Contact measures in isotropic spaces

TL;DR

This paper explores contact measures in isotropic spaces through valuation theory, linking kinematic formulas and integral geometry, and explicitly derives formulas for hermitian space.

Contribution

It introduces a valuation-theoretic perspective on contact measures, connecting existing kinematic formulas with the geometry of curved isotropic spaces and providing explicit formulas for hermitian space.

Findings

Established a link between contact measures and valuation theory.

Derived explicit kinematic formulas for surface area measures in hermitian space.

Connected kinematic formulas with integral geometry of curved isotropic spaces.

Abstract

We revisit the contact measures introduced by Firey, and further developed by Schneider and Teufel, from the perspective of the theory of valuations on manifolds. This reveals a link between the kinematic formulas for area measures studied by Wannerer and the integral geometry of curved isotropic spaces. As an application we find explicitly the kinematic formula for the surface area measure in hermitian space.