On the equivalence of Ricci-semisymmetry and semisymmetry

Arslan, Kadri; Çelik, Yunus; Deszcz, Ryszard; Ezentaş, Rıdvan

doi:10.4064/cm-76-2-279-294

Cited by 18 publications

(16 citation statements)

References 16 publications

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“…It can be easily shown that the scalar curvature of the resulting manifold 3 , which is non-vanishing and non-constant. We shall now show that R 4 is a GK 4 .…”

Section: Resultsmentioning

confidence: 99%

On generalized recurrent Riemannian manifolds

Arslan

Murathan

et al. 2008

Acta Math Hung

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The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero nonconstant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant scalar curvature, Weylsemisymmetry and semisymmetry are equivalent. Sucient condition for a generalized recurrent manifold to be a special quasi Einstein manifold is obtained.

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“…It can be easily shown that the scalar curvature of the resulting manifold 3 , which is non-vanishing and non-constant. We shall now show that R 4 is a GK 4 .…”

Section: Resultsmentioning

confidence: 99%

On generalized recurrent Riemannian manifolds

Arslan

Murathan

et al. 2008

Acta Math Hung

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“…It should be remarked here that some authors achieved Eq. (2.7) in their investigations about pseudo-Riemannian manifolds endowed with differential structures as [6,33,43,103]. However a study of its consequence on the structure of the Riemann and Weyl tensors and on Weyl's scalars and thus on the classification of space-times was pursued in [75-77, 83, 85].…”

Section: Proposition 22mentioning

confidence: 99%

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

Mantica

Suh

2016

Int. J. Geom. Methods Mod. Phys.

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In this paper we present some new results about [Formula: see text]-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for [Formula: see text] and a pp-wave space-time in [Formula: see text]. In all cases an algebraic classification for the Weyl tensor is provided for [Formula: see text] and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat [Formula: see text], [Formula: see text], space-time is conformal to the Robertson–Walker space-time.

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“…A semi-Riemannian manifold (M, g) is then said to be semisymmetric if it satisfies the condition R · R = 0. It is well known that the class of semisymmetric manifolds includes the set of locally symmetric manifolds (∇R = 0) as a proper subset [2]; here, we suppose that (M, g) is a Riemmanian manifold. If M satisfies the condition ∇R = 0, then M is called a locally symmetric manifold.…”

Section: Preliminariesmentioning

confidence: 99%

Pseudosymmetric lightlike hypersurfaces

Kazan¹,

Şahin²

2014

Turk J Math

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Abstract:We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. We obtain that a pseudosymmetric lightlike hypersurface is a semisymmetric lightlike hypersurface or totally geodesic under certain conditions. Moreover, we give an example of pseudosymmetric lightlike hypersurfaces and investigate pseudoparallel lightlike hypersurfaces. Furthermore, we introduce Ricci-pseudosymmetric lightlike hypersurfaces, obtain characterizations, and give an example for such hypersurfaces.

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On the equivalence of Ricci-semisymmetry and semisymmetry

Cited by 18 publications

References 16 publications

On generalized recurrent Riemannian manifolds

On generalized recurrent Riemannian manifolds

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

Pseudosymmetric lightlike hypersurfaces

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