Ricci decomposition

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

Letter: Four-dimensional Tensor Identities of Low Order for the Weyl and Ricci Tensors

Abstract: It is shown that the new tensor identityrecently discovered by Bonanos, and some other tensoridentities recently investigated, are consequences of avery simple and mathematically trivial (but subtle) identity highlighted some years ago byLovelock. Lovelock's identity gives a tensor identity offirst order in Weyl-like tensors, and a tensor identityof second order in Ricci-like tensors, from which higher order identities, such as those recentlystudied, can easily be constructed.

Best subspace tensor approximations

Abstract: In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition which allows to compute the best rank k approximations. For t-tensors with t > 2 many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. In this paper we will present a different approach which is based on best subspace approximations, which present an alternative generalization of the singular value decomposition to tensors.

Ricci Lower Bound for K\"ahler-Ricci Flow

Abstract: In this note, we provide some general discussion on the Ricci lower bound along K\"ahler-Ricci flow with singularity over closed manifold.

Spaces admitting gravitational fields

Abstract: We start with a modern version of Einstein's definition of a gravitational field. Tensors of curvature type and the curvature product of symmetric tensors are defined. The interaction tensor is defined as the curvature product of the fundamental tensor and the energy‐momentum tensor. The tensor W obtained by coupling the Riemann tensor and the interaction tensor is used to obtain a characterization of gravitational fields. The linear transformation of the space of second‐order differential forms, induced by W, is used to give a new definition of a gravitational field. The field equations are expressed in terms of the gravitational sectional curvature function f. Thorpe's theorem characterizing Einstein spaces is obtained as a corollary. New formulations of the field equations are used to solve the problem of classification of gravitational fields. The mathematical foundations of the theory of classification are examined and a geometric interpretation of classification is obtained by using the critical poi...

Algebraic classification of the Weyl tensor in higher dimensions based on its 'superenergy' tensor

Abstract: The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved.