Showing papers on "Ricci decomposition published in 1989"

Ricci curvature bounds and Einstein metrics on compact manifolds

Abstract: where f: MO M1 is a homeomorphism and dil f is the dilatation of f given by dil f = supX,X2 dist(f(x) , f(x2))/ dist(x1, x2) . If MO and M1 are not homeomorphic, define dL(MO, MI) = +oo. Gromov [20] proves the remarkable result that the space of compact Riemannian manifolds f((A, 3, D) of sectional curvature IKI 3 > 0, and diameter dM v, and diameter dM c(IKI , dm, VM1) In particular, Gromov's compactness theorem may be strengthened to the statement that f((A, v , D) is C1 'l compact in the Lipschitz topology. In this paper, we study the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in

Algebras of virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces

Abstract: The present paper is the direct continuation of the authors’ preceding papers [1, 2], in which the realization of the program of successive operator quantization of “multiloop diagrams” in the theory of boson strings was started. At the base of all our constructions lies the following function theoretic construction on a nonsingular Riemann surface Γ of genus g 1, if the triple Γ, P± is typical; and if g = 1, then this assertion is true for all n 6= 1/2. The definition of the tensors f n given generalizes to half-integral λ, where it depends in addition on the spinor structure. The cases λ = −1, 0, 1/2, 1, 2 (of vector fields, functions, spinors, differentials, quadratic differentials) are the most important. For them we use the special notation f−1 n = en, f 0 n = An, f 1/2 n = Φn, f n = dω−n, f 2 n = d Ω−n. The tensors f n have the following important multiplicative property of “almost gradedness:

Second order parallel tensor in real and complex space forms

Abstract: Levy's theorem ‘A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor’ has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form.

Local and global algebraic structures in general relativity

Abstract: The extent to which the well-known pointwise algebraic canonical forms used for the energy-momentum tensor, the Weyl tensor, etc., can be regarded as smooth relations over some open subset of (possibly the whole of) space-time is investigated.

Three-dimensional rotational averages in radiation-molecule interactions: an irreducible cartesian tensor formulation

Abstract: The authors present a new method for the calculation of the rotational averages which arise in the theory of spectroscopic radiation-molecule interactions in fluid media. Based upon the principles of irreducible cartesian tensor analysis, the method presented allows them to express results either in the usual reducible form, or directly in terms of linearly independent sets of irreducible tensor products. For interactions up to and including rank 3 in the molecular response (or non-linear susceptibility) tensor, the rotational averages cast in terms of irreducible tensor products are considerably simpler in structure than the corresponding results expressed in reducible form.

On the Einstein-Maxwell equations for a 'stiff' membrane

Abstract: The Einstein-Maxwell equations are examined for a distributional stress tensor depending on the mean shape of an immersion in a manifold with a piecewise smooth metric tensor. A solution is discussed that matches an exterior Reissner-Nordstrom metric to an interior Minkowski metric. Such a solution extends the Dirac particle model to incorporate both gravitational and electromagnetic interactions.

Every three-sphere of positive Ricci curvature contains a minimal embedded torus

Abstract: One of the most celebrated theorems of differential geometry is the 1929 theorem of Lusternik and Schnirelmann, which states that for every riemannian metric on the 2-sphere there exist at least three simple closed geodesies. Jurgen Jost [J] (following important work of Pitts [P] and Simon and Smith [SS]) has recently generalized this result by showing that for every riemannian metric on S, there exist at least 4 minimal embedded 2-spheres. This is optimal in that there are metrics for which the number of embedded minimal 2-spheres is exactly four [W2,4.5]. However, one can also ask about surfaces of higher genus, and here our knowledge is very incomplete. On the one hand, Lawson [L] showed that S with its standard metric contains embedded minimal surfaces of every orientable toplogical type, and recently Pitts and Rubinstein [PR] have discovered many new infinite families of examples. But for general metrics on S, no known theorem asserts the existence of any minimal surface other than a sphere. The present paper takes a first step in this direction by proving:

An expression of classical dynamics

Abstract: The proposed formulation extends the Euler variable approach, classical in Continuum Mechanics, up to make it valid for such singular systems as, for instance, a single mass-point. The key concept is the kinetic tensor measure of the investigated material, relative to some window in time-space. This is first developed in the framework of Galilean time-space. In that case, the fundamental equation involves the four-dimensional vector distribution divergence of the kinetic tensor measure. It is shown, in particular, how the initial conditions of an evolution problem or the confinement of the investigated system by a given boundary, possibly with shocks, may be described through adequate terms in the fundamental equation. In order to develop similar procedures in the Riemannian manifold setting of Analytical Dynamics, one introduces the differential operator equilibrium, acting on the doubly contravariant symmetric tensor measures of the manifold. This operator receives a variational interpretation, in terms of the transport by test flows. Thereby, the connection of the proposed formulation of Dynamics with Hamilton’s principle is explained.

Second‐order equations from a second‐order formalism

Abstract: It is often assumed that Lagrangians of gravitation that are quadratic in the curvature tensor produce field equations of fourth differential order in the metric tensor from a Hilbert variational principle. It is shown here, for the Lagrangian given by R+RμνRμν, that independent variations of the metric tensor and the torsion tensor produce gravitational field equations of second, not fourth, differential order.

Riemannian manifolds which admit a unique harmonic or killing tensor field

Abstract: The purpose of the present paper is to extend the previous results, in accordance to Lichnerowicz theory, to harmonic and Killing tensor fields of order p in a manifold M n . Also an extention of these results, in a conformally flat compact orientable Riemannian manifold is given, with the help of Weyl's conformal curvature tensor

Tensor Product of Finite and Infinite Representations in Physics

Abstract: In field theory, particles with gauge freedom, like the photon, are usually not described as unitary irreducible representations of the spacetime symmetry group, but they appear in indecomposable representations with indefinite invariant scalar product. We call such representations Gupta-Bleuler triplets. There is not yet a general representation theory, but some techniques are available to deal with them [1,2]. One—which is very useful in physical applications—stems from the observation, that the tensor product of finite and infinite representations may contain Gupta-Bleuler triplets in the reduction.