Showing papers on "Riemann curvature tensor published in 2011"

Frame-like geometry of double field theory

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.

Ricci curvature of graphs

Abstract: We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

Ricci curvature of finite Markov chains via convexity 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.

Beyond Einstein-Cartan gravity: quadratic torsion and curvature invariants with even and odd parity including all boundary terms

Abstract: Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein–Cartan(–Sciama–Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero–Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh–Yan form, can be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al and Baekler et al, emerges also in the framework of Diakonov et al. We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann–Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) In a Riemann–Cartan spacetime, that is, in a spacetime with torsion, parity-violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann–Cartan spacetime is a natural habitat for chiral fermionic matter fields.

Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics

Abstract: Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ -convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W 2,2 isometric immersion of a given 2 d metric into .

Rigidity of shrinking Ricci solitons

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.

Beyond Einstein-Cartan gravity: Quadratic torsion and curvature invariants with even and odd parity including all boundary terms

Abstract: Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al. and Baekler et al., emerges also in the framework of Diakonov et al.(2011). We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields.

Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications

Abstract: When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free tensors: (i) the Weyl tensor’s so-called electric part or tidal field Ɛ_(jk), which raises tides on the Earth’s oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ξ^k is Δa_j=-Ɛ_(jk)ξ^k), and (ii) the Weyl tensor’s so-called magnetic part or (as we call it) frame-drag field B_(jk), which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ξ^k, as measured using a local inertial frame at the tail of ξ^k, is ΔΩ_j=B_(jk)ξ^k). Being symmetric and trace-free, Ɛ_(jk) and B_(jk) each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of Ɛ_(jk)’s eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line’s eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for B_(jk) are frame-drag vortex lines or simply vortex lines, their vorticities, and their vortexes. These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side-by-side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing current-quadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outward-and-backward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing mass-quadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become current-quadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary’s tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary’s mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate). In Paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a brief overview of these applications in R. Owen et al. Phys. Rev. Lett. 106 151101 (2011). We expect these vortex and tendex concepts to become powerful tools for general relativity research in a variety of topics.

On locally conformally flat gradient shrinking ricci solitons

Abstract: In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

Structure of fundamental groups of manifolds with Ricci curvature bounded below

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

On conformal Killing symmetric tensor fields on Riemannian manifolds

Abstract: A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.

All unitary cubic curvature gravities in D dimensions

Abstract: We construct all the unitary cubic curvature gravity theories built on the contractions of the Riemann tensor in D-dimensional (anti)-de Sitter spacetimes. Our construction is based on finding the equivalent quadratic action for the general cubic curvature theory and imposing ghost and tachyon freedom, which greatly simplifies the highly complicated problem of finding the propagator of cubic curvature theories in constant curvature backgrounds. To carry out the procedure we have also classified all the unitary quadratic models. We use our general results to study the recently found cubic curvature theories using different techniques and the string generated cubic curvature gravity model. We also study the scattering in critical gravity and give its cubic curvature extensions.

Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem

Abstract: Analysis: Variational Problems Related to Sobolev Inequalities on Carnot Groups Groups of Heisenberg and Iwasawa Types Explicit Solutions to the Yamabe Equation Symmetries Solutions on Groups of Iwasawa Type Geometry: Quaternionic Contact Manifolds - Connection, Curvature and qc-Einstein Structures Quaternionic Contact Conformal Curvature Tensor The Quaternionic Contact Yamabe Pronlem and the Yamabe Constant of the qc Spheres CR Manifolds - Cartan and Chern-Moser Tensor and Theorem

Differential Geometry: Bundles, Connections, Metrics and Curvature

Abstract: 1 Smooth manifolds 2 Matrices and Lie groups 3 Introduction to vector bundles 4 Algebra of vector bundles 5 Maps and vector bundles 6 Vector bundles with fiber C]n 7 Metrics on vector bundles 8 Geodesics 9 Properties of geodesics 10 Principal bundles 11 Covariant derivatives and connections 12 Covariant derivatives, connections and curvature 13 Flat connections and holonomy 14 Curvature polynomials and characteristic classes 15 Covariant derivatives and metrics 16 The Riemann curvature tensor 17 Complex manifolds 18 Holomorphic submanifolds, holomorphic sections and curvature 19 The Hodge star Indexed list of propositions by subject Index

Effective gravitational wave stress-energy tensor in alternative theories of gravity

Abstract: The inspiral of binary systems in vacuum is controlled by the stress-energy of gravitational radiation and any other propagating degrees of freedom. For gravitational waves, the dominant contribution is characterized by an effective stress-energy tensor at future null infinity. We employ perturbation theory and the short-wavelength approximation to compute this stress-energy tensor in a wide class of alternative theories. We find that this tensor is generally a modification of that first computed by Isaacson, where the corrections can dominate over the general relativistic term. In a wide class of theories, however, these corrections identically vanish at asymptotically flat, future, null infinity, reducing the stress-energy tensor to Isaacson's. We exemplify this phenomenon by first considering dynamical Chern-Simons modified gravity, which corrects the action via a scalar field and the contraction of the Riemann tensor and its dual. We then consider a wide class of theories with dynamical scalar fields coupled to higher-order curvature invariants and show that the gravitational wave stress-energy tensor still reduces to Isaacson's. The calculations presented in this paper are crucial to perform systematic tests of such modified gravity theories through the orbital decay of binary pulsars or through gravitational wave observations.

The inverse mean curvature flow in rotationally symmetric spaces

Abstract: In this paper, the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied. It is proved that the flow converges to a unique geodesic sphere, i.e., every principle curvature of the hypersurfaces converges to a same constant under the flow.

Some aspects of field equations in generalized theories of gravity

Abstract: A class of theories of gravity based on a Lagrangian $L=L(R^{a}{}_{bcd},{g}^{ab})$ which depends on the curvature and metric---but not on the derivatives of the curvature tensor---is of interest in several contexts including in the development of the paradigm that treats gravity as an emergent phenomenon. This class of models contains, as an important subset, all Lanczos-Lovelock models of gravity. I derive several identities and properties which are useful in the study of these models and clarify some of the issues that seem to have received insufficient attention in the past literature.

Dirac fields in f(R)-gravity with torsion

Abstract: We study f(R)-gravity with torsion in the presence of Dirac massive fields. Using the Bianchi identities, we formulate the conservation laws of the theory and we check the consistency with the matter field equations. Further, we decompose the field equations in torsionless and torsional terms: we show that the nonlinearity of the gravitational Lagrangian reduces to the presence of a scalar field that depends on the spinor field; this additional scalar field gives rise to an effective stress?energy tensor and plays the role of a scale factor modifying the normalization of Dirac fields. Problems for fermions regarding the positivity of energy and the particle?antiparticle duality are discussed.

On Singular Semi-Riemannian Manifolds

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.

From Petrov-Einstein to Navier-Stokes in Spatially Curved Spacetime

Abstract: We generalize the framework in arXiv:1104.5502 to the case that an embedding may have a nonvanishing intrinsic curvature. Directly employing the Brown-York stress tensor as the fundamental variables, we study the effect of finite perturbations of the extrinsic curvature while keeping the intrinsic metric fixed. We show that imposing a Petrov type I condition on the hypersurface geometry may reduce to the incompressible Navier-Stokes equation for a fluid moving in spatially curved spacetime in the near-horizon limit.

Projective Metrizability and Formal Integrability

Abstract: The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P1 and a set of algebraic con- ditions on semi-basic 1-forms. We discuss the formal integrability of P1 using two sufficient conditions provided by Cartan{Kahler theorem. We prove in Theorem 4.2 that the symbol of P1 is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P1, and this obstruction is due to the curvature tensor of the in- duced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable.

Extended derdzinski-shen theorem for the riemann 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.

An exotic t1s 4 with positive curvature

Abstract: We construct a metric with positive sectional curvature on a 7-manifold which supports an isometry group with orbits of codimension 1. It is a connection metric on the total space of an orbifold 3-sphere bundle over an orbifold 4-sphere. By a result of S. Goette, the manifold is homeomorphic but not diffeomorphic to the unit tangent bundle of the 4-sphere. Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstructions to non-negative curvature which are: (1) the Betti number theorem of Gromov which asserts that the homology of a compact manifold with non-negative sectional curvature has an a priori bound on the number of generators depending only on the dimension; and (2) a result of Lichnerowicz and Hitchin implying that a spin manifold with non-trivial ˆ A genus or generalized a genus cannot admit a metric with non-negative curvature. One way to gain further insight is to construct and analyze examples. This is quite difficult and has been achieved only a few times. Aside from the classical rank- one symmetric spaces, i.e. the spheres and the projective spaces with their canoni- cal metrics, and the recently proposed deformation of the so-called Gromoll-Meyer sphere (PW2), examples were only found in the '60s by Berger (Ber), in the '70s by Wallach (W) and by Aloff and Wallach (AlW), in the '80s by Eschenburg (E1,2), and in the '90s by Bazaikin (Ba). The examples by Berger, Wallach and Aloff-Wallach were shown, by Wallach in even dimensions (W) and by Berard Bergery (Be) in odd dimensions, to constitute a classification of simply connected homogeneous mani- folds of positive curvature, whereas the examples due to Eschenburg and Bazaikin typically are non-homogeneous, even up to homotopy. All of these examples can be obtained as quotients of compact Lie groups G with a bi-invariant metric by a free isometric "two sided" action of a subgroup H ⊂ G × G. Since a Lie group with a bi- invariant metric has non-negative curvature so do such quotients, and in rare cases

The curvature of gradient Ricci solitons

Abstract: Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many of the techniques used in these works required some control of the Ricci curvature. For example, in [15] gradient shrinking Ricci solitons which are locally conformally flat were classified assuming an integral condition on the Ricci tensor. This condition and other integral estimates of the curvature were later proved in [12]. Without making the strong assumption of being conformally flat, it is natural to ask whether similar estimates are true for the Riemann curvature tensor. In this paper we are able to prove pointwise estimates on the Riemann curvature, assuming in addition that the Ricci curvature is bounded. We will show that any gradient shrinking Ricci soliton with bounded Ricci curvature has Riemann curvature tensor growing at most polynomially in the distance function. We note that by Shi’s local derivative estimates we can then obtain growth estimates on all derivatives of the curvature. This, in particular, proves weighted L estimates for the Riemann curvature tensor and its covariant derivatives. We point out that for self shrinkers of the mean curvature flow Colding and Minicozzi [8] were able to prove weighted L estimates for the second fundamental form, assuming the mean curvature is positive. These estimates were instrumental in the classification of stable shrinkers. Our estimates can be viewed as parallel to theirs, however the classification of gradient Ricci solitons is still a major open question in the field. A gradient shrinking Ricci soliton is a Riemannian manifold (M, g) for which there exists a potential function f such that

Simple counterterms for asymptotically AdS spacetimes in Lovelock gravity

Abstract: Although gravitational actions diverge in asymptotically anti-de Sitter spacetimes, boundary counterterms can be added in order to cancel out those divergences; such counterterms are known in general to third-order in the Riemann tensor for the Einstein-Hilbert action. Considering foliations of anti-de Sitter with an ${S}^{m}\ifmmode\times\else\texttimes\fi{}{H}^{d\ensuremath{-}m}$ boundary, we discuss a simple algorithm which we use to generate counterterms up to sixth order in the Riemann tensor, for the Einstein-Hilbert, Gauss-Bonnet and third-order-Lovelock Lagrangians. We also comment on other theories such as $F(R)$ gravity.

Structure of Lanczos-Lovelock Lagrangians in critical dimensions

Abstract: The Lanczos-Lovelock models of gravity constitute the most general theories of gravity in D-dimensions which satisfy (a) the principle of of equivalence, (b) the principle of general covariance, and (c) have field equations involving derivatives of the metric tensor only up to second order. The mth order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature tensor. The field equations resulting from it become trivial in the critical dimension D = 2m and the action itself can be written as the integral of an exterior derivative of an expression involving the vierbeins, in the differential form language. While these results are well known, there is some controversy in the literature as to whether the Lanczos-Lovelock Lagrangian itself can be expressed as a total divergence of quantities built only from the metric and its derivatives (without using the vierbeins) in D = 2m. We settle this issue by showing that this is indeed possible and provide an algorithm for its construction. In particular, we demonstrate that, in two dimensions, \({R \sqrt{-g} = \partial_j R^j}\) for a doublet of functions Rj = (R0, R1) which depends only on the metric and its first derivatives. We explicitly construct families of such Rj-s in two dimensions. We also address related questions regarding the Gauss-Bonnet Lagrangian in D = 4. Finally, we demonstrate the relation between the Chern-Simons form and the mth order Lanczos-Lovelock Lagrangian.

A second-order identity for the Riemann tensor and applications

Abstract: A second-order differential identity for the Riemann tensor is obtained, on a manifold with a symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors are derived from it. Applications to manifolds with recurrent or symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity due to Lovelock.