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

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: F...

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spelling irk-123456789-1485552019-02-19T01:26:07Z The Geometry of Almost Einstein (2,3,5) Distributions Sagerschnig, K. Willse, T. 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 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 c. Second, they are characterized by the existence of a holonomy reduction to SU(1,2), SL(3,R), or a particular semidirect product SL(2,R)⋉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; Kähler-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 Kähler-Einstein or para-Kähler-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. 2017 Article The Geometry of Almost Einstein (2,3,5) Distributions / K. Sagerschnig, T. Willse // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 67 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32Q20; 32V05; 53A30; 53A40; 53B35; 53C15; 53C25; 53C29; 53C55; 58A30 DOI:10.3842/SIGMA.2017.004 http://dspace.nbuv.gov.ua/handle/123456789/148555 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description 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 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 c. Second, they are characterized by the existence of a holonomy reduction to SU(1,2), SL(3,R), or a particular semidirect product SL(2,R)⋉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; Kähler-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 Kähler-Einstein or para-Kähler-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.
format Article
author Sagerschnig, K.
Willse, T.
spellingShingle Sagerschnig, K.
Willse, T.
The Geometry of Almost Einstein (2,3,5) Distributions
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Sagerschnig, K.
Willse, T.
author_sort Sagerschnig, K.
title The Geometry of Almost Einstein (2,3,5) Distributions
title_short The Geometry of Almost Einstein (2,3,5) Distributions
title_full The Geometry of Almost Einstein (2,3,5) Distributions
title_fullStr The Geometry of Almost Einstein (2,3,5) Distributions
title_full_unstemmed The Geometry of Almost Einstein (2,3,5) Distributions
title_sort geometry of almost einstein (2,3,5) distributions
publisher Інститут математики НАН України
publishDate 2017
url http://dspace.nbuv.gov.ua/handle/123456789/148555
citation_txt The Geometry of Almost Einstein (2,3,5) Distributions / K. Sagerschnig, T. Willse // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 67 назв. — англ.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 004, 56 pages The Geometry of Almost Einstein (2, 3, 5) Distributions Katja SAGERSCHNIG † and Travis WILLSE ‡ † Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: katja.sagerschnig@univie.ac.at ‡ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria E-mail: travis.willse@univie.ac.at Received July 26, 2016, in final form January 13, 2017; Published online January 19, 2017 https://doi.org/10.3842/SIGMA.2017.004 Abstract. We analyze the classic problem of existence of Einstein metrics in a given con- formal 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 struc- tures 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 c. Second, they are char- acterized by the existence of a holonomy reduction to SU(1, 2), SL(3,R), or a particular semidirect product SL(2,R) n Q+, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction par- titions 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 geome- tries – several classical geometries among them – including: Sasaki–Einstein geometry and its paracomplex and null-complex analogues in dimension 5; Kähler–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 projec- tive geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional Kähler–Einstein or para-Kähler–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. Key words: (2, 3, 5) distribution; almost Einstein; conformal geometry; conformal Killing field; CR structure; curved orbit decomposition; Fefferman construction; G2; holonomy reduction; Kähler–Einstein; Sasaki–Einstein; second-order ordinary differential equation 2010 Mathematics Subject Classification: 32Q20; 32V05; 53A30; 53A40; 53B35; 53C15; 53C25; 53C29; 53C55; 58A30 1 Introduction A (pseudo-)Riemannian metric gab is said to be Einstein if its Ricci curvature Rab is a multiple of gab. 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, New- man, 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. mailto:katja.sagerschnig@univie.ac.at mailto:travis.willse@univie.ac.at https://doi.org/10.3842/SIGMA.2017.004 2 K. Sagerschnig and T. Willse 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 ΘV0 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(S2T ∗M⊗E [2]), and a nowhere-vanishing section σ in the kernel of ΘV0 determines an Einstein metric σ−2g in c and vice versa. Writing the differential equation ΘV0 (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(S2V∗).) 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 ΘV0 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 geo- metric 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 distri- bution D on a 5-manifold which is maximally nonintegrable in the sense that [D, [D,D]] = TM . 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 G2 (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 ce- lebrated 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 G2 [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 G2 refers to the algebra automorphism group of the split octonions; this is a split real form of the com- plex simple Lie group of type G2.) 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 cD of signature (2, 3) onM , and it is these structures, which we call (2, 3, 5) conformal structures, whose almost Einstein geometry we investigate. For expository conve- nience, 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 cD for some (2, 3, 5) distibution D iff the holonomy group of ∇V is contained inside G2, or equivalently, iff there is a parallel tractor 3-form, that is, a section Φ ∈ Γ(Λ3V∗), compatible The Geometry of Almost Einstein (2, 3, 5) Distributions 3 with the conformal structure in the sense that the pointwise stabilizer of Φp in GL(Vp) (at any, equivalently, every point p) is isomorphic to G2 and is contained inside SO(Hp) [41, 52]. While the construction D cD 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 G2. 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 Λ3V∗ 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) n Q+, where Q+ < G2 is the connected, nilpotent subgroup of G2 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 Λ3V∗ 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) n Q+ refer to the common sta- bilizer 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 yΦ, 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 cD 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 cD′ = cD? Put another way, what are the fibers of the map D cD? 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 G2 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 cD. 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(Λ2T ∗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. Theorem B. Fix an oriented (2, 3, 5) distribution (M,D), denote by φ the corresponding generic conformal Killing 2-form, by Φ ∈ Γ(Λ3V∗) the corresponding parallel tractor 3-form, and by HΦ ∈ Γ(S2V∗) the parallel tractor metric associated to cD. 1Cf. [45, Corollary 2.4]. 4 K. Sagerschnig and T. Willse 1. Suppose (M, cD) 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 := −HΦ(S,S) ∈ {−1, 0,+1}. Then, for any (Ā, B) ∈ R2 such that −εĀ2 + 2Ā + B2 = 0 (there is a 1- parameter family of such pairs) the weighted 2-form φ′ab := φab + Ā [ 1 5σ 2 ( 1 3φab,c c + 2 3φc[a,b] c + 1 2φc[a, c b] + 4Pc[aφb]c ) − σσ,cφ[ca,b] (1.1) − 1 2σσ,[aφb]c, c − 1 5σσ,c cφab + 3σ,cσ,[cφab] ] +B[−1 4σφ cd, dφ[ab,c] + 3 4σ ,cφ[abφc]d, d] is a generic conformal Killing 2-form, and the oriented (2, 3, 5) distribution D′ it deter- mines induces the same oriented conformal structure that D does, that is cD′ = cD. 2. Conversely, all conformally isometric oriented (2, 3, 5) distributions arise this way: If an oriented (2, 3, 5) distribution D′ satisfies cD′ = cD (this condition is equality of oriented conformal structures), then there is an almost Einstein scale σ of cD (we may assume that the corresponding parallel tractor S satisfies ε := −HΦ(S,S) ∈ {−1, 0,+1}) and (Ā, B) ∈ R2 satisfying −εĀ2 + 2Ā + B2 = 0 such that the normal conformal Killing 2- form φ′ corresponding to D′ is given by (1.1). Herein, a comma , denotes the covariant derivative with respect to (any) representative g ∈ cD, and Pab denotes the Schouten tensor (2.19) of g. We say that the distributions in the 1-parameter family D determined by D and σ as in the theorem are related by σ. Theorem B is proved in Section 4.2. Different signs of the Einstein constant of the Einstein metric σ−2g, or equivalently, different causality types of the corresponding parallel tractor S, determine families of distributions with different qualitative behaviors. Section 4.2.1 gives simple parameterizations of the 1-parameter families of conformally isometric distributions, and Section 4.2.3 gives an explicit algorithm for recovering an almost Einstein scale σ of cD relating D and D′ whose existence is guaranteed by Part (2) of Theorem B. An immediate corollary of Theorem B is a natural geometric characterization of almost Einstein oriented (2, 3, 5) distributions: Theorem C. The conformal structure cD induced by an oriented (2, 3, 5) distribution (M,D) is almost Einstein iff there is a distribution (M,D′), D′ 6= D such that cD = cD′. Now, fix an oriented conformal structure c of signature (2, 3) and a nonzero almost Einstein scale σ of c. The conformal Killing field ξ of c determined together by σ and a choice of distribution D in the 1-parameter family D of oriented (2, 3, 5) distributions inducing c and related by σ turns out not to depend on the choice of D (Proposition 4.6), and we can ask for all of the geometric objects that (like ξ) are determined by σ and D. The almost Einstein scale σ alone partitions M into three subsets, M+, Σ, M−, according to the sign +, 0, − of σ at each point. By construction, σ determines an Einstein metric on the complement M −Σ = M+∪M−. If the Einstein metric determined by σ is not Ricci-flat (that is, if the parallel tractor S corresponding to σ is nonisotropic), the boundary Σ itself inherits a conformal structure cΣ that is suitably compatible with and that can be regarded as a natural compactifying structure for (M±, g±) along ∂M± = Σ [33]. Something similar but more involved occurs in the Ricci-flat case. This decomposition of M according to the geometry of the object σ – equivalently, the holon- omy reduction of ∇V determined by the parallel standard tractor S – along with descriptions of the geometry induced on each subset in the decomposition, is formalized by the theory of curved orbit decompositions [17]. Here, the involved geometric structures are encoded as Cartan geometries (Section 