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Showing papers on "Ricci decomposition published in 1999"


Journal ArticleDOI
TL;DR: In this article, the authors developed an analog of Riemannian geometry on finite and discrete sets, which is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set).
Abstract: Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set). Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (nonlocal) tensor product over the algebra of functions, as considered previously by several authors. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is, in general, nothing like a Ricci tensor or a curvature scalar. Because of the nonlocality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather nonlocal objects. But one can make use of the parallel transport associated with a connection to “localize” such objects, and in certain cases there is a distinguished way to achieve this. In particular, this leads to covariant components of the curvature tensor which allow a contraction to a Ricci tensor. Several examples are worked out to illustrate the procedure. Furthermore, in the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations.

74 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that a complete, complete, asymptotically  at manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, and that the manifold must be close to (IR 3, δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure.
Abstract: The Positive Mass Theorem implies that any smooth, complete, asymptotically flat3-manifold with non-negative scalar curvature which has zero total mass is isometricto (IR 3 ,δ ij ). In this paper, we quantify this statement using spinors and prove thatif a complete, asymptotically flat manifold with non-negative scalar curvature hassmall mass and bounded isoperimetric constant, then the manifold must be close to(IR 3 ,δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure. This curvatureestimate allows us to extend the case of equality of the Positive Mass Theorem toinclude non-smooth manifolds with generalized non-negative scalar curvature, whichwe define. 1 Introduction We introduce our problem in the context of General Relativity. Consider a 3 + 1 dimen-sional Lorentzian manifold N with metric g αβ of signature (− + ++). We denote theinduced Levi-Civita connection by ∇¯. Then the corresponding Ricci tensor R¯

61 citations


Journal ArticleDOI
TL;DR: The dominant superenergy property of the Bel-Robinson tensor has been shown to be non-negative in this article, which is a generalization of the positive energy theorem and singularity theorem.
Abstract: Several of the most important results in general relativity require or assume positivity properties of certain tensors. The positive energy theorem and the singularity theorems make assumptions about the energy-momentum tensor and Ricci tensor respectively. Positivity of the Bel–Robinson tensor is needed in the proof of the global stability of Minkowski spacetime. Senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor. For a Maxwell field or a scalar field the procedure yields the usual energy-momentum tensor, for the Weyl tensor and the Riemann tensor one obtains the Bel–Robinson tensor and Bel tensor respectively. In general, by considering any tensor as an r-fold n 1,…,n r )-form, one constructs a rank 2r superenergy tensor from it. By using spinor methods, we prove that the contraction of any such superenergy tensor with 2r future-pointing vectors is non-negative. We refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them. Some more examples are given and applications discussed.

40 citations


Journal ArticleDOI
TL;DR: In this article, the duality principle for the associated curvature tensor R of a pointwise Osserman Riemannian manifold was shown to hold for any manifold M = 0.

28 citations


Journal ArticleDOI
TL;DR: The canonical form of a single tensor is defined and it is shown that the problem of finding the canonical formof a generic tensor expression reduces to finding thecanon form of single tensors.
Abstract: We present efficient algorithms for simplifying tensor expressions that obey generic symmetries. We define the canonical form of a single tensor and we show that the problem of finding the canonical form of a generic tensor expression reduces to finding the canonical form of single tensors. Special symmetries are considered in order to push the efficiency further. We also present algorithms to address the cyclic symmetry of the Riemann tensor. With these algorithms it is possible to simplify generic Riemann tensor polynomials.

24 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied complete n-dimensional Riemannian manifolds with nonnegative Ricci curvatures and large volume growth and proved that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in the n-dimensions.
Abstract: In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space \( R^n \) if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in \( R^n \).

23 citations


Journal ArticleDOI
TL;DR: In this paper, the curvature description of geometries is used to investigate rigidly rotating axisymmetric perfect fluid spacetimes with magnetic Weyl tensors, and it is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution.
Abstract: Stationary axisymmetric perfect fluid spacetimes are investigated using the curvature description of geometries. We formulate the equations in terms of components of the Riemann tensor and the Ricci rotation coefficients in a comoving Lorentz tetrad. It is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution. Further, we find that all rigidly rotating axistationary fluids with magnetic Weyl tensor have local rotational symmetry. Rigidly rotating fluid spacetimes with purely electric or purely magnetic Weyl tensor are shown to be of Petrov type D.

