Ricci decomposition

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

On the Projective Curvature Tensor of Generalized Sasakian-Space-Forms

Abstract: The object of the present paper is to study the nature of generalized Sasakian-space-forms under some conditions regarding projective curvature tensor. All the results obtained in this paper are in the form of necessary and sufficient conditions. Keywords: Generalized Sasakian-space-forms; projectively flat; projectively-semisymmetric; projectively symmetric; projectively recurrent; Einstein manifold; scalar curvature Quaestiones Mathematicae 33(2010), 245–252

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)-,

On generalized Robertson--Walker spacetimes satisfying some curvature condition

Abstract: We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.

Twistor spaces with hermitian ricci tensor

Abstract: The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./"-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kahler manifold with Hermitian Ricci tensor is Kahlerian.

Algebraic computing and the Newman-Penrose formalism in general relativity

Abstract: The curvature tensor of space-time can be described most concisely by giving the components of the Weyl and Ricci tensors relative to a complex null tetrad. The Newman-Penrose equations provide a simple and direct algorithm for calculating these components. This paper describes a computer program, written in the symbolic manipulation language CAMAL, which performs this calculation. Comparisons are made with the classical tensorial method of calculation, and some applications are discussed.