Variations of Weyl's tube formula
aa r X i v : . [ m a t h . DG ] F e b VARIATIONS OF WEYL’S TUBE FORMULA
ANNEGRET BURTSCHER † AND GERT HECKMAN † Abstract.
In 1939 Weyl showed that the volume of spherical tubesaround compact submanifolds M of Euclidean space depends solely onthe induced Riemannian metric on M . Can this intrinsic nature of thetube 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 bound-ary) 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 > V M ( a ) = Ω m n X 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 | ≤ } in R m .The remarkable insight of Weyl is that the coefficients k d ( M ) are integralinvariants of M only determined by the intrinsic metric nature of M . Forexample, the initial coefficient k ( M ) = R M ds is the Riemannian volume of M and the next coefficient is k ( M ) = R M S ds with S the scalar curvatureof M . If M has empty boundary and is of even dimension it was provedby Allendoerfer and Weil [1] in their approach towards the Gauss–Bonnettheorem that the top coefficient k n ( M ) = (2 π ) n/ χ ( M ) with χ ( M ) the Eulercharacteristic of M is even of topological nature. See also the text books of Mathematics Subject Classification.
Key words and phrases.
Tubes, volumes, curvature, invariant theory, finite reflectiongroups. † Department of Mathematics, IMAPP, Radboud University Nijmegen, The Nether-lands,
Emails : [email protected] , [email protected] . Acknowledgements:
Research of the first author supported by the Dutch ResearchCouncil (NWO), Project number VI.Veni.192.208. Part of this material is based uponwork supported by the Swedish Research Council under grant no. 2016-06596 while thefirst author was in residence at Institut Mittag-Leffler in Djursholm, Sweden in the Fallof 2019.
Gray on tubes [6] and of Morvan on generalized curvatures [13] for furtherdetails.Due to the local nature of the tube formula we can assume that thesubmanifold M of R n + m comes with a chosen orthonormal frame in thenormal bundle N M of M in R n + m . In turn this gives for all r ∈ M anidentification of the normal space N r M with R m , and so for D m a compactdomain around 0 in R m we can consider the generalized tube { r + n ; r ∈ M, n ∈ a D m } around M of type a D m for a > n of M )symmetry requirements on the domain D m a similar intrinsic formula forthe volume V ( a ) of the above generalized tube remains valid as in Weyl’scase where D m equals the unit ball B m .Our generalized tubes share the feature that the domains D m are invariantunder the following subgroups of the orthogonal group. Definition 1.1.
A subgroup G m of the orthogonal group O m ( R ) on R m iscalled orthogonal of degree n if any polynomial p ( t ) ∈ R [ t ] on R m of degree ≤ n that is invariant under G m is in fact invariant under the full orthogonalgroup O m ( 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 R n + m . If the compact domain D m around in R m has a symmetry group G m inside O m ( R ) that is orthogonal of degree n then the volume of the generalized tube of type a D m for a > sufficientlysmall is given by V M ( a ) = n X d =0 { R D m | t | d dt } k d ( M ) a m + d m ( m + 2) · · · ( m + d −
2) ( d even) with intrinsic coefficients k d ( M ) = R M H d ds as specified in Theorem 4.1. In Sections 2–4 we shall review the proof of the tube formula followingWeyl’s original approach and along the way obtain variations of the tubevolume formula for polyhedral (and related) tubes rather than sphericaltubes. We discuss several examples in Section 5 and counterexamples inSection 6, as well as causal tubes in Minkowski space in Section 7.After this paper was finished we learned that tube volume formulas withmore 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 addSection 8 in order to briefly survey their approach and compare their resultswith ours. 2. The volume of tubes
