Showing papers on "Ricci decomposition published in 2013"

Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors

Abstract: We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434--454], [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360--2375]) to the recently proposed hierarchical Tucker (HT) tensor format (Hackbusch and Kuhn; see [J. Fourier Anal. Appl., 15 (2009), pp. 706--722]) and the tensor train (TT) format (Oseledets; see [SIAM J. Sci. Comput., 33 (2011), pp. 2295--2317]), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamical approximation approach, the time derivative of the tensor to be approximated is projected onto the time-dependent tangent space of the approximation manifold along the solution trajectory. This approach can be used to approximate the solutions to tensor differential equations in the HT or TT format and to compute updates in optimization algorithms within these reduced tensor formats. By deriving and analyzing th...

Towards an invariant geometry of double field theory

Abstract: We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an “index-free” proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.

A Liouville-Type Theorem for Smooth Metric Measure Spaces

Abstract: For smooth metric measure spaces (M,g,e −f d vol ) we prove a Liouville-type theorem when the Bakry–Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case f is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spaces with Bakry–Emery Ricci tensor bounded from below.

3-Manifolds with nonnegative Ricci curvature

Abstract: For a complete noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to ℝ3 or the universal cover splits. This confirms Milnor’s conjecture in dimension 3.

On Riemann and Weyl Compatible Tensors

Abstract: We investigate semi-Riemannian manifolds satisfying some curva- ture conditions. Those conditions are strongly related to pseudosymmetry.

Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces

Abstract: There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+ abla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the analysis of path space P(M) in a manner analogous to how lower Ricci curvature controls the analysis on M. In the second part of the paper we develop the analytic tools needed to in order to use these new characterizations to give a definition of bounded Ricci curvature on general metric measure spaces (X,d,m). We show that on such spaces many of the properties of smooth spaces with bounded Ricci curvature continue to hold on metric-measure spaces with bounded Ricci curvature.

Concentration, Ricci Curvature, and Eigenvalues of Laplacian

Abstract: In this paper we study the concentration behavior of metric measure spaces. We prove the stability of the curvature-dimension condition with respect to the concentration topology due to Gromov. As an application, under the nonnegativity of Bakry–Emery Ricci curvature, we prove that the kth eigenvalue of the weighted Laplacian of a closed Riemannian manifold is dominated by a constant multiple of the first eigenvalue, where the constant depends only on k and is independent of the dimension of the manifold.

Inside the degenerate horizons of regular black holes

Abstract: The regularized stress-energy tensor of the quantized massive scalar, spinor, and vector fields inside the degenerate horizon of the regular charged black hole in the (anti)-de Sitter universe is constructed and examined. It is shown that, although the components of the stress-energy tensor are small in the vicinity of the black hole degenerate horizon and near the regular center, they are quite big in the intermediate region. The oscillatory character of the stress-energy tensor can be ascribed to various responses of the higher curvature terms to the changes of the metric inside the (degenerate) event horizon, especially in the region adjacent to the region described by the nearly flat metric potentials. Special emphasis is put on the stress-energy tensor in the geometries being the product of the constant curvature two-dimensional subspaces.

Metric Ricci curvature for $PL$ manifolds

Abstract: We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds.

The mathematical foundations of general relativity revisited

Abstract: The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the " Riemann tensor " according to the " Vessiot structure equations " must not be identified with the quadratic terms appearing in the well known " Cartan structure equations " for Lie groups and a similar comment can be done for the " Weyl tensor ". In particular, " curvature+torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). Roughly, Cartan and followers have not been able to " quotient down to the base manifold ", a result only obtained by Spencer in 1970 through the "nonlinear Spencer sequence" but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The " Ricci tensor " only depends on the nonlinear transformations (called " elations " by Cartan in 1922) that describe the "difference " existing between the Weyl group (10 parameters of the Poincare subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework, though striking it may look like for certain apparently well established theories such as electromagnetism and general relativity. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.

