Showing papers on "Riemann curvature tensor published in 2010"
TL;DR: Using 3-algebras, the authors obtained a nonabelian system of equations that furnish a representation of the (2, 0)-supersymmetric tensor multiplet.
Abstract: Using 3-algebras we obtain a nonabelian system of equations that furnish a representation of the (2, 0)-supersymmetric tensor multiplet. The on-shell conditions are quite restrictive so that the system can be reduced to five-dimensional gauge theory along with six-dimensional abelian (2, 0) tensor multiplets. We briefly discuss possible applications to D4-branes using a spacelike reduction and M5-branes using a null reduction.
151 citations
Posted Content•
TL;DR: In this paper, the compatibility of two definitions of generalized curvature bounds, the lower bound for sectional curvature in the sense of Alexandrov, and the upper bound for Ricci curvature for Lott-Villani-Sturm, was shown.
Abstract: Here I show the compatibility of two definitions of generalized curvature bounds: the lower bound for sectional curvature in the sense of Alexandrov and the lower bound for Ricci curvature in the sense of Lott-Villani-Sturm.
144 citations
13 Jun 2010
TL;DR: This paper argues that the underlying geometry of the tensor space is an important property for action classification and characterize a tensor as a point on a product manifold and perform classification on this space.
Abstract: Videos can be naturally represented as multidimensional arrays known as tensors. However, the geometry of the tensor space is often ignored. In this paper, we argue that the underlying geometry of the tensor space is an important property for action classification. We characterize a tensor as a point on a product manifold and perform classification on this space. First, we factorize a tensor relating to each order using a modified High Order Singular Value Decomposition (HOSVD). We recognize each factorized space as a Grassmann manifold. Consequently, a tensor is mapped to a point on a product manifold and the geodesic distance on a product manifold is computed for tensor classification. We assess the proposed method using two public video databases, namely Cambridge-Gesture gesture and KTH human action data sets. Experimental results reveal that the proposed method performs very well on these data sets. In addition, our method is generic in the sense that no prior training is needed.
137 citations
TL;DR: In this paper, a canonical formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor is presented, which allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.
Abstract: We present a canonical formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor, which, for example, arises in the low-energy limit of superstring theories. Our approach allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.
126 citations
Book•
18 Jun 2010TL;DR: The companion to the CBMS lectures of Scott Wolpert at Central Connecticut State University as discussed by the authors provides a general self-contained course for graduate students and postgraduates, including a unified approach for an array of results.
Abstract: This book is the companion to the CBMS lectures of Scott Wolpert at Central Connecticut State University. The lectures span across areas of research progress on deformations of hyperbolic surfaces and the geometry of the Weil-Petersson metric. The book provides a generally self-contained course for graduate students and postgraduates. The exposition also offers an update for researchers; material not otherwise found in a single reference is included. A unified approach is provided for an array of results. The exposition covers Wolpert's work on twists, geodesic-lengths and the Weil-Petersson symplectic structure; Wolpert's expansions for the metric, its Levi-Civita connection and Riemann tensor. The exposition also covers Brock's twisting limits, visual sphere result and pants graph quasi isometry, as well as the Brock-Masur-Minsky construction of ending laminations for Weil-Petersson geodesics. The rigidity results of Masur-Wolf and Daskalopoulos-Wentworth, following the approach of Yamada, are included. The book concludes with a generally self-contained treatment of the McShane-Mirzakhani length identity, Mirzakhani's volume recursion, approach to Witten-Kontsevich theory by hyperbolic geometry, and prime simple geodesic theorem. Lectures begin with a summary of the geometry of hyperbolic surfaces and approaches to the deformation theory of hyperbolic surfaces. General expositions are included on the geometry and topology of the moduli space of Riemann surfaces, the $CAT(0)$ geometry of the augmented TeichmA A1/4ller space, measured geodesic and ending laminations, the deformation theory of the prescribed curvature equation, and the Hermitian description of Riemann tensor. New material is included on estimating orbit sums as an approach for the potential theory of surfaces. A co-publication of the AMS and CBMS.
