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

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

2006-11-09

arxiv.org/abs/math/0611264v2

Comm. Math. Helv. 84 (2009), 1-19

Mathematics

Differential Geometry

18 pages, to appear in Commentarii Mathematici Helvetici

Scientific paper

The Alesker-Poincare 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.

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