Showing papers on "Ricci decomposition published in 2016"

Tensor Complementarity Problem and Semi-positive Tensors

Abstract: In this paper, we prove that a real tensor is strictly semi-positive if and only if the corresponding tensor complementarity problem has a unique solution for any nonnegative vector and that a real tensor is semi-positive if and only if the corresponding tensor complementarity problem has a unique solution for any positive vector. It is shown that a real symmetric tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive.

η-Ricci solitons on Lorentzian para-Sasakian manifolds

Abstract: We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0 and S · R(ξ,X) = 0. We prove that on a Lorentzian para-Sasakian manifold (M, φ, ξ, η, 1), if the Ricci curvature satisfies one of the previous conditions, the existence of η-Ricci solitons implies that (M, 1) is Einstein manifold. We also conclude that in these cases there is no Ricci soliton on M with the potential vector field ξ. On the other way, if M is of constant curvature, then (M, 1) is elliptic manifold. Cases when the Ricci tensor satisfies different other conditions are also discussed.

On butterfly effect in higher derivative gravities

Abstract: We study butterfly effect in D-dimensional gravitational theories containing terms quadratic in Ricci scalar and Ricci tensor. One observes that due to higher order derivatives in the corresponding equations of motion there are two butterfly velocities. The velocities are determined by the dimension of operators whose sources are provided by the metric. The three dimensional TMG model is also studied where we get two butterfly velocities at generic point of the moduli space of parameters. At critical point two velocities coincide.

The Multi-Orientable Random Tensor Model, a Review

Abstract: After its introduction (initially within a group eld theory framework) in A. Tanasa, J. Phys. A 45 (2012) 165401, the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the 1=N expansion and of the large N limit (N being the size of the tensor), the combinatorial analysis of the various terms of this expansion and nally,

A distinguishing gravitational property for gravitational equation in higher dimensions

Abstract: It is well known that Einstein gravity is kinematic (meaning that there is no non-trivial vacuum solution; i.e. the Riemann tensor vanishes whenever the Ricci tensor does so) in 3 dimension because the Riemann tensor is entirely given in terms of the Ricci tensor. Could this property be universalized for all odd dimensions in a generalized theory? The answer is yes, and this property uniquely singles out pure Lovelock (it has only one Nth order term in the action) gravity for which the Nth order Lovelock–Riemann tensor is indeed given in terms of the corresponding Ricci tensor for all odd, \(d=2N+1\), dimensions. This feature of gravity is realized only in higher dimensions and it uniquely picks out pure Lovelock gravity from all other generalizations of Einstein gravity. It serves as a good distinguishing and guiding criterion for the gravitational equation in higher dimensions.

Local control on the geometry in 3D Ricci flow

Abstract: The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local C/t decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman and Tian-Wang in this dimension by not requiring the Ricci curvature to be almost-positive, and not asking the volume growth to be almost-Euclidean.

Generalized nonminimal couplings in Randall-Sundrum scenarios

Abstract: The geometrical localization mechanism in Randall-Sundrum (RS) scenarios is extended by considering the coupling between a quadratic mass term and geometrical tensors. Since the quadratic term is symmetric, tensors with two symmetric indices have to be taken into account. These are the Ricci and the Einstein tensors. For the Ricci tensor, it is shown that a localized zero mode exists while that is not possible for the Einstein tensor. It is already known that the Ricci scalar generates a localized solution but the metrics do not. Therefore, it can be concluded that divergenceless tensors do not localize the zero mode of the gauge fields. The result is valid for any warp factor recovering the RS metrics at the boundaries and, therefore, is valid for RS I and II models. We also compute resonances for all couplings. These are calculated using the transfer matrix method. The cases studied consider the standard RS with deltalike branes and branes generated by kinks and domain walls as well. The parameters are changed to control the thickness of the smooth brane. We find that, for all cases considered, geometrical coupling does not generate resonances. This enforces similar results for the coupling with the Ricci scalar and points to the existence of some unidentified fundamental structure of these couplings.

Synthetic theory of Ricci curvature bounds

Abstract: Synthetic theory of Ricci curvature bounds is reviewed, from the conditions which led to its birth, up to some of its latest developments.

The Ricci Curvature in Noncommutative Geometry

Abstract: Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. This Ricci functional uniquely determines a density element, called the Ricci density, which plays the role of the Ricci operator. The main result of this paper provides an explicit computation of the Ricci density when the conformally flat geometry of the noncommutative two torus is encoded by the modular de Rham spectral triple.

The W2-curvature tensor on warped product manifolds and applications

Abstract: The purpose of this paper is to study the W2-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the W2-curvature tensor on a warped product manifold in terms of its relation with W2-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate W2-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a W2-curvature flat warped product manifold are derived. Finally, we study the W2-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are W2-curvature flat.

Gradient Ricci solitons with vanishing conditions on Weyl

Abstract: We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of $N^{n-k}\times \mathbb{R}^k$, $(k > 0)$, the product of a Einstein manifold $N^{n-k}$ with the Gaussian shrinking soliton $\mathbb{R}^k$. The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton.

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

Abstract: In this paper we present some new results about n(≥ 4)-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for n > 4 and a pp-wave space-time in n = 4. In all cases an algebraic classification for the Weyl tensor is provided for n = 4 and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat (PZS)n, n ≥ 4, space-time is conformal to the Robertson–Walker space-time.

Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity

Abstract: In isotropic strain gradient elasticity, we decompose the strain gradient tensor into its irreducible pieces under the n-dimensional orthogonal group O(n). Using the Young tableau method for traceless tensors, four irreducible pieces (n>2), which are canonical, are obtained. In three dimensions, the strain gradient tensor can be decomposed into four irreducible pieces with 7+5+3+3 independent components whereas in two dimensions, the strain gradient tensor can be decomposed into three irreducible pieces with 2+2+2 independent components. The knowledge of these irreducible pieces is extremely useful when setting up constitutive relations and strain energy.

