Showing papers on "Riemann curvature tensor published in 2005"
TL;DR: In this article, the vanishing of the covariant divergence of the energy-momentum tensor in modified theories of gravity is presented, and the generalized Bianchi identity can also be deduced directly from the covariance of the extended gravitational action.
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, we demonstrate that 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. In the light of our interpretation of the independent connection as an auxiliary variable we can also reconsider some criticisms of the Palatini formulation originally raised by Buchdahl.
266 citations
TL;DR: In this paper, it was shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in 4D N = 4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes, which originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from nonholomorphic terms.
Abstract: It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N = 4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the macroscopic entropy, written as a function of the dilaton field, is stationary at the horizon by virtue of the attractor equations.
246 citations
TL;DR: A new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes based on the curvature tensor field analysis is presented, which decomposes the object into near constant curvature patches and corrects boundaries by suppressing their artefacts or discontinuities.
Abstract: This paper presents a new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes. The algorithm is based on the curvature tensor field analysis and presents two distinct complementary steps: a region based segmentation, which is an improvement of that presented by Lavoue et al. [Lavoue G, Dupont F, Baskurt A. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle, J WSCG 2004;12(2):245-52] and which decomposes the object into near constant curvature patches, and a boundary rectification based on curvature tensor directions, which corrects boundaries by suppressing their artefacts or discontinuities. Experiments conducted on various models including both CAD and natural objects, show satisfactory results. Resulting segmented patches, by virtue of their properties (homogeneous curvature, clean boundaries) are particularly adapted to computer graphics tasks like parametric or subdivision surface fitting in an adaptive compression objective.
219 citations
TL;DR: In this article, it was shown that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings.
Abstract: We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomologyclass. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.
172 citations
TL;DR: In this article, it was shown that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor are equivalent to multi-scalar-tensor gravities with four-derivative terms.
Abstract: We show that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor as well as the Ricci scalar are equivalent to multi-scalar–tensor gravities with four-derivative terms. By expanding the action around a vacuum spacetime, the action is reduced to that of the Einstein gravity with four-derivative terms, and consequently there appears a massive spin-2 ghost in such generalized gravity theories in addition to a massive spin-0 field.
168 citations
13 Jun 2005
TL;DR: In this paper, a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes is proposed, which is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting.
Abstract: We propose a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes. The crest lines, salient surface features defined via first- and second-order curvature derivatives, are widely used for shape matching and interrogation purposes. Their practical extraction is difficult because it requires good estimation of high-order surface derivatives. Our approach to the crest line detection is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting.Since the crest lines are not defined in the surface regions where the surface focal set (caustic) degenerates, we introduce a new thresholding scheme which exploits interesting relationships between curvature extrema, the so-called MVS functional of Moreton and Sequin, and Dupin cyclides,An application of the crest lines to adaptive mesh simplification is also considered.
168 citations
TL;DR: In this article, duality transformations of linearized gravity in four dimensions, i.e., rotations of the linearized Riemann tensor and its dual into each other, can be extended to the dynamical fields of the theory so as to be symmetries of the action and not just symmetry of the equations of motion.
Abstract: We show that duality transformations of linearized gravity in four dimensions, i.e., rotations of the linearized Riemann tensor and its dual into each other, can be extended to the dynamical fields of the theory so as to be symmetries of the action and not just symmetries of the equations of motion. Our approach relies on the introduction of two superpotentials, one for the spatial components of the spin-2 field and the other for their canonically conjugate momenta. These superpotentials are two-index, symmetric tensors. They can be taken to be the basic dynamical fields and appear locally in the action. They are simply rotated into each other under duality. In terms of the superpotentials, the canonical generator of duality rotations is found to have a Chern-Simons-like structure, as in the Maxwell case.
154 citations
TL;DR: In this article, the existence of conformally related metrics of zero scalar curvature and constant mean curvature on the boundary, under suitable hypotheses on the Weyl tensor, was proved.
Abstract: Given a compact Riemannian manifold with umbilic boundary, we prove the existence of conformally related metrics of zero scalar curvature and constant mean curvature on the boundary, under suitable hypotheses on the Weyl tensor. In order to carry out the estimates on the Sobolev quotient, we also prove the existence of conformal Fermi coordinates.
153 citations
TL;DR: In this paper, the notion of a connective structure for a gerbe on a space X is defined in a global manner, where the gerbe is endowed with trivializing data with respect to an open cover of X.
149 citations
TL;DR: In this article, the authors obtained a volume growth and curvature decay result for various classes of complete, non-compact Riemannian metrics in dimension 4; in particular, they applied to anti-self-dual or Kahler metrics with zero scalar curvature.
Abstract: We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.
119 citations
TL;DR: In this paper, a compactness result for various classes of Riemannian metrics in dimension four was obtained for anti-self-dual metrics, Kahler metrics with constant scalar curvature, and metrics with harmonic curvature.
TL;DR: In this article, the Ricci flat metrics with nonzero parallel spinors are shown to be stable in the direction of changes in conformal structures, which is a local version of the HMM03 result.
Abstract: Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.
TL;DR: In this paper, it was shown that globally Osserman manifolds are two-point homogeneous for n≠8, 16, and its pointwise version for n ≥ 2, 4, 8, 16.
Abstract: Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ T p M n , the Jacobi operator is defined by R X =R(X,·)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the Osserman Conjecture for n≠8, 16, and its pointwise version for n≠2, 4, 8, 16. Partial result in the case n=16 is also given.
TL;DR: In this paper, the authors studied the Ricci tensor invariance of the Riemannian curvature tensor of the Kenmotsu manifold, which is derived from the almost contact Ricci manifold with some special conditions.
Abstract: The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.
TL;DR: In this article, it was shown that if the Ricci curvature is uniformly bounded under the flow for all times t ∈ [0, T], then the curvature tensor has to be uniformly bounded as well.
Abstract: Consider the unnormalized Ricci flow (gjj)t = -2R ij for t ∈ [0, T), where T < oc. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T), then the solution can be extended beyond T. We prove that if the Ricci curvature is uniformly bounded under the flow for all times t ∈ [0, T), then the curvature tensor has to be uniformly bounded as well.
TL;DR: In this article, the authors determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.
Abstract: In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.
TL;DR: In this article, a non-minimal non-linear extension of the standard Einstein-Hilbert-Maxwell action is proposed to describe a spherically symmetric charged object.
Abstract: We establish a new self-consistent system of equations for the gravitational and electromagnetic fields. The procedure is based on a non-minimal non-linear extension of the standard Einstein-Hilbert-Maxwell action. General properties of a three-parameter family of non-minimal linear models are discussed. In addition, we show explicitly, that a static spherically symmetric charged object can be described by a non-minimal model, second order in the derivatives of the metric, when the susceptibility tensor is proportional to the double-dual Riemann tensor
TL;DR: In this article, the authors classify N(•)-contact metric manifolds which sat- isfy Z(»;X) ¢ Z = 0, Z( «;X] ¢ R = 0 or R(»,X) ǫ = 0.
Abstract: We classify N(•)-contact metric manifolds which sat- isfy Z(»;X) ¢ Z = 0, Z(»;X) ¢ R = 0 or R(»;X) ¢ Z = 0.
TL;DR: A three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated and defines the RadiusHit of a curvature Tensor to quantify estimation accuracy and applicability.
Abstract: Although curvature estimation from a given mesh or regularly sampled point set is a well-studied problem, it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise. Such input is ubiquitous in computer vision. In this paper, we propose a three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated. Our quantitative estimation is an improvement over the previous two-pass algorithm, where only qualitative curvature estimation (sign of Gaussian curvature) is performed. To overcome misalignment errors, our improved method automatically corrects input point locations at subvoxel precision, which also rejects outliers that are uncorrectable. To adapt to different scales locally, we define the RadiusHit of a curvature tensor to quantify estimation accuracy and applicability. Our curvature estimation algorithm has been proven with detailed quantitative experiments, performing better in a variety of standard error metrics (percentage error in curvature magnitudes, absolute angle difference in curvature direction) in the presence of a large amount of misalignment noise.
TL;DR: In this article, the authors extend the PN phenomenological framework by modifying the form of the coupling between curvature and stress tensors, and obtain a Pioneer-like anomaly for probes with an eccentric motion and a range dependence of Eddington parameter γ.
