Second Order Symmetries of the Conformal Laplacian

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisf...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2014
Hauptverfasser:	Michel, J.P., Radoux, F., Šilhan, J.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2014
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spelling	Michel, J.P. Radoux, F. Šilhan, J. 2019-02-11T17:00:31Z 2019-02-11T17:00:31Z 2014 Second Order Symmetries of the Conformal Laplacian / J.P. Michel, F. Radoux, J. Šilhan // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58J10; 53A30; 70S10; 53D20; 53D55 DOI:10.3842/SIGMA.2014.016 https://nasplib.isofts.kiev.ua/handle/123456789/146838 Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three. A special thanks is due to Jonathan Kress which points out a mistake in one of the example provided in the first version of this paper. It is a pleasure to acknowledge also Christian Duval and Galliano Valent for their constant interest in this work. Josef Silhan would like to thank ˇ Pawel Nurowski for helpful discussions. This research has been partially funded by the Interuniversity Attraction Poles Program initiated by the Belgian Science Policy Of fice. J. Silhan was supported by the grant agency of ˇ the Czech republic under the grant P201/12/G028. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Second Order Symmetries of the Conformal Laplacian Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Second Order Symmetries of the Conformal Laplacian
spellingShingle	Second Order Symmetries of the Conformal Laplacian Michel, J.P. Radoux, F. Šilhan, J.
title_short	Second Order Symmetries of the Conformal Laplacian
title_full	Second Order Symmetries of the Conformal Laplacian
title_fullStr	Second Order Symmetries of the Conformal Laplacian
title_full_unstemmed	Second Order Symmetries of the Conformal Laplacian
title_sort	second order symmetries of the conformal laplacian
author	Michel, J.P. Radoux, F. Šilhan, J.
author_facet	Michel, J.P. Radoux, F. Šilhan, J.
publishDate	2014
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146838
citation_txt	Second Order Symmetries of the Conformal Laplacian / J.P. Michel, F. Radoux, J. Šilhan // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 31 назв. — англ.
work_keys_str_mv	AT micheljp secondordersymmetriesoftheconformallaplacian AT radouxf secondordersymmetriesoftheconformallaplacian AT silhanj secondordersymmetriesoftheconformallaplacian
first_indexed	2025-11-27T00:43:21Z
last_indexed	2025-11-27T00:43:21Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 016, 26 pages Second Order Symmetries of the Conformal Laplacian Jean-Philippe MICHEL †, Fabian RADOUX † and Josef ŠILHAN ‡ † Department of Mathematics of the University of Liège, Grande Traverse 12, 4000 Liège, Belgium E-mail: jean-philippe.michel@ulg.ac.be, fabian.radoux@ulg.ac.be URL: http://www.geodiff.ulg.ac.be/recherche/fradoux/, http://www.geodiff.ulg.ac.be/recherche/michel/ ‡ Department of Algebra and Geometry of the Masaryk University in Brno, Janàčkovo nàm. 2a, 662 95 Brno, Czech Republic E-mail: silhan@math.muni.cz URL: http://www.math.muni.cz/~silhan/ Received October 25, 2013, in final form February 05, 2014; Published online February 14, 2014 http://dx.doi.org/10.3842/SIGMA.2014.016 Abstract. Let (M, g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M, g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three. Key words: Laplacian; quantization; conformal geometry; separation of variables 2010 Mathematics Subject Classification: 58J10; 53A30; 70S10; 53D20; 53D55 1 Introduction We work over a pseudo-Riemannian manifold (M, g) of dimension n ≥ 3, with Levi-Civita connection ∇ and scalar curvature Sc. Our main result is the classification of all the differential operators D1 of second order such that the relation ∆YD1 = D2∆Y (1.1) holds for some differential operator D2, where ∆Y := ∇agab∇b− n−2 4(n−1)Sc is the Yamabe Lapla- cian. Such operators D1 are called conformal symmetries of order 2 of ∆Y . They preserve the kernel of ∆Y , i.e. the solution space of the equation ∆Y ψ = 0, ψ ∈ C∞(M). Under a conformal change of metric, ĝ = e2Υg, Υ ∈ C∞(M), the Yamabe Laplacian transforms as ∆̂Y = e− n+2 2 Υ ◦∆Y ◦ e n−2 2 Υ, so that each conformal symmetry D1 of ∆Y gives rise to one of ∆̂Y given by D̂1 = e− n−2 2 Υ ◦D1 ◦ e n−2 2 Υ. This emphasizes the conformal nature of the problem and justify our choice of the Yamabe Laplacian, rather than the more usual Laplace–Beltrami one, ∆ := ∇agab∇b. Over flat pseudo- Euclidean space, the classification of conformal symmetries up to second order is due to Boyer, mailto:jean-philippe.michel@ulg.ac.be mailto:fabian.radoux@ulg.ac.be http://www.geodiff.ulg.ac.be/recherche/fradoux/ http://www.geodiff.ulg.ac.be/recherche/michel/ mailto:silhan@math.muni.cz http://www.math.muni.cz/~silhan/ http://dx.doi.org/10.3842/SIGMA.2014.016 2 J.-P. Michel, F. Radoux and J. Šilhan Kalnins and Miller [7], who use it to study the R-separation of variables of the Laplace equa- tion ∆Ψ = 0. More generally, Kalnins and Miller provide an intrinsic characterization for R-separation of this equation on (M, g) in terms of second order conformal symmetries [19]. Thus, classifying those symmetries happens to be a basic problem in the theory of separation of variables. A new input into the quest of conformal symmetries has been given by the work of Eastwood [15]. He classified indeed the conformal symmetries of any order over the confor- mally flat space and exhibited their interesting algebraic structure. This leads to a number of subsequent works, dealing with other invariant operators [16, 18, 31]. Using principal symbol maps, one can extract two informations from the equation (1.1): the operators D1 and D2 have the same principal symbol and the latter is a conformal Killing 2- tensor, i.e. a constant of motion of the geodesic flow, restricted to the null cone. One looks then for a right inverse to the principal symbol maps, called a quantization map, which associates with each conformal Killing tensor a conformal symmetry of ∆Y . For Killing vector fields this is trivial. If K is a 2-tensor, Carter proves that if the minimal prescription K 7→ ∇aKab∇b satisfies [∆Y ,∇aKab∇b] = 0, then K is Killing. Moreover, he shows that if (M, g) is Einstein, i.e. if Ric = 1 nScg with Ric the Ricci tensor, the fact that K is Killing is sufficient to ensure that the minimal prescription above is a symmetry of ∆Y (for application to the separation of variables, see [3]). Besides, in [15], Eastwood defines conformally invariant operators on an arbitrary pseudo-Riemannian manifold, which coincide with the conformal symmetries of ∆Y on the flat space. These operators are given by means of the natural and conformally invariant quantization Qλ0,λ0 (where λ0 = n−2 2n ), developed in [9, 24, 29, 30]. Explicitly, if X is a vector field and K a symmetric trace-less 2-tensor, Qλ0,λ0(X) and Qλ0,λ0(K) are differential operators acting between λ0-densities defined in the following way: Qλ0,λ0(X) = Xa∇a + n− 2 2n (∇aXa), Qλ0,λ0(K) = Kab∇a∇b + n n+2 ( (∇aKab)∇b + n−2 4(n+1) (∇a∇bKab) ) − n+2 4(n+1) RicabK ab. In the conformally flat case, all the conformal symmetries of second order are of the type Qλ0,λ0(K + X + c), where c ∈ R, X is a conformal Killing vector field and K is a conformal Killing 2-tensor. Thanks to the conformal covariance of ∆Y one can show that Qλ0,λ0(X) is still a conformal symmetry of ∆Y on an arbitrary pseudo-Riemannian manifold, if X is a conformal Killing vector field. However, as pointed out by Eastwood in [15], it is unclear whetherQλ0,λ0(K) is a conformal symmetry when K is a conformal Killing 2-tensor. Our strategy relies on the properties of the quantization map Qλ0,λ0 and on the classification of natural and conformally invariant operators acting on prescribed subspaces of symbols. This method has been developed first on conformally flat manifolds, in [26]. In that case, the map Qλ0,λ0 is a conformally equivariant quantization [12], and the author proved that it is precisely the bijective map between conformal Killing tensors and conformal symmetries of ∆Y , discovered by Eastwood. The description of conformal symmetries on arbitrary pseudo-Rieman- nian manifolds is more involved, even at order 2. Namely, there exists a conformal symmetry with principal symbol K if and only if K is a conformal Killing tensor and Obs(K)[ is an exact one-form. Here, Obs is a natural and conformally invariant operator which reads, in abstract index notation, as Obs(K)a = 2(n− 2) 3(n+ 1) (Cr st a∇r − 3Ast a)Kst, Second Order Symmetries of the Conformal Laplacian 3 where C denotes the Weyl tensor and A the Cotton–York tensor. If Obs(K)[ is equal to the exact