Showing papers on "Riemann curvature tensor published in 1978"

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

Gravity Lagrangian with ghost-free curvature-squared terms

Abstract: We construct a gravity theory using a set of Yang-Mills type gauge fields rather than the usual 10 metric fields ${g}_{\ensuremath{\mu}\ensuremath{ u}}$. This formalism, first proposed by Utiyama and Kibble, allows us to construct a gravity Lagrangian containing six invariants bilinear in the curvature tensor (as well as the usual invariant linear in $R$). The new terms are of interest since they may absorb some of the divergences which result when gravity is renormalized in the presence of matter. We study the graviton propagator in this theory. Usually, when curvature-squared terms are added to the Lagrangian, ghosts appear in the graviton propagator, and/or its high-energy behavior worsens. In the present case, we find we must drop four of the six invariants in order to avoid such difficulties. One of the surviving invariants predicts the existence of an extremely massive pseudoscalar particle. Neither surviving invariant is of the proper form to absorb renormalization divergences. The present investigation does not fully test the potentialities of the Kibble-Utiyama formalism, since the torsion tensor is nonzero in their framework, and our Lagrangian included no terms constructed from this tensor.

Symmetries in a unified gauge theory with torsion

Abstract: We consider aGL4×Un generalization of the Bonnor-Moffat-Boal formulation of Einstein unified field theory. Choosing a suitable form of the curvature tensor and introducing a phenomenological electric-current density, we show that charge conjugation corresponds to complex conjugation of the nonsymmetric unified fields. Making use of the complex tetrads underlying the fundamental Hermitian tensor field, we show that the gauge-covariant Dirac equation is obtained by minimal replacement of the Christoffel affinity of GR, with the generalized torsion containing connection. Finally, we consider spontaneous breaking of theUn gauge symmetry. The formalism splits into a vector sector and a tensor sector, the rotation angles\(\theta _W \) and\(\theta _G \) diagonalizing the vector and tensor mass matrices being related by\(\theta _G = \theta _W + \frac{1}{2}\pi \). TheUn components of the torsion vector describe the vector bosons mediating the electromagnetic and weak interactions, while theUn components of the symmetric part of the fundamental tensor describe the graviton and aSUn multiplet of tensor mesons.

The singular holonomy group

Abstract: The “fibre” of the extension of the frame-bundle of a space-time over ab-boundary pointp is a homogeneous space ℒ/Gp. It is shown thatGp can be found by a construction like that for a holonomy group, and that it contains a subgroup determined by the Riemann tensor. Near a curvature singularity one would expectGp = ℒ

Homogeneous and isotropic world models in the Yang–Mills dynamics of gravity. The structure of the adiabats

Abstract: The time evolution of homogeneous and isotropic matter distributions is analyzed for the restricted Yang–Mills curvature dynamics of gravity. This theory of gravity is a tidal dynamics for which relativistic matter in detailed balancing cannot produce tidal forces. It defines a dynamical system on the curvature plane spanned by the two components of the Riemann curvature of Robertson–Walker space–times; the essential features of the cosmological solutions are presented by means of their phase portraits in the curvature plane. In the asymptotic limit (S→∞) the phase portrait, which in general depends on the equation of state and on the change of the entropy per particle, is structurally stable under the transition from Einstein’s dynamics to the Yang–Mills dynamics for any realistic equation of state. The phase portraits are explicitly constructed for the equation of state p=nρ, 0≦n≦1, and constant entropy per particle. A criterion for the existence of regular trajectories is given for the full Yang–Mills ...

Riemannian Curvature and the Petrov Classification

Abstract: A connection is shown between the Riemannian Curvatures of the wave surfaces of a null vector I at a point p in space-time and the Petrov type of the space-time at p. Some other results on Riemannian Curvature are discussed

Remark on the equations ?R 2/?g ij = 0

Abstract: It is shown that the 4-dimensional equationsδR2/δgij=0 may be rewritten as 5-dimensional equations which are linear in the components of the Riemann tensor.

On the properties of torsions in Riemann-Cartan space-times. III: Classification of torsion and general discussion

Abstract: This is the third paper in a series of three papers which considers the physical properties of torsions in Riemann-Cartan space-timesU4 Paper III briefly considers some of the simpler generalizations of the ECSK theory together with their covering theories, and a simple classification scheme for torsions is introduced that depends in part on the Weyl conformal curvature tensor Moreover, further consideration is given to the possible physical role of the third-order «spin» tensor of Lanczos and to observations that could be made with spinning and nonspinning test particles In particular, we consider the possible importance of the LanczosV4 variational identity with Riemannian constraints in connection with speculations concerning the question of their microphysical significance

Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic spacetimes.

Abstract: We consider globally hyperbolic spacetimes (M, g) of dimension ≥ 3 satisfying the curvature condition Ric (g) (v, v) ≥ 0 for all non-spacelike tangent vectors v in TM. This curvature condition arises naturally as an energy condition in cosmology. Suppose (M, g) admits a smooth globally hyperbolic time function h: M → such that for some t0, the Cauchy surface h−1(t0) satisfies the strict curvature condition Ric (g) (v, v) > 0 for all non-spacelike v attached to h−1(t0). Then M admits a metric g′ conformal to g satisfying the strict curvature condition Ric (g′) (v, v) > 0 for all non-spacelike v in TM. If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.

The group structure of pseudo-Riemannian curvature spaces

Abstract: The linear conformal group G (pseudo-orthogonal automorphisms and dilatations) on a pseudo-orthogonal vector space induces an action in the space of pseudo-orthogonal curvature structures, which leaves Singer and Thorpe’s direct decomposition in trivial, non-Einsteinian and Weyl curvature structures invariant. It is shown that the condition of a curvature to be homogeneous, reductive, or symmetric is G-invariant. A condition for a non-Weyl curvature to be symmetric is formulated explicitly. Nomizu’s injection of the Jordan algebra of Lorentz-self-adjoint matrices is used to describe some G-orbits of non-Einsteinian curvatures. The Duffin–Kemmer–Petiau meson triple allows the construction of a cosmological model with trivial curvature.

Fermions and bosons in a unified framework. II. Interacting models

Abstract: In this paper we construct some fully interacting field theories. The first model has a colored, curved Weinberg-Salam-type action. It is formed by taking the Hilbert product of the (generalized) curvature of a (given) superalgebra with an auxiliary (generalized) curvature. Note that pieces of simple superalgebras are gauged; the effective superalgebra of gauge fields is not simple. The auxiliary curvature was needed to obtain the linear pieces of the action, and it thus appears to be somewhat ad hoc. In contrast we show how to construct an action using only the curvature of a local superalgebra without the auxiliary curvature (it is therefore quadratic). Nonetheless, linear terms arise as crossterms between pieces of the curvature. In fact, since we have chosen to use a special-unitary flavor algebra and four-component spinors, we discover we have already specified a unique simple supergroup whose other Bose gauge fields are in U(2,2), the Lie algebra formed by all the Dirac matrices. These fields gauge the spin structure of the fermions. Color causes certain complications discussed in the paper. The tensor piece of the U(2,2) curvature consists of the usual curvature plus a term identifiable as the old auxiliary tensor. Thus both linear and quadratic terms for the space-time curvature arise when the full curvature is squared. The field associated with the identity generator is electromagnetism; with the vector, torsion; with the tensor, curvature and auxiliary terms. We call the fields associated with the axial generators axial torsion and axial electromagnetism. When the fields which couple to Dirac spinors are assumed proportional to their scalar counterparts, an experimental value for a conserved axial electromagnetic coupling is ${10}^{\ensuremath{-}3}e$. We present a qualitative argument for the renormalizability of this action, since it is almost that of a standard Yang-Mills gauge theory, based on preservation of recoordinatization invariance by the quantization procedure.

Intrinsic Geometry of Surfaces: Local Theory

Abstract: We are now going to concentrate on the properties of a surface ƒ: U→ℝ3 which are intrinsic in the sense that they are definable in terms of tangent vectors to the surface and the first fundamental form and its derivatives. For example, the length of a vector or the length of a curve on a surface are intrinsic quantities. The Gauss curvature and the curvature tensor are also intrinsic since they may be defined in terms of the first fundamental form and its derivatives. In contrast, the second fundamental form is not intrinsic. It requires discussion of normal vector fields and cannot, in any case, be reduced to the first fundamental form. Also, principal curvatures are not intrinsic, even though their product, the Gauss curvature, is an intrinsic quantity.

On the affine connections that give rise to a given curvature

Abstract: A problem of both theoretical and practical importance is that of characterizing the collection of all affine connections that gives rise to a given curvature structure on a subset of a differentiable manifold of finite dimension. This problem is solved in closed form in Section three. We also show that the cardinality of the collection of all distinct connections that give the same curvature is that of the continuum, and that the connections of any two curvature structures can be brought into a1-to-1 correspondence.

Some conformally-flat space-times of a divergence-free Riemann tensor

Abstract: The authors present some interesting, conformally-flat metrics for a pseudo-Riemannian space-time with a divergence-free curvature tensor. The effective (gravitational) potentials for these metrics show both repulsive and periodic properties.

Curvature of product 3-manifolds

Abstract: Let M be a compact product 3-manifold without boundary. Let g be a Riemannian metric on M. If g has everywhere nonpositive sectional curvature, then g is locally diffeomorphic to a product metric. The proof is by the method of pseudoframes.