aa r X i v : . [ m a t h . DG ] O c t GL (2 , R ) GEOMETRY OF ODE'SMICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIAbstra t. We study (cid:28)ve dimensional geometries asso iated with the 5-dim-ensional irredu ible representation of GL (2 , R ) . These are spe ial Weyl geome-tries in signature (3 , having the stru ture group redu ed from CO (3 , to GL (2 , R ) . The redu tion is obtained by means of a onformal lass of totallysymmetri 3-tensors. Among all GL (2 , R ) geometries we distinguish a sub lasswhi h we term `nearly integrable GL (2 , R ) geometries'. These de(cid:28)ne a unique gl (2 , R ) onne tion whi h has totally skew symmetri torsion. This torsionsplits onto the GL (2 , R ) irredu ible omponents having respe tive dimensions3 and 7.We prove that on the solution spa e of every 5th order ODE satisfy-ing ertain three nonlinear di(cid:27)erential onditions there exists a nearly inte-grable GL (2 , R ) geometry su h that the skew symmetri torsion of its unique gl (2 , R ) onne tion is very spe ial. In ontrast to an arbitrary nearly integrable GL (2 , R ) geometry, it belongs to the 3-dimensional irredu ible representationof GL (2 , R ) . The onditions for the existen e of the stru ture are lower or-der equivalents of the Doubrov-Wil zynski onditions found re ently by BorisDoubrov [7℄.We provide nontrivial examples of 5th order ODEs satisfying the threenonlinear di(cid:27)erential onditions, whi h in turn provides examples of inhomo-geneous GL (2 , R ) geometries in dimension (cid:28)ve, with torsion in R .We also outline the theory and the basi properties of GL (2 , R ) geometriesasso iated with n -dimensional irredu ible representations of GL (2 , R ) in ≤ n ≤ . In parti ular we give onditions for an n th order ODE to possess thisgeometry on its solution spa e.MSC lassi(cid:28) ation: 53A40, 53B05, 53C10, 34C30Contents1. Introdu tion 22. A pe uliar third rank symmetri tensor 103. Irredu ible GL (2 , R ) geometries in dimension (cid:28)ve 133.1. Nearly integrable GL (2 , R ) stru tures 173.2. Arbitrary GL (2 , R ) stru tures 194. GL (2 , R ) bundle 205. 5th order ODE as nearly integrable GL (2 , R ) geometry with `small'torsion. Main theorem 225.1. 5th order ODE modulo onta t transformations 285.2. GL (2 , R ) bundle over spa e of solutions 295.3. Chara teristi onne tion with torsion in V GL (2 , R ) stru tures from 5th order ODEs 356.1. Torsionfree stru tures 356.2. Stru tures with vanishing Maxwell form 356.3. Simple stru tures with nonvanishing Maxwell form 386.4. A remarkable nonhomogeneous example 397. Higher order ODEs 407.1. Results from Hilbert's theory of algebrai invariants 427.2. Stabilizers of the irredu ible GL (2 , R ) in dimensions n < GL (2 , R ) geometries onthe solution spa e of ODEs 47Referen es 511. Introdu tionLet us start with an elementary algebrai geometry in R . Every point on a urve (1 , x, x ) in R de(cid:28)nes a straight line passing through the origin in the dualspa e ( R ) ∗ via the relation: θ + 2 θ x + θ x = 0 (1.1) θ + θ x = 0 . Here ( θ , θ , θ ) parametrize points of ( R ) ∗ . When moving along the urve (1 , x, x ) in R , the orresponding lines in the dual spa e ( R ) ∗ sweep out a ruled surfa ethere, whi h is the one(1.2) ( θ ) − θ θ = 0 with the tip in the origin. The points ( θ , θ , θ ) lying on this one may be thoughtas those, and only those, whi h admit a ommon root x for the pair of equations(1.1). A standard method for determining if two polynomials have a ommon rootis to equate to zero their resultant. In the ase of equations (1.1) the resultant is: R = det θ θ θ θ θ θ
00 0 θ θ θ θ θ θ θ . It vanishes if and only if ondition (1.2) holds.Before passing to R n with general n ≥ , it is instru tive to repeat the above onsiderations in the ases of n = 4 and n = 5 .A point on a urve (1 , x, x , x ) in R de(cid:28)nes a plane passing through the originin the dual spa e ( R ) ∗ via the relation: θ + 3 θ x + 3 θ x + θ x = 0 (1.3) θ + 2 θ x + θ x = 0 . Now ( θ , θ , θ , θ ) parametrize points of the dual ( R ) ∗ and when moving alongthe urve (1 , x, x , x ) in R , the orresponding planes in ( R ) ∗ sweep out a ruled L (2 , R ) GEOMETRY OF ODE'S 3hypersurfa e there, whi h is de(cid:28)ned by the vanishing of the resultant of the twopolynomials de(cid:28)ned in (1.3). This is given by(1.4) − θ ) ( θ ) + 4 θ ( θ ) + 4( θ ) θ − θ θ θ θ + ( θ ) ( θ ) = 0 , as an be easily al ulated.For n = 5 , a point on a urve (1 , x, x , x , x ) in R de(cid:28)nes a 3-plane passingthrough the origin in the dual spa e ( R ) ∗ via the relation: θ + 4 θ x + 6 θ x + 4 θ x + θ x = 0 (1.5) θ + 3 θ x + 3 θ x + θ x = 0 , where ( θ , θ , θ , θ , θ ) parametrize points of the dual ( R ) ∗ as before. And now,when moving along the urve (1 , x, x , x , x ) in R , the orresponding 3-planes in ( R ) ∗ sweep out a ruled hypersurfa e there, whi h is again de(cid:28)ned by the vanishingof the resultant of the two polynomials de(cid:28)ned in (1.5). The algebrai expressionfor this hypersurfa e in terms of the θ oordinates is quite ompli ated: − θ ) ( θ ) ( θ ) + 54 θ ( θ ) ( θ ) + 64( θ ) ( θ ) − θ θ θ ( θ ) +27( θ ) ( θ ) + 54( θ ) ( θ ) θ − θ ( θ ) θ − θ ) θ θ θ + (1.6) θ θ ( θ ) θ θ + 6 θ ( θ ) ( θ ) θ − θ ) θ ( θ ) θ + 27( θ ) ( θ ) − θ ( θ ) θ ( θ ) + 18( θ ) ( θ ) ( θ ) + 12( θ ) θ θ ( θ ) − ( θ ) ( θ ) = 0 , but easily al ulable.The beauty of the hypersurfa es (1.2), (1.4) and (1.6) onsists in this that theyare given by means of homogeneous equations, and thus they des end to the or-responding proje tive spa es. From the point of view of the present paper, evenmore important is the fa t, that they are GL (2 , R ) invariant. By this we mean thefollowing:Consider a real polynomial of ( n − -th degree(1.7) w ( x ) = n − X i =0 (cid:18) n − i (cid:19) θ i x i in the real variable x with real oe(cid:30) ients ( θ , θ , ..., θ n − ) . The n -dimensionalve tor spa e ( R n ) ∗ of su h polynomials may be identi(cid:28)ed with the spa e of their oe(cid:30) ients. Now, repla ing the variable x by a new variable x ′ su h that(1.8) x = αx ′ + βγx ′ + δ , αδ − βγ = 0 , we de(cid:28)ne a new ove tor ( θ ′ , θ ′ , ..., θ ′ n − ) whi h is related to ( θ , θ , ..., θ n − ) of(1.7) via n − X i =0 (cid:18) n − i (cid:19) θ ′ i x ′ i = ( γx ′ + δ ) n − w ( x ) . It is obvious that θ ′ = ( θ ′ , θ ′ , ..., θ ′ n − ) is linearly expressible in terms of θ =( θ , θ , ..., θ n − ) :(1.9) θ ′ = θ · ρ n ( a ) , a = (cid:18) α βγ δ (cid:19) . Here a orresponds to the GL (2 , R ) transformation (1.8), and the map ρ n : GL (2 , R ) → GL (( R n ) ∗ ) ∼ = GL ( n, R ) MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIde(cid:28)nes the real n -dimensional irredu ible representation of GL (2 , R ) . For example,if n = 2 , we have w ( x ) = θ + 2 θ x + θ x , and we easily get (cid:0) θ ′ θ ′ θ ′ (cid:1) = (cid:0) θ θ θ (cid:1) δ γδ γ βδ αδ + βγ αγβ αβ α , so that ρ is given by ρ (cid:18) α βγ δ (cid:19) = δ γδ γ βδ αδ + βγ αγβ αβ α . Now, let us de(cid:28)ne g ( θ, θ ) , I ( θ, θ, θ, θ ) and I ( θ, θ, θ, θ, θ, θ ) by g ( θ, θ ) = the left hand side of ( ) I ( θ, θ, θ, θ ) = the left hand side of ( ) (1.10) I ( θ, θ, θ, θ, θ, θ ) = the left hand side of ( ) . We will often abbreviate this notation to the respe tive: g ( θ ) , I ( θ ) and I ( θ ) .To explain our omment about the GL (2 , R ) invarian e of the respe tive hyper-surfa es g ( θ ) = 0 , I ( θ ) = 0 and I ( θ ) = 0 we al ulate g ( θ ′ ) , I ( θ ′ ) and I ( θ ′ ) with θ ′ as in (1.9). The result is g ( θ ′ ) = ( αδ − βγ ) g ( θ ) I ( θ ′ ) = ( αδ − βγ ) I ( θ ) I ( θ ′ ) = ( αδ − βγ ) I ( θ ) . Thus the vanishing of the expressions g ( θ ) , I ( θ ) and I ( θ ) is invariant under thea tion (1.9) of the irredu ible GL (2 , R ) on ( R n ) ∗ .We are now ready to dis uss the general ase n ≥ of the rational normal urve (1 , x, x , ..., x n − ) in R n . Asso iated with this urve is a pair of polynomials, namely w ( x ) as in (1.7), and its derivative d w d x . We onsider the relation(1.11) w ( x ) = 0 & d w d x = 0 . This gives a orresponden e between the points on the urve (1 , x, x , ..., x n − ) in R n and the ( n − -planes passing through the origin in the dual spa e ( R n ) ∗ parametrized by ( θ , θ , ..., θ n − ) . When moving along the rational normal urve in R n , the orresponding ( n − -planes in ( R n ) ∗ sweep out a ruled hypersurfa e there.This is de(cid:28)ned by the vanishing of the resultant, R ( w ( x ) , d w d x ) , of the two polyno-mials in (1.11). The algebrai expression for this hypersurfa e is the vanishing ofa homogeneous polynomial, let us all it I ( θ ) , of order n − , in the oordinates ( θ , θ , ..., θ n − ) . The hypersurfa e I ( θ ) = 0 in ( R n ) ∗ is GL (2 , R ) invariant, sin ethe property of the two polynomials w ( x ) and d w d x to have a ommon root is in-dependent of the hoi e (1.8) of the oordinate x . Thus GL (2 , R ) is in luded inthe stabilizer G I of I under the a tion of the full GL ( n, R ) group. This stabilizer,by de(cid:28)nition, is a subgroup of GL ( n, R ) with elements b ∈ G I ⊂ GL ( n, R ) su hthat I ( θ · b ) = (det b ) n − n I ( θ ) . Moroever, in n = 4 , , it turns out that G I is pre- isely the group GL (2 , R ) in the orresponding irredu ible representation ρ n . Thusif n = 4 , one an hara terize the irredu ible GL (2 , R ) in n dimensions as thestabilizer of the polynomial I ( θ ) . L (2 , R ) GEOMETRY OF ODE'S 5Cru ial for the present paper is an observation of Karl Wüns hmann that thealgebrai geometry and the orresponden es we were des ribing above, naturallyappear in the analysis of solutions of the ODE y ( n ) = 0 . Indeed, following Wün-s hmann1 (see the Introdu tion in his PhD thesis [21℄, pp. 5-6), we note the fol-lowing:Consider the third order ODE: y ′′′ = 0 . Its general solution is y = c +2 c x + c x , where c , c , c are the integration onstants. Thus, the solution spa e of the ODE y ′′′ = 0 is R with solutions identi(cid:28)ed with points c = ( c , c , c ) ∈ R . Thesolutions to the ODE y ′′′ = 0 may be also identi(cid:28)ed with urves y ( x ) = c +2 c x + c x , a tually parabolas, in the plane ( x, y ) . Suppose now, that we taketwo solutions of y ′′′ = 0 orresponding to two neighbouring points c = ( c , c , c ) and c + d c = ( c + d c , c + d c , c + d c ) in R . Among all pairs of neighbouringsolutions we hoose only those, whi h have the property that their orresponding urves y = y ( x ) and y + d y = y ( x ) + d y ( x ) tou h ea h other, at some point ( x , y ) in the plane ( x, y ) . If we do not require anything more about the properties of thisin iden e of the two urves, we say that solutions c and c + d c have zero order onta t at ( x , y ) .In this `baby' example everything is very simple:To get the riterion for the solutions to have zero order onta t we (cid:28)rst writethe urves y = c + 2 c x + c x and y + d y = c + d c + 2( c + d c ) x + ( c + d c ) x orresponding to c and c + d c . Thus the solutions have zero order onta t at ( x , y ( x )) provided that d y ( x ) = 0 , i.e. if and only if d c + 2 x d c + x d c = 0 . This shows that su h a onta t is possible if and only if the determinant g (d c , d c ) = (d c ) − d c d c is nonnegative, sin e otherwise the quadrati equation for x has no solutions. Un-expe tedly, we (cid:28)nd that the requirement for the two neighbouring solution urvesof y ′′′ = 0 to have zero order onta t at some point is equivalent to the requirementthat the orresponding two neighbouring points c and c + d c in R are spa elikeseparated in the Minkowski metri g in R . This is the dis overy of Wüns hmannthat is quoted in Elie Cartan's 1941 year's paper2 [5℄.Now we onsider the neighbouring solutions c and c + d c of y ′′′ = 0 whi h arenull separated in the metri d s . What we an say about the orresponding urvesin the plane ( x, y ) ?To answer this we need the notion of a (cid:28)rst order onta t: Two neighbouringsolution urves y = c + 2 c x + c x and y + d y = c + 2 c x + c x + (d c + 2 x d c + x d c ) of y ′′′ = 0 , orresponding to c and c + d c in R , have (cid:28)rst order onta t at ( x , y ) i(cid:27) they have zero order onta t at ( x , y ) and, in addition, their urves of1We are very grateful to Niels S human, who found a opy of Wüns hmann thesis in the itylibrary of Berlin and sent it to us. It was this opy, whi h after translation from German byDenson Hill, led us to write this introdu tion.2It is worthwhile to remark, that Wüns hmann thesis is dated `1905', the same year in whi hEinstein published his famous spe ial relativity theory paper [9℄. It was not until three years laterwhen Minkowski gave the geometri interpretation of Einstein's theory in terms of his metri [15℄.Perhaps Wüns hmann was the (cid:28)rst who ever wrote su h metri in a s ienti(cid:28) paper. This is avery interesting feature of Wüns hmann thesis: he alls the expressions like (d c ) − d c d c = 0 ,a Monges he Glei hung rather than a one in the metri , be ause the notion of a metri withsignature di(cid:27)erent than the Riemannian one was not yet abstra ted! MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKI(cid:28)rst derivatives, y ′ = 2 c + 2 c x and y ′ + d y ′ = 2( c + d c ) + 2( c + d c ) x , have zeroorder onta t at ( x , y ) . Thus the ondition of (cid:28)rst order onta t at ( x , y ( x )) is equivalent to d y ( x ) = 0 and d y ′ ( x ) = 0 , i.e. to the ondition that x is asimultaneous root for d c + 2 x d c + x d c = 0 (1.12) d c + x d c = 0 . Solving the se ond of these equations for x , and inserting it into the (cid:28)rst, afteran obvious simpli(cid:28) ation, we on lude that (d c ) − d c d c = 0 . Thus we getthe interpretation of the null separated neighbouring points in R as the solutionsof y ′′′ = 0 whose urves in the ( x, y ) plane are neighbouring and have (cid:28)rst order onta t at some point.Wüns hmann notes that the pro edure des ribed here for the equation y ′′′ = 0 an be repeated for the equation y ( n ) = 0 for arbitrary n ≥ . In the ases of n = 4 and n = 5 he however passes to the dis ussion of the solutions that have onta tof order ( n − rather then one. This is an interesting possibility, omplementaryin a sense to the one in whi h the solutions have (cid:28)rst order onta t. Wüns hmannspends rest of the thesis studying it. But we will not dis uss it here.Sin e Wüns hmann does not dis uss the (cid:28)rst order onta t of the solutions in n = 4 , , let us look loser onto these two ases:The general solution to y (4) = 0 is y = c + 3 c x + 3 c x + c x , and the generalsolution to y (5) = 0 is y = c +4 c x +6 c x +4 c x + c x . Thus now the solutions