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

  • Frank Bauer
  • Bobo Hua
  • Jürgen JostEmail author
  • Shiping Liu
  • Guofang Wang
Chapter
Part of the Lecture Notes in Mathematics book series (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.

Notes

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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© The Author(s) 2017

Authors and Affiliations

  • Frank Bauer
    • 1
    • 2
  • Bobo Hua
    • 3
  • Jürgen Jost
    • 2
    Email author
  • Shiping Liu
    • 4
  • Guofang Wang
    • 5
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiChina
  4. 4.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  5. 5.Department of MathematicsFreiburg UniversityFreiburg im BreisgauGermany

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