one-form −2df , with f ∈ C∞(M), then the operators Qλ0,λ0(K +X + c) + f are conformal symmetries of ∆Y for all conformal Killing vector field X and constant c ∈ R. As a consequence, Qλ0,λ0(K) is a conformal symmetry of ∆Y if and only if Obs(K) = 0. We illustrate our results on two examples in dimension three. In the first one, the space R3 is endowed with the most general Riemannian metric admitting a Killing 2-tensor K, which is diagonal in orthogonal coordinates [28]. Then, Obs(K)[ is a non-trivial exact 1-form and, up to our knowledge, the symmetry of ∆Y that we obtain is new. In the second one, we consider a conformal Stäckel metric g on R3 with one ignorable coordinate. Such a metric admits an irreducible conformal Killing tensor K. Using the generic form of g and K given in the reference [8], we obtain that Obs(K)[ is a non-exact 1-form in general. This means there are no conformal symmetries of ∆Y with principal symbol K in general. We detail now the content of the paper. In Section 2, we introduce the basic spaces: the one of tensor densities Fλ(M) of weight λ ∈ R, the one of differential operators Dλ,µ(M) acting between λ- and µ-densities, the one of symbols Sδ(M) with δ = µ − λ. Then, we define the Yamabe Laplacian ∆Y as an element of Dλ0,µ0(M), with λ0 = n−2 2n and µ0 = n+2 2n , so that it becomes a conformally invariant operator. Finally we introduce our main tool, namely the natural and conformally invariant quantization Qλ,µ : Sµ−λ(M) → Dλ,µ(M), and we provide explicit formulas for it. In Section 3, we classify the natural and conformally invariant operators between some sub- spaces of symbols. Among the operators we obtain (and which are crucial for understanding of 2nd order symmetries), one of them, G, is classical, whereas another one, Obs, acting on symbols of second degree, is new and admits no counterpart on flat space. We obtain also an analogous classification for higher order trace-free symbols where the situation is much more complicated. Note that the discovered operators act between source and target spaces of well- known conformally invariant operators, which appear in the generalized BGG sequence [10]. It would be interesting to understand better the relations between all these conformal operators. In Section 4 lies our main result. After defining the spaces of conformal symmetries and of conformal Killing tensors, we prove that, on symbols K of degree 2, we have (Qλ0,µ0)−1 (∆YQλ0,λ0(K)−Qµ0,µ0(K)∆Y ) = 2G(K) + Obs(K). The kernel of G is precisely the space of conformal Killing tensors, whereas Obs(K) is the obstruction for a conformal Killing 2-tensor to provide a conformal symmetry of the form Qλ0,λ0(K). The full description of conformal symmetries of 2nd order of ∆Y easily follows. Using that Qλ0,λ0(K) = Qµ0,µ0(K) for Killing 2-tensors, we deduce also the classification of second order symmetries of ∆Y , which satisfy by definition [∆Y , D1] = 0. In Section 5, we provide two examples illustrating our main result. In the first one, the Killing tensor K is such that Obs(K)[ is a non-vanishing but exact one-form. In the second example, we provide several conformal Killing tensors K such that Obs(K)[ is a non-exact one-form. Hence, there is no conformal symmetry with such K as principal symbols. 2 Conformal geometry, differential operators, and their symbols Throughout this paper, we employ the abstract index notation from [27]. That is, on a smooth manifold M , va denotes a section of the tangent bundle TM , va a section of the cotangent bundle 4 J.-P. Michel, F. Radoux and J. Šilhan T ∗M and e.g. vabc a section of TM ⊗ TM ⊗ T ∗M . The letters a, b, c, d and r, s, t are reserved for abstract indices. Repetition of an abstract index in the covariant and contravariant position means contraction, e.g. vabb is a section of TM . In few places we use concrete indices attached to a coordinate system. This is always explicitly stated and we denote such indices by letters i, j, k, l to avoid confusion with abstract indices. We always use the Einstein’s summation convention for indices, except if stated otherwise. 2.1 Basic objects Let M be a n-dimensional smooth manifold. If λ ∈ R, the vector bundle of λ-densities, Fλ(M)→M , is a line bundle associated with P 1M , the linear frame bundle over M : Fλ(M) = P 1M ×ρ R, where the representation ρ of the group GL(n,R) on R is given by ρ(A)e = \|detA\|−λe, ∀A ∈ GL(n,R), ∀ e ∈ R. We denote by Fλ(M) the space of smooth sections of this bundle. Since Fλ(M) is associated with P 1M , the space Fλ(M) is endowed with canonical actions of Diff(M) and Vect(M). If (x1, . . . , xn) is a coordinate system on M , we denote by \|Dx\|λ the local λ-density equal to [(Id, 1)], where Id is the identity frame in the coordinates system (x1, . . . , xn). Actually, a λ-density ϕ at a point x ∈ M can be viewed as a map on ∧nTxM with values in R such that ϕ(cX1 ∧ · · · ∧Xn) = \|c\|λϕ(X1 ∧ · · · ∧Xn) for all X1, . . . , Xn ∈ TxM and c ∈ R. The λ-density \|Dx\|λ is then the λ-density equal to one on ∂1∧· · ·∧∂n, where ∂1, . . . , ∂n denotes the canonical basis of TxM corresponding to the coordinate system (x1, . . . , xn). If a λ-density ϕ reads locally f \|Dx\|λ, where f is a local function, then the Lie derivative of ϕ in the direction of a vector field X reads locally LλXϕ = ( X.f + λ ( ∂iX i ) f ) \|Dx\|λ. (2.1) It is possible to define the multiplication of two densities. If ϕ1 reads locally f \|Dx\|λ and if ϕ2 reads locally g\|Dx\|δ, then ϕ1ϕ2 reads locally fg\|Dx\|λ+δ. On a pseudo-Riemannian manifold (M, g), it is possible to define in a natural way a λ-density. In a coordinate system, this λ-density reads \|Volg\|λ = \| det g\| λ 2 \|Dx\|λ, where \| det g\| denotes the absolute value of the determinant of the matrix representation of g in the coordinate system. We shall denote by Dλ,µ(M) the space of differential operators from Fλ(M) to Fµ(M). It is the space of linear maps between Fλ(M) and Fµ(M) that read in trivialization charts as differential operators. The actions of Vect(M) and Diff(M) on Dλ,µ(M) are induced by the actions on tensor densities: if LXD denotes the Lie derivative of the differential operator D in the direction of the vector field X, we have LXD = LµX ◦D −D ◦ L λ X , ∀D ∈ Dλ,µ(M) and ∀X ∈ Vect(M). φ ·D = φ ◦D ◦ φ−1, ∀D ∈ Dλ,µ(M) and ∀φ ∈ Diff(M). Second Order Symmetries of the Conformal Laplacian 5 The space Dλ,µ(M) is filtered by the order of differential operators. We denote by Dkλ,µ(M) the space of differential operators of order k. It is well-known that this filtration is preserved by the action of local diffeomorphisms. On a pseudo-Riemannian manifold (M, g), it is easy to build an isomorphism between Dλ,µ(M) and D(M), the space of differential operators acting between functions. Indeed, thanks to the canonical densities built from \|Volg\|, all operators D ∈ Dλ,µ(M) can be pulled-back on functions as follows Fλ(M) D // Fµ(M) C∞(M) \|Volg\|λ OO \|Volg\|−µ◦D◦\|Volg\|λ // C∞(M) \|Volg\|µ OO (2.2) The space of symbols is the graded space associated with Dλ,µ(M): it is then equal to grDλ,µ(M) := ∞⊕ k=0 Dkλ,µ(M)/Dk−1 λ,µ (M). The canonical projection σk : Dkλ,µ(M) → Dkλ,µ(M)/Dk−1 λ,µ (M) is called the principal symbol map. As the actions of Diff(M) and Vect(M) preserve the filtration of Dλ,µ(M), they induce actions of Diff(M) and Vect(M) on the space of symbols. Let δ = µ − λ be the shift of weights. If the sum of the k-order terms of D ∈ Dkλ,µ in a coordinate system (x1, . . . , xn) reads Di1...ik∂i1 · · · ∂ik and if (xi, pi) is the coordinate system on T ∗M canonically associated with (x1, . . . , xn), then we get the following identification: σk(D) ←→ Di1...ikpi1 · · · pik . Thus, the space of symbols of degree k can be viewed as the space Skδ (M) := Polk(T ∗M)⊗C∞(M) Fδ(M), where Polk(T ∗M) denotes the space of real functions on T ∗M which are polynomial functions of degree k in the fibered coordinates of T ∗M . The algebra S(M) := Pol(T ∗M) is clearly isomorphic to the algebra Γ(STM) of symmetric tensors and depending on the context we will refer to its elements as symbols, functions on T ∗M or symmetric tensors on M . Let us recall that, if S1, S2 ∈ S(M), then the Poisson bracket of S1 and S2, denoted by {S1, S2}, is defined in a canonical coordinate system (xi, pi) of T ∗M in the following way: {S1, S2} = (∂piS1) ( ∂xiS2 ) − (∂piS2)(∂xiS1). (2.3) We conclude this subsection by two properties of the principal symbol map linked to the composition and to the commutator of differential operators. For all k, l ∈ N, we have: σk+l(A ◦B) = σk(A)σl(B), (2.4) σk+l−1([A,B]) = {σk(A), σl(B)}, (2.5) where A and B are elements of D(M) of order k and l respectively. 6 J.