math.DG 2010-02-15 0

Metrisability of two-dimensional projective structures

We carry out the programme of R. Liouville \cite{Liouville} to construct an explicit local obstruction to the existence of a Levi--Civita connection within a given projective structure $[\Gamma]$ on a surface. The obstruction is of order 5 in the com…

Authors: Robert L. Bryant, Maciej Dunajski, Michael Eastwood

METRISABILITY OF TW O-DIMENSIO NAL P R OJECTIVE STR UCTURES ROBER T BR Y ANT, MAC IEJ D UNAJSKI , AND MICHA EL EASTWOOD Abstra ct. W e carry out the programme of R . Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a giv en p ro jective structure [Γ] on a su rface. The obstruction is of order 5 in the comp onents of a connection in a pro jectiv e class. It ca n be expressed as a p oint inv ari an t for a second order O D E whose integral curves are the geodesics of [Γ] o r as a wei ghted scalar pro jectiv e inv ari ant of the pro jectiv e class. If th e obstruction v anishes we fi nd the sufficient cond itions for the ex istence of a metric in th e real analytic case. In th e generic case they are expressed by the vanis hing of tw o inv arian ts of order 6 in the conn ection. In degenerate cases the sufficient obstruction is of order at most 8. 1. Introduction Recall that a pr oje ctive structur e [7, 22, 12] on an op en s et U ⊂ R n is an equiv alence class of torsion f r ee conn ections [Γ]. Two connections Γ and ˆ Γ are pro jectiv ely equiv alen t if they share the same u nparametrised geod esics. This means that the geod esic flo w s pro ject to the same foliation of P ( T U ). The analytic expression for this equiv alence class is ˆ Γ c ab = Γ c ab + δ c a ω b + δ c b ω a , a, b, c = 1 , 2 , ..., n (1.1) for some one-form ω = ω a dx a . A b asic unsolved pr oblem in pr o jectiv e d ifferen tial geometry is to determine the explicit criterion for the metrisability of pro jectiv e s tructure, i.e. a n sw er the follo wing question: • What are the n ecessary and sufficient lo cal conditions on a connection Γ c ab for the existence of a one form ω a and a symmetric non-degenerate tensor g ab suc h th at the pro jectiv ely equ iv alen t connection Γ c ab + δ c a ω b + δ c b ω a is the Levi-Civita connection for g ab . (W e are allo w ing Lorentz ian metrics.) W e shall fo cus on lo cal m etrisabilit y , i.e. the p air ( g , ω ) with det ( g ) no w here v an ish ing is required to exist in a neig h b ourho o d of a p oin t p ∈ U . This p r oblem leads to a v astly o ve r determined sy s tem of partial d ifferen tial equations for g and ω . There are n 2 ( n + 1) / 2 comp onent s in a connection, and ( n + n ( n + 1) / 2) comp onents in a pair ( ω , g ). One could therefore n aiv ely exp ect n ( n 2 − 3) / 2 cond itions on Γ. In this pap er we shall carry out the algorithm laid out b y R. Liouville [19] to solve this p roblem when n = 2 and U is a su rface 1 . In the tw o-dimensional case the pr o jectiv e structures are equ iv alen t to second order O DEs which are cu b ic in the fir s t deriv ativ es. T o see it consider the geo desic equations for x a ( t ) = ( x ( t ) , y ( t )) and eliminate the parameter t b et w een the t wo equations ¨ x c + Γ c ab ˙ x a ˙ x b = v ˙ x c . 1 Let us stress that th e ‘solution’ here means an explicit criterion, given by v anishing of a set of inv ariants, whic h can b e verified on any representati ve of [Γ]. 1 2 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD This yields the desired ODE for y as a fu nction of x d 2 y dx 2 = Γ 1 22  dy dx  3 + (2Γ 1 12 − Γ 2 22 )  dy dx  2 + (Γ 1 11 − 2Γ 2 12 )  dy dx  − Γ 2 11 . (1.2) Con versely , any second order ODEs cu b ic in the fi rst deriv ativ es d 2 y dx 2 = A 3 ( x, y )  dy dx  3 + A 2 ( x, y )  dy dx  2 + A 1 ( x, y )  dy dx  + A 0 ( x, y ) (1.3) giv es rise to some pr o jectiv e s tr ucture as the indep enden t comp onents of Γ c ab can b e read off from the A s up to the equ iv alence (1.1). The adv an tage of th is formulat ion is that the pro jectiv e am biguit y (1.1) has b een remo ved fr om the problem as th e combinatio n s of the connection symb ols in the ODE (1.2 ) are ind ep endent of the choice of the one form ω . There are 6 comp onen ts in Γ c ab and 2 components in ω a , b ut only 4 = 6 − 2 co efficien ts A α ( x, y ) , α = 0 , ..., 3. The diffeomorph isms of U can b e u sed to further eliminate 2 out of these 4 fun ctions (for example to mak e the equ ation (1.3) linear in the fir s t d eriv ativ es) so one can sa y that a general pro jectiv e stru cture in t wo dimensions dep end s on tw o arbitrary functions of t w o v ariables. W e are looking for inv arian t conditions, so we s hall not mak e use of this diffeomorphism freedom. W e shall state our first result. Consider the 6 by 6 matrix giv en in terms of its ro w v ectors M ([Γ]) = ( V , D a V , D ( b D a ) V ) (1.4) whic h dep ends on the fun ctions A α and their deriv ativ es up to order fiv e. The vec tor field V : U → R 6 is giv en b y (3.21), the expressions D a V = ∂ a V − VΩ a are computed using the righ t m ultiplication b y 6 by 6 mat rices Ω 1 , Ω 2 giv en b y (A5 3 ) and ∂ a = ∂ /∂ x a . W e a lso mak e a recursiv e d efi nition D a D b D c ...D d V = ∂ a ( D b D c ...D d V ) − ( D b D c ...D d V ) Ω a . Theorem 1.1. If the pr oje ctive structur e [Γ] is metrisable then det ( M ([Γ])) = 0 . (1.5) There is an immediate corollary Corollary 1.2. If the inte gr al curves of a se c ond or der ODE d 2 y dx 2 = Λ  x, y , dy dx  , (1.6) ar e ge o desics of a L evi-Civ i ta c onne ction then Λ i s at most cubic in dy /dx and (1.5) holds. The expression (1.5) is written in a relativ ely compact form usin g ( V , Ω 1 , Ω 2 ). All the algebraic manipu lations whic h are requ ired in expand ing the determin ant ha v e b een done using MAPLE co de whic h can b e obtained from us on request. W e sh all pro ve Theorem 1.1 in three steps. The first step, already tak en by Liouville [19], is to asso ciat e a linear system of four PDEs for three un kno wn functions with eac h metrisable connection. This will b e done in the next S ection. The second step will b e prolonging this linear system. Th is p oint was also under s to o d b y Liouville although he did not carry out the explicit compu tations. Geometrically this will come down to constructing a connection on a certain r ank six real vec tor bund le o ver U . The non-degenerate parallel sections of this b undle are in one to one corresp ondence the metrics whose geod esics are the geod esics of the give n pr o jectiv e structur e. In the generic case, the bu n dle has no parallel sections and hence the p r o jectiv e structure d o es n ot come f rom metric. In the real analytic case the pro jectiv e structure for which there is a single parallel section d ep ends on one arbitrary function of t wo v ariables, u p to d iffeomorphism. Finally we s hall obtain (1.5) as the int egrabilit y cond itions for the existence of a parallel section of this bu ndle. This will b e d one in Section 3. METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 3 In Secti on 4 we shall presen t some su fficien t conditions for metrisabilit y . All consider- ations here will b e in the real analytic catego r y . The p oint is that even if [Γ] is lo cally metrisable around every p oint in U , the global metric on U ma y not exist in the smo oth catego r y even in the simp ly-connected case. Thus no set of lo cal obstructions can guaran tee metrisabilit y of the w hole su rface U . Theorem 1.3. L et [Γ] b e a r e al an alytic pr oje c tiv e structur e such th at r ank ( M ([ Γ])) < 6 on U and ther e exist p ∈ U such that r ank ( M ( [Γ])) = 5 and W 1 W 3 − W 2 2 6 = 0 at p , wher e ( W 1 , W 2 , ..., W 6 ) sp ans the kernel of M ([Γ]) . Then [Γ] is metr isable in a sufficiently smal l neighb ourho o d of p if the r ank of a 10 by 6 matrix with the r ows ( V , D a V , D ( a D b ) V , D ( a D b D c ) V ) is e qual to 5 . M or e over this r ank c ondition holds if and only if two r elative i nv ariants E 1 , E 2 of or der 6 c onstructe d fr om the pr oje ctiv e structur e vanish. W e shall explain h o w to construct these tw o additional inv ariant s and sho w that the resulting set of conditions, a single 5th order equ ation (1.5) and t w o 6t h order equatio ns E 1 = E 2 = 0 form an in v olutive system whose general solution dep end s on three functions of t wo v ariables. In the degenerate cases wh en rank( M ([Γ])) < 5 h igher order obstructions will arise 2 : one condition of order 8 in the rank 3 case and one condition of order 7 in the rank 4 case. If rank ( M ([Γ])) = 2 there is alw a ys a four parameter family of metrics. If rank ( M ([Γ])) < 2 then [Γ] is pro jectiv ely flat in agreemen t with a theorem of Ko enigs [16]. In general we h a ve Theorem 1.4. A r e al analytic pr oje ctive structur e [Γ] i s metrisable in a sufficie ntly smal l neighb ourho o d of p ∈ U if and only i f the r ank of a 21 b y 6 matrix with the r ows M max = ( V , D a V , D ( a D b ) V , D ( a D b D c ) V , D ( a D b D c D d ) V , D ( a D b D c D d D e ) V ) is smal ler than 6 and ther e exists a ve ctor W in the kernel of this matrix su c h that W 1 W 3 − W 2 2 do es not vanish at p . The signature of the metric underlying a pr o jectiv e stru cture can b e Riemannian or Loren tzian dep end ing on the sign of W 1 W 3 − W 2 2 . In the generic case describ ed by Theorem 1.3 this sign ca n be foun d by ev aluating the p olynomial (4.28) of degree 10 in the entrie s of M ([Γ]) at p . In Section 5 w e shall construct v arious examples illustrating th e necessit y for the gener- icit y assumptions that w e hav e m ade. In Section 6 we shall d iscuss the t wistor app roac h to the p roblem. In this approac h a r eal analytic pr o jectiv e structure on U corresp ond s to a complex su rface Z h a ving a family of r ational curv es w ith s elf-in tersection num b er one. The metrisabilit y cond ition and the asso ciat ed linear sys tem are b oth deduced f r om the ex- istence of a certain ant i-canonical divisor on Z . In Section 7 w e sh all p resen t an alternativ e tensorial expression for (1.5) in terms of the curv ature of th e pro jectiv e connection and its co v arian t deriv ativ es. In particular we w ill shall sh o w that a section of the 14th p o w er of the canonical bund le of U det ( M )([Γ]) ( dx ∧ dy ) ⊗ 14 is a pro jectiv e inv ariant . T he appr oac h will b e that of tractor calculus [10]. In the der iv ation of the n ecessary condition (1.5) w e assume that the pr o jectiv e str u cture [Γ] adm its contin uous fifth deriv ativ es. The d iscussion of the sufficien t conditions and considerations in S ection 6 r equ ire [Γ] to b e real analytic. W e relegate some long formulae to the App endix. 