separability thesis

On the classification of the R-separable webs for the Laplace equation in E3 by Mark Chanachowicz A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2008 c/ci∇cleco√y∇tMark Chanachowicz 2008 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically availab le to the public. ii Abstract In the first two Chapters I outline the theory and background of separation of vari- ables as an ansatz for solving fundamental partial differentia l equations (pdes) in Mathematical Physics. Two fundamental approaches will be hig hlighted, and more modern approaches discussed. In Chapter 3 I calculate the gener al trace-free con- formal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determi ne the subcases with rotational symmetry and recover known examples pertaining t o classical rotational coordinates. In Chapter 5 I obtain the induced action of the co nformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of t he conformal group to characterize, up to equivalence, the symmetric R-separable webs in E3that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Lapl ace equation. iii Acknowledgments I thank my supervisor Ray McLenaghan for Ph.D funding, guidanc e and ideas in the two main research projects that comprise this thesis, and also for patient help during the thesis write-up prior to official submission. While on m y first exchange period abroad in Turin, Italy during the summer of 2005, in a hi gh intensity period when I had in a hurry to learn more background formalism and cur rent tools in the research on separation of variables theory - in parallel with a djusting to a new place, new culture and not to mention learning a new language, our co llaborator Claudia Chanu greatly assisted me by hours of patient guidance, tutorin g and inspiration which in a few month’s time put me on track. She is an excellent teacher and one to discuss ideas and problems in research with. Claudia was a mag nificent help to me during high stress upon seeing the wealth of formalism I had to w arm up to in so short a time. I wish to thank Luca Degiovanni and Giovanni Rastelli for helpf ul discussions on the background theory, as well as Luca’s patient explanat ion of first principles theory I still needed to learn. I appreciate the hospitality of the University of Turin during my three exchange periods (not to mention the pr ivilege of subsidized mensa that was extended to me, and accommodation in friendly st udent residences (especially Residenza Cavour) where I was exposed to the charm a nd openness of Italian culture through the many friends I met there which certainly boosted my energy of research during my stays), and the help of the exchan ge co-ordinate Lorenzo Fatibene and Enrico Bibbona in logistics during the c ourse of my research. I am also grateful to Stepf Czapor for providing me and my colla borators with useful theory of polynomial resultants that helped solidify ou r main conjecture of the research I collaborated on. I am grateful for the support my parents - Peter and Heidi - gave me throughout my Ph.D studies. Recently I owe my strong energy level to German Club and Intern ational Pub Night events at the University of Waterloo - the people and their ambiance certainly helped during the months I spent writing up this manuscript. I am grateful for additional funding which assisted me, in the Uni versity of Waterloo and in Universita di Torino, provided by numerous bur saries from the Graduate Student Office in UW. I am also indebted to C.M Lerici fou ndation in Stockholm, Sweden which provided me with a scholarship to assist m e during my 3rd exchange period in Turin. iv Contents 1 Introduction 1 1.1 Definitions of Separation of Variables . . . . . . . . . . . . . . . . . 2 1.2 The theory of St¨ ackel matrices . . . . . . . . . . . . . . . . . . . . . 5 1.3 Tensorial formulation of separation of variables theory . . . . . . . . 9 1.4 Example of R-separation of the Laplace equation for toroidal coor- dinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Symmetry operator approach to separation of variables th eory . . . 13 1.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Theory of separation of variables 16 2.1 Link between St¨ ackel formalism and conformal Killing te nsors . . . 16 2.2 Invariant theory of conformal Killing tensors . . . . . . . . . . . . . 24 2.3 Relation of CKT’s to existence of R-separable webs . . . . . . . . . 34 2.4 Conformal Killing tensors in spaces of zero curvature . . . . . . . . 36 3 Construction of the general CKT in E339 3.1 Killing vector formalism . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Compact and expanded form of the general CKT . . . . . . . . . . 4 2 4 The set of rotationally symmetric characteristic CKTs in E344 4.1 Definitions and constructions of rotationally symmetric w ebs . . . . 44 4.2 Characteristic Killing tensors for rotational R-separable coordinates 47 4.3 Jacobi elliptic coordinates . . . . . . . . . . . . . . . . . . . . . . . 55 5 Group actions preserving rotationally symmetric canonic al CKTs 63 5.1 Continuous group actions . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Discrete group transformations . . . . . . . . . . . . . . . . . . . . 64 5.3 Effect of continuous group actions on the Killing tensor co efficients and calculation of invariants . . . . . . . . . . . . . . . . . . . . . . 67 v 6 Classification of the symmetric R-separable webs 73 6.1 Classification of rotationally symmetric R-separable coordinates . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Classification of canonically centered rotationally symme tric webs . 74 6.3 The question of non-canonically centered rotationally sy mmetricR- separable coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.4 Canonically centered rotational coordinates related b y balanced com- bination of inversion and translation . . . . . . . . . . . . . . . . . 81 6.5 Classification scheme of non-canonically centered R-separable coor- dinates in E3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.6 Classifying the remaining symmetric R-separable coordinates in E3. . . . . . . . . . . . . . . . . . . . . 86 7 Asymmetric R-separable webs in E388 7.1 The conformally invariant Laplace equation . . . . . . . . . . . . . 88 7.2 Proof of conformal invariance . . . . . . . . . . . . . . . . . . . . . 8 9 7.3 The CI-Laplace equation and R-separation . . . . . . . . . . . . . . 92 7.4 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 94 7.5 Applications and examples . . . . . . . . . . . . . . . . . . . . . . . 98 8 Conclusion 102 A Proof of Levi-Civita’s criterion for separability 104 B Proof of the connection between St¨ ackel matrices and Killi ng ten- sors in Eisenhart’s formalism 108 C Construction of the St¨ ackel matrix associated with coord inates separating the HJ equation 112 D An equivalent property of the Schouten bracket 114 E Proof of the eigenvalue equations for characteristic conf ormal Killing tensors by construction from simple Killing tensors 115 vi F Alternate classification scheme using the invariants and c ovariants of biquartic polynomials 118 F.1 Characteristic CKTs of the known R-separable rotational coordinate systems . . . . . . . . . . . . . . . . . . . . . 120 F.2 Group action preserving rotationally symmetric CKTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 F.2.1 The group and its one-parameter subgroups . . . . . . . . . 12 2 F.2.2 Group action, invariants and canonical forms . . . . . . . . . 123 F.3 Invariant classification of the R-separable rotationally symmetric webs126 List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 vii List of Tables 5.1 Lie Commutator Table . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.1 Equivalence classes of R-separable webs . . . . . . . . . . . . . . . . 86 F.1 Characteristic CKT of rotationally symmetric R-separable webs . . 121 F.2 the nine types of inequivalent rotational R-separable webs . . . . . 128 F.3 Pairwise conformally equivalent webs . . . . . . . . . . . . . . . . . 129 F.4 Invariant classification of the webs . . . . . . . . . . . . . . . . . . . 129 viii NOTATIONS AND CONVENTIONS —————————– R: Set of real numbers M: Riemannian manifold [,] : Lie-Schouten bracket (,): Symmetrization of the indices [,]: Anti-symmetrization of the indices ∂i: Partial derivative w.r.t xi(also denoted by ,i) ∇i: Covariant differentiation operator (also denoted by ;i) ⊙: Symmetric tensor product g: contravariant metric tensor g: determinant of the metric tensor Γi jk: Christoffel symbol Γi: Contracted Christoffel symbol Rijkl: Riemann curvature tensor Xi;jk−Xi;kj=XlRl ijk: Ricci identity Rij=gklRkijl: Ricci tensor Rs=gijRij: Curvature scalar, usually in the literature denoted as R Rijk: Cotton tensor δij: Kroenicker delta Kij: Valence two Killing tensor ρ: Eigenvalue of a Killing tensor hi a: i’th component of the eigenvector corresponding to the eige nvalueρa ix p: Valence of Killing tensor CˆKp(M) : Linear space of valence ptrace-free conformal Killing tensors k: Some symmetric tensor of type ( p−1,0) or Killing vector, depending on context. S: Determinant of a St¨ ackel matrix Q: Conformal factor in St¨ ackel matrix R: Modulation factor of R-separability log : Natural logarithm function, base e Sij: St¨ ackel operator H: Hamiltonian of Classical System ∆ : Laplace-Beltrami operator W: Solution of the Hamilton-Jacobi equation E: Energy of a Classical System V: Potential of a Classical System E3: Euclidean space xi,ui,qi: i’th coordinate, depending on context pi: i’th component of the generalized momenta I: Invariant function C(M) : Conformal Group φt: One parameter group of transformations f∗: The push forward map of the diffeomorphism f:M→N L: Lie derivative operator G: Connected Lie group of transformations x v: Infinitesimal generator of Lie group action k: Jacobi Elliptic parameter 2: End of Proof symbol xi Chapter 1 Introduction The method of separation of variables is a classical tool to split a partial differen- tial equation into systems of ordinary differential equations, or decouple systems of partial differential equations (pdes) into classes of ordinary differential equations each depending on one variable only. This is a standard approa ch to solve impor- tant boundary value problems in mathematical physics. There a re many known examples of coordinate systems admitting separation of variab les for the Laplace equation. These include Cartesian, polar, spherical and cylin drical coordinates the choice of which depends on the type of initial conditions spec ified. Less known examples (applied especially in electromagnetic theory) in clude elliptic-hyperbolic, ellipsoidal, paraboloidal and conical coordinates to tackl e boundary problems that might otherwise be solved only numerically by those unaware of t he separation of variables properties such coordinates admit. Clearly an exhaustive search for other coordinate systems possessin g this prop- erty would facilitate calculations in many fields where anal ytic solutions of partial differential equations are desired over numerical ones, despit e the very complicated boundary value conditions that may arise. The standard theory of separation of variables has been broadened to include a weaker R-separation of variables method where the product solution ansatz contains a non-constant fact or (denoted by R, also called the modulation factor) depending on the coordina tes. This relaxation of strict separability permits a broader class of coordinate syst ems where equations can be separated in this way. The most familiar coordinate system admitting this property ar e toroidal coor- dinates. Less familiar are the Jacobi-elliptic coordinates t hat arise from the solu- tion of confocal quartics (defined in [39]) and contain degre e four surfaces. Other examples with rotational symmetry are bi-cyclide coordinat es, flat-ring cyclide co- ordinates and disk-cyclide coordinates. Two examples are kno wn to exist with no coordinate symmetries whatsoever - but have yet to be named. This thesis focuses on R-separation of variables of the Laplace equation in Eu- clidean space. The seventeen separable and R-separable coordinates in Euclidean space were first determined by Bˆ ocher, Eisenhart, Weinacht and Blaschke ([5], [20], 1 [49], [4]). Much later they were classified using group theoreti c methods by Boyer, Kalnins and Miller [6]. We provide an exhaustive classification of the extra set of R-separable coordinates based on differential invariants of va lence-two conformal Killing tensors under the action of the conformal group, whic h is an extension of the ordinary isometry group. The approach is an extension of th at employed by Horwood, McLenaghan and Smirnov [25] who give an invariant cl assification of the eleven simply separable coordinate systems (also known as coo rdinate webs) for the Hamilton Jacobi and Helmholtz equation in E3in terms of the invariants and reduced invariants (with respect to the isometry group) of valence-two Killing tensors. Associated to R-separable coordinates are certain valence-two conformal K illing tensors. Associated to each conformal Killing tensor is a trace-fr ee representa- tion, this subset has finite dimension and is the geometric objec t characterizing R-separation of variables. In this thesis these will be calculate d for all known rota- tionalR-separable coordinates and given in standard form (expressed in Cartesian coordinates) which will facilitate future researchers who wi sh to study boundary value problems with a potential. In the study of boundary value problems for Schr¨ odinger’s eq uation in E3, one would like to know whether the problem may be solved by separati on of variables. The potential function is usually given in terms of Cartesian c oordinates. The existence of a potential restricts the number of possible coordi nate systems with respect to which the equation separates or R-separates. This determination may be made in terms of the general valence-two Killing or confor mal Killing tensors admitted by E3. In this thesis we shall restrict ourselves to analysis of rotatio nally symmetric coordinate systems. 1.1 Definitions of Separation of Variables The linear partial differential equations (pdes) considered i n this thesis for sepa- ration of variables theory are the time-independent Schr¨ odinger (S) equation and the Helmholtz/Laplace (H/L) equation, defined on an ndimensional Riemannian manifold (M,g). These are often considered for separability in a class of coord inate systems to facilitate the solution of various boundary value pr oblems. All of the above pdes are special cases of the pde ∆ϕ+Cϕ= 0, (1.1.1) where ∆ is the Laplace-Beltrami operator defined by ∆ϕ=gij∇i∇jϕ=1√g∂ ∂xi/parenleftBigg√ggij∂ϕ ∂xj/parenrightBigg , (1.1.2) 2 where ∇iis the covariant derivative with respect to the Levi-Civita c onnection of the metric tensor g,gis the determinant of the metric tensor and Cis an in- finitely differentiable function defined on M. ForC/ne}ationslash=const, Eq. (1.1.1) yields the time-independent Schr¨ odinger equation. For C=const /ne}ationslash= 0, Eq. (1.1.1) re- duces to the Helmholtz equation. For C= 0, Eq. (1.1.1) reduces to the Laplace equation. Note that for the time-independent Schr¨ odinger e quation, it is customary to writeC=V−E, whereVdenotes the potential and Ethe energy of the system. Separation of variables theory for Eq. (1.1.1) is closely rel ated to that of the Hamilton-Jacobi (HJ) equation for a natural Hamiltonian define d on (M,g). Such an equation may be written as gij∂iW∂jW+V=E, (1.1.3) where∂idenotes∂ ∂xiandVdenotes the potential and Ethe energy of the system. In this thesis uiorqiorxiwill be understood to represent the ithcoordinate, this is because certain symbols for the coordinate are more prevale nt in certain proofs and definitions in the literature than others. The fundamental definition of R-separation of variables for the equation (1.1.1), given by Moon & Spencer [38] and Morse & Feshbach [39], is as foll ows: Definition 1.1.1 If the ansatz for trial solutions ϕ=U1(u1,a)·U2(u2,a)· ··· ·Un(un,a) R(u1,···,un)a∈R2n−1 for some analytic R/ne}ationslash=const permits the separation of ∆ϕ+Cϕ= 0 intonordinary differential equations, then the equation is said to be conformally orR-separable. The function Ris called the modulation factor . Remark 1.1.2 A completeness condition on the parameters aexists to ensure non- degeneracy of the solutions and is given in Chapter 2. If in th e above definition R=const, then the equation is said to be simply separable. One can des cribeR- separability as a relaxation of the ad-hoc simple separabil ity to permit solutions of partial differential equations in a broader class of coordina te systems - thus motivat- ing the research into classifying which coordinates admit th is property. Furthermore ifRis a product of functions of a single variable - that is ∂ijlog(R) = 0fori/ne}ationslash=j, we have the case of trivialR-separation [28]. Since coordinates are trivially R-separable iff they are separable, we regard trivial R-separation as equivalent to ordinary separation [31]. The fundamental definition of separation of variables for the equation (1.1.3) for fixedvalues of the energy is as follows: 3 Definition 1.1.3 If the ansatz for trial solutions W=U1(u1,a) +U2(u2,a) +···+Un(un,a)a∈Rn permits the separation of gij∂iW∂jW+V=E for fixed values of the energy Eintonordinary differential equations, then the equation is said to be conformally separable. Note that the HJ equation for free ranges of the energy admits sim ple (sum) sepa- ration of variables, where the parameter adepends on one more arbitrary constant - that is a∈Rn+1. The completeness condition on a, to ensure non-degeneracy of the solutions, is well known [23]. There is indeed a strong connection between the conditions of separability for the equations (1.1.1) and (1.1.3). Necessary conditions for separa tion of the HJ equa- tion are identical with those required for separation of the H/L /S equation, despite the fact that the HJ equation admits sum separability whereas the H/L/S equation admits product separability of the variables. The theory of separation of variables goes back well over 200 y ears. In 1905 a significant theorem by Levi-Civita was formulated [26]: Theorem 1.1.4 The HJ equation equation, which can be re-written as H(q1,...,qn,∂W ∂q1,...,∂W ∂qn) =E, whereHis the Hamiltonian operator and W(q1,...,qn)is the solution of the equa- tion for coordinates qi, admits (simple) sum separation in a coordinate system if and only if the following equation holds true: ∂H ∂pj∂H ∂pi∂2H ∂qi∂qj−∂H ∂pj∂H ∂qi∂2H ∂qj∂pi−∂H ∂qj∂H ∂pi∂2H ∂pj∂qi +∂H ∂qj∂H ∂qi∂2H ∂pj∂pi= 0, i/ne}ationslash=j (1.1.4) wherepi≡∂W ∂qirepresents the generalized momentum. Since the classical Levi-Civita criterion is almost always stat ed without a clear proof in the literature, an outline of a proof is provided in Appendi x A. The Levi-Civita criterion can be generalized to encompass the case of fixed ener gy, as derived in [3]. It can also be formulated to encompass R-separation of the L and S equation, as in [29]. However direct reference to it can be circumvented by co nsidering conformal transformations to metrics admitting simple sum separability o f the HJ equation; this result will be described in Section 2.1. 4 In this thesis we restrict ourselves to orthogonal separability . In fact in spaces of constant curvature separability of the HJ equation is necessar ily orthogonal, a proof in the literature is given in [26]. Orthogonal separabi lity was once assumed to be a strict condition for separation of the Helmholtz equatio n in any space, until Kalnins and Miller proved this to be false by a counter-exampl e, as well as B. Carter studying the HJ and S equation on the (non-orthogonal) Kerr met ric [10], [9]. In this thesis only orthogonal separability will be considered. I t should be noted that the general Laplace equation reduces for flat space (and Carte sian coordinates) to the well-known classical Laplace equation: ∆ϕ=n/summationdisplay i=1∂2ϕ ∂(ui)2= 0 (1.1.5) Clearly different orthogonal metrics determine the differen t forms that ∆ ϕtakes. R-separation is a weaker condition but the advantage is clearl y that a larger class of coordinate systems admit this property. Research is being un dertaken to charac- terize these extra coordinate systems based on a classification scheme - thi s thesis comprises such research for E3. In Euclidean space there are in fact eleven in- equivalent orthogonal coordinate systems affording simple sepa rability [25], and an additional six admitting R-separability which will be addressed in the coming chapters. Future research is required for higher dimensional fl at spaces pertinent to complicated boundary value problems in mechanics and ele ctromagnetism. 1.2 The theory of St¨ ackel matrices Powerful tools exist to determine whether or not a coordinate system admits simple separation of variables. One method was formulated by P. St¨ ac kel in 1896 [45]: Definition 1.2.1 Associated with each separable metric is a non-singular St¨ ackel matrix , which is an n×narray where the ithrow is a function of the coordi- nateqionly and the first row of its inverse yields the diagonal compone nts of the contravariant metric tensor. Proving the existence of a St¨ ackel matrix is non-algorithmi cal. However this matrix determines everything we need to know about separated equations [37]. Explicitly it is, for the general case of n-dimensions: [S] = φ11(q1)φ12(q1)... φ 1n(q1) φ21(q2)φ22(q2)... φ 2n(q2) ............ φn1(qn)φn2(qn)... φnn(qn) (1.2.1) 5 Moon and Spencer, in their papers [36], [37], formulated nec essary and sufficient conditions - connecting the metric tensor expressed in the separ able coordinates with the associated St¨ ackel matrix - for separation of the Helmh oltz equation: gii=S Mi1(1.2.2) g1 2 S=f1(q1)·f2(q2)·...·fn(qn), (1.2.3) whereMi1is the associated co-factor of the St¨ ackel matrix, and Sis its deter- minant. The first of the above conditions on the metric tensor is necessary and sufficient for (sum) separation of the HJ equation in Classical Mecha nics, whereas both conditions are required for product separation of the Hel mholtz as well as the Schr¨ odinger equation. A metric satisfying the first condit ion is also denoted to be in St¨ ackel form [31]. The second condition is also known as the Robertson condition - named after the mathematician who discussed it in a 1927 paper [ 44]. Remarkably there is a geometric relation with the associated R icci tensor of the metric (in separable coordinates) for the Robertson condition :Rij= 0, i/ne}ationslash=j[20], which was discovered by Eisenhart in 1934. For the case of the Helmholtz equation reducing to the (special c ase) Laplace equation, the following conditions are special cases of the pr evious: gii gjj=Mj1 Mi1(1.2.4) g1 2 gii=f1(q1)·f2(q2)·...·fn(qn)·Mi1 (1.2.5) The above equations become more complicated for the case of R-separation of variables, where for R/ne}ationslash=const simple separation of variables is no longer possible. An additional non-constant function Q(q1,q2,...,qn) must be introduced such that, forR-separation of both the Helmholtz and Laplace equation, gii=SQ Mi1(1.2.6) g1 2 S=f1(q1)·f2(q2)·...·fn(qn)·R2Q (1.2.7) α1≡−Q R3/summationdisplay i=11 fi(qi)gii∂ ∂qi/parenleftBigg fi(qi)∂R ∂qi/parenrightBigg =const (1.2.8) 6 A useful formula resulting from the above relates the determin ant of the St¨ ackel matrixSto the function Q[37]: S=giiMi1 Q(1.2.9) This may be seen as a definition for Q. Clearly the search for the ‘right’ function Q(q1,q2,...,qn) that satisfies all of the above conditions is also a non-algorit hmical endeavor. The function Qis sometimes referred to as the conformal factor . An orthogonal metric (with associated Riemann tensor not neccesar ily vanishing) ad- mitting separation of variables for the HJ equation with fixed e nergy is always conformal to an orthogonal metric admitting simple separabil ity of the HJ equa- tion (with free ranges of the energy). This is not necessarily t he case for the H/L/S equation. In exceptional circumstances when this is true, a si mple relationship exists between RandQ- a connection that becomes apparent when later the con- formally invariant Laplace equation is considered. We digre ss here to prove this relationship, which is a relationship between metrics admitt ingR-separation of variables for the H/L/S equation, and those satisfying Rij= 0, i/ne}ationslash=j. Theorem 1.2.2 In orthogonal metrics admitting R-separation of variables for the H/L/S equation that are conformal to orthogonal metrics adm itting simple separa- bility of the H/L/S equation, the modulation factor Rsatisfies R=Qn−2 4 (1.2.10) Proof: Letg′ ijbe the metric satisfying simple separability, and gijbe the metric admittingR-separability. Thus we have from this starting assumption: /radicalBig g′ S=f1(q1)·f2(q2)·...·fn(qn) (1.2.11) √g S=f1(q1)·f2(q2)·...·fn(qn)·R2Q (1.2.12) Now by definition gii=g′ iiQwhich implies, in general for n-dimensions, g=g′Qn org′=g/Qn. Therefore if√ g′ S=f1(q1)·f2(q2)·...·fn(qn) this implies√g Qn/2S= f1(q1)·f2(q2)·...·fn(qn). So√g S=f1(q1)·f2(q2)·...·fn(qn)·Qn/2, which im- pliesQn/2=R2Q. HenceQn−2 2=R2and after taking square roots we obtain the required formula. 2 In the special case of 3-dimensions, this reduces to R=Q1/4. Examples are numerous in [38]. With 6-sphere coordinates R= (u2+v2+w2)−1/2whereas Q= (u2+v2+w2)−2. The same pattern holds true for inverse oblate coordinates, inverse prolate coordinates and tangent sphere coordinates. T oroidal and bispher- ical coordinates satisfy this relationship too however they ar e unique in that they 7 are conformal to non-flat (albeit Ricci diagonal) metrics. Th e above relation does not hold for the cyclide coordinates, which are conformal to c oordinates that do not admit simple separability of the H/L/S equation - indeed the se coordinates do not admit an orthogonal Ricci tensor. Non-trivial R-separation for the Helmholtz equation does not occur in flat spa ces or spaces of constant curvature - only the Laplace equation R-separates. Also, any R-separable solution of the Laplace equation on a conformally fl at space corresponds to a regular separable solution of the Helmholtz equation on a spa ce of constant curvature [43]. An example of a metric admitting R-separation of the Helmholtz equation on a conformally flat space is given by [31]: ds2= (x+y+z)[(x−y)(x−z)dx2+ (y−z)(y−x)dy2 + (z−x)(z−y)dz2] R= (x+y+z)−1 4 (1.2.13) Another example is ds2=−dx2+dy2+ (y−x)−1dz2 R= (x−y)1 4 (1.2.14) Having introduced St¨ ackel matrices it must be pointed out tha t each coordinate system admitting separation of variables does not admit a uniqu e St¨ ackel matrix: they are not in 1:1 correspondence. Two such matrices are under stood to be equiv- alent if their ratiosS Mi1are the same - indeed if they generate the same metric tensor. This allows for a flexibility of operations on the colu mns except for one (usually the first or the last in the literature). Operations on t he rows are generally not permitted. The allowed actions on the columns are [37] 1. Interchanging of the ithand thejthcolumn, where we understand i, j= 2,3,···,n. 2. Multiplication of the jthcolumn by a non-zero constant c∈R. 3. Addition of each element in the ithcolumn by the corresponding element in thejthcolumn multiplied by a common non-zero factor c∈R. We conclude this section with a brief discussion of the equivalen ce between separa- bility of the HJ equation and the existence of a St¨ ackel matrix , which was achieved by Paul St¨ ackel in 1893. We start by constructing this non-sing ularn×nmatrix which we denote as ϕk i, the index klabeling the column and the index ilabeling the row. We define each row of the St¨ ackel matrix, of course, to only depend on the corresponding coordinate qi, thus∂ ∂qjϕk i= 0 ifi/ne}ationslash=j. This means then ϕk i=ϕk i(qi). For notational simplicity we denote S−1=ϕi k, taking care not to confuse this with the transpose operation. 8 Theorem 1.2.3 The HJ equation of a conservative Hamiltonian (no time-depe ndence) is separable in the qiiff: 1) (g11,g22,...,gnn) is a row of the inverse St¨ ackel matrix. 2) The potential V=g11U1(q1)+g22U2(q2)+...+gnnUn(qn)for some arbitrary set of functions Ui(qi). Proof: A known result is that separability of the Hamiltonian implies se parability of1 2giip2 i, the so called geodesic part. This means that since pi=∂W ∂qi=˜φi(qi,αk) that (pi)2=φi(qi,αk), which is a function of qionly. The squared variable is still only dependent on qi. So now we use the fact that, without loss of generality, we may set the total energy Eof the Hamiltonian to equal αn, one of the arbitrary constants [23]. Thus we have1 2giiφi(qi,αk) =E=αn. How the components of the St¨ ackel matrix are constructed, base d on the above, is given in Appendix C. 1.3 Tensorial formulation of separation of vari- ables theory A first principles approach to separation and conformal separat ion of variables for the HJ, H and L equation in terms of the separable coordinates wa s described in Section 1.2. In this section we shall discuss a coordinate invari ant approach based on the theory of valence-two symmetric Killing tensors and val ence-two symmetric conformal Killing tensors as discovered by Eisenhart [20], [21 ] during research in the 1930’s. Following [20] we make the definition, Definition 1.3.1 A valence-two symmetric tensor Kijis aKilling tensor if it satisfies: Kij;l+Kjl;i+Kli;j= 0, (1.3.1) where ;denotes the covariant derivative. Note that in Eisenhart’s 1934 paper Kijis denoted as aij. From the definition one can see that Killing tensors are unique up to a scalar ctimes the metric tensor. In the literature Killing tensors are often denoted as simple Killing tensors to dis- tinguish them clearly from conformal Killing tensors. If the Ki lling tensor has real pointwise simple eigenvalues and normal eigenvectors, then th ere exists a system of orthogonal coordinates such that [20]: Kij=ρigij(no sum), (1.3.2) 9 where the eigenvalues ρisatisfy the following system of equations (also known as theEisenhart equations ): ∂ρi ∂xi= 0 (1.3.3) ∂ρi ∂xj= (ρi−ρj)∂log(gii) ∂xj, i/ne}ationslash=j (1.3.4) The integrability conditions for the above equations (also c alled the Eisenhart in- tegrability conditions ) are ∂2log(gii) ∂xi∂xj+∂log(gii) ∂xj∂log(gjj) ∂xi= 0, i/ne}ationslash=j (1.3.5) and ∂2log(gii) ∂xj∂xk−∂log(gii) ∂xj∂log(gii) ∂xk+∂log(gii) ∂xj∂log(gjj) ∂xk +∂log(gii) ∂xk∂log(gkk) ∂xj= 0,i,j,k all distinct (1.3.6) Whether or not a particular metric admits separation of varia bles depends on whether or not the above two conditions are satisfied. Indeed it is proven in [2] and [3] that the HJ equation for null geodesics is separable in or thogonal coordi- nates on an n-dimensional Riemannian space if and only if there exist nKilling tensors (Ki) = (K1,K2,...,Kn) such that they are pointwise linearly independent and with common eigenvectors. As already mentioned the metric s admitting sepa- rability of the HJ equation for fixed values of the energy are co nformally related to metrics allowing separability of the HJ equation for free rang es of the energy via the conformal transformation equation ˜gii=e2σgii (1.3.7) for a well behaved function of the coordinates σ. Indeedσis related to the afore- mentioned Qfactor in determining a St¨ ackel matrix for proving R-separability of the Laplace equation. A coordinate invariant characterization of R-separability of the H/L/S equation requires the following definition: Definition 1.3.2 A valence-two symmetric tensor Kijis aconformal Killing tensor if it satisfies Kij;l+Kjl;i+Kli;j=kigjl+kjgli+klgij, (1.3.8) wherekiis some vector field. 