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Showing papers on "Riemann curvature tensor published in 1977"


Journal ArticleDOI
K.S. Stelle1
TL;DR: The necessary Slavnov identities are derived from Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields.
Abstract: Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields (and of the BRS transformations). Fortunately, these artifactual divergences may be eliminated by letting the coefficient of the harmonic gauge-fixing term tend to infinity, thus considerably simplifying the renormalization procedure. Coupling to other renormalizable fields may then be handled in a straightforward manner.

2,429 citations


Journal ArticleDOI
TL;DR: In this article, a unified geometric formulation of gravitation and supergravity is presented, which is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1, 4) for supergravity.
Abstract: A unified geometric formulation of gravitation and supergravity is presented. The action for these theories is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1,4) for supergravity. The requirement of invariance under reflections, local Lorentz transformations, and general coordinate transformations uniquely determines the action and ensures the existence of local supersymmetry in supergravity.

744 citations


Journal ArticleDOI
TL;DR: In this article, the authors study the regularization of the stress energy tensor for massless vector and massless and massive scalar particles propagating in a general background metric, using covariant point-separation techniques.

132 citations


Journal ArticleDOI
TL;DR: A necessary and sufficient condition for a Randers space to be of scalar curvature, found under some assumptions by this time, is given in this article in simple form with a geometrical meaning.

65 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the Chern-Moser invariants must vanish if the transformation group is noncompact and the hypersurface is compact, which is a Lie transformation group.
Abstract: The group of biholomorphic transformations leaving fixed a strongly pseudoconvex real hypersurface in a complex manifold is a Lie group. In this paper it is shown that the Chern-Moser invariants must vanish if this group is noncompact and the hypersurface is compact. Also considered are transformation groups of flat hypersurfaces and intransitive groups. Introduction. The purpose of this paper is to study the group of structure preserving, or pseudoconformal, transformations of a real hypersurface M (of dimension 2n + 1) in a complex n + 1 manifold, or more generally, of a manifold with the same C-R structure. We will always assume that the C-R structure is integrable and that the Levi form is nondegenerate. It follows from the work of Chern and Moser [2] that this group is a Lie transformation group. There is some similarity between the geometry of M and the conformal geometry of a Riemannian manifold. Over M there is a principal fibre bundle B (the pseudoconformal bundle), a connection on B, and curvature invariants the most important of which is the fourth order curvature tensor SAp, (see [2]). These have as analogues in the Riemannian case the bundle of conformal frames the conformal connection, and the Weyl conformal curvature tensor. If n > 2, S vanishes if and only if M is pseudoconformally flat, i.e., locally equivalent to the standard sphere S2n+1 in Cn+1 [8]. Locally the structure of M is given by a real one-form 9, n complex one-forms 9 a, and their complex conjugates 9a = 0a. They satisfy OAOlA . . . A on AOlA . . . A n :7 o and the equation d9 ig,,j9a A 9 mod 9. Here and throughout this paper Greek indices run from 1 to n and the summation convention is used. The nondegenerate hermitian matrix (ga) is Received by the editors February 12, 1976. AMS (MOS) subject classifications (1970). Primary 57E20, 53A55; Secondary 32F99.

57 citations


Journal ArticleDOI
TL;DR: A computer program is described, written in the symbolic manipulation language CAMAL, which performs this calculation of the curvature tensor of space-time, using the Newman-Penrose equations.
Abstract: The curvature tensor of space-time can be described most concisely by giving the components of the Weyl and Ricci tensors relative to a complex null tetrad. The Newman-Penrose equations provide a simple and direct algorithm for calculating these components. This paper describes a computer program, written in the symbolic manipulation language CAMAL, which performs this calculation. Comparisons are made with the classical tensorial method of calculation, and some applications are discussed.

50 citations


05 Sep 1977
TL;DR: In this paper, the local rate of massless particle production in a weakly anisotropic homogeneous cosmological model and a weak inhomogeneous gravitational field was calculated in which the condition q/sup i/q/sub i/> or = 0 is satisfied for the wave vectors of the nonzero Fourier components in second order perturbation theory.
Abstract: The local rate of production of massless particles is calculated in a weakly anisotropic homogeneous cosmological model, and also in a weak inhomogeneous gravitational field, in which the condition q/sup i/q/sub i/> or =0 is satisfied for the wave vectors of the nonzero Fourier components in second-order perturbation theory The rate turns out to be proportional to the local values of the invariants of the curvature tensor

44 citations



Journal ArticleDOI
TL;DR: In this article, the Ricci tensor from Phijk has been studied from the viewpoint of physics, and a special form of the curvature tensor Shijk is proposed and a problem relating to Phijk is studied.

