Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM USED BY THE ζ-PARALLEL STRUCTURE JACOBI OPERATOR

  • Kim, Nam-Gil (Department of Mathematics, Chosun University) ;
  • Ki, U-Hang (Department of Mathematics, Kyungpook National University) ;
  • Kurihara, Hiroyuki (Department of Computer and Media Science, Saitama Junior College)
  • Received : 2008.07.07
  • Accepted : 2008.09.01
  • Published : 2008.09.25
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Abstract

Let M be a real hypersurface of a complex space form with almost contact metric structure $({\phi},{\xi},{\eta},g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},{\xi}){\xi}$ is ${\xi}$-parallel. In particular, 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 and not equal to -c/24 on M, where c is a constant holomorphic sectional curvature of a complex space form.

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References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperblic spaces, J. Reine Angew. 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.
  3. 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
  4. J. T. Cho and U-H. Ki, 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. J. T. Cho and U-H. Ki, Real hypersurfaces in complex space form with Reeb flow symmetric structure Jacobi operetor, to appear in Canadian Math. Bull.
  6. U-H. Ki, Real hypersurfaces with pararell Ricci tensor of a complex space form, Tsukuba J. Math. 13 (1989) 73-81. https://doi.org/10.21099/tkbjm/1496161007
  7. U-H. Ki and H. Liu, Some characterizations of real hypersurfaces of type (A) in a nonflat complex space form, Bull. Korean. Math. Soc. 44 (2007) 152-157.
  8. U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math J. Okayama Univ. 32 (1990) 207-221.
  9. U. K. Kim, Nonexistence of Ricci-parallel real hypersurfaces in $P_2{\mathbb{C}}$ or $H_2{\mathbb{C}}$, Bull. Korean. Math. Soc. 41 (2004) 699-708. https://doi.org/10.4134/BKMS.2004.41.4.699
  10. M. Kimura, Real hypersurfaces and complex submanifold.s in complex projective space, Trans. Amer. Math. Soc. 296 (1986) 137-149. https://doi.org/10.1090/S0002-9947-1986-0837803-2
  11. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperblic space, Geom Dedicata 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, Rockey 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 of complex projective space 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,ll, J. Math. Soc. Japan 15 (1975) 43-53, 507-516.

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  1. CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A NONFLAT COMPLEX SPACE FORM WHOSE STRUCTURE JACOBI OPERATOR IS ξ-PARALLEL vol.31, pp.2, 2009, https://doi.org/10.5831/HMJ.2009.31.2.185