M.Sc Thesis | |

M.Sc Student | Rosmarin Yonatan |
---|---|

Subject | Perturbation Theory for Volume of Tubes |

Department | Department of Electrical and Computer Engineering |

Supervisor | PROFESSOR EMERITUS Robert Adler |

Full Thesis text |

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.