Commentarii Mathematici Helvetici
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Published online: 2009-03-31
A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic lineAndreas Bernig (1) J. W. Goethe-Universität, Frankfurt a.M., Germany
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.
Keywords: Valuations on manifolds, kinematic formulas, integral geometry, Alesker–Poincaré pairing
Bernig Andreas: A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line. Comment. Math. Helv. 84 (2009), 1-19. doi: 10.4171/CMH/150