19 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that there are no real hypersurfaces with recurrent Ricci tensors of the complex projective space Pn(C) under the condition that there is a principal curvature vector.
Abstract: Let M be a real hypersurface of the complex projective space Pn(C). The Ricci tensor S of M is recurrent if there exists a 1-form such that . In this paper we show that there are no real hypersurfaces with recurrent Ricci tensor of Pn(C) under the condition that is a principal curvature vector.1991 Mathematics Subject Classification 53C40 (53C25).

17 citations


Posted Content
TL;DR: In this paper, the authors present some generalizations and novel properties of the Bel-Robinson tensor in the context of constructing local invariants in D = 11 supergravity, which is a special case of our problem.
Abstract: We present some generalizations, and novel properties, of the Bel-Robinson tensor, in the context of constructing local invariants in D=11 supergravity.

16 citations



Journal ArticleDOI
TL;DR: In this article, complete conformally flat three dimensional Riemannian manifolds with constant scalar curvature and constant squared norm of Ricci curvature tensor were classified by applying the generalized maximum principle due to H. Omori.
Abstract: We classify complete conformally flat three dimensional Riemannian manifolds with constant scalar curvature and constant squared norm of Ricci curvature tensor by applying the Generalized Maximum Principle due to H. Omori.

Journal ArticleDOI
01 Jan 1999
TL;DR: In this paper, the authors generalize Weyl's work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group G2.
Abstract: Let V be the 7-dimensional irreducible representations of G2. We decompose the tensor power V ⊗n into irreducible representations of G2 and obtain all irreducible representations of G2 in the decomposition. This generalizes Weyl’s work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group G2.


Journal ArticleDOI
TL;DR: In this article, it was shown that a spacetime admitting Lie-group actions may be disjointly decomposed into a closed subset with no interior plus a dense finite union of open sets in each of which the character and dimension of the group orbits as well as the Petrov type are constant.
Abstract: Following a recent work in which it is shown that a spacetime admitting Lie-group actions may be disjointly decomposed into a a closed subset with no interior plus a dense finite union of open sets in each of which the character and dimension of the group orbits as well as the Petrov type are constant, the aim of this work is to include the Segre types of the Ricci tensor (and hence of the Einstein tensor) into the decomposition. We also show how this type of decomposition can be carried out for any type of property of the spacetime depending on the existence of a continuous endomorphism.

Journal ArticleDOI
TL;DR: In this paper, a four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space-times with vanishing scalar curvature.
Abstract: A four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space–times with vanishing scalar curvature. This tensor reduces in vacuum to the Bel–Robinson tensor. Furthermore, the completely timelike component referred to any observer is positive, and zero if and only if the space–time is flat (excluding some unphysical space–times). We also show that this tensor is the unique one that can be constructed with these properties. Such a tensor does not exist for general gravitational fields. Finally, we study this tensor in several examples: the Friedmann–Lemaitre–Robertson–Walker space–times filled with radiation, the plane–fronted gravitational waves, and the Vaidya radiating metric.

Journal ArticleDOI
TL;DR: In this article, it was shown that the new tensor identity recently discovered by Bonanos, and some other tensor identities recently investigated, are consequences of a simple and mathematically trivial (but subtle) identity highlighted some years ago by Lovelock.
Abstract: It is shown that the new tensor identityrecently discovered by Bonanos, and some other tensoridentities recently investigated, are consequences of avery simple and mathematically trivial (but subtle) identity highlighted some years ago byLovelock. Lovelock's identity gives a tensor identity offirst order in Weyl-like tensors, and a tensor identityof second order in Ricci-like tensors, from which higher order identities, such as those recentlystudied, can easily be constructed.

Journal ArticleDOI
TL;DR: In this article, the Ricci tensor has been computed in infinite dimensional situations for the case of the central extension of loop groups and in the asymptotic behaviour of the Riemannian metric on free loop groups.