92 citations
TL;DR: A feature-preserving denoising algorithm that is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions, and sharp features, such as edges and corners, are very well preserved.
Abstract: In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved.
87 citations
TL;DR: In this paper, a Riemannian material manifold is associated with the body, with a metric that explicitly depends on the temperature distribution, and a change in temperature corresponds to a change of the material metric.
Abstract: In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configura...
72 citations
TL;DR: In this paper, the Hawking-Penrose singularity theorem and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature Ric f m in timelike directions were studied.
65 citations
TL;DR: In this article, the authors consider the problem of quantum gravity and show that certain issues in quantum gravity transcend any of the current schemes and stem from a fundamental paradigm clash between general relativity and quantum physics.
Abstract: Introduction More than forty years have passed since I first became interested in the problem of quantum gravity. During that time, there have been many diversions and, perhaps, some advances. Certainly, the naively optimistic approaches have long been laid to rest, and the schemes that remain have achieved some degree of stability. The original “canonical” program evolved into loop quantum gravity, which has become one of the two major approaches. The other, of course, is string theory—a scheme whose roots lie in the old Veneziano model of hadronic interactions, but whose true value became apparent only after it had been reconceived as a theory of quantum gravity. However, notwithstanding these hard-won developments, certain issues in quantum gravity transcend any of the current schemes. These involve deep problems of both a mathematical and a philosophical kind and stem from a fundamental paradigm clash between general relativity—the apotheosis of classical physics—and quantum physics. In general relativity, spacetime “itself” is modeled by a differentiable manifold M , a set whose elements are interpreted as “spacetime points.” The curvature tensor of the pseudo-Riemannian metric on M is then deemed to represent the gravitational field. As a classical theory, the underlying philosophical interpretation is realist: both the spacetime and its points truly “exist,” as does the gravitational field.
65 citations
TL;DR: In this article, the authors investigated the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces and proved that a strict form of the condition is stable under C4 perturbation if the nonfocal domains are uniformly convex.
Abstract: We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger– Wang condition is stable under C4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.
58 citations
TL;DR: In this paper, the authors considered the generalized Born-Infeld electrodynamics with two parameters and obtained the canonical and symmetrical Belinfante energy-momentum tensors.
Abstract: The generalized Born–Infeld electrodynamics with two parameters is investigated. In this model the propagation of a linearly polarized laser beam in the external transverse magnetic field is considered. It was shown that there is the effect of vacuum birefringence, and we evaluate induced ellipticity. The upper bounds on the combination of parameters introduced from the experimental data of BRST and PVLAS Collaborations were obtained. When two parameters are equal to each other, we arrive at Born–Infeld electrodynamics and the effect of vacuum birefringence vanishes. We find the canonical and symmetrical Belinfante energy–momentum tensors. The trace of the energy–momentum tensor is not zero and the dilatation symmetry is broken. The four-divergence of the dilatation current is equal to the trace of the Belinfante energy–momentum tensor and is proportional to the parameter (with the dimension of the field strength) of the model. The dual symmetry is also broken in the model considered.
TL;DR: A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form as mentioned in this paper, which is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry.
TL;DR: In this paper, the Ricci flow is shown to preserve the cone of curvature operators with nonnegative isotropic curvature in dimensions greater than or equal to four, and it is shown that the nonlinearity is positive at a minimum.
Abstract: In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.
TL;DR: In this article, Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric metric connection were proved for the case of the Ricci curvature.
Abstract: In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.
TL;DR: In this article, the authors studied singularities in (3 + 1)-dimensional Horava-Lifshitz (HL) theory of gravity, which can be divided into scalar, non-scalar curvature, and coordinate singularities.
TL;DR: In this article, the authors studied the nature of generalized Sasakian-space-forms under some conditions regarding projective curvature tensor and obtained necessary and sufficient conditions for scalar curvature.
