Showing papers on "Ricci decomposition published in 2011"
TL;DR: In this paper, a self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar.
Abstract: We relate two formulations of the recently constructed double field theory to a frame-like geometrical formalism developed by Siegel. A self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar. This curvature scalar can be used to define an action, and it is shown that this action is equivalent to that of double field theory.
298 citations
TL;DR: In this paper, a new notion of Ricci curvature that applies to Markov chains on discrete spaces is introduced. But the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov Chains are gradient flows of the entropy.
Abstract: We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.
Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.
145 citations
01 Jun 2011
TL;DR: A survey of tensor triangular geometry can be found in this article, where the authors also discuss perspectives and suggest some problems in the early theory and first applications of the tensor triangle geometry.
Abstract: We survey tensor triangular geometry : Its examples, early theory and first applications. We also discuss perspectives and suggest some problems. Mathematics Subject Classification (2000). Primary 18E30; Secondary 14F05, 19G12, 19K35, 20C20, 53D37, 55P42.
106 citations
TL;DR: In this paper, it was shown that a Ricci soliton is rigid if and only if the Weyl conformal tensor of the Ricci tensor is harmonic, assuming that the curvature tensor has at most exponential growth.
Abstract: We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.
102 citations
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TL;DR: In this article, a generalized Margulis Lemma for manifolds with lower Ricci curvature bound was established for fundamental groups of compact $n$-manifolds.
Abstract: Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups
74 citations
01 Jun 2011
TL;DR: A survey of various results on the tensor product decomposition of irreducible finite-dimensional representations of a semisimple connected complex algebraic group can be found in this article.
Abstract: Let G be a semisimple connected complex algebraic group. We study the tensor product decomposition of irreducible finite-dimensional representations of G. The techniques we employ range from representation theory to algebraic geometry and topology. This is mainly a survey of author’s various results on the subject obtained individually or jointly with Belkale, Kapovich, Leeb, Millson and Stembridge.
65 citations
TL;DR: In this article, an invariant and canonical contraction between covariant indices was introduced for singular semi-Riemannian manifolds, which is applicable even for degenerate metrics.
Abstract: On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this article we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.
42 citations
TL;DR: In this paper, the authors extend the result of Derdzinski and Shen on the re- strictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor.
Abstract: We extend a classical result by Derdzinski and Shen, on the re- strictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of some well known differential structures.
38 citations
TL;DR: In this article, it was shown that any Weyl curvature model can be geometrically realized by a Weyl manifold, and that the manifold can be used to represent the Weyl curve.
37 citations
TL;DR: The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation and has been applied broadly to communication networks, medical physics, computer design and more as discussed by the authors.
Abstract: The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The three-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The four-dimensional Ric is the Einstein tensor for such spacetimes. More recently, the Ric was used by Hamilton to define a nonlinear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincar? conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area?an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimensions.
36 citations
TL;DR: In this paper, the L p essential spectra of the Laplacian on functions are [0, +∞] on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity.
TL;DR: In this paper, the authors established short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations.
Abstract: We establish short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations. We apply this theorem to flows by powers of the Laplacian of the Ricci tensor, and to flows generated by the ambient obstruction tensor. As a special case, we prove short-time existence for a type of Bach flow.
TL;DR: In this paper, the Eisenhart problem is interpreted in terms of Ricci solitons for the symmetric case and the generator of the manifold is assumed to have a Ricci-soliton.
Abstract: The Eisenhart problem of nding parallel tensors treated already in the framework
of quasi-constant curvature manifolds in [Jia] is reconsidered for the symmetric case
and the result is interpreted in terms of Ricci solitons. If the generator of the manifold
provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant
scalar curvature and vanishing D-concircular tensor eld and ii) shrinking on a class
of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space
form.
01 Jan 2011
TL;DR: The T-curvature tensor as mentioned in this paper is a new curvature tensors which is defined as the T-Curvature ten-plus tensor, and its properties for T-conservative and T-flat semi-Riemannian manifolds are given.
Abstract: We introduce a new curvature tensor named as the T-curvature ten- sor We show that the quasi-conformal, conformal, conharmonic, concircular, pseudo-projective, projective, M-projective, Wi-curvature tensors (i =0 ,, 9), W ∗ -curvature tensors (j =0 , 1) are the particular cases of the T-curvature tensor Some properties for the T-curvature tensor are given We obtain the results for T-conservative and T-flat semi-Riemannian manifolds
TL;DR: In this paper, the Paraholomorphic conditions for the complete lifts of vector fields are analyzed on the (1, 1) tensor bundle of a Riemannian manifold.
Abstract: Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kahler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.
TL;DR: In this paper, the authors studied limit spaces of a sequence of n-dimensional complete Riemannian manifolds whose Ricci curvatures have definite lower bounds, and gave several measure theoretical properties of such limit spaces.
Abstract: In this paper, we study limit spaces of a sequence of n-dimensional complete Riemannian manifolds whose Ricci curvatures have definite lower bound. We will give several measure theoretical properties of such limit spaces.
TL;DR: In this article, the authors consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics and propose necessary conditions on the discriminants of the associated characteristic equation which can be expressed in terms of polynomial scalar curvature invariants.
Abstract: We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restricted eigenvector structure for algebraic types II and D, which leads to necessary conditions on the discriminants of the associated characteristic equation which can be manifestly expressed in terms of polynomial scalar curvature invariants. The use of such necessary conditions in terms of scalar curvature invariants will be of great utility in the study and classification of black hole solutions in more than four dimensions.
TL;DR: A constructive proof based on an eigenvalue criterion is provided that shows when a 2’× 2 ×’2’–”2 tensor over ℝ is rank-3 and when it isRank-2 and the results are extended to show that n’s have maximum possible rank n + k where k is the number of complex conjugate eigen value pairs of the matrices forming the two faces of the
Abstract: As computing power increases, many more problems in engineering and data analysis involve computation with tensors, or multi-way data arrays. Most applications involve computing a decomposition of a tensor into a linear combination of rank-1 tensors. Ideally, the decomposition involves a minimal number of terms, i.e. computation of the rank of the tensor. Tensor rank is not a straight-forward extension of matrix rank. A constructive proof based on an eigenvalue criterion is provided that shows when a 2 × 2 × 2 tensor over ℝ is rank-3 and when it is rank-2. The results are extended to show that n × n × 2 tensors over ℝ have maximum possible rank n + k where k is the number of complex conjugate eigenvalue pairs of the matrices forming the two faces of the tensor cube.
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TL;DR: In this article, the Ricci flow on open manifolds of nonnegative complex sectional curvature was shown to have short-time existence and an optimal volume growth condition which guarantees long time existence.
Abstract: We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature We do not require upper curvature bounds By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold Furthermore, we find an optimal volume growth condition which guarantees long time existence, and we give an analysis of the long time behaviour of the Ricci flow Finally, we construct an explicit example of an immortal nonnegatively curved solution of the Ricci flow with unbounded curvature for all time
TL;DR: In this paper, the authors show that Yang-Mills gravity with translational gauge group T(4) in flat space-time implies a simple self-coupling of gravitons and a truly conserved energy-momentum tensor.
Abstract: Yang-Mills gravity with translational gauge group T(4) in flat space-time implies a simple self-coupling of gravitons and a truly conserved energy-momentum tensor. Its consistency with experiments crucially depends on an interesting property that an “effective Riemannian metric tensor” emerges in and only in the geometric-optics limit of the photon and particle wave equations. We obtain Feynman rules for a coupled graviton-fermion system, including a general graviton propagator with two gauge parameters and the interaction of ghost particles. The equation of motion of macroscopic objects, as an N -body system, is demonstrated as the geometric-optics limit of the fermion wave equation. We discuss a relativistic Hamilton-Jacobi equation with an “effective Riemann metric tensor” for the classical particles.
TL;DR: In this paper, a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold are obtained.
Abstract: We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.
TL;DR: In this paper, the authors studied the possibility of Lagrangian formulation for free higher spin bosonic totally symmetric tensor field on the background manifold characterizing by the arbitrary metric, vector and third rank tensor fields in the framework of BRST approach.
Abstract: We study a possibility of Lagrangian formulation for free higher spin bosonic totally symmetric tensor field on the background manifold characterizing by the arbitrary metric, vector and third rank tensor fields in framework of BRST approach. Assuming existence of massless and flat limits in the Lagrangian and using the most general form of the operators of constraints we show that the algebra generated by these operators will be closed only for constant curvature space with no nontrivial coupling to the third rank tensor and the strength of the vector fields. This result finally proves that the consistent Lagrangian formulation at the conditions under consideration is possible only in constant curvature Riemann space.
TL;DR: Physical decomposition of the non-Abelian gauge field has recently helped to achieve a meaningful gluon spin this paper, where the metric is unambiguously separated into a pure geometric term which contributes a null curvature tensor, and a physical term which represents the true gravitational effect and always vanishes in a flat space-time.
Abstract: Physical decomposition of the non-Abelian gauge field has recently helped to achieve a meaningful gluon spin. Here we extend this approach to gravity and attempt a meaningful gravitational energy. The metric is unambiguously separated into a pure geometric term which contributes a null curvature tensor, and a physical term which represents the true gravitational effect and always vanishes in a flat space-time. By this decomposition the conventional pseudotensors of the gravitational stress-energy are easily rescued to produce a definite physical result. Our decomposition applies to any symmetric tensor, and has an interesting relation to the transverse-traceless decomposition discussed by Arnowitt, Deser and Misner, and by York.
TL;DR: A physical interpretation for the Ricci scalar is derived and its significance in diffusion tensor imaging (DTI) is explored and extended to the case of high angular resolution diffusion imaging (HARDI) using Finsler geometry.
01 Mar 2011
TL;DR: In this paper, the authors studied the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary.
Abstract: In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is nonnegative-definite.
TL;DR: In this article, the authors define pure radiation metrics with parallel rays as n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K.
Abstract: We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays Then we derive conditions in terms of tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class We also give an analogous result for n-dimensional pseudo-Riemannian pp-waves
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TL;DR: In this article, a survey of Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations is presented, and it is shown that the vector field is an infiniteimal harmonic transformation.
Abstract: The concept of the Ricci soliton was introduced by Hamilton. Ricci soliton is defined by vector field and it's a natural generalization of Einstein metric. We have shown earlier that the vector field of Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations.
01 Jan 2011
TL;DR: In this paper, it was proved that a compact φ-conharmonically flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature.
Abstract: Some necessary and/or sufficient condition(s) for K-contact and/or Sasakian manifolds to be quasi conharmonically flat, ξ-conharmonically flat and φ-conharmonically flat are obtained. In last, it is proved that a compact φ-conharmonically flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature ( 3− 2 2n−1 ) . 2010 Mathematics Subject Classification: 53C25, 53D10, 53D15
Journal Article•
TL;DR: In this article, the Ricci tensor and the curvature tensor of a Riemannian manifold equipped with a semi symmetric metric connection were investigated and a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold was obtained.
Abstract: In this study, we consider a manifold equipped with semi symmetric metric connection whose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensor and the curvature tensor of this manifold . We obtain a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentioned above is conformally flat, then it is a mixed generalized quasi- Einstein manifold and we prove that if the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the special torsion tensor is independent from orientation chosen, then this manifold is of a mixed generalized quasi constant curvature.