Variations of Weyl's tube formula

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 D than the round ball? Under sufficiently strong symmetry conditions on D the answer turns out to be yes.


Introduction
Let us be given a compact connected manifold M (possibly with a boundary) of dimension n embedded in R n+m as submanifold of codimension m. For each r ∈ M we have an orthogonal decomposition T r M ⊕ N r M of R m+n into tangent space and normal space at r of M . It was shown by Weyl [15] that the Euclidean volume of the spherical tube {r + n ; r ∈ M, n ∈ N r M, |n| ≤ a} around M with radius a > 0 sufficiently small is equal to V M (a) = Ω m n d=0 k d (M ) a m+d (m + 2) · · · (m + d) (d even) with Ω m the volume of the unit ball B m = {t ; |t| ≤ 1} in 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 ) = M ds is the Riemannian volume of M and the next coefficient is k 2 (M ) = 1 2 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 [1] in their approach towards the Gauss-Bonnet theorem that the top coefficient k n (M ) = (2π) n/2 χ(M ) with χ(M ) the Euler characteristic of M is even of topological nature. See also the text books of Gray on tubes [6] and of Morvan on generalized curvatures [13] for further details.
Due to the local nature of the tube formula we can assume that the submanifold M of R n+m comes with a chosen orthonormal frame in the normal bundle N M of M in R n+m . In turn this gives for all r ∈ M an identification of the normal space N r M with R m , and so for D m a compact domain around 0 in R m we can consider the generalized tube {r + n ; r ∈ M, n ∈ aD m } around M of type aD 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 D m a similar intrinsic formula for the volume V (a) of the above generalized tube remains valid as in Weyl's case where D m equals the unit ball B m .
Our generalized tubes share the feature that the domains D m are invariant under the following subgroups of the orthogonal group. Our principal result is the following generalized tube formula. 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 D m had already been studied before by Domingo-Juan and Miquel [4]. 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.

The volume of tubes
Locally a submanifold M of dimension n in R n+m is given in the Gaussian approach by a parametrization r : U n → M ⊂ R n+m , u → r(u) with u = (u 1 , . . . , u n ) ∈ U n ⊂ R n and ∂ i = ∂/∂u i in the usual notation of differential geometry. A summation sign 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 with · the scalar product on the ambient Euclidean space R n+m and g ij = g ij (u) a positive definite symmetric matrix for all u ∈ U n . Let us choose an orthonormal frame field u → n 1 (u), . . . , n m (u) in the normal bundle of M , and so ∂ i r · n p = 0 and n p · n q = δ pq along M for all i = 1, . . . , n and p, q = 1, . . . , m. Let t = (t 1 , . . . , t m ) be Cartesian coordinates on R m . Let us be given a compact domain D m around 0 in R m such that the map is a diffeomorphism of U n ×D m onto its image in R n+m . This image is called a tube of type D m around r(U n ). We are interested in the Euclidean volume V U n (a) of the local tube of type aD m as a function of a small positive parameter a > 0. By the Jacobi substitution theorem we have with J the absolute value of the determinant det(∂ 1 x · · · ∂ n x n 1 · · · n m ) and ∂ i x = ∂ i r + t p ∂ i n p for i = 1, . . . , n.

Recall that
the Christoffel symbols and h p ij = h p ij (u) the coefficients of the second fundamental form h ij relative to the orthonormal normal frame n p . Here indices p, q = 1, . . . , m are coordinate indices in the normal direction, while the other indices i, j, k = 1, . . . , n are coordinate indices on the submanifold M . Since ∂ j r · n p = 0 we get with δ pq the Kronecker symbol. Writing t p = δ pq t q we find with g ij = ∂ i r · ∂ j r, g ij its inverse matrix, det g = det(g ij ) its determinant and . . . stands for a linear combination of the normal fields n p . If in the usual notation we write h jp i = g jk h p ik for i, j = 1, . . . , n and p = 1, . . . , m then we get det(∂ 1 x · · · ∂ n x n 1 · · · n m ) = det(δ j i − t p h jp i ) det(∂ 1 r · · · ∂ n r n 1 · · · n m ) which in turn implies for all a > 0 sufficiently small. For fixed u ∈ U n the integrand det(δ j i − t p h jp i ) is a polynomial in t of degree n and so, after integration and patching together the locally defined tubes, we conclude that In order to show that the volume V M (a) of a generalized tube of type D m depends only on intrinsic quantities of M , two steps are necessary. Firstly, by assuming that M is embedded in flat 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 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 B m ). These steps are carried out in Sections 3 and 4, respectively.

The Gauss equations
As before, we write ij the Christoffel symbols given by and h ij = h p ij 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 p ij , 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 In the normal directions this leads to the Codazzi-Mainardi equations for all i, j, k and all p. In the tangential directions this amounts to the Gauss equations the coefficients of the Riemann curvature tensor. As mentioned earlier, by raising indices h jp i = g jk h p ki and R kl ij = g ln R k nij the Gauss equations take the form

Averaging the integrand
For p(t) ∈ R[t 1 , · · · , t m ] a polynomial on Euclidean space R m and G m a closed subgroup of the orthogonal group O m (R) let us write for the average of p over G m , with µ the normalized Haar measure on G m . Clearly with t ·t = |t| 2 the norm squared of t ∈ 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 [15]).
with H d intrinsic functions on M given by for i, j, k, l = 1, . . . , n. In the expression for H d with d > 0 even the sum runs over all cardinality d subsets D of {1, . . . , n} and over all possible couplings of pairs 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(λδ
with the sum over all even d and all cardinality d subsets D of {1, . . . , n} and with A D (h j i ) given by . By the first fundamental theorem of invariant theory for O m (R) (see Corollary 4.2.3 of [5], which is a modern reincarnation of Weyl's classic [16] Moreover under the action of the symmetric group S d acting on both the lower and the upper indices B D (H kl ij ) transforms under the sign character. Therefore ji and by symmetry for S d we arrive at and all that is left is the computation of the constant c(m, d).
For this computation we take the special choice h jp i = δ j i for p = 1 and h jp i = 0 for p ≥ 2. In that case

In turn this implies
.
On the other hand R kl ij = δ k i δ l j − δ k j δ l i and so equal to ε kl ij if the pairs ij and kl coincide and 0 otherwise, and hence .
Recall that the volume ω m of the unit sphere S m−1 and the volume Ω m of the unit ball B m are related by Ω m = ω m /m. The tube formula of Weyl can now be easily derived. Proof. By Section 2 and the symmetry of B m we have and the result follows.
If we consider domains D m with symmetry groups G m < O m (R) such that the invariant polynomials of degree ≤ 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 with µ G , µ H and µ G/H the normalized invariant measures on G, H and G/H respectively. Hence by the assumption on G m we have 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 D m is centrally symmetric, the case already covered by Hotelling [10] if D m = B m . The following example shows that central symmetry is not a necessary condition.  Indeed for curves it is sufficient that the center of mass of D m is at the origin by the generalized Pappus centroid theorem. See Section 8 for the higher dimensional case.

Examples of polyhedral domains D m
If we are looking for domains D m in R m with a sufficiently large symmetry group G m < O m (R) it is natural to consider regular polytopes D m in 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 X m with X = A, B, D, E, F, H, I(k) for k ≥ 5. The corresponding reflection groups are denoted by G m = W (X m ).
It is a well known theorem due to Shephard and Todd [14] (with a case by case proof) and Chevalley [3] (with a proof from the Book) that the algebra of polynomial invariants for a finite reflection group W < O(R m ) is itself a polynomial algebra. 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 [2] or by Humphreys [11].  For example, if D 3 is an icosahedron with symmetry group W (H 3 ) then the tube formula is intrinsic for submanifolds M of dimension n ≤ 5 in R n+3 , and if D 4 is a 600-cell with symmetry group W (H 4 ) then the tube formula is intrinsic for n ≤ 11. For any dimension n of M ֒→ R n+2 with D 2 a regular k-gon with k > n the tube formula is intrinsic, since its symmetry group is W (I 2 (k)). 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 g, and let θ be a Cartan involution of G and g and g = k ⊕ p the decomposition in +1 and −1 eigenspaces of θ on g. If the domain D ⊂ p is the convex hull of a nonzero orbit of K = G θ on p then the tube formula of Theorem 1.2 does hold with intrinsic coefficients under the assumption that the dimension n of M ֒→ R n+m (with m = dim p) is strictly smaller than the second fundamental degree d 2 of the Weyl group W of the pair (g, θ).
Proof. The Killing form (·, ·) on p is positive definite and the fixed point group K = G θ of θ on G acts on p as a subgroup of SO(p). If a ⊂ p is a maximal Abelian subspace then each orbit of K on p intersects a in an orbit of the Weyl group W = N K (a)/Z K (a) of the pair (g, θ). Hence each invariant polynomial p ∈ R[p] K for K on p restricts to a Weyl group invariant polynomial on a. It is a theorem of Chevalley (see Lemma 7 in [7]) that the restriction map is an isomorphism of algebras. Since W acts on 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 Z. This means that H 3 and H 4 are excluded and only the dihedral types I 2 (k) = A 2 , B 2 , G 2 for k = 3, 4, 6 respectively are allowed. The text books [8] and [9] by Helgason give a thorough exposition of the theory. Using the convexity theorem of Kostant [12] it is easy to see that the convex hull of an orbit of K on p intersects a in the convex hull of an orbit of W on a.
For example, if G is the complex Lie group of type E 8 (and so K is the compact Lie group of type E 8 acting on p = ik) then we do find in this way examples of local submanifolds M of Euclidean space of dimension n ≤ 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.

No-go results for diamond domains D m
In this section we shall denote by D m the convex hull of the subset and averaging over the symmetry group G m yields and A + B = R 12 12 intrinsic. Thus by the above, if the integrals of R and S over D m agree, then the generalized tube volume is intrinsic as well.  and for the difference we find which is nonzero for m ≥ 3, as claimed. Hence the tube volume formula for a general surface M in R 2+m with diamond domain D m is no longer intrinsic for m ≥ 3. For m = 2 it still is intrinsic as should, because G 2 = W (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 D m ). Example 6.2. Let us next consider the case that n = 4 and m = 2. The symmetry group of the diamond domain D 2 = {(t 1 , t 2 ) ; |t 1 | + |t 2 | ≤ 1} is the dihedral group W (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 (B 2 ) are generated as an algebra by the quadratic invariant P (t) = t 2 1 + t 2 2 and the quartic invariant Q(t) = t 2 1 t 2 2 . Hence any quartic invariant is a unique linear combination of Q and R(t) = t 4 1 + t 4 2 = P 2 − 2Q. We would like to know if Weyl's averaging trick (over the dihedral group W (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 jp i = 0 for i = j and p = 1, 2. If we write h i1 i = a i and h i2 and averaging over the dihedral group W (B 2 ) yields with A, B, C homogeneous polynomials of degree 2, 4, 4 respectively. A direct calculation gives in the sum for B and R ij ij = h i i · h j j as before. Note that A as well as B + 6C are intrinsic quantities. For the integrals of Q and R over D 2 we find (put r = |t 1 | and s = |t Hence for fourfolds in R 6 with diamond domain 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 R n+m of dimension n with cross section the diamond D m will in general no longer be intrinsic, unless we are in one of the cases of the following table.
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.

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 R n+m , equipped with a nondegenerate but possibly indefinite scalar product denoted by a dot. Let D m be a compact domain around 0 in R m . Say we have a local parametrization around M given by with u = (u 1 , . . . , u n ) ∈ U n , t = (t 1 , . . . , t m ) ∈ D m while n 1 (u), . . . , n m (u) are vectors in R n+m depending smoothly on u ∈ U n and ∂ i r(u) · n p (u) = 0, n p (u) · n q (u) = η pq for all u ∈ U n , all i = 1, . . . , n, all p, q = 1, . . . , m and η pq a m × m diagonal matrix with entries ±1 (so in particular constant, that is independent of u ∈ U n ). Observe that the choice of such an orthonormal frame for the normal bundle of M in R n+m is in principle only possible locally. Indeed if 0 ∈ U n then by linear algebra we can choose a basis n 1 (0), . . . , n m (0) for the orthogonal complement of the tangent vectors ∂ 1 r(0), . . . , ∂ n r(0) with n p (0) · n q (0) = η pq and subsequently apply Gram-Schmidt to the vectors ∂ 1 r(u), . . . , ∂ n r(u), n 1 (0), . . . , n m (0) for u small. As in Section 2 we can write with second fundamental form normal vectors h j i = g jk h ik and the dots . . . stand for a linear combination of the normal fields n p . Likewise writing t p = η pq t q we arrive at the generalized tube volume formula  [10]. Also, if η pq = δ 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 R n+m−1,1 with scalar product · of signature (n+m−1, 1) and thus η pq = diag(1, . . . , 1, −1) in R m . If we denote by J = {x ∈ R n+m−1,1 ; x · x ≤ 0} the causal future and past of the origin then for e a unit timelike vector the domain D n+m (e) = {e + J} ∩ {−e + 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 O n+m−1,1 (R), while the symmetry group of a causal diamond is isomorphic to O n+m−1 (R) × O 1 (R). The set {r + n ; r ∈ M, n ∈ N r M ∩ a 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 with D m the diamond domain in R m in the notation of the previous section.
In accordance with Weyl's tube formula, apart from the ± sign, we obtain the following version of the tube formula for Riemannian hypersurfaces. There is yet another case, where the causal tube formula has an intrinsic form, namely in case M ֒→ R n+m−1 ֒→ 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 = codim M = 2 of Section 6 cannot be extended to the Lorentzian setting.