Anti-de Sitter-wave solutions of higher derivative theories.

Abstract: We show that the recently found anti--de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in $D$ dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-$N$ Weyl and type-$N$ traceless Ricci tensors.

The Ricci flow in a class of solvmanifolds

Abstract: In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time. (C) 2013 Elsevier B.V. All rights reserved.

On the Potential Function of Gradient Steady Ricci Solitons

Abstract: In this paper, we study the potential function of gradient steady Ricci solitons. We prove that the infimum of the potential function decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential function must be trivial, and that no gradient steady Ricci soliton admits uniformly positive scalar curvature.

$L^p$-Liouville theorems on complete smooth metric measure spaces

Abstract: We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Emery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of Bakry-Emery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Emery Ricci curvature case.

Pseudo-q-symmetric riemannian manifolds

Abstract: In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)n and pseudo-concircular symmetric manifolds is defined. This is named pseudo-Q-symmetric and denoted with (PQS)n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)n scalar field space-time is considered, and interesting properties are pointed out.

The Weyl Tensor of Gradient Ricci Solitons

Abstract: This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenbock type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.

Exact Mapping between Tensor and Most General Scalar Power Spectra

Abstract: We prove an exact relation between the tensor and the scalar primordial power spectra generated during inflation. Such a mapping considerably simplifies the derivation of any power spectra as they can be obtained from the study of the tensor modes only, which are much easier to solve. As an illustration, starting from the second order slow-roll tensor power spectrum, we derive in a few lines the next-to-next-to-leading order power spectrum of the comoving curvature perturbation in generalized single field inflation with a varying speed of sound.

A modified variational principle for gravity in the modified Weyl geometry

Abstract: The usual interpretation of the Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of the Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in the Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the action of the Weyl covariant derivative, we find a Weyl version of the Palatini identity for the Riemann tensor. To derive the energy–momentum tensor and equations of motion for gravity in the Weyl geometry, one naturally applies this identity at first, and then converts the variation of the additive Weyl connection to those of the metric tensor and Weyl gauge field. We also discuss possible connections to the current literature on the Weyl-invariant extension of massive gravity and the variational principles in f(R) gravity.

LP-sasakian manifolds with quasi-conformal curvature tensor

Abstract: The object of the paper is to study a Sasakian manifold with quasi-conformal curvature tensor.

On warped product spaces with a certain ricci condition

Abstract: In this paper, we obtain the criteria that the Riemannian manifold B is Einstein or a gradient Ricci soliton from the information of the second derivative of f in the warped product space R ×f B with gradient Ricci solitons. Moreover, we construct new examples of non- Einstein gradient Ricci soliton spaces with an Einstein or non-Einstein gradient Ricci soliton leaf using our main theorems. Finally we also get analogous criteria for the Lorentzian version.

A Generalized Mass Involving Higher Order Symmetric Functions of the Curvature Tensor

Abstract: We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the σk curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.

On the stability of Einstein manifolds

Abstract: Certain curvature conditions for stability of Einstein manifolds with respect to the Einstein-Hilbert action are given. These conditions are given in terms of quantities involving the Weyl tensor and the Bochner tensor. In dimension six, a stability criterion involving the Euler characteristic is given.

On Generalized Ricci Recurrent Manifolds with Applications To Relativity

Abstract: In the present paper we have obtained sufficient conditions for a generalized Ricci recurrent manifold to be a quasi-Einstein manifold. The existence of a generalized Ricci recurrent manifold have been proved by non-trivial examples. The semi-Riemannian GR n is also studied.

Pseudolocality of the Ricci Flow Under Integral Bound of Curvature

Abstract: We prove a pseudolocality type theorem for compact Ricci flows under local integral bounds on curvature. The main tool we use here is the local Ricci flow introduced by Deane Yang and the pseudolocality theorem due to Perelman. We also prove a theorem on the extension of the local Ricci flow.