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.Comment: 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 Analysi

Kinematic formulae for tensorial curvature measures

“…Various other groups of isometries, also in Riemannian isotropic spaces, have been studied in recent years. Major progress has been made, for instance, in Hermitian integral geometry (in curved spaces), where the interplay between global and local results turned out to be crucial (see [13,14,25,26,79,80,76] and the survey [11]), but various other group actions have been studied successfully as well (see [4,8,9,12,17,18,19,23]).…”

Crofton formulas and indefinite signature

“…The Lorentz group O(n−1, 1) was considered by S. Alesker and the author in [8], and the general signature was studied by A. Bernig and the author in [14]. There, the dimensions of the spaces of invariant valuations were computed, and a simple description was given in terms of their Klain sections.…”

“…In [14], a complete set of Crofton formulas was obtained for R 2,2 , while in [8] a complete set of Crofton distributions was constructed for the Lorentz signature (n − 1, 1). In both of those cases, the Crofton distributions could be chosen to be invariant.…”

“…Here the vertical arrows on the right are the corresponding maps for valuations as defined by Alesker [6], and extended to generalized valuations by Bernig and the author in [14]. The precise statements can be found in Appendix B.…”

“…Geometry of O(p, q). Let us start by introducing some notation and definitions which we will use throughout the paper, for details we refer to [14].…”