Almost Einstein and Poincare-Einstein manifolds in Riemannian signature
aa r X i v : . [ m a t h . DG ] M a r ALMOST EINSTEIN AND POINCARÉ-EINSTEINMANIFOLDS IN RIEMANNIAN SIGNATUREA. ROD GOVERAbstra t. An almost Einstein manifold satis(cid:28)es equations whi h area slight weakening of the Einstein equations; Einstein metri s, Poin aré-Einstein metri s, and ompa ti(cid:28) ations of ertain Ri i-(cid:29)at asymptoti- ally lo ally Eu lidean stru tures are spe ial ases. The governing equa-tion is a onformally invariant overdetermined PDE on a fun tion. Awayfrom the zeros of this the almost Einstein stru ture is Einstein, while thezero set gives a s ale singularity set whi h may be viewed as a onformalin(cid:28)nity for the Einstein metri . In this arti le we give a lassi(cid:28) ation ofthe possible s ale singularity spa es and derive geometri results whi hexpli itly relate the intrinsi onformal geometry of these to the on-formal stru ture of the ambient almost Einstein manifold. Classes ofexamples are onstru ted. A ompatible generalisation of the onstants alar urvature ondition is also developed. This in ludes almost Ein-stein as a spe ial ase, and when its urvature is suitably negative, is losely linked to the notion of an asymptoti ally hyperboli stru ture.The author gratefully a knowledges support from the Royal So iety ofNew Zealand via Marsden Grant no. 06-UOA-0291. Introdu tionA metri is said to be Einstein if its Ri i urvature is proportional to themetri [5℄. Despite a long history of intense interest in the Einstein equationsmany mysteries remain. In high dimensions it is not known if there are anyobstru tions to the existen e of Einstein metri . There are 3-manifolds and 4-manifolds whi h do not admit Einstein metri s and the situation is espe iallydeli ate in the latter ase, see [38℄ for an overview of some re ent progress.Here we onsider a spe i(cid:28) weakening of the Einstein ondition. By its naturethis provides an alternative route to studying Einstein metri s but, beyondthis, there are several points whi h indi ate that it may be a useful stru turein its own right. On the one hand the weakening is very slight, in a sense thatwill soon be lear. On the other it allows in some interesting ases: at leastsome manifolds satisfying these equations do not admit Einstein metri s,whi h suggests a role as a uniformisation type ondition; it in ludes in anatural way Poin aré-Einstein stru tures and onformally ompa t Ri i-(cid:29)at asymptoti ally lo ally Eu lidean (ALE) spa es, and so Einstein metri s,Poin aré-Einstein stru tures and these ALE manifolds are spe ial ases of auniform generalising stru ture.On a Riemannian manifold ( M d , g ) ( d ≥ here and throughout) theS houten tensor P (or P g ) is a tra e adjustment of the Ri i tensor given by Ric g = ( d − P g + J g g J g is the metri tra e of P g . Thus a metri is Einstein if and only if thetra e-free part of P g is zero. We will say that ( M, g, s ) is a dire ted almostEinstein stru ture if s ∈ C ∞ ( M ) is a non-trivial solution to the equation(1.1) A ( g, s ) = 0 where A ( g, s ) := trace − f ree ( ∇ g ∇ g s + sP g ) . Here ∇ g is the Levi-Civita onne tion for g , and the (cid:16)tra e-free(cid:17) means thetra e free part with respe t to taking a metri tra e. This is a generalisationof the Einstein ondition; we will see shortly that, on the open set where s isnon-vanishing, g o := s − g is Einstein; here Einstein is for ed as a onsisten y ondition for a solution to (1.1). On the other hand if g is Einstein then (1.1)holds with s = 1 . Any attempt to understand the nature and extent of thisgeneralisation should in lude a des ription of the possible lo al stru tures ofthe s ale singularity set, that is the set Σ where s is zero (and where g o = s − g is unde(cid:28)ned). The main results in this arti le are some answers to thisquestion and the development of a onformal theory to relate, quite dire tly,the intrinsi geometri stru ture of the singularity spa e Σ to the ambientstru ture. If s solves (1.1) then so does − s , and where s is non-vanishingthese solutions determine the same Einstein metri . We shall say that amanifold ( M, g ) is almost Einstein if it admits a overing su h that on ea hopen set U of the over we have that ( U, g, s U ) is dire ted almost Einsteinand on overlaps U ∩ V we have either s U = s V or s U = − s V . Althoughthere exist almost Einstein spa es whi h are not dire ted [23℄, to simplifythe exposition we shall assume here that almost Einstein (AE) manifoldsare dire ted. (So usually we omit the term (cid:16)dire ted(cid:17) but sometimes it isin luded in Theorems for emphasis.) In any ase the results apply lo ally onalmost Einstein manifolds whi h are not dire ted.On an Einstein manifold ( M, g ) the Bian hi identity implies that the s alar urvature Sc g (i.e. the metri tra e of Ric ) is onstant. Thus simply requiringa metri to be s alar onstant is another weakening of the Einstein ondition.On ompa t, onne ted oriented smooth Riemannian manifolds this may bea hieved onformally: this is the out ome of the solution to the (cid:16)Yamabeproblem(cid:17) due to Yamabe, Trudinger, Aubin and S hoen [51, 49, 3, 46℄. Justas almost Einstein generalises the Einstein ondition, there is an orrespond-ing weakening of the onstant s alar urvature ondition as follows. We willsay that ( M, g, s ) is a dire ted almost s alar onstant stru ture if s ∈ C ∞ ( M ) is a non-trivial solution to the equation S ( g, s ) = constant where(1.2) S ( g, s ) = 2 d s ( J g − ∆ g ) s − | ds | g . Away from the zero set (whi h again we denote by Σ ) of s we have S ( g, s ) =Sc g o /d ( d − where g o := s − g . In parti ular, o(cid:27) Σ , S ( g, s ) is onstant ifand only if Sc g o is onstant. The normalisation is so that if g o is the metri of a spa e form then S ( g, s ) is exa tly the se tional urvature. We shall saythat a manifold ( M, g ) is almost S alar onstant (ASC) if it is equipped witha overing su h that on ea h open set U of the over we have that ( U, g, s U ) is dire ted almost s alar onstant, and on overlaps U ∩ V we have either s U = s V or s U = − s V . In fa t, in line with our assumptions above andunless otherwise mentioned expli itly, we shall assume below that any ASCstru ture is dire ted.lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 3As suggested above, losely related to these notions are ertain lasses of so alled onformally ompa t manfolds that have re ently been of onsiderableinterest. We re all how these manifolds are usually des ribed. Let M d bea ompa t smooth manifold with boundary Σ = ∂M . A metri g o on theinterior M + of M is said to be onformally ompa t if it extends (with somespe i(cid:28)ed regularity) to Σ by g = s g o where g is non-degenerate up to theboundary, and s is a non-negative de(cid:28)ning fun tion for the boundary (i.e. Σ is the zero set for s , and ds is non-vanishing along M ). In this situationthe metri g o is omplete and the restri tion of g to T Σ in T M | Σ determinesa onformal stru ture that is independent of the hoi e of de(cid:28)ning fun tion s ; then Σ with this onformal stru ture is termed the onformal in(cid:28)nity of M + . (This notion had its origins in the work of Newman and Penrose, seethe introdu tion of [37℄ for a brief review.) If the de(cid:28)ning fun tion is hosenso that | ds | g = 1 along M then the se tional urvatures tend to − at in(cid:28)nityand the stru ture is said to be asymptoti ally hyperboli (AH) (see [41℄ wherethere is a detailed treatment of the Hodge ohomology of these stru turesand related spe tral theory). The model is the Poin aré hyperboli balland thus the orresponding metri s are sometimes alled Poin aré metri s.Generalising the hyperboli ball in another way, one may suppose that theinterior onformally ompa t metri g o is Einstein with the normalisation Ric( g o ) = − ng o , where n = d − , and in this ase the stru ture is said to bePoin aré-Einstein (PE); in fa t PE manifolds are ne essarily asymptoti allyhyperboli . Su h stru tures have been studied intensively re ently in relationto the proposed AdS/CFT orresponden e of Malda ena [40, 50℄, relatedfundamental geometri questions [1, 2, 6, 11, 32, 33, 39, 43℄ and through onne tions to the ambient metri of Fe(cid:27)erman-Graham [16, 17℄.For simpli ity of exposition we shall restri t our attention to smooth AEand ASC stru tures ( M d , g, s ) ; that is ( M, g ) is a smooth Riemannian man-ifold of dimension d ≥ and s ∈ C ∞ ( M ) satis(cid:28)es either (1.1) (the AE ase)or (1.2) (for ASC). Let us write M ± for the open subset of M on whi h s is positive or, respe tively, negative and, as above, Σ for the s ale sin-gularity set. The (cid:28)rst main results (proved in Se tion 2) are the following lassi(cid:28) ations for the possible submanifold stru tures of Σ .Theorem 1.1. Let ( M, d g, s ) be a dire ted almost s alar onstant stru -ture with g positive de(cid:28)nite and M onne ted. If S ( g, s ) > then s isnowhere vanishing and ( M, g o ) has onstant s alar urvature d ( d − S ( g, s ) .If S ( g, s ) < then s is non-vanishing on an open dense set and Σ is ei-ther empty or else is a hypersurfa e; On M \ Σ , Sc g o is onstant and equals d ( d − S ( g, s ) . Suppose M is losed (i.e. ompa t without boundary) with S ( g, s ) < and Σ = ∅ . A onstant res aling of s normalises S ( g, s ) to − ,and then ( M \ M − ) is a (cid:28)nite union of onne ted AH manifolds. Similar for ( M \ M + ) .By hypersurfa e we mean a submanifold of odimension 1 whi h may in ludeboundary omponents. In the following we will say that an ASC stru tureis s alar positive, s alar (cid:29)at, or s alar negative if, respe tively, S ( g, s ) ispositive, zero, or negative.It seems that almost Einstein manifolds, in the generality we des ribehere, were introdu ed in [19℄ and it was observed there that PE manifolds Goverare a spe ial ase; this was explained in detail in [20℄. Here, among otherthings, we see that PE manifolds arise automati ally in the s alar negative(i.e. S ( g, s ) < ) ase.Theorem 1.2. Let ( M, g, s ) be a dire ted almost Einstein stru ture with g positive de(cid:28)nite and M onne ted. Then s is non-vanishing on an open denseset and ( M, g, s ) is almost s alar onstant. Writing Σ for the s ale singular-ity set, on M \ Σ , g o is Einstein with s alar urvature d ( d − S ( g, s ) . Thereare three ases: • If S ( g, s ) > then the s ale singularity set Σ is empty. • If S ( g, s ) = 0 then Σ is either empty or otherwise onsists of isolated pointsand these points are riti al points of the fun tion s ; in this ase for ea h p ∈ M with s ( p ) = 0 , the metri g o is asymptoti ally lo ally Eu lidean (ALE)near p and the Weyl, Cotton, and Ba h urvatures vanish at p . • If S ( g, s ) < then Σ is either empty or else is a totally umbilli hyper-surfa e. In parti ular on a losed S ( g, s ) = − almost Einstein manifold ( M \ M − ) is a (cid:28)nite union of onne ted Poin aré-Einstein manifolds. Sim-ilar for ( M \ M + ) .The Cotton and Ba h urvatures are de(cid:28)ned in, respe tively, (4.6) and(4.10) below. Using ompa tness, the last statement is an easy onsequen eof Proposition 3.7. That AE implies ASC is part of Theorem 2.3. Given thisseveral parts of the Theorem are immediate from Theorem 1.1 above. Theremaining parts of the Theorem summarise Theorem 3.1, Proposition 3.3,Proposition 3.6, and parts of Proposition 4.3 and Corollary 4.4. We shall saythat the ALE stru tures arising as here are onformally onformally ompa tbe ause of the obvious link the term as used above.The equation (1.2) is onformally ovariant in the sense that for any ω ∈ C ∞ ( M ) we have S ( g, s ) = S ( e ω g, e ω s ) . Similarly for (1.1) we have e ω A ( g, s ) = A ( e ω g, e ω s ) and so if ( M, g, s ) is almost Einstein then so is ( M, e ω g, e w s ) . Evidently the notions of ASC and AE stru ture pass to the onformal geometry by taking a quotient of the spa e of all su h stru turesby the equivalen e relation ( M, g, s ) ∼ ( M, e ω g, e w s ) . This is the point ofview we wish to take, throughout g is to be viewed as simply a representativeof its onformal lass. (This shows that we should really view the fun tion s as orresponding, via the density bundle trivialisation a(cid:27)orded by the metri g , to a onformal density σ of weight 1 on the onformal manifold ( M, [ g ]) ,and A as a 2-tensor taking values in this density bundle. We shall post-pone this move until Se tion 2.) The onformal equivalen e lass of ( g, s ) (under ( g, s ) ∼ ( e ω g, e ω s ) ) is a stru ture whi h generalises the notion of ametri . This suggests a de(cid:28)nition whi h is onvenient for our dis ussions. ARiemannian manifold equipped with the onformal equivalen e lass (in thissense) of ( g, s ) , and where s is nowhere vanishing on an open dense set, isa well de(cid:28)ned stru ture that we shall term an almost Riemannian manifold.Of ourse the zero set of s is onformally invariant and so is a preferred set Σ ⊂ M . An almost Riemannian stru ture with Σ = ∅ is simply a Rieman-nian manifold. Note that in the other ases S ( g, s ) smoothly extends to M the natural s alar Sc g o /n ( n + 1) whi h is only de(cid:28)ned on M \ Σ . Similarly A ( g, s ) smoothly extends sP g o , where P g o is the tra e free part of P g o . Thuseven though the metri g o = s − g is not de(cid:28)ned along Σ , nevertheless A ( g, s ) lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 5and S ( g, s ) are de(cid:28)ned globally (at least if we view A ( g, s ) as representing adensity valued tensor) and it is natural to think of these as urvature quan-tities on almost Riemannian stru tures. It turns out that AE manifolds,and also the ases of ASC manifolds overed in Theorem 1.1, are ne essarilyalmost Riemannian.The stru tures we onsider here have an elegant and al ulationally ef-fe tive formulation in terms of onformal tra tor al ulus. On Riemannianmanifolds the metri anoni ally determines a onne tion on the tangentbundle, the Levi-Civita onne tion. On onformal stru tures we lose thisbut there is a anoni al onformally invariant onne tion ∇ T on the (stan-dard onformal) tra tor bundle T , as des ribed in the next se tion. On ( M d , [ g ]) this is a rank ( d + 2) bundle that ontains a onformal twisting ofthe tangent bundle as a subquotient. The bundle T also has a ( onformallyinvariant) tra tor metri h , of signature ( d + 1 , , that is preserved by ∇ T .On a given onformal stru ture we may ask if there is parallel se tion of T ; that is a se tion I of T satisfying ∇ T I = 0 . In fa t, as we see below(following [4℄), this equation is simply a prolongation of (1.1). In parti ular,on any open set, solving ∇ T I = 0 is equivalent to solving (1.1) and thereis an expli it 1-1 relationship between solutions. (We shall write s I for thesolution of (1.1) given by a parallel tra tor I .) Thus an almost Einsteinstru ture is a triple ( M, [ g ] , I ) where I is parallel for the standard tra tor onne tion determined by the onformal stru ture [ g ] . Sin e the tra tor on-ne tion preserves the metri h , the length (squared) of I , whi h we denote bythe shorthand | I | := h ( I, I ) , is onstant on onne ted AE manifolds (andwe hen eforth assume M is onne ted). In fa t S ( g, s I ) = −| I | . There is ahgeneralising result for ASC manifolds, see Proposition 2.2.The geometri study of PE manifolds has been driven by a desire to relatethe onformal geometry of the onformal in(cid:28)nity to the metri geometryon the interior. We may obviously extend this programme to the s alarnegative (i.e. S ( g, s ) < ) almost Einstein stru tures. As indi ated above,this is a ore aim here and in our treatment (Se tions 4 and 6) the tra torstru tures play a key role. The (cid:28)rst key result is Theorem 4.5 whi h shows,for example, that Σ satis(cid:28)es a onformal analogue of the Riemannian totallygeodesi ondition: the intrinsi tra tor onne tion of (Σ , [ g Σ ]) exa tly agreeswith a restri tion of the ambient tra tor onne tion. In fa t the results arestronger. Summarising part of Theorem 4.5 with Corollary 6.4, along thes ale singularity set Σ of a s alar negative AE stru ture we also have thefollowing:Theorem 1.3. Ω( u, v ) = Ω Σ ( u, v ) along Σ where u, v ∈ Γ( T Σ) . In dimensions d = 4 we have the stronger result Ω( · , · ) = Ω Σ ( · , · ) along Σ , where here, by trivial extension, we view Ω Σ as a se tion of Λ T ∗ M ⊗ End T .While in dimensions d ≥ we also have ( d − W | Σ = ( d − W Σ , Goverwhere W is the prolonged onformal urvature quantity (4.9) and again atrivial extension is involved.Here Ω is the urvature of the tra tor onne tion for ( M, [ g ]) while Ω Σ isthe urvature of the tra tor onne tion for the intrinsi onformal stru tureof Σ . W is the natural onformally invariant tra tor (cid:28)eld equivalent (in di-mensions d = 4 ) to the urvature of the Fe(cid:27)erman-Graham (ambient) metri over ( M, [ g ]) , while W Σ is the same for (Σ , [ g Σ ]) . In Se tion 6 Theorem 6.1we also show that the Fe(cid:27)erman-Graham (obstru tion) tensor must vanishon the s ale singularity hypersurfa e of an almost Einstein stru ture. Analternative dire t proof that Σ is Ba h-(cid:29)at, when n = 4 , is given in Corol-lary 4.8. A key tool derived in Se tion 6 is Theorem 6.3 whi h onstru tsa Fe(cid:27)erman-Graham ambient metri , formally to all orders, for the even di-mensional onformal stru ture of a s ale singularity set; this onstru tionwas heavily in(cid:29)uen ed by the model in Se tion 5.1. An important and en-tral aspe t of the works [16℄ and [17℄ is the dire t relationship between theFe(cid:27)erman-Graham (ambient) metri for onformal manifolds (Σ , [ g Σ ]) andsuitably even smooth formal Poin aré-Einstein metri s, with (Σ , [ g Σ ]) as the onformal in(cid:28)nity (see espe ially [17, Se tion 4℄); in Se tion 5.2 there is somedis ussion of the meaning of even in this ontext. Here, in ontrast, we workin one higher dimension and exploit the use of the ambient metri for thePoin aré-Einstein (or AE) spa e M itself as tool for studying the boundary Σ (or s ale singularity set); in this ase we may work globally on M andwith not ne essarily even PE (or AE) metri s.In Se tion 4.4 we des ribe equations ontrolling (at least partially) the on-formal urvature of almost Einstein stru tures. Importantly these are givenin a way that should be suitable for setting up boundary problems along Σ based around the onformal urvature quantities. For example in Proposi-tion 4.6 we observe that in this sense the Yang-Mills equations, applied tothe tra tor urvature, give the natural onformal equations for 4-dimensionalalmost Einstein stru tures. The anologue for higher even dimensions is givenin Proposition 4.10. In all dimensions we have the following result.Theorem 1.4. Let ( M d , [ g ] , I ) be an almost Einstein manifold then I A D/ A W = 0 . The operator I A D/ A has the form σ ∆ + lower order terms ex ept along Σ in dimensions d = 6 . The statement here is mainly interesting in dimensions d ≥ and is a part of Theorem 4.7. Sin e for d ≥ we have ( d − W | Σ = ( d − W Σ , for Poin aré-Einstein manifolds (and more generally s alar negativeAE stru tures) the Theorem suggests a Diri hlet type problem with the onformal urvature W Σ of Σ as the boundary (hypersurfa e) data. Theoperator I A D/ A is well de(cid:28)ned on almost Einstein manifolds and is linked tothe s attering pi ture of [33℄ as outlined in Corollary 4.9.As mentioned, almost Einstein stru tures provide a generalisation of thenotions of Einstein, Poin aré-Einstein and ertain onformally ompa t ALEmetri s. Aside from providing a new and uniform perspe tive on these spe- ialisations, the AE stru tures provide a natural uniformisation type prob-lem. We may ask for example whether any losed smooth manifold admitsan almost Einstein stru ture. While it is by now a lassi al result [5℄ thatlmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 7the sphere produ ts S × S and S × S do not admit Einstein metri s it isshown in [23℄ that these both admit almost Einstein stru tures; in fa t we onstru t these expli itly as part of a general onstru tion of losed mani-folds with almost Einstein stru tures. In this arti le we make just a smalldis ussion of examples in Se tion 5. This in ludes the onformal sphere asthe key model. It admits all s alar types of almost Einstein stru ture andhas a entral role in the onstru tion of other examples in [23℄. (In fa t thestandard onformal stru ture on the sphere admits a ontinuous urve of AEstru tures whi h in ludes the standard sphere metri , the Eu lidean metri pulled ba k by stereographi proje tion as well as negative S ( g, s ) AE stru -tures with g o hyperboli o(cid:27) the singularity set. See Corollary 2.4 and the(cid:28)nal omments in Se tion 5.1.) We on lude in Se tion 5.2 with a dis ussionof examples found by a doubling onstru tion. Non-Einstein almost Ein-stein metri s turn up in the onstru tions and lassi(cid:28) ations by Derdzinskiand Mas hler of Kähler metri s whi h are (cid:16)almost everywhere(cid:17) onformal toEinstein by a non- onstant re aling fa tor, see e.g. [13, 14℄ and referen estherein. Some of their examples were inspired by onstru tions known forsome time, su h as [12, 44℄. Examples of non-Einstein S ([ g ] , I ) = 0 AEstru tures are disussed in [36℄.It should also be pointed out that many of the te hniques and resultswe develop apply in other signatures. However there are also fundamentaldi(cid:27)eren es in the ase of non-Riemannian signature and so we on(cid:28)ne thestudy to the positive de(cid:28)nite setting.Conversations with Mi hael Eastwood, Robin Graham, Felipe Leitner, andPaul-Andi Nagy have been mu h appre iated. It should pointed out that theexisten e of AE stru tures whi h are not dire ted was observed in the jointwork [23℄ with Leitner and this in(cid:29)uen ed the presentation here.2. Almost Einstein stru tures and onformal tra tor al ulusAs above let M be a smooth manifold, of dimension d ≥ , equippedwith a Riemannian metri g ab . Here and throughout we employ Penrose'sabstra t index notation. We write E a to denote the spa e of smooth se tionsof the tangent bundle T M on M , and E a for the spa e of smooth se tionsof the otangent bundle T ∗ M . (In fa t we will often use the same symbolsfor the bundles themselves. O asionally, to avoid any onfusion, we write Γ( B ) to mean the spa e of se tions of a bundle B .) We write E for thespa e of smooth fun tions and all tensors onsidered will be assumed smoothwithout further omment. An index whi h appears twi e, on e raised andon e lowered, indi ates a ontra tion. The metri g ab and its inverse g ab enable the identi(cid:28) ation of E a and E a and we indi ate this by raising andlowering indi es in the usual way.With ∇ a denoting the Levi-Civita onne tion for g ab , and using that thisis torsion free, the Riemann urvature tensor R abcd is given by ( ∇ a ∇ b − ∇ b ∇ a ) V c = R abcd V d where V c ∈ E c . GoverThis an be de omposed into the totally tra e-free Weyl urvature C abcd andthe symmetri S houten tensor P ab a ording to(2.1) R abcd = C abcd + 2 g c [ a P b ] d + 2 g d [ b P a ] c , where [ · · · ] indi ates antisymmetrisation over the en losed indi es. Thus P ab is a tra e modi(cid:28) ation of the Ri i tensor Ric ab = R cacb : Ric ab = ( n − P ab + J g ab , J := P aa . In denoting su h urvature quantities we may write e.g.
