Weyl's tube formula

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 $\mathrm{\Sigma }$ be an oriented, closed, two-dimensional surface, and let $N_{\epsilon }(\mathrm{\Sigma })$ denote the set of all points within a distance $\epsilon$ of the surface $\mathrm{\Sigma }$. Then, for $\epsilon$ sufficiently small, the volume of $N_{\epsilon }(\mathrm{\Sigma })$ is

$V=2A(\mathrm{\Sigma })\epsilon +\frac{4\pi }{3}\chi (\mathrm{\Sigma }){\epsilon }^3,$

where $A(\mathrm{\Sigma })$ is the area of the surface and $\chi (\mathrm{\Sigma })$ is its Euler characteristic. This expression can be generalized to the case where $\mathrm{\Sigma }$ is a $q$-dimensional submanifold of $n$-dimensional Euclidean space ${ℝ}^n$.

References

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