Showing papers on "Riemann curvature tensor published in 2003"
01 Jul 2003
TL;DR: A novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry, and provides the flexibility to produce meshes ranging from isotropic to anisotropic, from coarse to dense, and from uniform to curvature adapted.
Abstract: In this paper, we propose a novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry. In particular, we use curvature directions to drive the remeshing process, mimicking the lines that artists themselves would use when creating 3D models from scratch. After extracting and smoothing the curvature tensor field of an input genus-0 surface patch, lines of minimum and maximum curvatures are used to determine appropriate edges for the remeshed version in anisotropic regions, while spherical regions are simply point sampled since there is no natural direction of symmetry locally. As a result our technique generates polygon meshes mainly composed of quads in anisotropic regions, and of triangles in spherical regions. Our approach provides the flexibility to produce meshes ranging from isotropic to anisotropic, from coarse to dense, and from uniform to curvature adapted.
614 citations
TL;DR: In this paper, the authors define and compute the energy of higher curvature gravity theories in arbitrary dimensions and show that these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant solutions with nontrivial energy properties.
Abstract: We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with nontrivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action is the square of the traceless Ricci tensor always has zero energy, unlike conformal (Weyl) gravity. We also study the string-inspired Einstein-Gauss-Bonnet model and show that both its flat and anti–de Sitter vacua are stable.
515 citations
08 Jun 2003
TL;DR: A definition of the curvature tensor for polyhedral surfaces is derived in a very simple and new formula that yields an efficient and reliable curvature estimation algorithm.
Abstract: We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a definition of the curvature tensor for polyhedral surfaces. This definition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an efficient and reliable curvature estimation algorithm. Moreover, we bound the difference between the estimated curvature and the one of the smooth surface in the case of restricted Delaunay triangulations.
510 citations
TL;DR: In this article, the Bakry-Emery tensor is shown to be non-decreasing under a Riemannian manifold with fiber transport preserving measures up to constants.
Abstract: The Bakry-Emery tensor gives an analog of the Ricci tensor for a
Riemannian manifold with a smooth measure. We show that some of
the topological consequences of having a positive or nonnegative
Ricci tensor are also valid for the Bakry-Emery tensor. We show
that the Bakry-Emery tensor is nondecreasing under a Riemannian
submersion whose fiber transport preserves measures up to constants.
We give some
relations between the Bakry-Emery tensor and measured
Gromov-Hausdorff limits.
332 citations
TL;DR: In this article, the authors studied cosmological braneworld models with a single timelike extra dimension and found that the universe either has a nonsingular origin or commences its expansion from a quasi-singular state during which both the Hubble parameter and the energy density and pressure remain finite while the curvature tensor diverges.
278 citations
TL;DR: A combinatorial analogue of Bochner's theorems is derived, which demonstrates that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature.
Abstract: . In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds. We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing the properties of the combinatorial Ricci curvature with those of its Riemannian avatar.
250 citations
TL;DR: A new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements is proposed, to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates.
Abstract: We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.
196 citations
Posted Content•
TL;DR: In this article, Hitchin and Altschuler derived a formula for the scalar curvature and Ricci curvature of a G2-structure in terms of its torsion.
Abstract: This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has since been subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature.
A formula is derived for the scalar curvature and Ricci curvature of a G_2-structure in terms of its torsion. When the fundamental 3-form of the G_2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. This version contains some new results on the pinching of Ricci curvature for metrics associated to closed G_2-structures.
Some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data.
124 citations
TL;DR: In this article, the cosmology of the Randall-Sundrum brane-world where the Einstein-Hilbert action is modified by curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term.
Abstract: We study the cosmology of the Randall-Sundrum brane-world where the Einstein-Hilbert action is modified by curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term. The combined effect of these curvature corrections to the action removes the infinite-density big bang singularity, although the curvature can still diverge for some parameter values. A radiation brane undergoes accelerated expansion near the minimal scale factor, for a range of parameters. This acceleration is driven by the geometric effects, without an inflaton field or negative pressures. At late times, conventional cosmology is recovered.
112 citations
TL;DR: In this paper, the authors generalized Smale's α theory to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds.
Abstract: In this paper, Smale's α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high-order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given.
109 citations
TL;DR: In this article, a conformal deformation involving a fully nonlinear equation in dimension 4 was presented, starting with a metric of positive scalar curvature, and a conformally invariant condition for positivity of the Paneitz operator.
Abstract: We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.
TL;DR: In this article, it was shown that the induced Ricci tensor of a totally umbilical light-like submanifold is symmetric if and only if its screen distribution is integrable.
Abstract: This paper provides new results on a class of totally umbilical lightlike submanifolds in semi-Riemannian manifolds of constant curvature. We prove that the induced Ricci tensor of any such submanifold is symmetric if and only if its screen distribution is integrable.
TL;DR: In this article, the flag curvature of a Finsler metric with isotropic S-curvature is studied and the curvature is partially determined when certain non-Riemannian quantities such as Cartan torsion, Landsberg curvature and S-Curvature vanish.
Abstract: The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.
TL;DR: In this paper, the authors considered universal lower bounds on the volume of a Riemannian manifold, given in terms of the volumes of lower dimensional objects (primarily the lengths of geodesics).
Abstract: In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By ‘universal’ we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary, allows the survey to be reasonably short. Although, even in this limited case the authors have left out many interesting results.
TL;DR: In this paper, the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero, which is a generalization of D-dimensional Lorentzian spacetimes, which have been of interest in the context of string theory in curved backgrounds in higher dimensions.
Abstract: We investigate D-dimensional Lorentzian spacetimes in which all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero. These spacetimes are higher-dimensional generalizations of D-dimensional pp-wave spacetimes, which have been of interest recently in the context of string theory in curved backgrounds in higher dimensions.
Posted Content•
TL;DR: In this paper, it was shown that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
Abstract: Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
Posted Content•
TL;DR: A survey of almost-Kahler manifolds with algebraic symmetries can be found in this paper, where the authors survey some recent results and constructions of almost Kahler manifold whose curvature tensors have certain algebraic properties.
Abstract: We survey some recent results and constructions of almost-K\"ahler manifolds whose curvature tensors have certain algebraic symmetries. This is an updated and corrected version of the (to be) published manuscript.
Journal Article•
TL;DR: In this paper, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained in a 3D trans-Sasakian manifold in cases of the manifold being η-Einstein or satisfying R (X, Y) · S = 0.
Abstract: In a 3-dimensional trans-Sasakian manifold, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained. In particular, expressions for Ricci tensor are obtained in a 3-dimensional trans-Sasakian manifold in cases of the manifold being η-Einstein or satisfying R (X, Y) · S = 0.
TL;DR: In this article, the scalar curvature and the boundary mean curvature of the standard half-three sphere were computed using blow-up analysis and minimax arguments, and some existence and compactness results were proved.
Abstract: We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.
TL;DR: In this article, it was shown that for most locally symmetric, non-positively curved Riemannian manifolds M, and for every continuous map f : N → M, the map f is homotopic to a smooth map with Jacobian bounded by a universal constant, depending only on Ricci curvature bounds of N.
Abstract: Let N be any closed, Riemannian manifold. In this paper we prove that, for most locally symmetric, nonpositively curved Riemannian manifolds M, and for every continuous map f : N → M, the map f is homotopic to a smooth map with Jacobian bounded by a universal constant, depending (as it must) only on Ricci curvature bounds of N. From this we deduce an extension of Gromov's Volume Comparison Theorem for negatively curved manifolds to (most) nonpositively curved, locally symmetric manifolds.
TL;DR: In this paper, it was shown that the fundamental group of a compact manifold M n with PIC, n ≥ 5, does not contain a subgroup isomorphic to ⊕.
