Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
aa r X i v : . [ m a t h - ph ] N ov Invariant classification of the rotationally symmetric R -separablewebs for the Laplace equationin Euclidean space Mark Chanachowicz , Claudia M. Chanu ,Raymond G. McLenaghan October 24, 2007
Abstract.
An invariant characterization of the rotationally symmetric R -separablewebs for the Laplace equation in Euclidean space is given in terms of invariants andcovariants of a real binary quartic canonically associated to the characteristic conformalKilling tensor which defines the webs. The theme of this paper is the study of R - separability for the Laplace equation∆ ψ = ∂ ψ∂x + ∂ ψ∂x + ∂ ψ∂x = 0 , (1.1)in three-dimensional Euclidean space E , that is the search for curvilinear coordinates ( q i )such that the equation can be split into a system of separated ordinary differential equationsby assuming a solution of the form ψ = R ( q , q , q ) Q i φ i ( q i , c a ) , where R is some nowhere vanishing function on E and φ i are functions of the single co-ordinate q i and a suitable set of constants ( c a ). Bˆocher [4] (see also [18, 15]) shows thatthe equation separates in seventeen types of cyclidic coordinate systems. These systems areclassified by group theoretic methods by Boyer, Kalnins and Miller [5]. They explicitly de-scribed the equivalence problem for the R -separable coordinate systems in terms of an orbitanalysis of the algebra of the second order symmetry operators of (1.1) under the action ofthe conformal group of E . Eleven of the coordinate systems are simply separable in thesense that they allow multiplicative separability in the ordinary sense (i.e., with R = 1) ofboth (1.1) and the Helmholtz equation∆ ψ + ω ψ = 0 , (1.2)while the remaining six coordinate systems afford R -separability of the Laplace equationonly for non-trivial functions R . Boyer et al. also solve the canonical forms problem by Department of Physics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada, email:[email protected] Dipartimento di Matematica, Universit`a di Torino, via Carlo Alberto 10, 10123 Torino, Italia, email:[email protected] Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada,email: [email protected] R -commuting symmetry operators from each of the seventeen orbit classesthey delineate.The purpose of the present paper is to consider the problem of the characterization ofthe R -separable coordinate systems for (1.1) from a geometric point of view. We present aninvariant classification scheme for the rotationally symmetric R -separable coordinate systemsin terms of invariants of valence two conformal Killing tensors (CKT). These tensors arethe geometric quantities in terms of which the second order symmetry operators of (1.1) aredefined. Such a tensor with point-wise distinct eigenvalues and normal eigenvectors, called a characteristic conformal Killing tensor , defines a geometric structure called an R -separableweb which is a set of three mutually orthogonal foliations of the space by two-dimensionalsurfaces. Each parameterization of an R -separable web defines an R -separable coordinatesystem the coordinate surfaces of which coincide with the leaves of the foliations. In thisperspective the characteristic CKTs play a central role in the theory of R -separation ofvariables.Our approach is an extension of that employed by Horwood, McLenaghan and Smirnov[14] who give an invariant classification of the eleven separable webs for the Hamilton-Jacobiequation for the geodesics and the Helmholtz equation in E in terms of the invariants andreduced invariants of valence two Killing tensors.The paper is organized as follows. In Sect. 2 we outline the theory of conformal Killingtensors defined on pseudo-Riemannian spaces and give an overview of invariant theory forvector spaces of CKTs defined on n -dimensional flat spaces. Subsequently, we specialize thetheory to the vector space of valence two trace-free CKTs defined on Euclidean space. In Sect.3 we describe how the R -separable webs for the Schr¨odinger equation may be characterizedby CKTs and define the concept of web symmetry in terms of any characteristic CKT whichdefines the web. In Sect. 4 we derive the general form of the rotationally symmetric CKTin Euclidean space in addition to giving the characteristic CKTs for each of the rotationallysymmetric R -separable webs listed in [18]. In Sect. 5 we give the group action, invariants andcanonical forms for the rotationally symmetric CKTs. We also show that the group action isequivalent to the classical action of GL (2 , R ) on real binary quartics. In Sect. 6 we give aninvariant classification of the rotationally symmetric R -separable webs in terms of invariantsand covariants of the real binary quartic corresponding to a rotationally symmetric CKT. InSect. 7 we show that there exist no symmetric R -separable webs in Euclidean space otherthan the rotationally symmetric ones characterized in Sect. 6 or those that are conformallyequivalent to simply separable symmetric webs. Sect. 8 contains the Conclusion. Let ( M, g ) be a pseudo-Riemannian manifold with metric tensor g . Definition 2.1.
A conformal Killing tensor of valence p defined in ( M, g ) is a symmetric ( p, tensor field K which satisfies the conformal Killing tensor equation [ g , K ] = k ⊙ g , (2.1)2 here [ , ] denotes the Schouten bracket, k some symmetric tensor of type ( p − , , and ⊙ denotes the symmetric tensor product. The tensor k can be determined by contracting (2.1) with the covariant metric tensor.When p = 1, K is said to be a conformal Killing vector and (2.1) reads L K g = f g , (2.2)where L denotes the Lie derivative operator. With respect to a local system of coordinates x i (2.1) may be written ∇ ( i K i ...i p +1 ) = k ( i ...i p − g i p i p +1 ) , (2.3)where ∇ denotes the covariant derivative with respect to the Levi-Cevita connection of g .If k = 0, in (2.1), then K is said to be a Killing tensor .It follows from the properties of the Schouten bracket that the set CK p ( M ) of all con-formal Killing tensors of type ( p,
