Showing papers on "Riemann curvature tensor published in 2004"

Symmetries and Curvature Structure in General Relativity

Abstract: Introduction topological spaces groups and linear algebra manifold theory transformation groups the Lorentz group general relativity theory space-time holonomy curvature structure in general relativity affine symmetries in space-time conformal symmetries in space-time curvature collineations sectional curvature structure.

Vanishing scalar invariant spacetimes in higher dimensions

Abstract: We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

The entropy formula for linear heat equation

Abstract: We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.

Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem

Abstract: This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts.

Membrane geometry with auxiliary variables and quadratic constraints

Abstract: Consider a surface described by a Hamiltonian which depends only on the metric and extrinsic curvature induced on the surface. The metric and the curvature, along with the basis vectors which connect them to the embedding functions defining the surface, are introduced as auxiliary variables by adding appropriate constraints, all of them quadratic. The response of the Hamiltonian to a deformation in each of the variables is determined and the relationship between the multipliers implementing the constraints and the conserved stress tensor of the theory established. For the purpose of illustration, a fluid membrane described by a Hamiltonian quadratic in curvature is considered.

Dirty black holes: spacetime geometry and near-horizon symmetries

Abstract: We consider the spacetime geometry of a static but otherwise generic black hole (that is, the horizon geometry and topology are not necessarily spherically symmetric). It is demonstrated, by purely geometrical techniques, that the curvature tensors, and the Einstein tensor in particular, exhibit a very high degree of symmetry as the horizon is approached. Consequently, the stress-energy tensor will be highly constrained near any static Killing horizon. More specifically, it is shown that—at the horizon—the stress-energy tensor block-diagonalizes into 'transverse' and 'parallel' blocks, the transverse components of this tensor are proportional to the transverse metric, and these properties remain invariant under static conformal deformations. Moreover, we speculate that this geometric symmetry underlies Carlip's notion of an asymptotic near-horizon conformal symmetry controlling the entropy of a black hole.

Asymptotic degeneracy of dyonic N=4 string states and black hole entropy

Abstract: It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N=4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the microscopic entropy, written as a function of the dilaton field, is stationary at the horizon by virtue of the attractor equations.

Normal based estimation of the curvature tensor for triangular meshes

Abstract: We introduce a new technique for estimating the curvature tensor of a triangular mesh. The input of the algorithm is only a single triangle equipped with its (exact or estimated) vertex normals. This way we get a smooth junction of the curvature tensor inside each triangle of the mesh. We show that the error of the new method is comparable with the error of a cubic fitting approach if the incorporated normals are estimated. If the exact normals of the underlying surface are available at the vertices, the error drops significantly. We demonstrate the applicability of the new estimation at a rather complex data set.

Osserman manifolds of dimension 8

Abstract: For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator RX is defined by RXY=R(X,Y)X. The manifold Mn is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues of the Jacobi operator RX do not depend of a unit vector X ∈ TpMn, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n≠8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.

Interior curvature bounds for a class of curvature equations

Abstract: We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Amp` ere type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C 1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

Penrose limits and spacetime singularities

Abstract: We give a covariant characterization of the Penrose plane wave limit: the plane wave profile matrix A(u) is the restriction of the null geodesic deviation matrix (curvature tensor) of the original spacetime metric to the null geodesic, evaluated in a comoving frame. We also consider the Penrose limits of spacetime singularities and show that for a large class of black hole, cosmological and null singularities (of Szekeres–Iyer 'power-law type'), including those of the FRW and Schwarzschild metrics, the result is a singular homogeneous plane wave with profile A(u) ~ u−2, the scale invariance of the latter reflecting the power-law behaviour of the singularities.

Topological complexity and the dynamics of coarsening.

