Separability and Symmetry Operators for Painlev\'e Metrics and their Conformal Deformations

Painlev\'e metrics are a class of Riemannian metrics which generalize the well-known separable metrics of St\"ackel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations which characterizes the St\"ackel case. Our goal in this paper is to carry out for Painlev\'e metrics the generalization of the analysis, which has been extensively performed in the St\"ackel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlev\'e metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the St\"ackel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for St\"ackel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist $r<n$ linearly independent second-order differential operators which commute with the Laplace-Beltrami operator and which are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations which are compatible with the separation into blocks of variables of the Helmholtz equation for Painlev\'e metrics, leading to solutions which are $R$-separable in blocks. The paper concludes with a set of open questions and perspectives.

1 Introduction and statement of results inlevé metris QID QP re lss of iemnnin metris tht provide rod generliztion of the wellEknown seprle metris of täkel ITD QS to the se in whih the rmiltonEtoi eqution for the geodesi )ow n e dditively seprted into prtil di'erentil equtions depending on groups of independent vriles rther ordinry di'erentil equtions resulting from omplete orthogonl seprE tionF sn prtiulrD while täkel metris in dimension n dmit n linerly independent oissonEommuting qudrti (rst integrls of the geodesi )owD inlevé metris in dimension n will dmit only r < n suh integrls in generlD where r denotes the numer of groups of vrilesF yur gol in this pper is to rry out the extension to inlevé metris of the wellEknown results ID PD QD ITD PID PPD PQ whih relte in the täkel se the dditive seprtion of vriles for the rmiltonE toi eqution to the multiplitive seprtion of vriles for the relmholtz equtionD nd the existene of qudrti (rst integrls of the geodesi )ow to tht of symmetry opertors for the vpleEfeltrmi opertorF e shll thus otin the generliztion to inlevé metris of the oertson seprility onditions QQ for the relmholtz eqution for täkel metrisD nd formultion of these generlized oertson onditions in terms of the vnishing of the o'Elok digonl omponents of the ii tensorD therey extending the lssil result proved y iisenhrt IT for täkel metrisF e shll lso show tht when the generlized oertson onditions re stis(edD there exist r < n linerly independent seondEorder di'erentil opertors whih ommute etween themselves nd with the vpleEfeltrmi opertorF hese opertors will e shown to dmit the lokEseprle solutions of the relmholtz eqution s forml eigenfuntionsD with the seprtion onstnts rising from the seprtion into groups vriles s eigenvluesF pinllyD we shll lso disuss onforml deformtions of inlevé metris stisfying further generliztion of the oertson onditionsD whih is omptile with the seprtion into loks of vriles of the relmholtz equtionD leding to solutions whih re REseprle in loksF P fefore desriing our results in further detilD we should remrk tht independently of the interest of inlevé metris from the point of view of seprtion of vrilesD key motivtion for our study lies in the gol of onstruting geometri models of mnifolds with oundry endowed with inlevé metrisD with the gol of investigting the nisotropi glderón prolem in this lss of geometriesF ell tht the nisotropi glderón prolem onsists in reovering the metri of iemnnin mnifold with oundry from the knowledge of the hirihletEtoExeumnn mp de(ned y the vpleEfeltrmi opertorF he nisotropi glderón prolem is t the enter of gret mount of urrent reserh tivityF e refer to IQD IWD PRD PSD PUD PVD QRD RHD RI nd the referenes therein for surveys of reent results on this prolemF e hve reently investigted the nisotropi glderón prolem t (xed energy for geometri models onsisting of lsses of täkel mnifolds with oundryD where the seprtion of vriles for the relmholtz eqution llows the deomposition of the hirihletEtoExeumnn mp onto sis of joint eigenfuntions of the symmetry opertors resulting from the omplete seprtion of vrilesD enling us to otin series of uniqueness nd nonEuniqueness results for the glderón prolemD with no E priori ssumptions of nlytiityD or on the existene of isometries IHD IQD IRD IID IPD IVF sn the se of inlevé metrisD the seprtion of the relmholtz into groups of vriles nd the onomitnt fmilies of ommuting symmetry opertors dmitted y these metris will serve s n e'etive strting point for the