-P. Michel, F. Radoux and J. Šilhan 2.2 Pseudo-Riemannian and conformal geometry Let (M, g) be a pseudo-Riemannian manifold. The isometries Φ of (M, g) are the diffeo- morphisms of M that preserve the metric g, i.e. Φ∗g = g. Their infinitesimal counterparts X ∈ Vect(M) are called Killing vector fields, they satisfy LXg = 0, with LXg the Lie derivative of g along X. Given the Levi-Civita connection ∇ corresponding to the metric g, the Riemannian curvature tensor, which reads as Rab c d in abstract index notation, is given by [∇a,∇b]vc = Rab c dv d for a tangent vector field vc. Then, one gets the Ricci tensor by taking a trace of the Riemann tensor, which is indicated by repeated indices: Ricbd = Rab a d. By contraction with the metric, the Ricci tensor leads to the scalar curvature Sc = gab Ricab. A conformal structure on a smooth manifold M is given by the conformal class [g] of a pseudo- Riemannian metric g, where two metrics g and ĝ are conformally related if ĝ = e2Υg, for some function Υ ∈ C∞(M). The conformal diffeomorphisms Φ of (M, [g]) are those which preserve the conformal structure [g], i.e. there exists Υ ∈ C∞(M) such that Φ∗g = e2Υg. Their infinitesimal counterparts X ∈ Vect(M) are called conformal Killing vector fields, they satisfy LXg = fXg, for some function fX ∈ C∞(M). Let (xi, pi) be a canonical coordinate system on T ∗M . If M is endowed with a metric g, we define the metric symbol and the trace operator by, respectively, H = gijpipj and Tr = gij∂pi∂pj . Note that the symbol \|Volg\|2/nH ∈ S2/n and the operator \|Volg\|−2/n Tr : Sδ → Sδ−2/n are conformally invariant. In consequence, we get a conformally invariant decomposition SkTM = ⊕ 0≤2s≤k Sk,sTM, (2.6) where S ∈ Sk,s(M) := Γ(Sk,sTM) is of the form S = HsS0 with TrS0 = 0. 2.3 The conformal Laplacian Starting from a pseudo-Riemannian manifold (M, g) of dimension n, one can define the Yamabe Laplacian, acting on functions, in the following way: ∆Y := ∇agab∇b − n− 2 4(n− 1) Sc, where∇ denotes the Levi-Civita connection of g and Sc the scalar curvature. For the conformally related metric ĝ = e2Υg, the associated Yamabe Laplacian is given by ∆̂Y = e− n+2 2 Υ ◦∆Y ◦ e n−2 2 Υ. According to the transformation law \|Volĝ\| = enΥ\|Volg\| and to the diagram (2.2), this translates into the conformal invariance of ∆Y viewed as an element of Dλ0,µ0(M), for the specific weights λ0 = n− 2 2n , µ0 = n+ 2 2n and δ0 = µ0 − λ0 = 2 n . (2.7) Thus, the data of a conformal manifold (M, [g]) is enough to define ∆Y ∈ Dλ0,µ0(M). We write it below as ∆M Y (g) and we refer to it as the Yamabe or conformal Laplacian. One easily gets Second Order Symmetries of the Conformal Laplacian 7 Proposition 2.1. The conformal Laplacian is a natural conformally invariant operator, i.e. • it satisfies the naturality condition: ∆N Y (Φ∗g) = Φ∗ ( ∆M Y (g) ) , (2.8) for all diffeomorphisms Φ : N →M and for all pseudo-Riemannian metric g on M , • it is conformally invariant, ∆M Y (e2Υg) = ∆M Y (g) for all Υ ∈ C∞(M). More generally, a natural operator over pseudo-Riemannian manifolds is an operator that acts between natural bundles, is defined over any pseudo-Riemannian manifold (M, g) and satisfies an analogue of the naturality condition (2.8). It is said to be conformally invariant if it depends only on the conformal class of g. For a general study of natural operators in the pseudo-Rie- mannian setting, see the book [20]. From Proposition 2.1, we deduce that the conformal Laplacian ∆Y is invariant under the action of conformal diffeomorphisms, which reads infinitesimally as Lµ0X ◦∆Y = ∆Y ◦ Lλ0X , (2.9) for all conformal Killing vector fields X. Here, as introduced in (2.1), Lλ0 and Lµ0 denote the Lie derivatives of λ0- and µ0-densities. If the manifold (M, [g]) is locally conformally flat, then, up to multiplication by a scalar, ∆Y is the unique second order operator acting on densities which is invariant under the action (2.9) of conformal Killing vector fields. 2.4 Natural and conformally invariant quantization Recall first the definition of a quantization on a smooth manifold M . Definition 2.2. Let λ, µ ∈ R and δ = µ − λ. A quantization on M is a linear bijection QMλ,µ from the space of symbols Sδ(M) to the space of differential operators Dλ,µ(M) such that σk ( QMλ,µ(S) ) = S, ∀S ∈ Skδ (M), ∀ k ∈ N. On locally conformally flat manifolds (M, [g]), for generic weights λ, µ, there exists a unique conformally equivariant quantization [12], i.e. a unique quantization which intertwines the ac- tions of the conformal Killing vector fields on Sδ(M) and on Dλ,µ(M). In the following, we need an extension of the conformally equivariant quantization to arbitrary conformal manifolds. This is provided by the notion of natural and conformally invariant quantization. The definition and the conjecture of the existence of such a quantization were given for the first time in [22]. Definition 2.3. A natural and conformally invariant quantization is the data for every pseudo- Riemannian manifold (M, g) of a quantization QMλ,µ(g), which satisfies • the naturality condition: QNλ,µ(Φ∗g)(Φ∗S) = Φ∗(QMλ,µ(g)(S)), ∀S ∈ Sδ(M), (2.10) for all diffeomorphisms Φ : N →M and for all pseudo-Riemannian metric g on M . • the conformal invariance: QMλ,µ(e2Υg) = QMλ,µ(g) for all Υ ∈ C∞(M). In the following we refer to a quantization map by Qλ,µ, the dependence in the chosen pseudo- Riemannian manifold (M, g) being understood. Accordingly, we drop the reference to M in the spaces of densities Fλ, symbols Sδ and differential operators Dλ,µ. 8 J.-P. Michel, F. Radoux and J. Šilhan The concept of natural and conformally invariant quantization is an extension to quantizations of the more usual one of natural conformally invariant operator, introduced in the previous section. Restricting to conformally flat manifolds (M, [g]) and to Φ ∈ Diff(M) preserving [g], the naturality condition (2.10) reads as conformal equivariance of the quantization map Qλ,µ. Thus, the problem of the natural and conformally invariant quantization on an arbitrary ma- nifold generalizes the problem of the conformally equivariant quantization on conformally flat manifolds. Remark that the bundles SkTM are natural bundles over (M, [g]). Hence, one can consider natural and conformally invariant quantization restricted to the subspaces of symbols Skδ or S≤kδ = ⊕ j≤k S j δ . In a first step, the proofs of the existence of a natural and conformally invariant quantization at the second and the third orders were given respectively in [13] and [23], together with explicit formulas. We provide the one at order 2, which we will need later on. Theorem 2.4 ([13]). Let δ /∈ { 2 n , n+2 2n , 1, n+1 n , n+2 n } . A natural and conformally invariant quanti- zation Qλ,µ : S≤2 δ → D2 λ,µ is provided, on a pseudo-Riemannian manifold (M, g) of dimension n, by the formulas Qλ,µ(f) = f, Qλ,µ(X) = Xa∇a + λ 1− δ (∇aXa), Qλ,µ(S) = Sab∇a∇b + β1(∇aSab)∇b + β2gab(∇a TrS)∇b + β3 ( ∇a∇bSab ) + β4gab∇a∇b(TrS),+β5 Ricab S ab + β6 Sc(TrS), (2.11) where f , X, S are symbols of degrees 0, 1, 2 respectively and TrS = gabS ab. Moreover the coefficients βi entering the last formula are given by β1 = 2(nλ+ 1) 2 + n(1− δ) , β2 = n(λ+ µ− 1) (2 + n(1− δ))(2− nδ) , β3 = nλ(nλ+ 1) (1 + n(1− δ))(2 + n(1− δ)) , β4 = nλ(n2µ(2− λ− µ) + 2(nλ+ 1)2 − n(n+ 1)) (1 + n(1− δ))(2 + n(1− δ))(2 + n(1− 2δ))(2− nδ) , β5 = n2λ(µ− 1) (n− 2)(1 + n(1− δ)) , β6 = n2λ(µ− 1)(nδ − 2) (n− 1)(n− 2)(1 + n(1− δ))(2 + n(1− 2δ)) . (2.12) In a second step, the proof of the existence of such a quantization, at an arbitrary order and for generic values of λ, µ, was given in [9, 25, 30] in different ways. We provide a slightly refined statement in the next section. 