Showing papers on "Riemann curvature tensor published in 1999"
TL;DR: In this article, the entropy of extremal black holes arising from terms quadratic in the Riemann tensor in N = 2, D = 4 supergravity theories was determined.
459 citations
TL;DR: In this article, the cosmological constant and the gravitational constant of topological black holes are reduced to two independent parameters, i.e., cosmologically constant and gravitational constant.
Abstract: We investigate topological black holes in a special class of Lovelock gravity. In odd dimensions, the action is the Chern-Simons form for the anti--de Sitter group. In even dimensions, it is the Euler density constructed with the Lorentz part of the anti--de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: the cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero, or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences due to the different curvatures of event horizons.
230 citations
TL;DR: In this paper, the notion of k-Ricci curvature of a Riemannian n-manifold was defined and sharp relations between the k-ricci curvatures and the shape operator were established.
Abstract: First we define the notion of k-Ricci curvature of a Riemannian n- manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean cur- vature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.
168 citations
TL;DR: The conformal normal curvature as mentioned in this paper provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness, and has been used to assess local influence of minor perturbations of statistical models.
Abstract: In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.
164 citations
137 citations
134 citations
TL;DR: In this article, the authors prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curve curvature, namely the one-sided affinity for curvature decreasing deformations.
Abstract: Scalar curvature is the simplest generalization of Gaussian curvature to higher dimensions However there are many questions open with regard to its relation to other geometric quantities and topology Here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature, namely the one-sided affinity for curvature decreasing deformations The first one is concerned with some prescribed decrease of the scalar curvature Scal(g) of some Riemannian metric g on a given manifoldMn of dimension≥ 3 We denote the e−neighborhood of some set U with respect to g by Ue
100 citations
TL;DR: In this paper, Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established.
Abstract: Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below
93 citations
TL;DR: This paper studies the convergence of general threshold dynamics type approx- imation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor to study the mean curvature evolution.
Abstract: We study the convergence of general threshold dynamics type approx- imation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature. In this paper we study the convergence of general threshold dynamics type ap- proximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. These schemes are generalizations and extensions of the threshold dynamics models introduced by Gravner and Grieath (GrGr) to study cellular automaton modeling of excitable media and by Bence, Merriman and Osher (BMO) to study the mean curvature evolution. Cellular automaton models are mathematical models used to understand the transmission of periodic waves through environments such as a network or a tissue. A common feature of many such models is that some threshold level of excitation must occur in a neighborhood of a location to become excited and conduct a pulse. Typical physical systems which exhibit such phenomenology are, among others, neural networks, cardiac muscle, Belousov-Zhabotinsky oscillating chemical reaction, etc. Interfaces (fronts, hypersurfaces) in R N evolving with normal velocity V ¼ vðDn;nÞ; ð0:1Þ
80 citations
TL;DR: In this article, the authors developed an analog of Riemannian geometry on finite and discrete sets, which is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set).
Abstract: Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set). Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (nonlocal) tensor product over the algebra of functions, as considered previously by several authors. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is, in general, nothing like a Ricci tensor or a curvature scalar. Because of the nonlocality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather nonlocal objects. But one can make use of the parallel transport associated with a connection to “localize” such objects, and in certain cases there is a distinguished way to achieve this. In particular, this leads to covariant components of the curvature tensor which allow a contraction to a Ricci tensor. Several examples are worked out to illustrate the procedure. Furthermore, in the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations.
74 citations
TL;DR: In this paper, it was shown that if the conformal class of the boundary contains a metric of positive scalar curvature, then the fundamental group of a Riemannian Einstein manifold is bounded by a conformal boundary.
Abstract: In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry.
TL;DR: In this paper, it was shown that a complete, complete, asymptotically at manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, and that the manifold must be close to (IR 3, δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure.
Abstract: The Positive Mass Theorem implies that any smooth, complete, asymptotically flat3-manifold with non-negative scalar curvature which has zero total mass is isometricto (IR 3 ,δ ij ). In this paper, we quantify this statement using spinors and prove thatif a complete, asymptotically flat manifold with non-negative scalar curvature hassmall mass and bounded isoperimetric constant, then the manifold must be close to(IR 3 ,δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure. This curvatureestimate allows us to extend the case of equality of the Positive Mass Theorem toinclude non-smooth manifolds with generalized non-negative scalar curvature, whichwe define. 1 Introduction We introduce our problem in the context of General Relativity. Consider a 3 + 1 dimen-sional Lorentzian manifold N with metric g αβ of signature (− + ++). We denote theinduced Levi-Civita connection by ∇¯. Then the corresponding Ricci tensor R¯
TL;DR: In this article, the authors developed an analogue of S-duality for linearized gravity in (3+1)-dimensions, and showed that strong-weak coupling duality is an exact symmetry and implies small-large duality for the cosmological constant.
TL;DR: In this paper, it was shown that the unit tangent bundle of S 4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere.
Abstract: We show that the unit tangent bundle of S 4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature.
TL;DR: In this article, the authors consider semi-regular metrics whose curvature tensor makes sense as a distribution and generalize their definition to form a wider class of regular metrics, which they call regular metrics.
Abstract: This paper considers metrics whose curvature tensor makes sense as a distribution. A class of such metrics, the regular metrics, was defined and studied by Geroch and Traschen. Here, we generalize their definition to form a wider class: semi-regular metrics. We then examine in detail two metrics that are semi-regular but not regular: (a) Minkowski spacetime minus a wedge and (b) a certain travelling wave metric.
