Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions

F ree n -distributions : holonom y , sub-Riemannian structures, F eﬀerman constructions and dual distributio ns. Stuart Armstrong 2007 Abstract This pap er analyses the parab olic geometries generated by a free n -distribution in the t angen t space of a manifold. It sh o ws that certain holonom y reductions of t h e associated normal T ra ctor connections, imply preferred connections with sp ecial prop erties, along with Riemannian or sub- Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, ho w ever, lie in th e free 3-distributions. In t h ese cases, there are normal F eﬀerman constructions o ver CR and Lagrangian con tact structures correspondin g to holonom y reductions to S O (4 , 2) and S O (3 , 3), re sp ectively . There is also a fascinating construction of a ‘dual’ distribution when the holonomy reduces to G ′ 2 . Con ten ts 1 In tro duction 2 2 The geometry of free n -dis tributions 3 2.1 Homogeneous mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Cartan connectio n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Harmonic curv ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The T r a ctor bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Preserv ed subundles 8 3.1 Sub-T racto r bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Twisted product 13 5 F eﬀerman constructions 16 5.1 Almost-spinoria l s tructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6 F ree 3 -di stributions 17 6.1 Geometric eq uiv alence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 G ′ 2 structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 CR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.4 Lagra ng ian contact str uctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 CONTENTS 1. Introduction 1 In tro duction On a manifold M , let H ⊂ T M be a distribution of rank n . Then there is a w ell deﬁned map L : H ∧ H → T M /H . F o r X, Y sections of H , it is g iven by the quotien ted Lie bra ck et X ∧ Y → [ X , Y ] / H . Then H is a fr e e n -distribution if L is a n isomor phism. The mo niker “ free” comes fro m the fact that there are no relations b et ween sections of H that w ould cause L to fail injectivity . This condition immediately implies that T M /H is of rank n ( n − 1) / 2, th us that M is of dimension m = n ( n + 1) / 2 . Br yan t [Bry05] ha s studied the case of n = 3 , m = 3, a free 3-distribution in a 6-manifold, but the general case r emains little studied. F ortunately , these s tr uctures lead themselves to be treated with the g eneral to ols of Ca rtan connec- tions on para b olic geo metries ([ ˇ CG02] and [ ˇ CS00]). The homogeneo us mo del is pr ovided b y the set of maximal isotropic pla nes in R n +1 ,n . The gr oup of transfor mations is G = S O ( n + 1 , n ) while the stabiliser of a po in t is P = GL ( n ) ⋊ R n ⋊ ∧ 2 R n . Its Lie alg ebra is p which has nilradical R n ⋊ ∧ 2 R n . These are pr ecisely the t wo-step free nilp otent Lie alge br as, with the Lie br ack et from R n ⊗ R n to ∧ 2 R n being given by taking the wedge. The fact this nil-radical is free is a consequence of the freeness of the n - distribution. W e do not in tro duce any ex tra infor ma tion, or make any choices by taking the Cartan co nnection, as the nor mal Ca r tan connection for a free n -distr ibution is determined entirely by H ([ ˇ Cap06]). The most natura l r estrictions to put on the holonomy o f a connection with s tructure algebra so ( n + 1 , n ) is to require that it preserves a subundle in the natural repres en tation bundle of that alg ebra – the standar d T r a ctor bundle T . This condition is a nalysed; it turns out it implies a class of preferr ed connections on M , which preser ve cer tain structure s o n the manifold, making them an example of sub-Riemannian manifolds. If the rank of the preser ved bundle V ⊂ T is n , ther e is a uniq ue preferr ed connection ∇ deﬁned by it, tha t ha s pro pe rties a nalogous to the Einstein condition in conformal and pro jective ge o metry . If V is further non-deg e ne r ate, there is a well-deﬁned metric on the manifold as a whole. In that case, it is a n Einstein inv o lution [Arm07]. Other issues worth lo ok ing into in any new g eometries is how the structures restrict to sub-manifolds; this is analy sed in the next sec tion. There is even a decomp ositio n/twisted pro duct result, similar to the Einstein pro duct result in con- formal geometr y ([Lei05] and [Arm05]) which applies to c e rtain very restr ictive holonomy alg e br as. In this case , there are explicit constr uctions o f manifolds with these proper ties , lea ving hope that manifolds with the weaker prop erties men tioned ab ov e will also exis t. The main results of this pap er will b e gleaned in the n = 3 , m = 6 case. The free 3-distribution has a F eﬀerman construction in to the conformal structure [Bry05]. W e will sho w this F eﬀerman construction is norma l for b oth T rac to r connec tio ns, mea ning that we hav e many known exa mples of holono m y reductions [Arm05]. Here, the normal Cartan connection is torsion- free, a nd the results of t he previous section can be applied to show that holonomy reductions to S U (2 , 2) ∼ = S pin (4 , 2) 0 and S L (4 , R ) ∼ = S pin (3 , 3) 0 do exist, and ar ise from their own F eﬀerman constructions – ov er integrable CR ma nifo lds and int egra ble Lagra ngian cont act structures, resp ectively . Here, normality of the under lying Cartan connections is equiv alent with no rmality of that generated by the free 3-distr ibutio n. The other interesting situation for a fre e 3-distribution is that of a r eduction of the ho lonomy of the T ractor co nnec tio n − → ∇ to G ′ 2 . This do es not ar ise fr om a n y F eﬀerman constr uction, but has a fascinating geometr y . On an open dense set of the manifold, there is a canonical W eyl structure ∇ . This determines a splitting of T = T − 2 ⊕ H , wher e H is the ca nonical free 3-distribution. 2 CONTENTS 2. The geometr y of free n -distributions Then H ′ = T − 2 is also a free 3- distribution, and the normal T r actor connection it generates is isomorphic with − → ∇ . Iterating this procedure gene r ates H a gain; thus H and H ′ are in some sense ‘dual’ distributions . 2 The geometry of free n - distributions 2.1 Homogeneous mo del The ho mogeneous mo del for a fr ee n -distribution is the space of isotropic n -planes inside R ( n +1 ,n ) . Let g = so ( n + 1 , n ), and put the metric on R ( n +1 ,n ) in the for m:   0 0 I d n 0 1 0 I d n 0 0   , where I d n is the identit y matrix on R n . The algebra g is then spanned by elements of the form:   A v B w 0 v t C w t − A t   , where A ∈ gl ( n ), B t = − B and C t = − C . Then the isotro pic plane V = { ( a 1 , a 2 , . . . , a n , 0 , 0 , . . . , 0) | a j ∈ R } is preserved by the subalg ebra p ⊂ g spa nned by elements o f the form   A v B 0 0 v t 0 0 − A t   , This alg e bra is isomorphic to gl ( n ) ⊕ ( R n ) ∗ ⊕ ( ∧ 2 R n ) ∗ , with the natural alge braic structure. The homogeneous manifold is M = G/P , where G = S O ( n + 1 , n ) 0 and P ⊂ G is the Lie group with Lie algebr a p . M is of dimension (2 n + 1)(2 n ) / 2 − ( n 2 + n + n ( n − 1) / 2) = n ( n + 1) / 2. There is a subspace g ( − 1) of g , consisting of those e lemen ts with C = 0. Right a ction on G generalis e s this to a distribution b H ⊂ T G . This distribution is preserved by P , a nd hence the map G → M maps b H to a dis tribution H ⊂ T M . T his distribution is evidently of r ank n . Now co nsider the diﬀerential brac ket of righ t-inv aria n t v ector ﬁelds which are sections of b H ; this matches up with the Lie brack et on g ( − 1) , thus [ b H , b H ] spans a ll of T G . C o nsequently , [ H , H ] must span all of T M , ma king H fre e b y dimensional co nsiderations. T o hav e an explicit mo del, let x j , 1 ≤ j ≤ n a nd x kl , 1 ≤ k < l ≤ n b e lo cal c o ordinates on M . Then we may deﬁne vector ﬁelds U kl via: U kl = ∂ ∂ x kl . Extend this deﬁnition by requiring U lk = − U kl , U kk = 0. Then w e complete the fra me with the vector ﬁelds X j = ∂ ∂ x j + 1 2 n X p =1 x p U pj . 3 CONTENTS 2.2 Cartan connection These vector ﬁe lds follow the co mm utator relations: [ X p , X j ] = U pj , [ U k 1 l 1 , U k 2 l 2 ] = 0 , [ U kl , X j ] = 0 . In this mo del, the distribution H is s imply the spa n of X j . 2.2 Cartan connection Given a semi-simple Lie algebra g with Killing for m ( − , − ), a subalgebra p ⊂ g is s aid to b e p ar ab olic iﬀ p ⊥ is the nilr adic al of p , i.e. its maximal nilpotent ideal. This gives ([CDS05], details are also av aila ble in the author’s thesis [Arm06]) a ﬁltr ation of g : g ( − k ) ⊃ g ( k − 1) ⊃ . . . ⊃ g (0) ⊃ g ( − 1) ⊃ . . . ⊃ g ( k ) , || || || g p p ⊥ such that { g ( j ) , g ( l ) } ⊂ g ( j + l ) . The as so ciated g r aded algebra is g r( g ) = L k − k g j , where g j = g ( j ) / g ( j +1) . By results fro m [CDS05 ], gr( g ) is isomor phic to g . F urthermore, there is a unique element ǫ 0 in g 0 such that { ǫ 0 , ξ } = j ξ for all ξ ∈ g j . The iso morphisms gr( g ) ∼ = g compatible with the ﬁltration ar e then given by a choice of lift ǫ of ǫ 0 with r esp ect to the exact seque nce 0 → p ⊥ → p → g 0 → 0 . (1) This means that the gra dings of g compa tible with the ﬁltration for m an aﬃne s pa ce mo delled o n the nilradical p ⊥ . Deﬁne P and G 0 as the subgr oups of G that preserve the ﬁltr ation and the gra ding, resp ectively . It is ea sy to see that their Lie algebra s ar e p a nd g 0 , and that the inclusio n G 0 ⊂ G is non-canonica l. Deﬁnition 2.1 (Cartan connection) . A Ca r tan connection on a manifold M for the parab olic subalgebra p ⊂ g is given by a principal P -bundle P → M , and a o ne fo r m ω ∈ Ω 1 ( P , g ) with v alues in the Lie a lgebra g such that: 1. ω is equiv aria n t under the P -action ( P a c ting b y Ad on g ), 2. ω ( σ A ) = A , where σ A is the fundamental vector ﬁeld of A ∈ p , 3. ω u : T P u → g is a linear isomo rphism for all u ∈ P . The inclusio n P ⊂ G generates a bundle inclusion P ⊂ G and ω is then the pull back of a unique G -equiv aria nt connection form on G which we will also desig nate by ω . Since ω takes v alue s in g , it g enerates a standa r d connection on any vector bundle associa ted to G . This connection is ca lle d the T r actor c onn e ction and will b e designa ted by − → ∇ . In the c a se we ar e lo ok ing at, g = so ( n + 1 , n ) and p = gl ( n ) ⊕ ( R n ) ∗ ⊕ ( ∧ 2 R n ) ∗ . g 0 is simply gl ( n ), and the nilradica l of p is p ⊥ = ( R n ) ∗ ⊕ ( ∧ 2 R n ) ∗ . 