investigtion of the nisotropi glderón prolem in this more generl settingF sn order to put the results of the present pper in ontextD we (rst rie)y rell some wellEknown de(nitions nd results pertining to täkel metris nd their seprility propertiesF hroughout the pperD we shll ssume for simpliity tht the mnifoldsD metris nd mps eing onsidered re smoothD lthough mny of the results tht we quote or otin n e shown to hold under weker di'erentiility propertiesF ell PD ITD PID QS tht Stäckel metric on n nEdimensionl mnifold M is iemnnin metri g for whih there exist lol oordintes (x 1 , . . . , x n ) in whih the metri hs the expression ds 2 = g ij dx i dx j = det S s 11 (dx 1 ) 2 + · · · + det S s n1 (dx n ) 2 , @IFIA where S is Stäckel matrixD tht is nonEsingulr n × n mtrix S = (s ij ) of the form S =    s 11 (x 1 ) . . . s 1n (x 1 ) F F F F F F s n1 (x n ) . . . s nn (x n )    , @IFPA nd s ij denotes the oftor of the omponent s ij of the mtrix SF täkel mtries thus hve the property tht for eh 1 ≤ i ≤ nD their iEth row depends only on the iEth lol oordinte x i D nd tht the oftor s ij is independent of the iEth lol oordinte x i F purthermoreD the digonl omponents of the täkel metri @IFIA re given y the inverses of the entries of the (rst row of the inverse täkel mtrix A = S −1 F he importne of täkel metris stems from the ft tht they onstitute the most generl lss of iemnnin metris dmitting orthogonl oordintes for whih the rmiltonEtoi eqution for the geodesi )ow of (M, g)D where E denotes positive rel onstntD dmits omplete integrl otined y dditive seprtion of vriles into ordinry di'erentil equtionsF st is useful t this stge to rell tht omplete integrl of @IFQA is de(ned s prmetrized fmily of solutions W = W (x 1 , . . . , x n ; a 1 , . . . , a n ) , a 1 := E , @IFRA Q de(ned over the domin U ⊂ M of the lol oordintes (x 1 , . . . , x n ) nd depending smoothly on n prmeters (a 1 , . . . , a n ) de(ned on n open suset A ⊂ R n D suh tht the rnk ondition det ∂ 2 W ∂x i ∂a j = 0 , @IFSA is stis(ed throughout the open set U × AF st is esily veri(ed tht the rmiltonEtoi eqution @IFQA will dmit n dditively seprle omplete integrl W (x 1 , . . . , x n ; a 1 , . . . a n ) of the form W = W 1 (x 1 ; a 1 , . . . , a n ) + · · · + W n (x n ; a 1 , . . . , a n ), if nd only if the summnds W α stisfy the set of seprted ordinry di'erentil equtions given y he n prmeters (a 1 , . . . , a n ) ppering in the expression of the dditively seprle omplete integrl @IFRA thus orrespond to the seprtion onstnts rising from the omplete seprtion of vriles of the rmiltonEtoi eqution into ordinry di'erentil equtionsF yne of the key onsequenes of this omplete seprtion of vriles property is tht the geodesi )ow of n nEdimensionl täkel metri dmits linerly independent set of n−1 mutully oissonEommuting qudrti (rst integrlsD given y where A = (a ij ) denotes s ove the inverse of the täkel mtrix S given y @IFPAF xote tht with the nottions of @IFTAD we hve K (1) = HD where denotes the rmiltonin for the geodesi )owF e question losely relted to the dditive seprtion of the rmiltonEtoi eqution is tht of the omplete multiplitive seprtion of the relmholtz eqution denotes the vpleEfeltrmi opertor on (M, g) nd λ denotes non vnishing rel onstntD into ordiE nry di'erentil equtionsF fy omplete multiplitive seprtionD we menD following PD the existene of prmetrized fmily of solutions u de(ned on domin U ⊂ M with lol oordintes (x 1 , . . . , x n ) of the form u(x 1 , . . . , x n ; a 1 , . . . a 2n ) = n i=1 u i (x 1 , . . . , x n ; a 1 , . . . a 2n ) , R depending smoothly on 2n prmeters (a 1 , . . . a 2n ) de(ned on on open suset A ⊂ R 2n D stisfying the rnk ondition his seprtion requires tht dditionl onditionsD known s the Robertson conditionsD nd given y , @IFIHA e stis(edF e refer to PD QD ITD PID PPD QQ for detiled proofs of the ft tht the oertson onditions re neessry nd su0ient for omplete multiplitive seprtion of the relmholtz equtionF he oertson onditions were interpreted y iisenhrt IT in terms of the vnishing of the o'Edigonl omponents of the ii tensor of the underlying täkel metriD tht is hen the oertson onditions re stis(edD the oissonEommuting qudrti (rst integrls of the geodesi )ow give rise to n − 1 linerly independent seondEorder di'erentil opertors whih ommute with the vpleEfeltrmi opertor nd lso ommute pirwiseF ewriting the qudrti (rst integrls K (l) de(ned y @IFTA in the form K (l) = K ij (l) p i p j , these ommuting opertorsD denoted y ∆ K (l) D re of the form where ∇ i denotes the veviEgivit onnetion on (M, g)F hese opertorsD whih re often referred to s symmetry operatorsD dmit the seprle solutions of the relmholtz