Riemannian curvature measures

TL;DR

This paper explores the connection between curvature measures, valuations, and Riemannian geometry, revealing a new algebraic structure that encodes geometric and kinematic properties of manifolds, especially complex space forms.

Contribution

It introduces a novel module structure over polynomial algebra that captures curvature invariants and kinematic formulas in Riemannian geometry.

Findings

Defines a new algebraic module reflecting curvature measures

Connects curvature invariants to Alesker valuations

Illustrates the structure in complex space forms

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.