Killing and Conformal Killing tensors
aa r X i v : . [ m a t h . DG ] M a r KILLING AND CONFORMAL KILLING TENSORS
KONSTANTIN HEIL, ANDREI MOROIANU, UWE SEMMELMANN
Abstract.
We introduce an appropriate formalism in order to study conformal Killing(symmetric) tensors on Riemannian manifolds. We reprove in a simple way some knownresults in the field and obtain several new results, like the classification of conformal Killing2-tensors on Riemannian products of compact manifolds, Weitzenb¨ock formulas leading tonon-existence results, and construct various examples of manifolds with conformal Killingtensors.2010
Mathematics Subject Classification : Primary: 53C25, 53C27, 53C40
Keywords : Killing tensors, conformal Killing tensors, St¨ackel tensors. Introduction
Killing p -tensors are symmetric p -tensors with vanishing symmetrized covariant derivative.This is a natural generalization of the Killing vector field equation for p = 1. Since manyyears Killing tensors, and more generally conformal Killing tensors, were intensively studiedin the physics literature, e.g. in [25] and [30]. The original motivation came from the fact thatsymmetric Killing tensors define (polynomial) first integrals of the equations of motion, i.e.functions which are constant on geodesics. Conformal Killing tensors still define first integralsfor null geodesics. Killing 2-tensors also appeared in the analysis of the stability of generalizedblack holes in D = 11 supergravity, e.g. in [11] and [22]. It turns out that trace-free Killing2-tensors (also called St¨ackel tensors) precisely correspond to the limiting case of a lowerbound for the spectrum of the Lichnerowicz Laplacian on symmetric 2-tensors. More recently,Killing and conformal Killing tensors appeared in several other areas of mathematics, e.g. inconnection with geometric inverse problems, integrable systems and Einstein-Weyl geometry,cf. [5], [8], [9], [13], [15], [23], [26].Any parallel tensor is in particular a Killing tensor. The simplest non-parallel examplesof Killing tensors can be constructed as symmetric products of Killing vector fields. Forthe standard sphere S n there is a direct correspondence between Killing tensors and algebraiccurvature tensors on R n +1 . Other interesting examples are obtained as Ricci tensors of certainRiemannian manifolds, e.g. of natural reductive spaces. Date : April 21, 2018.
The defining equation of trace-free conformal Killing tensors has the important property tobe of finite type (or strongly elliptic). This leads to an explicit upper bound of the dimensionof the space of conformal Killing tensors. In this respect, conformal Killing tensors are verysimilar to so-called conformal Killing forms, which were studied by the authors in severalarticles, e.g. [10], [20] and [27]. Moreover there is an explicit construction of Killing tensorsstarting from Killing forms, cf. Section 4.3 below.The existing literature on symmetric Killing tensors is huge, especially coming from theo-retical physics. One of the main obstacles in reading it is the old-fashioned formalism usedin most articles in the subject.In this article we introduce conformal Killing tensors in a modern, coordinate-free formal-ism. We use this formalism in order to reprove in a simpler way some known results, likeTheorem 8.1 saying that the nodal set of a conformal Killing tensor has at least codimension2, or Proposition 6.6 showing the non-existence of trace-free conformal Killing tensors oncompact manifolds of negative sectional curvature. In addition we give a unified treatmentof some subclasses of conformal Killing tensors, e.g. special conformal Killing tensors.We obtain several new results, like the classification of St¨ackel 2-tensors with at mosttwo eigenvalues, or the description of conformal Killing 2-tensors on Riemannian productsof compact manifolds (which turn out to be determined by Killing 2-tensors and Killingvector fields on the factors, cf. Theorem 5.1). We also prove a general Weitzenb¨ock formula(Proposition 6.1) leading to non-existence results on certain compact manifolds.
Acknowledgments.
This work was supported by the Procope Project No. 32977YJ.We are grateful to Mikko Salo who discovered an error in the proof of Proposition 6.6, and toGregor Weingart who helped us to correct this error. We also thank the anonymous refereefor having pointed out an error in the previous version of Theorem 5.1 and for several usefulsuggestions. 2.
Preliminaries
Let (
V, g ) be a Euclidean vector space of dimension n . We denote with Sym p V ⊂ V ⊗ p the p -fold symmetric tensor product of V . Elements of Sym p V are symmetrized tensor products v · . . . · v p := X σ ∈ S p v σ (1) ⊗ . . . ⊗ v σ ( p ) , where v , . . . , v p are vectors in V . In particular we have v · u = v ⊗ u + u ⊗ v for u, v ∈ V . Usingthe metric g , one can identify V with V ∗ . Under this identification, g ∈ Sym V ∗ ≃ Sym V can be written as g = P e i · e i , for any orthonormal basis { e i } .The direct sum Sym V := L p ≥ Sym p V is endowed with a commutative product makingSym V into a Z -graded commutative algebra. The scalar product g induces a scalar product, ONFORMAL KILLING TENSORS 3 also denoted by g , on Sym p V defined by g ( v · . . . · v p , w · . . . · w p ) = X σ ∈ S p g ( v , w σ (1) ) · . . . · g ( v p , w σ ( p ) ) . With respect to this scalar product, every element K of Sym p V can be identified with asymmetric p -linear map (i.e. a polynomial of degree p ) on V by the formula K ( v , . . . , v p ) = g ( K, v · . . . · v p ) . For every v ∈ V , the metric adjoint of the linear map v · : Sym p V → Sym p +1 V, K v · K is the contraction v y : Sym p +1 V → Sym p V, K v y K , defined by ( v y K )( v , . . . , v p − ) = K ( v, v , . . . , v p − ). In particular we have v y u p = pg ( v, u ) u p − , ∀ v, u ∈ V .We introduce the linear map deg : Sym V → Sym V , defined by deg( K ) = p K for K ∈ Sym p V . Then we have P e i · e i y K = deg( K ), where { e i } as usual denotes an or-thonormal frame. Note that if K ∈ Sym p T is considered as a polynomial of degree p then v y K corresponds to the directional derivative ∂ v K and the last formula is nothing else thanthe well-known Euler theorem on homogeneous functions.Contraction and multiplication with the metric g defines two additional linear maps:Λ : Sym p V → Sym p − V, K X e i y e i y K and L : Sym p − V → Sym p V, K X e i · e i · K , which are adjoint to each other. Note that L(1) = 2 g and Λ K = tr( K ) for every K ∈ Sym V .It is straightforward to check the following algebraic commutator relations(1) [ Λ , L ] = 2 n id + 4 deg , [ deg , L ] = 2 L , [ deg , Λ ] = − , and for every v ∈ V :(2) [ Λ , v · ] = 2 v y , [ v y , L ] = 2 v · , [ Λ , v y ] = 0 = [ L , v · ] . For V = R n , the standard O( n )-representation induces a reducible O( n )-representationon Sym p V . We denote by Sym p V := ker(Λ : Sym p V → Sym p − V ) the space of trace-freesymmetric p -tensors.It is well known that Sym p R n is an irreducible O( n )-representation and we have the fol-lowing decomposition into irreducible summandsSym p V ∼ = Sym p V ⊕ Sym p − V ⊕ . . . , where the last summand in the decomposition is R for p even and V for p odd. The summandsSym p − i V are embedded into Sym p V via the map L i . Corresponding to the decompositionabove any K ∈ Sym p V can be decomposed as K = K + L K + L K + . . . KONSTANTIN HEIL, ANDREI MOROIANU, UWE SEMMELMANN with K i ∈ Sym p − i V , i.e. Λ K i = 0. We will call this decomposition the standard decomposi-tion of K . In the following, the subscript 0 always denotes the projection of an element fromSym p V onto its component in Sym p V . Note that for any v ∈ V and K ∈ Sym p V we havethe following projection formula(3) ( v · K ) = v · K − n +2( p − L ( v y K ) . Indeed, using the commutator relation (1) we have Λ(L ( v y K )) = (2 n + 4( p − v y K ),since Λ commutes with v y and Λ K = 0. Moreover Λ( v · K ) = 2 v y K . Thus the right-handside of (3) is in the kernel of Λ, i.e. it computes the projection ( v · K ) .Recall the classical decomposition into irreducible O( n ) representations(4) V ⊗ Sym p V ∼ = Sym p +10 V ⊕ Sym p − V ⊕ Sym p, V , where V = R n is the standard O( n )-representation of highest weight (1 , , . . . , p V isthe irreducible representation of highest weight ( p, , . . . ,
0) and Sym p, V is the irreduciblerepresentation of highest weight ( p, , , . . . , p +10 V is the so-called Cartansummand. Its highest weight is the sum of the highest weights of V and Sym p V .Next we want to describe projections and embeddings for the first two summands. Theprojection π : V ⊗ Sym p V → Sym p +10 V onto the first summand is defined as(5) π ( v ⊗ K ) := ( v · K ) = v · K − n +2( p − L ( v y K ) . The adjoint map π ∗ : Sym p +10 V → V ⊗ Sym p V is easily computed to be π ∗ ( K ) = P e i ⊗ ( e i y K ). Note that for any vector v ∈ V the symmetric tensor v y K is again trace-free,because v y commutes with Λ. Since π π ∗ = ( p + 1) id on Sym p +10 V , we conclude that(6) p := p +1 π ∗ π : V ⊗ Sym p V → Sym p +10 V ⊂ V ⊗ Sym p V is the projection onto the irreducible summand of V ⊗ Sym p V isomorphic to Sym p +10 V .Similarly the projection π : V ⊗ Sym p V → Sym p − V onto the second summand in thedecomposition (4) is given by the contraction map π ( v ⊗ K ) := v y K . In this case the adjointmap π ∗ : Sym p − V → V ⊗ Sym p V is computed to be π ∗ ( K ) = X e i ⊗ ( e i · K ) = X e i ⊗ ( e i · P − n +2( p − L ( e i y P )) . It follows that π π ∗ = ( n + p −
1) id − p − n +2( p − id = ( n +2 p − n + p − n +2 p − id . Thus the projectiononto the irreducible summand in V ⊗ Sym p V isomorphic to Sym p − V is given by(7) p := n +2 p − n +2 p − n + p − π ∗ π : V ⊗ Sym p V → Sym p − V ⊂ V ⊗ Sym p V .
