The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form

Abstract

In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M2(c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.

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D. Lim, W. Sohn and H. Song, "The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 254-258. doi: 10.4236/apm.2013.32036.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] U.-H. Ki and Y. J. Suh, “On Real Hypersurfaces of a Complex Space Form,” Mathematical Journal of Okayama University, Vol. 32, 1990, pp. 207-221.
[2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 10, 1973, pp. 495-506.
[3] M. Kimura, “Real Hypersurfaces and Complex Submanifolds in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 296, 1986, pp. 137-149. doi:10.1090/S0002-9947-1986-0837803-2
[4] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Space,” Journal Für Die Reine und Angewandte Mathematik, Vol. 1989, No. 395, 1989, pp. 132-141.
[5] M. Okumura, “On Some Real Hypersurfaces of a Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 212, 1975, pp. 355-364. doi:10.1090/S0002-9947-1975-0377787-X
[6] S. Montiel and A. Romero, “On Some Real Hypersurfaces of a Complex Hyperbolic Space,” Geometriae Dedicata, Vol. 20, No. 2, 1986, pp. 245-261 doi:10.1007/BF00164402
[7] S. Maeda and T. Adachi, “Integral Curves of Characteristic Vector Fields of Real Hypersurfaces in Nonflat Complex Space Forms,” Geometriae Dedicata, Vol. 123, No. 1, 2006, pp. 65-72. doi:10.1007/s10711-006-9100-1
[8] W. H. Sohn, “Characterizations of Real Hypersurfaces of Complex Space Forms in Terms of Ricci Operators,” Bulletin of the Korean Mathematical Society, Vol. 44, No. 2007, pp. 195-202. doi:10.4134/BKMS.2007.44.1.195
[9] S. H. Kon and T. H. Loo, “Real Hypersurfaces of a Complex Space Form with η-Parallel Shape Operator,” Mathematische Zeitschrift, Vol. 269, No. 1-2, 2011, pp. 47-58. doi:10.1007/s00209-010-0715-4
[10] M. Kimura and S. Maeda, “On Real Hypersurfaces of a Complex Projective Space,” Mathematische Zeitschrift, Vol. 202, No. 3, 1989, pp. 299-311. doi:10.1007/BF01159962
[11] Y. J. Suh, “On Real Hypersurfaces of a Complex Space Form with η-Parallel Ricci Tensor,” Tsukuba Journal of Mathematics, Vol. 14, 1990, pp. 27-37.
[12] I.-B. Kim, K. H. Kim and W. H. Sohn, “Characterizations of Real Hypersurfaces in a Complex Space Form,” Canadian Mathematical Bulletin, Vol. 50, 2007, pp. 97-104. doi:10.4153/CMB-2007-009-5

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