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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: Tanasa et al. as discussed by the authors reviewed the most important results of the study of the multi-orientable tensor model, including the implementation of the 1=N expansion and of the large N limit (N being the size of the tensor).
Abstract: After its introduction (initially within a group eld theory framework) in A. Tanasa, J. Phys. A 45 (2012) 165401, the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the 1=N expansion and of the large N limit (N being the size of the tensor), the combinatorial analysis of the various terms of this expansion and nally,
57 citations
TL;DR: The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer ua is an eigendirection of eab as discussed by the authors.
Abstract: This paper contains an investigation of the algebraic structure and the analytic properties of a class of normal hyperbolic Riemannian 4‐spaces restricted by the following condition: There exists a timelike unit vector ua such that the Riemann tensor satisfies *Rabcdubud = 0. This condition is shown to be equivalent to the statement that the conform tensor is Petrov type I with real eigenvalues, ua being a principal vector and an eigenvector of the Ricci tensor. This means that there is no flux of nongravitational energy relative to an observer travelling with 4‐velocity ua.The eigen null directions (Debever vectors) of the conform tensor lie in a timelike hyperplane spanned by ua and the two eigenvectors of eac ≡ −Cabcdubud belonging to the eigenvalues with largest absolute value. The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer ua is an eigendirection of eab.The complete set of Bianchi identities is examined. It yields an expressio...
56 citations
TL;DR: Cao and Chen as discussed by the authors showed that any n-dimensional Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton.
Abstract: In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011)
56 citations
TL;DR: In this paper, the author applies the Excess Theorem of Abresch and Gromoll (1990) to prove two theorems: if such a manifold has small linear diameter growth then its fundamental group is finitely generated, and if it has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar.
Abstract: In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.
56 citations
TL;DR: A tensor is presented by combining Riemann–Christoffel curvature Tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases and is proved to have equivalency of different geometric structures.
Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2).
56 citations