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

### Andreas Bernig

J. W. Goethe-Universität, Frankfurt a.M., Germany

## 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 $SU(2)$- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line $H$ is stated and proved.

## Cite this article

Andreas Bernig, A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line. Comment. Math. Helv. 84 (2009), no. 1, pp. 1–19

DOI 10.4171/CMH/150