Showing papers on "Ricci decomposition published in 2014"

Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

Abstract: In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.

Born-Infeld gravity and its functional extensions

Abstract: We investigate the dynamics of a family of functional extensions of the (Eddington-inspired) Born-Infeld gravity theory, constructed with the inverse of the metric and the Ricci tensor. We provide a generic formal solution for the connection and an Einstein-like representation for the metric field equations of this family of theories. For particular cases we consider applications to the early-time cosmology and find that nonsingular universes with a cosmic bounce are very generic and robust solutions.

Consistency of universally nonminimally coupled $f(R,T,R_{\mu \nu}T^{\mu \nu})$ theories

Abstract: We discuss the consistency of a recently proposed class of theories described by an arbitrary function of the Ricci scalar, the trace of the energy-momentum tensor and the contraction of the Ricci tensor with the energy-momentum tensor. We briefly discuss the limitations of including the energy-momentum tensor in the action, as it is a non fundamental quantity, but a quantity that should be derived from the action. The fact that theories containing non-linear contractions of the Ricci tensor usually leads to the presence of pathologies associated with higher-order equations of motion will be shown to constrain the stability of this class of theories. We provide a general framework and show that the conformal mode for these theories generally has higher-order equations of motion and that non-minimal couplings to the matter fields usually lead to higher-order equations of motion. In order to illustrate such limitations we explicitly study the cases of a canonical scalar field, a K-essence field and a massive vector field. Whereas for the scalar field cases it is possible to find healthy theories, for the vector field case the presence of instabilities is unavoidable.

Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

Abstract: Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

Bach-flat gradient steady Ricci solitons

Abstract: In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011)

On equivalency of various geometric structures

Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2).

On generalized Robertson--Walker spacetimes satisfying some curvature condition

Abstract: We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.

Orthogonal Decomposition of Symmetric Tensors

Abstract: A real symmetric tensor is orthogonally decomposable (or odeco) if it can be written as a linear combination of symmetric powers of $n$ vectors which form an orthonormal basis of $\mathbb R^n$. Motivated by the spectral theorem for real symmetric matrices, we study the properties of odeco tensors. We give a formula for all of the eigenvectors of an odeco tensor. Moreover, we formulate a set of polynomial equations that vanish on the odeco variety and we conjecture that these polynomials generate its prime ideal. We prove this conjecture in some cases and give strong evidence for its overall correctness.

Tensor Spaces and Hierarchical Tensor Representations

Abstract: In the present report we provide a brief introduction into recently developed hierarchical tensor representations. The new formats extend the well-known Tucker format into a hierarchical framework, by combining its favourable characteristics with low-order scaling properties. We demonstrate the basic concept of subspace approximation and higher order SVD (HOSVD), and how to extend this in a hierarchical way. We highlight that the present tensor representations are constituting smooth manifolds, and give a perspective how these properties can be used to develop numerical solvers for tensor equations and tensor optimisation problems.

Examples of Algebraic Ricci Solitons in the Pseudo-Riemannian Case

Abstract: This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. We obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.

On generalized roter type manifolds

Abstract: The main object of the present paper is to study the geometric properties of a generalized Roter type semi-Riemannian manifold, which arose in the way of generalization to find the form of the Riemann-Christoffel curvature tensor R. Again for a particular curvature restriction on R and the Ricci tensor S there arise two structures, e. g., local symmetry (∇R = 0) and Ricci symmetry (∇S = 0); semisymmetry(R · R = 0) and Ricci semisymmetry (R · S = 0) etc. In differential geometry there is a very natural question about the equivalency of these two structures. In this context it is shown that generalized Roter type condition is a sufficient condition for various important second order restrictions. Some generalizations of Einstein manifolds are also presented here. Finally the proper existence of both type of manifolds are ensured by some suitable examples.

AdS-Plane wave and pp-wave solutions of generic gravity theories

Abstract: We construct the anti–de Sitter-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two-tensor built from the Riemann tensor and its derivatives can bewritten in terms of the traceless Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples of our general formalism, we work out the six-dimensional conformal gravity and its nonconformal deformation as well as the tricritical gravity, the Lanczos-Lovelock theory, and string-generated cubic curvature theory.

Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes

Abstract: Within the $\Gamma_2$-calculus of Bakry and Ledoux, we define the Ricci tensor induced by a diffusion operator, we deduce precise formulas for its behavior under drift transformation, time change and conformal transformation, and we derive new transformation results for the curvature-dimension conditions of Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results are based on new identities and sharp estimates for the $N$-Ricci tensor and for the Hessian. In particular, we obtain Bochner's formula in the general setting.