10 The property of being a conformal Killing tensor is preserved u nder addition of a smooth function ftimes the metric, a fact that will come into play in the sequel. Indeed, two conformal Killing tensors KandLare said to be equivalent , or in the sameequivalence class , ifKij=Lij+fgij, for some scalar function f. Not all conformal Killing tensors, with pointwise real and distinct eig envalues and normal eigenvectors, in an equivalence class are simple Killing tensor s with respect to some conformally related metric. However, there exists at least one suchrepresentative element in each equivalence class [3]. This will be proved in the next se ction. With respect to orthogonal coordinates, the diagonal components o f the conformal Killing tensor obey the following relations: Kij=ρigij,∂ρi ∂xi=ki(no sum), (1.3.9) ∂ρi ∂xj= (ρi−ρj)∂log(gii) ∂xj+∂ρj ∂xj(1.3.10) Eisenhart proved a fundamental theorem linking Killing tenso rs with St¨ ackel ma- trices [20]: Theorem 1.3.3 A necessary and sufficient condition for the existence of a St¨ ackel matrix in some coordinate system - making it a separable coord inate system [30], is that there exists a valence-two Killing tensor with real d istinct eigenvalues and normal eigenvectors. Remark 1.3.4 The existence of such a Killing tensor implies that there exi sts a system of coordinates such that the contravariant components of the diagonalized Killing tensor are elements in a row of the inverse St¨ ackel ma trix, defined for the same coordinates. These coordinates for which this property holds are often called separable coordinates . This is equivalent to the existence of nKilling tensors, in involution and pairwise commutation, associated with each separable metric - one member always being the metric tensor itself [3]. The above property is fundamental to this thesis, since we used th e connection between St¨ ackel matrices and Killing tensors to compute Kil ling tensors for known coordinate systems admitting separation of variables for the L aplace equation [27], [30]. In Appendix B is our own extension of the proof in Eisenhart ’s 1934 paper, noting that many steps were skipped or left to the reader to veri fy. 1.4 Example of R-separation of the Laplace equa- tion for toroidal coordinates A common boundary value problem in electromagnetic theory r equiringR-separation of variables are toroidal coordinates in E3, especially for problems with toroidal 11 boundary conditions such as the 3-Torus. In these coordinates, w herex1=η, x2=θandx3=ψ, the metric tensor components are g11=g22=a2 (cosh(η)−cos(θ))2 g33=a2sinh(η)2 (cosh(η)−cos(θ))2(1.4.1) The denominator in the metric proves an impossible difficulty in the search for a St¨ ackel matrix associated with simple separation for toroidal coordinates. It can be shown by first principles methods that simple separation of tor oidal coordinates is impossible - hence by extension no St¨ ackel matrix exists. Howev er, if we propose R-separation, and let Q=a2 (cosh(η)−cos(θ))2 then we obtainM11 S=M21 S= 1 andM31 S=1 sinh(η)2 The determinant Sof the St¨ ackel matrix is then simply unity and the matrix itsel f takes on the simple form:  1−1−1/sinh(η)2 0 1 0 0 0 1 (1.4.2) The condition α1≡−Q R3/summationdisplay i=11 fi(qi)gii∂ ∂qi/parenleftBigg fi(qi)∂R ∂qi/parenrightBigg =const (1.4.3) is satisfied since R2= (cosh(η)−cos(θ))−1 (recall the relationship between RandQin the sections prior). For the solutions (f1,f2,f3) in the condition g1 2 S=f1(q1)·f2(q2)·...·fn(qn)·R2Q we finally obtain α1= 1/4. Therefore requirements for R-separation of the Laplace equation are satisfied. 12 1.5 Symmetry operator approach to separation of variables theory In the classic papers by Boyer, Kalnins and Miller [6], [7], a co nnection with the existence of separable coordinates of the H/L equation and comm uting pairs of second order symmetry operators is elucidated. The separated solutions for or- thogonal coordinate systems are characterized as common eige nfunctions of pairs of commuting symmetry operators. These operators are linear an d differential. A first order symmetry operator is of the form L=/summationdisplay aj(x)∂j+b(x), (1.5.1) whereajandbare analytic functions of the coordinates in some domain Don the manifold such that Lψis a solution of the Helmholtz equation in Dfor any analytic solution ψof the Helmholtz equation in D. The set of all such symmetry operators forms a Lie algebra under the operations of scalar m ultiplication and commutator bracket [ L1,L2] =L1L2−L2L1. Second order symmetry operators are constructed from products of first order symmetry operators. Thi s is also known as anenveloping algebra of the space. Each separable system is associated with a two - dimensional subspace of commuting operators with S1,S2a non-unique basis for the subspace. The Euclidean group of isometries acts on the set of all two-dimensional subspaces of commuting operators and decomposes this set into or bits of equivalent subspaces. Separable coordinates associated with equivalent sub spaces are regarded as equivalent, as one can obtain any such system from any other by a Euclidean isometric transformation. To studyR-separable coordinates, the Euclidean group of isometries is e xtended to encompass the full conformal group ofE3, as well as the discrete inversion and space reflection. R-separable coordinates are regarded as equivalent to separab le coordinates if one can obtain the other and vice versa by a grou p transformation belonging to the above set. An R-separable coordinate is deemed ‘additional’ if this cannot be done. By this measure there are six additional R-separable coordinates found by [7] along with the eleven simply separable ones that E3admits with respect to the full conformal group as well as the discrete inversion an d reflection. It should be noted that the extended group of transformations does not m ake equivalent any of the eleven simply separable coordinates classified with respec t to the Euclidean isometry group. In their analysis Boyer, Kalnins and Miller used the isomorphism between the conformal group and the isometry group defined on Minkowski spac e of dimension (n+ 2) where the group action becomes linear [6]. This approach was anticipated by Bˆ ocher who constructed R-separable coordinates for the n-dimensional Laplace equation by a method in going up two dimensions and considering functions on the (n+ 2) cone. Modern conformal geometers have recently denoted this as the ambient space . This is beyond the scope of this thesis however, as we restrict ourselves directly to Euclidean space. 13 In the paper of [7], the above formalism is applied to the Helmho ltz equation defined on complex Riemannian manifolds. A discussion of this formalism is beyond the scope of this thesis. Results from this article will however b e correlated with the additionalR-separable coordinates found in the course of my research by mea ns of the invariance of conformal Killing tensors. Indeed the coeffic ientsAijof the second order part of the symmetry operators Scharacterizing each type of R-separable rotationally symmetric coordinates, with respect to Cartesian coordinates, listed in Table 2 of [6] when written as S=∂iAij∂j correspond to the components of conformal Killing tensors equ ivalent to those that will be calculated in this thesis for bi-cyclide, flat-ring cy clide, disk cyclide and toroidal coordinates. It should be added that the simple Killin g tensors found for the eleven simply separable orthogonal coordinates in Euclid ean space [25] corre- spond to the coefficients Aijof the second order part of the symmetry operators S listed in Table 1 of [6]. Referring to the previous example for toroidal coordinates, the two second order symmetry operators associated with that coordinate system are: S1= (x2∂1−x1∂2)2 S2=1 4(∂3+x3+ ((x3)2−(x1)2−(x2)2)∂3 + 2x3x1∂1+ 2x3x2∂2)2(1.5.2) 1.6 Outline of the thesis The remainder of the thesis will be organized as follows. In Cha pter 2 is an illus- tration of the modern tools used in the proof of the connection between St¨ ackel formalism and conformal Killing tensors, as well as the very imp ortant link be- tween conditions for sum separation of variables of the HJ equat ion and product separation of variables of the H/L and Schr¨ odinger equations. Killing vectors and conformal Killing vectors are also introduced, as well as the definition of isometries and group transformations. In Chapter 3, we confirm using symmetr ic products of conformal Killing vectors that the number of arbitrary co nstants for the most general trace-free conformal Killing tensor defined in E3is thirty five, as is stated in the literature [18, 46, 47]. All independent relations are listed. Initially there are twenty constants too many; this requires one to impose fourtee n conditions that result from the trace-free assumption. There are an additional six conditions that arise from relationships among the basis elements themselves. In Chapter 4, the ro- tationally invariant subset of all conformal Killing tensors i s given, using the known coordinate systems in [38] admitting conformal separation of v ariables. The gen- eral rotationally symmetric Killing tensor is deduced from th e most general Killing tensor by two equivalent means. These two approaches in consider ation of the nor- mality (integrability) of the eigenvectors (the TSN condit ions) of the coordinate 14 surfaces [48] are presented and make the study of the rotational webs considerably simpler. After this the characteristic Killing tensors of all th e knownR-separable coordinates are given. Their representations, in terms of symm etric tensor products of conformal Killing vectors, are also discussed and in difficult ca ses derived first (especially for fourth degree surfaces defined in terms of Jaco bi elliptic functions). In Chapter 5 the group transformations preserving the rotatio nally invariant subset of all conformal Killing tensors are given. The conforma l group, as well as discrete operations not continuously connected with the id entity, are discussed along with their effect on the transformed conformal Killing t ensor components. In Chapter 6, we present a proof based on the set of defined group tran sformations that theR-separable webs known thus far are either related to simple sepa rable webs or are otherwise inequivalent. The classical theory of invariant s [34] is a useful tool in this question and applied to characterize all the rotational ly symmetric coordinates tabulated in [38]. Although the main contents in Chapter 6 hav e been published in the Journal of Mathematical Physics [13], the formalism presen ted here differs from the formalism of the paper in that we gave an alternate charact erization conceived in 2006. For historical reasons, we chose to present this ‘first prin ciples’ approach which is admittedly not as compact as the formalism of invaria nts and covariants of bi-quartic polynomials which is given in the paper. A simpl e proof will be shown that the only additional R-separable coordinates admitting symmetries are the rotational ones, leaving the asymmetrical cases to be considere d next. In Chapter 7 the general Laplace equation is modified to include the prop erty of invariance of solutions under conformal transformations. This is also call ed the conformally invariant Laplace equation and an ansatz for a St¨ ackel matrix associated with it is used to derive metrics of asymmetric coordinates expressed in ca nonical Cartesian coordinates. These coordinates are discussed in light of the resul ts of [5] and [7]. Finally, we draw conclusions in Chapter 8 and discuss direction s for future research. Some classical proofs not easily found in the literature are giv en in the Appendices for the interested reader. The reader will no doubt realize that the classification of the c oordinate webs and the algorithm for determining characteristic conformal Killing tensors for all coordinates considered is highly computational. Neverthele ss, all computations are purely algebraic in nature, and this allowed all tasks to be performed in Maple 9 . 15 Chapter 2 Theory of separation of variables 2.1 Link between St¨ ackel formalism and confor- mal Killing tensors A beautiful geometric result is that all separable webs are defi ned by valence-two symmetric Killing tensors, with pointwise simple eigenvalues an d normal eigenvec- tors. Such Killing tensors are said to be characteristic. The sepa rable webs are the families of ( n−1)-dimensional hypersurfaces orthogonal to each eigenvecto r field of the Killing tensor. This geometrical property is why charact erizing all Killing ten- sors in a certain dimension is fundamental to this research. The goal is to express the Killing and conformal Killing tensors in canonical Carte sian coordinates, not in terms of canonical separable coordinates. This is because in physical problems involving potentials, where the method of separation of varia bles is used, the poten- tial is usually expressed in Cartesian coordinates. To this end th e Jacobian of the tensor transformation law must be calculated for every coordin ate system studied, and with the known (contravariant) Killing tensor diagonali zed in the separable coordinates, the following equation applied: K=JTDJ, (2.1.1) whereJis the Jacobian calculated from the coordinate transformati on from Carte- sian to separable coordinates and Dis the diagonalized Killing tensor. The tensorial expression can be written as Kij=∂xi ∂uk∂xj ∂ulDkl, (2.1.2) 16 wherexiare the canonical Cartesian coordinates and uiare the canonical separable coordinates. In component form the Killing tensor Kis:  ∂x1 ∂u1∂x1 ∂u2...∂x1 ∂un ∂x2 ∂u1∂x2 ∂u2...∂x2 ∂un ............ ∂xn ∂u1∂xn ∂u2...∂xn ∂un  ρ1g110... 0 0ρ2g22... 0 ............ 0 0 ... ρngnn  ∂x1 ∂u1∂x1 ∂u2...∂x1 ∂un ∂x2 ∂u1∂x2 ∂u2...∂x2 ∂un ............ ∂xn ∂u1∂xn ∂u2...∂xn ∂un T (2.1.3) wherexiare Cartesian coordinates in terms of the separable coordinat esui. Clearly to calculate the Jacobian explicitly the coordinate transfo rmation law must be known. After the above equation is applied, the result is initi ally expressed in sep- arable coordinates (albeit with the matrix form’s basis bein g the standard canonical basis) - the last step is to calculate or guess the result in Cartesian coordinates. To assist the reader in parallel calculations, for every separabl e coordinate system studied in this thesis, the coordinate transformation law and th e associated St¨ ackel matrix will be provided. Although the Killing tensors can be co mputed by solving the Eisenhart equations, we chose the route of first calculating t he diagonalized Killing tensors through St¨ ackel theory and then using the (pr oven) link given by Eisenhart. The recipe for finding conformal Killing tensors is precisely the same save for the different Jacobian arising from the conformal tran sformation law asso- ciated with simple separability - recall they share the same St¨ a ckel matrix modulo the functions RandQ. Recent research by S. Benenti, C. Chanu and G. Rastelli has yiel ded additional interpretations of conditions for metrics to admit separati on of variables of the H/L as well as the HJ equation. In [15] they showed that R-separation of the H/L equation is equivalent to additive R-separation of the HJ equation for fixed value of the energy (instead of a free range parameter of the energy) . Furthermore they show thatR-separation of variables corresponds to separability of the HJ e quation for fixed values of the energy, whereas simply separable coordi nates correspond to (simple) separability of the HJ equation for free ranges of th e energy. This gives a physical insight into what happens when simple separabil ity is relaxed. A fundamental concept they introduced to prove the above resul ts is the so called St¨ ackel operator : Definition 2.1.1 A St¨ ackel operator is a linear second order differential oper ator defined on any real function f(Q)→Rsuch that: Sij(f) =∂2 ijf−∂jln(gii)∂if−∂iln(gjj)∂jf (2.1.4) St¨ ackel operators satisfy the following properties [2], [3] : Sij(c) = 0 17 Sij(A+B) =Sij(A) +Sij(B) Sij(cA) =cSij(A) Sij(AB) =ASij(B) +BSij(A) +∂iA∂jB+∂jA∂iB Sij(A−1) = 2A−3∂iA∂jA−A−2Sij(A) (2.1.5) St¨ ackel operators ˜Sijcorresponding to a conformal orthogonal metric ˜ gii=e−2σgii satisfy the above as well as: ˜Sij(A) =Sij(A) +e−2σ·(∂ie2σ∂jA+∂iA∂je2σ) ˜Sij(˜gkk) =e−2σSij(gkk)−gkke−4σSij(e2σ) =e−2σgkk/parenleftBigg1 gkkSij(gkk)−e−2σSij(e2σ)/parenrightBigg (2.1.6) Proposition 2.1.2 An orthogonal coordinate system admitting simple separabil ity of the HJ equation satisfies Sij(ghh) = 0 (2.1.7) Proof: Consider a natural Hamiltonian in orthogonal coordinates of t he form: H(q,p) =1 2giip2 i+V(q)≡G+V (2.1.8) It can be shown, by some algebra, that the Levi-Civita separabil ity criterion on H is equivalent to the equation: Lij(H) =giigjjpipj/parenleftbigg1 2Sij(gkk)p2 k+Sij(V)/parenrightbigg = 0 (n.s) (2.1.9) and this is satisfied if and only if1 2Sij(gkk)p2 k+Sij(V) = 0. Indeed gkkis denoted as aSt¨ ackel metric iffSij(gkk) = 0, and a potential is simply separable in these coor- dinates iffSij(V) = 0. This completes the proof [3] that an orthogonal coordin ate system admitting simple separability of the HJ equation satisfies Eq . (2.1.7). 2 In the formalism of [28] and [33], Tis a St¨ ackel multiplier if Sij(T) = 0. Given a metric ds2in St¨ ackel form, the function T(x,y,z ) is a St¨ ackel multiplier if dˆs2=Tds2is also in St¨ ackel form. A St¨ ackel transform is one that is a co nformal transformation preserving the St¨ ackel form of the separable sy stem. Remark 2.1.3 In the previous formula about ˜Sij, if we in particular choose e2σto be any one of (g11,...,gnn), then one recovers a theorem of [7]: if giiis a St¨ ackel metric, then all of/parenleftBig gii g11,...,gii gnn/parenrightBig are St¨ ackel metrics. Remarkably the equation Sij(gkk) = 0is also equivalent to equations: ∂2 ij|gkk| −∂iln|gkk|∂jln|gkk|+∂iln|gkk|∂jln|gii| +∂jln|gkk|∂iln|gjj|= 0 (2.1.10) If in the above one makes the substitution gii=eiH2 i,ei=±1, one then recovers the famous Eisenhart’s equations. Thus we see the power of th e St¨ ackel operators. 18 Theorem 2.1.4 The HJ equation for fixed value of the energy, namely1 2giip2 i+ (V−E) = 0wherepi≡∂iWis separable in orthogonal coordinates, for E∈R, if and only if Sij(ghh) ghh−Sij(gkk) gkk= 0 (2.1.11) and Sij(V) =(V−E) ghhSij(ghh) (2.1.12) for all indices h, kandi/ne}ationslash=j[3]. Remark 2.1.5 This condition is conformally invariant, and will be exploit ed fully in Chapter 7. This is shown in [3] to be equivalent to the exist ence of a function e2σ such that the conformal metric ˜gii=e−2σgiiis a St¨ ackel metric, that is ˜Sij(˜gkk) = 0. Conformally separable coordinates are orthogonal coordinates q=qifor which Eq. (2.1.11) or Eq. (2.1.12) holds. Indeed the conformally s eparable coordinates are useful because they are the only ones in which a natural Ha miltonian with fixed value of the energy can be solved by additive separation of va riables. An important property is that coordinates qiare conformally separable if and only if there exists a St¨ ackel matrix, with elements of the in verse denoted by ϕi (n), such that ∃e2σ|e−2σgii=ϕi (n)⇔gii ϕi (n)=gjj ϕj (n)(2.1.13) for all indices iandj. The Eq. (2.1.11) and Eq. (2.1.12) of Theorem 2.1.4 for fixed va lue of the energy are useful in the proofs of the following two theorems: Theorem 2.1.6 The HJ equation 1 2giip2 i=E, (2.1.14) withE/ne}ationslash= 0fixed, is separable in orthogonal coordinates qiiffgiiis a St¨ ackel metric, that is iff it is separable in the ordinary sense for all values ofE. Proof: SinceV= 0, Eq. (2.1.12) yields Sij(gkk) = 0. For the other direction, if the equation is separable in the ordinary sense then Sij(gkk) = 0 and thus both Eq. (2.1.11) and Eq. (2.1.12) are trivially satisfied. 2 19 Theorem 2.1.7 The HJ equation of the null geodesics giip2 i= 0 (2.1.15) is separable in the orthogonal coordinates qiiff these coordinates are conformally separable. Proof: ForV=E= 0, the Eq. (2.1.12) is trivially satisfied, hence only Eq. (2.1.11) characterizes the equation. 2 Theorem 2.1.8 The HJ equation 1 2giip2 i+V−E= 0 (V−E)/ne}ationslash= 0, (2.1.16) is separable if and only if the conformal metric ˜gii=1 E−Vgii(2.1.17) is a St¨ ackel metric, or equivalently, if and only if for all in dicesh,kandi/ne}ationslash=j, 1 gkkSij(gkk) =1 V−ESij(V). (2.1.18) Thus the coordinates are conformally separable, but the con formal factor e2σmust be equal to the function V−E. Proof: Eq. (2.1.12) is equivalent to1 gkkSij(gkk) =1 V−ESij(V). Note this is just an instance of the St¨ ackel transform [8]. 2 Proposition 2.1.9 The HJ equation is separable for two distinct values of the energyEif and only if it is separable in the ordinary sense. Alternati vely - if a natural Hamiltonian H=G+Vis not simply separable, then there exists at most one value of the energy Esuch thatH=Eis separable. Consider one very important case of the HJ equation with fixed val ue of the energy, 1 2giip2 i+ (V−E) = 0,(V−E)/ne}ationslash= 0. (2.1.19) This is separable if and only if ˜ gii=1 (E−V)giiis a St¨ ackel metric, or equivalently Sij(gkk) gkk=1 (E−V)Sij(V). Coordinates are conformally separable, with conformal factore2σequal to (E−V) since with e2σ= (E−V), ˜Sij(˜gkk) =1 (E−V)Sij(gkk)−gkk (E−V)2Sij(E−V) (2.1.20) hence ˜Sij(˜gkk) = 0. The metric ˜ gii= (E−V)−1giiis called the Jacobi metric of the Hamiltonian H=G+Vwith fixed value of the energy E. 20 Proposition 2.1.10 If a conformal Jacobi metric is a St¨ ackel metric for two dis- tinct values E1/ne}ationslash=E2of the energy, then it is a St¨ ackel metric for all energy E. Proof: Indeed 1 (V−E1)Sij(V) =1 (V−E2)Sij(V)⇒Sij(V) = 0⇒Sij(gkk) = 0.2(2.1.21) Conditions for separability of the HJ equation for fixed energy correspond to those for separation of the Schr¨ odinger equation for fixed value of the energy. If one wishes to impose conditions on an arbitrary metric tensor, such that R-separation of the H/L equation is satisfied, one needs to consider the final ‘compati bility’ condition [14]: Sij(χ)g11=Sij(g11)χ= 0 Sij(χ)g22=Sij(g22)χ= 0 ... Sij(χ)gnn=Sij(gnn)χ= 0 i/ne}ationslash=j (2.1.22) where χ≡ghh 4/parenleftbigg 2∂hΓh−Γ2 h+1 2Rhh/parenrightbigg (2.1.23) Rhhare the diagonal components of the Ricci tensor associated with the orthogonal metricgij. Furthermore, Γh≡gihΓi,Γi≡ghhΓi hh, (2.1.24) where Γi hhis the standard Christoffel symbol of the metric. Remark 2.1.11 For the special case of three dimensions this yields nine pdes that the metric coefficients must satisfy - this is considered and w ith these tools the gen- eral metric of a totally asymmetric coordinate web is integra ted in the last chapter. TheΓhsymbols are useful in other respects as it can be shown that Rij=3 2∂jΓi. Hence the Robertson condition, namely Rij= 0fori/ne}ationslash=j, is equivalent to ∂iΓj= 0. TheΓhsymbols also share an explicit relationship with the metric te nsor that con- formally separable coordinates satisfy: ∂iΓj=∂jΓi⇔Sij(gjj) gjj=Sij(gii) gii(2.1.25) The material introduced in this section thus far is sufficient to prove two propo- sitions, one for the modulation factor Rand the other for the conformal factor Q: 21 Proposition 2.1.12 The modulation factor Rsatisfies the relation ∂ilog(R) =1 2·∂i/parenleftBigg log/parenleftBigggii√g/parenrightBigg/parenrightBigg +qi(xi), (2.1.26) whereqi(xi)is some arbitrary function of the ithcoordinate only, denoted here as xi. Proof: From St¨ ackel theory: √g ϕ=R2Qn/productdisplay i=1fi(xi), wherefiare functions of the coordinate xionly andR,Qare in general functions of all variables. It is sometimes customary to denote by ϕthe determinant of the non-singular St¨ ackel matrix, and gis the determinant of the non-singular metric tensor. The above expression can be inverted to give an equivale nt expression: ϕ√g=/producttextn i=1Ψi(xi) R2Q Here trivially Ψ i(xi)≡fi(xi)−1. Substituting the known expression for the deter- minantϕ, we arrive at: giiMi1 Q√g=/producttextn i=1Ψi(xi) R2Q Canceling out the common factor Qand taking the logarithm of both sides yields: log/parenleftBigggii√g/parenrightBigg + log(Mi1) =n/summationdisplay i=1log(Ψi(xi))−2 log(R) Performing the derivative of both sides with respect to xiand using the fact that the cofactor Mi1is independent of xiwe arrive at: ∂ilog/parenleftBigggii√g/parenrightBigg =−2∂ilog(R) +∂ilog(f−1 i(xi)) Re-arranging and division by two yields: ∂ilog(R) =−1 2∂ilog(fi(xi))−1 2∂ilog/parenleftBigggii√g/parenrightBigg A simple exercise with Christoffel symbols, assuming orthogonal met rics, will yield Γi=∂i/parenleftBig log/parenleftBig gii√g/parenrightBig/parenrightBig . We arrive then at the desired formula of separability conditi ons pertaining to an R-separability test: ∂ilog(R) =1 2Γi+qi(xi)2 22 Theqi(xi) are arbitrary functions to be determined. At first glance thi s appears contrary to the above formalism by a factor of minus unity, how ever in the literature formulae for Rare sometimes the reciprocal of the modulation factor defined in [38]. Thus the equation is satisfied from the above derivation and fur thermore the form of the arbitrary function qi(xi) is uniquely determined, up to the factor Qused in the St¨ ackel matrix definition, by the existence of the separ able functions fi(xi). Specifically the equation amounts to: qi(xi) =−1 2∂ilog(fi(xi)) Proposition 2.1.13 The reciprocal of the conformal Qfactor in St¨ ackel theory can be expanded in the following way: 1 Q=g11f1(u1) +g22f2(u2) +g33f3(u3) +···+gnnfn(un), (2.1.27) where each fiis a function of the ithcoordinate only, denoted here as ui. Proof: The conformal Qfunction is defined in [38] to satisfy gii=SQ Mi1⇒gii=Mi1 SQ⇒1 Q=giiS Mi1, (2.1.28) whereMi1denotes the determinant of the matrix co-factor. The determ inantS appearing in Eq.(2.1.28) can be expanded in terms of the elem ents of the first column of the St¨ ackel matrix, which we know from St¨ ackel th eory to be functions of the corresponding ithvariable only. Explicitly: giiS Mi1=giif1(u1)M11+f2(u2)M21+f3(u3)M31+···+fn(un)Mn1 Mi1(2.1.29) As we know from [38], the ratio of minors yields the inverse rati o of the correspond- ing covariant metric terms. This was used for studying separabil ity of the Laplace equation; now we use the fact that the ratio of the minors yield s the ratio of the corresponding contravariant metric terms. Explicitly: gii=Mi1 SQ⇒gii gjj=Mi1 Mj1 ⇒1 Q=giiS Mi1 =g11f1(u1) +g22f2(u2) +g33f3(u3) +···+gnnfn(un) (2.1.30) as is required to show. 2 23 The Schr¨ odinger equation can also be handled by the formalism introduced in this section. By [14], there is a one to one correspondence between t he solutions of −¯h2 2∆ψ+ (V−E)ψ= 0, (2.1.31) of the form ψ=R/producttext iφi(qi) and the additively separated solutions u= lnφof gijuiuj+giiuii−ˆΓiui+2 ¯h2E−U= 0, (2.1.32) whereui=∂iu,uii=∂2 iuandUis the modified potential U=−/parenleftbigg∆R R−2 ¯h2V/parenrightbigg , (2.1.33) and ˆΓi=gij(Γj−2∂jlnR). (2.1.34) The following proposition is proven in [14]: Theorem 2.1.14 Equation (2.1.32) is separable in orthogonal coordinates qiif and only if for all i/ne}ationslash=j ∂jˆΓi−ˆΓi∂jln(gii) = 0, Sij(ghh) ghh−Sij(gkk) gkk= 0,∀h,k, Sij(U)ghh−Sij(ghh)/parenleftbigg U−2 ¯h2E/parenrightbigg = 0,∀h. (2.1.35) Note that the potential Vis arbitrary: in Chapter 7 a very specific choice for Vis made to ensure conformal invariance, however the above condi tions will still hold. 2.2 Invariant theory of conformal Killing tensors In this section the theory of conformal Killing tensors defined on a Riemannian man- ifold (M,g) is described. We begin this section with a definition of the conformal group acting on this space. This group of transformations and the cor responding Lie algebra of infinitesimal transformations are fundamental for the classification scheme that will be constructed. Definition 2.2.1 A diffeomorphism φ:M→Mwith the property that φ∗g=fg, whereφ∗is the push forward of φandfsome positive function, is said to be conformal. 24 The set of all such transformations forms a Lie group of maximal d imension 1 2(n+ 1)(n+ 2), provided n≥3, called the conformal group of transformations of (M,g) which we’ll denote by C(M). Ifφis a homothetic transformation, then fis a positive number not equal to unity. If φis an isometry, then f= 1. Proposition 2.2.2 LetVbe an infinitesimal generator of the one-parameter group of conformal transformations φt. Then LVg=hg, (2.2.1) wherehis some function. Ifφtdenotes a one-parameter group of homothetic transformation s then the func- tionhis a non-zero constant. If φtdenotes a one-parameter group of isometries then the function his zero. We now proceed to give the general definition of a conformal Ki lling tensor on (M,g). Definition 2.2.3 A conformal Killing tensor of valence p defined on (M,g)is a symmetric (p,0)tensor Kwhich satisfies the conformal Killing tensor equation [g,K] = 2k⊙g, (2.2.2) where [,]denotes the Schouten bracket, kis some symmetric tensor of type (p−1,0) and⊙denotes the symmetric tensor product. The tensor kcan be determined by contracting Eq.