Integral geometry under $G_2$ and $Spin(7)$

Abstract

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.

arXiv comment13 pages

References in corpus (1)

• A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

Cited by in corpus (1)

• The product on smooth and generalized valuations