4 CONTENTS 2.2 Car tan connection And a ll the tw o-s1tep free-nilp otent alg ebras are pr ecisely of this form. In ter ms of the notation for parab olic subalgebr as intro duced [ ˇ CS00], this is: . . . > ◦ ◦ ◦ × . Deﬁne the Lie algebra bundle A = P × P g . This has a natural ﬁltration A = A ( − 2) ⊃ A ( − 1) ⊃ A (0) ⊃ A (1) ⊃ A (2) . Paper [ ˇ CG02] demonstrates that the tangent space T o f M is eq ual to the quo tien t bundle A / A (0) . The killing for m g ives an iso mo rphism A ( − 1) ∼ = ( A / A (0) ) ∗ = T ∗ . Hence there is a well deﬁned inclus ion T ∗ ⊂ A , and a well deﬁned pr o jection A → T . Deﬁnition 2.2 (W eyl structure) . A Weyl st ru ctur e o n ( M , P , ω ) is a P - equiv arient function η : P → p that is always a lift o f the grading element ǫ 0 , as in equa tion (1). A W eyl structure gives a splitting of g , and cons e quent ly allows a deco mp os ition of bo th the Lie algebra bundle a nd the Cartan co nnection: A = A − 2 ⊕ A − 1 ⊕ A 0 ⊕ A 1 ⊕ A 2 ω = ω − 2 + ω − 1 + ω 0 + ω 1 + ω 2 . Dividing out by th e action of p ⊥ gives a quotient map p → g 0 and hence a bundle map P → G 0 . There is a unique G 0 -equiv ar ient o ne-form o n G 0 of w hich ω 0 is the pull-back; we will ca ll it ω 0 as well. This is a principal connectio n on G 0 ; since G 0 acts on g − 2 + g − 1 ∼ = g / p , then T = A / A (0) is an ass o ciated bundle to G 0 . This implies that ω 0 generates an aﬃne connection ∇ on the tangen t bundle. These ∇ ’s are called preferr ed co nnections; they are in one-to-o ne cor resp ondence with W eyl struc- tures and henc e to compatible splittings of A . They consequent ly form an aﬃne space mo delled on A ( − 1) = T ∗ ; the relation betw een tw o preferre d co nnections ∇ and b ∇ is giv en ex plicitly by the one-form Υ o n M such that ∇ X Y = b ∇ X Y + { { X , Υ } , Y } − , with { , } the L ie brack et on A . The splitting o f A gives further splittings T = T − 2 ⊕ T − 1 and T ∗ 1 ⊕ T ∗ 2 . The bundles T − 1 and T ∗ 2 = ( T − 1 ) ⊥ are de ﬁned indep endently of the W ey l structure, since they ar e preserved by the action of T ∗ ⊂ A . Given a pr eferred ∇ , the T r actor connection on A and any asso ciated bundles is given by: − → ∇ X v = ∇ X v + X · v + P ( X ) · v , where P is the rho-tens o r, a section of T ∗ ⊗ T ∗ , a nd · is the action of T and T ∗ on A given by the Lie bra cket. Deﬁnition 2.3 (Curv ature) . The curv ature o f the Car tan connection is deﬁned to b e the tw o- form κ = dω + { ω , ω } ∈ Ω 2 ( P , g ). It is easy to see that κ v anis hes on vertical v ectors , and is P -equiv a riant; consequently dividing out by the action of P , κ may seen as an elemen t of Ω 2 ( M , A ); in this s e tting, it is the curv ature of the T ractor connec tion − → ∇ . Finally , the inclus ion T ∗ ⊂ A implies that κ is equiv alent to a P -eq uiv ariant function from P to ∧ 2 p ⊥ ⊗ g . W e shall use the designa tio n κ int erchangeably fo r these three equiv a lent deﬁnitions , thoug h it is generally the third one we shall be using. 5 CONTENTS 2.3 Har monic curv ature Given a grading on g , there is a decomp osition of a ny tensor product ⊗ c g = P g j 1 ,j 2 ,...j c where g j 1 ,j 2 ,...j c = g j 1 ⊗ g j 2 ⊗ . . . ⊗ g j c . The ho mogeneity of g j 1 ,j 2 ,...j c is deﬁned to b e the sum j 1 + j 2 + . . . + j c . An y elemen t of η of ⊗ c g can b e decomp osed into homogeneous element s η j 1 ,j 2 ,...,j c . The m inimal homo geneity of η is deﬁned to b e the lowest homogeneity amo ng the non-zer o η j 1 ,j 2 ,...,j c . Homogeneity is not preserved by the action of P ; howev er since p consists o f elements of ho mogeneity zero and ab ov e, the minimal homogeneity of any ele ment is preserved by the a c tion of P . Since κ is a map to ∧ 2 p ⊥ ⊗ g , the following deﬁnition ma kes sense: Deﬁnition 2.4 (Regularity) . A Car tan c o nnection is regula r iﬀ the minimal homogeneit y of κ is greater than z e ro. There a re well deﬁned Lie algebra diﬀeren tials ∂ : ∧ c p ⊥ ⊗ g → ∧ c +1 p ⊥ ⊗ g and co diﬀerentials ∂ ∗ : ∧ c p ⊥ ⊗ g → ∧ c − 1 p ⊥ ⊗ g . In ter ms of decomp osa ble elements, the co- diﬀerential is given by ∂ ∗ ( u 1 ∧ . . . ∧ u c ) ⊗ v = X j 6 = k u 1 ∧ . . . ∧ { u j , u k } ∧ . . . ∧ u c ⊗ v + X j u 1 ∧ . . . ˆ u j . . . ∧ u c ⊗ { u j , v } . Deﬁnition 2.5 (Norma lit y) . A Cartan connectio n is no rmal iﬀ ∂ ∗ κ = 0. If we let ( Z l ) b e a frame for T and ( Z l ) a dual frame for T ∗ , this condition is given, in terms o f κ an element of Ω 2 ( M , A ), as ( ∂ ∗ κ )( X ) = X l { Z l , κ ( Z l ,X ) } − 1 2 κ ( { Z l ,X } − ,Z l ) , for all X ∈ Γ( T ). Paper [ ˇ CS00] demonstrates tha t a regular, normal Ca rtan connectio n for these Lie g roups is de- termined entirely b y the distribution T − 1 . W e shall ca ll this distribution H , as b efore. Since ω is regular , the algebraic brac ket { , } matc hes up with g r aded v er sion of the Lie brack et [ , ] o f v ec to r ﬁelds; in o ther words, if X and Y ar e sec tions of H , [ X , Y ] − { X , Y } ∈ Γ( H ) , implying tha t H is ma ximally non-in tegrable. Indeed, any ma x imally no n-integrable H of correct dimension and co -dimension determines a unique normal Ca rtan co nnection as ab ov e. Deﬁnition 2.6. W e shall c a ll ( M , H ) a ma nifo ld with a free n -distribution. It is a lw ays of dimensio n m = n ( n + 1 ) / 2. 2.3 Harmonic curv ature A Ca rtan connec tion is s a id to b e torsion-fr e e if the curv a ture κ seen as a function P → ∧ 2 p ⊥ ⊗ g is actually a function P → ∧ 2 p ⊥ ⊗ p . If κ is instead seen as a sec tion of ∧ 2 T ∗ ⊗ A , tor sion-freeness implies that is a section o f ∧ 2 T ∗ ⊗ A (1) . Since ∂ ∗ κ = 0, κ m ust map into K er( ∂ ∗ ). This has a pro jection onto the homology component H 2 ( p ⊥ , g ) = Im( ∂ ∗ ) / Ker( ∂ ∗ ). The co mpo sition of κ with this pro jection is κ H , the harmo nic cur- vatur e . The Bia nci identit y for normal Ca rtan co nnec tions imply that it is a co mplete obstruction to int egra bilit y ([ ˇ CS00]). Indeed, pap er [CD01] demonstrates that κ H determines κ entirely . 6 CONTENTS 2.4 The T racto r bundle Now, Kos ta n t’s so lution of the Bott-Bore l- W eil theorem [Kos61] allows us to algo rithmically ca lculate H 2 ( p ⊥ , g ). The n = 1 ca se is triv ial, the n = 2 and n = 3 ca ses have H 2 ( p ⊥ , g ) contained inside ( g 1 ∧ g 2 ) ⊗ g 0 . The harmonic curv a ture must lie inside this comp onent, whic h is of homogeneit y t hree. Both the Bianci identit y fo r normal Cartan connectio ns [ ˇ CS00], and the cons tr uction of the full curv atur e from the ha r monic curv a ture [CD01] imply that the other comp onent of the cur v ature must hav e higher homogeneity . Since the tor sion compo ne nts hav e maximal homogeneity three (for ( ∧ 2 g 2 ) ⊗ g − 1 ), these tw o g eometries ar e tors io n-free. F or n ≥ 4, the harmonic curv atur e is contained inside ( g 1 ∧ g 2 ) ⊗ g − 2 , and thus these geometrie s are never torsio n-free (unless they ar e ﬂat). This will b e mos t evident when we lo ok at pr eserved substr uctures (see Section 3.5); while the tors ion-free geometries do not require extra conditions for these substructures to cor resp ond to structures o n immer s ed submanifolds, we will requir e extr a conditions on the c ur v ature κ in the g eneral case for this to b e true. 2.4 The T ractor bund le The standar d T ra ctor bundle T is bundle on which we will b e doing most o f o ur calculations. It is deﬁned to b e the bundle a sso ciated to P under the standard repr esentation of so ( n + 1 , n ): P × P R ( n +1 ,n ) . This makes T in to a rank 2 n + 1 bundle. A choice of preferred connectio n ∇ re duces the s tr ucture group of T to gl ( n ). Under this reduction, T = H ⊕ R ⊕ H ∗ . Changing the the choice of ∇ by a one-for m Υ changes this splitting as:   v τ X   →   v + τ Υ 1 − { Υ 2 , X } − 1 2 (Υ 1 ( X ))Υ 1 τ − Υ 1 ( X ) X   . (2) This demonstrates that the inclusions H ∗ ⊂ R ⊕ H ∗ ⊂ T are w ell deﬁned, as a re the pro jections T → H ⊕ R → H . The pr o jection that we sha ll b e using the mos t often is π 2 : T → H . The metric h on T , of signa ture ( n + 1 , 1) is g iven in this splitting by: h    v τ X   ,   w ν Y    = 1 2 ( w ( Y ) + v ( X ) + τ ν ) , while the T ractor deriv ative in the direction of an y Z ∈ Γ( T ) is − → ∇ Z = Z + ∇ Z + P ( Z ), or, more explicitly , − → ∇ Z   v τ X   =   ∇ Z v + τ P ( Z ) 1 − { P ( Z ) 2 , X } ∇ Z τ − v ( Z − 1 ) − P ( Z ) 1 ( X ) ∇ Z X + τ Z − 1 + { Z − 2 , v }   . 7 CONTENTS 3. Preser ved subundles Remark. Note that the Z term in Z + ∇ Z + P ( Z ) implies there cannot ex ist a section of H ∗ or R ⊕ H ∗ that is mapp ed into the sa me bundle by − → ∇ , at any po in t. The conv erse of that is that if V is any bundle preser ved by − → ∇ , then its intersection with H ∗ and R ⊕ H ∗ will be minimal on an op en, dense set. As we will gener ally b e dealing with s uch prese r ved V ’s and as our results ar e lo cal, we will generally ass ume that this in ters ection is minimal, b y restricting implicitly to the op en dense subset where it is true. 3 Preserv ed subundles W e sha ll b e lo oking a t the v a r ious implications of preserved subundles of the T ractor bundle T . W e shall always b e assuming that − → ∇ is normal, unless explicitly stated o therwise, but most o f these results do not need the no r mality condition. Recall that for any bundle A ⊂ B , the bundle A ⊥ ⊂ B ∗ is deﬁned to b e the m aximal bundle s uc h that the c ontraction A ⊥ x A is alwa ys zero . F o r any metric g on A , we deﬁne iso g ( A ) to b e A ∩ g ( A ⊥ ) – the maximal is otropic subspace of A . Now, let V b e a generic subundle of T of r a nk ≤ n . By g eneric we mean that the pro jection π 2 : T → H maps V injectively to a subundle A ⊆ H , on a o p en dense subset of M . The remark at the end of Section 2.4 implies tha t any bundle pres erved by − → ∇ is gener ic . The metric h , restric ted to V and then pr o jected to A , gives a (p o ssibly very degener ate or null) metric g on A . A choice of preferred connection ∇ giv es a splitting