Abstract: A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. By the results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC; namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds with PIC. An important open problem has been to understand the fundamental groups of manifolds with PIC. In this paper we prove a new result in this direction. We show that the fundamental group of a compact manifold M n with PIC, n ≥ 5, does not contain a subgroup isomorphic to ⊕ . The techniques used involve minimal surfaces.
TL;DR: In this paper, the authors considered spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection, and they introduced an action which is quadratic in curvature and studied the resulting system of Euler-Lagrange equations.
Abstract: We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with metric of a pp-wave and parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely Weyl) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in Minkowski space. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the graviton or the neutrino.
TL;DR: In this article, it was shown that a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional unit sphere Sm+1 without umbilics is Mobius equivalent to the image of the standard conformal map of a (m + 1)-dimensional hyperbolic space Hm+ 1 in the Euclidean space.
Abstract: Let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface in the (m+1)-dimensional unit sphere Sm+1 without umbilics. Four basic invariants of x under the Mobius transformation group in Sm+1 are a Riemannian metric g called Mobius metric, a 1-form Φ called Mobius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Mobius second fundamental form. In this paper, we prove the following classification theorem: let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface, which satisfies (i) Φ≡0, (ii) A+λg+μB≡0 for some functions λ and μ, then λ and μ must be constant, and x is Mobius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in Sm+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space Rm+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space Hm+1. This result shows that one can use Mobius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in Sm+1, Rm+1 and Hm+1.
TL;DR: In this article, the authors defined the total scalar curve plus total mean curve functional on the space of Riemannian metrics of a smooth compact manifold with boundary and characterized its critical points.
Abstract: The Total Scalar Curvature plus Total Mean Curvature functional is defined on the space of Riemannian metrics of a smooth compact manifold with boundary. We characterize its critical points restricted to spaces of Riemannian metrics satisfying various volume and area constraints, when the dimension of the manifold is n ≥ 3. In addition, we compute the second variation of said functional at critical points and exhibit directions in which it is positive, negative or zero. These results generalize to manifolds with boundary, well known results that hold in the case of manifolds without boundary.
TL;DR: In this paper, a precise correspondence between freely-acting orbifolds (Scherk-Schwarz compactifications) and string vacua with NSNS flux turned on is established using T-duality.
Abstract: A precise correspondence between freely-acting orbifolds (Scherk-Schwarz compactifications) and string vacua with NSNS flux turned on is established using T-duality.
We focus our attention to a certain non-compact Z_2 heterotic freely-acting orbifold with N=2 supersymmetry (SUSY). The geometric properties of the T-dual background are studied. As expected, the space is non-Kahler with the most generic torsion compatible with SUSY. All equations of motion are satisfied, except the Bianchi identity for the NSNS field, that is satisfied only at leading order in derivatives, i.e. without the curvature term. We point out that this is due to unknown corrections to the standard heterotic T-duality rules.
TL;DR: For a given positive measure μ on R n, the authors in this article consider integral functionals of the kind F(u)= ∫ R n f(x, ∇ u ∈ C 0 ∞ (R n ), and study their relaxation with respect to the L μ p topology, p being the growth exponent of f.
TL;DR: For a two-surface B tending to an infinite-radius round sphere at spatial infinity, this paper showed that the Brown-York boundary integral HB belongs to the energy sector of the gravitational Hamiltonian.
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TL;DR: In this paper, it was shown that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect, and this was generalized to show that in manifolds with positive curvature, in the integral sense, two minimal surfaces must be close to each other.
Abstract: In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci cur- vature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.
TL;DR: In this paper, the Ricci tensor of a slant submanifold in a complex space form is estimated in terms of the main extrinsic invariant, namely the squared mean curvature.
Abstract: B.-Y. 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. The Lagrangian version of this inequality was proved by the same author.
ewline In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold $M$ in a complex space form $\widetilde M(4c)$, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, $M$ is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal.