# Variations of Weyl’s tube formula

Annegret Burtscher† and  Gert Heckman†

###### Abstract.

In 1939 Weyl showed that the volume of spherical tubes around compact
submanifolds $M$ of Euclidean space depends solely on the induced Riemannian
metric on $M$. Can this intrinsic nature of the tube volume be preserved for
tubes with more general cross sections $\mathbb{D}$ than the round ball? Under
sufficiently strong symmetry conditions on $\mathbb{D}$ the answer turns out
to be yes.

###### Key words and phrases:

Tubes, volumes, curvature, invariant theory, finite reflection groups

###### 2020 Mathematics Subject Classification:

53A07 (primary), 53A55, 20G05 (secondary)

†Department of Mathematics, IMAPP, Radboud University Nijmegen, The
Netherlands, Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
_Acknowledgements:_ Research of the first author supported by the Dutch
Research Council (NWO), Project number VI.Veni.192.208. Part of this material
is based upon work supported by the Swedish Research Council under grant no.
2016-06596 while the first author was in residence at Institut Mittag-Leffler
in Djursholm, Sweden in the Fall of 2019.

##  1\. Introduction

Let us be given a compact connected manifold $M$ (possibly with a boundary) of
dimension $n$ embedded in $\mathbb{R}^{n+m}$ as submanifold of codimension
$m$. For each $r\in M$ we have an orthogonal decomposition $T_{r}M\oplus
N_{r}M$ of $\mathbb{R}^{m+n}$ into tangent space and normal space at $r$ of
$M$. It was shown by Weyl [Weyl1939a] that the Euclidean volume of the
spherical tube

$\\{r+n\,;r\in M,\,n\in N_{r}M,\,|n|\leq a\\}$

around $M$ with radius $a>0$ sufficiently small is equal to

$V_{M}(a)=\Omega_{m}\;\sum_{d=0}^{n}\frac{k_{d}(M)\,a^{m+d}}{(m+2)\cdots(m+d)}\qquad(d\;\mathrm{even})$

with $\Omega_{m}$ the volume of the unit ball $\mathbb{B}^{m}=\\{t\,;|t|\leq
1\\}$ in $\mathbb{R}^{m}$.

The remarkable insight of Weyl is that the coefficients $k_{d}(M)$ are
integral invariants of $M$ only determined by the _intrinsic_ metric nature of
$M$. For example, the initial coefficient $k_{0}(M)=\int_{M}ds$ is the
Riemannian volume of $M$ and the next coefficient is
$k_{2}(M)=\tfrac{1}{2}\int_{M}S\,ds$ with $S$ the scalar curvature of $M$. If
$M$ has empty boundary and is of even dimension it was proved by Allendoerfer
and Weil [Allendoerfer--Weil1943] in their approach towards the Gauss–Bonnet
theorem that the top coefficient $k_{n}(M)=(2\pi)^{n/2}\chi(M)$ with $\chi(M)$
the Euler characteristic of $M$ is even of topological nature. See also the
text books of Gray on tubes [Gray2004] and of Morvan on generalized curvatures
[Morvan2008] for further details.

Due to the _local_ nature of the tube formula we can assume that the
submanifold $M$ of $\mathbb{R}^{n+m}$ comes with a chosen orthonormal frame in
the normal bundle $NM$ of $M$ in $\mathbb{R}^{n+m}$. In turn this gives for
all $r\in M$ an identification of the normal space $N_{r}M$ with
$\mathbb{R}^{m}$, and so for $\mathbb{D}^{m}$ a compact domain around $0$ in
$\mathbb{R}^{m}$ we can consider the _generalized tube_

$\\{r+n\,;r\in M,\,n\in a\mathbb{D}^{m}\\}$

around $M$ of type $a\mathbb{D}^{m}$ for $a>0$ sufficiently small. The main
result of this paper is that under sufficiently strong (relative to the
dimension $n$ of $M$) symmetry requirements on the domain $\mathbb{D}^{m}$ a
similar intrinsic formula for the volume $V(a)$ of the above generalized tube
remains valid as in Weyl’s case where $\mathbb{D}^{m}$ equals the unit ball
$\mathbb{B}^{m}$.

Our generalized tubes share the feature that the domains $\mathbb{D}^{m}$ are
invariant under the following subgroups of the orthogonal group.

######  Definition 1.1.

A subgroup $G_{m}$ of the orthogonal group $\mathrm{O}_{m}(\mathbb{R})$ on
$\mathbb{R}^{m}$ is called _orthogonal of degree $n$_ if any polynomial
$p(t)\in\mathbb{R}[t]$ on $\mathbb{R}^{m}$ of degree $\leq n$ that is
invariant under $G_{m}$ is in fact invariant under the full orthogonal group
$\mathrm{O}_{m}(\mathbb{R})$.

Our principal result is the following generalized tube formula.

######  Theorem 1.2.

Let $M$ be a compact connected manifold of dimension $n$ embedded in Euclidean
space $\mathbb{R}^{n+m}$. If the compact domain $\mathbb{D}^{m}$ around $0$ in
$\mathbb{R}^{m}$ has a symmetry group $G_{m}$ inside
$\mathrm{O}_{m}(\mathbb{R})$ that is orthogonal of degree $n$ then the volume
of the generalized tube of type $a\mathbb{D}^{m}$ for $a>0$ sufficiently small
is given by

$V_{M}(a)=\sum_{d=0}^{n}\frac{\\{\int_{\mathbb{D}^{m}}|t|^{d}\,dt\\}\,k_{d}(M)\,a^{m+d}}{m(m+2)\cdots(m+d-2)}\qquad(d\;\mathrm{even})$

with intrinsic coefficients $k_{d}(M)=\int_{M}H_{d}\,ds$ as specified in
Theorem 4.1.

In Sections 2–4 we shall review the proof of the tube formula following Weyl’s
original approach and along the way obtain variations of the tube volume
formula for polyhedral (and related) tubes rather than spherical tubes. We
discuss several examples in Section 5 and counterexamples in Section 6, as
well as causal tubes in Minkowski space in Section 7.

After this paper was finished we learned that tube volume formulas with more
general cross sections $\mathbb{D}^{m}$ had already been studied before by
Domingo-Juan and Miquel [Domingo-Juan--Miquel2004]. We decided to leave our
paper as it was, but add Section 8 in order to briefly survey their approach
and compare their results with ours.

##  2\. The volume of tubes

Locally a submanifold $M$ of dimension $n$ in $\mathbb{R}^{n+m}$ is given in
the Gaussian approach by a parametrization

$r\colon U^{n}\rightarrow M\subset\mathbb{R}^{n+m},\quad u\mapsto r(u)$

with $u=(u^{1},\ldots,u^{n})\in U^{n}\subset\mathbb{R}^{n}$ and
$\partial_{i}=\partial/\partial u^{i}$ in the usual notation of differential
geometry. A summation sign $\sum$ without explicit mention of indices always
means a summation over all indices, which occur both as upper and as lower
index. The first fundamental form (or Riemannian metric) is given by

$ds^{2}=\sum g_{ij}du^{i}du^{j},\quad g_{ij}=\partial_{i}r\cdot\partial_{j}r,$

with $\cdot$ the scalar product on the ambient Euclidean space
$\mathbb{R}^{n+m}$ and $g_{ij}=g_{ij}(u)$ a positive definite symmetric matrix
for all $u\in U^{n}$.