eqution s @formlA eigenfuntionsF e will not give ny further detils on symmetry opertors t this stgeD nor shll we sy nything out the proofs of the results we hve just relled sine we shll shortly stte nd prove generliztions of these to the se of inlevé metrisD whih dmit ll täkel metris s speil seF e onlude these preliminries y remrking tht the ove setting my e expnded signi(ntly y onsidering onforml deformtions of täkel metris whih re omptile with the omplete seprtion of the relmholtz eqution into ordinry di'erentil equtionsD thus giving rise to the more generl notion of R-separability for the relmholtz equtionF eginD we shll not give ny dditionl detils on these topis t this stge sine onforml deformtions nd REseprility will e studied in the reminder of this pper in the more generl setting of inlevé metrisF e refer to PD QD RD VD PID PPD PQ for luid ounts S of the key results on the seprility nd REseprility properties of täkel metrisD their onnetion to uilling tensorsD qudrti (rst integrls of the geodesi )ow nd symmetry opertors for the vpleE feltrmi opertorF e lso refer to ID QH for reent surveys on seprility on iemnnin mnifolds nd to U for penetrting nlysis of the reltions etween qudrti (rst integrls of the geodesi )owD symmetry opertors nd onserved urrentsD in the generl setting of iemnnin or pseudoEiemnnin mnifoldsF ith these preliminries t hndD we re now redy to introdue the lss of inlevé metris QID QPF es stted oveD inlevé metris rise s nturl generliztion of täkel metris to the se in whih one no longer seeks omplete seprtion of the rmiltonEtoi eqution into ordinry di'erentil equtionsD ut rther seprtion into prtil di'erentil equtions involving groups of vrilesF he seprle oordintes dmitted y inlevé metris re thus generlly not orthogonalD lthough they re orthogonl with respet to groups of vrilesF vet us rell tht our gol in this pper is to rry out for inlevé metris the nlogue of the seprility nd REseprility nlyses of the relmholtz eqution whih hs een extensively worked out for täkel metris in PD QD VD PID PPD PQD nd to show tht the seprility into groups of vriles gives rise to vetor spes of mutully ommuting symmetry opertors for the vpleEfeltrmi opertorD the dimension of whih is determined y the numer of groups of vrilesF sn prtiulrD we will generlize to the se of inlevé metris the oertson onditions nd the hrteriztion thereof in terms of the ii tensorF e now proeed to de(ne the lss of inlevé metris long lines similr to the ones used ove for täkel metrisF vet (M, g) e n nEdimensionl iemnnin mnifold nd let x = (x 1 , . . . , x n ) denote set of lol oordintes on M F e shll onsider prtitions of x into r groups of lol oordintesD vtin indies 1 ≤ i, j . . . ≤ n will e used to lel the lol oordintes on M D greek indies α , β , . . . to lel the r groups of lol oordintesD nd hyrid indies i α , 1 α ≤ i α ≤ l α to denote the lol oordintes within the group x α F nless there is n miguity in the nottion eing usedD in whih se we will write out the summtion signs expliitlyD we shll pply the summtion onvention with the ove rnge of indiesF e generalized Stäckel matrix is nonEsingulr r × r mtrixEvlued funtion S on M of the form vet s αβ denote the oftor of the omponent s αβ of SF e note tht the oftor s βγ will not depend on the group of vriles x β = (x i β ) , 1 β ≤ i β ≤ l β F woreover we shll ssume tht det S s α1 > 0 , ∀ 1 ≤ α ≤ r , @IFIQA in order to work with Riemannian inlevé metrisF T Denition 1.1. Let S be a generalized Stäckel matrix satisfying (1.13). A Painlevé metric is a Riemannian metric g for which there exist local coordinates x = (x 1 , . . . , x r ) such that where each of the quadratic dierential forms is positive-denite in its arguments and depends only on the group of variables x α . e my thus write the metri @IFIRA in lokEdigonl form s st is importnt to note tht even though inlevé metris @IFIRA re lokEdigonlD nd eh qudrti di'erentil form @IFISA de(nes iemnnin metri on the sumnifolds de(ned y the level sets x β = c β , β = αD a Painlevé metric is generally not a direct sum of Riemannian metrics, nor a warped product, except for special non-generic casesF e lso note tht inlevé metris of semiEiemnnin @nd in prtiulr vorentzinA signture n redily e de(ned y modifying the requirement tht eh of the qudrti di'erentil forms G α given y @IFISA e positiveEde(nite to one in whih G α is ssumed to e of signture (p α , q α ) with p α + q α = l α F pinlly we lso remrk tht the inlevé form @IFIRA is oviously invrint under smooth nd invertile hnges of oordintes of the formx α = f α (x α )D where 1 ≤ α ≤ rF vet us ll block orthogonal coordinates system of oordintes (x α ) suh tht the metri g hs the where c α re nonEvnishing slr funtions on M nd the metris G α re given y @IFISAF sn nlogy with the täkel seD we hve Proposition 1.1. A metric g is of the Painlevé form (1.14) if and only if there exist block orthogonal coordinates such that the Hamilton-Jacobi equation admits a parametrized family of solutions which is sum-separable into groups of variables, of the form W = W 1 (x 1 ; a 1 , . . . , a r ) + · · · + W r (x r ; a 1 , . . . , a r ) , @IFIVA depending smoothly on r arbitrary real constants (a 1 := E, a 2 , . . . , a r ) dened on an open subset A ⊂ R r , and satisfying the rank condition U his roposition will e proved in etion R s well s other @intrinsiA hrteriztions of inlevé metrisF sn further nlogy with the täkel seD we now rell tht inlevé metris dmit r linerly indepenE dent qudrti (rst integrls of the geodesi )ow whih re oisson ommutingF sndeedD the summnds W α ppering in @IFIVA stisfy the following set of (rstEorder his QP nd where the (a α ) re ritrry rel seprtion onstntsF st follows now diretly from the seprted equtions @IFPHA nd from the ft tht the generlized täkel mtrix S is nonEsingulr tht one otins r linerly independent oissonEommuting qudrti (rst integrls K (α) of the geodesi )ow y solving for the r seprtion onstnts (a α ) from the seprted equtions @IFPHAF hese re expliitly given y @see QPA he ondition {K (α) , H} = 0 is equivlent to (K ij (α) ) eing symmetri Killing tensorD tht is he ommuttion reltions @IFPRA re thus equivlent to the vnishing of the houten rkets of the pirs of uilling tensors (K (α)ij ), (K (β)ij )F here exist few lssil exmples of inlevé metris in the littertureF hey inlude for instne the di Pirro metrics ISD QPD for whih the rmiltonin of the geodesi )ow is of the form . @IFPSA st my indeed e veri(ed diretly tht the funtion V oissonEommutes with HD nd thus de(nes uilling tensorD whih together with the metri tensor genertes mximl linerly independent set of uilling tensors for generi hoies of the metri funtions c 12 , a 1 , a 2 , a 3 , c 3 in @IFPSAF inlevé metris lso pper in the ontext of geodesilly equivlent metris s metris dmitting projetive symmetriesD see QUD QVD nd lso s instnes of REdimensionl vorentzin metris dmitting uilling tensor PH @see etion U for further remrks on the ltter pointAF et lstD we mention the reent pper y ghnu nd stelli W tht provides lssi(tion of inlevé metris with vnishing iemnn tensor in dimension 3D i.e. in E 3 F e will give some exmples of inlevé metris in ll dimensions stisfying the generlized oertson onditions @see elowA s well s tlogue of suh metris in dimensions 2, 3, 4 in etion PF es we stted oveD our min gol in this pper is to investigte for the lss of inlevé metris the losely relted question of produt seprility for the relmholtz eqution @IFUAD nd the reltionship etween qudrti (rst integrls of the geodesi )ow nd symmetry opertors for the vpleEfeltrmi equtionF he vpleEfeltrmi opertor for inlevé metri g given y @IFIRA n e expressed in terms of the generlized täkel mtrix S nd the vpleEfeltrmi opertors for the r iemnnin metris G β , 1 ≤ β ≤ rD de(ned y @IFISAD orresponding to the loks of vriles x β , 1 ≤ β ≤ rF e hve where ∆ G β denotes the vpleEfeltrmi opertor for the iemnnin metri G β D tht is . @IFPUA e now stte our min resultsD the proofs of whih will e given in etion TF e (rst de(ne the generalized Robertson conditionsD in nlogy with the lssil oertson onditions @IFIIA for täkel metrisFF Denition 1.2. A Painlevé metric g is said to satisfy the generalized Robertson conditions if and only if the dierential conditions he generlized oertson onditions @IFPVA imply tht W e shll e working with oth the forms @IFPVA nd @IFQHA of these onditionsF xote tht if the oertson onditions holdD then the positive vpleEfeltrmi opertor n e written in syntheti form s es will e seen in etion SD the generliztion of the notion of omplete multiplitive seprtion for the relmholtz eqution to the se of seprtion in terms of groups of vriles is given y onsidering prmetrized fmily of produtEseprle solutions of the form where we ssume tht u β = 0F yur (rst result sttes the seprility onditions for the relmholtz equtionD nd gives their interE prettion in terms of the vnishing of the o'Elok digonl omponents of the ii tensorX Theorem 1.1. 1) Given a Painlevé metric g of the form (1.14) satisfying the Robertson conditions (1.28), the Helmholtz equation where ∆ g denotes the Laplace-Beltrami operator (1.26) admits a solution that is product-separable in the r groups of variables (x 1 , . . . , x r ), u = r β=1 u β (x β ; a 1 := λ, a 2 , . . . , a r ) , @IFQTA and satises the rank condition (1.34).
2) The conditions (1.28) may be written equivalently in terms of the Ricci tensor of the Painlevé metric (1.14) as @IFQUA yur next result shows tht the vpleEfeltrmi opertor for inlevé metri stisfying the generE lized oertson onditions dmits r linerly independent mutully ommuting symmetry opertorsX Theorem 1.2. Consider a Painlevé metric (1.14) for which the generalized Robertson conditions (1.28), which imply the separability of the Helmholtz equation, are satised. Then the operators ∆ Kα dened for 2 ≤ α ≤ r by is dened by (1.23), commute with the Laplace-Beltrami operator ∆ g and pairwise commute and admit the separable solutions (1.36) as formal eigenfunctions with the separation constants a α arising from the separation of variables as eigenvalues, yur (nl result shows tht the ove frmework n e expnded just s in the täkel se y onE sidering onforml deformtions of the inlevé metris @IFIRA whih re omptile with the seprtion of the relmholtz eqution into groups of vrilesF his orresponds to generliztion of the importnt notion of REseprility RD VD PQ to the ontext of inlevé metrisF vet us (rst rell tht upon onforml resling of the metri given y where c denotes smooth positive funtionD the vpleEfeltrmi opertor ∆ g oeys the trnsformtion lw nd using the expression @IFPTA of the vpleEfeltrmi opertor for inlevé metri gD the relmholtz eqution −∆ c 4 g u = λ u , @IFRSA tkes the form Theorem 1.3. Let g be a Painlevé metric. Suppose furthermore that g is conformally rescaled by a factor c 4 as in (1.41), where c is chosen so as to satisfy the non-linear PDE and where a 1 is a constant and φ β = φ β (x β ) are arbitrary smooth functions. Then the Helmholtz equation (1.45) for the conformally rescaled metric c 4 g is R-separable in the r groups of variables (x 1 , . . . , x r ). More precisely, if u is given by which is separable in the r groups of variables (x 1 , . . . , x r ) in the sense of (1.33)-(1.34) with the operators B β replaced by the operators e remrk tht the hi of me type given y @IFRPA stis(ed y the onforml ftors c(x) n e viewed s n extension of the generlized oertson onditions to the setting of metris tht re conformally inlevéF woreoverD the existene of suh onforml ftors will e ddressed in etion P through roposition PFIF sn prtiulrD it will e shown there tht suh metris enlrge onsiderly the lss of inlevé metris stisfying the generlized oertson onditions @IFPVAF e onlude this setion y referring to the interesting reent pper y ghnu nd stelli W tht ws pulished during the elortion of the present pperF st turns out tht ghnu nd stelli de(ne the inlevé form of metris like our de(nition IFI in onnetion with the notion of seprility of the rmiltonEtoi equtions in groups of vrilesF hey provide severl intrinsi hrteriztions of inlevé metris extending the ones stted in our setion SF e refer for instne to the eutiful invrint hrteriztion of inlevé metris given in their roposition SFV tht llow them to lssi(y ll inlevé metris in E 3 F 2 Examples of Painlevé metrics satisfying the generalized Robertson conditions sn this setionD we provide severl exmples of inlevé metris stisfying the generlized oertson onditions @IFPVA in ll dimensionsF hen we try to give tlogue E s omplete s possile E of suh inlevé metris in dimensions 2D 3 nd 4F ell our exmples re lol in the sense tht they re de(ned in single oordinte hrtF ytining glol exmples of iemnnin or semiEiemnnin mnifolds dmitting n tls of oordinte hrts in whih the metri is in inlevé form ppers to e hllenging tskD well worthy of further investigtionF his point will e disussed s one of the perspetives listed in etion UF prom the nottions used in de(nition @IFIAD rell tht inlevé metri is given in lol oordintes (x 1 , . . . , x n ) = (x 1 , . . . , x r ) where x α denotes group of vriles indexed y 1 ≤ α ≤ r y IP prom @IFPVAD rell lso tht the oertson onditions red ine s β1 does not depend on the group of vriles x β D these onditions n e equivlently formulted s the lgerioEdi'erentil onditionX re ritrry funtions of the indited group of vrilesF e will use this lst expression of the oertson onditions to (nd di'erent exmples of inlevé metris in ll dimensions tht stisfy themF yur min exmples reX Example 2.1. sf r = 2 nd n = 2D then ny täkel mtrix stis(es utomtilly the usul oertson onditions @PFSRAF he orresponding täkel metris in Ph n e given the following normal form where f α , α = 1, 2 re ritrry funtions of x α suh tht f 1 + f 2 > 0F hus we reover the lssil viouville metrisF sf r = 2 nd n ≥ 3D then ny generlized täkel mtrix stisfy the generlized oertson onditions @PFSRAF he orresponding inlevé metris n e given the following normal form where G 1 , G 2 re iemnnin metris s in @PFSQA nd f 1 = f 1 (x 1 ) is ny positive funtionF xote tht the metris re lssil warped productsF Example 2.2. gonsider generlized täkel mtrix of the form where the entries a αβ , 2 ≤ α ≤ r , 1 ≤ β ≤ rD re rel onstnts hosen suh tht @IFIQA is stis(edF hen it is immedite tht

