Crofton formulas in pseudo-Riemannian space forms
Andreas Bernig, Dmitry Faifman, Gil Solanes
TL;DR
The work develops a distribution-based Crofton framework for pseudo-Riemannian space forms, connecting Crofton formulas to Alesker's Radon transform and Hadwiger-type classifications of intrinsic volumes. It constructs invariant Crofton distributions via holomorphic families and meromorphic continuation, yielding explicit, signature-independent Crofton formulas for generalized pseudospheres in flat, spherical, and hyperbolic geometries. The results unify diverse space forms under a single analytic scheme and provide tools for evaluating intrinsic volumes via Euler-characteristic integrals in the distributional setting. The approach advances integral geometry in indefinite signatures and offers computational templates for LC-regular domains and their Crofton wave fronts.
Abstract
Crofton formulas on simply-connected Riemannian space forms allow to compute the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners, 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.