3 in a complex projective space, Nihonkai Math. J. 14 (2003), 1-16
  10. U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221
  11. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), 137-149 https://doi.org/10.2307/2000565
  12. M. Kimura and S. Maeda, Lie derivatives on real hypersurfaces in a complex projective space, Czechoslovak Math. J. 45 (1995), 135-148
  13. S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515-535 https://doi.org/10.2969/jmsj/03730515
  14. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geometriae Dedicata 20 (1986), 245-261 https://doi.org/10.1007/BF00164402
  15. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, in Tight and Taut submanifolds, Cambridge Univ. Press (1998(T.E. Cecil and S.S. Chern, eds.)), 233-305
  16. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364 https://doi.org/10.2307/1998631
  17. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506
  18. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan 27 (1975), 43-53, 507-516 https://doi.org/10.2969/jmsj/02710043
  19. K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Birkhauser, 1983

Cited by

  1. The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space vol.94, pp.1-2, 2009, https://doi.org/10.1007/s00022-009-0006-6
  2. ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS vol.32, pp.4, 2010, https://doi.org/10.5831/HMJ.2010.32.4.747
  3. Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition vol.2016, 2016, https://doi.org/10.1155/2016/3089298