2.5 Adjoint operation and quantization For all weights λ ∈ R, there exists a non-degenerate symmetric bilinear pairing Fcλ ×Fc1−λ → R, (ϕ,ψ) 7→ ∫ M ϕψ, Second Order Symmetries of the Conformal Laplacian 9 where Fcλ is the space of compactly supported λ-densities. On a manifold M , this pairing is Diff(M)-invariant since 1-density is the right object for integration. In consequence, we can define an adjoint operation ∗ : Dλ,µ → D1−µ,1−λ by( ϕ,D∗ψ ) = (Dϕ,ψ), for all ϕ ∈ Fcλ and ψ ∈ Fc1−µ. We introduce the following subset of R2, I = { (λ, µ) ∈ R2 ∣∣µ− λ /∈ 1 2n (N \ {0}) } ∪ {(λ0, µ0)}, where λ0 = n−2 2n and µ0 = n+2 2n are the weights of the conformal Laplacian (see (2.7)). The set I is stable under the involutive map (λ, µ) 7→ (1− µ, 1− λ). Note that σ2(∆Y ) = \|Volg\|δ0H, where δ0 = µ0−λ0 and H = gijpipj in canonical coordinates. The proof of existence of a natural and conformally invariant quantization Qλ,µ in [30] leads easily to the following statement. Theorem 2.5. There exists a family (Qλ,µ)(λ,µ)∈I of natural and conformally invariant quan- tizations that satisfies: • the reality condition: Qλ,µ(S)∗ = (−1)kQ1−µ,1−λ(S), ∀S ∈ Skδ , ∀ (λ, µ) ∈ I, (2.13) • the factorization property: Qλ0,λ0 ( \|Volg\|δ0HS ) = Qµ0,λ0(S) ◦∆Y , ∀S ∈ Sk−δ0 , Qµ0,µ0 ( \|Volg\|δ0HS ) = ∆Y ◦ Qµ0,λ0(S), ∀S ∈ Sk−δ0 , (2.14) • the restriction of Qλ,µ to S≤2 δ is given by the formulas in (2.11) if (λ, µ) ∈ I \ {(λ0, µ0)}. Proof. We prove the theorem in four steps. In [30, Theorem 4.4], one of us determines that for (λ, µ) ∈ I there exists a natural and conformally invariant quantization map Q′′′λ,µ. From the above family of quantizations (Q′′′λ,µ)(λ,µ)∈I , we define (Q′′λ,µ)(λ,µ)∈I by Q′′λ,µ : S 7→ 1 2 ( Q′′′λ,µ(S) + (−1)kQ′′′1−µ,1−λ(S)∗ ) , ∀S ∈ Skδ , ∀ (λ, µ) ∈ I. The maps Q′′λ,µ are again natural and conformally invariant quantizations. Indeed, the adjoint operation ∗ is natural, does not depend of the choice of metric on M and satisfies σk ( D∗ ) = (−1)kσk(D), for all differential operators D of order k. The newly defined quantization maps clearly satisfy the property (2.13) since ∗ is an involution. For (λ, µ) ∈ I \{(λ0, λ0), (µ0, µ0)}, we define Q′λ,µ := Q′′λ,µ. On the space of traceless symbols we set Q′λ0,λ0 := Q′′λ0,λ0 and Q′µ0,µ0 := Q′′µ0,µ0 . We extend both maps to the whole symbol space by the formulas in (2.14). They are clearly still natural and conformally invariant and satisfy the reality condition (2.13). For all (λ, µ) ∈ I \{(λ0, µ0)}, we denote by Qλ,µ the natural and conformally invariant quan- tizations restricted to S≤2 δ , given by the formulas (2.11). A direct computation shows that they satisfy the reality condition (2.13) and the factorization property (2.14). For (λ, µ) = (λ0, µ0), we set Qλ0,µ0 := Q′λ0,µ0 on S≤2 δ0 . We extend then the quantizations Qλ,µ (where (λ, µ) ∈ I) to the whole symbol space by setting Qλ,µ := Q′λ,µ on S≥3 δ , for all (λ, µ) ∈ I. � 10 J.-P. Michel, F. Radoux and J. Šilhan In the following, the quantization maps Qλ,µ that we will use are always taken from a family (Qλ,µ)(λ,µ)∈I provided by Theorem 2.5. In fact, we will need only four of them, namely: Qλ0,λ0 , Qµ0,µ0 , Qλ0,µ0 , Qµ0,λ0 . With such a convention, it is worth noticing that the conformal Laplacian can be obtained as ∆Y = Qλ0,µ0 ( \|Volg\|δ0H ) . The conformal invariance of the symbol \|Volg\|δ0H translates into the conformal invariance of ∆Y . 3 On particular conformally invariant operators First, we introduce notation for classical objects of the pseudo-Riemannian and conformal geo- metries and recall basic facts about natural and conformally invariant operators. Then, we classify the natural conformally invariant operators between particular subspaces of symbols. 3.1 More on pseudo-Riemannian and conformal geometry We complete here Section 2.2, and use freely the notation introduced there. First, we work over a pseudo-Riemannian manifold. The Riemann tensor admits the following decomposition Rab c d = Cab c d + 2δc[aPb]d + 2gd[bPa] c, (3.1) where Cab c d is the totally trace-free Weyl curvature, Pab = 1 n−2 ( Ricab− 1 2(n−1) Sc gab ) is the Schouten tensor, δba is the Kronecker delta and square brackets denote antisymmetrization of enclosed indices. The Weyl tensor Cabcd is zero for the dimension n = 3. Note also that Cabcd obeys the same symmetries of indices as Rabcd does. Further curvature quantities we shall need are J = gabPab and Aabc = 2∇[bPc]a, where Aabc is the Cotton–York tensor and J is related to the scalar curvature via J = 1 2(n−1) Sc. Bianchi identities have the from R[abc]d = 0 and ∇[aRbc]de = 0 and lead to (n− 3)Aabc = ∇rCbc r a and ∇bPba = ∇aJ. Second, we consider a conformal manifold (M, [g]). The Weyl tensor Cab c d is a conformal invariant, i.e. it does not depend on the choice of the representative metric from [g]. The same is true for Aabc in the dimension 3. Further, a choice of metric provides a canonical trivialization of the bundle of λ-densities Fλ via the global section \|Volg\|λ (see Section 2.1). According to the transformation rule \|Volĝ\|λ = enλΥ\|Volg\| if ĝ = e2Υg, we have the conformally invariant object gab = gab⊗ \|Volg\|− 2 n , termed conformal metric, with the inverse gab in Γ(S2TM)⊗F2/n, see e.g. [1] for details. (Note that the space of densities E [w] in [1] corresponds to F−w/n in our notation.) The conformal metric gives a conformally invariant identification TM ∼= T ∗M ⊗ F−2/n. In other words, we can raise and lower indices, with expense of the additional density, in a conformally invariant way. For example, we get Cabcd ∈ Γ(S2(Λ2T ∗M)) ⊗ F−2/n. Note also that gab and gab are parallel for any choice of a Levi-Civita connection from the conformal class. Second Order Symmetries of the Conformal Laplacian 11 3.2 Description of natural conformally invariant operators Now we recall basic facts about natural and conformally invariant operators. Every natural operator on the Riemannian structure (M, g) between natural bundles V1 and V2 is a linear combination of terms of the form g−1 · · · g−1︸︷︷︸ r1 g · · · g︸︷︷︸ r2 (∇(i1)R) · · · (∇(is)R)︸︷︷︸ s ∇ · · ·∇︸︷︷︸ t f (3.2) to which one applies a GL(n)-invariant operation Γ (⊗ 2r1+sTM ⊗ ⊗ 2r2+3s+i+tT ∗M ⊗ V1 ) −→ Γ(V2). (3.3) Here f ∈ Γ(V1), g−1 stands for the inverse of the metric g, ∇(ij) denotes the ijth iterated covariant derivative where ij ≥ 0, abstract indices are omitted, i = i1 + · · · + is, and ∇ and R correspond to the choice of the metric g. The existence of a GL(n)-invariant operation (3.3) gives in general constraints on the possible values of r1, r2, s, t, i. See [20] for details. A natural operator on (M, g) is conformally invariant if it does not depend on the choice of metric in the conformal class. Then, it defines a natural operator on the conformal structure (M, [g]). It is convenient to use the conformal metric g instead of g and the inverse g−1 instead of g−1 in (3.2) since they are conformally invariant, namely g−1 · · ·g−1︸︷︷︸ r1 g · · ·g︸︷︷︸ r2 (∇(i1)R) · · · (∇(is)R)︸︷︷︸ s ∇ · · ·∇︸︷︷︸ t f ∈ Γ (⊗ 2r1+sTM ⊗ ⊗ 2r2+3s+i+tT ∗M ⊗ V1 ⊗ F 2 n (r1−r2) ) (3.4) for f ∈ Γ(V1). It is generally a difficult problem to determine which linear combinations of terms as in (3.2), together with suitable projections as in (3.3), give rise to a conformally invariant operator. We shall need details only in specific cases. 3.3 Conformally invariant operators on the symbol space This section concerns existence and uniqueness of natural and conformally invariant operators of certain type. The first one is well-known and can be obtained as an easy consequence of [17], or deduced from the general work [10] on curved BGG-sequences. We present a detailed proof to demonstrate the technique which is used (in much more complicated setting) later in the proof of Proposition 3.2. Recall first that Γ(Sk,0TM) is the space of trace-less symmetric k-tensors. In terms of the abstract index notation, a section f of ⊗k TM is denoted by fa1...ak . In the following, we write f [a1...ak], f (a1...ak) and f (a1...ak)0 for the projections of f to Γ(ΛkTM), Γ(SkTM) and Γ(Sk,0TM), respectively. Similar notation will be used for covariant indices. Proposition 3.1. Up to multiplication by a scalar, there exists a unique natural conformally invariant operator Sk0 → S k+1 2/n . It is given by the conformal Killing operator G, such that for all f ∈ Sk0 , (G(f))a0...ak = ∇(a0fa1...ak)0 . (3.5) Proof. Identifying Sk0 and Sk+1 2/n with corresponding spaces of sections of symmetric tensors, we consider natural and conformally invariant operators Γ(SkTM) → Γ(Sk+1TM ⊗ F2/n). By 12 J.