Abstract
The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Münzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n. These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n. In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.
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References
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)
T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space. Trans. Am. Math. Soc. 269, 481–499 (1982)
L. Conlon, Differentiable Manifolds: A First Course, 1st edn. (Birkhäuser, Boston, 1993)
M. Kimura, Sectional curvature of holomorphic planes on a real hypersurface in P n(C). Math. Ann. 276, 487–497 (1987)
N. Koike, The quadratic slice theorem and the equiaffine tube theorem for equiaffine Dupin hypersurfaces. Results Math. 47, 69–92 (2005)
S.H. Kon, T.-H. Loo, On Hopf hypersurfaces in a non-flat complex space form with η-recurrent Ricci tensor. Kodai Math. J. 33, 240–250 (2010)
S. Mullen, Isoparametric systems on symmetric spaces, in Geometry and Topology of Submanifolds VI, ed. by Dillen, F. et al. (World Scientific, River Edge, 1994), pp. 152–154
K. Nomizu, Fundamentals of Linear Algebra (McGraw-Hill, New York, 1966)
M. Ooguri, On Cartan’s identities of equiaffine isoparametric hypersurfaces. Results Math. 46, 79–90 (2004)
M. Ortega, Classifications of real hypersurfaces in complex space forms by means of curvature conditions. Bull. Belg. Math. Soc. Simon Stevin 9, 351–360 (2002)
J.D. Pérez, Cyclic-parallel real hypersurfaces of quaternionic projective space. Tsukuba J. Math. 17, 189–191 (1993)
Y.-S. Pyo, On real hypersurfaces in a complex space form (II). Commun. Korean Math. Soc. 9, 369–383 (1994)
R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures II. J. Math. Soc. Jpn. 27, 507–516 (1975)
C.-L. Terng, Isoparametric submanifolds and their Coxeter groups. J. Differ. Geom. 21, 79–107 (1985)
S.-T. Yau, Open problems in geometry, in Proceedings of Symposia in Pure Mathematics, vol. 54, Part 1 (American Mathematical Society, Providence, 1993), pp. 1–28
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© 2015 Thomas E. Cecil and Patrick J. Ryan
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Cecil, T.E., Ryan, P.J. (2015). Real Hypersurfaces in Complex Space Forms. In: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3246-7_6
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DOI: https://doi.org/10.1007/978-1-4939-3246-7_6
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