University of Illinois at Urbana-Champaign

Math 595 Section TF (Tube Formulas in Convex, Riemannian and Metric Geometry) Fall 2017

Course Description

Weyl's tube formula computes the volume of a small tubular neighborhood of a smooth submanifold of Euclidean space. Remarkably, the volume of the scale r neighborhood is a polynomial in r, whose coefficients encode various curvatures of the submanifold. There are similar formulas in convex geometry for the volumes of neighborhoods of convex sets, or more generally, Minkowski sums of convex sets. These sometimes go under the name quermassintegrals or mixed volumes, and go back to the work of Steiner. In this course we will discuss various tube formulas for neighborhoods of convex sets and neighborhoods of smooth submanifolds in Riemannian manifolds. At the end of the course, time permitting, we will indicate some recent extensions of these ideas to fractals and to sub-Riemannian manifolds, specifically, the sub-Riemannian Heisenberg group.