2 W e shall alw ays assume that the rank of M ([Γ]) is constant in a sufficiently small neighbourho o d of some p ∈ U . 4 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD W e shall fin ish this in tro duction with a comment ab out the formalism used in th e pap er. The linear system go ve r ning the metrisabilit y problem and its prolongation are constru cted in elemen tary wa y in S ections 2 –3 an d in tensorial tr actor formalism in Section 7. T h e resulting obstr u ctions are alwa ys giv en by inv arian t exp r essions. The mac h in ery of the Cartan connectio n could of course b e applied to do the ca lculations in v ariant ly fr om the v ery b eginnin g. Th is is in f act ho w some of the r esults ha ve b een obtained [2]. The readers familiar with the C artan approac h will r ealise that the rank six vec tor b undles used in our pap er are asso ciated to the SL(3 , R ) p rincipal bu ndle of Cartan. Suc h readers should b ewa re, ho wev er, that the connection D a that we naturally obtain on such a v ector bund le is n ot ind u ced by the Cartan connection of the und erlying pro jectiv e structure but is a minor mo dification thereof, as detailed for example in [11]. V arious w eigh ted inv arian ts on U , like (1.5), are pull-bac ks of f u nctions from the total space of C artan’s bun dle. Ac kno w ledgemen ts. The first author is supp orted b y the National Science F oun dation via grant DMS-0604195. The second au th or is grateful to Jeny a F erap onto v, Rod Go v er, Vladimir Matv eev an d Paul T o d for h elpful discuss ions . He also thanks BIRS in Banff and ESI in Vienna for h ospitalit y w here some of this research w as done. His work was p artly supp orted by Ro yal So ciet y an d Lond on Mathematical So ciet y gran ts. The third author is supp orted by the Australian Researc h Coun cil. 2. Linear Sys tem Let us assu me that the pro jectiv e structur e [Γ] is metrisable. Therefore there exist a symmetric b i-linear form g = E ( x, y ) dx 2 + 2 F ( x, y ) dxdy + G ( x, y ) dy 2 (2.7) suc h that the unp arametrised geod esics of g coincide with the in tegral cur v es of (1.3). The diffeomorphisms can b e used to eliminat e t wo arbitrary functions from g (for example to express g in isothermal co ordinates) but we shall not use this freedom. W e wan t to determine wh ether the four f u nctions ( A 0 , ..., A 3 ) arise from thr ee functions ( E , F , G ) so one might exp ect only one condition on the A s. This heur istic numerology is wrong and we shall d emonstrate in S ection 4 that three conditions are n eeded to establish sufficiency in the generic case 3 . W e c ho ose a d irect r oute and express the equation for n on-parametrised geo desics of g in the form (1.3). Using the Levi-Civita relation Γ c ab = 1 2 g cd  ∂ g ad ∂ x b + ∂ g bd ∂ x a − ∂ g ab ∂ x d  3 Additional conditions would arise if w e demand ed that t here b e more than one metric with the same unparametrised geo desics. I n our approach this situation corresponds to th e existence of tw o indep endent parallel sections of th e rank six bund le ov er U . The corresp onding metrics were, in the p ositive defi nite case, found by J. Liouville (the more famous of the tw o Liouvilles) and characterised by Dini. They are of the form (2.7) where F = 0 , E = G = u ( x ) + v ( y ) up to diffeomorphism. Roger Liouville whose steps w e follo w in t h is pap er w as a younger relative of Joseph and attended his lectures at th e Ecole Polytec h n ique. METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 5 and formulae (1.2), (1.3 ) yields the follo w ing expressions A 0 = 1 2 E ∂ y E − 2 E ∂ x F + F ∂ x E E G − F 2 , A 1 = 1 2 3 F ∂ y E + G∂ x E − 2 F ∂ x F − 2 E ∂ x G E G − F 2 , A 2 = 1 2 2 F ∂ y F + 2 G∂ y E − 3 F ∂ x G − E ∂ y G E G − F 2 , A 3 = 1 2 2 G∂ y F − G∂ x G − F ∂ y G E G − F 2 . (2.8) This giv es a fi r st order n onlinear d ifferen tial op erator σ 0 : J 1 ( S 2 ( T ∗ U )) − → J 0 (Pr( U )) (2.9) whic h carries the metric to its asso ciated pro jectiv e structur e. This op erator is defined on the first jet sp ace of s ymmetric tw o-forms as it dep ends on th e metric and its deriv ativ es. It tak es its v alues in th e affine rank 4 bun dle Pr( U ) of pr o jectiv e structures whose asso ciated v ector bu ndle Λ 2 ( T U ) ⊗ S 3 ( T ∗ U ) arises as a quotient in the exact sequence 0 − → T ∗ U − → T U ⊗ S 2 ( T ∗ U ) − → Λ 2 ( T U ) ⊗ S 3 ( T ∗ U ) − → 0 . This is a more abstract w a y of defining the equiv alence relation (1.1). W e will retur n to it in Section 4 . The op erator σ 0 is homogeneous of d egree zero so rescaling a metric by a constan t d o es not c h ange the resulting pro jectiv e s tructure. F ollo win g Liouville [19] we set E = ψ 1 / ∆ 2 , F = ψ 2 / ∆ 2 , G = ψ 3 / ∆ 2 , ∆ = ψ 1 ψ 3 − ψ 2 2 and substitute int o (2 .8 ). This yields an o v erd etermin ed system of four linear first order PDEs for three functions ( ψ 1 , ψ 2 , ψ 3 ) and pro ves the follo wing Lemma 2.1 (Liouville [19 ]) . A pr oje ctive structur e [Γ] c orr esp onding to the se c ond or der ODE (1.3) is metrisable on a neighb ourho o d of a p oint p ∈ U iff ther e exi sts functions ψ i ( x, y ) , i = 1 , 2 , 3 define d on a nei ghb ourho o d of p such that ψ 1 ψ 3 − ψ 2 2 do es not vanish at p and such that the e quations ∂ ψ 1 ∂ x = 2 3 A 1 ψ 1 − 2 A 0 ψ 2 , ∂ ψ 3 ∂ y = 2 A 3 ψ 2 − 2 3 A 2 ψ 3 , ∂ ψ 1 ∂ y + 2 ∂ ψ 2 ∂ x = 4 3 A 2 ψ 1 − 2 3 A 1 ψ 2 − 2 A 0 ψ 3 , ∂ ψ 3 ∂ x + 2 ∂ ψ 2 ∂ y = 2 A 3 ψ 1 − 4 3 A 1 ψ 3 + 2 3 A 2 ψ 2 (2.10) hold on the domain of definition. This lin ear system f orms a basis of our d iscussion of the metrisabilit y condition. It has recent ly b een used in [5] to construct a list of metrics on a t wo-dimensional surface admitting a tw o-dimensional group of pro jectiv e transformations. Its equiv alen t tensorial form, app licable in h igher d imensions, is pr esented for example in [11]. W e shall use this form in Section 7. 6 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD Here is a w a y to ‘remem b er’ (2.10). Int ro duce the symmetric pr o jectiv e connection ∇ Π with connection symb ols Π c ab = Γ c ab − 1 n + 1 Γ d da δ c b − 1 n + 1 Γ d db δ c a (2.11) where in our case n = 2. F orm u la (1.1) implies that the symb ols Π c ab do not dep end on a c hoice of Γ is a p ro jectiv e class. They are related to the second order ODE (1.3) by Π 1 11 = 1 3 A 1 , Π 1 12 = 1 3 A 2 , Π 1 22 = A 3 , Π 2 11 = − A 0 , Π 2 21 = − 1 3 A 1 , Π 2 22 = − 1 3 A 2 . The pro jectiv e co v arian t deriv ativ e is d efined on one-forms by ∇ Π a φ b = ∂ a φ b − Π c ab φ c with natural extension to other tensor bundles. The Liouville system (2.10) is then equiv alen t to ∇ Π ( a σ bc ) = 0 , (2.12) where th e round b rac ket s denote symmetrisation and σ bc is a rank 2 symmetric tensor with comp onent s σ 11 = ψ 1 , σ 12 = ψ 2 , σ 22 = ψ 3 . W e sh all end this S ection w ith a historical digression. The solution to th e metrisabilit y problem has b een reduced to find ing differen tial relations b et w een ( A 0 , A 1 , A 2 , A 3 ) when (2.8), or equiv alen tly (2.10), holds. These r elations are required to b e diffeomorphism in v ariant conditions, so w e are searching for invariants of th e ODE (1.3) u nder the p oin t transformations ( x, y ) − → ( ¯ x ( x, y ) , ¯ y ( x, y )) . (2.13) The p oin t inv arian ts of 2nd order ODEs hav e b een extensiv ely studied b y the classical differen tial geometers in late 19th and early 20th century . The earliest reference we are a wa re of is the wo r k of Liouville [18 , 19], who constructed p oin t inv arian ts of 2nd ord er ODEs cubic in the first deriv ativ es (it is easy to ve rify that the ‘cubic in the fir st deriv ativ e’ condition is itself in v ariant under (2.13)). Th e most complete work was pro d uced b y T resse (who w as a student of Sophus Lie) in his d issertation [23]. T resse stud ied the general case (1.6) and classified all p oin t in v ariants of a giv en differential order. The fi rst tw o inv arian ts are of order fou r I 0 = Λ 1111 , I 1 = D 2 x Λ 11 − 4 D x Λ 01 − Λ 1 D x Λ 11 + 4Λ 1 Λ 01 − 3Λ 0 Λ 11 + 6Λ 00 , where Λ 0 = ∂ Λ ∂ y , Λ 1 = ∂ Λ ∂ y ′ , D x = ∂ ∂ x + y ′ ∂ ∂ y + Λ ∂ ∂ y ′ . Strictly sp eaking these are only relativ e inv arian ts as they transform w ith a certain weigh t under (2.13). Th eir v anishing is ho w ever in v ariant. T r esse sh o wed that if I 0 = 0, then I 1 is lin ear in y ′ . T his is the case considered by Liouville. T o m ake con tact with th e w ork of Liouville w e note that I 1 = − 6 L 1 − 6 L 2 y ′ where the expressions L 1 = 2 3 ∂ 2 A 1 ∂ x∂ y − 1 3 ∂ 2 A 2 ∂ x 2 − ∂ 2 A 0 ∂ y 2 + A 0 ∂ A 2 ∂ y + A 2 ∂ A 0 ∂ y − A 3 ∂ A 0 ∂ x − 2 A 0 ∂ A 3 ∂ x − 2 3 A 1 ∂ A 1 ∂ y + 1 3 A 1 ∂ A 2 ∂ x , L 2 = 2 3 ∂ 2 A 2 ∂ x∂ y − 1 3 ∂ 2 A 1 ∂ y 2 − ∂ 2 A 3 ∂ x 2 − A 3 ∂ A 1 ∂ x − A 1 ∂ A 3 ∂ x + A 0 ∂ A 3 ∂ y + 2 A 3 ∂ A 0 ∂ y + 2 3 A 2 ∂ A 2 ∂ x − 1 3 A 2 ∂ A 1 ∂ y (2.14) w ere constructed by L iouville who has also pr o ve d that Y = ( L 1 dx + L 2 dy ) ⊗ ( dx ∧ dy ) METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 7 is a pro jectiv ely inv arian t tensor. The follo w ing result was known to b oth T resse and Liouville Theorem 2.2 ( Liouville [18 ], T resse [23]) . The 2nd or der ODE (1.6) is trivialisable by p oint tr ansformation (i . e. e quivalent to y ′′ = 0 ) iff I 0 = I 1 = 0 , or, e qui v alently, i f Λ is at most cu bic in y ′ and Y = 0 . W e note that the separate v anishing of L 1 or L 2 is not in v ariant . If b oth L 1 and L 2 v anish the p ro jectiv e structure is flat is the sens e describ ed in Section 7. 3. Prolonga tion and Cons istency Pro of of Theorem 1.1. T h e obstruction (1.5 ) will arise as the compatibilit y condition for the system (2.10). This system is o verdetermined, as there are more equations than unknowns. W e shall use the method of pr olonga tion and mak e (2.10) ev en m ore ov erde- termined 4 b y sp ecifying the d eriv ativ es of ψ i , i = 1 , 2 , 3 at an y giv en p oin t ( x, y , ψ i ) ∈ R 5 , th u s determining a tangent plane to a solution surface (if one exists) ( x, y ) − → ( x, y , ψ 1 ( x, y ) , ψ 2 ( x, y ) , ψ 3 ( x, y )) . F or this w e n eed six cond itions, an d the s y s tem (2.10) consist of four equations. W e need to add t wo conditions and we c h o ose ∂ ψ 2 ∂ x = 1 2 µ, ∂ ψ 2 ∂ y = 1 2 ν, (3.15 ) where µ, ν dep end on ( x, y ). Th e inte grabilit y cond itions ∂ x ∂ y ψ i = ∂ y ∂ x ψ i giv e thr ee PDEs for ( µ, ν ) of the form ∂ µ ∂ x = P , ∂ ν ∂ y = Q, ∂ ν ∂ x − ∂ µ ∂ y = 0 , (3.16) where ( P , Q ) give n by (A55) are expr essions linear in ( ψ i , µ, ν ) w ith co efficients dep ending on A α and their ( x, y ) deriv ativ es. The system (3.16) is again o verdetermined bu t we still need to pr olong it to sp ecify the v alues of all first deriv ativ es. It is immediate that the complex c haracteristic v ariet y of the system (2.1 0 ) is empt y , so the general theory (see Ch apter 5 of [3]) implies that, after a finite num b er of differen tiations of these equations (i.e., prolongations), all of the partials of the ψ i ab o ve a certain ord er can b e written in terms of low er ord er partials, i.e., th e prolonged system will b e complete. Alternativ ely , th e Liouville sy s tem w r itten in the form (2.12) is one of the simplest examples co vered by [1] in which the form of the prolongation is easily predicted. In an y case no app eal to the general theory is n eeded as it is easy to see that completion is reac hed by addin g one further equation ∂ µ ∂ y = ρ, (3.17) 4 Another approach more in th e sp irit of Liouville [19] would b e to eliminate ψ 2 and ψ 3 from (2.10) to obtain a system of tw o 3rd order PD Es for one function f := ψ 1 ( ∂ 3 x ) f = F 1 , ∂ y ( ∂ 2 x ) f = F 2 , where F 1 , F 2 are linear in f and its first and second deriva t ives with co efficients d ep en ding on A α ( x, y ) and their d eriv atives (the co efficient of ( ∂ 2 y ) f in F 1 is zero). The consistency ∂ y ( ∂ x ) 3 f = ∂ x ∂ y ( ∂ x ) 2 f give s a linear eq u ation for ∂ x ( ∂ y ) 2 f . Then ∂ x ( ∂ y ) 2 ∂ x f = ( ∂ y ) 2 ( ∂ x ) 2 f gives an eq uation for ( ∂ y ) 3 f . After this step the system is closed: all th ird order deriv atives are ex pressed in terms of low er order deriv atives. T o work out further consistencies imp ose ∂ x ( ∂ y ) 3 f = ( ∂ y ) 3 ∂ x f whic h gives ( when all 3rd order eq uations are u sed) a second order linear PDE for f . W e carry on d ifferentia t ing this second order relation to pro duce the remaining second order relations (b ecause we know all third order deriv atives), th en the first order relations and finally an algebraic relation whic h will constrain th e initial d ata unless (1.5) holds. 8 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD where ρ = ρ ( x, y ) and imp osing the consistency conditions on the system of f our PDEs (3.16, 3.17). This leads to ∂ ρ ∂ x = R, ∂ ρ ∂ y = S, (3.18) where R, S gi v en by (A55) are fun ctions of ( ρ, µ, ν, ψ i , x, y ) which are linear in ( ρ, µ, ν , ψ i ). After this step the prolongatio n process is finished and all the first d eriv ativ es ha v e b een determined. The final compatibilit y condition ∂ x ∂ y ρ = ∂ y ∂ x ρ for the sy s tem (3.18) yields ∂ R ∂ y − ∂ S ∂ x + S ∂ R ∂ ρ − R ∂ S ∂ ρ = 0 . (3.19) All th e fi rst deriv ativ es are no w determined, so (3.19) is an algebraic linear condition of the form V · Ψ := 6 X p =1 V p Ψ p = 0 , (3.20) where Ψ = ( ψ 1 , ψ 2 , ψ 3 , µ, ν, ρ ) T is a v ector in R 6 , and V = ( V 1 , ..., V 6 ) wh ere V 1 = 2 ∂ L 2 ∂ y + 4 A 2 L 2 + 8 A 3 L 1 , V 2 = − 2 ∂ L 1 ∂ y − 2 ∂ L 2 ∂ x − 4 3 A 1 L 2 + 4 3 A 2 L 1 , V 3 = 2 ∂ L 1 ∂ x − 8 A 0 L 2 − 4 A 1 L 1 , V 4 = − 5 L 2 , V 5 = − 5 L 1 , V 6 = 0 (3.21) and L 1 , L 2 are giv en b y (2.14) . W e collect the linear PDEs (2.10, 3.15 3.16, 3.17 , 3.18) as d Ψ + Ω Ψ = 0 , (3.22) where Ω = Ω 1 dx + Ω 2 dy and ( Ω 1 , Ω 2 ) are 6 by 6 matrices with co efficien ts dep endin g on A α and their fi rst and second d eriv ativ es (A53). No w differenti ate (3. 20) t wice with resp ect to x a = ( x, y ), and use (3.22). This y ields six linear conditions V · Ψ = 0 , (3.23) ( D a V ) · Ψ := ( ∂ a V − V Ω a ) · Ψ = 0 , ( D b D a V ) · Ψ := ( ∂ b ∂ a V − ( ∂ b V ) Ω a − ( ∂ a V ) Ω b − V ( ∂ b Ω a − Ω a Ω b )) · Ψ = 0 whic h m u st hold, or there are no solutions to (2.10). Therefore the determinant of th e asso- ciated 6 by 6 matrix (1.4) must v anish, thus giving our first desired metrisability condition (1.5). W e note that the expression ( D b D a V ) · Ψ in (3.23) is sym m etric in its indices. This symmetry condition redu ces to V F = 0 (where F is giv en by (A54)) and holds id entica lly . The expr ession det ( M ([Γ])) is 5th ord er in the d eriv ativ es of connection co efficien ts. I t do es not v anish on a generic pro jectiv e stru ctur e, bu t v anishes on metrisable conn ections (2.8) by construction. T his ends the pro of of Theorem 1.1.  In the next Section we shall need the follo wing generalisation of the symmetry pr op erties of (3.23). Let D a W = ∂ a W − WΩ a , where W : U → R 6 . Th en [ D a , D b ] W = ( W F ) ε ab = W 6 V ε ab , where ε 00 = ε 11 = 0 , ε 01 = − ε 10 = 1. Thus D a D b V = D ( a D b ) V , D a D b D c V = D ( a D b D c ) V + ε ab L c V , ... , D a 1 D a 2 ...D a k V = D ( a 1 D a 2 ...D a k ) V + o ( k − 2) (3.24) METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 9 where o ( k − 2) denotes terms linear in D ( a 1 D a 2 ...D a m ) where m ≤ k − 2. Thus w e can restrict ours elv es to the symmetrised expressions as the an tisymm etrisations d o not add an y new cond itions. 4. Sufficiency co nditions It is clear from the d iscussion in the pr eceding Section that the condition (1.5 ) is necessary for the existence of a metric in a giv en pro jectiv e class. It is ho wev er not sufficien t and in this Section w e shall establish some sufficiency conditions in th e real analytic case. W e require the real analyticit y in ord er to b e able to ap p ly the Cauc h y–Kow alewski Th eorem to the p r olonged system of PDEs. In p articular Theorem 4.1 whic h u nderlies our app roac h in this Section b uilds on the Cau ch y–Ko wale wski Theorem. Let us start off b y reph r asing the construction p resen ted in the last Section in the geo- metric language. T he exterior differentia l ideal I asso ciat ed to the prolonged system (3.22) consists of six one-forms θ p = d Ψ p + (( Ω a ) pq Ψ q ) dx a , p, q = 1 , ..., 6 a = 1 , 2 . (4.25) Tw o vecto r fields annihilating th e one-forms θ p span the solution su rface in R 8 . The closure of this id eal comes do w n to one compatibilit y (3.20). W e no w w ant to fin d one parallel section Ψ : U → E of a ran k six v ector bun dle E → U with a connection D = d + Ω . Lo cally the total space of this bund le is an op en set in R 8 . Differen tiating (3.22) and eliminating d Ψ y ields FΨ = 0, wher e F = d Ω + Ω ∧ Ω = ( ∂ x Ω 2 − ∂ y Ω 1 + [ Ω 1 , Ω 2 ]) dx ∧ dy = F dx ∧ dy is the curv ature of D . T h u s w e need F Ψ = 0 , (4.26) where F = F ( x, y ) is a 6 b y 6 matrix giv en by (A54). W e find that th is matrix is of rank one and in th e c hosen b asis its fi rst fiv e ro w s v anish and its b ottom ro w is giv en by th e v ector V with comp onents given b y (3.2 1). Therefore (4. 26 ) is equiv alent to (3.20). W e differen tiate the condition (4.26) and use (3.22 ) to pro d uce algebraic matrix equations F Ψ = 0 , ( D a F ) Ψ = 0 , ( D a D b F ) Ψ = 0 , ( D a D b D c F ) Ψ , ... where D a F = ∂ a F + [ Ω a , F ]. Using the symmetry argumen t (3.24) sho w s that after K differen tiations this leads to n ( K ) = 1 + 2 + 3 + ... + ( K + 1) linear equati ons whic h w e write as F K Ψ = 0 , where F K is a n ( K ) b y 6 matrix dep end ing on A s and their deriv ativ es. W e also set F 0 = F . W e con tinue different iating and adj oining the equations. Th e F r ob enius Theorem adapted to (4.26) and (3.22) tells us w hen we can stop the p ro cess. Theorem 4.1. Assume that the r anks of the matric es F K , K = 0 , 1 , 2 , ... ar e maximal and c onstant 5 . L et K 0 b e the smal lest natur al numb er such that r ank ( F K 0 ) = r ank ( F K 0 +1 ) . (4.27) If K 0 exists then r ank ( F K 0 ) = r ank ( F K 0 + k ) for k ∈ N and the sp ac e of p ar al lel se ctions (3.22) of d + Ω has dimension S ([Γ]) = 6 − r ank ( F K 0 ) . 5 This can alw ays b e ac h ieve d by restricting t o a su fficien tly small neighbourho od of some p oint p ∈ U . 10 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD The fi rst and second deriv ativ es of (4.26 ) w ill pro d u ce six indep enden t conditions on Ψ , and these conditions are precisely (3.23). Th us the necessary metrisabilit y condition (1.5) comes down to restricting the holonom y of the connection D on the rank six vec tor bun dle E . W e shall no w assume that (1. 