2.3.1) of an appropriate type, which are geometric structures modeled on appropriate homogeneous spaces G/P endowed with G-invariant geometric structures, and the decomposition of M in the presence of a holonomy reduction to a group H ≤ G is a natural The Geometry of Almost Einstein (2, 3, 5) Distributions 5 generalization of the H-orbit decomposition of G/P ; the subsets in the decomposition are ac- cordingly 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) distributions related by a Ricci-negative almost Einstein scale such that c = cD for all D ∈ D, as well as a decomposition M = M+ 5 ∪M − 5 ∪M4: • (Section 5.4) The orbits M±5 are open, and M5 := M+ 5 ∪M − 5 is equipped with a Ricci- negative Einstein metric g := σ−2g|M5. The pair (−g, ξ) is a Sasaki structure (see Sec- tion 5.4.3) on M5. Locally, M5 fibers along the integral curves of ξ, and the leaf space L4 inherits a Kähler–Einstein structure (ĝ, K̂). • (Section 5.5.3) The orbit M4 is a smooth hypersurface, and inherits a Fefferman confor- mal structure cS, which has signature (1, 3): Locally, cS arises from the classical Fef- ferman construction [16, 30, 51], which (in this dimension) canonically associates to any 3-dimensional CR structure (L3,H,J) a conformal structure on a circle bundle over L3. Again in the local setting, the fibers of the fibration M4 → L3 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 = cD for all D ∈ D, as well as a decomposition M = M+ 5 ∪M − 5 ∪M4 ∪M+ 2 ∪M − 2 : • (Section 5.4) The orbits M±5 are open, and M5 := M+ 5 ∪M − 5 is equipped with a Ricci- positive Einstein metric g := σ−2g|M5. The pair (−g, ξ) is a para-Sasaki structure (see Section 5.4.3) on M5. Locally, M5 fibers along the integral curves of ξ, and the leaf space L4 inherits a para-Kähler–Einstein structure (ĝ, K̂). 6 K. Sagerschnig and T. Willse • (Section 5.5.4) The orbit M4 is a smooth hypersurface, and inherits a para-Fefferman conformal structure cS, which has signature (2, 2): Locally, cS arises from the paracomplex analogue of the classical Fefferman construction, which (in this dimension) canonically associates to any Legendrean contact structure (L3,H+ ⊕H−) – or, locally equivalently, a point equivalence class of second-order ODEs ÿ = F (x, y, ẏ) – a conformal structure on a SO(1, 1)-bundle over L3. Again in the local setting, the fibers of the fibration M4 → L3 are the integral curves of ξ. • (Section 5.6.1) The orbits M±2 are 2-dimensional and inherit oriented projective structures. 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 D0. Let (M, c) be an oriented conformal structure of signature (2, 3). A holonomy reduction of c to SL(2,R) n Q+ determines a 1-parameter family D of oriented (2, 3, 5) distri- butions related by a Ricci-flat almost Einstein scale such that c = cD for all D ∈ D, as well as a decomposition M = M+ 5 ∪M − 5 ∪M4 ∪M2 ∪M+ 0 ∪M − 0 . • (Section 5.4) The orbits M±5 are open, and M5 := M+ 5 ∪M − 5 is equipped with a Ricci- flat metric g := σ−2g|M5. The pair (−g, ξ) is a null-Sasaki structure (see Section 5.4.3) on M5. Locally, M5 fibers along the integral curves of ξ, and the leaf space L4 inherits a null-Kähler–Einstein structure (ĝ, K̂). • (Section 5.5.5) The orbit M4 is a smooth hypersurface, and it locally fibers over a 3-mani- fold L̃ that carries a conformal structure c L̃ of signature (1, 2) and isotropic line field. • (Section 5.6.2) The orbit M2 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 con- sequences: 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 = cD. 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 M5 → L4 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 1- parameter 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 The Geometry of Almost Einstein (2, 3, 5) Distributions 7 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] ∼= Λ2D closely related to ΘV0 [55]. Its kernel can again be interpreted as the space of almost Einstein scales of cD, but it is a simpler object than ΘV0 , 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. 2 Preliminaries 2.1 ε-complex structures The ε-complex numbers, ε ∈ {−1, 0,+1}, is the ring Cε generated over R by the generator iε, which satisfies precisely the relations generated by i2ε = 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 K2 = ε idW;2; if ε = +1, we further require that the (±1)-eigenspaces both have dimension m. This identifies W with Cmε (as a free Cε-module) so that the action of K coincides with multiplication by iε, and the pair (W,K) is an ε-complex vector space. One specializes the names of structures to particular values of ε by omitting (−1)-, replacing (+1)- with the prefix para-, and replacing 0- with the modifier null -. See [59, Section 1]. 2.2 The group G2 2.2.1 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 Õ 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 ∈ Õ. In particular N(1) = 1, and polarizing N yields a nondegenerate symmetric bilinear form, which turns out for Õ 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 H(x× y, x) = 0 and H(x× y, x× y) = H(x, x)H(y, y)−H(x, y)2 (2.1) for all x, y ∈ I. 2In the case ε = 0, some references require additionally that a null-complex structure satisfy rankK = m [27] this anyway holds for the null-complex structures that appear in this article. 8 K. Sagerschnig and T. Willse 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 : I→ V such that A(x× y) = A(x)×A(y). 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 x× (x× y) := −H×(x, x)y +H×(x, y)x. (2.2) By (2.1), × is totally H×-skew, so lowering its upper index with H× yields a 3-form Φ ∈ Λ3V∗: Φ(x, y, z) := H×(x× y, z). 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 Λ3V∗. 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 ε× ∈ Λ7V∗ for H×: (ε×)ABCDEFG := 1 42ΦK[ABΦK CDΦEFG]. Thus, × determines an orientation [ε×] on V and Hodge star operators ∗ : ΛkV∗ → Λ7−kV∗. If V is a vector space endowed with a bilinear form H of signature (p, q) and an orienta- tion 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 = [ε×]. A split-generic 3-form Φ satisfies various contraction identities, including [12, equations (2.8) and (2.9)]: ΦE ABΦECD = (∗ΦΦ)ABCD +HACHBD −HADHBC , (2.3) ΦF AB(∗ΦΦ)FCDE = 3(HA[CΦDE]B −HB[CΦDE]A). (2.4) 2.2.2 The group G2 The (algebra) automorphism group of Õ is a connected, split real form of the complex Lie group of type G2, and so we denote it by G2. One can recover the algebra structure of Õ from (I,×), so G2 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 G2, see [45]. The action of G2 on V defines the smallest nontrivial, irreducible representation of G2, 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 G2-structure on a 7-dimensional real vector space V we mean either (1) a representation of G2 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 [ε×] on V, the induced actions of G2 preserve both, defining a natural embedding G2 ↪→ SO(H×) ∼= SO(3, 4). The Geometry of Almost Einstein (2, 3, 5) Distributions 9 Moreover, H× realizes V as the standard representation of SO(3, 4), and its restriction to G2 is the standard representation × defines. Like SO(3, 4), the G2-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 G2-structure on V defined in a basis (Ea) (with dual basis, say, (ea)) via the 3-form Φ := −e147 + √ 2e156 + √ 2e237 + e245 + e346, (2.5) where ea1···ak := ea1 ∧ · · · ∧ eak (cf. [41, equation (23)]). With respect to the basis (Ea), the induced bilinear form HΦ has matrix representation [HΦ] =  0 0 0 0 1 0 0 0 I2 0 0 0 −1 0 0 0 I2 0 0 0 1 0 0 0 0  , (2.6) where I2 denotes the 2× 2 identity matrix, and the induced volume form is εΦ = −e1234567. Let g2 denote the Lie algebra of G2. Differentiating the inclusion G2 ↪→ GL(V) yields a Lie algebra representation g2 ↪→ gl(V) ∼= End(V), and with respect to the basis (Ea) its elements are precisely those of the form trA Z s W> 0 X A √ 2JZ> s√ 2 J −W r − √ 2X>J 0 − √ 2ZJ s Y > − r√ 2 J √ 2JX −A> −Z> 0 −Y r −X> − trA  , (2.7) where A ∈ gl(2,R), W,X ∈ R2, Y, Z ∈ (R2)∗, r, s ∈ R, and J := ( 0 −1 1 0 ) . 2.2.3 Some G2 representation theory Fix a 7-dimensional real vector space V and a G2-structure Φ ∈ Λ3V∗. We briefly record the decompositions of the G2-representations Λ2V∗, Λ3V∗, and S2V∗ into irreducible subrepresenta- tions; we use and extend the notation of [12, Section 2.6]. In each case, Λkl denotes the irreducible subrepresentation of ΛkV∗ of dimension l, which is unique up to isomorphism. The representation Λ2V∗ ∼= so(HΦ) decomposes into irreducible subrepresentations as Λ2V∗ = Λ2 7 ⊕ Λ2 14; Λ2 7 ∼= V and Λ2 14 is isomorphic to the adjoint representation g2. Define the map ι27 : V→ Λ2V∗ by ι27 : SA 7→ SCΦCAB; (2.8) by raising an index we can view ι27 as the map V → so(HΦ) given by S 7→ (T 7→ T × S). It is evidently nontrivial, so by Schur’s lemma its (isomorphic) image is Λ2 7. Conversely, consider the map π2 7 : Λ2V∗ → V defined by π2 7 : AAB 7→ 1 6ABCΦBCA. (2.9) 10 K. Sagerschnig and T. Willse Raising indices gives a map Λ2V → 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 Λ2 14 and restricts to an isomorphism π2 7|Λ2 7 : Λ2 7 → V; we have chosen the coefficient so that π2 7 ◦ ι27 = idV and ι27 ◦ π2 7|Λ2 7 = idΛ2 7 . Since G2 is the stabilizer subgroup in SO(HΦ) of Φ, g2 is the annihilator in so(HΦ) ∼= Λ2V∗ of Φ. Expanding idV−ι27 ◦ π2 7 using (2.8) and (2.9) and applying (2.3) gives that under this identification, the corresponding projection π2 14 : so(V) ∼= Λ2V∗ → g2 is π2 14 : AAB 7→ 2 3A A B − 1 6(∗ΦΦ)D EA BADE . The representation Λ3V∗ decomposes into irreducible subrepresentations as Λ3V∗ = Λ3 1 ⊕ Λ3 7 ⊕ Λ3 27. Here, Λ3 1 is just the trivial representation spanned by Φ, and the map π3 1 : Λ3V∗ → R defined by π3 1 : ΨABC 7→ 1 42ΦABCΨABC (2.10) is a left inverse for the map R ∼=→ Λ3 1 ↪→ Λ3V∗ defined by a 7→ aΦ. The map ι37 : V→ Λ3V∗ defined by ι37 : SA 7→ −SD(∗ΦΦ)DABC = [∗Φ(S ∧ Φ)]ABC . is nonzero, so it defines an isomorphism V ∼= Λ3 7. The map π3 7 : Λ3V∗ → V defined by π3 7 : ΨABC 7→ 1 24(∗ΦΦ)BCDAΨBCD = 1 4 [∗Φ(Φ ∧Ψ)]A (2.11) is scaled so that π3 7 ◦ ι37 = idV and ι37 ◦ π3 7 = idΛ3 7 . The G2-representation S2V∗ decomposes into irreducible modules as R⊕ S2 ◦V∗, namely, into its HΦ-trace and HΦ-tracefree components, respectively. The linear map i : S2V∗ → Λ3V∗ defined by i : AAB 7→ 6ΦD [ABAC]D. (2.12) satisfies i(HΦ) = 6Φ and is nonzero on S2 ◦V∗, so i|S2 ◦ is an isomorphism S2 ◦ ∼=→ Λ3 27. The projection π3 27 : Λ3V∗ → S2 ◦V∗ π3 27 : ΨABC 7→ −1 8∗Φ[( · yΦ) ∧ ( · yΦ) ∧Ψ]AB − 3 4π 3 1(Ψ)HAB (2.13) is scaled so that π3 27 ◦ i|S2 ◦V∗ = idS2 ◦V∗ and i ◦ π3 27|Λ3 27 = idΛ3 27 . 2.3 Cartan and parabolic geometry 2.3.1 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 Rp(u) = u ·p for u ∈ G, p ∈ P . For each V ∈ p, the corresponding fundamental vector field ηV ∈ Γ(TG) is (ηV )u := ∂t|0[u · exp(tV )]. 