of T , and hence a ma p V → H ∗ . Since A a nd V are isomorphic, this gives a map µ ∇ : A → H ∗ . Finally , dividing out by the action of A ⊥ gives a map g ∇ : A → A ∗ . Deﬁnition 3 .1 ( V -preferred connections) . W e say that the connection ∇ is V -prefer red if g ∇ = g . W e say the connection ∇ is str o ngly V -pre ferred if it is prefer r ed a nd µ ∇ ( V ) ∩ A ⊥ = 0. The diﬀerence betw een preferred a nd strong ly preferr e d is sp ecious if g (hence V ) is no n-degenera te. F o r degenera te V , a V -pr eferred ∇ has a µ ∇ that maps iso tr opic elements of A to sections of A ⊥ , while a stro ngly V -pre fer red ∇ will have a µ ∇ that is zero on isotropic elements of A . F rom now on, w e will drop the s uper script fro m µ ∇ , referring to it as simply µ unless w e wan t to emphasis the dep endence. It is go o d to have a µ that is a metr ic on all o f T , not just on A ; we can extend it a s follows: If ∇ is stro ngly V -pre fer red, we can extend µ to a map H → H ∗ simply by deﬁning it to b e zer o on ( µ ( A )) ⊥ ⊂ H . Since µ ( A ) ∩ A ⊥ = 0, A + ( µ ( A )) ⊥ = H and µ is zer o on A ∩ ( µ ( A )) ⊥ , ma king this well deﬁned. This extended µ is still symmetric. If ∇ is just V -preferred, we ma y also extend µ , but w e hav e to make a choice of a bundle F transverse to A + ( µ ( A )) ⊥ ⊂ H . Then we ex tend µ by deﬁning it to b e zer o on ( µ ( A )) ⊥ , r equiring that µ ( F ) ⊂ F ⊥ , and requiring that µ remain symmetric. T he actua l choice of F will no t b e impo rtant. Finally , notice that the metric µ extends to T − 2 by the isomorphism T − 2 = { H , H } ∼ = H ∧ H . Deﬁning T − 2 and H to b e p erp endicular ex tends the deﬁnition o f µ to a metr ic on all of T . Theorem 3. 2. F or any generic V of ra nk ≤ n , ther e exists pr eferr e d and str ongly pr eferr e d c onn e c- tions ∇ for V . The V -pr eferr e d c onne ctions form an aﬃne sp ac e mo del le d on A ⊥ ⊕ { H, A ⊥ } ; the str ongly V -pr eferr e d form an aﬃne sp ac e mo del le d on A ⊥ ⊕ { ( iso µ ( A )) ⊥ , A ⊥ } . F urthermor e, the bund le B = { A, A } , is indep en dent of t he choic e of V -pr eferr e d c onne ct ion. If − → ∇ pr eserves V , these V - pr eferr e d c onne ctions ∇ have the fol lowing pr op erties: 8 CONTENTS 3. Preser ved subundles 1. A is pr eserve d by ∇ along dir e ctions in C = H ⊕ B + { µ ( A ) ⊥ , µ ( A ) ⊥ } ⊂ T , 2. ∇ µ is zer o on A ⊗ A , 3. P 11 = − µ + η with η a se ction of H ∗ ⊗ A ⊥ , 4. P 22 = − µ + η ′ , with η ′ a se ction of T ∗ 2 ⊗ { A ⊥ , H ∗ } , 5. P 21 and P 12 ar e se ctions of T ∗ 2 ⊗ A ⊥ and T ∗ 1 ⊗ { A ⊥ , H ∗ } , re sp e ctively. If ∇ is a ctual ly st r ongly V -pr eferr e d, then η ′ is a se ction of T ∗ 2 ⊗ { A ⊥ , iso g ( A ) ⊥ } and P 12 a se ction of T ∗ 1 ⊗ { A ⊥ , iso g ( A ) ⊥ } . The prop er ties of this theorem der ive from the following prop osition: Prop ositio n 3. 3 . L et V ⊂ T b e a generic bund le of r ank ≤ n . Then if ∇ determines a spli tting such that V has no R c omp onen t and µ is s ymmetric on A , then ∇ i s a V - pr eferr e d c onne ction. If V ∩ ( H , 0 , A ⊥ ) = 0 , it is str ongly V -pr eferr e d. And if V is pr eserve d by − → ∇ , ∇ has the pr op erties enumer ate d in The or em 3.2. Pr o of. Having no R comp onent is understo o d to mea n that given the splitting H ⊕ R ⊕ H ∗ , pro jection onto the c e n tral comp onent maps V to zer o. In this case, the map µ : A → H ∗ is deﬁned by ma pping the H compo ne nt o f V to its H ∗ comp onent. If ˇ X = ( X , 0 , µ ( X )) and ˇ Y = ( Y , 0 , µ ( Y )) a re s ections of V , the metric h o n these elements is given by: g ( X , Y ) = h ( ˇ X , ˇ Y ) = 1 2 ( µ ( X, Y ) + µ ( Y , X )) = µ ( X , Y ) , making ∇ V -preferr ed. µ | A ⊗ A remains constant under the action o f a section of A ⊥ ⊕ { H , A ⊥ } . F or the stro ngly V - preferred co nnections, V ∩ ( H , 0 , A ⊥ ) = 0 iﬀ µ ( A ) ∩ A ⊥ = 0 . Strong ly V - pr eferred connections are r elated b y the a ction of A ⊥ ⊕ { (iso µ ( A )) ⊥ , A ⊥ } , the subspace o f A ⊥ ⊕ { H , A ⊥ } that preserves the ab ov e rela tion. Now assume that − → ∇ prese r ves V . If Z is a section of C = H ⊕ B + { µ ( A ) ⊥ , µ ( A ) ⊥ } ⊂ T , then Z · (0 , 0 , µ ( A )) ⊂ ( A, R , 0). Since − → ∇ Z ( X, 0 , µ ( X )) = ( ∇ Z X + { Z − 2 , µ ( X ) } , . . . ) ∈ Γ( V ) , ∇ Z m ust map sections of A to sections of A . Now let Z b e an y section of T . Pick ˇ X = ( X , 0 , µ ( X )) and ˇ Y = ( Y , 0 , µ ( Y )) such that h ( ˇ X , ˇ Y ) = µ ( X, Y ) is a co ns tant . Then since − → ∇ pres erves h : h ( − → ∇ Z ˇ X , ˇ Y ) + h ( ˇ X , − → ∇ Z ˇ Y ) = 0 But this is: µ ( Y , ∇ Z X + { Z − 2 , µ ( X ) } ) + µ ( X , ∇ Z Y + { Z − 2 , µ ( Y ) } ) . The terms µ ( Y , { Z − 2 , µ ( X ) } ) + µ ( X , { Z − 2 , µ ( Y ) } ) m ust v a nish if Z − 2 is decomp osable under the ident iﬁcation T − 2 ∼ = ∧ 2 H – hence it must v anish for a ll Z , g iving 0 = µ ( Y , ∇ Z X ) + µ ( X , ∇ Z Y ) , implying that ∇ µ is zero on A ⊗ A . 9 CONTENTS 3. Preser ved subundles Now take the deriv ative of any section ˇ X of V : − → ∇ Z ˇ X = ( ∇ Z X + { Z − 2 · µ ( X ) } , − µ ( X , Z − 1 ) − P 11 ( Z, X ) − P 21 ( Z, X ) , ∇ Z µ ( X ) − { P ( Z ) 2 X } ) . Since this must be a section of V , the central term m ust v anish. Consequently P 21 v anishes on A , so is a s ection o f T ∗ 2 ⊗ A ⊥ , and P 11 = − µ + η with η a s ection of H ∗ ⊗ A ⊥ . Also since this is a section of V , µ ( ∇ Z X + { Z − 2 , µ ( X ) } ) = ∇ Z µ ( X ) − { P ( Z ) 2 , X } ) . Then s ince the µ ( ∇ Z X ) − ∇ Z µ ( X ) = ( ∇ Z µ )( X ) is a s ection of A ⊥ , s o m ust b e µ ( { Z − 2 , µ ( X ) } ) + { P ( Z ) 2 , X } . This implies that P 22 + µ and P 12 m ust, under the bracket action, map sections of A to sections of A ⊥ . This mea ns that P 22 + µ is a section of T ∗ 2 ⊗ { A ⊥ , H ∗ } and a section o f T ∗ 1 ⊗ { A ⊥ , H ∗ } . If ∇ is mor eov er strong ly V -preferr ed, then a ny X that is a section of iso g ( A ), must hav e µ ( X ) = 0, and µ ( A ) ∩ A ⊥ = 0 , implying that { P ( Z ) 22 + µ ( Z ) , X } = { P ( Z ) 12 , X } = 0. Consequently P 22 + µ and P 12 are sections o f T ∗ 2 ⊗ { A ⊥ , iso g ( A ) ⊥ } and T ∗ 1 ⊗ { A ⊥ , iso g ( A ) ⊥ } , resp ectively .  How ever, we ha ve not yet shown the exis tence of preferred c o nnections ∇ with the r e q uired prop er- ties. This done in the next lemma: Lemma 3 .4. F or any generic bund le V of r ank r < n such that π 2 : V → A ⊂ H is bije ctive, ther e lo c al ly exists a pr eferr e d c onne ction ∇ su ch t hat V has no R c omp onent, and µ is symmetric on A . Mor e over ther e exists such ∇ so tha t V ∩ (0 , R , A ⊥ ) = 0 . Pr o of of L emma. Pick an y frame ( v j ) for V , and an y preferred connection ∇ . Then, assuming without lo ss of ge ne r ality that v 1 has an R factor, we may c o nstruct ano ther frame ( v ′ j ) wher e v ′ 1 = ( X 1 , 1 , ω 1 ) is a sca ling of v 1 and the other v ′ j = ( X j , 0 , ω j ) hav e no R factors. Then pick a section α of H ∗ such that α ( X j ) = δ 1 j . Changing the preferred connection ∇ by the action of α gives a splitting wher e v ′ 1 = ( X 1 , 1 − α ( X 1 ) , ω ′ 1 ) = ( X 1 , 0 , ω ′ 1 ) and v ′ j = ( X j , − α ( X j ) , ω ′ j ) = ( X j , 0 , ω ′ j ). So V has no central comp onent. T o make µ symmetric, pick an or thogonal fra me ( w j ) of V , with w j = ( Y j , 0 , ν j ). I f w e can ensur e that ν j ( Y k ) = 0 whenever j 6 = k , then µ must b e symmetric on the frame ( Y j ) and hence on the whole of A ⊗ A . W e will pr o ceed by induction; assume that ν j ( Y k ) = 0 for j 6 = k , j ≤ l . Since the w j are o rthogonal, ν l +1 ( Y k ) = 0 for k < l . W e may now change the splitting by the action of the comp onent Υ = r X k = l +2 ν l +1 ( Y k ) { Y ∗ l +1 , Y ∗ k } where Y ∗ p is any section of H ∗ such tha t Y ∗ p ( Y q ) = δ pq . The action of Υ is trivial o n all Y k with k < j , so this change do es not change the rela tions ν j ( Y k ) = 0 for j 6 = k , j ≤ l . How ever, in the new splitting, ν l +1 ( Y k ) = 0 for k 6 = l + 1 . Since this pro cess works for l = 0 as well, by induction there exis ts a ∇ giving a splitting with no R comp onent such that µ is s y mmetric on A . Now, a ssume t hat we hav e a V -preferr ed connection ∇ as a b ove, and pick a new frame ( v j ) suc h that ( v j ) j ≤ p is a frame for iso h ( V ). The splitting we are lo oking for is o ne where v j = ( X j , 0 , 0) for j ≤ p . This result can also b e proved inductiv ely . W e shall b e using sections of { H ∗ , A ⊥ } to change the splitting, which will not a ﬀect the fact that ∇ is V - preferred. 10 CONTENTS 3.1 Sub-T ractor bundles Assume that w e ha ve a splitting wher e v j = ( X j , 0 , 0) for j ≤ l where 0 ≤ l < p . Then v l +1 = ( X l +1 , 0 , τ ). Since v l +1 is orthogo na l to all of V , τ must b e a section o f A ⊥ . Cho os e a section ξ of H ∗ , deﬁned so that ξ ( X j ) = δ j l . Then c hanging the splitting b y the action of { ξ , τ } giv es a new splitting where v j = ( X j , 0 , 0) for j ≤ l + 1. Since the co nstruction holds for l = 0 as well, by induction ther e e xists a str ongly V -pre fer red ∇ .  Theorem 3 . 5. If V is generic of r ank n , t he c onditions o f The or em 3.2 simpli fy c onsider ably. In these c ases, ther e exist a unique V -pr eferr e d c onn e ction ∇ (which is aut omatic al ly str ongly V - pr eferr e d). And if − → ∇ pr eserves V : • ∇ µ = 0 , • P 21 = P 12 = 0 , • P 11 = − µ on H , and P 22 = − µ on T − 2 , • henc e ∇ P = 0 . If V is a deﬁnite subspace of T , then this is in fac t an Einstein in volution, s o these pr op erties are exp ected – see pa per [Arm07]. By similar a rguments to ab ove, if V is pres e rved only in the directions along H , then w e ha ve an example of sub- Riemannian Geometry , for the metric µ on H . 