Abstract: The object of the present paper is to study the nature of generalized Sasakian-space-forms under some conditions regarding projective curvature tensor. All the results obtained in this paper are in the form of necessary and sufficient conditions. Keywords: Generalized Sasakian-space-forms; projectively flat; projectively-semisymmetric; projectively symmetric; projectively recurrent; Einstein manifold; scalar curvature Quaestiones Mathematicae 33(2010), 245–252
TL;DR: In this article, the determinantal form of the Born-Infeld gravity action is analyzed in terms of the metric fluctuations around constant curvature backgrounds, and it is shown that the unitarity of the theory depends only on finite number of terms built from the powers of the curvature tensor.
Abstract: We develop techniques of analyzing the unitarity of general Born-Infeld gravity actions in $D$-dimensional spacetimes. The determinantal form of the action allows us to find a compact expression quadratic in the metric fluctuations around constant curvature backgrounds. This is highly nontrivial since for the Born-Infeld actions, in principle, infinitely many terms in the curvature expansion should contribute to the quadratic action in the metric fluctuations around constant curvature backgrounds, which would render the unitarity analysis intractable. Moreover in even dimensions, unitarity of the theory depends only on finite number of terms built from the powers of the curvature tensor. We apply our techniques to some four-dimensional examples.
TL;DR: In this paper, the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives were shown to be locally characterized by scalar curvatures.
Abstract: We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime metric is either -non-degenerate, and hence locally characterized by its scalar polynomial curvature invariants, or is a degenerate Kundt spacetime. We present a number of results that generalize these results to higher dimensions and discuss their consequences and potential physical applications.
TL;DR: In this paper, it was shown that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants.
Abstract: We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterized by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of the alignment theory and the bivector form of the Weyl operator in higher dimensions and introduce the important notions of diagonalizability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.
TL;DR: In this article, the authors investigated the behaviour on approach to the initial singularity in higher order extensions of general relativity by finding exact cosmological solutions for a wide class of models in which the Lagrangian is allowed to depend nonlinearly upon the three possible linear and quadratic scalars built from the Riemann tensor: R, RabRab and RabcdRabcd.
Abstract: We investigate the behaviour on approach to the initial singularity in higher order extensions of general relativity by finding exact cosmological solutions for a wide class of models in which the Lagrangian is allowed to depend nonlinearly upon the three possible linear and quadratic scalars built from the Riemann tensor: R, RabRab and RabcdRabcd. We present new anisotropic vacuum solutions analogous to the Kasner solutions of general relativity and extend previous results to a much wider range of fourth-order theories of gravity. We discuss the implications of these results for the behaviour of the more general anisotropic Bianchi type VIII and IX cosmologies as the initial singularity is approached. Furthermore, we also consider the existence conditions for some other simple anisotropic Bianchi I vacuum solutions in which the expansion in each direction is of exponential, rather than power-law behaviour and their relevance for cosmic 'no-hair' theorems.
TL;DR: In this article, the authors introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannians case.
Abstract: We introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannian case. In particular, we classify contact pseudo-metric manifolds of constant sectional curvature, three-dimensional locally symmetric contact pseudo-metric manifolds and three-dimensional homogeneous contact Lorentzian manifolds.
TL;DR: The Wess-Zumino coupling on D-branes in string theory is known to receive higher derivative corrections which couple the Ramond-Ramond potential to terms involving the square of the spacetime curvature tensor.
Abstract: The Wess-Zumino coupling on D-branes in string theory is known to receive higher derivative corrections which couple the Ramond-Ramond potential to terms involving the square of the spacetime curvature tensor. Consistency with T-duality implies that the branes should also have four-derivative couplings that involve the NS-NS B-field. We use T-duality to predict some of these couplings. We then confirm these results with string worldsheet computations by evaluating disc amplitudes with insertions of one R-R and two NS-NS vertex operators.
01 Aug 2010
TL;DR: In this paper, the authors considered the conformal class of the Riemannian product go+g and showed that the number of metrics of constant scalar curvature grows at least linearly with respect to the square root of the curvature of g.
Abstract: We consider the conformal class of the Riemannian product go+g, where go is the constant curvature metric on S m and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of g. This is obtained by studying radial solutions of the equation Δu ― λu + λu p = 0 on S m and the number of solutions in terms of λ.
