Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM

  • Ki, U-Hang (The National Academy of Science) ;
  • Kim, In-Bae (Department of Mathematics, Hankuk University of Foreign Studies) ;
  • Lim, Dong-Ho (Department of Mathematics, Hankuk University of Foreign Studies)
  • Published : 2010.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let M be a real hypersurface with almost contact metric structure $(\phi,g,\xi,\eta)$ in a complex space form $M_n(c)$, $c\neq0$. In this paper we prove that if $R_{\xi}L_{\xi}g=0$ holds on M, then M is a Hopf hypersurface in $M_n(c)$, where $R_{\xi}$ and $L_{\xi}$ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field $\xi$ respectively. We characterize such Hopf hypersurfaces of $M_n(c)$.

Keywords

References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132–141. https://doi.org/10.1515/crll.1989.395.132
  2. 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), no. 1-2, 155–167. https://doi.org/10.1023/A:1006585128386
  3. U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207–221.
  4. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986), no. 2, 245–261. https://doi.org/10.1007/BF00164402
  5. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and taut submanifolds (Berkeley, CA, 1994), 233–305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997. https://doi.org/10.2277/0521620473
  6. 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
  7. M. Ortega, J. D. P´erez, and F. G. Santos, Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. Math. 36 (2006), no. 5, 1603–1613. https://doi.org/10.1216/rmjm/1181069385
  8. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506.

Cited by

  1. On characterizations of real hypersurface in complex space form with Codazzi type structure Lie operator vol.173, pp.3, 2014, https://doi.org/10.1007/s00605-013-0579-x
  2. On Characterizations of Hopf Hypersurfaces in a Nonflat Complex Space Form with Anti-commuting Operators vol.71, pp.1-2, 2017, https://doi.org/10.1007/s00025-016-0567-2
  3. Real hypersurfaces with Killing type operators in a nonflat complex space form 2017, https://doi.org/10.1007/s00022-017-0375-1