Pappus type theorems
Let us denote the graded commutative algebra R[t 1 , . . . , t m ] by P = ⊕ P d . The subalgebra of invariants for O m (R) is equal to R[t 2 1 + . . . + t 2 m ] and is denoted I = ⊕ I d . The graded subspace is the unique invariant complement of I in P . Here µ is the normalized Haar measure on O m (R). Hence P = I ⊕ C and clearly C d = P d for d odd while C d has codimension one in P d for d even.
If the compact domain D m has a symmetry group G m that is orthogonal of degree n (in the sense of our Definition 1.1) then for all polynomials p(t) of degree ≤ n. In particular, if the symmetry group G m of D m is orthogonal of degree n then the domain 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 D m being orthogonal of degree n replaced by the condition on D m being symmetric of degree n. This more general form of Theorem 1.2 was obtained as Theorem 4.4 in [4].
A compact domain D m in R m is symmetric of degree 1 if and only if the center of mass of D m lies at the origin. Hence the condition for 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 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 D 2 and for all n ≥ 1 the notion for D 2 to be symmetric of degree n is strictly weaker than the notion for the symmetry group G 2 of D 2 being orthogonal of degree n.
Example 8.2. Consider in polar coordinates t 1 = r cos φ, t 2 = r sin φ the planar domain D 2 = {(r, φ) ; 0 ≤ r ≤ a(φ), φ ∈ R/2πZ} for some continuous function a : R/2πZ → (0, ∞). The space C d is spanned by the functions r d cos(eφ) and r d sin(eφ) with 1 ≤ e ≤ d and e ≡ d (mod 2). The condition that D 2 is symmetric of degree n amounts to (a(φ)) d+2 cos(eφ) dφ = 2π 0 (a(φ)) d+2 sin(eφ) dφ = 0 for all 1 ≤ d ≤ n, 1 ≤ e ≤ d and e ≡ d (mod 2). Clearly these conditions are satisfied if for some k > n the function a(φ) is invariant under the cyclic group C k of order k acting on the circle R/2πZ by rotations. Indeed, in that case the Fourier coefficients of all functions a(φ) d+2 vanish for modes not contained in kZ. This is in accordance with our Theorem 1.2 since the symmetry group C k of this domain D 2 is orthogonal of degree k > n.
However, if for a fixed n ≥ 1 one chooses integers p > n and q > (n + 3)p then the function a(φ) = b(φ)(2 + cos(pφ)) with b > 0 invariant under C q has the property that the Fourier coefficients of all functions a(φ) d+2 for 1 ≤ d ≤ n vanish for modes ±1, . . . , ±n. Hence this domain 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 D 2 will be trivial in case b(φ) is chosen sufficiently general (so that the symmetry group for b(φ) is not larger than C q ), and G 2 = {1} is not orthogonal of any degree n ≥ 1.
The examples obtained in Proposition 4.3 of [4] of compact domains D m in R m that are symmetric of degree n are for n ≥ 2 domains D 2 with dihedral symmetry and for n = 2, 3 domains D m with hyperoctahedral symmetry, besides of course the unit ball 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 D m for which the intrinsic tube volume formula holds.