IQ
for some funtion f 1 depending only on x 1 F rene the oertson onditions @PFSRA re trivilly stis(edF he orresponding inlevé metris re of the generl form of multiply warped products where f α re ritrry positive funtions of x 1 nd G α re given y @PFSQAF e note tht the inverse nisotropi glderón prolem on lss of singulr metris of the form @PFSUA is studied in IIF Example 2.3. yur (nl lss of exmples is the most interesting one nd omes from the theory of geodesilly @or projetivelyA equivlent metris @see for instne TD QWD PWD QUD QVA nd its link to prtiulr täkel systems lled fenenti systems ID SF xote tht it only pplies to täkel metris stisfying the usul oertson onditionsD i.e. we ssume tht r = n in the followingF gonsider täkel mtrix S of ndermonde type @see hm VFS in IA where the funtions f α = f α (x α ) only depend on the vrile x α nd stisfy en esy lultion shows tht from whih we dedue tht the oertson onditions @PFSRA re stis(edF he orresponding täkel metris re given y vet us now use the ove lsses of exmples to give s exhustively s possile list of inlevé metris stisfying the generlized oertson onditions in dimensions n = 2, 3, 4F e lwys ssume tht 2 ≤ r ≤ nF 2D Painlevé metrics. vet n = 2 nd r = 2F hen ording to exmple PFID the only inlevé metris re täkel metris given y for some funtions f 1 nd f 2 suh tht f 1 + f 2 > 0F rene inlevé metris stisfying the oertson onditions in Ph re viouville metrisF 3D Painlevé metrics. vet n = 3F

IR
• sf r = 2 nd sy l 1 = 1, l 2 = 2D then ording to ixmple PFID inlevé metris stisfying the generlized oertson onditions re lssil wrped produtsY more preisely for some positive funtions f 1 nd f 2 depending only on the indited groups of vriles nd ny iemnnin metri • sf r = 3D then Qh inlevé metris re in ft täkel metrisF eording to ixmples PFP nd PFQD we hve the following possile expressions for täkel metris g stisfying the oertson onditions @see lso IVAX where f 1 , h 1 , k 1 re funtions of the vrile x 1 only nd f 2 , f 3 re funtions of the vriles x 2 nd x 3 respetively suh tht f 1 < f 2 < f 3 F e dd lst exmple to this list found y inspetion of the oertson onditions @PFSRAF gonsider the täkel mtrix where a is rel onstnt nd the s ij = s ij (x i ) re ritrry funtions of the indited vriles for whih det S = 0F hen we n hek diretly tht the oertson onditions re stis(ed nd we otin the following expression for the orresponding täkel metris g = (dx 1 ) 2 + 1 s 12 (s 22 s 33 − s 23 s 32 ) (dx 2 ) 2 s 32 − as 33 + (dx 3 ) 2 s 23 − as 22 . @PFTQA xote in prtiulr tht suh metris re wrped produts nd thus dmit onforml uilling vetor (eldF 4D Painlevé metrics. vet n = 4F • sf r = 2 nd l 1 +l 2 = 4D then ording to ixmple PFID inlevé metris tht stisfy the generlized oertson onditions re wrped produts of the type for some positive funtions f 1 nd f 2 depending only on the indited vriles nd ny iemnnin metris G 1 , G 2 of the type @PFSQAF • sf r = 3 nd l 1 = 2D l 2 = l 3 = 1 @the other ses re treted similrlyAD then ording to exmple PFPD we otin the following inlevé metris where h, k, l re positive funtions of the vriles x 1 , x 2 only nd G 1 = (G 1 ) ij (x 1 , x 2 )dx i dx j , i, j = 1, 2 is ny iemnnin metriF pollowing the sme proedure s in exmple @PFTQAD we lso otin the following lss of inlevé metris xote in prtiulr tht suh metris re wrped produts tht dmit onforml uilling vetor (eldF • sf r = 4D the inlevé metris re täkel metrisF eording to exmples PFP nd PFQD possile expressions for inlevé metris stisfying the oertson onditions re where s ij = s ij (x i ) ritrry funtions of the indited vrilesF hen we n hek diretly tht the generlized oertson onditions re stis(ed nd we otin the following expression for the orresponding täkel metris . @PFTWA xote tht suh metris re wrped produts nd tht the metris etween squre rkets re lso wrped produtsF e end this setion y giving some existene results for the onforml ftor c(x) ppering in heorem IFQD in the se in