Showing papers on "Riemann curvature tensor published in 2002"
Book•
08 Feb 2002
TL;DR: The local theory of surfaces and Riemannian geometry of surfaces were studied in this article, where the curvature tensor tensor is defined as a tensor of constant curvature.
Abstract: * Notations and prerequisites from analysis* Curves in $\mathbb{R}^n$* The local theory of surfaces* The intrinsic geometry of surfaces* Riemannian manifolds* The curvature tensor* Spaces of constant curvature* Einstein spaces* Solutions to selected exercises* Bibliography* List of notation* Index
443 citations
TL;DR: In this article, the authors studied the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary and showed that the integral of mean curvature of the boundary of the manifold cannot be greater than the integral integral of the mean curvatures of the embedded image as a hypersurface in Euclidean space.
Abstract: In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.
298 citations
TL;DR: In this article, the existence of a metric whose Schouten tensor satisfies a quadratic inequality was shown to imply that the eigenvalues of the Ricci tensor are positively pinched.
Abstract: We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
231 citations
TL;DR: In this article, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
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225 citations
TL;DR: The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases.
Abstract: A 4 x 4 complex curvature tensor M>(-1) is introduced to describe partially coherent anisotropic Gaussian-Schell model (GSM) beams. An analytical propagation formula for the cross-spectral density of partially coherent anisotropic GSM beams is derived. The propagation law of M(-1) that is also derived may be called partially coherent tensor ABCD law. The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases.
224 citations
Book•
13 Dec 2002
TL;DR: In this paper, the authors propose analytic methods in the initial value problem and estimate the connection coefficients and the curvature tensor tensor for the initial hypersurface and the last slice.
Abstract: Preface * Introduction * Analytic methods in the initial value problem * Definitions and results * Estimates for the connection coefficients * Estimates for the curvature tensor * The error estimates * The initial hypersurface and the last slice * Conclusions * Bibliography * Index
189 citations
TL;DR: In this paper, it was shown that a closed manifold with positive Ricci curvature must admit an action by a compact Lie group G with orbits of codimension one, and that this is also a sufficient condition for any manifold with Ricci or scalar curvature.
Abstract: One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.
171 citations
TL;DR: In this article, all Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined and the corresponding metrics in local coordinates are discussed.
Abstract: All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.
159 citations
25 Sep 2002
TL;DR: Two new approaches to quantifying the white matter connectivity in the brain using Diffusion Tensor Magnetic Resonance Imaging data are investigated, deriving from each tensor a local warping of space, and finding geodesic paths in the space.
Abstract: We investigate new approaches to quantifying the white matter connectivity in the brain using Diffusion Tensor Magnetic Resonance Imaging data. Our first approach finds a steady-state concentration/ heat distribution using the three-dimensional tensor field as diffusion/ conductivity tensors. Our second approach casts the problem in a Riemannian framework, deriving from each tensor a local warping of space, and finding geodesic paths in the space. Both approaches use the information from the whole tensor, and can provide numerical measures of connectivity.
156 citations
TL;DR: In this article, it was shown that the Ricci flow converges to a metric with constant bisectional curvature if and only if the curvature of the initial metric is positive.
Abstract: In this paper, we prove that if M is a Kahler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kahler-Ricci flow converges to a Kahler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kahler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kahler Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kahler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.
145 citations
TL;DR: In this paper, a curvature correction for explicit algebraic Reynolds stress models (EARSMs) is proposed, based on a formal derivation of the weak equilibrium assumption in a streamline oriented curvilinear co-ordinate syste...
TL;DR: A survey of the field can be found in this article, where Furstenberg, Varopoulos, Coulhon, Saloff-Coste, and others give a rough outline of the history of a specific point of view in this area, namely, the interplay between the geometry and the function theory.
