Valuation theory of indefinite orthogonal groups

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.

65 pages; Some details in proofs added and minor mistakes corrected, an appendix on wave front sets of G-invariant distributions and a section on the linear algebra of O(p,q) added. Accepted for publication in Journal of Functional Analysis

References in corpus (2)

• Classification of invariant valuations on the quaternionic plane
• Crofton Formulas and Indefinite Signature

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• The Hadwiger theorem on convex functions, I
• Crofton Formulas and Indefinite Signature
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• Contact integral geometry and the Heisenberg algebra
• Uniqueness of curvature measures in pseudo-Riemannian geometry
• Kinematic formulae for tensorial curvature measures
• Flag area measures
• Geometric valuation theory
• Contact measures in isotropic spaces
• Crofton formulas in pseudo-Riemannian space forms