Showing papers on "Riemann curvature tensor published in 2009"
TL;DR: In this paper, a notion of a length space X having nonnegative N-Ricci curvature, for N 2 [1;1], or having 1-RICci curvatures bounded below by K, for K2 R, was given.
Abstract: We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having \curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form
1,357 citations
TL;DR: In this article, the authors define the Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are.
728 citations
TL;DR: For Riemannian manifolds with a measure (M, g, edvolg) as mentioned in this paper showed that the Ricci curvature and volume comparison can be improved when the Bakry-Emery Ricci tensor is bounded from below.
Abstract: For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.
572 citations
TL;DR: In this paper, a necessary and sufficient condition on the cost function of a Riemannian manifold is given, expressed as a socalled cost-sectional curvature being non-negative.
Abstract: We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampere equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Holder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.
275 citations
TL;DR: In this article, the full set of equations governing the structure and the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity.
Abstract: The full set of equations governing the structure and the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, and the active gravitational mass. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through these scalars. Some solutions are exhibited to illustrate this point.
231 citations
TL;DR: The tensor computer algebra package xPert is presented, based on the combination of explicit combinatorial formulas for the nth order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica.
Abstract: We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for the nth order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica. We give examples of use and show the efficiency of the system with timings plots: it is possible to handle orders n = 4 or n = 5 within seconds, or reach n = 10 with timings below 1 h.
200 citations
Book•
16 Apr 2009TL;DR: In this article, a rigidity theorem for Riemannian fibrations of flat spaces over compact bases is proved and a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic RiemANNian foliations is given.
Abstract: We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.
167 citations
Book•
30 Jun 2009TL;DR: In this article, Bieri and Zipser provided two extensions to the Christodoulou-Klainerman result for the stability of Minkowski spacetime, and proved the existence of smooth, global solutions to the Einstein-Maxwell equations.
Abstract: This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of $r$ and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field $F$; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of $F$. In particular the Ricci curvature is a constant multiple of the stress-energy tensor for $F$. Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field.
154 citations
TL;DR: This Golden Oldie is a reprinting of a paper by F. A. Pirani as discussed by the authors first published in 1956 and is accompanied by a book by J. L. Synge and A. E. Trautman.
Abstract: This Golden Oldie is a reprinting of a paper by F. A. E. Pirani first published in 1956. It is accompanied by a reprinting of a paper by J. L. Synge first published in 1934. Together these papers pointed the way to the interpretation of geodesic deviation and its relation to the curvature tensor. These two Golden Oldies are accompanied by an Golden Oldie Editorial containing an editorial note written by A. Trautman, and by the biography of F. Pirani written by himself and commented by A. Trautman.
130 citations
TL;DR: In this article, the authors introduced the notion of -non-degenerate spacetime metric, which implies that the spacetime is locally determined by its curvature invariants.
Abstract: In this paper, we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an -non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is -non-degenerate. This enables us to prove our main theorem that a spacetime metric is either -non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.
123 citations
TL;DR: In this article, the evolution of the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion is studied and an analogue of the second Bianchi identity in G_2-geometry is derived.
Abstract: This is a foundational paper on flows of G_2 Structures. We use local coordinates to describe the four torsion forms of a G_2 Structure and derive the evolution equations for a general flow of a G_2 Structure on a 7-manifold. Specifically, we compute the evolution of the metric, the dual 4-form, and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernandez-Gray.
As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G_2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G_2 geometry.
TL;DR: This paper discusses Walker Structures, Lorentzian Walker Manifolds, and the Spectral Geometry of the Curvature Tensor.
Abstract: * Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds
TL;DR: In this paper, Sturm et al. introduced and studied rough curvature bounds for discrete spaces and graphs, and showed that the metric measure space which is approximated by a sequence of discrete spaces with rough curvatures ⩾ K will have curvature K in the sense of [J. Lott, C.Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. I, Acta Math.
TL;DR: In this article, the authors introduced the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime is locally determined by its curvature invariants.
Abstract: In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of \emph{strong} and \emph{weak} non-degeneracy.
TL;DR: In this article, a cubic four derivative 3 − 3 − 2 vertex for the interaction of spin 3 and spin 2 particles was constructed in terms of the (linearized) Riemann tensor.
Abstract: Recently Boulanger and Leclercq have constructed a cubic four derivative 3 − 3 − 2 vertex for the interaction of spin 3 and spin 2 particles. This vertex is trivially invariant under the gauge transformations of the spin 2 field, so it seemed that it could be expressed in terms of the (linearized) Riemann tensor. And indeed in this paper we managed to reproduce this vertex in the form R∂Φ∂Φ, where R is the linearized Riemann tensor and Φ is the completely symmetric third rank tensor. Then we consider the deformation of this vertex to (A)dS space and show that such deformation produces a 'standard' gravitational interaction for spin 3 particles (in the linear approximation) in agreement with general construction of Fradkin and Vasiliev. Then we turn to the massive case and show that the same higher derivative terms allow one to extend the gauge invariant description of a massive spin 3 particle from constant curvature spaces to arbitrary gravitational backgrounds satisfying Rμν = 0.
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TL;DR: In this article, a rigorous geometrical kinematical definition of the general Kundt spacetime in 4 dimensions is provided, where the degenerate Kundt spacetimes are defined as a null vector that admits a geodesic, expansion-free, shear-free and twist-free null vector.