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) ωu·p(TuR p · η) = Ad(p−1)(ωu(η)) for all u ∈ G, p ∈ P , η ∈ TuG, 2) (reproduction of fundamental vector fields) ω(ηV ) = V for all V ∈ p, and 3) (absolute parallelism) ωu : TuG → g is an isomorphism for all u ∈ G. The (flat) model of Cartan geometry of type (G,P ) is the pair (G→ G/P, ωMC), where ωMC is the Maurer–Cartan form on G defined by (ωMC)u := TuLu−1 (here Lu−1 : G→ G denotes left The Geometry of Almost Einstein (2, 3, 5) Distributions 11 multiplication by u−1). This form satisfies the identity dωMC + 1 2 [ωMC, ωMC] = 0. We define the curvature (form) of a Cartan geometry (G, ω) to be the section Ω := dω+ 1 2 [ω, ω] ∈ Γ(Λ2T ∗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 ωMC. 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 → Λ2(g/p)∗ ⊗ g, which we also call the curvature. 2.3.2 Holonomy Given any Cartan geometry (G →M,ω) of type (G,P ), we can extend the Cartan connection ω to a unique principal connection ω̂ on Ĝ := G ×P G characterized by (1) G-equivariance and (2) ι∗ω̂ = ω, where ι : G ↪→ Ĝ is the natural inclusion u 7→ [u, e]. Then, to any point û ∈ Ĝ we can associate the holonomy group Holû(ω̂) ≤ G. Different choices of û lead to conjugate subgroups of G, so the conjugacy class Hol(ω̂) thereof is independent of û, and we define the holonomy of ω (or just as well, of (G →M,ω)) to be this class. 2.3.3 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-representa- tion U. Then, for any Cartan geometry (π : G →M,ω) of type (G,P ), we can form the associated tractor bundle U := G×PU→M , which we can also view as the associated bundle Ĝ ×G U→M . Then, the principal connection ω̂ on Ĝ 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 ΠA0 : A → TM defined by (u, V ) 7→ Tuπ · ω−1 u (V ) descends to a natural isomorphism G ×P (g/p) ∼=→ TM , and via this identification ΠA0 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(Λ2T ∗M ⊗A), and again we call it the curvature of (G →M,ω). 2.3.4 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 con- dition) 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 [ga, gb] ⊆ ga+b and minimal in the sense that none of the summands ga, a = −k, . . . , k, is zero. The grading induces a P -invariant filtration (ga) of g, where ga := ga ⊕ · · · ⊕ g+k. In particular, p = g0 = g0 ⊕ · · · ⊕ g+k. We denote by G0 < P the subgroup of elements p ∈ P for which Ad(g) preserves the grading (ga) 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 (ga); 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 12 K. Sagerschnig and T. Willse the Lie algebra homology H•(p+,U) with the chain complex · · · → Λi+1(g/p)∗ ⊗ U ∂∗→ Λi(g/p)∗ ⊗ U→ · · · . The Kostant codifferential ∂∗ is P -equivariant, so it induces bundle maps ∂∗ : Λi+1T ∗M ⊗U → ΛiT ∗M ⊗ U between the associated bundles. The normalization conditions for a parabolic geometry (G →M,ω) are that 1) (normality) the curvature κ satisfies ∂∗κ = 0, and 2) (regularity) the curvature κ satisfies κ(u)(gi, gj) ⊆ gi+j+1 for all u ∈ G and all i, j. Finally, tractor bundles associated to parabolic geometries inherit additional natural struc- ture: Given a G-representation U, P determines a natural filtration (Ua) of U by successive action of the nilpotent Lie subalgebra p+ < g, namely U ⊇ p+ · U ⊇ p+ · (p+ · U) ⊇ · · · ⊇ {0}. (2.14) Since the filtration (Ua) of U is P -invariant, it determines a bundle filtration (Ua) of the tractor bundle U = G ×P U. For the adjoint representation g itself, this filtration (appropriately indexed) is just (ga), and the images of the filtrands A = G ×P g−k ) · · · ) G ×P g−1 under the projection ΠA0 comprise a canonical filtration TM = T−kM ) · · · ) T−1M of the tangent bundle. 2.3.5 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 subgroup P̄ of such a ray is parabolic. There is an equivalence of categories between regular, normal parabolic geometries of type (SO(p + 1, q + 1), P̄ ) 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0 0 1 0 Σ 0 1 0 0  . (2.15) (With respect to the basis (Ea), the matrix representation [HΦ] (2.6) of the bilinear form HΦ determined by the explicit expression (2.5) for Φ has the form (2.15).) The Lie algebra so(p+ 1, q + 1) consists of exactly the elements b Z 0 X B −Σ−1Z> 0 −X>Σ −b  , The Geometry of Almost Einstein (2, 3, 5) Distributions 13 where B ∈ so(Σ), X ∈ Rp+q, Z ∈ (Rp+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 g0 g+1 0 g−1 g0 g+1 0 g−1 g0  . (2.16) Since the grading on so(p+ 1, q+ 1) induced by P̄ has the form g−1⊕g0⊕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 4.1.2]. 2.3.6 Oriented (2, 3, 5) distributions The group G2 acts transitively on the space of HΦ-isotropic rays in V, and the stabilizer sub- group Q of such a ray is parabolic [54]. The subgroup Q is the intersection of G2 with the stabilizer subgroup P̄ < 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 g2 is the one defined by the block decomposition (2.7) and the labeling g0 g+1 g+2 g+3 0 g−1 g0 g+1 g+2 g+3 g−2 g−1 0 g+1 g+2 g−3 g−2 g−1 g0 g+1 0 g−3 g−2 g−1 g0  . There is an equivalence of categories between regular, normal parabolic geometries of type (G2, Q) and so-called oriented (2, 3, 5) distributions [18, Section 4.3.2]. On a manifold M , define the bracket of distributions E,F ⊆ TM to be the set [E,F] := {[α, β]x : x ∈M ;α ∈ Γ(E), β ∈ Γ(F)} ⊆ TM . Definition 2.4. A (2, 3, 5) distribution is a 2-plane distribution D on a 5-manifold M that is maximally nonintegrable in the sense that (1) [D,D] is a 3-plane distribution, and (2) [D, [D,D]] = TM . A (2, 3, 5) distribution is oriented iff the bundle D→M is oriented. An orientation of D determines an orientation of M and vice versa. The appropriate re- strictions of the Lie bracket of vector fields descend to natural vector bundle isomorphisms L : Λ2D ∼=→ [D,D]/D and L : D⊗ ([D,D]/D) ∼=→ TM/[D,D]; these are components of the Levi bracket. For a regular, normal parabolic geometry (G, ω) of type (G2, Q), the underlying (2, 3, 5) distribution D is T−1M = G ×Q (g−1/q), and [D,D] is T−2M = G ×Q (g−2/q). 2.4 Conformal geometry In this subsection we partly follow [4]. 2.4.1 Conformal density bundles A conformal structure (M, c) of signature (p, q) (denote n := p+q), determines a family of natural (conformal) density bundles on M : Denote by E [1] the positive (2n)th root of the canonically oriented line bundle (ΛnTM)2, and its respective wth integer powers by E [w]; E := E [0] is the trivial bundle with fiber R, and there are natural identifications E [w]⊗E [w′] ∼= E [w+w′]. Given 14 K. Sagerschnig and T. Willse any vector bundle B → M , we denote B[w] := B ⊗ E [w], and refer to the sections of B[w] as sections of B of conformal weight w. We may view c itself as the canonical conformal metric, gab ∈ Γ(S2T ∗M [2]). Contraction with gab determines an isomorphism TM → 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 gab and its inverse, gab ∈ Γ(S2TM [−2]). We call a nowhere zero section τ ∈ Γ(E [1]) a scale of c. A scale determines trivializations B[w] ∼=→ B, b 7→ b := τ−wb, of all conformally weighted bundles, and in particular a representative metric τ−2g ∈ c. 2.4.2 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 ∈ S2V∗ and a volume form ε ∈ Λn+2V∗; these respectively induce on V a parallel tractor metric H ∈ Γ(S2V∗) and parallel volume form ε ∈ Γ(Λn+2V∗). Consulting the block structure (2.16) of p̄+ < so(p+ 1, q + 1) gives that the filtration (2.14) of the standard representation V of SO(p+ 1, q + 1) determined by P̄ is ∗∗ ∗  ⊃  ∗∗ 0  ⊃  ∗0 0  ⊃  0 0 0  . (2.17) We may identify the composition series of the corresponding filtration of V as V ∼= E [1], TM [−1], E [−1]. We denote elements and sections of V using uppercase Latin indices, A,B,C, . . ., as SA ∈ Γ(V), and those of the dual bundle V∗ with lower indices, as SA ∈ Γ(V∗); we freely raise and lower indices using H. The bundle inclusion E [−1] ↪→ V determines a canonical section XA ∈ Γ(V[1]). Any scale τ determines an identification of V with the associated graded bundle determined by the above filtration, that is, an isomorphism V ∼= E [1] ⊕ TM [−1] ⊕ E [−1] [4]. So, τ also determines (non-invariant, that is, scale-dependent) inclusions TM [−1] ↪→ V and E [1] ↪→ V, which we can respectively regard as sections ZAa ∈ Γ(V ⊗ T ∗M [1]) and Y A ∈ Γ(V[−1]). So, for any choice of τ we can decompose a section S ∈ Γ(V) uniquely as SA τ = σY A + µaZAa + ρXA, where the notation τ = indicates that Y A and ZAa are the inclusions determined by τ . Reusing the notation of the filtration of V we write SA τ =  ρ µa σ  . (2.18) With respect to any scale τ , the tractor metric has the form (cf. (2.15)) HAB τ = 0 0 1 0 gab 0 1 0 0  . In particular, the filtration (2.17) of V is V ⊃ 〈X〉⊥ ⊃ 〈X〉 ⊃ {0}. The Geometry of Almost Einstein (2, 3, 5) Distributions 15 The normal tractor connection ∇V on V is [4] ∇Vb  ρ µa σ  τ =  ρ,b − Pbcµ c µa,b + Pabσ + δabρ σ,b − µb  ∈ Γ  E [−1] TM [−1] E [1] ⊗ T ∗M  . The subscript ,b denotes the covariant derivative with respect to g := τ−2g, and Pab is the Schouten tensor of g, which is a particular trace adjustment of the Ricci tensor Rab: Pab := 1 n− 2 ( Rab − 1 2(n− 1) Rccgab ) . (2.19) A section AA1···Ak of the tractor bundle ΛkV∗ associated to the alternating representa- tion ΛkV∗ decomposes uniquely as AA1···Ak τ = kφa2···akY[A1ZA2 a2 · · ·ZAk] ak + χa1···akZ[A1 a1 · · ·ZAk] ak + k(k − 1)θa3···akY[A1XA2ZA3 a3 · · ·ZAk] ak + kψa2···akX[A1ZA2 a2 · · ·ZAk] ak , which we write more compactly as AA1···Ak τ =  ψa2···ak χa1···ak | θa3···ak φa2···ak  ∈ Γ  Λk−1T ∗M [k − 2] ΛkT ∗M [k] | Λk−2T ∗M [k − 2] Λk−1T ∗M [k]  . The tractor connection ∇V induces a connection on ΛkV∗, and we denote this connection again by ∇V . In the special case k = 2, raising an index using H gives Λ2V∗ ∼= so(p+ 1, q + 1), so we can identify Λ2V∗ ∼= A. Any section AAB ∈ Γ(A) decomposes uniquely as AAB = ξa ( Y AZBa − ZAaYB ) + ζabZ A aZB b + α ( Y AXB −XAYB ) + νb ( XAZB b − ZAbXB ) , which we write as AAB τ =  νb ζab | α ξb  ∈ Γ  T ∗M Endskew(TM) | E TM  . Finally, p̄+ annihilates Λn+2V∗ ∼= R, yielding a natural bundle isomorphism Λn+2V∗ ∼= ΛnT ∗M [n]. This identifies the tractor volume form ε with the conformal volume form εg of g. 2.4.3 Canonical quotients of conformal tractor bundles For any irreducible SO(p + 1, q + 1)-representation U, the canonical Lie algebra cohomology quotient map U 7→ H0 := H0(p+,U) = U/(p̄+ · U) is P̄ -invariant and so induces a canonical bundle quotient map ΠU0 : U → H0 between the corresponding associated P̄ -bundles. (We reuse the notation ΠU0 for the induced map Γ(U) → Γ(H0) on sections.) Given a section A ∈ Γ(U), its image ΠU0 (A) ∈ Γ(H0) is its projecting part. For the standard representation V this quotient map is ΠV0 : V → E [1], ΠV0 : ∗∗ σ  7→ σ. 16 K. Sagerschnig and T. Willse For the alternating representation ΛkV∗, the quotient map is ΠΛkV∗ 0 : ΛkV∗ → Λk−1T ∗M [k], ΠΛkV∗ 0 :  ∗ ∗ | ∗ φa2···ak  7→ φa2···ak (2.20) For the adjoint representation so(p + 1, q + 1), the quotient map coincides with the map ΠA0 : A → TM defined in Section 2.3.3; in a splitting, it is ΠA0 :  ∗ ∗ | ∗ ξa  7→ ξa. 2.4.4 Conformal BGG splitting operators Conversely, for each irreducible SO(p+1, q+1)-representation U there is a canonical differential BGG splitting operator LU0 : Γ(H0)→ Γ(U) characterized by the properties (1) ΠU0 ◦ LU0 = idH0 and (2) ∂∗ ◦ ∇U ◦ LU0 = 0 [14, 19]. The only property of the operators LU0 we need here follows immediately from this characterization: If A ∈ Γ(U) is ∇U -parallel, then LU0 (ΠU0 (A)) = A. 2.5 Almost Einstein scales The BGG splitting operator LV0 : Γ(E [1])→ Γ(V) corresponding to the standard representation is [40, equation (114)] LV0 : σ 7→ − 1 n(σ,b b + Pbbσ) σ,a σ  . (2.21) Computing gives ∇Vb LV0 (σ)A τ =  ∗ (σ,ab + Pabσ)◦ 0  ∈ Γ  E [−1] T ∗M [1] E [1] ⊗ T ∗M  , (2.22) where (Tab)◦ denotes the tracefree part Tab − 1 nT c cgab of the (possibly weighted) covariant 2- tensor Tab, and where ∗ is some third-order differential expression in σ. Since the bottom component of ∇VLV0 (σ) is zero, the middle component, regarded as a (second-order) linear differential operator ΘV0 : Γ(E [1])→ Γ(S2T ∗M [1]), ΘV0 : σ 7→ (σ,ab + Pabσ)◦, is conformally invariant. The operator ΘV0 is the first BGG operator [19] associated to the standard representation V for (oriented) conformal geometry. We can readily interpret a solution σ ∈ ker ΘV0 geometrically: If we restrict to the complement M −Σ of the zero locus Σ := {x ∈M : σx = 0}, we can work in the scale of the solution σ itself: We have σ σ = 1 and hence 0 = ΘV0 (σ) = P◦. This says simply says that the Schouten tensor, P, of g := σ−2g|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]: Definition 2.5. An almost Einstein scale3 of an (oriented) conformal structure of dimension n ≥ 3 is a solution σ ∈ Γ(E [1]) of the operator ΘV0 . A conformal structure is almost Einstein if it admits a nonzero almost Einstein scale. 