A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line (Q1001220)

The Alesker theory of valuations on manifolds is developed. 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 is stated and proved.