Showing papers on "Riemann curvature tensor published in 2000"

On the structure of spaces with Ricci curvature bounded below. I

Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of

Gravity, stability, and energy conservation on the Randall-Sundrum brane world

Abstract: We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [L. Randall and R. Sundrum, Phys. Rev. Lett. $83,$ 4690 (1999)], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the ``electric'' part of the five-dimensional Weyl tensor, denoted by ${E}_{\ensuremath{\mu}\ensuremath{ u}},$ gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the five-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behavior in the direction of the fifth dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane world. We also obtain the asymptotic behavior of ${E}_{\ensuremath{\mu}\ensuremath{ u}}$ on the brane at spatial infinity, assuming that the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned.

The Classification of Spaces Defining Gravitational Fields

Abstract: In this paper written in 1954 Alexei Petrov describes his famous classification of spaces according to the algebraical structure of the curvature tensor, that determines the classes of the gravitational fields permitted therein. Now this classification of spaces (and, respectively, of the gravitational fields) is known as Petrov’s classification. This paper was originally published, in Russian, in Scientific Transactions of Kazan State University: Petrov A. Z. Klassifikazija prostranstv, opredelajuschikh polja tjagotenia. Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta, 1954, vol. 114, book 8, pages 55–69. Translated from Russian in 2008 by Vladimir Yershov, England–Pulkovo. In this paper, the detailed proof of results obtained and published by the author earlier in 1951 [1]. Namely, it is shown that by examining the algebraic structure of the curvature tensor V4 one can establish a classification of the gravitational fields defined by this tensor and given in the form ds = gij dx dx , (1) with the fundamental tensor satisfying the field equations

Derivative corrections to D-brane actions with constant background fields

Abstract: We study derivative corrections to the effective action for a single D-brane in type II superstring theory coupled to constant background fields. In particular, within this setting we determine the complete expression for the (disk level) four-derivative corrections to the Born-Infeld part of the action. We also determine 2n-form 2n-derivative corrections to the Wess-Zumino term. Both types of corrections involve all orders of the gauge field strength, F. The results are obtained via string sigma-model loop calculations using the boundary state operator language. The corrections can be succinctly written in terms of the Riemann tensor for a non-symmetric metric.

Collapsing Three-Manifolds Under a Lower Curvature Bound

Abstract: The purpose of this paper is to completely characterize the topology of three-dimensional Riemannian manifolds with a uniform lower bound of sectional curvature which converges to a metric space of lower dimension.

Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory

Abstract: We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics.

Complete Riemannian manifolds with pointwise pinched curvature

Abstract: An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M n . We prove that if n≥4 and the curvature operator of M n is pointwise pinched, or if n=3 and the Ricci curvature of M 3 is pointwise pinched, then M n is compact.

Construction of boundary invariants and the logarithmic singularity of the Bergman kernel

Abstract: in which case the coefficients aj are expressed, by the Weyl invariant theory, in terms of the Riemannian curvature tensor and its covariant derivatives. The Bergman kernel’s counterpart of the time variable t is a defining function r of the domain Ω. By [F1] and [BS], the formal singularity of K at a boundary point p is uniquely determined by the Taylor expansion of r at p. Thus one has hope of expressing φ modulo On+1(r) and ψ modulo O∞(r) in terms of local biholomorphic invariants of the boundary, provided r is appropriately chosen. In [F3], Fefferman proposed to find such expressions by reducing the problem to an algebraic one in invariant theory associated with CR geometry, and indeed expressed φ modulo On−19(r) invariantly by solving the reduced problem partially. The solution in [F3] was then completed in [BEG] to give a full invariant expression of φ modulo On+1(r), but the reduction is still

On fundamental groups of manifolds of nonnegative curvature

Abstract: We will characterize the fundamental groups of compact manifolds of (almost) nonnegative Ricci curvature and also the fundamental groups of manifolds that admit bounded curvature collapses to manifolds of nonnegative sectional curvature. Actually it turns out that the known necessary conditions on these groups are sufficient as well. Furthermore, we reduce the Milnor problem—are the fundamental groups of open manifolds of nonnegative Ricci curvature finitely generated?—to manifolds with abelian fundamental groups. Moreover, we prove for each positive integer n that there are only finitely many non-cyclic, finite, simple groups acting effectively on some complete n -manifold of nonnegative Ricci curvature. Finally, sharping a result of Cheeger and Gromoll [6], we show for a compact Riemannian manifold (M,g 0 ) of nonnegative Ricci curvature that there is a continuous family of metrics (g λ ),λ∈[0,1] such that the universal covering spaces of (M,g λ ) are mutually isometric and (M,g 1 ) is finitely covered by a Riemannian product N×T d , where T d is a torus and N is simply connected.

Noncollapsing examples with positive Ricci curvature and infinite topological type

Abstract: Abstract. ((Without Abstract)).