-P. Michel, F. Radoux and J. Šilhan naturality, such operators are linear combinations of terms in (3.2) with V1 = SkTM , composed with GLn-invariant maps⊗ 2r1+sTM ⊗ ⊗ 2r2+3s+i+tT ∗M ⊗ SkTM ⊗ F 2 n (r1−r2) −→ Sk+1TM ⊗ F2/n. Explicitly, those maps may consist of: contracting covariant and contravariant indices, project- ing the covariant and contravariant tensors on tensors of prescribed symmetry type (given by a Young diagram) and tensorizing with the density \|Volg\|δ for arbitrary δ ∈ R. The conformal in- variance does not allow for the last operation, hence r1−r2 = 1. The difference between the num- ber of covariant and contravariant indices is a constant therefore (2r1 +s+k)−(2r2 +3s+i+t) = k + 1, i.e. 2s+ i+ t = 1. This means s = i = 0 and t = 1. The sought operators are then first order (gradient) natural operators and using moreover the conformal invariance, the statement follows from the classification in [17]. � The next proposition is a crucial technical tool in the following. Proposition 3.2. Every natural conformally invariant operator Sk,00 → Sk−1 2/n has its target space in Sk−1,0 2/n ⊆ Sk−1 2/n . The space of natural conformally invariant operators Sk,00 → Sk−1,0 2/n on a pseudo-Riemannian manifold (M, g) is at most two-dimensional and depends on k ∈ N as follows. (i) This space is trivial for k = 1. (ii) If k = 2 or n = 3, this space is one-dimensional and generated by the operator F such that, for all f ∈ Sk,00 , (F(f))a1...ak−1 = Cr st (a1∇rfa2...ak−1)0st − (k + 1)Ast (a1fa2...ak−1)0st. (iii) If k = 3 and n > 3, this space is two-dimensional and generated by two operators, F1 and F2, such that for all f ∈ Sk,00 , (F1(f))a1...ak−1 = (F(f))a1...ak−1 + k − 2 n+ 2k − 2 C(a1 r a2 s∇tfa3...ak−1)0rst, (F2(f))a1...ak−1 = 4C(a1 r a2 s∇tfa3...ak−1)0rst + (n+ 2k − 2)(∇rCs (a1 t a2)fa3...ak−1)0rst + 2(n+ 2k − 2)Ars (a1fa2...ak−1)0rs. Remark 3.3. Let (xi, pi) be a canonical coordinate system on T ∗M . We can then write the operators G and F as follows on Sk0 G = Π0 ◦ ( gijpi∇j ) and F = Π0 ◦ gimpi∂pj∂pl ( Ck jlm∇k − (k + 1)Ajlm ) , (3.6) where Π0 : Sk−1 0 → Sk−1,0 2/n is the canonical projection on trace-less symbols. Actually, we will see in the sequel that the conformal Killing operator G can be used to define the conformal Killing tensors whereas the operator F occurs in the computation of the obstruction to the existence of conformal symmetries of ∆Y . Let us note that the proof of Proposition 3.2 is long, technical and interesting rather for experts in conformal geometry. The reader interested mainly in results about symmetries can continue the reading in Section 4 (details from the proof will not be needed there). Second Order Symmetries of the Conformal Laplacian 13 Proof. We study natural and conformally invariant operators Γ(Sk,0TM)→Γ(Sk−1TM⊗F2/n). In the first part of the proof we consider the naturality and in the second part the conformal invariance. I. Naturality. We start in a similar way as in the proof of Proposition 3.1. By naturality, the considered operators are linear combinations of terms in (3.2) composed with GLn-invariant maps⊗ 2r1+sTM ⊗ ⊗ 2r2+3s+i+tT ∗M ⊗ SkTM ⊗ F 2 n (r1−r2) −→ Sk−1TM ⊗ F2/n. The conformal invariance of the discussed operators leads to r1− r2 = 1 and the GLn-invariance of the maps above imposes (2r1 +s+k)−(2r2 +3s+ i+ t) = k−1, i.e. 2s+ i+ t = 3. This means either s = i = 0, t = 3 or s = i = 1, t = 0 or s = t = 1, i = 0. Hence, omitting abstract indices, the natural operators Γ(Sk,0TM)→ Γ(Sk−1TM ⊗ F2/n) are a linear combination of terms g−1 · · ·g−1︸︷︷︸ r+1 g · · ·g︸︷︷︸ r ∇∇∇f, g−1 · · ·g−1︸︷︷︸ r+1 g · · ·g︸︷︷︸ r R∇f, g−1 · · ·g−1︸︷︷︸ r+1 g · · ·g︸︷︷︸ r (∇R)f, (3.7) where r ≥ 0 and f ∈ Γ(Sk,0TM), each of which is followed by a GL(n)-invariant projection to Γ(Sk−1TM ⊗ F2/n). Irreducible components of the target bundle Sk−1TM ⊗ F2/n are Sk−1,0TM ⊗ F2/n, Sk−3,0TM, Sk−5,0TM ⊗ F−2/n, . . . , but since f is trace-free, one easily verifies from (3.7) that only possible target bundles are Sk−1,0TM ⊗ F2/n and Sk−3,0TM . In other words, in the expressions (3.7), one can restrict to r = 0. It remains to describe possible GL(n)-invariant projections of the terms in (3.7) in details. Using the decomposition (3.1) of R into Weyl and Schouten tensors, they split into five terms: g−1∇∇∇f , g−1C∇f , g−1(∇C)f , g−1P∇f and g−1(∇P)f . We shall start with natural operators Γ(Sk,0TM)→ Γ(Sk−1,0TM ⊗ F2/n). In this situation, at least one of the two indices above g−1 in the expression of the operator has to be contracted with a covariant index. For an operator of type g−1∇∇∇f , the two resulting operators are (up to the order of covariant derivatives) respectively ∇r∇r∇sfa1...ak−1s, ∇(a1∇s∇tfa2...ak−1)0st. (3.8) Since the change of the order of covariant derivatives gives rise to curvature operators of the form g−1R∇f and g−1(∇R)f , the previous display is sufficient for operators of type g−1∇∇∇f . Using that C is completely trace-free and (n − 3)Aabc = ∇rCbc r a, the different possibilities of contraction of indices for the expressions g−1C∇f and g−1(∇C)f lead to the operators Cr st (a1∇rfa2...ak−1)0st, C(a1 r a2 s∇tfa3...ak−1)0rst,( ∇rCs (a1 t a2)f a3...ak−1 ) 0 rst , Ars (a1fa2...ak−1)0rs. (3.9) Thanks to the decomposition of P into irreducible components and to the equality∇aPab = ∇bJ , we see that the different configurations of indices in the expressions g−1P∇f and g−1(∇P)f give rise to the operators P(rs)0∇ rfsa1...ak−1 , P(rt)0g t(a1∇sfa2...ak−1)0rs, Prs∇(a1fa2...ak−1)0rs, J∇rfa1...ak−1r, (∇(rPst)0)gt(a1fa2...ak−1)0rs, (∇rJ)fa1...ak−1r. (3.10) Hence all natural operators Γ(Sk,0TM)→ Γ(Sk−1,0TM⊗F2/n) are linear combinations of terms in (3.8)–(3.10). 14 J.-P. Michel, F. Radoux and J. Šilhan A similar discussion can be applied to natural operators Γ(Sk,0TM) → Γ(Sk−3,0TM). In this situation, none of the two indices of g−1 is contracted in the expressions (3.7). Reasoning as above, using the properties of symmetry of C and the fact that C and f are trace-free, we obtain (since the target bundle is now Sk−3,0TM) a simpler list of possible terms: ∇r∇s∇tfa1...ak−3rst, P(rs)0∇tf a1...ak−3rst, (∇rPst)fa1...ak−3rst, (3.11) for k ≥ 3. Hence all natural operators Γ(Sk,0TM) → Γ(Sk−3,0TM) are linear combinations of terms in (3.11). II. Conformal invariance. We shall denote quantities corresponding to the conformally related metric ĝ = e2Υg and the corresponding Levi-Civita connection ∇̂ by R̂abcd, P̂ab, Ĵ and Âabc. (The Weyl tensor is missing here since Ĉabcd = Cabcd.) This transformation is controlled by the one-form Υa = ∇aΥ, see e.g. [1] for details. Explicitly, one can compute that P̂ab = Pab −∇aΥb + ΥaΥb − 1 2 ΥrΥrgab, (3.12) Ĵ = J−∇rΥr − n− 2 2 ΥrΥr and Âabc = Aabc + ΥrCbc r a (3.13) and also that ∇̂(aP̂bc)0 = ∇(aPbc)0 −∇(a∇bΥc)0 + 4Υ(a∇bΥc)0 − 4Υ(aΥbΥc)0 − 2Υ(aPbc)0 , (3.14) ∇̂aĴ = ∇aJ−∇a∇rΥr − (n− 2)Υr∇rΥa + 2Υa∇rΥr − 2ΥaJ + (n− 2)ΥaΥ rΥr, (3.15) ∇̂( a Cb d c ) 0 e = ∇(aCb d c)0 e − 4Υ(aCb d c)0 e + 2Υrδ(a (dCb e) c) r. (3.16) We shall start with operators Γ(Sk,0TM) → Γ(Sk−1,0TM ⊗ F2/n). First observe that the space of such natural and conformally invariant operators is trivial in the flat case [4, 5] hence the two terms of (3.8) cannot appear. We need to know how remaining terms in (3.9) and (3.10) transform under the conformal rescaling ĝ = e2Υg. First observe that the rescaling of first order expressions we need is ∇̂rfa1...ak−1r = ∇rfa1...ak−1r + (n+ 2k − 2)Υrf a1...ak−1r, ∇̂[bfc] a1...ak−1 = ∇[bfc] a1...ak−1 + (k + 1)Υ[bfc] a1...ak−1 + (k − 1)Υrδ (a1 [b fc] a2...ak−1)r, ∇̂(a1fa2...ak−1)0 bc = ∇(a1fa2...ak−1)0 bc + 2Υ(a1fa2...ak−1)0 bc − 2Υ(bfc) a1...ak−1 + 2Υrδ (a1 (b fc) a2...ak−1)0r, ∇̂(bfc) a1...ak−1 = ∇(bfc) a1...ak−1 + (k − 1)Υ(bfc) a1...ak−1 − (k − 1)Υ(a1fa2...ak−1) bc + gbcΥrf a1...ak−1r + (k − 1)Υrδ (a1 (b fc) a2...ak−1)r. (3.17) We are interested in linear combinations of terms in (3.9) and (3.10) which are independent on the rescaling ĝ = e2Υg. Considering formulas (3.12)–(3.17), we observe that the term ∇(a∇bΥc)0 appears only in (3.14) and the term ∇a∇rΥr appears only on the right hand side of (3.15). This means, terms (∇rJ)fa1...ak−1r and (∇(rPst)0)gt(a1fa2...ak−1)0rs do not appear in the required linear combination. The Weyl tensor appears in the conformal transformation of the terms in (3.9) but not of the ones in (3.10). Therefore, we look for conformally invariant linear combinations x1Cr st (a1∇rfa2...ak−1)0st + x2C(a1 r a2 s∇tfa3...ak−1)0rst + x3 ( ∇rCs (a1 t a2)f a3...ak−1 ) 0 rst + x4Ars (a1fa2...ak−1)0rs (3.18) Second Order Symmetries of the Conformal Laplacian 15 and y1J∇rfa1...ak−1r + y2Prs∇(rfs)a1...ak−1 + y3Pr (a1∇sfa2...ak−1)0rs + y4Prs∇(a1fa2...ak−1)0rs, (3.19) where Prs = P(rs)0 denotes the trace-free part of P. In other words, we search for scalars xi, yj ∈ R such that both (3.18) and (3.19) are invariant independently. First we discuss (3.18) which is possible only for k ≥ 2 and some terms only for k ≥ 3. Assuming k ≥ 3, conformal transformations of these terms are Cr st (a1∇̂rfa2...ak−1)0st = Cr st (a1∇̂rfa2...ak−1)0st + (k + 1)ΥrC r st (a1fa2...ak−1)0st + (k − 2)C(a1 st a2Υrf a3...ak−1)0str, C(a1 r a2 s∇̂tfa3...ak−1)0rst = C(a1 r a2 s∇tfa3...ak−1)0rst − (n+ 2k − 2)C(a1 rs a2Υtf a3...ak−1)0rst,( ∇̂rCs (a1 t a2)f a3...ak−1 ) 0 rst = (∇rCs (a1 t a2)fa3...ak−1)0rst + 4C(a1 rs a2Υtf a3...ak−1)0rst − 2ΥrC r st (a1fa2...ak−1)0st, Ârs (a1fa2...ak−1)0rs = Ars (a1fa2...ak−1)0rs + ΥrC r st (a1fa2...ak−1)0st using (3.17), (3.16) and (3.13). Now, considering where the term ΥrC r st (a1fa2...ak−1)0st appears in the previous display, we see that (k + 1)x1 − 2x3 + x4 = 0. Considering the other term C(a1 rs a2Υtf a3...ak−1)0rst, we conclude that (k− 2)x1− (n+ 2k− 2)x2 + 4x3 = 0. Solutions of this pair of linear equations are generated by (x1, x2, x3, x4) = ( n+2k−2, k−2, 0,−(k+1)(n+2k−2) ) and (x1, x2, x3, x4) = ( 0, 4, n+2k−2, 2(n+2k−2) ) , therefore the space of corresponding invariant linear operators is generated by the operators F1 and F2 defined in the following way: (F1(f))a1...ak−1 = Cr st (a1∇rfa2...ak−1)0st − (k + 1)Ars (a1fa2...ak−1)0rs + k − 2 n+ 2k − 2 C(a1 r a2 s∇tfa3...ak−1)0rst, (F2(f))a1...ak−1 = 4C(a1 r a2 s∇tfa3...ak−1)0rst + (n+ 2k − 2)(∇rCs (a1 t a2)fa3...ak−1)0rst + 2(n+ 2k − 2)Ars (a1fa2...ak−1)0rs. This shows that the operators in the statement of the proposition for k ≥ 3 are invariant. In the case k = 2 only some terms from (3.18) can appear. Specifically, we study the conformal invariance of the linear combination x1Cr st a∇rfst + x4Ars af rs (3.20) for the section fab in Γ(S2,0TM). Since ∇̂af bc = ∇af bc + 2Υaf bc − 2Υ(bf c)a + 2δa (bΥrf c)r, conformal transformations of terms in the previous display are Cr st a∇̂rf st = Cr st a∇̂rf st + 3ΥrC r st afst and Ârs af rs = Ars af rs + ΥrC r st afst. By the same reasoning as in the case k ≥ 3, we obtain that the operator given in (3.20) is invariant if and only if (x1, x4) is a multiple of (1,−3). In the case k = 2, the only invariant operators are thus the multiples of the operator F defined by (F(f))a = Cr st a∇rfst − 3Ars af rs. Now we shall discuss terms (3.19) and we assume k ≥ 2 first. Consider an arbitrary but fixed point x ∈ M . We can choose the function Υ such that Υa(x) = 0, ∇(aΥb)0(x) = Φab(x) 16 J.