5 ) holds a nd use T heorem 4.1 to construct the s u fficien t conditions for the existence of a Levi–Civita connection in a given pr o jectiv e class. First of all there must exist a vec tor W = ( W 1 , ..., W 6 ) T in the k ernel of M ([Γ]), such that W 1 W 3 − ( W 2 ) 2 6 = 0. This will guaran tee that the corresp ondin g quad r atic form (if one exists) on U is non-degenerate. It is straightfo rw ard to v erify in the case when M ([Γ]) has rank 5 as then kernel ( M ([Γ])) is s p anned b y an y non-zero column of adj ( M ([Γ])) where the adjoin t of a matrix M is defined by M adj ( M ) = det ( M ) I . The entrie s of adj ( M ([Γ])) are determinants of th e co-factors o f M ([Γ]) and thus are p olynomials of degree 5 in the en tries of M ([Γ]) so P ([Γ]) = W 1 W 3 − ( W 2 ) 2 (4.28) is a p olynomial of degree 10 in the en tries of M ([Γ]). Definition 4.2. A pr oje c tive structur e for which (1 .5 ) holds is c al le d generic in a neigh- b ourho o d o f p ∈ U if r ank M ([Γ]) is maximal and e qual to 5 and P ([Γ]) 6 = 0 in this neighb ourho o d. In this generic case Th eorem 4.1 and Lemma 2.1 imply that there will exist a Levi–Civita connection in the pr o jectiv e class if the rank of the next der ived matrix F 3 do es not go up and is equ al to fiv e. W e shall see that this can be guarante ed by imp osing t wo more 6th order conditions on [Γ]. Pro of of Theorem 1.3 . First note that, in the generic case, the three vecto r s V , V a := ∂ a V − V Ω a , a = 1 , 2 m u s t b e linearly indep endent or otherwise the rank of M ([Γ]) would b e at most 3. No w pic k tw o indep end en t vec tors from the set V ab := ( ∂ b ∂ a V − ( ∂ b V ) Ω a − ( ∂ a V ) Ω b − V ( ∂ b Ω a − Ω a Ω b )) suc h that the r esulting set of fi v e v ectors is indep enden t. Say we h a ve pick ed V 00 and V 11 . W e no w tak e the third deriv ativ es of (3.20) with resp ect to x a and use (3.22) to eliminate deriv ativ es of Ψ . This adds f our vect ors to our set of five and so a priori w e need to satisfy four six order equations to ensure th at the r ank d o es n ot go up. Ho we ver only tw o of these are new and the other t wo are deriv ativ es of the 5th order condition (1.5). Before w e shall prov e this statemen t examining the images of linear op erators indu ced from (2.9) on j et s paces let us indicate why this count ing wo r ks. Let V ab...c denote the vecto r in R 6 annihilating Ψ (in th e sense of (3.20)) which is obtained b y eliminating the d eriv ativ es of Ψ fr om ∂ a ∂ b ...∂ c ( V · Ψ ) = 0. W e ha ve already argued in (3.24) that the an tisymmetrising o ve r an y pair of indices in V ab...c only adds lo wer order conditions. Th us w e shall alw ays assume that these expressions are symm etric. W e shall also set V 0 = V x , V 1 = V y . Our assump tions imply that V xy = c 1 V + c 2 V x + c 3 V y + c 4 V xx + c 5 V y y (4.29) METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 11 for some functions c 1 , ..., c 5 on U . The tw o six order conditions E 1 := det         V V x V y V xx V y y V xxx         , E 2 := det         V V x V y V xx V y y V y y y         , (4.30) ha ve to b e added for su fficiency . No w differenti ating (4.29) w .r .t x, y and using V xy y = V y y x , V xy x = V xxy (whic h hold mod ulo low er order terms ), implies th at V xy y and V xy x are in the span of { V , V x , V y , V xx , V y y , V xxx , V y y y } and n o additional cond itions need to b e add ed . T his pr o cedure can b e rep eate d if instead V y y b elongs to the span of { V , V x , V y , V xx , V xy } . No w we shall present the general argument. Consider the homogeneous differen tial op- erator (2.9). It maps the 1st jets o f metrics on U to the 0th jets of pro jectiv e structures. Differen tiating the relations (2.8) p rolongs this op erator to b undle maps σ k : J k +1 ( S 2 ( T ∗ U )) − → J k (Pr( U )) (4.31) from ( k + 1)-jets of metrics to k -jets of p ro jectiv e structur es. It has at least one d imensional fibre b ecause of the homogeneit y of σ 0 . The r ank of σ k is not constan t as we already kno w that the system (2.8 ) (or its equiv alen t linear form (2.10)) do es not ha ve to admit an y solutions in general but will admit at least one solutio n if the p r o jectiv e structure is metrisable. Th e table b elo w giv es the ranks of the jet b undles of metrics and pro jectiv e structures, the dimensions of the fib res of σ k and finally the image cod imension. The n u m b er of new conditions on [Γ] arising at eac h step is den oted b y a b old figure in the column co-rank(ker σ k ). k rank( J k +1 ( S 2 ( T ∗ U ))) ra nk( J k (Pr( U ))) rank(k er σ k ) co- rank(k er σ k ) − 1 3 − − − 0 9 4 5 0 1 18 12 6 0 2 30 24 6 0 3 45 40 5 0 4 63 60 3 0 5 84 84 1 1 = 1 6 10 8 112 1 5 = 3 + 2 7 13 5 144 1 10 = 6 + 6 − 2 There is no obstruction on a pro jectiv e s tr ucture b efore the ord er 5 so σ k are onto and generically submersive for k < 4. At k = 5 there h as to b e at least a 1-dimens ional fi b er, so the image of th e deriv ed map can at most b e 83-dimensional at its smo oth p oin ts. I n fact, w e ha ve sho wn that there is a condition there, giv en by (1 .5 ) , so it must define a co dimension 1 v ariet y that is generically smo oth. When the matrix M ([Γ]) has r ank 5, the equation (1.5) is regular, s o it f ollo ws that, outside the region wh ere (1.5) ceases to b e a regular 5th order PDE the solutions of this PDE will hav e their k -jets constrained b y the d eriv ativ es of (1.5) of order k − 5 or less. This sho w s that, a t k = 6, the 6-jets of the regular s olutions of (1.5) will hav e co d imension 3 in all 6-jets of pro jectiv e stru ctures, i.e., they will ha ve dimension 112 − 3 = 109. Ho wev er, we kn ow that the image of the 7-jets of metric s tr uctures can ha ve only dimension 108 − 1 = 107. Th us, th e 6-jets of regular metric structures hav e co dimension 2 in the 6-jets of regular solutions of (1.5). That is wh y there 12 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD ha ve to b e t w o m ore 6th order equations E 1 = 0 , E 2 = 0 . (4.32) The image in 6-jets has total co dimen s ion 5, i.e., it is cut out by a 5th order equation and four 6th order equations. Ho wev er, t w o of the 6th order equations a re obviously the deriv ativ es of the 5th order equation. The n ext line s h o ws that, at 7th order, the image has only co d imension 10, which m eans that there must b e 2 relations b etw een the first deriv ativ es of the 6th order equ ations and the second d eriv ativ es of the 5th order equation whic h imp lies that the resulting system of three equations is inv olutiv e. This ends the pro of of T h eorem 1.3.  The analysis of the non-generic cases w h ere the rank of M ([Γ]) < 5 is slightly m ore complicated. The argument based on the dimen sionalit y of jet bund les asso ciated to (4.31 ) breaks do w n as the PDE det M ([Γ]) = 0 is not regular and d o es n ot defi n e a smo oth co-dimension 1 v ariet y in J 5 (Pr( U )). Let S ([Γ]) b e the dimension of th e vec tor sp ace of solutions to the linear system (2.10). Some of these solutions ma y corresp ond to degenerate quadr atic forms on U bu t nev erth eless w e hav e Lemma 4.3. If S ([Γ]) > 1 then ther e ar e S ([Γ]) indep endent non-de gener ate quadr atic forms among the solutions to (2.10) . Pro of. Let us assume that at least one solution of (2.10) giv es rise to a quadratic form whic h is degenerate (rank 1) ev eryw here. W e can c h o ose co ordin ates su c h that this s olution is of th e form ( ψ 1 , 0 , 0). The statemen t of the Lemma will follo w if w e can sho w that th er e is n o ot h er s olution o f the form ( φ ( x, y ) ψ 1 , 0 , 0) where φ ( x, y ) is a non-constan t function. The Liouville system (2.10) is readily solv ed in this case to giv e A 1 = 3 2 1 ψ 1 ∂ ψ 1 ∂ x , A 2 = 3 4 1 ψ 1 ∂ ψ 1 ∂ y , A 3 ( x, y ) = 0 with A 0 unsp ecified. Thus for a giv en pro jectiv e class th e only freedom in this solution is to rescale ψ 1 b y a constan t.  Pro of of Theorem 1.4. W e shall list the num b er and the ord er of obstru ctions o ne can exp ect dep en d ing on the rank M ([Γ]). • If rank M ([Γ]) < 2 th e pro jectiv e str ucture is pro jectiv ely flat as L 1 = L 2 = 0, and the second order ODE is equiv alen t to y ′′ = 0 b y Theorem 2.2. This is obvi ous if rank M ([Γ]) = 0 as then V = 0 and f orm u la (3.21) gives L 1 = L 2 = 0. If rank M ([Γ]) = 1 then ∂ a V − VΩ a = γ a V (4.33) for some γ a . Using the expr essions (A53) for Ω a and the form ula (3.21) yields VΩ a = ( ∗ , ∗ , ∗ , ∗ , ∗ , 5 L a ) where ∗ are some terms wh ic h n eed not concern us an d L a are th e Liouville expres- sions (2.14). Com b in ing this with (4.33) yields L 1 = L 2 = 0. • If rank M ([Γ]) = 2 then V + c 1 V x + c 2 V y = 0 (4.34) for fu nctions c 1 , c 2 at least one of which do es not identic ally v anish. Differen tiating this r elation and u sing the fact that V ab ∈ span { V , V a } we see that no new relations METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 13 arise and so the system is closed at this lev el. In this case there exists a four dimensional family of metrics compatible with the giv en pro jectiv e str ucture. • If rank M ([Γ]) = 3 w e h a ve to consider t wo cases. If { V , V x , V y } are linearly indep end en t then reasoning as ab o ve shows that fu rther differen tiations do not add an y new conditions. The other p ossibilit y is that { V , V x , V xx } or { V , V y , V y y } are linearly indep enden t. Let us concen trate on the first case (or swa p x with y if n ecessary). T aking further x deriv ativ es ma y increase th e rank of the resulting system, bu t the y deriv ativ es will not yield an y new cond itions as ca n b e seen b y mixing the partial deriv ativ es and u s ing c 0 V + c 1 V x + c 2 V y = 0 , whic h is a consequence of the rank 3 condition. Let us assu m e that the rank incr eases to 5 by add ing tw o v ectors V xxx , V xxxx (otherwise the system is closed with r ank 3 or 4). T he rank will sta y 5 if one fu rther differen tiation do es not add new conditions. T h u s the first and only obstru ction in this case is of order 8 in the pro jectiv e structure det         V V x V xx V xxx V xxxx V xxxxx         = 0 . (4.35) • T he analogous p r o cedure can b e carried o ver if rank( M ([Γ])) = 4. Assu ming that the four linearly indep enden t v ectors are { V , V x , V y , V xx } leads to one obstr uction of order 7 det         V V x V xy V xx V xxx V xxxx         = 0 . This completes the pro of of Theorem 1.4.  