Locally a submanifold M of dimension n in R n + m is given in the Gaussianapproach by a parametrization r : U n → M ⊂ R n + m , u r ( u ) ARIATIONS OF WEYL’S TUBE FORMULA 3 with u = ( u , . . . , u n ) ∈ U n ⊂ R n and ∂ i = ∂/∂u i in the usual notationof differential geometry. A summation sign P without explicit mentionof indices always means a summation over all indices, which occur bothas upper and as lower index. The first fundamental form (or Riemannianmetric) is given by ds = X g ij du i du j , g ij = ∂ i r · ∂ j r, 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 ( u ) , . . . , n m ( u ) in thenormal bundle of M , and so ∂ i r · n p = 0 and n p · n q = δ pq along M forall i = 1 , . . . , n and p, q = 1 , . . . , m . Let t = ( t , . . . , t m ) be Cartesiancoordinates on R m . Let us be given a compact domain D m around 0 in R m such that the map x : U n × D m → R n + m , ( u, t ) x ( u, t ) = r ( u ) + X t p n p ( u )is a diffeomorphism of U n × D m onto its image in R n + m . This image is calleda tube of type 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 ) + X t p n p ( u ) ; u ∈ U n , t ∈ a D m } ⊂ R n + m of type a D m as a function of a small positive parameter a >
0. By theJacobi substitution theorem we have V U n ( a ) = Z U n { Z a D m J ( u, t ) dt } du with J the absolute value of the determinantdet( ∂ x · · · ∂ n x n · · · n m )and ∂ i x = ∂ i r + P t p ∂ i n p for i = 1 , . . . , n .Recall that ∂ i ∂ j r = X Γ kij ∂ k r + X h pij n p with Γ kij = Γ kij ( u ) the Christoffel symbols and h pij = h pij ( u ) the coefficients ofthe 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 direc-tion, while the other indices i, j, k = 1 , . . . , n are coordinate indices on thesubmanifold M . Since ∂ j r · n p = 0 we get ∂ i n p · ∂ j r = − n p · ∂ i ∂ j r = − X δ pq h qij with δ pq the Kronecker symbol. Writing t p = P δ pq t q we find ∂ i x = ∂ i r − X δ pq t p h qik g kj ∂ j r + . . . = X ( δ ji − X t p h pik g kj ) ∂ j r + . . . with g ij = ∂ i r · ∂ j r , g ij its inverse matrix, det g = det( g ij ) its determinantand . . . stands for a linear combination of the normal fields n p . If in the A. BURTSCHER AND G. HECKMAN usual notation we write h jpi = P g jk h pik for i, j = 1 , . . . , n and p = 1 , . . . , m then we get det( ∂ x · · · ∂ n x n · · · n m ) =det( δ ji − X t p h jpi ) det( ∂ r · · · ∂ n r n · · · n m )which in turn implies V U n ( a ) = Z U n { Z a D m det( δ ji − X t p h jpi ) dt } p det g ij du for all a > u ∈ U n the integrand det( δ ji − P t p h jpi ) is a polynomial in t ofdegree n and so, after integration and patching together the locally definedtubes, we conclude that V M ( a ) = n X d =0 v d a m + d is a polynomial in a of degree m + n and with coefficient v = vol( D m )vol( M ).If the domain D m is centrally symmetric with respect the origin, that is if − D m = D m , then the integrals of odd degree monomials in t over D m vanishand so v d = 0 for d odd.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, byassuming that M is embedded in flat R n + m , one observes that certain com-binations of the second fundamental forms are intrinsic curvature quantities(this was already done by Weyl). Secondly, by imposing certain symmetryconditions on D m , we show that only those intrinsic combinations remain inthe volume formula V M ( a ) for the generalized tube (done by Weyl for theball B m ). These steps are carried out in Sections 3 and 4, respectively.3. The Gauss equations
As before, we write ∂ i ∂ j r = X Γ kij ∂ k r + h ij with Γ kij the Christoffel symbols given by X g kl ( ∂ i g jl + ∂ j g il − ∂ l g ij )and h ij = P h pij n p the second fundamental form relative to the orthonormalframe n p in the normal bundle along M . Given scalar functions g ij and h pij ,the integrability conditions for the existence of an embedding of M into flatEuclidean space with these functions as coefficients of the first and secondfundamental forms are given by ∂ i ( X Γ ljk ∂ l r + X h pjk n p ) − ∂ j ( X Γ lik ∂ l r + X h pik n p ) = 0for all i, j, k (by working out ∂ i ( ∂ j ∂ k r ) − ∂ j ( ∂ i ∂ k r ) = 0).In the normal directions this leads to the Codazzi–Mainardi equations ∂ i h pjk − ∂ j h pik + X (Γ ljk h pil − Γ lik h pjl ) = 0 ARIATIONS OF WEYL’S TUBE FORMULA 5 for all i, j, k and all p . In the tangential directions this amounts to the Gaussequations R lkij = X δ pq g ln ( h pin h qjk − h pjn h qik )for all i, j, k, l with R lkij = ∂ i Γ lkj − ∂ j Γ lki + X (Γ mkj Γ lmi − Γ mki Γ lmj )the coefficients of the Riemann curvature tensor. As mentioned earlier, byraising indices h jpi = P g jk h pki and R klij = P g ln R knij the Gauss equationstake the form R klij = X δ pq ( h kpi h lqj − h kpj h lqi ) = h ki · h lj − h kj · h li for all i, j, k, l and h ji = P h jpi n p = P g jk h ki normal vectors along M .4. Averaging the integrand