3.1 Sub-T ractor bundles Here we attempt to grapple with the iss ue o f sub-s tr uctures co n tained within the total str ucture – sp eciﬁcally , of free n - distributions and their Cartan connections o n a distribution E ⊂ T of lo wer rank. Since t hese Cartan co nnections hav e to rsion in th e general case, the integrability of E m ust be address ed separ ately . W e wan t conditions for − → ∇ to des cend to a T ra ctor connectio n on a subdis tr ibution E ⊂ T . Assume there is a bundle A and a preferr ed connection ∇ suc h that ∇ pres erves A a long E = A + { A, A } . W e will work in the splitting g iven b y ∇ , a nd we wan t − → ∇ to des cend to a T ra ctor connectio n on E . There are tw o versions of this: a strong and a weak condition. Deﬁnition 3.6 (Strong condition) . − → ∇ pres erves E ⊕ ( A ⊗ A ∗ ) ⊕ E ∗ ⊂ g ( T ) along E for some choice of A ∗ ⊂ H ∗ (whic h then determines E ∗ ). This implies that ∇ preserves E ∗ along E . Deﬁnition 3.7 (W ea k conditio n) . − → ∇ preserves E along E for some choice of A ∗ ⊂ H ∗ (whic h then determines E ∗ ), where E ⊂ E ⊂ E ⊕ ( A ⊗ A ∗ ) ⊕ E ∗ ⊂ g ( T ) , for some choice of A ∗ ⊂ H ∗ . This implies that ∇ pre s erves E ∗ ∩ E alo ng E . The strong condition describ es a sub-tractor bundle A E contained in the algebra bun dle A . T he weak condition des c rib es a preser ved subundle of such a sub-T ra c tor bundle. Lemma 3.8. F or the str ong and we ak c onditions, E is inte gr able if and only if κ ( Z 1 , Z 2 ) j , j ≤ 1 is a se ction of E wheneve r Z 1 and Z 2 ar e se ctions of E . 11 CONTENTS 3.1 Sub-T ractor bundles Pr o of. By the prop erties ab ov e, P ( E ) preserve E . Hence if the “tor sion-terms” of κ ( Z 1 , Z 2 ) are sections of E , then the actua l torsio n of ∇ o n E ∧ E is a section of E . Since ∇ preser ves E along E , this implies that E is integrable.  Remark (Inclusio n) . If we have the strong condition, t hen deﬁne T E = A ⊕ R ⊕ A ∗ and we have an inclusion T E ⊂ T . This bundle must now b e preserved by − → ∇ along E . This inclusion is unchanged by changing ∇ by any section Υ o f E ∗ = A ∗ ⊕ { A ∗ , A ∗ } . Our work on pr eserved bundles in Section 3 g ives us a useful class of sub-T ractor bundles: Prop ositio n 3.9. Assume that we have a subun d le V ⊂ T of r ank n , pr eserve d by − → ∇ , with V 1 ⊂ V also pr eserve d. Then if A = π 2 ( V 1 ) and E = A ⊕ { A, A } , we have a we ak sub-T ra ctor c onne ction on V 1 . If µ is non-de gener ate on V 1 , we have the s t r ong c ondition, with a wel l deﬁne d inclusion T E ⊂ T . Pr o of. Pick the unique V -preferr ed co nnec tio n ∇ ; since this is also V 1 -preferred, it preserves A along E . Deﬁne A ∗ = µ ( A ) ⊂ H ∗ ; since ∇ µ = 0 this is preser ved by ∇ along E as w ell. Let Z b e any section of E ; consequent ly , P ( Z ) is a section of E ∗ 1 , a nd µ ( E ) is preserved along E . Th us star ting from E ⊂ A , rep eated diﬀerentiation along E generates the a lgebra E ⊕ ( A ⊗ µ ( A )) ⊕ µ ( E ) , and − → ∇ prese r ves that bundle along E . If µ is no n- degenerate, we have the strong condition, µ uniquely deﬁned b y the uniquenes s of ∇ . Moreover the inclusion T E = A ⊕ R ⊕ µ ( A ) ⊂ H R ⊕ H ∗ = T , is well deﬁned.  A slight weakening of the pr evious conditions give us : Prop ositio n 3.10. Assume we have a pr eserve d V of r ank r ≤ n . If ∇ Z µ = 0 for Z any se ction of E , we have a we ak sub-T r actor bund le. If also µ is of maximal ra nk, then we have the str ong c ondition. Pr o of. Same as previo us.  Finally , there is a consideratio n of normality . When is this s ub-T racto r bundle normal, assuming E is int egra ble? If Z is a s ection of E , then: 0 = ( ∂ ∗ κ )( Z ) = X Z j { κ ( Z j , Z ) , Z j } − 1 2 κ ( Z j ,Z ) − ,Z j = X Z ′ j { κ ( Z ′ j , Z ) , ( Z j ) ′ } − 1 2 κ (( Z j ) ′ ,Z ) − ,Z ′ j + X Z ′′ j { κ ( Z ′′ j , Z ) , ( Z j ) ′′ } − 1 2 κ (( Z j ) ′′ ,Z ) − ,Z ′′ j , 12 CONTENTS 4. Twisted pr oduct where ( Z ′ j ) is a fr ame for E , and ( Z ′′ j ) a frame for ( E ∗ ) ⊥ . The normality condition is then that P Z ′′ j { κ ( Z ′′ j , Z ) , ( Z j ) ′′ } − 1 2 κ (( Z j ) ′′ ,Z ) − ,Z ′′ j has a trivial action on E ⊥ ( A E for the strong condition). There is no r eason to s uppos e this is true in general, though Co rollary 4.4 in the next section demonstrates a pa rticular case where no rmality do es descend to E . 4 Twisted pro duct In this section we present a deco mpos ition and twisted pro duct construction for cer tain types of fr ee n -distributions, dep endent on the holo nomy and curv a ture of the T ractor co nnec tions. W e will ﬁrst need to introduce s ome terminology a nd deﬁnitions: Deﬁnition 4.1. A connection − → ∇ ﬁx es a v ector bundle V , if ther e is a frame ( v j ) of V such t hat − → ∇ v j = 0. A co-isotropic bundle V ⊂ T is a bundle such that W = h ( V ⊥ ) is contained in V . A minimal co-isotr o pic bundle is o ne of r ank n + 1 (which implies that W o f ra nk n is a maximal isotro pic bundle). If ( M 1 , H 1 ) and ( M 2 , H 2 ) are t wo free distributions, let N = H 1 N H 2 be the total space of the tensor pro duct of H 1 and H 2 ov er M 1 × M 2 (in other words there is a surjective map H → M 1 × M 2 and the v er tical ﬁbre of T N at the point ( x, y ) ∈ M 1 × M 2 is ( H 1 ) x × ( H 2 ) y ). The point of this construction is the following theor em: Theorem 4.2. If ( M 1 , H 1 ) and ( M 2 , H 2 ) ar e two manifolds c arrying fr e e d istributions of r anks n 1 and n 2 such that their T r actor c onne ctions − → ∇ 1 and − → ∇ 2 e ach ﬁx minimal c o-isotr opic bun d les V 1 and V 2 , then t her e exists a wel l-deﬁne d fr e e distribution on the total sp ac e of the ten s or pr o duct N = H 1 N H 2 , and a well- deﬁne d n ormal Cartan c onne ction − → ∇ 3 for this structur e – and also ﬁxes a minimal c o-isotr opic bund le V 3 , with an isomorphi sm W 3 ∼ = W 1 ⊕ W 2 . We c al l this t he twiste d pr o duct of ( M 1 , ∇ 1 ) and ( M 2 , ∇ 2 ) . If − → ∇ 1 ﬁxes V 1 , cho ose the unique W 1 -preferred co nnec tion ∇ 1 deﬁned by the bundle W 1 = h ( V ⊥ 1 ) (see Theorem 3.5). In this splitting, P = 0. Moreov er, since − → ∇ 1 ﬁxes V 1 , it must also ﬁx its isotro pic subspace, i.e. W 1 . Since the pro jectio n W 1 → H is bijective, this means that ∇ 1 is actually a ﬂa t connection. Let ( X j ) b e a lo ca l frame of H suc h that ∇ 1 X j = 0. Moreov er, − → ∇ 1 m ust also ﬁx another section of V . In the splitting deﬁned b y ∇ 1 , this m ust be of the for m ( A, 1 , 0) (since it is orthogona l to a ll the e lemen ts ( X j , 0 , 0)). By the for m ula for − → ∇ 1 1 of Equation (3 ), th is implies that ∇ 1 X j A = − X j and ∇ U A = 0 for U a section of ( T M 1 ) − 2 ; in other words, ∇ 1 A = I d H . Now A = x j X j for so me functions x j . The above formulas thus implies that X j · x k = − δ k j , and, furthermore, that U · x k = 0 for all sections U of ( T M 1 ) − 2 . Similar conclusion hold for V 2 , generating a ﬂat preferred co nnection ∇ 2 , a fra me ( Y k ) of H 2 and functions ( y k ) such that Y j · y k = δ k j . Now, if V is the vertical s ubundle of T N , there is a n exact sequence: 0 → V → T N → T M 1 × T M 2 → 0 . ∪ ∪ H 1 H 2 13 CONTENTS 4. Twisted pr oduct The preferr ed connectio ns ∇ 1 and ∇ 2 give a connection on N → M 1 × M 2 seen as a vector bundle and hence a splitting of the ab ov e sequence, g iven b y a map σ : T M 1 ⊕ T M 2 → T N . W e w ant to adjust this splitting so that the inclusion H 1 ⊕ H 2 → T N deﬁnes a maximally non- in tegra ble subundle of H 3 . T o do so, deﬁne sections e X j and e Y k of T N by e X j = σ ( X j ) − 1 2 X k y k X j ⊗ Y k e Y k = σ ( Y k ) + 1 2 X j x j X j ⊗ Y k . Deﬁne H 3 to b e the spa n o f these elements. The pro jection T N → T M 1 ⊕ T M 2 maps H 3 to H 1 ⊕ H 2 . Moreov er, since ∇ 1 and ∇ 2 resp ect the alg ebraic brack ets on M 1 and M 2 , we hav e the following commutator relations : [ e X j , e X k ] = σ ([ X j , X k ]) [ e Y j , e Y k ] = σ ([ Y j , Y k ]) [ e X j , e Y k ] = X j ⊗ Y k . The new splitting σ ′ of the sequence (3) is given my σ ′ ( X j ) = e X j , σ ′ ( Y k ) = e Y k and σ ′ = σ on ( T M 1 ) − 2 and ( T M 2 ) − 2 . W e now deﬁne ∇ 3 by re q uiring it to b e zero on a ll of e X j , e Y k and all of their Lie brack ets. Given ∇ 3 and σ ′ , the T r a ctor bundle T 3 is deﬁned a s H 3 ⊕ R ⊕ H ∗ 3 , and the T ractor connectio n as − → ∇ 3 Z = Z + ∇ 3 Z . T o show it is normal, we need the following prop ositio n: Prop ositio n 4. 3 . The curvatur e of − → ∇ 3 is the dir e ct sum of those of − → ∇ 1 and − → ∇ 2 , and henc e − → ∇ 3 is normal. On a op en, dense set, the lo c al holo nomy of − → ∇ 3 is the sum of those of − → ∇ 1 and − → ∇ 2 . Mor e over, − → ∇ 3 also ﬁ xes a maximal c o-isotr opic bund le. Pr o of. Both ∇ 1 and ∇ 2 are ﬂat, and have v anishing P . Hence the only terms in the cur v atures of − → ∇ 1 and − → ∇ 2 are the torsion t erms of ∇ 1 and ∇ 2 . W e need to show that ∇ 3 has exa ctly the same torsion t erms – which, since it is ﬂat, is equiv alent with demonstrating tha t the Lie brack et of its preserved sections is the same. First, it is easy to see that the splitting σ prese r ves the Lie bracket o n T M 1 and o n T M 2 . The splitting σ ′ adds extra ter ms to T M 1 , but all the extra terms are vertical vectors m ultiplied by a function whos e der iv ative v anishes along T M 1 and along ( T M 2 ) − 2 . This im plies that σ ′ preserves the Lie br a ck et on T M 1 and do es not introduce any extra tors io n b etw een T M 1 and ( T M 2 ) − 2 . The same argument shows that there are no e x tra torsion terms on T M 2 or b et ween T M 2 and ( T M 1 ) − 2 . The Lie bra ck et betw een horizontal sec tio ns a nd v er tical sections of T N is also t rivial, as it is on vertical sections of T N . Since we hav e deﬁned the algebr a ic brack et in such a wa y that it m atches the diﬀerential one on sections e X j and e Y k , this demonstra tes we hav e no extra torsio n ter ms. Since the c o-diﬀerential ∂ ∗ is C ∞ ( N )-linear, the normality of − → ∇ 1 and − → ∇ 2 imply the no rmality of − → ∇ 3 . Similarly , the inﬁnitesimal holo nomy of − → ∇ 3 – the span o f the image in A of the iterated deriv atives of the curv a ture of − → ∇ 3 – is the sum of the inﬁnitesimal ho lonomies of − → ∇ 1 and − → ∇ 2 . And on an o pen, dense set, the inﬁnitesimal holono m y of any co nnection matches up to its lo cal holo nomy . 14 CONTENTS 4. Twisted pr oduct The maximal co- isotropic bundle V ⊂ T 3 preserved by − → ∇ 3 is that spanned by ( e X j , 0 , 0) , ( e Y k , 0 , 0) and ( σ ′ ( A 1 ) + σ ′ ( A 2 ) , 1 , 0), whe r e A j is the section of H j deﬁned b y ∇ j A j = I d H j .  