TL;DR: In this paper, a lower bound for the first eigenvalue for compact manifolds with positive Ricci curvature is given. But the lower bound is not applicable to compact manifold with boundary, since the boundary is of nonnegative second fundamental form.
Abstract: We present some new lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature in terms of the diameter and the lower Ricci curvature bound of the manifolds. For compact manifolds with boundary, it is assumed that, with respect to the outward normal, it is of nonnegative second fundamental form for the first Neumann eigenvalue and the mean curvature of the boundary is nonnegative for the first Dirichlet eigenvalue.
TL;DR: In this article, it was shown that a compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature.
Abstract: In this paper we study a class of K-contact manifolds, namely φ-conformally flat K-contact manifolds and we show that a compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature.
01 Jan 1999
TL;DR: The idea that the Einstein action should be modified by the addition of interactions involving higher powers of the Riemann curvature tensor has a long history stretching back to the early days of general relativity as discussed by the authors.
Abstract: The idea that the Einstein action should be modified by the addition of interactions involving higher powers of the Riemann curvature tensor has a long history stretching back to the early days of general relativity. Such higher curvature theories originally appeared in proposals by Weyl and Eddington for a geometric unification of electromagnetism and gravity [1]. Much later, interest arose in higher derivative theories of gravity because they provided renormalizable quantum field theories [2, 3]. Unfortunately, the new massive spin-two excitations, which tame the ultraviolet divergences in such theories, result in the instability of the classical theory [4] and the loss of unitarity in the quantum theory [3, 5]. While higher curvature theories have thus proven inadequate as the foundation of quantum gravity, they still have a role to play within the modern paradigm of effective field theories [6].
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TL;DR: In this article, it was shown that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to (R^3,delta_{ij}), in the sense that there is an upper bound for the L^2 norm of the Riemannian curvature tensor over the manifold except for a set of small measure.
Abstract: The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to (R^3,delta_{ij}), in the sense that there is an upper bound for the L^2 norm of the Riemannian curvature tensor over the manifold except for a set of small measure. This curvature estimate allows us to extend the case of equality of the Positive Mass Theorem to include non-smooth manifolds with generalized non-negative scalar curvature, which we define.
TL;DR: In this article, the authors studied three and four-dimensional Riemannian manifolds with Ricci-curvature homogeneous eigenvalues, that is, having constant Ricci eigen values.
Abstract: The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podesta and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).
TL;DR: In this article, the authors studied the topological structure of all the limit spaces of the class of closed 2-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature and proved that these manifolds are precompact with respect to the Gromov-Hausdorff distance.
Abstract: We study the class of closed 2-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally 2-connected. We also study the limit of 2-manifolds with Lp-curvature bound for p ≥ 1.
TL;DR: In this article, the duality principle for the associated curvature tensor R of a pointwise Osserman Riemannian manifold was shown to hold for any manifold M = 0.
TL;DR: In this paper, Luck and Vishik proved a formula relating analytic torsion and Reidemeister torsions on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary.
Abstract: We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Lu\"ck and S. M. Vishik. We find that the extra term that comes in here in the nonproduct case is the transgression of the Euler class in the even dimensional case and a slightly more mysterious term involving the second fundamental form of the boundary and the curvature tensor of the manifold in the odd dimensional case.
TL;DR: In this paper, the authors consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure.
TL;DR: In this article, the authors studied complete n-dimensional Riemannian manifolds with nonnegative Ricci curvatures and large volume growth and proved that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in the n-dimensions.
Abstract: In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space \( R^n \) if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in \( R^n \).
TL;DR: In this article, the authors characterize two particular closed Finsler spaces in terms of the flag curvature tensor and give a unifying description for three special closed FINsler manifolds.
Abstract: In this article, we characterize two particular closed Finsler spaces in terms of the flag curvature tensor. That is, with respect to the \({\cal L}_{2}\)inner product of the space of all symmetric 2-tensor on the projective sphere bundle, a Finsler structure has Landsberg (Berwald resp.) type if and only if its flag curvature tensor is orthogonal to some given line (plane resp.). Using a result of Akbar-Zadeh we give a unifying description for three special closed Finsler manifolds.
TL;DR: In this paper, the authors construct the explicit form of three almost-complex structures that a Riemannian manifold with self-dual curvature admits and show that their Nijenhuis tensors vanish so that they are integrable.
Abstract: We construct the explicit form of three almost-complex structures that a Riemannian manifold with self-dual curvature admits and show that their Nijenhuis tensors vanish so that they are integrable. This proves that gravitational instantons with self-dual curvature admit hyper-Kahler structure. In order to arrive at the three vector-valued 1-forms defining almost-complex structure, we give a spinor description of real four-dimensional Riemannian manifolds with Euclidean signature in terms of two independent sets of two-component spinors. This is a version of the original Newman-Penrose formalism that is appropriate to the discussion of the mathematical, as well as physical properties of gravitational instantons. We shall build on the work of Goldblatt who first developed an NP formalism for gravitational instantons but we shall adopt it to differential forms in the NP basis to make the formalism much more compact. We shall show that the spin coefficients, connection 1-form, curvature 2-form, Ricci and Bianchi identities, as well as the Maxwell equations naturally split up into their self-dual and anti-self-dual parts corresponding to the two independent spin frames. We shall give the complex dyad as well as the spinor formulation of the almost-complex structures and show that they reappear under the guise of a triad basis for the Petrov classification of gravitational instantons. Completing the work of Salamon on hyper-Kahler structure, we show that the vanishing of the Nijenhuis tensor for all three almost-complex structures depends on the choice of a self-dual gauge for the connection which is guaranteed by virtue of the fact that the curvature 2-form is self-dual for gravitational instantons.