For p ( t ) ∈ R [ t , · · · , t m ] a polynomial on Euclidean space R m and G m aclosed subgroup of the orthogonal group O m ( R ) let us write h p ( t ) i G m = Z G m p ( gt ) dµ ( g )for the average of p over G m , with µ the normalized Haar measure on G m .Clearly h p ( t ) i O m ( R ) ∈ R [ t · t ]with t · t = | t | the norm squared of t ∈ R m . The crucial step for the intrinsicnature of the coefficients of the tube volume formula is the following result(see the Lemma on page 470 of Weyl’s paper [15]). Theorem 4.1.
We have (with ≤ i, j ≤ n and ≤ p ≤ m ) h det( δ ji − X t p h jpi ) i O m ( R ) = n X d =0 H d | t | d m ( m + 2) · · · ( m + d − with H d intrinsic functions on M given by H d = if d odd , if d = 0 , P ε j ...j d i ...i d R j j i i · · · R j d − j d i d − i d if d > even , and R klij = H klij − H klji , H klij = h ki · h lj = X δ pq h kpi h lqj for i, j, k, l = 1 , . . . , n . In the expression for H d with d > even the sum runsover all cardinality d subsets D of { , . . . , n } and over all possible couplingsof pairs j j i i | j j i i | · · · | j d − j d i d − i d | taken from D = { i , . . . , i d } = { j , . . . , j d } . Here a pair i i means twodistinct numbers i , i irrespective of their order. A. BURTSCHER AND G. HECKMAN
Proof.
Averaging the characteristic polynomial det( λδ ji − P t p h jpi ) over theorthogonal group O m ( R ) acting on t ∈ R m yields h det( λδ ji − X t p h jpi ) i O m ( R ) = n X d =0 λ n − d | t | d X D A D ( h ji )with the sum over all even d and all cardinality d subsets D of { , . . . , n } and with A D ( h ji ) given by h det( t · h ji ) i,j ∈D i O m ( R ) = | t | d A D ( h ji )as degree d polynomial on R m × d , which is invariant under the diagonalaction of O m ( R ). By the first fundamental theorem of invariant theory forO m ( R ) (see Corollary 4.2.3 of [5], which is a modern reincarnation of Weyl’sclassic [16]) we have A D ( h ji ) = B D ( H klij )with H klij = h ki · h lj and i, j, k, l ∈ D with i = j, k = l . From the explicitdeterminantal form it follows that B D ( H klij ) is in fact a linear combinationof monomials of the form H j j i i · · · H j d − j d i d − i d with D = { i , . . . , i d } = { j , . . . , j d } . Moreover under the action of thesymmetric group S d acting on both the lower and the upper indices B D ( H klij )transforms under the sign character. Therefore B D ( H klij ) = C D ( R klij ) with R klij = H klij − H klji and by symmetry for S d we arrive at C D ( R klij ) = c ( m, d ) X ε j ...j d i ...i d R j j i i · · · R j d − j d i d − i d with the sum over all possible couplings of pairs from D and c ( m, d ) aconstant depending solely on m and d . The conclusion is that h det( δ ji − X t p h jpi ) i O m ( R ) = n X d =0 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 jpi = δ ji for p = 1 and h jpi = 0 for p ≥
2. In that case A D ( h ji ) = R S m − t d dµ ( t ) R S m − dµ ( t )with µ the Euclidean measure on the unit sphere S m − in R m . The integralin the numerator becomes Z − r d (1 − r ) ( m − / dr = Z s ( d − / (1 − s ) ( m − / ds (apart from a factor volume ω m − of S m − ) and so equalsB(( d + 1) / , ( m − /
2) = Γ(( d + 1) / m − / d + m ) / . ARIATIONS OF WEYL’S TUBE FORMULA 7
In turn this implies A D ( h ji ) = Γ(( d + 1) / m/ / d + m ) /
2) = 1 · · · · ( d − m ( m + 2) · · · ( m + d − . On the other hand R klij = δ ki δ lj − δ kj δ li and so equal to ε klij if the pairs ij and kl coincide and 0 otherwise, and hence C D ( R klij ) = c ( m, d ) d !2 d/ ( d/ c ( m, d ) · · · · · ( d − . Hence c ( m, d ) = 1 /m ( m + 2) · · · ( m + d −
2) as desired. (cid:3)
Recall that the volume ω m of the unit sphere S m − and the volume Ω m of the unit ball B m are related by Ω m = ω m /m . The tube formula of Weylcan now be easily derived. Corollary 4.2.