This twisted pro duct has a cor ollary , a dec o mpo sition result: Corollary 4.4 . Given any manifold ( M , H ) with a fr e e n -distribut ion, with normal T r actor c onne c- tion − → ∇ that ﬁx es a minimal c o-isotr opic bund le V , let W 1 b e any pr eserve d su bsp ac e of W = h ( V ⊥ ) , and A 1 = π ( W 1 ) and E 1 = A 1 ⊕ { A 1 , A 1 } in the spli tting deﬁne d by t he W -pr eferr e d c onne ction ∇ . Then if κ ( E 1 , E 1 ) − ⊂ E 1 , E 1 is inte gr able. If, mor e over, ther e is a pr eserve d W 2 such that W = W 1 ⊕ W 2 , with the same pr op erties, and κ ( E j , T ) = κ ( E j , E j ) , for j = 1 , 2 then the (we ak) sub-T ra ctor c onne ctions on E 1 and E 2 ar e normal, and M is lo c al ly t he t wiste d pr o duct of the le aves of E 1 and E 2 . Pr o of. The ﬁrs t results ar e a direct conseq uence of the integrability and no rmality conditions of the previous section. T o see the last piece, recall that − → ∇ describ es H completely , and that since P = 0, ∇ describ es − → ∇ completely . Since ∇ is ﬂa t, it is en tirely described b y its preserved sections. Since P = 0, κ must be the torsion o f ∇ , mor e sp eciﬁcally the Lie brack et of the ﬂat sections o f T ﬁx e d b y ∇ . Now κ ( E j , T ) = κ ( E j , E j ) implies that κ ( E 1 , E 2 ) = 0, demonstrating that the ﬂat sections of E 1 ﬁxed b y ∇ co mm ute with the ﬂat sectio ns of E 2 . This makes ∇| E j inv aria n t along E k for j 6 = k . Moreover, these sections must commute with the ﬂat sections of B = { A 1 , A 2 } , meaning that ∇ is inv aria n t along B . So each leaf of E j carries the sa me free distribution ( M j , H j ). W e may div ide o ut by the action of B to get a pro duct manifold M 1 × M 2 . It is t hen easy to see that N = H 1 N H 2 caries the same T ractor connection − → ∇ a s M do es, thus implying they ar e lo cally isomor phic.  Prop ositio n 4.5. Th er e exists manifolds with non-ﬂ at − → ∇ that ﬁx a minimal c o-isotr opic subund le V , but only if the r ank n is ≥ 4 . Pr o of. If n = 2 , 3, ther e are no torsion terms, so the ﬂatness of ∇ implies the ﬂatness of − → ∇ . F or n ≥ 4, take the homog e ne o us mo del fro m Sec tion 2.1, a nd r eplace X 1 and X 2 with X ′ 1 = X 1 + x 12 U 34 + 1 2 x 2 U 21 , X ′ 2 = X 2 − 1 2 x 1 U 12 Then [ X ′ 1 , X ′ 2 ] = U 12 , a s b efore, and all the other brack ets are a s befor e (since U 12 only app ea r s in X 1 and X 2 ) except fo r [ U 12 , X ′ 1 ] = U 34 . Then deﬁne the ﬂat ∇ by a nnihilating this new frame for H (hence for T ∗ − 2 ), and − → ∇ X = X + ∇ X . The only piece o f curv a ture of − → ∇ is − → R U 12 ,X ′ 1 = T or ∇ U 12 ,X ′ 1 = U 34 . − → ∇ is nor mal since the only r elev ant terms in ∂ ∗ − → R are { U ∗ 12 , − → R U 12 ,X 1 } = { U ∗ 12 , U 23 } = 0 , { X ∗ 1 , − → R X 1 ,U 12 } = { X ∗ 1 , U 23 } = 0 , − → R { U ∗ 12 , −} − ,U 12 = − → R 0 ,U 12 = 0 , − → R { X ∗ 1 , −} − ,X 1 , 15 CONTENTS 5. Fefferman constr uctions and the last term is ze r o a s {− , −} − ⊂ H and − → R is zero on H ∧ H . Moreover, if is easy to see that X j · x k = δ j k , so the hypothes is of Theor em 4.2 a re fulﬁlled.  5 F eﬀerman constructions Consider a par ab olic geometry ( M , P , ω ) derived from the homog eneous mo del G/P . Assume that there is an inclusion G ⊂ b G with a para bo lic inclus ion b P ⊂ b G such that b P ∩ G ⊂ P . Assume further that the inclusion G/ ( b P ∩ G ) ⊂ b G/ b P is op en. Then we may do the F eﬀerman c onstruction o n this data. See for exa mple [ ˇ Cap02] for deta ils of the o riginal construction. Deﬁne c M as P / ( b P ∩ G ). The inclusion ( b P ∩ G ) ⊂ b P deﬁnes a principal bundle inclusio n i : P ֒ → b P ov er c M . Since g ⊂ b g , we may extend ω to a section ω ′ of ( T b P ∗ ⊗ b g ) | P by requiring that ω ′ ( σ A ) = A , for any element A ∈ b g and σ A the fundamental vector ﬁeld on b P deﬁned by A . W e ma y further extend ω ′ to all of b P by b P -equiv a riance. Since the inclusion G/ ( b P ∩ G ) ⊂ b G/ b P is open, the inclusion g ⊂ b g genera tes a linear isomo rphism g / ( b p ∩ g ) → b g / b p . A t any p oint u ∈ P , ω is a linear isomor phism T P u → g . The pr evious condition ensures that ω ′ is a linea r isomorphism T b P u → b g . This condition extends to all of P , then to all of b P by equiv aria nce. Co ns equently ω ′ is a Cartan co nnection. Dividing out by P / ( b P ∩ G ) makes c M in to a principal bundle ov er M . It is then easy to see that ω ′ is in v ar iant along the vertical vectors of c M and pro jects to ω on M . Th us ω ′ has holonomy gro up contained in G . Remark. The F eﬀerman co nstruction implies nothing a b o ut the re la tive normalities of ω and ω ′ . 5.1 Almost-spinorial structures There is a n evident inclusion o f S O ( n + 1 , n ) into S O ( n + 1 , n + 1 ). In terms of Dynkin diagr ams, . . . > ◦ ◦ ◦ ◦ ⊂ ✟ ✟ ❍ ❍ . . . ◦ ◦ ◦ ◦ ◦ . Prop ositio n 5.1. Ther e exists a F eﬀerman c onstruction for this inclusion, wh er e c M = M . In terms of Dynkin diagr ams with cr osse d no des (s e e [ ˇ CS00]), this i s . . . > ◦ ◦ ◦ × ⊂ ✟ ✟ ❍ ❍ . . . ◦ ◦ ◦ ◦ × and the other p ar ab olic ge ometry is an almost-spinorial ge ometry (se e [ ˇ CSS97]). Conversely, any almost spinorial ge ometry whose T r actor c onne ction pr eserves a se ct ion of t he stan- dar d T r actor bu nd le gener ates a fr e e n -distribut ion o n the m anifold. Pr o of. The homo geneous mo del for the a lmost-spinoria l geometry is b G/ b P where b G = S O ( n + 1 , n + 1) and b P is the stabilizer of a n isotropic n + 1 plane. The homoge ne o us model for a free n -distribution are given b y G = S O ( n + 1 , n ) and P the stabilizer of an isotropic n plane. Since the space R ( n +1 ,n ) ⊂ R ( n +1 ,n +1) m ust b e tra nsverse to every isotro pic n + 1 plane, b P ∩ G = P . The op en inclusion f or the F eﬀerman condition is equiv alent with the statement tha t g and b p are tr ansverse 16 CONTENTS 6. Free 3 -distributions inside b g . A simple compariso ns of the ranks o f all the algebras inv olved demonstrates that this is the case. This a llows us to do the F eﬀerman c onstruction. Since b P ∩ G = P , c M = b P / b P = P / b P ∩ G = P / P = M . So this almos t s pinorial structur e is on the same manifold as the fr ee n -distribution. Now let ( M , b P , − → ∇ as ) b e an almost-spino r ial g eometry . Let T as be its standard tra ctor bundle. Let − → ∇ τ = 0, for τ a non-degenerate section of T as . This giv es a reduction of the structure group of − → ∇ , making it onto a connection o n the principal bundle G ⊂ b G (where these principal bundles hav e structure gro ups G and b G resp ectively). On G , − → ∇ as is given b y ω as , a one- form with v alues in g . Lo cally on an o p en set U ⊂ M , G = G × U and b P = b P × U . Since G ∩ b P = P for every embedding G ⊂ b G given by the preser v ation of a non-deg enerate element, G ∩ b P = P , with structure gr oup P . Then ω as restricts to P , and be c omes a Cartan connectio n o n it, inheriting equiv aria nce and p oint-wise iso mo rphism.  The g eometric meaning of this is not hard to see. An almost-spinoria l geo metry has an is omorphism T M ∼ = ∧ 2 U , a nd there is a pro jection π : T as → U . If − → ∇ as τ = 0, this gives us a distribution H = π ( τ ) ∧ U ⊂ T M . The ra nk of U must b e n + 1 , so the rank of H is n ; this is our free n -distribution. There is a nother F eﬀerman co nstruction that seems relev ant here; that given by the inclusion ✟ ✟ ❍ ❍ . . . ◦ ◦ ◦ × × ⊂ . . . > ◦ ◦ ◦ × . But except when the ﬁrst alg e bra is D 4 or D 3 , par ab olic geometries of the the ﬁrst t yp e are ﬂat if regula r and normal (since all their harmonic cur v atures have zero ho mogeneity , see K ostant’s solution of the Bott-Bor el-W eil theo rem [Kos6 1]). The cas e of D 3 will b e dealt with in Section 6. 6 F ree 3 -distributions These ar e the g eometries detailed by Bryan t in [Bry 0 5]. They hav e tw o prop erties that distinguish them from the g eneral free n -distr ibutio n b ehaviour. First of all, they a re torsion free, see Section 2.3. Secondly , the almost spino rial s tructure of Sec tion 5 .1 is given by ✟ ✟ ❍ ❍ ◦ ◦ ◦ × . Ho wev er , triality implies that ✟ ✟ ❍ ❍ ◦ ◦ ◦ × ∼ = ✟ ✟ ❍ ❍ × ◦ ◦ ◦ , i.e. tha t the almost-s pinorial structure is a c tua lly a conformal structure, whenever the S O (4 , 3) structure lifts to a S pin (4 , 3) structure. This is always true lo ca lly . Paper [Bry 0 5] details the F eﬀer man constructio n explicitly . He further shows that if the T ractor connection for the free 3-distribution is r egular and normal, the conformal T ractor co nnection must be normal as well (re gularity is automatic since the co nformal para bo lic is | 1 | - graded. The holono m y of that confor mal T ractor co nnec tio n m ust evidently reduce to S pin (3 , 4). In fact, the conforma l str ucture is deter mined by the ﬁltra tion of T coming from t he T ractor con- nection of the 3-distribution (see nex t section). Consequen tly this lo ca l lift glo balises for all fre e 3-distributions. 17 CONTENTS 6.1 Geometr ic equiv alence Prop ositio n 6.1. Co nversely, if the normal c onformal holonomy of a six manifold M r e duc es t o S pin (4 , 3) , t his m anifold is the F eﬀerman c onst ruction of a r e gular normal fr e e 3 -distribution. Pr o of. Set b G = S O (4 , 4), with b P being C O (3 , 3) ⋊ R (3 , 3) , the confor mal para bo lic (deﬁned as th e stabiliser o f a nul-line in R (4 , 4) . G = S pi n (4 , 3) a nd P = GL (3) ⋊ R 3 ⋊ ∧ 2 R 3 as b efor e. Let − → ∇ c and ω c be the nor mal conformal T ra ctor and Cartan connectio ns. Let b P b e the co nformal b P bundle, b P ⊂ b G with b G the full structure bundle for ω c . The holo nomy reduction implies that ther e exists a G -bundle G ⊂ b G such that ω c reduces to a pr inc ipa l connection on G . The a c tio n of S pi n (4 , 3) on the nu l-lines o f R (4 , 4) is tr a nsitive; consequently G and b P alwa ys lie transitively in b G . This means that G ∩ b P = P , a P -bundle, so ω c reduces to a free 3- distribution Cartan connectio n – call it ω . It remains to sho w that this Ca rtan connection is normal. Lo o king at the ho mogeneous model, the conformal str ucture comes from the f act there is a unique co nformal class of b P -inv a riant inner pro ducts on b g / b p . This implies there is a unique conformal class of P inv ariant inner pro ducts on g / p . Since T M = G × P ( g / p ), this means that the conformal structure on T M dep e nds only on the negative homogeneity comp onents of ω – the soldering fo r m, ω − (see next sec tion for the geometric details of this). The curv a ture κ c of ω c can be seen