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Journal ArticleDOI

Real Hypersurfaces of Complex Space Forms in Terms of Ricci $*$-Tensor

01 Dec 2002-Tokyo Journal of Mathematics (Publication Committee for the Tokyo Journal of Mathematics)-Vol. 25, Iss: 2, pp 473-483
TL;DR: In this article, the authors classified the $*$-Einstein real hypersurfaces in complex space forms such that the structure vector is a principal curvature vector and the principal curvatures of the hypersurface can be computed with the K\"ahler metric.
Abstract: It is known that there are no Einstein real hypersurfaces in complex space forms equipped with the K\"ahler metric. In the present paper we classified the $*$-Einstein real hypersurfaces $M$ in complex space forms $M_{n}(c)$ and such that the structure vector is a principal curvature vector.
Citations
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Journal ArticleDOI
15 Apr 2021
TL;DR: In this article, it was shown that Bach flat almost coKahler manifold admits Ricci solitons, satisfying the critical point equation (CPE) or Bach flat.
Abstract: In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$ -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$ -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$ -almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$ -almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.

6 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of a proper Ricci soliton on a Kenmotsu 3-manifold with Codazzi type of Ricci tensor is proved.
Abstract: Abstract In the present paper we study η-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider η-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study φ-Ricci symmetric η-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition R.R = Q(S, R)is considered. Finally, an example is constructed to prove the existence of a proper η-Ricci soliton on a Kenmotsu 3-manifold.

6 citations


Cites background from "Real Hypersurfaces of Complex Space..."

  • ...Later, in [17] Hamada studied ∗-Ricci flat real hypersurfaces in non-flat complex space forms and Blair [4] defined ∗-Ricci tensor in contact metric manifolds by S∗(X,Y ) = g(Q∗X,Y ) = Trace{φ ◦R(X,φY )}, (1....

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Journal ArticleDOI
TL;DR: In this paper , the authors study the ∗-η-Ricci soliton and gradient almost ∗η−RICCI soliton within the framework of para-Kenmotsu manifolds as a characterization of Einstein metrics.
Abstract: The goal of the present study is to study the ∗-η-Ricci soliton and gradient almost ∗-η-Ricci soliton within the framework of para-Kenmotsu manifolds as a characterization of Einstein metrics. We demonstrate that a para-Kenmotsu metric as a ∗-η-Ricci soliton is an Einstein metric if the soliton vector field is contact. Next, we discuss the nature of the soliton and discover the scalar curvature when the manifold admits a ∗-η-Ricci soliton on a para-Kenmotsu manifold. After that, we expand the characterization of the vector field when the manifold satisfies the ∗-η-Ricci soliton. Furthermore, we characterize the para-Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies the gradient almost ∗-η-Ricci soliton.

5 citations

Journal ArticleDOI
18 Apr 2019-Symmetry
TL;DR: The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor, which are based on tools from differential geometry and solving systems of differential equations.
Abstract: In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

5 citations


Cites background from "Real Hypersurfaces of Complex Space..."

  • ..., the ∗-Ricci tensor satisfies S∗ = ρ∗g, with ρ∗ being constant, are introduced and the real hypersurfaces are studied with respect to the previous relations (see [10,11,15])....

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  • ...In [10] Hamada gave the definition of ∗-Ricci tensor S∗ on real hypersurfaces in Mn(c) in the following way g(S∗X, Y) = 1 2 trace(Z → R(X, φY)φZ),...

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References
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Journal ArticleDOI
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

Journal ArticleDOI
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 of Complex Space..." refers background in this paper

  • ... persurfaces in Pn(C), The proof can be found in [ 2 ]....

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  • ...that there are no Einstein real hypersurfaces of Pn(C) [ 2 ]....

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Journal ArticleDOI
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 of Complex Space..." refers background in this paper

  • ...Proposition 2.9. ([ 3 ]) Let M be a real hypersurface of Pn(C)....

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Journal ArticleDOI
TL;DR: In this article, the second fundamental tensor of a real hypersurface of a complex projective space (CPn) is compared with the corresponding hypersurfaces of a Riemannian manifold of constant curvature.
Abstract: A principal circle bundle over a real hypersurface of a complex projective space CPn can be regarded as a hypersurface of an odddimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of CPn into those imposed on a hypersurface of S2'+1. Some fundamental relations between the second fundamental tensor of a hypersurface of CPn and that of a hypersurface of S2n+1 are given. Introduction. As is well known a sphere S2n+1 of dimension 2n + 1 is a principal circle bundle over a complex projective space CPn and the Riemannian structure on CPn is given by the submersion ir: S2n+ 1 ~ CPn [4], [7]. This suggests that fundamental properties of a submersion would be applied to the study of real submanifolds of a complex projective space. In fact, H. B. Lawson [2] has made one step in this direction. His idea is to construct a principal circle bundle M2n over a real hypersurface M2n-1 of Cpn in such a way that M2n is a hypersurface of S2n + 1 and then to compare the length of the second fundamental tensors of M2n-1 and M2 n. Thus we can apply theorems on hypersurfaces of S2n+1. In this paper, using Lawson's method, we prove a theorem which characterizes some remarkable classes of real hypersurfaces of Cpn. First of all, in ?1, we state a lemma for a hypersurface of a Riemannian manifold of constant curvature for the later use. In ?2, we recall fundamental formulas of a submersion which are obtained in [4], [7] and those established between the second fundamental tensors of M and M. In ?3, we give some identities which are valid in a real hypersurface of CPn. After these preparations, we show, in ?4, a geometric meaning of the commutativity of the second fundamental tensor of M in Cp'n and a fundamental tensor of the submersion ir: M . M 1. Hypersurfaces of a Riemannian manifold of constant curvature. Let M be an (m + 1)-dimensional Riemannian manifold with a Riemannian metric G and i: M M be an isometric immersion of an m-dimensional differentiable Received by the editors March 25, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.

282 citations


"Real Hypersurfaces of Complex Space..." refers background in this paper

  • ...Proposition 2.3. ([ 6 ], [9]) Let M, where n ‚ 2, be a real hypersurface in Mn(c) of constant holomorphic sectional curvature 4c 6= 0. Then `A = A` if and only if M is an open subset of the following: (i) a geodesic hypersphere, (ii) a tube over totally geodesic complex space form Mk(c), where 0 < k • ni1....

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Journal Article
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).

253 citations


"Real Hypersurfaces of Complex Space..." refers background in this paper

  • ...Proposition 2.10. ([ 1 ]) Let M be a real hypersurface of Hn(C) of constant holomorphic sectional curvature 4c < 0. Then M has constant principal curvatures and » is a principal curvature vector if and only if M is locally congruent to the following:...

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