whih M is smooth ompt mnifold of dimension n ≥ 3D with smooth oundry ∂M F e rell from heorem IFQ tht the onforml ftor c(x) must stisfy nonEliner hi of me typeD given yX IT nd where φ β = φ β (x β ) re ritrry smooth funtionsF etting w = c n−2 D we re thus interested in solutions w = c n−2 of the nonEliner ellipti hiX where η is ny suitle smooth positive funtion on ∂M F e n solve @PFUPA y using the wellEknown tehnique of lower nd upper solutions whih we rie)y rell hereF etting we rell tht n upper solution w is funtion in C 2 (M ) ∩ C 0 (M ) stisfying Proof. IF e use the tehnique of lower nd upper solutionsF e de(ne w = where > 0 is smll enoughF husD w ≤ η on ∂M nd we hve so w is lower solutionF sn the sme wyD we de(ne w = C where C is su0iently lrgeF hus w ≥ η nd we hve st follows tht w is n upper solution nd lerly w ≤ wF husD there exist smooth positive solution w of @PFUPA stisfying ≤ w ≤ CF PF sn the se λ ≤ 0D f (x) < λ on M nd η ≤ 1 on ∂M D we de(ne w s the unique solution of the hirihlet prolem ∆ g w + f (x)w = 0, on M, w = η, on ∂M. @PFUUA IU he strong mximum priniple implies tht 0 < w ≤ max η on M F woreoverD g w + f (x, w) = −λ(w) n+2 n−2 ≥ 0F rene w is lower solution of @PFUPAF xowD we de(ne w s the unique solution of the hirihlet prolem on ∂M. @PFUVA eording to the mximum prinipleD we lso hve w ≥ 0 on M F etting v = w − max ηD we see tht on ∂M. @PFVIA henD the mximum priniple implies gin w ≥ wF hen ording to the lower nd upper solutions tehniqueD there exists smooth positive solution w of @PFUPAF 3 Generalized Killing-Eisenhart and Levi-Civita Conditions he proofs of the min results of our pperD tht is heorems IFID IFP nd IFQD mke use of generliztions to inlevé metris of the lssil uillingEiisenhrt equtions nd veviEgivit seprility onditions whih hold for täkel metris @see for exmple PD QD ITD PID PPD QPAF e present these generliztions in the form of the following two lemmsD eginning with the uillingEiisenhrt equtionsF hus in nlogy with the täkel seD we introdue the quntities ρ βγ := s γβ s γ1 . @QFVPA xote tht y the ssumption @IFIQAD we hve s γ1 = 0F he following lemm gives the generliztion to the se of inlevé metris of the uillingEiisenhrt equtions given in ITD PD QD PID PP for täkel metrisX Lemma 3.1. We have, for all 1 ≤ β, δ, γ ≤ r, the identities . @QFVQA e will refer to @QFVQA s the generalized Killing-Eisenhart equations.

@QFVWA
In particular, we have the identities e will likewise refer to the identities @QFVWA s the generalized Levi-Civita conditionsF Remark 3.1. 1) In the case of Stäckel metrics, that is when r = n, the conditions (3.89) reduce to the classical Levi-Civita conditions, given by We shall show at the end of the next section 4 that the generalized Levi-Civita conditions (3.89) hold in fact for all Painlevé metrics (1.14) without assuming our genericity hypothesis (3.88). Nevertheless, it is easier to obtain them from the Killing-Eisenhart equations under the assumption (3.88) as we do below.
Proof. he generl ide ehind the proof is similr to the one tht is used in the lssil täkel seD nd is sed on expressing the integrility onditions for the generlized uillingEiisenhrt @QFVQAD with dditionl twist resulting from the ft tht the seprtion is in groups of vriles onlyF e let 1 ≤ α = β ≤ r denote (xed indies nd introdue the simpli(ed nottion ρ δ := ρ βδ = s δβ s δ1 , @QFWPA so s to mke the expressions it more omptF he generlized uillingEiisenhrt equtions @QFVQA thus tke the form @QFWSA so tht @QFWRA eomes
e now prove the onverse sttementD nmely tht the existene of prmetrized fmily of lokE seprle solutions @SFIPHA of the relmholtz eqution stisfying prtil di'erentil equtions of the form r β=1 c β B β u = a 1 u , @SFIPRA PT nd the rnk ondition @SFIPIA implies tht the underlying metri must e of inlevé formF ustituting u of the form @SFIPHA into @SFIPRA gives hi'erentiting the ltter eqution with respet to a α D we otin we otin the expression of c β for inlevé metri s given in @RFIHUAF Remark 5.1. From (5.123) and the fact that the Stäckel matrix S is invertible, we conclude that the product separable solutions (1.36) satisfy eigenvalue equations of the form where the T α are the linear second order dierential operators given by They will be shown in Theorem 1.2 to be identical to the operators ∆ K (α) dened by (1.38). Hence the separation constants a 1 , . . . , a r can be understood as the natural eigenvalues of the operators ∆ K (α) .
