Real hypersurfaces and complex submanifolds in complex projective space
01 Jan 1986-Transactions of the American Mathematical Society (American Mathematical Society (AMS))-Vol. 296, Iss: 1, pp 137-149
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.
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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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Cites background from "Real hypersurfaces and complex subm..."
...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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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.
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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.
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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, the location of the focal points of a real submanifold is defined in terms of its second fundamental form, and the rank of a focal map onto a sheet of focal points corresponding to a principal curvature is computed using the Codazzi equation.
Abstract: Let M be a real submanifold of CPm, and let J denote the complex structure. We begin by finding a formula for the location of the focal points of M in terms of its second fundamental form. This takes a particularly tractable form when M is a complex submanifold or a real hypersurface on which Ji is a principal vector for each unit normal ( to M. The rank of the focal map onto a sheet of the focal set of M is also computed in terms of the second fundamental form. In the case of a real hypersurface on which JE is principal with corresponding principal curvature ,u, if the map onto a sheet of the focal set corresponding to ,u has constant rank, then that sheet is a complex submanifold over which M is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in cPm. First, there are no totally umbilic real hypersurfaces in cPm, but we show: THEOREM 3. Let M be a connected real hypersurface in CPm, m > 3, with at most two distinct principal curvatures at each point. Then M is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in cPm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4. One of the first results in the geometry of submanifolds is that an umbilic hypersurface M in Euclidean space must be an open subset of a hyperplane or sphere. The proof goes as follows: assume that the shape operator is a scalar multiple of the identity, A = AX, and use the Codazzi equation to show that X is constant. Then either X = 0, in which case M lies on a hyperplane, or the focal points fx(x) = x + (l/X)t, ( the unit normal, all coincide, and M lies on the sphere of radius 1 /X centered at the unique focal point. This simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures. Under fairly general conditions, the set of focal points corresponding to a principal curvature X can be given a differentiable Received by the editors December 19, 1980. Presented to the Society at its annual meeting in San Francisco, January 10, 1981. 1980 Mathematics Subject Classificatiom Primary 53B25, 53C40.
296 citations
"Real hypersurfaces and complex subm..." refers background or methods in this paper
...In this section, we review results about focal sets and tubes in P"(C), obtained by Cecil and Ryan....
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...Cecil and Ryan [4] studied real hypersurfaces in P"(C) on which J£ is principal....
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...Cecil and Ryan studied the focal points of M in the case where M is a complex submanifold or J£ is a principal vector of A^....
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...From the proof of Theorem 1 in [4], v(x) is a unit normal vector of M at 3c = <t>f(x) and 17 gives a local diffeomorphism from C = (<j>f)~x(x) to BXM, where BXM is the unit normal space of M at x....
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...Proposition 3.4 (due to Cecil-Ryan)....
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"Real hypersurfaces and complex subm..." refers background in this paper
...Maeda [5] proved that if 7£ is principal, then the corresponding principal curvature p is locally constant....
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