Valuation theory of indefinite orthogonal groups

TL;DR

This paper classifies all continuous and generalized valuations invariant under indefinite orthogonal groups, extending classical Euclidean valuation theory to more general geometric settings.

Contribution

It provides a complete description of invariant valuations for indefinite orthogonal groups and identifies the class of Klain-Schneider continuous valuations within translation-invariant valuations.

Findings

Classified invariant valuations under $ ext{SO}^+(p,q)$

Extended pull-back and push-forward operations to a broader valuation class

Identified Klain-Schneider continuous valuations as a strict subset

Abstract

Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain-Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.