(2.2.2) with the covari ant metric. LetKandLbe symmetric tensors of types ( p,0) and (q,0) respectively. The Schouten bracket of KandLdenoted by [ K,L] is a tensor of type ( p+q−1,0) and is defined in terms of local coordinates xi,i= 1,...,n , by [K,L]i1...ip+q−1=−qK(i1...ip,kLip+1...ip+q−1)k +pKk(i1...ip−1Lip...ip+q−1),k (2.2.3) It can be shown that [ K,L] has the following properties: [K,L] =−[L,K] [K,L+M] = [K,L] + [K,M] [K,L⊙M] = [K,L]⊙M+L⊙[K,M] [K,[L,M]] + [M,[K,L]] + [L,[M,K]] = 0 25 Special cases: If, in the definition of the Schouten bracket p=q= 1, then [K,L] is the standard Lie bracket of the vector fields KandL. For the case p= 1,qarbitrary: [K,L]i1...iq= (LKL)i1...iq which is the Lie derivative of Lwith respect to K. Whenp= 1,Kis said to be a conformal Killing vector (CKV) and Eq. (2.2.2) reads LKg=fg, (2.2.4) where Ldenotes the Lie derivative operator. With respect to a local sy stem of coordinates xiEq. (2.2.2) may be written as ∇(i1Ki2...ip+1)=k(i1...ip−1gipip+1), (2.2.5) where ∇denotes the covariant derivative with respect to the Levi-Ci vita connection ofg. Ifk= 0 in Eq.(2.2.2), then Kis said to be a Killing tensor . It follows from the properties of the Schouten bracket that t he setCKp(M) of all conformal Killing tensors of type ( p,0) forms a generally infinite dimensional vector space. However, it’s important to note that K′=K+l⊙g, (2.2.6) wherelis any symmetric tensor of type ( p−2,0), also defines a CKT. This property may be used to define the following equivalence relation on CKp(M): K′∼(K)⇔K′=K+l⊙g, (2.2.7) LetCˆKp(M) denote the set of equivalence classes of CKp(M). One may equip CˆKp(M) with the structure of a vector space over the reals. Let ˆK1andˆK2∈ CˆKp(M). Let K1andK2be representative elements of ˆK1andˆK2respectively. Then ˆK1+ˆK2is defined to be the equivalence class represented by K1+K2. LetK be representative of ˆKanda∈R. ThenaˆKis defined to be the equivalence class represented by aK. It is easy to check that these operations are well defined. Let TCKp(M) denote the vector space of trace-free conformal Killing ten sors of type (p,0). It is easily verified that TCKp(M) is canonically isomorphic to CˆKp(M). A necessary and sufficient condition for an element of CˆKp(M) to be represented by a Killing tensor is that there exists a type ( p−2,0) tensor lsuch that [l,g] = 2k. (2.2.8) Forp= 2 the above equation may be written as dl=−k. (2.2.9) The integrability condition for this equation is dk= 0. (2.2.10) 26 By solving Eq.(2.2.2) for kone may write the integrability condition in component form as Kk[i;k j]= 0. (2.2.11) This is a necessary and sufficient condition for ˆKto be represented by a Killing tensor. We now study the behavior of the conformal Killing tensor Kunder a conformal transformation, which is again ˜g=e−2σg. (2.2.12) By an easy calculation we find that [˜g,K] = 2˜k⊙˜g, (2.2.13) where ˜k= (k−[σ,K]). (2.2.14) This result shows that Kis also a conformal Killing tensor for the conformally related metric ˜g. Therefore we can prove the following: Proposition 2.2.4 A necessary and sufficient condition that Kis a Killing tensor with respect to the conformal metric, that is [˜g,K] = 0 (2.2.15) is that there exists a function σsuch that k= [σ,K]. (2.2.16) Note this proposition holds true for Killing tensors of any vale ncep. For the special casep= 2 we prove immediately Proposition 7.1 in [3]. For the remain der of this section and chapter we assume p= 2. Definition 2.2.5 A conformal Killing tensor Kis ofself-gradient type if there exists a continuous function Usuch that in the definition [K,g] = 2k⊙g,k= [K,U]. Indeed by Prop. (2.2.4) self-gradient conformal Killing ten sors are simple Killing tensors with respect to the conformally related metric ˜g=e−Ug. To determine the transformation equations of the eigenvecto rs and eigenvalues of the Killing tensor under a conformal transformation (2.2.12) , we present some gen- eral results that hold true for symmetric tensors and then successi vely add mathe- matical conditions to those corresponding to the conformal Ki lling tensor equation. 27 Proposition 2.2.6 A symmetric tensor satisfying ˜Kij=Kijunder a conformal transformation has the same eigenvectors with respect to th e conformally related metric and the corresponding eigenvalues satisfy ˜ρi=e2σρi. Proof: Note that implicit in Eq. (2.2.13) is that ˜Kij=Kij, however this identity could well be satisfied by other symmetric tensors of valence- two, or type (2 ,0), under a conformal transformation. Lowering an index to ge ner- ate a type (1 ,1) tensor we arrive at ˜Ki j=˜Kik˜gkj=e2σKikgkj. Therefore ˜Ki j=e2σKi j. Now we consider the eigenvalue problem for Ki j: Ki jXj=ρXi (e2σKi j)Xj=e2σρXi ˜Ki jXj= ˜ρXi(2.2.17) The conclusion is that Xjis an eigenvector of Ki jcorresponding to the eigenvalue ρ if and only if Xjis an eigenvector of ˜Ki jcorresponding to the eigenvalue ˜ ρ=e2σρ. 2 This property was arrived at by Eisenhart but with more mathem atical assump- tions. A proof outlining his method but with more steps is given in Appendix E. Now we assume that the symmetric tensor not only satisfies ˜Kij=Kij,but also has pointwise real and distinct eigenvalues. Proposition 2.2.7 A symmetric tensor Kij, with pointwise real and distinct eigen- values satisfies Kab=ρagab, where the components are with respect to a basis of normalized eigenvectors. Proof: Since the eigenvalues of Kabare real and distinct (point-wise Kabcan be described then as Hermitian), it admits northogonal eigenvectors hi a, wherei are the component indices and ais the label for the eigenvector. Thus Kijhj a=ρagijhj a, (2.2.18) wherehj ais the eigenvector corresponding to the eigenvalue ρa. Since the eigenval- ues are real and distinct, the eigenvectors are orthogonal wi th respect to the metric gij. Namely gijhi ahj b= 0, a/ne}ationslash=b (2.2.19) 28 The eigenvectors can be normalized, such that gijhi ahj a=ea, (2.2.20) wheree2 a= 1. For Riemannian geometry the e’s are always plus unity. Th us we can write gab=gijhi ahj b=eaδab (2.2.21) where there is no sum on the a. Note that the term hi amay be interpreted as a change of basis transformation from the natural basis∂ ∂xito the basis of eigenvectors Ea=hi a∂ ∂xi. Now contract (2.2.18) with hi bto obtain Kijhi bhj a=ρagijhi bhj a Kba=ρagba ⇒Kab=ρagab2 (2.2.22) Now we extend to the case of symmetric tensor fields withnormal eigenvectors: Proposition 2.2.8 LetKijbe a symmetric tensor field with pointwise real and distinct eigenvalues and normal eigenvectors. Then there ex ists a coordinate system uisuch that gij= 0, i/ne}ationslash=j Kij=ρigij (2.2.23) Proof: We assume that Kijis a symmetric tensor field with pointwise real distinct eigenvalues and normal (integrable) eigenvectors. Then Kijdefinesnmutually orthogonal eigenvector fields hi a. These can be written as Ea=hi a∂ ∂xi(2.2.24) with respect to a general coordinate system on M. TheEadefine a basis of the tangent space of Mat each point. Let Eadenote the dual basis of 1-forms. We can write Ea=ha idxi, (2.2.25) whereha iis the inverse of hi a. Since each eigenvector field is assumed normal (integrable), there exist functions faanduasuch that Ea=fadua, (2.2.26) where there is no sum assumed on the a, which ranges from 1 to n. Theuadefine a coordinate system on M. Write (2.2.26) as Ea=faδa idui, (2.2.27) 29 where again no sum on the ais assumed. Comparison with (2.2.25) yields ha i=faδa i (2.2.28) Now we compute the inverse of ha i: hi a=f−1 aδi a (2.2.29) This implies that Ea=hi a∂ ∂ui =f−1 a∂ ∂ua(2.2.30) We now need to prove the identity eif2 iδij=gij. To do so we write the metric in terms of the coordinates ui. Starting from (2.2.21), which is again gab=eaδab (2.2.31) contract this with ha ihb jto obtain gabha ihb j=eaδabfaδa ifbδb j ⇒gij=eiδijfifj gij=eif2 iδij (2.2.32) Thus the metric has the form ds2=gijduiduj =eif2 iδijduiduj =eif2 i(dui)2(2.2.33) We then write (2.2.22) in terms of the coordinate uiby contracting (2.2.22) with ha ihb jusing (2.2.28) Kabha ihb j=ρagabha ihb j Kij=ρagabfaδa ifbδb j Kij=ρieif2 iδij ⇒Kij=ρigij (2.2.34) Thus we have shown that for symmetric tensors with pointwise real and distinct eigenvalues, and normal (integrable) eigenvectors, there ex ists a coordinate system such that simultaneously Kij= 0 andgij= 0 fori/ne}ationslash=j.2 Definition 2.2.9 A conformal Killing tensor with pointwise real and distinct eigen- values and normal eigenvector fields is called a characteristic conformal Killing tensor. 30 We now impose the conformal Killing tensor equation (2.2.5) fo rp= 2, which reads: Kij;l+Kjl;i+Kli;j=kigjl+kjgli+klgij (2.2.35) The derivations outlined are therefore not valid for non cha racteristic Killing ten- sors. However, such tensors are not useful in the characterization of separable coor- dinates as will be explained later. In the remainder of this t hesis non-characteristic Killing tensors will not be considered. Proposition 2.2.10 The eigenvalues ρiof a characteristic conformal Killing ten- sorKijwhen expressed in terms of coordinates for which the conditi ons of Prop. 2.2.8 hold, satisfy the differential equations: ∂ρi ∂xi=ki ∂ρi ∂xj= (ρi−ρj)∂log(gii) ∂xj+∂ρj ∂xj(2.2.36) Proof: The first equation follows from setting i=j=lin the definition of the conformal Killing tensor equation which yields ∂Kii ∂xi−∂log(gii) ∂xiKii=kigii (2.2.37) Then substitute Kii=ρigiito get the required result. Note that in the simple Killing tensor case, where ki= 0, we obtain the Eisenhart result that the ith eigenvalue is independent of the ithcoordinate. An alternative proof of this is given in Appendix E for the interested reader. The second equation fol lows from setting j/ne}ationslash=i,l=jin the definition and arriving at ∂Kjj ∂xi−2∂log(gjj) ∂xiKjj+1 gii∂gjj ∂xiKii=kigjj (2.2.38) Substituting Kii=ρigiiandKjj=ρjgjjin the above yields the second formula. 2 Proposition 2.2.11 (i) A CKT Kwhich is diagonalized in orthogonal coordinates is equivalent to a CKT K′of self-gradient type. (ii) For any given orthogonal co- ordinate system there exists a function Usuch that any CKT Kwhich is diagonal- ized in these coordinates is equivalent to a CKT K′of self-gradient type such that [g,K′] = 2[K′,U]⊙g, that is to a simple Killing tensor of the conformal metric ˜g=e−Ug. (iii) The nfunctionsUk= log(gkk)satisfy (ii). Proof: Ifgij= 0 andKij= 0 fori/ne}ationslash=j, thenKii=ρigii. Furthermore, by the proof of Prop. 2.2.10 the CKT equation [ g,K] = 2k⊙gis equivalent to kj=∂ ∂xjρj and the formula ∂ρi ∂xj= (ρi−ρj)∂log(gii) ∂xj+∂ρj ∂xj. 31 Let us consider the equivalent tensor K′=K−ρngthat has eigenvalues ˜ ρi=ρi−ρn. By using the above, one can easily show that ∂˜ρi ∂xj= (˜ρi−˜ρj)∂log(gii) ∂xj+ ˜ρj∂log(gnn) ∂xj. This shows that K′is a CKT with ˜kj= ˜ρj∂log(gnn) ∂xj, thus of self-gradient type with U= log(gnn) and a simple Killing tensor for the conformal metric e−Ug.2 The connection with conformal Killing tensors and the existen ce ofR-separation of variables will now be described. As discussed before it is well known that Killing tensors are deepl y related with additive separation of variables for the HJ equation for the ge odesics or a natural Hamiltonian in orthogonal coordinates ([7], [1]) H=1 2giipipi+V=E, E ∈R, which reads 1 2gii/parenleftBigg∂W ∂xi/parenrightBigg2 +V=E. They are also connected to multiplicative separation of the Sc hr¨ odinger equation [7], [2] ∆ψ+ (E−V)ψ= 0, E ∈R, where ∆ is the Laplace-Beltrami operator. We have [25] Theorem 2.2.12 The Hamiltonian H= (1 2giipipi+V)is orthogonally separable if and only if there exists a valence-two characteristic Killi ng tensor K(the properties of which have been elucidated earlier) such that d(KdV) = 0. Note thatd(KdV) = 0 is equivalent to the formula Sij(V) = 0 in Prop. 2.1.7 and the metric components ghhin the above theorem must satisfy Sij(ghh) = 0. Finally, for the multiplicative separation of the Schr¨ odin ger equation the so- called Robertson condition must also hold: the Ricci tensor is di agonalized in the separable coordinates ([20]) (geometrically, this means th atKand the Ricci tensor share the same eigenvectors [2]). The condition that the eigen values are real is automatically satisfied for positive definite metrics; recently , KTs with complex conjugate eigenvalues have also been used to separate variable s for a natural HJ equation [17]. Similar results also hold for conformal Killing tensors. 32 Remark 2.2.13 Any CKT equivalent to a characteristic one is characteristic, which is a consequence of the eigenvectors remaining invaria nt within an equiv- alence class. Hence, it is always possible to choose a representativ e characteristic CKT which is trace-free. Furthermore any CKT Kwhich is characteristic with re- spect to the metric gis also characteristic with respect to any conformally related metric ˜g=e−2σg. This is a consequence of Prop. 2.2.6: clearly real and pointw ise distinct eigenvalues remain real and pointwise distinct after any conformal trans- formation. The invariance of the eigenvectors themselves gu arantees invariance of their normality. The following important result holds: Theorem 2.2.14 There exists an orthogonal coordinate system in which additi ve separation for the null geodesic HJ equation, gii(∂iW)2= 0 occurs, if and only if there exists a characteristic CKT KonM. By construction the coordinate hypersurfaces will be orthogonal to the eigen vectors of K. Proof: According to the intrinsic characterization of the orthogona l separation of a geodesic Hamiltonian [25], a metric ˜gdefined onMis orthogonally separable if and only if it admits a simple characteristic Killing tensor, note t his is a special case of Thm. 2.2.12. This simple Killing tensor is a conformal Killing tensor with respect to any conformally related metric galso defined on M, by Eq. (2.2.13). That this conformal Killing tensor is also characteristic has been expla ined in Remark 2.2.13, which summarizes Prop. 2.2.7 to Prop. 2.2.11, therefore the t heorem is proved. 2 Definition 2.2.15 We call a conformally separable web the set of hypersur- faces orthogonal to the eigenvectors of a characteristic CK T. Any coordinates as- sociated with a conformally separable web are called conformally separable co- ordinates . Remark 2.2.16 Note the connection here with conformally separable coordi nates defined in the previous section if Eq. (2.1.11) or Eq. (2.1.12 ) holds. Theorem 2.2.17 There exists an orthogonal coordinate system in which additi ve separation for the HJ equation with fixed value of the energy E, gii(∂iW)2+V−E= 0, occurs, if and only if there exists a characteristic CKT KonMsatisfying the compatibility condition [g,K] =1 E−V[K,V]⊙g. (2.2.39) 33 Note this is equivalent to the formula1 gkkSij(gkk) =1 V−ESij(V) in Thm. 2.1.8. Remark 2.2.18 In the compatibility condition (2.2.39) for the potential V, the characteristic CKT Kis in general not trace-free and the formula does not hold for all the CKT equivalent to K. Indeed, if we consider the equivalent characteristic CKT ˆK=K+fgthe compatibility condition becomes [g,ˆK] =1 E−V([K,V] + [f,g])⊙g. (2.2.40) In spite of the fact that the null geodesic equation is trivial f or a positive definite metric, the conformally separable coordinates are useful bec ause they are the only ones in which a natural Hamiltonian with fixed value of the ener gy can be solved by additive separation of variables. Moreover, they are the on ly ones in which R-separation of the Laplace equation can occur. This is a subjec t for the next section. Having discussed the uses of a single CKT, an important char acterization is associated with nCKTs, which is Theorem 7.2 in [3]: Theorem 2.2.19 Thencharacteristic conformal Killing tensors (Ki) = (K1,K2,...,Kn)associated with an orthogonal metric gijare (i) point-wise linearly independent, (ii) with common eigenvectors, (iii ) mutually commutative and (iv) in involution. Proof: Since the rows of the inverse St¨ ackel matrix are linearly ind ependent (this follows trivially from the definition that the determinant i s non-zero), by construc- tion thenKilling tensors produced (one of them being the metric tensor i tself) are point-wise linearly independent, being in the same (norma l) eigenbasis of the separable coordinates by Eisenhart theory. They are all then si multaneously diago- nalized. So are the nconformal Killing tensors conformally related to them, as we ll as thenequivalent conformal Killing tensors. Then by definition these conformal Killing tensors in any orthogonal coordinate system share commo n eigenvectors and by sharing eigenvectors they commute. That they are in involu tion is proven in [3]. 2 2.3 Relation of CKT’s to existence of R-separable webs Recall Definition 1.1.1 [14]: Definition 2.3.1 We say that multiplicative R-separation of the Laplace equation ∆ψ= 0or Schr¨ odinger equation −¯h2 2∆ψ+ (V−E)ψ= 0occurs in a coordinate 34 system (qi)if there exists a solution ψof the form ψ=R(q1,...,qn)/productdisplay iφi(qi,ca) (ca)∈R2n−1; (2.3.1) satisfying the completeness condition rank/bracketleftBigg∂ ∂ca/parenleftBiggφ′ i φ/parenrightBigg∂ ∂ca/parenleftBiggφ′′ i φ/parenrightBigg/bracketrightBigg = 2n−1, a = 1,...,2n−1, i= 1,...,n. From ([14]) we have Theorem 2.3.2 Necessary and sufficient conditions for R-separation of Schr¨ odinger’s equation −¯h2 2∆ψ+ (V−E)ψ= 0 (2.3.2) in a given coordinate system qiare: i: the coordinates are orthogonal; ii: the coordinates are conformally separable; iii: the function 2 ¯h2(E−V) +gii 4/parenleftBig 2∂iΓi−Γ2 i/parenrightBig (2.3.3) is a pseudo-St¨ ackel factor, in that it can be written in the f ormf=giiφi(qi)where giiis a conformal St¨ ackel metric. Furthermore in this case the modulation factor Ris any solution of 2∂ilnR= Γi−ξi(qi) (i= 1,...,n ), (2.3.4) whereξi(qi)is a function of one variable. For the proof, see ([14]). Furthermore we also have ([7], [14]) Theorem 2.3.3 On a flat manifold, R-separation of the Laplace equation occurs in a coordinate system (qi)if and only if the coordinates (qi)are orthogonal confor- mally separable coordinates. The function Ris (up to separated factors) a solution of the first order system ∂ilnR=1 2Γi, Remark 2.3.4 If the manifold is not flat the conformal separability is a nece ssary (but no longer sufficient) condition: to guarantee R-separation we also need that the function∆R Rbe of the form giifi(qi) for suitable functions of a single variable fi. 35 Definition 2.3.5 We call an R-separable web a conformally separable web if R- separation for the Laplace equation occurs in any associated coordinate system. Remark 2.3.6 InE3, every conformally separable web is an R-separable web for the Laplace equation. This means that R-separable webs are defined by any charac- teristic CKT. R-separable coordinates of E3have been extensively studied by many authors (see Bˆ ocher[5], Moon and Spencer [38], Boyer et al.[ 6]). The webs consist of families of confocal cyclides. In later chapters of this thesis we restrict ourselves to the web s and associated characteristic CKTs admitting a rotational symmetry. To make the notion of web- symmetry precise, we start with the definition of invariance of c onformal Killing tensors under one parameter groups of conformal transformati ons [13]. Definition 2.3.7 LetKdenote a characteristic conformal Killing tensor on (M,g). Letφtdenote a one parameter group of conformal transformations. The R-separable webs defined by Kare said to be φt-symmetric iff φt∗K=fK, (2.3.5) wherefis some function. The infinitesimal version of the above definition is given by the following proposition [13]: Proposition 2.3.8 LetVbe an infinitesimal generator of the one parameter group of conformal transformations φt. Thenφtis a web-symmetry of the R-separable web defined by a conformal Killing tensor Kif and only if LVK=hK, (2.3.6) wherehis some function. Ifφtdenotes a one-parameter group of homothetic transformation s then the func- tionsfandhare non-zero constants. If φtdenotes a one-parameter group of isometries then the functions fandhare zero. 2.4 Conformal Killing tensors in spaces of zero curvature We now assume that the Riemann curvature tensor Rijklofgvanishes. In this case it has been shown by Eastwood [19] that CˆKp(M) is finite dimensional and that its dimension dis given by d=(n+p−3)!(n+p−2)!(n+ 2p−2)(n+ 2p−1)(n+ 2p) p!(p+ 1)!(n−2)!n!(2.4.1) 36 forn≥3,p≥1. Thus the general element of CˆKp(M) is represented by darbitrary parameters a1,...,ad, with respect to an appropriate basis. Each element hof the conformal group C(M) induces, by a push forward map, a non-singular linear transformation ζ(h) ofCˆKp(M). It is implicit in the work of [19] that the map ζ:C(M)→GL(CˆKp(M)) (2.4.2) defines a representation of C(M). Once the form of the general element ˆKof CˆKp(M) is available with respect to some convenient coordinate system onM, the explicit form of the transformation ζ(h)ˆK(written more succinctly as h·ˆK) may be written in terms of the parameters a1,...,ad. We shall be particularly concerned with the smooth real-valued functions on CˆKp(M) that are invariant under the groupC(M). The precise definition of such C(M)-invariant functions of CˆKp(M) is as follows. Definition 2.4.1 Let(M,g)be a Riemannian manifold with zero curvature. Let p≥1be fixed. A smooth function F:CˆKp(M)→Ris said to be an C(M)- invariant of CˆKp(M)iff it satisfies the condition F(h·ˆK) =F(ˆK), (2.4.3) for all ˆK∈CˆKp(M)and for all h∈C(M). The above can also be formulated for pseudo-Riemannian manifo lds with zero cur- vature, however they are beyond the scope of this thesis. The mai n problem of invariant theory is to describe the whole space of invariants o f a vector space under the action of the group. To achieve this one has to determine t he set of fundamen- tal invariants with the property that any other invariant is an analytic fun ction of the fundamental invariants (see [41]). The fundamental theo rem of invariants for a regular Lie group action [41] determines the number of fund amental invariants needed to define the whole of the space of C(M)-invariants. Theorem 2.4.2 LetGbe a Lie group acting regularly on an n-dimensional mani- foldMwiths-dimensional orbits. Then, in a neighborhood Nof each point x∈M, there exist ( n−s) functionally independent G-invariants ∆1,...,∆n−s. Any other G-invariantIdefined near xcan be locally uniquely ex- pressed as an analytic function of the fundamental invariants namely I=F(∆1,...,∆n−s). One of the standard methods for determining the invariants of CˆKp(M) is to use the fact that the invariants of a function under an entire L ie group is equivalent to the invariants of the function under the infinitesimal tran sformation of the group given by the corresponding Lie algebra. The precise result is as follows [40]: 37 Proposition 2.4.3 LetGbe a connected Lie group of transformations acting reg- ularly on a manifold M. A smooth real valued function F:M→RisG-invariant iff v(F) = 0, (2.4.4) for allx∈Mand for every infinitesimal generator vofG. In our application Gis the representation ζdefined by Eq.(2.4.2) where the condition (2.4.4) reads Ui(F) = 0, i= 1,...,r, (2.4.5) where theUiare vector fields which form a basis of the Lie algebra of the rep re- sentation and r=dim C (M) =1 2(n+ 1)(n+ 2). This Lie algebra is isomorphic to the Lie algebra of C(M). Such a basis may be computed directly as the basis of the tangent space to ζ(C(M) at the identity if an explicit form of the represen- tation is available. According to Theorem 2.4.2 the general so lution of the system of first-order pdes (2.4.5) is an analytic function Fof a set of fundamental C(M)- invariants. The number of fundamental invariants is d−s, wheredis given by Eq. (2.4.1) and sis the dimension of the orbits of ζ(C(M)) acting regularly on the spaceCˆKp(M). 38 Chapter 3 Construction of the general CKT in E3 3.1 Killing vector formalism Now we wish to calculate the general thirty-five dimensional tra ce-free conformal Killing tensor Euclidean space admits, and express it in terms of Cartesian coor- dinates. To this end we now specialize the general theory of the previous chapter to the vector space CˆK2(M) of conformal Killing tensors of type (2 ,0) defined in Euclidean space E3. It is well known [42] that in E3, any conformal Killing tensor is expressible modulo a multiple of the metric as a sum of symmetrized products of conformal Killing vectors. A canonical basis of the Lie algebra of confor mal Killing vectors in E3with respect to a system of Cartesian coordinates ximay be written as Xi=∂ ∂xi Ri=ǫijkxjXk D=xiXi Ii= (2xixk−δikxjxj)Xk (3.1.1) fori= 1,2,3, and where ǫijkis the Levi-Civita tensor. We also note the commuta- tion relations [Xi,Xj] = 0 [Xi,Rj] =−ǫijkXk [Ri,Rj] =−ǫijkRk [Xi,D] =Xi [Ri,D] = 0 [Ii,Ij] = 0 39 [Xi,Ij] = 2(δijD−ǫijkRk) [Ri,Ij] =−ǫijkIk [D,Ii] =Ii (3.1.2) We now determine the form of the general element of TCK2(M). By Eq.(2.4.1) d= 35. It is clear that a sum of symmetrized products of conformal Killing vectors is a conformal Killing tensor. It will be shown that all trace-f ree conformal Killing tensors may be obtained in this way. One begins by writing K=AijXi⊙Xj+BijXi⊙Rj+CijRi⊙Rj+DiXi⊙D+EijXi⊙Ij +FiRi⊙D+GijRi⊙Ij+HD⊙D+LiD⊙Ii+MijIi⊙Ij(3.1.3) The coefficients in Eq.(3.1.3) obey the following symmetry rel ations Aij=Aji, Cij=Cji, Mij=Mji (3.1.4) Thus the apparent dimension of TCK2(M) is fifty five, which exceeds the required dimension by twenty. Indeed there exist the following six relat ions among the basis set of symmetric tensor products of Killing vectors: Xi⊙Ri= 0 Ri⊙Ii= 0 D⊙D=Xi⊙Ii+Ri⊙Ri 2Ri⊙D+ǫiklXk⊙Il= 0 (3.1.5) Consequently, the general element of TCK2(M) may be written as K=AijXi⊙Xj+BijXi⊙Rj+CijRi⊙Rj+DiXi⊙D +EijXi⊙Ij+GijRi⊙Ij+LiD⊙Ii+MijIi⊙Ij (3.1.6) where the coefficients BijandGijmay be chosen to satisfy Bii= 0 Gii= 0 (3.1.7) This follows from the fact that in the expression for the Killin g tensor K, one can add the terms kaXi⊙RiandkbRi⊙Iisince we know these are essentially zero for any arbitrary ka,kb∈R. Since by definition the Kronecker delta δijvanishes fori/ne}ationslash=j, the above expressions can be modified to kaδijXi⊙RjandkbδijRi⊙Ij. These will still vanish and hence can be added to the expression for the Killing tensor Kin terms of the basis of symmetric tensor products of Killing vec tors. Collecting coefficients in front of Xi⊙RjandRi⊙Ij, we define/tildewideBij=Bij+kaδij and/tildewideGij=Gij+kbδij. Settingi=jand enacting a summation, we arrive at /tildewideBii=Bii+3kaand/tildewideGii=Gii+3kb. Askaandkbare arbitrary, we have the freedom to set them such that/tildewideBii= 0 and/tildewideGii= 0, hence ka=−1 3Biiandkb=−1 3Gii. 40 Finally, the tilde sign is dropped and the trace-free result is proven for the Bijand Gijcoefficients. In terms of the natural basis, Xi⊙Xj, the components of Kare given by Kij=Aij+ (B(i|kǫkl|j)+D(iδj)l)xl + (Cmnǫmk(iǫ|nl|j)+ 2E(i|k|δj)l−E(ij)δlk)xlxk + (2Gmnǫmk(iδj)l−Gm(iǫj)mkδln+ 2Lkδinδjl−L(iδj)nδlk)xlxkxn + (4Mklδjn−4Mk(iδj)nδli+Mijδknδli)xlxkxnxi(3.1.8) Next we impose the trace-free condition namely Kii= 0. (3.1.9) This procedure yields the following additional fourteen re lations among the coeffi- cients of K: M11=−M22−M33 A11=−A22−A33 E12=C12−E21 E13=C13−E31 E23=C23−E32 E11=C11+ 1/2(C22+C33) E22=C22+ 1/2(C11+C33) E33=C33+ 1/2(C11+C22) L1=G32−G23 L2=G13−G31 L3=G21−G12 D1=B32−B23 D2=B13−B31 D3=B21−B12 (3.1.10) The above formulae may be written compactly as follows: Aii= 0, Di=Bjkǫkji E(ij)−1 3Ekkδij=1 2(Cij−1 3Ckkδij) Li=Glmǫmli, Mii= 0 Ekk= 2Ckk (3.1.11) We chose, among the Aii,Bii,GiiandMii, the coefficients with index (1,1) to be written in terms of the other two, for example: A11=−A22−A33. There are now twenty required relations among the coefficients. Implementi ng them, one obtains 41 the conformal Killing tensor in E3which we present first in compact form, and then in fully expanded form in components. We use the above conditio ns to remove the Di,Li, and (temporarily) Cij. Note that the matrix coefficients Aij,Mij,Bijand Gijmust be trace-free. 3.2 Compact and expanded form of the general CKT In terms of the natural basis the components of Kare given by: Kij=Aij+ (B(i|kǫkl|j)+Babǫba(iδj)l)xl + ((2E(mn)−1/2Eaaδmn)ǫmk(iǫ|nl|j)+ 2E(i|k|δj)l−E(ij)δlk)xlxk + (2Gmnǫmk(iδj)l−Gm(iǫj)mkδln+ 2Gabǫbakδinδjl−Gabǫba(iδj)nδlk)xlxkxn + (4Mklδjn−4Mk(iδj)nδli+Mijδknδli)xlxkxnxi(3.2.1) Moreover, any CKT of E3is equivalent to K∼AijXi⊙Xj+BijXi⊙Rj+EijXi⊙Ij+GijRi⊙Ij+MijIi⊙Ij, whereAij,Mij,BijandGijmust be trace-free matrices. The CKT coefficients Kij, found by collecting all polynomials with common factor Xi⊙Xj, may be written as follows: X1⊙X1: K11=−A22−A33−4M12xy3+(−G32+3G23)z2x+(−2M22−2M33)z2y2+(2C13− 2E31)zx−2C23yz+8M23zyx2+(6M22+2M33)x2y2+B12z−B13y+(B32−B23)x+ (−M22−M33)x4+(G32−G23)x3+4M13zx3+4M12yx3+(C11+1/2C22+1/2C33)x2+ (3G21−2G12)zx2+(6M33+2M22)z2x2+(−3G31+2G13)x2y+(−3G32+G23)xy2− 4M13zxy2+ (2C12−2E21)xy+ (2G22−2G33)zxy−4M12z2xy−4M13z3x−G21z3+ (−M22−M33)z4+G31y3+(−M22−M33)y4+(−C11−1/2C22+1/2C33)y2−G21zy2+ G31z2y+ (−C11+ 1/2C22−1/2C33)z2 X2⊙X2: K22=A22+ 4M12xy3+ (2C23−2E32)zy+ 2E21xy+ (C22+ 1/2C11+ 1/2C33)y2+ 4M23zy3+ (G13−G31)y3+ (−C22+ 1/2C33−1/2C11)x2−4M23zyx2+ (4M33− 2M22)z2y2+ (−3G12+ 2G21)zy2−4M12yx3+ 2M22z2x2+ 8M13zxy2−4M12z2xy+ G12zx2−B21z+B23x−G32z2x+ (B13−B31)y−2C13xz+ (−6M22−4M33)x2y2+ (−G13+3G31)x2y+(3G32−2G23)xy2+(4G33+2G22)zxy+M22x4−G32x3+M22z4+ G12z3+M22y4+ (−3G13+G31)z2y−4M23z3y+ (−C22+ 1/2C11−1/2C33)z2 42 X3⊙X3: K33= 2E32zy+A33+ (1/2C22−C33−1/2C11)x2−4M23zy3−4M23zyx2+G23x3+ 2E31xz+(−3G21+G12)zx2+(−G21+3G12)zy2−4M13zx3−4M13zxy2+8M12z2xy+ (1/2C11−C33−1/2C22)y2+4M13z3x+(−6M33−4M22)z2x2−2C12xy+4M23z3y− B32x+B31y+ (B21−B12)z+ (−4G22−2G33)zxy+ 2M33x2y2−G13x2y+M33x4+ (−3G23+ 2G32)z2x+ (4M22−2M33)z2y2+ (3G13−2G31)z2y+G23xy2+M33z4+ (G21−G12)z3+ (C33+ 1/2C11+ 1/2C22)z2+M33y4−G13y3 X1⊙X2: K12= (−G32+ 1/2G23)y3−2M23z3x+ (−1/2G13+G31)x3+ (2C13−E31)zy+ 3/2G23yz2+(B22+1/2B33)z+(−E21+1/2C12)y2+(−4M22−2M33)yx3−2M13z3y+ (E21−1/2C12)x2−2M13zy3+ 6M12y2x2−3/2G13xz2+ (3G32−3/2G23)yx2− 2M23zx3+ (1/2B32−B23)y−3/2G33zy2+ 3/2G33zx2+ (2C23−E32)zx+ (−G22− 1/2G33)z3+A12−3/2C12z2+6M33xyz2+6M13zyx2+(4M22+2M33)xy3+(3/2G13− 3G31)xy2+6M23zxy2+(−3G12+3G21)zxy−M12x4+(3/2C11+3/2C22)xy+(B13− 1/2B31)x−M12y4+M12z4 X1⊙X3: K13= (−1/2C13+E31)x2+ (−2M22−4M33)zx3+ (−G21+ 1/2G12)x3+ (3G21− 3/2G12)z2x+(2C12−E21)zy+3/2G12xy2+(1/2G22+G33)y3+(3/2C11+3/2C33)zx− 2M23xy3−3/2G32zy2+(1/2B21−B12)x+6M22xzy2−3/2C13y2+A13+3/2G22yz2− 2M23yx3+(3/2G32−3G23)zx2−3/2G22yx2+6M13z2x2−M13x4+6M12zyx2+(E32+ C23)xy+ (3G13−3G31)zxy+ 6M23z2xy+ (2M22+ 4M33)z3x−M13z4+M13y4− 2M12zy3−2M12z3y+ (B32−1/2B23)z+ (1/2C13−E31)z2+ (G23−1/2G32)z3+ (−1/2B22−B33)y X2⊙X3: K23=−3/2C23x2+ (3/2G22+ 3/2G33)z2x+ 3/2G31zx2−3/2G21yx2+ (1/2B33− 1/2B22)x−2M13yx3−2M12zx3−2M13xy3+(1/2G22−1/2G33)x3+(G12−1/2G21)y3− 2M12z3x+6M23z2y2+(3G13−3/2G31)zy2+(−1/2C23+E32)y2+(−2M33+2M22)zy3+ (1/2B13−B31)z+(−2M22+2M33)z3y+(3/2G21−3G12)z2y+(3/2C22+3/2C33)zy+ (B21−1/2B12)y+(−G13+1/2G31)z3+(−E32+1/2C23)z2+(−6M33−6M22)zyx2+ (−3/2G22−3/2G33)xy2+ 6M12zxy2+ (E31+C13)xy+M23x4+ (E21+C12)zx+ (3G32−3G23)zxy+ 6M13z2xy−M23z4−M23y4+A23 Knowing the form of the general conformal Killing tensor allo ws one to consider lower dimensional sub-sets. These are often representative of symm etries of coor- dinate webs which will be studied in the next chapters. 