as a b P -inv a riant map from b P to ∧ 2 ( b g / b p ) ∗ ⊗ b g . Simila r ly , the curv ature κ of ω is a P -inv ariant map from P to ∧ 2 ( g / p ) ∗ ⊗ g . On P , these tw o cur v atures are related by the commut ing dia gram: ∧ 2 g / p κ − → g ↑ ↓ ∧ 2 b g / b p κ c − → b g . Since ω c is normal, it is torsio n free (see [ ˇ CG02] or [Ar m05]), implying that it maps in to b p . This means that ω a lso maps into p – so is a lso torsion- free. This means that κ is of homogeneity ≥ 2, consequently – s ince ∂ ∗ resp ects homoge ne ity – ∂ ∗ κ is of homogeneity ≥ 2. Now, by [ ˇ CSed], any Ca rtan connection ω with cur v ature κ such that ∂ ∗ κ is of homogeneity ≥ l ≥ 0 diﬀers from the normal Cartan co nnection ω ′ by a section Φ ∈ Ω 1 ( P , g ) of ho mo geneity ≥ l . So here w e hav e ω + Φ = ω ′ , with ω ′ normal a nd Φ of homogeneity ≥ 2. This means that ω ′ and ω hav e the same soldering form (as the soldering form is of strictly negative homogeneity), thus that the confor mal structure that they b oth generate are the same. Since the conformal F e ﬀer man construction fo r ( P , ω ′ ) must b e normal (since ω ′ is), it must b e ( b P , ω c ). This means that Φ = 0 , hence that ω is nor mal.  6.1 Geometric equiv alence Given a free 3-distribution on manifold M , the co nfo r mal struc tur e can b e rec ov er ed directly fro m the decomp ositio n of T ∼ = T − 2 ⊕ H given by any W eyl str ucture. Let σ b e any lo ca l never-zero sec tion of ∧ 3 H . Then there is a map g : T − 2 ⊗ H → ∧ 3 H given by the isomor phism T − 2 ∼ = ∧ 2 H . E x tend g to a sectio n of ( ⊙ 2 T ∗ ) ⊗ ∧ 3 H by the inclusion T ∗ 2 ⊗ H ∗ ⊂ ⊙ 2 T ∗ . Then g σ − 1 is a metric o n M . This depe nds on the choice o f the section σ , so a ctually deﬁnes a conformal structure. It is then easy to see that g is inv aria nt under the action of a o ne-form Υ , (as g ( U + Y , X ) = g ( U, X ) + g ( X , Y ) = g ( U, X ) 18 CONTENTS 6.2 G ′ 2 structures for an y sections X a nd Y of H and any section U of T − 2 ). So this conformal structure do e s not depe nd on the choice of W eyl structur e , only on the ﬁltration of T (which de p ends on the Ca rtan connection). The a lgebra s pin (4 , 3) ⊂ so (4 , 4) is deﬁned b y preserving a generic four-form λ on V = R (4 , 4) , see [BK99]. Let T C be the standar d confor mal T racto r bundle o n M (see [ ˇ CG00] o r [Arm05] for more details on conformal geometries). If the conformal T ractor connection − → ∇ C has holonomy algebra reducing to spin (4 , 3), then there exists a generic four-form ν ∈ Γ( ∧ 4 T C ) such that − → ∇ C ν = 0 . There is a natural pro jection on T C , coming fro m its ﬁltration T C → E [1] ⊕ T [ − 1] → E [1 ] . Here E [1] is a densit y bundle, E [ α ] = ( ∧ 6 T ∗ ) α − 6 , and T [ − 1] = T ⊗ E [ − 1]. This implies that there is a well deﬁned pr o jection π : ∧ 4 T C → ( ∧ 3 T )[ − 2 ]. It tur ns o ut that π ( ν ) is decomp osable, and so deﬁnes a distribution H ∗ of ra nk three in T ∗ [2 / 3]. Since a distribution is unchanged b y a change of scale, this is actually a distribution in T ∗ , with dua l distribution H ⊂ T . This H is pr ecisely that deﬁning the Bry ant structure; the maximal non-integrability de r ives from the proper ties o f ν and − → ∇ C . 6.2 G ′ 2 structures The most natural subgroup of S pin (4 , 3) ⊂ S O (4 , 4) is G ′ 2 , deﬁned as the subgro up of S pi n (3 , 4) that preser ves a non-iso tropic element e in R (4 , 4) . There a re many equiv a le n t deﬁnitions. F or exa mples, G ′ 2 is equiv alently describ ed a s the subgro up of S O (4 , 3) that pres erves a generic three-for m θ on R (4 , 3) ; θ is just e x λ . Alterna tiv ely , it is the automorphism group of O ′ , the split Octonio ns. It acts irr educibly on the seven dimensional space V = I m O ′ . The split Octonio ns carr y a na tural inner pro duct N , gene r ated from the norm N ( x, x ) = x x . This quadratic for m is m ultiplicative, a nd is of signature (4 , 4). The iden tity elemen t 1 ∈ O ′ is of p ositive no r m squared, and is orthogo nal to V ; thus V is of signature (3 , 4). This algebr a is alternative; this mea ns that the alter nator [ x, y, z ] = ( xy ) z − x ( y z ) is tota lly an ti-symmetric in its three entries. W e ca n use N to make [ , , , ] in to an elemen t of ∧ 3 V ∗ ⊗ V ∗ ; it tur ns out to b e sk ew in all f our en tries, and equa l to ∗ θ where ∗ is the Hodg e s ta r generated by N . The three-for m θ its e lf is given by θ ( x, y , z ) = N ( xy , z ) . The prop er ties of the split O ctonions force this to b e skew in a ll three ar guments. Now ass ume that our free 3-distribution ha s a nor mal T ra ctor connection − → ∇ with a holo nomy reduction to G ′ 2 . By the conformal F eﬀerman constr uction, the conforma l s tr ucture will be given by a manifold that is confor mally Einstein and whose metric cone car ries a G ′ 2 structure (see [Ar m07]). Such ma nifo lds do ex ist – for insta nce, S L (3 , R ) /T 2 where T 2 is a maximal torus, is one ex ample [Bry87]. Here, the fr ee 3-distribution would b e chosen at I d ∈ S L (3 , R ) as the spa n of H I d =   0 a 0 0 0 b c 0 0   , 19 CONTENTS 6.2 G ′ 2 structures and extended to the whole manifold by Lie mu ltiplication. Note that { H I d , T 2 } ⊂ H I d , for T 2 the tangent space to the maximal torus , s o this extensio n is well deﬁned. It is not unique, howev er . W e could ha ve used the transpos e of H I d instead. Note that H t I d = { H I d , H I d } . This will b e a n impo rtant prop erty of G ′ 2 structures on a free 3-dis tr ibution. Prop ositio n 6.2. Ther e ar e thr e e orbi ts of isotr opic 3 -planes in R (4 , 3) under the a ction of G ′ 2 . – two op en, one close d. The close d orbit is distinguishe d by the fact that θ ( x, y , z ) = 0 for al l elements in an isotr opic 3 -plane inside this orbit. Pr o of. Let B ⊂ I m O ′ be an isotropic 3- pla ne. F or any element x of B , 0 = N ( x, x ) = x × x = − x × x . This implies that for any elements x and y of B , 0 = 2 N ( x, y ) = N ( x + y , x + y ) − N ( x, x ) − N ( y , y ) = ( x + y ) × ( x + y ) − x × x − y × y = − ( x + y ) × ( x + y ) = − x × y − y × x. Thu s a n isotropic 3 - plane is deﬁnes a s a subset of I m O ′ where every element squa r es to zero, and anti-comm ute. There a r e tw o situa tions to b e cov ered: 1. There ex ists a basis { x, y , z } for B such that λ ( x, y , z ) = 1. The set of all suc h B is evidently op en in the set of all isotropic 3-planes. W e aim to sho w G ′ 2 is transitive on this set. Lemma 6.3. The elements sp an of x , y and z under split Octonionic m u ltiplic ation gener ate al l of O ′ . Pr o of of L emma. Since the split O ctonions are alterna tiv e, th e m ultiplicative spa n of any tw o elements is a sso ciative. Hence ( xy )( xy ) = x ( xy ) y = − x ( xy ) y = − ( xx )( y y ) = 0 . This is true for any e le men ts x , y in B . Th us C = B × B is isotropic, so of max im um dimension three. The relation 1 = λ ( x, y , z ) = N ( xy , z ) , implies that xy is o rthogona l to x , y , but not to z . W e may cyclica lly p ermute x , y and z her e, thus demonstrating that C is of dimension thr ee. In fact N ( x ∧ y ∧ z , y z ∧ z x ∧ xy ) = 1, so λ ( y z , z x, xy ) = − 1 . Then deﬁne a = ( xy ) z − z ( xy ). Now a = z ( xy ) − ( xy ) z = z ( xy ) − ( xy ) z = − a , so a is purely imaginary . W e make the cla im that x, y , z , xy, z x, yz and a span I m O ′ and that the split Octonion m ultiplication of these elements is completely determined. First of all, the squares of x, y , z , x y , z x and yz are all zer o, as are a ll the terms x × xy , x × z x , y × y z , y × xy , z × z x and z × y z . Now cons ider b = ( xy ) × ( y z ). How N ( b, x ) = λ ( xy , y z , y ) = N (( y z ) y , xy ) = N (0 , xy ) = 0. Similar ly N ( b, z ) = N ( b, y ) = 0; th us b ∈ B ⊥ . Moreover N ( b , b ) = 20 CONTENTS 6.2 G ′ 2 structures N ( xy , xy ) N ( y z , y z ) = 0, so b ∈ B . T r ying to extract the x , y and z c o mpo nent s of b , w e ﬁnd N ( b, xy ) = 0, N ( b, z x ) = − 1 N ( b, y z ) = 0, so b = − y . Similar r easoning demonstrates tha t y z × z x = − z and z x × xy = − x. Similar manipulations, using the a s so ciator ∗ λ , show that xa = x = − a x, y a = y = − ay , z a = z = − z a a ( y z ) = xy = − ( y z ) a, a ( z x ) = z x = − ( z x ) a, a ( xy ) = xy = − ( xy ) a and a = ( y z ) x − x ( y z ) = ( z x ) y − y ( z x ) . And ﬁnally a × a = 1. It often helps to work with an explicit descr iption of s plit Octonio n m ultiplication. Here is one, due to Zorn. Here, a split Octonion is r epresented by the “matrix” x =  a v w b  with a and b rea l n umbers and v and w vectors in R 3 . The norm squa red N ( x, x ) is the “determinant” ab − v · w . Multiplication is given by  a v w b  ×  a ′ v ′ w ′ b ′  =  aa ′ + v · w ′ a v ′ + b ′ v + w ∧ w ′ a ′ w + aw ′ − v ∧ v ′ bb ′ + v ′ · w  . With · and ∧ the ordina ry dot a nd cross pro ducts on R 3 . The imag inary split Octonio ns are those where a = − b .  So if we call { x, y , z } an Octonionic triple , then an element g of G ′ 2 is entirely determined by { g ( x ) , g ( y ) , g ( z ) } . Con versely , fo r a n y tw o Octonionic triple, there is an element of G ′ 2 mapping one to the other. Moreov er, if G B ⊂ S L (7 , R ) is the stabiliser of B , G ′ 2 ∩ G B is the p ermutation g roup of the Octonionic triples in B – consequently G ′ 2 ∩ G B = S L (3 , R ). 2. F or all x, y , z ∈ B , λ ( x, y , z ) = 0. The set of all such B is closed in the set of all isotropic 3-planes, complemen tary to the previo us orbit, and with empty interior. W e aim to show G ′ 2 is transitive o n this set. As in the previous examples, ( xy )( xy ) = 0. And C = B × B is isotropic. Ho wev er 0 = λ ( x, y , z ) = N ( xy , z ) , implying that C ⊂ B ⊥ . Since C is iso tr opic, C ⊂ B . An insp ection of the explicit form of split Octonion multiplication demonstrates that there do es not e xist a three plane o n which × is totally degenerate. So C 6 = 0. Let z ∈ C . Now z = xy for elemen ts x and y in B . Since elemen ts of B square to zero, x 6 = y . Since the m ultiplicative span o f any tw o ele ments is assoc ia tive, z 6 = y and z 6 = . F urthermore, z cannot b e in the span of x a nd y , since x ( r 1 x + r 2 z ) = r 1 xx + r 2 x ( xy ) = 0 for all real r j . So x , y and z for m a basis for B , and the relations xy = z , xz = 0 , y z = 0 , xx = y y = z z = 0 , 21 CONTENTS 6.2 G ′ 2 structures determine multiplication o n B completely . In fact, B is deter mined by z . This can b e seen fro m the fact that G ′ 2 is transitive o n the set of iso tropic element o f I m O ′ , so we may set z =  0 e 1 0 0  , where e 1 , e 2 , e 3 is a basis for R 3 . Then the tw o sided kernel of the m ultiplications × z , z × : I m O ′ → O ′ is spanned by z ,  0 0 e 2 0  ,  0 0 e 3 0  . Since B is in the t wo-sided kernel of multiplication by z , and is isotropic, it m ust b e pre cisely the span o f these elements. Since B is determined by z , and s ince G ′ 2 is tra nsitive on isotro pic elements of I m O ′ , G ′ 2 m ust b e transitive on the set of iso tropic 3 -planes B on which λ v anishes.  