FOUR DIMENSIONAL CONFORMAL C-SPACES A.RODGOVERANDPAUL-ANDINAGY 7 0 0 Abstract. We investigate the structure of conformal C-spaces, a class of Riemmanian man- 2 ifolds which naturally arises as a conformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In di- n mension4weobtainafullanswertothisquestionandalsoinvestigatetheincidenceoftheBach a conditiononthisclassofmetrics. ThisisrelatedtoearlierresultsobtainedintheEinstein-Weyl J context. 8 1 ] G Contents D . 1. Introduction 1 h t 2. Conformal C-spaces and related structures 3 a m 3. Algebraic symmetries of Weyl curvature 5 3.1. The various equations 5 [ 3.2. The 4-dimensional case 10 1 v 4. A local classification 12 2 5. A Weitzenb¨ock formula and the unique continuation of the Weyl tensor 16 1 6. Bach flat manifolds 17 5 1 References 18 0 7 0 / h 1. Introduction t a m Let (Mn,g) be Riemannian manifold. The metric g is said to be Einstein if and : only if v i Ric = λg X r for some real constant λ, where Ric is the so-called Ricci curvature tensor of the a metric g. Einstein metrics have long had a priviliged role in geometry. Toward the study of Einstein structures, and also because Einstein metrics may be obstructed (for example topologocally), various generalisations of the Einstein condition are important [6]. One consists in requiring the so-called Cotton tensor C of the metric g to vanish and weakening further we might simply require that g be conformal to a Cotton metric. This is achieved if there is an gradient field ζ solving the equation (1.1) ι W +C = 0 , ζ where ι indicates insertion (or interior multiplication) ofξ and W denotes the Weyl ζ tensor of the metric g. Date:2ndFebruary2008. 2000 Mathematics Subject Classification. 53A30,53C24,53C55. Keywords and phrases. ConformalC-space,Einsteinmetric,Weyltensor. 1 2 A.R.GOVERANDP.-A.NAGY The requirement that ζ should be exact makes the condition equation (1.1) awk- ward to deal with, directly. It is not obvious, for example, how to give local con- formal invariants which characterise metrics which are locally conformally Cotton. This suggests that, in the first, instance one might consider a “Weyl analogue” of conformal Cotton equation (1.1) as follows. We will say (following [19]) that a Rie- mannian manifold is a conformal C-space, if there is a solution to (1.1), where ζ is any section of TM. Via a suitable (and natural) interpretation of ζ this is a con- formally invariant condition. In this context we term (1.1) the conformal C-space equation. This move is also motivated by the natural tractor/Cartan structures of Cartan and Thomas [10, 25] (see [5, 9, 15] for modern treatments) and the correspond- ing conformal holonomy [2, 23]. Dimension n conformal Riemannian manifolds are naturally equipped with a rank n + 2 vector bundle with a Lorentzian signature metric and compatible canonical connection. This is the tractor bundle and con- nection and the equivalent [9] principal bundle structure is the Cartan connection. An Einstein structure determines a parallel section I of the standard tractor bundle and, conversely, if a conformal structure admits a parallel standard tractor field I then this (parallel section) determines an Einstein metric on an open dense set (and we say