Ric g or simply Ric depending on whether there is a need to emphasise the metri involved. Alsoabstra t indi es will be displayed or suppressed as required for larity.Under a onformal res aling of the metri g g o = s − g, with s ∈ E non-vanishing, the Weyl tensor C abcd is is un hanged (and so wesay the Weyl tensor is onformally invariant) whereas the S houten tensortransforms a ording to(2.2) P g o ab = P gab + s − ∇ a ∇ b s − g cd s − ( ∇ c s )( ∇ d s ) g ab . Taking, via g o , a tra e of this we obtain J g o = s J g − s ∆ s − d | ds | g , where the ∆ is the (cid:16)positive energy(cid:17) Lapla ian. Note that the right hand sideis d S ( g, s ) , with S ( g, s ) as de(cid:28)ned in (1.2). Clearly this is well de(cid:28)ned forsmooth s even if s may be zero at some points. On the other hand the righthand side above (and hen e S ( g, s ) ) is learly invariant under the onformaltransformation ( g, s ) ( e ω g, e ω s ) : this is true away from the zeros of s sin e there J g o depends only on the 2-jet of g o = s − g , but the expli it onformal transformation of the right hand side is evidently polynomial in e ω and its 2-jet.Let us digress to prove Theorem 1.1 sin e it illustrates how an almostRiemannian stru ture may arise immediately from a formula polynomial inthe jets of s .Proof of Theorem 1.1: Under a dilation g µg ( µ ∈ R + ) we have S ( g, s ) µ − S ( g, s ) , so to prove the Theorem we may onsider just the ases S ( g, s ) = 1 and S ( g, s ) = − . Suppose that S ( g, s ) = 1 then if p ∈ M were to be a point where s p = 0 then at p we would have −| ds | g whi hwould be a ontradi tion. Suppose that S ( g, s ) = − . Then at any point p ∈ M where s p = 0 we have | ds | g = 1 . For the last statement of theTheorem assume that M is losed and the s ale singularity set Σ is notempty. Then Σ is a hypersurfa e whi h separates M a ording to the sign of s . The restri tion of g o to the interior of M \ M − (i.e. to M + ) is onformally ompa t sin e the restri tion of g to M \ M − extends s g o smoothly tothe boundary. Finally ( M \ M − , g, s ) is AH sin e | ds | g = 1 along Σ . By ompa tness this onsists of a (cid:28)nite union of onne ted AH omponents.The same analysis applies to M \ M + . (cid:3) .Note that although onstant S ( g, s ) is a weakening of the onstant s alar urvature ondition, the equation (1.2) is quite restri tive. For example, itlmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 9is evident that on losed manifolds with negative Yamabe onstant there areno non-trivial solutions with S ( g, s ) a non-negative onstant.The tensor A ( g, s ) de(cid:28)ned in the Introdu tion should be ompared to thetra e free part of the right hand side of (2.2) above. Arguing as for S ( g, s ) above, or by dire t al ulation, one (cid:28)nds that under ( g, s ) ( e ω g, e ω s ) wehave A ( g, s ) e ω A ( g, s ) as mentioned earlier. So both the AE onditionand the more general ASC ondition are best treated as stru tures on a onformal manifold. To obtain a lean treatment it is most e(cid:30) ient to drawin some standard obje ts from onformal geometry; for these further detailsand ba kground may be found in [10, 26℄. Clearly we may view a onformalstru ture on M is a smooth ray subbundle Q ⊂ S T ∗ M whose (cid:28)bre over x onsists of onformally related metri s at the point x . The prin ipal bundle π : Q → M has stru ture group R + , and so ea h representation R + ∋ x x − w/ ∈ End( R ) indu es a natural line bundle on ( M, [ g ]) that we termthe onformal density bundle E [ w ] . We shall write E [ w ] for the spa e ofse tions of this bundle. Note E [ w ] is trivialised by a hoi e of metri g fromthe onformal lass, and we write ∇ for the onne tion orresponding tothis trivialisation. It follows immediately that (the oupled) ∇ a preservesthe onformal metri . (Note on a (cid:28)xed onformal stru ture the onformaldensities bundle E [ − n ] may be identi(cid:28)ed in an obvious way with appropriatepowers of the 1-density bundle asso iated to the frame bundle through therepresentation | det( ) | − . See e.g. [10℄. Via this the onne tion we de(cid:28)nedon E [ w ] agrees with the usual Levi-Civita onne tion.)We write g for the onformal metri , that is the tautologi al se tion of S T ∗ M ⊗E [2] determined by the onformal stru ture. This will be hen eforthused to identify T M with T ∗ M [2] even when we have (cid:28)xed a metri from the onformal lass. (For example, with these onventions the Lapla ian ∆ isgiven by ∆ = − g ab ∇ a ∇ b = −∇ b ∇ b .) Although this is on eptually valuableand signi(cid:28) antly simpli(cid:28)es many al ulations, it is, however, a point wherethere is potential for onfusion. For example in the below, when we write J or J g we mean g ab P ab where P is the S houten tensor for some metri g .Thus J is a se tion of E [ − (whi h depends on g ).In this pi ture to study the ASC ondition we repla e s ∈ E with a se tion σ ∈ E [1] in (1.2) to obtain(2.3) S ([ g ] , σ ) = 2 d σ ( J g − ∆ g ) σ − |∇ σ | g , where we have written |∇ σ | g as a brief notation for g − ( ∇ σ, ∇ σ ) . When the onformal stru ture is (cid:28)xed we shall often denote the quantity displayed bysimply S ( σ ) . Similarly the onformally invariant version of A is the 2-tensorof onformal weight 1 given by A ([ g ] , σ ) := trace − f ree ( ∇ a ∇ b σ + P ab σ ) , again we may write simply A ( σ ) .The A ([ g ] , σ ) = 0 equation (i.e. (1.1)) be omes(2.4) ∇ a ∇ b σ + P ab σ + ρ g ab = 0 ρ is an unknown density (in E [ − ) to a ommodate the tra e-part.Here ∇ and P are given with respe t to some metri g in the onformal lass,but the equation is invariant under onformal res aling.We may repla e (2.4) with the equivalent (cid:28)rst order system ∇ a σ − µ a = 0 , and ∇ a µ b + P ab + g ab ρ = 0 , where µ a ∈ E a [1] := E a ⊗ E [1] . Di(cid:27)erentiating the se ond of these and onsidering two possible ontra tions yields ∇ a ρ − P ab µ b = 0 , when e we see that the system has losed up linearly. The equation (2.4) isequivalent to a onne tion and a parallel se tion for this; on any open set in M , a solution of (2.4) is equivalent to I := ( σ, µ a , ρ ) ∈ E [1] ⊕ E a [1] ⊕ E [ − satisfying ∇ T I = 0 where(2.5) ∇ T a σµ b ρ := ∇ a σ − µ a ∇ a µ b + g ab ρ + P ab σ ∇ a ρ − P ab µ b . The onne tion ∇ T onstru ted here (following [4℄) is the normal on-formal tra tor onne tion. We will often write simply ∇ for this when themeaning is lear by ontext. This is onvenient sin e we will ouple thetra tor onne tion to the Levi-Civita onne tion.Let us write J k E [1] for the bundle of k-jets of germs of se tions of E [1] .Considering, at ea h point of the manifold, se tions whi h vanish to (cid:28)rstorder at the given point point reveals a anoni al sequen e, → S T ∗ M ⊗ E [1] → J E [1] → J E [1] → . This is the jet exa t sequen e at 2-jets. Via the onformal metri g , thebundle of symmetri ovariant 2-tensors S T ∗ M de omposes dire tly intothe tra e-free part, whi h we will denote S T ∗ M , and a pure tra e partisomorphi to E [ − , hen e S T ∗ M ⊗ E [1] = ( S T ∗ M ⊗ E [1]) ⊕ E [ − . Thestandard tra tor bundle T may de(cid:28)ned as the quotient of J E [1] by the imageof S T ∗ M ⊗ E [1] in J E [1] . By onstru tion this is invariant, it depends onlyon the onformal stru ture. Also by onstru tion, it is an extension of the1-jet bundle(2.6) → E [ − X → T → J E [1] → . The anoni al homomorphism X here will be viewed as a se tion of T [1] = T ⊗ E [1] and, with the jet exa t sequen e at 1-jets, ontrols the (cid:28)ltrationstru ture of T .Next note that there is a tautologi al operator D : E [1] → T whi h is sim-ply the omposition of the universal 2-jet di(cid:27)erential operator j : E [1] → Γ( J E [1]) followed by the anoni al proje tion J E [1] → T . On the otherhand, via a hoi e of metri g , and the Levi-Civita onne tion it deter-mines, we obtain a di(cid:27)erential operator E [1] → E [1] ⊕ E [1] ⊕ E [ − by σ ( σ, ∇ a σ, d (∆ − J ) σ ) and this obviously determines an isomorphism(2.7) T g ∼ = E [1] ⊕ T ∗ M [1] ⊕ E [ − . lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 11In the following we shall frequently use (2.7). Sometimes this will be withoutany expli it omment but also we may write for example t g = ( σ, µ a , ρ ) ,or alternatively [ t ] g = ( σ, µ a , ρ ) , to mean t is an invariant se tion of T and ( σ, µ a , ρ ) is its image under the isomorphism (2.7). Changing to a onformally related metri b g = e ω g ( ω a smooth fun tion) gives a di(cid:27)erentisomorphism, whi h is related to the previous by the transformation formula(2.8) \ ( σ, µ b , ρ ) = ( σ, µ b + σ Υ b , ρ − g bc Υ b µ c − σ g bc Υ b Υ c ) , where Υ := dω . It is straightforward to verify that the right-hand-side of(2.5) also transforms in this way and hen e ∇ T gives a onformally invariant onne tion on T whi h we shall also denote by ∇ T . This is the tra tor onne tion. There is also a onformally invariant tra tor metri h on T given (as a quadrati form) by(2.9) ( σ, µ, ρ ) g − ( µ, µ ) + 2 σρ . This is preserved by the onne tion and learly has signature ( d + 1 , .Let us return to our study of the equations (2.4) and (1.2). First observethat, given a metri g , through (2.7) the tautologi al invariant operator D from above is given by the expli it formula(2.10) D : E [1] → T σ ( σ, ∇ a σ, d (∆ σ − J σ )) . This is a di(cid:27)erential splitting operator, sin e it is inverted by the anoni altra tor X : h ( X, Dσ ) = σ . (To see this one may use that in terms of thesplitting (2.7) X = (0 , , .) If a standard tra tor I satis(cid:28)es I = Dσ forsome σ ∈ E [1] then σ = h ( X, I ) and we shall term I a s ale tra tor. For thestudy of s ale tra tors the following result is useful.Lemma 2.1. For σ a se tion of E [1] we have(2.11) | Dσ | := h ( Dσ, Dσ ) = 2 d σ (∆ g − J g ) σ + |∇ g σ | g , where |∇ σ | g means g ab ( ∇ a σ ) ∇ b σ . In parti ular, if σ ( p ) = 0 , p ∈ M , then | Dσ | ( p ) = |∇ σ | g ( p ) . Proof: This follows easily from the formulae (2.9) and (2.10). (cid:3)
Using Lemma 2.1, we have the following.Proposition 2.2. If I is a s ale tra tor then | I | = − S ( σ ) , where σ = h ( X, σ ) . In parti ular o(cid:27) the zero set of σ we have | I | = − d J g o where g o = σ − g and J g o is the g o tra e of P g o . An ASC stru ture is a onformal manifold ( M, [ g ]) equipped with a s ale tra tor of onstant length.Proof: Everything is lear ex ept the point made in the se ond display.Re all that now, in ontrast to the Introdu tion, J g o denotes g ab P g o ab . So,writing g o for the inverse to g o , we have σ J g o = σ g ab P g o ab = g abo P ab =: J g o , g o tra e of the S houten tensor P g o . On the other hand,away from the zero set of σ , we may al ulate in the s ale σ and we have ∇ g o σ = 0 , when e − σ J g o /d is exa tly the right hand side of (2.11). (cid:3) Now olle ting our observations we obtain the basi elements of the tra torpi ture for AE stru tures, as follows.Theorem 2.3. A dire ted almost Einstein stru ture is a onformal manifold ( M n +1 , [ g ]) equipped with a parallel (standard) tra tor I = 0 . The mappingfrom non-trivial solutions of (2.4) to parallel tra tors is by σ Dσ withinverse I σ := h ( I, X ) . If I = 0 is parallel and σ := h ( I, X ) then thestru ture ( M, [ g ] , σ ) is ASC with S ([ g ] , σ ) = −| I | . On the open set where σ is nowhere vanishing g o := σ − g is Einstein with Ric g o = n | I | g o .Proof: The (cid:28)rst observation is immediate from the onstru tion in (2.5) ofthe tra tor onne tion as a prolongation of the equation (2.4) for an almostEinstein stru ture.Next observe that if I g = ( σ, µ a , ρ ) is a parallel se tion for ∇ T then itfollows immediately from the formula (2.5) that ne essarily(2.12) (cid:0) σ, µ a , ρ (cid:1) = ( σ, ∇ a σ, d (∆ σ − J σ )) , that is I is a s ale tra tor, I = Dσ . From the formula for the tra tor metri it follows that σ = h ( X, I ) .Sin e the tra tor onne tion preserves the tra tor metri it follows thatif I is a parallel tra tor then | I | := h ( I, I ) is onstant. Thus an almostEinstein stru ture is ASC as laimed.For the (cid:28)nal statement we use that