31 citations


Journal ArticleDOI
TL;DR: In this article, the Gauss-Codazzi-Ricci equations governing the local isometric embedding of Riemannian spaces are interrelated by the Bianchi identities.
Abstract: The Gauss-Codazzi-Ricci equations governing the local isometric embedding of Riemannian spacesV n ⊂vn (N=n + P, P > 0) are interrelated by the Bianchi identities inV n andV N. This leads to redundancies which permit great simplification in the embedding problem, i.e. allows a neglect of part of the equations. By transcription, to the case of semi-Riemannian spaces, of a result of R. Blum we obtain a number of theorems and corollaries expressing forV n ⊂ VN this interdependency of the Gauss-Codazzi-Ricci equations. They form a generalization of previous results and are felt to be useful for the study of the geometrical properties of space-time and its three-dimensional space sections.

21 citations


Journal ArticleDOI
01 Feb 1977
TL;DR: A complete Kihler metric of positive Riemannian sectional curvature on C n was constructed in this paper, which is the only known complete non-compact noncompact Kahler metric with positive Ricci curvature.
Abstract: A complete Kihler metric of positive curvature on C' is constructed and its importance is discussed. The purpose of this paper is to exhibit an example of a complete Kahler metric on Cn with strictly positive Riemannian sectional curvature. To accomplish this, let r2= S. 1ziz on Cn and consider metrics on Cn of the form g,-= a 2 (r2)/azij, r f(r2) E C'(R). We shall describe the conditions on f which make g,a complete metric of positive curvature, and then show that if f(x) = fx (ln(1 + T)/ T) dT, these conditions are satisfied, so for this f, gjis a C X complete Kahle1 metric of strictly positive curvature on Cn. Previously, there have been no known examples of complete noncompact Kahler manifolds of positive sectional curvature. Nevertheless several theorems have been proved regarding the structure of such manifolds. In particular, such manifolds are Stein manifolds [3], they are real diffeomorphic to R 2n [5], and they admit no nonconstant bounded holomorphic functions [6]. For these and other reasons, they have been conjectured to be biholomorphic to Cn [4]. Thus, the existence of a complete Kahler metric of positive sectional curvature on Cn is not surprising; but it is not trivial. One can easily show that C admits a complete Kahler metric of positive curvature: By the existence of isothermal coordinates [2] any complete metric on R2 of positive sectional curvature is a Hermitian metric of positive sectional curvature relative to some complex structure; the resulting complex manifold is in fact C as a consequence of the Blanc Fiala Theorem [1] or of the general result quoted on the nonexistence of nonconstant bounded holomorphic functions. If one then takes products of C with itself one obtains a complete Kahler metric of nonnegative sectional curvature and positive Ricci curvature on Cn. However, there is no obvious way to perturb this metric to obtain a complete Kahler metric of positive sectional curvature, for there is difficulty in finding a perturbation which gives a merely Hermitian metric of positive sectional curvature and, in addition, there is the difficulty of maintaining the Kahler condition, which is given, in effect, by a differential equation. Here, the problem of satisfying the Kahler condition is solved by considering only Received by the editors October 12, 1976 AMS (MOS) subject classifications (1970). Primary 53C55; Secondary 52E10.

Journal ArticleDOI
TL;DR: In this paper, the geometry of superspaces with Bose and Fermi-type coordinates is presented from a coordinate independent point of view, and various geometrical quantities of conventional manifolds are generalized so as to be applicable to superspaces.
Abstract: The geometry of superspaces with Bose‐ and Fermi‐type coordinates is presented from a coordinate independent point of view. Various geometrical quantities of conventional manifolds are generalized so as to be applicable to superspaces. It is shown that these generalizations can be basically arrived at algebraically by replacing, in the definitions of various geometrical quantities, the Lie derivative of the conventional manifolds with a generalized graded Lie bracket. Explicit expressions for connection coefficients, Riemann curvature tensor, etc., are derived. The general formalism is then applied to graded Lie bundles the relevance of which to supergravity theories is demonstrated.

Journal ArticleDOI
TL;DR: In this paper, the authors studied complete -dimensional surfaces of non-positive extrinsic 2-dimensional sectional curvature in Euclidean space, in the sphere, in the complex projective space, and in a Riemannian space.
Abstract: This article studies complete -dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive -dimensional curvature (for even) in Euclidean space , in the sphere , in the complex projective space , and in a Riemannian space . If the embedding codimension is sufficiently small, then a compact surface in or is a totally geodesic great sphere or complex projective space, respectively. If is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space , then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive -dimensional curvature cannot be isometrically immersed as a surface of small codimension. The order of growth of the volume of complete noncompact surfaces of nonpositive -dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.Bibliography: 25 titles.