3Our terminology follows that of the literature on almost Einstein scales, but this consistency entails a mild perversity, namely that, since they may vanish, almost Einstein scales need not be scales. The Geometry of Almost Einstein (2, 3, 5) Distributions 17 We denote the set ker ΘV0 of almost Einstein scales of a given conformal structure c by aEs(c). Since ΘV0 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, ΘV0 (σ). So, ∇V is a prolongation connection for the operator ΘV0 : Theorem 2.6 ([4, Section 2]). For any conformal structure (M, c), dimM ≥ 3, the restrictions of LV0 : Γ(E [1]) → Γ(V) and ΠV0 : Γ(V) → Γ(E [1]) comprise a natural bijective correspondence between almost Einstein scales and parallel standard tractors: aEs(c) LV0 � ΠV0 { ∇V-parallel sections of V } . 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 λ := −1 2H(LV0 (σ), LV0 (σ)) = 1 nσ(σ,a a + Paaσ)− 1 2σ,aσ ,a. (2.23) This definition is motivated by the following computation: On M − Σ the Schouten tensor of the representative metric g := σ−2g|M−Σ ∈ c|M−Σ determined by the scale σ|M−Σ is P = λg. Thus, the Ricci tensor of g is Rab = 2(n− 1)λgab, 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 2.6 Conformal Killing fields and (k − 1)-forms The BGG splitting operator LΛkV∗ 0 : Γ(Λk−1T ∗M [k])→ Γ(ΛkV∗) determined by the alternating representation ΛkV∗, 1 < k < n+ 1, is [40, equation (134)] LΛkV∗ 0 : φa2···ak 7→  ( 1 n [ − 1 kφa2...ak,b b + k−1 k φb[a3···ak,a2] b + k−1 n−k+2φb[a3···ak, b a2] +2(k − 1)Pb[a2φ|b|a3···ak] − Pbbφa2···ak ] ) φ[a2···ak,a1] | − 1 n−k+2φb[a3···aka2], b φa2···ak . (2.24) Proceeding as in Section 2.5, we find that ∇Vb LΛkV∗ 0 (φ)A1···Ak =  ∗ ∗ | ∗ φa2···ak,b − φ[a2···ak,b] − k−1 n−k+2gb[a2φ|c|a3···ak], c  , where each ∗ denotes some differential expression in σ. The bottom component defines an inva- riant conformal differential operator ΘΛkV∗ 0 : Γ(Λk−1T ∗M [k])→ Γ(Λk−1T ∗M � T ∗M [k]) (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 ΘΛkV∗ 0 (φ) does not in general imply the vanishing of the remaining components ∗; if they do vanish, that is, if∇VLΛkV∗ 0 (φ) = 0, φ is called a normal conformal Killing (k − 1)-form [40, Section 6.2], [50]. The BGG splitting operator LA0 : Γ(TM)→ Γ(A) for the adjoint representation so(p+1, q+1) is [40, equation (119)] LΛkV∗ 0 : ξb 7→  1 n ( −1 2ξb,c c + 1 2ξ c ,bc + 1 nξ c ,cb + 2Pbcξ c − Pccξb ) 1 2(−ξa,b + ξb, a) | − 1 nξ c ,c ξb  . 4The definition here of Einstein constant is consistent with some of the literature on almost Einstein conformal structures, but elsewhere this term is sometimes used for the quantity 2(n− 1)λ. 18 K. Sagerschnig and T. Willse So viewed, ΘA0 is the map Γ(TM)→ Γ(S2 ◦T ∗M [2]), ξa 7→ (ξ(a,b))◦ = (Lξg)ab. Thus, the solutions of ker ΘA0 are precisely the vector fields whose flow preserves c, and so these are called conformal Killing fields. If ∇VLA0 (ξ) = 0, we say ξ is a normal conformal Killing field. 2.7 (2, 3, 5) conformal structures About a decade ago, Nurowski observed the following: Theorem 2.7. A (2, 3, 5) distribution (M,D) canonically determines a conformal structure cD of signature (2, 3) on M . 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: Definition 2.8. A conformal structure c is a (2, 3, 5) conformal structure iff c = cD for some (2, 3, 5) distribution D. An oriented (2, 3, 5) distribution D determines an orientation of TM , and hence cD 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 cD 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 (G2, Q). Form the extended bundle Ḡ := G ×Q P̄ , and let ω̄ denote the Cartan connec- tion equivariantly extending ω to Ḡ. By construction (Ḡ, ω̄) is a parabolic geometry of type (SO(3, 4), P̄ ) (for which ω̄ 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 ×QU (here regarding U as a Q-representation) with the conformal tractor bundle Ḡ ×P̄ U, and so denote both of these bundles by U . Since ω̄ is itself normal, the (normal) tractor connections that ω and ω̄ induce on U coincide. 2.7.1 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 (G2, Q). In particular, this determines on the tractor bundle V = G ×QV a G2-structure Φ ∈ Γ(Λ3V∗) 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 bundle Ḡ ×P̄ V of cD and the normal conformal tractor connection. The G2-structure determines fiberwise a bilinear form HΦ ∈ Γ(S2V∗). Since this construction is algebraic, HΦ is parallel, and by construction it coincides with the conformal tractor metric on V determined by cD. Conversely, if an oriented, signature (2, 3) conformal structure c admits a parallel tractor G2-structure Φ whose restriction to each fiber Vx is compatible with the restriction Hx of the tractor metric (in which case we simply say that Φ is compatible with H), the distribution D underlying Φ satisfies c = cD. 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 = cD) iff the normal conformal tractor connection admits a holonomy reduction to G2, or equivalently, iff c admits a parallel tractor G2-structure Φ compatible with the tractor metric H. The Geometry of Almost Einstein (2, 3, 5) Distributions 19 2.7.2 The conformal tractor decomposition of the tractor G2-structure Fix an oriented (2, 3, 5) distribution (M,D), let Φ ∈ Γ(Λ3V∗) denote the corresponding parallel tractor G2-structure, and denote its components with respect to any scale τ of the induced conformal structure cD according to ΦABC τ =  ψbc χabc | θc φbc  ∈ Γ  Λ2T ∗M [1] Λ3T ∗M [3] | T ∗M [1] Λ2T ∗M [3]  . (2.25) In the language of Section 2.6, φ = ΠΛ3V∗ 0 (Φ) is a normal conformal Killing 2-form, and Φ = LΛ3V∗ 0 (φ). 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 φab ∈ Γ(Λ2TM [−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. Proposition 2.10. Let (M,D) be an oriented (2, 3, 5) distribution, let Φ ∈ Γ(Λ3V∗) denote the corresponding parallel tractor G2-structure, and denote its components with respect to an arbitrary scale τ as in (2.25). Then: 1. The distribution D is totally cD-isotropic; equivalently, φacφcb = 0. 2. The annihilator of φab (in TM) is [D,D], and hence D⊥ = [D,D] (here, D⊥ is the subbundle of TM orthogonal to D with respect to cD); equivalently, φbcχbca = 0. 3. The weighted vector field θb ∈ Γ(TM [−1]) is a section of [D,D][−1], or equivalently, the line field L that θ determines (which depends on τ) is orthogonal to D; equivalently, θbφba = 0. 4. The weighted vector field θb satisfies θbθ b = −1. In particular, the line field L is timelike. 5. Like φ, the weighted 2-form ψ is locally decomposable, that is, (ψ ∧ψ)abcd = 6ψ[abψcd] = 0. Since (by equation (2.26)) it vanishes nowhere, it determines a 2-plane distribution E (which depends on τ). 6. The distribution E is totally cD-isotropic; equivalently, ψacψcb = 0. 7. The line field L is orthogonal to E; equivalently, θbψba = 0. 8. The (weighted) conformal volume form εg ∈ Γ(Λ5T ∗M [5]) satisfies (εg)abcde τ = 1 2(φ ∧ θ ∧ ψ)abcde = 15φ[abθcψde]. (2.26) In particular, (8) implies that D, L, and E are pairwise transverse and hence span TM . More- over, (2) and (3) imply that D ⊕ L = [D,D] and so D ⊕ L ⊕ E is a splitting of the canonical filtration D ⊂ [D,D] ⊂ TM .5 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 G2-structure Φ. 5The splitting D⊕L⊕E determined by τ is a special case of a general feature of parabolic geometry, in which a choice of Weyl structure yields a splitting of the canonical filtration of the tangent bundle of the underlying structure [18, Section 5.1]. 20 K. Sagerschnig and T. Willse Proof of Proposition 2.10. Call a local frame (Ea) of V adapted to Φ iff (1) E1 is a local section of the line subbundle 〈X〉 determined by X, and (2) the representation of Φ in the dual coframe (ea) 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 := e7(X) ∈ E [1], and by construction it vanishes nowhere. Then, since 〈X〉⊥ = 〈E1, . . . , E6〉, the (weighted) vector fields Fa := Ea+〈E1〉, a = 2, . . . , 6 comprise a frame of 〈X〉⊥/〈X〉 which by Section 2.4.2 is canonically isomorphic to TM [−1]. Trivializing these frame fields (by multiplying by τ) yields a local frame (F 2, . . . , F 6) of TM ; denote the dual coframe by (f2, . . . , f6). One can read immediately from (2.5) that in an adapted local frame, (the trivialized) components of Φ are φ τ = √ 2f5 ∧ f6, χ τ = f2 ∧ f4 ∧ f5 + f3 ∧ f4 ∧ f6, θ τ = f4, ψ τ = √ 2f2 ∧ f3, and consulting the form of equation (2.6) gives that the (trivialized) conformal metric is g τ = f2f5 + f3f6 − ( f4 )2 . In an adapted frame, εΦ is given by −e1 ∧ · · · ∧ e7, so the (trivialized) conformal volume form is εg = f2 ∧ · · · ∧ f6. 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 = √ 2f5 ∧ f6 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. 1. The identity implies that the trace over the second and third indices of the tensor product φ]] ⊗ φ = (α ∧ β)⊗ (α[ ∧ β[) is zero, or, expanding, that 0 = −g(α, α)β ⊗ β[ + g(α, β)α⊗ β[ + g(α, β)β ⊗ α[ − g(β, β)β ⊗ β[. 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 cD to D in the given frame. 2. By Part (1), φ(α, · ) = φ(β, · ) = 0. Any local section η ∈ Γ([D,D]) can be written as η = Aα+Bβ+C[α, β] for some smooth functions A, B, C, giving φ(η, γ) = Cφ([α, β], γ). The invariant formula for the exterior derivative of a 2-form then gives φ([α, β], γ) = −dφ(α, β, γ). Now, in the chosen scale, dφ = χ so −[dφ(α, β, · )]a = −χbcaαbβc = −1 2χbca · 2α [bβc] = −1 2χbcaφ bc. � Since the tractor Hodge star operator ∗Φ is algebraic, ∗ΦΦ ∈ Γ(Λ4V∗) 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 ∗ : ΛlT ∗M [w]→ Λ5−lT ∗M [7− w] determined by g [50]: (∗ΦΦ)ABCD τ =  (∗ψ)fgh −(∗θ)efgh | (∗χ)gh −(∗φ)fgh  ∈ Γ  Λ3T ∗M [2] Λ4T ∗M [4] | Λ2T ∗M [2] Λ3T ∗M [4]  . (2.27) 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 ∗: (∗φ)fgh = 3φ[fgθh], (∗χ)gh = θiχigh, (∗θ)efgh = −3φ[efψgh], (∗ψ)fgh = 3ψ[fgθh]. (2.28) The Geometry of Almost Einstein (2, 3, 5) Distributions 21 3 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 cD; 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 cD is equal to G2 and hence cD admits no nonzero almost Einstein scales. Via the identification of the standard tractor bundles of D and cD, 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 G2 to the stabilizer subgroup S of a nonzero vector in the standard representation V. 3.1 Distinguishing a vector in the standard representation V of G2 In this subsection, let V denote the standard representation of G2 and Φ ∈ Λ3V∗ the correspond- ing 3-form. We establish some of the algebraic consequences of fixing a nonzero vector S ∈ V. 3.1.1 Stabilizer subgroups Recall from the introduction that the stabilizer group in G2 of S ∈ V is as follows: Proposition 3.1. The stabilizer subgroup of a nonzero vector S in the standard representation V of G2 is isomorphic to: 1) SU(1, 2), if S is spacelike, 2) SL(3,R), if S is timelike, and 3) SL(2,R) n Q+, where Q+ < G2 is the connected, nilpotent subgroup of G2 defined via Sections 2.3.4 and 2.3.6, if S is isotropic. 3.1.2 An ε-Hermitian structure Contracting a nonzero vector S ∈ V with Φ determines an endomorphism: KA B := −ι27(S)AB = −SCΦC A B ∈ so(3, 4). (3.1) We can identify K with the map T 7→ S×T, so if we scale S so that ε := −HΦ(S, S) ∈ {−1, 0, 1}, identity (2.2) becomes K2 = εidV + S⊗ S[. (3.2) By skewness, HACSAKC B = −SASDKDAB = 0, 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: K2 = ε idW. Thus, (HΦ|W,K) is an ε-Hermitian structure on W: this is a pair (g,K), where g ∈ S2W∗ 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). 22 K. Sagerschnig and T. Willse 3.1.3 Induced splittings and filtrations If S is nonisotropic, it determines an orthogonal decomposition V = W ⊕ 〈S〉. If S is isotropic, it determines a filtration (VaS) [37, Proposition 2.5]: −2 −1 0 +1 +2 +3 V ⊃ W ⊃ imK ⊃ kerK ⊃ 〈S〉 ⊃ {0} 7 6 4 3 1 0 (3.3) 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, imK = (kerK)⊥ (so kerK 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(VaS) = Va+2 S , where we set VaS = 0 for all a > 2. 3.1.4 The family of stabilized 3-forms For nonzero S ∈ V, elementary linear algebra gives that the subspace of 3-forms in Λ3V∗ fixed by the stabilizer subgroup S of S has dimension 3 and contains ΦI := S y (S[ ∧ Φ) ∈ Λ3 1 ⊕ Λ3 27, (3.4) ΦJ := −ι37(S) = −∗Φ ( S[ ∧ Φ ) = S y ∗ΦΦ ∈ Λ3 7, (3.5) ΦK := S[ ∧ (S yΦ) ∈ Λ3 1 ⊕ Λ3 27. (3.6) The containment ΦK ∈ Λ3 1 ⊕Λ3 27 follows from the fact that ΦK = 1 2 i(S [ ◦ S[), where i is the G2- invariant map defined in (2.12). The containment ΦI ∈ Λ3 1⊕Λ3 27 follows from that containment, the identity ΦI + ΦK = S y ( S[ ∧ Φ ) + S[ ∧ (S yΦ) = HΦ(S, S)Φ, (3.7) and the fact that Φ ∈ Λ3 1. It follows immediately from the definitions that S yΦI = S yΦJ = 0 and S yΦK = HΦ(S, S)S yΦ. (3.8) Since S annihilates ΦI but not Φ, the containments in (3.4), (3.5) show that {Φ,ΦI ,ΦJ} is a basis of the subspace of stabilized 3-forms. If HΦ(S, S) 6= 0, then (3.7) implies that {ΦI ,ΦJ ,ΦK} is also a basis of that space. If HΦ(S,S) = 0 then ΦK = −ΦI . It is convenient to abuse notation and denote by ΦI ,ΦJ the pullbacks to W of the 3-forms of the same names via the inclusion W ↪→ V. For nonisotropic S, define W1,0 ⊂W⊗RCε to be the (+iε)-eigenspace of (the extension of) K, and an ε-complex volume form to be an element of ΛmCε := Λm(W1,0)∗. Proposition 3.2. Suppose ε := −HΦ(S, S) ∈ {±1}. For each (A,B) such that A2 − εB2 = 1, Ψ(A,B) := [AΦI + εBΦJ ] + iε[BΦI +AΦJ ] ∈ Γ ( Λ3 CεW ) is an ε-complex volume form for the ε-Hermitian structure (HΦ|W,K) on W. Proposition 3.3. Suppose V′ is a 7-dimensional real vector space and H ∈ S2(V′)∗ is a symmet- ric 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 Hermi- tian structure on W, and fix a compatible ε-complex volume form Ψ ∈ Λ3 CεW ∗ satisfying the normalization condition Ψ ∧ Ψ̄ = −4 3 iεK ∧K ∧K. The Geometry of Almost Einstein (2, 3, 5) Distributions 23 Then, the 3-form Re Ψ + εS[ ∧K ∈ Λ3(V′)∗ is a G2-structure on V′ compatible with H. 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. 3.2 The canonical conformal Killing field ξ For this subsection, fix an oriented (2, 3, 5) distribution D, let Φ ∈ Γ(Λ3V∗) denote the corre- sponding parallel tractor G2-structure, and denote its components with respect to an arbitrary scale τ as in (2.25); in particular, φ := ΠΛ3V∗ 0 (Φ) is the underlying normal conformal Killing 2-form. Also, fix a nonzero almost Einstein scale σ ∈ Γ(E [1]) of cD, denote the correspon- ding parallel standard tractor by S := LV0 (σ), and denote its components with respect to τ as in (2.18). By scaling, we assume that −ε := HΦ(S,S) ∈ {−1, 0,+1}. We view the adjoint tractor KA B := −SCΦC A B ∈ Γ(A). as a bundle endomorphism of V (cf. (3.1)), and computing gives that the components of K with respect to τ are KA B τ =  µcψca − ρθa −σψab − µcχcab − ρφab | µcθc σθa + µcφ ca  . (3.9) We denote the projecting part of KA B by ξa := ΠA0 (K)a = σθa + µbφ ba ∈ Γ(TM); (3.10) because K is parallel, ξ is a normal conformal Killing field for cD. 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 = ι7(σ), where ι7 is the manifestly invariant differential operator ι7 := ΠA0 ◦ (−ι27) ◦ LV0 : Γ(E [1]) → Γ(TM). Here, ι27 is the bundle map V → Λ2V∗ associated to the algebraic map (2.8) of the same name, and we have implicitly raised an index with HΦ. Computing gives ξa = ι7(σ)a = −φabσ,b + 1 4φ ab ,bσ.6 Proposition 3.4. Given an oriented (2, 3, 5) distribution D, let φ denote the corresponding normal conformal Killing 2-form, and suppose the induced conformal class cD admits an almost Einstein scale σ. The corresponding vector field ξ := ι7(σ) is a section of [D,D]. Proof. By (3.10), φbaξ b = φba(σθ b + µcφ cb) = σφbaθ b + µcφ cbφbc, but the first and second term vanish respectively by Proposition 2.10(1),(3). Thus, ξ ∈ kerφ, which by Part (2) of that proposition is [D,D]. � 6This formula corrects a sign error in [41, equation (41)], and (3.11) below corrects a corresponding sign error in equation (40) of that reference. 24 K. Sagerschnig and T. Willse On the set Mξ := {x ∈M : ξx 6= 0}, ξ spans a canonical line field L := 〈ξ〉|Mξ , and by Proposition 3.4, L is a subbundle of [D,D]|Mξ . Henceforth we often suppress the re- striction notation |Mξ . 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). 3.3 Characterization of conformal Killing fields induced by almost Einstein scales Hammerl and Sagerschnig showed that for any oriented (2, 3, 5) distribution D, the Lie algebra aut(cD) of conformal Killing fields of the induced conformal structure cD admits a natural (vector space) decomposition, corresponding to the G2-module decomposition so(3, 4) ∼= g2 ⊕V into irreducible submodules, that encodes features of the geometry of the underlying distribution. Given an oriented (2, 3, 5) distribution M , a vector field η ∈ Γ(TM) is an infinitesimal symmetry of D iff D is invariant under the flow of η, and the infinitesimal symmetries of D comprise a Lie algebra aut(D) under the usual Lie bracket of vector fields. The construction D cD is functorial, so aut(D) ⊆ aut(cD). By construction, the map π7 given in (3.12) below is a left inverse for ι7, so in particular ι7 is injective. Theorem 3.5 ([41, Theorem B]). If (M,D) is an oriented (2, 3, 5) distribution, the Lie algebra aut(cD) of conformal Killing fields of the induced conformal structure cD admits a natural (vector space) decomposition aut(cD) = aut(D)⊕ ι7(aEs(cD)) (3.11) and hence an isomorphism aut(cD) ∼= aut(D)⊕ aEs(cD). The projection aut(cD) → aEs(cD) is (the restriction of) the invariant differential operator π7 := ΠV0 ◦ (−π2 7) ◦ LA0 : Γ(TM)→ Γ(E [1]), which is given by π7 : ηa 7→ 1 6φ abηa,b − 1 12φab, bηa. (3.12) The canonical projection aut(cD) → aut(D) is (the restriction of) the invariant differential operator π14 := idΓ(TM)−ι7 ◦ π7 : Γ(TM)→ Γ(TM). The map π2 7 is the bundle map A ∼= Λ2V∗ → V∗ associated to the algebraic map (2.9) of the same name. The conformal Killing fields in the distinguished subspace ι7(aEs(cD)), that is, those corre- sponding to almost Einstein scales, admit a simple geometric characterization: Proposition 3.6. Let (M,D) be an oriented (2, 3, 5) distribution. Then, a conformal Killing field of cD is in the subspace ι7(aEs(cD)) iff it is a section of [D,D]. Hence, the indicated restrictions of ι7 and π7 comprise a natural bijective correspondence aEs(cD) ι7 � π7 aut(cD) ∩ Γ([D,D]). Proof. Let q−3 denote the canonical projection TM → TM/[D,D] and the map on sections it induces. It follows from a general fact about infinitesimal symmetries of parabolic geometries [15] that an infinitesimal symmetry ξ of D can be recovered from its image q−3(ξ) ∈ Γ(TM/[D,D]) via a natural linear differential operator Γ(TM/[D,D]) → Γ(TM); in particular, if q−3(ξ) = 0 then ξ = 0, so aut(D) intersects trivially with ker q−3. On the other hand, Proposition 3.4 gives that the image of ι7 is contained in ker q−3 = [D,D]. The claim now follows from the decomposition in Theorem 3.5. � The Geometry of Almost Einstein (2, 3, 5) Distributions 25 3.4 The weighted endomorphisms I, J , K Since they are algebraic combinations of parallel tractors, the 3-forms ΦI ,ΦJ ,ΦK ∈ Γ(Λ3V∗) respectively defined pointwise by (3.4), (3.5), (3.6) are themselves parallel. Thus, their respective projecting parts, Iab := ΠΛ3V∗ 0 (ΦI)ab, Jab := ΠΛ3V∗ 0 (ΦJ)ab, Kab := ΠΛ3V∗ 0 (ΦK)ab, are normal conformal Killing 2-forms. The definitions of ΦI , ΦJ , ΦK , together with (2.27) and (2.28) give Iab = −σ2ψab − σµcχcab − 2σµ[aθb] + σρφab + 3µcµ[cφab]m, (3.13) Jab = −σθcχcab + 3µcφ[caθb], (3.14) Kab = σ2ψab + σµcχcab + 2σµ[aθb] + σρφab − 2µcµ[aφb]c. (3.15) Using the splitting operators LV0 (2.21) and LΛ3V∗ 0 (2.24), we can write these normal conformal Killing 2-forms as differential expressions in φ and σ: Iab = 1 5σ 2 ( 1 3φab,c c + 2 3φc[a,b] c + 1 2φc[a, c b] + 4Pc[aφb]c ) − σσ,cφ[ca,b] − 1 2σσ,[aφb]c, c − 1 5σσ,c cφab + 3σ,cσ,[cφab], (3.16) Jab = −1 4σφ cd, dφ[ab,c] + 3 4σ ,cφ[abφc]d, d, (3.17) Kab = −1 5σ 2 ( 1 3φab,c c + 2 3φc[a,b] c + 1 2φc[a, c b] + 4Pc[aφb]c + 2Pccφab ) + σσ,cφ[ab,c] + 1 2σσ,[aφb]c, c − 1 5σσ,c cφab − 2σ,cσ,[aφb]c. (3.18) Raising indices gives weighted g-skew endomorphisms Iab, J a b,K a b ∈ Γ(Endskew(TM)[1]). 3.5 The (local) leaf space Let L denote the space of integral curves of ξ in Mξ := {x ∈ M : ξx 6= 0}, and denote by πL : Mξ → L the projection that maps a point to the integral curve through it. Since ξ vanishes nowhere, around any point in Mξ there is a neighborhood such that the restriction of πL thereto is a trivial fibration over a smooth 4-manifold; henceforth in this subsection, we will assume we have replaced Mξ with such a neighborhood. 3.5.1 Descent of the canonical objects Some of the objects we have already constructed on M descend to L via the projection πL. 