The projection p onto the third irreducible summand in V ⊗ Sym p V is obviously given by p = id − p − p .Let ( M n , g ) be a Riemannian manifold with Levi-Civita connection ∇ . All the algebraicconsiderations above extend to vector bundles over M , e.g. the O( n )-representation Sym p V ONFORMAL KILLING TENSORS 5 defines the real vector bundle Sym p T M . The O( n )-equivariant maps L and Λ define bundlemaps between the corresponding bundles. The same is true for the symmetric product andthe contraction ι , as well as for the maps π , π and their adjoints, and the projection maps p , p , p . We will use the same notation for the bundle maps on M .Next we will define first order differential operators on sections of Sym p T M . We haved : Γ(Sym p T M ) → Γ(Sym p +1 T M ) , K X e i · ∇ e i K , where { e i } denotes from now on a local orthonormal frame. The formal adjoint of d is thedivergence operator δ defined by δ : Γ(Sym p +1 T M ) → Γ(Sym p T M ) , K
7→ − X e i y ∇ e i K ,
An immediate consequence of the definition is
Lemma 2.1.
The operator d acts as a derivation on the algebra of symmetric tensors, i.e.for any A ∈ Γ(Sym p T M ) and B ∈ Γ(Sym q T M ) the following equation holds d( A · B ) = (d A ) · B + A · (d B ) . An easy calculation proves that the operators d and δ satisfy the commutator relations:(8) [ Λ , δ ] = 0 = [ L , d ] , [ Λ , d ] = − δ, [ L , δ ] = 2 d . Lemma 2.2.
Let K = K +L K + . . . be the standard decomposition of a section of Sym p T M ,where K i ∈ Sym p − i T M . Then there exist real constants a i such that d K i − a i L δK i ∈ Sym p − i +10 T M .
The constants are given explicitly by a i := − n +2( p − i − . In particular, if K is a section of Sym p T M , it holds that (9) (d K ) = d K + n +2 p − L δK . Proof.
We write K = P i ≥ L i K i , where K i is a section of Sym p − i T M . Then d K i − a i L δK i is a section of Sym p − i +10 T M if and only if0 = Λ(d K i − a i L δK i ) = − δ K i − a i (2 n + 4( p − i − δ K i . Thus the constants a i are as stated above. In particular we have for i = 0 that the expressiond K + n +2 p − L δK is trace-free. This proves the last statement. (cid:3) The operators d and δ can be considered as components of the covariant derivative ∇ actingon sections of Sym p T M . To make this more precise we first note that(10) π ( ∇ K ) = (d K ) and π ( ∇ K ) = − δK , which follows from ∇ K = P e i ⊗ ∇ e i K and the definitions above. KONSTANTIN HEIL, ANDREI MOROIANU, UWE SEMMELMANN
Let K be a section of Sym p T M . Then ∇ K is a section of T M ⊗ Sym p T M and we maydecompose ∇ K corresponding to (4), i.e. ∇ K = P ( K ) + P ( K ) + P ( K ), where we use thenotation P i ( K ) := p i ( ∇ K ) , i = 1 , ,
3. Substituting the definition of the operators P i andapplying the resulting equation to a tangent vector X we obtain ∇ X K = p +1 π ∗ (d K ) ( X ) − n +2 p − n +2 p − n + p − π ∗ ( δK )( X ) + P ( K )( X )= p +1 X y (d K ) − n +2 p − n +2 p − n + p − ( X · δK ) + P ( K )( X )Using (3) and (9) we rewrite the formula for ∇ X K in terms of d K and δK and obtain ∇ X K = p +1 X y d K + (cid:16) n +2 p − p +1) + n +2 p − n + p − (cid:17) L ( X y δK )+ (cid:16) n +2 p − p +1) − n +2 p − n +2 p − n + p − (cid:17) X · δK + P ( K )( X )= p +1 X y d K + p +1)( n + p − L ( X y δK ) − p − p +1)( n + p − X · δK + P ( K )( X ) . Here we applied the commutator formula X y L K = L( X y K ) + 2 X · K . For later use we stillnote the formulas P ( K )( X ) = p +1 X y (d K ) and P ( K )( X ) = − n +2 p − n +2 p − n + p − ( X · δK ) . At the end of this section we want to clarify the relations between d , δ and P , P . Forconvenience we introduce the notation d K := (d K ) . The relation between d and d is givenin (9). An easy calculation shows that δ ∗ = d . Lemma 2.3.
On sections of
Sym p T M the following equations hold: d ∗ d = ( p + 1) P ∗ P and δ ∗ δ = ( n +2 p − n + p − n +2 p − P ∗ P . Proof.
Let EM be a vector bundle associated to the frame bundle via a SO( n ) representationE. The Levi-Civita connection induces a covariant derivative ∇ acting on sections of EM . IfT denotes the tangent representation, defining the tangent bundle, we have a decompositioninto irreducible summands: E ⊗ T = E ⊕ . . . ⊕ E N . Here the spaces E i are subspaces of E ⊗ T but often they appear also in other realizations, like the spaces Sym p +10 T and Sym p − Tin the decomposition of Sym p T ⊗ T considered above.Assume that ˜ E i are SO( n )-representations isomorphic to E i and that π i : E ⊗ T → ˜ E i arerepresentation morphisms with π i ◦ π ∗ i = c i id for some non-zero constants c i . Then we candefine projections p i : E ⊗ T → E i ⊂ E ⊗ T as above by p i := c i π ∗ i π i . From the conditionon π i we obtain p i = p ∗ i ◦ p i = p i . Now we define two sets of operators on sections of EM :d i := π i ◦ ∇ : Γ( EM ) → Γ( ˜ E i M ) and P i := p i ◦ ∇ : Γ( EM ) → Γ( EM ⊗ T M ). ONFORMAL KILLING TENSORS 7
We then have the general formula d ∗ i d i = c i P ∗ i P i . Indeed we haved ∗ i d i = ∇ ∗ π ∗ i π i ∇ = c i ∇ ∗ p i ∇ = c i ∇ ∗ p ∗ i p i ∇ = c i P ∗ i P i . The statement of the lemma now follows from (10) together with (6)–(7). (cid:3) Basics on Killing and conformal Killing tensors
Definition 3.1.
A symmetric tensor K ∈ Γ(Sym p T M ) is called conformal Killing tensor ifthere exists some symmetric tensor k ∈ Γ(Sym p − T M ) with d K = L( k ). Lemma 3.2.
The defining equation for conformal Killing tensors is conformally invariant.More precisely, a section K of Sym p T M is a conformal Killing tensor with respect to themetric g , if and only if it is a conformal Killing tensor with respect to every conformallyrelated metric g ′ = e f g .Proof. Let
X, Y be any vector fields. Then the Levi-Civita connection ∇ ′ for g ′ is given by ∇ ′ X Y = ∇ X Y + d f ( X ) Y + d f ( Y ) X − g ( X, Y ) grad g ( f )where grad g ( f ) is the gradient of f with respect to g (cf. [2], Th. 1.159). It immediatelyfollows that for any section K of Sym p T M we have ∇ ′ X K = ∇ X K + p d f ( X ) K + X · grad g ( f ) y K − grad g ( f ) · X y K .
Hence we obtain for the differential d ′ K = P i e ′ i · ∇ ′ e ′ i K = e − f P i e i · ∇ ′ e i K the equationd ′ K = e − f (d K + p grad g ( f ) · K + L (grad g ( f ) y K ) − p grad g ( f ) · K )= e − f d K + L ′ (grad g ( f ) y K ) . Hence if K is conformal Killing tensor with respect to the metric g , i.e. d K = L k for somesection k of Sym p − T M , then K is a conformal Killing tensor with respect to the metric g ′ ,too. Indeed d ′ K = L ′ ( k + grad g ( f ) y K ). (cid:3) Note that K is conformal Killing if and only if its trace-free part is conformal Killing.Indeed, since d and L commute, if K = P i ≥ L i K i , with K i ∈ Γ(Sym p − i T M ) is thestandard decomposition of K , then d K = P i ≥ L i d K i , so d K is in the image of L if andonly if d K is in the image of L. It is thus reasonable to consider only trace-free conformalKilling tensors. Lemma 3.3.