the manifold is almost Einstein [17].) This is the set where X(I) is non-vanishing, where X is a canonical homomorphism which takes sections of the tractor bundle to conformal densities. Writing Ω for the curvature of the tractor connection, I parallel clearly implies ΩI = 0. An obvious weakening of the almost Einstein condition is to require that there is a section I of the standard tractor bundle (now not necessarily parallel) satisfying ΩI = 0. On the open set where X(I) is non-vanishing, ΩI = 0 is exactly (1.1), the conformal C-space equation [19]. Thus in the sense of infinitesimal conformal holonomy the conformal C-space equation is a vastly weaker requirement than the Einstein condition (which would have I annihilate the full jet of the tractor curvature). Using related ideas it is straightforward to manufacture conformal obstructions to conformal C-space metrics. For example on Riemannian 4-manifolds with non- vanishing Weyl tensor the conformal invariant |W|2C +4WdijkC W abc ijk dabc (where we have used an obvious abstract index notation) vanishes if and only if the manifold is a conformal C-space; this result is easily recovered, or see Proposition 2.5 of [19]. Taking the conformal C-space equation as our basic generalisation of the Cotton and Einstein conditions, the fundamental question is then is how far we are from these “integrable cases”. In the case that the Weyl curvature is suitably non- degenerate, answers to this may be found in [19], and see also [22]. Here we initiate a study of this issue on Riemannian manifolds, but where the aim is to remove the assumption of local conditions on the Weyl curvature. One of the main results is the following. Theorem 1.1. If (M4,g) is a compact conformal C-space then it is locally confor- mally Cotton. FOUR DIMENSIONAL CONFORMAL C-SPACES 3 Recall that on 4-manifolds the Bach tensor B (of conformal relativity [4]) is a conformal invariant with leading term a divergence of the Cotton tensor. This vanishes on half-flat manifolds, on locally conformally Einstein structures [6] and also on the conformal classes of certain product manifolds [18]. Bringing this into the picture leads (see Section 6) to a stronger result. Theorem 1.2. If (M4,g) is a compact conformal C-space which is Bach-flat then it is locally conformally Einstein We note that in [22] the authors obtained a result that Bach flat conformal C- spaces are locally conformally Einstein in 4 dimensions, provided that the Weyl tensor satisfies non-degeneracy conditions. So in the compact Riemannian setting the Theorem improves their result by removing the need for a non-degeneracy as- sumption. Another generalisation of the Einstein condition which has been studied exten- sively (e.g. [14, 16, 13]) are the Einstein-Weyl equations. A conformal manifold is said to be Einstein-Weyl if it admits a compatible torsion-free connection that has vanishing trace-free symmetrised Ricci curvature. Writing h for the trace-adjusted (“reduced”) Ricci tensor and ∇ for the Levi-Civita connection, this problem is equivalent to finding a 1-form field ζ and a metric g so that the symmetric part of h−∇ζ +ζ ⊗ζ is pure trace. There is a close connection with conformal C-spaces and this plays a role in