Ricci decomposition

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

A Classification of Locally Homogeneous Affine Connections with Skew-Symmetric Ricci Tensor on 2-Dimensional Manifolds

Abstract: We classify, in an explicit form, the locally homogeneous torsionless affine connections as in the title We also give some motivation for this research coming from the study of Osserman spaces

Lovelock tensor as generalized Einstein tensor

Abstract: We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler–Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss–Bonnet gravity as an another version of string–inspired gravity.

Estimating the trace-free Ricci tensor in Ricci flow

Abstract: An important and natural question in the analysis of Ricci flow behavior in all dimensions n ≥ 4 is this: What are the weakest conditions that guarantee that a solution remains smooth? In other words, what are the weakest conditions that provide control of the norm of the full Riemann curvature tensor? In this short paper, we show that the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, for all compact solutions in all dimensions n ≥ 3, without any hypotheses on the initial data.

Ricci solitons in manifolds with quasi-constant curvature

Abstract: The Eisenhart problem of nding parallel tensors treated already in the framework of quasi-constant curvature manifolds in [Jia] is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing D-concircular tensor eld and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.

Tensor products of analytic continuations of holomorphic discrete series

Abstract: We give the irreducible decomposition of the tensor product of an an- alytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate. 0. Introduction. The work of Segal (IES) and Mautner (M) established the abstract Plancherel theorem for type I groups. This meant that for an arbitrary unitary represen- tation, one could find its spectral decomposition into irreducibles and a corresponding spectral measure. To make this program explicit on L 2 -spaces on homogeneous spaces is one of the main subjects of harmonic analysis. Another interesting case is that of decom- posing a tensor product of irreducible representations; our aim in this paper is to consider this for certain holomorphic representations. The problem of finding the irreducible decomposition of tensor products of holomor- phic discrete series of the group SL(2, ) has been studied by Repka (Re1). The results there were used by Howe (How) to give the decomposition of the metaplectic represen- tation for certain dual pairs. See also (OZ). For a general semisimple Lie group G of Hermitian type a similar problem is studied in (Re2). It is shown that the tensor prod- uct of a scalar holomorphic discrete series with its conjugate is unitarily equivalent to the L 2 -space on the corresponding Hermitian symmetric space, L 2 (G K). Therefore we know its decomposition from the known theory of Harish-Chandra; namely L 2 (G K) W H ( ) C( ) 2 d