Showing papers on "Ricci decomposition published in 1988"

Do non-linear metric theories of gravitation really exist?

Abstract: Earlier results of Higgs (1959), Stelle (1978) and Whitt (1984) on the dynamical equivalence between Einstein's theory and a class of quadratic theories of gravitation are analysed in view of a more general result, implying the same conclusion for a much larger class of theories (essentially all those which depend arbitrarily on the Ricci tensor). It is shown that all previously known cases of conformal equivalence follow from the prescription of a general Legendre transform, which is in fact suggested by an earlier idea of Einstein (1923) and Eddington (1924). The extension to cover also the dependence on Weyl's tensor (1950) is shortly discussed.

Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups

Abstract: On the other hand, every finitely generated subgroup of the fundamental group of any complete manifold with Ric > 0 {K > 0) is nilpotent (abelian) up to finite index [6, 5, 4]. PROOF OF THE THEOREM. Our construction is inspired by [2]. We first apply an observation in [3, pp. 126-127] to obtain a family of almost flat metrics gr on L, 0 < r < oo. Choose a triangular basis {Xi,...,Xn} for the Lie algebra / of L, i.e., [X,X^] € h-i whenever X € /, and U-i is spanned by X i , . . . ,X j_ i . For X = E ? = i « < * set ||X|| == £?=i*?(r)a?, where h{(r) = (1 + r 2 ) \" \" ' , and an — a > 0, 2ai — 4c*i+i = 1, 1 < i < n — 1. The above norm gives rise to a corresponding almost flat left invariant metric gr. Then (1) iRMX^cU+r)-,

Holonomy groups in general relativity

Abstract: The infinitesimal holonomy group structure of space‐time is discussed and related to the Petrov type of the Weyl tensor and the algebraic (Segre) type of the energy‐momentum tensor. The number of covariant derivatives of the curvature tensor required to determine the infinitesimal holonomy group is determined in each case and the complete classification scheme is tabulated. Some special cases of physical interest are investigated in more detail. A geometrical approach is followed throughout.

Real hypersurfaces with cyclic-parallel Ricci tensor of a complex projective space

Abstract: P_{n}C has two or three distinct constant principal curvatures, then M is locally congruent to one of the homogeneous ones of type A_{1} , A_{2} and B. This result is recently generalized by Kimura [4], who proves that a real hypersurface M of P_{n}C has constant principal curvatures and J\\xi is principal if and only if M is locally congruent to one of the homogeneous hypersurfaces, where \\xi denotes the unit normal and J is the complex structure of P_{n} C. In particular, real hypersurfaces of type A_{1} , A_{2} and B of P_{n}C have been studied by several authors (cf. Cecil and Ryan [2], Kimura [5], Maeda [6] and Okumura [10] ) . On the other hand, real hypersurfaces of a complex hyperbolic space H_{n} C have also been investigated from different points of view and there are some studies by Chen, Ludden and Montiel [3] and Montiel and Romero [9]. In particular, real hypersurfaces of H_{n}C , which are said of type A, similar to those of type A_{1} and A_{2} of P_{n}C were treated by Montiel and Romero [9]. Now, the Ricci tensor S is said to be cyclic-parallel if it satisfies

Weyl curvature tensor in static spherical sources.

Abstract: The role of the Weyl curvature tensor in static sources of the Schwarzschild field is studied. It is shown that in general the contribution from the Weyl curvature tensor (the ``purely gravitational field energy'') to the mass-energy inside the body may be positive, negative, or zero. It is proved that a positive (negative) contribution from the Weyl tensor tends to increase (decrease) the effective gravitational mass, the red-shift (from a point in the sphere to infinity), as well as the gravitational force which acts on a constituent matter element of a body. It is also proved that the contribution from the Weyl tensor always is negative in sources with surface gravitational potential larger than (4/9. It is pointed out that large negative contributions from the Weyl tensor could give rise to the phenomenon of gravitational repulsion. A simple example which illustrates the results is discussed.

Weyl tensor and gravitational entropy.

Abstract: There is a conjecture due to Penrose that there may be a relation between the Weyl curvature tensor and gravitational entropy. In this paper, this conjecture is studied in the context of a specific model, the Gowdy cosmology. The square of the curvature is calculated as an operator and its expectation values in states of clumped and unclumped gravitons are calculated. The results indicate that the curvature contains information about the entropy of the gravitational field.

The continuous determination of spacetime geometry by the Riemann curvature tensor

Abstract: It is shown that generically the Riemann tensor of a Lorentz (or positive definite) metric on an n-dimensional manifold (n>or=4) determines the metric up to a constant factor and hence determines the associated torsion-free connection uniquely. The resulting map from Riemann tensors to connections is continuous in the Whitney Cinfinity topology (1957) but, at least for some manifolds, constant factors cannot be chosen so as to make the map from Riemann tensors to metrics continuous in that topology. The latter map is, however, continuous in the compact open Cinfinity topology so that estimates of the metric and its derivatives on a compact set can be obtained from similar estimates on the curvature and its derivatives.

Dust EPL cosmologies

Abstract: A formalism based on the description of a geometry in terms of the curvature tensor and its covariant derivatives is used to show that there are no tilted dust exact power law (EPL) cosmologies.

The isometry groups of compact manifolds with negative Ricci curvature

Abstract: We estimate the order of the isometry groups of compact manifolds with negative Ricci curvature in terms of geometric quantities: the sectional curvature, the Ricci curvature, the diameter, and the injectivity radius.