# 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
• 88 4

#### 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 $M_{n}(C)$.

#### Keywords

real hypersurface;structure Jacobi operator;Ricci tensor;Hopf hypersurface

#### 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. 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
7. 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
8. 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
9. 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
10. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506
11. U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221
12. 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
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, to appear in Rocky Mountain J. Math

#### Cited by

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