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Showing papers on "Riemann curvature tensor published in 1984"


Journal ArticleDOI
TL;DR: In this paper, a new global idea was introduced to solve the Yamabe problem in dimensions 3, 4, and 5, and the existence of a positive solution u on M of the equation was proved in all remaining cases.
Abstract: A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation

1,303 citations


Journal ArticleDOI
TL;DR: In this paper, the structure of the gravitational field at infinity of asymptotically anti-de Sitter space-time is analyzed in detail using conformal techniques, and definitions of 'conserved' quantities at J in terms of the curvature tensor which are free of the ambiguities present in the previous definitions are introduced.
Abstract: The structure of the gravitational field at infinity of asymptotically anti-de Sitter space-time is analysed in detail using conformal techniques. It is found that the situation differs from that in the case of asymptotically Minkowskian space-times in a number of respects, primarily because J is now time-like rather than null. In particular, the asymptotic symmetry group is quite different from the BMS group, and there is no analogue of the Bondi news. The analysis also introduces definitions of 'conserved' quantities at J in terms of the curvature tensor which are free of the ambiguities present in the previous definitions based on the deviation of the physical metric from an anti-de Sitter background.

417 citations


Journal ArticleDOI
TL;DR: In this paper, the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces have been studied in the special case of scalar curvature.
Abstract: We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.

361 citations


Journal ArticleDOI
TL;DR: In this article, the quantization of antisymmetric tensor fields on an n-dimensional riemannian manifold is studied, and the connection between quantized tensor field of ranks k−1 and n−k−1 is analyzed.

51 citations


Journal ArticleDOI
TL;DR: In this article, the Ricci tensor uniquely determines the Riemannian structure and conditions that a doubly covariant tensor has to satisfy in order to be the Riccis tensor for a given structure.
Abstract: We investigate whether the Ricci tensor uniquely determines the Riemannian structure, and we give conditions that a doubly covariant tensor has to satisfy in order to be the Ricci tensor for some Riemannian structure.

41 citations



Journal ArticleDOI
TL;DR: In this article, a modified ansatz for the curvature tensor involving double duality was developed in order to reduce the two field equations of the Poincare gauge theory of gravity.
Abstract: A rather general procedure is developed in order to reduce the two field equations of the Poincare gauge theory of gravity by a modified ansatz for the curvature tensor involving double duality. In the case of quasilinear Lagrangians of the Yang–Mills type it is shown that nontrivial torsion solutions with duality properties necessarily ‘‘live’’ on an Einstein space as metrical background.

30 citations


Book ChapterDOI
Zhong Jia-Qing1
01 Jan 1984
TL;DR: In this paper, the authors proved the strong rigidity of compact quotients of irreducible bounded symmetric domains of dimension at least two, and proposed the following conjecture: complex-analyticity of harmonic maps between two Kahler manifolds under some conditions.
Abstract: In [1], [2] Siu discovered the complex-analyticity of harmonic maps between two Kahler manifolds under some conditions and prove the strong rigidity of compact quotients of irreducible bounded symmetric domains of dimension at least two. Furthermore, he proposed the following conjecture:

25 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the Riemann tensor admits a linear representation in terms of the covariant derivatives of a suitable potential tensor of rank 3, at least for a class of spacetime geometries including several physically significant ones.
Abstract: In recent years, following an earlier result of C. Lanczos concerning the representation of the Weyl tensor in arbitrary space-times, it has been conjectured that the Riemann tensor itself admits a linear representation in terms of the covariant derivatives of a suitable “potential” tensor of rank 3. This conjecture is shown to be false, at least for a class of spacetime geometries including several physically significant ones.

22 citations


Journal ArticleDOI
TL;DR: In this paper, a geometrical form of the d = 11 supergravity lagrangian is suggested, which depends on the vielbein eωm, on the spin connection wωmn and on the gravitino Ψωα.

