Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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TL;DR: In this article, it was shown that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature, which generalizes a result obtained for surfaces by Hamilton.
Abstract: In this short article we show that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature. This generalizes a result obtained for surfaces by Hamilton [4]. The proof involves a careful analysis of the ODE for the curvature which is associated to the Ricci flow.
343 citations
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TL;DR: In this article, Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies are derived for general relativistic spacetimes filled with an irrotational perfect fluid.
Abstract: For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an "effective equation of state" including kinematical as well as dynamical "backreaction" terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical "backreaction" terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a "stiff" fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form.
342 citations
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339 citations
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338 citations
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TL;DR: In this paper, the authors studied all four types of finite-time future singularities emerging in the late-time accelerating (effective quintessence/phantom) era from ℱ(R,G)-gravity, where R and G are the Ricci scalar and the Gauss-Bonnet invariant, respectively.
Abstract: We study all four types of finite-time future singularities emerging in the late-time accelerating (effective quintessence/phantom) era from ℱ(R,G)-gravity, where R and G are the Ricci scalar and the Gauss–Bonnet invariant, respectively. As an explicit example of ℱ(R,G)-gravity, we also investigate modified Gauss–Bonnet gravity, so-called F(G)-gravity. In particular, we reconstruct the F(G)-gravity and ℱ(R,G)-gravity models where accelerating cosmologies realizing the finite-time future singularities emerge. Furthermore, we discuss a possible way to cure the finite-time future singularities in F(G)-gravity and ℱ(R,G)-gravity by taking into account higher-order curvature corrections. The example of non-singular realistic modified Gauss–Bonnet gravity is presented. It turns out that adding such non-singular modified gravity to singular Dark Energy makes the combined theory a non-singular one as well.
337 citations