Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 081, 33 pages A Projective-to-Conformal Fefferman-Type Construction Matthias HAMMERL t1, Katja SAGERSCHNIG ^, Josef ŠILHÁN ^, Arman TAGHAVI-CHABERT114 and Vojtěch ŽÁDNÍK ? t University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1010 Vienna, Austria E-mail: matthias.hammerl@univie.ac.at t INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: katja.sagerschnig@univie.ac.at ^3 Masaryk University, Faculty of Science, Kotlářská 2, 61137 Brno, Czech Republic E-mail: silhan@math.muni.cz ' Universita di Torino, Dipartimento di Matematica "G. Peano", Via Carlo Alberto 10, 10123 Torino, Italy E-mail: ataghavi@unito.it -f-5 ' Masaryk University, Faculty of Education, Poříčí 31, 60300 Brno, Czech Republic E-mail: zadnik@mail.muni.cz Received February 09, 2017, in final form October 09, 2017; Published online October 21, 2017 https://doi.org/10.3842/SIGMA.2017.081 Abstract. We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into Spin(n + l,n + 1). The construction associates a split-signature (n, n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry. Key words: parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors 2010 Mathematics Subject Classification: 53A20; 53A30; 53B30; 53C07 1 Introduction In conformal geometry the geometric structure is given by an equivalence class of pseudo-Riemannian metrics: two metrics g and g are considered to be equivalent if they differ by a positive smooth rescaling, g = e2Jg. In projective geometry the geometric structure is given by an equivalence class of torsion-free affine connections: two connections D and D are considered as equivalent if they share the same geodesies (as unparametrised curves). While conformal and projective structures both determine a corresponding class of affine connections, neither of them induces a single distinguished connection on the tangent bundle. Instead, both structures have canonically associated Cartan connections that govern the respective geometries and encode 2 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik prolonged geometric data of the respective structures. It is therefore often useful when studying projective and conformal structures to work in the framework of Cartan geometries. The present paper investigates a geometric construction that produces a conformal class of split-signature metrics on a 2 n-dimensional manifold arising naturally from a projective class of connections on an n-dimensional manifold. Split-signature conformal structures of this type have appeared in several places in the literature before. The projective-to-conformal construction studied in this paper should be understood as a generalisation of the classical Riemann extensions of affine spaces by E.M. Patterson and A.G. Walker [26]. One of the main authors motivations for the present study was the article [15] by M. Dunajski and P. Tod, where the Patterson-Walker construction was generalised to a projectively invariant setting in dimension n = 2. On the other hand, in [25] conformal structures of signature (2, 2) were constructed using Cartan connections that contain the conformal structures arising from 2-dimensional projective structures as a special case. A generalisation of this Cartan-geometric approach to higher dimensions can be found in [24]. In this paper the construction is studied as an instance of a Fefferman-type construction, as formalised in [6, 11], based on an inclusion of the respective Cartan structure groups SL(n + l) ^ Spin(n + l,n + 1). We show that in the general situation n > 3 the induced conformal Cartan geometry is non-normal. To obtain information on the conformal structure it is thus important to understand how the normal conformal Cartan connection differs from the induced one, and the main part of the paper concerns the study of this modification. We may summarise the main contributions of the paper as follows: • A comprehensive treatment of the projective-to-conformal Fefferman-type construction including a discussion of the intermediate Lagrangean contact structure (Section 3) and a comparison with Patterson-Walker metrics (Section 6.1). • A thorough study of the normalisation process (Section 4) and an explicit formula for the modification needed to obtain the normal conformal Cartan connection (Section 5.2). • The characterisation of the conformal structures obtained via our Fefferman-type construction (culminating in Theorem 4.14). Let us comment upon the characterisation in more detail. This is formulated in terms of a conformal Killing field k and a twistor spinor x on the conformal space together with a (con-formally invariant) integrability curvature condition. In Theorem 4.14 the properties of k and x are specified in terms of corresponding conformal tractors, which nicely reflects the algebraic setup of the Fefferman-type construction in geometric terms. An alternative equivalent characterisation theorem was obtained by the authors in [20, Theorem 1] by different means, namely, by direct computations based on spin calculus in the spirit of [28, 29]. The conformal properties are given purely in underlying terms and do not refer to tractors. In Section 6.2 (Theorem 6.3) we indicate how this alternative characterisation can be obtained in the current framework. We remark that, to our knowledge, the present work is the first comprehensive treatment of a non-normal Fefferman-type construction and we expect that the techniques developed should have considerable scope for applications to other similar constructions. A particularly interesting case of this sort is the Fefferman construction for (non-integrable) almost CR-structures. Possible further applications concern relations between solutions of so-called BGG-equations and special properties of the induced conformal structures. Several such relationships were already obtained by the authors in [20]. For instance, we can give a full description of Einstein metrics contained in the resulting conformal class in terms of the initial projective structure. Moreover, in [21] we were able to show that the obstruction tensor of the induced conformal structure vanishes. A Projective-to-Conformal Fefferman-Type Construction 3 2 Projective and conformal parabolic geometries The standard reference for the background material on Cartan and parabolic geometries presented here is [11]. 2.1 Cartan and parabolic geometries Let G be a Lie group with Lie algebra g and P C G a closed subgroup with Lie algebra p. A Cartan geometry (Q,uj) of type (G,P) over a smooth manifold M consists of a P-principal bundle §->M together with a Cartan connection to £ J71(^,g). The canonical principal bundle G —> G/P endowed with the Maurer-Cartan form constitutes the homogeneous model for Cartan geometries of type (G,P). The curvature of a Cartan connection lj is the 2-form Ken\g,Q), K(£,ri):=dw&v) + MZ)Mv)], for alU,7?e£(£), which is equivalently encoded in the P-equivariant curvature function k: £ ^ A2(g/p)*®g, K(u)(X+p,Y + p):=K(u-1(u)(X),Lj-\u)(Y)). (2.1) The curvature is a complete obstruction to a local equivalence with the homogeneous model. If the image of k is contained in A2(g/p)* © p the Cartan geometry is called torsion-free. A parabolic geometry is a Cartan geometry of type (G, P), where G is a semi-simple Lie group and P C G is a parabolic subgroup. A subalgebra p C g is parabolic if and only if its maximal nilpotent ideal, called nilradical p+, coincides with the orthogonal complement p1- of p C g with respect to the Killing form. In particular, this yields an isomorphism (g/p)* = p+ of P-modules. The quotient go = P/P+ 1S called the Levi factor; it is reductive and decomposes into a semi-simple part Qff = [go, go] and the center 3(go)- The respective Lie groups are Gff C Go C P and P+ C P so that P = Go x P+ and P+ = exp(p+). An identification of go with a subalgebra in p yields a grading g = g_fc © ■ ■ ■ © g_i © g0 © fli © ■ ■ ■ © Qk, where p+ = gi © ■ ■ ■ © gfc. We set g_ = g_fc©- ■ -©g_i. If k is the depth of the grading the parabolic geometry is called \k\-graded. The grading of g induces a grading on A2p+®g = A2(g/p)*®g. A parabolic geometry is called regular if the curvature function k takes values only in the components of positive homogeneity. In particular, any torsion-free or |l|-graded parabolic geometry is regular. Given a g-module V, there is a natural p-equivariant map, the Kostant co-differential, d*: Afe(g/p)*®V^ Ak-\Q/p)*®V, (2.2) defining the Lie algebra homology of p+ with values in V; see, e.g., [11, Section 3.3.1] for the explicit form. For V = g, this gives rise to a natural normalisation condition: parabolic geometries satisfying d*(n) = 0 are called normal. The harmonic curvature kh of a normal parabolic geometry is the image of k under the projection ker<9* —> ker<9*/ im<9*. For regular and normal parabolic geometries, the entire curvature k is completely determined just by kjj. A Weyl structure j: Qq^-Q of a parabolic geometry (Q,uj) over M is a reduction of the P-principal bundle Q —> M to the Levi subgroup Go C P. The class of all Weyl structures, which are parametrised by one-forms on M, includes a particularly important subclass of exact Weyl structures, which are parametrised by functions on M: For |l|-graded parabolic geometries, these correspond to further reductions of Qq —> m just to the semi-simple part Gff of Go or, equivalently, to sections of the principal M+-bundle Qq/Gqs —> m. The latter bundle is called the bundle of scales and its sections are the scales. For a Weyl structure j: Qq ^ Q, the pullback j*uj = j*tu- + j*ujq + of the Cartan connection may be decomposed according to g = g_ © go © p+. The go-part j*u>o is a principal connection on the Go-bundle Qq —> M; it induces connections on all associated bundles, which are called (exact) Weyl connections. The p+-part j*uj+ is the so-called Schouten tensor. 