43 Chapter 4 The set of rotationally symmetric characteristic CKTs in E3 4.1 Definitions and constructions of rotationally symmetric webs Now we begin the task of finding characteristic conformal Killin g tensors corre- sponding to each of the three dimensional known rotational R-separable webs given in [38]. Later we address the question as to whether they describ e inequivalent coor- dinate webs or not. Because we restrict ourselves to the R-separable webs admitting a rotational symmetry, to describe them we must find the most gener al rotational conformal Killing tensor sub-set of the conformal Killing tenso r calculated in the previous chapter. Rotational coordinate webs means that on e foliation of the web consists of half planes with common intersection forming the z-axis. This we label the rotational axis. Without loss of generality we can restrict ourselves to the webs having the z-axis as their rotational axis. Up to an isometry, a characterist ic con- formal Killing tensor of such a web admits the Killing vector R3as an eigenvector. Thecontinuous operation to characterize a conformal tensor Trepresenting a symmetric web is given by the solutions of LkT=hT, (4.1.1) wherehis any real scalar and Lis the Lie derivative operator with respect to the conformal Killing vector kwhich generates a group action under which the web is invariant. Note the above is Prop. 2.3.8, where kis the infinitesimal generator of the one parameter group action. This is a property of all conf ormal Killing vectors. We use from now on the Lie derivative formula for contravarian t rank two tensors Twhich is: (LkT)ij=kl∂lTij−Tlj∂lki−Til∂lkj(4.1.2) 44 The condition (4.1.1) is not sufficient on its own because it does not imply normality of the eigenvectors of Tij. Hence the solution set is not the set of rotationally symmetric characteristic conformal Killing tensors. Thus not o nly must one set the rotational Lie derivative of the general conformal Killing tensor to zero, namely Eq. (4.1.1) for Killing vector k=R3, but in addition impose the three Tonolo- Schouten-Nijenhuis (TSN) conditions which are both necessary a nd sufficient for a given symmetric (Killing) tensor field to have integrable eig envectors. These conditions read Nl [jkgi]l= 0 Nl [jkKi]l= 0 Nl [jkKi]mKm l= 0, (4.1.3) where Ni jkare the components of the Nijenhuis tensor of Kijgiven by Ni jk=Ki lKl [j,k]+Kl [jKi k],l. (4.1.4) Lie differentiation leaves nine independent coefficients of t he conformal Killing ten- sor solution and the tensor (not characteristic yet) is: K11=−1 2A33+ 6G22xyz−M33x2y2+G12z3−B21z+G12zy2−2E21xy −1 2M33x4+ 5M33z2x2−1 2M33z4−1 2M33y4−5G12zx2−M33z2y2 + (−1/2C22−1/2C33)z2+ (3/2C22+ 1/2C33)x2 + (−3/2C22+ 1/2C33)y2 K22=−1 2A33−6G22xyz−M33x2y2+G12z3−B21z−5G12zy2 + 2E21xy−1 2M33x4−M33z2x2−1 2M33z4−1 2M33y4 +G12zx2+ 5M22z2y2+ (−3/2C22+ 1/2C33)x2+ (3/2C22+ 1/2C33)y2 + (−1/2C22−1/2C33)z2 K33=−C33y2+A33+ 2M33x2y2−2G12z3+ 2B21z−C33x2+ 4G12zy2 +M33x4−4M33z2x2+M33z4+M33y4+ 4G12zx2−4M33z2y2 + (C22+C33)z2 K12= 3C22xy−3G22zx2−6G12xyz+ 3G22y2z+ 6M33xyz2 +E21x2−E21y2 K13= (3/2C33+ 3/2C22)zx−9/2G12z2x+ 3M33z3x−E21zy + 3/2G12xy2−3M33xzy2+ 3/2G22yz2−3/2G22yx2−3M33zx3 + 3/2B21x+ 3/2G12x3−3/2G22y3+ 3/2B22y K23= 3M33z3y−9/2G12z2y−3/2G22z2x−3M33zy3+ 3/2G22xy2 + 3/2B21y−3/2B22x+E21xz+ 3/2G12yx2+ (3/2C33+ 3/2C22)zy −3M33zyx2+ 3/2G22x3+ 3/2G12y3(4.1.5) 45 The imposition of the TSN conditions implies that the coefficie ntsE21,B22and G22must vanish. The resulting six-dimensional rotational characte ristic Killing tensor thus takes the form: K11=−1/2M33x4−M33x2y2+ (3/2C22+ 1/2C33)x2−5G12zx2+ 5M33x2z2 −1/2M33y4+ (1/2C33−3/2C22)y2+G12zy2−M33y2z2−1/2A33−B21z + (−1/2C22−1/2C33)z2−1/2M33z4+G12z3 K22=−1/2M33x4−M33x2y2+ (1/2C33−3/2C22)x2+G12zx2−M33x2z2 −1/2M33y4+ (3/2C22+ 1/2C33)y2−5G12zy2+ 5M33y2z2−1/2A33−B21z + (−1/2C22−1/2C33)z2−1/2M33z4+G12z3 K33=M33x4+ 2M33x2y2−C33x2+ 4G12zx2−4M33x2z2+M33y4−C33y2+ 4G12zy2 −4M33y2z2+ 2B21z+ (C33+C22)z2−2G12z3+M33z4+A33 K12= 3C22yx−6G12xyz+ 6M33xyz2 K13= 3/2G12x3−3M33zx3+ 3/2G12xy2−3M33xzy2+ 3/2B21x+ 3M33z3x + (3/2C22+ 3/2C33)zx−9/2G12xz2 K23= 3/2G12yx2−3M33zyx2+ 3/2G12y3−3M33zy3+ 3/2B21y+ 3M33z3y + (3/2C22+ 3/2C33)zy−9/2G12yz2(4.1.6) A more compact way of expressing the above, in terms of symmetric tensor products of CKVs expressed as linear combinations of the chosen six paramet ers, is: K=−A33 2X1⊙X1−A33 2X2⊙X2+A33X3⊙X3 −B21X1⊙R2+B21X2⊙R1+C22R1⊙R1 +C22R2⊙R2+C33R3⊙R3+ 2B21X3⊙D + (3/2C22+C33/2)X1⊙I1+ (3/2C22+C33/2)X2⊙I2 + (C33+C22)X3⊙I3+G12R1⊙I2−G12R2⊙I1 −2G12D⊙I3−M33 2I1⊙I1−M33 2I2⊙I2+M33I3⊙I3(4.1.7) There is an elegant alternate approach which is computation ally easier than the method outlined above. Remark 4.1.1 The linear space of all possible CKTs which are characterist ic CKTs of rotational webs in Euclidean space is the subspace of the g eneral thirty five pa- rameter CKT defined by the discrete linear operation: (K·R3)×R3= 0 (4.1.8) The result by definition forces the third rotational Killing v ector to be an eigen- vector of the modified Killing tensor. Indeed the normality of the eigenvectors is ensured by the fact that R3is normal and that the second linearly independent eigenvector is tangent to the half-planes and can be consider ed planar. Clearly the 46 pair are surface forming and hence normal. In any case we checke d that the three conditions making up the Tonolo-Schouten-Nijenhuis test for integrability of the eigenvectors are satisfied. Note that Eq. (4.1.8) works only for 3-dimensional Eu- clidean space and not on higher dimensional manifolds. Thus the discrete method cannot be taken as a universal approach to find subsets of conform al Killing tensors indicative of symmetries of the corresponding coordinate web s. It was shown that each discrete operation analogous to Eq.(4.1 .8), but along all canonical conformal Killing vectors, results in Killing tensor subspaces o f dimen- sion six instead of thirty five that describes the most general CKT. Application of the condition Eq.(4.1.8) confirms that rotational webs ar e six dimensional webs characterized by the following conditions on the Killing te nsor coefficients: A22=−A33 2 B12=−B21 C11=C22 G12=−G21 M22=−M33 2 A12=A13=A23= 0 B13=B31=B23=B32= 0 B22=B33= 0 C12=C13=C23= 0 E12=E21=E13=E31=E23=E32= 0 G13=G31=G23=G32= 0 G22=G33= 0 (4.1.9) All other parameters vanish except for those that are linear com binations of the six free independent parameters ( A33,B21,C22,C33,G12andM33) as required by the trace-free condition explained in the previous section. The same general rotational conformal Killing tensor then results after applying the abov e criterion. This proves that the Lie derivative and discrete method of finding rotatio nally symmetric webs are equivalent for Euclidean space. We have verified that (4.1 .8) and (4.1.1) with TSN conditions are equivalent for all canonical conformal K illing vectors modulo cases of Killing tensors with constant components. 4.2 Characteristic Killing tensors for rotational R-separable coordinates In this subsection we discuss the derivation of characteristic Ki lling tensors for R-separable webs. As explained in the Introduction this method r elies on the 47 observation of Eisenhart that the associated St¨ ackel matrix co ntains in its inverse information about the characteristic Killing tensors unique to the coordinate system [20]. Namely, the three rows of the inverse St¨ ackel matrix are the contravariant components of the linearly independent Killing tensors expr essed in the eigenbasis that generates the coordinate web. One row is the usual contra variant metric tensor. This is a fundamental property of all St¨ ackel matrices and i n [38] the first row of the inverse is defined to represent the diagonalized contravarian t metric tensor. The second and third rows are the diagonalized Killing tensors; one will be common to all rotational systems but the other unique only to the coordin ate web. One is interested in Killing tensors expressed in Cartesian coord inates, thus the coordinate transformation law is needed to calculate the Jac obian matrix. Recall from Chapter 2 that when the Jacobian is left-multiplied by t he diagonalized Killing tensor and the transpose of the Jacobian, that this yields the te nsor expressed in Cartesian coordinates albeit with variables belonging to th e originalR-separable coordinate definition. So far the technique is algorithmic e specially when the as- sociated St¨ ackel matrix and coordinate transformation are a lready known (in this thesis we provide them for each coordinate case). The difficult st ep is guessing the Killing tensor in canonical Cartesian variables. Although an al gorithm is outlined in [25], it becomes very unwieldy for the cyclidic coordinat es where solving for one separable coordinate in terms of the Cartesian coordinates in volves solving quartic equations. This, as well as writing the tensor as a symmetrized pr oduct of Confor- mal Killing vectors (CKVs), will be discussed in each case. Of parti cular difficulty were the Jacobi-elliptic coordinate systems that comprised th e last four coordinate systems in Ch.4 of [38]. For6-sphere andtangent sphere coordinates the form of the characteristic tensor can be determined by inspection. 6-sphere coordinates are the only example in Ch.4 of [38] that are not rotational so we briefly digress from th e main theme of this chapter to discuss them. The coordinate transformation la w from Cartesian coordinates to canonical R-separable coordinates is given by x=u u2+v2+w2 y=v u2+v2+w2 z=w u2+v2+w2(4.2.1) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=g33=1 (u2+v2+w2)2(4.2.2) The associated St¨ ackel matrix is: 0−1−1 0 1 0 1 0 1 (4.2.3) 48 with conformal Q-factor (u2+v2+w2)−2and modulation R-factor the fourth root ofQ. Using this information and the Eisenhart theory the correspond ing conformal Killing tensors are:  4x2z24xyz22xz(−x2−y2+z2) 4xyz24y2z22yz(−x2−y2+z2) 2xz(−x2−y2+z2) 2yz(−x2−y2+z2) (−x2−y2+z2)2 (4.2.4)  4x2y2−2(x2−y2+z2)xy 4xy2z −2(x2−y2+z2)xy (x2−y2+z2)2−2(x2−y2+z2)yz 4xy2z −2(x2−y2+z2)yz 4z2y2 (4.2.5) They are not however trace-free. In order to determine the co efficients used in sym- metric tensor products of CKVs by comparing the above fourth de gree expressions with the general formula for the thirty five parameter confor mal Killing tensor - the trace must be removed. This is accomplished by addition of t he identity matrix multiplied by one third of the negative of the trace. For the a bove two tensors the coefficients of symmetric tensor products of CKVs are: M22=−1 3,M33=2 3and M22=2 3,M33=−1 3, respectively. All other coefficients are zero. With tangent sphere coordinates the coordinate transformation law from Cartesian coordinate s to canonical R-separable coordinates is given by x=µcosψ µ2+ν2 y=µsinψ µ2+ν2 z=ν µ2+ν2(4.2.6) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=1 (µ2+ν2)2 g33=µ2 (µ2+ν2)2(4.2.7) The associated St¨ ackel matrix is:  1−1−1/µ2 0 1 0 0 0 1 (4.2.8) 49 with conformal Q-factor (µ2+ν2)−2and modulation R-factor the fourth root of Q. From this the two conformal Killing tensors obtained are:  4x2z24xyz22xz(−x2−y2+z2) 4xyz24y2z22yz(−x2−y2+z2) 2xz(−x2−y2+z2) 2yz(−x2−y2+z2) (−x2−y2+z2)2 (4.2.9)  y2−xy0 −xy x20 0 0 0  (4.2.10) The second tensor is common to all coordinate webs invariant un der rotations. Its trace-free representation is  2y2 3−x2 3−xy 0 −xy2x2 3−y2 30 0 0 −y2 3−x2 3  (4.2.11) The basis in terms of symmetrized products of CKVs is easy to find fr om the general trace-free rotational CKT expressed in terms of six arbitrary c onstants, since this tensor is of second degree: C22=−1 3, C 33=1 3, A 33=B21=G12=M33= 0. (4.2.12) Although largely neglected in this chapter, this tensor will b e fundamental in discus- sions of group operations leaving rotational webs and algebr aic quantities invariant. It is required for classifying inequivalent coordinates in th e next section. The first characteristic tensor’s trace-free representation is purely degree four with basis: M33=2 3a2, A 33=B21=C22=C33=G12= 0. (4.2.13) Forcardioid coordinates the coordinate transformation law is: x=µνcosψ (µ2+ν2)2 y=µνsinψ (µ2+ν2)2 z=µ2−ν2 2(µ2+ν2)2(4.2.14) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=1 (µ2+ν2)3 g33=µ2ν2 (µ2+ν2)4(4.2.15) 50 The associated St¨ ackel matrix is:  µ2−1−1/µ2 ν21−1/ν2 0 0 1 (4.2.16) with conformal Q-factor (µ2+ν2)−4, and modulation Rfactor the fourth root of Q. The characteristic conformal Killing tensor, in canonical C artesian coordinates, could not be guessed but a formula relating the variables µ,νandψto Cartesian variablesx,yandzis given in [38] and can be inserted into the expression. After rearrangement one obtains the conformal Killing tensor writ ten in components:  −8z(x2−y2−z2) −16xyz 4x(x2+y2−3z2) −16xyz 8z(x2−y2+z2) 4y(x2+y2−3z2) 4x(x2+y2−3z2) 4y(x2+y2−3z2) 16z(x2+y2) (4.2.17) This is the only coordinate system yielding a purely degree thr ee characteristic tensor. The basis of symmetrized tensor products of CKVs is G12=8 3. A word of caution is required here. Although there is only one independ ent basis, this does not mean that G12is the only coefficient involved in the formula for symmetric tensor products of CKVs. Recall the trace-free condition on the CKT also requires thatLi=GlmαmliwhereLiwas defined as the tensor coefficient of the dilatation vector multiplied with the ithinversion vector. In the case of cardioid coordinates L3=−16 3,L1=L2= 0. The remaining rotational webs have the additional feature o f a parameter ‘ a’ which appears in the definition of the coordinates. This will b e present in the Jacobian and in the final characteristic conformal Killing te nsor. An interesting fact is that A33will always have ‘units’ a2andM33units1 a2. It will be clari- fied later that this parameter naturally arises from the dilat ation member of the conformal group acting on the coordinate web. C22andC33, the second degree terms, never depend on this parameter. The algebra of the char acteristic tensors (representing coordinates at least in canonical centered for m) will show that for all non-cardioid coordinates only four independent coeffici ents come into play and these areA33,C22,C33andM33withB21=G12= 0. This will be expanded on in Chapter 5 and 6. Fortoroidal coordinates the coordinate transformation law is: x=asinh(η) cosψ cosh(η)−cosθ y=asinh(η) sinψ cosh(η)−cosθ z=asinθ cosh(η)−cosθ(4.2.18) 51 The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2 (cosh(η)−cosθ)2 g33=a2sinh2(η) (cosh(η)−cosθ)2(4.2.19) The associated St¨ ackel matrix is:  1−1−1/sinh(η)2 0 1 0 0 0 1 (4.2.20) with conformal Qfactora2(cosh(η)−cosθ)−2and modulation R-factor the fourth root ofQ. The characteristic conformal Killing tensor in canonical Ca rtesian co- ordinates is found by solving quadratic equations relating ( η,θ,ψ), given in [38], with the Cartesian variables ( x,y,z ). One obtains  z2x2 a2xyz2 a2 −(x2+y2−z2−a2)zx 2a2 xyz2 a2z2y2 a2 −(x2+y2−z2−a2)zy 2a2 −(x2+y2−z2−a2)zx 2a2 −(x2+y2−z2−a2)zy 2a2(x2+y2−z2−a2)2 4a2 (4.2.21) The independent coefficients of symmetric tensor products of CK Vs are A33=a2 6, C 22= 0, C 33=1 3, M 33=1 6a2(4.2.22) Writing the complete expression in terms of symmetrized tensor p roducts of CKVs again requires the algebra derived in the previous section ba sed on the conditions that resulted from the trace-free assumption. Bispherical coordinates are handled in a similar way to toroidal coordinates. Their coordinate transformation to Cartesian coordinates ar e: x=asinθcosψ cosh(η)−cosθ y=asinθsinψ cosh(η)−cosθ z=asinh(η) cosh(η)−cosθ(4.2.23) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2 (cosh(η)−cosθ)2 g33=a2sin2θ (cosh(η)−cosθ)2(4.2.24) 52 The associated St¨ ackel matrix is: 1−1 0 0 1 −1/sinθ2 0 0 1 (4.2.25) with conformal Qfactora2(cosh(η)−cosθ)−2and modulation R-factor the fourth root ofQ. The characteristic conformal Killing tensor in Cartesian coo rdinates is expressed in components and is given by: K11=1 4a2(x4−2z2x2+ 2x2a2+ 2x2y2+z4−2z2a2+y4+ 2y2a2+a4+ 2z2y2) K22=1 4a2(x4+ 2x2y2+ 2x2a2+ 2z2x2+z4−2z2a2+y4+ 2y2a2+a4−2z2y2) K33=(x2+y2)z2 a2 K12=−xyz2 a2 K13=(x2+y2−z2+a2)zx 2a2 K23=(x2+y2−z2+a2)zy 2a2(4.2.26) The independent coefficients of symmetric tensor products of CK Vs are A33=−a2 6, C 22= 0, C 33=1 3, M 33=−1 6a2(4.2.27) These conditions are the same as those for toroidal coordinates except for a sign change inA33andM33; this subtlety will be revisited in the next section when considering inequivalence of coordinates. Inverse oblate spheroidal coordinates andinverse prolate spheroidal coordinates are handled in a similar manner: coordinate relations in [38] can be solved in terms of Cartesian variables by use of the quadratic formula. F orinverse oblate coordinates the coordinate transformation law is: x=acosh(η) sinθcosψ cosh2(η)−cos2θ y=acosh(η) sinθsinψ cosh2(η)−cos2θ z=asinh(η) cosθ cosh2(η)−cos2θ(4.2.28) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2(cosh2(η)−sin2θ) (cosh2(η)−cos2θ)2 g33=a2cosh2(η) sin2θ (cosh2(η)−cos2θ)2(4.2.29) 53 The associated St¨ ackel matrix is:  a2cosh2(η)−1 1/cosh2(η) −a2sin2θ1−1/sin2θ 0 0 1 (4.2.30) with conformal Qfactor (cosh2(η)−cos2θ)−2and modulation R-factor the fourth root ofQ. The characteristic conformal Killing tensor for inverse oblate coordinates is given in components, written in canonical Cartesian coordi nates, by: K11=(x4−2z2x2+ 2x2y2+y4+z4+z2a2+y2a2+ 2z2y2) a2 K22=(x4+ 2z2x2+x2a2+ 2x2y2+z2a2+y4+z4−2z2y2) a2 K33=(x2+y2)(4z2+a2) a2 K12=−(4z2+a2)xy a2 K13=−zx(a2−2x2−2y2+ 2z2) a2 K23=−zy(a2−2x2−2y2+ 2z2) a2(4.2.31) The independent coefficients of symmetric tensor products of CK Vs are A33= 0, C 22=−1 3, C 33=−1 3, M 33=−2 3a2(4.2.32) Forinverse prolate coordinates the coordinate transformation law is: x=asinh(η) sinθcosψ cosh2(η)−sin2θ y=asinh(η) sinθsinψ cosh2(η)−sin2θ z=acosh(η) cosθ cosh2(η)−sin2θ(4.2.33) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2(sinh2(η) + sin2θ) (cosh2(η)−sin2θ)2 g33=a2sinh2(η) sin2θ (cosh2(η)−sin2θ)2(4.2.34) 54 The associated St¨ ackel matrix is:  a2sinh2η−1−1/sinh2η a2sin2θ1−1/sin2θ 0 0 1 (4.2.35) with conformal Qfactor (cosh2(η)−sin2θ)−2and modulation R-factor the fourth root ofQ. The characteristic conformal Killing tensor for inverse prolate coordinates is given in components by: K11=(−x4+ 2z2x2−2x2y2+z2a2+y2a2−z4−y4−2z2y2) a2 K22=(−x4+x2a2−2x2y2−2z2x2+z2a2−y4−z4+ 2z2y2) a2 K33=(−4z2+a2)(x2+y2) a2 K12=−xy(−4z2+a2) a2 K13=−zx(a2+ 2x2+ 2y2−2z2) a2 K23=−zy(a2+ 2x2+ 2y2−2z2) a2(4.2.36) The independent coefficients of symmetric tensor products of CK Vs are A33= 0, C 22=−1 3, C 33=−1 3, M 33=2 3a2(4.2.37) 4.3 Jacobi elliptic coordinates The case of the Jacobi elliptic functions involved in the last f our rotational coor- dinate systems given in [38] are much more difficult to decipher t heir associated CKTs. Solving for µ,νandψdirectly in terms of x,yandzin [38] using informa- tion in [38] implies solving quartic equations. Closed form solu tions may exist in theory but are unwieldy. We had the idea of using numerical met hods to ‘measure’ what the unknowns ( A33,B21,C22,C33,G12,M33) must be by numerically evaluating the components of the characteristic Killing tensor and inver ting the ‘coefficient’ matrix defined in terms of the six unknown parameters. If the six parameter set were factored out into a column vector, the resulting six by six ‘coefficient’ matrix inverted and multiplied with the column vector (nu- merically estimated) of Killing tensor components (A33,B21,C22,C33,G12,M33) would be approximated. The hope was that the nu- merical output would be very close to repeating decimal expan sions hinting at 55 simple fractions. The fractions chosen would comprise an intell igent guess for the unknowns and their use in the general formula for symmetric ten sor products of CKVs would yield the Killing tensor automatically in Cartesian coordinates. The idea is sound except that the determinant of the coefficient mat rix is 0! In fact, the rank is only three. Even the guess B21=G12= 0 (which turned out to be correct) would leave us with one unknown too many. Moreover, it can be p roved that the singular condition implies any formula of symmetric tensor pro ducts of CKVs is unique up to a very general 3-parameter non-constant algebra ic expression. The only way around this difficulty was to proceed numerically : either to guess the CKT coefficients (some are now functions of the Jacobi-elli ptic parameter k) and judge by numerical estimates if the hypothesis was reasonab le or restrict the domain to extreme values such as µ=ν=ψ= 0. The coefficients are constant independently of where in the manifold the numerical evalua tions are done. How- ever, extreme cases vastly simplify the six by six matrix and step by step allowed one to numerically gauge certain unknowns one at a time. For bi-cyclide ,flat ring cyclide anddisk cyclide coordinates the above procedure was painstakingly applied to the limiting cases k= 1 andk= 0. For such limiting values the coordinates take on a simpler form, albeit the actual coordinate system in questio n is only defined forkbelonging to the open set (0 ,1). The coordinate parameter k′also belongs to the open set (0 ,1) with the relationship k′2= 1−k2. For interior values of k, we successfully conjectured - and then tested by the least squares met hod - that the coefficients involved in the symmetric tensor products of CKVs be have quadrati- cally inkwith the prior calculated ‘endpoints’. So in these last four coo rdinate systems calculating the characteristic Killing tensor did not p recede calculating its representation in terms of symmetrized tensor products of CKVs! A result was that the characteristic Killing tensor derived is automatically t race-free. Bi-cyclide coordinates have the coordinate transformation law: x=a Λcn(µ)dn(µ)sn(ν)cn(ν) cosψ y=a Λcn(µ)dn(µ)sn(ν)cn(ν) sinψ z=a Λsn(µ)dn(ν) Λ≡1−dn2(µ)sn2(ν) (4.3.1) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2(1−sn2(µ)dn2(ν))(dn2(ν)−k2sn2(µ)) Λ2 g33=a2cn2(µ)dn2(µ)sn2(ν)cn2(ν) Λ2(4.3.2) 56 The associated St¨ ackel matrix is:  −k2sn2(µ)−1−k′4sn2(µ) cn2(µ)dn2(µ) dn2(ν) 1 −dn2(ν) sn2(ν)cn2(ν) 0 0 1 (4.3.3) with conformal Qfactora2 Λ2(1−sn2(µ)dn2(ν)), and modulation R-factor equal to Λ−1/2. For the characteristic conformal Killing tensor of bi-cyclide coordinates the k= 0 limit has the following parameters: A33=−2a2 3,C22= 0,C33=2 3,M33= 0. Thek= 1 limit has the parameters: A33=−2a2 3,C22= 0,C33=4 3,M33=−2 3a2. Only theA33term is constant; conjecturing a quadratic dependence on the others to fit the ‘endpoints’, the coefficients of the Killing tensor for bi-cyclide coordinates are A33=−2a2 3, C 22= 0, C 33=2 3(1 +k2), M 33=−2k2 3a2(4.3.4) The conformal Killing tensor in components that results is: K11=1 3a2·(a2x2+a2x2k2+a2y2+a2y2k2−a2z2−a2z2k2+a4 +k2x4+ 2k2x2y2−10k2x2z2+k2y4+ 2k2y2z2+k2z4) K22=1 3a2·(a2x2+a2x2k2+a2y2+a2y2k2−a2z2−a2z2k2+a4 + 2k2x2y2+k2x4+ 2k2x2z2+k2y4−10k2y2z2+k2z4) K33=−2 3a2·(a2x2+a2x2k2+a2y2+a2y2k2−a2z2−a2z2k2+a4 −4k2x2z2−4k2y2z2+k2x4+ 2k2x2y2+k2y4+k2z4) K12=−4xyk2z2 a2 K13=−xz(−a2−a2k2−2k2x2−2k2y2+ 2k2z2) a2 K23=−yz(−a2−a2k2−2k2x2−2k2y2+ 2k2z2) a2 (4.3.5) Using Maple one verifies that this is indeed the trace-free char acteristic Killing tensor for bi-cyclide coordinates. Flat-ring cyclide coordinates have the coordinate transformation law: x=a Λsn(µ)dn(ν) cosψ y=a Λsn(µ)dn(ν) sinψ z=a Λcn(µ)dn(µ)sn(ν)cn(ν) Λ≡1−dn2(µ)sn2(ν) (4.3.6) 57 The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2(1−sn2(µ)dn2(ν))(dn2(ν)−k2sn2(µ)) Λ2 g33=a2sn2(µ)dn2(ν) Λ2(4.3.7) The associated St¨ ackel matrix is:  −k2sn2(µ)−1−(k2sn2(µ) + 1/sn2(µ)) dn2(ν) 1 (dn2(ν) +k2/dn2(ν)) 0 0 1 (4.3.8) with conformal Q-factora2 Λ2(1−sn2(µ)dn2(ν)) and modulation R-factor equal to Λ−1/2. For the characteristic conformal Killing tensor of flat-ring cyclide coordinates thek= 0 limit has the parameters: A33=2a2 3,C22=1 3,C33=1 3,M33= 0. The k= 1 limit has the parameters: A33=2a2 3,C22=2 3,C33=2 3,M33=2 3a2. Only the A33term is constant, again conjecturing a quadratic dependence on the others to fit the ‘endpoints’, the coefficients of the Killing tensor for flat- ring cyclide coordinates are A33=2a2 3, C 22=1 3(1 +k2), C 33=1 3(1 +k2), M 33=2k2 3a2(4.3.9) This also turned out to be correct. The conformal Killing tenso r in components that results is: K11=−1 3a2·(a2y2+a2z2−2a2x2+a2k2y2+a2k2z2−2a2k2x2+a4 +k2x4+ 2k2x2y2−10k2x2z2+k2y4+ 2k2y2z2+k2z4) K22=−1 3a2·(a2x2+a2z2−2a2y2+a2k2x2+a2k2z2−2a2k2y2+a4 + 2k2x2y2+k2x4+ 2k2x2z2+k2y4−10k2y2z2+k2z4) K33=1 3a2·(−a2x2−a2y2+ 2a2z2−a2k2x2−a2k2y2+ 2a2k2z2+ 2a4 −8k2x2z2−8k2y2z2+ 2k2x4+ 4k2x2y2+ 2k2y4+ 2k2z4) K12=xy(a2+a2k2+ 4k2z2) a2 K13=xz(a2+a2k2−2k2x2−2k2y2+ 2k2z2) a2 K23=yz(a2+a2k2−2k2x2−2k2y2+ 2k2z2) a2(4.3.10) 58 Disk cyclide coordinates have the following coordinate transformation law: x=a Λcn(µ)cn(ν) cosψ y=a Λcn(µ)cn(ν) sinψ z=a Λsn(µ)dn(µ)sn(ν)dn(ν) Λ≡1−dn2(µ)sn2(ν) (4.3.11) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=a2(sn2(ν) + sn2(µ)cn2(ν))(dn2(ν)−k2sn2(µ)) Λ2 g33=a2cn2(µ)cn2(ν) Λ2(4.3.12) The associated St¨ ackel matrix is:  −k2sn2(µ)−1 (k2cn2(µ)−k′2/cn2(µ)) dn2(ν) 1 (k′2cn2(ν)−k2/cn2(ν)) 0 0 1 (4.3.13) with conformal Q-factora2 Λ2(sn2(ν) + sn2(µ)cn2(ν)) and modulation R-factor equal to Λ−1/2. For the characteristic conformal Killing tensor of disk cyclide coordinates the k= 0 limit has the parameters: A33=2a2 3,C22=1 3,C33=1 3,M33= 0. The k= 1 limit has the parameters: A33= 0,C22=−1 3,C33=−1 3,M33=−2 3a2. This time no coefficient is constant with respect to the kcoordinate parameter but the quadratic dependence still holds; the coefficients of the Ki lling tensor for disk cyclide coordinates are A33=−a2 6(−4 + 4k2), C 22=1 3(1−2k2), C 33=1 3(1−2k2) M33=−2k2 3a2(4.3.14) The conformal Killing tensor in components is: K11=1 3a2·(2x2a2−4x2a2k2−y2a2+ 2y2a2k2−z2a2+ 2z2a2k2−a4 +a4k2+k2x4+ 2k2x2y2−10k2x2z2+k2y4+ 2k2y2z2+k2z4) K22=1 3a2·(−x2a2+ 2x2a2k2+ 2y2a2−4y2a2k2−z2a2+ 2z2a2k2 −a4+a4k2+ 2k2x2y2+k2x4+ 2k2x2z2+k2y4−10k2y2z2+k2z4) K33=−1 3a2·(x2a2−2x2a2k2+y2a2−2y2a2k2−2z2a2+ 4z2a2k2−2a4 59 + 2a4k2−8k2x2z2−8k2y2z2+ 2k2x4+ 4k2x2y2+ 2k2y4+ 2k2z4) K12=−xy(−a2+ 2a2k2+ 4k2z2) a2 K13=−xz(−a2+ 2a2k2−2k2x2−2k2y2+ 2k2z2) a2 K23=−yz(−a2+ 2a2k2−2k2x2−2k2y2+ 2k2z2) a2 (4.3.15) Cap-cyclide coordinates have the following coordinate transformation law: x=Λ aΥsn(µ)dn(ν) cosψ y=Λ aΥsn(µ)dn(ν) sinψ z=k1/2Π 2aΥ Λ≡1−dn2(µ)sn2(ν) Υ≡sn2(µ)dn2(ν) + [Λ√ k+ cn(µ)dn(µ)sn(ν)cn(ν)]2 Π≡Λ2 k−(sn2(µ)dn2(ν) + cn2(µ)dn2(µ)sn2(ν)cn2(ν)) (4.3.16) The resulting covariant metric coefficients in the separable co ordinates are g11=g22=Λ2(1−sn2(µ)dn2(ν))(dn2(ν)−k2sn2(µ)) a2Υ2 g33=Λ2sn2(µ)dn2(ν) a2Υ2(4.3.17) The associated St¨ ackel matrix is:  −k2sn2(µ)−1−(k2sn2(µ) + 1/sn2(µ)) dn2(ν) 1 (dn2(ν) +k2/dn2(ν)) 0 0 1 (4.3.18) with conformal Q-factorΛ2 a2Υ2(1−sn2(µ)dn2(ν)) and modulation R-factor Λ1/2Υ−1/2. The case of the characteristic conformal Killing tensor of cap-cyclide coordinates presented the biggest challenge to find the CKT coefficients. For o ne, thek= 0 limit was impossible to deal with numerically as the coordinat es suffer a singularity at that limiting case and consequently analyzing the six by six ma trix of coeffi- cients numerically meant dealing with ill conditioned system s. After many failed attempts, we realized the only way to proceed was to make numer ical estimations for several values of kclose to the known k= 1 limit. After seeing that the best 60 quadratic fit through the data points failed to be an accurate estimate for other measured points, the best cubic fit was conjectured and that turned out to give the answer. Furthermore some coefficients were defined in terms o f others and the overall dependence fails to be a polynomial function in k. One coefficient even undergoes a singularity for the k= 0 limit! The coefficients are: A33=a2(k+ 2k2+k3) 24, C 22=2k 3, C 33=−1 3·(1−4k+k2) M33= 2(1 +k)2 3a2k(4.3.19) The corresponding characteristic conformal Killing tensor in components is: K11=−1 48ka2·(k2+ 8x2a2k−8z2a2k+ 8x2a2k3+ 48z2a2k2−8z2a2k3 + 16a4x4+ 16a4x4k2+ 32a4x4k+ 16a4z4+k4+ 2k3−160a4x2z2 −160a4x2z2k2−320a4x2z2k+ 16a4z4k2+ 32a4z4k+ 16y2a2k2+ 8y2a2k3 + 16a4y4+ 16a4y4k2+ 32a4y4k−80x2a2k2+ 8y2a2k+ 32a4x2y2+ 32a4x2y2k2 + 64a4x2y2k+ 32a4y2z2+ 32a4y2z2k2+ 64a4y2z2k) K22=−1 48ka2·(k2+ 8x2a2k−8z2a2k+ 8x2a2k3+ 48z2a2k2−8z2a2k3+ 16a4x4 + 16a4x4k2+ 32a4x4k+ 16a4z4+k4+ 2k3+ 32a4x2z2 + 32a4x2z2k2+ 64a4x2z2k+ 16a4z4k2+ 32a4z4k−80y2a2k2+ 8y2a2k3 + 16a4y4+ 16a4y4k2+ 32a4y4k+ 16x2a2k2+ 8y2a2k+ 32a4x2y2+ 32a4x2y2k2 + 64a4x2y2k−160a4y2z2−160a4y2z2k2−320a4y2z2k) K33=1 24ka2·(k2+ 8x2a2k−8z2a2k+ 8x2a2k3+ 48z2a2k2−8z2a2k3 + 16a4x4+ 16a4x4k2+ 32a4x4k+ 16a4z4+k4+ 2k3−64a4x2z2 −64a4x2z2k2−128a4x2z2k+ 16a4z4k2+ 32a4z4k−32y2a2k2+ 8y2a2k3 + 16a4y4+ 16a4y4k2+ 32a4y4k−32x2a2k2+ 8y2a2k+ 32a4x2y2+ 32a4x2y2k2 + 64a4x2y2k−64a4y2z2−64a4y2z2k2−128a4y2z2k) K12=2xy(k2+ 4z2a2k+ 2a2z2+ 2z2a2k2) k K13=−1 2k·(xz(−6k2+k+k3+ 8x2a2k+ 4x2a2+ 4x2a2k2+ 8y2a2k + 4y2a2+ 4y2a2k2−8z2a2k−4a2z2−4z2a2k2)) K23=−1 2k·(yz(−6k2+k+k3+ 8x2a2k+ 4x2a2+ 4x2a2k2+ 8y2a2k + 4y2a2+ 4y2a2k2−8z2a2k−4a2z2−4z2a2k2)) (4.3.20) The algebra was too formidable to verify this for the general case except for the off-diagonal components. We verified the cases ψ= 0 andψ=π 2knowing that any angleψcan be redefined by an appropriate rotation in the x−yplane to yield the 61 simpler values tested above. This concludes the calculation o f the characteristic conformal Killing tensors associated with the R-separable coordinates in Ch. 4 of [38]. 62 Chapter 5 Group actions preserving rotationally symmetric canonical CKTs In this chapter we give explicitly the conformal transformat ions of Euclidean space that leave invariant the linear space of trace-free rotation ally symmetric conformal Killing tensors. We also give additional transformations define d directly on this space that are needed in the classification scheme. Invariants of these transfor- mations will be used to classify the rotationally symmetric R-separable coordinate webs of the Laplace equation in Chapter 6. 