Showing papers on "Riemann curvature tensor published in 2008"
TL;DR: In this article, electron backscattering diffraction was used to identify six components of the lattice curvature tensor, and improved lower bounds for the geometrically necessary dislocation content were obtained by linear optimization.
569 citations
TL;DR: In this article, the Ricci flow deforms a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures to a constant curvature metric.
Abstract: Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.
330 citations
TL;DR: The Ricci flow was introduced by Hamilton in 1982 as discussed by the authors in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form.
Abstract: The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following
286 citations
TL;DR: In this article, the authors proposed a Weyl-covariant formalism for the entropy current of an arbitrary conformal fluid in any spacetime (with 3$>d>3).
Abstract: In recent work [1, 2], the energy-momentum tensor for the = 4 SYM fluid was computed up to second derivative terms using holographic methods. The aim of this note is to propose an entropy current (accurate up to second derivative terms) consistent with this energy-momentum tensor and to explicate its relation with the existing theories of relativistic hydrodynamics. In order to achieve this, we first develop a Weyl-covariant formalism which simplifies the study of conformal hydrodynamics. This naturally leads us to a proposal for the entropy current of an arbitrary conformal fluid in any spacetime (with 3$>d>3). In particular, this proposal translates into a definite expression for the entropy flux in the case of = 4 SYM fluid. We conclude this note by comparing the formalism presented here with the conventional Israel-Stewart formalism.
213 citations
TL;DR: In this article, the equivalence of the Palatini and metric formulations at the level of equations of motion, which is a manifestation of the casuality based on Einstein relativity, is a physical criterion that restricts form of Lagrangians of modified gravity theories.
126 citations
TL;DR: In this paper, the authors compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives.
Abstract: We compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives. We find that there is a class of theories for which the two formalisms are equivalent. This class contains the Palatini version of Lovelock theory, but also more Lagrangians that are not Lovelock, but respect certain symmetries. For the general case, we find that imposing the Levi-Civita connection as an ansatz, the Palatini formalism is contained within the metric formalism, in the sense that any solution of the former also appears as a solution of the latter, but not necessarily the other way around. Finally we give the conditions the solutions of the metric equations should satisfy in order to solve the Palatini equations.
123 citations
TL;DR: The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ⋅ 10 23 objects with up to 12 derivatives of the metric.
90 citations
TL;DR: In this paper, the authors presented balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with nonzero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection.
Abstract: We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold $M_3$ satisfy the heterotic equations of motion.
89 citations
TL;DR: In this paper, the authors investigated solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor.
Abstract: We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor Tμν(gαβ, ∂τgαβ, ∂τ∂σgαβ, ...,) constructed from sums of terms, the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called universal if, when evaluated on that Einstein metric, Tμν is a multiple of the metric. A Ricci flat classical solution is called strongly universal if, when evaluated on that Ricci flat metric, Tμν vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalization; Einstein metrics with holonomy Sim(n − 2) in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalized Ghanam–Thompson metric is weakly universal and that the Goldberg–Kerr metric is strongly universal; indeed, we show that universality extends to all four-dimensional Sim(2) Einstein metrics. We also discuss generalizations to higher dimensions.
88 citations
TL;DR: In this paper, it was shown that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.
Abstract: Consider {(M n , g(t)), 0 ⩽ t < T < ∞} as an unnormalized Ricci flow solution: for t ∈ [0, T). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T) then the solution can be extended over T. Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.
82 citations
TL;DR: In this article, it was shown that all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes) in three dimensions.
Abstract: In this paper we study Lorentzian spacetimes for which all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes) in three dimensions. We determine all such CSI metrics explicitly, and show that for every CSI with particular constant invariants there is a locally homogeneous spacetime with precisely the same constant invariants. We prove that a three-dimensional CSI spacetime is either (i) locally homogeneous or (ii) it is locally a Kundt spacetime. Moreover, we show that there exists a null frame in which the Riemann (Ricci) tensor and its derivatives are of boost order zero with constant boost weight zero components at each order. Lastly, these spacetimes can be explicitly constructed from locally homogeneous spacetimes and vanishing scalar invariant spacetimes.