Let us choose an orthonormal frame field $u\mapsto n_{1}(u),\ldots,n_{m}(u)$
in the normal bundle of $M$, and so $\partial_{i}r\cdot n_{p}=0$ and
$n_{p}\cdot n_{q}=\delta_{pq}$ along $M$ for all $i=1,\ldots,n$ and
$p,q=1,\ldots,m$. Let $t=(t^{1},\ldots,t^{m})$ be Cartesian coordinates on
$\mathbb{R}^{m}$. Let us be given a compact domain $\mathbb{D}^{m}$ around $0$
in $\mathbb{R}^{m}$ such that the map

$x\colon U^{n}\times\mathbb{D}^{m}\to\mathbb{R}^{n+m},\quad(u,t)\mapsto
x(u,t)=r(u)+\sum t^{p}n_{p}(u)$

is a diffeomorphism of $U^{n}\times\mathbb{D}^{m}$ onto its image in
$\mathbb{R}^{n+m}$. This image is called a tube of type $\mathbb{D}^{m}$
around $r(U^{n})$.

We are interested in the Euclidean volume $V_{U^{n}}(a)$ of the local tube

$\\{x(u,t)=r(u)+\sum t^{p}n_{p}(u)\,;u\in U^{n},\;t\in
a\mathbb{D}^{m}\\}\subset\mathbb{R}^{n+m}$

of type $a\mathbb{D}^{m}$ as a function of a small positive parameter $a>0$.
By the Jacobi substitution theorem we have

$V_{U^{n}}(a)=\int_{U^{n}}\\{\int_{a\mathbb{D}^{m}}J(u,t)\,dt\\}\,du$

with $J$ the absolute value of the determinant

$\det(\partial_{1}x\;\cdots\;\partial_{n}x\;n_{1}\;\cdots\;n_{m})$

and $\partial_{i}x=\partial_{i}r+\sum t^{p}\partial_{i}n_{p}$ for
$i=1,\ldots,n$.

Recall that

$\partial_{i}\partial_{j}r=\sum\Gamma_{ij}^{k}\partial_{k}r+\sum
h_{ij}^{p}n_{p}$

with $\Gamma_{ij}^{k}=\Gamma_{ij}^{k}(u)$ the Christoffel symbols and
$h_{ij}^{p}=h_{ij}^{p}(u)$ the coefficients of the second fundamental form
$h_{ij}$ relative to the orthonormal normal frame $n_{p}$. Here indices
$p,q=1,\ldots,m$ are coordinate indices in the normal direction, while the
other indices $i,j,k=1,\ldots,n$ are coordinate indices on the submanifold
$M$. Since $\partial_{j}r\cdot n_{p}=0$ we get

$\partial_{i}n_{p}\cdot\partial_{j}r=-n_{p}\cdot\partial_{i}\partial_{j}r=-\sum\delta_{pq}h_{ij}^{q}$

with $\delta_{pq}$ the Kronecker symbol. Writing $t_{p}=\sum\delta_{pq}t^{q}$
we find

$\partial_{i}x=\partial_{i}r-\sum\delta_{pq}t^{p}h_{ik}^{q}g^{kj}\partial_{j}r+\ldots=\sum(\delta_{i}^{j}-\sum
t_{p}h_{ik}^{p}g^{kj})\partial_{j}r+\ldots$

with $g_{ij}=\partial_{i}r\cdot\partial_{j}r$, $g^{ij}$ its inverse matrix,
$\det g=\det(g_{ij})$ its determinant and $\ldots$ stands for a linear
combination of the normal fields $n_{p}$. If in the usual notation we write
$h_{i}^{jp}=\sum g^{jk}h_{ik}^{p}$ for $i,j=1,\ldots,n$ and $p=1,\ldots,m$
then we get

$\displaystyle\det(\partial_{1}x\;\cdots\;\partial_{n}x\;n_{1}\;\cdots\;n_{m})=$
$\displaystyle\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\det(\partial_{1}r\;\cdots\;\partial_{n}r\;n_{1}\;\cdots\;n_{m})$

which in turn implies

$V_{U^{n}}(a)=\int_{U^{n}}\\{\int_{a\mathbb{D}^{m}}\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\,dt\\}\sqrt{\det g_{ij}}\,du$

for all $a>0$ sufficiently small.

For fixed $u\in U^{n}$ the integrand $\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})$ is a polynomial in $t$ of degree $n$ and so, after
integration and patching together the locally defined tubes, we conclude that

$V_{M}(a)=\sum_{d=0}^{n}v_{d}\,a^{m+d}$

is a polynomial in $a$ of degree $m+n$ and with coefficient
$v_{0}=\mathrm{vol}(\mathbb{D}^{m})\mathrm{vol}(M)$. If the domain
$\mathbb{D}^{m}$ is centrally symmetric with respect the origin, that is if
$-\mathbb{D}^{m}=\mathbb{D}^{m}$, then the integrals of odd degree monomials
in $t$ over $\mathbb{D}^{m}$ vanish and so $v_{d}=0$ for $d$ odd.

In order to show that the volume $V_{M}(a)$ of a generalized tube of type
$\mathbb{D}^{m}$ depends only on intrinsic quantities of $M$, two steps are
necessary. Firstly, by assuming that $M$ is embedded in flat
$\mathbb{R}^{n+m}$, one observes that certain combinations of the second
fundamental forms are intrinsic curvature quantities (this was already done by
Weyl). Secondly, by imposing certain symmetry conditions on $\mathbb{D}^{m}$,
we show that only those intrinsic combinations remain in the volume formula
$V_{M}(a)$ for the generalized tube (done by Weyl for the ball
$\mathbb{B}^{m}$). These steps are carried out in Sections 3 and 4,
respectively.

##  3\. The Gauss equations

As before, we write

$\partial_{i}\partial_{j}r=\sum\Gamma_{ij}^{k}\partial_{k}r+h_{ij}$

with $\Gamma_{ij}^{k}$ the Christoffel symbols given by

$\tfrac{1}{2}\sum
g^{kl}(\partial_{i}g_{jl}+\partial_{j}g_{il}-\partial_{l}g_{ij})$

and $h_{ij}=\sum h_{ij}^{p}n_{p}$ the second fundamental form relative to the
orthonormal frame $n_{p}$ in the normal bundle along $M$. Given scalar
functions $g_{ij}$ and $h_{ij}^{p}$, the integrability conditions for the
existence of an embedding of $M$ into flat Euclidean space with these
functions as coefficients of the first and second fundamental forms are given
by

$\partial_{i}(\sum\Gamma_{jk}^{l}\partial_{l}r+\sum
h_{jk}^{p}n_{p})-\partial_{j}(\sum\Gamma_{ik}^{l}\partial_{l}r+\sum
h_{ik}^{p}n_{p})=0$

for all $i,j,k$ (by working out
$\partial_{i}(\partial_{j}\partial_{k}r)-\partial_{j}(\partial_{i}\partial_{k}r)=0$).

In the normal directions this leads to the Codazzi–Mainardi equations

$\partial_{i}h_{jk}^{p}-\partial_{j}h_{ik}^{p}+\sum(\Gamma_{jk}^{l}h_{il}^{p}-\Gamma_{ik}^{l}h_{jl}^{p})=0$

for all $i,j,k$ and all $p$. In the tangential directions this amounts to the
Gauss equations

$R_{kij}^{l}=\sum\delta_{pq}g^{ln}(h_{in}^{p}h_{jk}^{q}-h_{jn}^{p}h_{ik}^{q})$

for all $i,j,k,l$ with

$R_{kij}^{l}=\partial_{i}\Gamma_{kj}^{l}-\partial_{j}\Gamma_{ki}^{l}+\sum(\Gamma_{kj}^{m}\Gamma_{mi}^{l}-\Gamma_{ki}^{m}\Gamma_{mj}^{l})$

the coefficients of the Riemann curvature tensor. As mentioned earlier, by
raising indices $h_{i}^{jp}=\sum g^{jk}h_{ki}^{p}$ and $R_{ij}^{kl}=\sum
g^{ln}R_{nij}^{k}$ the Gauss equations take the form

$R_{ij}^{kl}=\sum\delta_{pq}(h_{i}^{kp}h_{j}^{lq}-h_{j}^{kp}h_{i}^{lq})=h_{i}^{k}\cdot
h_{j}^{l}-h_{j}^{k}\cdot h_{i}^{l}$

for all $i,j,k,l$ and $h_{i}^{j}=\sum h_{i}^{jp}n_{p}=\sum g^{jk}h_{ki}$
normal vectors along $M$.

