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Conformal Symmetries of the Yamabe and Paneitz Operators

We study first and second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. We show that first order conformal symmetries of the Yamabe operator induce second order conformal symmetries. We show that on an Einstein space every conformal Killing vector field induces a conformal symmetry of the Paneitz operator and, conversely, every first order conformal symmetry of the Paneitz operator arises from a conformal Killing vector field. Possible extensions of our main theorems are illustrated by examples. We correct formulas in a paper of B. Carter and a paper of Kamran and McLenaghan.


Introduction
In this paper, we study first and second order conformal symmetries of the Yamabe operator on an arbitrary pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. Specifically, we generalize part of a theorem of Gover andŠilhan [7] by proving that any pair of first order conformal symmetries of the Yamabe operator naturally yields a second order conformal symmetry. Moreover, in the context of Einstein spaces, we show that every conformal Killing vector induces a first order conformal symmetry of the Paneitz operator, and each of its first order conformal symmetries arise from a conformal Killing vector; this is an analogue of a special case of the same theorem of Gover anď Silhan. Finally we consider examples suggesting the possibility of extending this result to non-Einstein spaces, and of proving the aforementioned second order result for the Paneitz operator.
More precisely, consider a pseudo-Riemannian manifold (M, g) of dimension n ≥ 2 and a local Lie group G acting on M by local diffeomorphisms. We denote by g the Lie algebra of G, seen as a subalgebra of the Lie algebra X (M ) of smooth vector fields on M .
In the case G is a group of isometries, the corresponding vector fields on M , seen as a differential operators D acting on scalar fields, must satisfy where ∆ = g ij ∇ i ∇ j is the Laplacian on M . If G is a group of conformal transformations, the corresponding (first order) differential operators D will satisfy where ∆ Y = ∆ − n−2 4(n−1) R is the Yamabe Laplacian, for scalar fields q and r completely determined by D.
More generally, a differential operator D of any order satisfying (1) or (2) is called a symmetry of the Laplacian, or a conformal symmetry of the Yamabe Laplacian, respectively. One can further generalize these concepts. Given a differential operator P on M acting on scalar fields, a differential operator D is a conformal symmetry of P if there exists a differential operator B such that is an identity. Requiring that B be identically 0 gives the (exact) symmetries of P .
It is well known that the principal symbol T of a symmetry of the Laplacian must satisfy the Killing equation (see for example [4]); in local coordinates, if T = T i 1 ,...,i k then Second order symmetries of the Laplacian are of particular importance because of their relationship to separation of variables of the Hamilton-Jacobi and Helmholtz equations, see for example [13] and [9]. In [3] Carter characterized first and second order symmetries of the Yamabe Laplacian in a general pseudo-Riemannian manifold.
Second order conformal symmetries of the Laplacian and Yamabe Laplacian have been studied in relation to R-separation of variables of the Hamilton-Jacobi and Laplace equations in [1], [10], and to R-separation of variables of the (conformally invariant) Yamabe equation ∆ Y Ψ = 0 on a Lorentzian 4-dimensional manifold in [11]. In [12] the authors give a characterization of second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold in terms of quantizations maps.
Further, higher order conformal symmetries of the Laplacian have been studied by Eastwood [4] from a representation-theoretic perspective. Eastwood shows, in particular, that the principal symbol T of a conformal symmetry of ∆ must be a conformal Killing tensor. That is, T must satisfy the conformal Killing equation Since the symmetrized tensor product K of two conformal Killing vectors is a conformal Killing tensor, it is natural to ask whether K gives rise to a second order conformal symmetry of ∆ Y . One of our main results answers this question affirmatively: Theorem 4.4 Let (M n , g) be an arbitrary pseudo-Riemannian manifold. Let X 1 and X 2 be first order conformal symmetries of the Yamabe operator ∆ Y = ∆ − n−2 4(n−1) R on M . Then S := 1 2 {X 1 , X 2 }, where {·, ·} denotes the anti-commutator, is a second order conformal symmetry of ∆ Y .
The statement of Theorem 4.4 is not surprising at the level of principal symbols. However, the lower order terms impose additional conditions, which turn out to be identically satisfied. This "hereditary" behaviour of conformal symmetries of ∆ Y is not explicitly addressed in [12]. In the case of a locally conformally flat manifold the result in Theorem 4.4 follows from [7,Theorem 2.4].
Gover andŠilhan [7] studied higher order conformal symmetries of higher order GJMS operators [8] on locally conformally flat spaces. In particular, the authors establish that a (trace-free) conformal Killing tensor always gives rise to a conformal symmetry of every GJMS operator.
The fourth order GJMS operator is known as the Paneitz operator [14]. On an Einstein space M , the Paneitz operator is given by the expression [6] where c 1 and c 2 are explicit constants depending on the dimension of M . In this paper we initiate the study of first and second order conformal symmetries of the Paneitz operator on Einstein manifolds, extending some of the known results on conformally flat manifolds [7] to a more general context. Our main result in this direction is This paper is organized as follows. In Section 2 we introduce conformally covariant differential operators, GJMS operators and, in particular, the Yamabe and Paneitz operators. In Section 3 we cover technical lemmas and prove Theorem 3.4. We provide examples suggesting Theorem 3.4 could be extended to a wider class of manifolds. In Section 4 we correct formulas given in [3] and [11] and further discuss second order conformal symmetries of Laplacian operators. We prove Theorem 3.4 and we explore the possibility of an analogous result for the Paneitz operator via various examples.
Acknowledgements. The author would like to thank Niky Kamran and Dima Jakobson for many very helpful discussions.

