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Showing papers on "Ricci decomposition published in 1992"


Journal ArticleDOI
TL;DR: In this paper, the Riemann tensor is used to define a basis set of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric and exhibited a basis for these invariants up through order eight.
Abstract: Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.

207 citations



Journal ArticleDOI
TL;DR: In this article, a general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed, which is characterized by the tensors of Riemanian and non-Riemannians curvatures, an affine deformation tensor being the result of nonmetricity of one of the connections.
Abstract: A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.

127 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the vanishing of the trace free part Cμνρ of the second fundamental tensor Kμvρ is a sufficient condition for conformal flatness of the imbedded surface.

68 citations


Journal ArticleDOI
TL;DR: A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors.
Abstract: This paper essentially deals with the classification of a symmetric tensor on a four‐dimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.

33 citations


Journal ArticleDOI
TL;DR: In this paper, a gravitational analogue of Min-Oo's gap theorem for Yang-Mills fields is given, where the gap is defined as the distance from the Earth's center.
Abstract: A gravitational analogue is given of Min-Oo's gap theorem for Yang-Mills fields.

31 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that complex transformations can be applied on the parameters and coordinates entering a known curvature tensor in order to generate new curvatures which satisfy certain algebraic relationships following from the Einstein-Maxwell equations with cosmological constant.
Abstract: It is shown that complex transformations can be applied on the parameters and coordinates entering a known curvature tensor in order to generate new curvature tensors which, just as the seed tensor, possess the same symmetry properties and satisfy certain algebraic relationships following from the Einstein-Maxwell equations with cosmological constant.

29 citations


Journal ArticleDOI
TL;DR: In this article, the problem of choosing coordinates to diagonalize an n-dimensional Riemannian or Lorentzian metric with n) 3 is considered, and it is shown that diagonalizability of the metric generically imposes restrictions on the third derivative of the Weyl tensor when n = 4, the first derivative when n=5, and the tensor itself when n) 5.
Abstract: The problem of choosing coordinates to diagonalize an n-dimensional Riemannian or Lorentzian metric with n)3 is considered. It is shown that diagonalizability of the metric generically imposes restrictions on the third derivative of the Weyl tensor when n=4, the first derivative of the Weyl tensor when n=5 and the Weyl tensor itself when n)5. It is also shown that some of the plane-wave metrics provide examples of four-dimensional non-diagonalizable Lorentzian metrics.

19 citations


Journal ArticleDOI
TL;DR: A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature.
Abstract: A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor (of a symmetric connection) so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature. In this new result the conditions are imposed on a tensor of a more general character than the curvature tensor. In addition it is shown that once the symmetric connection is known to be metric, the metric is uniquely defined (up to a constant conformal factor). For those special curvature tensors which are excluded from the original result, supplementary conditions are suggested, which, alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also Riemann tensors.

17 citations


Journal ArticleDOI
TL;DR: In this article, a method of calculating the metric from the curvature of a tensor with the symmetry properties of a type D curvature tensor is given in an orthonormal tetrad.
Abstract: A method of calculating the metric from the curvature is presented. Assuming that a tensor with the symmetry properties of a type D curvature tensor is given in an orthonormal tetrad, we use the Bianchi identities and the relationship between the connection and the tetrad in order to calculate, under certain assumptions, the corresponding metric. Some well-known metrics are derived from the curvature by using the method given here.

17 citations


Journal ArticleDOI
01 Feb 1992
TL;DR: In this article, it was shown that two pointwise conformal metrics that have the same Ricci tensor must be homothetic, i.e., they are both Ricci-Ricci tensors.
Abstract: We show that two pointwise conformal metrics that have the same Ricci tensor must be homothetic

Journal ArticleDOI
TL;DR: In this paper, the authors classify Ricci tensors in terms of their collineations, and present an interesting case possessing six collineated tensors, which they worked out and discussed.
Abstract: In classifying Ricci tensors in terms of their collineations, an interesting case possessing six collineations arises. These collineations are worked out and discussed.

Journal ArticleDOI
TL;DR: In this paper, a local classification of all three-dimensional Riemannian manifolds whose Ricci tensor satisfies the equation ▿(ric-1 4 sg) = 1 20 ds ⊙ g is given.
Abstract: One derives a local classification of all three-dimensional Riemannian manifolds whose Ricci tensor satisfies the equation ▿(ric– 1 4 sg) = 1 20 ds ⊙ g.

01 Jan 1992
TL;DR: In this article, conditions for the existence of solitons for the Ricci flow were investigated and it was shown that globally defined soliton is harder to obtain on compact manifolds than locally defined soliton.
Abstract: In this thesis we investigate conditions for the existence of solitons for the Ricci flow. The Ricci flow, first introduced by Richard Hamilton, changes a Riemannian metric over time, in a way that the metric satisfies the partial differential equation ∂g/∂t = −2 Ric(g). “Solitons” for this flow are solutions of the equation where the metrics at different times differ by a diffeomorphism of the manifold. The soliton condition, sufficient for an initial metric to give rise to a soliton, is LX} = −∈Ric(}). We use the techniques of exterior differential systems to show that this condition is involutive, and gives an elliptic equation in harmonic coordinates. However, globally-defined solitons are harder to obtain on compact manifolds than locally-defined solitons: using techniques from Hamilton’s earlier papers, we show that the only solitons on compact three-manifolds are metrics of constant curvature. For compact manifolds of higher dimension we investigate the possibility of deforming an Einstein metric to a soliton, and we calculate the first-order deformation space explicitly in the case of symmetric spaces of compact type. Using warped products and ODE techniques, we also construct some examples of non-compact, complete solitons.



Journal ArticleDOI
TL;DR: In this article, a generalized Bianchi identity was proposed for the Nijenhuis tensor, which is a generalization of the Bianchi tensor identity for the Riemann tensor.
Abstract: Let M be a differentiable manifold with a (1,1) tensor field Sνμ(x). A notion of distorted torsion tensor is introduced that reduces to the ordinary torsion tensor when Sνμ=δνμ. It is then shown that the distorted torsion tensor satisfies a generalization of the first Bianchi identity. The new identity involves the Nijenhuis tensor in contrast to the standard Bianchi identity. Using this generalized formula, BRST‐like operators are constructed that satisfy a Lie‐super algebra, when both Nijenhuis and Riemann tensors vanish identically.

Journal ArticleDOI
TL;DR: The results of a complete enumeration of the scalers formed from the Riemann tensor by covariant differentiation, multiplication and contraction of order 14 in the derivatives of the metric are presented in this paper.
Abstract: The results of a complete enumeration of the scalers formed from the Riemann tensor by covariant differentiation, multiplication and contraction of order 14 in the derivatives of the metric are presented. The corresponding enumeration for the numbers of scalars constructed solely from the Weyl tensor is also given.


Journal ArticleDOI
TL;DR: In this article, the authors studied Ambrose-Singer connections with an algebraic curvature tensor on simply connected manifolds carrying a homogeneous Riemannian structure of class ℐ3.
Abstract: We study Ambrose-Singer connections with an algebraic curvature tensor on simply connected manifolds carrying a homogeneous Riemannian structure of class ℐ3 in the classification given by F. Tricerri and L. Vanhecke.