##  4\. Averaging the integrand

For $p(t)\in\mathbb{R}[t_{1},\cdots,t_{m}]$ a polynomial on Euclidean space
$\mathbb{R}^{m}$ and $G_{m}$ a closed subgroup of the orthogonal group
$\mathrm{O}_{m}(\mathbb{R})$ let us write

$\langle p(t)\rangle_{G_{m}}=\int_{G_{m}}\;p(gt)\,d\mu(g)$

for the average of $p$ over $G_{m}$, with $\mu$ the normalized Haar measure on
$G_{m}$. Clearly

$\langle p(t)\rangle_{\mathrm{O}_{m}(\mathbb{R})}\in\mathbb{R}[t\cdot t]$

with $t\cdot t=|t|^{2}$ the norm squared of $t\in\mathbb{R}^{m}$. The crucial
step for the intrinsic nature of the coefficients of the tube volume formula
is the following result (see the Lemma on page 470 of Weyl’s paper
[Weyl1939a]).

######  Theorem 4.1.

We have (with $1\leq i,j\leq n$ and $1\leq p\leq m$)

$\langle\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\rangle_{\mathrm{O}_{m}(\mathbb{R})}=\sum_{d=0}^{n}\frac{H_{d}\,|t|^{d}}{m(m+2)\cdots(m+d-2)}$

with $H_{d}$ intrinsic functions on $M$ given by

$\displaystyle H_{d}=\begin{cases}0&\text{if}~{}d~{}\text{odd},\\\
1&\text{if}~{}d=0,\\\ \sum\varepsilon_{i_{1}\ldots i_{d}}^{j_{1}\ldots
j_{d}}\;R_{i_{1}i_{2}}^{j_{1}j_{2}}\cdots
R_{i_{d-1}i_{d}}^{j_{d-1}j_{d}}&\text{if}~{}d>0~{}\text{even},\end{cases}$

and

$R_{ij}^{kl}=H_{ij}^{kl}-H_{ji}^{kl}\;\;,\;\;H_{ij}^{kl}=h_{i}^{k}\cdot
h_{j}^{l}=\sum\delta_{pq}h_{i}^{kp}h_{j}^{lq}$

for $i,j,k,l=1,\ldots,n$. In the expression for $H_{d}$ with $d>0$ even the
sum runs over all cardinality $d$ subsets $\mathcal{D}$ of $\\{1,\ldots,n\\}$
and over all possible couplings of pairs

${}_{i_{1}i_{2}}^{j_{1}j_{2}}\,|\,_{i_{3}i_{4}}^{j_{3}j_{4}}\,|\,\cdots\,|_{i_{d-1}i_{d}}^{j_{d-1}j_{d}}\,|$

taken from $\mathcal{D}=\\{i_{1},\dots,i_{d}\\}=\\{j_{1},\dots,j_{d}\\}$. Here
a pair $i_{1}i_{2}$ means two distinct numbers $i_{1},i_{2}$ irrespective of
their order.

###### Proof.

Averaging the characteristic polynomial $\det(\lambda\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})$ over the orthogonal group $\mathrm{O}_{m}(\mathbb{R})$
acting on $t\in\mathbb{R}^{m}$ yields

$\langle\det(\lambda\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\rangle_{\mathrm{O}_{m}(\mathbb{R})}=\sum_{d=0}^{n}\,{\lambda}^{n-d}\,|t|^{d}\sum_{\mathcal{D}}\,A_{\mathcal{D}}(h_{i}^{j})$

with the sum over all even $d$ and all cardinality $d$ subsets $\mathcal{D}$
of $\\{1,\ldots,n\\}$ and with $A_{\mathcal{D}}(h_{i}^{j})$ given by

$\langle\det(t\cdot
h_{i}^{j})_{i,j\in\mathcal{D}}\rangle_{\mathrm{O}_{m}(\mathbb{R})}=|t|^{d}A_{\mathcal{D}}(h_{i}^{j})$

as degree $d$ polynomial on $\mathbb{R}^{m\times d^{2}}$, which is invariant
under the diagonal action of $\mathrm{O}_{m}(\mathbb{R})$. By the first
fundamental theorem of invariant theory for $\mathrm{O}_{m}(\mathbb{R})$ (see
Corollary 4.2.3 of [Goodman--Wallach1998], which is a modern reincarnation of
Weyl’s classic [Weyl1939b]) we have

$A_{\mathcal{D}}(h_{i}^{j})=B_{\mathcal{D}}(H_{ij}^{kl})$

with $H_{ij}^{kl}=h_{i}^{k}\cdot h_{j}^{l}$ and $i,j,k,l\in\mathcal{D}$ with
$i\neq j,k\neq l$. From the explicit determinantal form it follows that
$B_{\mathcal{D}}(H_{ij}^{kl})$ is in fact a linear combination of monomials of
the form

$H_{i_{1}i_{2}}^{j_{1}j_{2}}\cdots H_{i_{d-1}i_{d}}^{j_{d-1}j_{d}}$

with $\mathcal{D}=\\{i_{1},\dots,i_{d}\\}=\\{j_{1},\dots,j_{d}\\}$. Moreover
under the action of the symmetric group $\mathfrak{S}_{d}$ acting on both the
lower and the upper indices $B_{\mathcal{D}}(H_{ij}^{kl})$ transforms under
the sign character. Therefore
$B_{\mathcal{D}}(H_{ij}^{kl})=C_{\mathcal{D}}(R_{ij}^{kl})$ with
$R_{ij}^{kl}=H_{ij}^{kl}-H_{ji}^{kl}$ and by symmetry for $\mathfrak{S}_{d}$
we arrive at

$C_{\mathcal{D}}(R_{ij}^{kl})=c(m,d)\sum\varepsilon_{i_{1}\ldots
i_{d}}^{j_{1}\ldots j_{d}}\;R_{i_{1}i_{2}}^{j_{1}j_{2}}\cdots
R_{i_{d-1}i_{d}}^{j_{d-1}j_{d}}$

with the sum over all possible couplings of pairs from $\mathcal{D}$ and
$c(m,d)$ a constant depending solely on $m$ and $d$. The conclusion is that

$\langle\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\rangle_{\mathrm{O}_{m}(\mathbb{R})}=\sum_{d=0}^{n}\,c(m,d)H_{d}\,|t|^{d}$

and all that is left is the computation of the constant $c(m,d)$.

For this computation we take the special choice $h_{i}^{jp}=\delta_{i}^{j}$
for $p=1$ and $h_{i}^{jp}=0$ for $p\geq 2$. In that case

$A_{\mathcal{D}}(h_{i}^{j})=\frac{\int_{S^{m-1}}\,t_{1}^{d}\,d\mu(t)}{\int_{S^{m-1}}\,d\mu(t)}$

with $\mu$ the Euclidean measure on the unit sphere $S^{m-1}$ in
$\mathbb{R}^{m}$. The integral in the numerator becomes

$\int_{-1}^{1}\,r^{d}(1-r^{2})^{(m-3)/2}dr=\int_{0}^{1}\,s^{(d-1)/2}(1-s)^{(m-3)/2}ds$

(apart from a factor volume $\omega_{m-1}$ of $S^{m-2}$) and so equals

$\mathrm{B}((d+1)/2,(m-1)/2)=\frac{\Gamma((d+1)/2)\Gamma((m-1)/2)}{\Gamma((d+m)/2)}\,.$

In turn this implies

$A_{\mathcal{D}}(h_{i}^{j})=\frac{\Gamma((d+1)/2)\Gamma(m/2)}{\Gamma(1/2)\Gamma((d+m)/2)}=\frac{1\cdot
3\cdots(d-1)}{m(m+2)\cdots(m+d-2)}\,.$

On the other hand
$R_{ij}^{kl}=\delta_{i}^{k}\delta_{j}^{l}-\delta_{j}^{k}\delta_{i}^{l}$ and so
equal to $\varepsilon_{ij}^{kl}$ if the pairs $ij$ and $kl$ coincide and $0$
otherwise, and hence

$C_{\mathcal{D}}(R_{ij}^{kl})=c(m,d)\,\frac{d!}{2^{d/2}(d/2)!}=c(m,d)\cdot
1\cdot 3\cdots(d-1)\,.$

Hence $c(m,d)=1/m(m+2)\cdots(m+d-2)$ as desired. ∎

Recall that the volume $\omega_{m}$ of the unit sphere $S^{m-1}$ and the
volume $\Omega_{m}$ of the unit ball $\mathbb{B}^{m}$ are related by
$\Omega_{m}=\omega_{m}/m$. The tube formula of Weyl can now be easily derived.