Abstract: Function theory on Euclidean domains in relation to potential theory, partial differential equations, probability, and harmonic analysis has been the target of investigation for decades. There is a wealth of classical literature in the subject. Geometers began to study function theory with the primary reason to prove a uniformization type theorem in higher dimensions. It was first proposed by GreeneWu and Yau to study the existence of bounded harmonic functions on a complete manifold with negative curvature. While uniformization in dimension greater than 2 still remains an open problem, the subject of function theory on complete manifolds takes on life of its own. The seminal work of Yau [Y1] provided a fundamental technique in handling analysis on noncompact, complete manifolds. It also opens up many interesting problems which are essential for the understanding of analysis on complete manifolds. Since Yau’s paper in 1975, there are many developments in this subject. The aim of this article is to give a rough outline of the history of a specific point of view in this area, namely, the interplay between the geometry – primarily the curvature – and the function theory. Throughout this article, unless otherwise stated, we will assume that M is an n-dimensional, complete, noncompact, Riemannian manifold without boundary. In this case, we will simply say that M is a complete manifold. One of the goal of this survey is to demonstrate, by way of known theorems, the two major steps which are common in many geometric analysis programs. First, we will show how one can use assumptions on the curvature to conclude function theoretic properties of the manifold M. Secondly, we will showed that function theoretic properties can in turn be used to conclude geometrical and topological statements about the manifold. In many incidents, combining the two steps will result in a theorem which hypothesizes on the curvature and concludes on either the topological, geometrical, or complex structure of the manifold. The references will not be comprehensive due to the vast literature in the subject. It is merely an indication of the flavor of the field for the purpose of whetting one’s appetite. As examples of areas not being discussed in this note are harmonic analysis (function theory) on symmetric spaces, Lie groups, and discrete groups. The contributors to this subject are Furstenberg, Varopoulos, Coulhon, Saloff-Coste, and
TL;DR: In this paper, the Ricci-Harmonic map flow is used to evolve non-smooth Riemannian metric tensors by the dual Ricci flow, up to a diffeomorphism.
Abstract: The purpose of this paper is to evolve non-smooth Riemannian metric tensors by the dual Ricci-Harmonic map flow. This flow is equivalent (up to a diffeomorphism) to the Ricci flow. One application will be the evolution of metrics which arise in the study of spaces whose curvature is bounded from above and below in the sense of Aleksandrov, and whose curvature operator (in dimension three Ricci curvature) is non-negative. We show that such metrics may always be deformed to a smooth metric having the same properties in a strong sense. §
TL;DR: In this paper, a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions is developed, based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances.
Abstract: When the interfacial energy is a nonconvex function of orientation, the anisotropic-curvature-flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions. The theory is based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances; a mechanical version of the second law that includes, via the configurational moments, work that accompanies changes in the curvature of the interface; a constitutive theory whose main ingredient is a positive-definite, isotropic, quadratic dependence of the interfacial energy on the curvature tensor. Two special cases are investigated: (i) the interface is a boundary between bulk phases or grains, and (ii) the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface.
TL;DR: In this article, it was shown that the Ricci curvature of the Schouten tensor is in a certain cone, Γ + k, which implies that the curvature is positive.
Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the kth elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ + k . We prove that this eigenvalue condition for k > n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.
TL;DR: In this article, the Kretschmann, Chern-Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman-Penrose formalism and in the framework of gravitoelectromagnetism are discussed.
Abstract: We discuss the Kretschmann, Chern–Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman–Penrose formalism and in the framework of gravitoelectromagnetism, using the Kerr–Newman geometry as an example. An analogy with electromagnetic invariants leads to the definition of regions of gravitoelectric or gravitomagnetic dominance.
Journal Article•
TL;DR: In this paper, it was shown that a manifold with constant positive flag curvature can be constructed from a hypersurface in a suitably general position in CPn by using a Riemannian Zoll metric of positive Gauss curvature.
Abstract: This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in CPn. The third remark is that there is a description of the Finsler metrics of constant curvature on S2 in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on S2 to construct a global Finsler metric of constant positive curvature on S2. The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1 > 2. It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S1·GL(n,R)structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. 1991 Mathematics Subject Classification. 53B40, 53C60, 58A15.
TL;DR: In this paper, generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry, and the notion of geodesics of a generalized metric is defined.
Abstract: Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a Fundamental Lemma of (pseudo-) Riemannian geometry in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.
TL;DR: In this paper, a new phenomenological deformation theory with strain gradient effects is proposed, which fits within the framework of general couple stress theory and involves a single material length scale.