Theorem 6.4. L et M b e a fr e e 3 -distribution manifold with normal T ra ctor c onne ction − → ∇ , with the holonomy gr oup of − → ∇ r e ducing to G ′ 2 . Then, on an op en, dense set of M , t her e is a unique Wey l structur e ∇ deﬁne d by this information. This Weyl structur e determines a splitting of T = T − 2 ⊕ H . Then H ′ = T − 2 is a fr e e 3 -distribution. And the normal T r actor c onne ction determine d by H ′ is isomorphi c to − → ∇ . Pr o of. If − → ∇ ha s holonomy contained in G ′ 2 , then it comes fro m a connection a n a pr inciple G ′ 2 -bundle G ′ 2 . Let A ′ = G ′ 2 × G ′ 2 g ′ 2 . The inclusion G ′ 2 ⊂ S O (3 , 4) gener ates inclus ions G ′ 2 ⊂ G and g ′ 2 ⊂ g , thus an inclusion A ′ ⊂ A . And by deﬁnition − → ∇ preserves A ′ , and L , a thr ee-form on T . Since − → ∇ has holonomy contained in G 2 , it a lso preserves split Oc tonionic multiplication on T . Designate this mu ltiplication b y × . By the pr evious prop os ition, on an op en, dense s ubs e t of M , the canonical H ∗ ⊂ T genera tes all of T b y × . W e hav e a well deﬁned subundle of T , K = H ∗ × H ∗ . Since K and H ∗ are transverse, the pro jectio n π 2 maps K isomor phica lly to H . Then let ∇ b e the (unique) strong ly K -prefer red connection. In the splitting it deﬁnes , set H ′ = T − 2 . Now consider A ′ 0 ⊂ A ′ , the subundle o f A ′ that stabilis es H ∗ (and K ). This m ust b e a sl (3 , R ) bundle, since the subgroup o f G ′ 2 that preserves a generic isotropic 3 -plane is S L (3 , R ). By the wa y we hav e chosen our curr ent splitting, A ′ 0 ⊂ A 0 . Consequently ∇ prese r ves a volume for m, and thus H ′ ∼ = H ∧ H ∼ = H ∗ . Th us under the action of A ′ 0 , A = H ′ ⊕ H ⊕ A ′ 0 ⊕ R ⊕ H ′ ⊕ H . Since A ′ is of rank 1 4, since G ′ 2 is fourteen dimensiona l, there are three po ssibilities for the structure of A ′ A ′ = A ′ 0 ⊕ H ⊕ H ′ A ′ = A ′ 0 ⊕ H ⊕ H A ′ = A ′ 0 ⊕ H ′ ⊕ H ′ . But the last t w o po ssibilities are not algebraica lly closed, so A ′ m ust b e of the ﬁrst t yp e. It is also simple, whic h means that it cannot b e of pur e p ositive or nega tive homogeneity . A bit of exp erimentation then s hows that the o nly pos sibilit y for A ′ is that it is comp osed of elements of the form ( X ′ , X, Θ , X ′ , X ) , 22 CONTENTS 6.2 G ′ 2 structures for X ∈ Γ( H ), X ′ ∈ Γ( H ′ ), Θ ∈ Γ( A ′ 0 ) (the sa me could instead b e deduced fro m the split Octonionic m ultiplication in this manifold). Since − → ∇ m us t pres erve this bundle, and that ∇ a lready do e s, P 12 m ust b e the ident ity on H , P 21 the iden tity on H ′ and P 11 = P 22 = 0. Notice that ∇ P = 0 – this is similar to, though not identical to, an Einstein inv olution [Arm07]. This implies that the Cartan co nnection ω decomp os e s as ω = ω − 2 + ω − 1 + ω 0 + ω 1 + ω 2 , with ω − 2 = ω 1 and ω − 1 = ω 2 . Since − → ∇ is torsion fr e e , κ ( T ∧ T ) must take v alues in A (0) ∩ A ′ = A ′ 0 . The har monic curv a ture comp onent of κ (see Section 2 .3) is in H ⊗ H ′ ⊗ A ′ 0 . The only other p ossible curv ature co mpo nent of κ is the hig her homogeneity H ′ ∧ H ′ ⊗ A ′ 0 . Since P 22 = 0, this is precisely the R ∇ 22 , wher e R ∇ is the curv a ture of ∇ . Lemma 6.5. κ 22 = R ∇ 22 = 0 . Pr o of of L emm a. Designate κ 22 by κ ′ . The Bia nc hi iden tit y for − → ∇ is d − → ∇ κ = 0 , where d − → ∇ is − → ∇ on A t wisted with the exterior deriv ative d on ∧ 2 T ∗ . F or X ′ and Y ′ sections of H ′ and with Z a section of H , 0 = ( d − → ∇ κ ) X ′ ,Y ′ ,Z = ( d ∇ R ∇ ) X ′ ,Y ′ ,Z + { κ, −} X ′ ,Y ′ ,Z + { κ, P ( − ) } X ′ ,Y ′ ,Z = { κ 12 , −} X ′ ,Y ′ ,Z + { κ 12 , P ( − ) } X ′ ,Y ′ ,Z + κ ′ X ′ ,Y ′ · Z + κ ′ X ′ ,Y ′ · P ( Z ) Now the expressio n κ ′ X ′ ,Y ′ · P ( Z ) is the o nly comp onent taking v alues in A 2 , so it must v a nish. This implies that κ ′ = 0.  Now we hav e κ as a section of H ⊗ H ′ ⊗ A ′ (0) . In particular κ ( H ∧ H ) = 0. Reca ll the deﬁnition o f normality; that ∂ ∗ κ = 0, where ( ∂ ∗ κ )( X ) = X l { Z l , κ ( Z l ,X ) } − 1 2 κ ( { Z l ,X } − ,Z l ) , for ( Z l ) a frame for T and ( Z l ) a dual frame for T ∗ . Now { Z l , X } − ∧ Z l is zero or a sec tio n o f H ∧ H for all X a nd Z l . Th us the nor mality of κ is ent irely enca psulated in X l { Z l , κ ( Z l ,X ) } , or, in o ther words, in the fact that κ is trace free. Now co nsider − → ∇ as a principal connection on G , forgetting ab out the inclusion P ⊂ G . W e ma y deﬁne an alternative inclusion P ′ ⊂ G by using K as the ca nonical subundle o f T . By our previo us results, the new solder ing form is now ω − 1 + ω − 2 rather tha n ω − 2 + ω − 1 – a nd this is a prop er soldering for m, mea ning that − → ∇ is a T ractor connection for the distribution H ′ . The curv ature of − → ∇ is still κ , though the new soldering form sends H ⊗ H ′ to H ′ ⊗ H . Under this new identiﬁcation, κ th us rema ins a tra c e-free se c tion o f H ′ ⊗ H ⊗ A ′ (0) . Th us if ∂ ∗ ′ is the op erato r for the new para bo lic, ∂ ∗ ′ κ = T race κ = 0 . Thu s − → ∇ is normal as the T ractor connectio n g enerated by H ′ .  23 CONTENTS 6.3 CR structures 6.3 CR structures W e aim to show here that ther e is a F eﬀerman constructio n for b G = S O (4 , 3), b P stabilis es an isotropic 3-plane, and G = S O (4 , 2) while P = ( S O (2) ⊕ GL (2)) ⋊ ( R 2 ⊗ R (2) ) ⋊ ∧ 2 R 2 stabilises an isotropic 2-plane. Let V = R (4 , 3) and B ⊂ W be an is otropic 3-plane whose inclusion deﬁnes b P ⊂ b G . Let W ∼ = R (4 , 2) , and ﬁx an inclusio n W ⊂ V that deﬁnes G ⊂ b G . Because of their sig natures, W and B must be transv erse, so their in ter section C = W ∩ B is an isotropic 2-plane. Deﬁning P a s the stabiliser of C , it is e v iden t that G ∩ b P ⊂ P . Now let B ′ be the orthog onal pro jection of B onto W . By cons tr uction, C ⊂ B ′ ⊂ C ⊥ . The bundle B ′ is equiv a lent ly deﬁned by a line through the origin in C ⊥ /C . The group P acts via S O (2) on this spa ce of lines. Thus G ∩ b P lie s as a co dimensio n one subg r oup in P . Then b G is of dimension 21, b P of dimens ion 15, G also of dimension 15, P o f dimensio n 10 and G ∩ b P of dimens ion 9. This implies that G and b P are transverse in b G , hence that the inclusio n g / ( g ∩ b p → b g / b p is open. Thus we may do the F eﬀerman c o nstruction. Deﬁnition 6.6 (CR) . A CR manifold is g iven by a contact distribution K ⊂ T N w ith a complex structure J on K . If Q = T N /K , and q : T N → Q is the obvious pro jection, there is a sk ew symmetric map L : K × K → Q given by L ( X , Y ) = q ([ X, Y ]) where X and Y are s ections of K . Int egra bilit y comes fro m using J to split K ⊗ C as K 1 , 0 ⊕ K 0 , 1 ; the CR structure is integrable if K 0 , 1 is closed under the Lie bracket. This implies that L is o f type (1 , 1), that is tha t L ( J X , J Y ) = L ( X , Y ). Theorem 6.7. The ge ometries mo del le d on G/P ar e the 5 dimensional spli t signatur e CR ge ome- tries. If the CR structur e is int e gr able and the Cartan c onn e ction is n ormal, the Cartan c onne ction on the fr e e 3 -distribut ion c oming fr om the F eﬀerman c onst ruction is also normal. Conversely, if the hol onomy gr oup of a normal Ca rtan c onn e ction for a fr e e 3 -distribution r e duc es to S O (4 , 2) , it is the F eﬀerman c onstruction over an int e gr able spli t signatur e CR ge ometry with normal Cartan c onne ction. The rest o f this section will b e devoted to proving that theo r em. The ﬁrst statement – that the G/ P g eometries are the CR geo metr ies – from the fact that the representation of P as a par ab olic is × ◦ × , the sa me as for CR structures , co mbin ed with the following lemma: Lemma 6.8. S pin (4 , 2) 0 = S U (2 , 2) . Pr o of of L emma. Cons ide r the action of S U (2 , 2) on V = C (2 , 2) ∧ C (2 , 2) . V carrie s a rea l structure on it f rom the action of the K¨ ahler form, and a natur al (4 , 2) signa ture metr ic. Since S U (2 , 2) is simple, and its action is no n-trivial on this spac e, there is an inclusio n su (2 , 2) ֒ → so (4 , 2) . And then dimensiona l c o nsiderations imply that this is an equality . The max imal co mpact s ubgroup o f S O (4 , 2) 0 is S ( O (4) × O (2)) 0 ; the ma ximal compact subgroup of S U (2 , 2) is S ( U (2 ) × U (2)). This means that the fundamental gro ups of the Lie gr o ups are: π 1 ( S O (4 , 2) 0 ) = Z 2 ⊕ Z π 1 ( S U (2 , 2)) = Z . 24 CONTENTS 6.3 CR structures Consequently S pin (4 , 2) 0 = S U (2 , 2).  