TL;DR: In this paper, the authors introduced the concept of almost paracontact manifold, and in particular, of ''para-Sasakian'' manifolds, and showed that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an''para'' Sasakian structure.
Abstract: We introduce the concept of ()-almost paracontact manifolds, and in particular, of ()-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of ()-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an ()-para Sasakian structure. We show that, for an ()-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) ()-para Sasakian manifold is locally isometric to a pseudohyperbolic space (resp., pseudosphere ). At last, it is proved that for an ()-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.
TL;DR: This work presents the geodesic curvature flow equation on general smooth manifolds based on an energy minimization of curves as dGCF, and applies it to three problems: the closed-curve evolution on manifolds, the discrete scale-space construction, and the edge detection of images painted on triangulated surfaces.
Abstract: Curvature flow (planar geometric heat flow) has been extensively applied to image processing, computer vision, and material science. To extend the numerical schemes and algorithms of this flow on surfaces is very significant for corresponding motions of curves and images defined on surfaces. In this work, we are interested in the geodesic curvature flow over triangulated surfaces using a level set formulation. First, we present the geodesic curvature flow equation on general smooth manifolds based on an energy minimization of curves. The equation is then discretized by a semi-implicit finite volume method (FVM). For convenience of description, we call the discretized geodesic curvature flow as dGCF. The existence and uniqueness of dGCF are discussed. The regularization behavior of dGCF is also studied. Finally, we apply our dGCF to three problems: the closed-curve evolution on manifolds, the discrete scale-space construction, and the edge detection of images painted on triangulated surfaces. Our method works for compact triangular meshes of arbitrary geometry and topology, as long as there are no degenerate triangles. The implementation of the method is also simple.
TL;DR: In this paper, the authors derived a formula for the spectral action of Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion and deduced the Lagrangian for the Standard Model of particle physics in the presence of torsions from the Chamseddine-Connes Dirac operator.
Abstract: We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in the presence of torsion from the Chamseddine-Connes Dirac operator.
TL;DR: In this paper, the general coupling between vector and scalar supermultiplets is encoded in an SU(4) tensor Wji which is a function of the matter fields and subject to a set of algebraic and super-differential relations.
Abstract: We present a systematic analysis of the \( \mathcal{N} = 6 \) superspace constraints in three space-time dimensions. The general coupling between vector and scalar supermultiplets is encoded in an SU(4) tensor Wji which is a function of the matter fields and subject to a set of algebraic and super-differential relations. We give a genuine \( \mathcal{N} = 6 \) classification for superconformal models with polynomial interactions and find the known ABJM and ABJ models.
TL;DR: In this article, the Laplace model was used to derive a two-dimensional Laplace system for the dynamics of a fluid film, where neither velocities nor deviation from the equilibrium are assumed small.
Abstract: Two-dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid film
Here δ/δ t is the invariant time derivative, ρ is the two-dimensional density of the film, C is the normal component of the velocity field, Vα are the tangential components, Bαβ is the curvature tensor, and ∇α is the covariant surface derivative. The surface energy density e(ρ) is a generalization of the common surface tension and eρ is its derivative. The Laplace model corresponds to e(ρ) =σ/ρ, where σ is the surface tension density. The proper choice of e(ρ) in paramount in capturing particular effects displayed by fluid films.
The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text.
TL;DR: In this article, a necessary condition for supersymmetrization of higher-order higher-dimensional gravities was formulated in terms of the Weyl anomaly coefficients, which can be computed holographically.
Abstract: Supersymmetric higher derivative gravities define superconformal field theories via the AdS/CFT correspondence. From the boundary theory viewpoint, supersymmetry implies a relation between the coefficients which determine the three point function of the stress energy tensor which can be tested in the dual gravitational theory. We use this relation to formulate a necessary condition for the supersymmetrization of higher derivative gravitational terms. We then show that terms quadratic in the Riemann tensor do not present obstruction to supersymmetrization, while generic higher order terms do. For technical reasons, we restrict the discussion to seven dimensions where the obstruction to supersymmetrization can be formulated in terms of the coefficients of Weyl anomaly, which can be computed holographically.