01 Jan 2008
TL;DR: Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas as mentioned in this paper, and it is completely parallel to conformal differential geometry, and there are direct applications within Riemannian differential geometry.
Abstract: Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ‘Bernstein-Gelfand-Gelfand machinery’. As such, it is completely parallel to conformal differential geometry. On the other hand, there are direct applications within Riemannian differential geometry. We shall soon see, for example, a good geometric reason why the symmetries of the Riemann curvature tensor constitute an irreducible representation of SL(n,ℝ) (rather than SO(n) as one might naively expect). Projective differential geometry also provides the simplest setting in which overdetermined systems of partial differential equations naturally arise.
TL;DR: In this article, the authors considered the stress energy tensor associated to the bienergy functional and showed that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four.
Abstract: Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map.
TL;DR: In this paper, a non-Riemannian quantity is closely related to the flag curvature, and it is shown that the flag's curvature is weakly isotropic if and only if this non-riemannians quantity takes a special form.
Abstract: In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.
Book•
01 Dec 2008
TL;DR: In this paper, Christoffel's Three-Index Symbols and Riemann's Curvature of a Curve have been used to represent the three-indexes of a curve.
Abstract: 1. Some Preliminaries 2. Coordinates, Vectors , Tensors 3. Riemannian Metric 4. Christoffel's Three-Index Symbols. Covariant Differentiation 5. Curvature of a Curve. Geodeics, Parallelism of Vectors 6. Congruences and Orthogonal Ennuples 7. Riemann Symbols. Curvature of a Riemannian Space 8. Hypersurfaces 9. Hypersurfaces in Euclidean Space. Spaces of Constant Curvature 10. Subspaces of a Riemannian Space.
TL;DR: In this article, it was shown that the Kahler-Ricci flow on a manifold with positive first Chern class converges to a Kahler−Einstein metric assuming positive bisectional curvature and certain stability conditions.
Abstract: We show that the Kahler–Ricci flow on a manifold with positive first Chern class converges to a Kahler–Einstein metric assuming positive bisectional curvature and certain stability conditions.
TL;DR: The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss−Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two-dimensional Gauss −Bonnet integral as mentioned in this paper.
Abstract: The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss–Bonnet curvature.
TL;DR: In this paper, it was shown that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations.
Abstract: We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations. This result extends previous work which required the conformal metric to be in the negative Yamabe class, and required the mean curvature function to be nonzero.
14 Mar 2008
TL;DR: In this article, a nonsingular asymptotic distribution is derived for a broad class of underlying distributions on a Riemannian manifold in relation to its curvature.
Abstract: In this article a nonsingular asymptotic distribution is derived for a broad class of underlying distributions on a Riemannian manifold in relation to its curvature. Also, the asymptotic dispersion is explicitly related to curvature. These results are applied and further strengthened for the planar shape space of k-ads.
Posted Content•
TL;DR: In this article, the existence of a new connection in a Riemannian manifold is proved and the curvature tensor of the new connection is also found, where the connection reduces to several symmetric, semi-symmetric and quarter symmetric connections; even some of them are not introduced so far.
Abstract: In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.
TL;DR: In this article, it was shown that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold.
Abstract: We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E
n+1 × S
n
(4) in higher dimensions.
TL;DR: In this article, the tetrad formulation of Chern-Simons (CS) modified gravity with a spacetime-dependent coupling field was studied and the result was that CS torsion vanishes only if the coupling vanishes, thus generically leading to a modification of gyroscopic precession.
Abstract: We study the tetrad formulation of Chern-Simons (CS) modified gravity, which adds a Pontryagin term to the Einstein-Hilbert action with a spacetime-dependent coupling field. We first verify that CS modified gravity leads to a theory with torsion, where this tensor is given by an antisymmetric product of the Riemann tensor and derivatives of the CS coupling. We then calculate the torsion in the far field of a weakly gravitating source within the parameterized post-Newtonian formalism, and specialize the result to Earth. We find that CS torsion vanishes only if the coupling vanishes, thus generically leading to a modification of gyroscopic precession, irrespective of the coupling choice. Perhaps most interestingly, we couple fermions to CS modified gravity via the standard Dirac action and find that these further correct the torsion tensor. Such a correction leads to two new results: (i) a generic enhancement of CS modified gravity by the Dirac equation and axial fermion currents; (ii) a new two-fermion interaction, mediated by an axial current and the CS correction. We conclude with a discussion of the consequences of these results in particle detectors and realistic astrophysical systems.