As a corollary from this analysis w e dedu ce the result of Ko enigs [16] Theorem 4.4. [16 ] The sp ac e of metrics c omp atible with a given pr oje ctive structur es c an have dimensions 0 , 1 , 2 , 3 , 4 or 6 . Our appr oac h to the Ko enigs’s th eorem is similar to that of Kr uglik o v’s [17] who has ho w- ev er constructed an additional set of in v arian ts determining whether a metrisable p ro jectiv e structure adm its more than one metric in its pro jectiv e class. 5. Examples It is p ossible that the determinant (1.4 ) v anishes and th e pro jectiv e structure [Γ] is non metrisable either b ecause th e further higher order o bstructions d o not v anish, or b ecause a solution to the Liouville system (2.10) is degenerate as a quadr atic form on T U . It can also happ en when the pro jectiv e str u cture fails to b e r eal analytic. In this section we sh all giv e four examples illustrating this. 14 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD 5.1. The imp ortance of 6th order conditions. Consider a one p arameter f amily of homogeneous pr o jectiv e structures corresp ondin g to the second ord er ODE d 2 y dx 2 = c e x + e − x  dy dx  2 . F or generic c the matrix M ([Γ]) h as rank six and the 5th order condition (1.5) h olds if ˆ c = 48 c − 11 is a ro ot of a quartic ˆ c 4 − 11286 ˆ c 2 − 8509 68 ˆ c − 19529 683 = 0 . (5.36) The 6th order conditions (4.32) are satisfied iff 3 ˆ c 5 + 529 ˆ c 4 + 222 ˆ c 3 − 2131102 ˆ c 2 − 1031968 49 ˆ c − 197790 0451 = 0 , ˆ c 3 − 213 ˆ c 2 − 7849 ˆ c − 19235 = 0 . It is easy to v erify that these three p olynomials d o not h av e a common ro ot. Cho osing ˆ c to b e a real ro ot of (5.36) we can mak e the 5th order obstruction (1.5) v anish, but the t wo 6th ord er obstru ctions E 1 , E 2 do not v anish. 5.2. The imp ortance of the non-degenerat e kernel. This example illustrates why we cannot hop e to c h aracterise the metrisabilit y condition purely b y v anish ing of an y set of in v ariant s. Let f b e a smo oth function on an op en set U ⊂ R 2 . Consider a one-parameter family of metrics g c = c exp ( f ( x, y )) dx 2 + dy 2 , where c ∈ R + . The corresp onding one-parameter family of pro jectiv e structures [Γ c ] is giv en by the O DE d 2 y dx 2 = c 2 ∂ f ∂ y exp ( f ) + 1 2 ∂ f ∂ x  dy dx  + ∂ f ∂ y  dy dx  2 . The 5th order obstruction (1.5) and 6th order conditions E 1 , E 2 of course v anish. Moreo ver rank M ([Γ c ]) = 5 for generic f ( x, y ). No w tak e the limit c = 0. The obstru ctions still v anish and rank M ([Γ 0 ]) = 5 but [Γ 0 ] is not metrisable. This is b ecause one can select a 3 by 3 lin ear su bsystem f M 0 φ = 0, w here φ = ( ψ 1 , ψ 2 , µ ) T , from the 6 b y 6 sy s tem (3 .22 ). The 3 b y 3 matrix f M 0 can b e read off (3.22). F or generic f the determinan t of f M 0 do es not v anish and so there d o es n ot exist a parallel sec tion Ψ of (3.22) such that ψ 1 ψ 3 − ψ 2 2 6 = 0. F or example f = xy giv es r ank M ([Γ 0 ]) = 5 and det ( f M 0 ) = 3 xy 4 − 9 2 . This n on-metrisable example fails the genericit y assu mption P ([Γ]) 6 = 0 where P ([Γ]) is giv en b y (4.28 ). The kernel of M ([Γ 0 ]) is sp anned by a vec tor (0 , 0 , 1 , 0 , 0 , 0) T and the corresp ondin g qu adratic form on T U is degenerate. 5.3. The imp ortance of real analyticity. This example illustrates why w e n eed to w ork in the r eal analytic case to get sufficien t conditions. W e sh all construct a simply connected pro jectiv e surf ace in wh ic h ev ery p oin t has a neigh b ourho o d on which there is a m etric compatible with th e giv en pro jectiv e structur e, but there is no m etric defined on the whole surface that is compatible with the pro jectiv e stru cture. Consider a plane U = R 2 with cartesian co ord inates ( x, y ). T ak e t wo constan t co efficien t metrics on the plane that are linearly indep enden t, say , g + and g − . No w consid er a mo d i- fication of g − in the half-plane x < − 1 such th at the mo d ified g − is the only global metric that is compatible with its underlying pro jectiv e structure. Similarly , consider a mo difi - cation of g + on the half-plane x > 1 suc h that the mo dified g + is the only global metric that is compatible w ith its un derlying pro jectiv e stru cture. The tw o p ro jectiv e structures METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 15 agree (with the fl at o ne) in the strip − 1 < x < 1, so let the new pro jectiv e structure b e the one that agrees w ith that of mo dified g − when x < 1 and with th e mo d ified g + when x > − 1. T his fi nal p r o jectiv e structure will h a ve compatible metrics lo cally near eac h p oint (sometimes, more than one, up to m u ltiples), bu t will not ha ve a compatible m etric globally . Thus, m etrisabilit y cannot b e detected lo cally in the smo oth category . 5.4. One more degenerate example. T ak e Γ 2 11 = A ( x, y ) and set all other comp onents of Γ a bc to zero. Eq u iv alen tly , ta ke A 1 = A 2 = A 3 = 0 , A 0 = − A ( x, y ) (the case A 0 = A 1 = A 2 = 0 is also degenerate and can b e obtained by rev ersin g the role of x and y ). F or this degenerate case the Liouville relativ e inv arian t [19] ν 5 = L 2 ( L 1 ∂ x L 2 − L 2 ∂ x L 1 ) + L 1 ( L 2 ∂ y L 1 − L 1 ∂ y L 2 ) + A 3 ( L 1 ) 3 − A 2 ( L 1 ) 2 L 2 + A 1 L 1 ( L 2 ) 2 − A 0 ( L 2 ) 3 v anishes. The matrix M ([Γ]) in (1.4) h as rank fiv e an d its determinan t v anishes identica lly . In this case we can nev ertheless analyse the linear system (2.10) directly without even prolonging it. W e solv e for ψ 2 = − (1 / 2) y α ( x ) + β ( x ) , ψ 3 = α ( x ) , where α an d β are some arbitrary functions of x , and cross-differentia te the remaining equations to find 2 β ′′ − y α ′′′ + 2( ∂ x A ) α − 2( ∂ y A ) β + (3 A + y ∂ y A ) α ′ = 0 . (5.37) No w assu m e further that 5 ∂ 2 y A + y ∂ 3 y A 6 = 0 , ∂ 3 y A 6 = 0 and p erform fur ther differen tiations to eliminate α, β from (5.37) and to find the necessary metrisable condition for A ( x, y ) 7( ∂ 3 y A ) ( ∂ 4 y A ) ( ∂ x ∂ 3 y A ) − 5( ∂ x ∂ 3 y A ) ( ∂ 5 y A ) ( ∂ 2 y A ) − 6( ∂ x ∂ 4 y A ) ( ∂ 3 y A ) 2 +6( ∂ 5 y A ) ( ∂ x ∂ 2 y A ) ( ∂ 3 y A ) − 7( ∂ 4 y A ) 2 ( ∂ x ∂ 2 y A ) + 5( ∂ x ∂ 4 y A ) ( ∂ 4 y A ) ( ∂ 2 y A ) = 0 . (5.38) The obstruction (5.38) is of the same different ial order as th e 6 by 6 matrix (1.4), and we c hec k ed that it arises as a v anishing of a determinant of some 5 by 5 minors of (1.4 ) (wh ic h factorise in this case with (5.38) as a common factor). W e hav e p oint ed out that further genericit y assumptions for A w ere needed to arr iv e at (5.38). T o construct an example of non-metrisable pro jectiv e connection wh ere these assumptions d o not h old consider the fi rst Pai nlev ´ e equation [15] d 2 y dx 2 = 6 y 2 + x, for which b oth (1.5) and (5.38) v anish. Ho wev er equation (5.37) im p lies that α ( x ) = β ( x ) = 0 s o no metric exists in this case. W e would ha ve reac hed the same conclusion by observing that in the Pai nlev ´ e I case rank( M )([Γ]) = 3 and v erifying th at the 6 by 6 m atrix in the 8th order obstruction (4.35) has rank 5. This obstruction therefore v anishes but the corresp ondin g one-dimen s ional k ern el is spann ed by (1 , 0 , 0 , 0 , 0 , 0) T and the corresp onding solution to the linear system (2.10) is d egenerate. In [13] it w as shown that the Liouville inv arian t ν 5 v anishes for all six Painlev ´ e equations, and we hav e verified that our in v arian t (1.5) also v anishes. The m etrisabilit y analysis w ould need to b e done on a case b y case b asis in a wa y analogous to our treatmen t of P ainlev´ e I. 16 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD 6. Twistor Theor y In this Sectio n we sh all giv e a t wistorial treatmen t of the problem, wh ic h clarifies the rather mysterious linearisation (2.10) of the non-linear sy s tem (2.8). In the real analytic case one complexifies the pro jectiv e structure, and establishes a one-to-one corresp ond ence b et we en holomorphic p r o jectiv e str u ctures ( U, [Γ]) and complex surfaces Z with r ational curves with s elf-in tersection num b er one [14]. The p oints in Z corresp ond to geod esics in U , and a ll geod esics in U passing through a p oin t u ∈ U form a rational curve ˆ u ⊂ Z with normal b undle N ( ˆ u ) = O (1). Here O ( n ) denotes th e n th tensor p o w er of the du al of the tautological line bu ndle O ( − 1) ov er P ( T U ) w hic h arises as a qu otient of T U − { 0 } by the Euler v ector fi eld. Restricting the canonical line bu ndle κ Z of Z to a t wistor line ˆ u = C P 1 giv es κ Z = T ∗ ( ˆ u ) ⊗ N ∗ ( ˆ u ) = O ( − 3) since the holomorph ic tangen t bund le to CP 1 is O (2). If U is a complex su rface with a holomorphic pro jectiv e structure, then its t w istor space Z is P ( T U ) /D x , where D x is the geod esic spra y of the pr o jectiv e connection (2.11) D x = z a ∂ ∂ x a − Π c ab z a z b ∂ ∂ z c (6.39) = ∂ ∂ x + ζ ∂ ∂ y + ( A 0 + ζ A 1 + ζ 2 A 2 + ζ 3 A 3 ) ∂ ∂ ζ . Here ( x a , z a ) are co ord inates on T U and the second line uses p ro jectiv e co ordinate ζ = z 2 /z 1 . Th is leads to the double fi bration U ← − P ( T U ) − → Z. All these structures sh ould b e in v arian t un d er an anti -holomorphic in vol ution of Z to reco ve r a real str ucture on U . This works in the real analytic case, but can in principle b e extend ed to the smo oth case usin g the holomorphic discs of LeBrun-Mason [20 ]. No w if the p ro jectiv e stru cture is metrisable, Z is equipp ed with a preferr ed s ection of the ant i-canonical d ivisor line bun dle κ Z − 2 / 3 [6, 20]. Th e zero set of th is section inte r sects eac h rational cur v e in Z at t wo p oints. The pu llbac k of th is section to T U is a homogeneous function of d egree t wo σ = σ ab z a z b , w here z a are h omogeneous