5
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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). $\endgroup$ Aug 6, 2019 at 19:53

3 Answers 3

4
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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).

$\endgroup$
1
  • 2
    $\begingroup$ Gray’s book does not contain an answer to my question. I should check the second reference you mentioned. $\endgroup$
    – asv
    Jul 31, 2019 at 7:23
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.