5.1 Continuous group actions The conformal transformations on E3induce linear transformations on the space of trace-free conformal Killing tensors. We now give conformal t ransformations that leave invariant the six dimensional space of trace-free confor mal Killing tensors that are invariant under rotations about the z-axis. We also give two additional transformations defined directly on this space that also leave i t invariant. We first consider the translations. Translations in the xyplane are ruled out since they will break the rotational symmetry about the z-axis. So are rotations about theyandxaxis or any combination thereof. However, a translation in the z-direction, of the form: ˜x=x ˜y=y ˜z=z+α, α∈R (5.1.1) is an allowed group action leaving the symmetry of the web inva riant. The other is the rotation of the coordinates in the xyplane; however, as the web itself is 63 rotationally symmetric this action leaves invariant all the coefficients of the tensor and is thus trivial. Clearly rotations in the xzandyzplanes are forbidden. A dilatation of the space described by the equations: ˜x=eαx ˜y=eαy α> 0 ˜z=eαz (5.1.2) also does not change the geometry of the coordinate surfaces. Her e we draw at- tention to two group transformations on the space of Killing te nsors which do not arise from the action of C(M). The first one is the dilatation of t he Killing tensor itself: ˜K=αK, α> 0 (5.1.3) The second arises by the addition of a scalar multiple of R3⊙R3to the Killing tensor. Thus ˜K=K+αR3⊙R3, α∈R (5.1.4) is another allowed functionally independent transformatio n from the first set. The last two transformations are not motivated by geometric consid erations, but rather by the allowed operations on St¨ ackel matrices which were use d to find the char- acteristic Killing tensor studied. Indeed, the last two group ac tions result directly from the allowed steps 2) and 3) on St¨ ackel matrices as explai ned on p. 593 of [36]. That St¨ ackel matrices are unique up to some defined oper ations translates to a characteristic Killing tensor being unique up to a multipl e of itself and the addition of a scalar multiple of R3⊙R3. Note that R3⊙R3is not a characteristic Killing tensor since two eigenvalues are zero and hence not all distinct. 5.2 Discrete group transformations The last transformation belonging to the conformal group to be considered is in- version about the unit sphere. It is a discrete transformation de fined by: x→˜x=x x2+y2+z2 y→˜y=y x2+y2+z2 z→˜z=z x2+y2+z2 (5.2.1) This transformation preserves the rotational web, inducing t he following simulta- neous interchanges on the CKT coefficients: ˜M33=A33,˜A33=M33,˜B21=−G12,˜G12=−B21 (5.2.2) 64 The second order terms C22andC33are left unchanged. Studying this effect of the discrete inversion on the 6-tuple of rotational CKT coefficient s reveals that some tuples (and hence CKTs) are mapped into one another - proving t hat their associ- ated coordinate webs are equivalent under that operation. I n particular this proves that 6-sphere coordinates result from the inverse of Cartesian c oordinates, tangent sphere coordinates are the inverse of circular cylindrical co ordinates, inverse prolate spheroidal coordinates are the inverse of prolate spheroidal c oordinates and inverse oblate spheroidal coordinates are the inverse of oblate sphero idal coordinates. The discrete inversion also puts cardioid coordinates in the same co ordinate class as paraboloidal coordinates since the coefficients after inversi on differ only by a factor of 4. As we have just discussed Killing tensors differing by a common n on-zero factor are deemed equivalent as they admit the same eigenvect ors. Therefore from the standpoint of group operations one might say that 6-sphere, tangent sphere, in- verse oblate spheroidal, inverse prolate spheroidal and cardio id coordinates are not inequivalent from the set of simple separable metrics. However, t heseR-separable coordinates are distinguished from the rest in that their A33coefficient is 0. ForA33/ne}ationslash= 0, a discrete inversion cannot map an R-separable web into a simply separable web, since after the interchange M33/ne}ationslash= 0 and this fourth order term is non-zero only for conformally separable metrics! Furthermor e, the claim in [38] that cap-cyclide coordinates are the (discrete) inverse of bi -cyclide coordinates is refuted by noting that interchanging A33andM33does not map the one set of coefficients into the other. Neither does inversion in the unit sp here map the one set of coordinate definitions to the other, as may be verified by Maple. Thus we had to study the five independent group actions permitted by th e geometry of the rotational webs. They can be used in various combination to classify the R- separable rotational webs, as well as the simple separable rotat ional cases already categorized in [25]. The restriction on the group parameter αfor the dilatation cases can be lifted if one allow the discrete ‘-1 switch’ of the Killing tensor and of the space, in the form: ˜K=−K,(˜x,˜y,˜z) = (−x,−y,−z) (5.2.3) Although these discrete operations preserve the rotational web , the entire group orbits will not all be continuously connected with the identi ty, since the singular caseα= 0 must always be avoided. The negative dilatation of the space has the following effect on the Killing tensor coefficients: ˜A33=A33,˜B21=−B21,˜C22=C22,˜C33=C33 ˜G12=−G12,˜M33=M33 (5.2.4) namely the first and third order terms are mapped to their negat ive inverses while the zeroth, second and fourth order terms are left unchanged. For the cases satisfy- ing (B21,G12) = 0, the negative dilatation of the space plays no further rol e. Note that the group operation ˜K=K+αg, α∈R (5.2.5) 65 is not allowed, where gis the contravariant metric tensor. Conformal Killing tensor s are of course only unique up to a functional multiple of the met ric tensor. However by setting the trace to vanish we have already fixed this free par ameter. The above group operation that was used in simple separable cases to disting uish between inequivalent coordinates [25] is now not available. Having defined the discrete inversion we next look at a continuou s group action that involves it. The definition is given on p.128 of [40] whic h defines the group as a composition of three maps: Discrete inversion, an infinitesimal translation along a pr eferred direction followed by the discrete inversion. Remark 5.2.1 Clearly this group action is connected with the identity. One should point out that this infinitesimal group transformation is no t independent of the preceding ones as it is constructed by an explicit compositi on of a discrete inversion and an infinitesimal translation. Unlike the discrete invers ion however this group action is continuous and described by a parameter making it m ore convenient in some cases of study than the discrete inversion. After some algebra one can show that the group action on the coord inates, along z, amounts to: x→˜x=x 1 + 2zα+α2(x2+y2+z2) y→˜y=y 1 + 2zα+α2(x2+y2+z2) z→˜z=z+α(x2+y2+z2) 1 + 2zα+α2(x2+y2+z2) (5.2.6) Formulae for group inversion along the directions xandyare similar. These group actions neatly coincide with the inversional conf ormal Killing vec- tors: indeed differentiations of the transformed variables at α= 0 yield the in- finitesimal inversions given in (3.1.1). As predicted these tran sformations along x andydo not preserve the rotational web which is defined about the zaxis. How- ever, the group inversion along zpreserves this rotational symmetry. This is a quasi extension of the existing four continuous group operations th at leave the rotational web considered invariant. 66 5.3 Effect of continuous group actions on the Killing tensor coefficients and calculation of invari- ants We begin this sub-section by giving a definition for an invarian t of a CKT under a continuous group action. Definition 5.3.1 An invariant of the rotationally symmetric CKT Kis an analytic functionFof the parameters defining Kwhich satisfies F(˜A33,˜B21,˜C22,˜C33,˜G12,˜M33) =F(A33,B21,C22,C33,G12,M33) (5.3.1) for all values of the parameters, where the tilded parameter s are related to the un- tilded parameters by the transformation laws induced by an a rbitrary group action. Note that the above is a very special case of (2.4.3). We first consid er an infinites- imal translation in the zdirection, which induces the following transformations: ˜A33=A33−2B21α+ (C22+C33)α2+ 2G12α3+M33α4 ˜B21=B21−(C22+C33)α−3G12α2−2M33α3 ˜C22=C22+ 2G12α+ 2M33α2 ˜C33=C33+ 4G12α+ 4M33α2 ˜G12=G12+ 2M33α ˜M33=M33 (5.3.2) The derivatives of the new coefficients with respect to the tran sformation parameter α, evaluated at α= 0, are given by ∂˜A33 ∂α=−2B21 ∂˜B21 ∂α=−(C22+C33) ∂˜C22 ∂α= 2G12 ∂˜C33 ∂α= 4G12 ∂˜G12 ∂α= 2M33 ∂˜M33 ∂α= 0 (5.3.3) 67 The derivative of Eq. (5.3.1) with respect to α, with the use of the chain rule yields ∂F ∂˜A33∂˜A33 ∂α+∂F ∂˜B21∂˜B21 ∂α+∂F ∂˜C22∂˜C22 ∂α+∂F ∂˜C33∂˜C33 ∂α+∂F ∂˜G12∂˜G12 ∂α +∂F ∂˜M33∂˜M33 ∂α= 0 (5.3.4) Evaluation at the identity ( α= 0) with the use of Eq. (5.3.3) gives, after dropping the tildes, our first determining pde namely 0 = −2∂F ∂A33B21−∂F ∂B21(C22+C33) + 2∂F ∂C22G12 + 4∂F ∂C33G12+ 2∂F ∂G12M33 (5.3.5) Remark 5.3.2 We note here that for all R-separable coordinates in [38], M33= 0 andG12/ne}ationslash= 0solely for cardioid coordinates. All other rotational coordinates satisfy M33/ne}ationslash= 0,G12= 0andB21= 0. This observation segregates cardioid coordinates from all the others. Using the tensor transformation law, it can be shown that transform ed coefficients resulting from a dilatation satisfy: ˜A33=e2αA33 ˜B21=eαB21 ˜C22=C22 ˜C33=C33 ˜G12=e−αG12 ˜M33=e−2αM33 (5.3.6) The derivatives of the tilded parameters at α= 0 are given by ∂˜A33 ∂α= 2A33 ∂˜B21 ∂α=B21 ∂˜C22 ∂α= 0 ∂˜C33 ∂α= 0 ∂˜G12 ∂α=−G12 ∂˜M33 ∂α=−2M33 (5.3.7) 68 The resulting determining pde associated to the dilatation is f ound to be 2∂F ∂A33A33+∂F ∂B21B21−∂F ∂G12G12−2∂F ∂M33M33= 0 (5.3.8) The third group transformation to be considered is the scalar mu ltiple of the Killing tensor itself, namely ˜K=αKforα/ne}ationslash= 0. The transformation of the coefficients and the resulting derivatives with respect to the parameter are to o trivial to tabulate. A subtle point is that evaluation at the identity element of the group transformation amounts to evaluation at α= 1, notα= 0 as in all the other examples. This subtlety is easily overlooked since the derivative of the coeffic ients are the unprimed coefficients themselves and have no dependence on α. The resulting determining pde follows easily: 0 =∂F ∂A33A33+∂F ∂B21B21+∂F ∂C22C22+∂F ∂C33C33 +∂F ∂G12G12+∂F ∂M33M33 (5.3.9) The fourth group transformation considered is the addition to the Killing tensor of a scalar multiple of the second rotational Killing tensor commo n to all rotational coordinates. Recall that this transformation has the form ˜K=K+αR3⊙R3. However, it should be pointed out that this is an abuse of notatio n, since the sym- metric tensor product indicated is only true for the non trace -free representation. Explicitly the tensor has two non-zero parameters with respec t to our basis set C22 andC33:C22=−1 3andC33=1 3. The resulting transformation of the coefficients is as follows: ˜A33=A33 ˜B21=B21 ˜C22=C22−1 3α ˜C33=C33+1 3α ˜G12=G12 ˜M33=M33 (5.3.10) All derivatives of the coefficients with respect to the transform ation parameter α are zero except for∂C22 ∂α=−1 3and∂C33 ∂α=1 3. Again the last result is from evaluation at the identity element of the group transformation. We thus a rrive at the fourth determining pde which is: ∂F ∂C22−∂F ∂C33= 0 (5.3.11) 69 We next look at the consequences of the continuous inversion. O ne can show the following transformation of the Killing tensor coefficients: ˜A33→A33 ˜B21→B21−2αA33 ˜C22→C22−2αB21+ 2α2A33 ˜C33→C33−4αB21+ 4α2A33 ˜G12→G12+ (C22+C33)α−3B21α2+ 2A33α3 ˜M33→M33+ 2G12α+ (C22+C33)α2−2B21α3+A33α4 (5.3.12) One sees immediately that the term A33is invariant; clearly the discrete inversion cannot be ‘reached’ by the continuous group inversion. This i s perhaps analogous to the fact that dilatation of the Killing tensor itself cannot attain multiplication of the Killing tensor by -1. The continuous inversion along zadmits a well defined pde which is: 0 = −2∂F ∂B21A33−2∂F ∂C22B21−4∂F ∂C33B21 +∂F ∂G12(C22+C33) + 2∂F ∂M33G12 (5.3.13) Now that the pdes resulting from all the infinitesimal group acti ons considered have been written down, it is instructive to tabulate the vectors o f the generators of the solution to verify that they are indeed linearly independent . In terms of the basis (∂ ∂A33,∂ ∂B21,∂ ∂C22,∂ ∂C33,∂ ∂G12,∂ ∂M33), (5.3.14) the vector generators of the five determining pdes are: V1= [−2B21,−(C22+C33),2G12,4G12,2M33,0] V2= [2A33,B21,0,0,−G12,−2M33] V3= [A33,B21,C22,C33,G12,M33] V4= [0,0,1,−1,0,0] V5= [0,2A33,2B21,4B21,−(C22+C33),−2G12] (5.3.15) The corresponding Lie algebra is: 70 [ , ]V1V2V3V4V5 V1 0V1002V2 V2−V1000V5 V3 0 00−V40 V4 0 0V40 0 V5−2V2−V500 0 Table 5.1: Lie Commutator Table The set of vectors ( V1, . . . ,V5) is generically linearly independent. However, there are special cases when it is not. In particular there are c oordinates such that the associated dimension of the space spanned by the Vis reduces to two. This closed commutator table proves that the system of pdes is completely i ntegrable. Actual integration of the above system to find group invariants is exce edingly difficult, especially after also imposing the discrete inversion not admitt ing a pde. It is best to first start with canonical examples of characteristic Killin g tensors which satisfy B21=G12= 0 for non-cardioid cases in [38]. The study of canonically cen tered coordinates admitting a restricted set of group transformatio ns will be the topic of the next chapter. The system of pdes was eventually integrated first by finding two f unctionally independent invariants with respect to the entire group modu lo the dilatation of the conformal Killing tensor. The method of undetermined pol ynomial coefficients, utilized especially in [25], was used since Maple 9 was unable to perform the inte- gration of Eq. (5.3.15). Thus the derived I1andI2are polynomial functions of the coefficients: I1=−72M33A33(C22+C33) + 108M33B2 21+ 2(C22+C33)3+ 108A33G2 12 + 36B21G12(C22+C33) I2= (C22+C33)2+ 12A33M33+ 12B21G12 (5.3.16) To construct the single functionally independent invariant w ith respect to the entire group action is now trivial. The square of I1is divided by the cube of I2to get the polynomial degrees (of the coefficients) to match (at 6). The re sulting quantity is then invariant under dilatation of the conformal Killing te nsor itself. This explicit form of the invariant is I= (−72M33A33(C22+C33) + 108M33B2 21+ 2(C22+C33)3+ 108A33G2 12 + 36B21G12(C22+C33))2/((C22+C33)2+ 12A33M33+ 12B21G12)3 (5.3.17) It is easy to see that the above is also invariant with respect to di screte inversion given the invariance of every term in the sum with respect to tha t discrete ac- tion. This solution will be discussed in the next chapter after fir st considering the reduced invariant on canonically centered coordinate webs which avoids the co n- tinuous translation and inversion group action. Unfortunatel y, it seems that Iis a 71 poor choice to discriminate between coordinate webs as it lum ps together simple and conformally separable cases whereas the reduced invarian t was defined only for ‘purely’ conformal coordinate webs. This will be elucidated in the next chapter. 72 Chapter 6 Classification of the symmetric R-separable webs 6.1 Classification of rotationally symmetric R-separable coordinates With the results and methods presented in the previous chapter s - we now proceed with the main goal of this thesis which is to classify the rotationally symmetric webs and partition them into equivalence classes under the Lie group of conformal transformations derived in the previous chapter. The questio n of the exhaustive- ness of rotationally symmetric R-separable coordinates will finally be answered. The lack of other types of symmetric coordinates admitting R-separation of the Laplace equation will be discussed at the end of this chapter. We begin this chapter by giving a definition of equivalence of characteristic CKTs under the action of the conformal group. Definition 6.1.1 Two characteristic CKTs are said to be equivalent if and only if there exists an element of the transformation group on the space of such tensors which maps one of the tensors into the other. The R-separable webs defined by equivalent tensors are said to be equivalent. For the classification of the rotationally symmetric R-separable webs the following transformations need to be considered: Translation along the z-axis Continuous inversion along the z-axis Dilatation of the space Discrete inversion Dilatation of the Killing tensor 73 Addition to the Killing tensor of a scalar multiple of the rotati onal Killing ten- sorR3⊙R3 The effect of these operations on the coefficients of the reduced Killing tensor has already been described in the previous section. Thus we have av ailable degrees of freedom for the change of coefficients, as well as an adjustment o f the parameter kthat appears in the definition of the four Jacobi-elliptic co ordinate systems. It is these last four coordinates in Ch. 4 of [38] that contain this arbitrary param- eter which is defined on the open set (0 ,1). Indeed kappears in the coefficients for symmetric tensor products of CKVs and its variation within t he allowed range must be taken into account. Recall that for elliptic-hyperbo lic coordinates in E3, the parameter used to describe the coordinate surface was define d to be the inter focal distance [25] whose range gives a related family of coord inates. Quantities invariant under group transformations leaving the web uncha nged are called ‘invari- ants’. However, usually the invariants are themselves function s of the parameters appearing in the coordinate definitions. In some of the cases pre sented in this sec- tion the invariants will differ for different values of k. If adjusting kin addition to the degrees of freedom afforded by all possible group transfor mations does not yield an equality of two sets of invariants, then it is reasonabl e to conclude that the two coordinate systems considered are inequivalent. Note we spec ifically omit the other coordinate parameter ‘ a’ appearing in most definitions in [38] because this is none other than the dilatation of the space which has already b een considered. The coordinates 6-Sphere, tangent sphere and cardioid coord inate systems are defined without any such parameters. All algebraic invariants w ill be constant but as mentioned before the discrete inversion puts these in the same equivalence class as simple separable coordinates in E3. 6.2 Classification of canonically centered rotation- ally symmetric webs We call attention to a curious paradox. Consider for the subset o f canonically centered webs satisfying B21= 0 andG12= 0, the three allowed functionally independent group actions acting on the coefficients. These are the dilatation of the Killing tensor, dilatation of the space and addition to the Killing tensor of a scalar multiple of the rotational Killing tensor R3⊙R3. Recall that the translation and continuous inversion along the z-axis changes the values of B21andG12which is unacceptable since both are zero and must remain so for canonica lly centered webs. Since there are only four independent unknowns in the coeffici entsA33,C22,C33and M33and three determining pdes, we obtain one functionally indep endent ( reduced ) invariant: /parenleftBiggC22+C33√A33M33/parenrightBigg (6.2.1) 74 Note that the product A33M33negates dependence on the coordinate parameter a: the invariant is thus constant for entire families of coordin ates. It is easy to see that the invariant is constant with respect to dilatation of th e space and dilatation of the Killing tensor itself. Furthermore the product A33M33is invariant under the discrete inversion since A33andM33are simply interchanged by this transformation. We now use this single invariant to partition the known canonic alR-separable webs into disjoint equivalence classes. Proposition 6.2.1 Inequality of invariants is a sufficient but not a necessary con- dition for inequivalence of any two R-separable coordinate systems. Contraposi- tively, equality of invariants to prove equivalence of webs is simply a necessary but not sufficient condition. Proof: This follows by analyzing two systems of coordinates. Recall th at the list of CKT coefficients identifying toroidal coordinates is: A33=a2 6, C 22= 0, C 33=1 3, M 33=1 6a2(6.2.2) The list of CKT coefficients identifying bispherical coordinates is: A33=−a2 6, C 22= 0, C 33=1 3, M 33=−1 6a2(6.2.3) Substituting these values into Eq.(6.2.1) gives the surprising result of +2 for both toroidal and bispherical coordinates. Is this proof that, wit h respect to the three group operations, toroidal and bispherical coordinates are e quivalent? A simple argument shows that they cannot be. Recall from the previous se ction that dilata- tion of the space amounts to multiplying M33by a positive quantity reciprocal to that which multiplies A33with the variables C22andC33being left unchanged. Di- latation of the Killing tensor multiplies all coefficients, inc ludingM33, equally by a greater-than-zero scalar. Addition of a scalar of the second rot ational Killing tensor leaves both M33andA33alone. Finally, it is noted that M33depends on the inverse square of the coordinate parameter a. Thus varying aamounts to multiplication ofM33by a positive scalar. It follows that no group transformation or coordinate adjustment can change the sign of M33andA33with respect to the other Ciico- efficients. Yet to transform the characteristic tensor for toroid al coordinates into the one for bispherical coordinates requires precisely this f orbidden operation! This reasoning is the ultimate ‘acid test’ for determining whether toroidal coordinates are indeed inequivalent to bispherical coordinates. Nonethel ess their invariants are precisely the same. What resolves this paradox? Note the factor ( A33M33) in the de- nominator completely eliminates information as to whether bothA33andM33were positive or negative. It is easy to see knowing the transformatio n of the coefficients that the invariant listed is indeed constant under the three gr oup transformations that generated it. This completes the proof by example for th e pair toroidal and bispherical coordinates. 2 75 The same proof could also be obtained by analyzing bi-cyclide a nd flat ring cy- clide coordinates. The CKT coefficients for bi-cyclide coordinates are given by: A33=−2a2 3, C 22= 0, C 33=2 3(1 +k2), M 33=−2k2 3a2(6.2.4) The corresponding coefficients for flat-ring cyclide coordinates are: A33=2a2 3, C 22=1 3(1 +k2), C 33=1 3(1 +k2), M 33=2k2 3a2(6.2.5) Substitution of these coefficients into the single invariant yie lds(1+k2) k∈(+2,∞) for both bi-cyclide and flat ring cyclide coordinates. Note aga in how their values forM33differ in sign. This criterion proves their inequivalence desp ite equality of their single invariant. On a reassuring note, for kdefined on the open set (0 ,1) the invariant(1+k2) kalways differs from +2 which is the invariant identifying tor oidal and bispherical coordinates, proving positively that these four ar e an inequivalent set. Of course Eq.(6.2.1) fails to discriminate between webs satisfy ingA33= 0, namely tangent sphere, inverse oblate spheroidal and inverse prolate sp heroidal coordinates. Inspection of the coefficients for the above (listed in the previ ous section) yields M33=−2 3a2for inverse oblate spheroidal coordinates and M33=2 3a2for inverse prolate spheroidal coordinates, while their Ciivariables are the same - thus setting them apart immediately. Recall from the last chapter that it i s not necessary to consider the A33= 0 cases alongside the canonical conditions B21=G12= 0 as the discrete inversion transformation places tangent sphere, inve rse oblate spheroidal and inverse prolate spheroidal coordinates into the same equiv alence class as simple separable webs already classified in [25]. Furthermore any cano nical Killing tensor identified by M33= 0 will admit a characteristic Killing tensor of degree two whi ch is already a subset of the simple separable cases. Cardioid coordin ates, with only G12/ne}ationslash= 0, is placed in the class of parabolic coordinates by the discr ete inversion, that is the degree three tensor is mapped to a degree one tensor a nd so is not an additionalR-separable coordinate system. Remark 6.2.2 The proof of Prop 6.2.1 relied on continuous group operation s only. It is true that no continuous group operation so far consider ed, connected with the identity, can map a Killing tensor to its additive inverse. How ever, if the three group operations can change the tensor coefficients to bring them to t he exact negative of another characteristic Killing tensor, then the Killing ten sor pairs are equivalent. The previous arguments must be modified somewhat to take this feature, the result of the inclusion of a discrete -1 switch, into account. For the coordinate pairs toroidal, bispherical, bi-cyclide and flat ring cyclide, A33,M33differ only by sign. Dilatation of the space, dilatation of the K illing tensor itself and addition of a scalar multiple of the second rotationa l tensor R3⊙R3as well as variation of the parameters aandkamount only to multiplication of A33 76 andM33by positive non-zero scalars. If the tensors are to be made equiva lent, thenA33andM33must be left alone. Once that is known, C22andC33must be simultaneously brought to their additive inverses. This is ea sily seen to be impossible, as one increases at the negative rate of the other un der the above group action. With toroidal and bispherical coordinates, C22= 0,C33/ne}ationslash= 0. This relation in size is ‘out of phase’ with the transformation of the variables u nder addition of a scalar multiple of R3⊙R3which implies ˜C22=C22+α 3and˜C33=C33−α 3. It is impossible to simultaneously bring both variables to their addi tive inverses - thus proving that bispherical and toroidal coordinates are inequ ivalent even with the discrete -1 switch added to the list of coefficient transformation s. With bi-cyclide and flat ring cyclide coordinates the A33andM33must be sim- ilarly left alone, but the C22coefficient is 0 for bi-cyclide coordinates and non-zero for flat ring cyclides. Adjusting C22accordingly will not make the C33pair additive inverses. This reasoning completes the discussion on possible equiv alence between bi-cyclide and flat-ring cyclide coordinates even with the - 1 switch degree of free- dom on the coefficients. Disk cyclide coordinates are described by the CKT coefficients: A33=−a2 6(−4 + 4k2), C 22=1 3(1−2k2), C 33=1 3(1−2k2) M33=−2k2 3a2(6.2.6) These are the only ones with the property A33M33<0. This makes Eq.(6.2.1) complex however its negative square value partitions it from the square value of other invariants discussed thus far. Another argument for their inequivalence to all other coordinate systems is this: A33andM33will always differ in sign as group transformations and adjustment of parameters amount to multi plications by non- zero constants. No operation exists to make their signs equal or th eir values vanish - so setting them apart from all coordinates considered. It shoul d be noted that this reasoning trivially explains why cardioid coordinates are distinguished from the rest. It is the only coordinate system admitting M33= 0 while every other member has M33/ne}ationslash= 0. Forcap-cyclide coordinates the invariant/parenleftBig C22+C33 √A33M33/parenrightBig is equal to(−2+12k−2k2) (1+k)2. With k∈(0,1) the range of the invariant is the open interval (-2,2). Thi s distinguishes cap-cyclide coordinates from toroidal and bispherical whic h is fixed at +2, and from bi-cyclide and flat ring cyclide coordinates whose range is (+ 2,∞). Studying the square of Eq.(6.2.1) now becomes useful. Recall t hat any contin- uous function of an invariant is an invariant. Firstly the squa re is invariant to the discrete ‘minus 1 switch’ that complements the dilatation of t he Killing tensor itself. Secondly it is then possible to consider disk cyclide coordinate s whereA33M33<0. 77 Disk cyclide coordinates will, from variation of k, admit an invariant in the range (−∞,0]. Cap-cyclide will have its invariant in the finite interva l [0,4). Toroidal and bispherical have their invariants fixed at +4, and finally b i-cyclide and flat-ring cyclides have their invariants in the infinite interval (4 ,+∞). Remark 6.2.3 For invariants with no intersection in their ranges, even aft er vary- ingaandk, it is sufficient to prove that the corresponding webs are ineq uivalent under dilations, discrete inversion and addition of R3⊙R3, since we have considered simultaneously the independent group actions on the subset of pertinent coefficients. Note that the singular infinities are excluded as they corres pond either to k= 0or k= 1which are not defined for the Jacobi-elliptic coordinates. Remark 6.2.4 The reader might well ask whether disk cyclide coordinates ar e only one of a pair of inequivalent coordinate systems since the pro ductA33M33<0 destroys information as to which coefficient is less than zero a nd which is greater than zero. Suppose one had M33<0andA33>0instead? Recall that dilatation of the space or variation of the coordi nate parameter amul- tipliesM33with the reciprocal of the positive quantity which multiplie sA33. Thus the set of coefficients can be continuously mapped to the case: A33<0 andM33>0 for dilatation parameter a2=A33 M33. In fact since B21=G12= 0 the discrete in- version can also be used to perform the same interchange and thus t he question of which coefficient is less than or greater than zero is meaningle ss. Recall once again the discrete inversion and dilatation of the space does not affec t the degree two Ciiparameters. Note that the above argument does not lump togethe r the cases where both A33andM33are either greater than or less than zero for fixed values ofC22andC33. Note further that the seemingly distinct cases M33<0,A33>0, (C22+C33)>0 andM33>0,A33<0, (C22+C33)<0 are related by the -1 switch and hence not representative of inequivalent webs. Thus the ra nge of the invariant (−∞,0) is indicative of only one equivalence class of coordinates. A similar argument (so far!) proves that cap-cyclide coordina tes does not appear to have a ‘twin’ inequivalent coordinate system either, as the reader could again point out that the case A33>0,M33>0 represents another inequivalent case as per the situation for the coordinate pairs toroidal, bispheri cal and bi-cyclide, flat ring cyclide coordinates. However the variation of the parame terkfor cap-cyclide coordinates can force ( C22+C33) to vanish and at that point the -1 switch can be applied to reverse the signs of A33andM33simultaneously without affecting the numerator of Eq.