Showing papers on "Ricci decomposition published in 2008"

Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness

Abstract: In this paper we introduce a new class of tensor decompositions. Intuitively, we decompose a given tensor block into blocks of smaller size, where the size is characterized by a set of mode-$n$ ranks. We study different types of such decompositions. For each type we derive conditions under which essential uniqueness is guaranteed. The parallel factor decomposition and Tucker's decomposition can be considered as special cases in the new framework. The paper sheds new light on fundamental aspects of tensor algebra.

Discrete Surface Ricci Flow

Abstract: This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Bounding scalar curvature and diameter along the kähler ricci flow (after perelman)

Abstract: In this short note we present a result of Perelman with detailed proof. The result states that if . We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kahler Ricci flow.

Rigidity of Quasi-Einstein Metrics

Abstract: We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.

On the Conditions to Extend Ricci Flow

Abstract: Consider {(M n , g(t)), 0 ⩽ t < T < ∞} as an unnormalized Ricci flow solution: for t ∈ [0, T). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T) then the solution can be extended over T. Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.

Metrics With Vanishing Quantum Corrections

Abstract: We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{\mu u}$ constructed from sums of terms the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called {\it universal} if, when evaluated on that Einstein metric, $T_{\mu u}$ is a multiple of the metric. A Ricci flat classical solution is called {\it strongly universal} if, when evaluated on that Ricci flat metric, $T_{\mu u}$ vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalisation; Einstein metrics with holonomy ${\rm Sim} (n-2)$ in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalised Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all 4-dimensional ${\rm Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.

The stress-energy tensor for biharmonic maps

Abstract: Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map.

Invariant Crease Lines for Topological and Structural Analysis of Tensor Fields

Abstract: We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. The key insight is that degenerate lines in tensor fields, as defined by the standard topological approach, are exactly crease (ridge and valley) lines of a particular tensor invariant called mode. This reformulation allows us to apply well-studied approaches from scientific visualization or computer vision to the extraction of topological lines in tensor fields. More generally, this main result suggests that other tensor invariants, such as anisotropy measures like fractional anisotropy (FA), can be used in the same framework in lieu of mode to identify important structural properties in tensor fields. Our implementation addresses the specific challenge posed by the non-linearity of the considered scalar measures and by the smoothness requirement of the crease manifold computation. We use a combination of smooth reconstruction kernels and adaptive refinement strategy that automatically adjust the resolution of the analysis to the spatial variation of the considered quantities. Together, these improvements allow for the robust application of existing ridge line extraction algorithms in the tensor context of our problem. Results are proposed for a diffusion tensor MRI dataset, and for a benchmark stress tensor field used in engineering research.

Contact metric manifolds with η-parallel torsion tensor

Abstract: We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E n+1 × S n (4) in higher dimensions.

Connections with Skew-Symmetric Ricci Tensor on Surfaces

Abstract: Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong’s canonical coordinate form of such connections is simplified

Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary

Abstract: We study the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensor in a negative convex cone on compact manifolds. A consequence of our main results is that any smooth bounded domain in Euclidean space of dimension greater or equal to 3 admits a complete conformally flat metric of negative Ricci curvature with .

Curvature estimates for the Ricci flow II

Abstract: In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kahler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.

Subelliptic Li-Yau estimates on three dimensional model spaces

Abstract: We describe three elementary models in three dimensional subelliptic geometry whichcorrespond to the three models of the Riemannian geometry (spheres, Euclidean spaces andHyperbolic spaces)which arerespectivelythe SU(2), Heisenbergand SL(2)groups. On thosemodels, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation.We use for that the Γ 2 techniques that we adapt to those elementary model spaces. Theimportant feature developed here is that although the usual notion of Ricci curvature ismeaningless (or more precisely leads to bounds of the form −∞ for the Ricci curvature), wedescribe a parameter ρ which plays the same rˆole as the lower bound on the Ricci curvature,and from which one deduces the same kind of results as one does in Riemannian geometry,like heat kernel upper bounds, Sobolev inequalities and diameter estimates. 1 Framework and Introduction The estimation of heat kernel measures is a topic which had been under thorough investigationfor the last thirty years at least, see [12, 8]. Among the many techniques developed for that,the famous Li-Yau parabolic inequality [12] is a very powerful tool, which relies in Riemanniangeometry bounds on the gradient on heat kernels to lower bounds on the Ricci curvature. Moreprecisely, in the simplest form, it asserts that, if E is a smooth Riemannian manifold withdimension n and non negative Ricci curvature, then if f is any positive solution of the heatequation∂

A second-order identity for the Riemann tensor

Abstract: A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock.

On weyl and browder spectra of tensor products

Abstract: Let A and B be Hilbert space operators. In this paper we explore the structure of parts of the spectrum of the tensor product A ⊗ B , and conclude some properties that follow from such a structure. We give conditions on A and B ensuring that σ w ( A ⊗ B ) =σ w ( A )ċσ( B ) ∪ σ( A )ċσ w ( B ), where σ(ċ) and σ w (ċ) stand for the spectrum and Weyl spectrum, respectively. We also investigate the problem of transferring Weyl and Browder's theorems from A and B to their tensor product A ⊗ B .

