Journal Article•
Real hypersurfaces with constant principal curvatures in complex hyperbolic space.
TL;DR: In this article, a complete classification of isoparametric hypersurfaces with constant principal curvatures has been obtained in the sphere, but the classification has not been obtained until now.
Abstract: 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).
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01 Jan 2015
TL;DR: 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 hypersurface in spheres as discussed by the authors.
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 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.
228 citations
TL;DR: In this article, it was shown that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field.
Abstract: We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.
154 citations
TL;DR: In this article, a new type of Riemannian curvature invariants are introduced, which have interesting applications to several areas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions.
Abstract: 1. IntroductionThe main purpose of this paper is to introduce a new type of Riemannian curvature invariants and to show that these new invariants have interesting applications to several areas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions.One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean space (or more generally, in a space form). According to a well-known theorem of J. F. Nash, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimension.In order to study this fundamental problem, in view of Nash's theorem, it is natural to impose a suitable condition on the immersions. For instance, if one imposes the minimality condition on the immersions, it leads toPROBLEM 1. Given a Riemannian manifold M, what are the necessary conditions for M to admit a minimal isometric immersion in a Euclidean m-space Em?It is well-known that for a minimal submanifold in Em, the Ricci tensor satisfies Ric_??_0. For many years this was the only known general necessary Riemannian condition for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space.The main results of this article were presented at the 3rd Pacific Rim Geometry Conference held at Seoul, Korea in December 1996; also presented at the 922nd AMS meeting held at Detroit, Michigan in May 1997.
143 citations
Journal Article•
TL;DR: In this paper, it was shown that in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurface.
Abstract: Obviously, every totally umbilical hypersurface of a Riemannian manifold is curvature-adapted. In spaces of constant sectional curvature every hypersurface is curvature-adapted. But in other ambient spaces our definition is restrictive. For example, in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurfaces (see [3] for the notion of Hopf hypersurfaces). In locally Symmetrie spaces it turns out that for the investigation of the geometry of curvature-adapted hypersurfaces Jacobi field theory may be very useful ( s can be seen in section 5).
113 citations
Cites background from "Real hypersurfaces with constant pr..."
...Its proof is entirely analogous to the one for the corresponding problem in complex space forms (see [2], p....
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Abstract: Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given
86 citations
Cites background from "Real hypersurfaces with constant pr..."
...Theorem 4 and Theorem 5). (d) Takagi ([20]) showed that every homogeneous real hypersurface of the complex projective space is a Hopf hypersurface, i.e., its structure field is principal (see Section 1). Moreover, Berndt ([ 2 ]) classified the homogeneous Hopf hypersurfaces in the complex hyperbolic space and asked the question ([3], Problem 4) whether homogeneous hypersurfaces exist in this space form which are not of Hopf type....
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References
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355 citations
TL;DR: In this paper, the authors considered the problem of determining 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-\\-\\)>.
Abstract: 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.
316 citations
TL;DR: In this paper, it was shown that a complex submanifold with constant principal curvatures is an open subset of a homogeneous hypersurface if and only if it has constant curvatures and Jt is principal.
Abstract: 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.
284 citations
"Real hypersurfaces with constant pr..." refers background in this paper
...Kimura [4] has given the classification of all hypersurfaces with constant principal curvatures in the m-dimensional complex projective space CP under the assumption: (C) For every unit normal of M the vector J is a, principal direction of M....
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229 citations
225 citations
"Real hypersurfaces with constant pr..." refers background in this paper
...Montiel [7] proved an analogous result in CH for the case...
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