The Geometry of Almost Einstein (2,3,5) Distributions

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures $\mathbf{c}$ that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce $\mathbf{c}$. Second, they are characterized by the existence of a holonomy reduction to $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; K\"ahler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional K\"ahler-Einstein or para-K\"ahler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative.


Introduction
A (pseudo-)Riemannian metric g ab is said to be Einstein if its Ricci curvature R ab is a multiple of g ab . The problem of determining whether a given conformal structure (locally) contains an Einstein metric has a rich history, and dates at least to Brinkmann's seminal investigations [9,10] in the 1920s. Other significant contributions have been made by, among others, Hanntjes and Wrona [39], Sasaki [57], Wong [66], Yano [67], Schouten [58], Szekeres [63], Kozameh, Newman, and Tod [47], Bailey, Eastwood, and Gover [4], Fefferman and Graham [29], and Gover and Nurowski [36]. Developments in this topic in the last quarter century in particular have stimulated substantial development both within conformal geometry and far beyond it.
In the watershed article [4], Bailey, Eastwood, and Gover showed that the existence of such a metric in a conformal structure (here, and always in this article, of dimension n ≥ 3) is governed by a second-order, conformally invariant linear differential operator Θ V 0 that acts on sections of a natural line bundle E [1] (we denote by E[k] the kth power of E [1]): Every conformal structure (M, c) is equipped with a canonical bilinear form g ∈ Γ(S 2 T * M ⊗E [2]), and a nowhere-vanishing section σ in the kernel of Θ V 0 determines an Einstein metric σ −2 g in c and vice versa. Writing the differential equation Θ V 0 (σ) = 0 as a first-order system and prolonging once yields a closed system and hence determines a conformally invariant connection ∇ V on a natural vector bundle V, called the (standard) tractor bundle. (The conformal structure determines a parallel tractor metric H ∈ Γ(S 2 V * ).) By construction, this establishes a bijective correspondence between Einstein metrics in c and parallel sections of this bundle satisfying a genericity condition. This framework immediately suggests a natural relaxation of the Einstein condition: A section of the kernel of Θ V 0 is called an almost Einstein scale, and it determines an Einstein metric on the complement of its zero locus and becomes singular along that locus. A conformal structure that admits a nonzero almost Einstein scale is itself said to be almost Einstein, and, somewhat abusively, the metric it determines is sometimes called an almost Einstein metric on the original manifold. The generalization to the almost Einstein setting has substantial geometric consequences: The zero locus itself inherits a geometric structure, which can be realized as a natural limiting geometric structure of the metric on the complement. This arrangement leads to the notion of conformal compactification, which has received substantial attention in its own right, including in the physics literature [62].
This article investigates the problem of existence of almost Einstein scales, as well as the geometric consequences of existence of such a scale, for a fascinating class of conformal structures that arise naturally from another geometric structure: A (2, 3, 5) distribution is a 2-plane distribution D on a 5-manifold which is maximally nonintegrable in the sense that [D, [D, D]] = T M . This geometric structure has attracted substantial interest, especially in the last few decades, for numerous reasons: (2,3,5) distributions are deeply connected to the exceptional simple Lie algebra of type G 2 (in fact, the study of these distributions dates to 1893, when Cartan [20] and Engel [28] simultaneously realized that Lie algebra as the infinitesimal symmetry algebra of a distribution of this type), they are the subject of Cartan's most involved application of his celebrated method of equivalence [21], they comprise a first class of distributions with continuous local invariants, they arise naturally from a class of second-order Monge equations, they arise naturally from mechanical systems entailing one surface rolling on another without slipping or twisting [1,2,7], they can be used to construct pseudo-Riemannian metrics whose holonomy group is G 2 [38,49], they are natural compactifying structures for indefinite-signature nearly Kähler geometries in dimension 6 [37], and they comprise an interesting example of a broad class of so-called parabolic geometries [18,Section 4.3.2]. (Here and henceforth, the symbol G 2 refers to the algebra automorphism group of the split octonions; this is a split real form of the complex simple Lie group of type G 2 .) For our purposes their most important feature is the natural construction, due to Nurowski [52,Section 5], that associates to any (2,3,5) distribution (M, D) a conformal structure c D of signature (2, 3) on M , and it is these structures, which we call (2,3,5) conformal structures, whose almost Einstein geometry we investigate. For expository convenience, we restrict ourselves to the oriented setting: A (2, 3, 5) distribution (M, D) is oriented iff D → M is an oriented bundle; an orientation of D determines an orientation of M and vice versa.
The key ingredient in our analysis is that, like almost Einstein conformal structures, (2,3,5) conformal structures can be characterized in terms of the holonomy group of the normal tractor connection, ∇ V (for any oriented conformal structure of signature (2,3), ∇ V has holonomy contained in SO(H) ∼ = SO (3,4)): An oriented conformal structure c of signature (2,3) coincides with c D for some (2,3,5) distibution D iff the holonomy group of ∇ V is contained inside G 2 , or equivalently, iff there is a parallel tractor 3-form, that is, a section Φ ∈ Γ(Λ 3 V * ), compatible with the conformal structure in the sense that the pointwise stabilizer of Φ p in GL(V p ) (at any, equivalently, every point p) is isomorphic to G 2 and is contained inside SO(H p ) [41,52]. While the construction D c D depends at each point on the 4-jet of D, the corresponding compatibility condition in the tractor setting is algebraic (pointwise), which reduces many of our considerations and arguments to properties of the algebra of G 2 .
With these facts in hand, it is immediate that whether an oriented conformal structure of signature (2,3) is both (2,3,5) and almost Einstein is characterized by the admission of both a compatible tractor 3-form Φ and a (nonzero) parallel tractor S ∈ Γ(V), which we may just as well frame as a reduction of the holonomy of ∇ V to the 8-dimensional common stabilizer S in SO (3,4) of a nonzero vector in the standard representation V of SO (3,4) and a 3-form in Λ 3 V * compatible with the conformal structure. The isomorphism type of S depends on the causality type of S: If the vector is spacelike, then S ∼ = SU (1,2); if it is timelike, then S ∼ = SL(3, R); if it is isotropic, then S ∼ = SL(2, R) Q + , where Q + < G 2 is the connected, nilpotent subgroup of G 2 defined via Sections 2.3.4 and 2.3.6. 1 Proposition A. An oriented conformal structure of signature (2, 3) is both (2,3,5) and almost Einstein iff it admits a holonomy reduction to the common stabilizer S of a 3-form in Λ 3 V * compatible with the conformal structure and a nonzero vector in V.
Throughout this article, S, SU(1, 2) SL(3, R), and SL(2, R) Q + refer to the common stabilizer of the data described above -that is, to any subgroup in a particular conjugacy class in SO(3, 4) (and not just to a subgroup of SO (3,4) of the respective isomorphism types).
Given an almost Einstein (2,3,5) conformal structure, algebraically combining Φ and S yields other parallel tractor objects. The simplest of these is the contraction K := −S Φ, which we may identify with a skew endomorphism of the standard tractor bundle, V. Underlying this endomorphism is a conformal Killing field ξ of the induced conformal structure c D that does not preserve any distribution that induces that structure. Thus, if ξ is complete (or alternatively, if we content ourselves with a suitable local statement) the images of D under the flow of ξ comprise a 1-parameter family of distinct (2,3,5) distributions that all induce the same conformal structure. This suggests -and connects with the problem of existence of an almost Einstein scale -a natural question that we call the conformal isometry problem for (2,3,5) distributions: Given a (2,3,5) distribution (M, D), what are the (2,3,5) distributions D on M that induce the same conformal structure, that is, for which c D = c D ? Put another way, what are the fibers of the map D c D ? By our previous observation, working in the tractor setting essentially reduces this to an algebraic problem, which we resolve in Proposition 4.1 (and which extends to the split real form of G 2 an analogous result of Bryant [12,Remark 4] for the compact real form). Translating this result to the tractor setting and then reinterpreting it in terms of the underlying data gives a complete description of all (2,3,5) distributions D that induce the conformal structure c D . In order to formulate it, we note first that, given a fixed oriented conformal structure c of signature (2,3), underlying any compatible parallel tractor 3-form, and hence corresponding to a (2,3,5) distribution D, is a conformally weighted 2-form φ ∈ Γ(Λ 2 T * M ⊗ E [3]), which in particular is a solution to the conformally invariant conformal Killing 2-form equation. The weighted 2-forms that arise this way are called generic. This solution turns out to satisfy φ ∧ φ = 0 -so it is locally decomposable -but vanishes nowhere. Hence, it defines an oriented 2-plane distribution, and this distribution is D.
1. Suppose (M, c D ) admits the nonzero almost Einstein scale σ ∈ Γ(E [1]), and denote by S ∈ Γ(V) the corresponding parallel tractor; by rescaling, we may assume that ε := −H Φ (S, S) ∈ {−1, 0, +1}. Then, for any (Ā, B) ∈ R 2 such that −εĀ 2 + 2Ā + B 2 = 0 (there is a 1parameter family of such pairs) the weighted 2-form generalization of the H-orbit decomposition of G/P ; the subsets in the decomposition are accordingly termed the curved orbits of the reduction. The curved orbits are parameterized by the intersections of H and P up to conjugacy in G, and H together with these intersections determine the respective geometric structures on each curved orbit. Section 5 carries out this decomposition for the Cartan geometry canonically associated to (M, c) determined by σ and D, that is, by a holonomy reduction to the group S. Besides elucidating the geometry of almost Einstein (2,3,5) conformal structures for its own sake, this serves three purposes: First, this documents an example of a curved orbit decomposition for which the decomposition is relatively involved. Second, and more importantly, we will see that several classical geometries occur in the curved orbit decompositions, establishing novel and nonobvious links between (2,3,5) distributions and those structures. Third, we can then exploit these connections to give new methods for construction of almost Einstein (2,3,5) conformal structures from classical geometries, and using these we produce, for the first time, examples both with negative (Example 6.1) and positive (Example 6.2) Einstein constants.
Different signs of the Einstein constant (equivalently, different causality types of the parallel tractor S corresponding to σ) lead to qualitatively different curved orbit decompositions, so we treat them separately. We say that an almost Einstein scale is Ricci-negative, -positive, or -flat if the Einstein constant of the Einstein metric it determines is negative, positive, or zero, respectively. See also Appendix A, which summarizes the results here and records geometric characterizations of the curved orbits.
In the Ricci-negative case, the decomposition of a manifold into submanifolds is the same as that determined by σ alone, but the family D determines additional structure on each closed orbit. (Herein, for readability we often suppress notation denoting restriction to a curved orbit.) Theorem D − . Let (M, c) be an oriented conformal structure of signature (2,3). A holonomy reduction of c to SU(1, 2) determines a 1-parameter family D of oriented (2,3,5)  • (Section 5.5.3) The orbit M 4 is a smooth hypersurface, and inherits a Fefferman conformal structure c S , which has signature (1, 3): Locally, c S arises from the classical Fefferman construction [16,30,51], which (in this dimension) canonically associates to any 3-dimensional CR structure (L 3 , H, J) a conformal structure on a circle bundle over L 3 . Again in the local setting, the fibers of the fibration M 4 → L 3 are the integral curves of ξ.
The Ricci-positive case is similar to the Ricci-negative case but entails 2-dimensional curved orbits that have no analogue there.
Theorem D + . Let (M, c) be an oriented conformal structure of signature (2,3). A holonomy reduction of c to SL(3, R) determines a 1-parameter family D of oriented (2, 3, 5) distributions related by a Ricci-positive almost Einstein scale such that c = c D for all D ∈ D, as well as • (Section 5.4) The orbits M ± 5 are open, and M 5 := M + 5 ∪ M − 5 is equipped with a Riccipositive Einstein metric g := σ −2 g| M 5 . The pair (−g, ξ) is a para-Sasaki structure (see Section 5.4.3) on M 5 . Locally, M 5 fibers along the integral curves of ξ, and the leaf space L 4 inherits a para-Kähler-Einstein structure (ĝ,K).
• (Section 5.5.4) The orbit M 4 is a smooth hypersurface, and inherits a para-Fefferman conformal structure c S , which has signature (2, 2): Locally, c S arises from the paracomplex analogue of the classical Fefferman construction, which (in this dimension) canonically associates to any Legendrean contact structure (L 3 , H + ⊕ H − ) -or, locally equivalently, a point equivalence class of second-order ODEsÿ = F (x, y,ẏ) -a conformal structure on a SO(1, 1)-bundle over L 3 . Again in the local setting, the fibers of the fibration M 4 → L 3 are the integral curves of ξ.
The descriptions of the geometric structures in the above two cases are complete in the sense that any other geometric data determined by the holonomy reduction to S can be recovered from the indicated data. We do not claim the same for the descriptions in the Ricci-flat case, which we can view as a sort of degenerate analogue of the other two cases.
Theorem D 0 . Let (M, c) be an oriented conformal structure of signature (2,3). A holonomy reduction of c to SL(2, R) Q + determines a 1-parameter family D of oriented (2, 3, 5) distributions related by a Ricci-flat almost Einstein scale such that c = c D for all D ∈ D, as well as • (Section 5.4) The orbits M ± 5 are open, and M 5 := M + 5 ∪ M − 5 is equipped with a Ricciflat metric g := σ −2 g| M 5 . The pair (−g, ξ) is a null-Sasaki structure (see Section 5.4.3) on M 5 . Locally, M 5 fibers along the integral curves of ξ, and the leaf space L 4 inherits a null-Kähler-Einstein structure (ĝ,K).
• (Section 5.5.5) The orbit M 4 is a smooth hypersurface, and it locally fibers over a 3-manifold L that carries a conformal structure c L of signature (1, 2) and isotropic line field.
• (Section 5.6. 2) The orbit M 2 has dimension 2 and is equipped with a preferred line field.
• (-) The orbits M ± 0 consist of isolated points and so carry no intrinsic geometry.
The statements of these theorems involve an expository choice that entails some subtle consequences: In each case, the holonomy reduction determines an almost Einstein scale σ and 1-parameter family D of oriented (2,3,5) distributions, but this reduction does not distinguish a distribution within this family. Alternatively, we could specify for c an almost Einstein scale and a distribution D such that c = c D . Such a specification determines a holonomy reduction to S as above, but the choice of a preferred D is additional data, and this is reflected in the induced geometries on the curved orbits.
Proposition 5.18 gives a partial converse to the statements in Theorems D − and D + about the ε-Sasaki-Einstein structures (−g, ξ) induced on the open orbits by the corresponding holonomy reductions: Any ε-Sasaki-Einstein structure (−g, ξ) (here restricting to ε = ±1) determines around each point a 1-parameter family of oriented (2,3,5) distributions related by the almost Einstein scale for [g] corresponding to g, and by construction the ε-Sasaki structure is the one induced by the corresponding holonomy reduction.
We also briefly present a generalized Fefferman construction that essentially inverts the projection M 5 → L 4 along the leaf space fibration in the non-Ricci-flat cases, in a way that emphasizes the role of almost Einstein (2, 3, 5) conformal structures (see Section 5.4.6). In particular, any non-Ricci-flat ε-Kähler-Einstein metric of signature (2,2) gives rise to a 1parameter family of (2, 3, 5) distributions. We treat this construction in more detail in an article currently in preparation [56].
As mentioned above we have for convenience formulated our results for oriented (2, 3, 5) distributions and conformal structures, but all the results herein have analogues for unoriented distributions and many of our considerations are anyway local. Alternatively one could further restrict attention to space-and time-oriented conformal structures (see Remark 4.11) or work with conformal spin structures, the latter of which would connect the considerations here more closely with those in [42].
Finally, we mention briefly one aspect of this geometry we do not discuss here, but which will be taken up in a shorter article currently in preparation: One can construct for any oriented (2,3,5) distribution D an invariant second-order linear differential operator that acts on sections of E[1] ∼ = Λ 2 D closely related to Θ V 0 [55]. Its kernel can again be interpreted as the space of almost Einstein scales of c D , but it is a simpler object than Θ V 0 , enough so that one can use it to construct new explicit examples of almost Einstein (2,3,5) distributions. Among other things, the existence of such an operator emphasizes that almost Einstein geometry of the induced conformal structure is a fundamental feature of the geometry of (2, 3, 5) distributions.
For simplicity of statements of results, we assume that all given manifolds are connected; we do not include this hypothesis explicitly in our statements of results.
We use both index-free and Penrose index notation throughout, according to convenience.

