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. 1–19DOI 10.4171/CMH/150