If the domain D m in R m is equal to the unit ball B m thenthe tube volume is given by V M ( a ) = Ω m n X d =0 k d ( M ) a m + d ( m + 2) · · · ( m + d ) ( d even) for a > small and k d ( M ) = R 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 B m we have V U n ( a ) = Z U n { Z a B m det( δ ji − X t p h jpi ) dt } p det g ij du = Z U n { Z a B m n X d =0 H d | t | d m ( m + 2) · · · ( m + d − dt } p det g ij du = Z U n { ω m Z a n X d =0 H d r m + d − m ( m + 2) · · · ( m + d − dr } p det g ij du = Ω m n X d =0 { R U n H d ds } a m + d ( m + 2) · · · ( m + d − m + d )and the result follows. (cid:3) If we consider domains D m with symmetry groups G m < O m ( R ) suchthat the invariant polynomials of degree ≤ n = dim M for both groupsagree, then we can prove Theorem 1.2. Proof of Theorem 1.2.
By the Fubini theorem we have for
H < G compactgroups and f a continuous function on G that Z G f ( g ) dµ G ( g ) = Z G/H { Z H f ( gh ) dµ H ( h ) } dµ G/H ( gH )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 h det( δ ji − X t p h jpi ) i G m = h det( δ ji − X t p h jpi ) i O m ( R ) and so we can just argue as in the previous proof. (cid:3) A. BURTSCHER AND G. HECKMAN
Using the discussion in Section 2, for n = 1 the tube formula is intrinsic aslong as D m is centrally symmetric, the case already covered by Hotelling [10]if D m = B m . Corollary 4.3. If M is a curve of finite length in R m +1 and D m is centrallysymmetric, then for a > sufficiently small V M ( a ) = length( M ) vol( D m ) a m and hence is intrinsic. (cid:3) The following example shows that central symmetry is not a necessarycondition.
Example 4.4.
Let D m be the union of half the unit ball {| t | ≤ , t ≤ } and the cone { t + · · · + t m ≤ (1 − t /b ) , ≤ t ≤ b } with top ( b, , . . . , forsome b > . By symmetry the average of any linear function of t , . . . , t m over D m equals zero. By direct computation the average of t over D m isequal to zero if b = √ m , and so the previous corollary remains valid for thisdomain as well. Indeed for curves it is sufficient that the center of mass of D m is at theorigin by the generalized Pappus centroid theorem. See Section 8 for thehigher dimensional case.5. Examples of polyhedral domains D m If we are looking for domains D m in R m with a sufficiently large symmetrygroup 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 irreduciblefinite reflection group. Such groups are classified by their Coxeter diagramsor by letters X m with X = A , B , D , E , F , H , I( k ) for k ≥
5. The correspondingreflection groups are denoted by G m = W (X m ).It is a well known theorem due to Shephard and Todd [14] (with a case bycase proof) and Chevalley [3] (with a proof from the Book) that the algebraof polynomial invariants for a finite reflection group W < O( R m ) is itself apolynomial algebra. Theorem 5.1.