ε-complex structures
The ε-complex numbers, ε ∈ {−1, 0, +1}, is the ring C ε generated over R by the generator i ε , which satisfies precisely the relations generated by i 2 ε = ε. An ε-complex structure on a real vector space W (necessarily of even dimension, say, 2m) is an endomorphism K ∈ End(W) such that K 2 = ε id W ; 2 ; if ε = +1, we further require that the (±1)-eigenspaces both have dimension m. This identifies W with C m ε (as a free C ε -module) so that the action of K coincides with multiplication by i ε , and the pair (W, K) is an ε-complex vector space.

Split cross products in dimension 7
The geometry studied in this article depends critically on the algebraic features of a so-called (split) cross product × on a 7-dimensional real vector space V. One can realize this explicitly using the algebra O of split octonions; we follow [37,Section 2], and see also [54]. This is a composition (R-)algebra and so is equipped with a unit 1 and a nondegenerate quadratic form N multiplicative in the sense that N (xy) = N (x)N (y) for all x, y ∈ O. In particular N (1) = 1, and polarizing N yields a nondegenerate symmetric bilinear form, which turns out for O to have signature (4,4). So, the 7-dimensional vector subspace I = 1 ⊥ of imaginary split octonions inherits a nondegenerate symmetric bilinear form H of signature (3,4), as well as a map × : I × I → I defined by x × y := xy + H(x, y)1; this is just the orthogonal projection of xy onto I. This map is a (binary) cross product in the sense of [11], that is, it satisfies for all x, y ∈ I. 2 In the case ε = 0, some references require additionally that a null-complex structure satisfy rank K = m [27] this anyway holds for the null-complex structures that appear in this article. Definition 2.1. We say that a bilinear map × : V × V → V on a 7-dimensional real vector space V is a split cross product iff there is a linear isomorphism A : A split cross product × determines a bilinear form H × (x, y) := − 1 6 tr(x × (y × · )); of signature (3,4) on the underlying vector space. For the split cross product × on I, H × = H. We say that a cross product × is compatible with a bilinear form iff× induces H, that is, iff H = H × . It follows from the alternativity identity (xx)y = x(xy) satisfied by the split octonions that (2.2) A 3-form is said to be split-generic iff it arises this way, and such forms comprise an open GL(V)-orbit under the standard action on Λ 3 V * . One can recover from any split-generic 3-form the split cross product × that induces it. A split cross product × also determines a nonzero volume form × ∈ Λ 7 V * for H × : Thus, × determines an orientation [ × ] on V and Hodge star operators * : If V is a vector space endowed with a bilinear form H of signature (p, q) and an orientation Ω, the subgroup of GL(V) preserving the pair (H, Ω) is SO(p, q), so we refer to such a pair as a SO(p, q)-structure on V. We say that a cross product × on V is compatible with an SO(p, q)structure (H, Ω) iff × induces H and Ω, that is, iff H = H × and Ω = [ × ].
A split-generic 3-form Φ satisfies various contraction identities, including [12, equations (2.8) and (2.9)]: The (algebra) automorphism group of O is a connected, split real form of the complex Lie group of type G 2 , and so we denote it by G 2 . One can recover the algebra structure of O from (I, ×), so G 2 is also the automorphism group of ×, and equivalently, the stabilizer subgroup in GL(V) of a split-generic 3-form on a 7-dimensional real vector space V. For much more about G 2 , see [45]. The action of G 2 on V defines the smallest nontrivial, irreducible representation of G 2 , which is sometimes called the standard representation. This action stabilizes a unique split cross product, or equivalently, a unique split-generic 3-form (up to a positive multiplicative constant). Thus, by a G 2 -structure on a 7-dimensional real vector space V we mean either (1) a representation of G 2 on V isomorphic to the standard one, or, (2) slightly abusively (on account of the above multiplicative ambiguity), a split cross product × on V, or equivalently, a split-generic 3-form Φ on V.
Since a split cross product × on a vector space V determines algebraically both a bilinear form H × and an orientation [ × ] on V, the induced actions of G 2 preserve both, defining a natural embedding Moreover, H × realizes V as the standard representation of SO (3,4), and its restriction to G 2 is the standard representation × defines. Like SO (3,4), the G 2 -action on the ray projectivization of V has exactly three orbits, namely the sets of spacelike, isotropic, and timelike rays [65,Theorem 3.1].
It is convenient for our purposes to use the G 2 -structure on V defined in a basis (E a ) (with dual basis, say, (e a )) via the 3-form where e a 1 ···a k := e a 1 ∧ · · · ∧ e a k (cf. [41, equation (23)]). With respect to the basis (E a ), the induced bilinear form H Φ has matrix representation where I 2 denotes the 2 × 2 identity matrix, and the induced volume form is Let g 2 denote the Lie algebra of G 2 . Differentiating the inclusion G 2 → GL(V) yields a Lie algebra representation g 2 → gl(V) ∼ = End(V), and with respect to the basis (E a ) its elements are precisely those of the form where A ∈ gl(2, R), W, X ∈ R 2 , Y, Z ∈ (R 2 ) * , r, s ∈ R, and J := 0 −1 1 0 .

Some G 2 representation theory
Fix a 7-dimensional real vector space V and a G 2 -structure Φ ∈ Λ 3 V * . We briefly record the decompositions of the G 2 -representations Λ 2 V * , Λ 3 V * , and S 2 V * into irreducible subrepresentations; we use and extend the notation of [12, Section 2.6]. In each case, Λ k l denotes the irreducible subrepresentation of Λ k V * of dimension l, which is unique up to isomorphism.
The representation Λ 2 V * ∼ = so(H Φ ) decomposes into irreducible subrepresentations as is isomorphic to the adjoint representation g 2 . Define the map ι 2 7 : V → Λ 2 V * by by raising an index we can view ι 2 7 as the map V → so(H Φ ) given by S → (T → T × S). It is evidently nontrivial, so by Schur's lemma its (isomorphic) image is Λ 2 7 . Conversely, consider the map π 2 7 : Λ 2 V * → V defined by Raising indices gives a map Λ 2 V → V, which up to the multiplicative constant, is the descent of × : V × V → V via the wedge product. In particular it is nontrivial, so it has kernel Λ 2 14 and restricts to an isomorphism π 2 7 | Λ 2 7 : Λ 2 7 → V; we have chosen the coefficient so that π 2 7 • ι 2 7 = id V and ι 2 7 • π 2 7 | Λ 2 7 = id Λ 2 7 . Since G 2 is the stabilizer subgroup in SO(H Φ ) of Φ, g 2 is the annihilator in so(H Φ ) ∼ = Λ 2 V * of Φ. Expanding id V −ι 2 7 • π 2 7 using (2.8) and (2.9) and applying (2.3) gives that under this identification, the corresponding projection π 2 14 : The representation Λ 3 V * decomposes into irreducible subrepresentations as Here, Λ 3 1 is just the trivial representation spanned by Φ, and the map π 3 1 : is a left inverse for the map R • V * , namely, into its H Φ -trace and H Φ -tracefree components, respectively. The linear map i :

Cartan geometry
In this subsubsection we follow [18,61]. Given a P -principal bundle π : G → M , we denote the (right) action of G × P → G by R p (u) = u · p for u ∈ G, p ∈ P . For each V ∈ p, the corresponding Definition 2.2. For a Lie group G and a closed subgroup P (with respective Lie algebras g and p), a Cartan geometry of type (G, P ) on a manifold M is a pair (G → M, ω), where G → M is a P -principal bundle and ω is a Cartan connection, that is, a section of T * G ⊗ g satisfying 1) (right equivariance) ω u·p (T u R p · η) = Ad(p −1 )(ω u (η)) for all u ∈ G, p ∈ P , η ∈ T u G, 2) (reproduction of fundamental vector fields) ω(η V ) = V for all V ∈ p, and 3) (absolute parallelism) ω u : T u G → g is an isomorphism for all u ∈ G.
The (flat) model of Cartan geometry of type (G, P ) is the pair (G → G/P, ω MC ), where ω MC is the Maurer-Cartan form on G defined by (ω MC ) u := T u L u −1 (here L u −1 : G → G denotes left multiplication by u −1 ). This form satisfies the identity dω MC + 1 2 [ω MC , ω MC ] = 0. We define the curvature (form) of a Cartan geometry (G, ω) to be the section Ω := dω+ 1 2 [ω, ω] ∈ Γ(Λ 2 T * G ⊗g), and say that (G, ω) is flat iff Ω = 0. This is the case iff around any point u ∈ G there is a local bundle isomorphism between G and G that pulls back ω to ω MC .
One can show that the curvature Ω of any Cartan geometry (G → M, ω) is horizontal (it is annihilated by vertical vector fields), so invoking the absolute parallelism and passing to the quotient defines an equivalent object κ : G → Λ 2 (g/p) * ⊗ g, which we also call the curvature.

