Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds
AALGEBRAIC INTEGRABILITY CONDITIONSFOR KILLING TENSORSON CONSTANT SECTIONAL CURVATURE MANIFOLDS
KONRAD P. SCHÖBEL
Abstract.
We use an isomorphism between the space of valence two Kil-ling tensors on an n -dimensional constant sectional curvature manifold andthe irreducible GL( n + 1) -representation space of algebraic curvature tensors[MMS04] in order to translate the Nijenhuis integrability conditions for a Kil-ling tensor into purely algebraic integrability conditions for the correspondingalgebraic curvature tensor, resulting in two simple algebraic equations of de-gree two and three. As a first application of this we construct a new family ofintegrable Killing tensors. Introduction
Besides the Euler-Lagrange formalism and the Hamilton formalism, the Hamilton-Jacobi equation is one of the three fundamental reformulations of classical Newto-nian mechanics with wide applications in physics as well as mathematics, rangingfrom classical mechanics over optics and semi-classical quantum mechanics to Rie-mannian geometry. In many cases this first-order non-linear partial differentialequation can be solved by a separation of variables after choosing appropriate coor-dinates. It is therefore a classical problem in Riemannian geometry to classify suchcoordinates [Stä97, LC04, Eis34, KJ80]. The Hamilton-Jacobi equation separates ina given system of orthogonal coordinates if and only if there exists an integrable va-lence two Killing tensor field with simple eigenvalues whose eigenvectors are tangentto the coordinate lines and such that the potential satisfies a certain compatibilitycondition involving this Killing tensor [Ben93]. Integrable Killing tensors are thusan important tool in the study of the seprarability of the Hamilton-Jacobi equation.The present work focusses on the integrability condition for Killing tensors.Killing tensors form a linear space which is invariant under the pullback actionof the manifold’s isometry group. In other words they constitute a representationspace of the isometry group. McLenaghan, Milson and Smirnov identified thisrepresentation in the case of constant sectional curvature manifolds as a certainirreducible representation of the general linear group [MMS04]. More precisely,they used the isometric embeddings of the standard models of constant sectionalcurvature manifolds as hypersurfaces M in a Euclidean vector space ( V, g ) in orderto write Killing tensors as restrictions of homogeneous polynomials on V , where thecoefficients obey certain symmetry relations. This yields in particular an explicitnatural isomorphism between the space of valence two Killing tensors on M and Mathematics Subject Classification.
Key words and phrases.
Killing tensors, integrability, constant sectional curvature manifolds.
E-mail. [email protected] . a r X i v : . [ m a t h . DG ] A p r KONRAD P. SCHÖBEL the irreducible
GL( V ) -representation space of algebraic curvature tensors on theambient space V . Algebraic curvature tensors are valence four tensors subject tothe symmetries of a Riemannian curvature tensor. Furthermore, this isomorphismis equivariant with respect to the action of the isometry group as a subgroup of GL( V ) .This is the starting point for the present work: If the Killing tensor fields on aconstant sectional curvature manifold correspond bijectively to algebraic curvaturetensors, i. e. simple algebraic objects, then their integrability must be expressible asa purely algebraic condition on algebraic curvature tensors. This idea leads finally –after some tensor calculus based on results from the theory of representations of thesymmetric and general linear groups – to the following simple algebraic integrabilityconditions: Main Theorem.
A Killing tensor on a constant sectional curvature manifold M is integrable if and only if the associated algebraic curvature tensor R on V satisfiesthe following two conditions: a b c d ¯ g ij R ib a b R jd c d = 0 (1.1a) a b c d a b c d ¯ g ij ¯ g kl R ib a b R j ka c R ld c d = 0 , (1.1b) where the operators on the left hand side are the Young symmetrisers for completeantisymmetrisation in the (underlined) indices a , b , c , d respectively completesymmetrisation in the indices a , b , c , d . The tensor ¯ g denotes the inner product g on V in case M is not flat. Otherwise, i. e. if M ⊂ V is a hyperplane, ¯ g is the(degenerated) pullback of g via the orthogonal projection V → M . This aproach to integrability of Killing tensor fields on constant sectional cur-vature manifolds has a certain number of advantages. The first and certainly themost important is, that we replace the Nijenhuis integrability conditions – a com-plicated non-linear system of partial differential equations for a tensor field on amanifold – by two simple algebraic equations for a tensor on a vector space. On theone hand this simplifies a numerical treatment considerably. Note that integrabil-ity can be checked by a simple evaluation of polynomials of degree two and three.On the other hand this opens the way for algebraic methods. Our formulation forexample allowed us to show that the third of the Nijenhuis integrability conditionsis redundant for Killing tensors on constant sectional curvature manifolds. In thespecial case of Euclidean -space, this result was already mentioned in a footnote of[HMS05], stating “Steve Czapor (private communication) has simplified the situa-tion considerably. Using Gröbner basis theory, he has shown that (4.4a) and (4.4b)imply (4.4c), for any Killing tensor K ∈ K ( E ) .” Moreover, we can exhibit a family of integrable Killing tensors which arises nat-urally from our algebraic description and extends a known family [IMM00, BM03]which is based on the work of Benenti [Ben92]: Equations (4.4) therein are the Nijenhuis integrability conditions, c. f. (2.3) here.
NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 3
Main Corollary.
There exists a family of integrable Killing tensors on a non-zeroconstant sectional curvature manifold, given by the algebraic curvature tensors R = λ h (cid:55) h + λ h (cid:55) g + λ g (cid:55) g h ∈ Sym V λ , λ , λ ∈ R . (1.2) Here
Sym V is the space of symmetric -tensors and (cid:55) denotes the Kulkarni-Nomizu product. The corresponding Killing tensors read in local coordinates K αβ = 2 λ ( h a a h b b − h a b h b a ) x a x a ∂ α x b ∂ β x b + λ ( h a a x a x a g αβ − h b b ∂ α x b ∂ β x b ) + 2 λ g αβ , where the vector components x i in V are regarded as functions on M by restriction. For Euclidean -space a complete description of integrable Killing tensors hasbeen obtained using computer algebra by Horwood, McLenaghan and Smirnovbased on the prior knowledge of the separable coordinate webs, but a general solu-tion of the integrability conditions has so far been considered intractable [HMS05].Our algebraic equations now render this feasible at least in dimension three. Thisgoes beyond the scope of this article and will be the subject of a forthcoming paper[Sch].In this context it is noteworthy that the first algebraic integrability conditioncan be recast into a variety of different forms. In terms of the curvature form Ω ∈ End( V ) ⊗ Λ V associated ot the algebraic curvature tensor R , condition (1.1a)reads Ω ∧ Ω = 0 , where the wedge denotes the ususal exterior product in the form component andmatrix multiplication in the endomorphism component. Another equivalent form,which makes more explicit the index symmetries in terms of GL( V ) -irrepresenta-tions, is b b d c d a g ij R ib a b R jd c d = 0 , (1.3)where the operator on the left is the Young symmetriser antisymmetrising first in a , b , c , d and then symmetrising in b , b , d . Similar forms can be obtained forthe second algebraic integrability condition (1.1b).The second and related advantage of our approach is, that the above algebraicformulation offers new insight into integrability from the perspectives of represen-tation theory and algebraic geometry as well as geometric invariant theory. Toillustrate this, regard the solutions of the first integrability condition as the alge-braic variety given as the vanishing locus of the following composed map π ◦ ν (wherethe spaces are denoted for convenience by their corresponding GL( V ) -isomorphismclass): Without loss of generality h can be supposed trace free. Note that dimension three of the manifold means dimension four of the ambient vector space.
KONRAD P. SCHÖBEL { } ν −−→ Sym { } π −−→ (cid:110) (cid:111) ν : R a b a b (cid:55)→ R a b a b R c d c d π : T a b a b c d c d (cid:55)→ b b d c d a g a c T a b a b c d c d . (1.4)The first space is the space of algebraic curvature tensors on V and the secondits symmetric product. The third space is the image of the Young symmetriserin V ⊗ . The map π is simply a projection given by an index contraction and aprojection to an irreducible GL( V ) -representation, both commuting. If we pass tothe projectivisation of π ◦ ν , P ( π ◦ ν ) : P { } P ν −−−→ P Sym { } P π −−−→ P (cid:110) (cid:111) , then the map P ν is nothing else than the Veronese embedding. This allows ageometric interpretation of the first integrability condition’s (projectivised) solutionspace as the intersection of a Veronese variety with a certain projective subspace.The same is true for the second integrability condition. Of course, every projectivevariety is isomorphic to an intersection of a Veronese variety with a linear space[Har92], but here all spaces and maps are given explicitly.Note also that in our algebraic setting an isometry invariant characterisation ofthe integrability of Killing tensors as in [HMS05] or [Hor07] reduces to chosing asuitable set of isometry invariants for algebraic curvature tensors and finding therestrictions imposed on them by the equations (1.1). This is essentially a problemin geometric invariant theory. Due to its importance in general relativity, a varietyof such sets have already been proposed in the case of four-dimensional Lorentzspace. Thirdly, we emphasise that our aproach is completely generic in the sense thatis does not depend neither on the dimension of the manifold, nor the value of theconstant sectional curvature nor the signature of the pseudo-Riemannian metric.This becomes manifest in the fact that these data enter the algebraic integrabilityconditions (1.1) only via the signature and the rank of ¯ g . We also remark that ourapproach is coordinate free. We do not rely on any particular choice of coordinatesneither on the manifold, nor on the space of Killing tensors.Finally, a last but not less important advantage comes from the fact that ouralgebraic equations are polynomials in a curvature tensor. Note that we owe thiscircumstance to the fact that Killing tensors on constant sectional curvature mani-folds are described by algebraic curvature tensors and not any other representationspace of the isometry group. This is a rather fortunate happenstance, because al-gebraic properties of curvature tensors are extensively studied – both in differentialgeometry as well as in mathematical physics. Especially the Lorentzian case, focusof interest in general relativity, is important here as it corresponds to hyperbolicspace. In the Riemannian case, corresponding to spheres, we even have a com-plete classification of the symmetry classes of the Riemann polynomials appearingin (1.1) with respect to the isometry group O( V, g ) ⊂ GL( V ) . Our methods areinspired by the corresponding article of Fulling et al. [FKWC92], although we donot rely on results presented there. See [LC04] and the references therein.
NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 5
We hope that our work will pave the way for an algebraic approach to the studyof integrable Killing tensors and seperable coordinates. To this end we list somesuggestions for future research based on our results: • Algebraic interpretation of other families of integrable Killing tensors, suchas the one arising from special conformal Killing tensors [CST00, CS01,Cra03a], cofactor systems [RWML99, Lun01, RW09] or bi-quasi-Hamiltoniansystems [CS02, Cra03b] (see also [Ben05]). We do not yet fully understandhow this – geometrically constructed – family translates to our algebraicframework, but we believe there is a simple algebraic interpretation. Viceversa, we neither know a geometric interpretation of the – algebraicallyconstructed – family we constructed in the present work. • An algebraic compatibility condition for the potential.
So far we disregardedthe compatibility condition for the potential in the Hamilton-Jacobi equa-tion. As integrability, it should be expressible entirely in algebraic termsas well. • Explicit solution of the algebraic integrability conditions.
It is possible tosolve the algebraic integrability conditions explicitely in dimension three.This has been done for -spheres in [Sch] and straightforwardly carriesover to Euclidean -space. In higher dimensions they can be solved usingcomputer algebra, by means of Gröbner bases for example. • Study of the algebraic variety of integrable Killing tensors, defined by thealgebraic integrability conditions, especially its dimension.This paper is organised as follows. In the next section we briefly recall Killingtensors, constant sectional curvature manifolds and the notion of integrability inthis context. In section 3 we regard a special family of integrable Killing tensors.This is followed by a short review of some necessary facts from the representationtheory of symmetric and linear groups in section 4, which can be skipped by areader familiar with them. After that we restate the algebraic characterisation ofKilling tensors in section 5 for our needs. Section 6 is the main part, where wederive the algebraic integrability conditions. As a first application of them, weextend the family from section 3 in the last section.2.
Preliminaries
Killing tensors.
Recall that a
Killing vector on a (pseudo-)Riemannian man-ifold ( M, g ) is a vector field K α on M satisfying the Killing equation ∇ ( α K β ) = 0 where ∇ is Levi-Civita connection of g and round brackets denote complete sym-metrisation in the enclosed indices. Definition 2.1. A Killing tensor on M is a symmetric (2 , -tensor field K αβ satisfying the generalised Killing equation ∇ ( α K βγ ) = 0 . (2.1)Geometrically, a tensor K αβ is a Killing tensor if and only if the amount K αβ ˙ x α ˙ x β is constant on geodesics. KONRAD P. SCHÖBEL
Examples 2.2. (i) The metric is a Killing tensor, since it is covariantly constant.(ii) As a consequence of the Leibniz rule the symmetrised tensor product of twoKilling vectors is a Killing tensor.(iii) The pullback of a Killing tensor under an isometry of M is again a Killingtensor. Integrability.
The metric establishes an isomorphism between the tangentspace and its dual. This identifies co- and contravariant tensor components vialowering or rising indices using the metric. In particular a Killing tensor can beidentified with a (0 , -tensor or a (1 , -tensor, the latter being an endomorphismfield on M . Definition 2.3. A (1 , -tensor field K on M is integrable , if almost every pointon M admits a neighbourhood with local coordinates x α such that the correspondingcoordinate vector fields ∂ α are eigenvector fields of K . Integrability can be characterised using the Nijenhuis torsion N ( X, Y ) := K [ X, Y ] − K [ KX, Y ] − K [ X, KY ] + [
KX, KY ] , given in local coordinates by N αβγ = K αδ ∇ [ γ K δβ ] + ∇ δ K α [ γ K δβ ] , (2.2)where square brackets denote complete antisymmetrisation in the enclosed indices[Nij51]: Theorem 2.4.
Let N be the Nijenhuis torsion of K . Then K is integrable if andonly if the following conditions hold: N δ [ βγ g α ] δ (2.3a) N δ [ βγ K α ] δ (2.3b) N δ [ βγ K α ] ε K εδ (2.3c)2.3. Manifolds with constant sectional curvature.
The Killing equation islinear, so the set of Killing tensors on M is a vector space. Its maximal dimensionis n ( n + 1) ( n + 2)12 n = dim M (2.4)and will be attained if and only if M has constant sectional curvature [Tho86,Wol98]. Every (pseudo-)Riemannian manifold with constant sectional curvature is(up to a rescaling) locally isometric to one of the standard models below. This factallows us to restrict all subsequent considerations to these standard models. Examples 2.5 (Standard models of manifolds with constant sectional curvature) . Let V be a vector space of dimension N := n + 1 , equipped with a non-degenerateinner product g of signature ( p, q ) . Then the (pseudo-)sphere M := { x ∈ V : g ( x, x ) = 1 } is an n -dimensional (pseudo-)Riemannian manifold of constant sectional curvaturewith respect to the metric obtained by restricting g to M . For ( p, q ) = ( n + 1 , this is the standard sphere and for ( p, q ) = ( n, the standard hyperbolic space. Forother choices of the signature we obtain the different de Sitter and anti de Sitter NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 7 spaces. The isometry group of M is the (pseudo-)orthogonal group O( V, g ) ⊂ GL( V ) acting on M by restriction.We can incorporate flat space in this pattern by embedding it as the hyperplane M := { x ∈ V : g ( x, u ) = 1 } in V for some fixed normal vector u ∈ V . The corresponding isometry group is thenembedded in GL( V ) as the semi-direct product M (cid:111) O( M, g ) . Although our approachgoes through for flat spaces as well, this case often has to be treated seperately. Benenti tensors
For the time being let the inner product g be positive definite so that M ⊂ V isthe unit sphere. Consider the diffeomorphism f A : M → Mx (cid:55)→ f A ( x ) := Ax (cid:107) Ax (cid:107) (3.1)for some fixed A ∈ GL( V ) . Since A is linear, f A maps great circles to greatcircles. This means that the metric g and its pullback g A := f ∗ A g have the same(unparametrised) geodesics, so we can apply the following theorem [MT98, MT00]: Theorem 3.1.
If two metrics g and g A on an n -dimensional (pseudo-)Riemannianmanifold have the same unparametrised geodesics, then K := (cid:18) det g det g A (cid:19) n +1 g A is a Killing tensor with respect to g . Corollary 3.2.
Let M ⊂ V be the unit sphere. Then the group GL( V ) generatesa family of Killing tensors on M given by K x ( v, w ) = g ( Ax, Ax ) g ( Av, Aw ) − g ( Ax, Av ) g ( Ax, Aw ) (3.2) for v, w ∈ T x M and A ∈ GL( V ) .Proof. Via the differential of (3.1), ( df A ) x v = g ( Ax, Ax ) Av − g ( Ax, Av ) Ax (cid:107) Ax (cid:107) , (3.3)one computes the pullback metric g A = f ∗ A g as ( g A ) x ( v, w ) = g ( Ax, Ax ) g ( Av, Aw ) − g ( Ax, Av ) g ( Ax, Aw ) (cid:107) Ax (cid:107) . To compute its determinant at a point x , choose an orthonormal basis e , . . . , e n of T x M and extend it with the vector x to an orthonormal basis of V . Since Ax/ (cid:107) Ax (cid:107) is a unit vector normal to T f A ( x ) M , we have (cid:18) det g A det g (cid:19) x = det ( df A ) x = det (cid:16) Ax (cid:107) Ax (cid:107) , ( df A ) x e , . . . , ( df A ) x e n (cid:17) = det (cid:16) Ax (cid:107) Ax (cid:107) , Ae (cid:107) Ax (cid:107) , . . . , Ae n (cid:107) Ax (cid:107) (cid:17) = (cid:18) det A (cid:107) Ax (cid:107) n +1 (cid:19) , KONRAD P. SCHÖBEL where we used (3.3) for the third equality. The claim now follows from theorem3.1, since (cid:18) det g det g A (cid:19) n +1 g A = (cid:18) (cid:107) Ax (cid:107) n +1 det A (cid:19) n +1 g ( Ax, Ax ) g ( Av, Aw ) − g ( Ax, Av ) g ( Ax, Aw ) (cid:107) Ax (cid:107) , differs from (3.2) by a constant. (cid:3) If g is not positive definite, the map (3.1) is not everywhere well defined, butformula (3.2) still gives a well defined Killing tensor, as we will see in section 5. Forflat space the result is analogous, only with more complicated expressions. Corollary 3.3.
Let M ⊂ V be a hyperplane with normal u . Then the group GL( V ) generates a family of Killing tensors on M given by K x ( v, w ) = g ( Ax, u ) g ( Ax, u ) g ( Av, Aw ) − g ( Ax, u ) g ( Ax, Av ) g ( Aw, u ) − g ( Ax, u ) g ( Ax, Aw ) g ( Av, u ) + g ( Ax, Ax ) g ( Av, u ) g ( Aw, u ) . (3.4) Proof.
