Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

Some Invariant Properties of Semi-symmetric Metric Recurrent Connections and Curvature Tensor Expressions

Abstract: The projective transformation of the special semi-symmetric metric recurrent connection is studied in this paper. First of all, an invariant under this transformation is granted; Secondly, by inducing of the invariant and making use of the properties that the corresponding covariant derivative keeps being fixed under the distinctness connection, the curvature tensor expression of the Riemannian manifold is posed at the same time.

Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions

Abstract: The elastic stiffness or compliance is a fourth-order tensor that can be expressed in terms of two second-order symmetric tensors A and B and a fourth-order completely symmetric and traceless tensor Z (or z) It is shown that the parts associated with A, B and Z (or z) are all structurally invariant under a three-dimensional transformation Thus a linear combination of the three parts gives a general expression for three-dimensional structural invariants All three-dimensional structural invariants available in the literature are shown to be special cases of this general expression Invariants that are inherited by each structural invariant are presented

A Generalization of the Goldberg-Sachs Theorem and its Consequences

Abstract: The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and admits a solution for the source-free Einstein-Maxwell equations.

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.