Complete Hypersurfaces with RS = 0 In E n+1

Matsuyama, Yoshio

doi:10.2307/2045122

Cited by 4 publications

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“…The manifold M is said to be Ricci symmetric if ∇S = 0 (see, e.g., [69]). The notion of Ricci symmetry was also weakend by various ways such as Ricci recurrent by Patterson [70], Ricci semisymmetric by Okumara [69], Roter [72], Tanno [112], Ryan [76], Matsuyama [66], Mirzoyan [68], Abdalla and Dillen [1] and others, Ricci pseudosymmetric by Deszcz [21], pseudo Ricci symmetric by Chaki [11], weakly Ricci symmetric by Tamássy and Binh [111].…”

Section: Preliminariesmentioning

confidence: 99%

Curvature properties of some 4-dimensional semi-Riemannian metrics

A.¹,

Matsuyama²

2017

Kragujevac J Math

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Section: Preliminariesmentioning

confidence: 99%

Curvature properties of some 4-dimensional semi-Riemannian metrics

A.¹,

Matsuyama²

2017

Kragujevac J Math

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“…In [1], Defever proved the existence of a hypersurface M 5 of E 6 which has principal curvatures (0, b, b, −b, −b) at every point p ∈ M 5 , for some function b on M 5 . In this section we give an easy explicit example of such a hypersurface.…”

Section: The Examplementioning

confidence: 99%

“…In his papers [7], [8], Ryan studies the relation between the two conditions, proving for instance that conditions (1.1) and (1.2) are equivalent for hypersurfaces in spheres and hyperbolic spaces and for hypersurfaces of Euclidean space with nonnegative scalar curvature. Another interesting result in this direction is proved in [5]: conditions (1.1) and (1.2) are equivalent for complete hypersurfaces of Euclidean space. Also it is proved in [2] that conditions (1.1) and (1.2) are equivalent for hypersurfaces of 5-dimensional semi-Riemannian space of constant curvature.…”

Section: Preliminaries and Introductionmentioning

confidence: 97%

A Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric

Abdalla

Dillen

2001

Proc. Amer. Math. Soc.

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Abstract. We construct the first explicit example of a Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric. Preliminaries and introductiondenote the Riemann-Christoffel curvature tensor of M n and let S, defined by) is said to be semi-symmetric ifIt is well known that the class of semi-symmetric manifolds includes the set of locally symmetric manifolds (∇R = 0) as a proper subset. A Riemannian manifold (M n , g) is said to be Ricci-semi-symmetric ifThe class of Ricci-semi-symmetric manifolds includes the set of Ricci-symmetric manifolds (∇S = 0) as a proper subset. Evidently, condition (1.1) implies condition (1.2). The converse is in general not true. However, under certain conditions (for instance if n = 3), (1.1) and (1.2) are equivalent; see [3] for an account of results in this direction.A long-standing open problem is the following: are conditions (1.1) and (1.2) equivalent for hypersurfaces in Euclidean space? This problem is known as Ryan's

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“…Ryan [18] raised the following question for hypersurfaces of Euclidean spaces in 1972: Are conditions R · R = 0 and R · Ric = 0 equivalent for hypersurfaces of Euclidean spaces? Although there are many results which contributed to the solution of the above question in the affirmative under some conditions (see [6], [7], [17] and references therein). In [1], the authors gave an explicit example of a hypersurface in Euclidean E n+1 (n 4) that is Ricci semi-symmetric but not semi-symmetric (see [5] for another example).…”

Section: Introductionmentioning

confidence: 99%

Equivalence Conditions of Symmetry Properties in Lightlike Hypersurfaces of Indefinite Kenmotsu Manifolds

Lungiambudila¹,

Massamba²,

Tossa³

2016

Bulletin of the Korean Mathematical Society

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Abstract. The paper deals with lightlike hypersurfaces which are locally symmetric, semi-symmetric and Ricci semi-symmetric in indefinite Kenmotsu manifold having constant φ-holomorphic sectional curvature c. We obtain that these hypersurfaces are totally goedesic under certain conditions. The non-existence condition of locally symmetric lightlike hypersurfaces are given. Some Theorems of specific lightlike hypersurfaces are established. We prove, under a certain condition, that in lightlike hypersurfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the parallel, semi-parallel, local symmetry, semi-symmetry and Ricci semi-symmetry notions are equivalent.

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Complete Hypersurfaces with RS = 0 In E n+1

Cited by 4 publications

References 0 publications

Curvature properties of some 4-dimensional semi-Riemannian metrics

Curvature properties of some 4-dimensional semi-Riemannian metrics

A Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric

Equivalence Conditions of Symmetry Properties in Lightlike Hypersurfaces of Indefinite Kenmotsu Manifolds

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