Crofton formulas in pseudo-Riemannian space forms

TL;DR

This paper develops a unified framework for Crofton formulas in pseudo-Riemannian space forms, enabling explicit computation of isometry-invariant valuations across various indefinite signature spaces using distributional methods.

Contribution

It introduces a distribution-based approach to Crofton formulas, extending classical results to all pseudo-Riemannian space forms and providing explicit formulas for invariant valuations.

Findings

Unified Crofton formulas for all pseudo-Riemannian space forms

Explicit valuations computed for pseudospheres and pseudo-Euclidean spaces

Formulas are signature-independent in a distributional sense

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.