0) forms a generally infinite dimensional vector space.However, it’s important to note that K ′ = K + l ⊙ g , (2.4)where l is any symmetric tensor of type ( p − , CK p ( M ): K ′ ∼ K ⇔ K ′ = K + l ⊙ g , (2.5)Let C ˆ K p ( M ) denote the set of equivalence classes of CK p ( M ). We may equip C ˆ K p ( M )with the structure of a vector space over the reals as follows. Let ˆ K and ˆ K ∈ C ˆ K p ( M ).Let K and K be representative elements of ˆ K and ˆ K respectively. Then ˆ K + ˆ K isdefined to be the equivalence class represented by K + K . Let K be representative of ˆ K and a ∈ R . Then a ˆ K is defined to be the equivalence class represented by a K . It is easyto check that these operations are well defined. Let T CK p ( M ) denote the vector space oftrace-free conformal Killing tensors of type ( p, T CK p ( M ) iscanonically isomorphic to C ˆ K p ( M ). A necessary and sufficient condition for an element of C ˆ K p ( M ) to be represented by a Killing tensor is that there exists a type ( p − ,
0) tensor l such that [ l , g ] = k . (2.6)For p = 2 the above equation may be written as d l = − k . (2.7)The integrability condition for this equation is d k = 0 . (2.8)By solving (2.3) for k we may write the integrability condition in component form as K k [ i ; kj ] = 0 . (2.9)3his is a necessary and sufficient condition for ˆ K to be represented by a Killing tensor.We now study the behavior of the conformal Killing tensor K under a conformal trans-formation of the metric, which may be written as˜ g = e − σ g . (2.10)By an easy calculation we find that [˜ g , K ] = k ′ ⊙ ˜ g , (2.11)where k ′ = e − σ ( k − σ, K ]) . (2.12)This result shows that K is also a conformal Killing tensor for the conformally related metric˜ g . A necessary and sufficient condition that K is a Killing tensor with respect to ˜ g is that[ σ, K ] = 12 k . (2.13) We now assume that the Riemann curvature tensor of g vanishes. In this case it has beenshown by Eastwood [11] that C ˆ K p ( M ) is finite dimensional and that its dimension d is givenby d = ( n + p − n + p − n + 2 p − n + 2 p − n + 2 p ) p !( p + 1)!( n − n ! , (2.14)for n ≥ , p ≥
1. Thus the general element of C ˆ K p ( M ) is represented by d arbitraryparameters a , . . . , a d , with respect to an appropriate basis.Each element h of the conformal group CI ( M ) induces, by the push forward map, anon-singular linear transformation ρ ( h ) of C ˆ K p ( M ). It is implicit in the work of [11] thatthe map ρ : CI ( M ) → GL ( C ˆ K p ( M )) (2.15)defines a representation of CI ( M ). Once the form of the general element ˆ K of C ˆ K p ( M ) isavailable with respect to some convenient coordinate system on M , the explicit form of thetransformation ρ ( h ) ˆ K (written more succinctly as h · ˆ K ) may be written explicitly in termsof the parameters a , . . . , a d . We shall be particularly concerned with the smooth real-valuedfunctions on C ˆ K p ( M ) that are invariant under the group CI ( M ). The precise definition ofsuch CI ( M )-invariant functions of C ˆ K p ( M ) is as follows. Definition 2.2.
Let ( M, g ) be a pseudo-Riemannian manifold with zero curvature. Let p ≥ be fixed. A smooth function F : C ˆ K p ( M ) → R is said to be a CI ( M ) -invariant of C ˆ K p ( M ) if it satisfies the condition F ( h · ˆ K ) = F ( ˆ K ) , (2.16) for all ˆ K ∈ C ˆ K p ( M ) and for all h ∈ CI ( M ) . fundamental invariants with the property that any other invariant is an analytic function ofthe fundamental invariants (see [20]). The fundamental theorem of invariants for a regularLie group action [20] determines the number of fundamental invariants needed to define thewhole of the space of CI ( M )-invariants. Theorem 2.3.
Let G be a Lie group acting regularly on an n-dimensional manifold M with s -dimensional orbits. Then, in a neighborhood N of each point p ∈ M , there exists n − s functionally independent G -invariants ∆ , . . . , ∆ n − s . Any other G -invariant I defined near p can be locally uniquely expressed asan analytic function of the fundamental invariants namely I = F (∆ , . . . , ∆ n − s ) . One of the standard methods for determining the invariants of C ˆ K p ( M ) is to use the factthat the invariants of a function under an entire Lie group is equivalent to the invariants ofthe function under the infinitesimal transformation of the group given by the correspondingLie algebra. The precise result is as follows [21] Proposition 2.4.
Let G be a connected Lie group of transformations acting regularly on amanifold M . A smooth real valued function F : M → R is G -invariant if and only if v ( F ) = 0 , (2.17) for all p ∈ M and for every infinitesimal generator v of G . In our application G is the representation ρ ( CI ( M )) defined by (2.15) where the condition(2.17) reads U i ( F ) = 0 , i = 1 , . . . , r, (2.18)where the U i are vector fields which form a basis of the Lie algebra of the representationand r = dim CI ( M ) = ( n + 1)( n + 2). This Lie algebra is isomorphic to the Lie algebra of CI ( M ). Such a basis may be computed directly as the basis of the tangent space to ρ ( CI ( M )at the identity if an explicit form of the representation is available. According to Theorem2.3 the general solution of the system of first-order PDEs (2.18) is an analytic function F of a set of fundamental CI ( M )-invariants. The number of fundamental invariants is d − s ,where d is given by (2.14) and s is the dimension of the orbits of ρ ( CI ( M )) acting regularlyon the space C ˆ K p ( M ).We are now ready to apply the above theory to the vector space C ˆ K ( E ). Recall thefollowing well-known result from invariant theory [20]. Theorem 2.5.
The orbits of a compact linear group acting in a real vector space are separatedby the fundamental (polynomial) invariants.
We first note that in our case the group is non-compact and so in order to distinguishbetween the orbits of CI ( E ) acting on the vector space C ˆ K ( E ) we need to employ a moreelaborate analysis than a mere computation of a set of fundamental invariants.5 .3 Construction of the general CKT in E We now specialize the general theory of the previous subsection to the vector space C ˆ K ( E )of conformal Killing tensors of type (2 ,