TL;DR: In this article, a modified gravity theory is proposed, which is based on the Plebanski formulation of gravity in terms of a triple Bi of two-forms, a connection Ai and a Lagrange multiplier field Ψij.
Abstract: We describe and study a certain class of modified gravity theories. Our starting point is the Plebanski formulation of gravity in terms of a triple Bi of two-forms, a connection Ai and a 'Lagrange multiplier' field Ψij. The generalization we consider stems from the presence in the action of an extra term proportional to a scalar function of Ψij. As in the usual Plebanski general relativity (GR) case, a certain metric can be constructed from Bi. However, unlike in GR, the connection Ai no longer coincides with the self-dual part of the metric compatible spin connection. Field equations of the theory are shown to be relations between derivatives of the metric and components of field Ψij, as well as its derivatives, the later being in contrast to the GR case. The equations are of second order in derivatives. An analog of the Bianchi identity is still present in the theory, as well as its contracted version tantamount to the energy conservation equation.
TL;DR: In this paper, the curvature tensor of the past (future) boundaries of points in regular, Einstein vacuum spacetimes has been investigated, and conditions, expressed in terms of a space-like foliation, have been derived for ensuring local nondegeneracy of these boundaries.
Abstract: We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries of the causal past (future) sets . Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on . Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations
TL;DR: In this paper, the authors analyzed the coupling of the electromagnetic field with the Riemann tensor and found that an amplification of the magnetic field occurs during the reheating phase of evolution of the universe.
Abstract: The inflationary amplification of magnetic field seeds for galaxies is discussed in the framework of the f(R) theories of gravity, where f(R) is of the form f(R)~Rn. The breaking of the conformal invariance necessary for the seeding of the primordial magnetic field from the vacuum is generated by means of the coupling of the electromagnetic field with curvature terms. We analyze the coupling of the electromagnetic field with the Riemann tensor, FαβFα'β'Rαβα'β'. We find that amplification of the magnetic field occurs during the reheating phase of evolution of the Universe. Estimates of the index n are derived by using the observed strength of the galactic magnetic fields. Moreover, the coupling of the electromagnetic field with a generic function of the scalar curvature, i.e. RmFαβFαβ, is discussed. In this case, we find that a growing of the primordial magnetic field during the reheating epoch may occur for an appropriate choice of the powers m and n.
TL;DR: In this article, Wong's canonical coordinate form of torsion-free connections with skew-symmetric Ricci tensors on surfaces is extended to connections with torsions.
Abstract: Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong’s canonical coordinate form of such connections is simplified
TL;DR: In this article, the authors prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F( x,u, du, d 2 u) = 0d ef ined on a Riemannian manifold M.
Journal Article•
TL;DR: In this article, a linear torsion-free connection for Finsler metrics is introduced, which is a family of connections for Riemannian curvature tensors.
Abstract: In this paper, we introduce a new family of linear torsion-free connections for
Finsler metrics. This family of connections de¯nes a Riemannian curvature tensor R
and a non-Riemannian quantity P. We show that P contains all the non-Riemannian
information, namely, P = 0 if and only if the Finsler metric is Riemannian. In fact, this
family of connections makes a systematical analysis of connections that characterize
Riemannian metrics.
30 Apr 2008
TL;DR: In this paper, the existence of a new connection in a Riemannian manifold is proved and a formula for curvature tensor of the new connection is also found, where the connection reduces to several symmetric, semi-symmetric, and quarter-switching connections, even some of them are not introduced so far.
Abstract: In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection. 2000 Mathematics Subject Classification: 53B15.
TL;DR: In this paper, the authors studied the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensors in a negative convex cone on compact manifolds, and showed that any smooth bounded domain in Euclidean space of dimension greater or equal to 3 admits a complete conformally flat metric of negative Ricci curvature with.
Abstract: We study the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensor in a negative convex cone on compact manifolds. A consequence of our main results is that any smooth bounded domain in Euclidean space of dimension greater or equal to 3 admits a complete conformally flat metric of negative Ricci curvature with .