Showing papers on "Riemann curvature tensor published in 2001"

Parallel spinors and connections with skew-symmetric torsion in string theory

Abstract: We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇-parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5,6 and 7.

Differential Geometry of Gerbes

Abstract: We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J.- L. Brylinski and N. Hitchin. We give a global definition of the 3-curvature of this connective structure as a 3-form on X with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebra-valued 3-forms. The Bianchi identity, which the curvature of a connection on a principal bundle satisfies, is replaced here by a more elaborate equation.

Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

Abstract: Algebraic curvature tensors the skew-symmetric curvature operator the Jacobi operator controlling the eigenvalue structure.

Combinatorial curvature for planar graphs

Abstract: Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001

Prescribing scalar curvature on Sn

Abstract: on S for n ≥ 3. In the case R is rotationally symmetric, the well-known Kazdan-Warner condition implies that a necessary condition for (1) to have a solution is: R > 0 somewhere and R′(r) changes signs. Then, (a) is this a sufficient condition? (b) If not, what are the necessary and sufficient conditions? These have been open problems for decades. In Chen & Li, 1995, we gave question (a) a negative answer. We showed that a necessary condition for (1) to have a solution is: R′(r) changes signs in the region where R is positive. (2)

Degeneration of Riemannian metrics under Ricci curvature bounds

Abstract: These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby Colding and myself, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the "non-collapsing" situation. A particularly interesting case is that in which the manifolds in question are Einstein (or Kahler-Einstein). Thus, the theory provides information on the manner in which Einstein metrics can degenerate.

D = 10 super-Yang-Mills at O(α′2)

Abstract: Using superspace techniques, the complete and most general action of D = 10 super--Yang-Mills theory is constructed at the α'2 level. No other approximations, e.g., keeping only a subset of the allowed derivative terms, are used. The Lorentz structure of the α'2 corrections is completely determined, while (depending on the gauge group) there is some freedom in the adjoint structure, which is given by a totally symmetric four-index tensor. We examine the second, non-linearly realised supersymmetry that may be present when the gauge group has a U(1) factor, and find that the constraints from linear and non-linear supersymmetry to a large extent coincide. However, the additional restrictions on the adjoint structure of the order α'2 interactions following from the requirement of non-linear supersymmetry do not completely specify the symmetrised trace prescription.

Poisson Equation, Poincaré-Lelong Equation and Curvature Decay on Complete Kähler Manifolds

Abstract: In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kahler manifolds. In particular, the Poincare-Lelong equation on complete noncompact Kahler manifolds with nonnegative holomorphic bisectional curvature is studied. Several applications are then derived, which include the Steinness of the complete Kahler manifolds with nonnegative curvature and the flatness of a class of complete Kahler manifolds satisfying a curvature pinching condition. Liouville type results for plurisubharmonic functions are also obtained.

Radion on the de Sitter Brane

Abstract: The radion on the de Sitter brane is investigated at the linear perturbation level, using the covariant curvature tensor formalism developed by Shiromizu, Maeda and Sasaki. 1) It is found that if there is only one de Sitter brane with positive tension, there is no radion, and thus ordinary Einstein gravity is recovered on the brane, with the exception of corrections due to the massive Kaluza-Klein modes. As a by-product of the covariant curvature tensor formalism, it is immediately seen that cosmological scalar, vector and tensor type perturbations all have the same Kaluza-Klein spectrum. On the other hand, if there are two branes, one with positive tension and another with negative tension, the gravity on each brane receives corrections from the radion mode in addition to the Kaluza-Klein modes, and the radion is found to have a negative mass-squared proportional to the curvature of the de Sitter brane. This is in contrast to the flat brane case, in which the radion mass vanishes and becomes degenerate with the 4-dimensional graviton modes. To relate our result with the metric perturbation approach, we derive the second-order action for the brane displacement. We find that the radion identified in our approach indeed corresponds to the relative displacement of the branes in the Randall-Sundrum gauge and describes the scalar curvature perturbations of the branes in Gaussian normal coordinates around the branes. The implications of our results with regard to the inflationary brane universe are briefly discussed.

Generalized pseudo-Riemannian geometry

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.

Torsion waves in metric-affine field theory

Abstract: The approach of metric-affine field theory is to define spacetime as a real oriented 4-manifold equipped with a metric and an affine connection. The 10 independent components of the metric tensor and the 64 connection coefficients are the unknowns of the theory. We write the Yang-Mills action for the affine connection and vary it both with respect to the metric and the connection. We find a family of spacetimes which are stationary points. These spacetimes are waves of torsion in Minkowski space. We then find a special subfamily of spacetimes with zero Ricci curvature; the latter condition is the Einstein equation describing the absence of sources of gravitation. A detailed examination of this special subfamily suggests the possibility of using it to model the neutrino. Our model naturally contains only two distinct types of particles which may be identified with left-handed neutrinos and right-handed antineutrinos.