0) defined in Euclidean space E .It is well-known [23] that in E , any CKT is expressible modulo a multiple of the metricas a sum of symmetrized products of conformal Killing vectors. A canonical basis of the Liealgebra of conformal Killing vectors in E with respect to a system of Cartesian coordinates( x i ) may be written as X i = ∂∂x i , R i = ǫ ijk x j X k , D = x i X i , I i = (2 x i x k − δ ik x j x j ) X k , (2.19)for i = 1 , ,
3, where ǫ ijk is the Levi-Cevita tensor. We also note the commutation relations[ X i , X j ] = 0 , [ X i , R j ] = − ǫ ijk X k , [ R i , R j ] = − ǫ ijk R k , [ X i , D ] = X i , [ R i , D ] = 0 , [ I i , I j ] = 0 , [ X i , I j ] = 2( δ ij D − ǫ ijk R k ) , [ R i , I j ] = − ǫ ijk I k , [ D , I i ] = I i . (2.20)We now determine the form of the general element of T CK ( E ). By (2.14) d = 35. It isclear that a sum of symmetrized products of conformal Killing vectors is a conformal Killingtensor. It will be shown that all trace free conformal Killing tensors may be obtained in thisway. We begin by writing K = A ij X i ⊙ X j + B ij X i ⊙ R j + C ij R i ⊙ R j + D i X i ⊙ D + E ij X i ⊙ I j + F i R i ⊙ D + G ij R i ⊙ I j + H D ⊙ D + L i D ⊙ I i + M ij I i ⊙ I j . (2.21)The coefficients in (2.21) obey the following symmetry relations A ij = A ji , C ij = C ji , M ij = M ji . (2.22)Thus an upper bound for the dimension of T CK ( E ) is fifty five, which exceeds the requireddimension. Indeed there exist the following six relations among the basis set of symmetrictensor products of conformal Killing vectors: X i ⊙ R i = 0 , I i ⊙ R i = 0 , D ⊙ D = X i ⊙ I i + R i ⊙ R i , R i ⊙ D + ǫ ikl X k ⊙ I l = 0 . (2.23)6onsequently, the general element of T CK ( E ) may be written as K = A ij X i ⊙ X j + B ij X i ⊙ R j + C ij R i ⊙ R j + D i X i ⊙ D + E ij X i ⊙ I j + G ij R i ⊙ I j + L i D ⊙ I i + M ij I i ⊙ I j , (2.24)where the coefficients B ij and G ij may be chosen to satisfy B ii = 0 ,G ii = 0 . (2.25)In terms of the natural basis, X i ⊙ X j , the components of K are given by K ij = A ij + ( B ( i | k ǫ kl | j ) + D ( i δ j ) l ) x l + ( C mn ǫ mk ( i ǫ | nl | j ) + 2 E ( i | k | δ j ) l − E ( ij ) δ lk ) x l x k + (2 G mn ǫ mk ( i δ j ) l − G m ( i ǫ j ) mk δ ln + 2 L k δ in δ jl − L ( i δ j ) n δ lk ) x l x k x n + (4 M kl δ jn − M k ( i δ j ) n δ li + M ij δ kn δ li ) x l x k x n x i . (2.26)Next we impose the trace free condition namely that K ii = 0 . (2.27)This procedure yields the following additional fourteen relations between the coefficients of K : A ii = 0 , D i = B jk ǫ kji , E kk = 2 C kk ,E ( ij ) − E kk δ ij = 12 ( C ij − C kk δ ij ) , L i = G lm ǫ mli , M ii = 0 . (2.28)We can use the above conditions to remove fourteen coefficients: to keep the expression of K ij as symmetric as possible, we chose to eliminate the D i , L i , and C ij . Moreover, thematrices A ij and M ij are necessarily trace free (we recall that also B ij and G ij are chosen tobe trace free).In terms of the natural basis, X i ⊙ X j , the components of K are given by: K ij = A ij + ( B ( i | k ǫ kl | j ) + B lm ǫ ml ( i δ j ) k ) x k + [(2 E ( mn ) − / E pp δ mn ) ǫ mk ( i ǫ | nl | j ) + 2 E ( i | k | δ j ) l − E ( ij ) δ lk ] x k x l + [2 G mn ǫ nk ( i δ j ) l − G n ( i ǫ j ) nk δ lm + 2 G np ǫ pnk δ im δ jl − G ab ǫ ba ( i δ j ) n δ lk ] x k x l x m + (4 M kl δ mn − M k ( i δ j ) l δ mn + M ij δ kl δ mn ) x k x l x m x n . (2.29)Moreover, any CKT of E is equivalent to A ij X i ⊙ X j + B ij X i ⊙ R j + + E ij X i ⊙ I j + G ij R i ⊙ I j + M ij I i ⊙ I j , where A ij , M ij , B ij and G ij must be trace free matrices.The condition (2.9) applied to (2.29) implies that E [ ij ] = 0 , G ij = 0 , M ij = 0 . (2.30)It follows from the above that (2.29) reduces to K ij = A ij + ( B ( i | k ǫ kl | j ) + B lm ǫ ml ( i δ j ) k ) x k + [(2 E mn − / E pp δ mn ) ǫ mk ( i ǫ | nl | j ) + 2 E k ( i δ j ) l − E ij δ lk ] x k x l , (2.31)which is the trace free part of an ordinary Killing tensor.7 Applications of CKTs to the geometric theory of sep-aration of variables
It is well known that Killing tensors are deeply related with additive separation of variablesfor the Hamilton-Jacobi equation for the geodesics or a natural Hamiltonian in orthogonalcoordinates (see [16], [2] and references therein) g ii ( ∂ i W ) + V = E, E ∈ R , as well as for multiplicative separation of the Schr¨odinger equation∆ ψ + ( E − V ) ψ = 0 , E ∈ R , where ∆ is the Laplace Beltrami operator. Indeed, the existence of a coordinate system inwhich separation of variable occurs is equivalent (for V = 0) to the existence of a Killingtensor K with real simple eigenvalues and normal eigenvectors, called a characteristic Killingtensor . The separable coordinate hypersurfaces are defined to be orthogonal to the eigen-vectors of K (the existence of these surfaces is equivalent to the normality of the eigenvec-tors). The set of the coordinate hypersurfaces is called an orthogonally separable web . Anyparametrization of it locally defines orthogonally separable coordinates . Moreover, if V = 0,the potential V must satisfy an additional compatibility condition also expressed in termsof the characteristic KT ([1]): d ( K dV ) = 0 , (3.32)where K is interpreted as a linear operator on one-forms. Finally, for the multiplicativeseparation of the Schr¨odinger equation the so called Robertson condition must also hold:the Ricci tensor is diagonalised in the separable coordinates ([12]). Geometrically, thismeans that K and the Ricci tensor have the same eigenvectors ([2]). The condition thatthe eigenvalues are real is automatically satisfied for positive definite metrics; recently, KT’swith complex conjugate eigenvalues have also been used to separate variables for a naturalHamilton-Jacobi equation [10].Similar results also hold for conformal Killing tensors. Definition 3.1.
A characteristic CKT is a valence two conformal Killing with real andsimple eigenvalues and normal eigenvectors.
Remark 3.2.
Any CKT equivalent to a characteristic one is characteristic. Hence, it isalways possible to choose a characteristic CKT which is trace free.The following result holds (see [3]):
Proposition 3.3.
There exists an orthogonal coordinate system in which additive separationfor the null geodesic Hamilton-Jacobi equation, g ii ( ∂ i W ) = 0 , occurs, if and only if there exists a characteristic CKT K on M . The coordinates hypersur-faces are orthogonal to the eigenvectors of K . Definition 3.4.
We call a conformally separable web the set of hypersurfaces orthogonalto the eigenvectors of a characteristic CKT. Any coordinates associated with a conformallyseparable web are called conformally separable coordinates. roposition 3.5. There exists an orthogonal coordinate system in which additive separationfor the Hamilton-Jacobi equation with fixed value of the energy E , g ii ( ∂ i W ) + V − E = 0 , occurs, if and only if there exists a characteristic CKT K on M satisfying the compatibilitycondition d (( E − V ) k ♭ + 2 K dV ) = 0 , (3.33) where k ♭ is the one-form corresponding to the vector field k such that [ g , K ] = k ⊙ g and K is considered as a linear operator on one-forms.Proof. According to [3] we have that the natural Hamilton-Jacobi equation with fixed value E of the energy is separable, if and only if there exist a function f and a characteristic CKT K such that [ g , K ] = 2 E − V ( K ∇ V + ∇ f ) ⊙ g , where ∇ is the gradient operator and K is considered as a linear operator on vector fields.By (2.1), we have that ( E − V ) k − K ∇ V must be the gradient of a function and (3.33)follows by passing from vector fields to one-forms. Remark 3.6. If K is a Killing tensor, the compatibility condition (3.33) reduces to (3.32)In spite of the fact that the null geodesic equation is trivial for a positive definite metric,conformally separable coordinates are useful because they are the only ones in which anatural Hamiltonian with fixed value of the energy can be solved by additive separation ofvariables. Moreover, they are the only ones in which R -separation of the Laplace equationcan occur. We recall the definition given in [8]: Definition 3.7.
We say that multiplicative R -separation of the Laplace equation ∆ ψ = 0 occurs in a coordinate system ( q i ) if there exists a solution ψ of the form ψ = R ( q , . . . , q n ) Y i φ i ( q i , c a ) ( c a ) ∈ R n − , (3.34) satisfying the completeness condition rank (cid:20) ∂∂c a (cid:18) φ ′ i φ (cid:19) ∂∂c a (cid:18) φ ′′ i φ (cid:19)(cid:21) = 2 n − , a = 1 , . . . , n − , i = 1 , . . . , n. The following theorem holds [8, 16]:
Theorem 3.8.
On a flat manifold, R -separation of the Laplace equation occurs in a coor-dinate system ( q i ) if and only if the coordinates ( q i ) are orthogonal conformally separablecoordinates. The function R is (up to separated factors) a solution of the first order system ∂ i ln R = 12 Γ i , where Γ i = g hk Γ hki denotes the contraction of the Christoffel symbols with the metric. Remark 3.9.
If the manifold is not flat, then conformal separability is a necessary, but nolonger sufficient condition. To guarantee R -separation we also require the function ∆ RR be ofthe form g ii f i ( q i ) for suitable functions of a single variable f i .9 efinition 3.10. We call an R -separable web a conformally separable web if R -separationfor the Laplace equation occurs in any associated coordinate system. In E , every conformally separable web is an R -separable web for the Laplace equation.This means that any characteristic CKT defines an R -separable web. R -separable coordi-nates of E have been extensively studied by many authors (see Bˆocher[4], Moon and Spencer[18], Boyer et al.[5]). The webs consist of families of confocal cyclides.In Sect. 4 to 6 we restrict ourselves to the webs and associated characteristic CKTsadmitting rotational symmetry. To make the notion of a web-symmetry precise, we give thedefinition of invariance of conformal Killing tensors under one parameter groups of conformaltransformations. Definition 3.11.
Let K denote a characteristic conformal Killing tensor on ( M, g ) . Let φ t denote a one parameter group of conformal transformations. The R -separable webs definedby K are said to be φ t -symmetric if and only if φ t ∗ K = f K, (3.35) where φ t ∗ denotes the push-forward map and f is some function. The infinitesimal version of the above definition is given by the following proposition:
Proposition 3.12.
Let V be an infinitesimal generator of the one parameter group of con-formal transformations φ t . Then φ t is a web-symmetry of the R -separable web defined by acharacteristic CKT K if and only if L V K = h K (3.36) where h is some function. If φ t denotes a one-parameter group of homothetic transformations then the functions f and h are constant. Moreover, if φ t denotes a one-parameter group of isometries then thefunctions f and h are identically zero. We apply (3.36) to compute the characteristic CKTs in E admitting a rotational symmetryaround the z -axis, associated with rotationally R -separable webs. Proposition 4.1.
A CKT K of E satisfies L R K = 0 , and has normal eigenvectors if and only if it is equivalent to M I ⊙ I + L D ⊙ I + H D ⊙ D + C R ⊙ R + D D ⊙ X + A X ⊙ X . (4.1)10 roof. The infinitesimal invariance condition is linear in the parameters and gives a ninedimensional linear subspace L of T CK ( E ). To check the normality of the eigenvectors weapply the Tonolo-Schouten-Nijenhuis conditions (TSN-conditions) [24, 22, 19] on the genericelement of L . The TSN conditions are both necessary and sufficient for a given symmetrictensor field to have normal eigenvectors. They read N l [ jk g i ] l = 0 ,N l [ jk K i ] l = 0 ,N l [ jk K i ] m K ml = 0 , (4.2)where N ijk are the components of the Nijenhuis tensor of K defined by N ijk = K il K l [ j,k ] + K l [ j K ik ] ,l . The TSN-conditions are verified in a six dimensional subspace L ′ ⊂ L of T CK ( E ). Howeverit appears that for the following calculations it is more effective to describe the elements of L ′ as linear combinations of symmetric tensor products of CKVs that are not trace freeCKTs.Let RCK ( E ) be the subspace of CK ( E ) of CKTs of the form (4.1). The free param-eters describing a general element K ∈ RCK ( E ) are( M , L , H, C , D , A ) (4.3)and all the other forty nine coefficients of the general linear combination of symmetric prod-ucts of CKVs (2.21) are null. Given any CKT in Cartesian coordinates satisfying L R K = 0,and the TSN-conditions, the value of the parameters (4.3) are determined as follows: • M is 1/4 of the coefficient of xyz in K ; • L is 1/2 of the coefficient of xyz in K ; • H is the coefficient of xz in K ; • H − C is the coefficient of xy in K ; • D is twice the coefficient of x in K ; • A is the constant term of K − K .Since we are considering components (or functions of the components) which are not affectedby the addition of a multiple of the metric f g , the six parameters are well defined, irrespectiveof whether one starts from a CKT in T CK ( E ) or not. Remark 4.2.