The algebra R [ R m ] W of polynomial invariants for W is of theform R [ p , . . . , p m ] with p , . . . , p m algebraically independent homogeneousinvariants of degrees d d , . . . , d m respectively. For each of the irreducible types these degrees can be calculated and aregiven in the next table. The proof of these results can be found in thestandard text books by Bourbaki [2] or by Humphreys [11].
ARIATIONS OF WEYL’S TUBE FORMULA 9 type m d , d , . . . , d m A m ≥ , , . . . , m + 1B m ≥ , , . . . , m D m ≥ , , . . . , m − , m E , , , , , , , , , , , , , , , , , , , , , , , , , , ( k ) 2 2 , k ≥ m ≥ W (X m ) < O( R m ) isorthogonal of degree d − Corollary 5.2. If D m is a domain in R m invariant under a finite reflectiongroup W (X m ) then the tube formula of Theorem 1.2 does hold with intrinsiccoefficients if n = dim M < d , that is, the dimension n of M is strictlysmaller than the second fundamental degree d . (cid:3) For example, if D is an icosahedron with symmetry group W (H ) thenthe tube formula is intrinsic for submanifolds M of dimension n ≤ R n +3 ,and if D is a 600-cell with symmetry group W (H ) then the tube formula isintrinsic for n ≤
11. For any dimension n of M ֒ → R n +2 with D a regular k -gon with k > n the tube formula is intrinsic, since its symmetry group is W (I ( k )). For dimension n = 2 or 3 we find in this way examples of intrinsictube formulas for arbitrary codimension m via domains D m with symmetrygroups W (A m ) ( n = 2) and W (B m ) ( n = 2 , n ≥ m ≤ m can be obtained by the followingconstruction. Corollary 5.3.
Let G be a noncompact simple Lie group acting on its Liealgebra g , and let θ be a Cartan involution of G and g and g = k ⊕ p thedecomposition in +1 and − eigenspaces of θ on g . If the domain D ⊂ p isthe convex hull of a nonzero orbit of K = G θ on p then the tube formula ofTheorem 1.2 does hold with intrinsic coefficients under the assumption thatthe dimension n of M ֒ → R n + m (with m = dim p ) is strictly smaller thanthe second fundamental degree d of the Weyl group W of the pair ( g , θ ) .Proof. The Killing form ( · , · ) on p is positive definite and the fixed pointgroup 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 inan orbit of the Weyl group W = N K ( a ) / Z K ( a ) of the pair ( g , θ ). Henceeach invariant polynomial p ∈ R [ p ] K for K on p restricts to a Weyl groupinvariant polynomial on a . It is a theorem of Chevalley (see Lemma 7 in [7])that the restriction map R [ p ] K → R [ a ] W is an isomorphism of algebras. Since W acts on a as a finite reflection groupthe latter algebra is described by Theorem 5.1. The possible finite reflectiongroups that can occur as such a Weyl group W are those reflection groups,which can be defined over Z . This means that H and H are excluded andonly the dihedral types I ( k ) = A , B , G for k = 3 , , K on p intersects a in the convex hull ofan orbit of W on a . (cid:3) For example, if G is the complex Lie group of type E (and so K is thecompact Lie group of type E acting on p = i k ) then we do find in this wayexamples of local submanifolds M of Euclidean space of dimension n ≤ m = 248 for which the tube formula of Theorem 1.2 hasintrinsic coefficients. Presumably this large codimension relative to the smalldimension of M allows for an abundance of room for isometric deformationsfor the embedding of M in such a Euclidean space.6. No-go results for diamond domains b D m In this section we shall denote by b D m the convex hull of the subset { t + . . . + t m − ≤ , t m = 0 } ⊔ { (0 , . . . , , ± } in R m , m ≥