TL;DR: It is shown by thermally equilibrating the many-body elastic energy using a Monte Carlo algorithm, that inclusions shaped as "saddles" attract each other and build an "egg-carton" structure, reminiscent of some patterns observed in membranes obtained from biological extracts.
TL;DR: In this article, the authors proposed a different approach that is based on an idea of approximation due to Airault [1], which has the advantage that a key property of the multiplicative function (i.e., the attendant Ito's formula for this functional) follows almost automatically from the approximate multiplicative functional without resorting to excursion theory, thus greatly simplifying this part of the theory.
Abstract: By the Weitzenbock formula relating the Hodge–de Rham Laplacian and the covariant Laplacian for differential forms on a Riemannian manifold, the heat equation for differential forms is naturally associated with a matrix-valued Feynman– Kac multiplicative functional determined by the curvature tensor. The case of a closed manifold (without boundary) is well known and will be briefly reviewed below. In constrast, the case of manifolds with boundary is not well known, and for good reasons. Because the absolute boundary condition on differential forms is Dirichlet in the normal direction and Neumann in the tangential directions, the associated multiplicative functional is discontinuous and much more difficult to handle. Ikeda and Watanabe [6; 7] have dealt with this situation by using an excursion theory (for reflecting Brownian motion) that seems to have been created especially for this problem. In this paper we suggest a different approach that is based on an idea of approximation due to Airault [1]. This construction has the advantage that a key property of the multiplicative functional (i.e., the attendant Ito’s formula for this functional) follows almost automatically from the approximate multiplicative functional without resorting to excursion theory, thus greatly simplifying this part of the theory; see Theorem 3.7. Before coming to another and more important raison d’etre for the present work, we briefly review some relevant facts for a closed manifold. LetM be a compact, closed Riemannian manifold and let α0 be a 1-form onM. Consider the following initial value problem: ∂α ∂t = 1 2 α,
TL;DR: An investigation of the second-order properties of such an objective function in a neighborhood of the minimizer opens an intuitive access to many traits of this approach to optimization of photon IMRT treatment plans.
Abstract: One approach to the computation of photon IMRT treatment plans is the formulation of an optimization problem with an objective function that derives from an objective density. An investigation of the second-order properties of such an objective function in a neighborhood of the minimizer opens an intuitive access to many traits of this approach. A general finding is that only a small subset of the parameter space has nonzero curvature, while the objective function is entirely flat in a neighborhood of the minimizer in most directions. The dimension of the subspace of vanishing curvature serves as a measure for the degeneracy of the solution. This finding is important both for algorithm design and evaluation of the mathematical model of clinical intuition, expressed by the objective function. The structure of the subspace of great curvature is found to be imposed on the problem by conflicts between objectives of target and critical structures. These conflicts and their corresponding modes of resolution form a common trait between all reasonable treatment plans of a given case. The high degree of degeneracy makes the use of a conjugate gradient optimization algorithm particularly favorable since the number of iterations to convergence is equivalent to the number of different eigenvalues of the curvature tensor and is hence independent from the number of optimization parameters. A high level of degeneracy of the fluence profiles implies that it should be possible to stipulate further delivery-related conditions without causing severe deterioration of the dose distribution.
TL;DR: In this article, the authors give necessary and sufficient conditions for tangent bundles having positive scalar curvature induced by the Cheeger-Gromoll metric on a Riemannian manifold of constant sectional curvature.
