On real hypersurfaces of a complex projective space with recurrent Ricci tensor
TL;DR: In this article, it was shown that there are no real hypersurfaces with recurrent Ricci tensors of the complex projective space Pn(C) under the condition that there is a principal curvature vector.
Abstract: Let M be a real hypersurface of the complex projective space
Pn(C). The Ricci tensor S
of M is recurrent if there exists a 1-form such that . In this paper we show that there
are no real hypersurfaces with recurrent Ricci tensor of
Pn(C) under the condition that is a principal curvature vector.1991 Mathematics
Subject Classification 53C40
(53C25).
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TL;DR: In this paper, the authors classify real hypersurfaces of complex projective space C P m, m ⩾ 3, with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurface with recurrent structure JacobI operator.
Abstract: We classify real hypersurfaces of complex projective space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.
32 citations
Cites background from "On real hypersurfaces of a complex ..."
...In [5] recurrent Ricci tensor S of a real hypersurface M in CP m is studied....
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TL;DR: In this paper, a reduction result for parallel symmetric covariant tensor fields of order two was obtained for the parallel Ricci solitons with regularity and regularity conditions.
Abstract: Torse-forming $\eta $-Ricci solitons are studied in the framework of almost paracontact metric $\eta $-Einstein manifolds. By adding a technical condition, called regularity and concerning with the scalars provided by the two $\eta $-conditions, is obtained a reduction result for the parallel symmetric covariant tensor fields of order two.
28 citations
TL;DR: In this paper, the authors give a non-existence theorem for Hopf hy-persurfaces in the complex two-plane Grassmannian G2(C m+2 ) with re-current normal Jacobi operator ¯ RN.
Abstract: In this paper we give a non-existence theorem for Hopf hy- persurfaces in the complex two-plane Grassmannian G2(C m+2 ) with re- current normal Jacobi operator ¯ RN.
10 citations
TL;DR: In this article, the authors introduced the notion of recurrent shape operator for real hypersurface M in the complex two-plane Grassmannians G2(Cm+2) and gave a non-existence property of real hypersuran surfaces in G 2 (Cm + 2) with the recurrent shape operators.
Abstract: We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G2(Cm+2) and give a non-existence property of real hypersurfaces in G2(Cm+2) with the recurrent shape operator.
9 citations
01 Jan 2008
TL;DR: In this paper, the authors introduced the notion of recurrent hypersurfaces in complex two-plane Grassmannians G2(C m+2 ) and gave a non-existence theorem for a Hopf hypersurface in G 2 (C m + 2 ) with recurrent shape operator.
Abstract: We introduce the notion of recurrent hypersurfaces in complex two-plane Grassmannians G2(C m+2 ) and give a non-existence theorem for a Hopf hypersurface in G2(C m+2 ) with recurrent shape operator.
7 citations
Cites background from "On real hypersurfaces of a complex ..."
...Recently, Hamada ([5] and [6]) applied such a notion of recurrent tensor to a shape operator or a Ricci tensor for real hypersurfaces M in complex projective space CP in such a way that ∇A = ω ⊗A or ∇S = ω ⊗ S for a certain 1-form ω defined on M , and proved the following : Theorem A....
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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
"On real hypersurfaces of a complex ..." refers background in this paper
...([13]) Let M be a homogeneous real hypersurface of Pn C....
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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
"On real hypersurfaces of a complex ..." refers background in this paper
...Cecil and Ryan proved that there are no Einstein real hypersurfaces of Pn C [1]....
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...([1]) Let M be a connected real hypersurface of Pn C, n 3, whose Ricci tensor S is pseudo-Einstein, i....
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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
"On real hypersurfaces of a complex ..." refers background in this paper
...([6]) Let M be a real hypersurface of Pn C....
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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
206 citations