2. Any multiple a b D m for a > G m of b D m is equal to O m − ( R ) × O ( R ) for m ≥ m = 2 the symmetry group G is equal to the dihedral group W (B ) of order 8. The essential point of Weyl’s argument for the intrinsicnature of the volume formula for tubes is the computation of the integral Z a b D m det( δ ji − X t p h jpi ) dt as a polynomial in a , by first averaging over the symmetry group G m ofthe domain b D m in R m . The outcome should hopefully be a polynomialexpression in Riemann curvature components R klij as in Theorem 4.1. Wewill work out two examples, one with n = 2 and m ≥ 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 . Since the symmetry group of b D m is then G m = O m − ( R ) × O ( R ) the invariants of degree in R [ t , . . . , t m ] are linear combinations of R =( t + . . . + t m − ) / ( m − and S = t m . The above determinant for n = 2 becomes det( δ ji − X t p h jpi ) = 1 − X t p ( h p + h p ) + X t p t q ( h p h q − h p h q ) and averaging over the symmetry group G m yields A ( h ji ) R ( t ) + B ( h ji ) S ( t ) ARIATIONS OF WEYL’S TUBE FORMULA 11 with A = m − X p =1 ( h p h p − h p h p ) B = ( h m h m − h m h m ) and A + B = R intrinsic. Thus by the above, if the integrals of R and S over b D m agree, then the generalized tube volume is intrinsic as well. Theintegrals of R ( t ) and S ( t ) over b D m amount respectively to (put r = √ R and s = √ S ) ω m − m − Z r r m − dr ds and 2 ω m − Z s r m − dr ds, integrated over the triangle { ( r, s ) ; r, s ≥ , r + s ≤ } , and we will show thatfor m ≥ these are distinct. Apart from the factor ω m − the left integralbecomes m − Z (1 − r ) r m dr = 1( m − m + 1)( m + 2) while the right integral equals Z
10 13 (1 − r ) r m − dr = 2( m − m ( m + 1)( m + 2) and for the difference we find m − m − m ( m + 1)( m + 2) which is nonzero for m ≥ , as claimed. Hence the tube volume formula for ageneral surface M in R m with diamond domain b D m is no longer intrinsicfor m ≥ . For m = 2 it still is intrinsic as should, because G = W (B ) isorthogonal of degree (in fact, it is not only intrinsic for n = 2 but also for n = 3 since odd exponents vanish for the centrally symmetric diamond b D m ). Example 6.2.
Let us next consider the case that n = 4 and m = 2 . Thesymmetry group of the diamond domain b D = { ( t , t ) ; | t | + | t | ≤ } is thedihedral group W (B ) of order generated by the two reflections s ( t , t ) =( − t , t ) and s ( t , t ) = ( t , t ) . The invariant polynomials for this group W (B ) are generated as an algebra by the quadratic invariant P ( t ) = t + t and the quartic invariant Q ( t ) = t t . Hence any quartic invariant is aunique linear combination of Q and R ( t ) = t + t = P − Q .We would like to know if Weyl’s averaging trick (over the dihedral group W (B ) 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 specialcase that h jpi = 0 for i = j and p = 1 , . If we write h i i = a i and h i i = b i we get det( δ ji − X t p h jpi ) = Y i =1 (1 − t a i − t b i ) and averaging over the dihedral group W (B ) yields A ( a, b ) P ( t ) + B ( a, b ) Q ( t ) + C ( a, b ) R ( t ) with A, B, C homogeneous polynomials of degree , , respectively. A directcalculation gives A = X i Let us suppose that M is a compact connected n -dimensional Riemanniansubmanifold of an ambient Cartesian space R n + m , equipped with a nonde-generate 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 parametrizationaround M given by x : U n × D m → R n + m , ( u, t ) x ( u, t ) = r ( u ) + X t p n p ( u )with u = ( u , . . . , u n ) ∈ U n , t = ( t , . . . , t m ) ∈ D m while n ( 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 diagonalmatrix with entries ± ARIATIONS OF WEYL’S TUBE FORMULA 13 u ∈ U n ). Observe that the choice of such an orthonormal frame for thenormal bundle of M in R n + m is in principle only possible locally. Indeedif 0 ∈ U n then by linear algebra we can choose a basis n (0) , . . . , n m (0) forthe orthogonal complement of the tangent vectors ∂ r (0) , . . . , ∂ n r (0) with n p (0) · n q (0) = η pq and subsequently apply Gram–Schmidt to the vectors ∂ r ( u ) , . . . , ∂ n r ( u ) , n (0) , . . . , n m (0) for u small.As in Section 2 we can write ∂ i x = X j ( δ ji − X t p n p · h ji ) ∂ j r + . . . with second fundamental form normal vectors h ji = P g jk h ik and the dots . . . stand for a linear