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

Abstract

The Alesker–Poincaré pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of $\mathrm{SU}(2)$- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line $H$ is stated and proved.