I parallel implies that σ satis(cid:28)es(2.4). On the set where σ is nowhere vanishing we may use the metri g o = σ − g . The orresponding Levi-Civita onne tion annihilates σ andthen (2.4) asserts that P g o is tra e-free. (cid:3) In view of the the Theorem we shall often use the notation ( M, [ g ] , I ) todenote a dire ted almost Einstein manifold. In this ontext I should betaken as parallel and non-zero.There is a useful immediate onsequen e of the Theorem, as follows.Corollary 2.4. On a (cid:28)xed onformal stru ture ( M, [ g ]) the set of dire ted AEstru tures is naturally a ve tor spa e with the origin removed. In parti ularif I and I are two linearly independent dire ted AE stru tures then for ea h t ∈ R I t := (sin t ) I + (cos t ) I is a dire ted AE stru ture. In this ase given p ∈ M there is t ∈ R so that σ t ( p ) := h ( X, I t ) p = 0 .One might suspe t that generi ally non-s alar positive AE manifolds willhave non-empty s ale singularity sets. The Corollary shows that this er-tainly is the ase on a (cid:28)xed onformal stru ture with two linearly independentAE stru tures.lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 133. Classifi ation of the s ale singularity setGiven a standard tra tor I and σ := h ( X, I ) let us write S ( I ) as analternate notation for S ( σ ) . As before we write Σ := { p ∈ M | σ ( p ) = 0 } and term this the s ale singularity set of I ; this is the set where g o = σ − g isunde(cid:28)ned. In this se tion we shall establish the following, and then ompleteto a proof of Theorem 1.2.Theorem 3.1. Let ( M, [ g ] , I ) be an almost Einstein stru ture. There arethree ases: • | I | < , whi h is equivalent to S ( I ) > , then Σ is empty and ( M, σ − g ) is Einstein with positive s alar urvature; • | I | = 0 , whi h is equivalent to S ( I ) = 0 , then Σ is either empty or onsistsof isolated points, and ( M \ Σ , σ − g ) is Ri i-(cid:29)at; • | I | > , whi h is equivalent to S ( I ) < , then the s ale singularity set Σ is either empty or else is a totally umbilli hypersurfa e, and ( M \ Σ , σ − g ) is Einstein of negative s alar urvature.The urvature statements follow from Theorem 2.3. Also from there we havethat an AE manifold is ASC. Thus from Theorem 1.1 we have at on e boththe (cid:28)rst result and also that if, alternatively, | I | > then the singularity setis either empty or is a hypersurfa e. The proof is ompleted via Propositions3.3 and 3.6 below.We shall make a general observation whi h sheds light on the s alar (cid:29)at ase. From Theorem 2.3, I parallel implies I = Dσ , for some density σ in E [1] . An obvious question is whether, at any point p ∈ M , we mayhave j p σ = 0 , i.e. whether the 1-jet of σ may vanish at p . Evidently thisis impossible if | I | = 0 . We observe here ( f. [19℄) that, in any ase, if I = Dσ = 0 is parallel then the zeros of j σ are isolated. In fa t we have aslightly stronger result. As usual here we write σ = h ( X, I ) .Lemma 3.2. Suppose that I = 0 is parallel and j p σ = 0 . Then there is aneighbourhood of p su h that, in this neighbourhood, σ is non-vanishing awayfrom p .Proof: Suppose that I = 0 is parallel and j p σ = 0 . Sin e I is paral-lel I = Dσ . This with j p σ = 0 implies that, at p , and in the s ale g ,we have I g = (0 , , ρ ) for some density ρ with ρ ( p ) = 0 . Thus from(2.5) (or equivalently (2.4)) we have ( ∇ a ∇ b σ )( p ) = − ρ ( p ) g ab ( p ) . Trivi-alising the density bundles using the metri g the latter is equivalent to ( ∇ a ∇ b s )( p ) = − r ( p ) g ab ( p ) where the smooth fun tion r satis(cid:28)es r ( p ) = 0 .(Here we use that g = τ − g for some non-vanishing τ in E [1] and s = τ − σ while r = τ ρ . Then sin e ∇ is the Levi-Civita for g we have ∇ τ = 0 .) So, interms of oordinates based at p , the (cid:28)rst non-vanishing term in the Taylorseries for s (based at p ) is − rg ij x i x j . (cid:3) Note that an ASC stru ture is s alar (cid:29)at if and only if g o is Ri i-(cid:29)at on M \ Σ . In the following σ := h ( X, I ) .4 GoverProposition 3.3. If ( M, [ g ] , I ) is an ASC stru ture with j p σ = 0 , at somepoint p , then ( M, [ g ] , I ) is s alar (cid:29)at. Conversely if ( M, [ g ] , I ) is a s alar (cid:29)atASC stru ture then, at any p ∈ M with σ ( p ) = 0 we have j p σ = 0 .If ( M, [ g ] , I ) is a s alar (cid:29)at AE stru ture then, at any p ∈ M with σ ( p ) = 0 we have j p σ = 0 and j p σ = 0 . For any AE manifold the s ale singularity set onsists of isolated points.Proof: Sin e by de(cid:28)nition I = Dσ , from Lemma 2.1 it is immediate that,at any point p with σ ( p ) = 0 , we have S ( σ )( p ) = 0 if and only if j p σ = 0 .(Alternatively this is visible dire tly from the formula 2.3.) The (cid:28)rst twostatements follow immediately, as by de(cid:28)nition S ( σ ) is onstant on an ASCmanifold.Now we onsider AE manifolds. These are ASC and so we have the (cid:28)rstresults. Sin e I is parallel, we have I = Dσ . If an AE manifold is s alar (cid:29)atthen, at a point p where σ ( p ) = 0 , we have j p σ = 0 and so from (2.12) thetra tor I is of the form I g = (0 , , ρ ) at p . On the other hand, sin e I = 0 isparallel, it follows that Dσ = I is nowhere zero on M . Hen e (sin e D is ase ond order di(cid:27)erential operator) j σ is non-vanishing. In fa t, from (2.10),at any point p where j p σ vanishes we have ρ ( p ) = d (∆ σ )( p ) = 0 . The laststatement is now an immediate onsequen e of Lemma (3.2). (cid:3) Remark: Note that j p σ = 0 means that when we work in terms of aba kground metri g we have j p s = 0 for the fun tion s orresponding to σ and so p is a riti al point of s . In fa t it is already lear from (1.2) that,even for ASC stru tures, if S ( g, s ) = 0 then s p = 0 implies p is a riti alpoint.3.1. Conformal hypersurfa es and the s ale singularity set. Let us(cid:28)rst re all some fa ts on erning general hypersurfa es in a onformal man-ifold ( M d , [ g ]) , d ≥ . If Σ is a boundary omponent of a Riemannian (or onformal) manifold then, without further omment, we will assume thatthe onformal stru ture extends smoothly to a ollar of the boundary. Ourresults will not depend on the hoi e of extension. So in the following wesuppose that Σ is an embedded odimension 1 submanifold of M .Let n a be a se tion of E a [1] su h that, along Σ , we have | n | g := g ab n a n b =1 . Note that the latter is a onformally invariant ondition sin e g − has onformal weight − . Now in the s ale g , the mean urvature of Σ is givenby H g = 1 n − (cid:0) ∇ a n a − n a n b ∇ a n b (cid:1) , as a onformal − -density. This is independent of how n a is extended o(cid:27) Σ . Now under a onformal res aling, g b g = e ω g , H transforms to H b g = H g + n a Υ a . Thus we obtain a onformally invariant se tion N of T | Σ N g = n a − H g , lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 15and from (2.9) h ( N, N ) = 1 along Σ . Obviously N is independent of any hoi es in the extension of n a o(cid:27) Σ . This is the normal tra tor of [4℄ andmay be viewed as a tra tor bundle analogue of the unit onormal (cid:28)eld fromthe theory of Riemannian hypersurfa es.Re all that a point p in a hypersurfa e is an umbilli point if at that pointthe se ond fundamental form is tra e free, this is a onformally invariant ondition. A hypersurfa e is totally umbilli if this holds at all points. Dif-ferentiating N tangentially along Σ using ∇ T , dire tly from (2.5) we obtainthe following result.Lemma 3.4. If the normal tra tor N is parallel, with respe t to ∇ T , alonga hypersurfa e Σ then the hypersurfa e Σ is totally umbilli .In fa t onstan y of N along a hypersurfa e is equivalent to total umbilli ity.This is (Proposition 2.9) from [4℄.Now let us return to the study of ASC and AE stru tures. First we see thatthe normal tra tor is linked, in an essential way, to the ambient geometryo(cid:27) the hypersurfa e.Proposition 3.5. Let ( M d , [ g ] , I ) be a s alar negative ASC stru ture withs ale singularity set Σ = ∅ and | I | = 1 . Then, with N denoting the normaltra tor for Σ , we have N = I | Σ .Proof: As usual let us write σ := h ( X, I ) . By de(cid:28)nition I = Dσ g = σ ∇ a σ d (∆ σ − J σ ) . Let us write n a := ∇ a σ . Along Σ we have σ = 0 , therefore I | Σ g = n a d ∆ σ , and from Lemma 2.1 | n | g = 1 , sin e | I | = 1 . So n a | Σ is a onformal weight1 onormal (cid:28)eld for Σ .Next we al ulate the mean urvature H in terms of σ . Re all ( d − H = ∇ a n a − n a n b ∇ b n a , on Σ . We al ulate the right hand side in a neighbourhoodof Σ . Sin e n a = ∇ a σ , we have ∇ a n a = − ∆ σ . On the other hand n a n b ∇ b n a = 12 n b ∇ b ( n a n a ) = 12 n b ∇ b (1 − d σ ∆ σ + 2 d J σ ) , where we used that | Dσ | = 1 . Now along Σ we have n a n a = n a ∇ a σ ,and so there this simpli(cid:28)es to n a n b ∇ b n a = − d ∆ σ. Putting these results together, we have ( d − H = 1 d (1 − d )∆ σ ⇒ H = − d ∆ σ . I | Σ g = n a − H , as laimed. (cid:3) A onsequen e for AE stru tures follows easily.Proposition 3.6. Let ( M d , [ g ] , I ) be a s alar negative almost Einstein stru -ture with s ale singularity set Σ = ∅ and | I | = 1 . Then Σ is a totally umbilli hypersurfa e with I | Σ = N , the normal tra tor for Σ .Proof: Sin e an AE stru ture ( M, [ g ] , I ) is ASC it follows from Proposition3.5 above that along the singularity hypersurfa e I agrees with the normaltra tor N . On the other hand, sin e I is parallel everywhere, it follows that N is parallel along Σ and so, from Lemma 3.4, Σ is totally umbilli . (cid:3) Proposition 2.8 of LeBrun's [37℄ also gives a proof that the onformalin(cid:28)nity of a PE metri is totally umbilli .Proof of Theorem 3.1: The remaining point is to show that if ( M, [ g ] , I ) is AE with | I | > and a singularity hypersurfa e Σ , then this is totallyumbilli . This is immediate from the previous Proposition as multiplying I with a positive onstant yields a yields a parallel tra tor with the samesingularity set. (cid:3) Most of Theorem 1.2 is simply repa kaging of the tra tor based statementsin Theorem 3.1 above. To omplete the proof of the former we simply needto des ribe PE manifolds in the same language, and this is our (cid:28)nal aim forthis se tion.Proposition 3.7. Suppose that M is a ompa t manifold with boundary Σ , and ( M, [ g ] , I ) is an almost Einstein stru ture with | I | = 1 , and su hthat the s ale singularity set is Σ . Then ( M, [ g ] , I ) is a Poin aré-Einsteinmanifold with the interior metri g o = σ − g , where σ := h ( X, I ) . ConverselyPoin aré-Einstein manifolds are s alar negative almost Einstein stru tures.Proof: Suppose that ( M, [ g ] , I ) is an AE stru ture as des ribed. Sin e AEmanifolds are ASC, with the parallel tra tor I giving the s ale tra tor of theASC stru ture, it follows from Theorem 1.1 that ( M, [ g ] , σ ) is AH. But I parallel means that g o = σ − g is Einstein on M \ Σ , and there | I | = 1 isequivalent to Ric( g o ) = − ng o . The onverse dire tion is also straightforward,or see [20℄. (cid:3)
