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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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TL;DR: In this article, a set of irreducible invariants involving the above mentioned tensor variables has been constructed, together with the invariants of the single argument tensors, the system of simultaneous or joint invariants is considered.
26 citations
TL;DR: In this paper, the anti-de Sitter-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives were constructed in analogy with the pp-wave solutions.
Abstract: We construct the anti–de Sitter-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two-tensor built from the Riemann tensor and its derivatives can bewritten in terms of the traceless Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples of our general formalism, we work out the six-dimensional conformal gravity and its nonconformal deformation as well as the tricritical gravity, the Lanczos-Lovelock theory, and string-generated cubic curvature theory.
26 citations
TL;DR: In this paper, the authors considered metric gab with conformally flat 3-space and the Ricci tensor has at most two different eigenvalues and the 4-velocity is an eigenvector of this tensor.
Abstract: Metrics of the form ds2=N2(x1, xn)(dx1)2+gab(xn) xadxb are considered which are subject to the conditions that the time-like 3-space (with metric gab) is conformally flat, that its Ricci tensor has at most two different eigenvalues and that the 4-velocity is an eigenvector of this Ricci tensor. The perfect fluid (or dust) solutions are necessarily of Petrov type D or O, and in the general case they do not admit a Killing vector. All rotating solutions are given explicitly. The non-rotating solutions are either conformally flat (and thus known) or (if of type D) contained in the class of solutions investigated, for example, by Szekeres (1975), Tomimura (1977), and Szafron and Wainwright (1977).
26 citations
26 citations
TL;DR: In this article, a new kind of Riemannian manifold is introduced, named (ZRF)n, which generalizes weakly Z-symmetric and pseudo-Z-Symmetric manifolds, and the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor is confirmed for(ZRFn)n with rank(Zkl) > 2.
Abstract: 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 (ZRF)n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank(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(ZRF)n manifolds with rank(Zkl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)n and weakly Z-symmetric manifolds (WZS)n in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)n and give a brief account of Z recurrent forms on Kaehler manifolds.
26 citations