Abstract: The general relativistic treatment of gravitation can be extended by preserving the geometrical nature of the theory but modifying the form of the coupling between curvature and stress tensors. The gravitation constant is thus replaced by two running coupling constants which depend on scale and differ in the sectors of traceless and traced tensors. When calculated in the solar system in a linearized approximation, the metric is described by two gravitation potentials. This extends the parametrized post-Newtonian (PPN) phenomenological framework while allowing one to preserve compatibility with gravity tests performed in the solar system. Consequences of this extension are drawn here for phenomena correctly treated in the linear approximation. We obtain a Pioneer-like anomaly for probes with an eccentric motion as well as a range dependence of Eddington parameter γ to be seen in light deflection experiments.
TL;DR: In this article, the singular behavior of singular solutions of the σk Yamabe equation has been studied in the context of conformal geometry, where the singular solution is considered to be a solution of a fully nonlinear elliptic PDE.
TL;DR: In this paper, the authors constructed all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries.
Abstract: We construct all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k . Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.
TL;DR: In this article, a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor is introduced, called the (p, q)-curvatures.
Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.
TL;DR: In this article, the authors considered hypersurfaces in the product manifold M × R with positive constant r-mean curvature and obtained height estimates of certain compact vertical graphs in the manifold with boundary in M ×{ 0}.
Abstract: Let M be an m-dimensional Riemannian manifold with sectional curvature bounded from below. We consider hypersurfaces in the (m + 1)-dimensional product manifold M × R with positive constant r-mean curvature. We obtain height estimates of certain compact vertical graphs in M × R with boundary in M ×{ 0}. We apply this to obtain topological obstructions for the existence of some hypersurfaces. We also discuss the rotational symmetry of some embedded complete surfaces in S 2 × R of positive constant 2-mean curvature.
TL;DR: In this paper, the robustness of biological systems by means of the eigenstructure of the deviation curvature tensor was studied by applying this theory to the Van der Pohl equations and some biological models, and examining the relationship between the linear stability of steady states and the stability of transient states.
Abstract: In this article, we study the robustness of biological systems by means of the eigenstructure of the deviation curvature tensor. This is the differential geometric theory of the variational equations for deviation of whole trajectories to nearby ones. We apply this theory to the Van der Pohl equations and some biological models, and examine the relationship between the linear stability of steady-states and the stability of transient states. The main application is the G 1 -model for the cell cycle, where Jacobi stability reveals the robustness and fragility of the cell arrest states and suggests the existence of more subtle checkpoints.
TL;DR: In this paper, it was shown that complete non-compact Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to two in which some Gagliardo-Nirenberg type inequality holds are not very far from the Euclidean space.
TL;DR: In this article, it was shown that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact.
Abstract: The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R n with negative flag curvature and constant S-curva- ture. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.
TL;DR: In this paper, it was shown that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If, and an explicit procedure for recovering the wavefront set was given.
Abstract: We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L2-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.
TL;DR: In this article, the authors generalize their work for finding exact minima of an approximate nonlinear free energy to the full, rotationally invariant smectic free energy, which exhibits the detailed connection between mean curvature, Gaussian curvature and layer spacing.
Abstract: Considerations of rotational invariance in one-dimensionally modulated systems such as smectics-A, necessitate nonlinearities in the free energy. The presence of these nonlinearities is critical for determining the layer configurations around defects. We generalize our recent construction for finding exact minima of an approximate nonlinear free energy to the full, rotationally invariant smectic free energy. Our construction exhibits the detailed connection between mean curvature, Gaussian curvature and layer spacing. For layers without Gaussian curvature, we reduce the Euler–Lagrange equation to an equation governing the evolution of a surface. As an example, we determine the layer profile and free energy of an edge dislocation.
Journal Article•
TL;DR: In this paper, the authors studied non-abelian extensions of a super Lie algebra and identified a cohomological obstruction to the existence, parallel to the known one for Lie algebras.
Abstract: We study (non-abelian) extensions of a super Lie algebra and identify a cohomological obstruction to the existence, parallel to the known one for Lie algebras. An analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is shown.