-P. Michel, F. Radoux and J. Šilhan and ∇rΥr(x) = Ψ(x) for any prescribed values of Φab(x) and Ψ(x). Therefore, the conformal transformation of terms in (3.19) is Ĵ∇̂rfa1...ak−1r = J∇rfa1...ak−1r −Ψ∇rfa1...ak−1r, P̂rs∇̂(rfs)a1...ak−1 = Prs∇(rfs)a1...ak−1 − Φrs∇(rfs)a1...ak−1 , P̂r (a1∇̂sfa2...ak−1)0rs = Pr (a1∇sfa2...ak−1)0rs − Φr (a1∇sfa2...ak−1)0rs, P̂rs∇̂(a1fa2...ak−1)0rs = Prs∇(a1fa2...ak−1)0rs − Φrs∇(a1fa2...ak−1)0rs at the point x (which is for simplicity omitted in the previous display). Choosing Ψ(x) 6= 0 and Φrs(x) = 0, the invariance of (3.19) means that y1 = 0. Henceforth we assume Ψ(x) = 0 and Φrs(x) 6= 0. To determine y2, y3 and y4, we shall test invariance of (3.19) for fa1...ak with specific properties at x. First assume that ∇bfa1...ak(x) = ∇(bfa1...ak)(x), or equivalently that ∇bfa1...ak(x) = ∇a1f ba2...ak(x). This in particular implies that ∇rfa1...ak−1r(x) = 0 and the invariance of (3.19) then means that y2+y4 = 0. Second, we assume ∇(bfa1...ak)(x) = 0 or equiv- alently 2∇(bf c)a1...ak−1(x)+(k−1)∇(a1fa2...ak−1)bc(x) = 0. This also implies ∇rfa1...ak−1r(x) = 0 and the invariance of (3.19) now means that −k−1 2 y2 + y4 = 0. Overall, this yields y2 = y4 = 0, and y3 = 0 follows. All scalars in (3.19) are thus equal to zero. This completes the proof of the part (ii) of the proposition. If k = 1, (3.19) reduces to the linear combination y1J∇rf ra + y2Prs∇rf sa. As above, the choice Ψ(x) 6= 0 and Φrs(x) = 0 shows that y1 = 0. Hence y2 = 0 and the part (i) follows. In order to complete the proof of the part (iii), it remains to describe natural and conformally invariant operators Γ(Sk,0TM) → Γ(Sk−3TM). The space of these operators is also trivial in the flat case [4, 5], hence the first term in (3.11) cannot appear. Thus the required operator is a linear combination of the form x1P(rs)0∇tf a1...ak−3rst + x2(∇rPst)fa1...ak−3rst, where x1, y1 ∈ R. Reasoning similarly as above, we observe that ∇(a∇bΥc)0 appears only in the conformal transformation of the second term in the previous display. Therefore x2 = 0, hence also x1 = 0 and the proposition follows. � 4 Classif ication of second order symmetries of ∆Y We start this section with the definition of the algebra A of conformal symmetries of the confor- mal Laplacian. Afterwards, we provide our main result: a complete description of the space A2 of second order conformal symmetries. 4.1 The algebra of symmetries of the conformal Laplacian Let (M, [g]) be a conformal manifold of dimension n. Fixing a metric g ∈ [g], we can regard the conformal Laplacian, ∆Y = ∇agab∇b − n−2 4(n−1) Sc, as acting on functions. The symmetries of ∆Y are defined as differential operators which commute with ∆Y . Hence, they preserve the eigenspaces of ∆Y . More generally, conformal symmetries D1 are defined by the weaker algebraic condition ∆Y ◦D1 = D2 ◦∆Y , (4.1) for some differential operator D2, so that they only preserve the kernel of ∆Y . The operator ∆Y can be considered in equation (4.1) as acting between different line bundles and in particular as an element of Dλ0,µ0 , where λ0 = n−2 2n , µ0 = n+2 2n . With this choice, ∆Y is conformally invariant Second Order Symmetries of the Conformal Laplacian 17 and the space of conformal symmetries depends only on the conformal class of the metric g. It is stable under linear combinations and compositions. The operators of the form P∆Y , i.e. in the left ideal generated by ∆Y , are obviously con- formal symmetries. Since they act trivially on the kernel of ∆Y , they are considered as trivial. Following [15, 18, 26], this leads to Definition 4.1. Let (M, [g]) be a conformal manifold with conformal Laplacian ∆Y ∈ Dλ0,µ0 . The algebra of conformal symmetries of ∆Y is defined as A := {D1 ∈ Dλ0,λ0 \| ∃D2 ∈ Dµ0,µ0 s.t. D2 ◦∆Y = ∆Y ◦D1}, and the subspace of trivial symmetries as (∆Y ) := {A∆Y \|A ∈ Dµ0,λ0}. Thus, A is a subalgebra of Dλ0,λ0 and (∆Y ) is the left ideal generated by ∆Y in Dλ0,λ0 . The filtration by the order on Dλ0,λ0 induces a filtration on A and we denote by Ak := A ∩Dkλ0,λ0 the algebra of conformal symmetries of order k. Obviously, A0 ' R is the space of constant functions, identified with zero order operators on λ0-densities. Moreover, the invariance of ∆Y under the action of conformal Killing vector fields, see (2.9), shows that A1 is the direct sum of A0 with the space of Lie derivatives Lλ0X ∈ D1 λ0,λ0 along conformal Killing vector fields X. Since A is an algebra, A2 contains in particular Lλ0X ◦ L λ0 Y for X, Y conformal Killing vector fields. 4.2 The algebra of symmetries of the null geodesic flow Let (M, g) be a pseudo-Riemannian manifold and (xi, pi) denote a canonical coordinate system on T ∗M . The inverse metric g−1 pertains to Γ(S2TM) and identifies with H := gijpipj ∈ S0, where S0 = Pol(T ∗M) ∼= Γ(STM) (see Section 2.1). Along the isomorphism T ∗M ∼= TM provided by the metric, the Hamiltonian flow of H corresponds to the geodesic flow of g. The symmetries of the geodesic flow are given by functions K ∈ S0 which Poisson commute with H. They coincide with the symmetric Killing tensors. The null geodesic flow, i.e. the geodesic flow restricted to the level set H = 0, depends only on the conformal class of g. It admits additional symmetries, namely all the functions K ∈ S0 such that {H,K} ∈ (H), where {·, ·} stands for the canonical Poisson bracket on T ∗M , defined in (2.3), and (H) for the ideal spanned by H in S0. The linearity and Leibniz property of the Poisson bracket ensure that the space of symmetries of the null geodesic flow is a subalgebra of S0. Besides, remark that all the functions in (H) are symmetries which act trivially on the null geodesic flow. Definition 4.2. Let (M, g) be a pseudo-Riemannian manifold and H ∈ S0 the function asso- ciated to g. The algebra of symmetries of the null geodesic flow of g is given by the following subalgebra of S0, K := {K ∈ S0 \| {H,K} ∈ (H)}. 18 J.-P. Michel, F. Radoux and J. Šilhan In particular, the algebra K contains the ideal (H) of trivial symmetries. It inherits the gradation of S0 by the degree, Kk := K ∩ Sk0 . The space K0 is the space of constant functions on T ∗M . The Hamiltonian flows of functions in K1 coincide with the Hamiltonian lift to T ∗M of the conformal Killing vectors on (M, [g]). For higher degrees, the elements in K are symmetric conformal Killing tensors whose Hamiltonian flows do not preserve the configuration manifold M . They are symmetries of the whole phase space but not of the configuration manifold and often named hidden symmetries by physicists. Proposition 4.3. The elements K ∈ Kk are symmetric conformal Killing k-tensors. They are characterized equivalently as: • symmetric tensors of order k s.t. ∇(a0Ka1...ak)0 = 0, • symbols of degree k satisfying {H,K} ∈ (H), • elements of Sk0 in the kernel of the conformal Killing operator G (see (3.5) or (3.6)). The proof is both classical and straightforward, we let it to the reader. The next proposition is essential to determine the algebra A of conformal symmetries. Proposition 4.4. If D1 ∈ Ak then σk(D1) ∈ Kk. Under the identification grDλ0,λ0 ∼= S0, the associated graded algebra grA becomes a subalgebra of K and gr(∆Y ) identifies with (H). Proof. Suppose that D1 is a conformal symmetry of order k, i.e. satisfies ∆Y ◦D1 = D2◦∆Y for some D2. Working in the algebra Dλ0,λ0 we deduce that [∆Y , D1] ∈ (∆Y ) and the property (2.5) leads then to {H,σk(D1)} ∈ (H), i.e. σk(D1) ∈ Kk. The inclusion grA ≤ K follows. As σ2(∆Y ) = H, the property (2.4) of the principal symbol maps implies that gr(∆Y ) ∼= (H). � 4.3 Second order conformal symmetries We adapt the strategy used in [26], dealing with conformally flat manifolds. Thanks to a natural and conformally invariant quantization, we get a first description of the potential obstruction for a conformal Killing tensor giving rise to a conformal symmetry of ∆Y . Theorem 4.5. Let Qλ,µ be a family of natural and conformally invariant quantizations as in Theorem 2.5. We get then ∆Y ◦ Qλ0,λ0(S)−Qµ0,µ0(S) ◦∆Y = Qλ0,µ0 ( 2G(S) + Obs(S) ) , ∀S ∈ S≤2 0 . (4.2) The operator Obs is the natural and conformally invariant operator defined by Obs = 2(n− 2) 3(n+ 1) F, where (F(S))a = Cr st a∇rSst − 3Ars aSrs for S ∈ S2 0 and we set F(S) = 0 for S ∈ S≤1 0 . Proof. According to (2.6), we have S2 0 = S2,0 0 ⊕ S2,1 0 and S ∈ S2,1 0 is of the following form S = (\|Volg\|δ0H)S0 with S0 ∈ F−δ0 . By Theorem 2.5, we have the identities Qλ0,λ0(S) = Qµ0,λ0(S0) ◦∆Y and Qµ0,µ0(S) = ∆Y ◦ Qµ0,λ0(S0). Besides, from the expressions of the operators G and F (see e.g. (3.6)), we deduce G(S) = 0 and Obs(S) = 0. Hence the equality (4.2) holds for all S ∈ S2,1 0 . Second Order Symmetries of the Conformal Laplacian 19 Next, we define a natural and conformally invariant operator QS on D2 λ0,λ0 by D 7→ ∆Y ◦ D −D ◦∆Y . Pulling this map back to trace-free symbols via the quantization maps, D2 λ0,λ0 QS // D3 λ0,µ0 (S≤1 0 ⊕ S2,0 0 ) Qλ0,λ0 OO CS // S≤3 δ0 Qλ0,µ0 OO this leads to a natural and conformally invariant operator CS on S≤1 0 ⊕S 2,0 0 . Since ∆Y is formally self-adjoint and the quantization maps satisfy the reality condition (2.13), we deduce that, for all S ∈ Sk,00 , ∆Y ◦ Qλ0,λ0(S)−Qµ0,µ0(S) ◦∆Y is of degree k + 1 and is formally skew-adjoint (resp. self-adjoint) if k is even (resp. odd). As such, it is of the form Qλ0,µ0(P ), with P ∈ S3 δ0 ⊕ S1 δ0 if S is of degree 2, P ∈ S2 δ0 ⊕ S0 δ0 if S is of degree 1 and P ∈ S1 δ0 if S is of degree 0. We can reduce accordingly the target space of CS restricted to homogeneous symbols. Applying Proposition 3.1 and Proposition 3.2, we deduce that CS = aG + bF for some real constants a, b. We have then ∆Y ◦ Qλ0,λ0(S)−Qµ0,µ0(S) ◦∆Y = Qλ0,µ0 ( aG(S) + bF(S) ) , ∀S ∈ S≤2 0 . (4.3) It is straightforward to prove that a = 2. To prove b = 2(n−2) 3(n+1) , we study a specific conformal symmetry of ∆Y . Lemma 4.6. Let η be the pseudo-Euclidean flat metric of signature (p, q), h a non-vanishing function on R2 and n = p + q + 2. Let (M0, g) be the pseudo-Riemannian manifold (R2 × Rn−2, g0 × η), where the metric on R2 is determined by (g0)−1 = h(x1, x2)p2 1 + p2 2 in canonical Cartesian coordinates (xi, pi) on T ∗Rn. Then, K = p2 3 is a Killing tensor on (M0, g), and we have the following relation: ∆Y ◦ Qλ0,λ0(K)−Qµ0,µ0(K) ◦∆Y = Qλ0,µ0(Obs(K)) 6= 0. Proof. Using the relation that links the coefficients of g and the Christoffel symbols Γijk of the associated Levi-Civita connection, it is obvious that Γijk = 0 if at least one of the indices i, j, k is greater than or equal to 3. Thus, the only non-vanishing components of the Riemann tensor and the Ricci tensor associated with g are given by the corresponding components of the Riemann tensor and the Ricci tensor of g0. In the same way, the scalar curvature of g is equal to the scalar curvature of g0. Using these facts and the formula for Qλ0,λ0(K) presented in the proof of Proposition 4.9, it is easy to see that Qλ0,λ0(K) = Qµ0,µ0(K) = ∂2 x3 + 1 2(n− 1)(n+ 1) Sc . By a direct computation, we obtain the following relation: ∆Y ◦ Qλ0,λ0(K)−Qµ0,µ0(K) ◦∆Y = [∆Y ,Qλ0,λ0(K)] = 1 (n− 1)(n+ 1) Qλ0,µ0(gij(∂i Sc)pj) + f, with f ∈ C∞(M). According to (4.3), the function f vanishes. 20 J.-P. Michel, F. Radoux and J. Šilhan Besides, we can compute easily the Cotton–York tensor A associated with g. Indeed, if P denotes the Schouten tensor, we have Aijk = 2∇[iPj]k = 2 n− 2 ∇[i ( Ricj]k− 1 2(n− 1) gj]k Sc ) . Using the peculiar form of K and the remark done previously about the Christoffel symbols and the curvature tensors of g, it is obvious that AijkK jk = − 1 2(n− 1)(n− 2) ∂i Sc for all i. The conclusion follows immediately. � By naturality of the map CS defined above, the coefficient b in (4.3) depends only on the signature of the metric. As b is equal to 2(n−2) 3(n+1) in the example presented in the previous lemma, where the dimension M0 is of arbitrary dimension n and g of arbitrary signature, we conclude that b = 2(n−2) 3(n+1) in (4.3). � Obviously, we have Obs(S) = 0 if S is a symbol of degree 0 or 1. Thus, we recover that A1 ∼= K1 ⊕ K0 and the isomorphism is provided by Qλ0,λ0 . Since the symmetric conformal Killing tensors K satisfy GK = 0, we deduce the following Corollary 4.7. Let (M, g) be a pseudo-Riemannian manifold of dimension n endowed with a symmetric conformal Killing 2-tensor K. The operator Qλ0,λ0(K) = Kab∇a∇b + n n+ 2 (∇aKab)∇b + n(n− 2) 4(n+ 2)(n+ 1) (∇a∇bKab)− n+ 2 4(n+ 1) RicabK ab, is a conformal symmetry of ∆Y if and only if Obs(K) = 0. Proof. Indeed, the condition is obviously sufficient. Next, the condition is necessary because if Qλ0,λ0(K) is a conformal symmetry of ∆Y , there exists a differential operator D such that ∆Y ◦ Qλ0,λ0(K) = D ◦∆Y . We have then successively, using Theorem 4.5: 0 = ∆Y ◦ Qλ0,λ0(K)−D ◦∆Y = (∆Y ◦ Qλ0,λ0(K)−Qµ0,µ0(K) ◦∆Y ) + (Qµ0,µ0(K) ◦∆Y −D ◦∆Y ) = Qλ0,µ0(Obs(K)) + (Qµ0,µ0(K)−D)∆Y . The operator Qλ0,µ0(Obs(K)) is of order one but not the operator (Qµ0,µ0(K)−D)∆Y , unless it vanishes. Hence, both terms Qλ0,µ0(Obs(K)) and (Qµ0,µ0(K) − D)∆Y have to vanish and then Obs(K) = 0. � In particular, on a conformally flat manifold, all the conformal Killing 2-tensors give rise to con- formal symmetries of ∆Y after quantization by Qλ0,λ0 , as proved in [26]. We are now in position to prove our main theorem, which provides a full description of the conformal symmetries of ∆Y given by second order differential operators. The isomorphism Γ(TM) ∼= Γ(T ∗M) provided by the metric is denoted by [. Second Order Symmetries of the Conformal Laplacian 21 Theorem 4.8. The second order conformal symmetries of ∆Y are classif ied as follows: (i) A1 = { Lλ0X + c \| c ∈ R and X ∈ K1 } , (ii) A2/A1 ∼= {K ∈ K2 \|Obs(K)[ is an exact 1-form}, and if K ∈ K2 satisfies Obs(K)[ = −2df , with f ∈ C∞(M), the corresponding element in A2/A1 is given by Qλ0,λ0(K) + f. Proof. We deduce from Proposition 4.4 that the principal symbolK of a second-order conformal symmetry D1 is a symmetric conformal Killing 2-tensor. Since quantization maps are bijective, the operator D1 reads as D1 = Qλ0,λ0(K +X + f), with f and X symbols of degree 0 and 1 respectively. Theorem 4.5 implies that ∆Y ◦D1 −Qµ0,µ0(K +X + f) ◦∆Y = Qλ0,µ0(2G(X) + Obs(K) + 2G(f)). Hence ∆Y ◦D1 ∈ (∆Y ) leads to G(X) ∈ (H). By definition of G, this means that G(X) = 0, i.e. X ∈ K1. As the symbols Obs(K) and G(f) are of degree 1, they cannot pertains to (∆Y ). Therefore, ∆Y ◦D1 ∈ (∆Y ) is equivalent to X ∈ K1 and Obs(K) + 2G(f) = 0. The items (i) and (ii) in the statement of the theorem are then easily proved. � 4.4 Second order symmetries The general formula (2.11) for the natural and conformally invariant quantization on symbols of degree 2 leads to the following result. Proposition 4.9. Let (M, g) be a pseudo-Riemannian manifold of dimension n endowed with a symmetric Killing 2-tensor K. The operator Qλ0,λ0(K) = Qµ0,µ0(K) = Kab∇a∇b + (∇aKab)∇b − n− 2 4(n+ 1) ( ∇a∇bKab ) (4.4) − n+ 2 4(n+ 1) RicabK ab + 1 2(n− 1)(n+ 1) Sc ( gabK ab ) , is a symmetry of ∆Y , i.e. [∆Y ,Qλ0,λ0(K)] = 0, if and only if Obs(K) = 0. Proof. Let (xi, pi) be a canonical coordinate system on T ∗M . The Killing equation satisfied by K reads as gijpi∇jK = 0. Applying the trace operator Tr = gij∂pi∂pj we deduce that gkl(∇k TrK)∇l = −2(∇iKil)∇l and gkl(∇k∇l TrK) = −2∇i∇lKil. Moreover, if λ = µ and δ = 0, we have β1− 2β2 = 1 and β3− 2β4 = n2λ(1−λ) (n+1)(n+2) , where the βi are defined in (2.12). The formula for the quantization Qλ,λ reduces then, for K a Killing tensor, to Qλ,λ(K) = Kab∇a∇b + ( ∇aKab ) ∇b − n2λ(1− λ) (n+ 1)(n+ 2) ( ∇a∇bKab ) − n2λ(λ− 1) (n− 2)(n+ 1) RicabK ab + 2n2λ(1− λ) (n− 2)(n− 1)(n+ 1)(n+ 2) Sc ( gabK ab ) . Since λ0 + µ0 = 1 we deduce that Qλ0,λ0(K) = Qµ0,µ0(K). In consequence, the equality [∆Y ,Qλ0,λ0(K)] = 0 is equivalent to the fact that Qλ0,λ0(K) is a conformal symmetry of ∆Y . By Corollary 4.7, this means that Obs(K) = 0. � 22 J.