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 Ξt of ξ and define the Lie derivative LξA to be LξA := ∂t|0Ξ∗tA [46]. Since the tractor projection ΠU0 is associated to a canonical vector space projection, it commutes with the Lie derivative. We exploit the following identity: Lemma 3.7 ([24, Appendix A.3]). Let (M, c) be a conformal structure of signature (p, q), p+ q ≥ 3, let U be a SO(p+ 1, q + 1)-representation, and denote by U → M the tractor bundle it induces. If ξ ∈ Γ(TM) is a conformal Killing field for c, and A ∈ Γ(U), then LξA = ∇ξA− LA0 (ξ) ·A, where · denotes the action on sections induced by the action so(p+ 1, q+ 1)×U→ U. In particular, if A is parallel, then LξA = −LA0 (ξ) · A. (3.19) 26 K. Sagerschnig and T. Willse Proposition 3.8. Suppose (M,D) is an oriented (2, 3, 5) distribution and let φ ∈ Γ(Λ2T ∗M [3]) denote the corresponding normal conformal Killing 2-form. Suppose moreover that the conformal structure cD induced by D admits a nonzero almost Einstein scale σ ∈ Γ(E [1]), let ξ ∈ Γ(TM) denote the corresponding normal conformal Killing field, and let I, J , K denote the normal conformal Killing 2-forms defined in Section 3.4. Then, Lξσ = 0, (Lξφ)ab = 3Jab, (LξI)ab = −3εJab, (LξJ)ab = 3Iab, (LξK)ab = 0. (3.20) As usual, we scale σ so that ε := −HΦ(S, S) ∈ {−1, 0,+1}, where SA := LV0 (σ)A is the parallel standard tractor corresponding to σ. In particular, σ and K descend via πL : Mξ → N to well-defined objects σ̂ and K̂, but φ and J do not descend, and when ε 6= 0 neither does I (recall that when ε = 0, I = −K). Proof. As usual, denote K := LA0 (ξ), Φ := LΛ3V∗ 0 (φ), ΦI := LΛ3V∗ 0 (I), ΦJ := LΛ3V∗ 0 (J), and ΦK := LΛ3V∗ 0 (K). Since ξ is a conformal Killing field, by (3.19) (LξS)A = −(LA0 (ξ) · S)A = −KA BSB = −SCΦC A BSB = 0. Applying ΠV0 yields Lξσ = LξΠV0 (S) = ΠV0 (LξS) = ΠV0 (0) = 0. The proofs for J and φ are similar, and use the identities (2.3), (2.4). By definition, LξΦK = Lξ[S[ ∧ (S yΦ)], and since LξS = 0, we have LξΦK = S[ ∧ (S yLξΦ) = S[ ∧ [S y (3ΦJ)], but by (3.5) this is 3S[ ∧ [S y (S y ∗ΦΦ)], which is zero by symmetry. App- lying ΠΛ3V∗ 0 gives LξK = 0. Finally, (3.7) gives LξΦI = Lξ(−εΦ − ΦK) = −εLξΦ − LξΦK = −ε(3ΦJ) − (0) = −3εΦJ , and applying ΠΛ3V∗ 0 gives LξI = −3εJ . � 4 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 con- formally isometric. This problem turns out to be intimately related to existence of a nonzero almost Einstein scale for cD. Approaching this question at the level of underlying structures is prima facie difficult: The value of the conformal structure cD 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 prolon- gations and normalizations), so analyzing directly the dependence of cD on D involves appre- hending high-order differential expressions that turn out to be cumbersome. We have seen that in the tractor bundle setting, however, this construction is essentially alge- braic: At each point, the parallel tractor G2-structure Φ ∈ Γ(Λ3V∗) determined by an oriented (2, 3, 5) distribution (M,D) determines the parallel tractor bilinear form HΦ ∈ Γ(S2V∗) and orientation [εΦ] canonically associated to the oriented conformal structure cD. So, the problem of determining the distributions (M,D′) such that cD′ = cD amounts to the corresponding algebraic problem of identifying for a G2-structure Φ on a 7-dimensional real vector space V the G2-structures Φ′ on V such that (HΦ′ , [εΦ′ ]) = (HΦ, [εΦ]). 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. 4.1 The space of G2-structures compatible with an SO(3, 4)-structure In this subsection, which consists entirely of linear algebra, we characterize explicitly the space of G2-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 G2-structure Φ. This char- acterization is essentially equivalent to that in [12, Remark 4] for the analogous inclusion of The Geometry of Almost Einstein (2, 3, 5) Distributions 27 the compact real form of G2 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 G2-structure Φ ∈ Λ3V∗. 1. Fix a nonzero vector S ∈ V; by rescaling we may assume that ε := −HΦ(S, S) ∈ {−1,0,+1}. For any (Ā, B) ∈ R2 such that −εĀ2 + 2Ā+B2 = 0 (there is a 1-parameter family of such pairs) the 3-form Φ′ := Φ + ĀΦI +BΦJ ∈ Λ3V∗ (4.1) is a G2-structure compatible with the SO(3, 4)-structure (HΦ, [εΦ]), that is, (HΦ′ , [εΦ′ ]) = (HΦ, [εΦ]). 2. Conversely, all compatible G2-structures arise this way: If a G2-structure Φ′ on V satisfies (HΦ, [εΦ]) = (HΦ′ , [εΦ′ ]), there is a vector S ∈ V (we may assume that ε := −HΦ(S, S) ∈ {−1, 0,+1}) and (Ā, B) ∈ R2 satisfying −εĀ2 +2Ā+B2 = 0 such that Φ′ is given by (4.1). Remark 4.2. Let V be the standard representation of SO(3, 4), and let S denote the intersection of (a copy of) G2 in SO(3, 4) and the stabilizer subgroup in SO(3, 4) of a nonzero vector S ∈ V. By Section 3.1.4 the space of 3-forms in Λ3V∗ stabilized by S is 〈Φ,ΦI ,ΦJ〉. By Proposition 4.1, this determines a 1-parameter family of copies of G2 containing S and contained in SO(3, 4), or equivalently, a 1-parameter family F of G2-structures Φ′ compatible with the SO(3, 4)-structure, but S does not distinguish a G2-structure in this family. Remark 4.3. We may identify the space of G2-structures that induce a particular SO(3, 4)- structure with the homogeneous space SO(3, 4)/G2. Since SO(3, 4) has two components but G2 is connected, this homogeneous space has two components; the G2-structures in one determine the opposite space and time orientations as those in the other. We can identify one component with the projectivization P(S3,4) of the cone of spacelike elements in R4,4 and the other as the projectivization P(S4,3) of the cone of timelike elements, and the homogeneous space SO(3, 4)/G2 (the union of these projectivizations) as the complement of the neutral null quadric in P(R4,4) [45, Theorem 2.1]. Henceforth denote by F [Φ;S] the 1-parameter family of G2-structures compatible with the SO(3, 4)-structure (HΦ, [εΦ]) defined by Proposition 4.1. By construction, if Φ′ ∈ F [Φ;S], then F [Φ′; S] = F [Φ; S]. Proposition 4.4. For any G2-structure Φ′∈F [Φ; S], the endomorphism (K′)AB := −SC(Φ′)C A B coincides with KA B := −SCΦC A B. Proof. Proposition 4.1 gives that Φ′ = Φ + ĀΦI +BΦJ for some constants Ā, B, so (3.8) gives that K′ := −S yΦ′ = −S y (Φ + ĀΦI +BΦJ) = −S yΦ = K. � We can readily parameterize the families F [Φ; S] of G2-structures. It is convenient henceforth to split cases according to the causality type of S, that is, according to ε. If ε 6= 0, then Φ = −ε(ΦI + ΦK), so in terms of A := Ā− ε and B, Φ′ = AΦI +BΦJ − εΦK and the condition on the coefficients is A2 − εB2 = 1. If ε = −1, then A2 +B2 = 1, and so we can parameterize F [Φ;S] by Φυ := (cos υ)ΦI + (sin υ)ΦJ + ΦK . (4.2) The parameterization descends to a bijection R/2πZ ∼= S1 ↔ F [Φ; S], and Φ0 = Φ. 28 K. Sagerschnig and T. Willse If ε = +1, then A2 −B2 = 1, and so we can parameterize F [Φ;S] by Φ∓t := (∓ cosh t)ΦI + (sinh t)ΦJ − ΦK (4.3) and Φ−0 = Φ. If ε = 0, the compatibility conditions simplify to 2Ā+B2 = 0, so we can parameterize F [Φ; S] by Φs := Φ− 1 2s 2ΦI + sΦJ (4.4) and Φ0 = Φ. Each of the above parameterizations Φu satisfies d du ∣∣ 0 Φu = ΦJ , and the parameterizations in the latter two cases are bijective. In the nonisotropic cases, we can encode the compatible G2-structures efficiently in terms of the ε-complex volume forms in Proposition 3.2. Proposition 4.5. Suppose S is nonisotropic, and let Ψ(A,B) ∈ Γ(Λ3 CεW), A2− εB2 = 1, denote the corresponding 1-parameter family of ε-complex volume forms defined pointwise in Proposi- tion 3.2. Then, F [Φ;S] consists of the G2-structures Φ(A,B) := Π∗W Re Ψ(A,B) + εΦK , A2 − εB2 = 1. (Here, ΠW is the orthogonal projection V→W.) Proof. This follows immediately from the appearance of the condition A2 − εB2 = 1 in the discussion after the proof of Proposition 4.4 and the form of Ψ(A,B). � 4.2 Conformally isometric (2, 3, 5) distributions We now transfer the results of Proposition 4.1 to the level of parallel sections of conformal tractor bundles, thereby proving Theorem B: Proof of Theorem B. The oriented (2, 3, 5) distributions D′ conformally isometric to D are precisely those for which the corresponding parallel tractor 3-form Φ is compatible with the parallel tractor metric HΦ and orientation [εΦ]. Transferring the content of Proposition 4.1 to the tractor setting gives that these are precisely the 3-forms Φ′ := Φ + ĀΦI +BΦJ ∈ Γ(Λ3V∗), and applying ΠΛ3V∗ 0 (2.20) gives that the underlying normal conformal Killing 2-forms are φ′ := φ + ĀI + BJ ∈ Γ(Λ2T ∗M [3]). Substituting for I, J respectively using (3.16) and (3.17) yields the formula (1.1). � We reuse the notation F [Φ; S] for the 1-parameter family of parallel tractor G2-structures defined pointwise by (4.1) by a parallel tractor 3-form Φ and a parallel, nonzero standard trac- tor S. By analogy, we denote by D[D;σ] the 1-parameter family of conformally isometric oriented (2, 3, 5) distributions determined by D and σ as in Theorem B; we say that the distributions in the family are related by σ. Again, if D′ ∈ D[D;σ], then by construction D[D′;σ] = D[D;σ]. Henceforth, D denotes a family D[D;σ] for some D and σ. Proposition 4.6. Let D be an oriented (2, 3, 5) distribution and σ an almost Einstein scale of cD. For any D′ ∈ D[D;σ], the conformal Killing fields ξ and ξ′ respectively determined by (D, σ) and (D′, σ) coincide. In particular, ξ, the line L ⊂ TM |Mξ its restriction spans, and its orthogonal hyperplane field C := L⊥ ⊂ TM |Mξ depend only on the family D[D;σ] and not on D. Proof. Let Φ,Φ′ denote the parallel tractor G2-structures corresponding respectively to D,D′. Translating Proposition 4.4 to the tractor bundle setting gives that K′ := −LV0 (σ) yΦ′ and K := −LV0 (σ) yΦ coincide, and hence ξ′ = ΠA0 (K′) = ΠA0 (K) = ξ. � Corollary 4.7. Let D be a 1-parameter family of conformally isometric oriented (2, 3, 5) distri- butions related by an almost Einstein scale. Every distribution D ∈ D satisfies D ⊂ C. The Geometry of Almost Einstein (2, 3, 5) Distributions 29 4.2.1 Parameterizations of conformally isometric distributions Now, given an oriented (2, 3, 5) distribution D and a nonzero almost Einstein scale σ of cD, we can explicitly parameterize the family D[D;σ] they determine by passing to the projecting parts φ′ := ΠΛ3V∗ 0 (Φ′) of the corresponding parallel tractor G2-structures Φ′ ∈ F [Φ; S]. To do so, it is convenient to split cases according to the sign of the Einstein constant (2.23) of σ. As usual we denote by Φ ∈ Γ(Λ3V∗) the parallel tractor G2-structure corresponding to D and scale σ (by a constant) so that S := LV0 (σ) satisfies ε := −HΦ(S, S) ∈ {−1, 0,+1}. For ε = −1, D[D; S] consists of the distributions Dυ corresponding to the normal conformal Killing 2-forms φυ := ΠΛ3V∗ 0 (Φυ) = (cos υ)I + (sin υ)J +K. (4.5) As for the corresponding family F [Φ; S] of parallel tractor G2-structures, this parameterization descends to a bijection R/2πZ ∼= S1 ↔ D[D;σ]. For ε = +1, D[D; S] consists of the distributions D∓t corresponding to φ∓t := ΠΛ3V∗ 0 (Φ∓t ) = (∓ cosh t)I + (sinh t)J −K. (4.6) Each value of the parameter (±, t) corresponds to a distinct distribution. For ε = 0, D[D; S] consists of the distributions Dυ corresponding to φs := ΠΛ3V∗ 0 (Φs) = φ− 1 2s 2I + sJ. (4.7) 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. Proposition 4.8. Let D be an oriented (2, 3, 5) distribution and σ an almost Einstein scale of cD. Denote ξ := ι7(σ) and denote its flow by Ξ•. Then, for each x ∈M there is a neighbor- hood U of x and an interval T containing 0 such that: 1) (TΞυ/3) ·D|U = Dυ|U for all υ ∈ T , if σ is Ricci-negative, 2) (TΞt/3) ·D|U = D−t |U and (TΞt/3) ·D+ 0 |U = D+ t |U for all t ∈ T , if σ is Ricci-positive, and 3) (TΞs/3) ·D|U = Ds|U for all s ∈ T , if σ is Ricci-flat. Proof. In the Ricci-negative case this follows immediately from the facts that the normal con- formal Killing 2-form φ corresponding to D satisfies Lξφ = 3J (3.20) and that the 1-parameter family of normal conformal Killing 2-forms φυ corresponding to the distributions Dυ satisfy d dυ ∣∣ 0 φυ = J . The other cases are analogous. � 4.2.2 Additional induced distributions An almost Einstein scale for an oriented (2, 3, 5) distribution (M,D) naturally determines one or more additional 2-plane distributions on M , depending on the sign of the Einstein constant. Definition 4.9. We say that two oriented (2, 3, 5) distributions D,D′ in a given 1-parameter family D of conformally isometric oriented (2, 3, 5) distributions are antipodal iff (1) they are distinct, and (2) their respective corresponding parallel tractor G2-structures, Φ, Φ′, together satisfy Φ ∧ Φ′ = 0. 