A volume comparison theorem for characteristic numbers

Daniel Luckhardt

Research output: Contribution to journalArticlepeer-review


We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

Original languageEnglish
Pages (from-to)687-708
Number of pages22
JournalAnnals of Global Analysis and Geometry
Issue number3
StatePublished - 1 Oct 2021


  • Betti numbers
  • Chern numbers
  • Euler characteristic
  • Hölder regularity
  • Pontryagin numbers
  • Ricci curvature

ASJC Scopus subject areas

  • Analysis
  • Political Science and International Relations
  • Geometry and Topology


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