Let K ∈ Γ(Sym p T M ) , then K is a conformal Killing tensor if and only if d K = − n +2 p − L δK or, equivalently, if and only if (d K ) = 0 . In particular, a trace-free symmetric tensor K ∈ Γ(Sym p T M ) is a conformal Killing tensor if and only if P ( K ) = 0 . KONSTANTIN HEIL, ANDREI MOROIANU, UWE SEMMELMANN
Proof.
We write K = P i ≥ L i K i , where K i is a section of Sym p − i T M . Because of [L , d] = 0we haved K = X i ≥ L i d K i = d K + X i ≥ L i d K i = (d K + n +2 p − L δ K ) − n +2 p − L δ K + X i ≥ L i d K i We know from (9) that the bracket on the right hand-side is the trace-free part of d K . Thusd K = − n +2 p − L δK holds if and only if d K = L( k ) for some section k of Sym p − T M , i.e.if and only if K is a conformal Killing tensor. (cid:3) Remark 3.4.
Since P ( K ) is the projection of the covariant derivative ∇ K onto the Cartansummand Sym p +10 T M ⊂ T M ⊗ Sym p T M , it follows that the defining differential equationfor trace-free conformal Killing tensors is of finite type, also called strongly elliptic (cf. [18]).In particular the space of trace-free conformal Killing tensors is finite dimensional on anyconnected manifold. Moreover one can show that a conformal Killing p -tensor has to vanishidentically on M if its first 2 p covariant derivatives vanish in some point of M , cf. [7]. Definition 3.5.
A symmetric tensor K ∈ Γ(Sym p T M ) is called Killing tensor if d K = 0 . A trace-free Killing tensor is called
St¨ackel tensor . Lemma 3.6.
A symmetric tensor K ∈ Γ(Sym p T M ) is a Killing tensor if and only if thecomplete symmetrization of ∇ K vanishes. A Killing tensor is in particular a conformalKilling tensor.Proof. Recall that d : Γ(Sym p T M ) → Γ(Sym p +1 T M ) was defined as d K = P i e i · ∇ e i K ,where { e i } is some local orthonormal frame. Thusd K ( X , . . . , X p +1 ) = X i X σ ∈ S p g ( e i , X σ (1) ) ( ∇ e i K )( X σ (2) , . . . , X σ ( p +1) )= X σ ∈ S p ( ∇ X σ (1) K )( X σ (2) , . . . , X σ ( p +1) ) . (cid:3) Proposition 3.7.
Let K = P i ≥ L i K i ∈ Γ(Sym p T M ) , with K i ∈ Γ(Sym p − i T M ) be asymmetric tensor, p = 2 l + ǫ with ǫ = 0 or . Then K is a Killing tensor if and only if thefollowing system of equations holds: d K = a L δK , d K = a L δK − a δK , ... = ... d K l = a l L δK l − a l − δK l − , δK l , ONFORMAL KILLING TENSORS 9 where a i are the constants of Lemma 2.2. In particular, the trace-free part K is a conformalKilling tensor.Proof. We write K = P i ≥ L i K i with K i ∈ Γ(Sym p − i T M ), then d K = 0 if and only if0 = X i ≥ L i d K i = X i ≥ L i (d K i − a i L δK i ) + a i L i +1 δK i = X i ≥ L i (d K i − a i L δK i + a i − δK i − ) , where we set a − = 0 and K − = 0 by convention. From Lemma 2.2 and [Λ , δ ] = 0 itfollows that d K i − a i L δK i + a i − δK i − is trace-free. We conclude that d K = 0 if and onlyif d K i − a i L δK i + a i − δK i − = 0 for all i . (cid:3) Example 3.8.
For p = 2 and K ∈ Γ(Sym T M ) we have K = K + 2 f g , for some function f = K ∈ C ∞ ( M ). Then K is a Killing tensor if and only if(11) d K = − n +2 L δ K and d f = n +2 δ K . The second equation can equivalently be written as d tr K = 2 δK . Example 3.9.
For p = 3 and K ∈ Γ(Sym T M ) we have K = K + L ξ , for some vector field ξ = K . Then K is a Killing tensor if and only ifd K = − n +4 L δK , d ξ = n +4 δK and δξ = 0 . Corollary 3.10. If K ∈ Γ(Sym p T M ) is a trace-free Killing tensor, then δK = 0 . In otherwords, St¨ackel tensors satisfy the equations d K = 0 = δK , or equivalently the equations P ( K ) = 0 = P ( K ) . Conversely we may ask what is possible to say about the components of divergence-freeKilling tensors. The result here is
Proposition 3.11.
Let K ∈ Γ(Sym p T M ) be a divergence-free Killing tensor. Then allcomponents K i ∈ Γ(Sym p − i T M ) of its standard decomposition K = P i ≥ L i K i are St¨ackeltensors.Proof. We first remark that iteration of the commutator formula (8) gives [ δ, L i ] = − i L i − dfor i ≥
1. Then δK = 0 implies 0 = δK + P i ≥ δ L i K i = δK − K + L k for somesymmetric tensor k . Substituting d K by the second equation of Proposition 3.7 we obtain0 = (1 + 2 a ) δK + L k for some symmetric tensor k . Since δK is trace-free and since thecoefficient (1 + 2 a ) is different from zero for n > δK = 0. But then thefirst equation of Proposition 3.7 gives d K = 0. Thus K is a St¨ackel tensor.We write K = K + L ˜ K with ˜ K = P i ≥ L i − K i . Since d commutes with L and multipli-cation with L is injective the equations d K = 0 = d K imply d ˜ K = 0. Similarly we obtain0 = δ (L ˜ K ) = L δ ˜ K −
2d ˜ K = L δ ˜ K . Thus δ ˜ K = 0 and we can iterate the argument above. (cid:3) Remark 3.12.
The above proof shows that the map L preserves the space of divergence-freeKilling tensors.In [26] the class of special conformal Killing tensors was introduced. These tensors weredefined as symmetric 2-tensors K ∈ Γ(Sym T M ) satisfying the equation(12) ( ∇ X K )( Y, Z ) = k ( Y ) g ( X, Z ) + k ( Z ) g ( X, Y )for some 1-form k . It follows that k = d tr( K ) and that d K = L( k ). Thus solutionsof Equation (12) are automatically conformal Killing tensors. The tensor k also satisfies δK = − ( n + 1) k and it is easily proved that ˆ K := K − tr( K ) g is a Killing tensor, which iscalled special Killing tensor in [26]. Moreover the map K ˆ K is shown to be injective andequivariant with respect to the action of the isometry group.We will now generalize these definitions and statements to Killing tensors of arbitrarydegree. We start with Definition 3.13.
A symmetric tensor K ∈ Γ(Sym p T M ) is called special conformal Killingtensor if the equation ∇ X K = X · k holds for all vector fields X and some symmetric tensor k ∈ Γ(Sym p − T M ).For p = 2, this is equivalent to Equation (12). Immediately from the definition it followsthat k = − n +1 δK and that d K = L k . Hence K is in particular a conformal Killing tensor.Using the standard decomposition K = P j ≥ L j K j and k = P j ≥ L j k j we can reformulatethe defining equation for special conformal Killing tensors into a system of equations forthe components K j and k j . Let K and k be symmetric tensors as above then ∇ X K = P j ≥ L j ∇ X K j and by (3) we have: X · k = X j ≥ L j ( X · k j ) = X j ≥ (cid:16) L j ( X · k j ) + n +2( p − − j ) L j +1 ( X y k j ) (cid:17) . Comparing coefficients of powers of L, we conclude that K is a special conformal Killingtensor if and only if the following system of equations is satisfied(13) ∇ X K j = ( X · k j ) + n +2( p − j ) X y k j − , j ≥ . With the convention k − = 0 this contains the equation ∇ X K = ( X · k ) for j = 0. Definition 3.14.