6 Proofs of the main Theorems 6.1 Proof of Theorem 1.1 he ft tht the generlized oertson onditions @IFPVA re su0ient onditions for the produt sepE rility of the relmholtz eqution @IFQSA in the groups of vriles ssoited to inlevé metri follows from roposition SFIF ht remins to e done in order to prove heorem IFI nd wht onstitutes our min tsk is therefore to show tht the generlized oertson onditions re equivlent to the onditions @IFQUA on the ii tensorD thus generlizing the lssil result of iisenhrt IT to inlevé metrisF sn order to do thisD we will show tht @TFIPUA e note tht the expression @TFIPTA of the o' lokEdigonl omponents of the ii tensor R jαk β is independent of the r iemnnin lok metris G β , 1 ≤ β ≤ r de(ned y @IFISAF e lso remrk tht the (rst term in the expression @TFIPUA of T jαk β D involving seond derivtivesD vnishes identilly in the speil se of täkel metris sine the preEftor l α + l β − 2 is zero in tht seF yne we will hve estlished @TFIPTAD it will then follow from vemm QFP nd more preisely from the generlized veviEgivit onditions @QFVWA nd @QFWHA tht the generlized oertson onditions @IFPVA re indeed equivlent to the vnishing onditions @IFQUA on the nonElok digonl omponents of the ii tensorF e therefore proeed to estlish the form @TFIPTA of the ii tensor for inlevé metriF he expression of the ii tensor in terms of the ghristo'el symols is given y where the summtion onvention is pplied with 1 ≤ l, m ≤ n = dim M F sn order to ompute the rightE hnd side of @TFIPVAD we will need expressions for the ghristo'el symols of inlevé metri @IFIRAF sing the stndrd formuls nd writing the inlevé metri @IFIRA in lokEdigonl form s . @TFIQPA sn view of the expressions @TFIQIA of the ghristo'el symolsD it is onvenient to split the sum over l ppering in @TFIPVA into three sumsD the (rst sum orresponding to the vlues of the summtion index l lying in groups of indies di'erent from the groups orresponding to α nd βD nd the remining two to the vlues of l elonging to the groups of indies lelled y α nd β respetivelyF hus we write PV vet us egin with the (rst termF e hveD for γ = α, βD

@TFIQRA
where the summtions hve een written out expliitly to void nottionl miguitiesF e egin with evluting the (rstEderivtive term on the rightEhnd side of @TFIQRA using the expressions @TFIQIA for the ghristo'el symolsD thus otiningD for γ = α, βD log(|g γ |) . @TFIQTA prom the inlevé form @IFIRAD we hve sing the ft tht |G γ | is funtion of the vriles x γ onlyD we otin . @TFIQWA e notie tht the ove expression is independent of the qudrti di'erentil forms G α de(ned y @IFISAF xext we evlute the terms qudrti in the ghristo'el symols in the rightEhnd side of @TFIQRAF eginD we use the ft tht γ = α, β nd the ft tht in the inlevé form @IFIRAD eh of the qudrti PW di'erentil forms G α de(ned y @IFISA depends on the group of vriles x α onlyF e hve lα mα=1α lγ pγ =1γ . @TFIRHA e notie tht the ove expression is gin independent of the qudrti di'erentil forms G α de(ned y @IFISAF vikewiseD we otin for the next qudrti term in the ghristo'el symols tht ppers in the rightEhnd side of @TFIQRAD . @TFIRIA por the third nd (nl qudrti termD we hve lγ mγ =1γ @TFIRPA whih is likewise independent of the qudrti di'erentil forms G α de(ned y @IFISAF utting together the expressions @TFIQSAD @TFIRHAD @TFIRIA nd @TFIRPA we otin @TFIRQA e still need to evlute the the urvture omponents R pα pαjαk β nd R p β p β jαk β D whih will require QH seprte lultionF e hve lα pα=1α where gin we hve written out the summtion signs expliitly to void nottionl miguitiesF e hveD using @TFIQIAD , @TFIRSA so using the ft tht the oftor s α1 is independent of x α D we otin . @TFIRUA e now evlute the qudrti terms in the ghristo'el symols tht pper in the urvture omponent @TFIRRAF sn order to do soD we sustitute into the expression @TFIQPA of the ghristo'el symols the expressions (g α ) iαjα = det S s α1 G α , @TFIRVA whih result from the inlevé form @IFIRAF e otin the following expressions for the ghristo'el symolsD