the proofs of the main theorems. Our paper is organised as follows. In section 2 we collect a number of basic facts of relevance for the study of conformal C-spaces. In section 3 we study suitablly defined symmetries of an algebraic Weyl curvature tensor and show how these can be fully understood in dimension 4. To prove Theorem 1.1 we first show in section 4 that it holds locally, that is on the open subset where the Weyl tensor does not vanish. We use a detailed analysis of the properties of the Weyl curvature of a 4-dimensional conformal C-space combined with some results from Hermitian ge- ometry [1]. In section 5 we establish that the unique continuation property holds for theclass ofconformalC-spaces andthis eventually leads totheproofinthecompact case. Finally, the last section of the paper is devoted to the proof of Theorem 1.2. 2. Conformal C-spaces and related structures This section is intended to recall a number of elementary facts concerning the objects we shall subsequently use. Let (Mn,g),n ≥ 3 be a Riemannian manifold, ∇ the Levi-Civita connection associated with the metric g and R the Riemannian curvature tensor, given by R(X,Y)Z = −∇2 Z + ∇2 Z, whenever X,Y,Z are X,Y Y,X vector fields on M. The Weyl tensor W is defined by the decomposition (2.1) R = W +S where the Schouten tensor S is given by S = h•g. Here h = 1 Ric − s g n−2 2(n−1) is the reduced Ricci tensor of the metric g, whilst the Kulkarni-Nom(cid:0)izu product o(cid:1)f two symmetric tensors h and k is defined by (h•k)(x,y,z,t) = h(x,z)k(y,t)+h(y,t)k(x,z)−h(x,t)k(y,z)−h(y,z)k(x,t). 4 A.R.GOVERANDP.-A.NAGY The Weyl tensor satisfies the first Bianchi identity (2.2) W(X,Y)Z +W(Y,Z)X +W(Z,X)Y = 0. The second Bianchi identity for W is slightly more complicated and depends on the Cotton tensor C, an element of TM ⊗Λ2(M). It is defined by C(U,X,Y) = (∇ h)(Y,U)−(∇ h)(X,U) X Y for all U,X,Y in TM and then (2.3) σ (∇ W)(Y,Z,U,T) +(C ∧T −C ∧U)(X,Y,Z) = 0 X,Y,Z X U T (cid:20) (cid:21) where σ stands for the cyclic sum. An appropriate contraction of the differential Bianchi identity (2.3) alternatively gives the Cotton tensor C from the Weyl tensor W as (2.4) δW = −(n−3)C n where δW = − (∇ W)(e ,·,·,·)for an arbitrarylocal orthonormalframe{e ,1 ≤ ei i i i=1 P i ≤ n} on M. The Bach tensor B of the Riemannian manifold (Mn,g) is defined by n hBX,Yi = ∇ (δW)(X,e ,Y)+W(X,e ,h(e ),Y) ei i i i Xi=1 for all X,Y in TM. As it is well known this is symmetric and tracefree, and moreover in dimension 4 it is conformally invariant. In dimension 4 it also vanishes on (anti)self-dual metrics [16]. In this paper we shall mainly study the following class of Riemannian manifolds. Definition 2.1. Let (Mn,g) be a Riemannian manifold and let ζ be a vector field on M. Then (Mn,g,ζ) is a conformal C-space if the equation (2.5) W(ζ,·,·,·)+C = 0. is satisfied. If ζ = 0 then (Mn,g) is called a Cotton space. It should be noted that conformal C-spaces are conformally invariant in the usual sense. Also, a natural sub-class to look at consists in closed conformal C-spaces, that is conformal C-spaces (Mn,g,ζ) such that dζ = 0, which is again a confor- mally invariant condition. Obviously, closedeness in the conformal C-space context rephrases globally that