Abstract: Coarsening or Ostwald ripening occurs in a vast array of two-phase systems. Coarsening results in a decrease in the interfacial area per unit volume and a concomitant increase in the size scale of the interfacial morphology. Much is known about the coarsening process in two-phase mixtures consisting of a polydisperse array of spherical particles1,2. In contrast, in many two-phase mixtures, such as those found in two-phase polymers3, ceramics4, dendritic solid–liquid mixtures5,6 and order–disorder transformations7, the interfaces are both interconnected and have a spatially varying mean curvature. Here we show that the morphological evolution of these topologically complex systems during coarsening can be quantified by measuring the probability of finding a patch of interface with a given curvature tensor. We find that the morphological evolution is described by the flow of probability density in this curvature space that is induced by the coarsening process. The hallmark of our approach is a close coupling between experiment and theory; we use the experimentally measured three-dimensional microstructure as an input to a phase-field calculation that then determines the flow in curvature space. The methodology is general, and applicable to many systems undergoing coarsening, regardless of their topology.

Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities

Abstract: In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli– Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.

Cohomogeneity One Manifolds of Even Dimension with Strictly Positive Sectional Curvature

Abstract: We show that the only compact positively curved Riemannian manifolds of even dimension acted on by a simple group with a codimension one orbit are the compact rank one symmetric spaces.

Addenda to "The Entropy Formula for Linear Heat Equation"

Abstract: We add two sections to (8) and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of (8), which reveals the relation between the entropy formula, (1.4) of (8), and the well-known Li-Yau's gradient estimate. As a by-product we obtain the sharp estimates on 'Nash's entropy' for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau's gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n .I n the second section we derive a dual entropy formula which, to some degree, connects Hamilton's entropy with Perelman's entropy in the case of Riemann surfaces. 1. The relation with Li-Yau's gradient estimates In this section we provide another derivation of Theorem 1.1 of (8) and discuss its relation with Li-Yau's gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash's 'entropy quantity' − M H log Hd vin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of (9). Let u(x, t) be a positive solution to ∂ ∂t − � u(x, t) = 0 with M ud v= 1. We define

Ricci flow and nonnegativity of sectional curvature

Abstract: In this paper, we extend the general maximum principle in (NT3) to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional cur- vature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.

Metrics of positive Ricci curvature on quotient spaces

Abstract: One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being understood. For example, so far the only obstructions to the existence of a metric with positive Ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature ([Li], [Hi], [Ro], [SchY], [Ta]) and the Bonnet-Myers theorem which implies that the fundamental group of a closed manifold with positive Ricci curvature must be finite. Fruitful constructions of metrics with positive Ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics ([Au], [Eh], [We]), Kahler geometry ([Yau1], [Yau2]), bundles and warping ([Po], [Na], [BB], [GPT]), special kinds of surgery ([SY], [Wr]), metrical glueing ([GZ2]), and Sasakian geometry ([BGN]). Particularly large classes of examples of manifolds with positive Ricci curvature are given by all compact homogeneous spaces with finite fundamental group ([Na]) and all closed cohomogeneity one manifolds with finite fundamental group ([GZ2]). In this article we present several new classes of manifolds which admit metrics of positive or nonnegative Ricci curvature. The idea is to consider quotients of manifolds (M,g) of positive or nonnegative Ricci curvature by a free isometric action. While taking such a quotient non-decreases the sectional curvature, it may well happen in general that the Ricci curvature of the quotient is inferior to the one of M . Our first results, however, state that the quotient of M does admit metrics of positive Ricci curvature if M belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action. More precisely, we prove:

Variational Approach to Differential Invariants of Rank 2 Vector Distributions

Abstract: In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n = 5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his ”reduction-prolongation” procedure (see [12]). After Cartan’s work the following questions remained open: first the geometric reason for existence of Cartan’s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan’s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in [4],[5]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ≥ 5. For n = 5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In the next paper [19] we show that in the case n = 5 our fundamental form coincides with Cartan’s tensor.

