Showing papers on "Riemann curvature tensor published in 2006"

A Riemannian Framework for Tensor Computing

Abstract: Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.

Lectures on the Ricci Flow

Abstract: 1. Introduction 2. Riemannian geometry background 3. The maximum principle 4. Comments on existence theory for parabolic PDE 5. Existence theory for the Ricci flow 6. Ricci flow as a gradient flow 7. Compactness of Riemannian manifolds and flows 8. Perelman's W entropy functional 9. Curvature pinching and preserved curvature properties under Ricci flow 10. Three-manifolds with positive Ricci curvature and beyond.

A note on covariant conservation of energy–momentum in modified gravities

Abstract: An explicit proof of the vanishing of the covariant divergence of the energy–momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy–momentum is built in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories.

On spacetimes with constant scalar invariants

Abstract: We study Lorentzian spacetimes for which all scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes). We obtain a number of general results in arbitrary dimensions. We study and construct warped product CSI spacetimes and higher-dimensional Kundt CSI spacetimes. We show how these spacetimes can be constructed from locally homogeneous spaces and VSI spacetimes. The results suggest a number of conjectures. In particular, it is plausible that for CSI spacetimes that are not locally homogeneous the Weyl type is II, III, N or O, with any boost weight zero components being constant. We then consider the four-dimensional CSI spacetimes in more detail. We show that there are severe constraints on these spacetimes, and we argue that it is plausible that they are either locally homogeneous or that the spacetime necessarily belongs to the Kundt class of CSI spacetimes, all of which are constructed. The four-dimensional results lend support to the conjectures in higher dimensions.

Holography of gravitational action functionals

Abstract: Einstein-Hilbert (EH) action can be separated into a bulk and a surface term, with a specific (``holographic'') relationship between the two, so that either can be used to extract information about the other. The surface term can also be interpreted as the entropy of the horizon in a wide class of spacetimes. Since EH action is likely to just the first term in the derivative expansion of an effective theory, it is interesting to ask whether these features continue to hold for more general gravitational actions. We provide a comprehensive analysis of Lagrangians of the form $\sqrt{\ensuremath{-}g}L=\sqrt{\ensuremath{-}g}Q_{a}{}^{bcd}R^{a}{}_{bcd}$, in which $Q_{a}{}^{bcd}$ is a tensor with the symmetries of the curvature tensor, made from metric and curvature tensor and satisfies the condition ${\ensuremath{ abla}}_{c}Q_{a}{}^{bcd}=0$, and show that they share these features. The Lanczos-Lovelock Lagrangians are a subset of these in which $Q_{a}{}^{bcd}$ is a homogeneous function of the curvature tensor. They are all holographic, in a specific sense of the term, and---in all these cases---the surface term can be interpreted as the horizon entropy. The thermodynamics route to gravity, in which the field equations are interpreted as $TdS=dE+pdV$, seems to have a greater degree of validity than the field equations of Einstein gravity itself. The results suggest that the holographic feature of EH action could also serve as a new symmetry principle in constraining the semiclassical corrections to Einstein gravity. The implications are discussed.

Constraints for the existence of flat and stable non-supersymmetric vacua in supergravity

Abstract: We further develop on the study of the conditions for the existence of locally stable non-supersymmetric vacua with vanishing cosmological constant in supergravity models involving only chiral super fields. Starting from the two necessary conditions for flatness and stability derived in a previous paper (which involve the Kahler metric and its Riemann tensor contracted with the supersymmetry breaking auxiliary fields) we show that the implications of these constraints can be worked out exactly not only for factorizable scalar manifolds, but also for symmetric coset manifolds. In both cases, the conditions imply a strong restriction on the Kahler geometry and constrain the vector of auxiliary fields de ning the Goldstino direction to lie in a certain cone. We then apply these results to the various homogeneous coset manifolds spanned by the moduli and untwisted matter fields arising in string compactications, and discuss their implications. Finally, we also discuss what can be said for completely arbitrary scalar manifolds, and derive in this more general case some explicit but weaker restrictions on the Kahler geometry.

Rigidity of Polyhedral Surfaces

Abstract: We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, Cohen-Kenyon-Propp, Leibon and Bobenko-Springborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi identity.