arepoints c in R and R , respe tively. The ondition that the neighbouring solutions c = ( c , c , c , c ) and c + d c = ( c + d c , c + d c , c + d c , c + d c ) of y (4) = 0 have(cid:28)rst order onta t at ( x , y ( x )) is equivalent to the requirement that the system d c + 3 x d c + 3 x d c + x d c = 0 (1.13) d c + 2 x d c + x d c = 0 has a ommon root x . Similarly, the ondition that the neighbouring solutions c = ( c , c , c , c , c ) and c + d c = ( c + d c , c + d c , c + d c , c + d c , c + d c ) of y (5) = 0 have (cid:28)rst order onta t at ( x , y ( x )) is equivalent to the requirementthat the system d c + 4 x d c + 6 x d c + 4 x d c + x d c = 0 (1.14) d c + 3 x d c + 3 x d c + x d c = 0 has a ommon root x . Cal ulating the resultants for the systems (1.12), (1.13),and (1.14) we get: • R = g (d c , d c )d c if n = 3 , • R = I (d c , d c , d c , d c )d c if n = 4 , • R = I (d c , d c , d c , d c , d c , d c )d c if n = 5 ,where g , I and I are as in (1.10).This on(cid:28)rms our earlier statement that two neighbouring solutions of y ′′′ = 0 have (cid:28)rst order onta t i(cid:27) g (d c , d c ) = 0 , sin e if d c = 0 the system (1.12) ollapsesto d c = d c = 0 . Similarly, one an prove that two neighbouring solutions of y (4) = 0 or y (5) = 0 have (cid:28)rst order onta t if and only if they are null separatedin the respe tive symmetri multilinear forms I or I . Our previous dis ussion ofthe invariant properties of these forms, shows that in the solution spa e of an ODE y ( n ) = 0 , for n ≥ , there is a naturally de(cid:28)ned a tion of the the GL (2 , R ) group. L (2 , R ) GEOMETRY OF ODE'S 7This group is the stabilizer of the invariant polynomial I (d c ) whi h distinguishesneighbouring solutions having (cid:28)rst order onta t.The main question one an ask in this ontext is if one an retain this GL (2 , R ) stru ture in the solution spa e for more ompli ated ODEs. In other words, onemay asks the following: What does one have to assume about the fun tion F ,de(cid:28)ning an ODE y ( n ) = F ( x, y, y ′ , . . . , y ( n − ) , in order to have a well de(cid:28)ned onformal tensor g , I or I , in the respe tive ases n = 3 , , , on the solution spa e of the ODE? The same question an be repeatedfor any n > and the invariant I .The answer to this question in the n = 4 ase was found by Robert Bryant in [2℄.Later, the answer for n > ase was given by Boris Doubrov [7℄ who establisheda onne tion between the Wil zynski invariants [20℄ for a linear ODE, and ertain onta t invariant onditions for a nonlinear ODE asso iated with it. For any n ≥ ,given F , Doubrov onditions are built from the Wil zynski invariants al ulated forthe linearization of y ( n ) = F about one of its solutions (see [7℄ for details). In a quitedi(cid:27)erent perspe tive, these onditions, were also dis overed by Ma iej Dunajski andPaul K Tod [8℄.Doubrov-Wil zynski onditions di(cid:27)er from Bryant ones for n = 4 . They alsodi(cid:27)er from the onditions we are going to dis uss in the present paper for n ≥ .Doubrov, Dunajski and Tod have ( n − nonlinear PDEs for F of ODE y ( n ) = F .Although this number, ( n − , agrees with the number of onditions we presenthere, there is an important di(cid:27)eren e: ea h of the ( n − onditions for F , de(cid:28)nedby the above authors, has a di(cid:27)erent order in the derivatives of F . When we arrangeDoubrov-Wil zynski onditions a ording to the order of the orresponding PDEsfor F , we (cid:28)nd that the (cid:28)rst ondition is of order , the se ond is of order , and soon, up to the order n of the ( n − -th ondition. On the ontrary ea h of our ( n − onditions is of the third order in the derivatives of F . The simple explanation ofthis dis repan y is as follows: We obtain our onditions, by applying a variant ofCartan's equivalen e method ; in the pro ess of extra ting them we obtain the (cid:28)rst ondition to be of the third order as everybody does. But the se ond onditionwhi h, if we were not applying Cartan's method, would be of order four, a tually ollapses in our derivation to a ondition of order three. This is be ause Cartan'smethod automati ally utilises the (cid:28)rst ondition of order three by di(cid:27)erentiatingit, and then eliminating the fourth derivative from the se ond ondition by meansof the fourth derivative from the di(cid:27)erentiated ondition of order three. The samesituation is automati ally a omplished for the ondition of order (cid:28)ve and so on.As a result we have ( n − onditions of order three. They are di(cid:27)erent from theDoubrov-Wil zynski onditions already for the ODE of order four. In the n = 3 ase all the onditions, namely those of Wüns hmann, Doubrov, Dunajski and Tod,and ours are the same. In dimension n = 4 our onditions agree with the Bryantones. Sin e Wüns hman was the (cid:28)rst who obtained these type of onditions in n = 3 and found method of their onstru ting for arbitrary n we all the onditionsdis ussed in this paper generalized Wüns hmann's onditions, or Wüns hmann's onditions, for short.Finding the Wüns hmann onditions for F of order n ≥ , although important,is only a byprodu t of our analysis. The present paper is devoted to a thoroughstudy of the irredu ible GL (2 , R ) geometry in dimension (cid:28)ve. This is done from MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKItwo points of view: (cid:28)rst as a study of an abstra t geometry on a manifold and,se ond, as a study of a onta t geometry of (cid:28)fth order ODEs.We de(cid:28)ne an abstra t 5-dimensional GL (2 , R ) geometry in two steps.First, in se tion 2, we show how to onstru t the algebrai model for the GL (2 , R ) geometry in dimension (cid:28)ve utilising properties of a rational normal urve. Se -ond, instead of obtaining the redu tion from GL (5 , R ) to GL (2 , R ) by stabiliz-ing the 6-tensor I , we get the desired redu tion by stabilizing a onformal met-ri g ij → e φ g ij of signature (3 , and a onformal totally symmetri 3-tensor Υ ijk → e φ Υ ijk . These tensors are supposed to be related by the following alge-brai onstraint:(1.15) g lm (Υ ijl Υ kmp + Υ kil Υ jmp + Υ jkl Υ imp ) = g ij g kp + g kl g jp + g jk g ip . It is worthwile to note that ondition (1.15) is a non-Riemannian ounterpart ofthe ondition onsidered by Elie Cartan in the ontext of isoparametri surfa es[3℄, [4℄.Our main obje t of study is then de(cid:28)ned as follows:De(cid:28)nition. An irredu ible GL (2 , R ) geometry in dimension (cid:28)ve is a 5-dimensionalmanifold M equipped with a lass of triples [ g, Υ , A ] su h that on M : ( a ) g is a metri of signature (3 , , ( b ) Υ is a tra eless symmetri 3rd rank tensor, ( c ) A is a 1-form, ( d ) the metri g and the tensor Υ satisfy the identity (1.15), ( e ) two triples ( g, Υ , A ) and ( g ′ , Υ ′ , A ′ ) are in the same lass [ g, Υ , A ] if andonly if there exists a fun tion φ : M → R su h that g ′ = e φ g, Υ ′ = e φ Υ , A ′ = A − φ. This de(cid:28)nition pla es GL (2 , R ) geometries in dimension (cid:28)ve among the Weylgeometries [ g, A ] . They are spe ial Weyl geometries i.e. su h for whi h the stru -ture group is redu ed from CO (3 , to GL (2 , R ) . A natural des ription of su hgeometries should be then obtained in terms of a ertain gl (2 , R ) -valued onne tion.However, unlike the usual Weyl ase, the hoi e of su h a onne tion is ambigu-ous, due to the fa t that su h a onne tion has non-vanishing torsion in general,and one must (cid:28)nd admissible onditions for the torsion that spe ify onne tionuniquely. Pursuing this problem in se tion 3 we (cid:28)nd an interesting sub lass of GL (2 , R ) geometries.Proposition. A GL (2 , R ) geometry [ g, Υ , A ] is alled nearly integrable if the Weyl onne tion W ∇ of [ g, A ] satis(cid:28)es ( W ∇ X Υ)(
X, X, X ) = − A ( X )Υ( X, X, X ) . It turns out, see se tion 3, that the nearly integrable GL (2 , R ) geometries uniquelyde(cid:28)ne a gl (2 , R ) onne tion D . This is hara terized by the following requirements: • it preserves the stru tural tensors: Dg ij = − Ag ij ,D Υ ijk = − A Υ ijk , • and it has totally skew symmetri torsion. L (2 , R ) GEOMETRY OF ODE'S 9We all this unique onne tion the hara teristi onne tion for the nearly inte-grable stru ture GL (2 , R ) .The rest of se tion 3 is devoted to study the algebrai stru ture of the torsionand the urvature of the hara teristi onne tion of a nearly integrable stru -ture. Sin e the tensor produ ts of tangent spa es are redu ible under the a tionof GL (2 , R ) , we de ompose the torsion and the urvature tensors into omponentsbelonging to the irredu ible representations. In parti ular, the skew symmetri tor-sion T has two omponents, T (3) and T (7) , lying in the three-dimensional and theseven-dimensional irredu ible representations respe tively. Likewise the Maxwell2-form d A and the Ri i tensor R de ompose a ording to d A = d A (3) + d A (7) and R = R (1) + R (3) + R (5) + R (7) + R (9) . The last problem we adress in se tion 3 on- erns with the properties of geometries whose hara teristi onne tions have `thesmallest possible' torsion, that is the torsion of the pure three-dimensional type. Inproposition 3.11 we prove that Ri i tensor for su h stru tures satis(cid:28)es remarkableidentities: R (3) = d A (3) , R (7) = d A (7) , R (9) = 0 . The third equation is equivalent to R ( ij ) = Rg ij + R kl Υ klm Υ ijm . In se tion 4 we brie(cid:29)y des ribe GL (2 , R ) geometry in the language of the bundle GL (2 , R ) → P → M . We also show how an appriopriate oframe de(cid:28)ned on a nine-dimensional manifold P turns this manifold into a bundle GL (2 , R ) → P → M and generates the GL (2 , R ) geometry on M . This onstru tion is the ore of theproof of the main theorem in se tion 5. This loses the part of the paper that isdevoted to abstra t GL (2 , R ) geometries.Se tion 5 ontains the main result of this paper, theorem 5.3, whi h links GL (2 , R ) geometry with the realm of ordinary di(cid:27)erential equations. It an be en apsulatedas follows.Theorem. A 5th order ODE y (5) = F ( x, y, y ′ , y ′′ , y ′′′ , y (4) ) that satis(cid:28)es three Wün-s hmann onditions de(cid:28)nes a nearly integrable irredu ible GL (2 , R ) geometry ( M , [ g, Υ , A ]) on the spa e M of its solutions. This geometry has the hara teristi onne tion with torsion of the `pure' type in the 3-dimensional irredu ible repre-sentation of GL (2 , R ) . Two 5th order ODEs that are equivalent under onta ttransformation of variables de(cid:28)ne equivalent GL (2 , R ) geometries.The theorem has numerous appli ations. For example, we use it to hara terisevarious lasses of Wüns hmann 5th order ODEs, by means of the algebrai typeof the tensors asso iated with the orresponding hara teristi onne tion. Forexample i(cid:27) F y (4) y (4) = 0 , the torsion of the hara tersti onne tion vanishes, andi(cid:27) F y (4) y (4) y (4) = 0 , then we have d A (7) = 0 .The proof of the theorem onsists of an appli ation of the Cartan method ofequivalen e. We write an ODE, onsidered modulo onta t transformation of vari-ables, as a G -stru ture on the four-order jet spa e. Starting from this G -stru turewe expli itly onstru t a 9-dimensional manifold P , whi h is a GL (2 , R ) bundleover the solution spa e and arries ertain distinguished oframe. This onstru -tion is only possible provided that the ODE satis(cid:28)es the Wüns hmann onditions,whi h we write down expli itly. By means of proposition 4.1 the oframe on P d A = 0 . We also give highly nontrivial examples of equations for whi h d A = 0 , in luding a family of examples with fun tion F being a solution of a ertainse ond order ODE.Finally, in se tion 7 we onsider ODEs of order n > . We apply results ofthe Hilbert theory of algebrai invariants, to de(cid:28)ne the tensors responsible for theredu tion GL ( n, R ) → GL (2 , R ) . We also give the expli it formulae for the ( n − third order Wüns hmann onditions for n = 6 and n = 7 .2. A pe uliar third rank symmetri tensorConsider R n equipped with a Riemannian metri g and a 3rd rank tra efreesymmetri tensor Υ ∈ S R n satisfying:(i) Υ ijk = Υ ( ijk ) - (symmetry)(ii) g ij Υ ijk = 0 - (tra efree)(iii) g lm (Υ ijl Υ kmp + Υ kil Υ jmp + Υ jkl Υ imp ) = g ij g kp + g kl g jp + g jk g ip .It turns out that the third ondition is very restri tive. In parti ular Cartan hasshown [3, 4℄ that for (iii) to be satis(cid:28)ed the dimension n must be one of the following: n = 5 , , , . Moreover Cartan onstru ted Υ in ea h of these dimensions and hasshown that it is unique up to an O ( n ) transformation. Restri ting to n = 5 , , , ,we onsider the stabilizer H n of Υ under the a tion of GL ( n, R ) : H n = { GL ( n, R ) ∋ a : Υ( aX, aY, aZ ) = Υ( X, Y, Z ) , ∀ X, Y, Z ∈ R n } . Then, one (cid:28)nds that: • H = SO (3) ⊂ SO (5) in the 5-dimensional irredu ible representation, • H = SU (3) ⊂ SO (8) in the 8-dimensional irredu ible representation, • H = Sp (3) ⊂ SO (14) in the 14-dimensional irredu ible representation, • H = F ⊂ SO (26) in the 26-dimensional irredu ible representation.The relevan e of onditions (i)-(iii) is that they are invariant under the O ( n ) a tionon the spa e of tra efree symmetri tensors S R n . Moreover they totally hara -terize the orbit O ( n ) /H n ⊂ S of the tensor Υ under this a tion [1, 16℄.The question arises if one an onstru t tensors satisfying (i)-(iii) for metri shaving non-Riemannian signatures. Below we show how to do it if n = 5 and themetri g has signature (3 , . This onstru tion des ribed to us by E. Ferapontov[10, 11℄ is as follows.Consider R with oordinates ( θ , θ , θ , θ , θ ) , and a urve γ ( x ) = (1 , x, x , x , x ) ⊂ R . Asso iated to the urve γ there are two algebrai varieties in R : • The bise ant variety. This is de(cid:28)ned to be a set onsisting of all the pointson all straight lines rossing the urve γ in pre isely two points. It is givenparametri ally as B ( x, s, u ) = (1 , x, x , x , x ) + u (0 , x − s, x − s , x − s , x − s ) , L (2 , R ) GEOMETRY OF ODE'S 11where x, s, u are three real parameters. • The tangent variety. This is de(cid:28)ned to be a set onsisting of all the pointson all straight lines tangent to the urve γ . It is given parametri ally as T ( x, s ) = (1 , x, x , x , x ) + s (0 , , x, x , x ) . One of many interesting features of these two varieties is that they de(cid:28)ne (up toa s ale) a tri-linear symmetri form(2.1) Υ( θ ) = 3 √ θ θ θ + 2 θ θ θ − ( θ ) − θ ( θ ) − θ ( θ ) ) and a bi-linear symmetri form(2.2) g ( θ ) = θ θ − θ θ + 3( θ ) . These forms are distinguished by the fa t that the bise ant and tangent varietiesare ontained in