A Weitzenböck formula for canonical metrics on four-manifolds

Abstract: We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds. As applications, we classify Einstein four-manifolds and conformally Einstein four-manifolds with half two-nonnegative curvature operator, which in some sense provides a characterization of Kähler-Einstein metrics and Hermitian, Einstein metrics with positive scalar curvature on four-manifolds, respectively. We also discuss the classification of four-dimensional gradient shrinking Ricci solitons with half two-nonnegative curvature operator and half harmonic Weyl curvature.

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold

Abstract: Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces , into tensor field analysis, based on the notion of eigenvalue manifold . Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches , to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

Generalized quasi-Einstein manifolds with harmonic anti-self dual Weyl tensor

Abstract: We prove that a 4-dimensional generalized m-quasi-Einstein manifold with harmonic anti-self dual Weyl tensor is locally a warped product with 3-dimensional Einstein fibers provided an additional condition holds.

Ricci inheritance collineations in Kantowski–Sachs spacetimes

Abstract: In this paper, we investigate Ricci Inheritance Collineations (RICs) in Kantowski–Sachs spacetimes. RICs are discussed in detail when Ricci tensor is degenerate and nondegenerate. In both the cases, RICs are obtained and it turns out that the dimension of Lie algebra of RICs is finite when Ricci tensor is nondegenerate. In the case when Ricci tensor is degenerate, we get finite as well as infinite dimensional group of RICs.

Ricci inheritance collineations in Bianchi type II spacetime

Abstract: In this paper, we present a complete classification of Bianchi type II spacetime according to Ricci inheritance collineations (RICs). The RICs are classified considering cases when the Ricci tensor is both degenerate as well as non-degenerate. In case of non-degenerate Ricci tensor, it is found that Bianchi type II spacetime admits 4-, 5-, 6- or 7-dimensional Lie algebra of RICs. In the case when the Ricci tensor is degenerate, majority cases give rise to infinitely many RICs, while remaining cases admit finite RICs given by 4, 5 or 6.

Existence and applications of Ricci flows via pseudolocality

Abstract: We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvature and injectivity radius bounded from below. We also study the short-time behaviour of these solutions which may have unbounded curvature at the initial time, and provide some applications. A key tool is Perelman's pseudolocality theorem.

Geometric Structures in Tensor Representations (Final Release)

Abstract: The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.

On Measure Contraction Property without Ricci Curvature Lower Bound

Abstract: Measure contraction properties M C P (K, N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then M C P (K, N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013) that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M C P (0,5). In this paper, we give sufficient conditions for a 2n+1 dimensional weakly Sasakian manifold to satisfy M C P (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013).

Ricci tensors on three-dimensional almost coKähler manifolds

Abstract: Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.

Ricci flows with bursts of unbounded curvature

Abstract: Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor ) to this hyperbolic metric, ie our Ricci flow geometrises the surface In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times In particular, when coupled with uniqueness, we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even

ON GENERALIZED QUASI-CONFORMAL N(k, μ)-MANIFOLDS

Abstract: . The object of the present paper is to introduce a new curva-ture tensor, named generalized quasi-conformal curvature tensor whichbridges conformal curvature tensor, concircular curvature tensor, pro-jective curvature tensor and conharmonic curvature tensor. Flatness andsymmetric properties of generalized quasi-conformal curvature tensor arestudied in the frame of (k,µ)-contact metric manifolds. 1. IntroductionIn 1968, Yano and Sawaki [27] introduced the notion of quasi-conformalcurvature tensor which contains both conformal curvature tensor as well asconcircular curvature tensor, in the context of Riemannian geometry. In tunewith Yano and Sawaki [27], the present paper attempts to introduce a newtensor field, named generalized quasi-conformal curvature tensor. The beautyof generalized quasi-conformal curvature tensor lies in the fact that it has theflavour of Riemann curvature tensor R, conformal curvature tensor C [8] con-harmonic curvature tensor Cˆ [9], concircular curvature tensor E [26, p. 84],projective curvature tensor P [26, p. 84] and m-projective curvature tensor H[15], as particular cases. The generalized quasi-conformal curvature tensor isdefined asW(X,Y)Z =2n−12n+1[(1−b+2na)−{1+2n(a+b)}c]C(X,Y )Z+[1−b+2na]E(X,Y)Z +2 n (b−a) P(X,Y )Z+2 n−12 n+1(1.1) (c −1){1+2 n(a +b)} Cˆ(X,Y)Zfor all X,Y,Z ∈ χ(M), the set of all vector field of the manifold M, where a,b and c are real constants. The above mentioned curvature tensors are defined

On the Weyl and Ricci tensors of Generalized Robertson-Walker space-times

Abstract: We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen's vector, and any of the two conditions is necessary and sufficient for the GRW space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen's vector. A GRW space-time in n = 4 with null conformal divergence is a Robertson-Walker space-time.

A symmetric 2-tensor canonically associated to Q-curvature and its applications

Abstract: In this article, we define a symmetric 2-tensor canonically associated to Q-curvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricci tensor. This tensor can also be used to understand Chang-Gursky-Yang's theorem on 4-dimensional Q-singular metrics. Moreover, we show an Almost-Schur Lemma holds for Q-curvature, which gives an estimate of Q-curvature on closed manifolds.

Ricci measure for some singular Riemannian metrics

Abstract: We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak notion of a Ricci flow solution on a manifold.