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∗Φπ3 7(Φ′), where π3 7 denotes the bundle map Λ3V∗ → V associated to the algebraic map of the same name in that equation. 30 K. Sagerschnig and T. Willse 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. • If the almost Einstein scale determining D is non-Ricci-flat, there is precisely one distri- bution 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 G2-structure corresponding to D and let S the par- allel standard tractor corresponding to the almost Einstein scale. Any D′ ∈ D corresponds to a compatible parallel tractor G2-structure Φ′ ∈ F [Φ; S] and by Proposition 4.1 we can write Φ′ = Φ + ĀΦI + BΦJ , so Φ ∧ Φ′ = 4∗Φπ3 7(Φ + ĀΦI + BΦJ). Since Φ ∈ Λ3 1, ΦI ∈ Λ3 7, and ΦJ ∈ Λ3 1 ⊕ Λ3 27 (see (3.4), (3.5)), where Λ3 • ⊂ Λ3V∗ denote the subbundles associated to the G2-representations Λ3 • ⊂ Λ3V∗, Schur’s Lemma implies that π3 7(Φ) = π3 7(ΦI) = 0. On the other hand, π3 7(ΦJ) = S 6= 0, so Φ ∧ Φ′ = 4B∗ΦS, which is zero iff B = 0. If ε = −1, then in the parameterization {Φυ} (4.2), the coefficient of ΦJ is sin υ, and this vanishes only for Φ0 = Φ = ΦI + ΦK and Φπ = −ΦI + ΦK , corresponding to the distributions D = D0 and E := Dπ. If ε = +1, then in the parameterization {Φ∓t } (4.3), the coefficient is sinh t, and this vanishes only for Φ+ 0 = Φ0 = ΦI −ΦK and Φ−0 = −ΦI −ΦK , corresponding to the distributions D = D−0 and E := D+ 0 . Finally, if ε = 0, then in the parameterization {Φs} (4.4), the coefficient is s, and this vanishes only for Φ0 = Φ itself, corresponding to D = D0. � Though in the Ricci-flat case there are no antipodal distributions (see Proposition 4.10), there is in that case a suitable replacement: Given an oriented (2, 3, 5) distribution (M,D) and an almost Einstein scale σ of cD, the family 2s−2φs converges to the normal conformal Killing 2-form φ∞ = −I = K as s → ±∞. By continuity, φ∞ is decomposable and hence defines on the set where φ∞ does not vanish a distinguished 2-plane distribution E called the null- complementary distribution (for D and σ), and the distribution so defined is the same for every D ∈ D[D;σ]. (This is analogous to the notion of antipodal distribution in that both antipodal and null-complementary distributions are spanned by the decomposable conformal Killing form −I − εK.) Corollary 5.7 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). Remark 4.11. Since G2 is connected, it is contained in the connected component SO+(3, 4) of the identity of SO(3, 4), and hence a G2-structure determines space- and time-orientations on the underlying vector space. If we replace SO(3, 4) with SO+(3, 4) in the description of the construction c cD in Section 2.7, the construction assigns to an oriented (2, 3, 5) distribu- tion D the (oriented) conformal structure cD along with space and time orientations. Suppose cD admits a nonzero almost Einstein scale σ. If σ is Ricci-negative or Ricci-flat, then the family D := D[D;σ] (parameterized respectively as in (4.5) or (4.7)) is connected, so the space and time orientations of cD determined by the distributions in D all coincide. If instead σ is Ricci-positive, then D consists of two connected components. Again, by connectness, the distributions D−t , which comprise the component containing D−0 = D (in the notation of Section 4.2.1) all determine the same space and time orientations of cD, but the distributions D+ t , which comprise the other component and which include the antipodal distribution D+ 0 = E, determine the space and time orientations opposite those determined by D. The Geometry of Almost Einstein (2, 3, 5) Distributions 31 In the case that σ is Ricci-positive, D and σ determine two additional distinguished distribu- tions: 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∓∞ := LΛ3V∗ 0 (φ∓∞) are not generic (they both annihilate LV0 (σ), that is, they are not G2-structures, and hence the distributions D∓∞ are not (2, 3, 5) distributions). 4.2.3 Recovering the Einstein scale relating conformally isometric distributions Given two distinct, oriented (2, 3, 5) distributions D, D′ for which cD = cD′ , we can reconstruct explicitly an almost Einstein scale σ ∈ Γ(E [1]) of cD for which D′ ∈ D[D;σ]. If we require that the corresponding parallel standard tractor S := LV0 (σ) satisfies ε := −HΦ(S,S) ∈ {−1, 0,+1}, then D and D′ together determine ε. If ε ∈ {±1}, σ is determined up to sign. If ε = 0, then we may choose σ so that D′ = D1, where the right-hand side refers to the parameterization in Sec- tion 4.2.1, and this additional condition determines σ. The maps π3 1 : Λ3V∗ → E , π3 7 : Λ3V∗ → V, π3 27 : Λ3V∗ → S2 ◦V∗ denote the bundle maps respectively associated to (2.10), (2.11), (2.13). Algorithm 4.12. Input: Fix distinct, oriented (2, 3, 5) distributions D, D′ on M such that cD = cD′. Let Φ,Φ′ denote the parallel tractor G2-structures respectively corresponding to D, D′, and define T := π3 7(Φ′). In each case, σ := ΠV0 (S). • If HΦ(T,T) < 0, set s := √ −HΦ(T,T), so that S := s−1T satisfies −HΦ(S, S) = +1. Then, φ′ = φarsinh s = ∓ √ s2 + 1I + sJ −K, where ∓ is the negative of the sign of π3 1(Φ′). • If HΦ(T,T) > 0, set s := − √ HΦ(T,T), so that S := s−1T satisfies −HΦ(S, S) = −1. Then, φ′ = φυ = cI + sJ + K, where c := 1 4 [7π3 1(Φ′) − 3] and υ is an angle that satisfies cos υ = c, sin υ = s. • If HΦ(T,T) = 0 but T 6= 0, set S := −1 4T, giving φ′ = φ1 = φ− 1 2I + J . • If T = 0, then (by definition) the distributions are antipodal. Now, π3 27(Φ′) + 1 7HΦ = ±S[ ⊗ S[ for a unique choice of ± and a parallel tractor S ∈ Γ(V) determined up to sign. If the equality holds for the sign +, then ε = −1 and φ′ = φπ. If the equality holds for −, then ε = 1 and φ′ = φ+ 0 . Since all of the involved tractor objects are parallel, the reconstruction problem is equivalent to the algebraic one recovering a normalized vector S from G2-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. 5 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 (G2, Q) to the stabilizer S of a nonzero ray in the standard representation V of G2. 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 32 K. Sagerschnig and T. Willse 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 D0. 5.1 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 connection ω̂ 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∗ω̂ is a principal connection on H. Such a reduction can be viewed equivalently as a section of the associated fiber bundle Ĝ/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 G- equivariant section s : Ĝ → O a holonomy reduction of type O. Note that we can identify Ĝ ×GO with G ×P O. 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(Gx) ⊆ O. This partitions M by P -type into a disjoint union⋃ a∈P\OMa 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 ofG/P intoH-orbits (for any particular choice of conjugacy class representativeH). 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 sub- groups Gx, Gx′ are related by Gx′ = gGxg −1. If x, x′ are in the same P -orbit, we can choose g ∈ P , and if we denote Px := Gx ∩ P , we likewise have Px′ = gPxg −1. Thus, as groups en- dowed with subgroups, (Gx, Px) ∼= (Gx′ , Px′). Given an orbit a ∈ P\O, we denote by (H,Pa) 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 ja : Ga ↪→ G|Ma, and (Ga →Ma, ωa) is a Cartan geometry of type (Ha, Pa) on the curved orbit Ma, where ωa := j∗aω. 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 := LV0 (σ) and hence determines a holonomy reduction to the stabilizer subgroup S̄ of a nonzero vector in the standard representation V of SO(p+ 1, q + 1); the conjugacy class of S̄ 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 : σx = 0}, and the re- duced Cartan geometries on these orbits are equivalent to the non-Ricci-flat Einstein metric σ−2g 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), P̄ ) ((SO(p+1, q), P̄ )), which corresponds to an oriented conformal structure cΣ of signature (p− 1, q) ((p, q − 1)). The Geometry of Almost Einstein (2, 3, 5) Distributions 33 If S is isotropic, then again there are two open orbits, and on the union M −Σ of these, the reduced Cartan geometry is equivalent to the Ricci-flat metric σ−2g of signature (p, q). In this case, Σ decomposes into three curved orbits: {x ∈ M : σx = 0, (∇σ)x 6= 0}, M+ 0 := {x ∈ M : σx = 0, (∇σ)x = 0, (∆σ)x < 0}, and M−0 := {x ∈ M : σx = 0, (∇σ) = 0, (∆σ)x > 0}; here ∆σ denotes the Laplacian σ,a a. The curved orbits M±0 are discrete, but the formermost curved orbit is a hypersurface that naturally (locally) fibers by the integral curves of the line field S spanned by σ,a, and the (local) leaf space thereof inherits a conformal structure of signature (p−1, q−1). Example 5.3 ((2, 3, 5) conformal structures). By Theorem 2.9, an oriented conformal structure (M, c) of signature (2, 3) is induced by a (2, 3, 5) distribution iff it admits a holonomy reduction to G2. Since G2 acts transitively on the flat model SO(3, 4)/P̄ ∼= S2×S3, the holonomy reduction to G2 determines only a single curved orbit. 5.2 The orbit decomposition of the flat modelM In this subsection, we determine the orbits and stabilizer subgroups of the action of S on the flat modelM := G2 /Q ∼= S2×S3, which by Theorem 5.1 determines the curved orbit decomposition of a parabolic geometry of type (G2, Q). Remark 5.4. Alternatively, as in the statements of Theorems D−, D+, and D0, we could fix a conformal structure c, that is, a normal parabolic geometry of type (SO(3, 4), P̄ ) (Section 2.3.5) equipped with a holonomy reduction to the intersection S of a copy of G2 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 ⊂ Λ3V∗ of compatible G2-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 = cD for all D ∈ D, but does not distinguish a distribution among them. As usual by scaling assume S satisfies ε := −HΦ(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 (VaX) of V determined via (3.3) by X: Explicitly this is −2 −1 0 +1 +2 +3 V ⊃ 〈X〉⊥ ⊃ im(X × · ) ⊃ ker(X × · ) ⊃ 〈X〉 ⊃ {0} (5.1) Here, X × · is the map −XCΦC A B ∈ End(V)[1], and by the comments after (3.3) it satisfies X × VaX = Va+2 X . Since Q preserves this filtration, the corresponding set differences {x ∈ M : Sx ∈ VaX − Va+1 X } are each unions of curved orbits. 