their null ones. In the homogeneous oordinates ( θ , θ , θ , θ , θ ) in R all the points θ of B ( x, s, u ) satisfy Υ( θ ) = 0 , whereas all the points θ of T ( x, s ) satisfy Υ( θ ) = 0 and g ( θ ) = 0 . Writing the forms as Υ( θ ) = Υ ijk θ i θ j θ k , g ( θ ) = g ij θ i θ j , i, j, k = 0 , , , , one an he k that so de(cid:28)ned g ij and Υ ijk satisfy relations (i)-(iii) of the previousse tion.Although it is obvious we remark that the above de(cid:28)ned metri g ij has signature (3 , .As we have already noted the forms Υ( θ ) and g ( θ ) are de(cid:28)ned only up to as ale. We were also able to (cid:28)nd a fa tor, the √ in expression (2.1), that makesthe orresponding g ij and Υ ijk satisfy (i)-(iii). Note that these onditions are onformal under the simultaneous hange: g ij → e φ g ij , Υ ijk → e φ Υ ijk . Thus it is interesting to onsider in R a lass of pairs [ g, Υ] , su h that: • in ea h pair ( g, Υ) (cid:21) g is a metri of signature (3 , ,(cid:21) Υ is a tra eless symmetri 3rd rank tensor,(cid:21) the metri g and the tensor Υ satisfy the identity g lm (Υ ijl Υ kmp + Υ kil Υ jmp + Υ jkl Υ imp ) = g ij g kp + g kl g jp + g jk g ip , • two pairs ( g, Υ) and ( g ′ , Υ ′ ) are in the same lass [ g, Υ] if and only if thereexists φ ∈ R su h that(2.3) g ′ = e φ g, Υ ′ = e φ Υ . Given a stru ture ( R , [ g, Υ]) we de(cid:28)ne a group CH to be a subgroup of thegeneral linear group GL (5 , R ) preserving [Υ] . This means that, hoosing a repre-sentative Υ of the lass [Υ] , we de(cid:28)ne CH to be: CH = { GL (5 , R ) ∋ a : Υ( ax, ax, ax ) = (det a ) (3 / Υ( x, x, x ) } . Note that the exponent in the above expression is aused by the fa t that ther.h.s. of the equation de(cid:28)ning the group elements must be homogeneous of degree3 in a , similarly as the l.h.s. is.2 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIThis de(cid:28)nition does not depend on the hoi e of a representative Υ ∈ [Υ] . Wehave the followingProposition 2.1. The set CH of × real matri es a ∈ GL (5 , R ) preserving [Υ] is the GL (2 , R ) group in its 5-dimensional irredu ible representation. Moreover,we have natural in lusions CH = GL (2 , R ) ⊂ CO (3 , ⊂ GL (5 , R ) , where CO (3 , is the -dimensional group of homotheties asso iated with the on-formal lass [ g ] .Remark 2.2. A ording to our Introdu tion, there is another GL (2 , R ) invariantsymmetri onformal tensor that stabilizes GL (5 , R ) to the irredu ible GL (2 , R ) .This is the tensor I ijklpq de(cid:28)ned via I ( θ ) = I ijklpq θ i θ j θ k θ l θ p θ q with I as in(1.10). We prefer however to work with a pair ( g ij , Υ ijk ) rather then with I ijklpq ,be ause of the lower rank, and more importantly, be ause of the evident onformalmetri properties of the ( g ij , Υ ijk ) approa h. Also, it is worthwhile to note thatthe invariants g ij , Υ ijk and I ijklpq are not independent. Indeed, one an easilly he k that I of (1.10), Υ of (2.1) and g of (2.2) are related by I = Υ − g . Weinterpret this relation as the de(cid:28)nition of I in terms of more primitive quantities g and Υ .The isotropy ondition for the group elements a of CH has its obvious ounter-part at the level of the Lie algebra gl (2 , R ) = ( R ⊕ sl (2 , R )) ⊂ co (3 , ⊂ gl (5 , R ) of CH = GL (2 , R ) . Writing a = exp( t Γ) we (cid:28)nd that the in(cid:28)nitesimal version of theisotropy ondition, written in terms of the × matri es Γ = (Γ ij ) is:(2.4) Γ li Υ ljk + Γ lj Υ ilk + Γ lk Υ ijl = T r (Γ)Υ ijk , where T r (Γ) = Γ mm . Given Υ ijk , these linear equations an be solved for Γ . Takingthe most general matrix Γ ∈ GL (5 , R ) and Υ ijk given by Υ( x, x, x ) = Υ ijk x i x j x k of (2.1) we (cid:28)nd the expli it realization of the 5-dimensional representation of the gl (2 , R ) Lie algebra as:(2.5)
Γ = Γ − E − + Γ + E + + Γ E + Γ E , where Γ − , Γ + , Γ , Γ are free real parameters, and ( E − , E + , E , E ) are × ma-tri es given by: E + = , E − = , (2.6) E = − − , E = − . The ommutator in gl (2 , R ) = Span R ( E − , E + , E , E ) L (2 , R ) GEOMETRY OF ODE'S 13is the usual ommutator of matri es. In parti ular, the non-vanishing ommutatorsare: [ E , E + ] = − E + , [ E , E − ] = 2 E − , [ E + , E − ] = − E . Note that sl (2 , R ) = Span R ( E − , E + , E ) is a subalgebra of gl (2 , R ) isomorphi to sl (2 , R ) . It provides the -dimensionalirredu ible representation of sl (2 , R ) .3. Irredu ible GL (2 , R ) geometries in dimension fiveIn this se tion we des ribe 5-dimensional manifolds whose tangent spa e at ea hpoint is equipped with the stru ture [ g, Υ] of the previous se tion. We will analyzesu h manifolds in terms of an appropriately hosen onne tion. We will des ribe onne tions on a manifold M in terms of Lie-algebra-valued 1-forms on M . To bemore spe i(cid:28) , let dim M = n and let g denote an n -dimensional representation ofsome Lie algebra. The onne tion 1-forms Γ ij on M are the matrix entries of anelement Γ ∈ g ⊗ Λ M . They de(cid:28)ne the ovariant exterior derivative D . This a tson tensor-valued-forms via the extension to the higher order tensors of the formula: Dv i = d v i + Γ ij ∧ v i . Now suppose that we have a 5-dimensional manifold M equipped with a lassof pairs [ g, Υ] su h that g is a metri , Υ is a 3rd rank tensor related to the metri via properties (i)-(iii) of the previous se tion, and two pairs ( g, Υ) and ( g ′ , Υ ′ ) are in the same pair i(cid:27) they are related by (2.3), where φ is now a fun tion on M . If we want to asso iate a onne tion with su h a stru ture we have to spe ifyhow this onne tion is related to the pair [ g, Υ] . A possible approa h is to hoose arepresentative ( g, Υ) of [ g, Υ] and de lare what is Dg and D Υ . A (cid:28)rst possible hoi e Dg = 0 or D Υ = 0 is de(cid:28)nitely not good sin e, in general, Dg ′ and D Υ ′ would notbe vanishing for another hoi e of the representative of [ g, Υ] . A remedy for thissituation omes from the Weyl geometry where, given a onformal lass ( M, [ g ]) , a1-form A is introdu ed so that the onne tion satis(cid:28)es Dg ij = − Ag ij . In our ase weintrodu e a 1-form A on M and require that Dg ij = − Ag ij and D Υ ijk = − A Υ ijk .Then, if we transform ( g, Υ) a ording to (2.3), the transformed obje ts will satisfy Dg ′ ij = − A ′ g ′ ij and D Υ ′ ijk = − A ′ Υ ′ ijk provided that A ′ = A − φ . This motivatesthe followingDe(cid:28)nition 3.1. An irredu ible GL (2 , R ) stru ture in dimension (cid:28)ve is a 5-dimensionalmanifold M equipped with a lass of triples [ g, Υ , A ] su h that on M : ( a ) g is a metri of signature (3 , , ( b ) Υ is a tra eless symmetri 3rd rank tensor, ( c ) A is a 1-form, ( d ) the metri g and the tensor Υ satisfy the identity g lm (Υ ijl Υ kmp + Υ kil Υ jmp + Υ jkl Υ imp ) = g ij g kp + g kl g jp + g jk g ip , ( e ) two triples ( g, Υ , A ) and ( g ′ , Υ ′ , A ′ ) are in the same lass [ g, Υ , A ] if andonly if there exists a fun tion φ : M → R su h that g ′ = e φ g, Υ ′ = e φ Υ , A ′ = A − φ. M was only equipped with a lass of pairs [ g, A ] satisfying onditions ( a ) , ( c ) and ( e ) (with Υ , Υ ′ omitted), then ( M , [ g, A ]) would de(cid:28)ne the Weyl geometry.Su h geometry, whi h has the stru ture group CO (3 , , is usually studied in termsof the Weyl onne tion. This is the unique torsionfree onne tion preserving the onformal stru ture [ g, A ] . It is de(cid:28)ned by the following two equations: W D g ij = − Ag ij (preservation of the class [ g, A ]) , (3.1) W D θ i = 0 (no torsion) , (3.2)where θ i is a oframe related to the representative g of the lass [ g ] by g = g ij θ i θ j .We des ribe the Weyl onne tion in terms of the Weyl onne tion 1-forms W Γ ij , i, j = 0 , , , , .Take a representative ( g, A ) of the Weyl stru ture [ g, A ] on M . Choose a oframe ( θ i ) , i = 0 , , , , , su h that g = g ij θ i θ j , with all the metri oe(cid:30) ients g ij being onstant. Then the above two equations de(cid:28)ne W Γ ij together with W Γ ij = g ik W Γ kj tobe 1-forms on M satisfying W Γ ij + W Γ ji = Ag ij (preservation of the class [ g, A ]) , (3.3) d θ i + W Γ ij ∧ θ j = 0 (no torsion) . (3.4)It follows that on e the representative ( g, A ) and the oframe θ i is hosen the aboveequations uniquely determine the Weyl onne tion 1-forms W Γ ij .We note that, due to ondition (3.3), matrix W Γ = ( W Γ ij ) of the Weyl onne tion1-forms belongs to the 5-dimensional de(cid:28)ning representation of the Lie algebra co (3 , ⊂ End (5 , R ) of the Lie group CO (3 , ⊂ GL (5 , R ) . Consequently, theWeyl onne tion oe(cid:30) ients W Γ ijk , de(cid:28)ned by W Γ ij = W Γ ijk θ k belong to the tensorprodu t co (3 , ⊗ R , the ve tor spa e of dimension (1+10)5=55.Now we assume that we have an irredu ible GL (2 , R ) stru ture [ g, Υ , A ] on a 5-manifold M . Forgetting about Υ gives the Weyl geometry as before. In parti ularthere is the unique Weyl onne tion W Γ asso iated with [ g, Υ , A ] . But the existen eof a metri ompatible lass of tensors Υ makes this Weyl geometry more spe ial.To analyze it we introdu e a new onne tion, whi h will be respe ting the entirestru ture [ g, Υ , A ] . This is rather a ompli ated pro edure whi h we des ribe below.Firstly we require that the new onne tion preserves [ g ] and [Υ] : Dg ij = − Ag ij (3.5) D Υ ijk = − A Υ ijk . (3.6)This does not determine the onne tion uniquely (cid:21) to have the uniqueness weneed to spe ify what the torsion of D is. We need some preparations to dis uss it.De(cid:28)nition 3.2. Let ( g, Υ , A ) be a representative of an irredu ible GL (2 , R ) stru -ture on a 5-dimensional manifold M . A oframe θ i , i = 0 , , , , , on M is alledadapted to the representative ( g, Υ , A ) if g = g ij θ i θ j = θ θ − θ θ + 3( θ ) and Υ = Υ ijk θ i θ j θ k = 3 √ θ θ θ + 2 θ θ θ − ( θ ) − θ ( θ ) − θ ( θ ) ) . L (2 , R ) GEOMETRY OF ODE'S 15Lo ally su h a oframe always exists and is given up to a GL (2 , R ) transforma-tion.Let us now hoose an adapted oframe θ i to a representative ( g, Υ , A ) of [ g, Υ , A ] .In this oframe equations (3.5)-(3.6) an be rewritten in terms of the onne tion1-forms Γ ij as Γ li g lj + Γ l jg li = Ag ij (3.7) Γ li Υ ljk + Γ lj Υ ilk + Γ lk Υ ijl = A Υ ijk . (3.8)When we ontra t the (cid:28)rst equation in indi es i and j we get(3.9) A = Γ ll = T r (Γ) . Inserting this into (3.8) we get(3.10) Γ li Υ ljk + Γ lj Υ ilk + Γ lk Υ ijl = Γ ll Υ ijk . Comparing this with (2.4) we see that the general solution for the onne tion 1-forms Γ ij are given by (2.5), i.e. Γ = Γ − E − + Γ + E + + Γ E + Γ E , where (Γ − , Γ + , Γ , Γ ) are four 1-forms on M su h that(3.11) Γ = − A. To (cid:28)x the remaining three 1-forms (Γ − , Γ + , Γ ) we introdu e an operator ¯Υ : co (3 , ⊗ R → S R de(cid:28)ned by: ¯Υ( W Γ) ijkm = Υ l ( ij W Γ lkm ) − W Γ ll ( m Υ ijk ) , and analyze its kernel ker ¯Υ .Writing equation (3.10) in terms of the oe(cid:30) ients Γ lim ∈ gl (2 , R ) ⊗ R andsymmetrizing it over the indi es { imjk } , we see that the whole gl (2 , R ) ⊗ R isin luded in ker ¯Υ .We use the metri to identify R with ( R ) ∗ , and more generally to identifytensor spa es N k ( R ) ∗ N l R with N ( k + l ) ( R ) ∗ . This enables us to identify theobje ts with upper indi es with the orresponding obje ts with lower indi es, e.g. T ijk = g il T ljk . Having in mind these identi(cid:28) ations we easily see that, due to theantisymmetry in the last two indi es, every 3-form T ijk = T [ ijk ] is in luded in ker ¯Υ .Thus we have: gl (2 , R ) ⊗ R ⊂ ker ¯Υ , V R ⊂ ker ¯Υ . The following proposition an be he ked by a dire t al ulation involving theexpli it form of the gl (2 , R ) representation given in (2.5), (2.6).Proposition 3.3. The ve tor spa e ker ¯Υ has the following properties: ker ¯Υ = ( gl (2 , R ) ⊗ R ) ⊕ V R and dim ker ¯Υ = 30 . W Γ lim ∈ ker ¯Υ , i. e. the equation(3.12) Υ l ( ij W Γ lkm ) = W Γ ll ( m Υ ijk ) , as a restri tion on possible Weyl onne tions. Let us assume that we have a stru -ture ( M , [ g, Υ , A ]) with the Weyl onne tion oe(cid:30) ients W Γ lim satisfying (3.12).The oe(cid:30) ients W Γ lim are written in a oframe adapted to some hoi e ( g, Υ , A ) .It is easy to see, using (3.3) and ontra ting (3.12) over all the free indi es with ave tor (cid:28)eld X i , that the restri tion on the Weyl onne tion (3.12) in oordinate-freelanguage is equivalent to(3.13) ( W ∇ X Υ)(
X, X, X ) = − A ( X )Υ( X, X, X ) . Here W ∇ denotes the Weyl onne tion in the Koszul notation.De(cid:28)nition 3.4. An irredu ible GL (2 , R ) stru ture ( M , [ g, Υ , A ]) is alled nearlyintegrable i(cid:27) its Weyl onne tion W ∇ asso iated to the lass [ g, A ] satis(cid:28)es (3.13).A ni e feature of nearly integrable stru tures ( M , [ g, Υ , A ]) is that they de(cid:28)nethe unique gl (2 , R ) -valued onne tion Γ . This follows from the above dis ussionabout the kernel of ¯Υ . Indeed, given a nearly integrable stru ture ( M , [ g, Υ , A ]) itis enough to hoose a representative ( g, Υ , A ) and to write the equation (3.13) forthe Weyl onne tion W Γ in an adapted oframe θ i . Then the uniquely given Weyl onne tion oe(cid:30) ients W Γ ijk are by de(cid:28)nition in ker ¯Υ = ( gl (2 , R ) ⊗ R ) ⊕ V R ,whi h means that they uniquely split onto Γ ijk ∈ gl (2 , R ) ⊗ R and T ijk ∈ V R .Thus, for all nearly integrable stru tures ( M , [ g, Υ , A ]) , in a oframe adapted to ( g, Υ , A ) , we have(3.14) W Γ ijk = Γ ijk + T ijk , and both Γ ijk ∈ gl (2 , R ) ⊗ R and T ijk ∈ V R are uniquely determined in termsof W Γ ijk . Now we rewrite the torsionfree ondition (3.4) for the Weyl onne tion inthe form(3.15) d θ i + Γ ij ∧ θ j = T ijk θ j ∧ θ k . It an be interpreted as follows: The nearly integrable stru ture ( M , [ g, Υ , A ]) ,via (3.14), uniquely determines gl (2 , R ) -valued onne tion Γ ijk whi h respe ts thestru ture [ g, Υ , A ] due to (3.5), (3.6), and has totally skew symmetri torsion T ijk due to (3.15).We summarize this part of our onsiderations in the followingProposition 3.5. Every nearly integrable GL (2 , R ) stru ture ( M , [ g, Υ , A ]) de-(cid:28)nes a unique gl (2 , R ) -valued onne tion whi h has totally skew symmetri torsion.Also the onverse is true:Proposition 3.6. Let ( M , [ g, Υ , A ]) be an irredu ible GL (2 , R ) stru ture and W Γ ijk be the Weyl onne tion oe(cid:30) ients asso iated, in an adapted oframe θ i , with theWeyl stru ture [ g, A ] . Assume that the Weyl stru ture [ g, A ] admits a split W Γ ijk = Γ ijk + T ijk , L (2 , R ) GEOMETRY OF ODE'S 17in whi h Γ ijk ∈ gl (2 , R ) ⊗ R and T ijk ∈ V R . Then [ g, Υ , A ] is nearly integrable,the split is unique and Γ ij = Γ ijk θ k is a gl (2 , R ) -valued onne tion with totally skewsymmetri