Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SOME CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE (A) IN A NONFLAT COMPLEX SPACE FORM

  • Ki, U-Hang (The National Academy of Sciences) ;
  • Liu, Hui-Li (Department of Mathematics Northeastern University)
  • Published : 2007.02.28
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Abstract

In this paper, we prove that if the structure Jacobi operator $R_{\xi}-parallel\;and\;R_{\xi}$ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form $M_{n}(C)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_{n}(C)$.

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References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, 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), no. 2, 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), no. 1-2, 155-167 https://doi.org/10.1023/A:1006585128386
  4. J. T. Cho and U-H Ki, Real hypersurfaces in a complex space form with the symmetric Reeb flow, preprint
  5. U-H. Ki, J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex space forms with $\xi$-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc. 44 (2007), no. 2, 307-326 https://doi.org/10.4134/JKMS.2007.44.2.307
  6. U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221
  7. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149 https://doi.org/10.2307/2000565
  8. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geometriae Dedicata 20 (1986), no. 2, 245-261 https://doi.org/10.1007/BF00164402
  9. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms. (English summary) Tight and taut submanifolds (Berkeley, CA, 1994), 233-305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997
  10. 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
  11. M. Ortega, J. D. Perez, and F. G. Santos, Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, to appear in Rocky Mountain J. Math
  12. J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\xi$-parallel, Diff. Geom. and its Appl. 22 (2005), no. 2, 181-188 https://doi.org/10.1016/j.difgeo.2004.10.005
  13. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506

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  1. Real Hypersurfaces in <i>CP<sup>2</sup></i> and <i>CH<sup>2</sup></i> Equipped With Structure Jacobi Operator Satisfying L<sub>ξ</sub>l =▽<sub>ξ</sub>l vol.02, pp.01, 2012, https://doi.org/10.4236/apm.2012.21001