19 citations






Journal ArticleDOI
TL;DR: Here the space of immersions Imm (M, N) where M is without boundary is considered, and the covariant derivative and the Riemannian curvature of one of these metrics, the non trivial one is computed.
Abstract: E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n−1) dimensional manifold into ℝn, and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.

Journal ArticleDOI
TL;DR: In this article, a modified version of the principle of inertia and equivalence is proposed, based on the simple de Sitter group instead of the Poincare group, which is equivalent to a particular Kaluza-Klein theory with the Lorentz group as gauge group.
Abstract: A discussion of the fundamental interrelation of geometry and physical laws with Lie groups leads to a reformulation and heuristic modification of the principle of inertia and the principle of equivalence, which is based on the simple de Sitter group instead of the Poincare group. The resulting law of motion allows a unified formulation for structureless and spinning test particles. A metrical theory of gravitation is constructed with the modified principle, which is structured after the geometry of the manifold of the de Sitter group. The theory is equivalent to a particular Kaluza-Klein theory in ten dimensions with the Lorentz group as gauge group. A restricted version of this theory excludes torsion. It is shown by a reformulation of the energy momentum complex that this version is equivalent to general relativity with a cosmologic term quadratic in the curvature tensor and in which the existence of spinning particle fields is inherent from first principles. The equations of the general theory with torsion are presented and it is shown in a special case how the boundary conditions for the torsion degree of freedom have to be chosen such as to treat orbital and spin angular momenta on an equal footing. The possibility of verification of the resulting anomalous spin-spin interaction is mentioned and a model imposed by the group topology ofSO(3,2) is outlined in which the unexplained discrepancy between the magnitude of the discrete valued coupling constants and the gravitational constant in Kaluza-Klein theories is resolved by the identification of identical fermions as one orbit. The mathematical structure can be adapted to larger groups to include other degrees of freedom.

Journal ArticleDOI
01 Feb 1984
TL;DR: The main theorem of as discussed by the authors states that every naturally reductive homogeneous Riemannian manifold of nonpositive Ricci curvature is symmetric, and as a corollary, every non-compact symmetric Eigen manifold is also symmetric.
Abstract: The main theorem states that every naturally reductive homogeneous Riemannian manifold of nonpositive Ricci curvature is symmetric. As a corollary, every noncompact naturally reductive Einstein manifold is symmetric.

Journal ArticleDOI
TL;DR: The analysis of the admissibility of a potential representation for the Riemann tensor is continued in this article, where it is shown that there never exist ordinary solutions in a four-dimensional manifold and the existence of singular solutions is established without requiring any integrability condition.
Abstract: The analysis of the admissibility of a potential representation for the Riemann tensor is here continued. As in the preceding paper, the starting point is to regard the relationship between the Riemann tensor and its possible potential as a system of partial differential equations determining the unknown potential. The first result, strengthening a previous conclusion, is that there never exist ordinary solutions. Surprisingly enough, in a four-dimensional Riemannian manifold the existence of singular solutions is established without requiring any integrability condition. Possible applications and generalizations are also suggested.

Book ChapterDOI
01 Jan 1984

Journal ArticleDOI
TL;DR: In this article, the curvature tensor is treated on a more primitive level, that is, if the curvatures is prescribed, what information does one have about the metric and associated connection of space-time?
Abstract: In General Relativity, one has several traditional ways of interpreting the curvature of spacetime, expressed either through the curvature tensor or the sectional curvature function. This essay asks what happens if curvature is treated on a more primitive level, that is, if the curvature is prescribed, what information does one have about the metric and associated connection of space-time? It turns out that a surprising amount of information is available, not only about the metric and connection, but also, through Einstein's equations, about the algebraic structure of the energy-momentum tensor.