a Riemannian metric is locally conformal to that of a Cotton space. Remark 2.1. Cotton spaces are known to have vanishing Pontriagin classes, fact used in [7] to show that in 4 dimensions the non-vanishing of the signature implies that the metric is Einstein, provided the manifold is compact. Further results and examples were obtained in dimension 4 [12] under degeneracy assumptions on the spectrum of the Weyl or the Schouten tensor. It is also known that-in the compact case-the metric has to be Einstein when in the presence of a compatible K¨ahler [6] or closed G structure [8],[11]. Despite constant interest a complete classification 2 seems to be still missing. FOUR DIMENSIONAL CONFORMAL C-SPACES 5 Notethatindimensionsn ≤ 3aconformalC-spaceisautomaticallyCottondueto theabsenceofalgebraicWeylcurvaturetensors, hencethefirstinterestingdimension in this context is when n = 4. Related to conformal C-spaces are Einstein-Weyl structures whose definition we give below. Definition 2.2. Let (Mn,g) be Riemannian. Then g is Einstein-Weyl if 1 h = ∇ζ −ζ ⊗ζ − dζ +fg 2 for some vector field ζ on M and some smooth function ζ on M. Moreover (g,ζ) is said to be a closed Einstein-Weyl structure if dζ = 0. Thequestion under study inthis paper consists ininvestigating uptowhat extent a conformal C-space must be closed. To appproach this, we consider for a given conformal C-space (Mn,g,ζ) the tensor h defined by ζ h = h−∇ζ +ζ ⊗ζ ζ and recall that Proposition 2.1. Let (Mn,g,ζ) be a conformal C-space. The tensor h belongs to ζ the space E . W Therefore one must first understand the algebraic structure of the space E and W then explore its geometric consequences. In the next section we gather a few gen- eral facts to this extent in arbitrary dimensions and explore thougroutly the four dimensional case. 3. Algebraic symmetries of Weyl curvature In this section we shall study various algebraic equations akin to produce sym- metries of an algebraic Weyl curvature tensor. These are actually insightful when studying various geometric structures, and the relevant connections will be made clear in the next section. 3.1. The various equations. Let (Vn,g),n ≥ 4 be a Euclidean vector space. In what follows shall use the metric to identity without further comment ⊗2V with End(V) using the convention β = g(h·,·), as well as vectors and 1-forms. As a point of notation, we shall use h·,·i for the form inner product induced by g. Let b : Λ2 ⊗Λ2 → Λ3 ⊗Λ1 be the Bianchi map given by 1 (b R)(x,y,z) = R(x,y)z +R(y,z)x+R(z,x)y 1 whenever x,y,z belong to V and for any R in Λ2⊗Λ2, where the standard notation applies. Considernowanon-vanishingalgebraicWeyl-curvaturetensorW onV that is an element W of Λ2 ⊗Λ2 satisfying the first Bianchi identity, that is b (W) = 0 1 n and which is moreover trace-free in the sense that W(·,e ,·,e ) = 0 for any i i i=1 P orthonormal frame {e ,1 ≤ i ≤ n}. Then we can extend W as a map W : ⊗2V → i ⊗2V by setting : n W(h) = W(e ,·,he ,·) i i Xi=1 6 A.R.GOVERANDP.