torsion Θ i = T ijk θ j ∧ θ k .De(cid:28)nition 3.7. The unique gl (2 , R ) -valued onne tion with totally skew symmet-ri torsion naturally asso iated with a nearly integrable stru ture ( M , [ g, Υ , A ]) is alled the hara teristi onne tion.In the next paragraph we analize algebrai stru ture of torsion and urvature of hara teristi onne tions.3.1. Nearly integrable GL (2 , R ) stru tures. Let ( M , [ g, Υ , A ]) be a nearlyintegrable GL (2 , R ) stru ture and let Γ be its hara teristi onne tion. Thenthe GL (2 , R ) invariant information about ( M , [ g, Υ , A ]) is en oded in its totallyskew symmetri torsion Θ i = T ijk θ i ∧ θ k and its urvature Ω ij = R ijkl θ k ∧ θ l = dΓ ij + Γ ik ∧ Γ kj . The spa es V R and gl (2 , R ) ⊗ V R are redu ible under the a tion of GL (2 , R ) .Their de ompositions into the GL (2 , R ) irredu ible omponents may be used to lassify the torsion types, in the ase of V R , and the urvature types, in the ase of gl (2 , R ) ⊗ V R . In parti ular, to de ompose V R we use the Hodge star operationasso iated with one of the metri s g from the lass [ g, Υ , A ] . This identi(cid:28)es V R with V R . The GL (2 , R ) invariant de omposition of V R is then transformed tothe de omposition of V R . This is a hieved in terms of the operator Y ijkl = 4Υ ijm Υ klp g mp . This, viewed as an endomorphism of N R given by Y ( w ) ik = g mj g pl Y ijkl w mp , has the following eigenspa es: J = { S ∈ N R | Y ( S ) = 14 · S } = { S = λ · g, λ ∈ R } , V = { F ∈ N R | Y ( F ) = 7 · F } = sl (2 , R ) , J = { S ∈ N R | Y ( S ) = − · S } , V = { F ∈ N R | Y ( F ) = − · F } , J = { S ∈ N R | Y ( S ) = 4 · S } . Here the index k in J k or V k denotes the dimension of the eigenspa e.The de omposition(3.16) N R = J ⊕ J ⊕ J ⊕ V ⊕ V is GL (2 , R ) invariant. All the omponents in this de omposition are GL (2 , R ) -irredu ible. We have the followingProposition 3.8. Under the a tion of GL (2 , R ) the irredu ible omponents of V R = ∗ V R are V R = V ⊕ V . GL (2 , R ) stru tures whose hara teristi onne tion has torsion of a`pure' type T ijk ∈ V ?In se tion 5 we give an a(cid:30)rmative answer to this question. Here we only state ausefulLemma 3.9. The 3-dimensional ve tor spa e V , when expressed in terms of anadapted oframe θ i of De(cid:28)nition 3.2 is V = Span R n θ ∧ θ − θ ∧ θ , θ ∧ θ − θ ∧ θ , θ ∧ θ − θ ∧ θ o . Similarly, in an adapted oframe θ i , the Hodge dual ∗ V of V is ∗ V = Span R n − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ , − θ ∧ θ ∧ θ + 8 θ ∧ θ ∧ θ , − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ o . In parti ular, torsion T ijk of the hara teristi onne tion Γ in system (3.15) is ofpure type in V if and only if, in an adapted oframe θ i , we have g il T ljk = T [ ijk ] ,and its orresponding 3-form T = g il T ljk θ i ∧ θ j ∧ θ k ∈ ∗ V .Now we pass to the analysis of the urvature. The urvature tensor R ijkl of a gl (2 , R ) onne tion de(cid:28)nes the following obje ts: R ij = R kikj the Ri i tensor, R = R ij g ij the Ri i s alar, R i v = Υ ijk R jk the Ri i ve tor, (d A ) ij = R kkij the Maxwell 2-form.The Ri i tensor belongs to the spa e N R and de omposes a ording to (3.16).The Ri i symmetri tensor reads(3.17) R ( ij ) = Rg ij + R k v Υ ijk + R (9) ij , where Rg ij is its J part, R k v Υ ijk is its J part and R (9) ij is its J part de(cid:28)nedby (3.17). The antisymmetri Ri i tensor de omposes into R [ ij ] = R (3) ij + R (7) ij with the respe tive V and V omponents given by R (3) ij = 815 R [ ij ] + Y ( R [ ] ) ij ,R (7) ij = 715 R [ ij ] − Y ( R [ ] ) ij . Here Y ( R [ ] ) denotes the value of the operator Y on R [ ij ] . Likewise, for the Maxwellform we have (d A ) ij = d A (3) ij + d A (7) ij and d A (3) ij = 815 (d A ) ij + Y (d A ) ij , d A (7) ij = 715 (d A ) ij − Y (d A ) ij . L (2 , R ) GEOMETRY OF ODE'S 19The Ri i tensor and and the Maxwell 2-form have
25 + 10 = 35 oe(cid:30) ients out oftotal number of oe(cid:30) ients of the urvature. Sin e, .f. [14℄, gl (2 , R ) ⊗ V R = J ⊕ V ⊕ J ⊕ V ⊕ J , the remaining parameters are related to the oe(cid:30) ients of a ve tor (cid:28)eld K m ,whi h is independent of the Ri i tensor. It is de(cid:28)ned in terms of the totally skewsymmetri part of the urvature. Using the volume form η ijklm , we have K m = R ijkl η ijklm , and the so de(cid:28)ned K m yields the missing (cid:28)ve omponents of the urvature. Thuswe have the followingProposition 3.10. The irredu ible omponents of the urvature R ijkl of a har-a teristi onne tion are given by R, R i v , R (9) ij , R (3) ij , R (7) ij , d A (3) ij , d A (7) ij , K i . It is interesting to ask what is the de omposition of the urvature if the hara -teristi onne tion has torsion in three-dimensional representation V . It appearstha it has a very spe ial algebrai form. Writing the stru tural equations for a hara teristi onne tion with the torsion in V T = t ( − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ ) ++ t ( − θ ∧ θ ∧ θ + 8 θ ∧ θ ∧ θ ) ++ t ( − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ ) and utilising Bian hi identites, we get the followingProposition 3.11. Let Γ be a hara teristi onne tion of a GL (2 , R ) stru turewith torsion in V given above. Then • The Ri i tensor omponent R (9) ij = 0 , whi h means that R ( ij ) = Rg ij + R k v Υ ijk . • The skew symmetri Ri i tensor and the Maxwell 2-form are related by d A (3) = 4 R (3) , d A (7) = R (7) . • The Ri i ve tor R v is fully determined by T : R i v =(40) ( ∗ T ) jk ( ∗ T ) lm g kl Υ jmi = 76 √ (cid:18) t , − t t , t t + 227 t , − t t , t (cid:19) . Thus, the urvature is fully des ribed by T , R , d A (3) , d A (7) ij and K i .3.2. Arbitrary GL (2 , R ) stru tures. So far we have been able to introdu e aunique GL (2 , R ) -valued onne tion for a nearly integrable ( M , [ g, Υ , A ]) only.Nevertheless su h a onne tion an be always introdu ed. To see this onsidera GL (2 , R ) -invariant onformal pairing in co (3 , ⊗ R given by ( W Γ , W Γ ′ ) = g il g jm g kp W Γ ijk W Γ ′ lmp , W Γ , W Γ ′ ∈ co (3 , ⊗ R . We use the orthogonal omplement of ker ¯Υ ⊂ co (3 , ⊗ R with respe t to this pairing: ker ¯Υ ⊥ = { W Γ ∈ co (3 , ⊗ R s . t (ker ¯Υ , W Γ) = 0 } . This ve tor spa e is 30-dimensional. It ontains a 5-dimensional subspa e spannedby g ij A m , whi h is related to the R fa tor in the split gl (2 , R ) = R ⊕ sl (2 , R ) ⊂ co (3 ,
2) = R ⊕ so (3 , . Thus it is reasonable to onsider the interse tion, say V ,of this 30-dimensional spa e with so (3 , ⊗ R . This 25-dimensional spa e V = ker ¯Υ ⊥ ∩ ( so (3 , ⊗ R ) has, in turn, zero interse tion with ( gl (2 , R ) ⊗ R ) ⊕ V R and provides the GL (2 , R ) invariant de omposition of co (3 , ⊗ R : co (3 , ⊗ R = ( gl (2 , R ) ⊗ R ) ⊕ V R ⊕ V . Therefore, if we hoose a oframe adapted to a representative ( g, Υ , A ) we anuniquely de ompose the Weyl onne tion oe(cid:30) ients W Γ ijk ∈ co (3 , ⊗ R of ourarbitrary GL (2 , R ) stru ture a ording to W Γ ijk = Γ ijk + B ijk . Now Γ ijk ∈ gl (2 , R ) ⊗ R , and they are interpreted as new onne tion oe(cid:30) ients;the tensor B ijk belongs to V R ⊕ V and its antisymmetrization T ijk = B i [ jk ] isnow interpreted as the torsion of Γ . Thus, every GL (2 , R ) stru ture ( M , [ g, Υ , A ]) uniquely de(cid:28)nes a gl (2 , R ) -valued onne tion with torsion in V R ⊕ V . The tor-sion is not totally skew anymore. Spa e V further de omposes onto the GL (2 , R ) -irredu ible omponents a ording to V = J ⊕ J ⊕ J . The GL (2 , R ) stru -tures equipped with the unique gl (2 , R ) onne tion whi h has torsion in V (cid:28)ndappli ation in the theory of integrable equations of hydrodynami type [11℄.4. GL (2 , R ) bundleFirst, we des ribe an irredu ible GL (2 , R ) stru ture [ g, Υ , A ] on M in the lan-guage of prin ipal bundles.Every irredu ible GL (2 , R ) stru ture [ g, Υ , A ] on a 5-manifold M de(cid:28)nes the9-dimensional bundle GL (2 , R ) → P → M , the GL (2 , R ) redu tion of the bundleof linear frames GL (5 , R ) → F ( M ) → M . If [ g, Υ , A ] is equipped with a gl (2 , R ) onne tion Γ then the stru tural equations on M read d ω i + Γ ij ∧ ω j = T ijk ω j ∧ ω k , dΓ ij + Γ ik ∧ Γ kj = R ijkl ω k ∧ ω l . Here ( ω i ) is an adapted oframe and Γ = (Γ ij ) is written in the representation(2.5). We lift these stru tural equations to P obtaining: d θ = 4(Γ + Γ ) ∧ θ − + ∧ θ + T ij θ i ∧ θ j , d θ = − Γ − ∧ θ + (4Γ + 2Γ ) ∧ θ − + ∧ θ + T ij θ i ∧ θ j , d θ = − − ∧ θ + 4Γ ∧ θ − + ∧ θ + T ij θ i ∧ θ j , d θ = − − ∧ θ + (4Γ − ) ∧ θ − Γ + ∧ θ + T ij θ i ∧ θ j , d θ = − − ∧ θ + 4(Γ − Γ ) ∧ θ + T ij θ i ∧ θ j , (4.1) L (2 , R ) GEOMETRY OF ODE'S 21 dΓ + = 2Γ ∧ Γ + + R + ij θ i ∧ θ j , dΓ − = − ∧ Γ − + R − ij θ i ∧ θ j , dΓ = Γ + ∧ Γ − + R ij θ i ∧ θ j , dΓ = R ij θ i ∧ θ j , with the forms θ i being the omponents of the anoni al R -valued form θ on P , .f. [13℄. In a oordinate system ( x, a ) on P , x ∈ M , a ∈ GL (2 , R ) , whi h is ompatible with the lo al trivialisation P ∼ = M × GL (2 , R ) they are given by θ i ( x, a ) = ( a − ) ij ω j ( x ) . The onne tion forms (Γ − , Γ + , Γ , Γ ) are de(cid:28)ned in terms of (2.6) via Γ − ( E − ) ij + Γ + ( E + ) ij + Γ ( E ) ij + Γ ( E ) ij = ( a − ) ik Γ kl ( x ) a lj + ( a − ) ik d a kj . Note that ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) is a oframe on P and the lass of 1-forms [ A ] lifts to a 1-form ˜ A = − .Se ond, we hange the point of view. Suppose that we are given a nine dimen-sional manifold P equipped with a oframe of nine 1-forms ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) on it. Suppose that these linearly indpendent forms, together withsome fun tions T ijk , R lijk , satisfy the system (4.1) on P . What we an say aboutsu h a 9-dimensional manifold P ?To answer this question onsider a distribution h on P whi h annihilates theforms ( θ , θ , θ , θ , θ ) : h = { X ∈ T P s . t . X − | θ i = 0 , i = 0 , , , , } . Then the (cid:28)rst (cid:28)ve equations of the system (4.1) guarantee that the forms ( θ , θ , θ ,θ , θ ) satisfy the Fröbenius ondition, d θ i ∧ θ ∧ θ ∧ θ ∧ θ ∧ θ = 0 , ∀ i = 0 , , , , and that, in turn, the distribution h is integrable. Thus manifold P is foliated by4-dimensional leaves tangent to the distribution h .Now on P we onsider two multilinear symmetri forms. The bilinear one, de(cid:28)nedby(4.2) ˜ g = θ θ − θ θ + 3( θ ) , and the three-linear one given by(4.3) ˜Υ = 3 √ θ θ θ + 2 θ θ θ − ( θ ) − θ ( θ ) − θ ( θ ) ) . Of ourse, sin e the 1-forms (Γ − , Γ + , Γ , Γ ) are not present in the de(cid:28)nitions(4.2), (4.3), then ˜ g and ˜Υ are degenerate. For example, the signature of the bilinearform ˜ g is (+ , + , + , − , − , , , , . The degenerate dire tions for these two formsare just the dire tions tangent to the leaves of the foliation generated by h . Letus denote by ( X , X , X , X , X , X , X , X , X ) the frame of ve tor (cid:28)elds on P dual to the 1-forms ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) . In parti ular ( X , X , X , X ) onstitutes a basis for h , and we have X µ − | θ i = 0 for ea h µ = 5 , , , and i = 0 , , , , . Using this, and the exterior derivatives of θ i given in the (cid:28)rst (cid:28)veequations (4.1), we easily (cid:28)nd the Lie derivatives of ˜ g and ˜Υ along the dire tionstangent to the leaves of h . These are: L X µ ˜ g = 8( X µ − | Γ )˜ g, L X µ ˜Υ = 12( X µ − | Γ ) ˜Υ , ∀ µ = 5 , , , . ˜ A = − , and we use the last of equations (4.1), we also (cid:28)nd that L X µ ˜ A = − X µ − | Γ ) , ∀ µ = 5 , , , . This is enough to dedu e that the obje ts (˜ g, ˜Υ , ˜ A ) des end to the 5-dimensionalleaf spa e M = P/ h . There they de(cid:28)ne a onformal lass of triples ( g, Υ , A ) withthe transformation rules g → e φ g , Υ → e φ Υ , A → A − φ . Due to the fa t that,when passing to the quotient M = P/ h , we redu ed the degenerate dire tionsof ˜ g and ˜Υ to points of M , the resulting des ended triples ( g, Υ , A ) have non-degenerate g of signature (3 , and non-degenerate Υ . It is lear that togetherwith A they de(cid:28)ne an irredu ible GL (2 , R ) stru ture on M : a se tion s : M → P is an adapted oframe on M , the triple ( s ∗ ˜ g, s ∗ ˜Υ , s ∗ ˜ A ) is a representative of thestru ture, the forms s ∗ Γ − , s ∗ Γ + , s ∗ Γ , s ∗ Γ are gl (2 , R ) onne tion 1-forms on M and s ∗ T , s ∗ R are torsion and urvature of this onne tion, respe tively. We havethe followingProposition 4.1. Every 9-dimensional manifold P equipped with nine 1-forms ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) whi h • are linearly independent at every point of P , • satisfy system (4.1) with some fun tions T ijk , R ijkl on P ,is foliated by 4-dimensional leaves over a 5-dimensional spa e M , whi h is thebase for the (cid:28)bration P → M . The manifold M is equipped with a natural ir-redu ible GL (2 , R ) stru ture [ g, Υ , A ] and a gl (2 , R ) onne tion ompatible with it.The torsion and the urvature of this onne tion is given by T ijk and R ijkl .5. 