On Higher Eta-Invariants and Metrics of Positive Scalar Curvature

Abstract: Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving π1(N) and dim N ,f orN to admit an infinite number of metrics of positive scalar curvature that are nonbordant.

HOLOGRAPHIC ENTROPY AND BRANE FRW DYNAMICS FROM AdS BLACK HOLE IN d5 HIGHER DERIVATIVE GRAVITY

Abstract: Higher derivative bulk gravity (without Riemann tensor square term) admits AdS–Schwarzschild black hole as an exact solution. It is shown that induced brane geometry on such background is open, flat or closed FRW radiation dominated universe. Higher derivative terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of five-dimensional gravitational constant and AdS scale parameter. These higher derivative terms do not destroy the AdS-dual description of radiation represented by strongly-coupled CFT. The Cardy–Verlinde formula which expresses cosmological entropy as the square root from other parameters and entropies is derived in R2 gravity. The corresponding cosmological entropy bounds are briefly discussed.

Dimensionally Dependent Tensor Identities by Double Antisymmetrisation

Abstract: Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p >= n$. We generalise Lovelock's results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrising over n+1 indices, we establish a very general 'master' identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner.

Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions.

Abstract: Constraints are found on the spatial variation of finite-time Lyapunov exponents of two- and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection–diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current.

Hypersurfaces with mean curvature given by an ambient sobolev function

Abstract: We consider n-hypersurfaces Σj with interior Ej whose mean curvature are given by the trace of an ambient Sobolev function uj ∈ W 1,p (R n+1 )

Curvature estimates in asymptotically flat manifolds of positive scalar curvature

Abstract: We consider an asymptotically flat Riemannian spin manifold of positive scalarcurvature. An inequality is derived which bounds the Riemann tensor in terms of thetotal mass and quantifies in which sense curvature must become small when the totalmass tends to zero. 1 Introduction Suppose that (M n ,g) is an asymptotically flat Riemannian spin manifold of positive scalarcurvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifoldis always positive, and is zero if and only if the manifold is flat. This result suggests thatthere should be an inequality which bounds the Riemann tensor in terms of the total massand implies that curvature must become small when the total mass tends to zero. In [4]such curvature estimates were derived in the context of General Relativity for 3-manifoldsbeing hypersurfaces in a Lorentzian manifold. In the present paper, we study the problemmore generally on a Riemannian manifold of dimension n≥ 3. Our curvature estimatesthen give a quantitative relation between the local geometry and global properties of themanifold.The main difficulty in higher dimensions is to bound the Weyl tensor (which for n= 3vanishes identically). Our basic strategy for controlling the Weyl tensor can be understoodfrom the following simple consideration. The existence of a parallel spinor in an open setU⊂ Mimplies that the manifold is Ricci flat in U. Thus it is reasonable that by gettingsuitable estimates for the derivatives of a spinor, one can bound all components of theRicci tensor. This method is used in [4], where a solution of the Dirac equation is analyzedusing the Weitzenbo¨ck formula. But the local existence of a parallel spinor does not implythat the Weyl tensor vanishes. This is the underlying reason why in dimension n>3,our estimates cannot be obtained by looking at one spinor, but we must consider a family(ψ

Lower Bounds for the Eigenvalues of the Dirac Operator: Part I. The Hypersurface Dirac Operator

Abstract: We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.

Topological obstructions to nonnegative curvature

Abstract: We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

Isometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant curvature

Abstract: This survey contains results on local and global isometric immersions of two-dimensional and multidimensional Riemannian and pseudo-Riemannian space forms into spaces of constant sectional curvature.

On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature

Abstract: This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper are the following. Theorem I: If M has positive Ricci curvature then it has the loops to infinity property. Theorem II: If M has nonnegative Ricci curvature then it either has the loops to infinity property or it is isometric to a flat normal bundle over a compact totally geodesic submanifold and its double cover is split. Theorem III: Let M be a complete riemannian manifold with the loops to infinity property along some ray starting at a point, p. Let D containing p be a precompact region with smooth boundary and S be any connected component of the boundary containing a point, q, on the ray. Then the map from the fundamental group of S based at q to the fundamental group of Cl(D) based at p induced by the inclusion map is onto.

The curvature invariant of a non-commutingn-tuple

Abstract: Non-commutative versions of Arveson's curvature invariant and Euler characteristic for a commutingn-tuple of operators are introduced. The non-commutative curvature invariant is sensitive enough to determine if ann-tuple is free. In general both invariants can be thought of as measuring the freeness or curvature of ann-tuple. The connection with dilation theory provides motivation and exhibits relationships between the invariants. A new class of examples is used to illustrate the differences encountered in the non-commutative setting and obtain information on the ranges of the invariants. The curvature invariant is also shown to be upper semi-continuous.