Integral geometry on 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.
Cite
@article{arxiv.2209.14979,
title = {Integral geometry on the octonionic plane},
author = {Jan Kotrbatý and Thomas Wannerer},
journal= {arXiv preprint arXiv:2209.14979},
year = {2022}
}
Comments
The main result (Theorem 1.3) was improved by adding the minimality statement; the background section was slightly compressed; several typos were corrected. 38 pages