Showing papers on "Riemann curvature tensor published in 1992"
TL;DR: In this article, the Riemann mapping theorem was generalized to higher dimensions and it was shown that a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways: it has zero scalar curvature and its boundary has constant mean curvature.
Abstract: One of the most celebrated theorems in mathematics is the Riemann mapping theorem. It says that an open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. In higher dimensions, very few regions are conformally diffeomorphic to the ball. However we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways, namely, it has zero scalar curvature and its boundary has constant mean curvature. In this paper we generalize the Riemann mapping theorem to higher dimensions in that sense.
401 citations
TL;DR: In this paper, the Riemann tensor is used to define a basis set of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric and exhibited a basis for these invariants up through order eight.
Abstract: Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.
207 citations
176 citations
TL;DR: In this article, a general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed, which is characterized by the tensors of Riemanian and non-Riemannians curvatures, an affine deformation tensor being the result of nonmetricity of one of the connections.
Abstract: A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.
127 citations
TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1992, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
116 citations
TL;DR: In this article, the authors generalized the topological massive gauge model of gravity by liberating its translational gauge degrees of freedom, and they uncovered a dynamical symmetry of the new theory by inquiring how the two Noether identities, the two Bianchi identities and the two field equations are interrelated to each other.
Abstract: Recently two of us generalized the topological massive gauge model of gravity of Deser, Jackiw, and Templeton (DJT) by liberating its translational gauge degrees of freedom. Consequently, the newR3◯SO(1,2) gauge model «lives» in a 3-dimensional Riemann-Cartan space-time with torsion. The extended Lagrangian consists, of the familiar Einstein-Cartan term, the Chern-Simons 3-form for the curvature, and, in addition, of a new translational Chern-Simons term. In this article we uncover a «dynamical symmetry» of the new theory by inquiring how the two Noether identities, the two Bianchi identities, and the two field equations are interrelated to each other. This includes two important subcases in which the first Bianchi identity is mapped into the second one and the first (energy-momentum) Noether into the second (angular-momentum) Noether identity. As a furtherexact result, the topological gauge field equations imply a covariant Proca-type field equation, for the translational gauge potential,i.e. the coframe. Thus the theory encompasses massive gravitons, as in the DJT model.
96 citations
Journal Article•
86 citations
TL;DR: In this article, the authors describe the geometry and topology of a compact simply connected positively curved Riemannian 6-manifold F′ which is related to the flag manifold F over C P2, and an infinite series of simply connected circle bundles over F′ with positive sectional curvature.
Abstract: We describe the geometry and the topology of a compact simply connected positively curved Riemannian 6-manifold F′ which is related to the flag manifold F over C P2, and an infinite series of simply connected circle bundles over F′, also with positive sectional curvature. All of these spaces are biquotients of the Lie group SU (3) and they are not homeomorphic to a homogeneous space of positive curvature.
80 citations
TL;DR: In this article, the irreducible decompositions of nonmetricity, torsion and curvature under the pseudo-orthogonal group as well as those of corresponding Bianchi identities are derived.
Abstract: The irreducible components of the curvature under the Lorentz group are of direct physical relevance in the four-dimensional Riemannian geometry of general relativity. The same is true for both curvature and torsion in the four-dimensional Riemann-Cartan geometry of the Poincare gauge theory of gravitation. In the latter theory a knowledge of these irreducible components is also extremely useful when setting up the Lagrangian and searching for exact solutions. The author deals with an n-dimensional metric-affine spacetime of arbitrary signature. In such a spacetime the connection is no longer metric so that there is an additional geometric object-the nonmetricity. The irreducible decompositions of nonmetricity, torsion and curvature under the pseudo-orthogonal group as well as those of the corresponding Bianchi identities are derived. Because of the increasing use made of it in the literature, the exterior form notation is used throughout.
75 citations
TL;DR: In this paper, it was shown that the vanishing of the trace free part Cμνρ of the second fundamental tensor Kμvρ is a sufficient condition for conformal flatness of the imbedded surface.
68 citations
Journal Article•
01 Apr 1992
TL;DR: In this paper, the Ricci curvature of an immersed submanifold in a Euclidean space has been shown to be a constant positive mean curvature in a non-compact hypersurface.
Abstract: We prove a best possible lower bound on the Ricci curvature of an immersed submanifold in a Euclidean space and apply it to study the size of the Gauss image of a complete noncompact hypersurface with constant positive mean curvature in a Euclidean space.
TL;DR: In this article, the Mandelstam covariant is introduced as an interpolating field constructed from geodesics, parallel transport, and the Riemann tensor, which is nonlocal in all gauges and therefore may not give the usual S -matrix beyond tree order.
TL;DR: Katzin et al. as discussed by the authors introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemanian space Vn and L ξ denotes the Lie derivative, and they proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field.
Abstract: Katzin et al. [G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10, 617 (1969)] introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemannian space Vn and Lξ denotes the Lie derivative. They proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field. Unfortunately, recent study indicates that the existence of a covariant constant vector restricts Vn to a very rare special case with limited physical use. In particular, for a fluid space time with special conformal motion, either stiff or unphysical equations of state are singled out. Moreover, perfect fluid space times do not admit special conformal motions. This information was not available, in 1969, when CC symmetry was introduced. In this paper, CC is generalized to another symmetry called ‘‘curvature inheritance’’ (CI) satisfying LξRbcda=2αRbcda, where α is a scalar function. We prove that a proper...
TL;DR: The N = 2 supersymmetric self-dual Yang-Mills theory and the N = 4 and n = 2 selfdual supergravities in 2 + 2 space-time dimensions are formulated for the first time in this paper.
TL;DR: In this paper, it was shown that the conformally flat radiation metric can admit at most one Killing vector and/or a homothety and that the true upper bound is three.
Abstract: It is shown that the conformally flat radiation metric found by Wils (1989) requires the fourth covariant derivative of the Riemann tensor for the Karlhede classification to terminate. This contradicts a widely held opinion that the true upper bound is three. The metric can admit at most one Killing vector and/or a homothety.
TL;DR: In this article, it is shown that complex transformations can be applied on the parameters and coordinates entering a known curvature tensor in order to generate new curvatures which satisfy certain algebraic relationships following from the Einstein-Maxwell equations with cosmological constant.
Abstract: It is shown that complex transformations can be applied on the parameters and coordinates entering a known curvature tensor in order to generate new curvature tensors which, just as the seed tensor, possess the same symmetry properties and satisfy certain algebraic relationships following from the Einstein-Maxwell equations with cosmological constant.
TL;DR: In this paper, the existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2.
Abstract: Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnerowicz-York equation of General Relativity); Ricci curvature in a weighted Kahler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.
16 Dec 1992
TL;DR: Using the Lagrangian framework and point transformations, an alternative derivation of an existing result on the special decomposition of the inertia matrix is presented and it is shown that a planar two-link manipulator cannot be linearized by point transformations only.
Abstract: Using the Lagrangian framework and point transformations, an alternative derivation of an existing result on the special decomposition of the inertia matrix is presented. The Riemann curvature tensor is introduced as a computational tool to test for this special decomposition. An example with configuration-dependent inertia which admits such a factorization is presented. For the cart-pole problem, it is shown that such a decomposition is possible and the linearizing transformation is computed. It is shown that a planar two-link manipulator cannot be linearized by point transformations only. >
TL;DR: In this article, an explicit form of the geodesic equation is presented for the fluid flow on a three-torus Riemannian connection, commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the Beltrami property.
Abstract: Motion of an ideal fluid is represented as geodesics on the group of all volume-preserving diffeomorphisms. An explicit form of the geodesic equation is presented for the fluid flow on a three-torus Riemannian connection, commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the Beltrami property. It is found that the curvature is non-positive for the section of two ABC flows with different values of the constants (A, B and C). The study is an extension of the Arnold's results (1989) in the two-dimensional case to three-dimensional fluid motions.
TL;DR: In this paper, the L2-index of the Dirac-Schr/Sdinger operator D + 2V is shown to be local at infinity, where V is a skew-adjoint potential.
Abstract: 1. Introduction. In [B2I a formula is given for evaluating the L2-index of a Dirac-type operator D on a certain class of (noncompact) complete Riemannian manifolds. Although in principle computable, especially in the Fredholm case, this formula contains terms reflecting the contribution of the small eigenvalues, which are difficult to evaluate. We show in this paper that the addition of a skew-adjoint potential V, satisfying reasonable assumptions at infinity, has the effect ofeventually overcoming the influence of the small eigenvalues of D. Thus, the L2-index of the "Dirac-Schr/Sdinger operator" D + 2V, for 2 sufficiently large, is given by an "adiabatic limit" ofr/-invariants and is therefore local at infinity. (See Theorem 3.2 below.) This generalizes and at the same time explains index formulae of Callias type. (See [C], [A].) Due in part to the nature of the problem, but mainly because of the limitations ofthe method we employ, the manifolds we are considering are subject to a number ofconstraints at infinity. Some of these conditions have a clear geometric meaning, but others do not. Thus, the class ofmanifolds to which our results apply is not easy to quantify. It seems possible to enlarge it to encompass all complete manifolds of strictly negative sectional curvature and finite volume; Theorem 3.5 below constitutes an important step in this direction. The L2-index theorem we prove in this paper can be used in conjunction with vanishing type arguments, much in the same way as the standard index theorem for Dirac operators and its relative version are employed in [GL], to gain information about the scalar curvature. To illustrate this, we discuss in Section 4 a "version with boundary" ofthe "conservation principle" for the scalar curvature ofperturbations ofthe standard metric on the n-sphere, suggested by Gromov i-G1 and proved in [L]. We wish to thank Maung Min-Oo for making us aware of these references
TL;DR: In this paper, the authors show that the energy-momentum tensor of the quadratic Yang-Mills action associated with the string reduces to that of an elementary fivebrane.
TL;DR: A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature.
Abstract: A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor (of a symmetric connection) so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature. In this new result the conditions are imposed on a tensor of a more general character than the curvature tensor. In addition it is shown that once the symmetric connection is known to be metric, the metric is uniquely defined (up to a constant conformal factor). For those special curvature tensors which are excluded from the original result, supplementary conditions are suggested, which, alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also Riemann tensors.
TL;DR: In this article, a method of calculating the metric from the curvature of a tensor with the symmetry properties of a type D curvature tensor is given in an orthonormal tetrad.
Abstract: A method of calculating the metric from the curvature is presented. Assuming that a tensor with the symmetry properties of a type D curvature tensor is given in an orthonormal tetrad, we use the Bianchi identities and the relationship between the connection and the tetrad in order to calculate, under certain assumptions, the corresponding metric. Some well-known metrics are derived from the curvature by using the method given here.
Journal Article•
TL;DR: Rouchon et al. as mentioned in this paper showed that for all flows, with the exception of the perfect eddy with constant vorticity (corresponding to a stationnary rotation around a fixed axis), there always exists small perturbations with strictly negative curvature.
Abstract: Following some Arnol’d results relative to the geometry underlying the dynamics of a perfect incompressible fluid [3] (geodesics of left-invariant metrics on Lie groups), the linear differential equation relative to a Lagrangian stability analysis are established. This differential equation, called Jacobi equation, describes, for the same fluid element, the time evolution of the difference between the trajectory starting from reference initial position in a reference flow and the trajectory starting from a perturbed initial position in a perturbed flow. Links with the linear differential equation relative to the classical Eulerian stability analysis are given. The stability of the solution of the Jacobi equation can be investigated through the sign of the Riemannian curvature. we prove here that, for all flows, with the exception of the perfect eddy with constant vorticity (corresponding to a stationnary rotation around a fixed axis), there always exists small perturbations with strictly negative curvature. If one assumes that negative curvature implies instability with exponential divergence of the geodesic flow, the above result proves effectively the instability, from a Lagrangian viewpoint, of all the solutions of the Euler equation, with the exception of the perfect eddy. An estimation of the most negative part of the curvature provides an interesting interpretation of the Kolmogorov timescale √ ν/e as the smallest time-constant relative to exponential divergence of fluid elements that are initially close. Running title: Jacobi equation and the motion of an incompressible fluid. ∗Ecole des Mines de Paris, Centre Automatique et Systemes, 60, Bd. Saint-Michel, 75006 Paris, FRANCE. Tel: (33) (1) 40 51 91 15. Fax: (33) (1) 43 54 18 93. Email: rouchon@cas.ensmp.fr .
TL;DR: In this paper, a complex curvature tensor concept and generalized tensor ABCD law were used to investigate three-dimensional astigmatic resonators. But the authors focused on the resonators bounded by cylindrical-spherical mirrors.
Abstract: Three-dimensional astigmatic resonators, typical examples of which are the resonators bounded by cylindrical-spherical mirrors and cylindrical-cylindrical mirrors oriented at an arbitrary crossed angle, are investigated in detail by using a complex curvature tensor concept and generalized tensor ABCD law. Computerized numerical calculations illustrate some interesting characteristics of these astigmatic resonators.