######  Corollary 4.2.

If the domain $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$ is equal to the unit ball
$\mathbb{B}^{m}$ then the tube volume is given by

$V_{M}(a)=\Omega_{m}\sum_{d=0}^{n}\frac{k_{d}(M)\,a^{m+d}}{(m+2)\cdots(m+d)}\qquad(d\;\mathrm{even})$

for $a>0$ small and $k_{d}(M)=\int_{M}H_{d}\,ds$ with $H_{d}$ the intrinsic
expression on $M$ in the previous theorem and $ds$ the Riemannian measure on
$M$.

###### Proof.

By Section 2 and the symmetry of $\mathbb{B}^{m}$ we have

$\displaystyle V_{U^{n}}(a)$
$\displaystyle=\int_{U^{n}}\\{\int_{a\mathbb{B}^{m}}\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\,dt\\}\sqrt{\det g_{ij}}\,du$
$\displaystyle=\int_{U^{n}}\\{\int_{a\mathbb{B}^{m}}\sum_{d=0}^{n}\frac{H_{d}\,|t|^{d}}{m(m+2)\cdots(m+d-2)}\,dt\\}\sqrt{\det
g_{ij}}\,du$
$\displaystyle=\int_{U^{n}}\\{\omega_{m}\int_{0}^{a}\sum_{d=0}^{n}\frac{H_{d}\,r^{m+d-1}}{m(m+2)\cdots(m+d-2)}\,dr\\}\sqrt{\det
g_{ij}}\,du$
$\displaystyle=\Omega_{m}\sum_{d=0}^{n}\frac{\\{\int_{U^{n}}H_{d}\,ds\\}\,a^{m+d}}{(m+2)\cdots(m+d-2)(m+d)}$

and the result follows. ∎

If we consider domains $\mathbb{D}^{m}$ with symmetry groups
$G_{m}<\mathrm{O}_{m}(\mathbb{R})$ such that the invariant polynomials of
degree $\leq n=\dim M$ for both groups agree, then we can prove Theorem 1.2.

###### Proof of Theorem 1.2.

By the Fubini theorem we have for $H<G$ compact groups and $f$ a continuous
function on $G$ that

$\int_{G}\,f(g)\,d\mu_{G}(g)=\int_{G/H}\\{\int_{H}\,f(gh)\,d\mu_{H}(h)\\}\,d\mu_{G/H}(gH)$

with $\mu_{G},\mu_{H}$ and $\mu_{G/H}$ the normalized invariant measures on
$G,H$ and $G/H$ respectively. Hence by the assumption on $G_{m}$ we have

$\langle\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\rangle_{G_{m}}=\langle\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\rangle_{\mathrm{O}_{m}(\mathbb{R})}$

and so we can just argue as in the previous proof. ∎

Using the discussion in Section 2, for $n=1$ the tube formula is intrinsic as
long as $\mathbb{D}^{m}$ is centrally symmetric, the case already covered by
Hotelling [Hotelling1939] if $\mathbb{D}^{m}=\mathbb{B}^{m}$.

######  Corollary 4.3.

If $M$ is a curve of finite length in $\mathbb{R}^{m+1}$ and $\mathbb{D}^{m}$
is centrally symmetric, then for $a>0$ sufficiently small

$V_{M}(a)=\mathrm{length}(M)\,\mathrm{vol}(\mathbb{D}^{m})\,a^{m}$

and hence is intrinsic. ∎

The following example shows that central symmetry is not a necessary
condition.

######  Example 4.4.

Let $\mathbb{D}^{m}$ be the union of half the unit ball $\\{|t|\leq
1,t_{1}\leq 0\\}$ and the cone
$\\{t_{2}^{2}+\dots+t_{m}^{2}\leq(1-t_{1}/b)^{2},0\leq t_{1}\leq b\\}$ with
top $(b,0,\ldots,0)$ for some $b>0$. By symmetry the average of any linear
function of $t_{2},\ldots,t_{m}$ over $\mathbb{D}^{m}$ equals zero. By direct
computation the average of $t_{1}$ over $\mathbb{D}^{m}$ is equal to zero if
$b=\sqrt{m}$, and so the previous corollary remains valid for this domain as
well.

Indeed for curves it is sufficient that the center of mass of $\mathbb{D}^{m}$
is at the origin by the generalized Pappus centroid theorem. See Section 8 for
the higher dimensional case.

##  5\. Examples of polyhedral domains $\mathbb{D}^{m}$

If we are looking for domains $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$ with a
sufficiently large symmetry group $G_{m}<\mathrm{O}_{m}(\mathbb{R})$ it is
natural to consider regular polytopes $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$. It
is well known that the symmetry group $G_{m}$ in that case is an irreducible
finite reflection group. Such groups are classified by their Coxeter diagrams
or by letters $\mathrm{X}_{m}$ with
$\mathrm{X}=\mathrm{A},\mathrm{B},\mathrm{D},\mathrm{E},\mathrm{F},\mathrm{H},\mathrm{I}(k)$
for $k\geq 5$. The corresponding reflection groups are denoted by
$G_{m}=W(\mathrm{X}_{m})$.

It is a well known theorem due to Shephard and Todd [Shephard--Todd1954] (with
a case by case proof) and Chevalley [Chevalley1955] (with a proof from the
Book) that the algebra of polynomial invariants for a finite reflection group
$W<\mathrm{O}(\mathbb{R}^{m})$ is itself a polynomial algebra.

######  Theorem 5.1.

The algebra $\mathbb{R}[\mathbb{R}^{m}]^{W}$ of polynomial invariants for $W$
is of the form $\mathbb{R}[p_{1},\ldots,p_{m}]$ with $p_{1},\ldots,p_{m}$
algebraically independent homogeneous invariants of degrees
$d_{1}\,d_{2},\ldots,d_{m}$ respectively.

For each of the irreducible types these degrees can be calculated and are
given in the next table. The proof of these results can be found in the
standard text books by Bourbaki [Bourbaki1968] or by Humphreys
[Humphreys1990].

$\mathrm{type}$ | $m$ | $d_{1},d_{2},\ldots,d_{m}$
---|---|---
$\mathrm{A}_{m}$ | $\geq 1$ | $2,3,\ldots,m+1$
$\mathrm{B}_{m}$ | $\geq 2$ | $2,4,\ldots,2m$
$\mathrm{D}_{m}$ | $\geq 4$ | $2,4,\ldots,2m-2,m$
$\mathrm{E}_{6}$ | $6$ | $2,5,6,8,9,12$
$\mathrm{E}_{7}$ | $7$ | $2,6,8,10,12,14,18$
$\mathrm{E}_{8}$ | $8$ | $2,8,12,14,18,20,24,30$
$\mathrm{F}_{4}$ | $4$ | $2,6,8,12$
$\mathrm{H}_{3}$ | $3$ | $2,6,10$
$\mathrm{H}_{4}$ | $4$ | $2,12,20,30$
$\mathrm{I}_{2}(k)$ | $2$ | $2,k\geq 5$

So for $m\geq 2$ the irreducible finite reflection group
$W(\mathrm{X}_{m})<\mathrm{O}(\mathbb{R}^{m})$ is orthogonal of degree
$d_{2}-1$ in the sense of Definition 1.1.

######  Corollary 5.2.

If $\mathbb{D}^{m}$ is a domain in $\mathbb{R}^{m}$ invariant under a finite
reflection group $W(\mathrm{X}_{m})$ then the tube formula of Theorem 1.2 does
hold with intrinsic coefficients if $n=\dim M<d_{2}$, that is, the dimension
$n$ of $M$ is strictly smaller than the second fundamental degree $d_{2}$. ∎

For example, if $\mathbb{D}^{3}$ is an icosahedron with symmetry group
$W(\mathrm{H}_{3})$ then the tube formula is intrinsic for submanifolds $M$ of
dimension $n\leq 5$ in $\mathbb{R}^{n+3}$, and if $\mathbb{D}^{4}$ is a
$600$-cell with symmetry group $W(\mathrm{H}_{4})$ then the tube formula is
intrinsic for $n\leq 11$. For any dimension $n$ of
$M\hookrightarrow\mathbb{R}^{n+2}$ with $\mathbb{D}^{2}$ a regular $k$-gon
with $k>n$ the tube formula is intrinsic, since its symmetry group is
$W(\mathrm{I}_{2}(k))$. For dimension $n=2$ or $3$ we find in this way
examples of intrinsic tube formulas for arbitrary codimension $m$ via domains
$\mathbb{D}^{m}$ with symmetry groups $W(\mathrm{A}_{m})$ ($n=2$) and
$W(\mathrm{B}_{m})$ ($n=2,3$), respectively. However, for dimension $n\geq 4$
we obtain in this way only examples of intrinsic tube formulas with relatively
small codimension $m\leq 8$.

Examples with larger codimension $m$ can be obtained by the following
construction.

######  Corollary 5.3.

Let $G$ be a noncompact simple Lie group acting on its Lie algebra
$\mathfrak{g}$, and let $\theta$ be a Cartan involution of $G$ and
$\mathfrak{g}$ and $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ the
decomposition in $+1$ and $-1$ eigenspaces of $\theta$ on $\mathfrak{g}$. If
the domain $\mathbb{D}\subset\mathfrak{p}$ is the convex hull of a nonzero
orbit of $K=G^{\theta}$ on $\mathfrak{p}$ then the tube formula of Theorem 1.2
does hold with intrinsic coefficients under the assumption that the dimension
$n$ of $M\hookrightarrow\mathbb{R}^{n+m}$ (with $m=\dim\mathfrak{p}$) is
strictly smaller than the second fundamental degree $d_{2}$ of the Weyl group
$W$ of the pair $(\mathfrak{g},\theta)$.

###### Proof.

The Killing form $(\cdot,\cdot)$ on $\mathfrak{p}$ is positive definite and
the fixed point group $K=G^{\theta}$ of $\theta$ on $G$ acts on $\mathfrak{p}$
as a subgroup of $\mathrm{SO}(\mathfrak{p})$. If
$\mathfrak{a}\subset\mathfrak{p}$ is a maximal Abelian subspace then each
orbit of $K$ on $\mathfrak{p}$ intersects $\mathfrak{a}$ in an orbit of the
Weyl group $W=\mathrm{N}_{K}(\mathfrak{a})/\mathrm{Z}_{K}(\mathfrak{a})$ of
the pair $(\mathfrak{g},\theta)$. Hence each invariant polynomial
$p\in\mathbb{R}[\mathfrak{p}]^{K}$ for $K$ on $\mathfrak{p}$ restricts to a
Weyl group invariant polynomial on $\mathfrak{a}$. It is a theorem of
Chevalley (see Lemma $7$ in [Harish-Chandra1958]) that the restriction map

$\mathbb{R}[\mathfrak{p}]^{K}\rightarrow\mathbb{R}[\mathfrak{a}]^{W}$

is an isomorphism of algebras. Since $W$ acts on $\mathfrak{a}$ as a finite
reflection group the latter algebra is described by Theorem 5.1. The possible
finite reflection groups that can occur as such a Weyl group $W$ are those
reflection groups, which can be defined over $\mathbb{Z}$. This means that
$\mathrm{H}_{3}$ and $\mathrm{H}_{4}$ are excluded and only the dihedral types
$\mathrm{I}_{2}(k)=\mathrm{A}_{2},\mathrm{B}_{2},\mathrm{G}_{2}$ for $k=3,4,6$
respectively are allowed. The text books [Helgason1978] and [Helgason1984] by
Helgason give a thorough exposition of the theory. Using the convexity theorem
of Kostant [Kostant1973] it is easy to see that the convex hull of an orbit of
$K$ on $\mathfrak{p}$ intersects $\mathfrak{a}$ in the convex hull of an orbit
of $W$ on $\mathfrak{a}$. ∎

For example, if $G$ is the complex Lie group of type $\mathrm{E}_{8}$ (and so
$K$ is the compact Lie group of type $\mathrm{E}_{8}$ acting on
$\mathfrak{p}=i\mathfrak{k}$) then we do find in this way examples of local
submanifolds $M$ of Euclidean space of dimension $n\leq 7$ and of codimension
$m=248$ for which the tube formula of Theorem 1.2 has intrinsic coefficients.
Presumably this large codimension relative to the small dimension of $M$
allows for an abundance of room for isometric deformations for the embedding
of $M$ in such a Euclidean space.

##  6\. No-go results for diamond domains $\widehat{\mathbb{D}}^{m}$

In this section we shall denote by $\widehat{\mathbb{D}}^{m}$ the convex hull
of the subset

$\\{t_{1}^{2}+\ldots+t_{m-1}^{2}\leq 1,t_{m}=0\\}\sqcup\\{(0,\ldots,0,\pm
1)\\}$

in $\mathbb{R}^{m}$, $m\geq 2$. Any multiple $a\widehat{\mathbb{D}}^{m}$ for
$a>0$ will be called a diamond domain. The symmetry group $G_{m}$ of
$\widehat{\mathbb{D}}^{m}$ is equal to
$\mathrm{O}_{m-1}(\mathbb{R})\times\mathrm{O}_{1}(\mathbb{R})$ for $m\geq 3$
while for $m=2$ the symmetry group $G_{2}$ is equal to the dihedral group
$W(\mathrm{B}_{2})$ of order $8$. The essential point of Weyl’s argument for
the intrinsic nature of the volume formula for tubes is the computation of the
integral

$\int_{a\widehat{\mathbb{D}}^{m}}\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\,dt$

as a polynomial in $a$, by first averaging over the symmetry group $G_{m}$ of
the domain $\widehat{\mathbb{D}}^{m}$ in $\mathbb{R}^{m}$. The outcome should
hopefully be a polynomial expression in Riemann curvature components
$R_{ij}^{kl}$ as in Theorem 4.1. We will work out two examples, one with $n=2$
and $m\geq 3$ and the other with $n=4$ and $m=2$, where this does not work.

######  Example 6.1.

Let us first consider the case of surfaces of codimension $m$ at least $3$.
Since the symmetry group of $\widehat{\mathbb{D}}^{m}$ is then
$G_{m}=\mathrm{O}_{m-1}(\mathbb{R})\times\mathrm{O}_{1}(\mathbb{R})$ the
invariants of degree $2$ in $\mathbb{R}[t_{1},\ldots,t_{m}]$ are linear
combinations of $R=(t_{1}^{2}+\ldots+t_{m-1}^{2})/(m-1)$ and $S=t_{m}^{2}$.
The above determinant for $n=2$ becomes

$\det(\delta_{i}^{j}-\sum t_{p}h_{i}^{jp})=1-\sum
t_{p}(h_{1}^{1p}+h_{2}^{2p})+\sum
t_{p}t_{q}(h_{1}^{1p}h_{2}^{2q}-h_{1}^{2p}h_{2}^{1q})$

and averaging over the symmetry group $G_{m}$ yields

$1+A(h_{i}^{j})R(t)+B(h_{i}^{j})S(t)$

with

$A=\sum_{p=1}^{m-1}(h_{1}^{1p}h_{2}^{2p}-h_{1}^{2p}h_{2}^{1p})\qquad
B=(h_{1}^{1m}h_{2}^{2m}-h_{1}^{2m}h_{2}^{1m})$

and $A+B=R_{12}^{12}$ intrinsic. Thus by the above, if the integrals of $R$
and $S$ over $\widehat{\mathbb{D}}^{m}$ agree, then the generalized tube
volume is intrinsic as well. The integrals of $R(t)$ and $S(t)$ over
$\widehat{\mathbb{D}}^{m}$ amount respectively to (put $r=\sqrt{R}$ and
$s=\sqrt{S}$)

$\frac{2\,\omega_{m-1}}{m-1}\int r^{2}r^{m-2}\,dr\,ds\quad\mathrm{and}\quad
2\,\omega_{m-1}\int s^{2}r^{m-2}\,dr\,ds,$

integrated over the triangle $\\{(r,s)\,;r,s\geq 0,r+s\leq 1\\}$, and we will
show that for $m\geq 3$ these are distinct. Apart from the factor
$2\,\omega_{m-1}$ the left integral becomes

$\frac{1}{m-1}\int_{0}^{1}(1-r)r^{m}\,dr=\frac{1}{(m-1)(m+1)(m+2)}$

while the right integral equals

$\int_{0}^{1}\tfrac{1}{3}(1-r)^{3}r^{m-2}\,dr=\frac{2}{(m-1)m(m+1)(m+2)}$

and for the difference we find

$\frac{m-2}{(m-1)m(m+1)(m+2)}$

which is nonzero for $m\geq 3$, as claimed. Hence the tube volume formula for
a general surface $M$ in $\mathbb{R}^{2+m}$ with diamond domain
$\widehat{\mathbb{D}}^{m}$ is no longer intrinsic for $m\geq 3$. For $m=2$ it
still is intrinsic as should, because $G_{2}=W(\mathrm{B}_{2})$ is orthogonal
of degree $3$ (in fact, it is not only intrinsic for $n=2$ but also for $n=3$
since odd exponents vanish for the centrally symmetric diamond
$\widehat{\mathbb{D}}^{m}$).

######  Example 6.2.

Let us next consider the case that $n=4$ and $m=2$. The symmetry group of the
diamond domain
$\widehat{\mathbb{D}}^{2}=\\{(t_{1},t_{2})\,;|t_{1}|+|t_{2}|\leq 1\\}$ is the
dihedral group $W(\mathrm{B}_{2})$ of order $8$ generated by the two
reflections $s_{1}(t_{1},t_{2})=(-t_{1},t_{2})$ and
$s_{2}(t_{1},t_{2})=(t_{2},t_{1})$. The invariant polynomials for this group
$W(\mathrm{B}_{2})$ are generated as an algebra by the quadratic invariant
$P(t)=t_{1}^{2}+t_{2}^{2}$ and the quartic invariant
$Q(t)=t_{1}^{2}t_{2}^{2}$. Hence any quartic invariant is a unique linear
combination of $Q$ and $R(t)=t_{1}^{4}+t_{2}^{4}=P^{2}-2Q$.

We would like to know if Weyl’s averaging trick (over the dihedral group
$W(\mathrm{B}_{2})$ this time) remains valid for any pencil of second
fundamental forms. In order to keep the calculation as simple as possible we
look at the special case that $h_{i}^{jp}=0$ for $i\neq j$ and $p=1,2$. If we
write $h_{i}^{i1}=a_{i}$ and $h_{i}^{i2}=b_{i}$ we get

$\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})=\prod_{i=1}^{4}\;(1-t_{1}a_{i}-t_{2}b_{i})$

and averaging over the dihedral group $W(\mathrm{B}_{2})$ yields

$1+A(a,b)P(t)+B(a,b)Q(t)+C(a,b)R(t)$

with $A,B,C$ homogeneous polynomials of degree $2,4,4$ respectively. A direct
calculation gives

$\displaystyle A$
$\displaystyle=\tfrac{1}{2}\sum_{i<j}\;(a_{i}a_{j}+b_{i}b_{j})=\tfrac{1}{2}\sum_{i<j}R_{ij}^{ij}$
$\displaystyle B$
$\displaystyle=\sum_{i<j,k<l}\;a_{i}a_{j}b_{k}b_{l}=\sum_{i<j,k<l}R_{ij}^{ij}R_{kl}^{kl}-3(a_{1}a_{2}a_{3}a_{4}+b_{1}b_{2}b_{3}b_{4})$
$\displaystyle C$
$\displaystyle=\tfrac{1}{2}(a_{1}a_{2}a_{3}a_{4}+b_{1}b_{2}b_{3}b_{4})$

with $\\{i,j,k,l\\}=\\{1,2,3,4\\}$ in the sum for $B$ and
$R_{ij}^{ij}=h_{i}^{i}\cdot h_{j}^{j}$ as before. Note that $A$ as well as
$B+6C$ are intrinsic quantities. For the integrals of $Q$ and $R$ over
$\widehat{\mathbb{D}}^{2}$ we find (put $r=|t_{1}|$ and $s=|t_{2}|$)

$\int
Q(t)\,dt_{1}\,dt_{2}=4\int_{0}^{1}\\{\int_{0}^{1-r}r^{2}s^{2}\,ds\\}\,dr=\tfrac{1}{45}$

and

$\int
R(t)\,dt_{1}\,dt_{2}=4\int_{0}^{1}\\{\int_{0}^{1-r}(r^{4}+s^{4})\,ds\\}\,dr=\tfrac{4}{15}\neq\tfrac{6}{45}.$

Hence for fourfolds in $\mathbb{R}^{6}$ with diamond domain
$\widehat{\mathbb{D}}^{2}$ we see that the tube volume formula need no longer
be intrinsic.

The conclusion therefore is that the tube formula for submanifolds $M$ in
$\mathbb{R}^{n+m}$ of dimension $n$ with cross section the diamond
$\widehat{\mathbb{D}}^{m}$ will in general no longer be intrinsic, unless we
are in one of the cases of the following table.

$m=\operatorname{codim}M$ | symmetry group of the diamond in $\mathbb{R}^{m}$ | $n=\dim M$
---|---|---
$1$ | $\mathrm{O}_{1}(\mathbb{R})$ | any
$2$ | $W(\mathrm{B}_{2})$ | $\leq 3$
any | $\mathrm{O}_{m-1}(\mathbb{R})\times\mathrm{O}_{1}(\mathbb{R})$ | $1$

Our motivation for looking at diamond tubes in a Euclidean vector space came
from the analogous causal tubes in a Lorentzian vector space, which are
discussed in the next section.

##  7\. Riemannian submanifolds of a Lorentzian vector space

Let us suppose that $M$ is a compact connected $n$-dimensional Riemannian
submanifold of an ambient Cartesian space $\mathbb{R}^{n+m}$, equipped with a
nondegenerate but possibly indefinite scalar product denoted by a dot. Let
$\mathbb{D}^{m}$ be a compact domain around $0$ in $\mathbb{R}^{m}$. Say we
have a local parametrization around $M$ given by

$x:U^{n}\times\mathbb{D}^{m}\rightarrow\mathbb{R}^{n+m},\quad(u,t)\mapsto
x(u,t)=r(u)+\sum t^{p}n_{p}(u)$

with $u=(u^{1},\ldots,u^{n})\in U^{n}$,
$t=(t^{1},\ldots,t^{m})\in\mathbb{D}^{m}$ while $n_{1}(u),\ldots,n_{m}(u)$ are
vectors in $\mathbb{R}^{n+m}$ depending smoothly on $u\in U^{n}$ and

$\partial_{i}r(u)\cdot n_{p}(u)=0,\quad n_{p}(u)\cdot n_{q}(u)=\eta_{pq}$

for all $u\in U^{n}$, all $i=1,\ldots,n$, all $p,q=1,\ldots,m$ and $\eta_{pq}$
a $m\times m$ diagonal matrix with entries $\pm 1$ (so in particular constant,
that is independent of $u\in U^{n}$). Observe that the choice of such an
orthonormal frame for the normal bundle of $M$ in $\mathbb{R}^{n+m}$ is in
principle only possible locally. Indeed if $0\in U^{n}$ then by linear algebra
we can choose a basis $n_{1}(0),\ldots,n_{m}(0)$ for the orthogonal complement
of the tangent vectors $\partial_{1}r(0),\ldots,\partial_{n}r(0)$ with
$n_{p}(0)\cdot n_{q}(0)=\eta_{pq}$ and subsequently apply Gram–Schmidt to the
vectors $\partial_{1}r(u),\ldots,\partial_{n}r(u),n_{1}(0),\ldots,n_{m}(0)$
for $u$ small.

As in Section 2 we can write

$\partial_{i}x=\sum_{j}(\delta_{i}^{j}-\sum t^{p}n_{p}\cdot
h_{i}^{j})\partial_{j}r+\ldots$

with second fundamental form normal vectors $h_{i}^{j}=\sum g^{jk}h_{ik}$ and
the dots $\ldots$ stand for a linear combination of the normal fields $n_{p}$.
Likewise writing $t_{p}=\sum\eta_{pq}t^{q}$ we arrive at the generalized tube
volume formula

$V_{U^{n}}(a)=\int_{U^{n}}\\{\int_{a\mathbb{D}^{m}}\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\,dt\\}\sqrt{\det g_{ij}}\,du$

with $\mathbb{D}^{m}$ a compact domain around $0$ and $a>0$ sufficiently
small. Hence $V_{M}(a)$ is a polynomial in $a$ of degree $m+n$ with
$\mathrm{vol}(M)\mathrm{vol}(\mathbb{D}^{m})a^{m}$ as lowest order term. The
Gauss equations

$R_{ij}^{kl}=h_{i}^{k}\cdot h_{j}^{l}-h_{j}^{k}\cdot h_{i}^{l}$

as derived in Section 3 remain valid for an indefinite scalar product.

For Riemannian curves $M$ of dimension $n=1$ and a centrally symmetric domain
$\mathbb{D}^{m}$ around $0$ in $\mathbb{R}^{m}$ we get

$V_{M}(a)=\mathrm{length}(M)\mathrm{vol}(\mathbb{D}^{m})a^{m}$

just like the original case of Hotelling [Hotelling1939]. Also, if
$\eta_{pq}=\delta_{pq}$ then we are essentially in the original setting of
Weyl and his spherical tube formula and our variations hold without change.

Let us suppose for the rest of this section that $M$ is a compact Riemannian
submanifold of a Lorentzian vector space $\mathbb{R}^{n+m-1,1}$ with scalar
product $\cdot$ of signature $(n+m-1,1)$ and thus
$\eta_{pq}=\mathrm{diag}(1,\ldots,1,-1)$ in $\mathbb{R}^{m}$. If we denote by
$\mathbb{J}=\\{x\in\mathbb{R}^{n+m-1,1}\,;x\cdot x\leq 0\\}$ the causal future
and past of the origin then for $e$ a unit timelike vector the domain
$\widehat{\mathbb{D}}^{n+m}(e)=\\{e+\mathbb{J}\\}\cap\\{-e+\mathbb{J}\\}$ is
called the _causal diamond_ around $0$ with unit timelike normal $e$. It is
the locus traced out by all causal curves between $e$ and $-e$. Any two causal
diamonds around $0$ can be transformed into each other by an element of the
Lorentz group $\mathrm{O}_{n+m-1,1}(\mathbb{R})$, while the symmetry group of
a causal diamond is isomorphic to
$\mathrm{O}_{n+m-1}(\mathbb{R})\times\mathrm{O}_{1}(\mathbb{R})$. The set

$\\{r+n\,;r\in M,\,n\in N_{r}M\cap a\widehat{\mathbb{D}}^{n+m}(n_{m}(r))\\}$

will be called the _causal tube_ with radius $a>0$ (sufficiently small) around
$M$ relative to the unit timelike normal field $n_{m}$. Its volume is given by

$V_{M}(a)=\int_{M}\\{\int_{a\widehat{\mathbb{D}}^{m}}\det(\delta_{i}^{j}-\sum
t_{p}h_{i}^{jp})\,dt\\}\,ds$

with $\widehat{\mathbb{D}}^{m}$ the diamond domain in $\mathbb{R}^{m}$ in the
notation of the previous section.

In accordance with Weyl’s tube formula, apart from the $\pm$ sign, we obtain
the following version of the tube formula for Riemannian hypersurfaces.

######  Corollary 7.1.

For a spacelike hypersurface $M$ of codimension $m=1$ in a Lorentzian vector
space $\mathbb{R}^{n,1}$ the causal tube volume formula takes the form

$V_{M}(a)=2\sum_{d=0}^{n}\frac{(-1)^{d/2}k_{d}(M)a^{1+d}}{3\cdot
5\cdots(1+d)}\qquad(d\;\mathrm{even}).$

Indeed if $h_{ij}$ is the scalar valued second fundamental form then
$H_{ij}^{kl}=-h_{i}^{k}h_{j}^{l}$ and so $R_{ij}^{kl}$ in Theorem 4.1 also
picks up a minus sign, that is $H_{d}$ and $k_{d}(M)=\int_{M}H_{d}\,ds$ pick
up a factor $(-1)^{d/2}$.

There is yet another case, where the causal tube formula has an intrinsic
form, namely in case
$M\hookrightarrow\mathbb{R}^{n+m-1}\hookrightarrow\mathbb{R}^{n+m-1,1}$. This
can be checked easily using Weyl’s tube formula in a straightforward way.

The next example shows, however, that the positive result for diamond tubes
for $\dim M=\operatorname{codim}M=2$ of Section 6 cannot be extended to the
Lorentzian setting.

######  Example 7.2.

If we specialize to the case $n=m=2$ and $\eta_{pq}=\mathrm{diag}(1,-1)$ of a
compact spacelike surface $M$ in Minkowski spacetime $\mathbb{R}^{3,1}$ then
the integrand

$\det(\delta_{i}^{j}-\sum t_{p}h_{i}^{jp})=1-\sum
t_{p}(h_{1}^{1p}+h_{2}^{2p})+\sum
t_{p}t_{q}(h_{1}^{1p}h_{2}^{2q}-h_{1}^{2p}h_{2}^{1q})$

averages over the symmetry group $W(\mathrm{B}_{2})$ of the square
$\widehat{\mathbb{D}}^{2}$ as in Example 6.1 to the expression

$1+(A(h_{i}^{j})+B(h_{i}^{j}))(t_{1}^{2}+t_{2}^{2})/2$

with $A=h_{1}^{11}h_{2}^{21}-h_{1}^{21}h_{2}^{11}$ and
$B=h_{1}^{12}h_{2}^{22}-h_{1}^{22}h_{2}^{12}$. Since

$\int_{a\widehat{\mathbb{D}}^{2}}dt_{1}\,dt_{2}=2a^{2},\quad\int_{a\widehat{\mathbb{D}}^{2}}t_{1}^{2}\,dt_{1}\,dt_{2}=\int_{a\widehat{\mathbb{D}}^{2}}t_{2}^{2}\,dt_{1}\,dt_{2}=a^{4}/3$

we find

$V_{M}(a)=\int_{M}\\{2a^{2}+(A+B)a^{4}/3\\}\,ds=\mathrm{area}(M)\,2a^{2}+\int_{M}(A+B)\,ds\,a^{4}/3$

for the volume of the causal tube along $M$.