Since E has dimension three, there is an equivalent way to characterizerotational R -separable webs. Any rotational web contains a family of hypersurfaces made ofhalf-planes issued from the rotation axis (the z -axis in our case). These planes are orthogonalto the Killing vector R . Hence R must be an eigenvector of the CKT defining the web.Moreover, this condition is also sufficient to ensure that the eigenvectors of K are normal.Indeed, one of them is the normal vector R and the other two are contained in the two-dimensional planes orthogonal to R and hence they are normal. By imposing the condition( K · R ) × R = 0 , we find again the six dimensional linear subspace described by (4.1).11inally, in order to prove that the general rotational CKT (4.1) is characteristic, we checkthat the eigenvalues are simple almost everywhere. Since R is orthogonal to I , D , X , wehave K · R = C ( x + y ) R . Hence, R = E is an eigenvector corresponding to the eigenvalue λ = C ( x + y ).The other two eigenvectors E and E are orthogonal to E ; they and their correspondingeigenvalues do not depend on C . Moreover, the associated eigenvalues are of the form λ , = A ± √ B , where A = r M + zr L + r H + zD + A , ( r = x + y + z ) (4.4) B = ( x + y ) (cid:20) r L + 2 zH + 4 z − r r D + 4 z (2 z − r ) r A (cid:21) + (4.5) (cid:20) r M + zr L + (2 z − r ) H + z (4 z − r ) r D + r − z ( r − z ) r A (cid:21) . Any change of the parameter C does not affect the web; indeed, E and E do not involve C (see also Sect. 5). Thus, it is always possible to choose C such that λ is different from λ and λ at any point outside of the z -axis. On the contrary for x = y = 0 we have λ = 0 , λ = ( q ( z ) + | q ( z ) | ) , λ = ( q ( z ) − | q ( z ) | ) , with q ( z ) = M z + L z + Hz + D z + A . (4.6)Thus, (at least) one of λ , λ identically vanishes and all points of the z -axis are singularpoints of all rotational webs. The singular points that are not on the rotation axis are thosesatisfying λ = λ , that is where B = 0. Remark 4.3.
The roots of (4.6) are points on the z -axis where the three eigenvalues coincideand K is proportional to the metric tensor. The number of the roots z of q in PR (so thatthe point at infinity is also considered) and their multiplicity characterize the web from ageometric point of view. Remark 4.4.
The knowledge of the eigenvalues of the characteristic tensor in a rotationalweb allows one to write the equations of the (not planar) hypersurfaces (see [9]) The hyper-surfaces S orthogonal to E satisfy the equation λ − λ x + y = h, h ∈ R , while the hypersurfaces S orthogonal to E satisfy the equation λ − λ x + y = h, h ∈ R . It follows that the hypersurfaces have the form2( h − C )( x + y ) + A = ±√ B, h − C )( x + y ) + A ] − B = 0 , (4.7)but for different ranges of the value of h : we have surfaces of S for h < h and surfaces of S for h > h , respectively, where h = C − A x + y ) = C − r M + zr L + r H + zD + A x + y . For h = h we do not obtain a surface of the web because this value of the parameter h would imply B = 0, that is λ = λ . Expanding the equations (4.7) we arrive at[4( H − C + h ) M − L ] r + [8 M D − C − h ) L ] r z +[2 L D − C − h ) H ] r + 16 M A z + 4( C − h ) ( x + y ) + (4.8)[8 L A − C − h ) D ] z − D + 4( H − C + h ) A = 0 , which represents two families of confocal cyclides, one for h > h and one for h < h . R -separablerotational coordinate systems Table 1 contains the parameters of a characteristic CKT corresponding to each of the rota-tional R -separable coordinates listed in Moon and Spencer’s book [18]. We briefly describehow they are determined (for further details, such as plots, transformation laws to Cartesiancoordinates, components of the metric tensor in these coordinates, separated equations etc.,see [18] or [4]). The CKTs are constructed from the St¨ackel matrices that are associatedwith each system of coordinates in [18].Recall that a St¨ackel matrix is a regular matrix of functions S ij depending on the singlevariable q i corresponding to the row index i of the element. One row (the first in the examplesin [18]) of the inverse of the St¨ackel matrix contains the components of the contravariantmetric tensor in the R -separable coordinates, while the other two rows are made of thecomponents of two CKTs with common eigenvectors orthogonal to the web hypersurfaces.Moreover, there is always a real linear combination of these two tensors which provides acharacteristic tensor of the web (see [3]).For each row of the inverse of the St¨ackel matrix we construct the conformal Killingtensors in the R -separable coordinates, then the parameters (4.3) are determined by trans-forming the tensor to Cartesian coordinates and comparing with the Cartesian componentsof the general rotationally symmetric CKT (4.1). For all the coordinate systems consideredin [18] the tensor corresponding to the third row of the inverse St¨ackel matrix is R ⊙ R . Inmost of the examples, the other tensor is a characteristic tensor of the web so its parametersappear unchanged in the Table 1. On the contrary, the tensors arising from the St¨ackel ma-trices given in [18] for Spherical, Tangent spheres and Cylindrical coordinates have C = 0,so they are not characteristic CKTs. In order to get a characteristic CKT associated withthese webs we add a suitable multiple of the tensor R ⊙ R : that is, we change the valueof C in Table 1.The first four coordinate systems have transformation laws to Cartesian coordinates in-volving Jacobi elliptic functions. The parameter a is a scaling parameter, while the parameter k ∈ (0 ,
1) is the parameter of the Jacobi elliptic functions.13oordinates M L H C D A Bi-cyclide − k a k k − a Flat-ringcyclide k a k a Disk cyclide − k a − k a (1 − k ) Cap cyclide a (1+ k ) k k − ( k − − ( k − k ( k +1) a Toroidal a
12 12 a Bispherical − a
12 12 − a Inverseprolatespheroidal a − a a Oblatespheroidal 0 0 1 0 0 a Spherical 0 0 1 -1 0 0Parabolical 0 0 0 0 1 0Cylindrical 0 0 0 -1 0 1Table 1: Characteristic CKT of rotationally symmetric R -separable webs14 Group action preserving rotationallysymmetric CKTs
In order to classify the different types of R -separable webs admitting a rotational symmetry,we consider transformations acting on CK ( E ) which preserve the space RCK ( E ) of therotationally symmetric CKTs defined in Sect. 4. For this purpose, we use a group G thatis generated by five one-parameter transformations and a discrete transformation. Three ofthe one-parameter transformations are induced on RCK ( E ) by conformal transformationsof E mapping the z -axis into itself. The other two are transformations of the CKT that donot change the corresponding web.The five continuous transformations to be taken into account are1. The change of the tensor under a continuous inversion along the z -axis parameterizedby a : φ : ( x, y, z ) → (cid:18) x a z + a r , y a z + a r , z + a r a z + a r (cid:19) , where r = x + y + z .2. The change of the tensor under a translation along the z -axis parameterized by a : φ : ( x, y, z ) → ( x, y, z + a ) .
3. The change of the tensor under a dilation of the space with singular point at the originparameterized by a : φ : ( x, y, z ) → ( a x, a y, a z ) , ( a = 0) .
4. The multiplication of the tensor by a non-zero scalar a : K → a K , ( a = 0) .
5. The addition to the tensor of a multiple of R ⊙ R : K → K + a R ⊙ R . Moreover, the discrete transformation considered, is the one induced by the inversion I withrespect to the unit sphere with centre at the origin I : ( x, y, z ) → (cid:18) xx + y + z , yx + y + z , zx + y + z (cid:19) . (5.9)Note that I − = I and that for the continous inversion φ we have φ = I − ◦ φ ◦ I , where φ is the transaltion along the z -axis. Remark 5.1.