combination of the normal fields n p . Likewise writing t p = P η pq t q we arrive at the generalized tube volume formula V U n ( a ) = Z U n { Z a D m det( δ ji − X t p h jpi ) dt } p det g ij du with D m a compact domain around 0 and a > V M ( a ) is a polynomial in a of degree m + n with vol( M )vol( D m ) a m as lowestorder term. The Gauss equations R klij = h ki · h lj − h kj · h li as derived in Section 3 remain valid for an indefinite scalar product.For Riemannian curves M of dimension n = 1 and a centrally symmetricdomain D m around 0 in R m we get V M ( a ) = length( M )vol( D m ) a m just like the original case of Hotelling [10]. Also, if η pq = δ pq then we areessentially in the original setting of Weyl and his spherical tube formula andour variations hold without change.Let us suppose for the rest of this section that M is a compact Riemanniansubmanifold of a Lorentzian vector space R n + m − , with scalar product · ofsignature ( n + m − , 1) and thus η pq = diag(1 , . . . , , − 1) in R m . If we denoteby J = { x ∈ R n + m − , ; x · x ≤ } the causal future and past of the originthen for e a unit timelike vector the domain b D n + m ( e ) = { e + J } ∩ {− e + J } is called the causal diamond around 0 with unit timelike normal e . It is thelocus traced out by all causal curves between e and − e . Any two causaldiamonds around 0 can be transformed into each other by an element of theLorentz group O n + m − , ( R ), while the symmetry group of a causal diamondis isomorphic to O n + m − ( R ) × O ( R ). The set { r + n ; r ∈ M, n ∈ N r M ∩ a b D n + m ( n m ( r )) } will be called the causal tube with radius a > M relative to the unit timelike normal field n m . Its volume is given by V M ( a ) = Z M { Z a b D m det( δ ji − X t p h jpi ) dt } ds with b 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 obtainthe following version of the tube formula for Riemannian hypersurfaces. Corollary 7.1. For a spacelike hypersurface M of codimension m = 1 in aLorentzian vector space R n, the causal tube volume formula takes the form V M ( a ) = 2 n X d =0 ( − d/ k d ( M ) a d · · · · (1 + d ) ( d even) . Indeed if h ij is the scalar valued second fundamental form then H klij = − h ki h lj and so R klij in Theorem 4.1 also picks up a minus sign, that is H d and k d ( M ) = R M H d ds pick up a factor ( − d/ .There is yet another case, where the causal tube formula has an intrinsicform, namely in case M ֒ → R n + m − ֒ → R n + m − , . This can be checkedeasily using Weyl’s tube formula in a straightforward way.The next example shows, however, that the positive result for diamondtubes for dim M = codim M = 2 of Section 6 cannot be extended to theLorentzian setting. Example 7.2. If we specialize to the case n = m = 2 and η pq = diag(1 , − of a compact spacelike surface M in Minkowski spacetime R , then the in-tegrand det( δ ji − X t p h jpi ) = 1 − X t p ( h p + h p ) + X t p t q ( h p h q − h p h q ) averages over the symmetry group W (B ) of the square b D as in Example 6.1to the expression A ( h ji ) + B ( h ji ))( t + t ) / with A = h h − h h and B = h h − h h . Since Z a b D dt dt = 2 a , Z a b D t dt dt = Z a b D t dt dt = a / we find V M ( a ) = Z M { a + ( A + B ) a / } ds = area( M ) 2 a + Z M ( A + B ) ds a / 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 = h · h − h · h = A − B. Since R M ( A + B ) ds enters in the tube volume formula while R M ( A − B ) ds isthe total Gauss curvature for M , the volume formula for causal tubes aroundsurfaces need not be intrinsic. The conclusion is that for spacelike submanifolds of Minkowski spacetime R , 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 volumeformulas was the starting point for our work. ARIATIONS OF WEYL’S TUBE FORMULA 15 Pappus type theorems Let us denote the graded commutative algebra R [ t , . . . , t m ] by P = ⊕ P d .The subalgebra of invariants for O m ( R ) is equal to R [ t + . . . + t m ] and isdenoted I = ⊕ I d . The graded