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real hypersurfaces; totally real bisectional curvature; sectional curvature; holomorphic sectional curvature
In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form $M_m(c)$, $c\ne 0$ as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].
[1] M. Barros and A. Romero: Indefinite Kaehler manifolds. Math. Ann. 281 (1982), 55–62. MR 0675207
[2] R. L. Bishop and S. I. Goldberg: Some implications of the generalized Gauss-Bonnet Theorem. Trans. Amer. Math. Soc. 112 (1964), 508–535. DOI 10.1090/S0002-9947-1964-0163271-8 | MR 0163271
[3] T. E. Cecil and P. J. Ryan: Focal sets and real hypersurfaces in complex projective space. Trans. A.M.S. 269 (1982), 481–499. MR 0637703
[4] S. I. Goldberg and S. Kobayashi: Holomorphic bisectional curvature. J. Diff. Geometry 1 (1967), 225–234. DOI 10.4310/jdg/1214428090 | MR 0227901
[5] C. S. Houh: On totally real bisectional curvature. Proc. Amer. Math. Soc. 56 (1976), 261–263. DOI 10.1090/S0002-9939-1976-0400128-2 | MR 0400128 | Zbl 0328.53012
[6] M. Kimura: Sectional curvatures of holomorphic planes on a real hypersurface in $P_nC$. Math. Ann. 276 (1987), 487–499. DOI 10.1007/BF01450843 | MR 0875342
[7] Y. Maeda: On real hypersurfaces of a complex projective space. J. Math. Soc. Japan 28 (1976), 529–540. DOI 10.2969/jmsj/02830529 | MR 0407772 | Zbl 0324.53039
[8] S. Montiel: Real hypersurfaces of complex hyperbolic space. J. Math. Soc. Japan. 37 (1985), 515–535. DOI 10.2969/jmsj/03730515 | MR 0792990
[9] R. Niebergall and P. J. Ryan: Real hypersurfaces in complex space forms. Tight and taut Submanifolds, edited by T. E. Cecil and S. S. Chern, Cambridge U. Press, 1997, pp. 233–305. MR 1486875
[10] M. Ortega and J. D. Pérez: Constant holomorphic sectional curvature and type number of real hypersurfaces of complex hyperbolic space. Proc. 4th Intnal. Congress of Geometry, Thessaloniki (1996). MR 1470995
[11] M. Ortega, J. D. Pérez and Y. J. Suh: Real hypersurfaces with constant totally real sectional curvature in a complex space form. Czechoslovak Math. J. 50 (2000), 531–537. DOI 10.1023/A:1022881510000 | MR 1777474
[12] J. D. Pérez and Y. J. Suh: Real hypersurfaces of quaternionic projective space satisfying ${\nabla }_{U_i}R=0$. Diff. Geom. and Its Appl. 7 (1997), 211–217. DOI 10.1016/S0926-2245(97)00003-X
[13] D. J. Sohn and Y. J. Suh: Classification of real hypersurfaces in complex hyperbolic space in terms of constant $\phi $-holomorphic sectional curvature. Kyungpook Math. J. 35 (1996), 801–819. MR 1678228
[14] Y. J. Suh: A characterization of ruled real hypersurfaces in $P_nC$. J. Korean Math. Soc. 29 (1992), 351–359. MR 1180662
[15] M. H. Vernon: Contact hypersurfaces of complex hyperbolic space. Tôhoku Math. J. 39 (1987), 215–222. DOI 10.2748/tmj/1178228324 | MR 0887937
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