Manifolds of Positive Scalar Curvature: A Progress Report

Abstract: In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a given manifold (always assumed C∞) admits a Riemannian metric with positive or non-negative scalar curvature. (If the manifold is non-compact, we require the metric to be complete; otherwise this is no restriction at all.) We will not go over the historical development of this subject or everything that is known about it; instead, our focus here will be on updating the existing surveys [20], [68], [69] and [58].

Geodesics in randers spaces of constant curvature

Abstract: Geodesies in Randers spaces of constant curvature are classified. Randers metrics have received much attention lately as solutions to Zermelo's problem of navigation; largely because this navigation structure provides the frame work for a complete classification of constant flag curvature Randers spaces. (Flag curvature is the Finslerian analog of Riemannian sectional curvature. See (BR04).) Briefly, a Randers metric is of constant flag curvature if and only if it solves Zer melo's problem of navigation on a Riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety W. See Subsection 1.1 for a sketch of the navigation problem, and Theorem 3 for an explicit statement of the classification result. The aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature. Intuitively, these paths minimize travel time across a Riemannian landscape under windy conditions. Presently we will show that these curves are given by composing geodesies of the Riemannian metric with the flow generated by W. This claim is formalized by Theorem 2. Geodesies on surfaces of constant, nonpositive curvature are illustrated in Section 3. We then turn, in Section 4, to the constant flag curvature K = 1 Randers metrics on Sn. The case of the sphere is especially interesting; it is possible to endow this closed manifold with a metric whose geodesies display distinctly non-Riemannian behaviors. For example: (1) A metric is projectively flat if every point admits coordinates in which the geodesies are straight lines. Belt r ami's theorem assures us that a Riemannian metric is of constant sectional curvature if and only if it is projectively flat. In contrast few Randers spaces of constant flag curvature are projectively flat. There are infinitely many nonisometric Randers metrics of constant, positive

Integral geometry problem for nontrapping manifolds

Abstract: We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.

Properties of a scalar curvature invariant depending on two planes

Abstract: Based on Schouten’s interpretation of the Riemann–Christoffel curvature tensor R, a geometrical meaning for the tensor R·R is presented. It follows that the condition of semi-symmetry, i.e. R·R = 0, can be interpreted as the invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram. Using the tensor R· R, and in analogy with the definition of the sectional curvature K(p,π) of a plane π, a scalar curvature invariant L(p,π, \({\overline{\pi}}\)) is constructed which in general depends on two planes π and \({\overline{\pi}}\) at the same point p. This invariant can be geometrically interpreted in terms of the parallelogramoids of Levi–Civita and it is shown that it completely determines the tensor R· R. Further it is demonstrated that the isotropy of this new scalar curvature invariant L(p,π, \({\overline{\pi}}\)) with respect to both the planes π and \({\overline{\pi}}\) amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.

The ambient obstruction tensor and the conformal deformation complex

Abstract: We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the (Fefferman?Graham) ambient obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence-free and vanishes identically for conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman?Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian-type operators that generalise the conformal Laplacians of Graham?Jenne?Mason?Sparling. We give an algorithm for calculating explicit formulae for these operators, and this is applied to give formulae for the obstruction tensor in dimensions 6 and 8. As background to these issues, we give an explicit construction of the deformation complex in dimensions n = 4, construct two related (detour) complexes, and establish essential properties of the operators in these.

Einstein-Cartan Theory

Abstract: The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum This modification was put forward in 1922 by Elie Cartan, before the discovery of spin Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium

Second fundamental measure of geometric sets and local approximation of curvatures

Abstract: Using the theory of normal cycles, we associate with each geometric subset of a Riemannian manifold a —tensor-valued— curvature measure, which we call its second fundamental measure. This measure provides a finer description of the geometry of singular sets than the standard curvature measures. Moreover, we deal with approximation of curvature measures. We get a local quantitative estimate of the difference between curvature measures of two geometric subsets, when one of them is a smooth hypersurface.

Fermi coordinates and Penrose limits

Abstract: We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in spacetime) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalizes the covariant description of the lowest order Penrose limit metric itself, obtained in Blau et al (2004 Class. Quantum Grav. 21 L43-9). We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS 5 x S 5 .

Curvature-Driven PDE Methods for Matrix-Valued Images

Abstract: Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo's concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts.

Eigenvalue Monotonicity for the Ricci-Hamilton Flow

Abstract: In this short note, we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow.

Statistical manifolds with almost contact structures and its statistical submersions

Abstract: In this paper, we discuss statistical manifolds with almost contact sturctures. We define a Sasaki-like statistical manifold. Moreover, we consider Sasaki-like statistical submersions, and we study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12).

