Integral geometry on the octonionic plane

TL;DR

This paper explicitly describes the algebra of Spin(9)-invariant valuations on the octonionic plane, introduces new invariant forms, and computes the principal kinematic formula, advancing integral geometry in octonionic settings.

Contribution

It extends invariant theory of Spin(7) to Spin(9) actions, providing a basis for valuations and computing the kinematic formula on the octonionic plane.

Findings

Explicit basis of invariant valuations constructed

Bernig-Fu convolution determined on this space

Principal kinematic formula computed for the octonionic plane

Abstract

We describe explicitly the algebra of Spin(9)-invariant, translation-invariant, continuous valuations on the octonionic plane. Namely, we present a basis in terms of invariant differential forms and determine the Bernig-Fu convolution on this space. The main technical ingredient we introduce is an extension of the invariant theory of the Lie group Spin(7) to the isotropy representation of the action of Spin(9) on the 15-dimensional sphere, reflecting the underlying octonionic structure. As an application, we compute the principal kinematic formula on the octonionic plane and express in our basis certain Spin(9)-invariant valuations introduced previously by Alesker.