Integral geometry of exceptional spheres

TL;DR

This paper explores the algebraic structure of valuations invariant under special symmetry groups on spheres, computes kinematic formulas, and extends classical valuation theorems, revealing deep geometric insights.

Contribution

It establishes isomorphisms of valuation algebras on spheres with tangent space valuations and computes comprehensive kinematic formulas for these spaces.

Findings

Isomorphism of valuation algebras on S^6 and S^7 with tangent space valuations.

Explicit computation of all kinematic formulas for invariant valuations.

Extension of classical theorems of Klain and Schneider to simple valuations.

Abstract

The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $\mathrm G_2$ and $\mathrm{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations.