Valuations on manifolds and Rumin cohomology

Abstract

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker's Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses Kähler identities, the Rumin-de Rham complex and spectral geometry.

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

“…Some more operators on V 1 .M / were introduced in [11]. For this, we suppose that M is a Riemannian manifold.…”

“…We will use some results of [11], which we would like to recall. The cosphere bundle S M is a contact manifold of dimension 2n 1 with a global contact form(˛i s not unique, but this will play no role here).…”

Theory of Valuations on Manifolds, IV. New Properties of the Multiplicative Structure

“…It was shown in [7], Theorem 1, that a pair (η, ψ) with η ∈ C ∞ (P + (T * R n ), Ω n−1 ), ψ ∈ C ∞ (R n , Ω n ) satisfies the equality (8.1.8) for any compact subanalytic subset P if and only if it satisfies the following two conditions (where π : P + (T * R n ) → R n is the canonical projection):…”

“…However in the proof of the "if" part of Theorem 1 in [7] the authors used equality (8.1.8) not for the whole class of compact subanalytic sets, but for the subclass of compact subanalytic submanifolds with boundary. Hence if (8.1.8) is satisfied for all P ∈ P(R n ) then the conditions (8.1.9), (8.1.10) are satisfied, and hence (8.1.8) is satisfied for an arbitrary compact subanalytic subset P ⊂ R n (again by Theorem 1 of [7]). …”

“…Milman for his attention to this work. I thank A. Bernig for sharing with me the recent preprint [7], J. Fu for very helpful explanations on the geometric measure theory, P.D. Milman for useful correspondences regarding subanalytic sets, and P. Schapira for useful discussions on constructible sheaves and functions.…”