5th order ODE as nearly integrable GL (2 , R ) geometry with`small' torsion. Main theoremA large number of examples of nearly integrable GL (2 , R ) stru tures in dimen-sion (cid:28)ve is related to 5th order ODEs. This is mainly due to the following, wellknown,Proposition 5.1. An ordinary di(cid:27)erential equation y (5) = 0 has GL (2 , R ) × ρ R as its group of onta t symmetries. Here ρ : GL (2 , R ) → GL (5 , R ) is the 5-dimensional irredu ible representation of GL (2 , R ) .To explain the above statement we onsider a general 5th order ODE(5.1) y (5) = F ( x, y, y ′ , y ′′ , y (3) , y (4) ) for a real fun tion R ∋ x y ( x ) ∈ R . Let us introdu e the notation y = y ′ , y = y ′′ , y = y (3) , y = y (4) and F i = ∂F∂y i , i = 1 , , , , F y = ∂F∂y . The fun tions ( x, y, y , y , y , y ) form a lo al oodinate system in the 4-order jet spa e J of urvesin R . De(cid:28)ne a total derivative, whi h is a ve tor (cid:28)eld in J (5.2) D = ∂ x + y ∂ y + y ∂ y + y ∂ y + y ∂ y + F ∂ y . With the help of D the derivatives are given by formulae y = D y/ D x , y = D y / D x and so on, up to y = D y / D x . L (2 , R ) GEOMETRY OF ODE'S 23A onta t transformation of variables in a 5-order ODE is a transformation thatmixes the independent variable x , the dependent variable y and the (cid:28)rst derivative y in su h a way that the meaning of the (cid:28)rst derivative is retained:De(cid:28)nition 5.2. A onta t transformation of variables is an invertible, su(cid:30) ientlysmooth transformation of the form(5.3) xyy ¯ x ¯ y ¯ y = ¯ x ( x, y, y )¯ y ( x, y, y )¯ y ( x, y, y ) satisfying the ondition ¯ y = D ¯ y D ¯ x . (preservation of (cid:28)rst derivative)The higher order derivatives are given by the iterative formula y n +1 ¯ y n +1 = D ¯ y n D ¯ x , i = 1 , , , . Let us now onsider the equation y (5) = 0 . We show how the (cid:29)at torsionfree5-dimensional irredu ible GL (2 , R ) stru ture is naturally generated on its spa e ofsolutions by means of the symmetry group. A solution to y (5) = 0 is of the form(5.4) y ( x ) = c x + 4 c x + 6 c x + 4 c x + c with (cid:28)ve integration onstants c , c , c , c , c . Then a solution of y (5) = 0 maybe ideanti(cid:28)ed with a point c = ( c , c , c , c , c ) T in R . A onta t symmetry of y (5) = 0 is a onta t transformation of variables that transforms its solutions intosolutions. Group of onta t symmetries of y (5) = 0 is generated by the followingone-parameter groups of transformations on the xy -plane: ϕ t ( x, y ) = ( x, y + t ) , ϕ t ( x, y ) = ( x, y + 4 xt ) ,ϕ t ( x, y ) = ( x, y + 6 x t ) , ϕ t ( x, y ) = ( x, y + 4 x t ) ,ϕ t ( x, y ) = ( x, y + x t ) , ϕ t ( x, y ) = ( xe t , ye t ) ,ϕ t ( x, y ) = ( x, ye t ) , ϕ t ( x, y ) = ( x + t, y ) ,ϕ t ( x, y ) = (cid:18) x xt , y (1 + xt ) (cid:19) and the transformation rules for y are given by ϕ A ( y ) = D ( ϕ A ( y )) / D ( ϕ A ( x )) , A = 0 , . . . , .Transforming (5.4) a ording to the above formulae we (cid:28)nd that ϕ t , . . . , ϕ t aretranslations in the spa e of solutions: ϕ t ( c ) = ( c − t, c , c , c , c ) T , . . . , ϕ t ( c ) = ( c , c , c , c , c − t ) T , while transformations ϕ t , . . . , ϕ t generate GL (2 , R ) and a t through the 5-dimensionalirredu ible representation (2.6): ϕ t ( c ) = exp( tE ) c, ϕ t ( c ) = exp( tE ) c,ϕ t ( c ) = exp( tE + ) c, ϕ t ( c ) = exp( tE − ) c. Of ourse, GL (2 , R ) stabilizes the origin (0 , , , , in R , thus the spa e of solu-tions is the homogeneous spa e GL (2 , R ) → GL (2 , R ) × ρ R → R . The total spa eof this bundle is equipped with the Maurer (cid:21) Cartan form ω MC of GL (2 , R ) × ρ R .4 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIChoosing an approriate basis in gl (2 , R ) and writing expli itly the stru tural equa-tions d ω MC + ω MC ∧ ω MC = 0 we get d θ = 4(Γ + Γ ) ∧ θ − + ∧ θ , d θ = − Γ − ∧ θ + (4Γ + 2Γ ) ∧ θ − + ∧ θ , d θ = − − ∧ θ + 4Γ ∧ θ − + ∧ θ , d θ = − − ∧ θ + (4Γ − ) ∧ θ − Γ + ∧ θ , d θ = − − ∧ θ + 4(Γ − Γ ) ∧ θ , dΓ + = 2Γ ∧ Γ + , dΓ − = − ∧ Γ − , dΓ = Γ + ∧ Γ − , dΓ = 0 , whi h is the system (4.1) with all the torsion and urvature oe(cid:30) ients equal to zero.A ording to proposition 4.1 it yields a (cid:29)at and torsionfree irredu ible GL (2 , R ) stru ture on the spa e of solutions of y (5) = 0 . Again, as in the ase of the algebrai geometri realization of se tion 2, we learned about that from E. V. Ferapontow[10℄.We now pass to a more general situation, namely to the equation (5.1) with ageneral F . The following questions are in order:What shall one assume about F to be able to onstru t an irredu ible GL (2 , R ) stru ture on the solution spa e of the orresponding ODE? Is the ase F = 0 veryspe ial, or there are other ODEs, onta t nonequivalent to the F = 0 ase, whi hde(cid:28)ne a GL (2 , R ) geometry on the solution spa e? If the answer is a(cid:30)rmative,how do we (cid:28)nd su h F s and what an we say about the orresponding GL (2 , R ) stru tures?Answer to these questions is given by the followingTheorem 5.3 (Main theorem). Every onta t equivalen e lass of 5th order ODEssatisfying the Wüns hmann onditions D F − D F + 50 F − F D F + 30 F F + 8 F = 0 , D F − D F + 350 D F + 1250 F − F D F + 200 F − F D F + 200 F F − F D F + 130 F F + 14 F = 0 , (5.5) D F − D F + 1750 D F D F − F D F − F D F + 1250 F F − F D F + 700( D F ) F +1250 F F − F F D F + 350 F F − F D F +550 F F − F D F + 210 F F + 28 F + 18750 F y = 0 de(cid:28)nes a nearly integrable irredu ible GL (2 , R ) geometry ( M , [ g, Υ , A ]) on thespa e M of its solutions. This geometry has the hara teristi onne tion withtorsion T of the `pure' type in the 3-dimensional irredu ible representation V .The (cid:28)rst stru tural equation for this onne tion are the following: d θ = 4(Γ + Γ ) ∧ θ − + ∧ θ + L (2 , R ) GEOMETRY OF ODE'S 25 − t θ ∧ θ − t θ ∧ θ − t θ ∧ θ + 2 t θ ∧ θ , d θ = − Γ − ∧ θ + (4Γ + 2Γ ) ∧ θ − + ∧ θ + − t θ ∧ θ − t θ ∧ θ − t θ ∧ θ , d θ = − − ∧ θ + 4Γ ∧ θ − + ∧ θ + − t θ ∧ θ + t θ ∧ θ − t θ ∧ θ − t θ ∧ θ , d θ = − − ∧ θ + (4Γ − ) ∧ θ − Γ + ∧ θ ++ t θ ∧ θ − t θ ∧ θ − t θ ∧ θ , d θ = − − ∧ θ + 4(Γ − Γ ) ∧ θ + − t θ ∧ θ + t θ ∧ θ − t θ ∧ θ − t θ ∧ θ with the torsion oe(cid:30) ients t = 6( α ) α F ,t = 9 α α ) [ α (10 D F + 3 F F ) + 5 α F ] ,t = [1000( α ) ] − × (cid:16) α ) F + 90 α α (10 D F + 3 F F ) + − α ) [20(5 D F + 20 F − F + 3 F D F − F F ) ++ F ( − D F + 340 F + 51 F )] (cid:17) , where ( y, y , y , y , y , x, α , α , α ) is a lo al oordinate system on GL (2 , R ) → P → M . The se ond stru tural equations are the following: dΓ + = 2Γ ∧ Γ + + (cid:0) b − t + c (cid:1) θ , θ + (cid:0) − t t − c + b (cid:1) θ ∧ θ ++ (cid:0) − t − t t + c − R + b − a (cid:1) θ ∧ θ ++ (cid:0) t t − a − c + b (cid:1) θ ∧ θ ++ (cid:0) − t − t t + R + 2 b + a (cid:1) θ ∧ θ ++ (cid:0) − t t + a + b (cid:1) θ ∧ θ + (cid:0) t + c + b (cid:1) θ ∧ θ ++ (cid:0) − t − c + b (cid:1) θ ∧ θ + b θ ∧ θ , dΓ − = − ∧ Γ − + b θ ∧ θ + (cid:0) b − t − c (cid:1) θ ∧ θ ++ (cid:0) − t t + c + b − a (cid:1) θ ∧ θ ++ (cid:0) t + b + c (cid:1) θ ∧ θ + (cid:0) t t + b + a (cid:1) θ ∧ θ ++ (cid:0) b − a + t t + t − c + R (cid:1) θ ∧ θ ++ (cid:0) t + t t − R + 2 b + a (cid:1) θ ∧ θ ++ (cid:0) t t + c + b (cid:1) θ ∧ θ + (cid:0) t − c + b (cid:1) θ ∧ θ + (2 t + t − t t − t ) θ ∧ θ + (4 t − t − t ) θ ∧ θ , dΓ = Γ + ∧ Γ − − b θ ∧ θ + (cid:0) − b − t + c (cid:1) θ ∧ θ ++ (cid:0) − t t − b + a (cid:1) θ ∧ θ + − (cid:0) t t + t + c + R (cid:1) θ ∧ θ + + (cid:0) t t − c − b − a (cid:1) θ ∧ θ ++ (cid:0) − t t − t + c + R (cid:1) θ ∧ θ ++ (cid:0) − t t − a + b (cid:1) θ ∧ θ ++ (cid:0) t t + a − c + b (cid:1) θ ∧ θ ++ (cid:0) − t + c + b (cid:1) θ ∧ θ + b θ ∧ θ , dΓ = − b θ ∧ θ − b θ ∧ θ − ( b + a ) θ ∧ θ − ( b + a ) θ ∧ θ ++ (cid:0) a − b (cid:1) θ ∧ θ + (cid:0) a − b (cid:1) θ ∧ θ − ( a + b ) θ ∧ θ ++ (cid:0) a − b (cid:1) θ ∧ θ − b θ ∧ θ − b θ ∧ θ , where a , a , a , b , b , b , b , b , b , b , c , c , c , c , c and R are fun tions.All of these fun tions but R are determined by the di(cid:27)erentials of torsions: d t = 2 t Γ − − t Γ − t Γ + b θ + (cid:0) b − t + 20 c (cid:1) θ ++ (cid:0) − t t − c + 3 b − a (cid:1) θ ++ (cid:0) − t t − t + 60 c + 6 b − a (cid:1) θ ++ (cid:0) − t t − a − c + b (cid:1) θ , d t = 3 t Γ − + t Γ + − t Γ + (cid:0) b + t − c (cid:1) θ ++ (cid:0) t t + 20 c + 2 b + a (cid:1) θ + 9 ( a + b ) θ ++ (cid:0) − t t + a − c + 2 b (cid:1) θ + (cid:0) − t + 10 c + b (cid:1) θ , d t = 2 t Γ + t Γ + − t Γ + (cid:0) t t + c + b − a (cid:1) θ ++ (cid:0) t t + t − c + 2 b − a (cid:1) θ ++ (cid:0) t t − a + 20 c + b (cid:1) θ + (cid:0) t − c + b (cid:1) θ + b θ . The fun tion R is the Ri i s alar for the onne tion.Before presenting the proof let us noti e several fa ts and orollaries.The theorem guarantees that every equivalen e lass of ODEs satisfying ondi-tions (5.5) has its orresponding nearly integrable GL (2 , R ) geometry ( M , [ g, Υ , A ]) with torsion in V . It may happen, however, that there are onta t non-equivalent lasses of ODEs de(cid:28)ning the same GL (2 , R ) geometries. (See also remark 5.9).The Wüns hmann onditions, although very ompli ated, possess nontrivial so-lutions. For example the equation y (5) = c (cid:16) y (3)3 (5 − cy ′′ )9(1 + cy ′′ ) + 10 y ′′ y (3) y (4) cy ′′ (cid:17) , where c = ± satis(cid:28)es the Wüns hmann onditions and is not onta t equivalentto F = 0 . Other examples are onsidered in se tion 6.The onne tion of theorem 5.3 is a hara teristi onne tion with torsion in V .If the Wüns hmann ODE is general enough, the torsion may be quite arbitrarywithin V . From proposition 3.11 we know that the independent omponents ofthe urvature of a hara teristi onne tion with T ∈ V are R , d A (3) , d A (7) and L (2 , R ) GEOMETRY OF ODE'S 27 K i . In the notation of theorem 5.3 they read: d A (3) = a a − a − a a a − a − a a a − a − a , d A (7) = b b b b − b b b b − b − b b b − b − b − b b − b − b − b − b ,K = √ (cid:0) c c c c c (cid:1) T , and, as we said above, the Ri i s alar is given by the fun tion R . The Ri i ve tor R i v = Υ ijk g jk is as follows R i v = 76 √ (cid:18) t , − t t , t t + 227 t , − t t , t (cid:19) . The Ri i tensor satis(cid:28)es the following equations R ( ij ) = Rg ij + R k v Υ ijk , d A (3) = 4 R (3) , d A (7) = R (7) . Using theorem 5.3 we an also express the Ri i tensor ( Ric ) ij = g ik R kj in termsof the endomorphisms E − , E , E + , E of (2.5):Corollary 5.4. The Ri i tensor of a hara teristi onne tion with torsion in V has the following form in any adapted oframe Ric = (cid:18) t + 136 t t − R (cid:19) E + 18 b E − + 1108 t E − ++ (cid:18) − t t + 18 a − b (cid:19) E − + 516 b E + (cid:18) t + 172 t t (cid:19) E ++ (cid:18) − b + 18 a (cid:19) E − b E + 112 t E + (cid:18) − a + 12 b − t t (cid:19) E + + − b E E + E + 18 b E + E E + + 154 t t E E − + 532 b E E − E ++ 18 b E − E E − − t t E E + . Of ourse, sin e the geometry is onstru ted from an ODE determined by the hoi e of F = F ( x, y, y , y , y , y ) , the oe(cid:30) ients a , . . . , a , b , . . . , b , R are ex-pressible in terms of F and its derivatives. Given the onne tion of theorem 5.3 we al ulated the expli it formulae for these oe(cid:30) ients and obtained the following8 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKICorollary 5.5. A GL (2 , R ) geometry generated by a 5th order ODE satisfyingWüns hmann onditions (5.5) has the following properties.The torsion T vanishes i(cid:27) F = 0 . The 2-form d A (3) vanishes i(cid:27) ( D F ) − ( D F ) − ( D F ) F − D F F + F F + F F ++ F F − F F + F F + F − F = 0 . The 2-form d A (7) vanishes i(cid:27) F = 0 . The Ri i ve tor R v is aligned with the ve tor K , i.e. K = uR v , u ∈ R , i(cid:27) ( D F ) − F − F F − F + 7 uF = 0 . We skip writing the formula for the Ri i s alar sin e it is very ompli ated.We now pass to the proof of theorem 5.3. On doing this we will apply a variantof the Cartan method of equivalen e. This will be a rather long and ompli atedpro edure. Thus, for the larity of the presentation, we will divide the proof intothree main steps, ea h of whi h will o uppy its own respe tive se tion 5.1, 5.2 and5.3. First, in se tion 5.1 we will prove lemma 5.6, whi h assures that a lass of onta t equivalent 5th order ODEs is a G -stru ture on a 4-order jet spa e J . Thus,we will have a bundle G → J × G → J , a redu tion of the frame bundle F ( J ) . Inthe se ond step, in se tion 5.2, we will use the Cartan method of equivalen e inorder to onstru t a submanifold P ⊂ J × G together with a oframe on P whi hful(cid:28)lls the requirements of proposition 4.1. This oframe, via proposition 4.1, willde(cid:28)ne an irredu ible GL (2 , R ) stru ture for us and simultanously will provide uswith a gl (2 , R ) onne tion on the spa e of solutions of the ODE. The obstru tionsfor an ODE to possess this stru ture, Wüns hmann's expressions for F , will appearautomati ally in the ourse of the onstru tion. This part of onsiderations issummarized in theorem 5.7. The GL (2 , R ) stru ture obtained in this way will turnout to be nearly integrable, but the onne tion onstru ted will di(cid:27)er from the hara teristi one. Therefore, in se tion 5.3, we will onstru t the hara teristi onne tion asso iated with the GL (2 , R ) stru ture obtained. This will have torsionin V . This onstru tion is des ribed by lemma 5.8.5.1. 5th order ODE modulo onta t transformations. Let us onsider a gen-eral 5th order ODE (5.1). We de(cid:28)ne the following oframe ω = d y − y d x,ω = d y − y d x,ω = d y − y d x, (5.6) ω = d y − y d x,ω = d y − F ( x, y, y , y , y , y )d x,ω + = d x on J . We see that every solution of (5.1) is a urve c ( x ) = ( x, y ( x ) , y ( x ) , y ( x ) , y ( x ) , y ( x )) ⊂ J and the ve tor (cid:28)eld D on J has urves c ( x ) as the integral urves. The1-forms ( ω , ω , ω , ω , ω ) annihilate D whereas D y ω + = 1 . The 5-dimensional L (2 , R ) GEOMETRY OF ODE'S 29spa e M of integral urves of D is learly the spa e of solutions of (5.1) and wehave a (cid:28)bration R → J → M .Suppose now, that equation (5.1) undergoes a onta t transformation (5.3),whi h brings it to ¯ y = ¯ F (¯ x, ¯ y, ¯ y , ¯ y , ¯ y , ¯ y ) . Then the oframe transforms a - ording to(5.7) ω ω ω ω ω ω + ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω + = α α α α α α α α α α α α α α α α α α ω ω ω ω ω ω + . Here α ij , i, j = 0 , , , , , , are real fun tions on J de(cid:28)ned by the formulae (5.3).They satisfy the nondegenera y ondition α α α α α α = 0 . The transformed oframe en odes all the onta t invariant information about theODE. In parti ular, it preserves the simple ideal ( ω , . . . , ω ) , from whi h we anre over solutions of the transformed equation. Hen e we haveLemma 5.6. A 5th order ODE y = F ( x, y, y , y , y , y ) onsidered modulo on-ta t transformations of variables is a G -stru ture on the 4-jet spa e J , su h thatthe oframe ( ω , ω , ω , ω , ω , ω + ) of (5.6) belongs to it and the group G is givenby the matrix in (5.7)5.2. GL (2 , R ) bundle over spa e of solutions. Using the Cartan method we ex-pli itly onstru t a submanifold P ⊂ J × G and a oframe ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) on P satisfying proposition 4.1. This part of the proof is divided into eight steps.1). We observe that there is a natural hoi e for the forms ( θ , θ , θ , θ , θ ) of the oframe. Sin e we are going to build a GL (2 , R ) stru ture on the spa e of solutions P must be a bundle over M and the forms ( θ , θ , θ , θ , θ ) must annihilate ve torstangent to leaves of the proje tion P → M . But on J × G there are six distinguished1-forms given by(5.8) θ θ θ θ θ θ + = α ω α ω + α ω α ω + α ω + α ω α ω + α ω + α ω + α ω α ω + α ω + α ω + α ω + α ω α ω + α ω + α ω + . These forms are the omponents of the anoni al R valued 1-form on J × G . Fiveamong these forms, θ , θ , θ , θ , θ also annihilate ve tors tangent to the proje -tion J × G → M . We hoose them to be the members of the sought oframe ( θ , θ , θ , θ , θ , Γ − , Γ + , Γ , Γ ) . Now we must onstru t a 9-dimensional subman-ifold P on whi h θ i satisfy equations (4.1) with some linearly independent forms Γ − , Γ + , Γ , Γ .0 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKI2). We al ulate d θ and get d θ = (cid:18) d α α − α α α θ + (cid:19) ∧ θ + α α α θ + ∧ θ − α α α θ ∧ θ For this equation to mat h (4.1) we de(cid:28)ne Γ + = θ + (5.9) + Γ ) = d α α − α α α θ + mod θ i , (5.10)with yet unspe i(cid:28)ed θ i terms in (5.10), and set(5.11) α = − α α to get − oe(cid:30) ient in the Γ + ∧ θ term. Thereby d θ = 4(Γ + Γ ) ∧ θ − + ∧ θ mod θ i ∧ θ j on the 23-dimensional subbundle of J × G → M given by (5.11). We see that theform θ + plays naturally the role of the onne tion 1-form Γ + .3). We al ulate d θ on the 23-dimensional bundle. In order to get d θ = − Γ − ∧ θ + (4Γ + 2Γ ) ∧ θ − + ∧ θ mod θ i ∧ θ j we set + 2Γ = d α α + α α − α α α α α θ + mod θ i , (5.12) Γ − = − d α α α + α d α α ) α (5.13) + ( α ) α + ( α ) α − α α α ) α ) α ( α ) θ + mod θ i , and(5.14) α = − α α obtaining a 22-dimensional subbundle of J × G → M on whi h d θ and d θ are inthe desired form.4). At this point all four onne tion 1-forms Γ − , Γ + , Γ , Γ are (cid:28)xed up to the θ i terms. They are determined by the equations (5.9), (5.10), (5.12), (5.13). Thus we an not introdu e any new 1-forms to bring d θ into the desired form. Now to get d θ in the form as in theorem 5.3, we may only use the yet unspe i(cid:28)ed oe(cid:30) ients α s. That is why d θ imposes more onditions on α s. It follows that for d θ , d θ and d θ to be of the form (4.1) the subbundle P must satisfy α = − α α ,α = − α ) + ( α ) ( − D F + 20 F + 7 F )300 α α ,α = − α + α F α ,α = − α α , L (2 , R ) GEOMETRY OF ODE'S 31 α = [1800( α α ) ] − × [1125( α ) + 45 α ( α ) (20 D F − F − F ) +2( α ) (100 D F − F − F D F − F F − F )] ,α = 225( α ) − α α F + ( α ) (80 D F − F − F )1200 α ( α ) , (5.15) α = 5 α − α F α ) ,α = α α ) ,α = [18000( α ) ( α ) ] − × [ − α ) + 225( α ) α F − α ( α ) (80 D F − F − F ) +( α ) ( − D F + 1400 F + 240 F D F + 180 F F + 11 F )] ,α = − α ) + 30 α α F + ( α ) ( − D F + 80 F + 17 F )600 α ( α ) ,α = − α + 3 α F α ) ,α = − α α ) . The ne essity of these onditions an be he ked by a dire t, quite lengthy al u-lations. We performed these al ulations using the symboli omputation programsMaple and Mathemati a.We stress that onditions (5.15) are only ne essary for d θ to satisfy (4.1). It isbe ause ertain unwanted terms annot be removed by any hoi e of subbundle P .Vanishing of these unwanted terms is a property of the ODE itself, and this is thereason for the Wüns hmann onditions to appear.More spe i(cid:28) ally, to a hieve d θ = − − ∧ θ + 4Γ ∧ θ − + ∧ θ mod θ i ∧ θ j on the bundle de(cid:28)ned by (5.11), (5.14) and (5.15) an ODE must satisfy(5.16) D F − D F + 50 F − F D F + 30 F F + 8 F = 0 . It follows from the onstru tion that this ondition, the (cid:28)rst of (5.5), is invariantunder the onta t transformation of variables.From now on we restri t our onsiderations only to onta t equivalen e lass ofODEs satisfying (5.16). If (5.15) and (5.16) are satis(cid:28)ed then the three di(cid:27)erentials d θ , d θ and d θ are pre isely in the form (4.1).5). The requirement that also d θ is in the form (4.1) is equivalent to the followingequation for α : α = [120000( α α ) ] − × [ − α ) − α α ) (20 D F − F − F ) − α ( α ) (50 D F − F + 30 F D F − F F − F ) + (5.17)2 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKI ( α ) (cid:16) − D F + 10 D F − D F ) + 10 F D F − F + F D F ) +120 F (7 D F − F ) − F (cid:17) ] . (d θ + 4Γ − ∧ θ − − Γ ) ∧ θ ) ∧ θ ∧ θ = 0 mod θ i . However, d θ ∧ θ ∧ θ ∧ θ ∧ θ = 0 if and only if se ond ondition of (5.5) is satis(cid:28)ed: D F − D F + 350 D F + 1250 F − F D F + 200 F − F D F + 200 F F − F D F + 130 F F + 14 F = 0 . (5.18)Again it follows from the onstru tion that ondition (5.18), onsidered simultane-ously with (5.16), is invariant under onta t transformations of the variables. Fromnow on, we assume that all our 5th order ODEs (5.1) satisfy both onditions (5.16),(5.18). It follows that it is still not su(cid:30) ient to for e d θ to satisfy the system (4.1),sin e without further assumptions on F , we do not have d θ ∧ θ ∧ θ ∧ θ ∧ θ = 0 .To a hieve this it is ne essary and su(cid:30) ient to impose the last restri tion on F : D F − D F + 1750 D F D F − F D F − F D F + 1250 F F − F D F + 700( D F ) F + (5.19) F F − F F D F + 350 F F − F D F +550 F F − F D F + 210 F F + 28 F + 18750 F y = 0 . F satis(cid:28)es onditions (5.5) and (cid:28)xing oe(cid:30) ients α ij a ordingto (5.15), (5.17) we are remained with a 11-dimensional subbundle of J × G → M parametrized by ( x, y, y , y , y , y , α , α , α , α , α ) . It follows that the forms Γ , Γ , Γ − , Γ + on this bundle are Γ + = θ + , Γ = d α α − α + α F α α θ + mod θ i , Γ = d α α − d α α + F α θ + mod θ i , (5.20) Γ − = d α α α − α d α α ) α − α ) + 10 α α F + ( α ) (20 D F − F − F )400( α α ) θ + mod θ i . θ i terms in (5.20) weneed to onsider the dΓ A part of equations (4.1). For ing dΓ A not to have Γ A ∧ θ i terms we uniquely spe ify the θ i terms in (5.20). This requirement, in parti ular,(cid:28)xes the oe(cid:30) ients α and α to be: α = α (10 D F + 5 F + 6 F F )50 , L (2 , R ) GEOMETRY OF ODE'S 33 α = α
250 [50( D F + 7 F − F ) + (5.21) F (6 D F − F ) + 2 F ( − D F + 145 F + 21 F )] . Now all the forms ( θ , θ , θ , θ , θ , Γ + , Γ − , Γ , Γ ) are well de(cid:28)ned and independenton a 9-dimensional manifold P parametrized by ( y, y , y , y , y , x, α , α , α ) .We al ulate stru tural equations (4.1) for these forms and have the followingTheorem 5.7. A 5th order ODE y (5) = F ( x, y, y ′ , y ′′ , y (3) , y (4) ) onsidered mod-ulo onta t transformation of variables has an irredu ible GL (2 , R ) stru ture onthe spa e of its solution M together with a gl (2 , R ) onne tion Γ if and only ifits de(cid:28)ning fun tion F = F ( x, y, y , y , y , y ) satis(cid:28)es the onta t invariant Wün-s hmann onditions (5.5). The bundle GL (2 , R ) → P → M is given by the equa-tions (5.15), (5.17) and (5.21). The (cid:28)rst stru tural equations for the onne tion Γ = (Γ + , Γ − , Γ , Γ ) on P read d θ = 4(Γ + Γ ) ∧ θ − + ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ , d θ = − Γ − ∧ θ + (4Γ + 2Γ ) ∧ θ − + ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ , d θ = − − ∧ θ + 4Γ ∧ θ − + ∧ θ + (5.22) t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ , d θ = − − ∧ θ + (4Γ − ) ∧ θ − Γ + ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ + t θ ∧ θ , d θ = − − ∧ θ + 4(Γ − Γ ) ∧ θ + t θ ∧ θ + t θ ∧ θ + 3 t θ ∧ θ , with the torsion oe(cid:30) ients t = 6( α ) α F ,t = 9 α α ) [ α (10 D F + 3 F F ) + 5 α F ] ,t = [1000( α ) ] − × (cid:16) α ) F + 90 α α (10 D F + 3 F F ) + − α ) [20(5 D F + 20 F − F + 3 F D F − F F ) ++ F ( − D F + 340 F + 51 F )] (cid:17) . Also the se ond stru tural equations are easily al ulable but we skip them dueto their omplexity.It is remarkable that the above gl (2 , R ) onne tion has torsion with not morethan three fun tionally independent oe(cid:30) ients t , t , t . This suggests that the4 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKI GL (2 , R ) geometry on the 5-dimensional solution spa e M of the ODE is nearlyintegrable with torsion in the irredu ible part V only. That it is really the asewill be shown below.5.3. Chara teristi onne tion with torsion in V . As we know from se -tion 3, given an irredu ible GL (2 , R ) -stru ture ( M , [ g, Υ , A ]) , we an ask if su ha stru ture is nearly integrable. A ording to propositions 3.5 and 3.6, the ne es-sary and su(cid:30) ient ondition for nearly integrability is that the stru ture admits a gl (2 , R ) -valued onne tion with totally skew symmetri torsion.In our ase of ODEs satisfying Wüns hmann onditions we have a gl (2 , R ) -valued onne tion of theorem 5.7, whose torsion is expressible in terms of three independentfun tions. This torsion, however, has quite ompli ated algebrai stru ture, inparti ular it is not totally skew symmetri .It appears that an irredu ible GL (2 , R ) stru ture ( M , [ g, Υ , A ]) asso iated withany 5th order ODE satisfying onditions (5.5) admits another gl (2 , R ) -valued on-ne tion that has totally skew symmetri torsion. Thus all stru tures ( M , [ g, Υ , A ]) originating from Wüns hmann 5th order ODEs are nearly integrable; the new on-ne tion is their hara teristi onne tion. Even more interesting is the fa t that itstorsion is still more spe ial: it is always in V .One way of seeing this is to al ulate the Weyl onne tion W Γ for the orresponding ( M , [ g, Υ , A ]) and to de ompose it a ording to (3.14). Here we prefer anothermethod (cid:21) the analysis in terms of the Cartan bundle P of theorem 5.7.Lemma 5.8. Consider a onta t equivalen e lass of 5th order ODEs satisfying onditions (5.5). Let θ , θ , θ , θ , θ , Γ + , Γ − , Γ , Γ and t , t , t be the obje ts oftheorem 5.7. Then there is a gl (2 , R ) onne tion ˜Γ = (˜Γ + , ˜Γ − , ˜Γ , ˜Γ ) whose torsion ˜ T ijk is totally skew symmetri and has its asso iated 3-form in ˜ T ∈ ∗ V . Expli itly: ˜ T = t ( − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ ) + t ( − θ ∧ θ ∧ θ + 8 θ ∧ θ ∧ θ ) + t ( − θ ∧ θ ∧ θ + 2 θ ∧ θ ∧ θ ) . Proof. Any gl (2 , R ) onne tion ˜Γ = (˜Γ + , ˜Γ − , ˜Γ , ˜Γ ) ompatible with the GL (2 , R ) stru ture of theorem 5.7 is given by ˜Γ A = Γ A + X i γ Ai θ i , A ∈ { + , , −} , i = 0 , . . . , , (5.23) ˜Γ = Γ with arbitrary fun tions γ Ai . We al ulate stru tural equations d θ + ˜Γ ∧ θ = ˜ T for ˜Γ utilising equations (5.22), and ask if there exists a hoi e of γ Ai su h that thenew torsion ˜ T ijk satis(cid:28)es g il ˜ T ljk = ˜ T [ ijk ] and ˜ T = g il ˜ T ljk θ i ∧ θ j ∧ θ k ∈ ∗ V . Usinglemma 3.9 we easily (cid:28)nd that the unique solution is given by ˜Γ + = Γ + − t θ − t θ − t θ , ˜Γ − = Γ − + t θ + t θ + t θ , ˜Γ = Γ − t θ − t θ − t θ , ˜Γ = Γ , (cid:3) L (2 , R ) GEOMETRY OF ODE'S 35Lemma 5.8 together with the results of se tion 4 prove theorem 5.3.Remark 5.9. Note that a passage from Γ + to ˜Γ + = Γ + − t θ − t θ − t θ belongs to a larger lass of transformations than the onta t transformations (5.7),(5.8); it involves a forbidden θ term. Thus it may happen that there are nonequiv-alent lasses of ODEs whi h de(cid:28)ne the same ( M , [ g, Υ , A ]) . To distinguish betweennonequivalent ODEs one has to use the onne tion of theorem 5.7.6. Examples of nearly integrable GL (2 , R ) stru tures from 5thorder ODEsIn this se tion we provide examples of Wüns hmann ODEs and nearly integrable GL (2 , R ) stru tures related to them. Sin e su h stru tures have the torsions of their hara teristi onne tions in V , then via proposition 3.11, they are hara terizedby the torsion T , the Ri i s alar R , the omponents of Maxwell 2-forms d A (3) , d A (7) , and the ve tor K ; all these obje ts being asso iated to the hara teristi onne tion Γ . There is also the unique Weyl onne tion W Γ asso iated with thesestru tures.6.1. Torsionfree stru tures. We see from orollary 5.5 that T ≡ ⇐⇒ F ≡ . Then W Γ = Γ and all the urvature omponents but the Ri i s alar ne essarilyvanish. The following proposition an be he ked by dire t al ulation.Proposition 6.1. The three nonequivalent di(cid:27)erential equations y (5) = c (cid:16) y (3)3 (5 − cy ′′ )9(1 + cy ′′ ) + 10 y ′′ y (3) y (4) cy ′′ (cid:17) , with c = +1 , , − , represent the only three onta t nonequivalent lasses of 5th or-der ODEs having the orresponding nearly integrable GL (2 , R ) stru tures ( M , [ g, Υ ,A ]) with the hara teristi onne tion with vanishing torsion. In all three ases theholonomy of the Weyl onne tion W Γ of stru tures ( M , [ g, Υ , A ]) is redu ed to the GL (2 , R ) . For all the three ases the Maxwell 2-form d A ≡ . The orrespond-ing Weyl stru ture is (cid:29)at for c = 0 . If c = ± , then in the onformal lass [ g ] there is an Einstein metri of positive ( c = +1 ) or negative ( c = − ) Ri is alar. In ase c = 1 the manifold M an be identi(cid:28)ed with the homogeneousspa e SU (1 , / SL (2 , R ) with an Einstein g des ending from the Killing form on SU (1 , . Similarly in c = − ase the manifold M an be identi(cid:28)ed with the ho-mogeneous spa e SL (3 , R ) / SL (2 , R ) with an Einstein g des ending from the Killingform on SL (3 , R ) . In both ases with c = 0 the metri g is not onformally (cid:29)at.6.2. Stru tures with vanishing Maxwell form. From now on we assume that F = 0 and onsider stru tures with vanishing Maxwell 2-form d A = 0 . For su h stru turesboth torsion and urvature have at most 9 independent oe(cid:30) ients ontained in T , K and the s alar R . The simplest geometries in this lass are those satisfying theadditional equality K i = uR i v , u ∈ R . d A = 0 and K i = uR i v into stru tural equations of theorem 5.3 andusing Bian hi identities we (cid:28)nd that either u = − , R = 3554 ( t − t t ) or u = 2105 , R = 1027 ( t − t t ) . Thus in these ases R is funtionally dependent on t , t , t and the only invariantsfor su h GL (2 , R ) stru tures are u and the sign of R . For ea h possible values of u and sgn R we found a generating ODE.Proposition 6.2. Consider the equations(6.1) F = 5 y y + ǫy / , ǫ = − , , , (6.2) F = 5 y y , and(6.3) F = 5(8 y − y y y + 3 y y )6(2 y y − y ) , where the sign of expression (2 y y − y ) is an invariant, and the singular lo us y y − y = 0 separates nonequivalent equations with ± signs. The equationsgenerate all the six GL (2 , R ) stru utres satisfying d A = 0 and K i = uR i v , u ∈ R .For (6.1) u = − and sgn R = ǫ, for (6.2) u = and R = 0 , for (6.3) u = and sgn R = sgn(3 y − y y ) . Morover, the above ODEs an be also des ribed in a geometri way by means ofthe symmetry group.Proposition 6.3. The equations (6.1), (6.2) and (6.3) are the only 5th orderWüns hmann ODEs satisfying F = 0 , F = 0 and possessing the maximalgroup of transitive onta t symmetries of dimension grater than (cid:28)ve. Equations F = y y and F = y y have 7-dimensional groups of symmetries, all the remaininghave 6-dimensional ones.Proof. The proof is based on further appli ation of the Cartan method of equiva-len e. Let us return to the oframe of theorem 5.7, whi h en odes all the onta tinvariant information about the ODE. If there are any non onstant oe(cid:30) ients inthe stru tural equations for this oframe we an use them for further redu tionof the group GL (2 , R ) and of the bundle P . For an ODE satisfying F = 0 wenormalize t = 1 , t = 0 , whi h implies α = 65 ( α ) F , α = −