4. Conformal geometry of Σ versus onformal geometry of M Here for almost Einstein manifolds we shall derive basi equations satis(cid:28)edby the onformal urvatures. In parti ular for Poin aré-Einstein manifolds,and more generally for s alar negative almost Einstein manifolds, we shallstudy the relationship between the onformal geometry of M and the intrinsi onformal geometry of the s ale singularity set Σ . Sin e Σ is a hypersurfa e,a (cid:28)rst step is to understand the onformal stru ture indu ed on an arbitrarylmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 17hypersurfa e in a onformal manifold and in parti ular the relationship be-tween the intrinsi onformal tra tor bundle of Σ and the ambient tra torbundle of M . This is the subje t of Se tion 4.1. On the other hand wehave already observed that on s alar negative AE manifolds the singularityset is umbilli . So the main aim of this se tion is to deepen this pi ture.We shall see that the along the singularity hypersurfa e the intrinsi tra -tor onne tion ne essarily agrees with an obvious restri tion of the ambienttra tor onne tion. This has immediate onsequen es for the relationshipbetween the intrinsi and ambient onformal urvature quantities, but weare able to also show that there is an even stronger ompatibility betweenthe onformal urvatures of (Σ , [ g Σ ]) and those of ( M, [ g ]) . Finally we shallderive equations on the latter that partly establish a Diri hlet type problembased dire tly on the onformal urvature quantities.4.1. Conformal hypersurfa es. Here we revisit ( f. Se tion 3.1) the studyof a general hypersurfa e Σ in a onformal manifold ( M d , [ g ]) , d ≥ . Thistime our aim is to see, in this general setting, how the onformal stru tureof the hypersurfa e is linked that of the ambient spa e.With respe t to the embedding map, ea h metri g from the onformal lass on M pulls ba k to a metri g Σ on Σ . Thus the ambient onformalstru ture of M indu es a onformal stru ture [ g Σ ] on Σ n ( n + 1 = d ); weshall refer to this as the intrinsi onformal stru ture of Σ . Given the re-lationship of the intrinsi and ambient onformal stru tures it follows easilythat the intrinsi onformal density bundle of weight w , E Σ [ w ] is anoni allyisomorphi to E [ w ] | Σ and we shall no longer distinguish these. It is also lear that sin e g Σ is determined by g the trivialisations they indu e on,respe tively, E Σ [ w ] and E [ w ] are onsistent. In parti ular the Levi-Civita onne tion on E Σ [ w ] from g Σ agrees with the restri tion of the onne tionon E [ w ] arising from the trivialisation due to g .If n ≥ then (Σ , [ g Σ ]) has an intrinsi tra tor bundle T Σ . We want torelate this to T along Σ . Note that T Σ has a anoni al rank n + 2 subbundle,viz. N ⊥ the orthogonal omplement (with respe t to h ) of the normal tra tor N . As noted in [7℄, there is a anoni al isomorphism(4.1) N ⊥ ∼ = −→ T Σ . To see this let n a denote a weight 1 onormal (cid:28)eld along Σ . There is a anoni al in lusion of T Σ in T M | Σ and we identify T ∗ Σ with the annihilatorsubbundle in T ∗ M | Σ of n a . These identi(cid:28) ations do not require hoosing ametri from the onformal lass. Now al ulating in a s ale g on M , T andhen e also N ⊥ , de omposes into a triple via (2.7). Then the mapping of theisomorphism is ( f. [34℄)(4.2) [ N ⊥ ] g ∋ σµ b ρ σµ b − Hn b σρ + H σ ∈ [ T Σ ] g Σ where, as usual, H denotes the mean urvature of Σ in the s ale g and g Σ is the pullba k of g to Σ . Sin e ( σ, µ b , ρ ) is a se tion of [ N ⊥ ] g we have n a µ a = Hσ . Using this one easily veri(cid:28)es that the mapping is onformally8 Goverinvariant: If we transform to b g = e ω g , ω ∈ E , then ( σ, µ b , ρ ) transformsa ording to (2.8). Using that b H = H + n a Υ a one al ulates that the imageof ( σ, µ b , ρ ) (under the map displayed) transforms by the intrinsi version of(2.8), that is by (2.8) ex ept where Υ a is repla ed by Υ Σ a = Υ a − n a n b Υ b (whi h on Σ agrees with d Σ ω , the intrinsi exterior derivative of ω ). Thissignals that the expli it map displayed des ends to a onformally invariantmap (4.1).So far we understand the tra tor bundle on Σ for n ≥ . In the aseof n = 2 , Σ does not in general have a preferred intrinsi onformal tra tor onne tion. There is mu h to be said in this ase but for our urrent purposesit will be most e onomi al to pro eed as follows. We shall de(cid:28)ne T Σ to bethe orthogonal omplement of N in T | Σ and in any dimension d ≥ letus write Proj Σ : T | Σ → T Σ for the orthogonal proje tion a(cid:27)orded by N .Then for d = 3 , equivalently n = 2 , we de(cid:28)ne the tra tor onne tion on Σ to be the orthogonal proje tion of the ambient tra tor onne tion. That is,working lo ally, for v ∈ Γ( T Σ) and T ∈ T Σ = N ⊥ we extend these smoothlyto v ∈ Γ( T M ) and T ∈ T . Then we de(cid:28)ne ∇ T Σ v T := Proj Σ ( ∇ T v T ) along Σ .It is veri(cid:28)ed by standard arguments that this is independent of the extension hoi es and de(cid:28)nes a onne tion on T Σ .Finally we observe a useful alternative approa h to the arguments abovevia a result that, for other purposes, we will all on later.Proposition 4.1. Let Σ be an orientable hypersurfa e in an orientable on-formal manifold ( M, [ g ]) . In a neighbourhood of Σ there is a metri b g in the onformal lass so that Σ is minimal, i.e. H b g = 0 . Proof: For simpli ity let al ulate in the metri g and write H g to be themean urvature of Σ as a fun tion along Σ . Take any smooth extension ofthis to a fun tion on M . By a standard argument one an show that in aneighbourhood of Σ there is a normal de(cid:28)ning fun tion s for Σ , that is Σ is the zero set of s , and along Σ the 1-form ds satis(cid:28)es | ds | g = 1 . Then n a := g ab ∇ b s is a unit normal ve tor (cid:28)eld along Σ . Re all the onformaltransformation of the mean urvature: If b g = e ω g , for some ω ∈ E then e ω H b g = H g + n a Υ a = H g + n a ∇ a ω . Thus if we take ω := − sH g then H b g = 0 . (cid:3) Dropping the `hat' on b g , we see that with su h g (satisfying H g = 0 ) themap (4.2) simpli(cid:28)es signi(cid:28) antly in this normalisation; the splittings of N ⊥ and T Σ then agree in the (cid:16)obvious way(cid:17). This is onsistent with onformaltransformation: The ondition H = 0 does not (cid:28)x the representative metri g , even along Σ . For example at the 1-jet level the remaining freedom along M is to onformally res ale by g e ω g where n a ∇ a ω = 0 . This is exa tlyas required to preserve the agreement of the splittings of N ⊥ and T Σ . In fa tthis was the point of view taken in [7℄. From there one easily re overs theformula (4.2).Finally we note here that the res aling involved in the proof of the propo-sition above is global and espe ially natural in the ase of dire ted ASC andAE stru tures.Corollary 4.2. Let ( M, [ g ] , I ) be a dire ted s alar negative ASC manifoldwith a s ale singularity set. Then there is a metri b g ∈ [ g ] with respe t tolmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 19whi h Σ is a minimal hypersurfa e. In parti ular if ( M, [ g ] , I ) is a dire tedAE manifold then Σ is totally geodesi with respe t to b g .Proof: Suppose that σ is the onformal weight 1 density de(cid:28)ning a S ( σ ) = − ASC stru ture with a non-trivial s ale singularity hypersurfa e Σ . Write H g (now as a − density) for the mean urvature of Σ with respe t to anarbitrary ba kground metri g and extend this smoothly to M . Then Σ hasmean urvature zero with respe t to the metri b g = e ω g where ω := − H g σ .For the last statement we re all that if σ satis(cid:28)es (2.4) then Σ is totallyumbilli and this is a onformally invariant ondition. (cid:3) ( M d , [ g ]) .It will be onvenient to introdu e the alternative notation E A for the tra -tor bundle T and its spa e of smooth se tions. Here the index indi ates anabstra t index in the sense of Penrose and so we may write, for example, V A ∈ E A to indi ate a se tion of the standard tra tor bundle. Using theabstra t index notation the tra tor metri is denoted h AB with inverse h BC .These will be used to lower and raise indi es in the usual way.In omputations, it is often useful to introdu e the `proje tors' from E A to the omponents E [1] , E a [1] and E [ − whi h are determined by a hoi eof s ale. They are respe tively denoted by X A ∈ E A [1] , Z Aa ∈ E Aa [1] and Y A ∈ E A [ − , where E Aa [ w ] = E A ⊗ E a ⊗ E [ w ] , et . Using the metri s h AB and g ab to raise indi es, we de(cid:28)ne X A , Z Aa , Y A . Then we immediately seethat Y A X A = 1 , Z Ab Z Ac = g bc and that all other quadrati ombinations that ontra t the tra tor indexvanish.Given a hoi e of onformal s ale we have the orresponding Levi-Civita onne tion on tensor and density bundles and we an use the oupled Levi-Civita tra tor onne tion to a t on se tions of the tensor produ t of a tensorbundle with a tra tor bundle and so forth. This operation is de(cid:28)ned via theLeibniz rule in the usual way. In parti ular we have(4.3) ∇ a X A = Z Aa , ∇ a Z Ab = − P ab X A − Y A g ab , ∇ a Y A = P ab Z Ab . The urvature Ω of the tra tor onne tion is de(cid:28)ned by(4.4) [ ∇ a , ∇ b ] V C = Ω abC E V E for V C ∈ E C . Using (4.3) and the usual formulae for the urvature of theLevi-Civita onne tion we al ulate ( f. [26℄)(4.5) Ω abCE = Z C c Z Ee C abce − X C Z Ee A eab + X E Z C e A eab where(4.6) A abc := 2 ∇ [ b P c ] a is the Cotton tensor.Next we note that there is a onformally invariant di(cid:27)erential operatorbetween weighted tra tor bundles D A : E B ··· E [ w ] → E AB ··· E [ w − , g , the tra tor- D operator by(4.7) D A V := ( d + 2 w − wY A V + ( d + 2 w − Z Aa ∇ a V + X A (∆ − wJ ) V. This is the (Thomas) tra tor-D operator as re overed in [4℄; see [21, 18℄ foran invariant derivation. The onformal operator D from Se tion 2 is simply d times D applied to E [1] . (It is onvenient to retain the two notations,rather than arry the fa tor /d into many al ulations.) Using D we obtain(following [21, 18℄) a onformally invariant urvature quantity as follows(4.8) W BC EF := 3 d − D A X [ A Ω BC ] EF , where Ω BC EF := Z Aa Z Bb Ω bcEF . In a hoi e of onformal s ale, W ABCE isgiven by(4.9) ( d − (cid:0) Z Aa Z Bb Z C c Z Ee C abce − Z Aa Z Bb X [ C Z E ] e A eab − X [ A Z B ] b Z C c Z Ee A bce (cid:1) + 4 X [ A Z B ] b X [ C Z E ] e B eb , where(4.10) B ab := ∇ c A acb + P dc C dacb . is known as the Ba h tensor or the Ba h urvature. From the formula (4.9)it is lear that W ABCD has Weyl tensor type symmetries. It is shown in [10℄and [26℄ that the tra tor (cid:28)eld W ABCD has an important relationship to theambient metri of Fe(cid:27)erman and Graham. See also Se tion 4.4 below.For later use we re all here some standard identities whi h arise fromthe Bian hi identity ∇ a R a a de = 0 , where sequentially labelled indi es areskewed over:(4.11) ∇ a C a a cd = g ca A da a − g da A ca a ; (4.12) ( n − A abc = ∇ d C dabc ; (4.13) ∇ a P ab = ∇ b J ; (4.14) ∇ a A abc = 0 . I A ∈ E A expands to I E = Y E σ + Z Ed µ d + X E ρ, where, for example, σ = X A I A . Hen e Ω abCE I E = σZ C c A cab + Z C c µ d C abcd − X C µ d A dab . Now assume that I A = 0 is parallel (of any length). As a point of notation:in this ase we shall write I E = Y E σ + Z Ed n d + X E ρ . That is n d = ∇ d σ .Then the left-hand-side of the last display vanishes, when e the oe(cid:30) ientsof Z C c and X C must vanish, i.e., σA cab + n d C abcd = 0 and n d A dab = 0 . Away from the zero set of σ , we have that σ − n d = σ − ∇ d σ is a gradient andthe (cid:28)rst equation of the display is the ondition that the metri is onformallmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 21to a Cotton metri ( f. e.g. [5, 25, 35℄). On the other hand at a point p where σ ( p ) = 0 the same equation shows that(4.15) C abcd ∇ d σ = C abcd n d = 0 at p . On e again using the formulae (4.3) for the tra tor onne tion we obtain(4.16) ∇ a Ω acDE = ( d − Z Dd Z Ee A cde − X D Z Ee B ec + X E Z De B ec , where B ab := ∇ c A acb + P dc C dacb is the Ba h tensor. This too is annihilatedby ontra tion with the parallel tra tor I E and so we obtain ( d − Z Dd n e A cde − X D n e B ec + σZ Dd B dc = 0 . From the oe(cid:30) ient of Z Dd we have σB dc + ( d − n e A cde = 0 . In dimension four B dc is onformally invariant and this re overs the wellknown result that, in this dimension, it vanishes on the onformally Einsteinpart of M . But then by ontinuity it follows that the Ba h tensor vanisheseverywhere on M . In other dimensions the last display shows that n e A cde = 0 at any zeros of σ . This with (4.15) gives the (cid:28)rst part of the following.Proposition 4.3. Consider an almost Einstein manifold ( M, [ g ] , I ) and let σ := I A X A . We have σA cab + n d C abcd = 0 , ⇒ n c A cab = 0 , and σB ac + ( d − n e A cae = 0 ⇒ n a B ab = 0 , everywhere on M . Hen e for any point p with σ ( p ) = 0 we have n a C abcd = 0 at p. In dimension d = 4 we have C abcd ( p ) = 0 , while B ab = 0 on M . In dimen-sions d = 4 we have: n a Ω abCD = 0 at p. In any dimension, if j p σ = 0 then C abcd = 0 = A bcd ⇔ Ω abCD = 0 at p, and W ABCD ( p ) = 0 . Proof: The displayed impli ations follow by ontra ting n a into the equa-tions and using