Ricci decomposition

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

A modified variational principle for gravity in the modified Weyl geometry

Abstract: The usual interpretation of the Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of the Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in the Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the action of the Weyl covariant derivative, we find a Weyl version of the Palatini identity for the Riemann tensor. To derive the energy–momentum tensor and equations of motion for gravity in the Weyl geometry, one naturally applies this identity at first, and then converts the variation of the additive Weyl connection to those of the metric tensor and Weyl gauge field. We also discuss possible connections to the current literature on the Weyl-invariant extension of massive gravity and the variational principles in f(R) gravity.

Conformally flat manifolds with nonnegative Ricci curvature

Abstract: We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works by Q.-M. Cheng, M.H. Noronha, B.-L. Chen and X.-P. Zhu, and S. Zhu.

On 3-dimensional normal almost contact metric manifolds satisfying certain curvature conditions

Abstract: The object of the present paper is to study 3-dimensional normal almost contact metric manifolds satisfying certain curvature con- ditions. Among others it is proved that a parallel symmetric (0;2) tensor fleld in a 3-dimensional non-cosympletic normal almost contact metric manifold is a constant multiple of the associated metric tensor and there does not exist a non-zero parallel 2-form. Also we obtain some equivalent conditions on a 3-dimensional normal almost contact metric manifold and we prove that if a 3-dimensional normal almost contact metric manifold which is not a fl-Sasakian manifold satisfles cyclic parallel Ricci tensor, then the manifold is a manifold of constant curvature. Finally we prove the existence of such a manifold by a concrete example.

A Note on Ricci Solitons

Abstract: In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.

Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors

Abstract: Many idealized problems in signal processing, machine learning, and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximation (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation problem. In practice, however, the inevitable errors (say) from estimation, computation, and modeling necessitate that the input tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor slightly perturbed from the underlying SOD tensor. This paper shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor. It is shown that when the perturbation error is small enough, the approximation errors do not accumulate with ...