Holonomy
Given any Cartan geometry (G → M, ω) of type (G, P ), we can extend the Cartan connection ω to a unique principal connectionω onĜ := G × P G characterized by (1) G-equivariance and (2) ι * ω = ω, where ι : G →Ĝ is the natural inclusion u → [u, e]. Then, to any pointû ∈Ĝ we can associate the holonomy group Holû(ω) ≤ G. Different choices ofû lead to conjugate subgroups of G, so the conjugacy class Hol(ω) thereof is independent ofû, and we define the holonomy of ω (or just as well, of (G → M, ω)) to be this class.

Tractor geometry
Fix a pair (G, P ) as in Section 2.3.1, denote the Lie algebra of G by g, and fix a G-representation U. Then, for any Cartan geometry (π : G → M, ω) of type (G, P ), we can form the associated tractor bundle U := G × P U → M , which we can also view as the associated bundleĜ × G U → M . Then, the principal connectionω onĜ determined by ω induces a vector bundle connection ∇ U on U.
Of distinguished importance is the adjoint tractor bundle A := G × P g. The canonical map Π A 0 : A → T M defined by (u, V ) → T u π · ω −1 u (V ) descends to a natural isomorphism G × P (g/p) ∼ = → T M , and via this identification Π A 0 is the bundle map associated to the canonical projection g → g/p.
Since the curvature Ω of (G, ω) is also P -equivariant, we may regard it as a section K ∈ Γ(Λ 2 T * M ⊗ A), and again we call it the curvature of (G → M, ω).

Parabolic geometry
In this article we will mostly (but not exclusively) work with geometries that can be realized as a special class of Cartan geometries that enjoy additional properties, most importantly suitable normalization conditions on ω that guarantee (subject to a usually satisfied cohomological condition) a correspondence between Cartan geometries satisfying those conditions and geometric structures on the underlying manifold. We say that a Cartan geometry (G → M, ω) of type (G, P ) is a parabolic geometry iff G is semisimple and P is a parabolic subgroup. For a detailed survey of parabolic geometry, including details of the below, see the standard reference [18].
Recall that a parabolic subgroup P < G determines a so-called |k|-grading on the Lie algebra g of G: This is a vector space decomposition g = g −k ⊕ · · · ⊕ g +k compatible with the Lie bracket in the sense that [g a , g b ] ⊆ g a+b and minimal in the sense that none of the summands g a , a = −k, . . . , k, is zero. The grading induces a P -invariant filtration (g a ) of g, where g a := g a ⊕ · · · ⊕ g +k . In particular, p = g 0 = g 0 ⊕ · · · ⊕ g +k . We denote by G 0 < P the subgroup of elements p ∈ P for which Ad(g) preserves the grading (g a ) of g, and by P + < P the subgroup of elements p ∈ P for which Ad(p) ∈ End(g) have homogeneity of degree > 0 with respect to the filtration (g a ); in particular, the Lie algebra of P + is p + = g +1 = g +1 ⊕ · · · ⊕ g +k .
Since g is semisimple, its Killing form is nondegenerate, and it induces a P -equivariant identification (g/p) * ↔ p + . Via this identification, for any G-representation U we may identify the Lie algebra homology H • (p + , U) with the chain complex The Kostant codifferential ∂ * is P -equivariant, so it induces bundle maps ∂ * : Λ i+1 T * M ⊗ U → Λ i T * M ⊗ U between the associated bundles.
Finally, tractor bundles associated to parabolic geometries inherit additional natural structure: Given a G-representation U, P determines a natural filtration (U a ) of U by successive action of the nilpotent Lie subalgebra p + < g, namely (2.14) Since the filtration (U a ) of U is P -invariant, it determines a bundle filtration (U a ) of the tractor bundle U = G × P U.
For the adjoint representation g itself, this filtration (appropriately indexed) is just (g a ), and the images of the filtrands A = G × P g −k · · · G × P g −1 under the projection Π A 0 comprise a canonical filtration T M = T −k M · · · T −1 M of the tangent bundle.

Oriented conformal structures
The group SO(p+1, q+1), p+q ≥ 3, acts transitively on the space of isotropic rays in the standard representation V, and the stabilizer subgroupP of such a ray is parabolic. There is an equivalence of categories between regular, normal parabolic geometries of type (SO(p + 1, q + 1),P ) and oriented conformal structures of signature (p, q) [18, Section 4.1.2]. Definition 2.3. A conformal structure (M, c) is an equivalence class c of metrics on M , where we declare two metrics to be equivalent if one is a positive, smooth multiple of the other. The signature of c is the signature of any (equivalently, every) g ∈ c, and we say that (M, c) is oriented iff M is oriented. The conformal holonomy of an oriented conformal structure c is Hol(c) := Hol(ω), where ω is the normal Cartan connection corresponding to c.
We can choose a basis of V for which the nondegenerate, symmetric bilinear form H preserved by SO(p + 1, q + 1) has block matrix representation where B ∈ so(Σ), X ∈ R p+q , Z ∈ (R p+q ) * . The first element of the basis is isotropic, and if we take choose the preferred isotropic ray in V to be the one determined by that element, the corresponding Lie algebra grading on so(p + 1, q + 1) is the one defined by the labeling Since the grading on so(p + 1, q + 1) induced byP has the form g −1 ⊕ g 0 ⊕ g +1 , any parabolic geometry of this type is regular. The normality condition coincides with Cartan's normalization condition for what is now called a Cartan geometry of this type [18,Section 4

) distributions
The group G 2 acts transitively on the space of H Φ -isotropic rays in V, and the stabilizer subgroup Q of such a ray is parabolic [54]. The subgroup Q is the intersection of G 2 with the stabilizer subgroupP < SO (3,4) of the preferred isotropic ray in Section 2.3.5. In particular, the first basis element is isotropic, and if we again choose the preferred isotropic ray to be the one determined by that element, the corresponding Lie algebra grading on g 2 is the one defined by the block decomposition (2.7) and the labeling There is an equivalence of categories between regular, normal parabolic geometries of type (G 2 , Q) and so-called oriented (2,3,5)  For a regular, normal parabolic geometry (G, ω) of type (G 2 , Q), the underlying (2,

Conformal geometry
In this subsection we partly follow [4]. We may view c itself as the canonical conformal metric, g ab ∈ Γ(S 2 T * M [2]). Contraction with g ab determines an isomorphism T M → T * M [2], which we may use to raise and lower indices of objects on the tangent bundle at the cost of an adjustment of conformal weight. By construction, the Levi-Civita connection ∇ g of any metric g ∈ c preserves g ab and its inverse,

Conformal density bundles
We call a nowhere zero section , of all conformally weighted bundles, and in particular a representative metric τ −2 g ∈ c.

Conformal tractor calculus
For an oriented conformal structure (M, c) of signature (p, q), n := p + q ≥ 3, the tractor bundle V associated to the standard representation V of SO(p + 1, q + 1) is the standard tractor bundle. It inherits from the normal parabolic geometry corresponding to c a vector bundle connection ∇ V . The SO(p + 1, q + 1)-action preserves a canonical nondegenerate, symmetric bilinear form H ∈ S 2 V * and a volume form ∈ Λ n+2 V * ; these respectively induce on V a parallel tractor metric H ∈ Γ(S 2 V * ) and parallel volume form ∈ Γ(Λ n+2 V * ).
Consulting the block structure (2.16) ofp + < so(p + 1, q + 1) gives that the filtration (2.14) of the standard representation V of SO(p + 1, q + 1) determined byP is We may identify the composition series of the corresponding filtration of V as We denote elements and sections of V using uppercase Latin indices, A, B, C, . . ., as S A ∈ Γ(V), and those of the dual bundle V * with lower indices, as S A ∈ Γ(V * ); we freely raise and lower indices using H. The bundle inclusion E[−1] → V determines a canonical section X A ∈ Γ(V [1]). Any scale τ determines an identification of V with the associated graded bundle determined by the above filtration, that is, a are the inclusions determined by τ . Reusing the notation of the filtration of V we write With respect to any scale τ , the tractor metric has the form (cf. (2.15)) In particular, the filtration The subscript ,b denotes the covariant derivative with respect to g := τ −2 g, and P ab is the Schouten tensor of g, which is a particular trace adjustment of the Ricci tensor R ab : A section A A 1 ···A k of the tractor bundle Λ k V * associated to the alternating representation Λ k V * decomposes uniquely as which we write more compactly as The tractor connection ∇ V induces a connection on Λ k V * , and we denote this connection again by ∇ V .
In the special case k = 2, raising an index using H gives Λ 2 V * ∼ = so(p + 1, q + 1), so we can identify Λ 2 V * ∼ = A. Any section A A B ∈ Γ(A) decomposes uniquely as which we write as . This identifies the tractor volume form with the conformal volume form g of g.

Canonical quotients of conformal tractor bundles
For any irreducible SO(p + 1, q + 1)-representation U, the canonical Lie algebra cohomology quotient map U → H 0 := H 0 (p + , U) = U/(p + · U) isP -invariant and so induces a canonical bundle quotient map Π U 0 : U → H 0 between the corresponding associatedP -bundles. (We reuse the notation Π U 0 for the induced map Γ(U) → Γ(H 0 ) on sections.) Given a section A ∈ Γ(U), For the alternating representation Λ k V * , the quotient map is For the adjoint representation so(p + 1, q + 1), the quotient map coincides with the map

Conformal BGG splitting operators
Conversely, for each irreducible SO(p + 1, q + 1)-representation U there is a canonical differential BGG splitting operator [14,19]. The only property of the operators L U 0 we need here follows immediately from this characterization:

Almost Einstein scales
The BGG splitting operator (2.21) Computing gives where (T ab ) • denotes the tracefree part T ab − 1 n T c c g ab of the (possibly weighted) covariant 2tensor T ab , and where * is some third-order differential expression in σ. Since the bottom component of ∇ V L V 0 (σ) is zero, the middle component, regarded as a (second-order) linear differential operator is conformally invariant. The operator Θ V 0 is the first BGG operator [19] associated to the standard representation V for (oriented) conformal geometry.
We can readily interpret a solution σ ∈ ker Θ V 0 geometrically: If we restrict to the complement M − Σ of the zero locus Σ := {x ∈ M : σ x = 0}, we can work in the scale of the solution σ itself: We have σ σ = 1 and hence 0 = Θ V 0 (σ) = P • . This says simply says that the Schouten tensor, P, of g := σ −2 g| M −Σ is a multiple of g, and hence so is its Ricci tensor, that is, that g is Einstein. This motivates the following definition [32]: 5. An almost Einstein scale 3 of an (oriented) conformal structure of dimension n ≥ 3 is a solution σ ∈ Γ(E [1]) of the operator Θ V 0 . A conformal structure is almost Einstein if it admits a nonzero almost Einstein scale.
We denote the set ker Θ V 0 of almost Einstein scales of a given conformal structure c by aEs(c). Since Θ V 0 is linear, aEs(c) is a vector subspace of Γ(E [1]). The vanishing of the component * in (2.22) turns out to be a differential consequence of the vanishing of the middle component, Θ V 0 (σ). So, ∇ V is a prolongation connection for the operator Θ V 0 : ) comprise a natural bijective correspondence between almost Einstein scales and parallel standard tractors: In particular, if σ is an almost Einstein scale and vanishes on some nonempty open set, then σ = 0. In fact, the zero locus Σ of σ turns out to be a smooth hypersurface [17]; see Example 5.2.
We define the Einstein constant of an almost Einstein scale σ to be This definition is motivated by the following computation: On M − Σ the Schouten tensor of the representative metric g : Thus, the Ricci tensor of g is R ab = 2(n − 1)λg ab , so we say that σ (or the metric g it induces) is Ricci-negative, -flat, or -positive respectively iff λ < 0, λ = 0, or λ > 0. 4

Conformal Killing f ields and (k − 1)-forms
The BGG splitting operator Proceeding as in Section 2.5, we find that where each * denotes some differential expression in σ. The bottom component defines an inva- ) (here denotes the Cartan product) and elements of its kernel are called conformal Killing (k − 1)forms [60]. Unlike in the case of almost Einstein scales, vanishing of Θ Λ k V * 0 (φ) does not in general imply the vanishing of the remaining components * ; if they do vanish, that is, if [50].
The BGG splitting operator Thus, the solutions of ker Θ A 0 are precisely the vector fields whose flow preserves c, and so these are called conformal Killing fields. If ∇ V L A 0 (ξ) = 0, we say ξ is a normal conformal Killing field.

(2, 3, 5) conformal structures
About a decade ago, Nurowski observed the following: This construction has since been recognized as a special case of a Fefferman construction, so named because it likewise generalizes a classical construction of Fefferman that canonically assigns to any nondegenerate hypersurface-type CR structure on a manifold N a conformal structure on a natural circle bundle over N [30]. In fact, this latter construction arises in our setting, too; see Section 5.5.3.
We use the following terminology: An oriented (2, 3, 5) distribution D determines an orientation of T M , and hence c D is oriented (henceforth, that symbol refers to an oriented conformal structure).
Because we will need some of the ingredients anyway, we briefly sketch a construction of c D using the framework of parabolic geometry: Fix an oriented (2, 3, 5) distribution (M, D), and per Section 2.3.6 let (G → M, ω) be the corresponding regular, normal parabolic geometry of type (G 2 , Q). Form the extended bundleḠ := G × QP , and letω denote the Cartan connection equivariantly extending ω toḠ. By construction (Ḡ,ω) is a parabolic geometry of type (SO (3,4),P ) (for whichω turns out to be normal, see [41,Proposition 4]), and hence defines an oriented conformal structure on M .
For any (2, 3, 5) distribution (M, D) and for any representation U of SO(p + 1, q + 1), we may identify the associated tractor bundle G × Q U (here regarding U as a Q-representation) with the conformal tractor bundleḠ ×P U, and so denote both of these bundles by U. Sinceω is itself normal, the (normal) tractor connections that ω andω induce on U coincide.

Holonomy characterization of oriented (2, 3, 5) conformal structures
An oriented (2, 3, 5)-distribution D corresponds to a regular, normal parabolic geometry (G, ω) of type (G 2 , Q). In particular, this determines on the tractor bundle parallel with respect to the induced normal connection on V, and again we may identify V and the normal connection thereon with the standard conformal tractor bundleḠ ×P V of c D and the normal conformal tractor connection. The G 2 -structure determines fiberwise a bilinear form H Φ ∈ Γ(S 2 V * ). Since this construction is algebraic, H Φ is parallel, and by construction it coincides with the conformal tractor metric on V determined by c D .
Conversely, if an oriented, signature (2, 3) conformal structure c admits a parallel tractor G 2 -structure Φ whose restriction to each fiber V x is compatible with the restriction H x of the tractor metric (in which case we simply say that Φ is compatible with H), the distribution D underlying Φ satisfies c = c D . This recovers a correspondence stated in the original work of Nurowski [52] and worked out in detail in [41]: Theorem 2.9. An oriented conformal structure (M, c) (necessarily of signature (2, 3)) is induced by some (2, 3, 5) distribution D (that is, c = c D ) iff the normal conformal tractor connection admits a holonomy reduction to G 2 , or equivalently, iff c admits a parallel tractor G 2 -structure Φ compatible with the tractor metric H. Fix an oriented (2, 3, 5) distribution (M, D), let Φ ∈ Γ(Λ 3 V * ) denote the corresponding parallel tractor G 2 -structure, and denote its components with respect to any scale τ of the induced conformal structure c D according to In the language of Section 2.
An argument analogous to that in the proof of Proposition 2.10(5) below shows that φ is locally decomposable, and Proposition 2.10(8) shows that it vanishes nowhere, so the (weighted) bivector field φ ab ∈ Γ(Λ 2 T M [−1]) determines a 2-plane distribution on M , and this is precisely D [41].
We collect for later some useful geometric facts about D and encode them in algebraic identities in the tractor components φ, χ, θ, ψ. Parts (1) and (2) of the Proposition 2.10 are well-known features of (2, 3, 5) distributions.
denote the corresponding parallel tractor G 2 -structure, and denote its components with respect to an arbitrary scale τ as in (2.25). Then: , or equivalently, the line field L that θ determines (which depends on τ ) is orthogonal to D; equivalently, In particular, the line field L is timelike.
Since (by equation (2.26)) it vanishes nowhere, it determines a 2-plane distribution E (which depends on τ ).
In particular, (8) implies that D, L, and E are pairwise transverse and hence span T M . Moreover, (2) and ( It is possible to give abstract proofs of the identities in Proposition 2.10, but it is much faster to use frames of the standard tractor bundle suitably adapted to the parallel tractor G 2 -structure Φ. Proof of Proposition 2.10. Call a local frame (E a ) of V adapted to Φ iff (1) E 1 is a local section of the line subbundle X determined by X, and (2) the representation of Φ in the dual coframe (e a ) is given by (2.5); it follows from [65, Theorem 3.1] that such a local frame exists in some neighborhood of any point in M .
Any adapted local frame determines a (local) choice of scale: Since X ∈ Γ(V[1]), we have τ := e 7 (X) ∈ E [1], and by construction it vanishes nowhere. Then, since X ⊥ = E 1 , . . . , E 6 , the (weighted) vector fields F a := E a + E 1 , a = 2, . . . , 6 comprise a frame of X ⊥ / X which by Section 2.4.2 is canonically isomorphic to T M [−1]. Trivializing these frame fields (by multiplying by τ ) yields a local frame (F 2 , . . . , F 6 ) of T M ; denote the dual coframe by (f 2 , . . . , f 6 ). One can read immediately from (2.5) that in an adapted local frame, (the trivialized) components of Φ are and consulting the form of equation (2.6) gives that the (trivialized) conformal metric is In an adapted frame, Φ is given by −e 1 ∧ · · · ∧ e 7 , so the (trivialized) conformal volume form is All of the identities follow immediately from computing in this frame. For example, to compute (1), we see that raising indices gives φ = √ 2F 2 ∧ F 3 , and that contracting an index of this bivector field with φ = √ 2f 5 ∧ f 6 yields 0. It remains to show that the geometric assertions are equivalent to the corresponding identities; these are nearly immediate for all but the first two parts. For both parts, pick a local frame (α, β) of D around an arbitrary point; by scaling we may assume that φ = α ∧ β.

The identity implies that the trace over the second and third indices of the tensor product
Since α, β are linearly independent, the four coefficients on the right-hand side vanish separately, but up to sign these are the components of the restriction of c D to D in the given frame.
Since the tractor Hodge star operator * Φ is algebraic, is parallel. We can express its components with respect to a scale τ in terms of those of Φ and the weighted Hodge star operators * : Computing in an adapted frame as in the proof of Proposition 2.10 yields some useful identities relating the components of Φ and their images under * : The global geometry of almost Einstein (2,3,5) distributions In this section we investigate the global geometry of (2, 3, 5) distributions (M, D) that induce almost Einstein conformal structures c D ; naturally, we call such distributions themselves almost Einstein.
Almost Einstein (2,3,5) distributions are special among (2,3,5) conformal structures: In a sense that can be made precise [38, Theorem 1.2, Proposition 5.1], for a generic (2,3,5) distribution D the holonomy of c D is equal to G 2 and hence c D admits no nonzero almost Einstein scales.
Via the identification of the standard tractor bundles of D and c D , Theorem 2.6 gives that an oriented (2, 3, 5) distribution is almost Einstein iff its standard tractor bundle V admits a nonzero parallel standard tractor S ∈ Γ(V), or equivalently, iff it admits a holonomy reduction from G 2 to the stabilizer subgroup S of a nonzero vector in the standard representation V.

Distinguishing a vector in the standard representation V of G 2
In this subsection, let V denote the standard representation of G 2 and Φ ∈ Λ 3 V * the corresponding 3-form. We establish some of the algebraic consequences of fixing a nonzero vector S ∈ V.

Stabilizer subgroups
Recall from the introduction that the stabilizer group in G 2 of S ∈ V is as follows: Proposition 3.1. The stabilizer subgroup of a nonzero vector S in the standard representation V of G 2 is isomorphic to:

An ε-Hermitian structure
Contracting a nonzero vector S ∈ V with Φ determines an endomorphism: , so the image of K is contained in W := S ⊥ , and hence we can regard K| W as an endomorphism of W, which by abuse of notation we also denote K. Restricting (3.2) to W gives that this latter endomorphism is an ε-complex structure on that bundle: is an ε-Hermitian structure on W: this is a pair (g, K), where g ∈ S 2 W * is a symmetric, nondegenerate, bilinear form and K is an ε-complex structure on W compatible in the sense that g( · , K · ) is skew-symmetric. If K is complex, g has signature (2p, 2q) for some integers p, q; if K is paracomplex, g has signature (m, m).

Induced splittings and f iltrations
If S is nonisotropic, it determines an orthogonal decomposition V = W ⊕ S . If S is isotropic, it determines a filtration (V a S ) [37, Proposition 2.5]: The number above each filtrand is its filtration index a (which are canonical only up to addition of a given integer to each index) and the number below its dimension. Moreover, im K = (ker K) ⊥ (so ker K is totally isotropic). If we take Q to be the stabilizer subgroup of the ray spanned by S, then the filtration is Q-invariant, and checking the (representation-theoretic) weights of V as a Q-representation shows that it coincides with the filtration (2.14) determined by Q. The map K satisfies K(V a S ) = V a+2 S , where we set V a S = 0 for all a > 2.