The addition of the metric g and the transformation induced by the rotationaround the z -axis are not relevant, since they do not modify the parameters (4.3) definingthe tensor. 15 .2 Group action, invariants and canonical forms Let G be the group generated by the above described transformations. Since the discreteinversion is included, G is not connected. Moreover, two of the continuous one-parametertransformations are defined only for values of the parameter in R −{ } , so that the connectedcomponent of G containing the identity is characterized by a > a >
0. Two otherdiscrete transformations are implicitly included in G : the change of sign of the tensor (for a = −
1) and the transformation induced by the symmetry around the origin in E (for a = − K ∈ RCK ( E ) is given by ˜ M = A , ˜ L = D , ˜ H = H, ˜ C = C , ˜ D = L , ˜ A = M . (5.10)The equations of the action generated by the five continuous transformations acting on (4.3)are ˜ M = a P ( a ) a , ˜ L = a − a P ( a ) − a P (1) ( a ) a , ˜ H = a a P ( a ) + 3 a a P (1) ( a ) + a P (2) ( a ) a , ˜ C = a + a C + a a P ( a ) + 3 a a P (1) ( a ) + a ( P (2) ( a ) − H )3 a , ˜ D = a − a P ( a ) − a a P (1) ( a ) − a a P (2) ( a ) − a P (3) ( a ) a , ˜ A = a a P ( a ) + a a P (1) ( a ) + . . . + a a P (3) ( a ) + a P (4) ( a ) a , where P ( a ) = A a − D a + Ha − L a + M , (5.11)and P ( n ) = 1 n ! d n P ( da ) n . Since C and a are involved only with ˜ C , and C is unchanged by the discrete inversion(5.10), we can disregard C (which can be made equal to any fixed constant by choosinga particular value for a ). Then we consider the reduced action on the vector subspace of RCK ( E ) defined by the five parameters( M , L , H, D , A ) (5.12)16f the subgroup G ′ of G defined by a = 0:˜ M = a P ( a ) a , (5.13)˜ L = a − a P ( a ) − a P (1) ( a ) a , (5.14)˜ H = a a P ( a ) + 3 a a P (1) ( a ) + a P (2) ( a ) a , (5.15)˜ D = a − a P ( a ) − a a P (1) ( a ) − a a P (2) ( a ) − a P (3) ( a ) a , (5.16)˜ A = a a P ( a ) + a a P (1) ( a ) + . . . + a a P (3) ( a ) + a P (4) ( a ) a . (5.17)It appears that the building blocks of the action equation is the polynomial (5.11) and itsderivatives. Remark 5.2.
If we denote the parameters (5.12) by α i ( i = 0 , . . . , α = M , α = L , α = H , α = D , α = A ), then their transformation laws under the action canbe written in a compact formal way as˜ α − i = a a i X h =0 ( − i (cid:18) − hi (cid:19) P ( h ) ( a ) a i − h a h , i = 0 , . . . , . Theorem 5.3.
Let G be the subgroup of G ′ defined by a > . Then, the action of G on ( ) given by ( − − ) and ( ) is equivalent to the classical action of GL (2 , R ) on real binary quartics.Proof. Consider the following binary quartic constructed from the five coefficients (5.12) ofthe CKT: Q ( X, Y ) = M X + L X Y + HX Y + D XY + A Y . (5.18)By inserting the linear transformation of the variables ( X, Y ) X = α ¯ X + β ¯ Y , Y = γ ¯ X + δ ¯ Y , with ( αδ − βγ ) = 0, in (5.18), we obtain a new quartic ¯ Q ( ¯ X, ¯ Y ) whose coefficients ¯ M , . . . , ¯ A depend on the GL (2 , R ) matrix M = (cid:20) α βγ δ (cid:21) and on the coefficients of Q ( M , . . . , A ). Since we assume a >
0, by setting α = q a a , β = − a q a a , γ = − a q a a , δ = ( a a + a ) q a a , we obtain equations (5.13 −− M follows from ( αδ − βγ ) = √ a a | a | 6 =0, since a a = 0. Furthermore, setting α = γ = 0, and β = δ = 1, we recover (5.10). Con-versely, we prove that for any transformation of the quartic we can associate a transformationof G . We distinguish two cases: for α = 0, by setting a = − γα − , a = − βα − , a = ( αδ − βγ ) α − , a = ( αδ − βγ ) , − − M on the quartic form. The fact that a a = 0follows from the regularity of the matrix M . If α = 0, we apply first the discrete inversion(5.10) on the parameters of the CKT, that is we multiply M by (cid:20) (cid:21) on the left. In thisway we obtain a new matrix M with α = γ = 0 since M is regular, and thus revert to theprevious case.As an immediate consequence of the theorem we are able to determine the invariant ofthe action and the list of canonical forms which are given in the following propositions. Proposition 5.4.
The only independent differential invariant of the action of G on RCK ( E ) is F = I J , where the functions I = 12 A M − L D + H ,J = 72 A M H − A L − D M + 9 D L H − H , are relative invariants of the action of G and independent differential invariants for theaction of the subgroup of G defined by a = 1 .Proof. The functions I and J are the fundamental invariants (of weight 4 and 6 respectively)of the binary quartic form (5.18) [13],[20]. Proposition 5.5.
Each CKT of
RCK ( E ) is equivalent under the action of G to one ofthe following representatives: I. I ⊙ I + µ D ⊙ D + X ⊙ X , µ ∈ R , (5.19) II. I ⊙ I + µ D ⊙ D − X ⊙ X , µ ∈ R , (5.20) III. I ⊙ I + ν D ⊙ D , ν = ± , (5.21) IV. D ⊙ I , (5.22) V. I ⊙ I . (5.23) Proof.
Starting from the list of canonical forms of real binary quartics (given for instance in[13]), we combine those differing only by sign. We remark that for µ = 2 the canonical form I. is equivalent to D ⊙ D . Remark 5.6.
The action of G over RCK ( E ) has infinitely many orbits. However, thetensors in (5.19) and (5.20) are not pairwise inequivalent for all values of µ : for µ = ± µ ′ such that the corresponding tensors are pairwise equivalent(see [13]). R -separable rotation-ally symmetric webs The polynomial P defined in (5.11) as the building block of the action equations (5.13–5.17)is deeply related to the polynomial q (4.6). Indeed, we have P ( X ) = X q ( − /X ) . Moreover q is the inhomogeneous polynomial corresponding to the quartic binary form Q (5.18).18he roots of q are the points on the z -axis where all the eigenvalues of K coincide (seeRemark 4.3). The conformal transformations φ , φ , φ and I described in Sect. 5.1 mapthe z -axis to itself with a one to one correspondence (if we include also the point at infinity).Thus two distinct points cannot be made coincident or removed. This provides the geometricinterpretation of the fact that the invariants of Q are invariants of the CKT defining theweb. The meaning of q in terms of invariant theory is made more precise in the followingproposition. Proposition 6.1.
