Crofton formulas in pseudo-Riemannian space forms

Andreas Bernig, Dmitry Faifman, Gil Solanes

Research output: Contribution to journalArticlepeer-review

Abstract

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-Type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

Original languageEnglish
Pages (from-to)1935-1979
Number of pages45
JournalCompositio Mathematica
Volume158
Issue number10
DOIs
StatePublished - 28 Oct 2022

Keywords

  • Crofton formula
  • Lipschitz-Killing curvature measures
  • pseudo-Riemannian space form
  • valuation

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