Real Hypersurfaces in *CP*^{2} and *CH*^{2} Equipped With Structure Jacobi Operator Satisfying L_{ξ}l =▽_{ξ}l

ABSTRACT

Recently in [1], Perez and Santos classified real hypersurfaces in complex projective space CP^{n} for n ≥ 3, whose Lie derivative of structure Jacobi operator in the direction of the structure vector field coincides with the covariant derivative of it in the same direction. The present paper completes the investigation of this problem studying the case n = 2 in both complex projective and hyperbolic spaces.

Recently in [1], Perez and Santos classified real hypersurfaces in complex projective space CP

Cite this paper

K. Panagiotidou and P. Xenos, "Real Hypersurfaces in*CP*^{2} and *CH*^{2} Equipped With Structure Jacobi Operator Satisfying L_{ξ}l =▽_{ξ}l," *Advances in Pure Mathematics*, Vol. 2 No. 1, 2012, pp. 1-5. doi: 10.4236/apm.2012.21001.

K. Panagiotidou and P. Xenos, "Real Hypersurfaces in

References

[1] J. D. Perez and F. G. Santos, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Satisfies ,” Rocky Mountain Journal of Mathematics, Vol. 39, No. 4, 2009, pp. 1293-1301. doi:10.1216/RMJ-2009-39-4-1293

[2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 10, 1973, pp. 495-506.

[3] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Space,” Journal fur die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132-141. doi:10.1515/crll.1989.395.132

[4] 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

[5] 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

[6] U.-H. Ki and H. Liu, “Some Characterizations of Real Hypersurfaces of Type (A) in Nonflat Complex Space Form,” KMSBulletin of the Korean Mathematical Society, Vol. 44, No. 1, 2007, pp. 157-172. doi:10.4134/BKMS.2007.44.1.157

[7] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is Lie -Parallel,” Differential Geometry and Its Applications, Vol. 22, No. 2, 2005, pp. 181-188. doi:10.1016/j.difgeo.2004.10.005

[8] T. A. Ivey and P. J. Ryan, “The Structure Jacobi Operator for Real Hypersurfaces in and ,” Results in Mathematics, Vol. 56, No. 1-4, 2009, pp. 473-488. doi:10.1007/s00025-009-0380-2

[9] M. Ortega, J. D. Perez and F. G. Santos, “Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms,” Rocky Mountain Journal of Mathematics, Vol. 36, No. 5, 2006, pp. 1603-1613. doi:10.1216/rmjm/1181069385

[10] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is D-Parallel,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 13, No. 3, 2006, pp. 459-469.

[11] U.-H. Ki, J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Space Forms with ξ-Parallel Ricci Tensor and Structure Jacobi Operator,” Journal of the Korean Mathematical Society, Vol. 44, No. 2, 2007, pp. 307-326. doi:10.4134/JKMS.2007.44.2.307

[12] N.-G. Kim and U.-H. Ki, “ξ-Parallel Structure Jacobi Operators of Real Hypersurfaces in a Nonflat Complex Space Form,” Honam Mathematical Journal, Vol. 28, No. 4, 2006, pp. 573-589.

[13] D. E. Blair, “Riemannian Geometry of Contact and Symplectic Manifolds,” Progress in Mathematics 203, Birkh?user, Boston, 2002.

[14] R. Niebergall and P. J. Ryan, “Real Hypersurfaces in Complex Space Forms,” Publications of the Research Institute for Mathematical Sciences, Vol. 32, 1997, pp. 233- 305.

[15] T. Cecil and P. J. Ryan, “Focal Sets and Real Hypersurfaces in Complex Projective Spaces,” Transactions of the American Mathematical Society, Vol. 269, No. 2, 1982, pp. 481-499.

[1] J. D. Perez and F. G. Santos, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Satisfies ,” Rocky Mountain Journal of Mathematics, Vol. 39, No. 4, 2009, pp. 1293-1301. doi:10.1216/RMJ-2009-39-4-1293

[2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 10, 1973, pp. 495-506.

[3] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Space,” Journal fur die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132-141. doi:10.1515/crll.1989.395.132

[4] 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

[5] 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

[6] U.-H. Ki and H. Liu, “Some Characterizations of Real Hypersurfaces of Type (A) in Nonflat Complex Space Form,” KMSBulletin of the Korean Mathematical Society, Vol. 44, No. 1, 2007, pp. 157-172. doi:10.4134/BKMS.2007.44.1.157

[7] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is Lie -Parallel,” Differential Geometry and Its Applications, Vol. 22, No. 2, 2005, pp. 181-188. doi:10.1016/j.difgeo.2004.10.005

[8] T. A. Ivey and P. J. Ryan, “The Structure Jacobi Operator for Real Hypersurfaces in and ,” Results in Mathematics, Vol. 56, No. 1-4, 2009, pp. 473-488. doi:10.1007/s00025-009-0380-2

[9] M. Ortega, J. D. Perez and F. G. Santos, “Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms,” Rocky Mountain Journal of Mathematics, Vol. 36, No. 5, 2006, pp. 1603-1613. doi:10.1216/rmjm/1181069385

[10] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is D-Parallel,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 13, No. 3, 2006, pp. 459-469.

[11] U.-H. Ki, J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Space Forms with ξ-Parallel Ricci Tensor and Structure Jacobi Operator,” Journal of the Korean Mathematical Society, Vol. 44, No. 2, 2007, pp. 307-326. doi:10.4134/JKMS.2007.44.2.307

[12] N.-G. Kim and U.-H. Ki, “ξ-Parallel Structure Jacobi Operators of Real Hypersurfaces in a Nonflat Complex Space Form,” Honam Mathematical Journal, Vol. 28, No. 4, 2006, pp. 573-589.

[13] D. E. Blair, “Riemannian Geometry of Contact and Symplectic Manifolds,” Progress in Mathematics 203, Birkh?user, Boston, 2002.

[14] R. Niebergall and P. J. Ryan, “Real Hypersurfaces in Complex Space Forms,” Publications of the Research Institute for Mathematical Sciences, Vol. 32, 1997, pp. 233- 305.

[15] T. Cecil and P. J. Ryan, “Focal Sets and Real Hypersurfaces in Complex Projective Spaces,” Transactions of the American Mathematical Society, Vol. 269, No. 2, 1982, pp. 481-499.