co ordinates on the fib r es of P ( T U ) → U , and σ ab with a, b = 1 , 2 is a symmetric 2-tensor on U . This fun ction L ie derive s along the spr a y (6.39) and th is giv es the o verdetermined linear system as th e v anishing of a p olynomial h omogeneous of degree 3 in z a : The condition D x ( σ ) = 0 imp lies the equation (2.12) wh ic h is equ iv alen t to (2.10). W e can u nderstand the equ ation (2.12) using any connection Γ in a p ro jectiv e class instead of th e pro jectiv e connection ∇ Π . T o see it we need to introdu ce a concept of pro jectiv e weigh t [10]. First recall th at the co v arian t deriv ativ e of the p ro jectiv e connection acting on vec tor fields is giv en by ∇ a X c = ∂ a X c + Γ c ab X b and on 1-forms b y ∇ a φ b = ∂ a φ b − Γ c ab φ c . Let ǫ ab = ǫ [ ab ] b e a v olume form on U . Changing a represen tativ e of the pro jectiv e class yields ˆ ∇ a ǫ bc = ∇ a ǫ bc − 3 ω a ǫ bc . (6.40) Let E (1) b e a line bund le o ver U such that the 3rd p ow er of its d ual bund le is the canonical bu ndle of U . The bund les E ( w ) = E (1) ⊗ w ha ve a fl at connection in duced fr om [Γ]. It c h anges according to ˆ ∇ a h = ∇ a h + w ω a h under (1.1), where h is a s ection of E ( w ). METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 17 Definition 6.1. The weighte d ve ctor field with pr oje ctive weight w is a se ction of a bund le T U ⊗ E ( w ) . This d efinition naturally extends to other tensor bun d les. No w w e shall c ho ose a con ve- nien t normalisat ion of [Γ]. F or any choice of ǫ ab w e must hav e ∇ a ǫ bc = θ a ǫ bc for s ome θ a . W e can change the p ro jectiv e r ep resen tativ e with ω a = θ a / 3 and use (6.40) to set θ a = 0 so that ǫ ab is parallel. Let us assume that su c h a choic e has b een made. W e sh all u se the v olume forms to raise and lo we r indices according to z a = ǫ ba z b , z a = z b ǫ ba . T he residual freedom in (1.1) is to use ω a = ∇ a f where f is any function on U . I f ∇ a ǫ bc = 0 then ˆ ∇ a ˆ ǫ bc = 0 , if ˆ ǫ ab = e 3 f ǫ ab . (6.41) Th us if h ∈ E ( w ) is a scalar of w eight w and we change the vo lu me form as in (6.41) then w e must r escale h − → ˆ h = e w f h with natur al extension to other tensor bun dles. T hus ǫ ab has weig ht − 3. Let us n o w come bac k to equation (2.12) where the Πs are rep laced b y comp onen ts of some connection in [Γ] ∇ ( a σ bc ) = 0 . If we c hange the representati v e of th e pro jectiv e class by (1.1) with ω a = ∇ a f the equation D x ( σ ) = 0 stays in v arian t if σ ab − → ˆ σ ab = e 4 f σ ab . This argument sho ws that the linear op erator σ ab − → ∇ ( a σ bc ) is pr o jectiv ely in v ariant on symm etric tw o-tensors with weigh t 4. No w σ ab := ǫ ac ǫ bd σ cd is a section of S 2 ( T U ) ⊗ E ( − 2) an d satisfies ∇ a σ bc = δ b a µ c + δ c a µ b (6.42) for some µ b . The Liouville lemma 2.1 implies that if σ ab satisfies this equation then g ab = (det σ ) σ ab is a metric in the p ro jectiv e class. The expression (6.42 ) is the tensor ve r sion of the first prolongation of th e linear system (2.10). In the next sectio n we sh all carry o v er the prolongation in the in v arian t manner and expr ess the 5th order obstruction (1.5) as a w eighte d pro jectiv e scalar inv ariant . 7. An Al terna tive Der iv a tion In this section, we use the approac h of [11] to deriv e the obstruction det M ([Γ]) of Theorem 1.1. On e adv an tage of th is approac h is that (1.5) may th en b e written in terms of the cu r v ature of the connection and its co v arian t deriv ativ es for an y connection in the giv en p ro jectiv e class. The s ymmetric form σ ab used in this Section is p rop ortional to the quadratic form (2.7) and the ob jects ( µ a , ρ ) are r elated bu t not equal to ( µ, ν, ρ ) defi ned by (3.15) and (3.17) from Section 3. Similarly the 6 by 6 matrix (7.47) is related but not equal to M ([Γ]) giv en by (1.4). This is b ecause the choice s mad e in the prolongation pro cedur e leading (7.44) are d ifferent than those made in Section 3. The resulting obstructions (1.5) and (7.48) do n ot dep end on these c h oices and are the same up to a non-zero exp on ential factor. Let Γ ∈ [Γ] b e a connection in th e pr o jectiv e class. Its cu r v ature is defined by [ ∇ a , ∇ b ] X c = R c abd X d 18 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD and can b e u niquely d ecomp osed as R c abd = δ c a P bd − δ c b P ad + β ab δ c d (7.43) where β ab is sk ew. In dimensions higher than 2 there w ould b e another term (the W eyl tensor) in this curv ature but dimension in 2 it v anishes identic ally . If we change the conn ection in the pro jectiv e class using (1.1) then ˆ P ab = P ab − ∇ a ω b + ω a ω b , ˆ β ab = β ab + 2 ∇ [ a ω b ] . The Bianc hi iden tity implies that β ab is closed and so lo cally it is clear that we can alwa ys c ho ose a connection in our pro jectiv e class w ith β ab = 0 (in fact, this also true globally on an orien ted manifold). Th e residu al freedom in c hanging the represent ativ e of the equ iv alence class (1.1) is giv en by gradients ω a = ∇ a f , where f is a fu nction on U . No w P ab = P ba and the Ricci tensor of Γ is s y m metric. The Bianc hi id en tit y implies that Γ is flat on a bund le of v olume forms on U . Thus the normalisation of ∇ a ma y , equiv alen tly , b e stated as requiring the existence of a vo lu me f orm ǫ ab suc h that ∇ a ǫ bc = 0 . Lo cally , such a vo lume form is u nique u p to scale: let us fi x one. This is the normalisation used in the p revious Section. The linear system and its prolongation dev elop ed in § 2 and § 3 is assem bled in [11] in to a single connection on a r ank 6 ve ctor bundle o v er U . Sp ecifically , sections of this bund le comprise triples of contra v ariant tensors ( σ ab , µ a , ρ ) with σ ab b eing symmetric. The connection is giv en by          σ bc µ b ρ          ∇ a 7− →          ∇ a σ bc − δ b a µ c − δ c a µ b ∇ a µ b − δ b a ρ + P ac σ bc ∇ a ρ + 2P ab µ b − 2 Y abc σ bc          , (7.44) where Y abc = 1 2 ( ∇ a P bc − ∇ b P ac ), the Cotton tensor. T h e follo wing is prov ed in [11]. Theorem 7.1. The c onne ction ∇ a is pr oje ctively e quivalent to a L evi–Civita c onne ction if and only if ther e is a c ovariantly c onstant se ction ( σ ab , µ a , ρ ) of the bund le with c onne ction (7.44) for which σ ab is non-de gener ate. It is also shown in [11] ho w the rank 6 b undle itself an d its connection (7.44) ma y b e view ed as pro jectiv ely in v ariant . In any case, obstructions to th e existence of a cov arian tly constan t section may b e obtained from the curv ature of this connection, whic h we now compute. ∇ a ∇ b          σ cd µ c ρ          = ∇ a          ∇ b σ cd − δ c b µ d − δ d b µ c ∇ b µ c − δ c b ρ + P bd σ cd ∇ b ρ + 2P bc µ c − 2 Y bcd σ cd          =          ∇ a ( ∇ b σ cd − δ c b µ d − δ d b µ c ) − δ c a ( ∇ b µ d − δ d b ρ + P be σ de ) − δ d a ( ∇ b µ c − δ c b ρ + P be σ ce ) ∇ a ( ∇ b µ c − δ c b ρ + P bd σ cd ) − δ c a ( ∇ b ρ + 2P bd µ d − 2 Y bde σ de ) + P ad ( ∇ b σ cd − δ c b µ d − δ d b µ c ) ∇ a ( ∇ b ρ + 2 P bc µ c − 2 Y bcd σ cd ) + 2 P ac ( ∇ b µ c − δ c b ρ + P bd σ cd ) − 2 Y acd ( ∇ b σ cd − δ c b µ d − δ d b µ c )          =          ∇ a ∇ b σ cd − δ c a P be σ de − δ d a P be σ ce + ⋆⋆ ∇ a ∇ b µ c − δ c a P bd µ d + ( ∇ a P bd ) σ cd + 2 δ c a Y bde σ de + ⋆⋆ ∇ a ∇ b ρ + 2( ∇ a P bc ) µ c − 2( ∇ a Y bcd ) σ cd + 2 Y abd µ d + 2 Y acb µ c + ⋆⋆          , where ⋆⋆ denotes expressions th at are manifestly symmetric in ab . Also notice that ( ∇ [ a P b ] d ) σ cd + 2 δ c [ a Y b ] de σ de = δ c d Y abe σ de + 2 δ c [ a Y b ] de σ de = 3 δ c [ a Y bd ] e σ de = 0 , METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 19 and that Y [ abc ] = 0 = ⇒ Y acb − Y bca = Y abc . Therefore, ( ∇ a ∇ b − ∇ b ∇ a )          σ cd µ c ρ          =          0 0 10 Y abc µ c − 4( ∇ [ a Y b ] cd ) σ cd          . (7.4 5) Denoting the triple ( σ ab , µ b , ρ ) by Σ α , we are seeking a sect ion Σ α of our rank 6 bundle so that ∇ a Σ α = 0 and ha ve found the explicit form of the eviden t necessary condition ( ∇ a ∇ b − ∇ b ∇ a )Σ α = 0. W e may rewrite our necessary condition as ǫ ab ∇ a ∇ b Σ α = 0. Notice, ho wev er, that th er e is only one non -zero entry on the r igh t hand side of (7.45). Ou r necessary cond ition analogous to (3.20) b eco mes Ξ α Σ α = 0 (7.46) for Ξ α ≡          0 5 Y a Z ab          , where Y c ≡ ǫ ab Y abc and Z cd ≡ − 2 ǫ ab ∇ a Y b ( cd ) = ∇ ( c Y d ) . Eviden tly , the qu antit y Ξ α is a section of a rank 6 bun dle du al to our p r evious on e. Its sections consist of triples of co v ariant tensors ( κ, λ a , τ ab ) with τ ab b eing symmetric and it inherits a connection dual to the previous one. Sp ecificall y ,          κ λ b τ bc          ∇ a 7− →          ∇ a κ + λ a ∇ a λ b + 2 τ ab − 2P ab κ ∇ a τ bc − P a ( b λ c ) + 2 Y a ( bc ) κ          , where          κ λ b τ bc                   σ bc µ b ρ          ≡ κρ + λ b µ b + τ bc σ bc is the dual p airing. By d ifferen tiating our necessary condition f or ∇ a Σ γ = 0 w e obtain Ξ γ Σ γ = 0 ( ∇ a Ξ γ )Σ γ = 0 ( ∇ ( a ∇ b ) Ξ γ )Σ γ = 0 . Since Σ α is sup p osed to b e a non-zero s ection, it follo ws that the 6 × 6 matrix                   0 5 Y c Z cd          , ∇ a          0 5 Y c Z cd          , ∇ ( a ∇ b )          0 5 Y c Z cd                   (7.47) m u s t b e singular. Its determinant is the obstruction from Th eorem 1.1. W e compute ∇ a          0 5 Y c Z cd          =          5 Y a 5 ∇ a Y c + 2 Z ac ∇ a Z cd − 5P a ( c Y d )          and ∇ a ∇ b          0 5 Y c Z cd          =          5 ∇ a Y b + 5 ∇ b Y a + 2 Z ba ∇ a (5 ∇ b Y c + 2 Z bc ) + 2 ∇ b Z ac − 10P b ( a Y c ) − 10P ac Y b ∇ a ( ∇ b Z cd − 5P b ( c Y d ) ) − 5P a ( c ∇ | b | Y d ) − 2P a ( c Z | b | d ) + 10 Y a ( cd ) Y b          20 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD so ∇ ( a ∇ b )          0 5 Y c Z cd          =              12 Z ab 5 ∇ ( a ∇ b ) Y c + 4 ∇ ( a Z b ) c − 5P ab Y c − 15P c ( a Y b ) ∇ ( a ∇ b ) Z cd − 5( ∇ ( a P b )( c ) Y d ) − 5P c ( a ∇ b ) Y d − 5P d ( a ∇ b ) Y c − P c ( a Z b ) d − P d ( a Z b ) c + 10 Y ( a Y b )( cd )              . T o compute the d eterminan t of the 6 × 6 matrix (7.47) we m ay use th e follo wing. Lemma 7.2. L et ǫ ab denote the skew form in two dimensions normalise d as ǫ 00 = 0 ǫ 01 = 1 ǫ 10 = − 1 ǫ 11 = 0 . Then the determinant of the 6 × 6 matrix                    0 P 0 P 1 Q 00 Q 01 Q 11 R 0 S 00 S 01 T 000 T 001 T 011 R 1 S 10 S 11 T 100 T 101 T 111 U 00 V 000 V 001 X 0000 X 0001 X 0011 U 01 V 010 V 011 X 0100 X 0101 X 0111 U 11 V 110 V 111 X 1100 X 1101 X 1111                    is ǫ ab ǫ cd ǫ ef ǫ g h ǫ ij ǫ k l ǫ mn ǫ pq         Q g i S mp T nj k U ac V deq X bf hl − 1 6 P p R m S nq X acg i X behk X d f j l − 1 2 P p S mq T nj l U ce X adgk X bf hi − 1 2 P p T mg i T nj k U ac V deq X bf hl + 1 2 P p R m T ng i V acq X dej k X bf hl − 1 2 Q g i R m S np V acq X dej k X bf hl − 1 2 Q g i R m T nj k V acp V deq X bf hl − 1 4 Q g i S mp S nq U ac X dej k X bf hl − 1 4 Q g i T mj k T nhl U ac V dep V bf q         (7.48) wher e Q ab = Q ( ab ) , T cab = T c ( ab ) , U cd = U ( cd ) , V cda = V ( cd ) a , and X cdab = X ( cd )( ab ) . Pro of. A tedious compu tation.  Ev ery tensor Q, S, T , ... in this expression is constructed u sing one ǫ ab . Th u s coun ting the total num b er of ǫ ab s s ho ws that the determinant has a total pro jectiv e w eigh t − 42 in a sense of Definition 6.1 wh ic h al s o means th at it represen ts a section of th e 14th p o w er of the canonical bu ndle of U . T o su m marise, we hav e p ro ved the follo w ing alternativ e form ulation of Theorem 1.1. Theorem 7.3. Supp ose that ∇ a is a torsion-fr e e c onne ction in two-dimensions and that ǫ bc is a volume form such that ∇ a ǫ bc = 0 . Define the Schouten tensor P ab by (7.43) and Y abc ≡ 1 2 ( ∇ a P bc − ∇ b P ac ) Y c ≡ ǫ ab ∇ a P bc Z ab ≡ ∇ ( a Y b ) . L et P a ≡ 5 Y a Q ab ≡ 12 Z ab R c ≡ 5 Y c S ca ≡ 5 ∇ a Y c + 2 Z ac T cab ≡ 5 ∇ ( a ∇ b ) Y c + 4 ∇ ( a Z b ) c − 5P ab Y c − 15P c ( a Y b ) U cd ≡ Z cd V cda ≡ ∇ a Z cd − 5P a ( c Y d ) X cdab ≡ ∇ ( a ∇ b ) Z cd − 5( ∇ ( a P b )( c ) Y d ) − 5P c ( a ∇ b ) Y d − 5P d ( a ∇ b ) Y c − P c ( a Z b ) d − P d ( a Z b ) c + 10 Y ( a Y b )( cd ) and define D (Γ) b y the formula (7.48) . If ∇ a is pr oje ctively e quiv alent to a L evi–Civita c onne ction, then D (Γ) = 0 . METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 21 In add ition to giving an explicit form u la for D (Γ), there are sev eral other consequences of this theorem, whic h we shall now discuss. W e ha ve foun d that D (Γ) = det        0 P a Q ab R c S ca T cab U cd V cda X cdab        where Q ab = Q ( ab ) , T cab = T c ( ab ) , U cd = U ( cd ) , V cda = V ( cd ) a , X cdab = X ( cd )( ab ) and the precise meaning of determinan t is giv en b y Lemma 7.2. Though it mak es n o difference to the determinant and seemingly giv es a more complicated expression, it is more con venien t to write D (Γ) = 1 4320 det ¯ Θ wh ere ¯ Θ ≡        0 12 P a Q ab 30 R c 12 S ca T cab − 5P ab R c 30 U cd 12 V cda X cdab − 5P ab U cd        , where the und erlying matrix is eviden tly obtained by column op erations from the previous one. The reason is that th is matrix b etter trans f orms u nder p ro jectiv e change of connection. Sp ecifically , if we w r ite ¯ Θ =        0 ¯ P a ¯ Q ab ¯ R c ¯ S ca ¯ T cab ¯ U cd ¯ V cda ¯ X cdab        , then un der the change in connection b ∇ a φ b = ∇ a φ b − ω a φ b − ω b φ a induced by (1.1 ), we find b ¯ Θ =         0 ˜ P a ˜ Q ab − ˜ P ( a ω b ) ˜ R c ˜ S ca − 2 ˜ R c ω a ˜ T cab − ˜ S c ( a ω b ) + ˜ R c ω a ω b ˜ U cd ˜ V cda − 2 ˜ U cd ω a ˜ X cdab − ˜ V cd ( a ω b ) + ˜ U cd ω a ω b         (7.49) where e Θ =         0 ˜ P a ˜ Q ab ˜ R c ˜ S ca ˜ T cab ˜ U cd ˜ V cda ˜ X cdab         =        0 ¯ P a ¯ Q ab ¯ R c ¯ S ca − 2 ω c ¯ P a ¯ T cab − 2 ω c ¯ Q ab ¯ U cd − ω ( c ¯ R d ) ¯ V cda − ω ( c ¯ S d ) a + ω c ω d ¯ P a ¯ X cdab − ω ( c ¯ T d ) ab + ω c ω c ¯ Q ab        . (7.50) Notice that e Θ is obtained from ¯ Θ b y col umn op erations and then b ¯ Θ is obtained from e Θ by row op erations. It follo ws that determinant do es n ot change, i.e. D ( b Γ) = D (Γ) is a pro jectiv e inv arian t (from the formula (7.48) it is already apparent that D (Γ) is indep enden t of c h oice of co¨ ordinates). Thus w e u s e the notation D ([Γ]). The argument follo win g the formula (4.31) sh o ws that th ere is only one obs tr uction to the metrisabilit y at order 5 so d et M ([Γ]) = 0 iff D ([Γ]) = 0. T h u s det M ([Γ])( dx ∧ dy ) ⊗ 14 is indeed a p ro jectiv e inv arian t as claimed in the Introdu ction. A more in v arian t viewp oint on these matters is as follo ws . T he formula (7.44 ) is f or a connection on an in v ariantl y defin ed v ector bu ndle, denoted by E ( B C ) in [11]. It arises f rom a representa tion of SL(3 , R ) and the connection (7.44) is closely related (b ut not equal to) 22 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD the pro jectiv e Cartan connection ind u ced on bund les so arising. T h e bu ndle is canonically filtered with comp osition series E AB = E bc ( − 2) + E b ( − 2) + E ( − 2) as detai led in [11]. Strictly sp eaking the quant it y Ξ γ is not a section of th e dual bu ndle E ( C D ) but r ather the pro jectiv ely weig h ted bun d le E ( C D ) ( − 5) with comp osition series E ( C D ) ( − 5) = E ( − 3) + E c ( − 3) + E ( cd ) ( − 3) and ¯ Θ is then obtained by ap p lying the inv arian tly defined ‘splitting op erator’ E ( − 5) ∋ ξ 7→        30 ξ 12 ∇ a ξ ∇ ( a ∇ b ) ξ − 5P ab ξ        ∈ E ( AB ) ( − 7) coupled to the pro jectiv ely inv arian t connection (7.44). The upsh ot is that ¯ Θ is an in- v arian tly defined section of E ( C D )( AB ) ( − 7). Indeed, th e form ulae (7. 49) and (7.5 0 ) giving b ¯ Θ in te r ms of ¯ Θ are precisely h o w sections of E ( C D )( AB ) or E ( C D )( AB ) ( − 7) transform un- der pro jectiv e c hange. Consequent ly , the obstruction D (Γ) is an in v ariant of pro jectiv e w eight − 42. 8. Outlook In the language of C artan [7, 4], th e general 2nd order O DE (1.6) defi n es a p ath ge ometry , and the paths are geo desics of pro jectiv e connection if the ODE is of th e form (1.3). In this pap er we ha ve shown u nder what conditions the p aths in this geometry are un p arametrised geod esics of some metric. In case of higher d imensional p ro jectiv e str u ctures the link with ODEs is lost, b ut n ev ertheless one could searc h f or conditions obstru cting the metrisability in a wa y analogous to w h at w e did in Section (7). The results will ha v e a d ifferen t c h aracter, ho wev er, o wing to the presence of the W eyl curv ature wh ic h will mo dify the connection (7.44) as explained in [11]. Th e fi rst n ecessary condition analogous to (1.5 ) o ccurs already at order 2. Sp ecifically , it is shown in [11] that the cu r v ature of the r elev ant connection in n dimen sions is giv en by ( ∇ a ∇ b − ∇ b ∇ a )          σ cd µ c ρ          =          W c abe σ de + W d abe σ ce + 2 n δ c [ a W d b ] ef σ ef + 2 n δ d [ a W c b ] ef σ ef ∗ ∗          where W c abd is th e W eyl curv ature and ∗ denotes expressions that we shall not need. Since w e are searc hing for co v ariant constant sections with n on-degenerate σ cd , in particular it follo ws that the linear transf ormation σ ef 7→ Ξ cd abef σ ef where Ξ cd abef := W c ab ( e δ d f ) + W d ab ( e δ c f ) + 2 n δ c [ a W d b ]( ef ) + 2 n δ d [ a W c b ]( ef ) (8.51) is obliged to hav e a non-trivial kernel. Regarding Ξ cd abef as a matrix representing this linear transformation, it should h a ve n ( n + 1) / 2 columns accounting for the symmetric indices ef . In its r emainin g indices it is sk ew in ab , sym metric cd , and trace-free. Th ese symmetries sp ecify an irreducible representa tion of GL( n, R ) of dimension ( n 2 − 1)( n 2 − 4) / 4, whic h we ma y regard as the num b er of rows of th e m atrix Ξ cd abef . Notice that wh en n = 2 this matrix is zero b ut as so on as n ≥ 3 it has more ro ws than col umns (for example, it is a 10 × 6 matrix in dimension 3). W e claim that ha ving a n on-trivial k ernel is a genuine condition and therefore an obstr u ction to metrisabilit y . F or this, we need to show that Ξ cd abef can METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 23 ha ve maximal rank ev en when it is of th e sp ecial form (8.51) for some W c abd ha ving the symmetries of a W eyl tensor, namely W c abd = − W c bad , W c [ abd ] = 0 , W a abd = 0 . (8.52) Cho ose a frame and, for n ≥ 3, consid er the particular tensor W c abd ha ving as its only non-zero comp onen ts (no sum mation) W 1 121 = − W 1 211 = 3( n 2 − n − 1) W 2 122 = − W 2 212 = 3 W 3 123 = − W 3 213 = − ( n − 1)(2 n + 3) W c 12 c = − W c 21 c = − ( n − 1) , ∀ c ≥ 4 W 3 132 = − W 3 312 = − ( n 2 − n − 3) W c 1 c 2 = − W c c 12 = n + 2 , ∀ c ≥ 4 W 3 231 = − W 3 321 = n ( n + 2) W c 2 c 1 = − W c c 21 = 2 n + 1 , ∀ c ≥ 4 . It is readily verified that the symmetries (8.52) are satisfied. F orm the corresp ondin g Ξ cd abef according to (8.51 ) and consider Ξ cd 12 ef . Being sym m etric in cd and ef , w e ma y regard it as a squ are matrix of size n ( n + 1) / 2 and it s uffices to sho w that this matrix is inv ertible. In fact, it is easy to c heck that it is diagonal with non- zero en tries along its d iagonal. The twisto r analysis of Section (6) su ggest that there is some analogy b etw een the metris- abilit y pr oblem w e stud ied in t wo dimensions and existence of (p ossibly indefin ite) K¨ ahler structure in a giv en anti -self-dual (ASD) conformal class c in on a f our-manifold M . A K¨ ahler structure corresp onds to a pr eferred section of ant i-canonical divisor κ B − 1 / 2 , w here κ B is the canonical bun dle of the t wistor space [21] B (a complex three-fold with an em b ed- ded rational cu r v e w ith normal bu ndle O (1) ⊕ O (1)). Not all ASD structures