A symmetric tensor ˆ K ∈ Γ(Sym p T M ) is called special Killing tensor if itis a Killing tensor satisfying the additional equation ∇ X ˆ K = X · ˆ k + X y ˆ l for all vector fields X and some symmetric tensors ˆ k ∈ Γ(Sym p − T M ) and ˆ l ∈ Γ(Sym p +1 T M ).From the definition it follows directly that the tensors ˆ k and ˆ l are related by the equationsˆ l = − p +1 Lˆ k and δ ˆ K = − ( n + p − k − Λˆ l . ONFORMAL KILLING TENSORS 11
Hence ˆ K is a special Killing tensor if and only if for all vector fields X we have ∇ X ˆ K = X · ˆ k − p +1 X y Lˆ k = p − p +1 X · ˆ k − p +1 L X y ˆ k. As for special conformal Killing tensors we can reformulate the defining equation for specialKilling tensors as a system of equations for the components ˆ K j and ˆ k j . We find X j ≥ L j ∇ X ˆ K j = X j ≥ p − p +1 X · L j ˆ k j − p +1 L X y L j ˆ k j = X j ≥ p − − jp +1 L j X · ˆ k j − p +1 L j +1 X y ˆ k j = X j ≥ p − − jp +1 L j ( X · ˆ k j ) + (cid:16) p − − j ( p +1)( n +2( p − j − − p +1 (cid:17) L j +1 X y ˆ k j = X j ≥ p − − jp +1 L j ( X · ˆ k j ) − n + p − j − p +1)( n +2( p − j − L j +1 X y ˆ k j Hence ˆ K is a special Killing tensor if and only if the components ˆ K j and ˆ k j of the standarddecomposition satisfy the equations(14) ∇ X ˆ K j = p − − jp +1 ( X · ˆ k j ) − n + p − j − p +1)( n +2( p − j )) X y ˆ k j − j ≥ . With the convention ˆ k − = 0 we have for j = 0 the equation ∇ X ˆ K = p − p +1 ( X · ˆ k ) .For a given special conformal Killing tensor K we now want to modify its components K j by scalar factors in order to obtain a special Killing tensor ˆ K . This will generalize thecorrespondence between special conformal and special Killing 2-tensors obtained in [26].We are looking for constants a j , such that ˆ K = P j ≥ ˆ K j with ˆ K j = a j K j is a specialKilling tensor, where ˆ k = P j ≥ ˆ k j with ˆ k j = b j k j for a other set of constants b j . Consideringfirst the special Killing equation for j = 0 we have ∇ X a K = p − p +1 ( X · b k ) . Hence wecan define a = 1 and b = p +1 p − . Then writing (14) with the modified tensors ˆ K j and ˆ k j and comparing it with (13) multiplied by a j , we get the condition a j = p − − jp +1 b j for the firstsummand on the right hand side and a j n +2( p − j ) = − ( n + p − j − p +1)( n +2( p − j )) b j − = − ( n + p − j − p +1)( n +2( p − j )) p +1 p +1 − j a j − , for the second summand. Hence a j = − n + p − j − p +1 − j a j − and in particular a = − n + p − p − . Bythis recursion formula the coefficients a j and b j are completely determined and indeed ˆ K defined as above is a special Killing tensor. As an example we consider the case p = 2. Let K = K + L K be a special conformalKilling 2-tensor. Then ˆ K = K + a L K = K − ( n − K = K − n L K = K − tr( K ) g .Indeed tr( K ) = Λ K = 2 nK and L = 2 g . Hence we obtain for special conformal Killing2-tensors the same correspondence as in [26].Special conformal Killing 2-tensors have the important property of being integrable. Indeedtheir associated Nijenhuis tensor vanishes (cf. [26], Prop. 6.5). Recall that if A is anyendomorphism field on M , its Nijenhuis tensor is defined by N A ( X, Y ) = − A [ X, Y ] + A [ AX, Y ] + A [ X, AY ] − [ AX, AY ]= A ( ∇ X A ) Y − A ( ∇ Y A ) X − ( ∇ AX A ) Y + ( ∇ AY A ) X .
Lemma 3.15.
Any special conformal Killing -tensor has vanishing Nijenhuis tensor andthus is integrable.Proof. Let A be a special conformal Killing 2-tensor. Then ∇ X A = X · ξ for some vector field ξ . Then ( ∇ X A ) Y = g ( X, Y ) ξ + g ( ξ, Y ) X and it follows N A ( X, Y ) = g ( X, Y ) Aξ + g ( ξ, Y ) AX − g ( Y, X ) Aξ − g ( ξ, X ) AY − g ( AX, Y ) ξ − g ( ξ, Y ) AX + g ( AY, X ) ξ + g ( ξ, X ) AY = 0 . (cid:3) It is easy to check that special Killing 2-tensors associated to special conformal Killingtensors as above, have non-vanishing Nijenhuis tensor, therefore the statement in [26], Prop.6.5 is not valid for special Killing tensors.From Proposition 3.7 it follows that St¨ackel tensors are characterized among trace-freetensors by the vanishing of the two operators P and P . It is natural to consider trace-freesymmetric tensors with other vanishing conditions. Here we mention two other cases: Lemma 3.16.
For a trace-free symmetric tensor K ∈ Γ(Sym p T M ) the relations P ( K ) = 0 and P ( K ) = 0 hold if and only if there exists a section k of Sym p − T M with ∇ X K = ( X · k ) .In this case k is uniquely determined and ∇ X K = − n +2 p − n +2 p − n + p − ( X · δK ) . It follows from (13) for j = 0 that the trace-free part K of a special conformal Killingtensor K satisfies P ( K ) = 0 and P ( K ) = 0.For p = 2 consider a section K of Sym T M with P ( K ) = 0 and P ( K ) = 0. Thenthere exists a 1-form k with ∇ X K = ( X · k ) = X · k − n +2 p − L( X y k ) = X · k − n k ( X ) g .Substituting two tangent vectors Y and Z we obtain the equation(15) ( ∇ X K )( Y, Z ) = g ( X, Y ) k ( Z ) + g ( X, Z ) k ( Y ) − n k ( X ) g ( Y, Z ) . For the sake of completeness, we also consider the case of trace-free symmetric tensors K ∈ Γ(Sym p T M ) satisfying the vanishing conditions P ( K ) = 0 = P ( K ). Equivalently ONFORMAL KILLING TENSORS 13 the covariant derivative ∇ K is completely symmetric, i.e. ∇ K ∈ Γ(Sym p +10 T M ). Thiscan also be written as ( ∇ X K )( Y, . . . ) = ( ∇ Y K )( X, . . . ) for all tangent vectors X , Y or asd ∇ K = 0, where K is considered as a 1-form with values in Sym p − T M , i.e. a section ofT ∗ M ⊗ Sym p − T M . For p = 2, a tensor K with d ∇ K = 0 is called a Codazzi tensor .4.
Examples of manifolds with Killing tensors
Any parallel tensor is tautologically a Killing tensor. By the conformal invariance of theconformal Killing equation, a parallel tensor defines conformal Killing tensors for any con-formally related metric. These are in general no Killing tensors. There are several explicitconstructions of symmetric Killing tensors which we will describe in the following subsections.4.1.
Killing tensors on the sphere.
Let Curv( n +1) denote the space of algebraic curvaturetensors on R n +1 . Then any symmetric Killing 2-tensor K on S n is given in a point p ∈ S n by K ( X, Y ) = R ( X, p, p, Y ) for some algebraic curvature tensor R ∈ Curv( n + 1). Thesubspace of Weyl curvature tensors in Curv( n + 1) corresponds to the trace-free (and hencedivergence-free) Killing tensors on S n , cf. [19].The dimension of the space of Killing tensors on S n gives an upper bound for this dimensionon an arbitrary Riemannian manifold [29], Theorem 4.7.4.2. Symmetric products of Killing tensors.
Let ( M n , g ) be a Riemannian manifold withtwo Killing vector fields ξ , ξ . We define a symmetric 2-tensor h as the symmetric product h := ξ · ξ . Then h is a Killing tensor. Indeed d( ξ · ξ ) = (d ξ ) · ξ + ξ · (d ξ ) = 0 holdsbecause of Lemma 2.1. More generally the symmetric product of Killing tensors defines againa Killing tensor. Conversely, it is known that on manifolds of constant sectional curvature,any Killing tensor can be written as a linear combination of symmetric products of Killingvector fields, cf. [29], Theorem 4.7. Lemma 4.1.
For any two Killing vector fields ξ , ξ it holds that δ ( ξ · ξ ) = d g ( ξ , ξ ) . Proof.
We compute δ ( ξ · ξ ) = − P e i y ∇ e i ( ξ · ξ ) = − P e i y (( ∇ e i ξ ) · ξ + ξ · ( ∇ e i ξ )). Sincethe Killing vector fields ξ , ξ are divergence free we obtain δ ( ξ · ξ ) = −∇ ξ ξ − ∇ ξ ξ . Usingagain that ξ , ξ are Killing vector field we have X ( g ( ξ , ξ )) = g ( ∇ X ξ , ξ ) + g ( ξ , ∇ X ξ ) = − g ( ∇ ξ ξ + ∇ ξ ξ , X ) = g ( δ ( ξ · ξ ) , X ) . (cid:3) Corollary 4.2.
Let ξ , ξ be two Killing vector fields with constant scalar product g ( ξ , ξ ) .Then ξ · ξ is a divergence free Killing tensor and h := ξ · ξ − g ( ξ , ξ ) n g is a trace-free, divergence-free Killing -tensor.Proof. Indeed the trace of the symmetric endomorphism h = ξ · ξ ∈ Γ(Sym T M ) is givenas tr( h ) = 2 g ( ξ , ξ ). Hence the tensor h defined as above is a trace-free and divergence-freeKilling tensor. (cid:3) Example 4.3.
Let ( M n , g, ξ ) be a Sasakian manifold. Then ξ · ξ − n g is a trace-free Killing2-tensor. On a 3-Sasakian manifold one has three pairwise orthogonal Killing vector fields ofunit length defining a six-dimensional space of trace-free Killing 2-tensors. Example 4.4.
On spheres S n with n ≥ n + 1 defines a pair of orthogonal Killing vector fields on S n .4.3. Killing tensors from Killing forms.