Conformally covariant operators
A differential operator P g on a manifold (M, g) is said to be conformally covariant if, under a conformal change of metric g →ĝ := e 2f g, f ∈ C ∞ (M, R), it transforms according to for some constants γ, γ ′ . The equation P g ψ = 0 is therefore conformally invariant in the sense that P g ψ = 0 if and only if Pĝe −γf ψ = 0. An important class of conformally covariant operators are the so-called GJMS operators introduced in [8]. The GJMS operator of order 2k transforms as Note that if k = n 2 the kernel of P k,g is preserved under any conformal change of metric.

First order conformal symmetries
Throughout this paper we will assume that (M, g) is a smooth pseudo-Riemannian manifold of dimension n. We will adopt the following sign convention: given a vector field (v k ), the Riemann curvature tensor is defined by lij v l and the Ricci tensor by R ij = R k ikj . In [3] Carter derives the necessary and sufficient conditions for N k ∇ k to be a first order symmetry of ∆ + U, where U is any scalar field. First order conformal symmetries can easily be derived and we do so in Lemma 3.2. We will need the following general facts about conformal Killing vectors.
Lemma 3.1. Let (M n , g) be an arbitrary pseudo-Riemannian manifold and let N = N k ∇ k be a conformal Killing vector. Then Lemma 3.2. Let (M n , g) be an arbitrary pseudo-Riemannian manifold and let U be a scalar field on M . There exists a scalar field r such that Equating same degree (symmetrized) covariant derivatives we conclude that N k ∇ k must be a conformal Killing vector. Using (a) from Lemma 3.1 the rest of the conditions easily follow. Proposition 3.3. Let (M n , g) be an arbitrary pseudo-Riemannian manifold. Then N k ∇ k + Q is a conformal symmetry of the Yamabe operator if and only if N k ∇ k is a conformal Killing vector and Q = n−2 2n N k ;k . Proof. Taking U = −c 1 R in Lemma 3.2, condition 3 rewrites as which, by Lemma 3.1, is always satisfied by any conformal Killing vector.
A result analogous to Proposition 3.3 holds for the Paneitz operator on Einstein spaces.
Theorem 3.4. Let (M n , g) be an Einstein manifold with R ij = cg ij and n ≥ 3. Then N k ∇ k + Q is a conformal symmetry of the Paneitz operator if and only if N k ∇ k is a conformal Killing vector and Q = n−4 2n N k ;k .
Proof. We want to solve Since, After symmetrizing derivatives we equate same-order terms in (8). The fourth order terms give which holds if and only if N k ∇ k is a conformal Killing vector and r = 4 n N k ;k . Indeed, If N is a conformal Killing vector (10) becomes 4 3 1 n N k ;k g ij g kl + g ik g jl + g il g jk = 4 n N k ;k g (ij g kl) .
Conversely, contracting the i and j indices in (9) 2 3 (n + 4)N (k ; l) + g kl N j ;j = r and contracting the two remaining indices in (11) we obtain r = 4 n N k ;k . Substituting r back into (11) gives Using that N is a conformal Killing vector the third order terms become 4 − n n N k ;k Remark 3.5. Let (M n , g) be an Einstein manifold of dimension n ≥ 3. The first order differential operator N k ∇ k + Q commutes with the Paneitz operator if and only if N k ∇ k is a Killing vector field and Q is a constant.
Remark 3.6. Let (M n , g) be an Einstein manifold of dimension n ≥ 3. There is a correspondence between the first order conformal symmetries of the Yamabe operator and the first order conformal symmetries of the Paneitz operator. Namely, N k ∇ k + Q is a conformal symmetry of ∆ Y if and only if N k ∇ k + n−4 n−2 Q is a conformal symmetry of P.
The case of a 2-dimensional manifold is quite special since the algebra of conformal symmetries is infinite dimensional. We can still establish a correspondence between conformal symmetries of ∆ and ∆ 2 .
Corollary 3.7. Let (M 2 , g) be 2-dimensional. There is a correspondence between the first order conformal symmetries of ∆ and those of ∆ 2 . Namely, N k ∇ k is a conformal symmetry of ∆ if and only if N k ∇ k − 1 2 N k ;k is a conformal symmetry of ∆ 2 .
One could ask whether Theorem 3.4 can be extended to hold on a general pseudo-Riemannian manifold. We present examples in spaces that are neither Einstein nor locally conformally flat where the conclusion of Theorem 3.4 holds, thus suggesting an affirmative answer to this question.
Example 3.8. Consider R 4 endowed with the pp-wave metric . This metric is Lorentzian and Ricci-flat. Note that g is not locally conformally flat since the Weyl tensor has non-zero components, for example C 1212 = − 1 u 4 . Let us conformally rescale the metric byĝ = e 2f g so as to obtain a metric that is not Einstein. For instance take, , soĝ is not Einstein as long as f 2 v + f vv ≡ 0. These metrics have maximal conformal group admitting seven conformal Killing vectors. One can check that all of them give rise to conformal symmetries of the Paneitz operator. For instance let L = u 2 ∇ 0 + x 2 +y 2 2 ∇ 1 + ux∇ 2 + uy∇ 3 . We have Pĝ = e −4f (v) ∆ 2 g . Then Example 3.9. Let H 2 be the 5-dimensional Heisenberg group and let Γ ⊂ H 2 be a uniform discrete subgroup as in [5]. Then M = Γ\H 2 is a compact manifold. Let (x 1 , x 2 , y 1 , y 2 , t) be local coordinates. We endow M with the left-invariant metric g = dx 1 dx 1 + dx 2 dx 2 + (1 − x 2 1 )dy 1 dy 1 − 2x 1 x 2 dy 1 dy 2 + 2x 1 dy 1 dt + (1 − x 2 2 )dy 2 dy 2 + 2x 2 dy 2 dt − dtdt. This metric is neither Einstein nor locally conformally flat. The frame is orthonormal for this metric. The Laplacian on scalar fields is therefore given by