625 ( α ) (10 D F + 3 F F ) . Now the oframe of theorem 5.7 is redu ed to a 7-dimensional manifold P param-eterized by ( x, y, y , y , y , y , α ) , three 1-forms (Γ , Γ − , Γ ) be ome dependenton ea h other and we an use only one of them, our hoi e is Γ , to supplement ( θ , θ , θ , θ , θ , Γ + ) to an invariant oframe on P . Next we al ulate stru tural L (2 , R ) GEOMETRY OF ODE'S 37equations for the new oframe. The oe(cid:30) ients in these equations are built from α and 16 fun tions f , . . . , f of x , y , y , . . . , y . In parti ular d θ = 6Γ ∧ θ − + ∧ θ + f ( α ) θ ∧ θ ++ f α θ ∧ θ + f θ ∧ θ + f α θ ∧ θ , where for example f = − F F + 10 DF F + 6 F F F F , f = 5 F F . Let us assume F = 0 and onsider two possibilities: f = const and f = const . If f = const then it follows from the equations d θ i = 0 , d Γ A = 0 that f may not be a onstant. Thus f /α and f are two fun tionally independent oe(cid:30)- ients in stru tural equations for the 7-dimensional oframe ( θ , θ , θ , θ , θ , Γ + , Γ ) .A ording to the pro edure of (cid:28)nding symmetries of ODEs, whi h is des ribed in[19℄, the dimension of the group of onta t symmetries of a orresponding 5-orderODE is not larger than the dimension of the oframe minus the number of theindependent oe(cid:30) ients in the stru tural equations, that is − . It followsthat ODEs possessing onta t symmetry group greater than 5-dimensional ne es-sarily satisfy f = const . Let us assume f = const then and we get from identities d θ i = 0 , d Γ A = 0 that (i) either f = 2 or f = and (ii) for both admissiblevalues of f all the remaining nonvanishing fun tions f j are expressible by f . Forexample, the system orresponding to f = is the following d θ = 6Γ ∧ θ − + ∧ θ + f ( α ) θ ∧ θ + θ ∧ θ d θ = 4Γ ∧ θ + f α ) Γ + ∧ θ − + ∧ θ + f α ) θ ∧ θ + θ ∧ θ d θ = 2Γ ∧ θ + f α ) Γ + ∧ θ − + ∧ θ − f α ) θ ∧ θ + f α ) θ ∧ θ + f α ) θ ∧ θ + θ ∧ θ + θ ∧ θ d θ = f α ) Γ + ∧ θ − Γ + ∧ θ − f α ) θ ∧ θ + f α ) θ ∧ θ + f α ) θ ∧ θ + θ ∧ θ d θ = − ∧ θ + f α ) Γ + ∧ θ − f α ) θ ∧ θ + f α ) θ ∧ θ + θ ∧ θ dΓ + = 2Γ ∧ Γ + + f α ) θ ∧ θ + f α ) θ ∧ θ + θ ∧ θ dΓ = f α ) Γ + ∧ θ − Γ + ∧ θ + f α ) θ ∧ θ + f α ) θ ∧ θ + f α ) θ ∧ θ + θ ∧ θ . f = 0 then to this system there orresponds a unique equivalen e lass ofODEs satisfying Wüns hmann onditions and having 7-dimensional transitive on-ta t symmetry group. The lass is represented by F = 5 y y . In the ase f = 0 we have next two nonequivalent lasses of ODEs enumeratedby the sign of f and possessing 6-dimensional transitive onta t symmetry groups.Representatives of these lasses are F = 5 y y ± y / , where ± f .In the similar vein we (cid:28)nd that the only ODEs related to the ase f = 2 are(6.2) and (6.3). (cid:3) F = 0 . In this paragraph wegive examples of Wüns hmann ODEs with F = 0 . As su h they will lead to the GL (2 , R ) stru tures with the Maxwell form having a nonzero d A (7) part. First andthe simplest example of su h equations is(6.4) F = ( y ) (5 / . The GL (2 , R ) stru ture asso iated with this ODE has the following properties d A (3) = 0 , R = 0 , K = 2105 R v . It is then an example of a stru ture with nonvanishing d A belonging to the 7-dimensional irredu ible representation.Next example is the ODE given by the formula F = 19( y + y ) (cid:16) w (cid:0) y + 3 y y + 9 y y − y − y y + 12 y y y + 4 y − y ( y + y ) (cid:1) +45 y ( y + y )(2 y y + y ) − y − y y − y y − y y + 270 y y + (6.5) y y + 45 y y y − y y y + 45 y y + 60 y y − y y y − y (cid:17) , where3 w = y + 3 y y + 9 y y − y − y y + 12 y y y + 4 y − y y − y y . Torsion and urvature for the orresponding GL (2 , R ) stru ture are ompli atedand are of general algebrai form. Both these examples have 6-dimensional transi-tive group of onta t symmetries.3Note that w = 0 also gives rise to F satisfying onditions (5.5). But sin e su h F has onlyquadrati y -dependen e it is equivalent to one of proposition 6.1. A tually the one with c < . L (2 , R ) GEOMETRY OF ODE'S 396.4. A remarkable nonhomogeneous example. Finally, we present an exampleof 5th order ODEs satisfying Wüns hmann onditions (5.5), whi h are generi , ina sense that the fun tion F representing it satis(cid:28)es F = 0 , but whi h have the orresponding group of transitive symmetries of dimension D < . We onsider anansatz in whi h fun tion F depends in a spe ial way on only two oordinates y and y . Expli itly:(6.6) F = ( y ) / q (cid:16) y y (cid:17) , where q = q ( z ) is a su(cid:30) iently di(cid:27)erentiable real fun tion of its argument z = y y . It is remarkable that the above F satis(cid:28)es all Wüns hmann onditions providedthat • either q ( z ) = z / • or fun tion q ( z ) satis(cid:28)es the following se ond order ODE:(6.7) z / (3 q − z / ) q ′′ − z / q ′ + 30 z / (6 q − z / ) q ′ − q = 0 . In the (cid:28)rst ase F = y y , and we re over fun tion (6.1) with 7-dimensional groupof symmetries. Note that one of the solutions of equation (6.7) is q = z / , whi h orresponds to F = y y . Thus also the other solution with seven symmetries, thesolution (6.2), is overed by this ansatz.We observe that if fun tion q ( z ) satis(cid:28)es(6.8) q − zq ′ + 27 z / q ′ = 0 , then it also satis(cid:28)es the redu tion (6.7) of onditions (5.5). Equation (6.8) an besolved by (cid:28)rst putting it in the form q ′ = 5(2 z / ± p (4 z / − q ))9 z / and then by integrating, a ording to the sign ± . In the upper sign ase theintegration gives q in an impli it form: (2 z / + p (4 z / − q )) (2 p (4 z / − q ) − z / ) (2 p (4 z / − q ) + z / ) (5 z / − q ) = const . In the lower sign ase the impli it equation for q is: (2 z / + p (4 z / − q )) (2 p (4 z / − q ) − z / ) (5 z / − q ) (2 p (4 z / − q ) + z / ) q = const . Inserting these q s into (6.6) we have a quite nontrivial Wüns hmann ODE F = F ± .We lose this se tion with a remark that other solutions to the se ond order ODE(6.7) also provide examples of 5th order Wüns hmann ODEs.0 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKI7. Higher order ODEsAll our onsiderations about GL (2 , R ) stru tures asso iated with ODEs of 5thorder an be repeated for other orders. This is due to the following well known fa tgeneralizing proposition 5.1:Proposition 7.1. For every n ≥ , the ordinary di(cid:27)erential equation y ( n ) = 0 has GL (2 , R ) × ρ n R n as its group of onta t symmetries. Here ρ n : GL (2 , R ) → GL ( n, R ) is the n -dimensional irredu ible representation of GL (2 , R ) .The representation ρ n , at the level of Lie algebra gl (2 , R ) , is given in terms ofthe Lie algebra generators E + = n − ... n − ... ... ... ... ... ... , E − = ... ... ... ... ... ... n − ... n − ,E = − n ... − n ... − n ... ... ... n − ... n − ... n − , E = (1 − n ) , where is the n × n identity matrix. In ase of dimension n = 5 these matri es oin ide with (2.6). They also satisfy the same ommutation relations [ E , E + ] = − E + , [ E , E − ] = 2 E − , [ E + , E − ] = − E , where the ommutator in the gl (2 , R ) = Span R ( E − , E + , E , E ) ⊂ End ( R n ) is theusual ommutator of matri es.Now, we onsider a general n -th order ODE(7.1) y ( n ) = F ( x, y, y ′ , y ′′ , y (3) , ..., y ( n − ) , and as before, to simplify the notation, we introdu e oordinates x, y, y = y ′ , y = y ′′ , y = y (3) , ..., y n − = y ( n − on the ( n + 1) -dimensional jet spa e J . Introdu ingthe n onta t forms ω = d y − y d x,ω = d y − y d x, ... ω i = d y i − y i +1 d x, (7.2) ... ω n − = d y n − − y n − d x,ω n − = d y n − − F ( x, y, y , y , ..., y n − )d x L (2 , R ) GEOMETRY OF ODE'S 41and the additional 1-form w + = d x, we de(cid:28)ne a onta t transformation to be a di(cid:27)eomorphism φ : J → J whi h trans-forms the above n + 1 one-forms via: φ ∗ ω i = i X k =0 α ik ω k , i = 0 , , ...n − ,φ ∗ w + = α n ω + α n ω + α nn w + . Here α ij are fun tions on J su h that n Y i =0 α ii = 0 at ea h point of J .Therefore, as in the ase of n = 5 , the onta t equivalen e problem for the n thorder ODEs (7.1) an be studied in terms of the invariant forms ( θ , θ , ...., θ n − , Γ + ) de(cid:28)ned by θ i = i X k =0 α ik ω k , i = 0 , , ...n − , (7.3) Γ + = α n ω + α n ω + α nn w + . These forms initially live on an n +3 n +82 -dimensional manifold G → J × G → J ,with the G -fa tor parametrized by α ij , su h that n Y i =0 α ii = 0 .Introdu ing gl (2 , R ) -valued forms(7.4) Γ = Γ − E − + Γ + E + + Γ E + Γ E , where (Γ + , Γ − , Γ , Γ ) are 1-forms on J × G , we an spe ialize to F ≡ , andreformulate proposition 7.1 toProposition 7.2. If F ≡ then one an hose n ( n +1)2 parameters α ij , as fun tionsof x , y , y , ..., y n − and the remaining three α s, say α i j , α i j , α i j , so that the ( n + 4) -dimensional manifold P parametrized by ( x, y, y , ..., y n − , α i j , α i j , α i j ) is lo ally the onta t symmetry group, P ∼ = GL (2 , R ) × ρ n R n , of equation y ( n ) = 0 .Forms (7.3), after restri tion to P , an be supplemented by three additional 1-forms (Γ − , Γ , Γ ) , so that ( θ , θ , ..., θ n − , Γ + , Γ − , Γ , Γ ) onstitute a basis of theleft invariant forms on the Lie group P . The hoi e of α s and Ω s is determined bythe requirement that basis ( θ , θ , ..., θ n − , Γ + , Γ − , Γ , Γ ) satis(cid:28)es d θ + Γ ∧ θ = 0 , (7.5) dΓ + Γ ∧ Γ = 0 , where θ = ( θ , θ , ..., θ n − ) T is a olumn n -ve tor, and Γ is given by (7.4).The de(cid:28)ning equations (7.5) of the left invariant basis, when written expli itlyin terms of θ i s and Γ s, read d θ = ( n − + Γ ) ∧ θ + (1 − n )Γ + ∧ θ , d θ = − Γ − ∧ θ + [( n − + ( n − ] ∧ θ + (2 − n )Γ + ∧ θ , ... d θ k = − k Γ − ∧ θ k − + [( n − + ( n − k − ] ∧ θ k + +(1 + k − n )Γ + ∧ θ k +1 , (7.6) ... d θ n − = (1 − n )Γ − ∧ θ n − + ( n − − Γ ) ∧ θ n − , dΓ + = 2Γ ∧ Γ + , dΓ − = − ∧ Γ − , dΓ = Γ + ∧ Γ − , dΓ = 0 . This system an be analyzed in the same spirit as system (4.1) of se tion 4. Thus,we (cid:28)rst onsider the distribution h = { X ∈ T P s . t . X − | θ i = 0 , i = 0 , , , .., n − } annihilating θ .Then the (cid:28)rst n equations of the system (7.6) guarantee that forms ( θ , θ , θ , , ..., θ n − ) satisfy the Fröbenius ondition, d θ i ∧ θ ∧ θ ∧ θ ∧ ... ∧ θ n − = 0 , ∀ i = 0 , , , ...n − and that, in turn, the distribution h is integrable. Thus manifold P is foliatedby 4-dimensional leaves tangent to the distribution h . The spa e of leaves of thisdistribution P/ h an be identi(cid:28)ed with the solution spa e M n = P/ h of equation y ( n ) = 0 . This in parti ular means, that all equations (7.5) an be interpretedrespe tively as the (cid:28)rst and the se ond stru ture equations for a gl (2 , R ) -valued onne tion Γ having vanishing torsion and and vanishing urvature. This gl (2 , R ) -valued onne tion originates from a ertain GL (2 , R ) ( onformal) stru ture on thesolution spa e M n .To make this last statement more pre ise we have to invoke a few results fromHilbert's theory of algebrai invariants [12℄ adapted to our situation of ODEs.7.1. Results from Hilbert's theory of algebrai invariants. First we ask iffor a given order n ≥ of an ODE (7.1) with F = 0 there exists a bilinear form ˜ g on P of proposition 7.2 su h that it proje ts to a nondegenerate onformal metri on M n . This is answered, in a bit more general form, by applying the re ipro itylaw of Hermite (see [12℄, p. 60), and its orollaries, due to Hilbert (see [12℄, p. 60).To adapt Hilbert's results to our paper we introdu e a de(cid:28)nition of an invariantof degree q . Let ˜ t be a totally symmetri ovariant tensor (cid:28)eld of rank q de(cid:28)ned onthe group manifold P of proposition 7.2.De(cid:28)nition 7.3. The tensor (cid:28)eld ˜ t is alled a GL (2 , R ) -invariant of degree q , ifand only if, it is degenerate on h and if for every X ∈ h , there exists a fun tion c ( X ) on P su h that L X ˜ t = c ( X )˜ t. The degenera y ondition means that ˜ t ( X, ... ) = 0 , for all X ∈ h .In the following we will usually abbreviate the term `a GL (2 , R ) -invariant' to:`an invariant'.The (cid:28)rst result from Hilbert's theory, adapted to our situation, is given by thefollowing L (2 , R ) GEOMETRY OF ODE'S 43Proposition 7.4. For every n = 2 m + 1 , m = 2 , , ... there exists a unique, up toa s ale, invariant ˜ g of se ond degree on P . This invariant, a degenerate symmetri onformal bilinear form ˜ g of signature ( m + 1 , m, , , , on P , satis(cid:28)es L X ˜ g = 2( n − X − | Γ )˜ g, for all X ∈ h .In ase of even orders n = 2 m , Hilbert's theory gives the followingProposition 7.5. For n = 2 m every GL (2 , R ) -invariant has degree q ≥ .Thus, if n = 2 m , we do not have a onformal metri on the solution spa e M n .Returning to odd orders, we present the quadrati invariants ˜ g , of proposition7.4, for n < : ˜ g = 3( θ ) − θ θ + θ θ , if n = 5 , ˜ g = − θ ) + 15 θ θ − θ θ + θ θ if n = 7 , (7.7) ˜ g = 35( θ ) − θ θ + 28 θ θ − θ θ + θ θ if n = 9 . These expressions an be generalized to higher (odd) n s. We have the followingProposition 7.6. If n = 2 m + 1 and m ≥ , the invariant ˜ g of proposition 7.4 isgiven by: ˜ g = m − X j =0 ( − j (cid:18) mj (cid:19) θ j θ m − j + ( − m (cid:18) mm (cid:19) ( θ m ) . Remark 7.7. This proposition is also valid for m = 1 . For su h m , the value of n is n = 3 , and we are in the regime of third order ODEs. Su h ODEs were onsidered by Wüns hmann [21℄. Sin e ˜ g = θ θ − ( θ ) is the only invariant inthis ase, the ounterpart of the bundle P of proposition 7.2 is a 10-dimensionalbundle P ∼ = O (2 , , the full onformal group in Lorentzian signature (1 , . The ounterpart of system (7.5)/(7.6) is given by Maurer-Cartan equations for O (3 , : d θ = 2(Γ + Γ ) ∧ θ − + ∧ θ , d θ = − Γ − ∧ θ + 2Γ ∧ θ − Γ + ∧ θ , d θ = − θ + (2Γ − ) ∧ θ , dΓ + = 2Γ ∧ Γ + + Γ ∧ θ + Γ ∧ θ , dΓ − = − ∧ Γ − + Γ ∧ θ + Γ ∧ θ , dΓ = Γ + ∧ Γ − − Γ ∧ θ + Γ ∧ θ , dΓ = − Γ ∧ θ − Γ ∧ θ − Γ ∧ θ , dΓ = − Γ ∧ Γ − + 2Γ ∧ Γ + 2Γ ∧ Γ , dΓ = − ∧ Γ + − ∧ Γ − + 2Γ ∧ Γ , dΓ = − ∧ Γ − Γ ∧ Γ + + 2Γ ∧ Γ . Here, apart from θ , θ , θ and Γ + , Γ − , Γ , Γ we have also left invariant forms Γ , Γ , Γ .Now we pass to the invariants of degree q = 3 . The question of their existen ewas again determined by Hilbert (see [12℄, p. 60), in terms of the re ipro ity law ofHermite. In the language of our paper we have the following4 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIProposition 7.8. An invariant of third degree ˜Υ exists on P if and only if n = 4 µ + 1 , µ ∈ N . Hilbert's theory, [12℄, p. 60, implies also the following:Proposition 7.9. In low dimensions n = 4 µ + 1 , the unique up to a s ale ubi invariant is given by • n = 5 : ˜Υ = ( θ ) − θ θ θ + θ ( θ ) − θ θ θ + ( θ ) θ • n = 9 : ˜Υ = 15( θ ) − θ θ θ + 24 θ ( θ ) + 24( θ ) θ − θ θ θ − θ θ θ + 3 θ ( θ ) − θ θ θ + 12 θ θ θ − θ θ θ +3( θ ) θ − θ θ θ + θ θ θ . The rough statement about the even orders, n = 2 m , des ribed in proposition7.5, an be again re(cid:28)ned in terms of the re ipro ity law of Hermite. FollowingHilbert we haveProposition 7.10. If ≤ n = 2 m the lowest order invariant tensor ˜Υ on P hasdegree four. This is unique (up to a s ale) only if n = 4 , , , . If n = 10 or n = 14 we have two independent quarti invariants ˜Υ ; if n = 16 , , we havethree independent quarti invariants; and so on.Proposition 7.11. In low dimensions n = 2 m , the quarti invariant tensor ˜Υ on P is given by • n = 4 : ˜Υ = − θ ) ( θ ) + 4 θ ( θ ) + 4( θ ) θ − θ θ θ θ + ( θ ) ( θ ) • n = 6 : ˜Υ = − θ ) ( θ ) + 48 θ ( θ ) + 48( θ ) θ − θ θ θ θ − θ ( θ ) θ + 9( θ ) ( θ ) + 16 θ θ ( θ ) − θ ( θ ) θ +16( θ ) θ θ + 4 θ θ θ θ − θ θ θ θ + ( θ ) ( θ ) . • n = 8 : ˜Υ = − θ ) ( θ ) + 600 θ ( θ ) + 600( θ ) θ − θ θ θ θ − θ ( θ ) θ + 81( θ ) ( θ ) + 360 θ θ ( θ ) − θ ( θ ) θ + 360( θ ) θ θ +50 θ θ θ θ + 40 θ ( θ ) θ − θ θ θ θ − θ θ θ θ + 25( θ ) ( θ ) +24 θ θ ( θ ) + 40 θ ( θ ) θ − θ θ θ θ − θ θ θ θ + 24( θ ) θ θ +18 θ θ θ θ − θ θ θ θ + ( θ ) ( θ ) . Among the small dimensions n = 7 is quite spe ial, sin e here the next invariantlinearly and fun tionally independent of the metri ˜ g has q = 4 . We have thefollowingProposition 7.12. In dimension n = 7 , the invariant of the lowest degree isthe metri ˜ g . There are no invariants of degree q = 3 and only two linearly L (2 , R ) GEOMETRY OF ODE'S 45independent, invariants of degree q = 4 . The (cid:28)rst of them is ˜ g . The se ond anbe hosen to be ˜Υ = 160( θ ) − θ ( θ ) θ + 1035( θ ) ( θ ) − θ θ ( θ ) + 540 θ ( θ ) − θ ) θ θ + 1920 θ ( θ ) θ − θ θ θ θ − θ θ θ θ − θ ) ( θ ) +540 θ θ ( θ ) + 540( θ ) θ − θ θ θ θ + 400 θ ( θ ) θ + 540( θ ) θ θ − θ θ θ θ − θ θ θ θ + 7( θ ) ( θ ) . GL (2 , R ) in dimensions n < . In di-mensions n ≤ the GL (2 , R ) invariant tensors of low order q ≤ turn out to besu(cid:30) ient to redu e the GL ( n, R ) group to GL (2 , R ) in its irredu ible n -dimensionalrepresentation.Given an invariant tensor ˜ t = 1 q ! t i i ...i q θ i ...