-P. Michel, F. Radoux and J. Šilhan As a straightforward consequence, we get Corollary 4.10. Let (M, g) be a conformally flat manifold and K be a Killing 2-tensor. Then, we have [Qλ0,λ0(K),∆Y ] = 0. This corollary enlights some of the results obtained in [2]. As for conformal symmetries, we provide a full description of the symmetries of ∆Y given by second order differential operators. Theorem 4.11. The second order symmetries of ∆Y are exactly the operators Qλ0,λ0(K +X) + f, where X is a Killing vector field, K is a Killing 2-tensor such that Obs(K)[ is an exact one-form and f ∈ C∞(M) is defined up to a constant by Obs(K)[ = −2df . Proof. Let D1 be a second order symmetry of ∆Y . In view of (2.5), we can deduce from [∆Y , D1] = 0 that {H,σ2(D1)} = 0. This means that K = σ2(D1) has to be a symmetric Killing 2-tensor. Since quantization maps are bijective, the operator D1 reads as D1 = Qλ0,λ0(K +X + f), with f and X symbols of degree 0 and 1 respectively. Theorem 4.5 implies that [∆Y , D1] = Qλ0,µ0(2G(X) + Obs(K) + 2G(f)) + ( Qµ0,µ0(K +X + f)−Qλ0,λ0(K +X + f) ) ◦∆Y . We have shown that Qµ0,µ0(K) = Qλ0,λ0(K) in Proposition 4.9. Moreover, the general formulas in Theorem 2.4 prove that Qµ0,µ0(f) = Qλ0,λ0(f) and Qµ0,µ0(X)−Qλ0,λ0(X) = 2 n∇aX a. Hence, we get [∆Y , D1] = Qλ0,µ0(2G(X)) + 2 n (∇aXa)∆Y +Qλ0,µ0(Obs(K) + 2G(f)) and σ2([∆Y , D1]) = 2G(X) + 2 n (∇aXa)H. As S2TM = S2,0TM ⊕ S2,1TM , each of the two terms in the right hand side of the second equation are independent. Therefore, [∆Y , D1] = 0 is equivalent to G(X) = 0, ∇aXa = 0 and Obs(K) + 2G(f) = 0. The equations G(X) = 0 and ∇aXa = 0 mean that X is a conformal Killing vector field with vanishing divergence, i.e. X is a Killing vector field. Applying the met- ric, the equation Obs(K)+2G(f) = 0 translates into Obs(K)[ = −2df . The result follows. � For comparison, we recall the alternative classification obtained in [3]. Theorem 4.12 ([3]). Let K be a Killing 2-tensor and put I(K)ab = Kac Ricbc−RicacKb c . Then, we have[ ∇aKab∇b + f,∆ + V ] = 0 ⇐⇒ Kab(∇aV )− 1 3 ( ∇bI(K)ab ) = ∇af, where ∆ = ∇agab∇b and f, V ∈ C∞(M). As an advantage of our method, the obtained condition to get a symmetry (namely Obs(K)[ exact one-form) is conformally invariant and obviously vanishes on conformally flat manifolds. As an advantage of the approach used in [3] and initiated by Carter [11], one recovers easily that [∆Y ,∇aKab∇b] = 0, for all Killing 2-tensors K on an Einstein manifold. Second Order Symmetries of the Conformal Laplacian 23 4.5 Higher order conformal symmetries Up to now we discussed symbols of order ≤ 2. The more general version (which we shall state without proof) of Theorem 4.5 is as follows. Assume that Qλ,µ is a family of natural and conformally invariant quantizations as in Theorem 2.5 and let S be a trace-free symbol S ∈ Sk,00 . Then we get ∆Y ◦ Qλ0,λ0(S)−Qµ0,µ0(S) ◦∆Y = Qλ0,µ0 ( 2G(S) + xF1(S) + yF2(S) + Φ(S) ) , (4.5) where operators F1,F2 : Sk,00 → Sk−1,0 δ0 are defined in Proposition 3.2, scalars x and y have the value x = k(k − 1)(n+ 2k − 6) 3(n+ 2k − 2)(n+ 2k − 3) and y = k(k − 1)(k − 2)(n+ 2k) 12(n+ 2k − 2)(n+ 2k − 3) , and Φ is a natural and conformally invariant operator Φ : Sk,00 → S≤k−3 δ0 . For k ≤ 2 this recovers Theorem 4.5, the general case k ≥ 3 can be shown by a direct (but tedius) computation. Using (4.5) we can formulate a higher order version of Corollary 4.7: If K ∈ Kk is a conformal Killing k-tensor such that the operatorQλ0,λ0(K) is a conformal symmetry of ∆Y , then xF1(K)+ yF2(K) = 0. Moreover, the same reasoning as in the proof of Theorem 4.8 yields a higher order analogue of this theorem, i.e. Ak/Ak−1 ⊆ { K ∈ Kk \|xF1(K) + yF2(K) = G(K) for some K ∈ Sk−1,0 0 } . 5 Examples in dimension 3 In this section, we consider the space R3 endowed successively with two types of metrics: the conformal Stäckel metrics and the Di Pirro metrics. The conformal Stäckel metrics are those for which the Hamilton–Jacobi equation gij(∂iW )(∂jW ) = E admits additive separation in an orthogonal coordinate system for E = 0 (see [8] and references therein). They are conformally related to the Stäckel metrics, for which the additive separation of the Hamilton–Jacobi equation holds for all E ∈ R. Moreover, the separating coordinates, called (conformal) Stäckel coordinates are characterized by two commuting (conformal) Killing 2-tensors. Except for the Stäckel metrics, every diagonal metric on R3 admitting a diagonal Killing tensor is a Di Pirro metric g (see [28, p. 113]), whose corresponding Hamiltonian is (see e.g. [14]) H = g−1 = 1 2(γ(x1, x2) + c(x3)) ( a(x1, x2)p2 1 + b(x1, x2)p2 2 + p2 3 ) , (5.1) where a, b, c and γ are arbitrary functions and (xi, pi) are canonical coordinates on T ∗R3. 5.1 An example of second order symmetry The Di Pirro metrics defined via equation (5.1) admit diagonal Killing tensors K given by K = 1 γ(x1, x2) + c(x3) ( c(x3)a(x1, x2)p2 1 + c(x3)b(x1, x2)p2 2 − γ(x1, x2)p2 3 ) . For generic functions a, b, c and γ, the vector space of Killing 2-tensors is generated by H and K. However, for some choices of functions, this metric can admit other Killing tensors. For example, if (r, θ) denote the polar coordinates in the plane with coordinates (x1, x2), if the functions a, b, γ depend only on r and if a = b, then the metric is Stäckel and admits p2 θ as additional Killing tensor. 24 J.-P. Michel, F. Radoux and J. Šilhan Proposition 5.1. On the space R3, endowed with the metric g defined by (5.1), there exists a symmetry D of ∆Y whose principal symbol is equal to the Killing tensor K. In terms of the conformally related metric ĝ := 1 2(γ(x1, x2) + c(x3)) g, this symmetry is given by: D = Qλ0,λ0(K) + 1 16(3R̂icab − Ŝcĝab)K ab, i.e. by: D = ∇̂aKab∇̂b − 1 16 (∇̂a∇̂bKab)− 1 8 R̂icabK ab, where ∇̂, R̂ic and Ŝc represent respectively the Levi-Civita connection, the Ricci tensor and the scalar curvature associated with the metric ĝ. Proof. We use Theorem 4.11. In order to compute the obstruction Obs(K)[, we used a Math- ematica package called “Riemannian Geometry and Tensor Calculus”, by Bonanos [6]. This obstruction turns out to be an exact one-form equal to d(−1 8(3R̂icab − Ŝcĝab)K ab). The first expression of the symmetry D follows, the second one is deduced from (4.4), giving Qλ0,λ0(K). � 5.2 An example of obstructions to symmetries If written in conformal Stäckel coordinates, the conformal Stäckel metrics g on R3 admit four possible normal forms, depending on the numbers of ignorable coordinates (see [8]). A coordi- nate x is ignorable if ∂x is a conformal Killing vector field of the metric. Thus, if x1 is an ignorable coordinate, the conformal Stäckel metrics g read as g = Q ( (dx1)2 + ( u(x2) + v(x3) )( (dx2)2 + (dx3)2 )) , (5.2) where Q ∈ C∞(R3) is the conformal factor and where u and v are functions depending respec- tively on the coordinates x2 and x3. Such metrics admit ∂x1 as conformal Killing vector field and K = (u(x2) + v(x3))−1 ( v(x3)p2 2 − u(x2)p2 3 ) (5.3) as conformal Killing 2-tensor. Proposition 5.2. On R3, there exist metrics g as in (5.2) whose conformal Laplacian ∆Y admits no conformal symmetry with principal symbol K. Proof. Indeed, the obstruction associated with K, Obs(K)[, is generally not closed. Thanks to the Mathematica package “Riemannian Geometry and Tensor Calculus”, by Bonanos [6], we can actually compute that dObs(K)[ = −1 4 ( ∂2 x2 + ∂2 x3 ) ∂x2∂x3 log(u(x2) + v(x3))dx2 ∧ dx3, where the symbol ′ denotes the derivatives with respect to the coordinates x2 and x3. This expression does not vanish e.g. for the functions u(x2) = x2 and v(x3) = x3. We conclude then using Theorem 4.8. � Second Order Symmetries of the Conformal Laplacian 25 An example of a metric of the form (5.2) is provided by the Minkowski metric on R4 reduced along the Killing vector field X = x3∂t + t∂x3 + a(x1∂x2 − x2∂x1), a ∈ R (see [21]). In the time-like region of X and in appropriate coordinates (r, φ, z), the reduced metric is equal to g = dr2 + r2z2 z2 − a2r2 dφ2 + dz2 and admits ∂φ as Killing vector field. Moreover, after reduction, the Killing tensor p2 x1 + p2 x2 is equal to K = p2 r + 1 r2 p2 φ. Notice that the metric g is a Stäckel metric with one ignorable coordinate. Indeed, the metric takes the form (5.2), with Q(r, z) = r2z2 z2−a2r2 , u(r) = 1/r2 and v(z) = −a2/z2, whereas the conformal Killing tensor K − z2 z2−a2r2H can be written as in (5.3). Here, H = g−1 is the metric Hamiltonian. In this situation, there is no conformal symmetry of ∆Y with principal symbol K if a 6= 0. Indeed, the one-form Obs(K)[ is then non-exact, as shown by Mathematica computations dObs(K)[ = 3 2 ( a+ a3 )( 1 (z + ar)4 − 1 (z − ar)4 ) dr ∧ dz. Remark 5.3. Extending the metric (5.2) to Rn as g = Q ( (dx1)2 + (u(x2) + v(x3)) ( (dx2)2 + (dx3)2 ) + (dx4)2 + · · ·+ (dxn)2 ) , one can check that K, given in (5.3), is again a conformal Killing tensor and that the one-form Obs(K)[ is in general non-exact. Thus, there is no conformal symmetry of ∆Y with principal symbol K. 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Second Order Symmetries of the Conformal Laplacian

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