Journal ArticleDOI
TL;DR: A general discussion of the recurrence properties of the Riemann, Ricci and Weyl tensors is given in this paper, where space-times possessing these properties are classified according to the Petrov type of the Ricci tensor and the Legre tensor.
Abstract: A general discussion of the recurrence properties of the Riemann, Ricci and Weyl tensors is given Space-times possessing these properties are classified according to the Petrov type of the Weyl tensor and the Legre type of the Ricci tensor The proofs of some known theorems are shortened and some new results are given



Journal ArticleDOI
TL;DR: In this article, the authors define curves on a Riemannian manifold as integrals of generalized Jacobi fields and show that the force term that deviates the trajectory from the geodesic motion can be constructed as a functional of the metric tensor.
Abstract: We define curves on a Riemannian manifold as integrals of generalized Jacobi fields. We show that the force term that deviates the trajectory from the geodesic motion can be constructed as a functional of the metric tensor. These curves can be interpreted as particles (observers) coupled nonminimally with gravitation that can provide a class of residual observers for the inevitable singularity—as shown in the text.

Journal ArticleDOI
TL;DR: In this article, the Riemann tensor is found to admit a one-parameter group of motions and the Hamilton-Jacobi equation is (partially) separable.
Abstract: All empty space-times admitting a one-parameter group of motions and in which the Hamilton-Jacobi equation is (partially) separable are obtained Several different cases of such empty space-times exist and the Riemann tensor is found to be either type D or N The results presented here complete the search for empty space-times with separable Hamilton-Jacobi equation

Dissertation
01 Jan 1977

Journal ArticleDOI
TL;DR: In this paper, it was shown that both the torsion tensor as well as the Riemann-Christoffel curvature tensor are necessary in order to describe completely the closure failure associated with a Burgers circuit.
Abstract: The methods of differential geometry have been applied to the Burgers circuit taken about a wedge diaclination. It is shown that both the torsion tensor as well as the Riemann-Christoffel curvature tensor are necessary in order to describe completely the closure failure associated with such a circuit. In particular, both tensors measure the dislocation contribution associated with the disclination.

Journal ArticleDOI
TL;DR: In this paper, it was pointed out that the Yang theory of gravitation is not mathematically well founded, and the Palatini variational method used by Yang in a gauge theory of gravity was criticised.
Abstract: Attention is drawn to difficulties associated with gravitational theories based on Lagrangians formed from quadratic invariants of the Riemann tensor, and in particular with the Palatini variational method used by Yang in a gauge theory of gravitation. It is pointed out that the Yang theory is not mathematically well founded.

Journal ArticleDOI
TL;DR: In this article, the Ricci tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate, and linear, second-order differential expressions have been found for these two functions.
Abstract: The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.


Book ChapterDOI
01 Jan 1977
TL;DR: In this paper, the authors first proved the reduction of codimension of minima-1 immersions to a (n+l)-dimensiona1 space of constant curvature.
Abstract: In this paper we first prove the fo11owing theorem on reduction of codimension of minima1 immersions: Theorem 1 - Let x: Mn→X be a minima1 immersion of an n-dimensiona1 connected manifold Mn into an (n+l)-dimensiona1 space X of constant curvature. Assume that the curvature tensor of the norma1 connexion is paral1e1 in the norma1 bundle and the first norma1 space of the immersion has constant dimension k.




Journal ArticleDOI
TL;DR: In this paper, a first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensorRμνkλ in terms of the affine connection and metric, and the definition of a set of gravitational superpotentials closely connected with the Komar conservation laws.
Abstract: A first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensorRμνkλ in terms of the affine connection and metric; the definition of the affine connection in terms of the metric; the Einstein field equations; and the definition of a set of gravitational “superpotentials” closely connected with the Komar conservation laws [7]. Substitution of the definition of the affine connection into this Lagrangian results in a second-order Lagrangian, from which follow the definition of the fully covariant Riemann tensor in terms of the metric, the Einstein equations, and the definition of the gravitational “superpotentials”.


Journal ArticleDOI
Ho Jung Paik1
TL;DR: In this paper, the response of a disk antenna to a Riemann tensor with six possible modes of polarization was analyzed and only the monopole and quadrupole modes of the antenna were found to couple to an arbitrary gravitational wave.
Abstract: Response of a disk antenna to a completely general Riemann tensor with six possible modes of polarization is analyzed. Only the monopole and quadrupole modes of the antenna are found to couple to an arbitrary gravitational wave. The absorption cross sections of these modes for scalar and tensor waves are calculated numerically. It is pointed out that, with two local disk detectors oriented 90degree with respect to each other, one can not only determine the incident angles and polarization of the wave but also eliminate spurious non-gravitational-wave signals. (AIP)