On the other hand, the Gauss equation (for n=2 there is just a single one) in
this particular case of spacelike surfaces in Minkowski spacetime becomes

$R_{12}^{12}=h_{1}^{1}\cdot h_{2}^{2}-h_{1}^{2}\cdot h_{2}^{1}=A-B.$

Since $\int_{M}(A+B)\,ds$ enters in the tube volume formula while
$\int_{M}(A-B)\,ds$ is the total Gauss curvature for $M$, the volume formula
for causal tubes around surfaces need not be intrinsic.

The conclusion is that for spacelike submanifolds of Minkowski spacetime
$\mathbb{R}^{3,1}$ the causal tube volume formula will in general no longer be
intrinsic, except for the obvious cases of spacelike curves ($n=1$) or
hypersurfaces ($m=1$). This question about the intrinsic nature of causal tube
volume formulas was the starting point for our work.

##  8\. Pappus type theorems

Let us denote the graded commutative algebra $\mathbb{R}[t_{1},\ldots,t_{m}]$
by $P=\oplus\,P^{d}$. The subalgebra of invariants for
$\mathrm{O}_{m}(\mathbb{R})$ is equal to
$\mathbb{R}[t_{1}^{2}+\ldots+t_{m}^{2}]$ and is denoted $I=\oplus\,I^{d}$. The
graded subspace

$C=\\{p\in
P\,;\int_{\mathrm{O}_{m}(\mathbb{R})}\,g(p)\,d\mu(g)=0\\}=\oplus\,C^{d}$

is the unique invariant complement of $I$ in $P$. Here $\mu$ is the normalized
Haar measure on $\mathrm{O}_{m}(\mathbb{R})$. Hence $P=I\,\oplus\,C$ and
clearly $C^{d}=P^{d}$ for $d$ odd while $C^{d}$ has codimension one in $P^{d}$
for $d$ even.

######  Definition 8.1.

A compact domain $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$ is called _symmetric of
degree $n$_ if

$\int_{\mathbb{D}^{m}}\,p(t)\,dt=0$

for all polynomials $p\in C^{1}\oplus\ldots\oplus C^{n}$.

If the compact domain $\mathbb{D}^{m}$ has a symmetry group $G_{m}$ that is
orthogonal of degree $n$ (in the sense of our Definition 1.1) then

$\int_{\mathbb{D}^{m}}\,p(t)\,dt=\int_{\mathbb{D}^{m}}\,\langle
p(t)\rangle_{G_{m}}\,dt=\int_{\mathbb{D}^{m}}\,\langle
p(t)\rangle_{\mathrm{O}_{m}(\mathbb{R})}\,dt$

for all polynomials $p(t)$ of degree $\leq n$. In particular, if the symmetry
group $G_{m}$ of $\mathbb{D}^{m}$ is orthogonal of degree $n$ then the domain
$\mathbb{D}^{m}$ is necessarily symmetric of degree $n$. From the discussions
in Section 2 and Section 4 it follows that our Theorem 1.2 holds with the
condition on the symmetry group $G_{m}$ of $\mathbb{D}^{m}$ being orthogonal
of degree $n$ replaced by the condition on $\mathbb{D}^{m}$ being symmetric of
degree $n$. This more general form of Theorem 1.2 was obtained as Theorem 4.4
in [Domingo-Juan--Miquel2004].

A compact domain $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$ is symmetric of degree
$1$ if and only if the center of mass of $\mathbb{D}^{m}$ lies at the origin.
Hence the condition for $\mathbb{D}^{m}$ to be symmetric of degree $1$ is a
good deal more general than the condition for the symmetry group $G_{m}$ to be
orthogonal of degree $1$. If the manifold $M$ is a circle in $\mathbb{R}^{3}$
then the tube volume formula boils down to the ancient Pappus’s centroid
theorem. For this reason the higher dimensional tube volume formulas are
sometimes also called Pappus type theorems.

The next example shows that for a planar domain $\mathbb{D}^{2}$ and for all
$n\geq 1$ the notion for $\mathbb{D}^{2}$ to be symmetric of degree $n$ is
strictly weaker than the notion for the symmetry group $G_{2}$ of
$\mathbb{D}^{2}$ being orthogonal of degree $n$.

######  Example 8.2.

Consider in polar coordinates $t_{1}=r\cos\phi,t_{2}=r\sin\phi$ the planar
domain $\mathbb{D}^{2}=\\{(r,\phi)\,;0\leq r\leq
a(\phi),\phi\in\mathbb{R}/2\pi\mathbb{Z}\\}$ for some continuous function
$a\colon\mathbb{R}/2\pi\mathbb{Z}\rightarrow(0,\infty)$. The space $C^{d}$ is
spanned by the functions $r^{d}\cos(e\phi)$ and $r^{d}\sin(e\phi)$ with $1\leq
e\leq d$ and $e\equiv d$ (mod $2$). The condition that $\mathbb{D}^{2}$ is
symmetric of degree $n$ amounts to

$\int_{0}^{2\pi}\int_{0}^{a(\phi)}\,r^{d+1}\,dr\cos(e\phi)\,d\phi=\int_{0}^{2\pi}\int_{0}^{a(\phi)}\,r^{d+1}\,dr\sin(e\phi)\,d\phi=0$

or equivalently

$\int_{0}^{2\pi}(a(\phi))^{d+2}\cos(e\phi)\,d\phi=\int_{0}^{2\pi}(a(\phi))^{d+2}\sin(e\phi)\,d\phi=0$

for all $1\leq d\leq n$, $1\leq e\leq d$ and $e\equiv d$ (mod $2$). Clearly
these conditions are satisfied if for some $k>n$ the function $a(\phi)$ is
invariant under the cyclic group $C_{k}$ of order $k$ acting on the circle
$\mathbb{R}/2\pi\mathbb{Z}$ by rotations. Indeed, in that case the Fourier
coefficients of all functions $a(\phi)^{d+2}$ vanish for modes not contained
in $k\mathbb{Z}$. This is in accordance with our Theorem 1.2 since the
symmetry group $C_{k}$ of this domain $\mathbb{D}^{2}$ is orthogonal of degree
$k>n$.

However, if for a fixed $n\geq 1$ one chooses integers $p>n$ and $q>(n+3)p$
then the function $a(\phi)=b(\phi)(2+\cos(p\phi))$ with $b>0$ invariant under
$C_{q}$ has the property that the Fourier coefficients of all functions
$a(\phi)^{d+2}$ for $1\leq d\leq n$ vanish for modes $\pm 1,\dots,\pm n$.
Hence this domain $\mathbb{D}^{2}$ is certainly symmetric of degree $n$. On
the other hand, if we pick $p$ and $q$ relatively prime then the symmetry
group $G_{2}$ of $\mathbb{D}^{2}$ will be trivial in case $b(\phi)$ is chosen
sufficiently general (so that the symmetry group for $b(\phi)$ is not larger
than $C_{q}$), and $G_{2}=\\{1\\}$ is not orthogonal of any degree $n\geq 1$.

The examples obtained in Proposition 4.3 of [Domingo-Juan--Miquel2004] of
compact domains $\mathbb{D}^{m}$ in $\mathbb{R}^{m}$ that are symmetric of
degree $n$ are for $n\geq 2$ domains $\mathbb{D}^{2}$ with dihedral symmetry
and for $n=2,3$ domains $\mathbb{D}^{m}$ with hyperoctahedral symmetry,
besides of course the unit ball $\mathbb{B}^{m}$ for all $n$. Hence apart from
giving a pedestrian exposition of Weyl’s tube volume formula and also a
discussion of tube volume formulas for Riemannian submanifolds of a Lorentzian
vector space our paper gives a more complete and transparent discussion in
Section 5 of examples based on symmetry of cross sections $\mathbb{D}^{m}$ for
which the intrinsic tube volume formula holds.

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