the symmetries of A and C . In dimension 4 n a C abcd = 0 at p implies C abcd ( p ) = 0 . When I is not null (and so n a ( p ) = 0 ), this uses thefa t that we are in Riemannian signature and is an immediate onsequen eof the well known dimension 4 identity C abcd C ebcd = δ ea | C | . It remainsto establish the (cid:28)nal laims. If at some point p we have j p σ = 0 , then, at p we have I A = ρX A with ρ ( p ) = 0 . So from I E ∇ a Ω bcDE = 0 it followsthat X E ∇ a Ω bcDE = 0 at p . But, ∇ a X E = Z Ea and from (4.5) we have X E Ω bcDE = 0 everywhere. So Z Dd C bcda − X D A abc = Z Ea Ω bcDE = 0 at p .This immediately yields the result of the last display. But we also have that X A W ABCD and I A W ABCD are zero everywhere, so an easy variation of thelast argument also shows that W ABCD vanishes at p . (cid:3) In the ase that I is null g o = σ − g is Ri i (cid:29)at on M \ Σ . So, from the lastpart of the Proposition, it follows that g o is asymptoti ally (cid:29)at (lo ally) aswe approa h any points of Σ . Following [25℄ let us say a onformal manifoldof dimension d ≥ is weakly generi at p ∈ M if the only solution at p to2 Gover C abcd v d = 0 is v dp = 0 ; then say that ( M, [ g ]) is weakly generi if this holds atall points of M . From the Proposition above and Corollary 2.4 we see thatthe rank of the Weyl tensor obstru ts ertain AE stru tures. Summarisingwe have the following.Corollary 4.4. Let ( M d , [ g ] , I ) be an almost Einstein stru ture with S ( I ) = −| I | = 0 . Then ( M, g o ) is asymptoti ally lo ally Eu lidean as we approa hany point p with σ ( p ) = 0 . If ( M d ≥ , [ g ] , I ) is an AE stru ture with s alesingularity set Σ = ∅ then ( M, [ g ]) is not weakly generi . If ( M, [ g ]) admitsany two linearly independent AE stru tures then it is nowhere weakly generi . S ( I ) = 0 AE stru tures were studied via a di(cid:27)erent approa h in [36℄; aswell as some of the results mentioned here they show that if M is losed,or ( M \ Σ , g o ) is omplete, then Σ = ∅ implies that ( M, g ) is onformallydi(cid:27)eomorphi to the standard sphere. They also dis uss the asymptoti (cid:29)atness in preferred oordinates based at p .Now we spe ialise to the ase of a s alar negative almost Einstein mani-fold ( M, [ g ] , I ) , with a non-empty s ale singularity set Σ . We may suppose,without loss of generality, that | I | = 1 . From Corollary 4.2 we may alsoassume that g is a metri in the onformal lass so that H g = 0 , where H g is the mean urvature of the hypersurfa e Σ .As usual we identify T Σ with its image in T M | Σ under the obvious in- lusion and T ∗ Σ with the orthogonal omplement of n a . In our al ulationshere we will reserve the abstra t indi es i, j, k, l for T Σ ⊂ T M | Σ and itsdual. For example R ijcd means the restri tion of the Riemannian urvature R abcd = R gabcd to tangential (to Σ ) dire tions in the (cid:28)rst two slots. Now, al ulating in the metri g , re all that the Riemannian urvature R abcd de- omposes into the totally tra e-free Weyl urvature C abcd and a remainingpart des ribed by the S houten tensor P ab , a ording to (2.1). It follows thatalong Σ R ijkl = C ijkl + 2 g Σ k [ i P j ] l + 2 g Σ l [ j P i ] k , where we have used that the intrinsi onformal metri on Σ is just therestri tion of the ambient onformal metri . The Levi-Civita onne tion ∇ on ( M, g ) indu es a onne tion on T Σ (this is by di(cid:27)erentiating tangentiallyfollowed by orthogonal proje tion into Γ( T Σ) ). It is easily veri(cid:28)ed thatthe indu ed onne tion is torsion free and on the other hand, sin e Σ istotally geodesi for g , it follows that the indu ed onne tion preserves theindu ed metri g Σ . Thus we (cid:28)nd the standard result that for totally geodesi hypersurfa es the indu ed parallel transport agrees with the intrinsi paralleltransport. It follows immediately that R ijkl = R Σ ijkl , where by R Σ we meanthe intrinsi Riemannian urvature of (Σ , g | Σ ) . But sin e n a C abcd = 0 wehave that C ijkl | Σ is ompletely tra e-free with respe t to g Σ and so hasWeyl-tensor type symmetries, as a tensor on Σ . It follows easily that, for d ≥ , the right-hand-side of the last display ne essarily gives the anoni alde omposition of R Σ ijkl into its Weyl and S houten parts. On the other handusing again (2.1), but now applied to R ijcd we see that P ib n b = 0 . That isalong Σ (4.17) C Σ ijkl = C ijkl , P Σ ij = P ij and P ib n b = 0 lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 23Note that sin e the Weyl urvature of any 3 manifold is identi ally zero, inthe ase of dimension d = 4 we have C Σ ≡ . Thus in this dimension thedisplay is onsistent with Proposition 4.3 where we observed that C | Σ = 0 .As dis ussed in Se tion 4.1, T Σ may be identi(cid:28)ed with N ⊥ (i.e. the or-thogonal omplement of the normal tra tor) in T | Σ and we shall ontinueto make this identi(cid:28) ation. Sin e N is parallel along Σ , ∇ T preserves thissubbundle. Now re all that the onformal density bundles on Σ are justthe restri tions of their ambient ounterparts: E Σ [ w ] = E [ w ] | Σ . When wework with the metri g , whi h has H g = 0 , then the splittings of the tra torbundles also oin ide in the obvious way (see the Remark on luding Se tion4.1), and in parti ular (via the intrinsi version of (2.7)) T Σ de omposes to E Σ [1] ⊕ E Σ i [1] ⊕ E Σ [ − where the weight one 1-forms on Σ , E Σ i [1] may beidenti(cid:28)ed with n ⊥ in E a [1] | Σ . It follows from these observations, the expli itformula (2.5) expressed with respe t to the metri g , and the se ond resultin the display (4.17), that the tra tor parallel transport on Σ n ≥ is just therestri tion of the ambient. Although we used spe ial s ales for the argumentit su(cid:30) es to use any metri from the onformal lass to verify the agree-ment sin e the onne tions are onformally invariant. Let us summarise the onsequen es.Theorem 4.5. Let ( M d ≥ , [ g ] , I ) be a s alar negative almost Einstein stru -ture with a non-empty s ale singularity hypersurfa e Σ . The tra tor onne -tion of ( M, [ g ]) preserves the intrinsi tra tor bundle of Σ , where the latteris viewed as a subbundle of the ambient tra tors: T Σ ⊂ T . Furthermore theintrinsi tra tor parallel transport of ∇ T Σ oin ides with the restri tion ofthe parallel transport of ∇ T .We have Ω( u, v ) = Ω Σ ( u, v ) along Σ where u, v ∈ Γ( T Σ) . In dimensions d = 4 we have the stronger result Ω( · , · ) = Ω Σ ( · , · ) along Σ , where here, by trivial extension, we view Ω Σ as a se tion of Λ T ∗ M ⊗ End T .Proof: In the ase of d = 3 the agreement of the parallel transport isimmediate from the de(cid:28)nition of the tra tor onne tion ∇ T Σ and that thenormal tra tor N A is parallel along Σ . In the remaining dimensions this wasestablished immediately above. From this, and the fa t that on Σ we have Ω( u, v ) N = 0 , it follows at on e that Ω( u, v ) = Ω Σ ( u, v ) along Σ , as laimed.For dimensions d = 4 we have from Proposition 4.3 above that Ω( n, · ) = 0 ,when e the (cid:28)nal laim. (cid:3) Remark: To obtain the result that the intrinsi tra tor parallel transportof ∇ T Σ oin ides with the restri tion of the ambient parallel transport of ∇ T to se tions of T Σ uses that Σ is totally umbilli and that n a C abcd = 0 along Σ . These onditions are su(cid:30) ient for the agreement of the onne tions. ||||||| Σ . Given a onformal manifold (Σ , [ g Σ ]) we may ask ifthis an arise as the s ale singularity set of a s alar negative almost Einsteinmanifold. Narrowing the problem, we may begin with a (cid:28)xed smooth (orwith spe i(cid:28)ed regularity) odimension 1 embedding of Σ in a manifold M and onsider the Diri hlet-type problem of (cid:28)nding a dire ted AE stru ture4 Gover ( M, [ g ] , I ) with (Σ , [ g Σ ]) as the s ale singularity set; issues in lude whetheror not there is any solution and, if there is, then whether (Σ , [ g Σ ]) determines ( M, [ g ] , I ) uniquely. This is exa tly the problem of (cid:28)nding on M a onformalstru ture [ g ] and on this a solution σ to the onformally invariant equation ∇ a ∇ b σ + P ab σ + ρ g ab = 0 (i.e. (2.4)) su h that Σ is the zero set of σ (andthen there is the question of whether the pair ([ g ] , σ ) is unique). We want toderive onsequen es of this equation that make the nature of this problemmore transparent. We have seen already that this may be viewed as (cid:28)ndingon M a onformal stru ture admitting a parallel tra tor parallel tra tor I with I | Σ agreeing with the normal tra tor N along Σ .The data on Σ is a onformal stru ture, and, for any solution [ g Σ ] , is simplythe pull ba k of the ambient onformal stru ture [ g ] on M . By Theorem4.5 we know (at least to some order along Σ ) how the ambient onformal urvature is related to the intrinsi onformal urvature of (Σ , [ g Σ ]) . Thusit seems natural to derive the equations whi h ontrol how this extends o(cid:27) Σ . With less ambition we shall not attempt here to study the full boundaryproblem. Rather we seek to (cid:28)nd equations whi h ontrol the onformal urvature quantities o(cid:27) Σ and whi h are also well de(cid:28)ned along Σ .First note that it follows from the Bian hi identity (4.13) that Einsteinmanifolds ( M d , g o ) are Cotton, i.e. A g o abc = 0 . In dimension d = 3 the Weyltensor vanishes identi ally and the Cotton tensor is onformally invariant.Thus almost Einstein 3 manifolds are Cotton and hen e onformally (cid:29)at. Soif ( M , [ g ] , I ) is s alar positive then it is a positive se tional urvature spa eform. If ( M , [ g ] , I ) has S ( I ) ≤ then I may have a s ale singularity set Σ ,but o(cid:27) this the stru ture ( M, g o ) is either hyperboli (if S ( I ) < ) or lo allyEu lidean (if S ( I ) = 0 ).From (4.16) one easily on ludes that on an Einstein manifold ( M d , g o ) thetra tor urvature satis(cid:28)es the (full) Yang-Mills equations, that is ∇ a Ω abC D =0 (see also [28℄) (where the onne tion ∇ is in the s ale g o ). In dimension d =4 this equation is onformally invariant. Thus almost Einstein 4 manifoldsare globally Yang-Mills. Combining with relevant results from Proposition4.3 and Theorem 4.5, let us summarise .Proposition 4.6. Let ( M , [ g ] , I ) be an almost Einstein manifold. Then thetra tor urvature satis(cid:28)es the onformally invariant Yang-Mills equations, ∇ a Ω abC D = 0 . If I is s alar negative then along any singularity hypersurfa e Σ of I we have C abcd = 0 and Ω( u, v ) = Ω Σ ( u, v ) along Σ where u, v ∈ Γ( T Σ) .Note that in dimension 4 the tra tor urvature is Yang Mills if and only ifthe onformal stru ture is Ba h (cid:29)at. However the Proposition suggests thatit is useful to view the Ba h (cid:29)at ondition as a Yang-Mills equation in orderto formulate an extension problem (or boundary problem in the PE ase).The additional data required in ludes n a Ω abDF along Σ whi h is equivalentto n b A abc | Σ . We note that in [37℄ LeBrun established the existen e anduniqueness of a real analyti self-dual Poin aré-Einstein metri in dimension4 de(cid:28)ned near the boundary with pres ribed real analyti onformal in(cid:28)nity.lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 25If a 4-dimensional metri is self-dual then so is its tra tor urvature andhen e the tra tor onne tion is Yang-Mills.Before we ontinue we need some further notation. Let us write (hash)for the natural tensorial a tion of se tions A of End( T ) on tra tor se tions.For example, on an ovariant 2-tra tor T AB , we have A♯T AB = − A C A T CB − A C B T AC . If A is skew for h , then at ea h point, A is so ( h ) -valued. The hash a tionthen ommutes with the raising and lowering of indi es and preserves the SO ( h ) -de omposition of tra tor bundles.As a se tion of the tensor square of the h -skew bundle endomorphismsof T , the urvature quantity W has a double hash a tion on tra tors T ; wewrite W ♯♯T for this. Now for dimensions d = 4 we use this to onstru t aLapla ian operator on (possibly onformally weighted) tra tor se tions. For T a se tion of ( ⊗ k T )[ w ] and d = 4 we make the de(cid:28)nition (cid:3) / T := (∆ − wJ ) T − d − W ♯♯T.
Then from this we obtain a variant of the usual tra tor-D operator as follows: D/ A T := ( d + 2 w − wY A T + ( d + 2 w − Z Aa ∇ a T + X A (cid:3) / T. In terms of this operator we have,Theorem 4.7. Let ( M d , [ g ] , I ) be an almost Einstein manifold. Then if d = 4 we have W BCDE = 0 . In dimension 6 the onformally invariantequation (cid:3) / W A A B B = 0 holds. In dimensions d = 4 we have(4.18) I A D/ A W BCEF = 0 . Also(4.19) W BCEF I F = 0 = I B W BCEF . In parti ular if I is s alar negative and Σ the singularity hypersurfa e for I then W BCEF N F = 0 = N B W BCEF along Σ .Proof: From (4.9) it follows that, in dimension 4, W = 0 is equivalentsetting the Ba h tensor to zero and, as noted earlier, this is equivalent to the onformal tra tor onne tion being Yang Mills. We have this, in parti ular,on almost Einstein manifolds. Sin e I is parallel it annihilates the tra tor urvature, i.e. Ω bcEF I E = 0 . But sin e it is parallel and has onformalweight 0, I ommutes with the tra tor-D operator D . It follows from (4.8)that W BCEF I F = 0 . But W has Weyl tensor type symmetries, so (4.19)follows. We also note here that sin e I ommutes with D , and, on theother hand, any ontra tion of I with W is zero, it follows by an elementaryargument that I ommutes with D/ .In dimension 6 we have from [27℄ that (cid:3) / W A A B B = KX A Z A a X B Z B b B ab , K is a nonzero onstant and B ab is the Fe(cid:27)erman-Graham (obstru -tion) tensor (see also [19℄). The sequentially labelled indi es here are im-pli itly skewed over. But the onformal invariant B ab is zero on Einsteinmanifolds [16, 31, 27℄ and hen e also (by ontinuity) on almost Einsteinmanifolds.It remains to establish (4.18). Sin e W has onformal weight − , it followsthat when d = 6 we have I A D/ A W BCEF = σ (cid:3) / W A A B B , where as usual σ denotes the onformal density X A I A . Thus (4.18) holds in dimension 6.Let us suppose now that d = 4 , . Here we will use the link between thestandard tra tor bundle on ( M, [ g ]) and the Fe(cid:27)erman-Graham (FG) metri of [16, 17℄. This link was developed in [10, 26, 8℄ but here we use espe iallythe notation and results from [27℄. (It should be noted however that here weuse the opposite sign for the Lapla ian.) The arguments we use below are aminor variation of similar developments from those sour es.For a Riemannian onformal manifold ( M d , [ g ]) the ambient manifold [16℄is a signature ( d + 1 , pseudo-Riemannian manifold with Q as an embeddedsubmanifold. There is some further ba kground on the FG metri in Se -tion 6. Suitably homogeneous tensor (cid:28)elds on the ambient manifold, uponrestri tion to Q , determine tra tor (cid:28)elds on the underlying onformal man-ifold. In parti ular, in dimensions other than 4, W ABCD is the tra tor (cid:28)eldequivalent to ( d − R ABCD | Q where R is the urvature of the FG ambientmetri . Under this orresponden e the FG ambient metri applied to tra -tors along Q , des ends to the tra tor metri . Ambient di(cid:27)erential operatorsthat are suitably tangential and homogeneous (see e.g. [8, 27℄) also des endto operators between tra tor bundles or subquotients thereof. For examplethe tra tor onne tion arises from ambient parallel transport along Q .On the FG ambient manifold let us de(cid:28)ne a Lapla ian operator ∆ / by theformula ∆ / := ∆ − R . Then in all dimensions d = 4 , , ∆ / R | Q = 0 , [27, Se tion 3.2℄. On the otherhand D/ A orresponds to the ambient operator ( d + 2 w − ∇ + X ∆ / =: D/ : T Φ ( w ) → T ⊗ T Φ ( w − where T Φ ( w ) indi ates the spa e of se tions, homogeneous of weight w , ofsome ambient tensor bundle. (NB: An ambient tensor T is homogeneousof weight w if and only if ∇ X T = wT .) From the Bian hi identity on theFG ambient manifold, and the fa t that ∆ / R | Q = 0 , it follows that on theambient manifold we have D/ [ A R BC ] DE = 0 , along Q . This des ends to D/ [ A W BC ] DE = 0 . So we have I A D/ [ A W BC ] DE = 0 . But sin e I A W ABDE = 0 and I ommutes with D/ (4.18) follows. (cid:3) lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 27From the Theorem we may on lude some restri tions on the intrinsi onformal stru ture. For example we have the following.Corollary 4.8. Let ( M , [ g ] , I ) be a s alar negative almost Einstein mani-fold with s ale singularity set Σ = ∅ . Then the indu ed onformal stru ture (Σ , [ g Σ ]) is Ba h (cid:29)at.Proof: In dimensions d = 6 the equation (4.18), i.e. I A D/ A W = 0 , on M implies that along any s ale singularity subspa e we Σ have δW = 0 where δ is the ( onformally invariant) tra tor twisted onformal Robin op-erator [7, 20℄ applied to W ; in terms of g we have δW = n a ∇ ga + 2 H g W where H g is the mean urvature of Σ and ∇ is the usual (density oupled)the tra tor onne tion.To simplify the presentation let us temporarily display the (cid:28)rst two ab-stra t indi es of the tra tor W , but suppress the last pair; we shall write W BC rather than W BCDE . From the de(cid:28)ning formula (4.8) for W BC it fol-lows easily that(4.20) W BC = ( d − Z Bb Z C c Ω bc − X B Z C b ∇ a Ω ab + X C Z Bb ∇ a Ω ab . This is expression (13) from [26℄. Exploiting Corollary 4.2, let us al ulatein a metri g with respe t to whi h Σ is totally geodesi . Setting d = 5 ,applying δ = n a ∇ a to (4.20), and using the tra tor onne tion formulae(4.3) we see that the oe(cid:30) ient of Z Bb Z C c is n a ∇ a Ω bc − n b ∇ a Ω ac + n c ∇ a Ω ab , where Ω bc is the tra tor urvature of the ambient onformal stru ture ( M, [ g ]) (where we have suppressed the tra tor indi es). Evidently a part of the on-dition δW | Σ = 0 is that the last display is zero along Σ . Thus, in parti ular, n b ontra ted into this must vanish, that is n b n a ∇ a Ω bc − ∇ a Ω ac + n c n b ∇ a Ω ab = 0 along Σ . But using that Σ is totally geodesi and, from Theorem 4.5, that Ω( u, v ) =Ω Σ ( u, v ) along Σ where u, v ∈ Γ( T Σ) , this exa tly states that g ij Σ ∇ Σ i Ω Σ jk = 0 , where g Σ is the intrinsi metri on Σ indu ed by g and Ω Σ jk is the tra tor ur-vature of its onformal lass. Thus the onformally invariant intrinsi tra tor urvature of the (Σ , g Σ ) satis(cid:28)es the Yang-Mills equations. As mention ear-lier, in dimension 4 these are onformally invariant and are equivalent to thestru ture (Σ , [ g Σ ]) being Ba h-(cid:29)at. (cid:3) There is an analogue of this result for higher odd d , see Theorem 6.1 below.It is likely that there is a proof of Theorem 6.1 using only equation (4.18),but ertainly approa hing this dire tly (as in the proof for d = 5 above)would rapidly be ome te hni al for in reasing dimension. Se tion 6 gives asimple and on eptual treatment, using the Fe(cid:27)erman-Graham metri .Remarks: From the equation (4.18) it follows that the onformal aspe ts ofthe asymptoti s of Poin aré-Einstein metri s are ontrolled by the operator I A D/ A .8 GoverIn dimension 6 the main equation (4.18) (or equivalently (cid:3) / W A A B B = 0 )is equivalent to requiring ( M, [ g ]) to have vanishing Fe(cid:27)erman-Graham ten-sor.In dimensions other than 3,4, and 6, and o(cid:27) Σ , a key part of (4.18)is the harmoni equation ∆ C − R♯♯C = 0 on the Weyl urvature whi hholds on Cotton (and hen e Einstein) manifolds, as follows easily from theBian hi identities (4.11) and (4.12). However in dimensions other than 3 we annot on lude that there is a s ale for whi h an AE manifold is Cotton(everywhere). On the other hand the equation (4.18) holds globally on anAE manifold ( d = 4 ). ||||||| The following sheds some light on the meaning of the equation (4.18)and its relation to possible boundary problems. This follows easily from theTheorem and the de(cid:28)nitions of the operators involved, ex ept we have also alled on Corollary 6.4 below.Corollary 4.9. On an Einstein manifold ( M ( n +1) ≥ , g o ) we have (∆ g o + 4 J g o n + 1 ( n − W − n − W ♯♯W = 0 . In parti ular this holds on an almost manifold ( M ( n +1) ≥ , [ g ] , I ) o(cid:27) the zeroset Σ of σ = h ( X, I ) . If | I | = 1 and Σ is non-empty, then on M \ Σ wehave (∆ g o − n − W − n − W ♯♯W = 0 , while along the hypersurfa e Σ we have N − | W = 0 , and, if n ≥ , ( n − W | Σ = ( n − W Σ , while, if n = 5 , δW = 0 along Σ , where δ is the onformal Robin operator applied to W ; in terms of g we have δW = n a ∇ ga + 2 H g W is the where H g is the mean urvature of Σ .It is shown in [20℄ that on densities I A D A agrees with the Lapla iansarising in the s attering problems treated in [33℄. The operator (∆ g o − n − here is a tra tor twisted version of su h. We have used the n = d − here to simplify omparisons with [20℄ and [33℄.We have seen in dimension 3,4 and 6 that there are onformally invariantequations ontrolling the onformal urvature of an AE manifold. This isa hieved trivially in dimension 3. As a (cid:28)nal note for this se tion we pointout that there is an analogue of the results for dimensions 4 and 6 to highereven dimensions.Proposition 4.10. Almost Einstein manifolds ( M d even , [ g ] , I ) , d ≥ , sat-isfy the onformally invariant equation that the Fe(cid:27)erman-Graham tensorvanishes. This may be expressed in the form(4.21) (cid:3) / d/ − W = ∆ d/ − W + lower order terms , where by (cid:3) / and ∆ we mean the operator given by multipli ation by 1.lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 29The linear operator (cid:3) / d/ − is onstru ted in [27℄, and the result here is aneasy onsequen e of the results there for Einstein manifolds. On e again ona s alar negative AE manifold with a s ale singularity set Σ , (4.21) expressesthe vanishing Fe(cid:27)erman-Graham tensor ondition in a form suitable to linkwith the onformal urvature data on Σ (using Corollary 6.4, or for n = 3 , Theorem 4.5). It should be interesting to onstru t ompatible onformalboundary operators for W along embedded submanifolds Σ so that theseyield a well posed and onformal ellipti problem for the onformal urvature Ω . Close analogues of the onformal boundary operators developed in [7℄should play a role.Remark: Note that onformal equations, su h as (4.21), o(cid:27)er the han e tosplit the problem of (cid:28)nding almost Einstein stru tures (or Poin aré-Einsteinmetri s) into a onformal problem, say ontrolled by (4.21) with furtherboundary operators along Σ , and a se ond part where one would (cid:28)nd a ompatible (cid:16)s ale(cid:17) σ . We should expe t that a solution to the onformalproblem is ne essary but in general not su(cid:30) ient. However one may ask if(in Riemannian signature and say on losed even manifolds) (4.21) plus the( learly ne essary) vanishing of the onformal invariant Ω abC F Ω cdDF · · · Ω ef EF d +1 , where the sequentially labelled indi es are skewed over, is su(cid:30) ient for a on-formal manifold to ne essarily admit an almost Einstein stru ture lo ally. A orresponding global question is whether a smooth se tion K of T satisfying Ω abC D K D = 0 plus (4.21) is su(cid:30) ient to on lude that the onformal stru -ture on a losed even manifold admits a dire ted almost Einstein stru ture.In dimension 4 there is a positive answer to this if we restri t to K su h that h ( X, K ) is non-vanishing [24℄; in this ase the stru ture must be Einstein. |||||||
