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Suh, Young Jin,Mantica, Carlo Alberto,De, Uday Chand,Pal, Prajjwal World Scientific 2017 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODE Vol.14 No.9
<P>In this paper, we introduce a new tensor named <TEX>$ B$</TEX>-tensor which generalizes the <TEX>$ Z$</TEX>-tensor introduced by Mantica and Suh [Pseudo <TEX>$ Z$</TEX> symmetric Riemannian manifolds with harmonic curvature tensors, <I>Int. J. Geom. Methods Mod. Phys.</I><B>9</B>(1) (2012) 1250004]. Then, we study pseudo-<TEX>$ B$</TEX>-symmetric manifolds <TEX>$ (PBS)_{n}$</TEX> which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo <TEX>$ Z$</TEX> symmetric Riemannian manifolds with harmonic curvature tensors, <I>Int. J. Geom. Methods Mod. Phys.</I><B>9</B>(1) (2012) 1250004]. At first, we prove the existence of a <TEX>$ (PBS)_{n}$</TEX>. Next, we prove that a pseudo-Riemannian manifold is <TEX>$ B$</TEX>-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a <TEX>$ (PBS)_{n}$</TEX> to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a <TEX>$ (PBS)_{n}$</TEX> if the <TEX>$ B$</TEX>-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-<TEX>$ B$</TEX>-symmetric manifolds and prove that a <TEX>$ (PBS)_{n}(n > 3)$</TEX> spacetime is a <TEX>$ pp$</TEX>-wave under certain conditions.</P>
Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves
Mantica, Carlo Alberto,Suh, Young Jin World Scientific 2016 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODE Vol.13 No.2
<P>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.</P>
A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMES
Mantica, Carlo Alberto,Molinari, Luca Guido Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.2
In this note we show that Lorentzian Concircular Structure manifolds $(LCS)_n$ coincide with Generalized Robertson-Walker space-times.
On weakly conformally symmetric pseudo-Riemannian manifolds
Mantica, Carlo Alberto,Suh, Young Jin World Scientific Publishing Company 2017 Reviews in mathematical physics Vol.29 No.3
<P>In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the <TEX>$ 4$</TEX>-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in <TEX>$ n \geq 5$</TEX>. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric <TEX>$ 4$</TEX>-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.</P>