4 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik 2.2 Tractor bundles and BGG operators Every Cartan connection lj on Q —> M naturally extends to a principal connection Cj on the G-principal bundle Q := Q x p G —?> M, which further induces a linear connection Vv on any associated vector bundle V := Q XpV = Q V for a G-representation V. Bundles and connections arising in this way are called tractor bundles and tractor connections. The tractor connections induced by normal Cartan connections are called normal tractor connections. In particular, for the adjoint representation we obtain the adjoint tractor bundle AM := Q Xp g. The canonical projection g —> g/p and the identification TM = Q Xp (g/p) yield a bundle projection II: AM —> TM; the inclusion p+ C g and the identification p+ = (g/p)* yield a bundle inclusion T*M ^ AM. This allows us to interpret the Cartan curvature k from (2.1) cis db 2-form on M with values in AM. The holonomy group of the principal connection Cj is by definition the holonomy of the Cartan connection lj, i.e., Hol(w) := Hol(ci)) C G. By the holonomy of a geometric structure we mean the holonomy of the corresponding normal Cartan connection. In [12], and later in a simplified manner in [4], it was shown that for a tractor bundle V = Q Xp V one can associate a sequence of differential operators, which are intrinsic to the given parabolic geometry (Q,oj), r(?*o) 4 r(?*i) 4 ■■■ 41 v(nn). The operators are the BGG-operators and they operate between the sections of subquotients Uk = ker a*/im d* of the bundles of V-valued /c-forms, where d* : AkT*M V ->■ Ak~xT*M <8> V denotes the bundle map induced by the Kostant co-differential (2.2). The first BGG-operator 0q : r(%o) ~~> r(?^i) is constructed as follows. The bundle T-Lq is simply the quotient V/V, where V' C V is the subbundle corresponding to the largest P-invariant filtration component in the G-representation V. It turns out, there is a distinguished differential operator that splits the projection Bio : V —> ~Ho, namely, the splitting operator, which is the unique map Lq : r(%o) -> B(V) satisfying IIo(LoV)) = CT> d*(dvVL%(a)) = 0, for all a £ r(W0)- The latter condition allows to define the first BGG-operator by Oq := ni o dvV o Lq , where ni: ker d* —> T(Hi). The first BGG-operator defines an overdetermined system of differential equations on a £ T^Ho), Oq (K(w), ^[ab][w] '■= E[a6] <8> E[w], and the like. 2.4 Projective structures Let M be a smooth manifold of dimension n > 2. A projective structure on M is given by a class, p, of torsion-free projectively equivalent affine connections: two connections D and D A Projective-to-Conformal Fefferman-Type Construction 5 are projectively equivalent if they have the same geodesies as unparametrised curves. This is the case if and only if there is a one-form G r(E^) such that, for all £A G r(E^), Da? = DA? + TA? + TP?5AB. An oriented projective structure (M, p), which is a projective structure p on an oriented manifold M, is equivalently encoded as a normal parabolic geometry of type (G,P), where G = SL(n + 1) and P = GL+(n) x W1* is the stabiliser of a ray in the standard representation Rn+1. Affine connections from the projective class p are precisely the Weyl connections of the corresponding parabolic geometry. Exact Weyl connections are those D G p which preserve a volume form — these are also known as special affine connections. In particular, a choice of D G p reduces the structure group to Go = GL+(n), if D is special, the structure group is further reduced to Gff = SL(n). For later purposes we now give explicit expressions of the main curvature quantities, cf., e.g., [2, 17]. For D G p, the Schouten tensor is determined by the Ricci curvature of D; if D is special, then the Schouten tensor is Pab = '^l^p? b^ m particular, it is symmetric. The projective Weyl curvature and the Cotton tensor are c~* c~* c~* c~* wab £> = rab ^D + PADS b ~ PbdS 'A, Ycab = 2D[APB}C. Henceforth, we use a suitable normalisation of densities so that the line bundle associated to the canonical one-dimensional representation of P has projective weight —1. Hence, comparing with the usual notation, the density bundle of projective weight w, denoted by E(iu), is just the bundle of ordinary (^pj)-densities. As an associated bundle to Q —> M, K(w) corresponds to the 1-dimensional representation of P given by GL+(n) x Rn* -> R+, (A, X) ^ det(A)w. (2.3) The projective standard tractor bundle is the tractor bundle associated to the standard representation of G = SL(n + 1). The projective dual standard tractor bundle is denoted by T*, i.e., T* := G Xp Rra+1*. With respect to a choice of D G p, we write 7 -\E(1))> Vc [o-J - { Dco--pc J' 2.5 Conformal spin structures and tractor formulas Let M be a smooth manifold of dimension 2n > 4. A conformal structure of signature in, n) on M is given by a class, c, of conformally equivalent pseudo-Riemannian metrics of signature (n, n): two metrics g and g are conformally equivalent if g = f2g for a nowhere-vanishing smooth function / on M. It may be equivalently described as a reduction of the frame bundle of M to the structure group CO(n,n) = R+ X SO(n, n). An oriented conformal structure of signature in, n) is a conformal structure of signature in, n) together with fixed orientations both in time-like and space-like directions, equivalently, a reduction of the frame bundle to the group C0o(n,n) = R+ X S0o(n,n), the connected component of the identity. An equivariant lift of such a reduction with respect to the 2-fold covering CSpin(n, n) = R+ X Spin(n, n) —> C0o(n, n) is referred to as a conformal spin structure (M,c) of signature in,n). A conformal spin structure of signature in,n) is equivalently encoded as a normal parabolic geometry of type (G,P), where G = Spin(n + l,n + 1) and P = CSpin(n,n) x R™'™* is the stabiliser of an isotropic ray in the standard representation 6 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnfk A general Weyl connection is a torsion-free affine connection D such that Dg £ c for any g £ c. If Dg = 0, i.e., D is the Levi-Civita connection of a metric g £ c, it is an exact Weyl connection. A choice of Weyl connection reduces the structure group to Go = CSpin(n, n). If the Weyl connection is exact the structure group is further reduced to Gqs = Spin(n,n). Now we briefly introduce the main curvature quantities of conformal structures, cf., e.g., [16]. For g £ c, the Schouten tensor, i 2n ( Ric(ff) Sc(ff) 2(2r is a trace modification of the Ricci curvature Ric(g) by a multiple of the scalar curvature Sc(g); its trace is denoted J = gpqPpq. The conformal Weyl curvature and the Cotton tensors are Wahcd = Rahcd - 25c[aPb]d + 2gd[aPh], Yc cab 2D[aPb]c. As for projective structures, we will employ a suitable parametrisation of densities so that the canonical 1-dimensional representation of P has conformal weight —1. Hence, the density bundle of conformal weight w, denoted as E[it>], is just the bundle of ordinary (-^f)-densities. As an associated bundle to the Cartan bundle Q —> M, it corresponds to the 1-dimensional representation of P given by (R+ x Spin(n, n)) x i>2n* ->K+, (a,A,Z)i-^a~ (2.4) In particular, the conformal structure may be seen as a section of E(a;,)[2], which is called the conformal metric and denoted by gab. The spin bundles corresponding to the irreducible spin representations of Spin(n, n) are denoted by and S_, and S = © S_. We employ the weighted conformal gamma matrix 7 £ T(EQ ® (EndS)[l]) such that lplq + lqlp = -2gpq. For £ £ X(M) and X S T(S), the Clifford multiplication of £ on x is then written as £ ■ x = £,Pr7pX- The conformal standard tractor bundle is the associated bundle T := Q X; &n+l,n+l with respect to the standard representation. It carries the canonical tractor metric h and the con- formal standard tractor connection V^~, which preserves h. have With respect to a metric g £ c, we /E[-l]^ 0 1\ ~~(p\ i T = EQ[1] , h = ° 9 ^ E[l] J V 0 0/ The BGG-s plitting operator is g iven by Dcp ~PC r(XT), p := 7P-Dp, is the Dirac operator. The first BGG-operator associated to S± is the twistor operator ef: r(s±[i]) ->r(Ea®s±[i]), x ^ 5aX + ^7a|)x, cf., e.g., [3]. Elements in the kernel of ©q are called twistor spinors. It is well known that IIq induces an isomorphism between V^-parallel sections of S with ker ©q . The adjoint tractor bundle is the associated bundle AM := Q Xp§ with respect to the adjoint representation of G on q = so(n + l,n + l) = A2Rra+1'ra+1. The standard pairing on AM induced by the Killing form on g is denoted as (■, ■): AM x AM —> R. Henceforth we identify AM with A27~. With respect to a metric g £ c, AM ( Ea[0] \ E[aoai][2] I E[l] V Ea[2] / The standard representation of q on gives rise to the map • : AM®f^f, I = \ P iür — LpV PbTUJr ~ O-pb ~ vßh ßrüür + (pa (2.9) The normal tractor connection is given by am ( p% \ ]Paoai \ V = ( V Dcpa ~ Pc ppa ~ Pca

