The product on smooth and generalized valuations

TL;DR

This paper develops a new geometric framework for the product of smooth and generalized valuations on manifolds, extending classical concepts and applying them to derive a general kinematic formula on symmetric spaces.

Contribution

It introduces a differential forms and Gelfand transform-based description of valuation products, extending to generalized valuations and linking to transversal intersections.

Findings

Product extends to generalized valuations

Provides a geometric interpretation via transversal intersections

Derives a general kinematic formula for symmetric spaces

Abstract

The product of smooth valuations on manifolds is described in terms of differential forms, Gelfand transforms and blow-up spaces. It is shown that the product extends partially to generalized valuations and corresponds geometrically to transversal intersections. This result is used to prove a general kinematic formula on compact rank one symmetric spaces.