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

Abstract

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.

Spin-invariant valuations on the octonionic plane

“…[35]. Since dim Val 2 (H n ) Sp(n)·Sp(1) = 2 [15], we obtain by comparing the Klain functions T 2 p µ sec = µ 2 + 3τ, where µ 2 is the second intrinsic volume (whose Klain function is 1). In particular, K(E (1,0),(i,0) ) = 4, K(E (1,0),(0,1) ) = 1.…”

A Hadwiger-Type Theorem for the Special Unitary Group

“…In order to compute the product structure on Val SU (n) , we need the following corollary of Theorem 4.1. from [11]. Proposition 6.1.…”

Integral geometry under G2 and Spin(7)

“…In particular, we will need that G 2 and Spin(7) act 2-transitively on the unit sphere. This is not the case for G = Spin (9), which is the reason why other ideas have to be used in the study of the integral geometry under this group.…”

“…The algebra structure was computed in [9]. The kinematic formulas were known before by work of Tasaki [27].…”