There is a well-known relation between Killingforms and Killing tensors (e.g. cf. [3]). Let u ∈ Ω p ( M ) be a Killing form, i.e. a p -formsatisfying the equation X y ∇ X u = 0 for any tangent vector X . We define a symmetric bilinearform K u by K u ( X, Y ) = g ( X y u, Y y u ). Then K u is a symmetric Killing 2-tensor (for p = 2this fact was also remarked in [10], Rem. 2.1). Indeed it suffices to show ( ∇ X K u )( X, X ) = 0for any tangent vector X , which is immediate:( ∇ X K u )( X, X ) = ∇ X ( K u ( X, X )) − K u ( ∇ X X, X ) = 2 g ( X y ∇ X u, X y u ) = 0 . Since Killing 1-forms are dual to Killing vector fields, this construction generalizes the the onedescribed in Section 4.2. If u is a Killing 2-form considered as skew-symmetric endomorphism,then the associated symmetric Killing tensor K u is just − u . In this case K u commutes withthe Ricci tensor, since the same is true for the skew-symmetric endomorphisms correspondingto Killing 2-forms, cf. [3]. Examples of manifolds with Killing forms are: the standard sphere S n , Sasakian, 3-Sasakian, nearly K¨ahler or weak G manifolds [27].A related construction appears in the work of V. Apostolov, D.M.J. Calderbank and P.Gauduchon [1], in particular Appendix B.4. They prove a 1–1 relation between symmetric J - invariant Killing 2-tensors on K¨ahler surfaces and Hamiltonian 2-forms. In contrast toKilling forms there are many examples known of Hamiltonian 2-forms, thus providing a richsource of symmetric Killing tensors.4.4. The Ricci curvature as Killing tensor.
Special examples of Killing tensors arise asRicci curvature of a Riemannian metric. Notice that if Ric is a Killing tensor then the scalarcurvature scal is constant. Riemannian manifolds whose Ricci tensor is Killing were studied in[2] Ch. 16.G, as a class of generalized Einstein manifolds. In the same context this conditionwas originally discussed by A. Gray in [12]. It can be shown shown that all D’Atri spaces, i.e.Riemannian manifolds whose local geodesic symmetries preserve, up to a sign, the volumeelement, have Killing Ricci tensor (cf. [2], [6]). This provides a wide class of examples, manyof them with non-parallel Ricci tensor. In particular naturally reductive spaces are D’Atri,and thus have Killing Ricci tensor. Here we want to present a direct argument.
ONFORMAL KILLING TENSORS 15
Proposition 4.5.
The Ricci curvature of a naturally reductive space is a Killing tensor.Proof.
Naturally reductive spaces are characterized by the existence of a metric connection ¯ ∇ with skew-symmetric, ¯ ∇ -parallel torsion T and parallel curvature ¯ R . This gives in particularthe following equations g ( T X Y, Z ) + g ( Y, T X Z ) = 0 and T X Y + T Y X = 0 . The condition ¯ ∇ ¯ R = 0 can be rewritten as the following equation for the Riemannian curva-ture R ( ∇ X R ) Y,Z = [ T X , R Y,Z ] − R T X Y,Z − R Y,T X Z . The Ricci curvature is defined as Ric(
X, Y ) = P g ( R X,e i e i , Y ), where { e i } is an orthonormalframe. Its covariant derivative is given by( ∇ Z Ric)(
X, Y ) = X g (( ∇ Z R ) X,e i e i , Y ) . The Ricci curvature Ric ∈ Γ(Sym T M ) is a Killing tensor if and only if ( ∇ X Ric)(
X, X ) = 0for all tangent vectors X , which is equivalent to X g (( ∇ X R ) X,e i e i , X ) = 0 . If R is the Riemannian curvature tensor of naturally reductive metric this curvature expressioncan be rewritten as X g (( ∇ X R ) X,e i e i , X ) = X g ([ T X , R X,e i ] e i − R T X X,e i e i − R X,T X e i e i , X )= X g ( T X R X,e i e i , X ) − g ( R X,e i T X e i , X ) − g ( R X,T X e i e i , X )= X − g ( R X,e i e i , T X X ) − g ( R X,e i T X e i , X )= 0Here we used T X X = 0 and the equation g ( R X,e i T X e i , X ) = 0, which holds because of g ( R X,e i T X e i , X ) = − g ( R e i ,T X e i X, X ) − g ( R T X e i ,X e i , X ) = g ( R X,T X e i e i , X )= − g ( R X,e i T X e i , X ) . (cid:3) We define a modified Ricci tensor f Ric ∈ Sym T M as f Ric := Ric − n +2 id . If Ric is a Killingtensor then the same is true for f Ric, but not conversely.
Lemma 4.6.
The modified Ricci tensor f Ric is Killing tensor if and only if it is a conformalKilling tensor. Moreover f Ric is Killing if and only if ( ∇ X Ric)(
X, X ) = n +2 X (scal) g ( X, X ) for all vector fields X . Proof.
A symmetric 2-tensor is a Killing tensor if and only if the two equations of (11) aresatisfied. The first equation characterizes conformal Killing 2-tensors. Hence we only have toshow that the second equations holds for f Ric, i.e. we have to show that d tr( f Ric) = 2 δ f Ric.Since tr( f Ric) = − nn +2 scal , the well-known relation δ Ric = − d scal implies δ f Ric = − d scal + n +2 d scal = − n +22( n +2) d scal , d tr( f Ric) = − nn +2 d scal . Hence d tr( f Ric) = 2 δ f Ric and the modified Ricci tensor f Ric is a Killing tensor if it is aconformal Killing tensor. The other direction and the equation for Ric are obvious. (cid:3)
Remark 4.7.
A. Gray introduced in [12] the notation A , for the class of Riemannian man-ifolds with Killing Ricci tensor, and C for the class of Riemannian manifolds with constantscalar curvature. The class of Riemannian manifolds whose modified Ricci tensor is Killingwas studied by W. Jelonek in [15] under the name A ⊕ C ⊥ . Finally we note that there aremanifolds with Killing Ricci tensors which are neither homogeneous nor D’Atri. Exampleswere constructed by H. Pedersen and P. Tod in [24] and by W. Jelonek in [14].5. Conformal Killing tensors on Riemannian products
Let ( M , g ) and ( M , g ) be two compact Riemannian manifolds. The aim of this sectionis to prove the following result, which reduces the study of conformal Killing 2-tensors onRiemannian products M := M × M to that of Killing tensors on the factors. Theorem 5.1.