Then it is easy to s ee that P is the stabiliser of a complex nul-line in C (2 , 2) , demons tr ating that these ar e split signature CR geo metries (see [ ˇ Cap02]). In o rder to demonstrate the no rmality conditions , we s hall use b oth this F eﬀerman co nstruction a nd the confor mal F eﬀerman construction. Let b b G = S O (4 , 4), with b b P the stabiliser o f a n ul line. In details, if ( P , ω ) is a split sig nature CR geometry , we hav e thr ee related structures : ( P , ω ) ( b P , b ω ) ( b b P , b b ω ) , where b ω is the T ractor connection for a fr ee 3-distribution while b b ω is a co nfo r mal T ractor connection. W e know that b ω is normal if a nd only if b b ω is normal. But now consider the total T r actor connection, from G to b b G . This is determined b y how G lies in the la rger gro up. If w e complexify everything, w e ha ve S pin (6) ⊂ S pin (7 ) ⊂ S O (8 , C ). The spin represent ations of S pin (6) are isomor phic with C 4 , so deco mpo s e C 8 int o tw o distinct co mpo nent s. This implies that the action of S pin (4 , 2) ⊂ S pin (4 , 3) ⊂ S O (4 , 4) on R (4 , 4) either decomp oses it int o t wo four dimensional comp onents, or is irreducible on it (a nd preser ves a complex structure o n it). How ever, S U (2 , 2) = S pin (4 , 2) do es not have any four dimensio nal re a l representations (apart from the trivia l one). Consequently the inclusion S U (2 , 2) ⊂ S O (4 , 4) is the standard inclusion. This means that the inclusion G ⊂ b b G is the standard one. This g enerates b b ω via the F eﬀerman construction. But this F eﬀerman construction has to be the standa r d o ne. This implies that b b ω is no rmal if and only if ω is normal and the CR structure is integrable (see [ ˇ Cap02], [Lei06b] a nd [Lei06a]). Consequently , b ω is nor mal if and o nly if ω is normal and the CR structure is int egra ble. Remark. The inclusion S U (2 , 2) ⊂ S pin (4 , 3) ca n b e seen directly . S pin (4 , 3) is deﬁned as preser v - ing a generic four-fo rm λ on R (4 , 4) (see [BK99]). S U (2 , 2) o n the other ha nd, pr eserves a K ¨ ahler form µ , which ca n b e seen a s a section of ∧ 2 V that is co njugate linear with resp ect to the volume form. It also pr e s erve a co mplex volume fo rm v ∈ Γ( ∧ (4 , 0) V C ). The inclusion o f S U (2 , 2) into S pi n (3 , 4) is given by the generic four for m: Re ( v ) − ( µ ) 2 . (3) Now w e need to show the conv erse. Let ( M , b P , − → ∇ ) b e a normal Car tan co nnection for a free 3- distribution. Assume the holonom y group o f − → ∇ reduces to S O (4 , 2) – equiv a lent ly , that there is a section τ of T , of negative no rm squared, such that − → ∇ τ = 0. Deﬁne R ∈ Γ( H ) as π 2 ( τ ); since τ is of nega tive nor m squared, R is never-zero. Deﬁne the bundle e K as H ⊕ [ H , R ]. It is a bundle of ra nk ﬁve. Let N b e the manifold got fro m M by pro jecting along the ﬂow of R . Prop ositio n 6.9. N c arries a CR st ructur e, and the c ontact distribution K in T N is the pr oje ction of e K t o N . The c omplex struct ur e J on K is given by the action of R . Pr o of. W e ﬁrst need to s how that [ e K , R ] = e K . Let L b e the line subundle of T generated by τ . Pick any L -preferre d connection ∇ . This ob eys the pro pe r ties of Theorem 3.2 – τ = ( α, 0 , R ), α ( R ) = 1 and ∇ X α = ∇ X R = 0 for a n y section X of H while ∇ U R = −{ U, α } for U a section of T − 2 . W e may c ho ose sections X and Y o f H that ob ey the following prop erties: 1. X , Y and R for m a frame of H , 25 CONTENTS 6.4 Lag rangia n contact structures 2. ∇ R X = ∇ R Y = 0, 3. α ( X ) = α ( Y ) = 0, (for instance , we co uld deﬁne X and Y ob eying the algebr aic prop erties o n a submanifold tr ansverse to R , and extend b y parallel transp ort along R ; then the r elation ∇ R R = ∇ R α = 0 ensur es the algebraic prop er ties ar e preserved). Since − → ∇ is tor sion-free, [ R, X ] = ∇ R X − ∇ X R − { R, X } = { R, X } . Similarly , [ R, [ R, X ]] = ∇ R { R, X } − ∇ { R,X } R − 0 = {{ R , X } , α } = − X . The same hold for Y and { R , Y } . Cons equently [ R , e K ] = e K and e K pro jects to a distr ibution K in N = M /R . This distribution must be a con tact distr ibution, b y the pr op erties o f t he Lie bracket on e K . Let r be any co ordinate o n M such that R · r = 1. Then the vector ﬁelds cos( r ) X − sin( r ) { R, X } , sin( r ) X − c os( r ) { R, X } (4) cos( r ) Y − sin( r ) { R, Y } , sin( r ) Y − cos( r ) { R, Y } (5) are R -inv ariant, hence lifts of vector ﬁelds in K . Since we hav e this explicit form, we ca n see that the Lie brack et o f X and Y with R genera tes an endomorphism of e K that descends to a n automorphism J of K , sq uaring to minus the identit y . Changing to another L -prefer red connection will c hange X and Y by adding m ultiples of R . This will change neither their pr o jections nor the properties of J . This mea ns that the CR structure is well deﬁned. And M must b e the F eﬀerman construction over this CR structure. This implies that CR structure m ust b e in tegra ble and that − → ∇ descends to a nor mal CR T ractor connectio n on N .  One interesting consider ation: though the fr e e 3-dis tribution deﬁnes the CR structure uniquely , the conv erse is o nly true up to isomorphism. A ny diﬀeomo rphism φ : M → M generated by a ﬂow on R will change the distribution H , but since φ pr o jects to the iden tity on N , it leaves the underly ing CR structure inv ariant. The for going means that all the results on CR holonomy (equiv alently , co nformal holonomy contained in su (2 , 2)) hav e equiv a lent formulations in terms of free 3-distributions. See pap ers [Lei0 6 b], [ ˇ Cap02] and [Lei06a]; pa pe r [Arm05] has some Einstein examples as w ell. This implies, for insta nc e , that holonomy reductions to S U (2 , 1) exis t (whenever N is a Sa saki-Einstein manifold with the correct signature and sign o f the E instein c o eﬃcien t). F r o m the free 3-distr ibution p oint of v ie w , this corres p o nds to a co mplex structure o n the complement of τ in T . Similar considera tion e xist fo r a holono m y reduction to S p (1 , 1) ⊂ S U (2 , 2), with the qua ternionic analogue of CR spa ces. 6.4 Lagrangian contact structures Lagra ng ian contact structures (see for exa mple [ ˇ Cap05]) geometries generated by another real for m of the pa rab olic that mo dels CR s tr uctures. 26 REFERENCES REFERENCES Deﬁnition 6.10. A Lag rangian contact structure is given by a co n tact distribution K on a manifold of dimension 2 m + 1 , toge ther with t wo bundles E and F of ra nk m such that K = E ⊕ F , [ E , E ] ⊂ K and [ F, F ] ⊂ K . The str ucture is integrable if b oth E and F are integrable. The para bo lics are g iven b y G = S L ( m, R ) while P = ( R ⊕ GL ( m − 2 , R ) ⋊ ( R m ⊕ R m ∗ ) ⋊ R . Then we have: Lemma 6.11. S pin (3 , 3) 0 = S L (4 , R ) Pr o of of L emma. Consider the action of S L (4 , R ) o n V = ∧ 2 R 4 . Since S L (4 , R ) preserves a volume form which is a n element of ∧ 4 R 4 ∗ ⊂ ⊙ 2 ( ∧ 2 R 4 ) ∗ , it pr e serves a metric on V , of split sig nature. Then since S L (4 , R ) is simple and acts non-tr ivially , we get an algebra inclusion sl (4 , R ) ⊂ so (3 , 3) and the dimensions imply equality . The maximum compact subgro up of S L (4 , R ) is S O (4) while the maximum compa ct subgroup of S O (3 , 3) 0 is S ( O (3) × O (3)) 0 . Consequently π 1 ( S L (4 , R )) = Z 2 , π 1 ( S O (3 , 3) 0 ) = Z 2 × Z 2 , demonstrating the result.  Given this, the results for CR structures go thr ough almo st verbatim to this new setting. There is one subtlet y , howev er: R (3 , 3) need not be transverse to a given isotropic 3-plane in R (4 , 3) . So w e may often need to restr ict our results to open dense subsets of our manifolds. Also the inclusion GL (4 , R ) ⊂ S pin (4 , 3) ⊂ S O (4 , 4) is now the sta ndard inclusio n tha t de c ompo ses R (4 , 4) as R 4 ⊕ R 4 ∗ . Summarising all the results: Theorem 6. 12. L et N b e a ﬁve dimensional inte gr able L agr angian c ontact manifold. Then ther e is a F eﬀerman c onstruction for N to a f r e e 3 -distribution o n a manifold M . The T r actor c onne ction on M is norma l i f and only if t he T r actor c onne ction on N is normal. Conversely, if M is a f r e e 3 -distribution ge ometry with normal T r actor c onne ction − → ∇ , and the holonomy gr oup of − → ∇ r e duc es to S O (3 , 3) , then an op en dense set of M is t he F eﬀerman sp ac e of a inte gr able, normal L agr angian c ontact manifold. References [Arm05] Stuart Armstrong. Deﬁnite signature conformal holonomy: a complete classiﬁca tio n. arXiv , 2005. [Arm06] Stuart Armstro ng . T r actor holonomy classiﬁcatio n for pro jective and conforma l structur es. Do ctor al Thesis, Bo delian Libr ary, Oxfor d Unive rsity , 2 0 06. [Arm07] Stuart Armstrong. Generalis ed einstein co ndition and c o ne construction for pa rab olic geometries. arXiv , 200 7. [BK99] Helga Baum and Ines Kath. Parallel s pino rs a nd holonomy groups on pseudo - riemannian spin manifolds. Ann. Glob al Anal. Ge om. , 17 (1):1–17, 19 99. [Bry87] Ro be r t Bry ant. Metr ics with exceptional holono m y . A nn. of Math. (2) , 126(3):525 – 576, 1987. 27 REFERENCES REFERENCES [Bry05] Ro be r t Bryant. Conformal geometry and 3 -plane ﬁelds on 6-manifolds. arXiv , 2005 . [ ˇ Cap02] Andr e as ˇ Cap. Parab olic geometries, cr-tr actors, and the feﬀerman constr uction. Diﬀer en- tial Ge om. A ppl. , 17(2-3 ):123–1 38, 2 002. [ ˇ Cap05] Andr e as ˇ Cap. Inﬁtesimal a utomorphisms and defor mations o f pa rab olic geometry . Vienna, Pr eprint ESI , 1684 , 200 5. [ ˇ Cap06] Andr e as ˇ Cap. Two constructions with parab olic geometr ies. R end. Cir c. Mat. Palermo (2) , 79:1 1–37, 2 006. [CD01] David Calderbank and T ammo Diemer. Diﬀerential in v ariants and curv ed ber nstein- gelfand-gelfand seq uences. J. R eine Angew. Math. , 5 37:67 – 103, 2001. [CDS05] David Calderbank, T ammo Diemer , a nd Vladimir Souˇ cek. Ricci-corrected deriv atives and inv aria n t diﬀerential op era tors. Diﬀer ential Ge om. App l. , 2 3(2):149– 175, 2005. [ ˇ CG00] Andr e a s ˇ Cap and Ro d Gov er. T r a ctor bundles for irr educible parab olic ge o metries. S .M.F. Col lo ques, Seminair es & C ongr es , 4:129 –154, 200 0. [ ˇ CG02] Andr e a s ˇ Cap and Ro d Gover. T rac tor calculi for par a bo lic geometrie s . T r ans. Amer. Math. So c. , 35 4(4):151 1–154 8, 2002 . [ ˇ CS00] Andreas ˇ Cap a nd Herma nn Schic hl. Parab olic geo metries and canonica l cartan connectio ns. Hokkaido Math. J. , 2 9(3):453– 505, 2000. [ ˇ CSS97] Andreas ˇ Cap, Jan Slov´ ak, and Vladimir So uˇ cek. Inv ar iant o pe rators on manifolds with almost hermitian symmetric structures. ii. normal cartan connectio ns. A cta Math. Univ. Comenian. (N.S.) , 66 (2):203–2 20, 1997 . [ ˇ CSed] Andreas ˇ Cap and Jan Slo v ´ ak. Par ab olic Ge ometries I: Backgr ound & Gener al The ories . T o b e published. [Kos61 ] Bertr am Kos tant . Lie a lgebra cohomology and th e generalized b orel-weil theorem. A nn. of Math. (2) , 7 4:329– 387, 1961. [Lei05] F elipe Leitner . Conformal killing forms with nor malisation condition. R end. Cir c. Mat. Palermo (2) , 75:2 79–2 9 2, 2005. [Lei06a] F elipe Le itner . Abo ut complex structures in conforma l tractor calculus. arXiv , 2 0 06. [Lei06b] F elip e Leitner. A remar k o n c o nformal su ( p, q )-holo nomy . arXiv , 200 6. 28

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