By [2] the Paneitz operator is given by
The metric g does not admit conformal Killing vectors, other than exact Killing vectors. Consider insteadĝ = e 2f (x 1 ) g where f is a smooth function of x 1 only. Now, Rĝ = −e 2f (x 1 ) (12f 2 x 1 + 8f x 1 x 1 − 1), soĝ will not Einstein if Rĝ is not a constant. The metricsĝ admits 4 conformal Killing vectors. The Paneitz operator in the metricĝ is given by and each of the conformal Killing vectors can be checked to induce a first order symmetry of Pĝ. For instance, let

Second order conformal symmetries
We now deal with second order differential operators on M that satisfy for some vector field C = C k ∇ k and scalar field D.
In [3] Carter shows the following result.
Proposition 4.1 (Carter [3]). [ The expression given in [3], however, contains a typographical mistake where the term is missing. In [11] Kamran and McLenaghan derive the necessary and sufficient conditions for ∇ i K ij ∇ j to be a symmetry of ∆+U . The omitted term in [3] is carried over to [11]. The correct expression, in a slightly more general context, is given in the following proposition.
Again, we may take U = − (n−2) 4(n−1) R to obtain the second order conformal symmetries of the Yamabe operator. A characterization of such symmetries has also been given in [12]. Remark 4.3. Let L and M be conformal Killing vectors. Then the symmetrized tensor product K ij := L (i M j) is a conformal Killing tensor. Indeed, K (ij ; Since the property of being conformal Killing is inherited under symmetrized tensor products, a natural question is whether two first order conformal symmetries induce a second order conformal symmetry. Let {·, ·} denote the anti-commutator. Note that . We answer this question in the following theorem. is not, in general, a second order symmetry of ∆ + U . However, S + q is a second order symmetry if and only if q is a scalar field satisfying (1) q ; In such case, Proof. Let Then K (ij ; k) = 1 2 C (i g jk) and 3K (ij and Finally, the terms of order zero require The case of the Yamabe Laplacian is again special. As stated in the introduction, we are able to show that two conformal Killing vectors always induce a second order conformal symmetry of ∆ Y . Theorem 4.5. Let (M n , g) be an arbitrary pseudo-Riemannian manifold. If L k ∇ k + A and M k ∇ k + B are conformal symmetries of ∆ Y , then S : Proof. We will show that q, as defined in Theorem 4.4, is constant. First, Therefore, we have Then, so it follows that Next, we ask whether first order conformal symmetries of the Paneitz operator induce second order symmetries. This is much harder to obtain by direct computation. If M is locally conformally flat, a positive answer follows from [7]. The following examples suggest that first order conformal symmetries of P would also induce second order conformal symmetries in geometries that are not locally conformally flat.

Conclusions
A number of questions that naturally follow from this work remain to be settled. First, we described all the first order symmetries of the Paneitz operator on Einstein manifolds starting from the fact that, on such manifolds, the Paneitz operator can be written as a polynomial in the Laplacian with constant coefficients. It is not clear whether the Einstein condition is essential for the vector space of first order conformal symmetries to be isomorphic to the space of conformal Killing vectors, our examples suggest it is not. The nature of the interplay between curvature properties and the conditions for the existence of first order symmetries remains to be fully understood. Further, higher order GJMS operators might share an analogous behaviour on Einstein or perhaps more general spaces.
Second, we showed how first order conformal symmetries of the Yamabe operator induce second order symmetries and suggested this could also be true for the Paneitz operator. Again, it is not evident whether the curvature could be an obstruction for such a hereditary property to hold for the Paneitz operator. Higher order GJMS operator might also admit a similar hereditary property. We hope to investigate these questions in the future.