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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: This paper employs a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau to derive lower RicCI curvature bounds on graphs in terms of such local clustering coefficients.
Abstract: In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.

185 citations

Journal ArticleDOI
TL;DR: In this article, a local algebraic function of the boundary metric and Ricci curvature is used to construct a variational principle for asymptotically flat spacetimes in any spacetime dimension d ≥ 4.
Abstract: A new local, covariant 'counter-term' is used to construct a variational principle for asymptotically flat spacetimes in any spacetime dimension d ≥ 4. The new counter-term makes direct contact with more familiar background subtraction procedures, but is a local algebraic function of the boundary metric and Ricci curvature. The corresponding action satisfies two important properties required for a proper treatment of semi-classical issues and, in particular, to connect with any dual non-gravitational description of asymptotically flat space. These properties are that (1) the action is finite on-shell and (2) asymptotically flat solutions are stationary points under all variations preserving asymptotic flatness, i.e., not just under variations of compact support. Our definition of asymptotic flatness is sufficiently general to allow the magnetic part of the Weyl tensor to be of the same order as the electric part and thus, for d = 4, to have non-vanishing NUT charge. Definitive results are demonstrated when the boundary is either a cylindrical or a hyperbolic (i.e., de Sitter space) representation of spacelike infinity (i0), and partial results are provided for more general representations of i0. For the cylindrical or hyperbolic representations of i0, similar results are also shown to hold for both a counter-term proportional to the square-root of the boundary Ricci scalar and for a more complicated counter-term suggested previously by Kraus, Larsen and Siebelink. Finally, we show that such actions lead, via a straightforward computation, to conserved quantities at spacelike infinity which agree with, but are more general than, the usual (e.g., ADM) results.

185 citations

Journal ArticleDOI
TL;DR: In this article, the state-finder diagnostic is applied to a holographic dark energy model from Ricci scalar curvature, called the Ricci dark energy models (RDE), and the evolutionary trajectories of this model are plotted in the statefinder parameter planes.

183 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023268
2022536
2021505
2020448
2019424
2018433