Abstract: Kundt spacetimes are of great importance in general relativity in 4 dimensions and have a number of topical applications in higher dimensions in the context of string theory The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in 4 dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned The degenerate Kundt spacetimes are the only spacetimes in 4 dimensions that are not $\mathcal{I}$-non-degenerate, so that they are not determined by their scalar polynomial curvature invariants We first discuss the non-aligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which $
abla(Riem),
abla^{(2)}(Riem)$ are also of type D We discuss other local characteristics of the degenerate Kundt spacetimes Finally, we discuss degenerate Kundt spacetimes in higher dimensions
TL;DR: In this article, the existence of embedded spheres with large constant mean curvatures in any compact Riemannian manifold (M, g) was proved and this result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold has non-degenerate critical points.
Abstract: We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.
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TL;DR: In this article, it was shown that all manifolds with affine connection are globally projectively equivalent to some space with equiaffine connection (e.g., a symmetric Ricci tensor).
Abstract: In this paper we prove that all manifolds with affine connection are globally projectively equivalent to some space with equiaffine connection (equiaffine manifold). These manifolds are characterised by a symmetric Ricci tensor.
TL;DR: In this article, the authors presented balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity.
Abstract: We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold M
3 satisfy the heterotic equations of motion.
TL;DR: In this article, the results of using the computer algebra program Cadabra to develop Riemann normal coordinate expansions of the metric and other geometrical quantities, in particular the geodesic arc length, are given to sixth order in the curvature tensor.
Abstract: We present the results of using the computer algebra program Cadabra to develop Riemann normal coordinate expansions of the metric and other geometrical quantities, in particular the geodesic arc length. All of the results are given to sixth order in the curvature tensor.
TL;DR: The structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field are described.
Abstract: The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectory-based vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field. To illustrate the structures in asymmetric tensor fields, we introduce the notions of eigenvalue and eigenvector manifolds. These concepts afford a number of theoretical results that clarify the connections between symmetric and antisymmetric components in tensor fields. In addition, these manifolds naturally lead to partitions of tensor fields, which we use to design effective visualization strategies. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparameterization with physical meaning. This allows us to relate our tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide physical interpretation of our tensor-driven vector field analysis in the context of fluid mechanics. To demonstrate the utility of our approach, we have applied our visualization techniques and interpretation to the study of the Sullivan vortex as well as computational fluid dynamics simulation data.
23 Jun 2009
TL;DR: In this paper, some properties of Riemannian curvature tensors and curvature scalars of Kahler-Norden manifolds using the theory of Tachibana operators are presented.
Abstract: This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kahler-Norden manifolds using the theory of Tachibana operators is presented.
TL;DR: In this paper, the authors formulate a nonlinear theory of thermal stresses and obtain the explicit form of the governing partial differential equations for this equilibrium change, and show that geometric linearization leads to governing equations that are identical to those of the classical linear theory.
Abstract: In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change of temperature corresponds to a change of the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change of the material manifold, i.e. a change of the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change of temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.
TL;DR: In this article, the Ricci tensor tensor equation and the Einstein equation were considered in the pseudo-Euclidean space (Rn, g), where n ≥ 3 and gij = δij ei, ei = ± 1, where at least one ei is 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij depends on xi and xj.
Abstract: We consider the pseudo-Euclidean space (Rn, g), with n ≥ 3 and gij = δij ei, ei = ±1, where at least one ei = 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij (xi, xj) depends on xi and xj. We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on Rn, on the n-dimensional torus Tn and on cylinders Tk×Rn-k, that solve the Ricci equation or the Einstein equation.
TL;DR: In this article, the necessary and sufficient conditions for a metric tensor to be the Kerr solution were given, and these conditions exclusively involve explicit concomitants of the Riemann tensor.
Abstract: We give the necessary and sufficient (local) conditions for a metric tensor to be the Kerr solution. These conditions exclusively involve explicit concomitants of the Riemann tensor.
01 Jan 2009
TL;DR: For Riemannian manifolds with a measure (M, g, e f dvolg) as mentioned in this paper showed that the Ricci curvature and volume comparison can be improved when the Bakry-Emery Ricci tensor is bounded from below.
Abstract: For Riemannian manifolds with a measure (M, g, e f dvolg) we prove mean curvature and volume comparison results when the ∞Bakry-Emery Ricci tensor is bounded from below and f or |∇f | is bounded, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.
TL;DR: In this paper, the curvature of the B-compatible SO(3) connection is assumed to be self-dual, with the notion of selfduality defined by the B field.
Abstract: In Plebanski's self-dual formulation general relativity becomes SO(3) BF theory supplemented with the so-called simplicity (or metricity) constraints for the B-field. The main dynamical equation of the theory states that the curvature of the B-compatible SO(3) connection is self-dual, with the notion of self-duality being defined by the B-field. We describe a theory obtained by dropping the metricity constraints, keeping only the requirement that the curvature of the B-compatible connection is self-dual. It turns out that the theory one obtains is to a very large degree fixed by the Bianchi identities. Moreover, it is still a gravity theory, with just two propagating degrees of freedom as in GR.
TL;DR: In this paper, the relativistic significance of concircular curvature tensors has been explored and the existence of Killing and conformal Killing vectors has been established for spacetimes satisfying Einstein field equations.
Abstract: In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.
TL;DR: In this article, Chen-Yokota's argument was used to obtain a local lower bound of the scalar curvature for Ricci flow on complete manifolds. And if the curvature attains its minimum value at some point, then the manifold is Einstein.
Abstract: In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.
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TL;DR: In this paper, a local test function construction was used to seattle the most cases left by Escobar's and Marques's works and reduce the remaining case to the positive mass theorem.
Abstract: In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test function construction, we are able to seattle the most cases left by Escobar's and Marques's works. Moreover, we reduce the remaining case to the positive mass theorem. In this proof, we use the method developed in previous works by Brendle and by Brendle and the author.