Showing papers on "Ricci decomposition published in 2002"
TL;DR: In this article, the existence of a metric whose Schouten tensor satisfies a quadratic inequality was shown to imply that the eigenvalues of the Ricci tensor are positively pinched.
Abstract: We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
231 citations
TL;DR: In this article, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
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225 citations
TL;DR: In this paper, it was shown that a closed manifold with positive Ricci curvature must admit an action by a compact Lie group G with orbits of codimension one, and that this is also a sufficient condition for any manifold with Ricci or scalar curvature.
Abstract: One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.
171 citations
TL;DR: In this article, it was shown that the Ricci flow converges to a metric with constant bisectional curvature if and only if the curvature of the initial metric is positive.
Abstract: In this paper, we prove that if M is a Kahler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kahler-Ricci flow converges to a Kahler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kahler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kahler Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kahler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.
145 citations
TL;DR: In this article, it was shown that the Ricci curvature of the Schouten tensor is in a certain cone, Γ + k, which implies that the curvature is positive.
Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the kth elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ + k . We prove that this eigenvalue condition for k > n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.
103 citations
TL;DR: In this article, the authors classified the $*$-Einstein real hypersurfaces in complex space forms such that the structure vector is a principal curvature vector and the principal curvatures of the hypersurface can be computed with the K\"ahler metric.
Abstract: It is known that there are no Einstein real hypersurfaces in complex space forms equipped
with the K\"ahler metric. In the present paper we classified the $*$-Einstein real
hypersurfaces $M$ in complex space forms $M_{n}(c)$ and such that the structure vector is
a principal curvature vector.
76 citations
TL;DR: In this paper, the authors generalize Lovelock's results to trace-free (k, l)-forms, and give direct application to Maxwell, Lanczos, Ricci, Bel, and Bel-Robinson tensors.
Abstract: Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p⩾n. We generalize Lovelock’s results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n+1 indices, we establish a very general “master” identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel, and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner.
59 citations
TL;DR: In this paper, the authors investigated curvature properties of semi-Riemannian manifolds (M,g), n/4, whose Weyl curvature tensor C can be expressed by a KulkarniNomizu square of the tensor S - j9-
Abstract: We investigate curvature properties of semi-Riemannian manifolds (M,g), n/4, whose Weyl curvature tensor C can be expressed by a KulkarniNomizu square of the tensor S - j9- We investigate also the problem of isometric immersion of such manifolds into space forms.
56 citations
TL;DR: In this article, the authors present a series of papers on the metric theory of tensor products according to Grothendieck's "Resume de la theorie des produits tensoriels topologiques".
Abstract: This paper presents the first of a multi-part
series of papers on the metric theory of tensor products according to Grothendieck's “Resume de la theorie
metrique des produits tensoriels topologiques” It
contains the basics on tensor norms: a discussion of the special character of
the injective and the projective norms, fundamental operations on tensor norms,
extension of tensor norms to spaces of infinite dimensions. Mathematics Subject
Classification (2000): 46B28, 46B07, 46B04.
Key words: Reasonable crossnorms;
projective norm; injective norm; integral bilinear form; tensor norm; (metric)
accessibility.
Quaestiones
Mathematicae 25 (2002), 37-72
49 citations
TL;DR: In this article, the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted γ-law perfect fluid is studied.
Abstract: We study the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted γ-law perfect fluid. We investigate all the Bianchi and Thurston type universe models and calculate the asymptotic evolution of Weyl curvature invariant for generic solutions to the Einstein field equations. The influence of compact topology on Bianchi types with hyperbolic space sections is also considered. Special emphasis is placed on the late-time behaviour where several interesting properties of the Weyl curvature invariant occur. The late-time behaviour is classified into five distinctive categories. It is found that for a large class of models, the generic late-time behaviour of the Weyl curvature invariant is to dominate the Ricci invariant at late times. This behaviour occurs in universe models which have future attractors that are plane-wave spacetimes, for which all scalar curvature invariants vanish. The overall behaviour of the Weyl curvature invariant is discussed in relation to the proposal that some function of the Weyl tensor or its invariants should play the role of a gravitational 'entropy' for cosmological evolution. In particular, it is found that for all ever-expanding models the measure of gravitational entropy proposed by Gron and Hervik increases at late times.
48 citations
TL;DR: In this paper, a closed-form representation for the derivative of non-symmetric tensor power series is proposed for the special case of repeated eigenvalues, which is the only possibility to calculate the derivative in a closed form.