5. Examples and the model5.1. The model (cid:21) almost Einstein stru tures on the sphere.Proposition 5.1. The d -sphere, with its standard onformal stru ture, ad-mits a ( d +2) -dimensional spa e of ompatible dire ted almost Einstein stru -tures. For ea h S ∈ R there is an almost Einstein stru ture I on S d with S ( I ) = S .The AE stru tures on the sphere also may be viewed as examples of ASCstru tures on the sphere. In any ase we shall see that, in a sense, (cid:16)most(cid:17) ofthese are s alar negative (whi h might at (cid:28)rst seem ounterintuitive).Before we prove this let us re all the onstru tion of the standard onfor-mal stru ture on the sphere. Consider a ( d +2) -dimensional real ve tor spa e V equipped with a non-degenerate bilinear form H of signature ( d + 1 , .The null one N of zero-length ve tors form a quadrati variety. Let us write N + for the forward part of N \ { } . Under the ray proje tivisation of V theforward one N + is mapped to a quadri in P + ( V ) ∼ = S d +1 . This image istopologi ally a sphere S d and we will write π for the submersion N + → S d .Ea h point p ∈ N + determines a positive de(cid:28)nite inner produ t on T x = πp S d by g x ( u, v ) = H p ( u ′ , v ′ ) where u ′ , v ′ ∈ T p N + are lifts of u, v ∈ T x S d . For agiven ve tor u ∈ T x S d two lifts to p ∈ N + di(cid:27)er by a verti al ve tor (cid:28)eld.0 GoverSin e any verti al ve tor is normal (with respe t to H ) to the one it followsthat g x is independent of the hoi es of lifts. Clearly then, ea h se tion of π determines a metri on S and by onstru tion this is smooth if the se tionis. (Evidently the metri agrees with the pull-ba k of H via the se tion on- erned.) Now, viewed as a metri on T R d +2 , H is homogeneous of degree 2with respe t to the standard Euler ve tor (cid:28)eld E on V , that is L E H = 2 H ,where L denotes the Lie derivative. In parti ular this holds on the one,whi h we note is generated by E .Write g for the restri tion of H to ve tor (cid:28)elds in T N + whi h are the liftsof ve tor (cid:28)elds on S d . Then for any pair u, v ∈ Γ( T S d ) , with lifts to ve tor(cid:28)elds u ′ , v ′ on N + , g ( u ′ , v ′ ) is a fun tion on N + homogeneous of degree 2, andwhi h is independent of how the ve tor (cid:28)elds were lifted. Evidently N + maybe identi(cid:28)ed with the total spa e of a bundle of onformally related metri s.Thus g ( u ′ , v ′ ) may be identi(cid:28)ed with a onformal density of weight on S d .That is, this onstru tion determines a se tion of S T ∗ S d ⊗ E [2] that we shallalso denote by g . By onstru tion this is a onformal metri (see Se tion2) on S d . Fix a future pointing ve tor I in V with | I | := H ( I, I ) = − .Regarding V as an a(cid:30)ne spa e, view I as a onstant se tion of T V . Write X A for standard oordinates on V (i.e. via an isomorphism V ∼ = R d +2 ). Itis straightforward to verify that the H ( I, X ) = 1 hyperplane meets N + in a opy of S d and the metri indu ed by this se tion of π is the standard metri on S d . Thus g is a standard onformal stru ture on the sphere. We are readyto prove the Proposition.Proof of Proposition 5.1: It is easily veri(cid:28)ed that G := SO ( H ) ∼ = SO ( d + 1 , (the identity onne ted omponent of the Lorentz group) a tstransitively on the sphere. Thus the onformal sphere may be identi(cid:28)ed with G/P where P is the paraboli subgroup of G whi h stabilises a nominatedray in N + . Now G → G/P may be viewed as a (cid:29)at Cartan bundle over
G/P = S d and the standard tra tor bundle T is G × P V where V is viewedas a P -module, by restri tion. Here G × P V = G × V / ∼ where the equiva-len e relation is ( gp, v ) ∼ ( g, p · v ) with g ∈ G , p ∈ P and where (cid:16) · (cid:17) indi atesthe standard representation of G on V . The bundle G × P V is trivialised anoni ally by the map ( g, v ) ( gP, g · v ) and so we have a onne tion ∇ T on T indu ed from the trivial onne tion on ( G/P ) × V . It is straightforwardto verify that this is the normal tra tor onne tion. (In fa t this is essentiallya tautology; one view the idea of a normal onformal onne tion tra tor asmodelled on this homogeneous ase.) Thus in this ase the tra tor onne -tion is globally (cid:29)at, with the bundle T admits ( d + 2) linearly independentparallel se tions. (cid:3) Using the embedding of N + in V we an expli itly des ribe the almostEinstein stru tures of the Proposition. For example we may onstru t as alar negative AE stru ture on S d as follows. Take a ve tor I ∈ V of length1 (i.e. | I | = 1 ). We shall use the same notation for the ove tor H ( I, · ) .By the standard parallel transport (of V viewed as an a(cid:30)ne stru ture) viewthis as a onstant se tion of T ∗ V . Then as above, writing X A for standard oordinates on V , the interse tion of the hyperplane I A X A = 1 with N + ,whi h we shall denote S + , is a se tion of π over an open ap C + of thesphere. Similarly the interse tion of the hyperplane I A X A = − with N + ,lmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 31whi h we shall denote S − , is a se tion of π over another open ap C − ofthe sphere. On the other hand the hyperplane I A X A = 0 (parallel to theprevious) interse ts N + in a one of one lower dimension. The image Σ ofthis under π is a opy of S n embedded in S d (where as usual d = n + 1 ). It iseasily dedu ed that S d is the union of the three submanifolds C − , Σ , and C + and that, for example, with respe t to (a restri tion of) the smooth stru turestru ture on S d , the embedded Σ is a boundary for its union with C + . Thisfollows be ause any forward null ray though the origin and parallel to the I A X A = 1 hyperplane lies in the hyperplane I A X A = 0 , whereas every otherforward null ray through the original meets either the I A X A = 1 hyperplaneor the I A X A = − hyperplane. Let us write g o for the metri that these tions S ± give on C ± . Note that the hypersurfa e S n anoni ally has nomore than a onformal stru ture. This may obviously be viewed as arisingas a restri tion of the onformal stru ture on S d . Equivalently we may viewits onformal stru ture as arising in the same way as the onformal stru tureon S d , ex ept in this ase by the restri tion of π to the sub- one I A X A = 0 in N + , and from (the restri tion of) H along this sub- one. In the followingwe write g to denote any metri from the standard onformal lass on S d .Note that on C ± this is onformally related to g o .Now let us hen eforth identify, without further mention, ea h fun tionon N + whi h is homogeneous of degree w ∈ R with the orresponding onformal density of weight w . With σ := I A X A , as above, note that σ − g is homogeneous of degree 0 on N + and agrees with the restri tion of H along S ± . Thus on C ± we have σ − g = g o ; σ − g re overs the metri determined by S ± . Similarly on S d we have g = τ − g , where τ is a non-vanishing onformaldensity of weight 1. So on C + ∪ C − , g o = s − g where s is the fun tion σ/τ .We see that g o is onformally ompa t on S d \ C + , and also on S d \ C − .We may now understand this stru ture via the tra tor bundle on S d . Letus write ρ t for the natural a tion of R + on N + and then ρ t ∗ for the derivativeof this. Now modify the latter a tion on T V by res aling: we write t − ρ t ∗ forthe a tion of R + on T V whi h takes u ∈ T p V to t − ( ρ t ∗ u ) ∈ T ρ t ( p ) V . Notethat u and t − ( ρ t ∗ u ) are mutually parallel, a ording to the a(cid:30)ne stru tureon V . It is easily veri(cid:28)ed that the quotient of T V |N + by the R + a tionjust de(cid:28)ned is a rank d + 2 ve tor bundle T on M . Obviously the paralleltransport of V determines a parallel transport on T , that is a onne tion ∇ T .Sin e V is totally parallel this onne tion is (cid:29)at. The twisting of ρ t ∗ to t − ρ t ∗ is designed so that the metri H on R d +2 also des ends to give a (signature ( d + 1 , ) metri h on T and learly this is preserved by the onne tion. Infa t ( T , h, ∇ T ) is the usual normal standard tra tor bundle. This is provedunder far more general ir umstan es in [10℄ (see also [26℄); it is shownthere that the tra tor bundle may be re overed from the Fe(cid:27)erman-Grahamambient metri by an argument generalising that above. In this pi ture theEuler ve tor (cid:28)eld E = X A ∂/∂X A (using the summation onvention), whi hgenerates the (cid:28)bres of π , des ends to the anoni al tra tor (cid:28)eld X ∈ T [1] .It follows from these observations that, sin e the ve tor (cid:28)eld I is parallelon V , its restri tion to N + is equivalent to a parallel se tion of T ; we shallalso denote this by I . So this is an almost Einstein stru ture on S d ; | I | = 1 means that the almost Einstein stru ture we re over has S ( σ ) = − , when e2 Goverhas Ric( g o ) = − ng o on C ± . The zero set for σ = h ( X, I ) is exa tly Σ . Sowe see that ( S d , [ g ] , I ) is an almost Einstein manifold. Sin e it is onformally(cid:29)at with S ( σ ) = − it is what may be termed an almost hyperboli stru tureon the sphere. The fa t that along Σ the parallel tra tor I gives the normaltra tor N is espe ially natural in this pi ture sin e Σ is determined by ahyperplane orthogonal to I . Finally we observe that it follows from Propo-sition 3.7 that the spa es ( S d \ C ± , [ g ] , I ) are Poin aré-Einstein manifolds,in fa t ea h equivalent to the onformal ompa ti(cid:28) ation of the hyperboli ball.Sin e the group G a ts transitively on length 1 spa elike ve tors, from thepi ture above we see that any s alar negative AE stru ture on the sphere isrelated to the one onstru ted by a onformal transformation after an R + a tion on the parallel tra tor I .The s alar (cid:29)at almost Einstein stru tures are obtained by a similar on-stru tion to the s alar negative ase above. Note that if I is a non-zero nullve tor in V then the hyperplane H ( I, X ) = 1 meets all future null rays in N + ex ept the one parallel to I . So the almost Einstein stru ture determined by I has a single isolated point of s ale singularity. The Einstein metri g o is onformally related to the round metri , and | I | = 0 means that S ( I ) = 0 and so g o is (cid:29)at; this is the usual Eu lidean stru ture on the sphere minusa point. It is straightforward to on lude that the map, along null gener-ators, relating this Eu lidean almost Einstein stru ture and the standardsphere embedded in the one (as des ribed earlier) is the usual stereographi proje tion.In a partial summary then, if I and I are onstant ve tors in V with | I | = − and | I | = 1 then, as parallel tra tors on S d these determine,respe tively the standard sphere metri and almost hyperboli stru tures.We an interpolate between these via Corollary 2.4 and we note that forsome t ∈ R the parallel tra tor I t := (sin t ) I + (cos t ) I is null and sodetermines a Ri i (cid:29)at stru ture in the onformal lass, that is a Eu lideanmetri on the sphere minus a point. For ea h t ∈ R the isotropy subgroup G I t of G = SO ( H ) (cid:28)xing the ve tor I t learly a ts transitively and by isometrieson the onne ted omponents of S d \ Σ t , where Σ t is the s ale singularity setof I t .5.2. Doubling and almost hyperboli onstru tions. One route to on-stru ting further ompa t almost Einstein manifolds is via the doubling of ompa t Poin aré-Einstein manifolds. So suppose that M is a ompa tPoin aré-Einstein with onformal in(cid:28)nity Σ . The double we seek is a gluingalong Σ , M (2) := ( M ⊔ M ) / Σ where the identi(cid:28) ation of the two opies of Σ is the obvious one. As pointedout in [42℄, for example, this may be equipped with a smooth stru ture om-patible with the smooth stru ture on M and so that the natural involutionex hanging the fa tors is also smooth. Now extend the PE metri g o of M to a metri on M (2) by symmetry. This will be smooth if g o is even in thesense of [17, Se tion 4℄ (following [32℄): Lo ally along the ollar the metri may be put in normal form, relative to some g Σ from the onformal lass onlmost Einstein and Poin aré-Einstein manifolds in Riemannian signature 33 Σ ,(5.1) g o = s − ( ds + g Σ s ) where s satis(cid:28)es | ds | g = 1 and g s is a 1-parameter family of metri s on Σ su h that g Σ0 = g Σ . The metri is even if for ea h point of Σ , and with themetri g o in this form, we have that ds + g s is the restri tion to M × [0 , ∞ ) of a smooth metri g on a neighbourhood U ⊂ Σ × ( −∞ , ∞ ) su h that U and g are invariant under the map s