Journal ArticleDOI
TL;DR: In this paper, a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature was established.
Abstract: In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature. Our result will be a consequence of an integral formula involving the Ricci curvature and the scalar curvature of the hypersurface. We also derive some other consequences and applications of this formula.

Journal ArticleDOI
TL;DR: In this paper, the Hilbert C*-module representation associated with completely positive linear maps was used to construct covariant representations of covariant group systems on Hilbert C *-modules without bridging maps.
Abstract: Using the Hilbert C*-module representation associated with completely multi-positive linear maps [Heo], we give another representation on Haagerup tensor product without the bridging maps. We also construct covariant representations of covariant group systems on Hilbert C *-modules. §

Journal ArticleDOI
TL;DR: In this paper, the Cayley-Hamilton theorem was shown to be applicable to the problem of finding a complete set of symmetric matrices, in terms of which all symmetric matrix polynomials in these two matrices can be expressed.
Abstract: A large number of Riemann tensor invariants can be written as traces of products of two 3×3 matrices, representing the Weyl tensor and the Weyl-like square of the Ricci tensor. It is pointed out that finding a complete set, I, for all of these invariants is a simple consequence of the more general problem of finding a complete set of symmetric matrices, M, in terms of which all symmetric matrix polynomials in these two matrices can be expressed. Such a set is constructed and a formal proof of its completeness is given. Several matrix identities and a scalar syzygy, obtained recently by Sneddon, are rederived and their interrelationships clarified. They are shown to be, ultimately, consequences of the Cayley–Hamilton theorem. A “minimal set” of invariants, that must be contained in the complete set of invariants of the general problem, is identified and it is concluded that no set proposed so far is complete.

Journal ArticleDOI
TL;DR: In this article, it was shown that a conformally flat contact strongly pseudo-convex integrable manifold is locally isometric to a unit sphere, provided the characteristic vector field is an eigenvector of the Ricci tensor at each point.
Abstract: It is shown that a locally symmetric contact strongly pseudo-convex integrableCR manifold of dimension greater than 3 and other than 7 is locally isometric to a unit sphere or the Riemannian product of an (n + 1)-dimensional Euclidean space and a sphere. A conformally flat contact strongly pseudo-convex integrableCR manifold is locally isometric to a unit sphere, provided the characteristic vector field is an eigenvector of the Ricci tensor at each point.

Book ChapterDOI
01 Jan 1999
TL;DR: In this article, the Riemann tensor is computed and used to characterize flat space-times with the Lorentz metric g = (dx − vdt)2 − dt2.
Abstract: Given a Newtonian velocity field v(x, t), one considers the manifold R with the Lorentz metric g = (dx − vdt)2 − dt2. The Riemann tensor is computed and used to characterize flat space-times with g of this form. Among non-flat solutions of Einstein’s equations for such a g there are some cosmological models, the Schwarzschild and Kasner metrics and their generalizations to include matter fields and the cosmological constant. If |v| = 1, then the vector field ∂/∂t is null and has vanishing divergence; it is geodetic and shear-free if, and only if, ∂v/∂t is parallel to v. Department of Mathematical Methods in Physics, Warsaw University, Hoża 74, Warszawa, Poland. E-mail: nurowski@fuw.edu.pl. Present address: Dipartimento di Scienze Matematiche, Universita degli Studi di Trieste, Piazzale Europa 1, 34127 Trieste, Italy. E-mail: nurowski@mathsun1.univ.trieste.it Department of Physics, New York University, 4 Washington Pl., New York, NY 10003, USA. Institute of Theoretical Physics, Warsaw University, Hoża 69, Warszawa, Poland. Email: amt@fuw.edu.pl


Journal ArticleDOI
TL;DR: A generalized version of the Einstein equations in the 4-index form, containing the Riemann tensor linearly, is derived in this article, and it is shown that the gravitational energy-momentum density tensor outside a source is represented across the Weyl tensor vanishing at the 2-index contraction.
Abstract: A generalized version of the Einstein equations in the 4-index form, containing the Riemann tensor linearly, is derived. It is shown, that the gravitational energy-momentum density tensor outside a source is represented across the Weyl tensor vanishing at the 2-index contraction. The 4-index energy-momentum density tensor for matter also is constructed.