TL;DR: In this paper, it was shown that any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional space with a non-egenerate Ricci tensor which is equal, up to a local analytic diffeomorphism.
Abstract: We discuss and prove a theorem which asserts that any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional space with a nondegenerate Ricci tensor which is equal, up to a local analytic diffeomorphism, to the Ricci tensor of an arbitrary specified space. This may be regarded as a further extension of the Campbell–Magaard theorem. We highlight the significance of embedding theorems of increasing degrees of generality in the context of higher dimensional space–times theories and illustrate the new theorem by establishing the embedding of a general class of Ricci-flat space–times.
TL;DR: In this paper, Li-Yau-Hamilton inequalities for Ricci flow with nonnegative curvature operators were shown to be equivalent to the linear trace inequalities of Hamilton and one of the authors.
Abstract: We prove Li–Yau–Hamilton inequalties that extend Hamilton’s matrix inequality for solutions of the Ricci flow with nonnegative curvature operators. To obtain our extensions, we apply the space-time formalism of S.-C. Chu and one of the authors to solutions of the Ricci flow modified by a cosmological constant. Then we adjoin to the Ricci flow the evolution of a 1-form and a 2-form flowing by a system of heat-type equations. By a rescaling argument, the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of Hamilton and one of the authors.
01 Jan 2002
TL;DR: In this article, a new description of the moment polytope associated with a complex projective variety acted on by a reductive group is presented, and a short proof of certain inequalities due to Manivel and Strassen concerning the decomposition of (inner) tensor products of irreducible representations of the symmetric group is given.
Abstract: We present a new description of the moment polytope associated with a complex projective variety acted on by a reductive group. We apply this to give a short proof of certain inequalities due to Manivel and Strassen concerning the decomposition of (inner) tensor products of irreducible representations of the symmetric group, and to exhibit, in a concrete example, a complete system of inequalities.
01 Sep 2002
TL;DR: In this article, an isolation theorem of Weyl conformal tensor tensor of positive Einstein manifolds is given, when its $L n/2$-norm is small.
Abstract: An isolation theorem of Weyl conformal tensor of positive Einstein manifolds is given, when its $L^{n/2}$-norm is small.
TL;DR: In this article, it was shown that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and gave an elementary proof that any local 2-point homogeneous Lorenzian manifold has constant curvature.
Abstract: We show that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szab\'o Lorentzian covariant derivative algebraic curvature tensor vanishes.
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TL;DR: In this paper, the null-shell formalism is used to define a preferred flow to which the flow of matter can be compared on the null hypersurface, and the authors derive the stress-energy tensor with a simple characterization in terms of a mass density, a mass current, and an isotropic pressure.
Abstract: We consider a situation in which two metrics are joined at a null hypersurface. It often occurs that the union of the two metrics gives rise to a Ricci tensor that contains a term proportional to a Dirac delta-function supported on the hypersurface. This singularity is associated with a thin distribution of matter on the hypersurface, and following Barrabes and Israel, we seek to determine its stress-energy tensor in terms of the geometric properties of the null hypersurface. While our treatment here does not deviate strongly from their previous work, it offers a simplification of the computational operations involved in a typical application of the formalism, and it gives rise to a stress-energy tensor that possesses a more recognizable phenomenology. Our reformulation of the null-shell formalism makes systematic use of the null generators of the singular hypersurface, which define a preferred flow to which the flow of matter can be compared. This construction provides the stress-energy tensor with a simple characterization in terms of a mass density, a mass current, and an isotropic pressure. Our reformulation also involves a family of freely-moving observers that intersect the surface layer and perform measurements on it. This construction gives operational meaning to the stress-energy tensor by fixing the argument of the delta-function to be proper time as measured by these observers.
TL;DR: In this paper, the Riemann-Christoffel tensor, Ricci tensor and other tensors of a quasi-Sasakian manifold were studied on this basis.
Abstract: The full system of structure equations of a quasi-Sasakian structure is obtained. The structure of the main tensors on a quasi-Sasakian manifold (the Riemann-Christoffel tensor, the Ricci tensor, and other tensors) is studied on this basis. Interesting characterizations of quasi-Sasakian Einstein manifolds are obtained. Additional symmetry properties of the Riemann-Christoffel tensor are discovered and used for distinguishing a new class of quasi-Sasakian manifolds. An exhaustive description of the local structure of manifolds in this class is given. A complete classification (up to the -transformation of the metric) is obtained for manifolds in this class having additional properties of the isotropy kind.