The family of stabilized 3-forms
For nonzero S ∈ V, elementary linear algebra gives that the subspace of 3-forms in Λ 3 V * fixed by the stabilizer subgroup S of S has dimension 3 and contains 3 27 follows from that containment, the identity and the fact that Φ ∈ Λ 3 1 . It follows immediately from the definitions that Since S annihilates Φ I but not Φ, the containments in (3.4), (3.5) show that {Φ, Φ I , Φ J } is a basis of the subspace of stabilized 3-forms. If H Φ (S, S) = 0, then (3.7) implies that {Φ I , Φ J , Φ K } is also a basis of that space. If H Φ (S, S) = 0 then Φ K = −Φ I . It is convenient to abuse notation and denote by Φ I , Φ J the pullbacks to W of the 3-forms of the same names via the inclusion W → V. For nonisotropic S, define W 1,0 ⊂ W⊗ R C ε to be the (+i ε )-eigenspace of (the extension of) K, and an ε-complex volume form to be an element of Λ m Cε W is an ε-complex volume form for the ε-Hermitian structure (H Φ | W , K) on W. Proposition 3.3. Suppose V is a 7-dimensional real vector space and H ∈ S 2 (V ) * is a symmetric bilinear form of signature (3,4). Now, fix a vector S ∈ V such that −ε := H(S, S) ∈ {±1}, denote W := S ⊥ , fix an ε-complex structure K ∈ End(W) such that (H| W , K) is a Hermitian structure on W, and fix a compatible ε-complex volume form Ψ ∈ Λ 3 Cε W * satisfying the normalization condition Then, the 3-form Here, Re Ψ and K are regarded as objects on V via the decomposition V = W ⊕ S .
This proposition can be derived, for example, from [23, Proposition 1.12], since, using the terminology of the article, (Re Ψ, K) is a compatible and normalized pair of stable forms.
We view the adjoint tractor as a bundle endomorphism of V (cf. (3.1)), and computing gives that the components of K with respect to τ are (3.9) We denote the projecting part of K A B by because K is parallel, ξ is a normal conformal Killing field for c D . By (3.2) K is not identically zero and hence neither is ξ. This immediately gives a simple geometric obstruction -nonexistence of a conformal Killing field -for the existence of an almost Einstein scale for an oriented (2, 3, 5) conformal structure. By construction, ξ = ι 7 (σ), where ι 7 is the manifestly invariant differential operator Here, ι 2 7 is the bundle map V → Λ 2 V * associated to the algebraic map (2.8) of the same name, and we have implicitly raised an index with H Φ . Henceforth we often suppress the restriction notation | M ξ . We will see in Proposition 5.8 that L coincides with the line field of the same name determined via Proposition 2.10 by the preferred scale σ (on the complement of its zero locus).  The projection aut(c D ) → aEs(c D ) is (the restriction of) the invariant differential operator
The map π 2 7 is the bundle map A ∼ = Λ 2 V * → V * associated to the algebraic map (2.9) of the same name.

The (local) leaf space
Let L denote the space of integral curves of ξ in M ξ := {x ∈ M : ξ x = 0}, and denote by π L : M ξ → L the projection that maps a point to the integral curve through it. Since ξ vanishes nowhere, around any point in M ξ there is a neighborhood such that the restriction of π L thereto is a trivial fibration over a smooth 4-manifold; henceforth in this subsection, we will assume we have replaced M ξ with such a neighborhood.

Descent of the canonical objects
Some of the objects we have already constructed on M descend to L via the projection π L . One can determine which do by computing the Lie derivatives of the various tensorial objects with respect to the generating vector field ξ, but again it turns out to be much more efficient to compute derivatives in the tractor setting. Since any conformal tractor bundle U → M is a natural bundle in the category of conformal manifolds, one may pull back any section A ∈ Γ(U) by the flow Ξ t of ξ and define the Lie derivative L ξ A to be L ξ A := ∂ t | 0 Ξ * t A [46]. Since the tractor projection Π U 0 is associated to a canonical vector space projection, it commutes with the Lie derivative. We exploit the following identity: where · denotes the action on sections induced by the action so(p + 1, q + 1) × U → U. In particular, if A is parallel, then As usual, we scale σ so that ε := −H Φ (S, S) ∈ {−1, 0, +1}, where S A := L V 0 (σ) A is the parallel standard tractor corresponding to σ.
In particular, σ and K descend via π L : M ξ → N to well-defined objectsσ andK, but φ and J do not descend, and when ε = 0 neither does I (recall that when ε = 0, I = −K).

The conformal isometry problem
In this section we consider the problem of determining when two distributions (M, D) and (M, D ) induce the same oriented conformal structure; we say two such distributions are conformally isometric. This problem turns out to be intimately related to existence of a nonzero almost Einstein scale for c D .
Approaching this question at the level of underlying structures is prima facie difficult: The value of the conformal structure c D at a point x ∈ M induced by a (2, 3, 5) distribution (M, D) depends on the 4-jet of D at x [52, equation (54)] (or, essentially equivalently, multiple prolongations and normalizations), so analyzing directly the dependence of c D on D involves apprehending high-order differential expressions that turn out to be cumbersome.
We have seen that in the tractor bundle setting, however, this construction is essentially algebraic: At each point, the parallel tractor G 2 -structure Φ ∈ Γ(Λ 3 V * ) determined by an oriented (2, 3, 5) distribution (M, D) determines the parallel tractor bilinear form H Φ ∈ Γ(S 2 V * ) and orientation [ Φ ] canonically associated to the oriented conformal structure c D . So, the problem of determining the distributions (M, D ) such that c D = c D amounts to the corresponding algebraic problem of identifying for a G 2 -structure Φ on a 7-dimensional real vector space V the ). We solve this algebraic problem in Section 4.1 and then transfer the result to the setting of parallel sections of conformal tractor bundles to resolve the conformal isometry problem in Section 4.2.

The space of G 2 -structures compatible with an SO(3, 4)-structure
In this subsection, which consists entirely of linear algebra, we characterize explicitly the space of G 2 -structures compatible with a given SO(3, 4)-structure on a 7-dimensional real vector space V, or more precisely, the SO(3, 4)-structure determined by a reference G 2 -structure Φ. This characterization is essentially equivalent to that in [12, Remark 4] for the analogous inclusion of the compact real form of G 2 into SO(7, R). The following proposition can be readily verified by computing in an adapted frame. (Computer assistance proved particularly useful in this verification.) Proposition 4.1. Let V be a 7-dimensional real vector space and fix a G 2 -structure Φ ∈ Λ 3 V * .
For any (Ā, B) ∈ R 2 such that −εĀ 2 + 2Ā + B 2 = 0 (there is a 1-parameter family of such pairs) the 3-form 2. Conversely, all compatible G 2 -structures arise this way:   We can readily parameterize the families F[Φ; S] of G 2 -structures. It is convenient henceforth to split cases according to the causality type of S, that is, according to ε. If ε = 0, then Φ = −ε(Φ I + Φ K ), so in terms of A :=Ā − ε and B, Φ = AΦ I + BΦ J − εΦ K and the condition on the coefficients is A 2 − εB 2 = 1.
If ε = +1, then A 2 − B 2 = 1, and so we can parameterize F[Φ; S] by and Φ − 0 = Φ. If ε = 0, the compatibility conditions simplify to 2Ā+B 2 = 0, so we can parameterize F[Φ; S] by and Φ 0 = Φ. Each of the above parameterizations Φ u satisfies d du 0 Φ u = Φ J , and the parameterizations in the latter two cases are bijective. In the nonisotropic cases, we can encode the compatible G 2 -structures efficiently in terms of the ε-complex volume forms in Proposition 3.2.

Conformally isometric (2, 3, 5) distributions
Each value of the parameter s corresponds to a distinct distribution. These parameterizations are distinguished: Locally they agree (up to an overall constant) with the flow of the distinguished conformal Killing field ξ determined by D and σ.
Proof . In the Ricci-negative case this follows immediately from the facts that the normal conformal Killing 2-form φ corresponding to D satisfies L ξ φ = 3J (3.20) and that the 1-parameter family of normal conformal Killing 2-forms φ υ corresponding to the distributions D υ satisfy d dυ 0 φ υ = J. The other cases are analogous.  This condition is visibly symmetric in D, D . Note that rearranging (2.11) and passing to the tractor bundle setting gives that Φ ∧ Φ = 4 * Φ π 3 7 (Φ ), where π 3 7 denotes the bundle map Λ 3 V * → V associated to the algebraic map of the same name in that equation. Proposition 4.10. Let D be a 1-parameter family of conformally isometric oriented (2,3,5) distributions related by an almost Einstein scale and fix D ∈ D.

Additional induced distributions
• If the almost Einstein scale determining D is non-Ricci-flat, there is precisely one distribution E antipodal to D.
• If the almost Einstein scale determining D is Ricci-flat, there are no distributions antipodal to D.
Proof . Let Φ denote the parallel tractor G 2 -structure corresponding to D and let S the parallel standard tractor corresponding to the almost Einstein scale. Any D ∈ D corresponds to a compatible parallel tractor G 2 -structure Φ ∈ F[Φ; S] and by Proposition 4.1 we can write 7 gives a precise description of the set on which φ ∞ does not vanish and hence on which E is defined; this set turns out to be the complement of a set that (if nonempty) has codimension ≥ 3. Corollary 5.17 below shows that E is integrable (and hence not a (2, 3, 5) distribution). In the case that σ is Ricci-positive, D and σ determine two additional distinguished distributions: In the notation of Section 4.2.1, the family (sech t)φ ∓ t converges to the normal conformal Killing 2-form ∓I ± J as t → ± ∞. By continuity φ ∓∞ := ±I + J are decomposable and hence determine distributions D ∓∞ on the sets where they respectively do not vanish. Proposition 5.28 below describes precisely these sets (their complements, if nonempty, have codimension 3). By construction D ∓∞ depend only on the family D[D; σ] and not D itself. Computing in an adapted frame shows that the corresponding parallel tractor 3-forms Φ ∓∞ := L Λ 3 V * 0 (φ ∓∞ ) are not generic (they both annihilate L V 0 (σ), that is, they are not G 2 -structures, and hence the distributions D ∓∞ are not (2, 3, 5) distributions).
Since all of the involved tractor objects are parallel, the reconstruction problem is equivalent to the algebraic one recovering a normalized vector S from G 2 -structures Φ, Φ on a 7-dimensional real vector space inducing the same SO(3, 4)-structure such that Φ ∈ F[Φ; S]. One can thus verify the algorithm by computing in an adapted basis.

The curved orbit decomposition
In this section, we treat the curved orbit decomposition of an oriented (2, 3, 5) distribution determined by an almost Einstein scale, that is of a parabolic geometry of type (G 2 , Q) to the stabilizer S of a nonzero ray in the standard representation V of G 2 .
In Section 5.1 we briefly review the general theory of curved orbit decompositions and the decomposition of an oriented conformal manifold determined by an almost Einstein scale. In Section 5.2 we determine the orbit decomposition of the flat model. In Section 5.3 we state and prove geometric characterizations of the curved orbits, both in terms of tractor data and in terms of data on the base manifold. In the remaining subsections we elaborate on the induced geometry determined on each of the curved orbits, which among other things yields proofs of Theorems D − , D + , and D 0 .

The general theory of curved orbit decompositions
Here we follow [17]. If the holonomy Hol(ω) of a Cartan geometry (G, ω) is a proper subgroup of G, the principal connectionω extending ω (see Section 2.3.2) can be reduced: If H ≤ G is a closed subgroup that contains any group in the conjugacy class Hol(ω),Ĝ := G × P G admits a reduction j : H →Ĝ of structure group to H, and j * ω is a principal connection on H. Such a reduction can be viewed equivalently as a section of the associated fiber bundlê G/H :=Ĝ × G (G/H). We henceforth work with an abstract G-homogeneous space O instead of G/H, which makes some exposition more convenient, and we call the corresponding Gequivariant section s :Ĝ → O a holonomy reduction of type O. Note that we can identifyĜ Given a Cartan geometry (G → M, ω) of type (G, P ) and a holonomy reduction thereof of type O corresponding to a section s :Ĝ → O, we define for each x ∈ M the P -type of x (with respect to s) to be the P -orbit s(G x ) ⊆ O. This partitions M by P -type into a disjoint union a∈P \O M a of so-called curved orbits parameterized by the space P \O of P -orbits of O. By construction, the P -type decomposition of the flat model G/P coincides with the decomposition of G/P into H-orbits (for any particular choice of conjugacy class representative H). Put another way, the P -types correspond to the possible intersections of H and P in G up to conjugacy.
The central result of the theory of curved orbit decompositions is that each curved orbit inherits from the Cartan connection ω and the holonomy reduction s an appropriate Cartan geometry: We need some notation to state the result: Given a G-homogeneous space O and elements x, x ∈ O, we have x = g · x for some g ∈ G, and their respective stabilizer subgroups G x , G x are related by G x = gG x g −1 . If x, x are in the same P -orbit, we can choose g ∈ P , and if we denote P x := G x ∩ P , we likewise have P x = gP x g −1 . Thus, as groups endowed with subgroups, (G x , P x ) ∼ = (G x , P x ). Given an orbit a ∈ P \O, we denote by (H, P a ) an abstract representative of the isomorphism class of groups so endowed.
Theorem 5.1 ([17, Theorem 2.6]). Let (G → M, ω) be a parabolic (more generally, Cartan) geometry of type (G, P ) with a holonomy reduction of type O. Then, for each orbit a ∈ P \O, there is a principal bundle embedding j a : G a → G| Ma , and (G a → M a , ω a ) is a Cartan geometry of type (H a , P a ) on the curved orbit M a , where ω a := j * a ω.
Informally, since each P -type corresponds to an intersection of H and P up to conjugacy in G, for each such intersection H ∩ P (up to conjugacy) the induced Cartan geometry on the corresponding curved orbit has type (H, H ∩ P ).
Example 5.2 (almost Einstein scales, [17,Theorem 3.5]). Given a conformal structure (M, c) of signature (p, q), n := p + q ≥ 4, by Theorem 2.6 a nonzero almost Einstein scale σ ∈ Γ(E[1]) corresponds to a nonzero parallel standard tractor S := L V 0 (σ) and hence determines a holonomy reduction to the stabilizer subgroupS of a nonzero vector in the standard representation V of SO(p + 1, q + 1); the conjugacy class ofS depends on the causality type of S.
If S is nonisotropic, there are three curved orbits, characterized by the sign of σ. The union of the open orbits is the complement M − Σ of the zero locus Σ := {x ∈ M : σ x = 0}, and the reduced Cartan geometries on these orbits are equivalent to the non-Ricci-flat Einstein metric σ −2 g of signature (p, q). If S is spacelike (timelike) the reduced Cartan geometry on the hypersurface curved orbit Σ is a normal parabolic geometry of type (SO(p, q +1),P ) ((SO(p+1, q),P )), which corresponds to an oriented conformal structure c Σ of signature (p − 1, q) ((p, q − 1)).

The orbit decomposition of the f lat model M
In this subsection, we determine the orbits and stabilizer subgroups of the action of S on the flat model M := G 2 /Q ∼ = S 2 × S 3 , which by Theorem 5.1 determines the curved orbit decomposition of a parabolic geometry of type (G 2 , Q).
Remark 5.4. Alternatively, as in the statements of Theorems D − , D + , and D 0 , we could fix a conformal structure c, that is, a normal parabolic geometry of type (SO (3,4),P ) (Section 2.3.5) equipped with a holonomy reduction to the intersection S of a copy of G 2 in SO (3,4) and the stabilizer of a nonzero vector S ∈ V (where we now temporarily view V as the standard representation of SO (3,4)). By Remark 4.2, this determines a 1-parameter family F ⊂ Λ 3 V * of compatible G 2 -structures but does not distinguish an element of this family. Transferring this statement to the setting of tractor bundles and then translating it into the setting of a tangent bundle, such a holonomy reduction determines a 1-parameter family D of conformally isometric oriented (2, 3, 5) distributions for which c = c D for all D ∈ D, but does not distinguish a distribution among them.
As usual by scaling assume S satisfies ε := −H Φ (S, S) ∈ {−1, 0, +1} and denote W := S ⊥ . The parabolic subgroup Q preserving any ray x ∈ M (spanned by the isotropic weighted vector X ∈ V[1]) preserves the filtration (V a X ) of V determined via (3.3) by X: Explicitly this is Here, X × · is the map −X C Φ C A B ∈ End(V) [1], and by the comments after (3.3) it satisfies X × V a X = V a+2 X . Since Q preserves this filtration, the corresponding set differences {x ∈ M : S x ∈ V a X − V a+1 X } are each unions of curved orbits.
If σ > 0, then σ −1 w ∈ W satisfies H Φ (σ −1 w, σ −1 w) = −1. But the set of vectors w 0 ∈ W satisfying H(w 0 , w 0 ) = −1 is just the sphere S 2,3 , and SU(1, 2) acts transitively on this space, and hence on the 5-dimensional space M + 5 ∼ = S 2,3 of rays in M it subtends. The isotropy group of the ray spanned by w 0 + S preserves the appropriate restrictions of H, K, Ψ to the four-dimensional subspace w 0 , Kw 0 ⊥ ⊂ W, so that space is neutral Hermitian and admits a complex volume form, and hence the isotropy subgroup is contained in SU(1, 1) ∼ = SL(2, R). On the other hand, the isotropy subgroup has dimension dim S − dim M + 5 = 8 − 5 = 3, so it must coincide with SU(1, 1).
If σ = 0, then w = X ∈ W is isotropic. The set of such vectors is the intersection of the null cone of H with W. Again, SU(1, 2) acts transitively on the ray projectivization M 4 ∼ = S 3 × S 1 of this space. By construction, the isotropy subgroup P − , which (since dim M 4 = 4) has dimension four, is contained in the 5-dimensional stabilizer subgroup P SU (1,2) in SU(1, 2) of the complex line X, KX ⊂ W generated by X; this latter group is (up to connectedness) the only parabolic subgroup of SU (1, 2).

Ricci-positive case
In this case, which corresponds to S timelike (ε := +1), S ∼ = SL(3, R), and W inherits a paracomplex structure K and a paracomplex volume form Ψ. This case is similar to the Ricci-negative one, and we omit details that are similar thereto. Since all of the vectors in im X − ker X are timelike, however, that set difference corresponds to a union of orbits that has no analogue for the other causality types of S.
Similarly to the Ricci-negative case, V decomposes (as an SL(3, R)-module) as W ⊕ S , we If σ > 0, then the resulting curved orbit is M + 5 ∼ = S 2,3 , and the stabilizer subgroup is isomorphic to SL(2, R). If σ = 0, then w = X ∈ W is isotropic. As mentioned at the beginning of this subsubsection, unlike in the Ricci-negative case, this subcase entails more than one orbit. To see this, let E denote the (+1)-eigenspace of K. Using (the restriction of) H Φ we may identify the −1eigenspace with E * , and so we can write X as (0, e, β) ∈ S ⊕ E ⊕ E * . Since X is isotropic, 0 = 1 2 H Φ (X, X) = β(e), and we can identify the ray projectivization of the set of such triples with S 2 × S 2 . Now, the action of S preserves whether each of the components e, β is zero, giving three cases.
One can readily compute that S acts transitively on pairs (e, β) of nonzero elements with isotropy group P + ∼ = R + R 3 , which is characterized by its restriction to E, and which in turn is given in a convenient basis by So, the corresponding orbit M 4 has dimension 4, and it follows from the remaining two cases that M 4 ∼ = S 2 ×(S 2 −{± * }) ∼ = S 2 ×S 1 ×R for some point * ∈ S 2 . By construction, P + is contained in the 5-dimensional stabilizer subgroup P 12 in SL(3, R) of the paracomplex line X, KX ⊂ W generated by X, and we may identify P 12 with the subgroup of the stabilizer subgroup in SL(3, R) of a complete flag in W that preserves either (equivalently, both of) the rays of the 1-dimensional subspace in the flag.
In the case that e = 0 but β = 0, S again acts transitively, and this time the isotropy subgroup is isomorphic to the (parabolic) stabilizer subgroup P 1 := GL(2, R) (R 2 ) * of a ray in E, which is the first parabolic subgroup in SL(3, R), so the corresponding orbit is M + The dual case e = 0, β = 0 is similar: S acts transitively, and we can identify the isotropy subgroup with the (parabolic) stabilizer subgroup P 2 := GL(2, R) R 2 ∼ = P 1 of a dual ray in E * , so the corresponding orbit is M − 2 ∼ = S 2 .
Finally, the case σ < 0 is essentially identical to the case σ > 0 and we denote the corresponding orbit by M − 5 ∼ = S 2,3 .

Ricci-f lat case
In this case, which corresponds to S isotropic (ε = 0), S ∼ = SL(2, R) Q + and W inherits a endomorphism K whose square is zero. Since S is isotropic, it determines a filtration (V a S ) of V. By symmetry, we may identify the sets {x ∈ M : If σ > 0, S acts transitively on the 5-dimensional space of rays. Computing directly shows that the isotropy subgroup is conjugate to the Levi factor SL(2, R) < G 2 , so we may identify If σ = 0, we see there are several possibilities. Since every vector in im K − ker K is timelike, {x ∈ M : X ∈ S ⊥ − im K} is the set of points x ∈ M such that H Φ (X, S) = 0 but X × S = 0. Again, S acts transitively on the 4-dimensional space M 4 of rays. In this case, computing gives that the isotropy subgroup is isomorphic to R R 3 .
Next, we consider the set of points {x ∈ M : X ∈ ker K− S }. Since ker K is totally isotropic, ker K − X is the set difference of a 3-dimensional affine space and a linear subspace, so the corresponding orbit M 2 of rays is a twice-punctured 2-sphere. Again, computing directly gives that S acts transitively on this space, and the stabilizer subgroup is a certain 6-dimensional solvable group.
When X ∈ S , either S is in the ray determined by X or its opposite, and these correspond respectively to 0-dimensional orbits M + 0 and M − 0 . Finally, the case σ < 0 is again essentially identical to the case σ > 0, and again we denote the corresponding orbit M − 5 ∼ = R 5 .