The polynomial q ( z ) = M z + L z + Hz + D z + A is a relativecovariant of the induced extended action on C ˆ K ( E ) × E restricted on the invariant subset S = { x = y = 0 } .Proof. The equations of the extended action are (5.13–5.17) together with˜ x = a x ( a z + 1) + a ( x + y ) , ˜ y = a y ( a z + 1) + a ( x + y ) , ˜ z = a z + a ( x + y + z )( a z + 1) + a ( x + y ) + a . The subset S = { x = y = 0 } is an invariant subset of the extended action. Moreover, on S the transformation law for z reduces to the linear fractional transformation˜ z = ( a + a a ) z + a a z + 1 ( a = 0) , (6.24)which is the general linear transformation on RP (see [20]). Let ˜ q (˜ z ) be the polynomial weobtain by inserting (5.13–5.17) and (6.24) in (4.6). We obtain( a z + 1) ˜ q (˜ z ) = a a q ( z ) , (6.25)that is (up to a ) a covariant of weight two of the action. For the discrete inversion (mapping z into ˜ z = 1 /z ), we immediately see that it maps q ( z ) to ˜ q (˜ z ) = q ( z ) z Equation (6.25) shows that the number and multiplicity of the real roots of q ( z ) (that isthe number and multiplicity of the real linear factors of Q ) are invariant with respect to thegroup action. Hence they can be used to define and classify the different types of webs. Definition 6.2.
We say that two rotationally symmetric R -separable webs are of the sametype if the polynomials associated with the corresponding characteristic CKT have the samenumber and multiplicity of real roots. Thus we have reduced the classification of rotational R -separable webs to the classicalclassification of real binary quartics (see [20], [13]).We have nine types of webs, listed in Table 2.The remaining coordinates systems of Table 1 are equivalent to one of the coordinateslisted above (correcting a typographical error in [18], where Cap cyclide coordinates are saidto be equivalent to Bicyclide coordinates), as it is described in Table 3.19ssociated web roots of q canonical form of K Bi-cyclide 4 real distinct roots I. for µ < − I. for µ > − µ = 2Disk cyclide 4 distinct roots,2 real, 2 complexconjugate II.
Inverse prolatespheroidal 1 double real root,2 distinct real roots
III. for ν = − III. for ν = 1Toroidal 2 double complexconjugate roots I. for µ = 2Bispherical 2 double real roots I. for µ = − IV.
Tangentsphere 1 quartuple (real) V. Table 2: the nine types of inequivalent rotational R -separable webs Remark 6.3.
The number of the types of rotationally R -separable coordinate systems agreewith the results of [5], where the subject is examined from the point of view of symme-try operators. The coefficients A ij of the second order part of the symmetry operators S characterizing each type of R -separable rotationally symmetric coordinates, with respect toCartesian coordinates, listed in Table 2. of Boyer, et al. [5] when written as S = A ij ∂ i ∂ j + B i ∂ i correspond to the components of CKTs equivalent to those listed in Table 1 for Bi-cyclide,Flat-ring cyclide, Disk cyclide and Toroidal coordinates, respectively.Finally, we provide algebraic conditions on the parameters (5.12) in order to determine thetype of the corresponding web. In order to obtain these conditions, we solve the equivalentproblem of determining the number and multiplicity of the linear factors of the correspondingbinary quartic form Q which can be done by applying the classical algorithm (see for example[13]) based on the sign and vanishing of relative invariants and covariants of Q .Together with I and J , the following invariant and covariants are used in the classificationscheme: the discriminant of the form (a relative invariant which vanishes if and only if thequartic has a multiple root) ∆ = I − J , the Hessian of the form (a covariant which vanishes if and only if the quartic has a quadrupleroot) H ( X, Y ) = ( ∂ XX Q ) · ( ∂ Y Y Q ) − ( ∂ XY Q ) ;the covariants L ( X, Y ) = IH ( X, Y ) − J Q ( X, Y ) , M ( X, Y ) = 12 H ( X, Y ) − IQ ( X, Y ) . We summarize the classification in Table 4.Web Algebraic conditionDisk cyclide ∆ < > H ( X, Y ) <
0, and M ( X, Y ) > > H ( X, Y ) > M ( X, Y ) > L ( X, Y ) < L ( X, Y ) > L ( X, Y ) = 0 and H ( X, Y ) > L ( X, Y ) = 0 and H ( X, Y ) < I = J = 0 and H ( X, Y ) = 0Tangent sphere H ( X, Y ) = 0Table 4: Invariant classification of the websThe following simple example shows how the above method can be used.
Example 6.4.
Let us consider the natural Hamiltonian scalar potential H = 12 ( p x + p y + p z ) − c ( x + y + z − c ) + 4 c z . (6.26)Although the potential clearly admits a rotational symmetry around the z -axis, it is straight-forward to check that the Hamilton-Jacobi equation H = h does not admit separation ofvariables in any simply separable coordinate system of E . This is because the compatibilitycondition (3.32) is never satisfied except for KTs of the form K = a R ⊙ R + b g , none ofwhich are characteristic. However, by imposing the conformal compatibility condition (3.33)21o the general rotationally symmetric CKT (4.1) we get, that for E = 0, the conditions onthe parameters of K are M = 12 c H, L = 0 , D = 0 , A = c H. The associated polynomial and binary quartic are z c + z + c (cid:18) z c + c (cid:19) , Q ( X, Y ) = X c + X Y + c Y , respectively. The polynomial has two double imaginary roots and - according to Table 2 -the associated web is the Toroidal web. From the invariant point of view, the discriminant∆ and the covariant L ( X, Y ) vanish and the Hessian of Q is H ( X, Y ) = 12 ( X + Y · c ) c > H = 0, for the Hamiltonian (6.26) admits separationof variables in Toroidal coordinates centered at the origin. R -separable webs in E z -axis X , the dilatation D and the infinitesimal inversion along the z -axis I . The three types of CKVs have beenput in the above simple canonical forms by use of the isometry group SE (3).In all cases the infinitesimal invariance together with the TSN conditions are equivalentto the fact that the infinitesimal symmetry is an eigenvector of the CKT (see Remark 4.2).The results obtained are given in the following proposition. Proposition 7.1.
There exist no R -separable webs in E that possess translational, dilationalor inversional symmetry which are not separable webs or can be obtained from one by theinversion I .Proof. We consider separately the three possible cases.
