The Proof of Weyl’s Tube Formula

Abstract

In Theorem 3.15 we wrote down a formula for the volume $V_P^{\mathbb{R}^n } (r)$ of a tube about a submanifold P of Euclidean space $\mathbb{R}^n.$ Although this formula has a great deal -of interest, it is not our principal concern, because the integrand is a function of the second fundamental form T of P. As Weyl says in [Weyll]:

So far we have hardly done more than what could have been accomplished by any student in a course of calculus.