Characterizations of the curved orbits
In this subsection we give geometric characterizations of the curved orbits M • determined by the holonomy reduction to S.
For the rest of this section, let D be an oriented (2, 3, 5) distribution, denote the corresponding parallel tractor G 2 -structure by Φ, and denote its components with respect to a scale τ by φ, χ, θ, ψ as in (2.25). Also fix an almost Einstein scale σ, denote the corresponding parallel tractor by S := L V 0 (σ), and denote its components with respect to τ by σ, µ, ρ as in (2.18). On the zero locus Σ := {x ∈ M : σ x = 0} of σ, µ is invariant (that is, independent of the choice of scale τ ), so on the set where moreover µ = 0, µ determines a line field S. See also Appendix A. M a ε tractor condition Proof . The characterizations of M ± 5 are immediate from the descriptions in Section 5.2. Passing to the tractor setting, a point x ∈ M is in M 4 if S x ∈ S x ⊥ − im(X x × · ) (for readability, in this proof we herein sometimes suppress the subscript x ). By the discussion after (5.1), X × (im X) = X , so S ∈ im(X × · ) iff (X × S) ∧ X = 0, yielding the tractor characterization of M 4 . Next, x ∈ M ± 2 if S ∈ im(X × · ) − ker(X × · ), so this curved orbit is characterized by X × S ∈ X − {0}; in fact, since X × S = −S × X = K(X), X is an eigenvalue of K, and we acutally have In the splitting determined by a scale τ , So, the only nonzero component of (X × S) ∧ X is ξ a , which together with the tractor characterization gives the tangent bundle characterization of For M 2 , if S ∈ ker(X × · ) we have σ = 0, ξ a = 0, and α = 0, so by the argument in the previous case we have S ⊂ D. Since S ∈ X , we have S ∧ X = 0, which is equivalent to µ a = 0, and by (2.21) µ = ∇σ.
Finally, if S ∈ X , which by the previous case comprises the points where µ = 0. Again using L V 0 (and that σ = 0) gives that ±ρ = ∓ 1 5 ∆σ, and so ±ρ > 0 (and hence x ∈ M ± 0 ) iff ∓∆σ > 0. Since σ is nowhere zero on M 5 , we can work in the scale σ| M 5 itself, in which many earlier formulae simplify. (Henceforth in this subsection, we suppress the restriction notation | M 5 .) As usual, by rescaling we may assume that the parallel tractor S corresponding to σ satisfies ε := −H Φ (S, S) ∈ {−1, 0, +1}. Then, from the discussion after Theorem 2.6, g ab := σ −2 g ab is almost Einstein and has Schouten tensor P ab = 1 2 εg ab , or equivalently, Ricci tensor R ab = 4εg ab , and hence scalar curvature R = 20ε. In the scale σ, the components of the parallel tractor S itself are and substituting in (3.9) gives that We have introduced the 2-formφ ab := −2ψ ab ∈ Γ(Λ 2 T * M [1]) for notational convenience. Note that ξ b σ = θ b . Substituting (5.2) in (3.13), (3.14), (3.15) gives that I, J, K simplify to The endomorphism component of K in the scale σ coincides with −K a b .

The canonical splitting
The components of canonical tractor objects on M 5 in the splitting determined by σ are themselves canonical (and so are, just as well, their trivializations); in particular this includes the components χ abc , θ c , ψ bc of Φ ABC . Moreover, via Proposition 2.10, D and the scale σ together determine a canonical splitting of the canonical filtration D ⊂ [D, D] ⊂ T M 5 : The fact that ξ = θ gives the following:

The canonical hyperplane distribution
Denote by C ⊂ T M ξ the hyperplane distribution orthogonal to L := ξ | M ξ .
Proposition 5.11. Let D be a family of conformally isometric oriented (2,3,5) distributions related by an almost Einstein scale σ. Then, on M 5 : 1. The pullback g C of the metric g := σ −2 g to the hyperplane distribution C has neutral signature.

2.
The hyperplane distribution C is a contact distribution iff σ is not Ricci-flat.
3. If we fix D ∈ D, then C = D ⊕ E, where E is the distribution antipodal or nullcomplementary to D.

4.
The canonical pairing D × E → R is nondegenerate, and the bilinear form it induces on C via the direct sum decomposition C = D ⊕ E is g C .
(3) By Theorem 5.9, D and E are transverse, and by Corollary 4.7 they are both contained in C, so the claim follows from counting dimensions.
(4) This follows from computing in an adapted frame.
Computing in an adapted frame gives the following pointwise description: Proposition 5.12. Let D be a 1-parameter family of conformally isometric oriented (2,3,5) distributions related by an almost Einstein scale σ. Then, for any x ∈ M 5 = {x ∈ M : σ x = 0}, the family D x := {D x : D ∈ D} is precisely the set of totally isotropic 2-planes in C x self-dual with respect to the (weighted) bilinear form g C and the orientation determined by C .
Computing in an adapted frame shows that the images of I, J, K ∈ Γ(End(T M ) [1]) are contained in C [1], so they restrict sections of End(C) [1], which by mild abuse of notation we denote I α β , J α β , K α β (here and henceforth, we use lowercase Greek indices α, β, γ, . . . for tensorial objects on M ). It also gives that these maps satisfy, for example, I α γ I γ β = −εσ 2 δ α β ∈ Γ(End(C) [2]) and Proposition 5.13 below. In the scale σ, this and the remaining equations become: In the non-Ricci-flat case, we can identify the pointwise U(1, 1)-structure on M 5 as follows: Proposition 5.14. Let D be a 1-parameter family of conformally isometric oriented (2,3,5) distributions related by a non-Ricci-flat almost Einstein scale σ.
1. For any D ∈ D, the endomorphisms I, J, K determine an almost split-quaternionic structure on (the restriction to M 5 of) C, that is, an injective ring homomorphism H → End(C x ) for each x ∈ M 5 , where H is the ring of split quaternions.
2. The almost split-quaternionic structure in (1) depends only on D.
(2) This follows from computing in an adapted frame.

The ε-Sasaki structure
The union M 5 of the open orbits turns out also to inherit an ε-Sasaki structure, the odddimensional analogue of a Kähler structure. 2) ξ (a,b) = 0 (or equivalently, (L ξ h) ab = 0, that is, ξ is a Killing field for h), and An ε-Sasaki-Einstein structure is an ε-Sasaki structure (h, ξ) for which h is Einstein.
It follows quickly from the definitions that the restriction of ξ a ,b is an almost ε-complex structure on the subbundle ξ ⊥ , and that if ε = +1, the (±1)-eigenbundles of this restriction (which have equal, constant rank) are integrable and totally isotropic. Proof . Set c = [−g], let σ ∈ Γ(E [1]) be the unique section such that −g = σ −2 g, and S := L V 0 (σ) the corresponding parallel tractor. Define the adjoint tractor K := L A 0 (ξ). It is known that a Sasaki-Einstein metric g satisfies P ab = 1 2 εg ab , and so by (2.23) S satisfies H(S, S) = −ε, where H is the tractor metric determined by c. Thus, the proof of Theorem 5.16 gives Transferring the content of Section 3.1.2 to the tractor bundle setting then shows that the parallel subbundle W := S ⊥ ⊂ V inherits a parallel almost ε-Hermitian structure. Denote the curvature of the normal tractor connection by Ω ab defines a parallel G 2 -structure on V compatible with H. By the discussion before Proposition 2.10, its projecting slot defines a (2, 3, 5) distribution with associated conformal structure c = [−g]. Finally, parallel sections of Λ 3 C W satisfying Ψ ∧Ψ = − 4 3 i ε K ∧ K ∧ K are parametrized by {z ∈ C ε : zz = 1} (that is, S 1 if ε = −1 and SO(1, 1) if ε = 1).

Projective geometry
On the complement M 5 of the zero locus Σ of σ, we may canonically identify (the restriction of) the parallel subbundle W := S ⊥ with the projective tractor bundle of the projective structure [∇ g ], where g is the Einstein metric σ −2 g, and the connection ∇ W that ∇ V induces on W with the normal projective tractor connection [34,Section 8].
This compatibility determines a holonomy reduction of the latter connection to S, and one can analyze separately the consequences of this projective reduction. For example, if σ is non-Ricci-flat, then lowering an index of the parallel complex structure K ∈ Γ(End(W)) with H| W yields a parallel symplectic form on W. A holonomy reduction of the normal projective tractor connection on a (2m + 1)-dimensional projective manifold M to the stabilizer Sp(2m + 2, R) of a symplectic form on a (2m + 2)-dimensional real vector space determines precisely a torsionfree contact projective structure [18, Section 4.2.6] on M suitably compatible with the projective structure [31].
This also leads to an alternative proof that the open curved orbits inherit a Sasaki-Einstein structure in the Ricci-negative case: The holonomy of ∇ W is reduced to SU(1, 2), but [3, Section 4.2.2] identifies su(p , q ) as the Lie algebra to which the projective holonomy connection determined by an Sasaki-Einstein structure is reduced.
The upcoming article [35] discusses the consequences of a holonomy reduction of (the normal projective tractor connection of) a (2m + 1)-dimensional projective structure to the special unitary group SU(p , q ), p + q = m + 1.

The open leaf space L 4
As in Section 3.5, we assume that we have replaced M by an open subset so that π L is a locally trivial fibration over a smooth 4-manifold. Define L 4 := π L (M 5 ): By Corollary 5.6(2) M 5 is a union of π L -fibers, so L 3 := π L (M 4 ) = L − L 4 is a hypersurface.
Since ξ| M 5 is a nonisotropic Killing field, −g := −σ −2 c D | M 5 descends to a metricĝ on L 4 (henceforth in this subsection we sometimes suppress the restriction notation | M 5 ). By Proposition 3.8 L ξ σ = 0 and L ξ K = 0, so the trivialization K ∈ Γ(End(T M 5 )) is invariant under the flow of ξ. Since it annihilates ξ, it descends to an endormorphism field we denoteK ∈ End(T L 4 ). Then, Proposition 5.6 impliesK 2 = −ε id T L 4 , that is, K is an almost ε-complex structure on L 4 . This yields a specialization to our setting of a well-known result in Sasaki geometry.
Proof . Since K a b ξ b = 0, the g-skewness of K implies theĝ-skewness ofK. Thus,ĝ andK together comprise an almost Kähler structure on L 4 ; the integrability ofK is proved, for example, in [5], so they in fact consistute a Kähler structure. Since π L | L 4 is a (pseudo-)Riemannian submersion, we can relate the curvatures of g andĝ via the O'Neill formula, which gives thatĝ is Einstein and determines the Einstein constant.

The ε-Kähler-Einstein Fef ferman construction
The well known construction of Sasaki-Einstein structures from Kähler-Einstein structures immediately generalizes to the ε-Kähler-Einstein setting; see, for example, [44] (in this subsubsection, we restrict to ε ∈ {±1}). Here we briefly describe the passage from ε-Kähler-Einstein structures to almost Einstein (2, 3, 5) conformal structures as a generalized Fefferman construction [18,Section 4.5] between the respective Cartan geometries. Further details will be discussed in an article in preparation [56].
We realize A within S as block diagonal matrices detA −1 0 0 A . The action of A preserves the decomposition s = a ⊕ m = a m m a and is given on m ⊂ s by X → det(A)AX; in particular it preserves an ε-Hermitian structure (unique up to multiples) on m and we fix a (standard) choice. The m-part θ of a Cartan connectionω of type (S, A) determines an isomorphism T L 4 ∼ = S × A m and (via this isomorphism) an ε-Hermitian structure on T L 4 . The a-part γ of the Cartan connection defines a linear connection ∇ preserving this ε-Hermitian structure. If ω is torsion-free then ∇ is torsion-free, and thus the ε-Hermitian structure is ε-Kähler. Conversely, given an ε-Kähler structure, the Cartan bundle S → L 4 is the reduction of structure group of the frame bundle to A ⊂ SO(2, 2) defined by the parallel ε-Hermitian structure and the (reductive) Cartan connectionω ∈ Ω 1 (S, s) is given by the sumω = γ + θ of the pullback of the Levi-Civita connection form γ ∈ Ω 1 (S, a) and the soldering form θ ∈ Ω 1 (S, m).
For the construction we first build the correspondence space where A 0 = SU(1, 1) if ε = −1 and A 0 = SL(2, R) if ε = 1. Then, CL 4 → L 4 is an S 1 -bundle if ε = −1 and an SO(1, 1)-bundle if ε = 1. We can viewω ∈ Ω 1 (S, s) as a Cartan connection on the A 0 -principal bundle S → CL 4 . Next we fix inclusions field ξ. Since ξ is a normal conformal Killing field, it inserts trivially into the curvature of ω 5 . Since ξ spans V M 5 , by [18,Theorem 1.5.14] this guarantees that on a sufficiently small leaf space L 4 one obtains a Cartan geometry of type (S, A) such that the restriction of (G 5 → M 5 , ω 5 ) is locally isomorphic to the canonical geometry on the correspondence space over L 4 . Normality of the conformal Cartan connection implies that the Cartan geometry of type (S, A) is torsionfree and the corresponding ε-Kähler metric is non-Ricci-flat Einstein.
Remark 5.21. It is interesting to note the following geometric interpretation of the correspondence spaces: If ε = −1, the bundle CL 4 → L 4 can be identified with the twistor bundle TL 4 → L 4 whose fiber over a point x ∈ L 4 comprises all self-dual totally isotropic 2-planes in T x CL 4 . If ε = 1, CL 4 → L 4 can be identified with the subbundle of the twistor bundle whose fiber over a point x ∈ L 4 comprises all self-dual 2-planes in T x CL 4 except the eigenspaces of the endomorphismK x . The total space CL 4 carries a tautological rank 2-distribution obtained by lifting each self-dual totally isotropic 2-plane horizontally to its point in the fiber, and it was observed [2,6] that, provided the self-dual Weyl tensor of the metric on L 4 vanishes nowhere, this distribution is (2, 3, 5) almost everywhere. This suggests a relation of the present work to the An-Nurowski twistor construction (and recent work of Bor and Nurowski).

