Showing papers on "Ricci decomposition published in 1968"
TL;DR: In this article, the authors conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3, where m is a complete hypersurface in a Euclidean space.
Abstract: where the endomorphism R(X, Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (•*) on the curvature tensor field R imply that M is locally symmetric (i.e. Vi? = 0) ? We conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3 . For partial and related results, see [4], p.ll, [9], Theorem 8, and [6]. The main purpose of the present paper is to give an affirmative answer in the case where M is a complete hypersurface in a Euclidean space. More precisely, we prove
92 citations
TL;DR: In this paper, the classification of symmetric second-rank tensors in Minkowski space and its application to the Einstein tensor is reviewed; it is shown that, for spherically symmetric metrics, the EPT always has a spacelike double eigenvector; and the possible types of EPTs that this degeneracy allows are discussed.
Abstract: The classification of symmetric second‐rank tensors in Minkowski space and its application to the Einstein tensor is reviewed. It is shown that, for spherically symmetric metrics, the Einstein tensor always has a spacelike double eigenvector; and the possible types of Einstein tensor that this degeneracy allows are discussed. A complete classification of all spherically symmetric metrics with two double eigenvalues is given. A study of the timelike eigencongruence, in the case when one timelike and two spacelike eigenvectors exist, is carried out. Canonical forms for the metric, the Einstein tensor, and the Weyl tensor (which is always of type D) are given for each of the various possible types.
44 citations
TL;DR: In this paper, the reduction of n-fold tensor products of induced unitary representations of noncompact groups into irreducible constituents is shown. And the coefficients of the Clebsch-Gordon coefficients are calculated.
Abstract: The reduction ofn-fold tensor products of induced unitary representations of noncompact groups into irreducible constituents is shown. Clebsch-Gordon coefficients are then calculated. The technique is applied to then-fold tensor products of the positive mass representations of the Poincare group.
12 citations
TL;DR: In this paper, the topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied.
Abstract: The topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied. It is shown that for such space-times of Bel Case III the metric tensor is singularity-free and that the Pontrjagin number identically vanishes. Bel Cases I and II are examined and conditions are given for which the metric is singularity-free and the Pontrjagin number vanishes. Applications to gravitional radiation in general relativity are discussed.
2 citations
TL;DR: In this paper, a pictorial representation of the Bel-Petrov-Penrose classification of the Weyl conformal tensor is presented in the form of a series of intersecting manifolds nested in a four-dimensional projective space.
Abstract: A (2j+1)-spinor formalism is used to discuss the Bel-Petrov-Penrose classification of the Weyl conformal tensor A convenient pictorial representation of this classification is presented in the form of a series of intersecting manifolds nested in a four-dimensional projective space The relation to other formalisms is considered briefly