Hypersurfaces of prescribed curvature in Lorentzian manifolds

Abstract: The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved, provided there are barriers. . 0 INTRODUCTION Consider the problem of finding a closed hypersurface of prescribed curvature F in a complete (n+1)-dimensional manifold N. To be more precise, let Ω be a connected open subset of N, f ∈ C2,α(Ω̄), F a smooth, symmetric function defined in an open cone Γ ⊂ Rn, then we look for a hypersurface M ⊂ Ω such that F|M = f(x) for all x ∈ M, (0.1) where F|M means that F is evaluated at the vector (κi(x)) the components of which are the principal curvatures of M. The prescribed function f should satisfy natural structural conditions, e.g., if Γ is the positive cone and the hypersurface M is supposed to be convex, then f should be positive, but no further, merely technical, conditions should be imposed. If N is a Riemannian manifold, then the problem has been solved in the case when F = H, the mean curvature, where in addition n had to be small, and N conformally flat, cf. [7], and for curvature functions F of class (K), no restrictions on n, cf. [4, 6]. We also refer to [5], where more special situations are considered, and the bibliography therein. 1125 Indiana University Mathematics Journal c ©, Vol. 49, No. 3 (2000)

Three-Dimensional Lorentz Metrics and Curvature Homogeneity of Order One

Abstract: In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k ∈ N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.

Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups

Abstract: In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.

Torsion-free path geometries and integrable second order ODE systems

Abstract: A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segre varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segre varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segre varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces.

Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

Abstract: It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8.

Rigidity of Einstein 4-manifolds with positive curvature

Abstract: An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥e0, e0≡( -23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S 4, RP 4 with constant sectional curvature K=1/3, or CP 2 with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S 4 and CP 2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S 4, RP 4, or CP 2 but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions.

Generalized scalar curvature type equations on compact Riemannian manifolds

Abstract: The paper is concerned with nonlinear equations of critical Sobolev growth involving the p-Laplace operator. These equations generalize the more classical scalar curvature equation.

Almost Kähler 4-manifolds with J-invariant ricci tensor and special weyl tensor

Abstract: We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.

An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci $\ge 0$

Abstract: By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades.

Manifolds with quadratic curvature decay and slow volume growth

Abstract: We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

The geometry of the Barbour-Bertotti theories: I. The reduction process

Abstract: The dynamics of N 3 interacting particles is investigated in the non-relativistic context of the Barbour-Bertotti theories. The reduction process on this constrained system yields a Lagrangian in the form of a Riemannian line element. The involved metric, degenerate in the flat configuration space, is the first fundamental form of the space of orbits of translations and rotations (the Leibniz group). The Riemann tensor and the scalar curvature are computed using a generalized Gauss formula in terms of the vorticity tensors of generators of the rotations. The curvature scalar is further given in terms of the principal moments of inertia of the system. Line configurations are singular for N 3. A comparison with similar methods in molecular dynamics is traced.

On the Uniqueness of Curvature Features 1 2

Abstract: Curved image features like corners are good features to track and also the onlyimage regions where motion can be estimated reliably. In addition, such featurescontain higher amounts of information than straight edges and uniform regions [7, 2].Here we extend previous results related to the uniqueness of curvature features [2] andproof that curved image regions contain all the information in an image. The proof isconstructive and we present first reconstruction results. Finally, an extension of theproof for the case of image sequences and 3D curvature is outlined. 1 Introduction It is well known that curvature features like corners and junctions are important for humanand computer vision, see e.g. [1, 7], and the limits of linear models for dealing with suchphenomena have been shown [6]. It has also been argued that differential geometry canprovide a unified framework for the processing of both motion and form [4, 3]. Sucha framework reveals subtle relations between orientation, endstopping, form and motionwhich are otherwise studied in isolation. Indeed, curved regions of the image are the onlypoints where the optical flow can be estimated reliably, and it has been shown that underthe assumption of slowly varying velocities, the flow field can be computed from the ratioof the curvature tensor components, and that, in addition, the geometric approach canexplain some further properties of motion-selective neurons in the area MT [3]. Thus,differential geometry can provide an elegant and well founded theory for mid-level vision.In this work, we study the uniqueness of curvature features and proof that imageregions that are not curved are redundant. Due to this result, curvature becomes anoutstanding nonlinear image property. Since curvature is unique, other image regions thatare not curved are not needed for further processing. In some cases, however, it may beof interest to recover images from curvature, e.g., for image compression. The inversion ofcurvature operators remains a difficult problem, but we present preliminary reconstructionresults. Further results that give approximate reconstruction had been presented in [2].Our results are of particular interest for cortical models since they show that end-stopped neurons can provide a complete representation of the visual input.

Volume growth and positive scalar curvature

Abstract: Abstract. ((Without Abstract)).

Review of some classical gravitational superenergy tensors using computational techniques

Abstract: We use computational algorithms recently developed by ourselves to study completely four index divergence-free quadratic in Riemann tensor polynomials in GR. Some results are new and others reproduce and/or correct known ones. The algorithms are part of a Mathematica package called Tools of Tensor Calculus (TTC).

The geometry of the Barbour-Bertotti theories: II. The three-body problem

Abstract: We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analysed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for N 3, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there.