The canonical lattices of hypersurface distributions
Recall that if S is nonisotropic (if σ is not Ricci-flat) it determines a direct sum decomposition V = W ⊕ S , where W := S ⊥ , and if S is isotropic (if σ is Ricci-flat), it determines a filtration associated to (3.3): , so X × S ∈ ker(X × · ) − X . In particular, X × S is isotropic but nonzero, and hence it determines an analogous filtration of V.
Forming the intersections and spans of the components of the filtrations determined by S and X × S gives a lattice of vector subbundles of V under the operations of span and intersection (in fact, it is a graded lattice graded by rank). It has 22 elements in the non-Ricci-flat case and 26 in the Ricci-flat case, so for space reasons we do not reproduce these here. However, since W/ X ∼ = T M [−1], the sublattice of vector bundles N satisfying X N W descends to a natural lattice of subbundles of T M | M 4 . We record these lattices (they are different in the Ricci-flat and non-Ricci-flat cases), which efficiently encode the incidence relationships among the subbundles, in the following proposition. (We omit the proof, which is tedious but straightforward, and which can be achieved by working in an adapted frame.) Proposition 5.23. The bundle T M | M 4 admits a natural lattice of vector subbundles under the operations of span and intersection: If σ is non-Ricci-flat, the lattice is In the lattices, all bundles are implicitly restricted to M 4 , the numbers indicate the ranks of the bundles in their respective columns, and (in the case of the two large lattices) the diagram is arranged so that each bundle is positioned horizontally opposite its g-orthogonal bundle.
Recall that L 3 := π L (M 4 ). Since S is spanned by the invariant component µ of S and L ξ S = 0, S descends to a line fieldŜ ⊂ T L| L 3 . This line field is contained in T L 3 iff S is contained in T M 4 = S ⊥ , that is (by Proposition 5.23) iff σ is Ricci-flat.
Similarly, since the flow of ξ preserves g, it also preserves C ∩ T M 4 = (L ⊕ S) ⊥ . Then, because L ⊂ C ∩ T M 4 ⊂ S ⊥ = T M 4 , C ∩ T M 4 descends to a 2-plane distribution H ⊂ T L 3 .
Proposition 5.24. The 2-plane distribution H ⊂ T L 3 defined as above is contact.
We show that this map is well-defined, that it is independent of the choice of τ , and that we may regard it as an endomorphism of H: By Lemma 3.7, L ξ K = −[L A 0 (ξ), K] = −[K, K] = 0, so K and hence ζ is itself invariant under the flow of ξ, and hence that ζ is independent of choice of basepoint x of the lift. Now, any two lifts η, η ∈ T x M 4 differ by an element of ker T x π L = ξ x ; on the other hand, expanding (3.2) in terms of the splitting determined by τ , taking a particular component equation, and evaluating at σ = 0 gives the identity ζ b a ξ a = αξ b for some smooth function α, so T x π L · ζ(ξ) = 0, and hence J is well-defined. Finally, under a change of scale, ζ is transformed to ζ a b → ζ a b + Υ a ξ b − ξ a Υ b for some form Υ a ∈ Γ(T * M ) [4]. A lift η b ofη ∈ H is an element of C ∩ T M 4 ⊂ C = ker ξ , Υ a ξ b η b = 0. The term ξ a Υ b η b is again in ker T x π L , and we conclude that J is independent of the scale τ . Now, in the notation of the previous paragraph, we have µ b ζ b c η c = µ b (−µ d χ db c − ρφ b c )η c = −ρν b φ b c η c . This is −ρξ c η c , and we saw above that ξ c η c = 0, so µ b ζ b c η c = 0, that is, ζ b c η c ∈ ker µ = S ⊥ . Using the g-skewness of ζ gives ξ b ζ b c η c = −η b ζ b c ξ c = −η b (αξ b ) = −αη b ξ b , but again η b ξ b = 0, so we also have ζ b c η c ∈ ker ξ = L ⊥ . Thus, ζ(η) ∈ L ⊥ ∩ S ⊥ = C ∩ T M 4 , and pushing forward by π L gives J(η) ∈ H, so we may view J as an endomorphism of H.
Proof . In the above notation, unwinding (twice) the definition of J gives that J 2 (η) = T x π L · ζ 2 (η). Now, another component equation of (3.2) is ζ a c ζ c b − ξ a ν b − ν a ξ b = εδ a b + µ a µ b for some ν ∈ Γ(T * M ). The above observations about the terms Υ a ξ b and ξ a Υ b apply just as well to ξ a ν b and ν a ξ b , and since η ∈ C ∩ T M 4 ⊂ S ⊥ = ker µ, we have µ a µ b η b = 0, so the above component equation implies J 2 = ε id H . In the case ε = +1 one can verify that the (±1)-eigenspaces of J are both 1-dimensional, that is, J is an almost ε-complex structure on H.
In the Ricci-negative case, this shows precisely that (L 3 , H, J) is an almost CR structure (in fact, it turns out to be integrable, see the next subsubsection), and one might call the resulting structure in the general case an almost ε-CR structure. The three signs of ε (equivalently, the three signs of the Einstein constant) give three qualitatively distinct structures, so we treat them separately.
The geometry of 3-dimensional Legendrean contact structures admits another concrete, and indeed classical (local) interpretation, namely as that of second-order ordinary differential equations (ODEs) modulo point transformations: We can regard a second-order ODEÿ = F (x, y,ẏ) as a function F (x, y, p) on the jet space J 1 := J 1 (R, R), and the vector fields D x := ∂ x + p∂ y + F (x, y, p)∂ p and ∂ p span a contact distribution (namely the kernel of dy − p dx ∈ Γ(T * J 1 )), so D x ⊕ ∂ p is a Legendrean contact structure on J 1 . Point transformations of the ODE, namely those given by prolonging to J 1 (local) coordinate transformations of R 2 xy , are precisely those that preserve the Legendrean contact structure (up to diffeomorphism) [25].

Ricci-f lat case: A f ibration over a special conformal structure
In this case, Example 5.2 gives that the hypersurface curved orbit M 4 locally fibers over the space L of integral curves of S (nota bene the fibrations π L | M 4 : M 4 → L 3 in the non-Ricciflat cases above are instead along the integral curves of L), and that L inherits a conformal structure c L of signature (1,2). Considering the sublattice of the last lattice in Proposition 5.23 of the distributions containing S and forming the quotient bundles modulo S yields a complete flag field of T L that we write as 0 ⊂ E/S ⊂ E ⊥ /S ⊂ T L it depends only on D. Since E is totally c-isotropic, the line field E/S is c L -isotropic, and by construction it is orthogonal to E ⊥ /S with respect to c L . Thus, we may regard the induced structure on L as a Lorentzian conformal structure equipped with an isotropic line field.
Similarly, the fibration along the integral curves of L determines a complete flag field that we denote 0 ⊂ E/L ⊂ H ⊂ T L 3 . Computing in a local frame gives E/L = ker J = im J, and this line the kernel of the (degenerate) conformal (negative semidefinite) bilinear form c determines on H. Recall that on these orbits, ξ = 0 and K = 0, and hence E is not defined. Recall also that if σ is Ricci-negative, all three of these curved orbits are empty. If σ is Ricci-flat, only M 2 and M ± 0 occur, and if σ is Ricci-positive, only M ± 2 occur. Since the curved orbits M ± 0 are 0-dimensional, they inherit no structure. 5.6.1 The curved orbits M ± 2 : Projective surfaces By Theorem 5.1 and the orbit decomposition of the flat model in Section 5.2, the holonomy reduction determines parabolic geometries of type (SL(3, R), P 1 ) and (SL(3, R), P 2 ) on M ± 2 . Torsion-freeness of the normal conformal Cartan connection immediately implies that these parabolic geometries are torsion-free and hence determine underlying torsion-free projective structures (that is, equivalence classes of torsion-free affine connections having the same unparametrized geodesics).
One can compute the tractor connection explicitly (the explicit expression is unwieldy, so we do not reproduce it here) and use it to compute that the conformal holonomy Hol([−g]) is the full group SU (1,2). In particular, this shows that in the Ricci-negative case the holonomy reduction considered in this case does not automatically entail a holonomy reduction to a smaller group.
Since almost Einstein scales are in bijective correspondence with parallel standard tractors, the space of Einstein scales is 1-dimensional (as an independent parallel standard tractor would further reduce the holonomy). One can compute that aut(D υ ) ∼ = aut(ĝ,K) ∼ = so(3, R) ⊕ sl(2, R) and aut([−g]) ∼ = aut(g) ∼ = aut(ĝ,K) ⊕ ξ ∼ = so(3, R) ⊕ sl(2, R) ⊕ R. One can show that every distribution D υ is equivalent to the so-called rolling distribution for the Riemannian surfaces (S 2 , g + ) and (H 2 , g − ). The underlying space of this distribution, which we can informally regard as the space of relative configurations of S 2 and H 2 in which the surfaces are tangent at a single point, is the twistor bundle [2] over S 2 × H 2 whose fiber over (x + , x − ) is the circle Iso(T x + S 2 , T x − H 2 ) ∼ = S 1 of isometries. The distribution is the one characterized by the so-called no-slip, no-twist conditions on the relative motions of the two surfaces [13,Section 3].
We can produce a para-Sasaki analogue of this example, which in particular has full holonomy group SL(3, R) and hence shows the holonomy reduction to that group again does not automatically entail a reduction to a smaller group. Let (L 2 , h, J) denote the para-Kähler Lorenztian surface with h := 2 3 r 2 + 1 2 −dr 2 + r 2 dϕ 2 , J := r∂ r ⊗ dϕ + 1 r ∂ ϕ ⊗ dr.
Then, the triple (L 2 × L 2 , h ⊕ h, J ⊕ J), is a suitably normalized para-Kähler structure and we can proceed as before. Every (2, 3, 5) distribution in the determined family is diffeomorphic to the Lorentzian analogue of the rolling distribution for the surfaces (L 2 , h) and (L 2 , −h). 8 Example 6.2 (a cohomogeneity 1 distribution from a homogeneous projective surface). We construct an example of a Ricci-positive almost Einstein (2, 3, 5) conformal structure by specifying a para-Fefferman conformal structure c N on a 4-manifold N and solving a natural geometric Dirichlet problem: We produce a conformal structure c on N × R equipped with a holonomy reduction to SL(3, R) for which the hypersurface curved orbit is N and the induced structure there is c N . In particular, this yields an example of an almost Einstein (2, 3, 5) distribution for which the zero locus of the almost Einstein scale is nonempty, and hence for which the curved orbit decomposition has more than one nonempty curved orbit. Consider the projective structure [∇] on R 2 xy containing the torsion-free connection ∇ characterized by A Tabular summary of the curved orbit decomposition. --X ∧ S = 0 -- † This is the solvable 5-dimensional group R ((R 2 R) ⊕ R). ‡ The leaf space also inherits a degenerate conformal bilinear form with respect to which the line field is isotropic and the plane distribution is isotropic but not totally isotropic.