Let M be a real hypersurface in /"'(C), J be the complex structure and | denote a unit normal vector field on M. We show that M is (an open subset of) a homogeneous hypersurface if and only if M has constant principal curvatures and Jt, is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, /""(C) (totally geodesic), Q", Pl(C) x P"(C). SU{5)/S(U{2) X (7(3)) and SO(10)/t/(5) are the only complex submani- folds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.

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Real hypersurfaces and complex submanifolds in complex projective space

Since E. Cartan's work in the late 30's the classification problem of hypersurfaces with constant principal curvatures is known to be far from trivial. In real space forms it leads to the well-known classification problem of isoparametric hypersurfaces, which has been solved in euclidean space by T. Levi-Civita [6] and B. Segre [11] and in hyperbolic space by E. Cartan [1]; in the sphere, however, a complete classification has not been obtained until now (for essential results see [1], [2], [3], [8], [9], [10], and the literature cited there).

Real hypersurfaces with constant principal curvatures in complex hyperbolic space.

On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables

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Real hypersurfaces in a complex projective space with constant principal curvatures II

The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’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.

Real Hypersurfaces in Complex Space Forms

The purpose of this paper is to determine those homogeneous real hypersurfaces in a complex projective space Pn(C) of complex dimension n(^>2) which are orbits under analytic subgroups of the projective unitary group PU(n-\\-\\)> and to give some characterizations of those hypersurfaces. In § 1 from each effective Hermitian orthogonal symmetric Lie algebra of rank two we construct an example of homogeneous real hypersurface in Pn(C)y which we shall call a model space in Pn(C). In §2 we show that the class of all homogeneous real hypersurfaces in Pn{C) that are orbits under analytic subgroups of PU(n-\\-l) is exhausted by all model spaces. In §§3 and 4 we give some conditions for a real hypersurface in Pn(C) to be an orbit under an analytic subgroup of PU(n-\\-l) and in the course of proof we obtain a rigidity theorem in Pn(C) analogous to one for hypersurfaces in a real space form. The author would like to express his hearty thanks to Professor T. Takahashi for valuable discussions with him and his constant encouragement, and to Professor M. Takeuchi who made an original complicated proof of Lemma 2.3 short and clear.

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On homogeneous real hypersurfaces in a complex projective space

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Degree of symmetric Kählerian submanifolds of a complex projective space

Generic submanifold have been investigated by many authors (e.g. [5], [7], [8], [9], [21]). Here a submanifold M in a Kaehlerian manifold is called generic if each normal space of M is mapped into the tangent space of M by the complex structure of the ambient space (cf. [2], [4], [22]). Any real hypersurface in a Kaehlerian manifold is a typical example of the generic submanifold. In particular, the model space of the so called AI , A^ B, C, D and E-type are typical examples of a real hypersurface in a complex projective space P(C). Recently, the third named author, B. H. Kim and I.-B. Kim [19] proved that those model spaces exhaust all intrinsic homogeneous real hypersurfaces in P(C). On the other hand, the model spaces of the type AI and A 2 was frist introduced by Law son [13], and he gave a characterization of them. Moreover, Choe and Okumura [5] gave a generalization of Lawson's theorem in [13] from a viewpoint of the CR-submanifold (see §1 for the definiton). The purpose of the present paper is to give another generalization (Theorem A) of Lawson's theorem, from a viewpoint of the generic submanifold, and to give new examples of a generic submanifold in P(C). The authors would like to thank the refree for his suggestions, which resulted in many improvments of the present paper.

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Compact minimal generic submanifolds with parallel normal section in a complex projective space

Let (C) be a complex hyperbolic space of complex dimension ( ≧ 2) endowed with the metric of constant holomorphic sectional curvatur e 4 , and be the identity component of the group of all isometries of ( C). A submanifold in ( C) is said to be extrinsically homogeneous if is an orbit under a closed subgroup of . As proposed also in R. Niebergall and P.J. Ryan ([7]), the fol lowing is an open problem:Classify all extrinsically homogeneous real hypersurface s in (C). As a partial answer of this problem, J. Berndt ([1]) classified all ex trinsically homogeneous real hypersurfaces in ( C) whose structure vector fields are principal, where an eigen vector of the shape operator is called principal. Recently he constructed in [2] a subgroup of for each ( ≧ 2) such that a certain orbit under in ( C) has three distinct principal curvatures 1, −1 and 0 with multiplicities 1, 1 and 2− 3 respectively and the structure vector field on is not principal. We shall call this group the Berndt subgroupof . The following is due to J. Berndt and H. Tamaru.

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Ryoichi Takagi

Papers

On homogeneous real hypersurfaces in a complex projective space

Degree of symmetric Kählerian submanifolds of a complex projective space

Compact minimal generic submanifolds with parallel normal section in a complex projective space

Extrinsically homogeneous real hypersurfaces with three distinct principal curvatures in Hn(C)