The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

Abstract: The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M 4 × ℝ1. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, −, −, −) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M 4 × S 1, where S 1 is the circle. For such functions, electric charges are topologically quantized.

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

Abstract: We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prufer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17–28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R 3 , having the prescribed principal Ricci curvatures.

A connection with skew symmetric torsion and k ahler curvature tensor on quasi-k ahler manifolds with norden metric

Abstract: (Submitted by Corresponding Member S. Dodunekov on July 17, 2008)AbstractThere is considered a connection with skew symmetric torsion on a quasi-Kahler manifold with Norden metric. Some necessary and sucient conditionsare derived for the corresponding curvature tensor to be Kahlerian. In the casewhen this tensor is Kahlerian, some relations are obtained between its scalarcurvature and the scalar curvature of other curvature tensors. Conditions aregiven for the considered manifolds to be isotropic-Kahler.Key words:almost complex manifold, Norden metric, nonintegrable struc-ture, skew symmetric torsion, quasi-Kahler manifolds2000 Mathematics Subject Classi cation: 53C15, 53C50

Segmenting Fiber Bundles in Diffusion Tensor Images

Abstract: We consider the problem of segmenting fiber bundles in diffusion tensor images. We cast this problem as a manifold clustering problem in which different fiber bundles correspond to different submanifolds of the space of diffusion tensors. We first learn a local representation of the diffusion tensor data using a generalization of the locally linear embedding (LLE) algorithm from Euclidean to diffusion tensor data. Such a generalization exploits geometric properties of the space of symmetric positive semi-definite matrices, particularly its Riemannian metric. Then, under the assumption that different fiber bundles are physically distinct, we show that the null space of a matrix built from the local representation gives the segmentation of the fiber bundles. Our method is computationally simple, can handle large deformations of the principal direction along the fiber tracts, and performs automatic segmentation without requiring previous fiber tracking. Results on synthetic and real diffusion tensor images are also presented.

A sharp stability estimate in tensor tomography

Abstract: Let (M, g) be a simple compact Riemannian manifold with boundary. We give a sharp stability estimate for the problem of determining the solenoidal part of a tensor field from its integral along geodesics of the metric g.

Tensor analysis with applications

Abstract: 1. Tensors and their Algebra 1.1 Introduction 1.2 Transformation of Coordinates 1.3 Summation Convention 1.4 Kronecker Delta 1.5 Scalars, Contravariant and Covariant Vectors 1.6 Tensors of Higher Rank 1.7 Symmetry of Tensors 1.8 Algebra of Tensors, Addition and Subtraction, Equality of Tensors, Inner and Outer Products, Contraction, The Quotient Law 1.9 Irreducible Tensors, Exercises 2. Riemannian Space and Metric Tensor 2.1 Introduction 2.2 The Metric Tensor 2.3 Raising and Lowering of Indices-Associated Tensor 2.4 Vector Magnitude 2.5 Relative and Absolute Tensors 2.6 The Levi-Civita Tensor, Exercises 3. Christoffel Symbols and Covariant Differentiation 3.1 Introduction 3.2 Christoffel Symbols 3.3 Transformation Law for Christoffel Symbols 3.4 Equation of a Geodesic 3.5 Affine Parameter 3.6 Geodesic Coordinate System 3.7 Covariant Differentiation, Covariant Derivatives of Contravariant and Covariant Vectors, Covariant Derivatives of Rank Two Tensors, Covariant Derivatives of Tensors of Higher Rank 3.8 Rules for Covariant Differentiation 3.9 Some Useful Formulas, Divergence of a Vector Field, Gradient of a Scalar and Laplacian, Curl of a Vector Field, Divergence of a Tensor Field 3.10 Intrinsic Derivative-Parallel Transport 3.11 Null Geodesics 3.12 Alternative Derivation of Equation of Geodesic, Exercises 4. The Riemann Curvature Tensor 4.1 Introduction 4.2 The Riemann Curvature Tensor 4.3 Commutation of Covariant Derivatives 4.4 Covariant Form of the Riemann Curvature Tensor 4.5 Properties of the Riemann Curvature Tensor 4.6 Uniqueness of the Riemann Curvature Tensor 4.7 The Number of Algebraically Independent Components of the Riemann Curvature Tensor 4.8 The Ricci Tensor and the Scalar Curvature 4.9 The Einstein Tensor 4.10 The Integrability of the Riemann Tensor and the Flatness of the Space 4.11 The Einstein Spaces 4.12 Curvature of a Riemannian Space 4.13 Spaces of Constant Curvature, Exercises 5. Some Advanced Topics 5.1 Introduction, 5.2 Gewodesic Deviation 5.3 Decomposition of Riemann Curvature Tensor 5.4 Electric and Magnetic Parts of the Riemann and Weyl Tensors 5.5 Classification of Gravitational Fields 5.6 Invariants of the Riemann Curvature Tensor 5.7 Curvature Tensors Involving the Riemann Tensor, Space-matter Tensor, Conharmonic Curvature Tensor 5.8 Lie Derivative 5.9 The Killing Equation 5.10 The Curvature Tensor and Killing Vector, Exercises

Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

Abstract: In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one-forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss’ theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations.

Curvature structure of self-dual 4-manifolds

Abstract: We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.