The proof follows the lines of the proof in the non-flat case, considering thediffeomorphism f A : M → Mx (cid:55)→ f A ( x ) := Axg ( u,Ax ) (3.5)instead of (3.1). This map is line preserving, so we can again apply theorem 3.1.The differential ( df A ) x v = g ( u, Ax ) Av − g ( u, Av ) Axg ( u, Ax ) of (3.5) yields the pullback metric ( g A ) x ( v, w )= 1 g ( u, Ax ) (cid:104) g ( u, Ax ) g ( u, Ax ) g ( Av, Aw ) − g ( u, Av ) g ( u, Ax ) g ( Ax, Aw ) − g ( u, Ax ) g ( u, Aw ) g ( Av, Ax ) + g ( u, Av ) g ( u, Aw ) g ( Ax, Ax ) (cid:105) . As above, we compute the determinant of ( g A ) x using an othonormal basis e , . . . , e n of T x M , (cid:18) det g A det g (cid:19) x = det ( df ) x = det (cid:0) u, ( df ) x e , . . . , ( df ) x e n (cid:1) = det (cid:0) Axg ( Ax,u ) , ( df ) x e , . . . , ( df ) x e n (cid:1) = det (cid:16) Axg ( Ax,u ) , Ae g ( Ax,u ) , . . . , Ae n g ( Ax,u ) (cid:17) = (cid:18) det Ag ( Ax, u ) n +1 det (cid:0) x, e , . . . , e n (cid:1)(cid:19) = (cid:18) det Ag ( Ax, u ) n +1 (cid:19) , and the corollary follows from theorem 3.1. (cid:3) Killing tensors of type (3.2) respectively (3.4) coincide with those introduced inlocal coordinates in [Ben92]. Following [IMM00, BM03] we will call them
Benentitensors . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 9 Some facts from representation theory
We briefly collect some facts we need from the representation theory of symmetricand general linear groups. More details can be found in standard textbooks.4.1.
Young symmetrisers and the Littlewood-Richardson rule.
The iso-morphism classes of irreducible representations (“irreps”) of S d are labelled by par-titions of d , i. e. integers d (cid:62) d (cid:62) . . . (cid:62) d r > with d + . . . + d r = d . It is usefulto depict partitions as so called Young frames , as in the following example:
Example 4.1.
The partitions of are , , , and with corresponding Young frames . The dimension of an irrep corresponding to a Young frame is given by dividing d ! by the product of the hook lengths of all boxes of the frame, where the hooklength of a box is ( the number of boxes to the right ) + 1 + ( the number of boxes below ) . This is the so called hook formula . Example 4.2.
The hook lengths of the boxes of the Young frames in example 4.1are . The corresponding dimensions are thus , , , , . The irreps of S d can be realised on subspaces of the group algebra R S d = (cid:40) (cid:88) π ∈ S d λ π π : λ π ∈ R (cid:41) . This is the free real vector space over S d as a set, endowed with the obvious multi-plication given by extending the group multiplication in S d linearly. Multiplicationwith elements of S d from the left defines a representation of S d on R S d . One thenconstructs projectors onto irreducible subspaces as follows.Let τ be a Young frame whose d boxes are filled with the integers , . . . , d withoutrepetition in an arbitrary order. We call this a Young tableau . For each row r of τ let S r ⊂ S d be the symmetric group of permutations of the labels in r and similarly S c ⊂ S d for each column c of τ . The Young symmetriser corresponding to τ is thenthe element in R S d defined by (cid:89) r row of τ (cid:16) (cid:88) π ∈ S r π (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) row symmetriser of r (cid:89) c column of τ (cid:16) (cid:88) π ∈ S c (sign π ) π (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) column antisymmetriser of c . (4.1)We will often identify a Young tableau with its corresponding Young symmetriseras in the following example. Example 4.3. (cid:0) e + (12) (cid:1)(cid:0) e + (34) (cid:1)(cid:0) e − (13) (cid:1)(cid:0) e − (24) (cid:1) (4.2) = e + (12) + (34) − (13) − (24) + (12)(34) + (13)(24) − (132) − (234) − (124) − (143) + (1423) + (1324) − (1234) − (1432) + (14)(23) If we denote by h τ the product of the hook lengths of all boxes in a Youngtableau τ , the corresponding Young symmetriser satisfies τ = h τ τ . (4.3)Every element in R S d is at the same time a linear operator on R S d via multipli-cation from the right. Then the image of τ is an irreducible subrepresentation of R S d whose isomorphism class is given by the Young frame of τ . Rewriting (4.3)we get a projector onto the corresponding subspace: (cid:18) τh τ (cid:19) = τh τ . (4.4)The group algebra R S d carries a natural S d -invariant inner product defined bytaking the basis S d to be orthonormal. With respect to this inner product theadjoint of an element in R S d is (cid:16) (cid:88) π ∈ S d α π π (cid:17) (cid:63) = (cid:88) π ∈ S d α π π − . Note that the column symmetrisers of a Young tableau are self-adjoint and com-mute and likewise for its row antisymmetrisers. Taking the adjoint of a Youngsymmetriser therefore simply exchanges the two products in (4.1).
Example 4.4. (cid:63) = 13 24 1 2 3 4 = (cid:0) e − (13) (cid:1)(cid:0) e − (24) (cid:1)(cid:0) e + (12) (cid:1)(cid:0) e + (34) (cid:1) (4.5)We see that Young symmetrisers are in general not self-adjoint, so that thecorresponding Young projectors (4.4) will not be orthogonal. However, a Youngprojector and its adjoint project onto isomorphic irreps [Ful97]. From (4.1) and(4.3) follows that up to an apropriate factor the element τ τ (cid:63) is an orthogonalprojector onto the image of τ and similarly τ (cid:63) τ onto the image of τ (cid:63) .Two representations λ and λ of S d on V respectively S d on V determine arepresentation of S d × S d on V ⊗ V given by ( g × g )( v ⊗ v ) := ( g v ) ⊗ ( g v ) g ∈ S d , g ∈ S d . Via the inclusion S d × S d (cid:44) → S d + d this induces a representation λ (cid:2) λ of S d + d on V ⊗ V , called the exterior tensor product of λ and λ . The Littlewood-Richardson rule tells us how this product decomposes into irreps: Theorem 4.5 (The Littlewood-Richardson rule) . The decomposition of the exte-rior tensor product λ (cid:2) λ of two irreps λ of S d and λ of S d into irreps of S d + d is given by the following algorithm. First label all the boxes in λ with theircorresponding row number. Then add the labelled boxes of λ to λ – one by onefrom top to bottom – respecting at each step the following rules: NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 11 (i) The obtained frame is a legitimate Young frame.(ii) No two boxes in the same column are labelled equally.(iii) If the labels are read off from right to left along the rows from top to bottom,one never encounters more 1’s than 2’s, and so on.Each of the distinct Young frames constructed in this way specifies an irreduciblesum term in the decomposition of λ (cid:2) λ with the correpsonding multiplicity, sincethe same shaped Young frame may arise in more than one way. Since the exteriortensor product is commutative, one can choose the simpler Young frame for λ . Example 4.6. (cid:2) ∼ = 1 1 ⊕ ⊕ (4.6) (cid:2) ∼ = 12 ⊕ ⊕ (4.7) (cid:2) ∼ = 1 1 1 ⊕ (4.8)4.2. Weyl’s construction and algebraic curvature tensors.
Every Youngtableau τ gives rise to a GL( V ) -irrep in the following way, called Weyl’s construc-tion. Consider the d -fold tensor product V ⊗ d as a representation space for both GL( V ) and S d with respect to the commuting actions g ( v i ⊗ . . . ⊗ v i d ) := ( gv i ) ⊗ . . . ⊗ ( gv i d ) g ∈ GL( V ) π ( v ⊗ . . . ⊗ v d ) := v π − (1) ⊗ . . . ⊗ v π − ( d ) π ∈ S d . Each element in R S d gives a linear operator on V ⊗ d by linearly extending theaction of S d . The image of a Young symmetriser τ ∈ R S d is then an irreducible GL( V ) -subrepresentation. Considering instead the dual action of GL( V ) on thedual ¯ V of V yields the (non-isomorphic) dual representation on ¯ V ⊗ d . Example 4.7.
The Young symmetriser (4.2) determines the following operator on ¯ V ⊗ whose image constitutes a GL( V ) -irrep: a a b b T a a b b = T a a b b − T a a b b − T a a b b + T a a b b + T b a a b − T a b a b − T b a b a + T a b b a + T a b b a − T b a b a − T a b a b + T b a a b + T b b a a − T b b a a − T b b a a + T b b a a . (4.9) Note that on the level of tensor components one gets the correct action of S d bypermuting index names, not index positions. In the same way we can construct an irreducible
GL( V ) -subrepresentation fromthe adjoint τ (cid:63) of a Young tableau τ . Example 4.8. a a b b (cid:63) T a b a b = T a b a b + T a b a b + T a b a b + T a b a b − T b a a b − T a a b b − T b b a a − T a b b a − T a b b a − T b b a a − T a a b b − T b a a b + T b a b a + T b a b a + T b a b a + T b a b a . (4.10) Example 4.9 (Algebraic curvature tensors) . An algebraic curvature tensor on V isan element R ∈ ¯ V ⊗ satisfying the symmetry relations of a Riemannian curvaturetensor, i.e.: antisymmetry: R b a a b = − R a b a b = R a b b a (4.11a) pair symmetry: R a b a b = R a b a b (4.11b) Bianchi identity: R a b a b + R a a b b + R a b b a = 0 (4.11c) A little computation shows, that on one hand any tensor of the form (4.10) hasthese symmetries and that on the other hand any tensor having these symmetriesverifies a a b b (cid:63) R a b a b = R a b a b . This means that algebraic curvature tensors form an irreducible
GL( V ) -representa-tion space. Example 4.10 (Symmetrised algebraic curvature tensors) . A symmetrised alge-braic curvature tensor on V is an element S ∈ ¯ V ⊗ satisfying the following sym-metry relations: symmetry: S a a b b = + S a a b b = S a a b b (4.12a) pair symmetry: S b b a a = S a a b b (4.12b) Bianchi identity: S a a b b + S a b a b + S a b b a = 0 (4.12c) As in the previous example, this is equivalent to a a b b S a a b b = S a a b b so that symmetrised algebraic curvature tensors form another irreducible GL( V ) -representation space. Remark 4.11 (Bianchi identity) . In presence of the first two symmetries, there areseveral equivalent forms of the Bianchi identity in both cases. First, we can write itas vanishing cyclic sum over any three of the four indices, for example as R a b a b + R b a a b + R a a b b = 0 . Second, for (symmetrised) algebraic curvature tensors theBianchi identity is equivalent to the vanishing of the complete antisymmetrisation(symmetrisation) in any three of the four indices, for example to b a b R a b a b = 0 or a b b S a a b b = 0 . (4.13) In the following we will refer to all these forms as “Bianchi identity”.
NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 13
The
GL( V ) -irreps constructed from τ and τ (cid:63) are isomorphic. Example 4.12.
An explicit isomorphism between the irreps of
GL( V ) on algebraiccurvature tensors respectively on symmetrised algebraic curvature tensors is givenby S a a b b = 1 √ (cid:0) R a b a b + R a b a b (cid:1) (4.14a) R a b a b = 1 √ (cid:0) S a a b b − S a b b a (cid:1) , (4.14b) which is easily checked using the symmetries (4.11) and (4.12) . Two Young tableaux determine isomorphic
GL( V ) -representations if and only ifthey fill the same Young frame λ . Their dimension can be computed by labellingeach box of λ with ( number of the box’ column ) + N − ( number of the box’ row ) , taking the product of these labels and dividing by the product h λ of the hooklengths of λ . It is standard to denote the isomorphism class obtained from λ viathe Weyl construcction by { λ } . Example 4.13.
The isomorphism class of the irrepresentations given by (sym-metrised) algebraic curvature tensors is { } an has dimension dim (cid:110) (cid:111) = ( N − N ( N + 1)12 . (4.15)The dual pairing between ¯ V ⊗ d and V ⊗ d is given by index contraction. From theidentity S i ··· i d T i π (1) ··· i π ( d ) = S i π − ··· i π − d ) T i ··· i d we deduce S i ··· i d (cid:0) πT i ··· i d (cid:1) = (cid:0) π − S i ··· i d (cid:1) T i ··· i d = (cid:0) π (cid:63) S i ··· i d (cid:1) T i ··· i d . This means that with respect to the dual pairing the adjoint of the linear operatoron V ⊗ d given by an element τ ∈ R S d acting on upper indices is given by τ (cid:63) actingon lower indices. To save notation we will use parentheses as above to indicatewhether a given element of R S d acts on upper or lower indices.5. An algebraic characterisation of Killing tensors
Recall that we consider standard models of constant sectional curvature mani-folds M , embedded isometrically as hypersurfaces in a Euclidean vector space ( V, g ) .As common in relativity, we distinguish coordinates on M and V by index types: Convention 5.1.
Throughout this exposition we use latin indices a, b, c, . . . forcomponents in V (ranging from to n ) and greek indices α, β, γ, . . . for local coor-dinates on M (ranging from to n ). We can then denote both the inner producton V as well as the induced metric on M by the same letter g and distinguish bothvia the type of indices. Consequently, latin indices are rised and lowered using g ab and greek ones using g αβ . The key result for our algebraic characterisation of integrability stems from Mc-Lenaghan, Milson & Smirnov and is a special case of theorem 3.5 in [MMS04]:
Theorem 5.2.
Let M ⊂ V be one of the standard models for constant sectionalcurvature manifolds as in example 2.5.(i) There is an isomorphism between the irreducible GL( V ) -representation spaceof antisymmetric tensors A ab on V and the vector space of Killing vectors K on M , given by K x ( v ) := A ab x a v b x ∈ M, v ∈ T x M , when K is written covariantly.(ii) There is an isomorphism between the irreducible GL( V ) -representation spaceof algebraic curvature tensors R a b a b on V and the vector space of Killingtensors K on M , given by K x ( v, w ) := R a b a b x a v b x a w b x ∈ M, v, w ∈ T x M , (5.1) when K is written covariantly.Both isomorphisms are equivariant with respect to the action of the isometry groupof M as a subgroup of GL( V ) . First note that due to the term x a x a the tensor R a b a b in (5.1) is implicitelysymmetrised in the indices a , a and can therefore be replaced by the correspondingsymmetrised algebraic curvature tensor (4.14a). Since this will simplify subsequentcomputations considerably, we reformulate the the second part of the theorem: Corollary 5.3.
Let M ⊂ V be one of the standard models as in example 2.5. Then K ( v, w ) := S a a b b x a x a v b w b x ∈ M, v, w ∈ T x M (5.2) defines an isomorphism between the irreducible GL( V ) -representation space of sym-metrised algebraic curvature tensors S a a b b and the vector space of Killing tensors K on M , which is equivariant with respect to the action of the isometry group. We include a short proof here, because some ideas and intermediate results willbe useful in subsequent computations.
Remark 5.4.
If we consider the standard coordinates x a of a vector x ∈ V asfunctions on M ⊂ V by restriction, then we can write for any tangent vector u ∈ T x M ⊂ V with coordinates u a in V : ∇ u x a = u a , (5.3) where ∇ denotes the Levi-Civita connection of the metric on M .Proof (of corollary 5.3). We first show that the tensor (5.2) actually is a Killingtensor. Extend the vectors u, v, w ∈ T x M to arbitrary vector fields ¯ u, ¯ v, ¯ w on M .Using (5.2) and (5.3), we compute ∇ u K ( v, w ) = ∇ ¯ u (cid:0) K (¯ v, ¯ w ) (cid:1) − K (cid:0) ∇ ¯ u ¯ v, ¯ w (cid:1) − K (cid:0) ¯ v, ∇ ¯ u ¯ w (cid:1) = S a a b b (cid:16) u a x a v b w b + x a u a v b w b + x a x a ( ∇ ¯ u ∇ ¯ v − ∇ ∇ ¯ u ¯ v ) x b w b + x a x a v b ( ∇ ¯ u ∇ ¯ w − ∇ ∇ ¯ u ¯ w ) x b (cid:17) . (5.4)The operator H ( u, v ) = ∇ ¯ u ∇ ¯ v − ∇ ∇ ¯ u ¯ v is the Hesse operator and does not dependon the extensions ¯ u and ¯ v of u and v . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 15
Lemma 5.5.
The Hesse form of the function x b is given by H ( u, v ) x b = − g ( v, w ) x b if M is not flat and zero otherwise.Proof. Extend the vector fields ¯ u, ¯ v on M further to all of V and denote the Levi-Civita connection in V by ¯ ∇ . Then, using (5.3), H ( u, v ) x a = ∇ ¯ u ∇ ¯ v x a − ∇ ∇ ¯ u ¯ v x a = ∇ ¯ u (¯ v a ) − (cid:0) ∇ ¯ u ¯ v (cid:1) a = ¯ ∇ ¯ u (¯ v a ) − (cid:0) ∇ ¯ u ¯ v (cid:1) a = (cid:2) ¯ ∇ ¯ u ¯ v − ∇ ¯ u ¯ v (cid:3) a = (cid:2) II( u, v ) (cid:3) a It is not difficult to show that the second fundamental form of M ⊂ V is II x ( u, v ) = − g ( u, v ) x if M is not flat. Otherwise the lemma is trivial. (cid:3) We resume the proof of corollary 5.3. Together with the Bianchi identity thelemma shows that the terms in (5.4) containing the Hesse form H ( u, v ) x b vanish.Using the symmetry of S a a b b in a , a we get ∇ u K ( v, w ) = 2 S a a b b x a u a v b w b . We reformulate the results obtained so far in local coordinates, using (5.3) again: K αβ = S a a b b x a x a ∇ α x b ∇ β x b (5.5a) ∇ γ K αβ = 2 S c c d d x c ∇ γ x c ∇ α x d ∇ β x d (5.5b)That K satisfies the Killing equation (2.1) is now a direct consequence of the Bianchiidentity.We continue the proof by showing that the map defined by (5.2) is injective.Suppose S a a b b x a x a v b w b = 0 (5.6)for all x, v, w ∈ V with x ∈ M and v, w ∈ T x M . From the Bianchi identity we seethat (5.6) is trivially satisfied if v = x or w = x . We can thus drop the condition v, w ∈ T x M by decomposing v, w ∈ V according to the splitting V = T x M ⊕ R x . Wecan also drop the condition x ∈ M , since R M is dense in V . Finally, by polarisationwe obtain S a a b b x a y a v b w b = 0 for all x, y, v, w ∈ V which is equivalent to S = 0 .Bijectivity now follows from the fact that the dimensions (4.15) and (2.4) of bothspaces coincide for N = n + 1 . Equivariance is evident. (cid:3) Definition 5.6.
The
Kulkarni-Nomizu product h (cid:55) k of two symmetric tensors h and k is the algebraic curvature tensor ( h (cid:55) k ) a b a b := h a a k b b − h a b k b a − h b a k a b + h b b k a a = 14 a a b b (cid:63) h a a k b b . In the language of representation theory this product corresponds to the projectionof (4.6) to the -component.
