In 1939 H. Weyl proved the following non-trivial theorem. Let $(M^n, g)$ be a **closed** smooth Riemannian manifold. Fix an isometric imbedding $\iota\colon M\to \mathbb{R}^N$ into a Euclidean space (now such an imbedding is known to exist e.g. by Nash Theorem; Weyl actually used weaker results on local imbeddings). Consider the volume in $\mathbb{R}^N$ of the $\varepsilon$-neighborhood of $\iota(M)$. The first (elementary) observation of Weyl was that this volume is a polynomial in $\varepsilon$ for small values of $\varepsilon$:
$$vol\_N(\iota(M)\_\varepsilon)=\varepsilon^{N-n}\sum\_{i=0}^n \kappa\_{i,n,N} V\_i(M)\varepsilon^{n-i},$$
where $\kappa\_{i,n,N}$ are appropriately chosen normalizing coefficients depending only on $i,n,N$ but not on $(M,g)$. **The non-trivial result of Weyl says that $V\_i(M)$ are independent of the imbedding $\iota$.** More precisely he has shown that $V\_i(M)$ equals to the integral over $M$ of some explicit polynomial in the Riemann curvature tensor. 



> 
> Now let $(M,g)$ be a compact smooth Riemannian manifold **with boundary**. One can chose as previously an isometric imbedding $\iota\colon M\to \mathbb{R}^N$. It is well known (and not hard to show) that, as previously, volume of the $\varepsilon$-neighborhood of $\iota(M)$ is a polynomial for small $\varepsilon$. **Is it true that its coefficients (appropriately normalized) are independent of the imbedding $\iota$?** If yes, what are explicit expressions for them? A reference would be most welcome. 
> 
> 
> 



Weyl tube formula for manifolds with boundary
I was shown that the positive answer to my question (in fact even for manifolds not only with boundary but even with corners) follows from ther recent stronger result: Theorem 3.11 in the paper by J. Fu and T. Wannerer
<https://arxiv.org/abs/1711.02155>