Let h ∈ Γ(Sym ( T M )) be a trace-free conformal Killing tensor. Then thereexist Killing tensors K i ∈ Γ(Sym ( T M i )) , i = 1 , , and Killing vector fields ξ , . . . , ξ k on M and ζ , . . . , ζ k on M such that h = ( K + K ) + k X i =1 ξ i · ζ i . Conversely, every such tensor on M is a trace-free conformal Killing tensor.Proof. We denote by n , n and n the dimensions of M , M and M . Consider the natural de-composition h = h + h + φ , where h , h and φ are sections of π ∗ Sym ( T M ), π ∗ Sym ( T M ),and π ∗ T M ⊗ π ∗ T M respectively. We consider the lifts to M of the operators d i , δ i , L i , Λ i on the factors. Clearly two such operators commute if they have different subscripts, andsatisfy the relations (1) and (8) if they have the same subscript. We define f i := Λ i ( h i ). Since h is trace-free, we have f + f = 0. The conformal Killing equationd h = − n + 2 L δh ONFORMAL KILLING TENSORS 17 reads d h + d h + d φ + d h + d h + d φ = − n + 2 L ( δ h + δ φ + δ h + δ φ ) − n + 2 L ( δ h + δ φ + δ h + δ φ ) . Projecting this equation onto the different summands of Sym ( T M ) yields the followingsystem:(16) d h = − n +2 L ( δ h + δ φ )d h = − n +2 L ( δ h + δ φ )d h + d φ = − n +2 L ( δ h + δ φ )d h + d φ = − n +2 L ( δ h + δ φ )Applying Λ to the first equation of (16) and using (8) gives − δ h + d f = − n + 2) n + 2 ( δ h + δ φ ) , whence(17) δ h = n + 2 n δ φ + n + 22 n d f . Similarly, applying Λ to the second equation of (16) gives(18) δ h = n + 2 n δ φ + n + 22 n d f . Replacing δ i h i in the right hand side of (16) using (17) and (18), yields(19) d h = − n L (2 δ φ + d f )d h = − n L (2 δ φ + d f )d h + d φ = − n L (2 δ φ + d f )d h + d φ = − n L (2 δ φ + d f )We now apply δ to the third equation of (19) and use (8) and (18) together with the factthat f = − f to compute: δ d φ = − δ d h + 1 n d (2 δ φ + d f )= − d (cid:18) n + 2 n δ φ + n + 22 n d f (cid:19) + 2 n d δ φ + 1 n d d f = − n + 2 n d δ φ + 2 n d δ φ + n ( n + 2)2 n n d d f . Similarly, applying δ to the fourth equation of (19) and using (8) and (17) yields:(20) δ d φ = − n + 2 n d δ φ + 2 n d δ φ + n ( n + 2)2 n n d d f . In order to eliminate the terms involving f and f in these last two formulas, we multiplythe first one with n + 2 and add it to the second one multiplied with n + 2, which yields( n + 2) δ d φ + ( n + 2) δ d φ + ( n + 2)d δ φ + ( n + 2)d δ φ = 0 , which after a scalar product with φ and integration over M gives d φ = d φ = 0 and δ φ = δ φ = 0. Plugging this back into (20) also shows that d d f = 0. In other words,there exist functions ϕ i ∈ C ∞ ( M i ) such that f = ϕ + ϕ , and correspondingly f = − ϕ − ϕ (we identify here ϕ i with their pullbacks to M in order to simplify the notations).The system (19) thus becomes:(21) d h = − n L d f = n L d ϕ d h = − n L d f = − n L d ϕ d h = − n L d f = n L d ϕ d h = − n L d f = − n L d ϕ This system shows that the tensors K := h + 12 n L ϕ − n L ϕ and K := h + 12 n L ϕ − n L ϕ verify d K = d K = 0 and d K = d K = 0. The first two equations show that K and K are pull-backs of symmetric tensors on M and M , and the last two equations show thatthese tensors are Killing tensors of the respective factors M and M . Moreover, we have K + K = h + h + L (cid:18) n ϕ − n ϕ (cid:19) and thus ( K + K ) = h + h .Finally, we claim that d φ = 0 and d φ = 0 imply that φ has the form stated in thetheorem. Let T := π ∗ ( T M ) and T := π ∗ ( T M ) denote the pull-backs on M of the tangentbundles of the factors. Then φ is a section of T ⊗ T , and d φ = 0 and d φ = 0 imply that ∇ φ = ∇ M φ + ∇ M φ is a section of Λ T ⊗ T + T ⊗ Λ T . In some sense, one can view φ as a Killing vector field on M twisted with T , and also as a Killing vector field on M twisted with T . Let us denote by φ := ∇ M φ ∈ Γ(Λ T ⊗ T ), φ := ∇ M φ ∈ Γ( T ⊗ Λ T )and φ := ∇ M ∇ M φ = ∇ M ∇ M φ ∈ Γ(Λ T ⊗ Λ T ). The usual Kostant formula for Killingvector fields immediately generalizes to ∇ X φ = R X φ, ∇ X φ = R X φ, ∇ X φ = R X φ , ∇ X φ = R X φ , ONFORMAL KILLING TENSORS 19 where for any vector bundle F , i ∈ { , } , and X ∈ T i , R X : T i ⊗ F → Λ T i ⊗ F is definedby R X ( Y ⊗ σ ) := R X,Y ⊗ σ .We thus get a parallel section Φ := ( φ, φ , φ , φ ) of the vector bundle E := ( T ⊗ T ) ⊕ (Λ T ⊗ T ) ⊕ ( T ⊗ Λ T ) ⊕ (Λ T ⊗ Λ T )with respect to the connection defined on vectors X i ∈ T i by˜ ∇ X ( φ, φ , φ , φ ) := ( ∇ X φ − φ ( X ) , ∇ X φ − R X φ, ∇ X φ − φ ( X ) , ∇ X φ − R X φ )and˜ ∇ X ( φ, φ , φ , φ ) := ( ∇ X φ − φ ( X ) , ∇ X φ − φ ( X ) , ∇ X φ − R X φ, ∇ X φ − R X φ ) . We now define for i = 1 , ∇ i on E i := T M i ⊕ Λ T M i by˜ ∇ iX i ( α i , β i ) := ( ∇ M i X i α i − β i ( X i ) , ∇ M i X i β i − R M i X i ( α i )) , and notice that E = π ∗ ( E ) ⊗ π ∗ ( E ) and that ˜ ∇ coincides with the tensor product connectioninduced by ˜ ∇ and ˜ ∇ on E . It follows that the space of ˜ ∇ -parallel sections of E is the tensorproduct of the spaces of ˜ ∇ -parallel sections of E and of ˜ ∇ -parallel sections of E . Takingthe first component of these sections yields the desired result. (cid:3) Remark 5.2.
Note that the compactness assumption in Theorem 5.1 is essential. Thereare many non compact products, with trace-free conformal Killing 2-tensors which are notdefined by Killing tensors of the factors. The simplest example is the flat space R n .6. Weitzenb¨ock formulas
Let ( M n , g ) be an oriented Riemannian manifold with Riemannian curvature tensor R . Thecurvature operator R : Λ T M → Λ T M is defined by g ( R ( X ∧ Y ) , Z ∧ V ) = R ( X, Y, Z, V ).With this convention we have R = − id on the standard sphere.Let P = P SO( n ) be the frame bundle and EM a vector bundle associated to P via a SO( n )-representation ρ : SO( n ) → Aut( E ). Then the curvature endomorphism q ( R ) ∈ End EM isdefined as q ( R ) := 12 X i,j ( e i ∧ e j ) ∗ ◦ R ( e i ∧ e j ) ∗ . Here { e i } , i = 1 , . . . n , is a local orthonormal frame and for X ∧ Y ∈ Λ T M we define( X ∧ Y ) ∗ = ρ ∗ ( X ∧ Y ), where ρ ∗ : so ( n ) → End E is the differential of ρ . In particular, thestandard action of Λ T M on T M is written as ( X ∧ Y ) ∗ Z = g ( X, Z ) Y − g ( Y, Z ) X =( Y · X y − X · Y y ) Z . This is compatible with g (( X ∧ Y ) ∗ Z, V ) = g ( X ∧ Y, Z ∧ V ) = g ( X, Z ) g ( Y, V ) − g ( X, V ) g ( Y, Z ) . For any section ϕ ∈ Γ( EM ) we have R ( X ∧ Y ) ∗ ϕ = R X,Y ϕ . It is easy to check that q ( R ) acts as the Ricci tensor on tangent vectors. The definition of q ( R ) is independent ofthe orthonormal frame of Λ T M , i.e. q ( R ) can be written as q ( R ) = P ω i ∗ ◦ R ( ω i ) ∗ for any orthonormal frame of Λ T M . Moreover it is easy to verify that q ( R ) is a symmetricendomorphism of the vector bundle EM .The action of q ( R ) on a symmetric p -tensors K can be written as q ( R ) K = P e j · e i y R e i e j K .On symmetric 2-tensors h the curvature endomorphism q ( R ) is related to the classical cur-vature endomorphism ˚ R (cf. [2, p. 52]), which is defined by( ˚ Rh )( X, Y ) = X h ( R X,e i Y, e i ) . If h is considered as a symmetric endomorphism the action of ˚ R on h can be written as˚ R ( h )( X ) = − P R X,e i h ( e i ) .The action of Ric is extended to symmetric 2-tensors h as a derivation, i.e. it is defined asRic( h )( X, Y ) = − h (Ric X, Y ) − h ( X, Ric Y ) . Then the following formula holds on Sym T M :(22) q ( R ) = 2 ˚ R − Ric . If h is the metric g then ( ˚ Rg )( X, Y ) = − Ric(
X, Y ) and Ric( g )( X, Y ) = − X, Y ) .As seen above, the covariant derivative ∇ on Sym p T M decomposes into three componentsdefining three first order differential operators: P i ( K ) := p i ( ∇ K ) , i = 1 , ,
3, where p i are theorthogonal projections onto the three summands in the decomposition (4). The operators P ∗ i P i , i = 1 , , p T M . These threeoperators are linked by a Weitzenb¨ock formula: Proposition 6.1.
Let K be any section of Sym p T M , then: q ( R ) K = − p P ∗ P K + ( n + p − P ∗ P K + P ∗ P K .
Proof.
The stated Weitzenb¨ock formula can be obtained as a special case of a general proce-dure described in [28]. However it is easy to check it directly using the following remarks.Let E be any SO( n )-representation defining a vector bundle EM and let T be the standardrepresentation defining the tangent bundle T M . Then any p ∈ End( T ⊗ E ) can be interpretedas an element in Hom( T ⊗ T ⊗ E, E ) defined as p ( a ⊗ b ⊗ e ) = ( a y ⊗ id) p ( b ⊗ e ), for a, b ∈ Tand e ∈ E . Important examples of such endomorphisms are the orthogonal projections p i , i = 1 , . . . , N , onto the summands in a decomposition T ⊗ E = V ⊕ . . . ⊕ V N . Anotherexample is the so-called conformal weight operator B ∈ Hom( T ⊗ T ⊗ E, E ) defined as B ( a ⊗ b ⊗ e ) = ( a ∧ b ) ∗ e . As an element in End( T ⊗ E ), the conformal weight operator canbe written as B ( b ⊗ e ) = P e i ⊗ ( e i ∧ b ) ∗ e .Let K be a section of EM , then ∇ K = P e i ⊗ e j ⊗ ∇ e i ,e j K is a section of the bundleHom( T M ⊗ T M ⊗ EM, EM ). Using the remark above we can apply elements of the bundleEnd( T M ⊗ EM ) to ∇ K . It is then easy to check that B ( ∇ K ) = q ( R ) K, id( ∇ K ) = − ∇ ∗ ∇ K, p i ( ∇ K ) = − P ∗ i P i K where P i , i = 1 , . . . , N are the first order differential operators P i ( K ) := p i ( ∇ K ). Hence inorder to prove the Weitzenb¨ock formula above it is enough to verify the following equation ONFORMAL KILLING TENSORS 21 for endomorphisms of T ⊗ E in the case E = Sym p T: B = p p − ( n + p − p − p = ( p + 1) p − ( n + p − p − id= π ∗ π − n +2 p − n +2 p − π ∗ π − id . This is an easy calculation using the explicit formulas for π ∗ i and π i , i = 1 , (cid:3) Eigenvalue estimates for the Lichnerowicz Laplacian.