Example 5.7 (The metric) . If M is not flat, the metric as a Killing tensor on M is represented by the algebraic curvature tensor g (cid:55) g . This follows from (5.1) and (cid:0) g (cid:55) g (cid:1) a b a b = 18 a a b b (cid:63) g a a g b b = g a a g b b − g a b g a b , (5.7) since g a a x a x a = 1 and g a b x a v b = 0 . In the flat case the metric is represented by ( u ⊗ u ) (cid:55) g , given by (cid:0) ( u ⊗ u ) (cid:55) g (cid:1) a b a b = 14 a a b b (cid:63) u a u a g b b = u a u a g b b − u a u b g b a − u b u a g a b − u b u b g a a , (5.8) since u a x a = 1 and u b v b = 0 . The isometry group of M is a subgroup of GL( V ) . As a consequence of theorem5.2 its action on the space of Killing tensors extends to a natural action of GL( V ) . Example 5.8 (Benenti tensors) . Rewriting (3.2) respectively (3.4) in the form (5.1) shows that Benenti tensors are represented by the algebraic curvature tensors ( Ag ) (cid:55) ( Ag ) respectively ( Au ⊗ Au ) (cid:55) Ag if M is flat. Here Ag denotes the image of g under the action of A ∈ GL( V ) onsymmetric tensors on V . We can interpret this by saying that Benenti tensors formthe orbit of the metric under the natural action of GL( V ) on Killing tensors. The algebraic integrability conditions
We saw that Killing tensors on a constant sectional curvature manifold corre-spond to algebraic curvature tensors. The aim of this section is to translate theNijenhuis integrability conditions for such Killing tensors into algebraic integrabilityconditions on the corresponding algebraic curvature tensors.First note that in the integrability conditions (2.3) the Nijenhuis torsion (2.2)appears only antisymmetrised in its two lower indices β , γ . To simplify computationswe will thus replace the Nijenhuis torsion N αβγ in the integrability conditions bythe tensor ¯ N αβγ := (cid:0) K αδ ∇ γ K δβ + K δβ ∇ δ K αγ (cid:1) ¯ N α [ βγ ] = N αβγ . Together with (5.5) this can be written as ¯ N αβγ = S a a b b S c c d d x a x a x c ∇ α x b ∇ δ x b ∇ γ x c ∇ δ x d ∇ β x d + S a a b b S c c d d x a x a x c ∇ δ x b ∇ β x b ∇ δ x c ∇ α x d ∇ γ x d . Lemma 6.1.
Let M be one of the standard models for constant sectional curvaturemanifolds as in example 2.5. Then ∇ δ x a ∇ δ x b = (cid:40) g ab − u a u b if M is flat g ab − x a x b otherwise . Proof.
Let e , . . . , e n be a basis of T x M and complete it with a unit normal vector e := u to a basis of V . Then on one hand n (cid:88) i,j =0 g ( e i , e j ) ∇ e i x a ∇ e j x b = n (cid:88) i,j =1 g ( e i , e j ) ∇ e i x a ∇ e j x b + ∇ u x a ∇ u x b = g αβ ∇ α x a ∇ β x b + u a u b . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 17
On the other hand, choosing the standard basis of V instead, the left hand side isjust g ab . This proves the lemma, remarking that we can choose u = x if M is notflat. (cid:3) For flat M the lemma yields ¯ N αβγ = ¯ g b d S a a b b S c c d d x a x a x c ∇ α x b ∇ β x d ∇ γ x c + ¯ g b c S a a b b S c c d d x a x a x c ∇ α x d ∇ β x b ∇ γ x d , (6.1)where ¯ g := g ab − u a u b . In all other cases we have ¯ N αβγ = (cid:0) g b d − x b x d (cid:1) S a a b b S c c d d x a x a x c ∇ α x b ∇ β x d ∇ γ x c + (cid:0) g b c − x b x c (cid:1) S a a b b S c c d d x a x a x c ∇ α x d ∇ β x b ∇ γ x d . But here the two subtracted terms vanish by the Bianchi identity because theycontain the terms x a x a x b S a a b b respectively x a x a x b S a a b b . This allowsus to use (6.1) for all models M if we define ¯ g ab := (cid:40) g ab − u a u b if M is flat g ab otherwise . (6.2)In the case of a hyperplane M ⊂ V , the tensor ¯ g ab is the pullback of the metric on M via the orthogonal projection V → M and thus degenerated. Note that we stilllower and rise indices with the metric g ab and not with ¯ g ab .In (6.1) the lower indices b , d respectively b , c are contracted with ¯ g . We canmake use of the symmetries of S a a b b to bring these indices to the first position: ¯ N αβγ = ¯ g b d S b b a a S d d c c x a x a x c ∇ α x b ∇ β x d ∇ γ x c + ¯ g b c S b b a a S c c d d x a x a x c ∇ α x d ∇ β x b ∇ γ x d . Renaming, lowering and rising indices appropriately finally results in ¯ N αβγ = ¯ g ij (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) x b x b x d ∇ α x a ∇ β x c ∇ γ x d . (6.3)In what follows we will substitute this expression together with (5.5a) into each ofthe three integrability conditions (2.3) and transform them into purely algebraicintegrability conditions.6.1. The first integrability condition.
The first integrability condition (2.3a)can be written as ¯ N [ αβγ ] = 0 . For the expression (6.3) this is equivalent to thevanishing of the antisymmetrisation in the upper indices a , c , d : ¯ g ij (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) x b x b x d ∇ α x [ a ∇ β x c ∇ γ x d ] = 0 . Due to the symmetry (4.12a) the second term to vanishes. If we write u , v and w for the tangent vectors ∂ α , ∂ β respectively ∂ γ and use (5.3) in order to get rid ofthe indices and ∇ ’s, we get the condition ¯ g ij S ia b b S jc d d x b x b x d u [ a v c w d ] = 0 ∀ x ∈ M ∀ u, v, w ∈ T x M (6.4)on the symmetrised algebraic curvature tensor S a a b b .Note that tensors of the form x b x b x d u [ a v c w d ] are completely symmetricin the indices b , b , d and completely antisymmetric in the remaining indices a , c , d . On the level of isomorphism classes the decomposition of the correspond-ing GL( V ) -representation space Sym V ⊗ Λ V into irreducible components resultsfrom (4.8). The following lemma gives an explicit realisation of this decompositionin terms of orthogonal projection operators: Lemma 6.2. q ! a ... a q · p ! s ··· s p (6.5) = pq +1 ( p + q ) p ! q ! s ··· s p a ... a q s ··· s p a ... a q (cid:63) + qp +1 ( p + q ) p ! q ! a s ··· s p ... a q (cid:63) a s ··· s p ... a q In particular, for p = q = 3 : c d a · b b d = 12 b b d c d a b b d c d a (cid:63) + 12 c b b d d a (cid:63) c b b d d a . (6.6) Proof.
Write (6.5) as P = P + P . Decomposing temporarily the Young sym-metrisers on the right hand side as in (4.1) into a product of a symmetriser and anantisymmetriser and using (4.3), one easily checks that P , P and P are orthog-onal projectors verifying P P = 0 = P P , P P = P and P P = P . Therefore P + P is an orthogonal projector with image im P ⊕ im P ⊆ im P . The decom-position of the isomorphism class of im P into irreducible components is given bythe Littlewood-Richardson rule as q (cid:40) ... (cid:2) p (cid:122) (cid:125)(cid:124) (cid:123) ··· ∼ = p (cid:122) (cid:125)(cid:124) (cid:123) ··· ... ⊕ ··· ... (cid:41) q . The Young frames on the right hand side are those appearing in the expression for P respectively P . Hence they describe the isomorphism classes of im P and im P .This shows that im P and im( P + P ) = im P ⊕ im P have the same dimensionand are thus equal. This implies P = P + P . (cid:3) Remark 6.3.
The lemma can be interpreted as an explicit splitting of the terms inthe long exact sequence → Sym d V → . . . → Sym p V ⊗ Λ q V → Sym p − ⊗ Λ q +1 → . . . → Λ d V → , known as the Koszul complex . Applying (6.6) to the tensor x b x b x d u [ a v c w d ] , we conclude that x b x b x d u [ a v c w d ] = 12 b b d b c d a b b d + c d a c b b d c d a x b x b x d u [ a v c w d ] = constant · b b d c d a + c b b d d a (cid:63) x b x b x d u a v c w d . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 19
In the last step we omitted an explicit antisymmetrisation in a , c , d , since thisis already carried out implicitely by each of the Young symmetrisers. Accordinglythe left hand side of (6.4) splits into two terms. The following lemma shows thatthe second of them, namely ¯ g ij S ia b b S jc d d (cid:32) c b b d d a (cid:63) x b x b x d u a v c w d (cid:33) = (cid:32) c b b d d a ¯ g ij S ia b b S jc d d (cid:33) x b x b x d u a v c w d , vanishes identically. Lemma 6.4. c b b d d a ¯ g ij S ia b b S jc d d = 0 (6.7)Before we prove the lemma, we mention an identity which we will frequently useand which is obtained from symmetrising the Bianchi identity (4.12c) in b , b : b b S ia b b = − b b S ib b a . (6.8)We refer to this identity as symmetrised Bianchi identity . Proof. c b b d d a ¯ g ij S ia b b S jc d d = c b b d c d a ¯ g ij S ia b b S jc d d = c b b d g ij (cid:0) S ia b b S jc d d − S id b b S jc d a + S id b b S ja d c − S ia b b S jd d c + S ic b b S jd d a − S ic b b S ja d d (cid:1) Regard the parenthesis under complete symmetrisation in c , b , b , d . The lasttwo terms vanish due to the Bianchi identity (4.13). Renaming i, j as j, i in thethird term shows that it cancels the fourth. That the first two also cancel eachother can be seen by applying twice the symmetrised Bianchi identity (6.8), onceto S ia b b and once to S ic d d . (cid:3) Remark 6.5.