Charged axially symmetric solution, energy and angular momentum in tetrad theory of gravitation

Abstract: Charged axially symmetric solution of the coupled gravitational and electromagnetic fields in the tetrad theory of gravitation is derived. The metric associated with this solution is an axially symmetric metric which is characterized by three parameters "the gravitational mass M, the charge parameter Q and the rotation parameter a." The parallel vector fields and the electromagnetic vector potential are axially symmetric. We calculate the total exterior energy. The energy–momentum complex given by Moller in the framework of the Weitzenbock geometry "characterized by vanishing the curvature tensor constructed from the connection of this geometry" has been used. This energy–momentum complex is considered as a better definition for calculation of energy and momentum than those of general relativity theory. The energy contained in a sphere is found to be consistent with pervious results which is shared by its interior and exterior. Switching off the charge parameter, one finds that no energy is shared by the exterior of the charged axially symmetric solution. The components of the momentum density are also calculated and used to evaluate the angular momentum distribution. We found no angular momentum contributes to the exterior of the charged axially symmetric solution if zero charge parameter is used.

Existence of an asymptotic velocity and implications for the asymptotic behaviour in the direction of the singularity in T3-Gowdy

Abstract: This is the first of two papers that together prove strong cosmic censorship in T-3-Gowdy space-times In the end, we prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with respect to the C-infinity topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, ie, the Riemann tensor contracted with itself, blows up in the incomplete direction In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions In this paper, we shall, however, focus on the concept of asymptotic velocity Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint The target of the wave map is the hyperbolic plane There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy We define the asymptotic velocity v(infinity) to be the nonnegative square root of the limit of the kinetic energy density The asymptotic velocity has some very important properties In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v(infinity) It also has properties such that if 0 1 and v(infinity) is continuous at theta(0), then v(infinity) is smooth in a neighborhood of theta(0) Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to the C-2 X C-1 topology on initial data

Fermi Coordinates and Penrose Limits

Abstract: We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalises the covariant description of the lowest order Penrose limit metric itself, obtained in hep-th/0312029. We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5.

Post-Einsteinian tests of gravitation

Abstract: Einstein gravitation theory can be extended by preserving its geometrical nature but changing the relation between curvature and energy–momentum tensors. This change accounts for radiative corrections, replacing the Newton gravitation constant by two running couplings which depend on scale and differ in the two sectors of traceless and traced tensors. The metric and curvature tensors in the field of the Sun, which were obtained in previous papers within a linearized approximation, are then calculated without this restriction. Modifications of gravitational effects on geodesics are then studied, allowing one to explore phenomenological consequences of extensions lying in the vicinity of general relativity. Some of these extended theories are able to account for the Pioneer anomaly while remaining compatible with tests involving the motion of planets. The PPN ansatz corresponds to peculiar extensions of general relativity which do not have the ability to meet this compatibility challenge.

Spectral line broadening and angular blurring due to spacetime geometry fluctuations

Abstract: We treat two possible phenomenological effects of quantum fluctuations of spacetime geometry: spectral line broadening and angular blurring of the image of a distance source. A geometrical construction will be used to express both effects in terms of the Riemann tensor correlation function. We apply the resulting expressions to study some explicit examples in which the fluctuations arise from a bath of gravitons in either a squeezed state or a thermal state. In the case of a squeezed state, one has two limits of interest: a coherent state which exhibits classical time variation but no fluctuations, and a squeezed vacuum state, in which the fluctuations are maximized.

Averaging spherically symmetric spacetimes in general relativity

Abstract: We discuss the averaging problem in general relativity, using the form of the macroscopic gravity equations in the case of spherical symmetry in volume preserving coordinates. In particular, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. On cosmological scales, the correlation tensor in a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background is found to be of the form of a spatial curvature. On astrophysical scales the correlation tensor can be interpreted as the sum of a spatial curvature and an anisotropic fluid. We briefly discuss the physical implications of these results.

Removability of singularities for a class of fully non-linear elliptic equations

Abstract: In this paper we address the problem of understanding the singularities of the fully non-linear elliptic equation σk(v) = 1. These σk curvature are defined as the symmetric functions of the eigenvalues of the Schouten tensor of a Riemannian metric and appear naturally in conformal geometry, in fact, σ1 is just the scalar curvature.