M.Sc Thesis
M.Sc Student Rosmarin Yonatan Perturbation Theory for Volume of Tubes Department of Electrical Engineering Professor Emeritus Robert Adler

Abstract

In 1939, Hermann Weyl derived a formula for the volume of tubular sets of radius ρ around manifolds embedded in Euclidean spaces or spheres. This now classic “tube formula”, which gave rise to an entire research area, is in the form of a finite order polynomial in ρ, with coefficients that can be expressed as curvature integrals and are intrinsic to the underlying manifold.

In the study presented in this thesis, a perturbation theory for the Euclidean version of Weyl's tube formula was developed. The approach to the problem takes two directions. Initially, a general manifold is perturbed by moving each of its points x a short distance, z(x), along a normal originating from x. The first objective of this study was to find an exact expression for the volume of the gap between the original and perturbed manifolds, involving an integral of z over the manifold, and thus, effectively obtain an extension of Weyl's original formula for tubes of non-constant radius. Using this expression, together with expressions for the first and second fundamental forms of the perturbed manifold, an expression for the volume around the perturbed manifold as an integral over the original one was obtained, which was then exploited to obtain a series expansion of this volume in terms of small parameters.

The second direction considers manifolds defined via the level sets of smooth functions. The perturbed manifolds now come from the level sets of perturbations for the initial functions. As in the first direction, a volume integral of the tube is written as an integral over the original level set and then as a series of one small parameter multiplied by the integral functionals of the initial functions and the perturbations.