(6.2.1) since it is 0. Note this subtlety could n ot hold for invariants not spanning the null element as the -1 switch would reverse all t he four signs of the CKT coefficients simultaneously and not two in isolation. The invariant ranges from cap-cyclide and disk-cyclide coordinates seem to interse ct at the null element. However, as mentioned before no group action can make A33andM33switch from being equal in sign to opposite. This is yet another example of w here equality of invariants is only a necessary condition for equivalence of co ordinates. 78 6.3 The question of non-canonically centered ro- tationally symmetric R-separable coordinates We have, with respect to the invariant in Eq.(6.2.1), a total o f six additional in- equivalentR-separable webs to the known set of eleven simple separable webs i nE3. The entire real line of the invariant in Eq.(6.2.1) is exhaust ed as well as ambiguities that resulted from the product factor. We must now address the qu estion as to whether this represents the totality of distinct R-separable webs especially since we have restricted ourselves to the canonically centered cases B21=G12= 0. What if one was given a 6-tuple of random numbers representing the v alues for the six CKT coefficients where the previous condition no longer holds? This spells trouble since an addition of two coefficients even with the inclusion of t he translation group operation implies that an additional invariant has to be con sidered. This immedi- ately means there is ‘more room’ for inequivalent coordinat es which will be hard to classify not knowing a priori (i.e from solving the Eisenhart e quations) similar formulae for the coordinate parameter k. An alternative is to consider the translation and continuous inversion group actions as a set in its own right used simply to bring the two coeffic ientsB21and G12, one or both assumed non-zero, to vanish irrespective of the effec t on the other terms. Definition 6.3.1 Any 6-tuple of numbers representing the CKT coefficients are equivalent to some 4-tuple of numbers representing coordina tes in canonical centered form if any pair ( B21,G12) belonging to the original 6-tuple can be mapped to (0,0) by a composition of continuous inversion and translation gro up actions. If the above holds uniquely then the previous section provides proof that the coor- dinates in [38] represent an exhaustive list of all rotationally symmetric R-separable coordinates in E3, up to equivalence. The issue of uniqueness is a subtle one and the consequence for non-uniqueness will be addressed later on. We have proceeded with this calculation and deduced that for the above to hold, the following pair of equations resulting from the compositio n of an inversion with a translation must have a real solution for the unknown transform ation parameters (a,b) respectively with given arbitrary B21andG12: 0 =B21−2aA33−3(C22+C33)ab2−(C22+C33)b−2(C22+C33)b3a2 + 4B21a3b3−6A33ba2−2M33b3−2A33a4b3+ 9B21a2b2+ 6B21ba −4G12ab3−3G12b2−6A33a3b2 0 =G12+ 2(C22+C33)a2b+ 2A33a3+ 2A33a4b+ 2M33b−4B21a3b −3B21a2+ (C22+C33)a+ 4G12ab (6.3.1) Solving for the group parameter variables ( a,b) results however in solutions given in terms of roots of a degree six polynomial expression divided by a quartic polynomial. 79 By classical algebra solutions are guaranteed but over the comp lex fieldZ. We performed stochastic numerical tests to state with a degree of ce rtainty that the roots of the sextic polynomial always contain at least one real pair. The presence of the quartic polynomial in the denominator poses a potentia l problem, since one must take into account the possibility of a pathological case whe re the numerator and the denominator share real roots leaving only complex roo ts for the remainder. Even one such case would signify a new R-separable web. However, its Lebesgue measure is zero in the abstract space of the coefficients, and beyon d the reach of random numerical tests. The special case where only the pair B21,G12is not trivial was verified to yield real solutions for ( a,b) provided the product G12B21is non-zero. Recall that this subcase either represents a simply sep arable web or can be mapped to one by a discrete inversion, as is the case for cardio id coordinates. Due to this special case yielding a real solution and the numerou s numerical tests performed to verify the existence of real solutions to the degr ee six polynomial, we present Proposition 6.3.2 Eq.(6.3.1) contains at least one pair of real roots. Proof: Using resultant theory of polynomials one can prove the proposition for the special case of C22,C33,B21,G12arbitrary and A33=M33= 0. This is made possible since Maple 9 and 10 both give two classes of solutions to ( a,b) in solving for B21 andG12= (0,0). Where one class yields only complex or undefined soluti ons, the other does. This may be proved with the help of resultant theory of polynomials. By extension one can prove the conjecture for A33= 0 orM33= 0, since infinitesimal inversion or translation can bring the other term to zero since t he equation involved is of degree three in the parameter guaranteeing a real soluti on. ForA33,M33both non-zero however a pure inversion or translation cannot guara ntee solving for either of them (the equations involved are of degree four in the para meters). However, a composition of inversion and translation yields the transforma tion equation for A33 being degree four but bivariate: ˜A33=A33+ (−2B21+ 4A33a)b+ (C22+C33−6B21a+ 6A33a2)b2 + (2C33a+ 2C22a−6B21a2+ 4A33a3+ 2G12)b3 + (2G12a−2B21a3+A33a4+C33a2+M33+C22a2)b4(6.3.2) Stochastic tests using fifty million random cases have verified th at for any 6-tuple of coefficients it is very likely that ˜A33can be made zero in the general case. No pathological case of Lebesgue measure zero exists and so Prop. 6.3 .2 appears true for arbitrary values of ( A33,B21,C22,C33,G12,M33).2 The transformation equations for the other rotational CKT co efficients under com- position of inversion and translation parameters ( a,b) respectively are given here for completeness, where for compactness C≡C22+C33: ˜C=C+ 6A33a2−6B21a + (−18B21a2+ 12A33a3+ 6Ca+ 6G12)b + (6Ca2+ 6M33+ 6A33a4−12B21a3+ 12G12a)b2(6.3.3) 80 and lastly, ˜M33=M33+ 2G12a+Ca2−2B21a3+A33a4(6.3.4) Should any reader wish to convert a random 6-tuple of rotation al CKT coefficients to canonically centered form, namely ( B21,G12) = (0,0) by inversion parameter a and translation parameter b, the other coefficients can then be calculated using the information presented above. 6.4 Canonically centered rotational coordinates related by balanced combination of inversion and translation With the previously discussed ‘projection’ from 6-tuple to 4-tu ple space of Killing tensor coefficients, one must be aware of the non-uniqueness of rea l roots to Eq.(6.3.1). This suggests the possibility of applying the continuous inversi on and translation group action on a priori canonically centered webs to yield o ther canonically cen- tered webs. Namely it is a question of the existence of real solutio ns to Eq.(6.3.1) whenB21andG12are zero a priori while the other four coefficients are arbitra ry. Of course the trivial solution ( a,b) = (0,0) representing the identity transformation is always present but ignored. Solving for Eq.(6.3.1) yields for inversion parameter a: a=4/radicalBigg M33 A33(6.4.1) The translation parameter bin terms of the inversion parameter ais given by: b=−a 2((C22+C33) + 2A33a2) (2M33+ (C22+C33)a2). (6.4.2) One sees that all coordinates could be affected by the above tra nsformations save for disk-cyclide which is the only one where the product A33M33<0 and so only the trivial case a=b= 0 is possible which amounts to the identity transformation. Fo r all other canonically centered rotationally symmetric coor dinates one must verify whether the transformed values for A33,C22,C33andM33correspond to the same equivalence class or not by studying the square of the invariant given by Eq.(6.2.1). In so doing we take into account all the members of the conforma l group used to classify coordinates. The transformation for A33from canonical form to canonical form is given in Eq.(6.3.2) by setting both B21andG12to zero. We list here the transformation of the other coefficients as a result of the above process, in terms of the solved values for inversion parameter aand translation parameter b: ˜M33=M33+ (C22+C33)a2+A33a4 81 ˜C22=C22+ 2A33a2+ (4A33a3+ 2C33a+ 2C22a)b + (2C33a2+ 2C22a2+ 2A33a4+ 2M33)b2 ˜C33=C33+ 4A33a2+ (4C22a+ 4C33a+ 8A33a3)b + (4M33+ 4A33a4+ 4C33a2+ 4C22a2)b2(6.4.3) Applying this to toroidal coordinates one finds that M33is multiplied by 4 while A33is divided by 4, which is merely the effect of a dilation of the sp ace! Toroidal coordinates are therefore not affected by this procedure. Bi spherical coordinates are another matter: the inversion parameter bin Eq.(6.4.2) has an indeterminate form0 0; if one labels this indeterminate fraction as cone obtains - after applying canonical form to canonical form mapping: ˜A33=−k2 6+k2c 3−k2c2 6˜C22=−1 3˜C33=−1 3˜M33= 0 (6.4.4) Note that all cases with respect to the indeterminate constant care simple separa- ble cases, but if M33= 0 the only way to invert the procedure back to bispherical coordinates is to choose the constant csuch thatA33= 0. IfA33/ne}ationslash= 0 andM33= 0 the canonical form to canonical form mapping will never map t hat simple sep- arable coordinate web to an R-separable coordinate web. Therefore c≡1 and we prove that bispherical coordinates are indeed conformall y related to spheroidal coordinates, as is claimed in the literature, such as [6]. Reca ll that spheroidal coordinate webs are invariant to the discrete inversion - this is the case when A33=M33=B21=G12= 0. Bi-cyclide coordinates are mapped to bi-cyclide coordinat es given that the trans- formed coefficients satisfy the same invariant (with respect to th e three other group transformations) as the original ones do as a function of k. However, one can show that flat ring cyclide coordinates and cap-cyclide coordina tes are interchanged (by studying the range of the invariants after transformation) an d vice-versa as a re- sult of the above process. Hence one loses either flat-ring cyclide or cap-cyclide coordinates as an additional R-separable coordinate system. We prove finally that cap-cyclides are related by inversion to other cyclides alth ough not in the manner implied by [38]. The number of additional rotationally symmetric R-separable coordinates seems to agree, after consideration of canonical form to canonical form mapping, with the results of Miller et al. In p. 70 and 71 of [6] toroidal and three additional cyclide coordinates are listed as exhausting the possibilities for rota tionally symmetric R- separable webs. One should add that canonical form to canonical form mapping d oes not reduce the set of inequivalent simply separable rotational coordinat es classified previously [25], despite that additional transformation group actions c ould very well do this. Indeed, the property of simple separable webs, namely M33= 0, implies that the inversion parameter avanishes and bis either 0 or undefined - meaning the coeffi- cients are left invariant. The case of paraboloidal coordina tes, withA33=M33= 0 82 butB21/ne}ationslash= 0, cannot be mapped to any other simple separable rotational c oordinate system using the algebraic knowledge that no inversion and/or tr anslation can map B21andG12simultaneously to zero, as mentioned previously. 6.5 Classification scheme of non-canonically cen- teredR-separable coordinates in E3 Now that we have provided an existential proof of the maximal nu mber of inequiv- alent rotational coordinates, we discuss here a classification of arbitrary 6-tuples of conformal Killing tensor coefficients without a priori mapping to canonical centered form which has been dealt with in the previous sections. Given a 6-tuple of CKT coefficients, the first step in the classificati on scheme is to compute the full invariant I=I2 1 I3 2. A simple calculation reveals that I∈(−∞,0] andI∈(4,+∞) for disk cyclide coordinates. I∈[0,4) for the inequivalent twin bi-cyclide and flat-ring cyclide coordinates, and finally I= +4 for the remaining rotationally symmetric coordinates except for the pairs car dioid-paraboloidal and tangent sphere-circular cylindrical coordinates where its value is indeterminate. As it stands, the full group invariant lumps together toroidal, b ispherical-spheroidal, prolate-inverse prolate spheroidal and oblate-inverse oblat e spheroidal coordinates when its value is computed to be +4. For the case I1= 0, it is possible to distinguish between disk-cyclide and the pairs bi-cyclide/flat-ring cyclide coordinates by the obser vation that in the case for disk-cyclide coordinates I2<0 and in the other case I2>0. Note that although on its ownI2is not invariant to dilation of the Killing tensor, it is by vir tue of being a quadratic invariant to the discrete -1 switch and so the sign can not be altered by positive dilation of the Killing tensor alone! Thus one is ab le to discriminate disk-cyclide coordinates from all the rest. Another classificati on scheme is needed to discriminate between the other case which share the same value for the full group invariant. A first idea was to use the assumption of undetermined polynomial c oefficients to a fixed degree in the CKT coefficients, as in [25], to compute th e invariants to continuous inversion, translation and addition of R3⊙R3. This was motivated by studying how such invariants behave under dilation of the space and thereby see if different cases could be discriminated. The result is the vani shing of the purely linear and purely quartic ansatz, and the quadratic and cubic ansatz being a linear combination of the already found I2andI1respectively! Under the polynomial ansatz, one arrives at the interesting (but not useful) fact tha t an invariant to inversion and translation is simultaneously an invariant with r espect to dilation of the space. This approach reveals a deeper pattern. Computing also by und etermined poly- nomial coefficients the invariant with respect to infinitesimal translation and dila- tion of the space yields an invariant to infinitesimal inversion . Similarly an invariant 83 with respect to infinitesimal inversion and dilation of the space is invariant with respect to infinitesimal translation. Thus the hope of ignoring one group action when finding invariants, and using that group action to split deg enerate pairs of cases fails when the answer is again the total invariant. Thus one is left with having to consider ignoring two group act ions and finding invariants with respect to the remainder. The logical choice was to ignore trans- lation and infinitesimal inversion but consider ratios such asB21G12 A33M33,B21G12 (C22+C33)2and A33M33 (C22+C33)2and see how the transformed coefficients affect the values of these ratios when arbitrarily applying inversion and translation to known canonically centered cases. The first such ratio, denoted by I3, seemed to bear fruit in discriminating some cases. A simple calculation reveals that for paraboloidal/cardioi d coordinates I3is in- determinate and for tangent sphere/circular cylindrical co ordinatesI3=−4 (the ratio is constant and defined for non-zero inversion/translati on; thus it has a lim- iting value for the null group actions). If I3lies in the span of −4(ba2+a+b)2 (a2+ 1)(1 + 2ab+b2a2+b2)(6.5.1) for non-zero inversion parameter aand/or translation parameter b(where the span explicitly lies in the open interval ( −4,0)), the coordinates are either toroidal, oblate/inverse oblate spheroidal or prolate/inverse prolate spheroidal. If I3lies in the (disjoint from all previous) span of −4(ba2+a−b)2 (a2−1)(1 + 2ab+b2a2−b2)(6.5.2) for non-zero inversion parameter aand/or translation parameter b(where the span explicitly lies in the open intervals ( −∞,−4) and (0,∞)), the coordinates are either bispherical/spheroidal, oblate/inverse oblate spheroidal or prolate/inverse prolate spheroidal. In the case of bispherical and toroidal coordinate s,I3= 0 corresponds uniquely to the case a=b= 0. The invariance of the rankof the group action can also be utilized to discrimi- nate between entangled coordinate pairs. A closer inspection o f the transformation equations of the coefficients of the CKT reveals that C22,C33always appear in the sum (C22+C33) which is invariant to the group action of addition of R3⊙R3. This motivates one to consider ignoring that group action on a reduced number of coefficients, namely defining C≡(C22+C33) and studying only four group actions on the reduced set ( A33,B21,C,G 12,M33). A simple calculation reveals that the modified pde array of this reduced group action set becomes: V1= [−2B21,−C,6G12,2M33,0] V2= [0,2A33,6B21,−C,−2G12] 84 V3= [A33,B21,C,G 12,M33] V4= [2A33,B21,0,−G12,−2M33] (6.5.3) This is advantageous because now not all canonically centere d coordinates will obey the maximal rank condition (as they did with the full gro up action on six coefficients). In particular, the rank of the pde set for the obla te/inverse oblate spheroidal, prolate/inverse prolate spheroidal coordinates is maximal at +4. How- ever the rank for toroidal and bispherical/spheroidal coordi nates is not maximal – at +3. This value is also shared for paraboloidal/cardioid coo rdinates. Finally for tangent sphere/circular cylindrical coordinates the rank i s even further reduced at +2. Thus one is able with the invariance of rank under the grou p action, which was explicitly checked in our case, to discriminate between th e pairs inverse oblate spheroidal, inverse prolate spheroidal and toroidal, bispheri cal coordinates. Unfortunately for the pairs oblate/prolate spheroidal and bi -cyclide/flat-ring cyclide coordinates, one must use the mapping to canonical cent ered form, and then use the reduced invariant to classify the transformed coeffic ients (˜A33,˜C22,˜C33,˜M33) - it will be an either/or scenario. We have tried plotting qua n- tities such as ˜B21˜G12and˜A33˜M33to distinguish a pair of coordinates by the sign of the graph which is invariant. Unfortunately one member of e ach pair does not admit a graph of the above, as a function of inversion paramete raand translation parameterb, with a unique sign. The other member however admits a negativ e graph for all a,b. Hence for some 6-tuples, the above plots might ascertain which member of the coordinate pair the tuple is associated with by ev aluation of the listed products and observation that that product is positive. When t his criterion fails, one must then map to canonically centered form to complete the discrimination. For a complete classification without transformation to canonically centered form, recent research was completed using the invariance of th e roots of binary quartics and the equivalence of the discussed group actions on t he coefficients with the general linear transformation on binary quartics studied extensively in the lit- erature, for example in [41]. For details, see the material pla ced in Appendix F which is based on [13]. 85 The results of the classification given in this chapter of the rot ationally sym- metric webs listed by Moon and Spencer are summarized in the fol lowing table: Coordinate web Equivalent to Transformation Cap cyclide Flat-ring cyclide cont. inversion + trans. Inverse Prolate Spheroidal Prolate Spheroidal discrete inversion Inverse Oblate Spheroidal Oblate Spheroidal discrete inversion Bispherical Spheroidal cont. inversion + trans. Cardiod Paraboloidal discrete inversion Tangent sphere Circular cylindrical discrete inversion Toroidal - - Bi-cyclide - - Flat-ring cyclide - - Table 6.1: Equivalence classes of R-separable webs 6.6 Classifying the remaining symmetric R-separable coordinates in E3 Proposition 6.6.1 Aside from rotational cases just discussed, there are no add i- tional symmetric R-separable coordinates in E3. Proof: We have applied Eq.(4.1.8) on the general conformal Killing tensor us- ing, instead of the rotational Killing vector R3, the translational, dilatation and inversion Killing vector. The result after some algebra analog ous to Eq.(4.1.9) is a degree two, six coefficient subset of the general Killing tensor fo r webs admitting a translational symmetry along x. This result holds whether the discrete or contin- uous method is carried out to characterize translational symm etry. In components the resulting tensor is: K11=−A22−A33+ 1/2B21z−1/2B31y+C33(y2+z2) K22=A22−B21z−1/2B31y+C33(y2−2z2) K33=A33+ 1/2B21z+B31y+C33(z2−2y2) K12= 0 86 K13= 0 K23=A23−3/4B31z+ 3/4B21y+ 3C33zy (6.6.1) For dilatational webs, where the Lie derivative of the genera l conformal Killing tensor is set equal to a real scalar of the Killing tensor before im posing the TSN integrability conditions, we found five integer cases where th e result is non-trivial. Each case corresponds to a single degree expression for the Killin g tensor subset. The degree two case, after imposing the TSN criterion – is in exa ct agreement with the discrete operation, Eq.(4.1.8), using the dilational Kil ling vector. The other cases however, either correspond directly to simple separable w ebs or can be mapped into one by the discrete inversion, namely all degree four term s to degree zero and all degree three terms to degree one. Immediately one can con clude there are no additional conformal coordinates admitting a translationa l or dilational symmetry, irrespective of whether one characterizes the symmetries usin g the standard Lie derivative method or the discrete formula! Note this is in agre ement with [6] where the only non-rotationally symmetric R-separable coordinates are asymmetric cases not studied in this chapter. Recall that on p. 234 and 235 of [37 ] a first principles proof was given that R-separability of the Helmholtz and Laplace equations is never possible for a cylindrical coordinate system. Thus we are in agre ement with known results in the literature for null cases of translational confo rmal coordinates. Repeating the above procedure using the vector generator of t he continuous inversion, for both the discrete and Lie derivative method (set equal to zero for all components) coupled with the TSN impositions, yields a degr ee four, six coef- ficient subset of the general Killing tensor with no presence of ze ro or first order terms. It was checked that the discrete inversion maps this web t o that generated by Eq.(4.1.8) using the translational Killing vector, and vic e versa. Setting the Lie derivative, with respect to the inversional conformal Killin g vector, of the general conformal Killing tensor equal to a non-zero scalar of the Kill ing tensor yields the trivial result even before applying the TSN conditions. This i s analogous to the dilational Lie derivative set equal to an arbitrary multiple of the conformal Killing tensor for values not equal the five integer cases found. Our con clusion is that the only additional conformal coordinates are either rotation ally symmetric or asym- metric admitting no symmetries. 2 The difficult task of characterizing asymmetric webs remains to be studied. This adds more emphasis on the need to generate the class of R-separable webs by first principles from the method of Eisenhart. 87 Chapter 7 Asymmetric R-separable webs in E3 In this chapter we study the remaining case of R-separable webs in E3that admit no symmetry. This case is not considered in [38]. However, it has b een studied in [6] and [7]. We adopt a different starting point than these pap ers by studying theconformally invariant (CI) Laplace equation rather that the ordinary Laplace equation which is not conformally invariant. It will be seen t hat there is a close relation of the conformal invariance property of the equati on andR-separability which is also a conformally invariant property. Our approach which has been de- scribed in [11], [12] is based on the theory of R-separability explained in Chapter 2. 7.1 The conformally invariant Laplace equation The subject of this chapter is the study of R-separation of variables for the confor- mally invariant (CI) Laplace equation on an n-dimensional Riemannian manifold (M,g), which is: Hϕ:= ∆ϕ+Cϕ= 0, (7.1.1) where we make a very specific choice of the constant C: C=n−2 4(n−1)Rs (7.1.2) whereRsis the Ricci scalar and nis the dimension of the space. As mentioned before, the closely related problem is additive separation of variables for the HJ equation with null geodesics, which like the classical Laplace e quation can also be extended to the pseudo-Riemannian case namely gij∂iW∂jW= 0. (7.1.3) 88 The crucial property of both (7.1.1) and (7.1.3) is invarian ce under conformal trans- formation of the metric. From this it follows that if ϕis any solution of Hϕ= 0, then ˜ϕ=e2−n 2ϕis a solution of ˜H˜ϕ= 0 on any conformally related manifold. Consequently, R-separability of the CI-Laplace equation is a conformally in variant property, which is not shared by the classical Laplace equation introduced at the start of this thesis: ∆ϕ= 0, (7.1.4) which is the equation most often studied in this regard [5, 7, 15 ]. We digress here to give a proof of the conformal invariance pro perty of the Laplace-Beltrami operator. 7.2 Proof of conformal invariance Consider a pseudo-Riemannian manifold ( M,g) with corresponding Levi-Civita connection Γ. The covariant derivative ∇may be written in local coordinates {xi} as follows: ∇iAj=∂iAj+ Γj ikAk(7.2.1) for a contravariant vector field Aj, where Γj ikdenotes the Christoffel symbols of the second kind. Contracting Eq. (7.2.1) over iandjwe obtain the divergence of Aj namely: ∇iAi=∂iAi+ Γi ikAk(7.2.2) By a standard result, [22] we have equivalently: Γi ik= 2∂klog√g (7.2.3) Thus (7.2.2) takes the form ∇iAi=∂iAi+Ai∂ilog√g =∂iAi+1√gAi∂i√g ∇iAi=1√g(√g∂iAi+Ai∂i√g) ∇iAi=1√g∂i(√gAi) (7.2.4) Letφbe a function defined on M. The Laplace-Beltrami operator on Mwas defined as the divergence of the vector field gij∂jφ. In local coordinates we thus have: ∆φ=1√g∂i(√ggij∂jφ) (7.2.5) 89 Given conformal transformations of the pseudo-Riemannian me tric it follows that the determinant of gijtransforms as: ˜g=e2nσg, (7.2.6) wheren= dim(M). Thus /radicalBig ˜g=enσ√g (7.2.7) Suppose that φtransforms as ˜φ=emσφ, (7.2.8) wherem∈R. We are now in a position to compute the transformation law for t he Laplace-Beltrami operator ∆ acting on φ. ˜∆˜φ=1√g∂i(/radicalBig ˜g˜gij∂j˜φ) =e−nσ √g∂i(e(n−2)σ√ggij∂j(emσφ)) =e−nσ √g∂i(e(n−2)σ√ggij(emσ(∂jφ+mφ∂jσ))) =e−nσ √g∂i(e(m+n−2)σ√g(gij∂jφ+mφgij∂jσ))) =e−nσ √g[(m+n−2)e(m+n−2)σ∂iσ√g(gij∂jφ+mφgij∂jσ) +e(m+n−2)σ∂i(√ggij∂jφ +m√gφgij∂jσ)] ˜∆˜φ=e(m−2)σ √g[(m+n−2)√ggij∂iσ∂jφ+m(m+n−2)√gφgij∂iσ∂jσ+∂i(√ggij∂jφ) +m√ggij∂iφ∂jσ+mφ∂i(√ggij∂jσ)] ˜∆˜φ=e(m−2)σ[(m+n−2)gij∂iσ∂jφ+m(m+n−2)φgij∂iσ∂jσ+1√g∂i(√ggij∂jφ) +mgij∂iφ∂jσ+mφ1√g∂i(√ggij∂jσ)] ˜∆˜φ=e(m−2)σ[∆φ+mφ(∆σ+ (m+n−2)gij∂iσ∂jσ) + (m+n−2)gij∂iσ∂jφ+mgij∂iφ∂jσ] (7.2.9) The next step results from the fact that: gij∂iφ∂jσ=gij∂iσ∂jφ, which follows from the symmetry of gijin the indices iandj. The final formula is: ˜∆˜φ=e(m−2)σ[∆φ+ (2m+n−2)gij∂iσ∂jφ+mφ(∆σ+ (m+n−2)gij∂iσ∂jσ)] 90 The second last term on the RHS may be removed by choosing m=1 2(2−n) (7.2.10) With this choice Eq. (7.2.9) reduces to ˜∆˜φ=e−1 2(2+n)σ[∆φ+1 2(2−n)φ(∆σ+1 2(n−2)gij∂iσ∂jσ)] (7.2.11) Now we consider the quantity ˜R˜φ, recalling that the Ricci scalar Rtransforms under conformal transformation as [22]: ˜R=e−2σ[R+ 2(n−1)∆σ+ (n−1)(n−2)gijσiσj] and remembering that φwas defined to transform as: ˜φ=emσφ. This yields: ˜R˜φ=e(m−2)σφ[R+ 2(n−1)∆σ+ (n−1)(n−2)gijσiσj] (7.2.12) Adding (7.2.11) and ktimes (7.2.12) we obtain the new operator ˜∆˜φ+k˜R˜φ=e(m−2)σ[∆φ+1 2(2−n)φ(∆σ+1 2(n−2)gijσiσj) +kφ(R+ 2(n−1)∆σ+ (n−1)(n−2)gijσiσj)] =e(m−2)σ[∆φ+kRφ+φ((1−n 2+ 2k(n−1))∆σ + (−1 4(n−2)2+k(n−1)(n−2))gijσiσj)] (7.2.13) To remove the term containing ∆ σwe must choose k=1 4(n−2) (n−1)(7.2.14) Fortuitously this choice of kalso removes the term containing gijσiσj! We conclude that the operator (7.2.13) with kgiven by Eq. (7.2.14) has the transformation law: ˜∆˜φ+1 4(n−2) (n−1)˜φ˜R=e−1 2(n+2)σ(∆φ+1 4(n−2) (n−1)Rφ) (7.2.15) whereφand˜φare related by: ˜φ=emσφ=e1 2(2−n)σφ (7.2.16) The operator is thus invariant under a conformal transformat ion, as desired. We now state some special cases for lower dimensions: 91 n= 3 ∆ φ+1 8Rφ , ˜φ=e−σ 2φ (7.2.17) n= 4 ∆ φ+1 6Rφ , ˜φ=e−σφ (7.2.18) n= 5 ∆ φ+3 16Rφ , ˜φ=e−3 2σφ (7.2.19) This concludes the proof of conformal invariance of the oper ator (7.1.1) with Cgiven by (7.1.2). For the rest of this chapter the advantages of study ingR-separability for the CI-Laplace equation continuing the work begun in [32 ] are illustrated. For the flat case (in which the CI-Laplacian reduces to the classical one) we recover the results given by Bˆ ocher [5] and Boyer et al. [7]. Furtherm ore, these results are applied to provide CI-Laplace R-separable coordinates on other conformally flat manifolds. 