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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.

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  • $\begingroup$ Yes it is true, but do not ask me to write a formula. Look at the proof of Weyl tube formula and try to modify it (it is straightforward). $\endgroup$ – Anton Petrunin Aug 6 '19 at 19:53
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A manifold with boundary is a set with positive reach. The tube formula and kinematic formulas for such sets were described by Federer 60 years ago in this paper.

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I think Alfred Gray's book "Tubes" https://www.springer.com/gp/book/9783764369071 is relevant. See also https://www.sciencedirect.com/science/article/pii/0040938382900052 (Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, by the same author).

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    $\begingroup$ Gray’s book does not contain an answer to my question. I should check the second reference you mentioned. $\endgroup$ – makt Jul 31 '19 at 7:23
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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

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