Journal ArticleDOI
TL;DR: In this paper, it was shown that all equations for particles of spinS⩾3/2 which may be said to conform to a strong principle of equivalence are compatible if and only if the V4 is of constant Riemannian curvature.
Abstract: On a previous occasion it was shown that the « natural generalization » to a Riemann spaceV4 of a certain set of flat-space free-field equations for particles of spinS=3/2 is internally consistent if and only if theV4 is an Einstein space. It is now shown that, this case apart, all equations for particles of spinS⩾3/2 which may be said to conform to a « strong principle of equivalence » are compatible if and only if theV4 is of constant Riemannian curvature. The corresponding second-order wave equations in such a space are written down. Certain modified first-order equations for the caseS=2 which involve the curvature tensor explicitly are shown to be consistent in an Einstein space.

Journal ArticleDOI
TL;DR: In this article, the algebraic structure of the energy-momentum tensor and the Weyl tensor on a space-time M calculated from given space time metric components gab is determined.
Abstract: Consider the curvature tensor components Rbcda on a space-time M calculated from given space-time metric components gab. The number of alternative space-time metrics for M which yield the same curvature components is known to be heavily restricted, the exact ambiguity in the metric being dependent on the form of the curvature components. In this paper it is shown that, in spite of these ambiguities, the algebraic structure of the energy-momentum tensor and the Weyl tensor on M are essentially determined.

Book ChapterDOI
Yum-Tong Siu1
01 Jan 1984
TL;DR: In this paper, the similarity between the methods used in investigating the rigidity of compact complex manifolds and those employed in studying their strong rigidity was pointed out, and it was shown that these methods can be used to obtain strong rigidness results for Einstein manifolds.
Abstract: The purpose of this talk is to point out the similarity between the methods used in investigating the rigidity of compact complex manifolds and those employed in studying their strong rigidity We hope that in understanding more fully this similarity we can obtain strong rigidity results for Einstein manifolds by using the techniques used in obtaining known results on the rigidity of Einstein manifolds

Book ChapterDOI
09 Jul 1984
TL;DR: Practical algorithms for determination of the multiplicities of their roots and hence for the classification of Riemann tensors are considered.
Abstract: The Petrov classification of the Weyl conformal curvature and the Plebanski or Segre classification of the Ricci tensor of spacetimes in general relativity both depend on multiplicities of the roots of quartic equations. The coefficients in these quartic equations may be complicated functions of the space-time coordinates. We review briefly the general theory of quartic equations and then consider practical algorithms for determination of the multiplicities of their roots and hence for the classification of Riemann tensors. Preliminary results of tests of computer implementations of these algorithms, using the computer algebra system SHEEP, are reported.

Journal ArticleDOI
TL;DR: The solutions of the two-dimensional Einstein equation or Ernst equation, parametrized by arbitrary functions and generated by the solution of the corresponding O(2,1) σ-model and by a special choice of the determinant of the metric are presented in this paper.

Journal ArticleDOI
TL;DR: A simple algorithm is presented which extends and improves the Levenberg-Marquardt minimization procedure and permits efficient application to regions where there is negative curvature along some lines.


Journal ArticleDOI
TL;DR: As further illustrations of how the programming routines developed in the present work can be applied to solve practical problems in continuum mechanics, the plate bending problem formulated within the framework of tensor calculus and the problem of finding the total surface area of a general curved surface were solved.


Journal ArticleDOI
TL;DR: In this paper, the scaling behavior of the stress tensor of a scalar quantum field in curved space-time was examined using the idea of metric scaling, and it was shown that the cosmological constant and the gravitational constant approach UV fixed points.
Abstract: Using the idea of metric scaling we examine the scaling behavior of the stress tensor of a scalar quantum field in curved space-time. The renormalization of the stress tensor results in a departure from naive scaling. We view the process of renormalizing the stress tensor as being equivalent to renormalizing the coupling constants in the Lagrangian for gravity (with terms quadratic in the curvature included). Thus the scaling of the stress tensor is interpreted as a nonnaive scaling of these coupling constants. In particular, we find that the cosmological constant and the gravitational constant approach UV fixed points. The constants associated with the terms which are quadratic in the curvature logarithmically diverge. This suggests that quantum gravity is asymptotically scale invariant.