7.3 The CI-Laplace equation and R-separation The study of R-separation of the CI-Laplace equation, instead of the classica l equa- tion, is more general [32]. Indeed, the existence of a complet eR-separated solution of the CI-Laplace equation is a conformally invariant prope rty that holds on the whole class of conformally related metrics. This follows dire ctly from the form of the solution ansatz and the conformal invariance of Eq. (7.1.1 ). The techniques giving differential conditions for the R-separation of a single pde [15] were outlined in Chapter 2 and are repeated here: Theorem 7.3.1 Equation (7.1.1)admitsR-separation in the coordinates (qi)if and only if 1. the coordinates are orthogonal: gij= 0,i/ne}ationslash=j; 2. the coordinates are conformally separable; 3. the contravariant components (gii)satisfy the differential condition Sij(ghh) ghh=Sij(gkk) gkk,(∀h,k,∀i/ne}ationslash=j, i,j n.s.) (7.3.1) whereSijare the second order St¨ ackel operators, 92 4. the function Ris (up to separated factors) a solution of ∂ilnR=1 2Γi, (7.3.2) where Γi=ghkΓhki; Remark 7.3.2 Recall from Chapter 2 that orthogonal coordinates satisfying con- dition (7.3.1) are called conformally separable (see [3]), while orthogonal coordinates satisfyingSij(ghh) = 0 are said to be simply separable . The additive separation of variables for the null geodesic HJ equation in orthogonal coor dinates, gii(∂iW)2= 0, (7.3.3) and for the geodesic HJ equation 1 2gii(∂iW)2=E,(E∈R), (7.3.4) occurs if and only if the coordinates are conformally separab le and simply separable, respectively. This fact shows an important link between Eq. (7 .1.1) and Eq. (7.1.3). Also in the Riemannian case, even if the null geodesics are trivia l, the study of conformal separation can be applied effectively to the CI-Lap lace equation. Indeed, as for the Laplace equation ∆ ψ= 0, we have that Corollary 7.3.3 A necessary condition for R-separation of the CI-Laplace equa- tion(7.1.1)in a given coordinate system is that the null geodesic equatio n(7.1.3) is additively separable in the same coordinates. Remark 7.3.4 Two conditions equivalent to (7.3.1) are •gis conformal to a metric which is separable for the geodesic HJ eq uation (7.3.4) in the same coordinates; •there exists a St¨ ackel matrix Ssuch that [35] gii gjj=Min Mjn, (7.3.5) whereMinis the minor of Sobtained by eliminating the i-th row and the n-th column. We remark that the elements of the last column of th e St¨ ackel matrix are not involved in (7.3.5). 93 7.4 The three-dimensional case Definition 7.4.1 A coordinate qiis said to be conformally ignorable if it ap- pears in the conformal factor of the metric only, that is if ∂i(ghh/gkk) = 0, for allh,k. We call a coordinate system general if it does not contain any confor- mally ignorable coordinates. Up to a coordinate transformation of the form ˜ qi(qi), a coordinate qiis confor- mally ignorable if and only if ∂iis a conformal Killing vector, that is an infinites- imal conformal symmetry. In this chapter only general coordi nate systems are considered, leaving as a further research project the analysis of the cases involving conformal symmetries. The form of the general conformally separable coordinates in a three dimensional manifold is given in the following proposition (see [7]) Proposition 7.4.2 In general conformally separable coordinates (qi), the form of the (contravariant) metric on a three dimensional manifold is given by gii=Qhi(qi)(qi+2−qi+1), i= 1, . . .,3 (mod3), (7.4.1) whereQis the conformal factor, and hithree arbitrary functions of a single variable. Proof: Following [7], without loss of generality, we may choose Sto be a 3 ×3 St¨ ackel matrix with third column set equal to unity. S= φ1ψ11 φ2ψ21 φ3ψ31  (7.4.2) Then, we have g11=Q(ψ3φ2−ψ2φ3), g22=Q(ψ1φ3−ψ3φ1), g33=Q(ψ2φ1−ψ1φ2).(7.4.3) In the general case, we can assume that none of the ψiandφiis identically null. Thus g11=Qφ2φ3/parenleftBiggψ3 φ3−ψ2 φ2/parenrightBigg , g22=Qφ1φ3/parenleftBiggψ1 φ1−ψ3 φ3/parenrightBigg , g33=Qφ1φ2/parenleftBiggψ2 φ2−ψ1 φ1/parenrightBigg . (7.4.4) By transforming each coordinate ˜ qi= ˜qi(qi) and the conformal factor such that ˜gii→φigii, ˜Q→Qφ1φ2φ3, 94 we obtain ˜gii=˜Q(Fi+2−Fi+1) withFi(qi) =ψi φi. If none of the Fiis a constant, then we can use them as coordinates; thus we obtain g11=˜Qh1(q1)(q3−q2), g22=˜Qh2(q2)(q1−q3), g33=˜Qh3(q3)(q2−q1), where thehiare the reciprocal of φi, and the tilde symbol can be dropped for the conformal factor. 2 Remark 7.4.3 If one of the elements of the St¨ ackel matrix is zero or one of t he functionsFiis a constant then, up to a coordinate transformation ˜ qi(qi), one of the coordinates is conformally ignorable. By Proposition 7.4.2 and Theorem 7.3.1 we obtain Theorem 7.4.4 The form of the metric in general R-separable coordinates for the CI-Laplace equation is gii=QP(qi)·(qi+2−qi+1), i= 1, . . .,3 (mod3), (7.4.5) wherePis an arbitrary fifth-degree polynomial. Proof: Computing the modified potential χfor the general conformal separable metric (7.4.1) and imposing the compatibility condition Sij(χ)g11=Sij(g11)χ= 0 Sij(χ)g22=Sij(g22)χ= 0 Sij(χ)g33=Sij(g33)χ= 0 i/ne}ationslash=j, (7.4.6) where χ≡ghh(2∂hΓh−Γ2 h+1 2Rhh) (7.4.7) we obtain three additional independent differential condit ions (out of the nine equa- tions) on the functions hithat form a linear second order ODE system in three unknowns. Performing the calculation in Maple, where for con venience we denote q1=u, q2=v, q3=wandh1=U(u), h2=V(v), h3=W(w) - we write the covariant metric components as the Tensor Package requires t he metric tensor to be written with both indices down. The conformal freedom in t he metric ansatz allows us now to drop the conformal factor Qand write: g11=1 U(u)(v−w), g22=1 V(v)(w−u), g33=1 W(w)(u−v)(7.4.8) 95 The reduced Christoffel symbols that follow are, where we write ( U, V, W ) instead of ( U(u), V(v), W(w)): Γ1=−Uu 2U+(v−2u+w) 2(v−u)(u−w) Γ2=−Vv 2V+(2v−u−w) 2(v−u)(v−w) Γ3=−Ww 2W+(2w−u−v) 2(u−w)(v−w)(7.4.9) The diagonal Ricci tensor components Rhhand Ricci scalar Rneeded for a future calculation are easily attained using the Tensor Package and ar e omitted here. Finally the differential equations that follow are eq1=−(v−w)Uu,u 4(v−u)+(u−w)Vv,v 4(v−u)+(v−w)(v−4u+ 3w)Uu 2(v−u)2(u−w) −(u−w)(4v−3w−u)Vv 2(v−u)2(v−w)+(v−u)2W 2(u−w)2(v−w)2 −(v−w)(v2−5uv+ 3vw+ 10u2−15wu+ 6w2)U 2(v−u)3(u−w)2 +(u−w)(u2−5uv+ 3wu+ 10v2−15vw+ 6w2)V 2(v−u)3(v−w)2 (7.4.10) eq2=(v−w)Uu,u 4(u−w)−(v−u)Ww,w 4(u−w)−(v−w)(3v+w−4u)Uu 2(v−u)(u−w)2 −(v−u)(3v−4w+u)Ww 2(u−w)2(v−w)−(u−w)2V 2(v−u)2(v−w)2 +(v−w)(6v2+ 3vw−15uv+w2−5wu+ 10u2)U 2(v−u)2(u−w)3 −(v−u)(6v2−15vw+ 3uv+ 10w2−5wu+u2)W 2(u−w)3(v−w)2 (7.4.11) eq3=−(u−w)Vv,v 4(v−w)−(v−u)Ww,w 4(v−w)+(u−w)(4v−3u−w)Vv 2(v−u)(v−w)2 −(v−u)(v−4w+ 3u)Ww 2(v−w)2(u−w)+(v−w)2U 2(v−u)2(u−w)2 −(u−w)(6u2+ 3uw−15uv+w2−5vw+ 10v2)V 2(v−u)2(v−w)3 −(v−u)(6u2−15uw+ 3uv+ 10w2−5vw+v2)W 2(v−w)3(u−w)2(7.4.12) Maple successfully integrated the above yielding U(u) =c1+c2u+c3u2+c4u3+c5u4+c6u5 96 V(v) =c1+c2v+c3v2+c4v3+c5v4+c6v5 W(w) =c1+c2w+c3w2+c4w3+c5w4+c6w5(7.4.13) Going back to the older notation, we see that the hi=P(qi), wherePis an arbi- trary fifth-degree polynomial. 2 Note that the compatibility condition has an intriguing geom etrical interpretation. Theorem 7.4.5 On a three dimensional manifold, R-separation of the CI-Laplace equation occurs in general conformal separable coordinate s if and only if the metric is conformally flat. Proof: The conformal flatness conditions for a 3-dimensional Riemann ian man- ifold are (see for example [22]) Rijk=Rij;k−Rik;j+1 4(gikRs;j−gijRs;k) = 0, (7.4.14) where ; denotes the covariant derivative and Rijthe covariant Ricci tensor. By imposing these conditions on the general conformally separabl e metric (7.4.1), we arrive at nine linear second order ODEs in the hi. Three of them are trivially zero, while another three are equivalent to the remaining three wh ich are equal to those allowingR-separation. Thus, the conformal flatness condition is equival ent to the compatibility condition for R-separation. 2 Remark 7.4.6 If one or more conformally ignorable coordinates appears, th en being conformally flat is a sufficient but no longer a necessary con dition forR- separation. Hence, in particular, equations (7.1.1) and (7.1 .3) separate in the same orthogonal coordinates for all conformally flat 3-manifold s. We can apply these results to the study of R-separation for the classical Laplace equation. Indeed if a three dimensional manifold satisfies Rs= 0, thenR-separation of the Laplace equation ∆ ψ= 0 occurs in general coordinates if and only if the manifold is conformally flat. This is because since Rs= 0, the CI-Laplace equation and Laplace equation coincide. Furthermore on a conformall y flat three dimen- sional manifold, R-separation of the Laplace equation ∆ ψ= 0 occurs in general conformally separable coordinates if and only if the Ricci sca larRssatisfies the compatibility condition Sij(ghh)Rs=Sij(Rs)ghh. This follows since the manifold is conformally flat, the CI-La place equation admits R-separation of variables in general conformally separable co ordinates. 97 7.5 Applications and examples A fundamental example is the flat case, where the Laplace equat ion and the CI- Laplace equation become the same. This has been studied by sever al authors (see [5, 39, 35, 7]). Example 7.5.1 In order to determine the expression of general R-separable co- ordinates on E3we need to compute the conformal factor Qsuch that the metric (7.4.5) is flat and the coordinate transformations from a Cart esian coordinate sys- tem. Let us denote the R-separable coordinates ( q1,q2,q3) by (u,v,w ) and by e1< e2< e3< e4< e5the five zeros of the polynomial P(we restrict ourselves to the special case where all zeros eiare real and distinct, as we use tools from classical Riemannian geometry and not those pertaining to comp lex manifolds). We now discuss the special case of gii=QP(qi)·(qi+2−qi+1), i= 1, . . .,3 (mod3) whenP(qi) can be factored into five real and distinct factors which we de note as in the literature by ei. An explicit formula for the (covariant) conformal factor ˜Q is given on Eq.(4.31) of [7], as well as coordinate transforma tions from Cartesian toR-separable ones. The conformal factor ˜Qis written 1 /λ2where: λ=/radicaltp/radicalvertex/radicalvertex/radicalbt(q1−e1)·(q2−e1)·(q3−e1) (e1−e2)·(e1−e3)·(e1−e4)·(e1−e5) +/radicaltp/radicalvertex/radicalvertex/radicalbt−(q1−e5)·(q2−e5)·(q3−e5) (e5−e1)·(e5−e2)·(e5−e3)·(e5−e4)(7.5.1) This is an adaptation from the pentaspherical coordinate rep resentation given by the formula for ds2in Euclidean space on p. 89 of [5], where the (covariant) metric coefficients gii=˜Q·P−1(qi)·(qi+1−qi)·(qi+2−qi+1) appear explicitly - with polynomial P(qi) described on p. 87. To reconcile with our contravariant formalism for Q, we note the relation Q=˜Q−1/(qi+1−qi). By using pentaspherical coordinates we can derive the followi ng relations linking Cartesian coordinates to the R-separable ones given in [7]: λ·x=/radicaltp/radicalvertex/radicalvertex/radicalbt(q1−e2)·(q2−e2)·(q3−e2) (e2−e1)·(e2−e3)·(e2−e4)·(e2−e5) λ·y=/radicaltp/radicalvertex/radicalvertex/radicalbt(q1−e3)·(q2−e3)·(q3−e3) (e3−e1)·(e3−e2)·(e3−e4)·(e3−e5) λ·z=/radicaltp/radicalvertex/radicalvertex/radicalbt(q1−e4)·(q2−e4)·(q3−e4) (e4−e1)·(e4−e2)·(e4−e3)·(e4−e5), (7.5.2) 98 where the following (not unique) relations on the (assumed rea l and distinct) set of roots is assumed: e1<e2, e2<e3, e3<e4, e4<e5 e1<u<e 2, e2<v<e 3, e3<w<e 4 (7.5.3) As expected the metric giiresulting from the coordinate transforms of Eq.(7.5.2) is orthogonal and flat, and conformal to the metric ˜ gii=P(qi)·(qi+2−qi+1) but with conformal factor ˜Q= 1/(4λ2) instead of the given 1 /(λ2) in [5] on p. 89. This is clearly a trivial error as a spatial dilatation of two units wo uld ensure consistency of the formulae and of course not affect the flatness condition. Therefore there is agreement, assuming the polynomial P(qi) has been factored into five real and distinct factors, between this derived form o f the metric and results given in the literature. The proof of the above metric coefficients resulting from the co ordinate trans- formation equations given by Kalnins and Miller could not be verified by ‘brute force’ in Maple 9. This is due to the irrational factors appea ring in the algebra which Maple handles poorly. We digress here to provide a simpli fication of the algebra involved, should the reader wish to reproduce the abov e results. Let us denote by ( qi) = (u,v,w ), (i= 1,...,3) and (eh) = (e1,...e5), (h= 1,...,5). We start from the assumption e1<u<e 2<v<e 3<w<e 4<e5and Lh=(u−eh)(v−eh)(w−eh) /producttext k/negationslash=h(eh−ek)=/producttext3 i=1(qi−eh) /producttext k/negationslash=h(eh−ek) We haveL1>0,L2>0,L3>0,L4>0 butL5<0. Then we let ( xi) = (x,y,z ) be Cartesian coordinates, so that the transformation rules by Ka lnins and Miller are in compact form: xi=√Li+1√L1+√−L5, where theλfactor isλ=√L1+√−L5. Let us compute the derivatives of Lhand√Lh: ∂ ∂qiLh=/producttext j/negationslash=i(qj−eh) /producttext k/negationslash=h(eh−ek)=Lh (qi−eh) ∂ ∂qi/radicalBig Lh=1 2√Lh∂ ∂qiLh=1 2√LhLh (qi−eh)=√Lh 2(qi−eh), h = 1,...,4 and ∂ ∂qi/radicalBig −L5=−1 2√−L5∂ ∂qiL5=−1 2√−L5L5 (qi−e5)=√−L5 2(qi−e5) Moreover ∂ ∂qixj=∂i/radicalBig Lj+1(√L1+√−L5)−/radicalBig Lj+1(∂i√L1+∂i√−L5) (√L1+√−L5)2= 99 =√ Lj+1 (qi−ej+1)(√L1+√−L5)−/radicalBig Lj+1(√L1 (qi−e1)+√−L5 (qi−e5) 2(√L1+√−L5)2 =/radicalBig Lj+1 2(√L1+√−L5)2/parenleftBigg/radicalBig L1(ej+1−e1) (qi−ej+1)(qi−e1)+/radicalBig −L5(ej+1−e5) (qi−ej+1)(qi−e5)/parenrightBigg Hence /parenleftBigg∂ ∂qixj/parenrightBigg2 =Lj+1(qi−ej+1)−2 4(√L1+√−L5)4· /parenleftBigg L1(ej+1−e1)2 (qi−e1)2−L5(ej+1−e5)2 (qi−e5)2+ 2/radicalBig −L1L5(ej+1−e1)(ej+1−e5) (qi−e1)(qi−e5)/parenrightBigg (7.5.4) Note thatgii(√L1+√−L5)4contains only the irrational term√−L1L5and it is a first order polynomial in it (with coefficients that are ration al functions of qi andeh). Thus Maple is able to simplify it (always as a first order polyn omial in√−L1L5). Finally by comparing the expansion of (/radicalBig L1+/radicalBig −L5)2=L1−L5+ 2/radicalBig −L1L5 as a polynomial in√−L1L5, we recover the answer. Remark 7.5.2 The conformal metric ˜gii=(qi−qi+1)(qi−qi+2) P(qi) is the general three-dimensional conformally flat metric all owing multiplicative sep- aration of the Helmholtz equation computed by Eisenhart [21]. The formulae for the flat case can be adapted to a general confor mally flat manifold (M,gM). SinceMis conformally flat, there exists a coordinate system (Xi) such that gM=Q−1 ME/summationdisplay idXi⊗dXi, whereQMEis the conformal factor transforming gMinto the flat Euclidean metric gE. Then, if we formally replace ( x1,x2,x3) by (X1,X2,X3) in the transformations (7.5.2) we obtain the coordinate transformations from ( Xi) to theR-separable coordinates ( qi). Indeed, by inserting these relations in the metric gM, we have gM=Q−1 ME/summationdisplay idXi⊙dXi=Q−1 MEQ−1 E/summationdisplay i[P(qi)·(qi+2−qi+1)dqi⊙dqi]. Hence,QM=QMEQEis the conformal factor that transforms the general confor- mally flat metric (7.4.5) into a metric on the specific conforma lly flat manifold M. Then, in order to compute the conformal factor and the coordi nate transformation, we only need to know the coordinates XionMcorresponding to the Cartesian coordinates on E3. In the following example we develop explicitly the case of S3 100 Example 7.5.3 Let (X1,X2,X3) be stereographic coordinates on S3, considered as a sub manifold of E4. They are related to the Cartesian coordinates ( x1,...,x4) ofE4by the following equations xa=2r2Xa r2+/summationtext3 i=1(Xi)2, a = 1,...,3 x4=r−2r3 r2+/summationtext3 i=1(Xi)2, whereris the radius of the sphere. The components of the metric of S3in the coordinates ( Xi) are (see also [22]) gii=4r4 (r2+/summationtext3 i=1(Xi)2)2. Hence, the function QSE= (r2+/summationtext3 i=1(Xi)2)2/4r4is the conformal factor relating S3toE3. Then, QS=λ2[(r2+/summationtext3 i=1(Xi(q1,q2,q3))2)2] r4/producttext h(qh−qh+1) is the conformal factor which makes (7.4.5) the metric of S3. The coordinates (q1,q2,q3),related to the stereographic coordinates ( X1,X2,X3) by λ·X1=/radicalbigg (q1−e2)·(q2−e2)·(q3−e2) (e2−e1)·(e2−e3)·(e2−e4)·(e2−e5), λ·X2=/radicalbigg (q1−e3)·(q2−e3)·(q3−e3) (e3−e1)·(e3−e2)·(e3−e4)·(e3−e5), λ·X3=/radicalbigg (q1−e4)·(q2−e4)·(q3−e4) (e4−e1)·(e4−e2)·(e4−e3)·(e4−e5), withλgiven by Eq. (7.5.1), are coordinates on S3. 101 Chapter 8 Conclusion We have shown in this thesis that the known R-separable coordinate webs with symmetry form an exhaustive set of additional coordinates admi ttingR-separation of variables for the Laplace equation in Euclidean space. The conformal Killing tensors derived for each case expressed in canonical Cartesian co ordinates can now be used by future researchers who wish to include a potential in b oundary value problems that involve the use of Killing tensors written in can onical Cartesian coor- dinates. Using geometrical methods very different from the lite rature we have also independently derived the form of the asymmetric metric tenso r forR-separable co- ordinates of the conformally invariant Laplace equation in Euclidean space admit- ting no symmetry. The associated Killing tensors for them have ye t to be computed. The canonical rotational R-separable webs known thus far form an exhaustive self contained set based on the study of the square of Eq.(6.2.1). The se emingly infinite room for more inequivalent coordinates was ‘filled up’ by the infinite ranges of the invariants resulting from variation of the coordinate param eterkappearing in the definitions for Jacobi-elliptic coordinate systems. Without a priori knowledge of these specific examples it would be hard to partition the invari ant defined over R into a finite number of partitions of Rrepresenting disjoint equivalence classes. This provides a clear motive to solve for the R-separable coordinates in E3 directly using the method of Eisenhart [20] along the lines of h is classic 1935 paper on conformal separation in Euclidean space. Clearly more work is needed in this area. We have been able to discriminate between coordinates k nown a priori, as well as put others into same equivalence classes or with simple sep arable cases previously studied in [25]. All this was achieved without systema tically solving for all cases of the Eisenhart conditions in conformal Euclide an space which could comprise a future research project of searching for the remaini ng two asymmetric R-separable webs in Euclidean space. The last chapter of this thesis gives a derivation of the genera l asymmetric met- rics inE3, and associated coordinate transforms to Cartesian coordinate s. However, they have not been classified under the full conformal group lik e the known sym- 102 metric and conformally symmetric webs have been. Such an inva riant classification using the approach employed in Chapter 6 would be a difficult task ; all the invari- ants of the general characteristic conformal Killing tensor c omponents under the entire conformal group would have to be computed. The TSN con ditions would have to be solved on the entire thirty five parameter CKT, not just on n ine-dimensional subsets that followed Lie differentiations. A direction for fur ther research lies in the cases of metrics with one or more conformal symmetries; these can be found in principle by using the techniques discussed prior. The main ob stacle at present in this calculation are the coupled non-linear pdes resultin g from solving for the conformal factor representing transformation to flat space. Wo rk has been done in this direction in [7] but using the formalism of complex Rieman nian space which is beyond the scope of this thesis. 103 Appendix A Proof of Levi-Civita’s criterion for separability For the proof in one direction, we assume sum separability of the so lutionWto the HJ equation. Assume a conservative system, namely that the Hamilt onian is time independent and equal to a constant which is the system’s total energy. Equivalently we assume the Hamiltonian does not explicitly dep end on time: H= H(qi,pi). Hence dE dqi=dH dqi= 0⇒∂H ∂qj∂qj ∂qi+∂H ∂pj∂pj ∂qi+∂H ∂t∂t ∂qi= 0 (A.0.1) As the coordinate system is linearly independent, it is clear th at∂qj ∂qi=δj,iand the sum∂H ∂qj∂qj ∂qireduces to one term only:∂H ∂qi. In general,∂t ∂qi/ne}ationslash= 0 however we have assumed∂H ∂t= 0. Finally∂pj ∂qi= 0 fori/ne}ationslash=jby our assumption of separability. This is so because from HJ theory pj=∂W ∂qjand since∂W ∂qjis a function of qjonly, due to our starting assumption that W≡/summationtextn i=1Wi(qi,c1,c2,...,cn),∂2W ∂qi∂qj= 0,i/ne}ationslash=j. Thus we obtain ∂H ∂qi+∂H ∂pi∂pi ∂qi= 0 ⇒∂pi ∂qi=−∂H ∂qi ∂H ∂pi=∂2W ∂(qi)2,∂ ∂qj(∂pi ∂qi) = 0. (A.0.2) It is clear that if piis a function of qionly, then∂pi ∂qiis a function of qionly, hence the second statement in the above formula for i/ne}ationslash=j. Let us denote S≡∂pi ∂qi. We have then dS dqj=∂S ∂ql∂ql ∂qj+∂S ∂pl∂pl ∂qj+∂S ∂t∂t ∂qj ⇒dS dqj=∂S ∂qj+∂S ∂pj∂pj ∂qj= 0, (A.0.3) 104 where we have∂S ∂t= 0 sinceWhas no explicit dependence on time t. Performing the calculations we obtain: ∂S ∂qj=−∂H ∂pi∂2H ∂qj∂qi+∂H ∂qi∂2H ∂qj∂pi (∂H ∂pi)2 ∂S ∂pj=−∂H ∂pi∂2H ∂pj∂qi+∂H ∂qi∂2H ∂pj∂pi (∂H ∂pi)2 ∂pj ∂qj=−∂H ∂qj ∂H ∂pj(A.0.4) The last line follows from Hamilton’s canonical equations. Cle arly in the identity ∂S ∂qj= 0 one can multiply away the (∂H ∂pi)2term, and furthermore multiply both sides of the equation by −∂H ∂pjobtaining the classical Levi-Civita criterion, as required t o show. 2 Next we must show the sufficiency of the Levi-Civita conditions. Let us define Ri=−∂H ∂qi ∂H ∂pi, or in more common notation: Ri=−∂iH ∂piHasH=H(ph,qh). We recognizeRias a general function of phandqh. If we assume the Levi Civita conditions hold true, we obtain ∂jRi+Rj∂ ∂pjRi= 0,i/ne}ationslash=j. We seek a possible existence of a continuous, well defined solution to the followin g ansatz of a first order system: ∂iPh=δi,hRh⇒∂iPh= 0,i/ne}ationslash=jand∂iPh=Rh,i=h. Consider the more generic first order system of the form ∂ ∂xiyh(x1,...,xn) =Fh i(x1,...,xn,y1,...,yn) (A.0.5) A continuous solution yhmust satisfy d dxi(∂yh ∂xj) =d dxj(∂yh ∂xi) (A.0.6) note that total derivatives are needed because Fh iis defined to depend also on y1,...,yn- this will be useful since Riactually depends on both sets qhandph, not on one set alone! But since complete integrability conditions refers to the demand that∂2yh ∂xi∂xj=∂2yh ∂xj∂xi, we require that d dxi(∂yh ∂xj) =∂ ∂xi(∂yh ∂xj) (A.0.7) namely that x1,...,xnform an independent set. We thus assume that no xican be functionally dependent on xj,j/ne}ationslash=i. Proceeding from Eq. (A.0.6) we have d dxi(Fh j(x1,...,xn,y1,...,yn)) =d dxj(Fh i(x1,...,xn,y1,...,yn)) (A.0.8) 105 Expanding the total derivatives, we obtain ∂ ∂xiFh j+/summationdisplay a∂Fh j ∂ya∂ya ∂xi=∂ ∂xjFh i+/summationdisplay a∂Fh i ∂ya∂ya ∂xj(A.0.9) or in more simple Einstein summation notation (where a repeated index above and below implies summation unless stated otherwise) we derive, as a test for complete integrability: ∂ ∂xiFh j+∂Fh j ∂yaFa i=∂Fh i ∂xj+∂Fh i ∂yaFa j (A.0.10) We want the L.H.S = R.H.S for Fh j=δj,hRhandFh i=δi,hRh, andya=paon account that we seek a solution to ∂iPh=δi,hRh. AlsoFa i=δi,aRaandFa j=δj,aRa. Expanding, we obtain: ∂iδj,hRh+∂ ∂pa(δj,hRh)δi,aRa=∂jδi,hRh+∂ ∂pa(δi,hRh)δj,aRa (A.0.11) The sums collapse down to one term each: ∂iδj,hRh+Ri∂ ∂pi(δj,hRh) =∂jδi,hRh+Rj∂ ∂pj(δi,hRh) (A.0.12) There are distinct cases of indices to consider on both sides of th e equation. For the casei/ne}ationslash=h, clearly the δi,hterm in the R.H.S are 0, hence ∂j(0)+Rj∂ ∂pj(0) = 0. For the casei=h,j/ne}ationslash=iwe have, on the R.H.S, ∂jRi+Rj∂ ∂pjRi= 0 by the Levi-Civita assumption. For the case i=h=jthis is the only chance of R.H.S not vanishing; it takes on the form ∂iRi+Ri∂ ∂piRiwhich is not necessarily zero. For the L.H.S, case j/ne}ationslash=hyields 0 as clearly the δj,hterms all vanish. Hence ∂i(0)+Ri∂ ∂pi(0) = 0. For case j=h,i/ne}ationslash=jwe have, on the L.H.S, ∂iRj+Ri∂ ∂piRj= 0 also by the Levi-Civita assumption. Case j=h,i=jis the only chance for the L.H.S not to vanish; it takes on the form ∂iRi+Ri∂ ∂piRiwhich is not necessarily zero. Thus for all possible nfunctionsyh,h= (1,2,...,n ) and all possible partial derivatives denoted by∂ ∂xiand∂ ∂xj, the complete integrability conditions are satis- fied. Hence ∃nfunctionsQi(qi) such that Pi=Qi(qi) are solutions of ∂iPh=δi,hRh. As∂jPi= 0,j/ne}ationslash=i, it is clear that each Piis a function of qialone, hence we write Pi=Qi(qi). The next step is to verify that H(q1,...,qn,Q1(q1),...,Qn(qn)) is equal to a constant, namely the total energy, which must be a constant for a conservative system. Then we needdH dqj= 0 for every j∈(1,2,...,n ). dH dqj=∂H ∂qj+∂H ∂Pl∂Ql(ql) ∂qj=∂H ∂qj+∂H ∂Pj∂Qj(qj) ∂qj =∂H ∂qj+∂H ∂PjRj=∂H ∂qj+∂H ∂Pj(−∂H ∂qj ∂H ∂pj) = 0. (A.0.13) 106 As the only independent variables of Hinvolve (q1,q2,...,qn), and since we prove thatdH dqj= 0∀j, we can clearly write H(q1,...,qn,Q1(q1),...,Qn(qn)) =E (A.0.14) But we know that H(q1,...,qn,∂W ∂q1,...,∂W ∂qn) =E (A.0.15) Taking ∂W ∂q1=Q1(q1) =P1 ∂W ∂q2=Q2(q2) =P2 ... ∂W ∂qn=Qn(qn) =Pn (A.0.16) we have constructed a separated solution to the HJ equation of the form: W=/integraldisplay Q1(q1)dq1+/integraldisplay Q2(q2)dq2+...+/integraldisplay Qn(qn)dqn−Et (A.0.17) satisfying∂W ∂qi=Pi, wherePiis a function of qionly. The solution Wis unique up to an arbitrary constant, which is fixed by the system’s total energy. Thus, specifying E, we found a unique solution to the HJ equation that is separable, assuming the L-C criterion holds as well as conservation of energy and independence of the variables ( q1,q2,...,qn). This completes the proof of sufficiency of the Levi-Civita conditions. 2 107 Appendix B Proof of the connection between St¨ ackel matrices and Killing tensors in Eisenhart’s formalism The proof I present here starts with the fact, proven in Eisenhar t’s 1934 paper, that the matrix elements ϕi1defined in Eq. (2.1) are known to be functions of xi at most. By definition in [20], ρα i≡ψiα ψi1is independent of xi. The quantity denoted byϕiαis the co-factor (determinant of the minor) of ϕij. For two dimensions let the arbitrary matrix ϕijbe denoted by:  f1(x1)a f2(x2)b  (B.0.1) At this stage a and b are any arbitrary functions over all varia blesx1and x2. However, from Eisenhart’s definitions, ρ2 1is independent of x1; thus in two dimensions ρ2 1is a function of x2at most. Now ρ2 1=ϕ12 ϕ11=−f2(x2) b⇒b=−f2(x2) ρ2 1; the last equality being a function of x2only. Similarlyρ2 2=ϕ22 ϕ21=f1(x1) −a⇒a=−f1(x1) ρ2 2; the last equality being a function ofxionly. This easy analysis completes the proof for two dimensions that then2 ϕijfunctions defined by ( Hi)2=ϕ ϕi1,ρα i=ϕiα ϕi1form a St¨ ackel matrix whenever the total determinant ϕ/ne}ationslash= 0. Recall that in a St¨ ackel matrix the functions inside the ithrow must be functions of xionly. In three dimensions let the general ‘St¨ ackel’ matrix be deno ted by:  f1(x1)a b f2(x2)c d f3(x3)e f (B.0.2) 108 Knowing that the expressions ρ2 1andρ3 1are independent of x1, we arrive in particular atd·f3(x3)−f2(x2)·f c·f−d·enot depending on x1. Hence: (c·f−d·e)[f3(x3)∂d ∂x1−f2(x2)·∂f ∂x1] −[d·f3(x3)−f·f2(x2)][∂c ∂x1·f+c·∂f ∂x1−∂d ∂x1·e−d·∂e ∂x1] = 0 (B.0.3) Sinceg11is finite and non-zero, clearly ( c·f−d·e)/ne}ationslash= 0 and one can divide by the quantity, obtaining: f3(x3)∂d ∂x1−f2(x2)∂f ∂x1−f3(x3)d((cf−de)′ (cf−de)) +ff2(x2)((cf−de)′ (cf−de)) = 0 (B.0.4) This can be re-arranged to yield the more familiar looking fo rm: f3(x3)[∂d ∂x1−d((cf−de)′ (cf−de))]−f2(x2)[∂f ∂x1−f((cf−de)′ (cf−de))] = 0 (B.0.5) This expression must be satisfied for any arbitrary functional for m off3(x3) andf2(x2), so it is clear that the expressions in parenthesis must identica lly vanish. Say the second one is evaluated for some fixed function fand a fixed value for x1, denoted for now by x0: [∂f ∂x1|x0−f|x0∂ ∂x1log(cf−de)|x0] = 0 (B.0.6) This must hold true no matter what functional form c,dandemay take. In particular hold fixed any two of the ( c,d,e) triplet and vary the third. For concep- tual simplicity, denote log( cf−de) instead by z(c,d,e,f ), wherezis a continuous function of the arbitrary functions c,d,eandf.fis already fixed, now fix say the functionsdande. Thus∂ ∂x1z(c(x1,x2,x3),d0,e0,f0) = 0 implies the possible variations in the ar- bitrary functions ccannot involve the independent variable x1, for then each dif- ferentiation with respect to x1, evaluated at the particular value x0, would yield a different numerical answer which we hold as impossible. The var iation is thus limited tox2andx3, hencec=c(x2,x3) alone and the argument is unchanged for e andd; simply fix the other two and perform the above steps. Note all of th e above steps necessitated the arbitrary function fbeing fixed; otherwise the reader may well ask if the possibility exists for the two separate terms in par enthesis to negate each other - but that is overrided by the fixing of fat one particular function it can take. 109 To provef=f(x2,x3) we now only consider the previous argument applied to the first expression in parenthesis in equation (B.0.5): [∂d ∂x1|x0−d|x0∂ ∂x1log(cf−de)|x0] = 0 (B.0.7) Fixd,candeto ascertain that variations in fcannot involve the variable x1. Note there is no need to consider the quantity ρ3 1for the same conclusions will follow identically. Now that the general method has been presented, the reader may w ell inquire about special cases where the method may break down. For instanc e, what if the factor (cf−de) = 0 leading to log(0)? This case is impossible since Eisenhart already built into the matrices the definition gii=ϕi1 ϕand we know that the metric components can neither vanish nor be infinite, thus ( cf−de)/ne}ationslash= 0. What if (cf−de)<0? Then the minus sign can be absorbed, to yield say [∂f ∂x1+ f[−(cf−de)′ (cf−de)] but the final argument remains unchanged. Now consider the cha in rule application.∂ ∂x1z(c(x1,x2,x3),d0,e0,f0) =z′(c(x1,x2,x3),d0,e0,f0)·∂c(x1,x2,x3) ∂x1by the usual chain rule of the composition of two continuous funct ions, of which log( z) is one of them. The first derivative in the above product cannot be zero as that would imply ( cf−de) =∞, leading to a contradiction considering the non-singular element of the metric component g11. Thus we finally prove c,d,eandf- all belonging to the second and third row - cannot be functions of x1. By manipulating the second row, we get by similar argumentation (i.e ( af−be)/ne}ationslash= 0) thata,b,eandfcannot be functions of x2. This proves immediately that e,fare functions of x3at most; similar steps prove the analogous statements for ( a,b) and (c,d). Thus each row of the matrix defined by Eisenhart’s paper is a function of one variable only, as requi red to show. Note also that by fixing functions, one must choose a form for them an d values ofx0such that the determinant ϕof the matrix defined never vanishes. This technically limits the independence of the n2arbitrary functions somewhat, but innumerable functions exist within this constraint for which all the above facts hold, and must hold for the ‘special cases’ considered, hence validatin g the conclusion. For example, set f3(x3) =x3andf2(x2) =x2, and functions to be fixed like ex1, e(x2)2ande(x3)2ensuring that the determinant can never vanish for non-trivia l values of the independent variables. If all of the c,d,eandfare independent of x1, then clearly∂ ∂x1(cf−de) = 0 and individually∂ ∂x1c,∂ ∂x1d,∂ ∂x1eand∂ ∂x1f= 0 thus the coefficients of f3(x3) and f2(x2) reduce to zero, as needed. Note that metrics and St¨ ackel matrices very well exist where o ne off1(x1), f2(x2) andf3(x3) vanish, hence negating the necessity of all the coefficients in t he above parenthesis vanishing. These are special cases only howeve r; the formulae must work for all of the f1(x1),f2(x2),f3(x3) and all possible metrics (of which 110 the specific components g11,g22andg33can be functions of all the coordinates) thus to encompass all cases satisfied simultaneously the coefficient functions must all vanish. This completes the proof, in two and three dimensio ns, of the St¨ ackel form of the matrices defined by Eisenhart – namely that the first r ow of the inverse form the components of the contravariant metric, and the rem aining rows form the diagonal components of the remaining contravariant Killin g tensors characteristic to the metric. 