subspace C = { p ∈ P ; Z O m ( R ) g ( p ) dµ ( g ) = 0 } = ⊕ C d is the unique invariant complement of I in P . Here µ is the normalized Haarmeasure 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. Definition 8.1. A compact domain D m in R m is called symmetric of degree n if Z D m p ( t ) dt = 0 for all polynomials p ∈ C ⊕ . . . ⊕ C n . If the compact domain D m has a symmetry group G m that is orthogonalof degree n (in the sense of our Definition 1.1) then Z D m p ( t ) dt = Z D m h p ( t ) i G m dt = Z D m h p ( t ) i O m ( R ) dt 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 necessarilysymmetric of degree n . From the discussions in Section 2 and Section 4it follows that our Theorem 1.2 holds with the condition on the symmetrygroup 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 1.2was obtained as Theorem 4.4 in [4].A compact domain D m in R m is symmetric of degree 1 if and only if thecenter of mass of D m lies at the origin. Hence the condition for D m to besymmetric of degree 1 is a good deal more general than the condition forthe symmetry group G m to be orthogonal of degree 1. If the manifold M is a circle in R then the tube volume formula boils down to the ancientPappus’s centroid theorem. For this reason the higher dimensional tubevolume formulas are sometimes also called Pappus type theorems.The next example shows that for a planar domain D and for all n ≥ D to be symmetric of degree n is strictly weaker than thenotion for the symmetry group G of D being orthogonal of degree n . Example 8.2. Consider in polar coordinates t = r cos φ, t = r sin φ theplanar domain D = { ( r, φ ) ; 0 ≤ r ≤ a ( φ ) , φ ∈ R / π Z } for some continuousfunction a : R / π Z → (0 , ∞ ) . The space C d is spanned by the functions r d cos( eφ ) and r d sin( eφ ) with ≤ e ≤ d and e ≡ d (mod ). The conditionthat D is symmetric of degree n amounts to Z π Z a ( φ )0 r d +1 dr cos( eφ ) dφ = Z π Z a ( φ )0 r d +1 dr sin( eφ ) dφ = 0 or equivalently Z π ( a ( φ )) d +2 cos( eφ ) dφ = Z π ( a ( φ )) d +2 sin( eφ ) dφ = 0 for all ≤ d ≤ n , ≤ e ≤ d and e ≡ d (mod ). Clearly these conditions aresatisfied if for some k > n the function a ( φ ) is invariant under the cyclicgroup C k of order k acting on the circle R / π Z by rotations. Indeed, inthat case the Fourier coefficients of all functions a ( φ ) d +2 vanish for modesnot contained in k Z . This is in accordance with our Theorem 1.2 since thesymmetry group C k of this domain D is orthogonal of degree k > n .However, if for a fixed n ≥ one chooses integers p > n and q > ( n + 3) p then the function a ( φ ) = b ( φ )(2 + cos( pφ )) with b > invariant under C q has the property that the Fourier coefficients of all functions a ( φ ) d +2 for ≤ d ≤ n vanish for modes ± , . . . , ± n . Hence this domain D is certainlysymmetric of degree n . On the other hand, if we pick p and q relativelyprime then the symmetry group G of D will be trivial in case b ( φ ) is chosensufficiently general (so that the symmetry group for b ( φ ) is not larger than C q ), and G = { } is not orthogonal of any degree n ≥ . 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 ≥ D with dihedralsymmetry and for n = 2 , D m with hyperoctahedral symmetry,besides of course the unit ball B m for all n . Hence apart from giving apedestrian exposition of Weyl’s tube volume formula and also a discussionof tube volume formulas for Riemannian submanifolds of a Lorentzian vec-tor space our paper gives a more complete and transparent discussion inSection 5 of examples based on symmetry of cross sections D m for which theintrinsic tube volume formula holds. References [1] C. B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhe-dra , Trans. Amer. Math. Soc. (1943), 101–129.[2] N. Bourbaki, ´El´ements de math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie.Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´espar des r´eflexions. 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