On compact manifolds admitting indefinite metrics with parallel Weyl tensor

Derdziński, Andrzej; Roter, W.

doi:10.1016/j.geomphys.2008.03.011

Cited by 24 publications

(53 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(6)]). We refer to [43][44][45][46] for the recent results on semi-Riemannian manifolds with parallel Weyl conformal curvature tensor and, in particular, for classification results. Semi-Riemannian warped products having parallel Weyl conformal curvature tensor were investigated in [47].…”

Section: Proposition 1 (Cf [40]) For Any Semi-riemannian Manifoldmentioning

confidence: 99%

Curvature properties of Gödel metric

Deszcz

Hotloś

Jełowicki

et al. 2014

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

show abstract

Section: Proposition 1 (Cf [40]) For Any Semi-riemannian Manifoldmentioning

confidence: 99%

Curvature properties of Gödel metric

Deszcz

Hotloś

Jełowicki

et al. 2014

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

show abstract

“…The proposition is relevant in the study of Pontryagin forms. Given orthonormal tangent vectors X r , Pontryagin forms p k are antisymmetric combinations p k (X 1 ...X 4k ) = P (−1) P ω k (X i1 ...X i 4k ) of forms ω k (see [11] [12] [29] [36] [38]):…”

Section: A Related Class Of Manifoldsmentioning

confidence: 99%

“…, etc. It was shown by Avez [2] (see also [11]) that forms are unchanged if Riemann's tensor is replaced by Weyl's tensor, for example: Proof. By prop.4.1 it is C lmj k C pqk j = 0, then ω 1 = 0 and p 1 = 0.…”

Section: A Related Class Of Manifoldsmentioning

confidence: 99%

On weakly conformally symmetric pseudo-Riemannian manifolds

Mantica

Suh

2017

Rev. Math. Phys.

View full text Add to dashboard Cite

Conformally quasi-recurrent (CQR)n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was introduced recently by the authors and applies to significative space-times), (CQR)n manifolds with concircular fundamental vector are quasi-Einstein. For 4-dimensional (CQR) 4 Lorentzian manifolds the fundamental vector is null and unique up to a scaling, it is an eigenvector of the Ricci tensor, and its integral curves are geodesics. Such space-times are Petrov type-N with respect to the fundamental null vector.

show abstract

“…(8.11)]; the recurrence properties of the Weyl curvature tensor of pp-waves are reported in [116]: under certain conditions they are complex recurrent space-times (see [59,61,89]) with closed recurrence parameter. The condition A i C jklm + A j C kilm + A k C ijlm = 0 on a pseudo Riemannian manifold has been extensively studied among others in [33,45,46,48,78,85] [38,36,76,78,91]):…”

Section: Remark 45mentioning

confidence: 99%

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

Mantica

Suh

2016

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

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.

show abstract

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

Cited by 24 publications

References 18 publications

Curvature properties of Gödel metric

Curvature properties of Gödel metric

On weakly conformally symmetric pseudo-Riemannian manifolds

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

Contact Info

Product

Resources

About