2 111 Appendix C Construction of the St¨ ackel matrix associated with coordinates separating the HJ equation We start with the formalism of the components of the St¨ ackel ma trix introduced in Chapter 1, whereby ϕk i=1 2∂φi ∂αk. This satisfies∂ ∂qjϕk i= 0 ifi/ne}ationslash=j, sinceφionly depends on qi. Note that since φi= (pi)2, we obtain∂φi ∂αk= 2pi∂pi ∂αk. We know that all the pi/ne}ationslash= 0 since if one were it could not have been a canonical variable [23]. Criti- callydet(∂pi ∂αk)/ne}ationslash= 0,since we have assumed a complete solution of the HJ equation necessitating that det(∂2W ∂qi∂αk)/ne}ationslash= 0⇒det(∂pi ∂αk)/ne}ationslash= 0. Specifically det(∂φi ∂αk) =kdet(∂pi ∂αk)/producttextn ipi/ne}ationslash= 0 as/producttextn ipi/ne}ationslash= 0. Each piacts on the ithrow of the non-singular matrix∂pi ∂αkhence the presence of the/producttextn ipiformula. We will construct S−1by defining, without loss of generality, ( g11,g22,...,gnn) to be the last row. Hence∂ ∂αk(1 2giiφi(qi,αk)) =∂αn ∂αk=δn k, which equals1 2gii∂φi ∂αk= /summationtext igiiϕk i=δn k. This proves that ( g11,g22,...,gnn) was the last row of the inverse St¨ ackel matrix. Now, if the potential Vis separable we want to prove condition 2). Note this condi- tion follows trivially if V= 0; simply declare U1(q1) =U2(q2) =...=Un(qn) = 0. Now thatV=−1 2giip2 i+E, construct Ui=−1 2φi+Eϕn i.ϕn iis simply a column, n, of the St¨ ackel matrix S. We have/summationtextgiiUi=−1 2/summationtextgiiφi+/summationtextgiiEϕn i=−1 2giip2 i+E, asgiiis thenthrow ofS−1, clearlyE/summationtextgiiϕn i=E. Thus since V=−1 2giip2 i+E, it follows that V=g11U1(q1) +g22U2(q2) +...+gnnUn(qn), as required to show. 2 112 Now for the other direction, assume we are given a St¨ ackel matri x such that (g11,g22,...,gnn) is thenthrow of its inverse. Let ϕk i(qi) be defined, with gii=ϕi n. We then express V=/summationtextgiiUi(qi). Thus we write (where the Einstein summation rule is assumed to apply over i): 1 2giip2 i+giiUi(qi) =E 1 2gii[p2 i+ 2Ui(qi)] =E 1 2ϕi n[p2 i+ 2Ui(qi)] =E (C.0.1) Next letϕi k[p2 i+ 2Ui(qi)] = 2αk,k= 1,...,n hence we have αn=E, and each αk∈R. Now ϕk jϕi k(p2 i+ 2Ui(qi)) = 2ϕk jαk δi j(p2 i+ 2Ui(qi)) = 2αkϕk j(qj) ⇒p2 j+ 2Uj(qj) = 2αkϕk j(qj) p2 j= 2αkϕk j(qj)−2Uj(qj) (C.0.2) The last line clearly shows that pjis a function of qjonly, showing that W(qi,αk) is indeed sum separable as required to show. 2 113 Appendix D An equivalent property of the Schouten bracket LetKandLbe symmetric tensors of types ( p,0) and (q,0) respectively. The Schouten bracket of KandLdenoted by [ K,L] is a tensor of type ( p+q−1,0). An equivalent property of [ K,L] = 0, also known as an involution ofKandL, is that their contraction into quadratic polynomials in the mo menta commute in the standard classical Poisson bracket: {K,L}=n/summationdisplay i=1(∂K ∂pi∂L ∂qi−∂K ∂qi∂L ∂pi) (D.0.1) wherebyPKh=Kij hpipjforh/ne}ationslash=nandPKn=gijpipjsatisfies, for ( h,j) = 1,...,n : /braceleftBig PKh,PKj/bracerightBig = 0 (D.0.2) The above is the equivalent formulation of the standard Lie-S chouten bracket. 114 Appendix E Proof of the eigenvalue equations for characteristic conformal Killing tensors by construction from simple Killing tensors Consider again the basic eigenvalue equation from Eisenhart th eory: Kii=giiρi, (E.0.1) whereKiiis understood to be the diagonal components of the Killing ten sor al- ready diagonalized in the normal eigenbasis of its eigenvect ors. The proof of this is instructive, however it relies on more mathematical assumptio ns than in the text and is hence presented here instead. Ordinarily the eigenvect or fields of tensors are notnormal - namely that these eigenvector fields admit a family of hypersu rfaces orthogonal to them. In such cases the vector fields are deemed no n-integrable or non-surface-forming. Eisenhart assumed that the Killing tensor eigenvectors are normal and hence the hypersurfaces can be taken as parametric . Furthermore he assumed the symmetric Killing tensor to have real simple eigenvalues , namelyndis- tinct eigenvalues admitting northogonal eigenvectors xithat are surface-forming: ds2=e1g11(dx1)2+e2g22(dx2)2+...+engnn(dxn)2(E.0.2) where for Riemannian manifolds all the e’s are unity, and for non-Riemannian manifolds the e’s may take on plus/minus unity. That the funda mental form above can be expressed in terms of the eigenvectors xidepends crucially on the normality assumption as well as the fact that there are northogonal eigenvectors the Killing tensor admits. We perform a coordinate transformation to the co ordinates defined by the normal eigenvectors such that Kij= 0,i/ne}ationslash=jand ˜gij= 0,i/ne}ationslash=j. The tilde can be dropped and the Killing tensor equation for i=j=lreduces to ∂log(√Kii) ∂xi=∂log(√gii) ∂xi(E.0.3) 115 This can be easily integrated to log(/radicalBig Kii) = log(√gii) +a(x1,x2,...,xi−1,xi+1,...,xn) (E.0.4) After performing exponentials the form Kii=giiρiis arrived at, where ∂ρi ∂xi= 0. (E.0.5) This eigenvalue property of the diagonal components of the K illing tensor is actually true in general for tensors, without the assumption of the specific Killing tensor equation. Knowing that gii= ˜giie−2σ, we can rewrite the previous equation as: Kii= ˜giie2σρi Kii= ˜gii˜ρi, (E.0.6) where ˜ρi≡e2σρi. Thus we arrive at two fundamental properties [3] of conform al Killing tensors: Proposition E.0.4 The eigenvectors of Kare the same with respect to both met- rics˜gandg. Furthermore if ˜ρiare the eigenvalues with respect to (contravariant) ˜g, then the eigenvalues with respect to gareρ=e2σ˜ρi. Note that eigenvalues of two equivalent conformal Killing te nsors differ only by the scalar function f; going from one equivalent tensor to another means simply addi ng or subtracting ffrom all its eigenvalues, namely: ˜ρi=ρi±f (E.0.7) These transformations still leave invariant the form Kii= ˜gii˜ρi, as well as its eigen- vectors, for allconformal Killing tensors in an equivalence class. Remark E.0.5 In words a special conformal Killing tensor, within its equiv alence class, is in terms of some specially conformally related met ric a simple Killing tensor, and to calculate the eigenvalues of this conformal K illing tensor, the tools for simple Killing tensors apply provided the conformal fact or is a priori known. This was especially important for this thesis, since the con nection between St¨ ackel matrices and the eigenvalues of the ordinary Killing tensors in terms of the separable coordinates are employed at length. Knowing how the eigenvalues transform allows us to deduce the differential relationship they satisfy for conformal Killing tensors. The seco nd identity for eigenvalues from [20] is ∂ ∂xjlog((ρi−ρj) gii) = 0, i/ne}ationslash=j, (E.0.8) 116 which is a result for simple Killing tensors. The proof comes from the Killing tensor equation (on the diagonalized Killing tensor in terms of the n ormal eigenbasis) for the casej/ne}ationslash=i,l=j: ∂Kii ∂xj−2Kii∂log(gii) ∂xj+Kjj1 gjj∂gii ∂xj= 0. (E.0.9) Considering both transformation rules on the eigenvalues, we se e immediately that the same form in Eq. (E.0.8) is preserved for transformed me tric coefficients and eigenvalues with the only change being that ρican depend on xidue to the general form of the conformal factor e2σ. Note trivially that the equivalence function fcancels in the numerator of Eq. (E.0.8). Thus we are able to pr ove an important proposition in [3]: Proposition E.0.6 The eigenvalues ρiof a conformal Killing tensor satisfy the set of coupled linear partial differential equations: ∂ρi ∂xj= (ρi−ρj)∂log(gii) ∂xj+∂ρj ∂xj(E.0.10) Proof: 0 =∂ ∂xjlog((ρi−ρj) gii) =gii (ρi−ρj)((∂ρi ∂xj−∂ρj ∂xj)1 gii−((ρi−ρj) g2 ii∂gii ∂xj)) (E.0.11) multiplying through by giiand discarding the ( ρi−ρj)−1common factor yields the required result. 2 A crucial subtlety in the above reasoning is that our proof is re stricted to confor- mal Killing tensors with normal eigenvectors and real simple e igenvalues, allowing them to be diagonalized in orthogonal coordinates. Indeed th is is the hypothesis of Proposition 7.2 in [3], which among other statements reads: Proposition E.0.7 A CKT Kwhich is diagonalized in orthogonal coordinates (that isgij= 0andKij= 0fori/ne}ationslash=j) is equivalent to a CKT K′that is a simple Killing tensor with respect to a conformally related metric. Our proof of the properties of the eigenvalues of a conformal Killing tensor is essen- tially the reverse direction of Proposition 7.2 in [3], as we fir st started from a simple diagonalized Killing tensor and then enacted a conformal tra nsformation bringing us to a representative of an equivalence class of characteristi c conformal Killing ten- sors. The proof is therefore not valid for non characteristic Ki lling tensors, however they are not useful for characterizing separable coordinates anyhow. 117 Appendix F Alternate classification scheme using the invariants and covariants of biquartic polynomials Here is material from [13], firstly please note the change in dict ionary of the co- efficients of the rotational Killing tensor which is outlined b elow. This rotational characteristic conformal Killing tensor is equivalent to M33I3⊙I3+L3D⊙I3+HD⊙D+C33R3⊙R3+D3D⊙X3+A33X3⊙X3.(F.0.1) LetRCK2(E3) be the subspace CK2(E3) of CKTs of the form (F.0.1). The free parameters describing a general element K∈RCK2(E3) are (M33,L3,H,C 33,D3,A33) (F.0.2) and all the other forty nine coefficients of the general linear combination of sym- metric products of CKVs (3.1.3) are null. Given any CKT in Cart esian coordinates satisfying LR3K= 0, and the TSN-conditions, the value of the parameters (F.0.2 ) are determined as follows: •M33is 1/4 of the coefficient of xyz2inK12; •L3is 1/2 of the coefficient of xyzinK12; •His the coefficient of xzinK13; •H−C33is the coefficient of xyinK12; •D3is twice the coefficient of xinK13; •A33is the constant term of K33−K22. Since we are considering components (or functions of the comp onents) which are not affected by the addition of a multiple of the metric fg, the six parameters are well defined, irrespective of whether one starts from a CKT in TCK(E3) or not. 118 Remark F.0.8 SinceE3has dimension three, there is an equivalent way to char- acterize rotational R-separable webs. Any rotational web contains a family of hypersurfaces made of half-planes issued from the rotation axi s (thez-axis in our case). These planes are orthogonal to the Killing vector R3. Hence R3must be an eigenvector of the CKT defining the web. Moreover, this condi tion is also sufficient to ensure that the eigenvectors of Kare normal. Indeed, one of them is the normal vector R3and the other two are contained in the two-dimensional planes orthogonal toR3and hence they are normal. By imposing the condition (K·R3)×R3= 0, we find again the six dimensional linear subspace described by (F.0 .1). Finally, in order to prove that the general rotational CKT (F .0.1) is charac- teristic, we check that the eigenvalues are simple almost every where. Since R3is orthogonal to I3,D,X3, we have K·R3=C33(x2+y2)R3. Hence, R3=E1is an eigenvector corresponding to the eigenvalue λ1=C33(x2+y2). The other two eigenvectors E2andE3are orthogonal to E1; they and their corre- sponding eigenvalues do not depend on C33. Moreover, the associated eigenvalues are of the form λ2,3=A±√ B 2, where A=r4M33+zr2L3+r2H+zD3+A33,(r2=x2+y2+z2) (F.0.3) B= (x2+y2)/bracketleftBigg r2L3+ 2zH+4z2−r2 r2D3+4z(2z2−r2) r4A33/bracketrightBigg2 + (F.0.4) /bracketleftBigg r4M33+zr2L3+ (2z2−r2)H+z(4z2−3r2) r2D3+r4−8z2(r2−z2) r4A33/bracketrightBigg2 . Any change of the parameter C33does not affect the web; indeed, E2andE3do not involveC33(see also Sect. F.2). Thus, it is always possible to choose C33such that λ1is different from λ2andλ3at any point outside of the z-axis. On the contrary forx=y= 0 we have λ1= 0, λ 2=1 2(q(z) +|q(z)|), λ 3=1 2(q(z)− |q(z)|), with q(z) =M33z4+L3z3+Hz2+D3z+A33. (F.0.5) Thus, (at least) one of λ2,λ3identically vanishes and all points of the z-axis are singular points of all rotational webs. The singular points tha t are not on the rotation axis are those satisfying λ2=λ3, that is where B= 0. 119 Remark F.0.9 The roots of (F.0.5) are points on the z-axis where the three eigen- values coincide and Kis proportional to the metric tensor. The number of the roots z0ofqinPR1(so that the point at infinity is also considered) and their multi plicity characterize the web from a geometric point of view. Remark F.0.10 The knowledge of the eigenvalues of the characteristic tensor in a rotational web allows one to write the equations of the (not planar) hypersurfaces (see [16]) The hypersurfaces S2orthogonal to E2satisfy the equation λ1−λ3 x2+y2=h, h ∈R, while the hypersurfaces S3orthogonal to E3satisfy the equation λ1−λ2 x2+y2=h, h ∈R. It follows that the hypersurfaces have the form 2(h−C33)(x2+y2) +A=±√ B, that is they are both described by the equation [2(h−C33)(x2+y2) +A]2−B= 0, (F.0.6) but for different ranges of the value of h: we have surfaces of S2forh < h 0and surfaces of S3forh>h 0, respectively, where h0=C33−A 2(x2+y2)=C33−r4M33+zr2L3+r2H+zD3+A33 x2+y2. Forh=h0we do not obtain a surface of the web because this value of the par ameter hwould imply B= 0, that is λ2=λ3. Expanding the equations (F.0.6) we arrive at [4(H−C33+h)M33−L2 3]r4+ [8M33D3−4(C33−h)L3]r2z+ [2L3D3−4(C33−h)H]r2+ 16M33A33z2+ 4(C33−h)2(x2+y2) + (F.0.7) [8L3A33−4(C33−h)D3]z−D2 3+ 4(H−C33+h)A33= 0, which represents two families of confocal cyclides, one for h>h 0and one for h<h 0. F.1 Characteristic CKTs of the known R-separable rotational coordinate systems Table 1 contains the parameters of a characteristic CKT corre sponding to each of the rotational R-separable coordinates listed in Moon and Spencer’s book [38] . 120 Coordinates M33L3HC33D3A33 Bi-cyclide −k2 a2 01 +k21 +k20−a2 Flat-ring cyclidek2 a2 01 +k20 0a2 Disk cyclide −k2 a2 01−2k20 0a2(1−k2) Cap cyclidea2(1+k)2 k04k−(k−1)2 2−(k−1)2 20k(k+1)2 16a2 Toroidal1 4a2 01 21 20a2 4 Bispherical −1 4a201 21 20−a2 4 Inverse prolate spheroidal1 a2 0 -1 0 0 0 Inverse oblate spheroidal−1 a2 0 -1 0 0 0 Tangent spheres1 0 0 0 0 0 Cardioid 0 1 0 0 0 0 Prolate spheroidal0 0 -1 0 0a2 Oblate spheroidal0 0 1 0 0a2 Spherical 0 0 1 -1 0 0 Parabolical 0 0 0 0 1 0 Cylindrical 0 0 0 -1 0 1 Table F.1: Characteristic CKT of rotationally symmetric R-separable webs We briefly describe how they are determined (for further detai ls, such as plots, transformation laws to Cartesian coordinates, components of t he metric tensor in these coordinates, separated equations etc., see [38] or [5]) . The CKTs are constructed from the St¨ ackel matrices that are associated wit h each system of coordinates in [38]. Recall that a St¨ ackel matrix is a regular matrix of functions Sijdepending on the single variable qicorresponding to the row index iof the element. One row (the first in the examples in [38]) of the inverse of the St¨ ackel matrix contains the components of the contravariant metric tensor in the R-separable coordinates, while the other two rows are made of the components 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 a chara cteristic tensor of the web (see [3]). For each row of the inverse of the St¨ ackel matrix we construct t he conformal Killing tensors in the R-separable coordinates, then the parameters (F.0.2) are 121 determined by transforming the tensor to Cartesian coordinate s and comparing with the Cartesian components of the general rotationally sym metric CKT (F.0.1). For all the coordinate systems considered in [38] the tensor corr esponding to the third row of the inverse St¨ ackel matrix is R3⊙R3. In most of the examples, the other tensor is a characteristic tensor of the web so its paramete rs appear unchanged in the Table 1. On the contrary, the tensors arising from the St¨ ackel matrices given in [38] for Spherical, Tangent spheres and Cylindrical coord inates have C33= 0, so they are not characteristic CKTs. In order to get a characterist ic CKT associated with these webs we add a suitable multiple of the tensor R3⊙R3: that is, we change the value of C33in Table 1. The first four coordinate systems have transformation laws to Car tesian coor- dinates involving Jacobi elliptic functions. The parameter ais a scaling parameter, while the parameter k∈(0,1) is the parameter of the Jacobi elliptic functions. F.2 Group action preserving rotationally symmetric CKTs F.2.1 The group and its one-parameter subgroups In order to classify the different types of R-separable webs admitting a rotational symmetry, we consider transformations acting on CK2(E3) which preserve the space RCK2(E3) of the rotationally symmetric CKTs previously discussed in chap ter five. For this purpose, we use a group Gthat is generated by five one-parameter transfor- mations and a discrete transformation. Three of the one-param eter transformations are induced on RCK2(E3) by conformal transformations of E3mapping the z-axis into itself. The other two are transformations of the CKT that d o not change the corresponding web. The five continuous transformations to be taken into account a re 1. The change of the tensor under a continuous inversion along t hez-axis pa- rameterized by a0: φ0: (x,y,z )→/parenleftBiggx 1 + 2a0z+a2 0r2,y 1 + 2a0z+a2 0r2,z+a0r2 1 + 2a0z+a2 0r2/parenrightBigg , wherer2=x2+y2+z2. 2. The change of the tensor under a translation along the z-axis parameterized bya1: φ1: (x,y,z )→(x,y,z +a1). 3. The change of the tensor under a dilation of the space with sing ular point at the origin parameterized by a2: φ2: (x,y,z )→(a2x,a2y,a2z),(a2/ne}ationslash= 0). 122 4. The multiplication of the tensor by a non-zero scalar a3: K→a3K,(a3/ne}ationslash= 0). 5. The addition to the tensor of a multiple of R3⊙R3: K→K+a4R3⊙R3. Moreover, the discrete transformation considered is the one in duced by the inversion Iwith respect to the unit sphere with center at the origin I: (x,y,z )→/parenleftBiggx x2+y2+z2,y x2+y2+z2,z x2+y2+z2/parenrightBigg . (F.2.8) Note thatI−1=Iand that for the continuous inversion φ0we haveφ0=I−1◦φ1◦I, whereφ1is the translation along the z-axis. Remark F.2.1 The addition of the metric gand the transformation induced by the rotation around the z-axis are not relevant, since they do not modify the pa- rameters (F.0.2) defining the tensor. F.2.2 Group action, invariants and canonical forms LetGbe the group generated by the above described transformations. Since the discrete inversion is included, Gis not connected. Moreover, two of the continuous one-parameter transformations are defined only for values of the parameter in R− {0}, so that the connected component of Gcontaining the identity is characterized bya2>0 anda3>0. Two other discrete transformations are implicitly include d inG: the change of sign of the tensor (for a3=−1) and the transformation induced by the symmetry around the origin in E3(fora2=−1). The effect of the inversion around the unit sphere on the coefficie nts (F.0.2) of K∈RCK2(E3) is given by ˜M33=A33, ˜L3=D3, ˜H=H, ˜C33=C33, ˜D3=L3, ˜A33=M33.(F.2.9) The equations of the action generated by the five continuous t ransformations acting on (F.0.2) are ˜M33=a3P(a0) a2 2, ˜L3=a3−4a1P(a0)−a2P(1)(a0) a2 2, 123 ˜H=a36a2 1P(a0)+3a1a2P(1)(a0)+a2 2P(2)(a0) a2 2, ˜C33=a4+a3C33+a36a2 1P(a0)+3a1a2P(1)(a0)+a2 2(P(2)(a0)−H) 3a2 2, ˜D3=a3−4a3 1P(a0)−3a2 1a2P(1)(a0)−2a1a2 2P(2)(a0)−a3 2P(3)(a0) a2 2, ˜A33=a3a4 1P(a0)+a3 1a2P(1)(a0)+...+a1a3 2P(3)(a0)+a4 2P(4)(a0) a2 2, where P(a0) =A33a4 0−D3a3 0+Ha2 0−L3a0+M33, (F.2.10) and P(n)=1 n!dnP (da0)n. SinceC33anda4are involved only with ˜C33, andC33is unchanged by the discrete inversion (F.2.9), we can disregard C33(which can be made equal to any fixed constant by choosing a particular value for a4). Then we consider the reduced action on the vector subspace of RCK2(E3) defined by the five parameters (M33,L3,H,D 3,A33) (F.2.11) of the subgroup G′ofGdefined bya4= 0: ˜M33=a3P(a0) a2 2, (F.2.12) ˜L3=a3−4a1P(a0)−a2P(1)(a0) a2 2, (F.2.13) ˜H=a36a2 1P(a0)+3a1a2P(1)(a0)+a2 2P(2)(a0) a2 2, (F.2.14) ˜D3=a3−4a3 1P(a0)−3a2 1a2P(1)(a0)−2a1a2 2P(2)(a0)−a3 2P(3)(a0) a2 2, (F.2.15) ˜A33=a3a4 1P(a0)+a3 1a2P(1)(a0)+...+a1a3 2P(3)(a0)+a4 2P(4)(a0) a2 2. (F.2.16) It appears that the building blocks of the action equation is the polynomial (F.2.10) and its derivatives. Remark F.2.2 If we denote the parameters (F.2.11) by αi(i= 0,...,4), setting α4=M33,α3=L3,α2=H,α1=D3,α0=A33), then their transformation laws under the action can be written in a compact formal way as ˜α4−i=a3 a2 2i/summationdisplay h=0(−1)i/parenleftBigg 4−h i/parenrightBigg P(h)(a0)ai−h 1ah 2, i = 0,...,4. Theorem F.2.3 LetG1be the subgroup of G′defined bya3>0. Then, the action ofG1on(F.2.11)given by (F.2.12− −F.2.16)and(F.2.9)is equivalent to the classical action of GL(2,R)on real binary quartics. 124 Proof: Consider the following binary quartic constructed from the fiv e coefficients (F.2.11) of the CKT: Q(X,Y) =M33X4+L3X3Y+HX2Y2+D3XY3+A33Y4. (F.2.17) By inserting the linear transformation of the variables ( X,Y) X=α¯X+β¯Y, Y =γ¯X+δ¯Y, with (αδ−βγ)/ne}ationslash= 0, in (F.2.17), we obtain a new quartic ¯Q(¯X,¯Y) whose coefficients ¯M33,..., ¯A33depend on the GL(2,R) matrix M=/bracketleftBigg α β γ δ/bracketrightBigg and on the coefficients of Q(M33,...,A 33). Since we assume a3>0, by setting α=4/radicalBig a3a2 2, β=−a14/radicalBig a3a2 2, γ=−a04/radicalBig a3a2 2, δ= (a1a0+a2)4/radicalBig a3a2 2, we obtain equations ( F.2.12− −F.2.16). The regularity of Mfollows from ( αδ− βγ) =√a3a2|a2| /ne}ationslash= 0, since a2a3/ne}ationslash= 0. Furthermore, setting α=γ= 0, and β=δ= 1, we recover (F.2.9). Conversely, we prove that for any tran sformation of the quartic we can associate a transformation of G1. We distinguish two cases: for α/ne}ationslash= 0, by setting a0=−γα−1, a 1=−βα−1, a 2= (αδ−βγ)α−1, a 3= (αδ−βγ)2, into (F.2.12− −F.2.16) we obtain the action of Mon the quartic form. The fact thata2a3/ne}ationslash= 0 follows from the regularity of the matrix M. Ifα= 0, we apply first the discrete inversion ( F.2.9) on the parameters of the CKT, that is we multiply M by/bracketleftBigg 0 1 1 0/bracketrightBigg on the left. In this way we obtain a new matrix M1withα1=γ/ne}ationslash= 0 sinceMis regular, and thus revert to the previous case. 2 As an immediate consequence of the theorem we are able to determ ine the invariant of the action and the list of canonical forms which a re given in the following propositions. Proposition F.2.4 The only independent differential invariant of the action of G onRCK2(E3)is F=I3 J2, where the functions I= 12A33M33−3L3D3+H2, J= 72A33M33H−27A33L2 3−27D2 3M33+ 9D3L3H−2H3, are relative invariants of the action of Gand independent differential invariants for the action of the subgroup of Gdefined bya3= 1. 125 Proof: The functions IandJare the fundamental invariants (of weight 4 and 6 respectively) of the binary quartic form (F.2.17) [24],[41] .2 Proposition F.2.5 Each CKT of RCK2(E3)is equivalent under the action of G to one of the following representatives: I.I3⊙I3+µD⊙D+X3⊙X3, µ ∈R, (F.2.18) II.I3⊙I3+µD⊙D−X3⊙X3, µ ∈R, (F.2.19) III. I3⊙I3+νD⊙D, ν =±1, (F.2.20) IV. D⊙I3, (F.2.21) V. I3⊙I3. (F.2.22) Proof: Starting from the list of canonical forms of real binary quart ics (given for instance in [24]), we combine those differing only by sign. We remark that for µ= 2 the canonical form I.is equivalent to D⊙D.2 Remark F.2.6 The action of GoverRCK2(E3) has infinitely many orbits. How- ever, the tensors in (F.2.18) and (F.2.19) are not pairwise ine quivalent for all values ofµ: forµ/ne}ationslash=±2 there exists a finite number of µ′such that the corresponding tensors are pairwise equivalent (see [24]). F.3 Invariant classification of the R-separable ro- tationally symmetric webs The polynomial Pdefined in (F.2.10) as the building block of the action equati ons (F.2.12–F.2.16) is deeply related to the polynomial q(F.0.5). Indeed, we have P(X) =X4q(−1/X).Moreoverqis the inhomogeneous polynomial corresponding to the quartic binary form Q(F.2.17). The roots of qare the points on the z-axis where all the eigenvalues of Kcoincide (see Remark F.0.9). The conformal transformations φ0,φ1,φ2andIdescribed in Sect. F.2.1 map the z-axis to itself with a one to one correspondence (if we include also the point at infinity). Thus two distinct points cannot be m ade coincident or removed. This provides the geometric interpretation of the fact that the invariants ofQare invariants of the CKT defining the web. The meaning of qin terms of invariant theory is made more precise in the following proposi tion. Proposition F.3.1 The polynomial q(z) =M33z4+L3z3+Hz2+D3z+A33is a relative covariant of the induced extended action on CˆK2(E3)×E3restricted on the invariant subset S0={x=y= 0}. 126 Proof: The equations of the extended action are (F.2.12–F.2.16) to gether with ˜x=a2x (a0z+1)2+a2 0(x2+y2), ˜y=a2y (a0z+1)2+a2 0(x2+y2), ˜z=a2z+a2 0(x2+y2+z2) (a0z+1)2+a2 0(x2+y2)+a1. The subset S0={x=y= 0}is an invariant subset of the extended action. Moreover, on S0the transformation law for zreduces to the linear fractional trans- formation ˜z=(a2+a1a0)z+a1 a0z+ 1(a2/ne}ationslash= 0), (F.3.23) which is the general linear transformation on RP1(see [41]). Let ˜ q(˜z) be the polynomial we obtain by inserting (F.2.12–F.2.16) and (F.3. 23) in (F.0.5). We obtain (a0z+ 1)4˜q(˜z) =a3a2 2q(z), (F.3.24) that is (up to a3) a covariant of weight two of the action. For the discrete inve rsion (mappingzinto ˜z= 1/z), we immediately see that it maps q(z) to ˜q(˜z) =q(z) z42 Equation (F.3.24) shows that the number and multiplicity of t he real roots of q(z) (that is the number and multiplicity of the real linear fact ors ofQ) are invariant with respect to the group action. Hence they can be used to define a nd classify the different types of webs. Definition F.3.2 We say that two rotationally symmetric R-separable webs are of the same type if the polynomials associated with the correspon ding characteristic CKT have the same number and multiplicity of real roots. Thus we have reduced the classification of rotational R-separable webs to the classical classification of real binary quartics (see [41], [24]) . We have nine types of webs, listed in Table 2. The remaining coordinates systems of Table 1 are equivalent to one of the co- ordinates listed above (correcting a typographical error in [38], where Cap cyclide coordinates are said to be equivalent to Bi-cyclide coordina tes), as it is described in Table 3. Remark F.3.3 The number of the types of rotationally R-separable coordinate systems agree with the results of [6], where the subject is examin ed from the point of view of symmetry operators. The coefficients Aijof the second order part of the symmetry operators Scharacterizing each type of R-separable rotationally sym- metric coordinates, with respect to Cartesian coordinates, list ed in Table 2. of Boyer, et al. [6] when written as S=Aij∂i∂j+Bi∂i correspond to the components of CKTs equivalent to those listed in Table 1 for Bi-cyclide, Flat-ring cyclide, Disk cyclide and Toroidal co ordinates, respectively. 127 Associated web roots ofq canonical form of K Bi-cyclide 4 distinct real roots I.forµ<−2 Flat-ring cyclide4 distinct complex conjugate rootsI.forµ>−2,µ/ne}ationslash= 2 Disk cyclide4 distinct roots, 2 real, 2 complex conjugateII. Inverse prolate spheroidal1 double real root, 2 distinct real rootsIII.forν=−1 Inverse oblate spheroidal1 double real root, 2 distinct complex conjugate rootsIII.forν= 1 Toroidal2 double complex conjugate rootsI.forµ= 2 Bispherical 2 double real roots I.forµ=−2 Cardioid1 triple (real) root 1 simple real rootIV. Tangent sphere1 quartuple (real) root V. Table F.2: the nine types of inequivalent rotational R-separable webs Finally, we provide algebraic conditions on the parameters (F.2.11) in order to determine the type of the corresponding web. In order to obtai n these conditions, we solve the equivalent problem of determining the number and multiplicity of the linear factors of the corresponding binary quartic form Qwhich can be done by applying the classical algorithm (see for example [24]) based on the sign and vanishing of relative invariants and covariants of Q. Together with IandJ, the following invariant and covariants are used in the classification scheme: the discriminant of the form (a relative i nvariant which van- ishes if and only if the quartic has a multiple root) ∆ =I3−27J2, the Hessian of the form (a covariant which vanishes if and only if t he quartic has a quadruple root) H(X,Y) = (∂2 XXQ)·(∂2 YYQ)−(∂2 XYQ)2; the covariants L(X,Y) =IH(X,Y)−6JQ(X,Y), and M(X,Y) = 12H2(X,Y)−IQ2(X,Y). We summarize the classification in Table 4. 128 Web equivalent to transformation Cap cyclide Flat-ring cyclide cont. inversion + trans. Prolate Spheroidal Inverse Prolate Spheroidal discrete inversion Oblate Spheroidal Inverse Oblate Spheroidal discrete inversion Spherical Bispherical cont. inversion + trans. Parabolical Cardioid discrete inversion Circular Cylindrical Tangent sphere discrete inversion Table F.3: Pairwise conformally equivalent webs Web Algebraic condition Disk cyclide ∆<0 Bi-cyclide ∆>0 andH(X,Y)<0, andM(X,Y)>0 Flat-ring cyclide ∆>0 and (H(X,Y)>0 orM(X,Y)>0) Inverse prolate spheroidal ∆ = 0 and L(X,Y)<0 Inverse oblate spheroidal ∆ = 0 and L(X,Y)>0 Toroidal L(X,Y) = 0 andH(X,Y)>0 Bispherical L(X,Y) = 0 andH(X,Y)<0 Cardioid I=J= 0 andH(X,Y)/ne}ationslash= 0 Tangent sphere H(X,Y) = 0 Table F.4: Invariant classification of the webs 129 List of References [1] S. Benenti. Separability in Riemannian manifolds. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 2004. [2] S. Benenti, C. Chanu, and G. Rastelli. 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