On some curvature conditions of pseudosymmetry type

Deszcz, Ryszard; Głogowska, Małgorzata; Hotloś, Marian; Zafindratafa, Georges

doi:10.1007/s10998-014-0081-9

Cited by 35 publications

(64 citation statements)

References 41 publications

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“…For example, the Robertson-Walker space-times are quasi-Einstein (see [10,116] and references therein for details on these space-times). For recent results on quasi-Einstein manifolds we refer to [22,51,52,50]. An example of a warped product manifold of dimension ≥ 4, with 1-dimensional fiber, is presented in [43,Example 4.1].…”

Section: Divergence-free Weyl Tensor and Pp-wavesmentioning

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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Section: Divergence-free Weyl Tensor and Pp-wavesmentioning

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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“…Further, it seems that there is no essential result on manifolds satisfying (vi) with non-zero tensors R · C − C · R, Q(g, g ∧ S) and Q(S, g ∧ S). Manifolds satisfying (iii) or (vi) were investigated in [29]. Next, manifolds satisfying (v) were investigated in [1] and [57].…”

Section: Some Generalized Einstein Metric Conditionsmentioning

confidence: 99%

“…An example of a warped product manifold satisfying (iv) is given in [39,Example 5.1]. For further results on manifolds satisfying (iv) we refer to [29]. We also mention that warped product manifolds satisfying (i), (ii), (iv) and (v) are quasi-Einstein or not.…”

Section: Some Generalized Einstein Metric Conditionsmentioning

confidence: 99%

On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type

Deszcz

Petrović–Torgašev

Verstraelen

et al. 2015

Bull. Malays. Math. Sci. Soc.

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In this paper, we study Chen ideal submanifolds M n of dimension n in Euclidean spaces E n+m (n ≥ 4, m ≥ 1) satisfying curvature conditions of pseudosymmetry type of the form: the difference tensor R · C − C · R is expressed by some Tachibana tensors. Precisely, we consider one of the following three conditions: R ·C −C · R is expressed as a linear combination of Q(g, R) and Q(S, R), R ·C −C · R is expressed as a linear combination of Q(g, C) and Q(S, C) and R · C − C · R is expressed as a linear combination of Q(g, g∧S) and Q(S, g∧S). We then characterize Dedicated to the memory of Professor Franki Dillen. Communicated by Young Jin Suh. B Miroslava Petrović-Torgasev mirapt@kg.ac.rs Ryszard Deszcz Ryszard.Deszcz@up.wroc.pl

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“…We also mention that semi-Riemannian manifolds with the curvature tensor R expressed by a linear combination of the tensors g ∧g, g ∧S and S ∧S were introduced and investigated in [10]. For further results on this class of manifolds we refer to [13,17,19,20,24,27,29,31,36,42].…”

Section: Hypersurfaces With Three Principal Curvaturesmentioning

confidence: 99%