If the inner product g is positive definite, then M is the unit sphere.In this case the lemma above also follows from the symmetry classification of Rie-mann tensor polynomials [FKWC92] , since the tensor g ij R ia b b R jc d d has sym-metry type ⊕ ⊕ ⊕ and g ij S ia b b S jc d d can be expressed in terms of this tensor via (4.14a) . Resuming, the first integrability condition is equivalent to ¯ g ij S ia b b S jc d d b b d c d a x b x b x d u a v c w d = 0 ∀ x ∈ M ∀ u, v, w ∈ T x M . (6.9)
We can drop the restriction u, v, w ∈ T x M in (6.9). Indeed, decomposing u, v, w ∈ V according to the splitting V = T x M ⊕ R x shows b b d c d a x b x b x d u a v c w d = 0 if u = x or v = x or w = x . This follows from Dirichlet’s drawer principle. This trick is crucial, as it allowsus to deal with
GL( N ) -representations instead of the more complicated O( N ) -representations. Obviously, we can also drop the restriction x ∈ M since R M isdense in V .Next we use the fact that tensors of the form x b x b x d and tensors of the form b b d x b y b z d both span the same space, namely Sym V . With this remarkcondition (6.9) is now equivalent to ¯ g ij S ia b b S jc d d b b d c d a b b d x b y b z d u a v c w d = 0 ∀ x, y, z, u, v, w ∈ V .
But the operator b b d c d a b b d = b b d b c d a b b d = 14! b b d b c d a b c d a b b d = 14! b b d c d a b b d c d a (cid:63) is self-adjoint and hence ¯ g ij S ia b b S jc d d b b d c d a b b d x b y b z d u a v c w d = 14! b b d c d a b b d c d a (cid:63) ¯ g ij S ia b b S jc d d x b y b z d u a v c w d . Now recall that V ⊗ is spanned by tensors of the form x b y b z d u a v c w d and thatthe dual pairing ¯ V ⊗ × V ⊗ → R is non-degenerate. This allows us finally to writethe first integrability condition in the purely algebraic form b b d c d a b b d c d a (cid:63) ¯ g ij S ia b b S jc d d = 0 , (6.10)which is independent of x, y, z, u, v, w . We will now give a number of equivalentformulations. Proposition 6.6 (First integrability condition) . The following conditions are equiv-alent to the first integrability condition (2.3a) for a Killing tensor on a constantsectional curvature manifold M : NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 21 (i) The corresponding symmetrised algebraic curvature tensor S satisfies P ¯ g ij S ia b b S jc d d = 0 , (6.11) where P is any of the following symmetry operators ( a ) b b d c d a (cid:63) ( b ) c d a b b d ( c ) b c d a ( d ) b b d c d a (6.12) (ii) The corresponding algebraic curvature tensor R satisfies P ¯ g ij R ib a b R jd c d = 0 , (6.13) where P is any of the symmetry operators (6.12) .If M is not flat, this is is in addition equivalent to:(iii) The curvature form Ω ∈ End( V ) ⊗ Λ V of R , defined by Ω a b := R a b a b dx a ∧ dx b satisfies Ω ∧ Ω = 0 , (6.14) where the wedge product is defined by taking the exterior product in the Λ V -component and usual matrix multiplication in the End( V ) -component. Remark 6.7. In (6.12) we can permute the labels a , b , c , d arbitrarily as wellas exchange the labels b , d . This follows from the integrability condition in theform (6.11c) . In particular, in (6.11b) one can antisymmetrise in any three of thefour indices a , b , c , d and symmetrise in the remaining three.Proof. We showed that the first integrability condition (2.3a) is equivalent to (6.10).But this is equivalent to condition (6.11a) since the kernels of
P P (cid:63) and P (cid:63) coincide: P P (cid:63) v = 0 ⇔ (cid:107) P (cid:63) v (cid:107) = (cid:104) v | P P (cid:63) v (cid:105) = 0 ⇔ P (cid:63) v = 0 . (6.15)The equivalence (6.11a) ⇔ (6.11b) follows from (6.6) combined with (6.7).The implication (6.11c) ⇒ (6.11d) is trivial. We finish the proof of part (i) byproving (6.11d) ⇒ (6.11b) ⇒ (6.11c) through a stepwise manipulation of b b d c d a ¯ g ij S ia b b S jc d d = b b d b c d a ¯ g ij S ia b b S jc d d . (6.16a)In order to sum over all q ! permutations of q indices, one can take the sum over q cyclic permutations, chose one index and then sum over all ( q − permutations ofthe remaining ( q − indices. Apply this to the antisymmetrisation in a , b , c , d (fixing b ):(6.16a) = b b d c d a ¯ g ij (cid:16) S ia b b S jc d d − S ib b c S jd d a + S ic b d S ja d b − S id b a S jb d c (cid:17) . (6.16b) Up to a constant they are all projectors.
For a better readability we underlined each antisymmetrised index. Now use thesymmetrised Bianchi identity (6.8) to bring the index c from the fourth to thesecond index position:(6.16b) = b b d c d a ¯ g ij (cid:16) S ia b b S jc d d + S ic b b S jd d a + S ic b d S ja d b + S id b a S jc d b (cid:17) . (6.16c)Then rename i, j as j, i in the last two terms:(6.16c) = b b d c d a ¯ g ij (cid:16) S ia b b S jc d d + S ic b b S jd d a + S ia d b S jc b d + S ic d b S jd b a (cid:17) . (6.16d)Finally use the symmetrisation in b , b , d and the antisymmetrisation in c , d , a to bring each term to the same form:(6.16d) = b b d c d a ¯ g ij (cid:16) S ia b b S jc d d (cid:17) . (6.16e)This proves (6.11d) ⇔ (6.11b). To continue, antisymmetrise c d a b b d ¯ g ij S ia b b S jc d d = 2 c d a ¯ g ij (cid:16) S ia b b S jc d d + S ia b d S jc b d + S ia d b S jc b d (cid:17) in a , b , c , d . Then the last term vanishes by the symmetry (4.12a), yielding a b c d ¯ g ij (cid:16) S ia b b S jc d d + S ia d b S jc b d (cid:17) . Both sum terms are equal under antisymmetrisation in a , b , c , d and contractionwith ¯ g ij . Indeed, exchanging b and d is tantamount to exchanging a with c and b with d and renaming i, j as j, i . This proves (6.11b) ⇒ (6.11c).From the correspondence (4.14) between R and S we conclude the equivalence(6.11c) ⇔ (6.13c). The proof of the remaining part of (ii) is completely analogous tothe proof of (i), so we leave it to the reader. Condition (6.14) is just a reformulationof (6.13c). This finishes the proof. (cid:3) Remark 6.8.
In the preceeding proof we made use of a particular notation as wellas some particular tensor index manipulations. We will do this several times inmore complex computations during the next two sections, so we would like to makethis explicit. • First, we call a Young symmetriser as in (6.16a) , which is the product ofa symmetriser and an antisymmetriser sharing a common label (and thusnot commuting) a hook symmetriser . Note that (6.16a) and (6.16b) aremerely different ways to write down the same term, using a smaller anti-symmetriser but applied to more terms. We call this to reduce an antisym-metriser by a label ( b in this case). This works likewise for a symmetriser NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 23 and allows us to replace any hook symmetriser by a product of a symmetriserand an antisymmetriser with disjoint label sets (and thus both commuting).The latter are more easy to deal with. We call this procedure splitting ahook symmetriser . • Second, for better readability we stick to the above notation and underlineantisymmetrised tensor indices as in (6.16c) . • Third, regard the manipulations from (6.16c) to (6.16e) . What we didis to bring the indices of every term in (6.16c) to the same order as in g ij S ia b b S jc d d by using: – the symmetry in i, j under contraction with ¯ g ij , – the (anti)symmetry under the (anti)symmetriser and – the symmetries of S itself, especially the symmetrised Bianchi identity (6.8) .We will call this procedure reordering indices . The second integrability condition.
The proceeding for the remainingtwo integrability conditions is similar as for the first one, only longer. We thereforetreat both in parallel as far as possible and shorten the explications where theyare analogous. We begin by substituting the expressions (6.3) and (5.5a) into thetensors appearing in (2.3b) and (2.3c): ¯ N δβγ K δα = ¯ g ij (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) x b x b x d ∇ δ x a ∇ β x c ∇ γ x d S e e f f x e x e ∇ δ x f ∇ α x f ¯ N δβγ K εα K εδ = ¯ g ij (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) x b x b x d ∇ δ x a ∇ β x c ∇ γ x d S e e f f x e x e ∇ ε x f ∇ α x f S g g h h x g x g ∇ ε x h ∇ δ x h . As before, we replace the contractions over δ and ε according to lemma 6.1 andomit the terms that vanish according to the Bianchi identity: ¯ N δβγ K δα = ¯ g ij ¯ g a f (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) S e e f f x b x b x d x e x e ∇ β x c ∇ γ x d ∇ α x f ¯ N δβγ K εα K εδ = ¯ g ij ¯ g a h ¯ g f h (cid:0) S ia b b S jc d d + S ic b b S jd a d (cid:1) S e e f f S g g h h x b x b x d x e x e x g x g ∇ β x c ∇ γ x d ∇ α x f . The integrability conditions (2.3b) and (2.3c) are equivalent to the vanishing of theantisymmetrisation of the above tensors in α, β, γ . As before, this can be written as ¯ g ij ¯ g kl (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S lf e e x b x b x d x e x e u [ c v d w f ] = 0¯ g ij ¯ g kl ¯ g mn (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S mf e e S nlg g x b x b x d x e x e x g x g u [ c v d w f ] = 0 ∀ x ∈ M ∀ u, v, w ∈ T x M . (6.17)The tensors x b x b x d x e x e u [ c v d w f ] x b x b x d x e x e x g x g u [ c v d w f ] are antisymmetric in c , d , f and symmetric in the remaining indices. We decom-pose them according to lemma 6.2. This yields x b x b x d x e x e u [ c v d w f ] = constant · b b d e e c d f x b x b x d x e x e u c v d w f + constant · c b b d e e d f (cid:63) x b x b x d x e x e u c v d w f and x b x b x d x e x e x g x g u [ c v d w f ] = constant · b b d e e g g c d f x b x b x d x e x e x g x g u c v d w f + constant · c b b d e e g g d f (cid:63) x b x b x d x e x e x g x g u c v d w f . The following lemma shows that, when substituted into (6.17), only the first termis relevant in each case:
Lemma 6.9. c b b d e e d f ¯ g ij ¯ g kl S ikb b S jc d d S lf e e = 0 (6.18a) c b b d e e d f ¯ g ij ¯ g kl S ic b b S j kd d S lf e e = 0 (6.18b) c b b d e e g g d f ¯ g ij ¯ g kl ¯ g mn S ikb b S jc d d S mf e e S nlg g = 0 (6.19a) c b b d e e g g d f ¯ g ij ¯ g kl ¯ g mn S ic b b S j kd d S mf e e S nlg g = 0 (6.19b) NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 25
Proof.
Expanding the antisymmetriser of the Young symmetriser on the left handside of (6.18a) yields c b b d e e ¯ g ij ¯ g kl S ikb b (cid:0) S jc d d S lf e e − S jc d f S ld e e + S jf d c S ld e e − S jd d c S lf e e + S jd d f S lc e e − S jf d d S lc e e (cid:1) . Now regard the parenthesis under complete symmetrisation in b , b , c , d , e , e .The last two terms vanish by the Bianchi identity. Renaming i, j, k, l as k, l, i, j in the third term shows that it cancels the fourth due to the contraction with ¯ g ij ¯ g kl S ikb b . That the first two also cancel each other can be seen after applyingtwice the symmetrised Bianchi identity, once to S jc d d and once to S lf e e .In the same way, the left hand side of (6.18b), written without terms vanishingby the Bianchi identity, is c b b d e e ¯ g ij ¯ g kl S j kd c (cid:0) S id b b S lf e e − S if b b S ld e e (cid:1) . Renaming i, j, k, l as l, k, j, i in the first term shows that this is zero too. Theproof of (6.19) is straightforward, using the same arguments. We leave this to thereader. (cid:3) Remark 6.10.
For the unit sphere, the lemma also follows from the symmetryclassification of Riemann tensor polynomials [FKWC92] . Indeed, the tensors un-der the Young symmetriser in (6.18) and (6.19) can be expressed in terms of thecorresponding algebraic curvature tensor via (4.14a) and the resulting tensors haveno respectively component.
We have shown the equivalence of the second and third integrability conditionto ¯ g ij ¯ g kl (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S lf e e b b d e e c d f x b x b x d x e x e u c v d w f = 0¯ g ij ¯ g kl ¯ g mn (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S mf e e S nlg g b b d e e g g c d f x b x b x d x e x e x g x g u c v d w f = 0 ∀ x ∈ M ∀ u, v, w ∈ T x M respectively. As before, the restrictions ∀ u, v, w ∈ T x M and ∀ x ∈ M can be droppedand this allows us to write both conditions independently of the vectors x, u, v, w as b b d e e c d f (cid:63) ¯ g ij ¯ g kl (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S lf e e = 0 (6.20) b b d e e g g c d f (cid:63) ¯ g ij ¯ g kl ¯ g mn (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S mf e e S nlg g = 0 . (6.21)In order to simplify these conditions we need the following two lemmas. Lemma 6.11.
The first integrability condition is equivalent to b c d b d ¯ g ij (cid:0) S i kb b + 2 S i kb b (cid:1) S jc d d = 0 . (6.22) Proof.
Take the first integrability condition in the form (6.11c) and reduce theantisymmetriser by the label a : b c d ¯ g ij (cid:0) S ia b b S jc d d − S ib b c S jd d a + S ic b d S ja d b − S id b a S jb d c (cid:1) = 0 . If we symmetrise this expression in b , d , the first and third as well as the secondand fourth term become equal. Permuting indices, we get b c d b d ¯ g ij (cid:0) S ia b b − S ib a b (cid:1) S jc d d = 0 . (6.23)If we now symmetrise in a , b , d and apply the symmetrised Bianchi identity to S ia b b , we get back the first integrability condition in the form (6.11b). Thisproves its equivalence to (6.23). Applying now the Bianchi identity to the firstterm in (6.23) yields (6.22) with the index k lowered and renamed as a . (cid:3) Lemma 6.12.
The following identity is a consequence of the first integrabilitycondition: c d b b d ¯ g ij (cid:0) S i kb b S jc d d − S i kd d S jc b b − S i kd d S jc b b (cid:1) = 0 . (6.24) Proof.
Reduce the antisymmetriser in (6.22) by the index b , c d b d ¯ g ij (cid:0) S i kb b S jc d d +2 S i kb b S jc d d + S i kb c S jd d b +2 S i kc b S jd d b + S i kb d S jb d c +2 S i kd b S jb d c (cid:1) = 0 , and then symmetrise in b , b , d . In the last line we can then apply the symmetrisedBianchi identity in order to move the antisymmetrised index c from the fourth to NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 27 the second position: c d b b d ¯ g ij (cid:0) S i kb b S jc d d +2 S i kb b S jc d d − S i kb d S jc d b − S i kd b S jc d b − S i kb d S jc d b − S i kd b S jc d b (cid:1) = 0 . After permuting indices under symmetrisation appropriately, we get the desiredresult. (cid:3)
Proposition 6.13 (Second integrability condition) . Suppose a Killing tensor ona constant sectional curvature manifold satisfies the first integrability condition (2.3a) . Then the following conditions are equivalent to the second integrability con-dition (2.3b) :(i) The corresponding symmetrised algebraic curvature tensor S satisfies oneof the following two equivalent conditions: b b d e e c d f (cid:63) ¯ g ij ¯ g kl S ic d d S j kb b S lf e e = 0 (6.25a) b b d e e c d f (cid:63) ¯ g ij ¯ g kl S ic b b S j kd d S lf e e = 0 . (6.25b) (ii) The corresponding symmetrised algebraic curvature tensor S satisfies oneof the following two equivalent conditions: c d f b b d e e ¯ g ij ¯ g kl S ic d d S j kb b S lf e e = 0 (6.26a) c d f b b d e e ¯ g ij ¯ g kl S ic b b S j kd d S lf e e = 0 . (6.26b) (iii) The corresponding symmetrised algebraic curvature tensor S satisfies oneof the following three equivalent conditions: ¯ g ij ¯ g kl S ic d d S j kb b S lf e e = 0 (6.27a) b c d f b d e e ¯ g ij ¯ g kl S ic d d S j kb b S lf e e = 0 (6.27b) ¯ g ij ¯ g kl S ic d d S j ke e S lf b b = 0 . (6.27c) (iv) The corresponding algebraic curvature tensor R satisfies b c d f b d e e ¯ g ij ¯ g kl R id c d R j ke e R lb f b = 0 . (6.28) Remark 6.14.
To facilitate the reading of this and subsequent proofs, note thatthe names of symmetrised indices are completely irrelevant.
Proof. (i) Contract (6.24) with ¯ g kl S lf e e , antisymmetrise in c , d , f and sym-metrise in b , b , d , e , e . This yields c d f b b d e e ¯ g ij ¯ g kl (cid:0) S i kb b S jc d d − S i kd d S jc b b − S i kd d S jc b b (cid:1) S lf e e = 0 . Reordering indices shows that the third term differs from the second by a factorof minus two. Indeed, exchanging d and d in the third term is tantamount toexchanging the upper indices i and k , due to the pair symmetry of S i kd d . Butunder contraction with ¯ g ij ¯ g kl this is tantamount to exchanging the upper indices j and l . This in turn is tantamount to exchanging c , b , b with f , e , e which,under symmetrisation and antisymmetrisation, is tantamount to a sign change.Therefore c d f b b d e e ¯ g ij ¯ g kl (cid:0) S i kb b S jc d d + S i kd d S jc b b (cid:1) S lf e e = 0 . (6.29)Applying the symmetrised Bianchi identity to S i kb b and antisymmetrising in b , c , d , f yields b b d e e c d f (cid:63) ¯ g ij ¯ g kl (cid:0) S ikb b S jc d d − S i kd d S jc b b (cid:1) S lf e e = 0 . We have derived this identity from the first integrability condition via lemma 6.12.Comparing it with condition (6.20) shows that (6.20) is equivalent to (6.25b) and,after using once again the symmetrised Bianchi identity, also to (6.25a). This proves(i), since we have already shown that the second integrability condition is equivalentto (6.20).(ii) Condition (6.26a) is equivalent to (6.25a). This results from (6.5) whentaking (6.18a) and (6.15) into account. In the same way (6.26b) is equivalent to(6.25b), using (6.18b).(iii) We will prove the equivalence of (6.25a) to each of the equations (6.27).To this aim we establish three linearly independent homogeneous equations for thethree tensors on the left hand side of (6.27). For the first equation we split thehook symmetriser in (6.25a) at the label b and get b c d f b d e e ¯ g ij ¯ g kl (cid:0) S ic d d S j kb b S lf e e + S ic e d S j kd b S lf e b + S ic e d S j ke d S lf b b + S ic b d S j ke e S lf b d + S ic b d S j kb e S lf d e (cid:1) = 0 . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 29
The fourth term vanishes by the Bianchi identity and the second term is equal tothe third. Therefore (6.25a) is equivalent to b c d f b d e e ¯ g ij ¯ g kl S ic d d (cid:0) S j kb b S lf e e + 2 S j ke e S lf b b + S j kb b S lf e e (cid:1) = 0 . (6.30a)This is our first equation. The other two equations follow from the first integrabilitycondition as follows. The second equation is obtained from (6.22) by contractingwith ¯ g kl S lf e e , antisymmetrising in b , c , d , f and symmetrising in b , d , e , e : b c d f b d e e ¯ g ij ¯ g kl (cid:0) S i kb b + 2 S i kb b (cid:1) S jc d d S lf e e = 0 . This can be rewritten as b c d f b d e e ¯ g ij ¯ g kl S ic d d (cid:0) S j kb b S lf e e + 2 S j kb b S lf e e (cid:1) = 0 (6.30b)and is our second equation. For the third equation, we rename b , b in (6.24) as e , e , contract with ¯ g kl S lf b b , antisymmetrise in b , c , d , f and symmetrise in b , d , e , e : b c d f b d e e ¯ g ij ¯ g kl S lf b b (cid:0) S i ke e S jc d d − S i kd d S jc e e − S i kd d S jc e e (cid:1) = 0 . This can be rewritten as b c d f b d e e ¯ g ij ¯ g kl S ic d d (cid:0) S j ke e S lf b b − S j kb b S lf e e − S j kb b S lf e e (cid:1) = 0 (6.30c)and is our last equation. Clearly, the resulting homogeneous system (6.30) implies(6.27). On the other hand, any of the equations (6.27) together with (6.30b) and(6.30c) implies (6.30a) and therefore (6.25a).(iv) Condition (6.28) is equivalent to (6.27c) via (4.14). (cid:3) Redundancy of the third integrability condition.
