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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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TL;DR: In this article, the authors conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3, where m is a complete hypersurface in a Euclidean space.
Abstract: where the endomorphism R(X, Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (•*) on the curvature tensor field R imply that M is locally symmetric (i.e. Vi? = 0) ? We conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3 . For partial and related results, see [4], p.ll, [9], Theorem 8, and [6]. The main purpose of the present paper is to give an affirmative answer in the case where M is a complete hypersurface in a Euclidean space. More precisely, we prove
92 citations
TL;DR: In this paper, the authors studied the Ricci tensor invariance of the Riemannian curvature tensor of the Kenmotsu manifold, which is derived from the almost contact Ricci manifold with some special conditions.
Abstract: The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.
90 citations
TL;DR: The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ⋅ 10 23 objects with up to 12 derivatives of the metric.
90 citations
TL;DR: In this paper, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space and the equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms.
Abstract: By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Rimannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovauum source becoming Isaacson's stress tensor for gravitational waves.
87 citations
TL;DR: In this article, the authors investigate the dynamics of a family of functional extensions of the Born-Infeld gravity theory, constructed with the inverse of the metric and the Ricci tensor.
Abstract: We investigate the dynamics of a family of functional extensions of the (Eddington-inspired) Born-Infeld gravity theory, constructed with the inverse of the metric and the Ricci tensor. We provide a generic formal solution for the connection and an Einstein-like representation for the metric field equations of this family of theories. For particular cases we consider applications to the early-time cosmology and find that nonsingular universes with a cosmic bounce are very generic and robust solutions.
87 citations