Showing papers on "Ricci decomposition published in 1980"

Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor

Abstract: S being the Ricci tensor. While every manifold with parallel Ricci tensor has harmonic curvature, i.e., satisfies fiR=O, there are examples ([3], Theorem 5.2) of open Riemannian manifolds with fiR=O and VS+O. In [1] Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this paper is to give examples (see Remark 2) answering this question in the negative. All our examples are conformally flat (Corollary 1). Moreover, we obtain some classification results, restricting our consideration to Riemannian manifolds with fiR = O, VS + 0 and such that the Ricci tensor S has at any point less than three distinct eigenvalues. Starting from a description of their local structure at generic points (Theorem 1), we find all four-dimensional, analytic, complete and simply connected manifolds of this type (Theorem 2). They are all non-compact, but some of them do possess compact quotients. Next we prove (Theorem 3) that all compact four-dimensional analytic Riemannian manifolds with the above properties are covered by S 1 x S 3 with a metric of an explicitly described form. Throughout this paper, by a manifold we mean a connected paracompact manifold of class C ~ or analytic. By abuse of notation, concerning Riemannian manifolds we often write M instead of (M,g) and @ , v ) instead of g(u,v) for tangent vectors u, v.

Tensor products of unitary representations of the three-dimensional lorentz group

Abstract: The author obtains a decomposition into irreducible representations of the tensor product of any two irreducible unitary representations of the group SO0(1,2). An explicit construction of this decomposition is given, and the corresponding Plancherel measure is found. Bibliography: 13 titles.

Torsion and Related Concepts: An Introductory Overview

Abstract: The aim of this paper is to provide an overview of all the basic aspects of the torsion of a manifold, with particular stress on the expressions in an anholonomic basis. After a brief review of anholonomic bases and Koszul covariant derivative, we show how the expressions for the torsion and the Riemann tensors in a general (anholonomic) basis arise from their expressions in a coordinate basis. We further derive the expression for the contortion tensor, which arises from the requirement that an affine connection with torsion be metric (preserving). The latter requirement is related to the equivalence principle, whose mathematical aspects in a manifold with torsion are discussed next. Finally, we derive the expression for the distortion tensor, which is an analog of the curvature tensor but arising from the torsion rather than the metric tensor.

Conformal coupling of gravitational wave field to curvature

Abstract: Conformal properties of the equations for weak gravitational waves in a curved space–time are investigated. The basic equations are derived in the linear approximation from Einstein's equations. They represent, in fact, the equations for the second-rank tensor field hαβ, restricted by the auxiliary conditions hαβ;α=0, h=γαβhαβ=0, and embedded into the background space–time with the metric tensor γαβ. It is shown that the equations for h are not conformally invariant under the transformations γαβ=e2σγαβ and hαβ=eσhhαβ, except for those metric rescalings which transform the Ricci scalar R of the original background space–time into e−2σR, where R is the Ricci scalar of the conformally related background space–time. The general form of the equations for hαβ which are conformally invariant have been deduced. It is shown that these equations cannot be derived in the linear approximation from any tensor equations which generalize the Einstein equations.

On a coordinate-invariant description of Riemannian manifolds

Abstract: A coordinate-invariant description of a Riemannian manifold is known to be furnished by the curvature tensor and a finite number of its covariant derivatives relative to a field of orthogonal frames. These tensors are closer to measurements than the metrical tensor is. The present article discusses this description's usefulness in general relativity and the redundancy among the curvature tensor and its derivatives.

A natural scalar field in the einstein gravitational theory

Abstract: In a Riemannian space-time, the difference between the third-order tensor potentialH αβλ of the Riemann tensor (presented in a precedent paper) and the Lanczos generating function of the Weyl tensor is here shown to be characterized by a vectorV α , obtained by contractionH αβλ . The significant role of such a vector, in the context of general relativity, is then discussed. Particular attention is paid to the scalar potential ϑ which characterizes the irrotational part ofV α : such a scalar field satisfies a space-time wave equation of the Poisson type. Weak fields are also considered: in the particular case of a static metric, the scalar ϑ is found to be proportional to the classic Newtonian potential.

Four-dimensional sigma-model coupled to the metric tensor field

Abstract: We discuss the four-dimensional nonlinear sigma-model with an internalO(n) invariance coupled to the metric tensor field satisfying Einstein equations. We derive a bound on the coupling constant between the sigma-field and the metric tensor using the theory of harmonic maps. A special attention is paid to Einstein spaces and some new explicit solutions of the model are constructed.

On algebraically special space–times with nontwisting rays

Abstract: With the aid of the Newman–Penrose formalism and Penrose’s conformal technique Einstein’s gravitational field equations are first solved exactly as far as is possible for arbitrary sources. It is assumed, however, that space–time is algebraically special with hypersurface‐orthogonal geodesic and shear‐free rays. Special cases are considered. Next, the asymptotic behavior of the components of the metric tensor, the Weyl tensor, the Ricci tensor and the spin coefficients is determined in a suitable frame. Einstein–Maxwell space–times with the above properties are treated in some detail.

Einstein Tensor in Scalar Curvature Finsler Spaces

Abstract: Einstein tensor of Finsler space which has internal coordinate dependent scalar curvature is presented. Einstein Tensor in Finsler-Raumen mit skalarer Krummung Im folgenden wird der Einstein-Tensor in Finsler-Raumen mit von internen Koordinaten abhangiger skalarer Krummung dargestellt.

Classification of Ricci tensor in space-times with a four- parameter group of motions acting on null hypersurfaces

Abstract: The Plebanski type of the Ricci tensor is given for all Bianchi type metrics admitting a four-parameter group of motions acting on null hypersurfaces using the Ludwig-Scanlan classification method and the Newman-Penrose formalism. A class of solutions is found for which the Ricci tensor has a null eigenvector.

Asymptotic behavior of gravitational fields in a type II coordinate system

Abstract: With the aid of Penrose’s conformal technique the asymptotic behavior of the components of the metric tensor, the Weyl tensor, the Ricci tensor and the spin‐coefficients is calculated for a large class of space‐times that includes the NUT (Newman–Unti–Tamburino) solution as well as all asymptotically flat space‐times. The calculations are done in a coordinate system associated not with null hypersurfaces but with an asymptotically shearfree twisting null congruence. For vacuum the results presented here reduce to those of Aronson and Newman to the order given in their paper.

Classification of the Ricci tensor in space-times admitting a four-parameter group of motions acting on non-null hypersurfaces

Abstract: The Plebanski type of the Ricci tensor for all metrics admitting a four-parameter group of motions acting on non-null (space-like or time-like) hypersurfaces is determined by using the Newman-Penrose formalism and a classification method due to Ludwig and Scanlan. For each metric, all possible degeneracies of the Plebanski type are considered.