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Showing papers on "Ricci decomposition published in 1979"


Journal ArticleDOI
TL;DR: In this paper, the decomposition of tensor products of holomorphic discrete series representations was studied in the context of generalized Verma modules, and the results of this paper can be read off as easy corollaries.
Abstract: We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL 2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions. For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors. The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

78 citations




Journal ArticleDOI
TL;DR: In this paper, it was shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2form and the exterior derivative of related 2-forms are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group.
Abstract: It is shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2-form and the exterior derivative of a related 2-form are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group with nonsingular orbits

32 citations


Journal ArticleDOI
TL;DR: In this paper, the curvature tensor is expressed as a function of the motion of the source of this curvature, and the behavior of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is used to compute the asymptotic behavior of corresponding curvatures along past-directed null straight lines.
Abstract: The standard weak-field, slow-motion approximation to Einstein's relativistic theory of gravitation is used to express the curvature tensor, up to order ${r}^{\ensuremath{-}5}$ on a flat background space-time, as a functional of the motion of the source of this curvature. The behavior, in the distant past, of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is then used to compute the asymptotic behavior of the corresponding curvature tensor along past-directed null straight lines in the flat background. It is found, on the one hand, that the falloff of the curvature is fast enough to guarantee satisfaction of a condition to exclude incoming gravitational radiation. On the other hand, the falloff is slower than would have been expected if the conformally rescaled curvature tensor had been regular on the hypersurface at past null infinity of the flat background.

20 citations


Journal ArticleDOI
TL;DR: In this paper, an algebraic classification of the Ricci tensor is given in terms of its invariant two-space structure, which involves classifying a complex fourth-order tensor which is algebraically equivalent to the trace-free Ricci Tensor and which has all the algebraic symmetries of the (complex) Riemann tensor.
Abstract: An algebraic classification of the Ricci tensor is given in terms of its invariant two-space structure. The method involves classifying a complex fourth-order tensor which is algebraically equivalent to the trace-free Ricci tensor and which has all the algebraic symmetries of the (complex) Riemann tensor.

19 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that Cotton's tensor density is not the Euler-Lagrange expression corresponding to a scalar density built from one metric tensor.
Abstract: It is well known that a necessary and sufficient condition for the conformal flatness of a three‐dimensional pseudo‐Riemannian manifold can be expressed in terms of the vanishing of a third‐order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.

16 citations


Journal ArticleDOI
TL;DR: In this article, the connection between the critical point structure of the Riemannian curvature function and the Petrov classification of the Ricci tensor has been investigated, and a similar function is defined whose critical point structures are connected with the algebraic classification of RicCI tensors.
Abstract: Some theorems proved by Thorpe concerning the connection between the critical point structure of the Riemannian (sectional) curvature function and the Petrov classification are extended. A similar function is defined whose critical point structure is connected with the algebraic classification of the Ricci tensor.

12 citations



Journal ArticleDOI
TL;DR: In this paper, the behavior of the electromagnetic field tensor is examined under symmetry mappings, including Ricci collineations, curvature and Ricci curvature, and the Ricci Ricci Collineations family.
Abstract: General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.

8 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of locally imbedding a null hypersurface in a Riemannian manifold was studied and the generalized Gauss-Codazzi equations were derived.
Abstract: This paper is concerned with the problem of locally imbedding a null hypersurface in a Riemannian manifold. More precisely, on a one‐parameter family of null hypersurfaces, rigged by an arbitrary null vector field, in a four‐dimensional space–time manifold, a particular symmetric affine connection is used to derive the corresponding generalized Gauss–Codazzi equations. In addition, expressions are obtained for the projections of the Ricci tensor, which are relevant to the characteristic initial‐value problem of general relativity.


Journal ArticleDOI
C. Hoenselaers1
TL;DR: In this article, the invariants of the Weyl tensor for the Tomimatsu-Sato δ = 3 solution were calculated for the 3-approximation.
Abstract: The calculation of the invariants of the Weyl tensor for the Tomimatsu-Sato δ = 3 solution is reported.

Journal ArticleDOI
01 Sep 1979
TL;DR: The present paper has studied some recurrent properties of pseudocurvature tensor in ann-dimensional Finsler space Fn.
Abstract: Sinha [2] has defined pseudocurvature tensor field in ann-dimensional Finsler spaceF n. In the present paper we have studied some recurrent properties of pseudocurvature tensor.

Journal ArticleDOI
TL;DR: In this article, the well-known algebraic classification of the electromagnetic tensor field is used to provide the space-time manifold of general relativity with the latest technique of differential geometry.
Abstract: The well-known algebraic classification of the electromagnetic tensor field is used to provide the space-time manifold of general relativity with the latest technique of differential geometry.


Journal ArticleDOI
TL;DR: In this article, it was shown that a square invariant Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation.
Abstract: It is shown that a square invariant of the Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation. Moreover we show that such a Lagrangian implicitly includes a conformally invariant theory characterized by two gauge fields and the metric tensor.