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

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

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

References

• Weyl, Hermann (1939). "On the volume of tubes". American Journal of Mathematics. 61: 461–472. JSTOR 2371513.
• Gray, Alfred (2004). "An introduction to Weyl's Tube Formula". Tubes. Progress in Mathematics, volume 221. Springer Science+Business Media. doi:10.1007/978-3-0348-7966-8_1. ISBN 978-3-0348-9639-9.
• Willerton, Simon (2010-03-12). "Intrinsic Volumes and Weyl's Tube Formula". The n-Category Café. Retrieved 2018-03-10.