Title: Riemannian curvature measures,
Abstract: A famous theorem of Weyl states that if $M$ is a compact submanifold of
euclidean space, then the volumes of small tubes about $M$ are given by a
polynomial in the radius $r$, with coefficients that are expressible as
integrals of certain scalar invariants of the curvature tensor of $M$ with
respect to the induced metric. It is natural to interpret this phenomenon in
terms of curvature measures and smooth valuations, in the sense of Alesker,
canonically associated to the Riemannian structure of $M$. This perspective
yields a fundamental new structure in Riemannian geometry, in the form of a
certain abstract module over the polynomial algebra $\mathbb R[t]$ that
reflects the behavior of Alesker multiplication. This module encodes a key
piece of the array of kinematic formulas of any Riemannian manifold on which a
group of isometries acts transitively on the sphere bundle. We illustrate this
principle in precise terms in the case where $M$ is a complex space form.