Geodesic
Real hypersurfaces in complex space forms concerned with the local symmetry
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 885-905 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m(4\epsilon )$. In the second, we give a complete classification of real hypersurfaces in $M_m(4\epsilon )$ which satisfy the above geometric facts.
This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m(4\epsilon )$. In the second, we give a complete classification of real hypersurfaces in $M_m(4\epsilon )$ which satisfy the above geometric facts.
Classification : 53C15, 53C40, 53D15
Keywords: real hypersurfaces; local symmetry; derivations; Kulkarni-Nomizu product 
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Lyu, Seon Mi; Pérez, Juan de Dios; Suh, Young Jin. Real hypersurfaces in complex space forms concerned with the local symmetry. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 885-905. http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a8/

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

[2] J. Berndt: Real hypersurfaces in quaternionic space forms. J.  Reine Angew. Math. 419 (1991), 9–26. | MR | Zbl

[3] J.  Berndt, Y. J.  Suh: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1–14. | DOI | MR

[4] J.  Berndt, Y. J.  Suh: Isometric flows on real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87–98. | DOI | MR

[5] B. Y. Chen, G. D. Ludden, and S. Montiel: Real submanifolds of a Kaehler manifold. Algebras Groups Geom. 1 (1984 1984), 176–212. | MR

[6] U.-H. Ki, Y. J. Suh: On real hypersurfaces of a complex space form. Math. J.  Okayama Univ. 32 (1990), 207–221. | MR

[7] U.-H. Ki, Y. J. Suh: On a characterization of real hypersurfaces of type  $A$ in a complex space form. Canad. Math. Bull. 37 (1994), 238–244. | DOI | MR

[8] S.  Kobayashi, K.  Nomizu: Foundations of Differential Geometry. Interscience, New York-London-Sydney, 1963. | MR

[9] M. Kimura: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Amer. Math. Soc. 296 (1986), 137–149. | DOI | MR | Zbl

[10] M. Kimura, S. Maeda: On real hypersurfaces of complex projective space  II. Tsukuba J.  Math. 15 (1994), 547–561. | MR

[11] S. M.  Lyu, J. D.  Pérez, and Y. J.  Suh: On real hypersurfaces in a quaternionic hyperbolic space in terms of the derivative of the second fundamental tensor. Acta Math. Hungarica 97 (2002), 145–172. | DOI | MR

[12] Y. Maeda: On real hypersurfaces of a complex projective space. J. Math. Soc. Japan 28 (1976 1976), 529–540. | MR

[13] S. Montiel: Real hypersurfaces of a complex hyperbolic space. J.  Math. Soc. Japan 37 (1985), 515–535. | DOI | MR | Zbl

[14] S. Montiel: Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter space. J.  Math. Soc. Japan 55 (2003), 915–938. | DOI | MR

[15] S. Montiel, A. Romero: On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20 (1986), 245–261. | DOI | MR

[16] M. Okumura: On some real hypersurfaces of a complex projective space. Trans. Amer. Math. Soc. 212 (1975), 355–364. | DOI | MR | Zbl

[17] B. O’Neill: The fundamental equation of a submersion. Michigan Math.  J. 13 (1966), 459–469. | DOI | MR

[18] M. Ortega, J. D. Pérez, and Y. J. Suh: Real hypersurfaces with constant totally real sectional curvatures in a complex space form. Czech. Math.  J. 50(125) (2000), 531–537. | DOI | MR

[19] M.  Ortega, J. D. Pérez, and Y. J.  Suh: Real hypersurfaces in quaternionic projective space with commuting tangent Jacobi operators. Glasgow Math.  J. 45 (2003), 51–65. | DOI | MR

[20] J. D.  Pérez, Y. J.  Suh: Real hypersurfaces of quaternionic projective space satisfying $\nabla _{U_i}R=0$. Diff. Geom. Appl. 7 (1997), 211–217. | DOI | MR

[21] D. J. Sohn, Y. J. Suh: Classification of real hypersurfaces in complex hyperbolic space in terms of constant $\phi $-holomorphic sectional curvatures. Kyungpook Math.  J. 35 (1996), 801–819. | MR

[22] Y. J. Suh: A characterization of ruled real hypersurfaces in $P_m(\mathbb{C})$. J.  Korean Math. Soc. 29 (1992), 351–359. | MR

[23] Y. J. Suh, R. Takagi: A rigidity for real hypersurfaces in a complex projective space. Tohoku Math.  J. 43 (1991), 501–507. | DOI | MR

[24] R. Takagi: On homogeneous real hypersurfaces of a complex projective space. Osaka J.  Math. 10 (1973), 495–506. | MR