八元数平面上的Integral geometry
Integral geometry on the octonionic plane
摘要：
我们明确地描述了八元数平面上的Spin（9）-不变，连续赋值的代数。也就是说，我们提出了一个基不变的微分形式和确定的Bernig-Fu卷积在这个空间。我们介绍的主要技术成分是将李群Spin（7）的不变理论扩展到Spin（9）在15维球体上作用的各向同性表示，反映了潜在的八元数结构。作为应用，我们计算了八元数平面上的主运动学公式，并在此基础上给出了Alesker先前引入的某些Spin（9）不变赋值。
英文摘要：
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.
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