-A.NAGY foranyhin⊗2V andforsomearbitraryorthonormalbasis{e }inV. Thisextension i of W preserves the tensor type, that is it preserves the splitting ⊗2V = Λ2 ⊕S2 ⊕ 0 Rg. Moreover, the restriction of W to Λ2(V) is given by < W(v ∧ w),u ∧ q >= W(v,w,u,q) for all v,w,u,q in V. Using the first Bianchi identity W has to satisfy this can also be rephrased to say that n 1 W(α) = W(e ,Fe ) i i 2 Xi=1 for an arbitrary orthonomal basis {e ,1 ≤ i ≤ n} and for all 2-forms α with associ- i ated skew-symmetric endomorphism F, that is α = g(F·,·). We shall be interested in what follows in the space E of tensors h in End(V) such that W (3.1) W(x,y,hz,·)+W(y,z,hx,·)+W(z,x,hy,·) = 0 for all x,y,z in V. We also define the spaces S = E ∩ S2,A = E ∩Λ2 and W W W W point out that, a priori, E is not the direct sum of S and A . The space S has W W W W been studied in detail in [7]. In dimension 4, as we shall recall later on, additional information is available [12]. Lemma 3.1. Let h be in End(V). The following hold: (i) if b (W(·,·,h·,·)) belongs to Λ4 then h satisfies (3.1) and 1 (3.2) W(hx,y,z,u)+W(x,hy,z,u) = W(x,y,hz,u)+W(x,y,z,hu) whenever x,y,z,u belong to V. (ii) h satisfies (3.1) if and only if it satisfies (3.2). Proof. (i) Let us set T = b (W(·,·,h·,·)). Then 1 W(x,y,hz,u)+W(y,z,hx,u)+W(z,x,hy,u) = T(x,y,z,u) for all x,y,z,u in V. We anti-symmetrise in z,u hence W(x,y,hz,u)+W(x,y,z,hu)+(W(y,z,hx,u)−W(y,u,hx,z)) +(W(z,x,hy,u)−W(u,x,hy,z)) = 2T(x,y,z,u) and further W(x,y,hz,u)+W(x,y,z,hu)−W(hx,y,z,u)−W(x,hy,z,u) = 2T(x,y,z,u) aftermaking use oftheBianchi identity. Since T isa fourform, itbelongstoS2(Λ2), but since the l.h.s in the equation above belongs to Λ2(Λ2), it must vanish and the claim follows. (ii) follows from the Bianchi identity when taking the cylic sum upon x,y,z in (3.2). (cid:3) Lemma 3.2. The following hold: (i) Suppose that h is in E . Then W (3.3) W(hF −Fh⋆) = W(F)h−h⋆W(F) and (3.4) W(hF +Fh⋆) = W(F)h+h⋆W(F) FOUR DIMENSIONAL CONFORMAL C-SPACES 7 whenever F is a skew symmetric endomorphism of V and where h⋆ stands for the adjoint of h with respect to the metric g. (ii) if h in End(V) satisfies both of (3.3) and (3.4) then it satisfies (3.1) as well. (iii) the identity (3.4) is equivalent with (3.1). Proof. (i) Let us fix an orthonormal basis {e } in V. From (3.1) we obtain i W(e ,Fe ,hv,w)+W(Fe ,v,he ,w)+W(v,e ,hFe ,w) = 0 i i i i i i whenever v,w belong to V. After summation, we obtain n 2 < W(F)hv,w > + W(Fe ,v,he ,w)+W(v,e ,hFe ,w) = 0 i i i i Xi=1 n n Since W(Fe ,v,he ,w) = − W(e ,v,hFe ,w) = −W(hF)(v,w) we find i i i i i=1 i=1 P P (3.5) W(F)h = W(hF) whenever F belongs to Λ2. Now the equations in (3.3), (3.4) follow when using that W respects the splitting ⊗2V = S2 ⊕Λ2. (ii) follows when rewriting (3.5) using elements of the form F = v ∧w where v,w belong to V. (iii) again by rewritting (3.4) by means of decomposable elements of the form F = z ∧u in Λ2 we find W(z,x,hu,y)−W(u,x,hz,y)+ W(hz,x,u,y)−W(hu,x,z,y) = W(z,u,hx,y)+W(z,u,x,hy) whenever x,y,z,u belong to V. The use the Bianchi identity upon the first and third respectively second and fourth terms in the l.h.s. of the equation above shows that h satisfies the identity in Lemma 3.1 and therefore it belongs to E . (cid:3) W It remains now to understand up to what extent (3.3) and (3.4) are equivalent. Lemma 3.3. Let W be an algebraic Weyl curvature tensor and let h in ⊗2V satisfy (3.3). Then h equally satisfies (3.1). Proof. As in the proof of (iii) of the Proposition above we rewrite (3.3) for decom- posable F’s in Λ2, of the form F = x∧y, where x,y in V. Since < W(hF)z,u >= W(x,z,hy,u)−W(y,z,hx,u) and < W(Fh⋆)z,u >= W(hx,z,y,u)−W(hy,z,x,u) for all z,u in V, we arrive at − < W(x,z)u+W(x,u)z,hy > (3.6) + < W(y,z)u+W(y,u)z,hx >= W(x,y,hz,u)−W(x,y,z,hu) for all x,y,z,u in V. Let T in Λ3 ⊗ Λ1 be defined by T = b (W(·,·,h·,·)). We 1 rewrite then (3.6) as T(x,y,z,u) =W(x,y,z,hu)−W(x,u,z,hy)+W(y,u,z,hx) =−T(x,y,u,z) 8 A.R.GOVERANDP.