TL;DR: In this article, the authors classify all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure.
Abstract: We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure.
TL;DR: For every nonholonomic manifold, i.e., manifold with nonintegrable distribution the analog of the Riemann tensor is introduced in this paper, which is interpreted as modiÞcations of the Spencer cohomology.
Abstract: For every nonholonomic manifold, i.e., manifold with nonintegrable distribution the analog of the Riemann tensor is introduced. It is calculated here for the contact and Engel structures: for the contact structure it vanishes (another proof of Darboux’s canonical form); for the Engel distribution the target space of the tensor is of dimension 2. In particular, the Lie algebra preserving the Engel distribution is described. The tensors introduced are interpreted as modiÞcations of the Spencer cohomology and, as such, provide with a new way
TL;DR: In this article, the contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components.
Abstract: The contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components. The relations of the wave-propagation metric tensor to the curvature matrix and Gaussian curvature of the slowness surface and to the curvature matrix and Gaussian curvature of the ray-velocity surface are demonstrated with the help of ray-centred coordinates.
01 Dec 2002
TL;DR: For a positive definite fundamental tensor, all known examples of Osserman algebraic curvature tensors have a typical structure They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Cliord commutation relations as mentioned in this paper.
Abstract: For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Cliord commutation relations We show by means of Young symmetrizers and a theorem of S A Fulling, R C King, B G Wybourne and C J Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors We verify our results by means of the Littlewood-Richardson rule and plethysms For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS
TL;DR: In this article, a regularization procedure that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed.
Abstract: A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density , associated to the Einstein tensor of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the (2 + 1)-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works.
TL;DR: In this paper, it was shown that any interaction Lagrangian, depending on a collection of fields and on the metric field on a (space-time) manifold whose energy-momentum tensor depends on at most first derivatives of the metric tensor, is of a certain polynomial character in these derivatives.
TL;DR: In this paper, the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to xa for a model of string cloud and string fluid stress tensor in the context of general relativity are studied.
Abstract: We study the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to xa for a model of string cloud and string fluid stress tensor in the context of general relativity. Necessary and sufficient conditions are derived for a spacetime with a model of string cloud and string fluid stress tensor to admit a SpRIV, and a SpRIV which is also a spacelike conformal Killing vector. Also, some results are obtained.
TL;DR: In this article, it was shown that every irreducible compact compact Kahler surface with δW − = 0 is a Kahler-Einstein surface and is biholomorphic to a ruled extremal kahler surface.
Abstract: We prove that every irreducible compact Kahler surface with δW − =0 is Kahler–Einstein or is biholomorphic to a ruled extremal Kahler surface.
TL;DR: In this paper, a complete determination is given of those pairs of finite-dimensional irreducible representations whose tensor products (or squares) may be resolved into irreduceible representations that are multiplicity free, i.e. such that no irrawucible representation occurs in the decomposition of the tensor product more than once.
Abstract: For each of the exceptional Lie groups, a complete determination is given of those pairs of finite-dimensional irreducible representations whose tensor products (or squares) may be resolved into irreducible representations that are multiplicity free, i.e. such that no irreducible representation occurs in the decomposition of the tensor product more than once. Explicit formulae are presented for the decomposition of all those tensor products that are multiplicity free, many of which exhibit a stability property.
TL;DR: In this article, the authors construct metrics of positive Ricci curvature on some vector bundles over tori, which are homotopy equivalent but not homeomorphic to manifolds of non-negative sectional curvature.
Abstract: We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of non-negative sectional curvature.
TL;DR: In this paper, the authors examined algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrization, and obtained new identities for superenergy tensors of these general (nonsimple) forms.
Abstract: The classical Rainich(–Misner–Wheeler) theory gives necessary and sufficient conditions on an energy–momentum tensor T to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations, these conditions can be expressed in terms of the Ricci tensor, thus providing conditions for a spacetime geometry to be an Einstein–Maxwell spacetime. One of the conditions is that T2 is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple p-form. Here we examine algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrization. Using antisymmetrization techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse: that the identities are sufficient to determine the form. As an example we obtain the complete generalization of the classical Rainich theory to five dimensions.
TL;DR: In this paper, the first eigenvalue of Dirac operator on a compact Riemannian spin manifold was proved by refined Weitzenbock techniques and applied to manifolds with harmonic curvature tensor.
Abstract: We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenbock techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.