θ i q of degree q on P and a GL ( n, R ) -valued fun tion a = ( a ij ) on P , at every point p ∈ P , we have a GL ( n, R ) -a tion ( a ij , ˜ t i i ...i q ) ( ρ n ( a )˜ t ) j j ...j q = a i j a i j ...a i q j q ˜ t i i ...i q . A subgroup G ˜ t of GL ( n, R ) onsisting of a = ( a ij ) su h that ρ n ( a )˜ t = (det a ) q/n ˜ t, is the stabilizer of ˜ t at p ∈ P . Sin e ˜ t is an invariant then, obviously GL (2 , R ) ⊂ G ˜ t .This leads to the following question: how many invariants is needed in dimension n so that its ommon stabilizer is pre isely GL (2 , R ) in its n dimensional irredu iblerepresentation?Inspe ting Hilbert's results we he ked that in dimensions ≤ n ≤ we haveTheorem 7.13. For ea h n = 4 , , , , , , the full stabilizer group of the respe -tive invariant tensor n ˜Υ of propositions 7.9, 7.11, 7.12, is the group GL (2 , R ) in the n -dimensional irredu ible representation ρ n . In parti ular, if n = 5 , , these stabi-lizers are subgroups of the respe tive pseudohomotheti groups CO (3 , , CO (4 , and CO (5 , , ea h in its de(cid:28)ning representation.Thus in ea h of these dimensions it is the lowest order nonquadrati invariantwhat is responsible for the full redu tion from GL ( n, R ) to GL (2 , R ) .Remark 7.14. In dimension n = 5 , using (7.7) and proposition 7.9 we de(cid:28)ne a onformal metri [ g ij ] represented by g ij = 12 ∂ ∂θ i ∂θ j (cid:0) ˜ g (cid:1) , i, j = 0 , , , , and a onformal symmetri tensor of third degree [ Υ ijk ] represented by Υ ijk = − √ ∂ ∂θ i ∂θ j ∂θ k (cid:0) ˜Υ (cid:1) , i, j, k, l = 0 , , , , . The onvenient fa tor − √ in the expression for Υ ijk was hosen so that thepair ( g ij , Υ ijk ) satis(cid:28)es Cartan's identities (i)-(iii) of se tion 2. This leads to the GL (2 , R ) geometries in dimension 5 onsidered in se tions 3-5.6 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKIRemark 7.15. In the next odd dimension situation is quite similar, but now we havea quarti invariant ˜Υ . Thus apart from the onformal metri [ g ij ] represented by g ij = 12 ∂ ∂θ i ∂θ j (cid:0) ˜ g (cid:1) , i, j = 0 , , , , , , we have a onformal symmetri tensor of fourth degree [ Υ ijkl ] represented by(7.8) Υ ijkl = 124 ∂ ∂θ i ∂θ j ∂θ k ∂θ l (cid:0) ˜Υ (cid:1) , i, j, k, l = 0 , , , , , , . Note that ˜Υ of proposition 7.12 was hosen in su h a way that the fourth order Υ ijkl satis(cid:28)ed g ij Υ ijkl = 0 , where g ij g jk = δ ik . This hoi e of the fourth order invariant is nevertheless arbitrary, sin e we analways get another invariant of the fourth order by repla ing Υ with ¯Υ ijkl = c ˜Υ ijkl + c ˜ g ( ij ˜ g kl ) . It is interesting to note that the hoi e c = 2 √ √ , c = 34 √ applied to ¯Υ , leads, via formula like (7.8), to ¯Υ ijkl satisfying Cartan-like identity: g ih g ef ¯Υ ie ( jk ¯Υ lm ) fh = g ( jk g lm ) and g ij ¯Υ ijkl = c g kl , where g ij g jk = δ ik . Note also that the above Cartan(cid:21)like identities are preserved under the onformaltransformation ( g ij , ¯Υ ijkl ) ( g ′ ij , ¯Υ ′ ijkl ) = (e φ g ij , e φ ¯Υ ijkl ) , where φ ∈ R .Thus the GL (2 , R ) geometries in dimension n = 7 may be de(cid:28)ned by a onformal lass of pairs of tensors [ g ij , ¯Υ ijkl ] with the properties and transformations asabove.Remark 7.16. By analogy, in dimensions n = 4 , , , the irredu ible GL (2 , R ) ge-ometries may be des ribed in terms of a onformal tensor [ n Υ ijkl ] represented by n Υ ijkl = 124 ∂ ∂θ i ∂θ j ∂θ k ∂θ l (cid:0) n ˜Υ (cid:1) , i, j, k, l = 0 , , , ..., n − , and obtained in terms of the respe tive quarti invariants n ˜Υ of proposition 7.11.Remark 7.17. Dimension n = 9 is similar to dimension n = 5 . A periodi ity withperiod four is a remarkable feature of Hilbert's theory of algebrai invariants [12℄,p. 60. L (2 , R ) GEOMETRY OF ODE'S 477.3. Wüns hmann onditions for the existen e of GL (2 , R ) geometries onthe solution spa e of ODEs. An invariant tensor ˜ t , by its very de(cid:28)nition, has aproperty that it des ends to a nondegenerate onformal tensor [ t ] on the solutionsspa e M n = P/ h of the equation y ( n ) = 0 . In parti ular in dimensions ≤ n ≤ the onformal lass [ n Υ] , orresponding to invariant tensors n ˜Υ redu es the stru turegroup of M n to GL (2 , R ) de(cid:28)ning an irredu ible GL (2 , R ) geometry there. We donot know how many invariant tensors are needed to a hieve this redu tion for n > ,but it is obvious that for a given n this number is (cid:28)nite, say w n . Thus for ea h n ≥ we have a (cid:28)nite number of invariants n ˜Υ I , I = 1 , , ...w n , whi h des end to thesolution spa e M n of the equation y ( n ) = 0 equipping it with a GL (2 , R ) stru ture.It is important that ea h of the invariants n ˜Υ I has only onstant oe(cid:30) ients whenexpressed in terms of the invariant oframe ( θ , ..., θ n − ) on P (see, for example,every n ˜Υ of the pre eding se tion).Now, we return to a general n -th order ODE (7.1). Thus we now have ageneral fun tion F ( x, y, y ′ , y ′′ , y (3) , ..., y ( n − ) , whi h determines the onta t forms ( ω , ω , ..., ω n − , w + ) by (7.2). Corresponding to these forms we have the invariantforms ( θ , ..., θ n − , Γ + ) of (7.3), whi h live on bundle J × G over J . We an nowask the following question (this generalizes to arbitrary n > the similar questionof se tion 5): What shall we assume about F de(cid:28)ning the onta t equivalen e lassof ODEs (7.1) that there exists a (4 + n ) -dimensional subbundle P of J × G onwhi h the forms ( θ , ..., θ n − , Γ + ) satisfy: d θ = ( n − + Γ ) ∧ θ + (1 − n )Γ + ∧ θ + 12 T ij θ i ∧ θ j , d θ = − Γ − ∧ θ + [( n − + ( n − ] ∧ θ ++(2 − n )Γ + ∧ θ + 12 T ij θ i ∧ θ j , ... d θ k = − k Γ − ∧ θ k − + [( n − + ( n − k − ] ∧ θ k ++(1 + k − n )Γ + ∧ θ k +1 + 12 T kij θ i ∧ θ j , (7.9) ... d θ n − = (1 − n )Γ − ∧ θ n − + ( n − − Γ ) ∧ θ n − + 12 T n − ij θ i ∧ θ j , dΓ + = 2Γ ∧ Γ + + 12 R + ij θ i ∧ θ j , dΓ − = − ∧ Γ − + 12 R − ij θ i ∧ θ j , dΓ = Γ + ∧ Γ − + 12 R ij θ i ∧ θ j , dΓ = 12 R ij θ i ∧ θ j . As (cid:28)rst observed by Wüns hmann [21℄ and then su essively used by Newman and ollaborators [18℄ this question an be reformulated into a ni er one. To make thisreformulation we repeat our arguments from se tion 7.1.8 MICHAŠ GODLI‹SKI AND PAWEŠ NUROWSKISuppose that we are able to satisfy system (7.9) by forms (7.3). Consider thedistribution h = { X ∈ T P s . t . X − | θ i = 0 , i = 0 , , , .., n − } annihilating θ s. Despite of the fa t that system (7.9) involves new terms, when ompared with system (7.6), they do not destroy the integrability of the distribution h ; the (cid:28)rst n equations (7.9) still guarantee that h is integrable. Thus manifold P isfoliated by 4-dimensional leaves tangent to the distribution h . The spa e of leavesof this distribution P/ h an be identi(cid:28)ed with the solution spa e M n = P/ h ofequation (7.1). Now, on manifold P of system (7.9), we de(cid:28)ne w n tensors n ˜Υ I ,whi h formally are given by the same formulae that de(cid:28)ned the w n invariants n ˜Υ I of the (cid:29)at system (7.6) needed to get the full redu tion to GL (2 , R ) . So,when de(cid:28)ning the present n ˜Υ I , we use the same formulae as for the y ( n ) = 0 ase, repla ing forms θ of the (cid:29)at ase, with forms θ satisfying system (7.9). It isnow easy to verify that the question about the onditions on F to admit P withsystem (7.9) is equivalent to the requirement that all w n tensors n ˜Υ I transform onformally when Lie transported along the leaves of distribution h . In(cid:28)nitesimallythis ondition is equivalent to the existen e of fun tions c I ( X ) on P su h that L X ( n ˜Υ I ) = c I ( X ) n ˜Υ I , ∀ X ∈ h , and ∀ I = 1 , , ...w n . If this is satis(cid:28)ed then tensors n ˜Υ I des end to a onformal lass of tensors [ n Υ , n Υ , ..., n Υ w n ] on the solution spa e M n de(cid:28)ninga GL (2 , R ) there.We know that in dimension n = 5 the onformal preservation of ˜ g and ˜Υ isequivalent to the requirement on fun tion F = F ( x, y, y , y , y , y ) to satisfy Wün-s hmann onditions (5.5). The generalization of this fa t to other low dimensions ≤ n < is given by the followingTheorem 7.18. Let M n be the solution spa e of n th order ODE(7.10) y ( n ) = F ( x, y, y ′ , y ′′ , y (3) , ..., y ( n − ) , with ≤ n < , and let D = ∂ x + y ∂ y + y ∂ y + . . . + y n − ∂ y n − + F ∂ y n − be the total derivative. The ne essary onditions for a onta t equivalen e lass ofODEs (7.10) to de(cid:28)ne a prin ipal GL (2 , R ) -bundle GL (2 , R ) → P → M n with in-variants forms ( θ , ..., θ n − , Γ + , Γ − , Γ , Γ ) satisfying system (7.9) is that the de(cid:28)n-ing fun tion F of (7.10) satis(cid:28)es n − Wüns hmann onditions given below: • n = 4 : D F − D F + 8 F − D F F + 4 F F + F = 0 , D F − D F + 144( D F ) − D F F + 144 F − D F F + 160 F F − D F F + 88 F F + 9 F + 16000 F y = 0 , • n = 5 : D F − D F + 50 F − F D F + 30 F F + 8 F = 0375 D F − D F + 350 D F + 1250 F − F D F + 200 F − F D F + 200 F F − F D F + 130 F F + 14 F = 0 L (2 , R ) GEOMETRY OF ODE'S 49 D F − D F + 1750 D F D F − F D F − F D F + 1250 F F − F D F + 700( D F ) F +1250 F F − F F D F + 350 F F − F D F +550 F F − F D F + 210 F F + 28 F + 18750 F y = 0 . • n = 6 : D F − D F + 27 F − D F F + 18 F F + 5 F D F − D F + 900( D F ) + 1575 F − D F F + 333 F − D F F + 315 F F − D F F + 225 F F + 25 F = 02835 D F − D F + 4320 D F D F + 14175 F − D F F − D F F + 1863 F F − D F F + 1800( D F ) F + 1575 F F − D F F F + 576 F F − D F F + 855 F F − D F F + 360 F F + 50 F = 014175 D F − D F + 6480( D F ) + 16200 D F D F − D F F − D F F + 3645 F − D F F +5400 D F F + 11475 F F − D F F + 864 F − D F F +10800 D F D F F + 14175 F F − D F F F − D F F F +5940 F F F − D F F + 4500( D F ) F + 5175 F F − D F F F + 2340 F F − D F F + 1800 F F − D F F +1050 F F + 125 F + 297675 F y = 0 • n = 7 : D F − D F + 98 F − D F F + 70 F F + 20 F = 06860 D F − D F + 6615( D F ) + 6860 F − D F F +1715 F − D F F + 1568 F F − D F F + 1190 F F + 135 F = 09604 D F − D F + 15435 D F D F + 24010 F − D F F − D F F + 4459 F F − D F F + 6615( D F ) F + 3430 F F − D F F F + 1470 F F − D F F + 2107 F F − D F F + 945 F F + 135 F = 0336140 D F − D F + 180075( D F ) + 432180 D F D F +2352980 F − D F F − D F F + 64827 F − D F F + 154350( D F ) F + 192080 F F − D F F +17150 F − D F F + 308700 D F D F F + 192080 F F − D F F F − D F F F + 113190 F F F − D F F +132300( D F ) F + 89180 F F − D F F F + 47775 F F − D F F + 35280 F F − D F F + 22050 F F + 2700 F = 02352980 D F − D F + 1512630 D F D F + 2268945 D F D F − D F F − D F F − D F F + 648270( D F ) F +907578 F F − D F F + 1080450 D F D F F + 1596665 F F − D F F F − D F F + 288120 F F − D F F +540225( D F ) F + 1296540 D F D F F + 2352980 F F − D F F F − D F F F + 324135 F F − D F F F +926100( D F ) F F + 756315 F F F − D F F F + 154350 F F − D F F + 926100 D F D F F + 732305 F F − D F F F − D F F F + 524790 F F F − D F F + 396900( D F ) F +231525 F F − D F F F + 209475 F F − D F F +119070 F F − D F F + 75600 F F + 8100 F + 65883440 F y = 0 . Remark 7.19. Although we al ulated the Wüns hmann onditions for n = 8 and n = 9 , we do not present them here due to their length. We remark, however that inany order n ≥ , the n − Wüns hmann onditions, whi h by the very de(cid:28)nition are onditions needed for an ODE to de(cid:28)ne a GL (2 , R ) geometry on its solution spa e,are always of the third order in the derivatives of the fun tion F whi h de(cid:28)nes anODE. In this sense they di(cid:27)er from the generalizations of Wüns hmann onditionsobtained by [7℄ and [8℄.Remark 7.20. If n = 3 we have only one Wüns hmann ondition [6, 21℄: D F − D F − D F F + 18 F F + 4 F + 54 F y = 0 . and, if it satis(cid:28)ed, a onformal Lorentzian geometry asso iated with a metri g = θ θ − ( θ ) is naturally de(cid:28)ned on the solution spa e.Remark 7.21. If n = 4 the ODEs satisfying the two Wüns hmann onditions leadto very nontrivial geometries on 4-dimensional solution spa es. These are a sortof onformal Weyl geometries, whi h instead of a metri are de(cid:28)ne in terms of the onformal rank four tensor Υ . These geometries de(cid:28)ne a hara teristi onne tion,whi h is gl (2 , R ) valued and has an exoti holonomy [2℄. By this we mean that theholonomy of this nonmetri but torsionless onne tion does not appear on theBerger's list [2℄. See also our a ount on this subje t in [17℄.Remark 7.22. Our studies of the ODES with n = 3 , , , and the preliminary resultsabout the ases with n ≥ , make us to onje ture that if n ≥ then the n − Wüns hmann onditions are too stringent to admit many solutions for F . Thus,we strongly believe, that if n ≥ the orresponding GL (2 , R ) geometries on thesolution spa es of the Wüns hmann ODEs are very spe ial, su h that, for example,their hara teristi onne tions have identi ally vanishing urvatures. We intendto dis uss these matters in a subsequent paper. 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