Search

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the -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 . 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 -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. [ABSTRACT FROM AUTHOR]

In this paper we present some new results about -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 and a pp-wave space-time in . In all cases an algebraic classification for the Weyl tensor is provided for 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 , , space-time is conformal to the Robertson-Walker space-time. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński-Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds ()n and pseudo-concircular symmetric manifolds is defined. This is named pseudo-Q-symmetric and denoted with ()n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A 44 (1992) 1-34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski-Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat ()n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of ()n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a ()n scalar field space-time is considered, and interesting properties are pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named ()n. The main result of the paper is that the closedness property of the associated covector is achieved also for (Zkl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for()n manifolds with (Zkl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric ()n and weakly Z-symmetric manifolds ()n in the case of non-singular Z tensor. In the last sections we study special conformally flat ()n and give a brief account of Z recurrent forms on Kaehler manifolds. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15-26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1-34]). Finally, we study some properties of (PZS)4 spacetime manifolds. [ABSTRACT FROM AUTHOR]

In order to give a new proof of a theorem concerned with conformally symmetric Riemannian manifolds due to Roter and Derdzinsky [8], [9] and Miyazawa [15], we have adopted the technique used in a theorem about conformally recurrent manifolds with harmonic conformal curvature tensor in [3]. In this paper, we also present a new proof of a successive refined version of a theorem about conformally recurrent manifolds with harmonic conformal curvature tensor. Moreover, as an extension of theorems mentioned above we prove some theorems related to quasi conformally recurrent Riemannian manifolds with harmonic quasi conformal curvature tensor. [ABSTRACT FROM AUTHOR]