The aim of this sectionis to prove the following:
Proposition 6.15 (Third integrability condition) . For a Killing tensor on a con-stant sectional curvature manifold the third of the three integrability conditions (2.3) is redundant.
We have already shown that the third integrability condition is equivalent to(6.21). As before we can infer from (6.5) together with (6.19) and (6.15) that(6.21) is equivalent to c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S ikb b S jc d d + S ic b b S j kd d (cid:1) S mf e e S nlg g = 0 . (6.31) The proceeding to prove this equation is similar to the proof of part (iii) in propo-sition 6.13. From the first two integrability conditions we will deduce the followingthree equations (6.33a) − (6.33b) − (6.33c) =0 (6.32a) (6.33b) + (6.33c) =0 (6.32b)(6.33b) − (6.33c) =0 (6.32c)for the tensors ¯ g ij ¯ g kl ¯ g mn S ic d d S j kb b S l mg g S nf e e (6.33a) c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn S ic b b S j kd d S l mg g S nf e e (6.33b) ¯ g ij ¯ g kl ¯ g mn S ic b b S j kd d S l mg g S nf e e . (6.33c)The system (6.32) shows that each of the tensors (6.33) is zero. In particularthis proves our claim, since (6.31) can be written as a linear combination of thesetensors.6.3.1. First equation.
Contract (6.24) with ¯ g kl ¯ g mn S l mg g S nf e e , antisymmetrisein c , d , f and symmetrise in the remaining seven indices: c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S i kb b S jc d d − S i kd d S jc b b − S i kd d S jc b b (cid:1) S l mg g S nf e e = 0 . Renaming i, j as j, i , this can be written as c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S ic d d S j kb b − S ic b b S j kd d − S ic b b S j kd d (cid:1) S l mg g S nf e e = 0 . This is our first equation (6.32a).6.3.2.
Second equation.
Reduce the antisymmetriser in (6.24) completely, b b d ¯ g ij (cid:0) S i kb b S jc d d − S i kd d S jc b b − S i kd d S jc b b − S i kb b S jd d c + S i kd c S jd b b + S i kc d S jd b b (cid:1) = 0 , raise the index c , rename it as m and bring the last two terms to the right handside, b b d ¯ g ij (cid:0) S i kb b S jmd d − S i kd d S jmb b − S i kd d S jmb b − S i kb b S j md d (cid:1) = − b b d ¯ g ij (cid:0) S i kmd S jd b b + S imkd S jd b b (cid:1) , contract with ¯ g kl ¯ g mn S lf e e S nh g g , antisymmetrise in d , f , h and symmetrisein the remaining seven indices: d f h b b d e e g g ¯ g ij ¯ g kl ¯ g mn S lf e e S nh g g (cid:0) S i kb b S jmd d − S i kd d S jmb b − S i kd d S jmb b − S i kb b S j md d (cid:1) = − d f h b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S i kmd S jd b b + S imkd S jd b b (cid:1) S lf e e S nh g g . NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 31
On the right hand side the upper indices j, n, l are implicitely antisymmetrised bythe symmetrisation in b , b , e , e , g , g and the antisymmetrisation in d , f , h .Due to the term ¯ g ij ¯ g kl ¯ g mn the same holds for the upper indices i, m, k . The Bianchiidentity therefore implies that the right hand side is zero. On the left hand side, theBianchi identity allows us to bring the index m in each term to the third position: d f h b b d e e g g ¯ g ij ¯ g kl ¯ g mn S lf e e (cid:0) − S i kb b S j md d + S i kd d S j mb b − S i kb b S j md d +2 S i kd d S j mb b (cid:1) S nh g g = 0 . Using pair symmetry and renaming the indices i, j, k, l as k, l, j, i , this can be writtenas d f h b b d e e g g ¯ g ij ¯ g kl ¯ g mn S if e e (cid:0) − S j kb b S l md d + S j kd d S l mb b − S j kb b S l md d +2 S j kd d S l mb b (cid:1) S nh g g = 0 Renaming i, j, k, l, m, n in reverse order, the first term can be seen to differ fromthe second by a sign. The same is true for the third and fourth term, resulting in d f h b b d e e g g ¯ g ij ¯ g kl ¯ g mn S if e e (cid:0) S j kd d S l mb b + 2 S j kd d S l mb b (cid:1) S nh g g = 0 . This is our second equation (6.32b).6.3.3.
Third equation.
Take the second integrability condition in the form (6.26b),reduce the antisymmetriser by the label f , c d b b d e e ¯ g ij ¯ g kl (cid:0) S ic b b S j kd d S lf e e + S id b b S j kd f S lc e e + S if b b S j kd c S ld e e (cid:1) = 0 , raise the index f , rename it as m and bring the second term to the right hand side, c d b b d e e ¯ g ij ¯ g kl (cid:0) S ic b b S j kd d S lme e + S imb b S j kd c S ld e e (cid:1) = − c d b b d e e ¯ g ij ¯ g kl S id b b S j kmd S lc e e , contract with ¯ g mn S nf g g , antisymmetrise in c , d , f and symmetrise in the re-maining seven indices: c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S ic b b S j kd d S lme e + S imb b S j kd c S ld e e (cid:1) S nf g g = − c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn S id b b S j kmd S lc e e S nf g g . As before, the right hand side is zero. On the left hand side use the symmetrisedBianchi identity to move the upper index m to the third position: c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S ic b b S j kd d S l me e + S i mb b S j kd c S ld e e (cid:1) S nf g g = 0 . Using pair symmetry and renaming the upper indices in the second term, this canbe written as c d f b b d e e g g ¯ g ij ¯ g kl ¯ g mn (cid:0) S ic b b S j kd d S l me e + S id e e S j kc d S l mb b (cid:1) S nf g g = 0 . This is our third and last equation (6.32c).7.
Application
Finally, we show that the family (1.2) satisfies the algebraic integrability condi-tions (1.1) and therefore describes integrable Killing tensors.
Proof of the Main Corollary.
We will write a dot in place of each index whose nameis irrelevant for our considerations. Written in components, the algebraic curvaturetensor R = λ h (cid:55) h + λ h (cid:55) g + λ g (cid:55) g is then a linear combination of tensors of the form h ·· h ·· , h ·· g ·· and g ·· g ·· or, writtenin another way, of the form A ·· B ·· with A, B ∈ { g, h } . Then R a b a b R c d c d is a linear combination of terms of the form A ·· B ·· C ·· D ·· with A, B, C, D ∈ { g, h } and thus g ij R ib a b R id c d is a linear combination of terms of the form A ·· B ·· C ·· with A, B, C ∈ { g, h, h } . Here g , h and h are symmetric tensors, where h denotes the tensor g ij h ia h jb . Therefore the antisymmetrisation of A ·· B ·· C ·· in fourof the six indices vanishes by Dirichlet’s drawer principle. This proves that the firstintegrability condition (1.1a) is satisfied.In the same way, the tensor R ···· R ···· R ···· is a linear combination of terms of theform A ·· B ·· C ·· D ·· E ·· F ·· with A, . . . , F ∈ { g, h } . Hence the tensor g ij g kl R ib a b R j ka c R ld c d is a linear combination of terms of the form A ·· B ·· C ·· D ·· with either A, B, C, D ∈{ g, h, h } or A, B, C, D ∈ { g, h, h } , where h denotes the symmetric tensor g ij g kl h ia h jk h lb . Without loss of generality we may suppose D = C , owing to the drawer principle.Consider therefore the tensor A ·· B ·· C ·· C ·· under antisymmetrisation in four of itsindices and symmetrisation in the remaing four. The result vanishes trivially if theantisymmetrisation includes an index pair of one of the symmetric tensors A, B, C .Otherwise it can be written as a b c d a b c d A a a B b b C c c C d d NTEGRABILITY OF KILLING TENSORS ON CONSTANT CURVATURE MANIFOLDS 33 and vanishes too, which becomes evident when c , c is renamed as d , d . Thisdemonstrates that the second algebraic integrability condition (1.1b) is also satis-fied. (cid:3) Remark 7.1.
The family (1.2) properly extends Benenti tensors, given by λ = λ = 0 λ = h = Ag A ∈ GL( V ) . Indeed, a Killing tensor corresponding to h (cid:55) g with tr h < for example is not aBenenti tensor, as can be seen by comparing the scalar curvatures Scal (cid:0) ( Ag ) (cid:55) ( Ag ) (cid:1) = tr ( Ag ) − tr( Ag ) = tr ( A T A ) − tr( A T AA T A )= (cid:107) A (cid:107) − (cid:107) A T A (cid:107) (cid:62) and Scal( h (cid:55) g ) = 2( N −
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