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.
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TL;DR: In this article , the authors studied the Riemannian geometry of normal metric contact pair manifolds under certain conditions related to the Ricci tensor and proved that a concircular flat NMCP manifold is locally isometric to (2 p + 2 q + 2)-dimensional hyperbolic space.
Abstract: . In this paper we study NMCP manifolds, which are (2 p +2 q + 2) − dimensional differential manifolds with an normal metric contact pair structure. The aim of the study is to examine the Riemannian geometry of normal metric contact pair manifolds under certain conditions related to the ⋆ − Ricci tensor. We prove that a ⋆ − Ricci-semi-symmetric NMCP manifold is ⋆ − Ricci-flat, and that a ⋆ − generalized quasi-Einstein NMCP manifold cannot be ⋆ − Ricci-semi-symmetric. Finally, we consider the concircular curvature tensor on NMCP manifolds and prove that a concircular flat NMCP manifold is locally isometric to (2 p +2 q +2)-dimensional hyperbolic space.
1 citations
TL;DR: In this paper , the authors introduce the notion almost ⁎-Ricci-Bourguignon solitons and find its geometric characterizations on Sasakian manifolds.
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TL;DR: In this paper , the authors characterized Sasakian manifolds endowed with a set of solitons and proved the existence of Ricci-Yamabe solitions in a 5-dimensional Sasakians manifold.
Abstract: In this note, we characterize Sasakian manifolds endowed with
-
-Ricci-Yamabe solitons. Also, the existence of
-
-Ricci-Yamabe solitons in a 5-dimensional Sasakian manifold has been proved through a concrete example.
1 citations
TL;DR: In this paper, the authors studied the Weyl curvature tensor under the framework of Sasakian and contact manifolds, and showed that it is possible to construct a Weyl tensor from contact manifold.
Abstract: The object of the present paper is to study $*$-Weyl curvature tensor within the framework of Sasakian and $(\kappa,\mu)$-contact manifolds
1 citations
TL;DR: In this article , the authors introduce the notion of semi-symmetric Ricci tensors and illustrate that a non-Sasakian $ (\kappa, \mu) $-contact manifold is $ \ast $-Ricci semi symmetric or has parallel Ricci operator if and only if it is
Abstract: We introduce the notion of semi-symmetric $ \ast $-Ricci tensor and illustrate that a non-Sasakian $ (\kappa, \mu) $-contact manifold is $ \ast $-Ricci semi-symmetric or has parallel $ \ast $-Ricci operator if and only if it is $ \ast $-Ricci flat. Then we find that among the non-Sasakian $ (\kappa, \mu) $-contact manifolds with the same Boeckx invariant $ I_M $, only one is $ \ast $-Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct $ \ast $-Ricci flat $ (\kappa, \mu) $-contact manifolds.
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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 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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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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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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