Matter may matter

Abstract: We propose a gravitational theory in which the effective Lagrangian of the gravitational field is given by an arbitrary function of the Ricci scalar, the trace of the matter energy–momentum tensor, and the contraction of the Ricci tensor with the matter energy–momentum tensor. The matter energy–momentum tensor is generally not conserved, thus leading to the appearance of an extra-force, acting on massive particles in a gravitational field. The stability conditions of the theory with respect to local perturbations are also obtained. The cosmological implications of the theory are investigated, representing an exponential solution. Hence, a Ricci tensor–energy–momentum tensor coupling may explain the recent acceleration of the universe, without resorting to the mysterious dark energy.

The Riemann extensions with cyclic parallel Ricci tensor

Abstract: The property of being a D'Atri space (i.e., a Riemannian manifold with volume-preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L3 is said to be an L3-space. This definition extends easily to the affine case. Here we investigate the torsion-free affine manifolds and their Riemann extensions as concerns heredity of the condition L3. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.

Nonnegative Tensor Decomposition

Abstract: It is more and more common to encounter applications where the collected data is most naturally stored or represented in a multi-dimensional array, known as a tensor. The goal is often to approximate this tensor as a sum of some type of combination of basic elements, where the notation of what is a basic element is specific to the type of factorization employed. If the number of terms in the combination is few, the tensor factorization gives (implicitly) a sparse (approximate) representation of the data. The terms (e.g. vectors, matrices, tensors) in the combination themselves may also be sparse. This chapter highlights recent developments in the area of non-negative tensor factorization which admit such sparse representations. Specifically, we consider the approximate factorization of third and fourth order tensors into non-negative sums of types of outer-products of objects with one dimension less using the so-called t-product. A demonstration on an application in facial recognition shows the potential promise of the overall approach. We discuss a number of algorithmic options for solving the resulting optimization problems, and modification of such algorithms for increasing the sparsity.

A theorem of Ambrose for Bakry–Emery Ricci tensor

Abstract: In this short note, we prove a theorem of Ambrose (or Myers) for the Bakry–Emery Ricci tensor with the potential function at most linear growth. We also prove a complete manifold \((M, g, f)\) with the Bakry–Emery Ricci tensor bounded from below by a uniform positive constant and the potential function at most quadratic growth is compact.

A note on locally conformally flat gradient Ricci solitons

Abstract: We show that locally conformally flat gradient Ricci solitons, possibly incomplete, are locally isometric to a warped product of an interval and a space form. Consequently, we get that complete gradient shrinking and steady Ricci solitons with vanishing Weyl tensor are rotationally symmetric, from which their classification follows.

Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below

Abstract: We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.

Tensor Triangular Geometry for Classical Lie Superalgebras

Abstract: Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional ${\mathfrak g}$-modules which are semisimple over ${\mathfrak g}_{\bar{0}}$.

Structure Tensor Image Filtering using Riemannian L_1 and L_∞ Center-of-Mass

Abstract: Structure tensor images are obtained by a Gaussian smoothing of the dyadic product of gradient image. These images give at each pixel a n×n symmetric positive definite matrix SPD(n), representing the local orientation and the edge information. Processing such images requires appropriate algorithms working on the Riemannian manifold on the SPD(n) matrices. This contribution deals with structure tensor image filtering based on Lp geometric averaging. In particular, L1 center-of-mass (Riemannian median or Fermat-Weber point) and L∞ center-of-mass (Riemannian circumcenter) can be obtained for structure tensors using recently proposed algorithms. Our contribution in this paper is to study the interest of L1 and L∞ Riemannian estimators for structure tensor image processing. In particular, we compare both for two image analysis tasks: (i) structure tensor image denoising; (ii) anomaly detection in structure tensor images.

A metric Ricci flow for surfaces and its applications

Abstract: Motivated largely by Perleman’s work, the Ricci flow has become lately an object of interest and study in Graphics and Imaging. Various approaches have been suggested previously, ranging from classical approximation methods of smooth differential operators to discrete, combinatorial methods. In this paper we introduce a metric Ricci flow for surfaces and we investigate its properties: existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the metric one and that, moreover, the same results hold for a more general, metric notion of curvature. Furthermore, using the metric curvature approach, we show the existence of the Ricci flow for polyhedral 2-manifolds of piecewise constant curvature. We also study the problem of the realizability of the said flow in R.

On the 1+3 Formalism in General Relativity

Abstract: We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence (namely, the spatial metric tensor, the extrinsic curvature tensor and the Riemann curvature tensor), we derive the Gauss, Codazzi and Ricci equations, along with the evolution equation for the spatial metric. In the present framework, the spatial frames do not form any hypersurfaces as we allow the congruence to exhibit vorticity. The splitting procedure is then applied to the Einstein field equation and it results in an equivalent set of constraint and evolution equations. We discuss the resulting systems and compare them with the ones obtained from the 3+1 formalism, where the manifold is foliated by means of a family of three-dimensional space-like surfaces.

The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons

Abstract: We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.

Completely positive tensor decomposition

Abstract: A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a symmetric tensor is completely positive. If it is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.