The Lichnerowicz Lapla-cian ∆ L is a Laplace-type operator acting on sections of Sym p T M . It can be defined by∆ L := ∇ ∗ ∇ + q ( R ). On symmetric 2-tensors it is usually written as ∆ L = ∇ ∗ ∇ + 2 ˚ R − Ric,which is the same formula, by (22).
Proposition 6.2.
Let ( M n , g ) be a compact Riemannian manifold. Then ∆ L ≥ q ( R ) holds on the space of divergence-free symmetric tensors. Equality ∆ L h = 2 q ( R ) h holds for adivergence free tensor h if and only if h is a Killing tensor.Proof. Directly from the definition we calculated δh = − X e i · ∇ e i ( e j y ∇ e j h ) = − X e i · ( e j y ∇ e i ∇ e j h ) . Similarly we have δ d h = − X e i y ∇ e i ( e j · ∇ e j h ) = − X e i y ( e j · ∇ e i ∇ e j h )= − X ∇ e i ∇ e i h − X e j · ( e i y ∇ e i ∇ e j h )= ∇ ∗ ∇ h − X e i · ( e j y ∇ e j ∇ e i h )Taking the difference we immediately obtain δ d h − d δh = ∇ ∗ ∇ h − X e i · e j y R e j ,e i h = ∇ ∗ ∇ h − q ( R ) h = ∆ L h − q ( R ) h . Thus, if h is divergence free we have (∆ L − q ( R )) h = δ d h and the inequality follows aftertaking the L product with h . The equality case is clearly characterized by d h = 0. (cid:3) Remark 6.3.
For symmetric 2-tensors this estimate for ∆ L was proven in [11]. Remark 6.4.
As a consequence of Proposition 6.2 we see that divergence free Killing tensorson compact Riemannian manifolds are characterized by the equation ∇ ∗ ∇ h = q ( R ) h . Thisgeneralizes the well known characterization of Killing vector fields as divergence free vectorfields ξ with ∇ ∗ ∇ ξ = Ric( ξ ). Remark 6.5.
Recall that q ( R ) is a symmetric endomorphism. The eigenvalues of q ( R )are constant on homogeneous spaces. On symmetric spaces M = G/K the LichnerowiczLaplacian ∆ L can be identified with the Casimir operator Cas G of the group G and q ( R ) withthe Casimir operator Cas K of the group K . Non-existence results.
In [5] Dairbekov and Sharafutdinov show the non-existence oftrace-free conformal Killing tensors on manifolds with negative sectional curvature. In thissection we will give a short new proof of this result.
Proposition 6.6.
On a compact Riemannian manifold ( M, g ) of non-positive sectional cur-vature any trace-free conformal Killing tensor has to be parallel. If in addition there exists apoint in M where the sectional curvature of every two-plane is strictly negative, then M doesnot carry any (non-identically zero) trace-free conformal Killing tensor.Proof. On sections of Sym p T M we consider the Weitzenb¨ock formula of Proposition 6.1: q ( R ) = − p P ∗ P + ( n + p − P ∗ P + P ∗ P . Trace-free conformal Killing tensors are characterized by the equation P K = 0. In particularwe obtain for the L -scalar product:(23) ( q ( R ) K, K ) L = ( n + p − k P K k + k P K k ≥ , where K is a trace-free conformal Killing tensor. We will show that ( q ( R ) K, K ) L ≤ P K = 0 and P K = 0 and thus that K has to be parallel.For any x ∈ M and any fixed tangent vector X ∈ T x M we consider the symmetricbilinear form B X ( Y, Z ) := g ( R X,Y
X, Z ), defined on tangent vectors
Y, Z ∈ T x M . Since thesectional curvature is non-positive, this bilinear form is positive semi-definite. Hence there isan orthonormal basis e , . . . , e n of T x M (depending on X ) with B X ( e i , e j ) = 0 for i = j and B X ( e i , e i ) = a i ( X ) ≥ i .A symmetric tensor K ∈ Γ(Sym p T M ) can also be considered as a polynomial map onT M by the formula K ( X ) := g ( K, X p ). In particular we have for the Riemannian curvature( R Y,Z K )( X ) = ( P R Y,Z e k · e k y K )( X ) = p P g ( R Y,Z e k , X ) ( e k y K )( X ).Let T be the tangent space T = T x M for some x ∈ M . Then we can define a scalar producton Sym p T by ˜ g ( K , K ) := R S T K ( X ) K ( X )d µ , where S T is the unit sphere in T and d µ denotes the standard Lebesgue measure on S T . From Schur’s Lemma it follows the existenceof a non-zero constant c such that ˜ g ( K , K ) = c g ( K , K ) holds for all K , K ∈ Sym p T .Since both scalar products are positive definite the constant c has to be positive. We nowcompute the scalar product g ( q ( R ) K, K ) at some point x ∈ M . From the remarks above we ONFORMAL KILLING TENSORS 23 obtain: g ( q ( R ) K, K ) = X g ( e j · e i y R e i ,e j K, K ) = X g ( R e i ,e j K, e i · e j y K )= c Z S T X ( R e i ,e j K )( X ) · ( e i · e j y K )( X ) d µ = c p Z S T X g ( R e i ,e j e k , X ) · ( e k y K )( X ) · g ( e i , X ) · ( e j y K )( X ) d µ = − c p Z S T X g ( R X,e j X, e k ) · ( e k y K )( X ) · ( e j y K )( X ) d µ = − c p Z S T X B X ( e j , e k ) · ( e k y K )( X ) · ( e j y K )( X ) d µ = − c p Z S T X a j ( X ) (( e j y K )( X )) d µ ≤ . This proves that for every trace-free symmetric tensor K , on a manifold with non-positivesectional curvature, the inequality g ( q ( R ) K, K ) ≤ K is conformal Killing, then K has to be parallel.If in addition there is a point x ∈ M where all sectional curvatures are negative, thenthe symmetric form B X is positive definite for all X ∈ S T , so its eigenvalues are positive: a j ( X ) >
0. The computation above shows that ( Y y K )( X ) = 0 for every X ∈ S T and for alltangent vectors Y orthogonal to X . This is equivalent to(24) 0 = g ( K, ( | X | Y − h X, Y i X ) · X p − ) = | X | g ( K, Y · X p − ) − h X, Y i g ( K, X p )for all tangent vectors X, Y ∈ T x M . On the other hand, from (2) we immediately get forevery X, Y ∈ T x M g ( Y · K, X p +1 ) = ( p + 1) h X, Y i g ( K, X p )and g (L · ( Y y K ) , X p +1 ) = p ( p + 1) | X | g ( K, Y · X p − ) . From (24) we thus obtain L · ( Y y K ) = p Y · K, ∀ Y ∈ T x M .
Applying Λ and using (1) and (2) together with the fact that Λ K = 0, yields(2 n + 4( p − Y y K = 2 p Y y K at x , whence K x = 0. As K is parallel from the first part of the proof, this shows that K ≡ (cid:3) Corollary 6.7.
Let Σ g be a compact Riemannian surface of genus g ≥ . Then M admits notrace-free conformal Killing tensors. More generally, there are no trace-free conformal Killingtensors on compact quotients of symmetric spaces of non-compact type. Remark 6.8.
Note that this result was also obtained by D.J.F. Fox in [9], Corollary 3.1.7.
Killing tensors with two eigenvalues
Let K ∈ Γ(Sym T M ) be a non-trivial trace-free Killing (i.e. St¨ackel) tensor on a connectedRiemannian manifold ( M n , g ). We assume throughout this section that K has at most twoeigenvalues at every point of M .The following result was proved by W. Jelonek (cf. [15, Theorem 2.1]): Lemma 7.1.
