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The Geometric Meaning of Curvature: Local and Nonlocal Aspects of Ricci Curvature

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Modern Approaches to Discrete Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2184))

Abstract

Curvature is a concept originally developed in differential and Riemannian geometry. There are various established notions of curvature, in particular sectional and Ricci curvature. An important theme in Riemannian geometry has been to explore the geometric and topological consequences of bounds on those curvatures, like divergence or convergence of geodesics, convexity properties of distance functions, growth of the volume of distance balls, transportation distance between such balls, vanishing theorems for Betti numbers, bounds for the eigenvalues of the Laplace operator or control of harmonic functions. Several of these geometric properties turn out to be equivalent to the corresponding curvature bounds in the context of Riemannian geometry. Since those properties often are also meaningful in the more general framework of metric geometry, in recent years, there have been several research projects that turned those properties into axiomatic definitions of curvature bounds in metric geometry. In this contribution, after developing the Riemannian geometric background, we explore some of these axiomatic approaches. In particular, we shall describe the insights in graph theory and network analysis following from the corresponding axiomatic curvature definitions.

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Notes

  1. 1.

    The indices k and l appear in different orders on the two sides of (1.46), according to the convention of Jost [47] that made an attempt to mediate between the different conventions in use in Riemannian geometry.

  2. 2.

    The Borel sigma algebra is the set of all subsets of X that are obtained from the open balls by taking complements, finite intersections and countable unions. For the sets in the Borel sigma, one can then define their volumes w.r.t. to a Radon probability measure. The technical details are not so important for understanding the essence of the subsequent constructions.

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Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Advanced Investigator Grant Agreement no. 267087. Frank Bauer was partially supported by the Alexander von Humboldt foundation and partially supported by the NSF Grant DMS-0804454 Differential Equations in Geometry. Shiping Liu was partially supported by the EPSRC Grant EP/K016687/1 Topology, Geometry and Laplacians of Simplicial Complexes. Bobo Hua was supported by the NSFC 11401106.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Frank Bauer

  2. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Frank Bauer & Jürgen Jost

  3. School of Mathematical Sciences, Fudan University, Shanghai, China

    Bobo Hua

  4. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Shiping Liu

  5. Department of Mathematics, Freiburg University, Freiburg im Breisgau, Germany

    Guofang Wang

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  1. Frank Bauer
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Correspondence to Jürgen Jost .

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  1. Laboratoire d’Informatique Gaspard Monge, Université Paris-Est - ESIEE Paris, Noisy-le-Grand, France

    Laurent Najman

  2. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Marne-la-Vallée, Marne-la-Vallée, France

    Pascal Romon

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Bauer, F., Hua, B., Jost, J., Liu, S., Wang, G. (2017). The Geometric Meaning of Curvature: Local and Nonlocal Aspects of Ricci Curvature. In: Najman, L., Romon, P. (eds) Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, vol 2184. Springer, Cham. https://doi.org/10.1007/978-3-319-58002-9_1

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