Abstract: . Let (M,g) be aRiemannian manifold of constant sectional curvature κ and (TM,g ˜ ) be the tangentbundle of M equipped with the Cheeger-Gromoll metric induced by g . We give necessary and sufficient conditionsfor TM having positive scalar curvature. This gives counterexamples to a stated theorem of Sekizawa. 1. Introduction. ARiemannianmetric g onasmoothmanifold M givesrisetoseveralnaturalRiemannianmetrics on the tangent bundle TM of M . The best known example is the Sasaki metric g ˆintroduced in [6], see also [2]. In the present paper we study tangent bundles equipped withthe so called Cheeger-Gromoll metric. Its construction was suggested in [1] but the firstexplicit description was given by Musso and Tricerri in [5].In [7] Sekizawa calculates the Levi-Civita connection ∇˜, the curvature tensor R ˜,thesectional curvatures K ˜ and the scalar curvature S ˜ of the Cheeger-Gromoll metric. He thenstates in his Theorem 6.3 that if (M,g) is an m -dimensional manifold of constant sectionalcurvature
TL;DR: In this paper, the authors investigated curvature properties of semi-Riemannian manifolds (M,g), n/4, whose Weyl curvature tensor C can be expressed by a KulkarniNomizu square of the tensor S - j9-
Abstract: We investigate curvature properties of semi-Riemannian manifolds (M,g), n/4, whose Weyl curvature tensor C can be expressed by a KulkarniNomizu square of the tensor S - j9- We investigate also the problem of isometric immersion of such manifolds into space forms.
TL;DR: In this article, it was shown that there are no complete graphs in the Euclidean space with positive constant 2-mean curvature, which is a generalization of a result first proved by Chern.
Abstract: Hypersurfaces of constant 2-mean curvature in spaces of constant sectional curvature are known to be solutions to a variational problem. We extend this characterization to ambient spaces which are Einstein. We then estimate the 2-mean curvature of certain hypersurfaces in Einstein manifolds. A consequence of our estimates is a generalization of a result, first proved by Chern, showing that there are no complete graphs in the Euclidean space with positive constant 2-mean curvature.
TL;DR: In this article, the problem of uniformization of general Riemann surfaces through consideration of the curvature equation was considered, and in particular of constructing Poincare metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e2u=Ko(z).
Abstract: We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincare metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e2u=Ko(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore2u giving the Poincare metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.
TL;DR: In this article, the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted γ-law perfect fluid is studied.
Abstract: We study the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted γ-law perfect fluid. We investigate all the Bianchi and Thurston type universe models and calculate the asymptotic evolution of Weyl curvature invariant for generic solutions to the Einstein field equations. The influence of compact topology on Bianchi types with hyperbolic space sections is also considered. Special emphasis is placed on the late-time behaviour where several interesting properties of the Weyl curvature invariant occur. The late-time behaviour is classified into five distinctive categories. It is found that for a large class of models, the generic late-time behaviour of the Weyl curvature invariant is to dominate the Ricci invariant at late times. This behaviour occurs in universe models which have future attractors that are plane-wave spacetimes, for which all scalar curvature invariants vanish. The overall behaviour of the Weyl curvature invariant is discussed in relation to the proposal that some function of the Weyl tensor or its invariants should play the role of a gravitational 'entropy' for cosmological evolution. In particular, it is found that for all ever-expanding models the measure of gravitational entropy proposed by Gron and Hervik increases at late times.
TL;DR: In this article, an estimator for local curvature of space curves embedded in n-dimensional (n-D) grey-value images is presented. But this estimator works on the orientation field of the space curve and a description of local structure is obtained by the gradient structure tensor.
Abstract: Local curvature represents an important shape parameter of space curves which are well described by differential geometry. We have developed an estimator for local curvature of space curves embedded in n-dimensional (n-D) grey-value images. There is neither a segmentation of the curve needed nor a parametric model assumed. Our estimator works on the orientation field of the space curve. This orientation field and a description of local structure is obtained by the gradient structure tensor. The orientation field has discontinuities; walking around a closed contour yields two such discontinuities in orientation. This field is mapped via the Knutsson (1985) mapping to a continuous representation; from a n-D vector to a symmetric n/sup 2/-D tensor field. The curvature of a space curve, a coordinate invariant property, is computed in this tensor field representation. An extensive evaluation shows that our curvature estimation is unbiased even in the presence of noise, independent of the scale of the object and furthermore the relative error stays small.
TL;DR: In this article, the Ricci curvature is shown to be conformally equivalent to a compact surface with a finite number of points removed, and the curvature assumption must be strengthened in order to get an analogous conclusion in higher dimensions.
Abstract: It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite
$ L^{n/2} $
-norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions.
TL;DR: In this article, the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms were shown to be equal in terms of the squared Mean curvature.
Abstract: Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms. In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.