Book Review: Metric structures for Riemannian and non-Riemannian spaces

Grove, Karsten

doi:10.1090/s0273-0979-01-00904-1

Cited by 5 publications

(2 citation statements)

References 39 publications

(41 reference statements)

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“…However, the spectrum of possible sets of weights that are admissible in the definition of Forman curvature is far more extensive and general. An important if not the main motivation behind considering weighted manifolds (or, as in Forman's work, CW complexes), stems from the observation made by Cheeger, Gromov and others [34,35] that to control collapse (degeneracy) of manifolds under curvature bounds (mainly, Ricci curvature bounds), one has to consider the volume and also more general measures. For other motivations, such as appertaining to minimal surfaces, we refer the reader to [36].…”

Section: Forman Curvature -A Brief Overviewmentioning

confidence: 99%

Systematic evaluation of a new combinatorial curvature for complex networks

Sreejith

Jost

Saucan

et al. 2017

Chaos, Solitons & Fractals

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We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a complex network. In this contribution, we perform a comparative analysis of Forman curvature with other edge-based measures such as edge betweenness, embeddedness and dispersion in diverse model and real networks. We find that Forman curvature in comparison to embeddedness or dispersion is a better indicator of the importance of an edge for the large-scale connectivity of complex networks. Based on the definition of the Forman curvature of edges, there are two natural ways to define the Forman curvature of nodes in a network. In this contribution, we also examine these two possible definitions of Forman curvature of nodes in diverse model and real networks. Based on our empirical analysis, we find that in practice the unnormalized definition of the Forman curvature of nodes with the choice of combinatorial node weights is a better indicator of the importance of nodes in complex networks.

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Section: Forman Curvature -A Brief Overviewmentioning

confidence: 99%

Systematic evaluation of a new combinatorial curvature for complex networks

Sreejith

Jost

Saucan

et al. 2017

Chaos, Solitons & Fractals

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“…Alexandrov spaces have yielded major insights into classical Riemannian geometry ( [Per93], cf. [Kap07,Gro01]). Generalized cones and warped products provide examples and counterexamples in Alexandrov geometry (cf.…”

Section: Introductionmentioning

confidence: 99%

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

Alexander,

Graf,

Kunzinger

et al. 2019

Preprint

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We study generalizations of Lorentzian warped products with onedimensional base of the form I × f X, where I is an interval, X is a length space and f is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [KS18], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber X. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

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Hacking the quantum revolution: 1925–1975

Schweber

2015

EPJ H

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Book Review: Metric structures for Riemannian and non-Riemannian spaces

Cited by 5 publications

References 39 publications

Systematic evaluation of a new combinatorial curvature for complex networks

Systematic evaluation of a new combinatorial curvature for complex networks

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

Hacking the quantum revolution: 1925–1975

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