-A.NAGY for all x,y,z,u in V. It follows that T belongs to Λ4 and we conclude by means of Lemma 3.1. (cid:3) Remark 3.1. It is easy to see that (3.6), and therefore (3.1), is yet equivalent with (3.7) W(hS −Sh⋆) = W(S)h−h⋆W(S) for all S in S2. Lemma 3.4. We have π−E ⊆ Ker(W ), where π− : ⊗2V → Λ2 is the orthogonal W |Λ2 projection. Proof. Follows by taking the trace of (3.1) in the last two arguments. (cid:3) Therefore Ker(W ) appears as a first obstruction to the equality of E and S |Λ2 W W for when W is injective the latter spaces coincide. We shall show now that the |Λ2 algebraic structure of E is related to the symmetry group W G = {γ ∈ GL(V) : W(γ·,γ·,γ·,γ·) = W} W of the Weyl tensor W. The Lie algebra g of G consists in the space of tensors W W h in ⊗2V satisfying W(hx,y,z,u)+W(x,hy,z,u)+ (3.8) W(x,y,hz,u)+W(x,y,z,hu) = 0 for all x,y,z,u in V. Before making explicit the relationship between E and g W W we need to establish to reinterpret the identity (3.8) as it has been done for (3.1) and establish the analogous equivalences. Lemma 3.5. The following are equivalent: (i) h belongs to g W (ii) W(hF +h⋆F) = −W(F)h−h⋆W(F) for all F in Λ2 Proof. Follows by using, with minor changes, the same ingredients as in the proof of Lemma 3.2. Details are left to the reader. (cid:3) Proposition 3.1. We have [E ,E ] ⊆ g . W W W Proof. We shall make essentially use of the equation (3.5), all tensors in E must W satisfy. Letthereforeh andh belongtoE . Using (3.5)we haveW(h π−(h F)) = 1 2 W 1 2 W(π−(h F))h for all F in Λ2. Since π−(h F) = 1(h F +Fh⋆) we get 2 1 2 2 2 2 W(h h F)+W(h Fh⋆) =W(h F +Fh⋆)h 1 2 1 2 2 2 1 = W(F)h +h⋆W(F) h (cid:20) 2 2 (cid:21) 1 after making use of (3.5) for h . Similarly, W(h h F)+W(h Fh⋆) = W(F)h + 2 2 1 2 1 (cid:20) 1 h⋆W(F) h whence 1 (cid:21) 2 W([h ,h ]F)+W(h Fh⋆ −h Fh⋆) = −W(F)[h ,h ]+h⋆W(F)h −h⋆W(F)h 1 2 1 2 2 1 1 2 2 1 1 2 FOUR DIMENSIONAL CONFORMAL C-SPACES 9 for all F in Λ2. But h Fh⋆ − h Fh⋆ and h⋆W(F)h − h⋆W(F)h are symmetric 1 2 2 1 2 1 1 2 tensors therefore π−W([h ,h ]F) = −π−W(F)[h ,h ] and this leads to 1 2 1 2 W([h ,h ]F +F[h ,h ]⋆) = −W(F)[h ,h ]−[h ,h ]⋆W(F) 1 2 1 2 1 2 1 2 for all F in Λ2. Therefore, by using the equivalence of (ii) and (i) in Lemma 3.5 we find that [h ,h ] belongs to g . (cid:3) 1 2 W As it has been done for the space E the obstruction for g to be contained in W W so(V) is measured as follows. Lemma 3.6. We have π+g ⊆ Ker(W ) where π+ : ⊗2V → S2 denotes the W |S2 orthogonal projection. Proof. Follows by taking the trace of the identity (3.8) in the variables x and z. (cid:3) The space E has also an algebraic structure of its own, though different than W that of g . Let {·,·} : ⊗2V ×⊗2V → ⊗2 denote the anti-commutator. W Proposition 3.2. Let W be an algebraic Weyl curvature tensor. Then {E ,E } ⊆ W W E . W Proof. Given h in E it is enough to show that h2 still belongs to E . Now if F W W belongs to Λ2 and since h satisfies (3.3) we have W(hF−Fh⋆) = W(F)h−h⋆W(F) and using this for π−(hF) = 1(hF +Fh⋆) we obtain 2 W(h(hF +Fh⋆)−(hF +Fh⋆)h⋆) =W(hF +Fh⋆)h−h⋆W(hF +Fh⋆) =(W(F)h+h⋆W(F))h−h⋆(W(F)h+h⋆W(F)) =W(F)h2 −(h2)⋆W(F) for all F in Λ2, where we have used that h satisfies (3.4). It follows that h2 satisfies (3.3) hence the claim follows by making use of Lemma 3.3. (cid:3) Corollary 3.1. Let h belong to E . The following hold: W (i) W(hFh⋆) = h⋆W(F)h for all F in Λ2. (ii) W(hx,hy,z,u) = W(x,y,hz,hu) whenever x,y,z,u belong to V. Proof. (i) By Lemma 3.2 we know that h satisfies (3.4), that is W(hF + Fh⋆) = W(F)h+h⋆W(F)forallF inΛ2. Using thisforπ−(hF) = 1(hF+Fh⋆)we compute 2 W(h(hF +Fh⋆)+(hF +Fh⋆)h⋆) =W(hF +Fh⋆)h+h⋆W(hF +Fh⋆) =(W(F)h+h⋆W(F))h+h⋆(W(F)h+h⋆W(F)) =W(F)h2 +(h2)⋆W(F)+2h⋆W(F)h for all F in Λ2. Since h2 belongs to E by Proposition 3.2and therefore satisfies W (3.4) the claim follows. (ii) follows when rewriting (i) by means of decomposable elements of Λ2. (cid:3) 10 A.R.GOVERANDP.-A.NAGY 3.2. The 4-dimensional case. Let (V4,g) be a four dimensional, oriented, Eu- clidean vector space together with an algebraic Weyl tensor W. We consider the splitting Λ2(V) = Λ+⊕Λ− in its self-dual resp. anti-self-dual components. Accord- ingly, we have the splitting of the Weyl tensor as W = W+ +W− in its self-dual, resp. anti-self-dual parts. Then W± belong to S2(Λ±) and let 0 us denote by Σ± = {λ±,1 ≤ k ≤ 3} their spectra. Of course λ± + λ± + λ± = k 1 2 3 0. Consider now the corresponding (normalized) system of eigenforms W±ω± = k λ±ω±,k = 1,2,3. These forms are associated to g-compatible almost complex k k structures J±,1 ≤ k ≤ 3 that is ω± = g(J±·,·). The almost complex structures k k k satisfy the quaternion identities i.e. J±J± +J±J± = 0,J± = J±J± and moreover 1 2 2 1 3 1 2 [J+,J−] = 0 for all 1 ≤ k,p ≤ 3. k p Lemma 3.7. Let h in S2(V). The {h,J±} belongs to Λ∓ for all 1 ≤ k ≤ 3. 0 k Itwillbeimportantforsubsequent computationstonotethat|ω±| = 2,1 ≤ k ≤ 3 k (here we use the norm on forms). Define now the endomorphisms σ = J+J−,1 ≤ i,j i j i,j ≤ 3; then the σ ’s are orthogonal involutions of V, producing and orthogonal i,j basis in S2(V). Note also that |σ | = 2,1 ≤ i,j ≤ 3, where the inner product on 0 i,j 4 S2(V) is define as usually : < S ,S >= < S e ,S e >, for some orthonormal 1 2 1 i 2 i i=1 P basis {e ,1 ≤ i ≤ 4} in V. i Lemma 3.8. We have W(σ ) = (λ+ +λ−)σ for all 1 ≤ i,j ≤ 3. i,j i j i,j Proof. Let S be in S2(V). Let {e ,1 ≤ i ≤ 4} be an orthonormal basis in V and 0 i let v,w be arbitrary vectors in V. We compute by expanding ei ∧ v in the basis ω±,1 ≤ k ≤ 3 k 3 1 W(e ,v,Se ,w) = ω+(e ,v) < W(ω+)Se ,w > +ω−(e ,v) < W(ω−)Se ,w > i i 2 k i k i k i k i Xk=1 3 1 = λ+ω+(e ,v)ω+(Se ,w)+λ−ω−(e ,v)ω−(Se ,w). 2 k k i k i k k i k i Xk=1 Summing now over i we obtain that 4 W(S) = W(e ,·,Se ,·) i i Xi=1 3 1 = − λ+J+SJ+ +λ−J−SJ− 2 k k k k k k Xk=1 We now take S = σ ,1 ≤ i,j ≤ 3 to arrive, after a short computation to the proof i,j of the Lemma. (cid:3) Corollary 3.2. Any algebraic Weyl tensor is, in 4-dimensions, subject to the alge- braic identities W{F,G} = {W(F),G} +{F,W(G)} 0 0