Translational symmetry :The equation (3.36) with V = X , K given by (2.29), and h = 0, implies that the onlynon-zero components of the CKT are given by K = − A − A + 1 / B z − / B y + C ( y + z ) K = A − B z − / B y + C ( y − z ) K = A + 1 / B z + B y + C ( z − y ) K = A − / B z + 3 / B y + 3 C zy (7.27)By comparison with (2.31) we conclude that K is the trace free part of an ordinary KTand hence defines the simply separable webs which admit a translational symmetry alongthe z-axis [14]. This result agrees with that in [17] where a first principles proof is giventhat non-trivial R -separability of the Laplace and Helmholtz equations is never possible ina cylindrical coordinate system. Dilatational symmetry : 22he equation (3.36) with V = D , K given by (2.29), and h assumed constant, implies that h has five possible integer values. The components of the CKT corresponding to each valueof h are polynomials of the same degree. The imposition of the TSN condition yields anadditional restriction only in the case when the components are second degree polynomials.All cases correspond either to simply separable webs or can be mapped into one by theinversion I . Thus we conclude that modulo equivalence with separable webs, there are no R -separable webs which admit a dilatational symmetry. Inversional symmetry :The equation (3.36) with V = I , K given by (2.29), and h some non-zero function impliesthat K = 0. If h = 0, it may be shown by use of the discrete inversion that this case is equiv-alent to that of the translational symmetry considered above. Thus, modulo equivalence,there are no R -symmetric webs admitting an inversional symmetry.We note that this proposition is in agreement with the results of Table 2 of [5] where theonly non-rotationally symmetric R -separable coordinate systems listed are the asymmetriccases not studied in this paper. We gave a classification of the rotationally symmetric R -separable webs for the Laplaceequation on E in terms of the invariants of characteristic conformal Killing tensors underthe action of the conformal group. Our method shows that there are exactly nine inequivalenttypes of webs, five of which are conformally equivalent to separable webs, in agreement withthe results of B¨ocher [4] and Boyer, Kalnins and Miller [5]. An invariant classification ofthe asymmetric webs based on the results of [6] and [7] and following the approach of thepresent article will be the subject of a subsequent paper. The authors wish to thank G. Rastelli and L. Degiovanni for helpful discussions on thebackground theory. MC and RGM also wish to thank the the Department of Mathematics,Universit`a di Torino for hospitality during which part of this paper was written. The researchwas supported in part by a Natural Sciences and Engineering Research Council of CanadaDiscovery Grant and by Senior Visiting Professorships of the Gruppo di Fisica Matematicadell’Italia (RGM) and by a PRIN project of the Ministero dellUniversit‘a e della Ricerca(CMC).
References [1] Benenti S., “Intrinsic characterization of the variable separation in the Hamilton-Jacobiequation,” J. Math. Phys. , 65786602 (1997).[2] Benenti S., Chanu C., Rastelli G., “Remarks on the connection between the additiveseparation of the Hamilton-Jacobi equation and the multiplicative separation of theSchroedinger equation. 1 The completeness and Robertson conditions,” J. Math. Phys. , 11, 5183-5222 (2002). 233] Benenti S., Chanu C., Rastelli G., “Variable-separation theory for the null Hamilton-Jacobi equation,” J. Math. Phys. , 042901 (2005).[4] Bˆocher M., ¨Uber die Reihenentwickelungen der Potentialtheorie (mit einem Vorwort vonFelix Klein) (Teubner, Leipzig, 1894).[5] Boyer C.P., Kalnins E.G., Miller W. Jr., “Symmetry and Separation of variables for theHelmholtz and Laplace Equations,” Nagoya J. Math. , 35-80 (1976).[6] Boyer C.P., Kalnins E.G., Miller W. Jr., “R-separable coordinates for three-dimensionalcomplex Riemannian spaces,” Trans.Amer.Math.Soc. , 355-376 (1978).[7] Chanachowicz M., Chanu C.M., McLenaghan R.G., “R-separation of variables for theconformally invariant Laplace equation,” preprint 2007, arXiv:0708.2163.[8] Chanu C., Rastelli G., “Fixed-Energy R -separation for Schr¨odinger Equation,” Inter-national Journal on Geometric Methods in Modern Physics, , 489-508, (2006).[9] Chanu C., Rastelli G., “Eigenvalues of Killing tensors and separable webs on Rieman-nian and pseudo-Riemannian manifolds,” SIGMA Symmetry Integrability Geom. Meth-ods Appl. , Paper 021, 21 pp. (electronic) (2007).[10] Degiovanni L., Rastelli G., “Complex variables for separation of the Hamilton–Jacobiequation on real pseudo-Riemannian manifolds,” J. Math. Phys. , 073519 (2007).[11] Eastwood M., “Higher symmetries of the Laplacian,” Ann. of Math. , 1645-1665(2005).[12] Eisenhart L.P., “Separable Systems of St¨ackel,” Ann. of Math., , 284-305 (1934).[13] Gurevich G.B., Foundations of the Theory of algebraic invariants , (P. Noordhoff Ltd.,Groningen, 1964).[14] Horwood J.T., McLenaghan R.G., Smirnov R.G., “Invariant Classification of Orthogo-nally Separable Hamiltonian Systems in Euclidean Space,” Commun. Math. Phys. ,679-709 (2005).[15] Miller W. Jr.,
Symmetry and Separation of Variables , (Addison-Wesley Publishing Co.,Mass, U.S.A, 1977).[16] E. G. Kalnins, “Separation of Variables for Riemannian spaces of constant curvature”,
Pitman Monographs and Surveys in Pure and Applied Mathematics, 28 , (Longman Sci-entific & Technical, Essex, England, 1986).[17] Moon P., Spencer D.E., “Separability in a class of coordinate systems,” J. Franklin Inst , 227-242 (1952).[18] Moon P., Spencer D.E.,
Field Theory Handbook , (Springer-Verlag OHG, Berlin, 1961).[19] Nijenhuis A., “ X n − -forming sets of eigenvectors,” Neder. Akad. Wetensch. Proc. ,200-212 (1951). 2420] Olver P.J., Classical Invariant Theory , London Math. Soc. Stud. Texts 44, (CambridgeUniversity Press, Cambridge, 1999).[21] Olver P.J.,
Applications of Lie groups to Differential Equations , Grad. Texts in Math.107, 2nd Edition, (Springer Verlag, New York, 1993).[22] Schouten J.A., “ ¨Uber Differentalkomitanten zweier kontravarianter Gr¨ossen,” Proc.Kon. Ned. Akad. Amsterdam , 449-452 (1940).[23] Rani, R., Edgar, S.B., Barnes, A. “Killing tensors and conformal Killing tensors fromconformal Killing vectors,” Class. Quantum Grav. , 19231942 (2003).[24] Tonolo A., “Sulle variet`a Riemanniane normali a tre dimensioni,” Pont. Acad. Sci. Acta13