Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator

Cho, Jong Taek ; Ki, U-Hang

Canadian mathematical bulletin, Tome 51 (2008) no. 3, pp. 359-371

Résumé

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $\left( A \right)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Détail

DOI : 10.4153/CMB-2008-036-7

Mots-clés : 53B20, 53C15, 53C25, complex space form, real hypersurface, structure Jacobi operator

Bibliographie

[1] [1] Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395(1989), 132–141. Google Scholar

[2] [2] Berndt, J. and Vanhecke, L., Two natural generalizations of locally symmetric spaces. Differential Geom. Appl. 2(1992), no. 2, 57–80. Google Scholar

[3] [3] Cecil, T. E. and Ryan, P. J., Focal sets and real hypersurfaces in complex projective space. Trans. Amer. Math. Soc. 269(1982), no. 2, 481–499. Google Scholar

[4] [4] Cho, J. T., On some classes of almost contact metric manifolds. Tsukuba J. Math. 19(1995), no. 1, 201–217. Google Scholar

[5] [5] Cho, J. T. and Ki, U-H., Real hypersurfaces of a complex projective space in terms of the Jacobi operators. Acta Math. Hungar 80(1998), no. 1–2, 155–167. Google Scholar

[6] [6] Cho, J. T. and Ki, U-H., Jacobi operators on real hypersurfaces of a complex projective space. Tsukuba J. Math. 22(1998), no. 1, 145–156. Google Scholar

[7] [7] Ki, U-H. Real hypersurfaces with parallel Ricci tensor of a complex space form. Tsukuba J. Math. 13(1989), no. 1, 73–81. Google Scholar

[8] [8] Ki, U-H. and Suh, Y. J., On real hypersurfaces of a complex space form. Math. J. Okayama Univ. 32(1990), 207–221. Google Scholar

[9] [9] Ki, U-H., Kim, H.-J., and Lee, A.-A., The Jacobi operator of real hypersurfaces of a complex space form. Commun. Korean Math. Soc. 13(1998), no. 3, 545–560. Google Scholar

[10] [10] Kim, U. K., Nonexistence of Ricci-parallel real hypersurfaces in P ℂ or H ℂ . Bull. Korean Math. Soc. 41(2004), no. 4, 699–708. Google Scholar

[11] [11] Kimura, M., Real hypersurfaces and complex submanifolds in complex projective space. Trans. Amer. Math. Soc. 296(1986), no. 1, 137–149. Google Scholar

[12] [12] Montiel, S. and Romero, A., On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20(1986), no. 2, 245–261. Google Scholar

[13] [13] Okumura, M., On some real hypersurfaces of a complex projective space. Trans. Amer. Math. Soc. 212(1975), 355–364. Google Scholar

[14] [14] de Dios Pérez, J., On parallelness of structure Jacobi operator of a real hyper-surface in complex projective space. In: Proceedings of the Eight International Workshop on Differential Geometry. Kyungpook Nat. Univ., Taegu, 2004, pp. 47–55. Google Scholar

[15] [15] Ortega, M., de Dios Pérez, J., and Santos, F. G., Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms. Rocky Mountain J. Math. 36(2006), 1603–1614. Google Scholar

[16] [16] de Dios Pérez, J., Santos, F. G., and Suh, Y. J., Real hypersurfaces of complex projective space whose structure Jacobi operator is -parallel. Bull. Belg.Math. Soc. Simon Stevin 13(2006), no. 3, 459–469. Google Scholar

[17] [17] Takagi, R., On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10(1973), 495–506. Google Scholar

[18] [18] Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures. I. II. J. Math. Soc. Japan 15(1975), 43–53, no. 4, 507–516. Google Scholar