T(AM) 8 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnfk is determined by Paoai = D[aokai], (p =-^-gpqDpkq, (2.12) Pa = -^DPDpka + ^-DPDakp + -±-DaDPkp + -PPakp - ^-Jka, and the corresponding first BGG-operator of AM is computed as r(EQ[2]) ->. r(E(a6)o[2]), ia b(cia)o, where the subscript 0 denotes the trace-free part. Thus ©q*m is the conformal Killing operator and solutions to the first BGG-equation are conformal Killing fields. In a prolonged form, the conformal Killing equation is equivalent to v£ms = zanab, (2.13) where s = L$M(£), see [7, 18]. 3 The Fefferman-type construction The construction of split-signature conformal structures from projective structures discussed in this section fits into a general scheme relating parabolic geometries of different types. Namely, it is an instance of the so-called Fefferman-type construction, whose name and general procedure is motivated by Fefferman's construction of a canonical conformal structure induced by a CR structure, see [6] and [11] for a detailed discussion. 3.1 General procedure Suppose we have two pairs of semi-simple Lie groups and parabolic subgroups, (G, P) and (G, P), and a Lie group homomorphism i: G —> G such that the derivative i': g —> g is injective. Assume further that the G-orbit of the origin in G/P is open and that the parabolic P C G contains Q := i_1(P), the preimage of P C G. Given a parabolic geometry (Q —> M, lj) of type (G, P), one first forms the Fefferman space M :=g/Q = g xPP/Q. (3.1) Then (Q —> M,w) is automatically a Cartan geometry of type (G,Q). As a next step, one considers the extended bundle Q := Q Xq P with respect to the homomorphism Q —?> P. This is a principal bundle over M with structure group P and j: Q ^ Q denotes the natural inclusion. The equivariant extension of w G J71(^,g) yields a unique Cartan connection ujlnd £ J71(^,g) of type (G, P) such that j*£ind = i> o CO. Altogether, one obtains a functor from parabolic geometries (Q —> M,lu) of type (G, P) to parabolic geometries (Q —> M,cDmd) of type (G,P). The relation between the corresponding curvatures is as follows: The previous assumptions yield a linear isomorphism g/p = g/q and an obvious projection g/q —> g/p, where q C p is the Lie algebra of Q C P. Composing these two maps one obtains a linear projection g/p —> g/p, whose dual map is denoted as ip: (g/p)* —> (fl/p)*- Since i': g —> g is a homomorphism of Lie algebras, the curvature function «md: Q —> A2(g/p)* <8> g is related to k: Q —> A2(g/p)* <8> g by ~ind 0 j = (A2(p ® i') o k. We note that «ind is fully determined by this formula. Since i' is an embedding, the notation is in most cases simplified such that we write g C g, q = g n p, etc. A Projective-to-Conformal Fefferman-Type Construction 9 3.2 Algebraic setup and the homogeneous model Here we specify the general setup for Fefferman-type constructions from Section 3.1 according to the description of oriented projective and conformal spin structures given in Sections 2.4 and 2.5, respectively. Let be the real vector space R2ra+2 with an inner product, h, of split-signature. Let an(j ^e £ne irreducible spin representations of G := Spin(n + 1, n + 1) as in Section 2.5. We fix two pure spinors sj? £ ^n+i,n+i an^ SE £ A«+i,ra+i w^n non-trivial pairing, which is assigned for later use to be (sp,sp) = —\- Note that sp lies in if n A71+1,71+1 • c ■ J J _ it n is odd. Let us denote by E, F C the kernels of se, sf with respect to the Clifford multi- plication, i.e., E:= {X £ : X ■ sE = 0}, F := [X G : X ■ sF = 0}. The purity of s# and sp means that E and F are maximally isotropic subspaces in The other assumptions guarantee that E and F are complementary and dual each other via the inner product h. Hence we use the decomposition M„+l,n+l =E@Fg± Rn+1 0 Rn+1* (32^ to identify the spinor representation = 0 with the exterior power algebra A'E = A*Rn+1, whose irreducible subrepresentations are A™+1'n+1 = AevenMn+1 and A„+i,„+i ^ ^ddj^n+i^ When n ig eveil) respectiveiy? odd? we Can identify (a™+1'™+1)* ^ A™+i,77+i) reSpeCtively (A™+1'n+1)* 9* A^+1,Il+1. Now, let us consider the subgroup in G defined by G := {g G Spin(n + 1, n + 1): g ■ sE = sp, g ■ sF = sF}. This subgroup preserves the decomposition (3.2) so that the restriction of the action to F is dual to the restriction to E. It further preserves the volume form on E, respectively F = E*, which is determined by sp and sp according to the previous identifications. Hence G = SL(n+ 1) and this defines an embedding i: SL(n + 1) ^+ Spin(n + l,n + l).1 The G-invariant decomposition (3.2) determines a G-invariant skew-symmetric involution K Gso(n + l,n + l) acting by the identity on E and minus the identity on F. The relationship among K, sp and sp may be expressed as h(X, K(Y)) = -h(K(X), Y) = 2(sE, (X A Y) ■ sF), (3.3) where {X A Y) ■ sF = ^(X Y ■ sF -Y ■ X ■ sF) = X Y ■ sF + h(X, Y)sF. The spin action of g is denoted by •, and thus A • s = —\A ■ s, for any A G 0 and s G A. In particular, K»sp = — ^(n + l)sp and K • sp = ^(n + l)sp. Here we identify g = so(n + l,n + l) with A2Rn+1'n+1. It is convenient to split q in terms of irreducible g-modules as g = A2{E © F) = {E ® F)0 ®(E® F)Tr © A2E © A2F, (3.4) "-v-' "-v-' g=s((n+i) gx 1Instead of the embedding SL(n + 1) Spin(n + 1, n + 1) we could also consider the embedding SL(n +1) SO(n + l,n + 1). The advantage of employing the embedding into the spin group is two-fold: on the one hand, it is then seen directly that the induced conformal structure has a canonical spin structure, and, on the other hand, we can then use convenient spinorial objects for its characterisation. 10 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik where (E © F)Tr = RK, and K acts as [K, 4>] = 2(f), [K, ip] = -2ip, [K, A] = 0, for any 4> G A2E, i\) G A2F and A G E © F. Further, the annihilators of sF and sj? in g are the subalgebras kers£ = sl(n + 1) © h?E and ker sF = sl(n + 1) © A2F. The homogeneous model for conformal spin structures of signature (n,n) is the space of isotropic rays in ]^Jl+1>Jl+1) G/P = Sn x Sn. The subgroup G Q G does not act transitively on that space. According to the decomposition (3.2), there are three orbits: the set of rays contained in E, the set of rays contained in F, and the set of isotropic rays that are neither contained in E nor in F. Note that only the last orbit is open in G/P, which is one of the requirements from Section 3.1. Therefore, we define P C G to be the stabiliser of a ray through a light-like vector v G \ (EUF). Denoting by Q = i~x(P) the stabiliser_of the ray R+v in G, we have the identification of G/Q with the open orbit of the origin in G/P. The subgroup Q, which is not parabolic, is contained in the parabolic subgroup P C G defined as the stabiliser in G of the ray through the projection of v to E. In particular, G/P is the standard projective sphere Sn, the homogeneous model of oriented projective structures of dimension n, and G/Q —> G/P is the canonical fibration with the standard fibre P/Q, whose total space is the model Fefferman space. Let us denote by L = the line spanned by the light-like vector v and let L1- be the orthogonal complement in with respect to h. The tangent space of G/Q at the origin can be seen in three different ways, namely, (LVL)[l]-0/q-0/p. The latter isomorphism is induced by the embedding g C §, the former one by the standard action of g C g on the vector v G Both these identifications are Q-equivariant. There are several natural Q-invariant objects that in turn yield distinguished geometric objects on the general Fefferman space. The n-dimensional Q-invariant subspace / := ((F + L)/L)[l] C (1^/1)11], where F := F D L^, which is isomorphic to p/q Cg/q, the kernel of the projection g/q —> g/p. Another n-dimensional Q-invariant subspace is e := ({E + L)/L)[V\ C (L±/L)[l], where E:=EC\L^. The intersection e n / is 1-dimensional with a distinguished Q-invariant generator that corresponds to the G-invariant involution K G g, k := K + p G g/p. Note that all these objects are isotropic with respect to the natural conformal class induced by the restriction of h to L1- C jn particular, both e and / are maximally isotropic subspaces such that k£eDfCk± = e + f. (3.5) In Section 3.1 we introduced a map ip: (g/p)* —> (fl/p)*, the dual map to the projection g/p=g/q—^g/p. The kernel of this projection is just / and the image of ip is identified with its annihilator, which will be denoted by f°. Since / is a maximally isotropic subspace in g/p = g/q, f° = /[-2]. Since (g/p)* = p+, we may conclude with the help of explicit matrix realisations from Appendix A that f° = p+ n ker sp. Moreover, we note that (p+nkeisF)mF = p+, (pnkeisF)mF = p, (3.6) A2F n p = A2F C go, [P+, A2F] = /°, [/°, A2F] = 0. (3.7) A Projective-to-Conformal Fefferman-Type Construction 11 3.3 The Fefferman space and induced structure The pairs of Lie groups (G, P) and (G, P) from the previous subsection satisfy all the properties to launch the Fefferman-type construction. Proposition 3.1. The Fefferman-type construction for the pairs of Lie groups (G,P) and (G, P) yields a natural construction of conformal spin structures (M, c) of signature (n, n) from n-dimensional oriented projective structures (M, p). The Fefferman space M is identified with the total space of the weighted cotangent bundle without the zero section T*M(2)\{0}. Proof. The first part of the statement is obvious from the general setting for Fefferman-type constructions and the Cartan-geometric description of oriented projective and conformal spin structures. The second part is shown due to two natural identifications: On the one hand, the Fefferman space is by (3.1) equal to the total space of the associated bundle M = Q Xp P/Q over M. On the other hand, the weighted cotangent bundle to M is identified with