Curvature tensor based triangle mesh segmentation with boundary rectification

Abstract: This paper presents a new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes into surface patches. The algorithm is based on the curvature tensor field analysis and presents two distinct complementary steps: a region based segmentation, which is an improvement of that presented by [G. Lavoue et al., (2004)] and which decomposes the object into known and near constant curvature patches, and a boundary rectification based on curvature tensor directions, which corrects boundaries by suppressing their artifacts or discontinuities. Experiments were conducted on various models including both CAD and natural objects, results are satisfactory. Resulting segmented patches, by virtue of their properties (known curvature, clean boundaries) are particularly adapted to computer graphics tasks like parametric or subdivision surface fitting in an adaptive compression objective

The curvature of a hessian metric

Abstract: Inspired by Wilson's paper on sectional curvatures of Kahler moduli, we consider a natural Riemannian metric on a hypersurface {f=1} in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question posed by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R3 such that the associated metric has constant negative curvature. We ask if our examples, together with one example by Dubrovin, are the only ones with constant negative curvature. This question can be rephrased as an appealing question in classical invariant theory, involving the "Clebsch covariant". We give a positive answer for polynomials of degree at most 4, as well as a partial result in any degree.

Rozansky-Witten invariants of hyperkähler manifolds

Abstract: We investigate invariants of compact hyperk{\"a}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifold, arriving at a number. For certain graph homology classes, the invariants we get are Chern numbers, and in fact all characteristic numbers arise in this way. We use relations in graph homology to study and compare these hyperk{\"a}hler manifold invariants. For example, we show that the norm of the Riemann curvature can be expressed in terms of the volume and characteristic numbers of the hyperk{\"a}hler manifold. We also investigate the question of whether the Rozansky-Witten invariants give us something more general than characteristic numbers. Finally, we introduce a generalization of these invariants which incorporates holomorphic vector bundles into the construction.

Statistics on Multivariate Normal Distributions: A Geometric Approach and its Application to Diffusion Tensor MRI

Abstract: This report is dedicated to the statistical analysis of the space of multivariate normal probability density functions. It relies on the differential geometrical properties of the underlying parameter space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on non-linear spaces. We will first proceed to the state of the art in section 1, while expressing some quantities related to the structure of the manifold of interest, and then focus on the derivation of closed-form expressions for the mean, covariance matrix, modes of variation and normal law between multivariate normal distributions in section 2. We will also address the derivation of accurate and efficient numerical schemes to estimate the proposed quantities. A major application of the present work is the statistical analysis of diffusion tensor Magnetic Resonance Imaging. We show promising results on synthetic and real data in section 3

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Abstract: In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kahler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kahler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

Variational Properties of the Gauss-Bonnet Curvatures

Abstract: The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss-Bonnet curvature.

Heat Kernel Comparison on Alexandrov Spaces with Curvature Bounded Below

Abstract: In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with lower curvature bound in the sense of Alexandrov and with sufficiently fast asymptotic decay of the volume of small geodesic balls. As corollaries we recover Varadhan's short time asymptotic formula for the heat kernel (1967) and Cheng's eigenvalue comparison theorem (1975). Finally, we derive an integral inequality for the distance process of a Brownian Motion on (X,d) resembling earlier results in the smooth setting by Debiard, Geavau and Mazet (1975).

Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds

Abstract: The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.

Curvature Functionals, Optimal Metrics, and the Differential Topology of 4-Manifolds

Abstract: This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor methods, optimal metrics of the second type are constructed on the connected sums kCP_2 for k > 5. However, related constructions also show that large classes of simply connected 4-manifolds do not admit any optimal metrics at all. Interestingly, the difference between existence and non-existence turns out to delicately depend on one's choice of smooth structure; there are smooth 4-manifolds which carry optimal metrics, but which are homeomorphic to infinitely many distinct smooth 4-manifolds on which no optimal metric exists.

On isometric immersions of a riemannian space with little regularity

Abstract: We consider a Riemannian metric in an open subset of the d-dimensional Euclidean space and assume that its Riemann curvature tensor vanishes. If the metric is of class C2, a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the d-dimensional Euclidean space. We establish that if the metric belongs to the Sobolev space W1,∞ and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the d-dimensional Euclidean space.