Weyl's tube formula

From HandWiki

Weyl's tube formula gives the volume of an object defined as the set of all points within a small distance of a manifold. Let Σ be an oriented, closed, two-dimensional surface, and let Nε(Σ) denote the set of all points within a distance ε of the surface Σ. Then, for ε sufficiently small, the volume of Nε(Σ) is

V=2A(Σ)ε+4π3χ(Σ)ε3,

where A(Σ) is the area of the surface and χ(Σ) is its Euler characteristic. This expression can be generalized to the case where Σ is a q-dimensional submanifold of n-dimensional Euclidean space n.

References