are K¨ ahler and th e existence of the divisor sh ould lead to v anishing of some conformal inv ariant s con- structed out of the ASD W eyl tensor (to the b est of our knowle dge they ha ve n ev er b een written down. If one adds the Ricci fl at condition, some of the inv ariants are known and can b e expr essed in terms of the Bac h tensor). These t wo constructions (ASD+K¨ ahler in four dimensions and p ro jectiv e + metrisable in tw o d imensions) are linked in the follo win g wa y: ev ery ASD stru cture in (2 , 2) signature with a co nformal null Killing v ector indu ces a p r o jectiv e stru cture on a t wo-dimensional space U of the β surfaces (null ASD surfaces) in M . Conv ersely an y tw o-dimensional pro jectiv e stru cture giv es rise to (a class of ) ASD structures with null conform al symmetry [8, 6]. Consider B to b e a holomorphic fibr e bu ndle o v er Z w ith one dimen s ional fi bres, where Z is the twistor space of ( U, [Γ]) in tro duced in Section 6. Let ˆ u ⊂ Z b e rational curve in Z corresp onding to u ∈ U . The three-fold B will b e a t wistor space of an ASD co n formal structure if B restricts to O (1) on eac h t wistor line ˆ u ⊂ Z . If Z corresp ond s to a metrisable pro jectiv e structure then the divisor σ lifts to a section of κ B − 1 / 2 , th u s giving a (2 , 2) K¨ ahler class. If th e conformal Killing vect or is not h yp er-surface orthogonal th e lo cal expression for th e conformal class is c = dz a ⊗ dx a − Π c ab z c dx a ⊗ dx b , where Π c ab are comp onen ts of the pro jectiv e connection (2.11). The conformal Killing vect or is a homothet y z a /∂ z a . This form ula for c is equiv alen t to a sp ecia l case of expression (1.3) in [8] after a c hange of co ordin ates and a conformal rescaling (set z a = ( − z e t , e t ) and tak e G = z 2 / 2 + γ ( x, y ) z + δ ( x, y ) for certain γ , δ in [8]). It is a pr o jectiv ely inv arian t mo d ification of th e Riemann ian extensions of spaces with affine conn ection studied b y W alk er [24]. Th e conformal class c is conformally fl at iff [Γ] is pro jectiv ely fl at, i.e. its C otton tensor v anishes. This in turn is equiv alen t to the v anishing of the Liouville expressions (2.14). 24 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD The metrisable pr o jectiv e structures will therefore giv e rise to (2, 2) ASD K¨ ahler metric with conformal null symm etry . Ultimately , the metrisabilit y in v ariant (1.5 ) in t w o dimen- sions will hav e its coun terpart: a conformal inv arian t in f our d imensions. Some p rogress in this direction has b een made in [9]. Appendix The connection D = d + Ω 1 dx + Ω 2 dy on the rank six ve ctor bund le E → U is Ω 1 =         − 2 3 A 1 2 A 0 0 0 0 0 0 0 0 − 1 2 0 0 − 2 A 3 − 2 3 A 2 4 3 A 1 0 1 0 ( Ω 1 ) 41 ( Ω 1 ) 42 ( Ω 1 ) 43 − 1 3 A 1 − 3 A 0 0 0 0 0 0 0 − 1 ( Ω 1 ) 61 ( Ω 1 ) 62 ( Ω 1 ) 63 ( Ω 1 ) 64 ( Ω 1 ) 65 ( Ω 1 ) 66         , Ω 2 =         − 4 3 A 2 2 3 A 1 2 A 0 1 0 0 0 0 0 0 − 1 2 0 0 − 2 A 3 2 3 A 2 0 0 0 0 0 0 0 0 − 1 ( Ω 2 ) 51 ( Ω 2 ) 52 ( Ω 2 ) 53 3 A 3 1 3 A 2 0 ( Ω 2 ) 61 ( Ω 2 ) 62 ( Ω 2 ) 63 ( Ω 2 ) 64 ( Ω 2 ) 65 ( Ω 2 ) 66         . (A53) Let V 1 , ..., V 6 b e giv en by (3.21). The cur v ature of D is F = d Ω + Ω ∧ Ω = F dx ∧ dy =         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V 1 V 2 V 3 V 4 V 5 V 6         dx ∧ dy . (A54) METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 25 The matrix elemen ts of the connection are ( Ω 1 ) 41 = − 4 3 ∂ x A 2 + 4 A 0 A 3 + 2 3 ∂ y A 1 , ( Ω 1 ) 42 = − 2 ∂ y A 0 + 2 3 ∂ x A 1 + 4 A 2 A 0 − 4 9 ( A 1 ) 2 , ( Ω 1 ) 43 = 2 ∂ x A 0 − 4 A 0 A 1 , ( Ω 1 ) 61 = − 4 3 ∂ x ∂ y A 2 − 20 3 A 0 A 2 A 3 + 2 3 ( ∂ y ) 2 A 1 + 4 A 3 ∂ y A 0 − 2 A 0 ∂ y A 3 − 16 9 A 2 ∂ x A 2 + 8 9 A 2 ∂ y A 1 , ( Ω 1 ) 62 = 2 3 ∂ x ∂ y A 1 − 4 3 A 1 ∂ y A 1 + 2 A 0 ∂ y A 2 − 2( ∂ y ) 2 A 0 + 4 A 2 ∂ y A 0 + 4 A 3 ∂ x A 0 +6 A 0 ∂ x A 3 + 8 9 A 1 ∂ x A 2 + 4 3 A 0 A 1 A 3 − 4 3 A 0 ( A 2 ) 2 , ( Ω 1 ) 63 = 2 ∂ x ∂ y A 0 + 2 3 A 0 ∂ x A 2 − 4 3 A 2 ∂ x A 0 − 4 A 1 ∂ y A 0 − 4 3 A 0 ∂ y A 1 + 8 3 A 0 A 1 A 2 + 4 A 3 ( A 0 ) 2 , ( Ω 1 ) 64 = 4 3 ∂ x A 2 − ∂ y A 1 + 5 A 0 A 3 , ( Ω 1 ) 65 = 1 3 ∂ x A 1 − 4 ∂ y A 0 + 3 A 2 A 0 − 2 9 ( A 1 ) 2 , ( Ω 1 ) 66 = − 1 3 A 1 , ( Ω 2 ) 51 = − 2 ∂ y A 3 − 4 A 3 A 2 , ( Ω 2 ) 52 = 2 ∂ x A 3 − 2 3 ∂ y A 2 + 4 A 1 A 3 − 4 9 ( A 2 ) 2 , ( Ω 2 ) 53 = 4 3 ∂ y A 1 − 2 3 ∂ x A 2 + 4 A 0 A 3 , ( Ω 2 ) 61 = − 2 ∂ x ∂ y A 3 − 4 A 2 ∂ x A 3 − 4 3 A 3 ∂ x A 2 + 2 3 A 3 ∂ y A 1 − 4 3 A 1 ∂ y A 3 − 4 A 0 ( A 3 ) 2 − 8 3 A 1 A 2 A 3 , ( Ω 2 ) 62 = 2( ∂ x ) 2 A 3 − 4 3 A 2 ∂ x A 2 − 4 3 A 0 A 2 A 3 − 2 3 ∂ x ∂ y A 2 +4 A 1 ∂ x A 3 + 4 A 0 ∂ y A 3 + 6 A 3 ∂ y A 0 + 4 3 A 3 ( A 1 ) 2 + 2 A 3 ∂ x A 1 + 8 9 A 2 ∂ y A 1 , ( Ω 2 ) 63 = − 2 3 ( ∂ x ) 2 A 2 + 8 9 A 1 ∂ x A 2 + 4 A 0 ∂ x A 3 − 2 A 3 ∂ x A 0 − 16 9 A 1 ∂ y A 1 + 4 3 ∂ x ∂ y A 1 + 20 3 A 0 A 1 A 3 , ( Ω 2 ) 64 = 4 ∂ x A 3 − 1 3 ∂ y A 2 + 3 A 1 A 3 − 2 9 ( A 2 ) 2 , ( Ω 2 ) 65 = ∂ x A 2 − 4 3 ∂ y A 1 + 5 A 0 A 3 , ( Ω 2 ) 66 = 1 3 A 2 . 26 ROBER T BR Y ANT, M ACIEJ DUNAJSKI, AND MICHAEL EASTWOOD The pr olongatio n formula e are P = − ( Ω 1 ) 41 ψ 1 − ( Ω 1 ) 42 ψ 2 − ( Ω 1 ) 43 ψ 3 − ( Ω 1 ) 44 µ − ( Ω 1 ) 45 ν, Q = − ( Ω 2 ) 51 ψ 1 − ( Ω 2 ) 52 ψ 2 − ( Ω 2 ) 53 ψ 3 − ( Ω 2 ) 54 µ − ( Ω 2 ) 55 ν, R = − ( Ω 1 ) 61 ψ 1 − ( Ω 1 ) 62 ψ 2 − ( Ω 1 ) 63 ψ 3 − ( Ω 1 ) 64 µ − ( Ω 1 ) 65 ν − ( Ω 1 ) 66 ρ, S = − ( Ω 2 ) 61 ψ 1 − ( Ω 2 ) 62 ψ 2 − ( Ω 2 ) 63 ψ 3 − ( Ω 2 ) 64 µ − ( Ω 2 ) 65 ν − ( Ω 2 ) 66 ρ. (A55) Referen ces [1] Branson, T. P ., ˇ Cap, A ., Eastw o od , M. G., and Gov er, A . R., Pr olongations of ge ometric over deter- mine d systems , Internat. J. Math. 17 (2006) 641–664. [2] Bryant, R. L. Homogeneous Pro jective Stru ct u res, unp u blished notes. [3] Bryant R. L., Chern S. S., Gardner R. B., Goldschmidt H . L. & Griffiths P . A., (1991) Exterior differential systems, Mathematical Sciences Researc h Institute Publications, v ol. 18, Springer-V erlag, New Y ork. [4] Bryant, R., Griffiths, P ., Hsu, L. T o ward a Geometry of Differential Eq uations, in Ge ometry, T op ol- o gy, and Physics, Conf. Pr o c. L e ctur e Notes Ge om. T op olo gy, e di te d by S. -T. Y au, vol. IV (1995), pp. 1-76, Internat. Press, Cambridge, MA. [5] Bryant, R. L., Manno, G. & Matveev, V. (2008) A solution of a problem of Sophus Lie. Math. Ann. 340 , 437–463. [6] Calderbank, D. M. J. Selfdual 4-manifolds, pro jective stru ctures, and the Duna jski-West construc- tion. math.DG/0606754 . [7] Cartan, E. (1924) Su r les v ari´ et´ es ` a conn exion pro jective , Bull. So c. Math. F rance 52 , 205–241. [8] D una jski, M., & W est, S. (2007) Anti-self-dual conforma l structures from pro jective structures. Comm. Math. Phys. 272 , 85–118. [9] D una jski, M., & T o d, K. P . (2010) F our Dimensional Metrics Conformal to K¨ ahler, Math. Pro c. Cam b. So c. arXiv:0901.2261 v1 . [10] Eastw o od , M. G. ( 2007) Notes on pro jectiv e differential geometry , in Symmetries and Over deter- mine d Systems of Partial Di ffer ential Equations , I MA V olumes in Mathematics and its Applications 144, Springer V erlag 2007, pp. 41–60. [11] Eastw o od , M. G. & Matveev, V (2007) Metric connections in pro jectiv e differential geometry , in Symmetries and Over determine d Systems of Partial Di ffer ential Equations , IMA V olumes in Math- ematics and its Ap plications 144, Sp ringer V erlag 2007, pp. 339–350. [12] Eisenhart, L. P . Riemannian Ge ometry , Princeton. [13] Hietarinta, J. & Dryum a, V. ( 2002) Is my ODE a Pa in lev´ e equation in disguise? J. Nonlinear Math. Phys. 9 , sup pl. 1, 67–74. [14] Hitchin, N. J. (1982) Complex manifolds and Einstein’s eq uations. In Twistor ge ometry and Non- Line ar Systems , Lecture N otes in Math., 970, Springer, pp . 73–99. [15] In ce, E. L. ( 1956) Or dinary Differ ential Equations . Dover Publications, New Y ork. [16] Ko enigs, M. G. Su r les g´ eodesiques a int ´ egrales quadratiqu es. N ote I I from Darb oux’ L e¸ cons sur la th ´ eorie g ´ en ´ er ale des surfac es , V ol. I V, Chelsea Pu blishing, 1896. [17] Kru glik ov, B. (2008) Inv arian t c h aracterisation of Liouville metrics and polynomial integr als. J. Geom. Phys. 58 , 979–995 . [18] Liouville, R. ( 1887) Sur u ne classe d’´ equations diff´ eren tielles, parmi lesquelles, en p articulier, toutes celles des lignes g´ eod´ esiques se trouvent comprises, Comptes rendus heb d omadaires des seances de l’Academie des sciences 105 , 1062-1064 . [19] Liouville, R . (1889) Sur les inv ariants de certaines ´ equations diff´ erentiell es et sur leurs applications, Jour. de l’Ecole Poli technique, Cah.59 , 7–76. [20] LeBrun, C. & Mason L. J. (2002) Zoll Manifolds and Complex S u rfaces J. Differential Geom. 61 , 453–535 . [21] Pon tecorvo, M. (1992) On tw istor spaces of anti-sel f-dual hermitian surfaces. T rans. Am. Math. So c. 331 , 653–661. [22] Thomas, T. Y . (1925) On pro jectiv e end equi–pro jective geometry of path s, Pro c. Nat. Acad. S ci. 11 , 207–209. [23] T resse, A. (1896) D´ etermination des inv arian ts p onctuels de l’ ´ equation diff´ erentiel le ordinair e du second ordre y ′′ = ω ( x , y , y ′ ) . Leipzig 1896. 87 S. gr. 8 o . F ¨ urstl. Jablonow sk i’sc hen Gesellsc haft zu Leipzig. N r. 32 ( 13 der math.-naturw. Section). M´ emoire couronn´ e p ar l’Acad´ emie Jablonows k i; S. Hirkel, Leipzig. METRISABILITY OF TWO-DIMENSI ONAL PROJECTIVE STRUCTURES 27 [24] W alk er, A. G. (1953) Riemann exten sions of n on-Riemannian spaces. In Conve gno di Ge ometria Differ enziale . V enice. The Ma thema tical Scie nces Research Institute, 1 7 Ga uss W a y, Berkeley, CA 94720-5070, USA. E-mail addr ess : bryant@m sri.org Dep ar tment of Applied Ma them a tics and The oretical Physics, University of Cambridge, Wilberfor ce Ro ad, Cambridge CB 3 0W A, UK. E-mail addr ess : m.dunajs ki@damtp.c am.ac.uk School of Ma thema tical S ciences, Australian Na tional Uni versity, ACT 0200, Australia. E-mail addr ess : meastwoo @member.am s.org

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