Variations of Weyl's tube formula

TL;DR

This paper investigates whether Weyl's tube formula, originally dependent only on the Riemannian metric for spherical tubes, can be extended to tubes with more general cross sections under symmetry conditions.

Contribution

It demonstrates that Weyl's intrinsic tube volume formula can be generalized to non-round cross sections given certain symmetry conditions.

Findings

Weyl's formula extends to symmetric cross sections

Intrinsic volume depends on the Riemannian metric under conditions

Generalization broadens applicability of tube volume calculations

Abstract

In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. Can this intrinsic nature of the tube volume be preserved for tubes with more general cross sections D than the round ball? Under sufficiently strong symmetry conditions on D the answer turns out to be yes.