The multiplicities of the eigenvalues of K are constant on M , so the eigenspacesof K define two distributions T M = E ⊕ E . If n , n denote the dimensions of E , E and π i denote the orthogonal projection onto E i for i = 1 , , then K is a constant multiple of n π − n π .Proof. Since K is trace-free, the eigenvalues of K are distinct at every point p ∈ M where K p = 0. Every such point p has a neighborhood U on which the multiplicities of the eigen-values of K are constant. The eigenspaces of K define two orthogonal distributions E and E along U such that T M | U = E ⊕ E . Then the Killing tensor K can be written as K = f π + hπ . Since K is trace-free, we have 0 = n f + n h . The covariant derivative of K can be written as g (( ∇ X K ) Y, Z ) = g ( ∇ X ( KY ) − K ( ∇ X Y ) , Z )= g ( X ( f ) π ( Y ) + f ( ∇ X π ) Y + X ( h ) π ( Y ) + h ( ∇ X π ) Y, Z ) . Note that for any vector X and vector fields X i , Y i ∈ E i for i = 1 , g (( ∇ X π ) X i , Y i ) = 0and similarly for π . For X ∈ E , the Killing tensor equation gives ( ∇ X K )( X, X ) = 0 and itfollows from the formula above that g ( X ( f ) π ( X ) , X ) = 0. Thus X ( f ) = 0 for all X ∈ E and similarly X ( h ) = 0 for all X ∈ E . It follows that f and h are constant on U , since f and h are related via n f + n h = 0. The eigenvalues of K are thus constant on U . Since this istrue on some neighbourhood of every point p where K p = 0, we deduce that the eigenvaluesof K , and their multiplicities, are constant on M . This proves the lemma. (cid:3) We will now characterize orthogonal splittings of the tangent bundle which lead to trace-freeKilling tensors.
Proposition 7.2.
Let E and E be orthogonal complementary distributions on M of dimen-sions n and n respectively. Then the trace-free symmetric tensor K = n π − n π is Killingif and only if the following conditions hold: (26) ∇ X X ∈ Γ( E ) ∀ X ∈ Γ( E ) and ∇ X X ∈ Γ( E ) ∀ X ∈ Γ( E ) . ONFORMAL KILLING TENSORS 25
Proof.
Assume first that K = n π − n π is a Killing tensor. Since π + π = id is parallel,we see that π and π are Killing tensors too. Let X ∈ Γ( E ) and X ∈ Γ( E ). As π is aKilling tensor, we get from (25):0 = g (( ∇ X π ) X , X ) + g (( ∇ X π ) X , X ) + g (( ∇ X π ) X , X )= 2 g (( ∇ X π ) X , X )= 2 g ( ∇ X X , X ) , and similarly 0 = g ( ∇ X X , X ), thus proving (26).Conversely, if (26) holds, then for every vector field X on M we can write X = X + X ,where X i := π i ( X ) for i = 1 , g (( ∇ X π ) X, X ) = 2 g (( ∇ X π ) X , X ) = 2 g ( ∇ X X , X )= 2 g ( ∇ X X , X ) + 2 g ( ∇ X X , X )= 2 g ( ∇ X X , X ) − g ( ∇ X X , X ) = 0 . (cid:3) Pairs of distributions with this property were studied in [21] by A. Naveira under the nameof almost product structures of type D . Note that Killing tensors with two eigenvalueswere intensively studied by W. Jelonek in [15], [16], [17] and also by B. Coll et al. in [4].In particular W. Jelonek proves that a Killing tensor with constant eigenvalues satisfies ourcondition (26). Thus our Proposition 7.2 is in some sense a converse to his result. Example 7.3. If M → N is a Riemannian submersion with totally geodesic fibers and V and H denote its vertical and horizontal distributions, then E := V and E := H satisfy(26), by the O’Neill formulas. It turns out that this generalizes several examples of Killingtensors appearing in the physics literature, e.g. in [11]. Remark 7.4.
Note that (26) does not imply the integrability of E or E . However, assumingthat (26) holds and that one of the distributions, say E , is integrable, then there existslocally a Riemannian submersion with totally geodesic fibers whose vertical and horizontaldistributions are E and E respectively.8. Conformal Killing tensors on hypersurfaces
In this last section we give a short proof, using the formalism developed above, of a van-ishing result of Dairbekov and Sharafutdinov:
Theorem 8.1 ([5]) . Let ( M n , g ) be a connected Riemannian manifold and let H ⊂ M be ahypersurface. If a trace-free conformal Killing tensor K vanishes along H , then K vanishesidentically on M .Proof. Let K ∈ Γ(Sym p T M ) be a trace-free conformal Killing tensor vanishing along ahypersurface H ⊂ M . Starting with K := K we recursively define tensors K l := ∇ K l − , which are sections of T M ⊗ l ⊗ Sym p T M . We claim that all tensors K i vanish along H . Sincethe conformal Killing equation is of finite type this will imply that K is identically zero on M .Consider the natural extensions d : Γ( T M ⊗ l ⊗ Sym p T M ) → Γ( T M ⊗ l ⊗ Sym p +1 T M ), δ : Γ( T M ⊗ l ⊗ Sym p T M ) → Γ( T M ⊗ l ⊗ Sym p − T M ) and ∇ : Γ( T M ⊗ l ⊗ Sym p T M ) → Γ( T M ⊗ ( l +1) ⊗ Sym p T M ) of d , δ and ∇ , defined on decomposable tensors byd( T ⊗ K ) := X i e i · ∇ e i ( T ⊗ K ) = T ⊗ d K + X i ∇ e i T ⊗ e i · K ,δ ( T ⊗ K ) := − X i e i y ∇ e i ( T ⊗ K ) = T ⊗ δK − X i ∇ e i T ⊗ e i y K , ∇ ( T ⊗ K ) := ∇ T ⊗ K + X i ( e i ⊗ T ) ⊗ ∇ e i K , where { e i } denotes as usual a local orthonormal basis of T M . A straightforward computationshows that(27) [d , ∇ ] = R + , [ δ, ∇ ] = R − , where R + : T M ⊗ l ⊗ Sym p T M → T M ⊗ ( l +1) ⊗ Sym p +1 T M is defined by R + ( T ⊗ K ) := X i,j ( e i ⊗ T ) ⊗ ( e j · R e j ,e i K ) + ( e i ⊗ R e j ,e i T ) ⊗ ( e j · K ) , and R − : T M ⊗ l ⊗ Sym p T M → T M ⊗ ( l +1) ⊗ Sym p − T M is defined by R − ( T ⊗ K ) := − X i,j ( e i ⊗ T ) ⊗ ( e j y R e j ,e i K ) + ( e i ⊗ R e j ,e i T ) ⊗ ( e j y K ) . Since K is trace-free conformal Killing, Lemma 3.3 shows that(28) d K = − n +2 p − L δK . (Note that since K is trace-free, the notation K from Lemma 3.3 coincides with our notation K = K above). We will prove by induction that there exist vector bundle morphisms F i,l : T M ⊗ i ⊗ Sym p T M → T M ⊗ l ⊗ Sym p +1 T M such that(29) d K l = − n +2 p − L δK l + l − X i =0 F i,l ( K i ) , where here L : T M ⊗ i ⊗ Sym p T M → T M ⊗ i ⊗ Sym p +2 T M denotes the natural extension ofL, which of course commutes with ∇ . For l = 0, this is just (28). Assuming that the relation ONFORMAL KILLING TENSORS 27 holds for some l ≥ K l +1 = d ∇ K l = ∇ d K l + R + K l = ∇ − n +2 p − L δK l + l − X i =0 F i,l ( K i ) ! + R + K l = − n +2 p − L ∇ δK l + l − X i =0 (( ∇ F i,l )( K i ) + (id ⊗ F i,l )( K i +1 )) + R + K l = − n +2 p − (cid:0) L δK l +1 + L R − K l (cid:1) + l − X i =0 (( ∇ F i,l )( K i ) + (id ⊗ F i,l )( K i +1 )) + R + K l , which is just (29) for l replaced by l + 1 and F i,l +1 := ( ∇ F i,l + (id ⊗ F i − ,l ) , i ≤ l − − n +2 p − L R − + (id ⊗ F l − ,l ) + R + , i = l . This proves (29) for all l .Assume now that K , . . . , K l vanish along H for some l ≥
0. We claim that K l +1 is alsovanishing along H . Take any point x ∈ H and choose a local orthonormal frame { e i } suchthat e =: N is normal to H and e , . . . , e n are tangent to H at x . From (29) we haved K l = − n +2 p − L δK l at x . Moreover, ∇ e i K l vanishes at x for every i ≥
2. The previousrelation thus reads(30) N · ∇ N K l = n +2 p − L N y ∇ N K l . Writing ∇ N K l = X I ∈{ ,...,n } l e I ⊗ S I , with e I := e i ⊗ . . . ⊗ e i l , and S I ∈ Sym p T M , (30) becomes(31) N · S I = n +2 p − L N y S I for every I . It is easy to check that this implies S I = 0 for every I . Indeed, if i ∈ { , . . . , p } denotes the largest index i such that the coefficient C i of N i in S I is non-zero, comparing thecoefficients of N i +1 in (31) yields C i = n +2 p − i C i , which is clearly impossible for n > ∇ N K l = 0 at x , and since we already noticed that ∇ e i K l vanishes at x forevery i ≥
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Commun. Math.Phys. (1975), no. 1, 9–38. Konstantin Heil, Institut f¨ur Geometrie und Topologie, Fachbereich Mathematik, Uni-versit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
E-mail address : [email protected] Andrei Moroianu, Laboratoire de Math´ematiques de Versailles, UVSQ, CNRS, Universit´eParis-Saclay, 78035 Versailles, France
E-mail address : [email protected] Uwe Semmelmann, Institut f¨ur Geometrie und Topologie, Fachbereich Mathematik, Uni-versit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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