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.