the associated bundle T*M(2) = Q Xp (g/p)*(2) with respect to action of P induced by the adjoint action and the representation (2.3) for w = 2. Hence it remains to verify that the action of P on (g/p)*(2) \ {0} is transitive and Q is a stabiliser of a non-zero element. But this is a purely algebraic task, which may be easily checked in a concrete matrix realisation. ■ From the algebraic setup in Section 3.2 we easily conclude number of specific features of the induced conformal structure on M: Proposition 3.2. The conformal spin structure (M, c) induced from an oriented projective structure (M, p) by the Fefferman-type construction admits the following tractorial objects that are all parallel with respect to the induced tractor connection: (a) pure tractor spinors sp £ r( M, (d) k G r(TM) is a nowhere-vanishing light-like vector field generating the intersection en/. Proof. The G-invariant spinor sp G A± gives rise to the tractor spinor sp £ r( A±. Hence s# is automatically parallel with respect to the induced tractor connection on g/p, the corresponding subbundle / is identified with the vertical subbundle of the projection M —> M. The intersection e n / is 1-dimensional and it is generated by the projection of K G 0 to g/p. Indeed, K cannot be contained in p, since K acts by the identity on E and minus the identity on F and p is the stabiliser of a line that is neither contained in E nor in F. Altogether, the corresponding vector field k on M is a nowhere-vanishing generator of e n /, in particular, it is light-like. ■ 3.4 Relating tractors, Weyl structures and scales As a technical preliminary for further study we now relate natural objects associated to the original projective Cartan geometry (G,u>) on M and the induced conformal geometry (C/,cDmd) on the Fefferman space M. Since G C G, any G-representation V is also a G-representation, which yields compatible tractor bundles over M and M with compatible tractor connections: V = Q Xp V —?> M with the tractor connection V induced by u and V = G Xp V = G Xq V —> m with the tractor connection Vmd induced by ujlnd. Sections of V bijectively correspond to P-equivariant functions ip: Q —> V, while sections of V correspond to Q-equivariant functions ip: Q —> V. Since Q C P, every section of V gives rise to a section of V, and we can view T(V) C r(V). Now, Proposition 3.2 in [8] admits a straightforward generalisation to Fefferman-type constructions for which P/Q is connected and thus, in particular, to the one studied in this article: Proposition 3.3. (a) A section s G T(V) is contained in T(V) (i.e., the corresponding Q-equivariant function p is indeed P-equivariant) if and only if Vmds is strictly horizontal (i.e., waV™ds = 0 for all (b) The restriction of Vmd to T(V) C T(V) coincides with the tractor connection V. Remark 3.4. Another instance of compatible bundles over M and M is provided by the density bundles K(w) and K[w], which are defined via the representation of P and P as in (2.3) and (2.4), respectively. Restricting these representations to Q, it easily follows that the notation is indeed compatible so that we can view r(E(io)) C r(EH). Both projective and conformal density bundles can be described as associated bundles to the respective bundles of scales. Hence everywhere positive sections of density bundles are considered as scales. In particular, the inclusion T(E+(1)) C r(E+[l]) may be interpreted so that any projective scale induces a conformal one. Such conformal scales will be called reduced scales. An intrinsic characterisation of reduced scales among all conformal ones is formulated in Proposition 5.2. The previous remark yields that any projective exact Weyl structure on M induces a conformal exact Weyl structure on M. This fact can be generalised as follows: Proposition 3.5. Any projective (exact) Weyl structure on M induces a conformal (exact) Weyl structure on the Fefferman space M. Proof. A version of this result in a more general context was proved in [1, Proposition 6.1]: any Weyl structure for lu induces a Weyl structure for ujind if P+ C P and (Go HP) C Gq. But both A Projective-to-Conformal Fefferman-Type Construction 13 these conditions are satisfied as follows from the setup in Section 3.2 and explicit realisations in Conformal Weyl structures induced by projective ones as above will be called reduced Weyl structures. 3.5 Normality Here we show that our Fefferman-type construction does not preserve the normality in general, see Proposition 3.8. This can be shown directly as we did in a previous version of the article, see arXiv:1510.03337v2. Alternatively, we can treat the construction as the composition of two other constructions via a natural intermediate Lagrangean contact structure. A Lagrangean contact structure on M' consists of a contact distribution ~H C TM' together with a decomposition ~H = e' @ f into two subbundles that are maximally isotropic with respect to the Levi form ~H x ~H —> TM'/%. Such structure on a manifold M' of dimension 2n — 1 is equivalently encoded as a normal parabolic geometry of type (G, P'), where G = SL(n + 1) and P' C G is the stabiliser of a flag of type line-hyperplane in the standard representation Rn+1. For n > 2 there are three harmonic curvatures, two of which are torsions whose vanishing is equivalent to the integrability of the respective subbundles e', f C ~H. For n = 2 there are two harmonic curvatures of homogeneity 4, hence the Cartan connection is torsion-free. In that case both e' and /' are 1-dimensional and thus automatically integrable. On the one hand, P' is contained in P, where P C G is the stabiliser of a ray in Rn+1. For suitable choices as in Appendix A, the Lie algebra to P' consists of matrices of the form Given a projective Cartan geometry (Q —> M, lj) of type (G, P), it turns out that the correspondence space M' := Q/P' can be identified with the projectivised cotangent bundle V(T*M). The Cartan geometry (Q —> M',lj) of type (G,P') is regular and thus it covers a natural Lagrangean contact structure on M'. In particular, the canonical contact distribution on V(T*M) coincides with ~H and the vertical subbundle of the projection M' —> M coincides with one of the two distinguished subbundles, say /' C ~H. As in general, this construction preserves normality. In accord with [5], respectively [11, Section 4.4.2] we may state: Proposition 3.6. Let (Q —> M,lj) be a normal projective parabolic geometry and let (Q —> M', lj) be the corresponding normal Lagrangean contact parabolic geometry. The latter geometry is torsion-free if and only if n = 2 or it is flat, i.e., the initial projective structure is flat. On the other hand, P' contains Q, where Q = G n P as before. This allows us to consider the Fefferman-type construction for the pairs (G,P') and (G,P). Given a Lagrangean contact structure on M', it induces a conformal spin structure on M = Q/Q. This construction is indeed very similar to the original Fefferman construction; one deals with different real forms of the same complex Lie groups in the two cases. That is why the following statement and its proof is analogous to the one for the CR case. Following [8], respectively [11, Section 4.5.2] we may state: Proposition 3.7. Let (Q —> M', lj) be the normal Lagrangean contact parabolic geometry and let (Q —> M,cDmd) be the conformal parabolic geometry obtained by the Fefferman-type construction. Then cDmd is normal if and only if lu is torsion-free. Appendix A. 14 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik Altogether, composing the two previous steps we obtain our projective-to-conformal Feffer-man-type construction with the desired control of the normality. Note that from (3.5) and the respective matrix realisations it follows that the induced objects on M = T*M(2) \ {0} from Proposition 3.2 correspond to the induced objects on M' = V(T*M). In particular, the vertical subbundle of the projection M —> M' is spanned by k and the decomposition k1- = e©/ C TM descends to the decomposition ~H = e' © /' C TM' M. Proposition 3.8. Let (Q —> M,u) be a normal projective parabolic geometry and let (Q —> M,cDmd) be the conformal parabolic geometry obtained by the Fefferman-type construction. (a) If dim M = 2 then ujlnd is normal. (b) If dim M > 2 then ujlnd is normal if and only if lu is flat. Moreover, independently of the dimension of M, wind is flat if and only if lu is flat. 3.6 Remarks on torsion-free Lagrangean contact structures At this stage it is easy to formulate a local characterisation of split-signature conformal structures arising from torsion-free Lagrangean contact structures, see Proposition 3.10. As before, the results and their proofs are very analogous to those in the CR case, therefore we just quickly indicate the reasoning and point to differences. As in Proposition 3.2, the G-invariant algebraic objects induce the tractor fields se,&f and K on the conformal Fefferman space that are parallel with respect to the induced tractor connection and have the required compatibility properties. But, starting with a torsion-free Lagrangean contact structure, the induced connection is already normal. In particular, the corresponding underlying objects 77 and k are pure twistor spinors and a light-like conformal Killing field, respectively. The existence of parallel tractors se, sf and K with the algebraic properties as in Proposition 3.2 are by no means independent conditions: Proposition 3.9. Let (M, c) be a conformal spin structure of split-signature (n,n). Then the following conditions are locally equivalent: (a) The spin tractor bundle admits two pure parallel tractor spinors se G r( 3, the induced conformal Cartan connection associated to a non-flat n-dimensional projective structure differs from the normal conformal Cartan connection for the induced conformal structure. In this section we will analyse the form of the difference and thus derive properties of the induced conformal structures. Furthermore, we will show that any split-signature conformal manifold having these properties is locally equivalent to the conformal structure on the Fefferman space over a projective manifold. 