Journal ArticleDOI
TL;DR: In this paper, the leaves of a harmonic, Riemannian 3-dimensional foliation are obtained as locally homogeneous manifolds with negative scalar curvature, whose Ricci tensor satisfies (S) = 0 for all tangent vector fields.
Abstract: Examples of slant submanifolds in the Sasakian space R2n+1 are obtained as the leaves of a harmonic, Riemannian 3-dimensional foliation. With the exception of the anti-invariant ones, these leaves are all locally homogeneous manifolds with negative scalar curvature, whose Ricci tensor satisfies (▿S)(X, X) = 0 for all tangent vector fields.

Journal ArticleDOI
TL;DR: In this paper, the Riemann tensor of a submanifold of Euclidean space is expressed in terms of the derivatives of the defining functions and the components of the tangent vectors.
Abstract: In the paper the Riemann tensor of a submanifold of Euclidean space is expressed in terms of the derivatives of the defining functions and the components of the tangent vectors. The codimension 2 case, in particular the case of a two-dimensional surface inE 4, is treated in detail. As an example, the Gaussian curvature of the intersection of hyperquadrics with common axes is found.

Journal ArticleDOI
TL;DR: In this article, the affine curvature tensor R∗ is characterized for which R ∗ is parallel relative to the induced connection and it is shown that the rank of h is independent of the choice of a transversal vector field.
Abstract: In [OV] we introduced an affine curvature tensor R∗. Using it we characterized some types of hypersurfaces in the affine space Rn+1. In this paper we study hypersurfaces for which R∗ is parallel relative to the induced connection. 1. Let M be an n-dimensional connected manifold and f : M → R its immersion into the standard affine space R. Denote by D the standard connection in R. If ξ is an equiaffine transversal vector field for f , that is, Dξ is tangential to f , then the formulas of Gauss and Weingarten can be written as follows: DXf∗Y = f∗∇XY + h(X,Y )ξ, (1.1) DXξ = −f∗SX, (1.2) where X,Y are tangent vector fields on M , ∇ is the induced connection on M , h the second fundamental form and S the shape operator. It is known that the rank of h is independent of the choice of a transversal vector field. If the rank is equal to n everywhere on M , then f is called nondegenerate. For a nondegenerate hypersurface there exists a unique (up to a constant) equiaffine transversal vector field such that (1.3) trh(∇Xh)(·, ·) = 0 for every X ∈ TM . This transversal vector field is called the affine normal. Throughout the paper we shall study nondegenerate hypersurfaces endowed with equiaffine transversal vector fields. The relationship between ∇, h and S is given by the fundamental equations (1.4) R(X,Y )Z = h(Y,Z)SX − h(X,Z)SY (Gauss), 1991 Mathematics Subject Classification: Primary 53A15.

Journal Article
TL;DR: In this article, the character of a geodesic without conjugate points in a complete Riemannian manifold with nonnegative Ricci Curvdere was discussed, and it was shown that every simply connected 3-dimensional open manifold M, which Ricci curvature is nonnegative and moreover exists a point P M,Ric(P)0, thenM is contractable.
Abstract: In the paper, we discuss the character of a geodesic without conjugate points in a completeRiemannian manifold with nonnegative Ricci Curvdere, Prove that every simply connected 3-dimensional open manifold M, which Ricci curvature is nonnegative and moreover exists a point P M,Ric(P)0, thenM is contractable. we also prove that the first Betti number b1≤n-3 for a completeRiemannian maFnifold with positive Ricci curvature.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kahler manifold.
Abstract: We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kahler manifold. We also obtain analogous results on almost Kahler 4-manifolds.

Journal ArticleDOI
TL;DR: In this paper, the tensor product of two modules, q a root of 1, is decomposed into indecomposable summands for both irreducible and indecompositionable modules, and the Clebsch-Gordan coefficients in the general case are computed.
Abstract: The tensor product of two modules, q a root of 1, is decomposed into indecomposable summands for both irreducible and indecomposable modules. Clebsch-Gordan coefficients in the general case are computed. An apparently new identity is derived and some possible applications are conjectured.