# On pseudo Ricci symmetric manifolds

01 Jan 1988-Vol. 15, pp 526-531

About: The article was published on 1988-01-01 and is currently open access. It has received 126 citations till now.

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TL;DR: WeakWeakly Z-symmetric (WZS) as mentioned in this paper is a Riemannian manifold that includes weakly-, pseudo-and pseudo projective Ricci symmetric manifolds.

Abstract: We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n .I f theZ tensor is singular we give condi- tions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally har- monic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.

70 citations

### Cites background from "On pseudo Ricci symmetric manifolds..."

...In [2] (PRS)n with harmonic curvature tensor (i....

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01 Jun 2010

TL;DR: In this paper, the Eisenhart problem of finding parallel tensors for the symmetric case in the regular f-Kenmotsu framework is solved for the Ricci tensors.

Abstract: The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular f-Kenmotsu framework. In this way, the Olszack-Rosca example of Einstein manifolds provided by f-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our result is interpreted in terms of Ricci solitons and special quadratic first integrals.

59 citations

### Cites background from "On pseudo Ricci symmetric manifolds..."

...The same condition was sometimes called generalized pseudo-Ricci symmetric manifold ([12]) or simply pseudo-Ricci symmetric manifold ([4])....

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TL;DR: In this article, the authors introduced a new notion of Z-tensor and a new kind of Riemannian manifold called pseudoZ symmetric manifold and denoted by (PZS)n.

Abstract: 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.

51 citations

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TL;DR: In this article, the curvature tensors of Ricci solitons in a perfect fluid spacetime are described in terms of different curvatures tensors and conditions for the Ricci Solitons to be steady, expanding or shrinking are also given.

Abstract: Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and η-Ricci and η-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field ξ of the soliton is of gradient type, ξ:= grad(f), we derive a Poisson equation from the soliton equation.

40 citations

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TL;DR: In this paper, it was shown that a general relativistic space-time with covariant-constant energy-momentum tensors is Ricci symmetric, and two particular types of such general space-times were considered and determined.

Abstract: It is shown that a general relativistic space-time with covariant-constant energy-momentum tensor is Ricci symmetric. Two particular types of such general relativistic space-times are considered and the nature of each is determined.

37 citations