4.1 The normalisation process We are going to normalise the conformal Cartan connection ujind £ J71(^,g) that is induced by a normal projective Cartan connection ui £ 0,^(0, q). Any other conformal Cartan connection to' differs from ujind by some ^ £ J71(^,g) so that ui' = ujind + ^. This ^ must vanish on vertical fields and be P-equivariant. The condition on ui' to induce the same conformal structure on M as ujind is that ^ has values in p C g. One can therefore regard ^ as a P-equivariant function ^'■ Q —> (fl/p)* © P- According to the general theory as outlined in [11, Section 3.1.13] there is a unique such ^ such that the curvature function k' of ui' satisfies d*H' = 0, and then ui' is the normal conformal Cartan connection ujuot . The failure of ujind to be normal is given by d*Kind: Q —> (g/p)* ©p. The normalisation of ujind proceeds by homogeneity of (g/p)* p, which decomposes into two homogeneous components according to the decomposition p = go © P+- In the first step of normalisation one looks for a "J1 such that uj1 =uj + has c^k1 taking values in the highest homogeneity, i.e., c^k1 : Q —> (g/p)*©p+. To write down this first normalisation we employ Weyl structures Qq^s-Q. By Proposition 3.5 we can take a reduced Weyl structure, i.e., one that is induced by a reduction Qq^-Q ^ Q with respect to the structure group Qq := QnGo- This allows us to project d*kind to (<9*kmd)q: Qq —> (fl/p)* ® 00 and to employ the Go-equivariant Kostant Laplacian □: (g/p)* © go —> (fl/p)* © 80, □ := d o d* + d* o d. For the first normalisation step we need to form a map "J1: Q —> (g/p)* © p that agrees with —□_1(9*kmd)0 in the go-component. If we have formed any such ty1 along Qq^-Q we can just equivariantly extend this to all of Q. To proceed with the analysis of the normalisation we need to establish a couple of technical lemmas. As before, we denote by f° C p+ = (g/p)* the annihilator of / = p/q C g/q = g/p. Recall that f° = ip(p+) S f[-2}. Lemma 4.1. Let V be a ^-representation contained in a ^-representation V and denote by 4> i-» 4> the inclusion Akp+ © V ^ Akp+ © V induced by ip: p+ -> p+ and V ^ V. Then, for any 4> £ Akp+ © V, d*j) - d*4> £ A*'1/0 © (A2P . V) C Ak-1:p+ © V. In particular, for the adjoint representations, d*cj) = 0 if and only if d*cj) £ Ak~1f° © A2P. Proof. For the sake of presentation, assume that cf> is decomposable, i.e., of the form cf> = Z\ A ■ ■ ■ A Zfc © v, where Zi £ p+ and v £ V. Let us denote by the same symbols also the images of these elements under the inclusion g ^ g and V ^ V, i.e., Zi £ p and v £ V, respectively. Let Zi £ f° be the images of Zi under the inclusion p: p+ —> p+. Now, by definition of the A Projective-to-Conformal Fefferman-Type Construction 17 Kostant co-differential, the difference d*cf> — d*cf> evaluated on any k — 1 elements from g/p is a linear combination of terms of the form {Zi-Zi)mv. (4.1) However, the differences Z^ — Zi G p are represented by the matrices as in (A.l) in the Appendix where only the Z-entries are non-vanishing and hence contained in A2F Hp = A2F. Thus (4.1) belong to the image of •: A2F x V —> V and the first claim follows. For the second claim we use that A2F • g = [A2F,0] C A2F and A2F n 0 = 0: since d*(f> (evaluated on any k — 1 elements from g/p) has values in 0 C 0, vanishing of d*cf> is equivalent to d*4> having values in A2F. But d*4> has generally values in p and A2F Hp = A2F, hence the claim follows. ■ Lemma 4.2. If ip G p+ A f° ®A2F C A2p+®p then d*ip G f° 0 f° C p+ 0 p+. Proof. i\) is a sum of terms of the form Z\ A Z2 0 A, where Z\ G p+, Z2 G /° and A G A2F. Applying the Kostant co-differential gives d\Zx l\Z2® A) = Zx® [Z2,A] -Z2® [Zi, A]. Now [Z2,A] belongs to [/°,A2F] = 0 and [ZltA] belongs to [p+,A2F] = f°, hence the claim follows. ■ The following lemma contains the crucial information which is necessary to perform our normalisation. We are going to specify the curvature function kmd (later also knor) by describing its values along the natural Q-reduction Q ^+ Q over M. Recall from Section 3.2 that A2F is a Q-invariant subspace in go, which can be identified with (A2/)[-2]. Lemma 4.3. For any u G Q, we have d*Kind(u) G/°®A2FCp+®0O. Identifying A2F ^ (A2/) [-2] and f° ^ f[-2], we have in fact d*Kind(u) G (/ 0 A2/) [-4], i.e., d*Klnd(u) is contained in the kernel of the alternation map alt: (/®A2/)[-4]+(A3/)[-4]. Proof. It is a general assumption that wind is induced by a normal projective Cartan connection on Q, i.e., d*K,{u) = 0, for any u G Q. Hence it follows from Lemma 4.1 that d*Kind(u) belongs to f°®A2F^ (/0A2/)[-4]. Further we need a finer discussion involving the properties oi k: Q —> A2p+ 0 0 to show that k(u) belongs to the kernel of the Q-equivariant map A2p+ 0 0 —> (A3/) [—4] given by ^ alt (d*4>). (4.2) Note that any element (f> G A2p+ 0 p+ for which d*4> = 0 is mapped to zero: since (f> G A2p+ 0 p and = p+, the co-differential d*cf> has values in f° 0 p+. But, by Lemma 4.1, it also has values in f° 0 A2F and p+ n A2F = 0. Thus it suffices to consider the harmonic elements from A2p+00o, i.e., the ones corresponding to the projective Weyl tensor. For that purpose we consider the simple part of Qo = QnGo which is isomorphic to SL(n— 1), cf. the matrix realisation (A.2) in the Appendix where it corresponds to the A-block. Considering both A2p+ 0 0O = A2Rn* 0 Rn* 0 Rn and (A3/) [-4] S A3Rn* as 18 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik representations of SL(n— 1), the map (4.2) is either trivial or an isomorphism on each SL(n— 1)-irreducible component. One can check that there is only one SL(n — l)-irreducible component that occurs in both spaces, and it is isomorphic to A2Rra_1*. Hence it suffices to compute (4.2) on one element contained in such component. Let Xn £ g_ and Zn £ p+ be the two dual basis vectors stabilised by SL(n — 1) and consider an element (f> = Z\ A Z2 ® Xn ® Zn - Z\ A Z2 ® Ai ® Z\ + Zn A Z2 ® Xn ® Z\. Indeed cf> is completely trace-free, satisfies the algebraic Bianchi identity and the SL(n— l)-orbit of is isomorphic to A2Rra_1*. Now, d*4> = -Zx ®ZnAZ2-Zn®Z1A Z2, which indeed lies in the kernel of the alternation map. Hence the statement follows. ■ We can now determine the form of the normal conformal Cartan connection: Proposition 4.4. The normal conformal Cartan connection is of the form -nor = ~ind + ^1 + ^ where ty1 = — ^d*Hlnd £ ^nor(^>p) and ^2 £ ^hor(^'P+)' Furthermore, along the reduction Q^Qwe have tf1 £ ^or(£,A2F), *2 £ (fl/p)* ® P and 4>2- G -> (fl/p)* p+ such that, for any u £ Q, ^x = ^(u) o ujind and 3>2 = (f)2(u) ocDmd. In these terms, the proposition means that along the reduction Q ^ Q these maps restrict to Q-equivariant functions 01: G^f°®A2F, cp2: G ->• f° ® f'■ Further we put ^ = ^x + 3>2 and (f> = (f)1 + (f>2. Proof. The Kostant Laplacian □ restricts to an invertible endomorphism of ((g/p)*®go)nim d* that acts by scalar multiplication on each of the Go-irreducible components. Now, restricting to G ^ G and suppressing all arguments u £ Q, it was shown in Lemma 4.3 that d* kind is contained in one of the irreducible components, namely in (/ 0 A2/) [—4]. On this component □ acts by multiplication by 2. Thus, the modification map accomplishing the first normalisation step is 01: G -> f° ® A2F, 4>x := -^d*kind = -CHd*^. Now, let uj1 := cDmd + 01 o ujind be the modified Cartan connection. The corresponding curvature function k1 can be expressed in terms of kmd, (p1 and its differential dcf)1 so that Ti\X,Y) = kind(A,F)+ [X^\Y)] - [Y^\X)] + dcp\i){Y) - d4\r,){X) - 01([A, Y]) + [J>\X), c/>\Y)], (4.3) A Projective-to-Conformal Fefferman-Type Construction 19 where A, Y G 0 and £ = (wind) 1(A), 77 = (wind) cf. [11, formula (3.1)]. For the last term we have [01(A), 01(F)] = 0 since 0X(A) has values in A2F. The first three terms are «ind(A,F) + [A,0x(F)] - [Y^\X)] = ^A{X,Y) + d^\X,Y), which by construction vanishes upon application of the Kostant co-differential, i.e., c?*(/?nd + dcf)1) = 0. The remaining terms in (4.3) can be combined into a map A20 —> A2F, (A, F) d4\£){Y) - d4\rj){X) - \[X, Y]), which vanishes upon insertion of two elements X,Y G p. Therefore, applying Lemma 4.2, we conclude that d*^1 has values in /° © /°. Thus the second modification map is 4>2- G^f°®f°, 4>2 ■= -fi"1^1. ■ 4.2 Properties The information provided in the previous proposition allows us to determine the properties satisfied by the normal conformal Cartan curvature: Proposition 4.6. The normal conformal Cartan curvature Knor restricts to a map Knor: g -> A2(0/p)*® (sl(n + l)©A2F). (4.4) Moreover, the following integrability condition holds: ixknm(u) G f° ® (A2F © /°), for allX ef,ue Q. (4.5) Proof. Let Knor be the curvature function of the normal Cartan connection cDnor = cDmd + o ujind, where = cf)1 + 2. With the same conventions as in the proof of Proposition 4.4, [11, formula (3.1)] yields Knor(A, Y) = «ind(A, Y) + [X, 0(F)] - [Y, 0(A)] + " MvKX) - 0([A, Y]) + [0(A), 0(F)]. Clearly, Kmd(A, F) has values in sl(n + 1) and vanishes upon insertion of A G p. A term of the form [A, 0(F)] vanishes if F G p and has values in [p, A2F © /°] C A2F © f° for A G p. A term of the form V, u 1—> v, along g. Then the covariant derivative Vnorv corresponds to the Q-equivariant function g f° ®V, u h-> 01(w) »v + (f>2(u) • v. 20 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik Proof. The covariant derivative Vnorv corresponds to the map legn) (wnor)_1(A) ■ v + X »v. (4.6) The first term in (4.6) vanishes since it is the directional derivative of the constant function v. Now wnor = cDmd + (f)1 + (f)2, and since X • v = 0 the claim follows. ■ We now show that the distinguished tractors sp, sf and K on the Fefferman space are all given as BGG-splittings from their underlying objects. Moreover, several stronger properties hold: Proposition 4.8. Let Se G r(A2F. (4.7) Proof, (a) Since cj)1, cf)2 have values in kers^ we have cf)1 • sp + cf)2 • sp = 0. Thus, according to Lemma 4.7, we have VnorSi7 = 0 and the rest is obvious. (b) The spinor sp is of the form sp = (^). According to Lemmas 4.3 and 4.7, (f)1 has values in (/©A2/)[-4] and d*(VnoTsE) corresponds to The projection (01 • 77) g can be realised as the full (triple) Clifford action on ^(u) G (®3/)[—4], where u G Q. Now it is easy to see that this action must vanish for a ^(u) G (/ © A2/) [—4]: We realise ^(u) equivalently in (S2f <8> /) [—4] by symmetrisation in the first two slots, then the complete Clifford action on 77 vanishes because the action of the first two slots is just a (trivial) trace multiplication. (c) According to Lemma 4.7, VnorK corresponds to cf)1 • K + cf)2 • K. Since K/p = k G /, the previous element lies in p. In particular, VnorK has trivial projecting slot, and thus k = LTo(K) is a conformal Killing field. Since cf)2 • K G p+, we have that <9*(VnorK) corresponds to d*(4>1 • K). Now cf)1 • K = -K • cf)1 = 201, since K acts by multiplication with -2 on A2F. But cf)1 G im d* C ker<9*, and the expression • K~j therefore vanishes. The equality ynorK = -fc^nor ig jugt (2.13) for the conformal Killing field k with its BGG-splitting K. In terms of the Q-equivariant functions (f> = cf)1 + cf)2 and Knor along Q ^ Q, this can be expressed as cj) • K = ikkno1, which yields (4.7). ■ We now collect the essential information about the induced conformal structure (M, c) which we derived: Proposition 4.9. Let (M, c) be the conformal spin structure induced from an oriented projective structure (M, p) via the Fefferman-type construction. Then the following properties are satisfied: A Projective-to-Conformal Fefferman-Type Construction 21 (a) (M, c) admits a nowhere-vanishing light-like conformal Killing field k such that the corresponding tractor endomorphism K = Lq-m(k) is an involution, i.e., K2 = id^. (b) (M,c) admits a pure twistor spinor \ £ T(X_ [^]) with k £ r(ker%) swc/i £/ia£ the corre-sponding parallel tractor spinor sp = Lg~(x) is pure. (c) K acts by minus the identity on kers^. (d) The following integrability condition holds: vawcWabcd = 0, for all v, we T(ker x) • (W) The only thing left to show for Proposition 4.9 is that the integrability condition (4.5) is equivalent to the condition (W) on the Weyl tensor: Lemma 4.10. Let (M,c) be a split-signature conformal spin structure endowed with tractors se, sf and K satisfying conditions (a) and (b) from Proposition 4.9. Then condition (4.5) is equivalent to (W). Proof. The implication (4.5) ==> (W) is obvious. It remains to prove the converse implication (W) (4.5). By (W), one has that (iXKUOT)~o(u) £ (/® A2/)[-4] C f°®A2F for X £ /, u £ Q. Since sF is parallel with respect to Vnor, we have Knoi(u) £ A2(g/p)* (p n kers/r). The projection of pnkersf to p+ is precisely /°, hence it follows that (ixKnor)~+(w) £ (fl/p)*<8>/°, and we obtain iXKnor(^) S (g/p)* ® (A2F © /°). We now prove that ix1ix2^n°I = 0 f°r an Ai,X2 £ /. For this purpose it will be useful to work with the curvature form S7nor, which we can represent as in (2.11). By (W) and the algebraic Bianchi identity, Wabcd vanishes upon insertion of v,w £ r(ker%) into any two slots, and in particular vawhWabcd = 0. Thus, it remains to check that vawhY,ia,b = 0. As in the proof of Proposition 3.2, a vector field w £ r(ker%) corresponds to a section (^d) £ ^(.T-"). According to (2.9), aonor v Kb • Since ivQ,no1 annihilates J7, it follows that vawrYrafJ = 0. Using Yrab = —Y{^a — Yabr, we obtain also vawhYrah = 0. ■ (*\ f-vaYrabwr\ | = | | er(E6®r; w ^ o [ 4.3 Characterisation We are now going to characterise the induced conformal structures. For this purpose we will introduce the following ('intermediate') Cartan connection form: ^=£nOT-^fcKnor. (4.8) The following observation then follows immediately from Proposition 4.4 and formula (4.7): Lemma 4.11. The pullbacks of the Cartan connection forms to' £ ^^(QiO), wnor £ ^^OI{G,0) and £ind £ n{OI(g , g) to Q ^ Q agree modulo forms with values in p+ C sl(n + 1) C g. 22 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik For the rest of this section, we will start with a given split-signature conformal spin structure (M, c) satisfying all the properties of Proposition 4.9. In particular, M is endowed with a conformal Killing field k G r(TM), and we can still use formula (4.8) to define a Cartan connection uj'. The corresponding tractor connection will be denoted by V' and the curvature by Q' or k'. The following proposition now shows that the so constructed Cartan connection uj' is in fact an SL(n + l)-connection. Proposition 4.12. Let (M, c) be a split-signature conformal (spin) structure satisfying all the properties of Proposition 4.9. Then the sections sp and k are parallel with respect to the tractor connection V', i.e., V's/r = 0 and V'k = 0. In particular, Ho1(uj') C SL(n + 1) C Spin(n + l,n + 1) and uj' pulls back to a Cartan connection of type (SL(n + 1), Q) with respect to the Q-reduction Q ^ Q. Along that reduction, the curvature functions k' and Knor are related according to k' = (^n°r),s[(ra+i) an<^ ^' sa^sfies the following integrability condition: ixk(u) G f © p+, for allX G /, u G Q. (4.9) Proof. A tractor connection induced by uj' can be written as V' = Vnor + V with $ = -\ikO}. That V'sF = 0 follows immediately from the fact that V' - Vnor = -\ik£lWI has values in A2T. Since k is a conformal Killing field we have Vnork = ikO,no1. By definition V'k = Vnork - ^ikQnm • k, which vanishes, since ifcfJnor has values in A2J7 and therefore ^ifcfJnor • k = ifcfJnor. As in Proposition 4.9, we write the decomposition of T into maximally isotropic eigenspaces of k with eigenvalues ±1 as E © J-. Since k is V'-parallel, it follows that this decomposition is preserved by V'. Moreover, since J7 is the kernel of the pure tractor spinor sp it follows that Hol(w') C SL(n + 1). In particular, lj' reduces to a Cartan connection of type (SL(n + 1), Q) on a Q-principal bundle Q C Q. We further compute that n' = nnm - -d^noiiknno1 = nnm - -d^noivnoik 2 2 = ^nor - -nnm. k = nnm + -k» nnm = (nnoi)/?^, , 2 2 v HS^W where we are again using Vnork = ikQno1 for the conformal Killing field k and that J7nor has values in 8 J- © A2J7. Stated for the corresponding curvature functions, this yields k' = (Knor)s(^+1^. Moreover, since Knor has values in kers^ n p, it follows from (3.6) that (p n ker SF)s((n+i) = thus k' has values in p. We know from (4.5) that ixknot has values in A2F©/° for X G /. But since (A2F)s((n+1^ = 0 and n ker sp)si(n+i) = P+; we obtain that (ix^WI)^(n+i) has values in p+. Finally, (*Xi*X2^nor)s[(ra+i) = 0 f°r X\,X2 G / follows immediately from (4.5), and altogether we obtain (4.9). ■ Next, before proving the main characterisation Theorem 4.14, we will show the following proposition on factorisations of particular Cartan geometries. This proposition can be understood as an adapted variant of [5, Theorem 2.7]. Proposition 4.13. Let (Q —> M,u) be a Cartan geometry of type (SL(n +1), Q) with curvature k: Q —> A2(g/q)* © q and let the following conditions be satisfied: iXlix2K(u) S p, for all Xi,X2 G fl/q, u G Q, A Projective-to-Conformal Fefferman-Type Construction 23 iXlix2k(u) G p+, for all Xx G p/q, X2 G g/q, u G G, ix1ix2K(u) = 0> for all Xi,X2 E p/q, ueG- Then G is locally a P-bundle over M = G/P and lu defines a canonical projective structure on M. Proof. The third of the above listed conditions implies that G is locally a P-bundle G —> M by [5]. We will restrict Q to assume this globally. We define M = Q/P and Go = G/P+ ■ Let a: Go —> G be a Go-equivariant splitting. It follows from the second of the above listed conditions that £(xlu = —ad(A) o lu mod p+, for all X G p. Now define 9 G ^(Go, Q-), 7 £ ^(GotQo) and p G Q1(Go,P+) via the decomposition a*uu = #©7 ©p. Since a is Go-equivariant and the Lie derivative is compatible with pullbacks it follows that % (0 © 7) = -ad(A) 0(997), for all X G g0. In particular, 6 and 7 are Go-equivariant and define a (reductive) Cartan geometry (Go ~> M, 6 © 7) of type (W1 x SL(n), SL(n)), i.e., an affine connection on M. Since by assumption 0, has values in p, 9 © 7 is torsion-free and so is the affine connection. Now take another splitting a' = o ■ exp T, for some T: G —> p+- Since Ad(exp T) acts by the identity on g_ = g/p, one has (rexpT)*w = lu mod p, and thus 6 is independent of the choice of splitting. Then a'*uj = 6@^'@p' and #©7' = Ad(expT)o(6>©7), projected to 0-ffigo- But since expT G P+, this shows that 7' is projectively equivalent to 7. We thus obtain a well-defined projective structure on M. Since lu is P-torsion-free and P-equivariant modulo p+, it can be (uniquely) modified to a normal Cartan connection lu1101 G J71(^,g) with lu1101 — lu G Q1(G,p+)- In particular, each splitting a: Go ~^ G is in fact a Weyl structure of the projective structure on M. ■ Theorem 4.14. A split-signature (n,n) conformal spin structure c on a manifold M is (locally) induced by an n-dimensional projective structure via the Fefferman-type construction if and only if the following properties are satisfied: (a) (M, c) admits a nowhere-vanishing light-like conformal Killing field k such that the corresponding tractor endomorphism K = Lq-m(k) is an involution, i.e., K2 = id^. (b) (M,c) admits a pure twistor spinor \ £ T(X_ [^]) with k G T(kerx) such that the corre-sponding parallel tractor spinor sp = L0~ (x) is pure. (c) K acts by minus the identity on kersp. (d) The following integrability condition holds: vawcWabcd = 0, for allv,w G T(ker X) ■ Proof. Starting with a projective structure (M, p), it follows from Proposition 4.9 that the induced conformal structure (M, c) has all the stated properties. On the other hand, let (M, c) be a conformal structure with the stated properties. Then, by Proposition 4.12, lu' restricts to a Q-equivariant Cartan connection form with values in sl(n + 1) on the reduction G ^ G■ The corresponding curvature k' takes values in p and for X G / we have that ix^' takes values in p+. It follows from Proposition 4.13 that lu' factorises to a projective structure p on the leaf space M. 24 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik Let us now show that the two constructions are inverse to each other. Assume first that a conformal structure (M,c) is induced by a projective structure (M, p). Then according to Lemma 4.11 uj1 and ujind agree modulo p+. This implies that the projective structure defined by u' is equal to the original projective structure. Conversely, assume now that (M, p) is a projective structure with associated Cartan geometry (Q, lj') that is induced from a conformal structure (M, c) with associated Cartan geometry (Q, wnor). Since uj' is not normal, but torsion-free, there is ip £ ^hor(^' P+) sucn that uj' + tp is the normal projective Cartan connection. Since p+ C p the induced conformal structure on m agrees with the original conformal structure. Thus, the Fefferman-type construction (with normalisation) and the described factorisation are (locally) inverse to each other. ■ For a reformulation of the characterisation theorem in terms of underlying objects, see Section 6.2. 5 Reduced scales and explicit normalisation Although we obtained the desired characterisation in Theorem 4.14, we do not yet know the explicit relationship between the induced Cartan connection form ujlnd and the normal conformal Cartan connection form uJno1. One of the aims of the present section is to obtain a formula for this difference, which is achieved in Theorem 5.7. As a consequence, we also obtain an explicit formula for the curvature fjmd in terms of the normal conformal Cartan curvature fjn°r in Corollary 5.8. In this more refined analysis, reduced scales will play an important role. 5.1 Characterisation of reduced scales The notion of reduced Weyl structures and reduced scales is introduced in Section 3.4. Here we shall find an intrinsic characterisation (i.e., using the conformal structure only) of reduced scales and discuss their properties. As the scale bundle on the projective manifold m we may consider the positive elements in the density bundle E(l), which is the projecting part of the dual standard tractor bundle T*, see Section 2.4. Similarly, on the Fefferman space m we take the positive elements in the density bundle E(l), the projecting part of the conformal standard tractor bundle T ■ Hence for a projective scale p £ r(E+(l)) we have the tractor Lq* (p) £ T(T*); similarly, for a conformal scale cr G r(E+(l)) we have the tractor Lq( m seen as a subset of all sections of E+(l) —> m, see Remark 3.4. On the other hand, sections of T* —> m are understood as specific sections of the bundle J->M, which is a subbundle in T —> m, see the generalities in Section 3.4 and the setup of our construction in Section 3.2. It follows that these two natural inclusions commute with the BGG-splitting operators. Lemma 5.1. The full arrows in the following diagram commute: r(T*) c-^r(f) Lf LI n0 n0 r(E+(i)) c—^r(E+[i]). Proof. Consider a projective density p £ T(E+(1)) on m, the corresponding tractor Lq*(p) £ r(T*), and its extension to J7 C T, which is denoted by s'. The extension of p £ T(E+(1)) to E+[l] obviously coincides with the projection no(s')> and it is denoted by cr. We need to show A Projective-to-Conformal Fefferman-Type Construction 25 that s' = LT((j), i.e., that <9*Vnors' = 0. According to Proposition 4.4, cDnor = wind + ^ + ^2 with tf1 G fi^M,A2J), $2 G fi^M,/0), hence we have VnoV = Vinds' + • a' + 2 • a'. Since A2J7 acts trivially on T C T, we have • s' = 0. Since /° C T*M, it follows that 0*($2 • a') = 0. It thus follows that <9*(VnoV) = <9*(Vinds'). Let 0 be the frame form of VLq*(p). Then, according to Lemma 4.1, we have that d*4> = 0 since A2F • F = 0, and m particular <9*(VnoV) =0. ■ We can now characterise reduced scales in terms of the corresponding scale tractors: Proposition 5.2. Let (M,c) be a conformal spin structure associated to an oriented projective structure (M, p) via the Fefferman-type construction. Let a G T(E+[1]) be a conformal scale and let s := L0r(cr) G r(T) be the corresponding scale tractor. Then the following statements are equivalent: (a) The scale a is reduced. (b) The tractor s is a section of J- C T. (c) The twistor spinor \ *s parallel with respect to the Levi-Civita connection D of the metric corresponding to the scale a. Furthermore, in a reduced scale, the Schouten tensor is strictly horizontal, i.e., it satisfies vaPah = 0, for all va G T(kerx), (5.1) and the scalar curvature J vanishes. Proof, (a) => (b): This follows from definitions and Lemma 5.1. (b) => (c): The condition (b) means that s ■ sp = 0. According to (2.6), (2.8) and (2.7), this condition expanded in slots yields where we use the Levi-Civita connection D corresponding to a. In particular, px = 0 and, since x is a twistor spinor, the condition (c) follows. (c) => (b): The condition (c) yields sp = (£) according to (2.8). The fact that Vqsf = 0 yields Pac7cX = 0 according to (2.7). Hence (5.1) holds, which in particular means J = 0. Summarising, we have s = 4( (a): According to the previous reasoning and (2.5), we have Hence Vnors is strictly horizontal, i.e., j;°V™s = 0 for every va G T(kerx). Since Vnor = ymd_l_\j; anci \j> is horizontal, the horizontality of Vnors is equivalent to the horizontality of Vmds. Altogether, the condition (a) follows from Proposition 3.3 and Lemma 5.1. ■ 26 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik We will need some finer discussion on the slots of the distinguished tractor Pa K= ( pab\v ) eT(AM) (5.3) ka in reduced scales. From Proposition 4.8 we know that K is the BGG-splitting L04M(/c), which in particular means that pat, = D^kf^ and p = — <^iDrkr according to (2.12). However, in the following statement we only exploit the algebraic properties of K, namely, that it acts by minus and plus the identity on J- and £, respectively. Lemma 5.3. Let us fix a reduced scale. Then the expression o/K as in (5.3) satisfy pa = 0, p = -1, parvr = ~va for every va G T(kerx) and pj Prb = 9ab- Further we have pah = {^[a^b}X-,V) for some fj G T (S^ [— ^]). Proof. Firstly, we use K • s = — s for any s G r(j^). The scale tractor s = L^^cr) of a reduced scale a is a section of J7 and it has the form as in (5.2). Thus it follows from (2.9) that pa = 0 and p = — 1. Next, for every va G T(ker x), the tractor s = ) is clearly a section of J7, since s ■ sp = 0. Thus it follows from (2.9) that pjvr = —va. Secondly, we use K • s = s for any s G T (<£"). Considering the tractor s = ^ wa ^ with arbitrary coa G r(TM), the tractor s:=s + K»sisa section of £, whose middle slot is coa + pjujr. It follows again from (2.9) that parPrb = 9ab- Thirdly, we use (3.3) which shows how K is built from sp and sp. Since the top slot of sp vanishes, middle slots of K are given by a suitable tensor product of \ and the top slot of sp. ■ We will also need more properties of conformal curvature quantities in reduced scales. Lemma 5.4. In a reduced scale, Wahcdpcd = 0, where pah = D{akh], (5.4) vcYahc = 0, for all va G T(ker x) ■ (5.5) Proof. In slots, the condition Q^1 • sp = 0 implies that WabcdlcldX = 0- Pairing both sides of the latter equality with a spinor 77 from Lemma 5.3 yields (5.4). Consider two arbitrary sections va,wh of / = ker%. Conditions (W) and (5.1) imply Rabcdv(lwd = 0. Now, inserting va and we into the Bianchi identity -D[a-R&c]de = 0> we obtain vaweDaRtJaie = 0, where we used the fact that / is parallel. Since we can always regard gah as a section of / <8> /*, this implies 0 = gecvaDaR()C(ie = vaDaRic()C where Ricf,c is the Ricci tensor of Da. Since J = 0, we have that Pat, is proportional to Ricaf, by a constant factor. Thus vaDaP)JC = 0. From (5.1) and since / is parallel we also have vhDaP)JC = 0. Altogether, (5.5) follows by the definition of the Cotton tensor. ■ 5.2 Explicit normalisation formula So far we discussed three Cartan connections on the Fefferman space M: the induced one ujind (Section 3.3), the corresponding normal one ujuot (Section 4.1) and the modified auxiliary one uj' (Section 4.3). Various properties of these and derived objects are enumerated in Propositions 4.9 and 4.12. The following proposition refines the integrability conditions included there. A Projective-to-Conformal Fefferman-Type Construction 27 Proposition 5.5. Let (M, c) be the conformal spin structure induced from an oriented projective structure (M, p) via the Fefferman-type construction. Then, along the reduction Q ^ Q, ixKnm(u) G f° ® A2F, for allX G /, it G 0, (5.6) ixk'(u) = 0, /or allX £ f, u G 0. (5.7) Proof. From (4.5) we already know that ixKnm has values in /° (A2F©/°). We note that the top slot of sections of A2 J7 vanishes in reduced scales, cf. (3.7). Thus the part in f° corresponds to vrYafjr for a v G T(/), which however has to vanish by (5.5). Hence (5.6) follows. The last condition (5.7) follows from k' = (/snor)sl^n+1y cf- Proposition 4.12. ■ Since uj' is an SL(n+ l)-connection on Q —> M, it is the extension of a Cartan connection lj', on Q —> M. Now, due to (5.7), any section v G r(ker%) inserts trivially into its curvature. But this is the standard condition on the connection lj' to be a Cartan connection also on the bundle Q —> M, i.e., to be a projective Cartan connection, cf. [5]. Furthermore, we will show that the descended Cartan connection is normal, i.e., lj' = lj. To do this, we first compute d*H' and then use the relation between the co-differentials d* on M and d* on M discussed in Lemma 4.1. Proposition 5.6. The curvature k' satisfies d*7i'(u) = ik7inoT(u) G f° ® A2F, for all u G Q. (5.8) Proof. We shall compute d*Q! directly. First observe that using Proposition 4.12 we have fi' = (^nor)s((n+i) = ^n°r + 5K • ^n°r- Hence = ^d*(K • ttnm), since d*ttnm = 0. Using (5.3) and (2.11), we compute K • S7°£r as _PrWahrc - HcrYrab + ipYcab ^ N [c0Mci]r + ^[cfy^ci\ab I ^ Yrab V krWahrc J In a reduced scale, from the previous display together with Lemmas 5.3 and 5.4 we compute r(K.^x) = ^wraC0C11 oj =2krnz, which yields (5.8). ■ Theorem 5.7. Let (Q,lj) be a projective normal Cartan geometry over M and let (£,wind) be the conformal Cartan geometry over M induced via the Fefferman-type construction. Then (a) wind = u' = wnor - \ikKnoT, (b) wnor = wind + tf1, where tf1 = -\d*kina = \ikHwl'. Proof, (a) We use that ixfi' = 0 for all X G / according to (5.7). Then Proposition 5.6 together with Lemma 4.1 imply that d*k' = 0. Thus lj' is projectively normal, and therefore we obtain lj' = wind. (b) The normalisation process of Proposition 4.4 provides ^ = "J1 + ^2 such that cDnor = ujind + ^, where ^2 are the first and second normalisation steps. However since lj' = cDmd, it follows from Proposition 5.6 and (4.8) that 3*k' = d*kind is, up to a constant multiple, the difference between wnor and wmd. Therefore already the first normalisation step completes the normalisation, i.e., ^2 = 0. ■ • Wabdodj 0 = V kc J ^ o ) 28 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik Using the explicit relationship provided in Theorem 5.7 we can also obtain a detailed description of the difference between the induced and the normal Cartan curvatures: Corollary 5.8. In a reduced scale, we have the following relation between the curvatures of the induced and the normal conformal Cartan connection: Qind ab -K. n°?r 2 ab ( cab \ Wabcoa - Wabr[copcl]r + k[CQYcl}ab | 0 V 2 krWabrc (5.9) In particular, hikW is the torsion of the induced Cartan connection uo J ind Proof. We obtained the concrete expression of K • S7nor in the proof of Proposition 5.6. Now Lemmas 5.3 and 5.4, and a short computation yields (5.9). ■ 6 Comparison with Patterson—Walker metrics and alternative characterisation In this section we will show that the Fefferman-type construction studied in this article is closely related to the construction of so-called Patterson-Walker metrics. These are the Riemann extensions of affine connected spaces, firstly described in [26]. A conformal version of this construction was obtained by [15] for dimension n = 2, and was treated by the authors of the present article in general dimension in [20]. 6.1 Comparison Let M be a smooth manifold and p: T*M —?> M its cotangent bundle. The vertical subbundle V C T(T*M) of this projection is canonically isomorphic to T*M. An affine connection D on M determines a complementary horizontal distribution H C T(T*M) that is isomorphic to TM via the tangent map of p. Definition 6.1. The Riemann extension or the Patterson-Walker metric associated to a torsion-free affine connection D on M is the pseudo-Riemannian metric g on T*M fully determined by the following conditions: (a) both V and H are isotropic with respect to g, (b) the value of g with one entry from V and another entry from H is given by the natural pairing between V = T*M and H = TM. It follows that V is parallel with respect to the Levi-Civita connection of the just constructed metric. Hence Patterson-Walker metrics are special cases of Walker metrics, i.e., metrics admitting a parallel isotropic distribution. The explicit description of the metric g in terms of the Christoffel symbols of D can be found in [20, 26]. The previous definition can be adapted to weighted cotangent bundles T*M(w) = T*M E(if), provided that M is oriented and D is special, i.e., preserving a volume form on M, which allows a trivialisation of K(w). It turns out that Patterson-Walker metrics induced by projectively equivalent connections are conformally equivalent if and only if w = 2 (interpreted as a projective weight according to the conventions from Section 2.4). Altogether, we have a natural split-signature conformal structure on T*M(2) induced by an oriented projective structure (M, p). From Section 3.3 we know that M = T*M(2) \ {0} is the Fefferman space. Special affine connections from p are just the exact Weyl connections of the corresponding parabolic geometry. A Projective-to-Conformal Fefferman-Type Construction 29 The corresponding objects on M are the reduced Weyl connections, respectively reduced scales, which correspond to distinguished metrics in the conformal class, see Section 3.4. We are going to show that these metrics are just the Patterson-Walker metrics. Proposition 6.2. Let (M, c) be the conformal structure of signature (n,n) associated to an Tridimensional projective structure (M, p) via the Fefferman-type construction. Then any metric in c corresponding to a reduced scale is a Patterson-Walker metric. Proof. Within the proof we refer to the notation and explicit matrix realisations from Appendix A. By definition, the Fefferman space is M = Q/Q, which yields TM = Q Xq g/q. Conformally invariant objects on M, respectively objects related to a choice of reduced scale, correspond to data on g/q = g/p that are invariant under the action of Q, respectively Gff n Q. Elements in g/q will be represented by matrices of the form z 2 * X * w Y' where z,w G R and X, Y G Rn_1. Firstly, one verifies that YlX - zw, (6.1) is the only quadratic form that is invariant under Gqs fl Q. Hence any reduced-scale metric in c corresponds to the quadratic form (6.1) in a suitable frame. Secondly, the vertical subbundle V C TM corresponds to the Q-invariant subspace / = p/q C g/q given by X = 0 and w = 0. The horizontal distribution H C TM induced by a linear connection from p corresponds to the unique (Gff fl <5)-invariant subspace h e2n+2) and the decomposition (3.4) can be written as E®F A2E A2F E ® F ) ' q = A2{E®F) : For v := e\ + e2n+2, the Lie algebra p of the parabolic subgroup P C G is of the following form (A.l) (a Ul w 0 -Wl -b X B V w C -X 0 yt c b X1 0 0 -Yl -d — a -X1 0 Y D -Z -U —Bl —Y \d Zl 0 —w -V1 —c where a,b, c, d,w £ with a so(n — 1). The nilradical p+ = p1- is then of the form P+ d-c,U, V, W, X,Y,Z G B G gl(n - 1) and C, D G (a Ul w 0 -V1 -a\ 0 0 V V 0 0 0 0 a a 0 0 0 0 — a —a 0 0 0 0 -U -U 0 0 \a u* 0 —w -V* -a) A choice of Levi subalgebra go Q P determines a grading g = g_ © go © p+- We shall choose go = p n pop7 where pop C g is the stabiliser of the light-like vector en+2- Explicitly, 00 a 0 0 0 0 0 X B V 0 C —X 0 yt c 0 X1 0 0 _yt —a — c — a -X1 0 Y D -Z 0 —Bl —Y a + c Zl 0 0 -V1 —c 32 M. Hammerl, K. Sagerschnig, J. Silhan, A. Taghavi-Chabert and V. Zadnik The embedding i': g g of Lie algebras has the form Ah> (|J _°At). The subgroup Q = i~1(P) is contained in P, the stabiliser in G of v = {v)e = ei; the inclusion of corresponding Lie algebras is fa ul w \ i j fa C7* ° A v 1 0 B A Vo 0 -a) 1 1 X1 c q = 0np= 0 A V C 0 5 v j = p, \0 0 -a) \0 X1 c) where tr(A) = 0 and a + ti{B) + c = 0. The standard projective grading g = g_ © go © P+> 0 °) I j 0 °\ \x 0 o1 00 = 0 V) \y 0 1 1 A* is compatible with the previous conformal grading so that the reduced Lie subalgebra qo := qHgo coincides with the intersection of go n go- Explicitly, qo = 0 A V I , (A.2) \0 C where tr(A) = 0. A cknowledgement s The authors express special thanks to Maciej Dunajski for motivating the study of this construction and for a number of enlightening discussions on this and adjacent topics. KS would also like to thank Pawel Nurowski for drawing her interest to the subject and for many useful conversations. MH gratefully acknowledges support by project P23244-N13 of the Austrian Science Fund (FWF) and by 'Forschungsnetzwerk Ost' of the University of Greifswald. KS gratefully acknowledges support from grant J3071-N13 of the Austrian Science Fund (FWF). JS was supported by the Czech science foundation (GACR) under grant P201/12/G028. AT-C was funded by GACR post-doctoral grant GP14-27885P. 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