5.2.1 Ricci-negative case In this case, which corresponds to S spacelike (ε = −1), S ∼= SU(1, 2) (Proposition 3.1), and W inherits a complex structure K (3.1) and a complex volume form Ψ (Proposition 3.2). Since kerX is isotropic and HΦ has signature (3, 4), imX = (kerX)⊥ is negative-semidefinite, so the only unions of orbits determined by the filtrations are {x : S ∈ V−〈X〉⊥} and {x : S ∈ 〈X〉⊥− imX}. Pick a nonzero vector X ∈ V in the ray determined by X ∈ V[1]. Since S is nonisotropic, V decomposes (as an SU(1, 2)-module) as W ⊕ 〈S〉, and with respect to this decomposition, X decomposes as w+σS ∈W⊕〈S〉, where σ := HΦ(X, S); so, 0 = HΦ(X,X) = HΦ(w,w) +σ2. If σ > 0, then σ−1w ∈ W satisfies HΦ(σ−1w, σ−1w) = −1. But the set of vectors w0 ∈ W satisfying H(w0, w0) = −1 is just the sphere S2,3, and SU(1, 2) acts transitively on this space, 34 K. Sagerschnig and T. Willse and hence on the 5-dimensional space M+ 5 ∼= S2,3 of rays in M it subtends. The isotropy group of the ray spanned by w0 + S preserves the appropriate restrictions of H, K, Ψ to the four-dimensional subspace 〈w0,Kw0〉⊥ ⊂ 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 dimS − dimM+ 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 projectivizationM4 ∼= S3×S1 of this space. By construction, the isotropy subgroup P−, which (since dimM4 = 4) has dimension four, is contained in the 5-dimensional stabilizer subgroup PSU(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). The case σ < 0 is essentially identical to the case σ > 0, and we denote the correponding orbit M−5 ∼= S2,3. 5.2.2 Ricci-positive case In this case, which corresponds to S timelike (ε := +1), S ∼= SL(3,R), and W inherits a paracom- plex 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 imX − kerX 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 can decompose X = w + σS ∈ W ⊕ 〈S〉, where σ = HΦ(X, S), and we have 0 = HΦ(X,X) = HΦ(w,w)− σ2. If σ > 0, then the resulting curved orbit is M+ 5 ∼= S2,3, 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 subsubsec- tion, 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) HΦ we may identify the −1- eigenspace with E∗, and so we can write X as (0, e, β) ∈ 〈S〉 ⊕ E ⊕ E∗. Since X is isotropic, 0 = 1 2HΦ(X,X) = β(e), and we can identify the ray projectivization of the set of such triples with S2×S2. 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+ nR3, which is characterized by its restriction to E, and which in turn is given in a convenient basis by a b c 0 1 d 0 0 a−1  : a ∈ R+; b, c, d ∈ R  . So, the corresponding orbit M4 has dimension 4, and it follows from the remaining two cases thatM4 ∼= S2×(S2−{±∗}) ∼= S2×S1×R for some point ∗ ∈ S2. By construction, P+ is contained in the 5-dimensional stabilizer subgroup P12 in SL(3,R) of the paracomplex line 〈X,KX〉 ⊂W generated by X, and we may identify P12 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 6= 0 but β = 0, S again acts transitively, and this time the isotropy subgroup is isomorphic to the (parabolic) stabilizer subgroup P1 := GL(2,R) n (R2)∗ of a ray in E, which is the first parabolic subgroup in SL(3,R), so the corresponding orbit is M+ 2 ∼= S2. The Geometry of Almost Einstein (2, 3, 5) Distributions 35 The dual case e = 0, β 6= 0 is similar: S acts transitively, and we can identify the isotropy subgroup with the (parabolic) stabilizer subgroup P2 := GL(2,R)nR2 ∼= P1 of a dual ray in E∗, so the corresponding orbit is M−2 ∼= S2. Finally, the case σ < 0 is essentially identical to the case σ > 0 and we denote the corre- sponding orbit by M−5 ∼= S2,3. 5.2.3 Ricci-f lat case In this case, which corresponds to S isotropic (ε = 0), S ∼= SL(2,R) n Q+ and W inherits a endomorphism K whose square is zero. Since S is isotropic, it determines a filtration (VaS) of V. By symmetry, we may identify the sets {x ∈M : Sx ∈ VaX −Va+1 X } that occur in this case with {x ∈ M : Xx ∈ V− 〈S〉⊥}, {x ∈ M : Xx ∈ 〈S〉⊥ − imK}, {x ∈ M : Xx ∈ kerK− 〈S〉}, and {x ∈ M : Xx ∈ 〈S〉 − {0}}. (The difference {x ∈ M : Xx ∈ imK − kerK} does not occur here, as every vector in imK− kerK is timelike.) 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) < G2, so we may identify M+ 5 ∼= (SL(2,R) nQ+)/ SL(2,R) ∼= Q+ ∼= R5. If σ = 0, we see there are several possibilities. Since every vector in imK− kerK is timelike, {x ∈ M : X ∈ 〈S〉⊥ − imK} is the set of points x ∈ M such that HΦ(X,S) = 0 but X × S 6= 0. Again, S acts transitively on the 4-dimensional spaceM4 of rays. In this case, computing gives that the isotropy subgroup is isomorphic to Rn R3. Next, we consider the set of points {x ∈M : X ∈ kerK−〈S〉}. Since kerK is totally isotropic, kerK − 〈X〉 is the set difference of a 3-dimensional affine space and a linear subspace, so the corresponding orbit M2 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 ∼= R5. 5.3 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 G2-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 := LV0 (σ), and denote its components with respect to τ by σ, µ, ρ as in (2.18). On the zero locus Σ := {x ∈ M : σx = 0} of σ, µ is invariant (that is, independent of the choice of scale τ), so on the set where moreover µ 6= 0, µ determines a line field S. See also Appendix A. Proposition 5.5. The curved orbits are characterized (separately) by the following conditions on tractorial and tangent data. The bullets • indicate which curved orbits occur for each causality type. For the curved orbits M±2 , (∗) indicates the following: On M+ 2 ∪ M − 2 we have µx ∈ [D,D]x[−1] −Dx[−1], so projecting to ([D,D]x/Dx)[−1] gives a nonzero element and via the Levi bracket we can regard this as an element of Λ2Dx[−1]. Then, x ∈ M−2 (M+ 2 ) iff this (weighted) bivector is oriented (anti-oriented). Also, ∆σ := σ,a a. 36 K. Sagerschnig and T. Willse Ma ε tractor condition tangent condition−1 0 +1 M±5 • • • ±HΦ(X,S) > 0 ±σ > 0 M4 • • • HΦ(X,S) = 0 (X × S) ∧X 6= 0 σ = 0 ξ 6= 0 M±2 • X × S = ±X ξ = 0 (∗) M2 • X × S = 0 X ∧ S 6= 0 S ⊂ D M±0 • S ∈ (±X · R+)[−1] σ = 0 ∇σ = 0 ∓∆σ > 0 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 M4 if Sx ∈ 〈Sx〉⊥ − im(Xx × · ) (for readability, in this proof we herein sometimes suppress the subscript x). By the discussion after (5.1), X × (imX) = 〈X〉, so S ∈ im(X × · ) iff (X × S) ∧ X = 0, yielding the tractor characterization of M4. 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 X × S = ±X. Finally, x ∈M±0 iff S ∈ 〈X〉, that is, if X ∧ S = 0, so x ∈M2 = kerX − 〈X〉 iff X × S = 0 but X ∧ S 6= 0. In the splitting determined by a scale τ , (X × S)A = KA BX B τ = α ξa 0  =  µcθc σθa + µcφ ca 0  ∈ Γ  E TM E [2]  . So, the only nonzero component of (X × S) ∧X is ξa, which together with the tractor charac- terization gives the tangent bundle characterization of M4. Since σ = 0 on M4, we have on that curved orbit that 0 6= ξa = µcφ ca, so µb 6∈ [D,D] and hence S∩ [D,D] = {0} (including this one, the assertions about the components of S and Φ in this proof all follow from Proposition 2.10). For M±2 , comparing the components of X × S = ±X gives ξa = 0 and α = µcθc = ±1. Together with σ = 0 the first condition gives φabµb = 0, which is equivalent to µb ∈ [D,D][−1]; on the other hand, µcθc 6= 0 gives that µb 6∈ D[−1], so µb projects to a nonzero element of ([D,D]/D)[−1] ∼= Λ2D[−1] (the isomorphism is the one given by the Levi bracket). Since θc ∈ [D,D][−1] −D[−1], it also determines a nonzero (and by construction, oriented) element of Λ2D[−1]. Thus, since θcθc = −1, we have µcθc = −1 (and hence x ∈ M−2 ) iff the element of Λ2D[−1] determined by µ is oriented. (The above gives S ⊂ [D,D] and S ∩D = {0}.) For M2, if S ∈ ker(X × · ) we have σ = 0, ξa = 0, and α = 0, so by the argument in the previous case we have S ⊂ D. Since S 6∈ 〈X〉, we have S∧X 6= 0, which is equivalent to µa = 0, and by (2.21) µ = ∇σ. Finally, if S ∈ 〈X〉, which by the previous case comprises the points where µ = 0. Again using LV0 (and that σ = 0) gives that ±ρ = ∓1 5∆σ, and so ±ρ > 0 (and hence x ∈ M±0 ) iff ∓∆σ > 0. � Corollary 5.6. Let D be a 1-parameter family of conformally isometric oriented (2, 3, 5) dis- tributions related by the almost Einstein scale σ, and let ξ denote the corresponding conformal Killing field. Then: The Geometry of Almost Einstein (2, 3, 5) Distributions 37 1. The set Mξ := {x ∈M : ξx 6= 0} is the union of the open and hypersurface curved orbits: Mξ = M5 ∪M4 ∪M−5 . In particular, (a) Mξ ⊇ M+ 5 ∪M − 5 , (b) the complement M −Mξ (if nonempty) has codi- mension 3, and (c) if σ is Ricci-negative then Mξ = M . 2. The curved orbits M±5 and M4 are preserved by the flow of ξ. 3. If x ∈ M+ 5 ∪M − 5 , then for every distribution D ∈ D, Lx ⊂ [D,D]x and Lx is transverse to Dx. In particular, Lx is timelike. 4. If x ∈M4, then Lx ⊂ Dx for every distribution D ∈ D. In particular, Lx is isotropic. Proof. Only (2) is not immediate: It follows from the characterizations M±5 = {±σ > 0} and M4 = {σ = 0} ∩Mξ together with the fact (3.20) that the flow of ξ preserves σ. � Corollary 5.7. Let D be a 1-parameter family of conformally isometric oriented (2, 3, 5) distri- butions related by the almost Ricci-flat scale σ. Then, the limiting distribution D∞ vanishes pre- cisely on M2∪M+ 0 ∪M − 0 , which (if nonempty) has codimension 3, and so the null-complementary distribution E it determines is defined precisely on Mξ = M+ 5 ∪M4 ∪M−5 . Proof. On M+ 5 ∪M − 5 , (5.4) below gives φ∞ = I σ = −ψ , which vanishes nowhere by Proposi- tion 2.10(8) (here, I is the 2-form determined by any D ∈ D; in the Ricci-flat case, it does not depend on that choice). On M − (M+ 5 ∪M − 5 ), (5.10) below gives that I = ξ[ ∧ µ[. In the proof of Proposition 5.5 we saw that on M4, ξ ∈ D and µ 6∈ [D,D] ⊃ D, so ξ[ and µ[ are linearly independent there. On M2∪M+ 0 ∪M − 0 , that proposition gives ξ = 0, and hence I = 0 there. � 5.4 The open curved orbits M± 5 In this section, we fix an oriented (2, 3, 5) distribution D and a nonzero almost Einstein scale σ of cD and restrict our attention to the union M5 := M+ 5 ∪M − 5 = {x ∈M : σx 6= 0} = M − Σ determined by the corresponding holonomy reduction; in particular M5 is open, and moreover, by the discussion after Theorem 2.6, it is dense in M . Since σ is nowhere zero on M5, we can work in the scale σ|M5 itself, in which many earlier formulae simplify. (Henceforth in this subsection, we suppress the restriction notation |M5 .) As usual, by rescaling we may assume that the parallel tractor S corresponding to σ satisfies ε := −HΦ(S, S) ∈ {−1, 0,+1}. Then, from the discussion after Theorem 2.6, gab := σ−2gab is almost Einstein and has Schouten tensor Pab = 1 2εgab, or equivalently, Ricci tensor Rab = 4εgab, and hence scalar curvature R = 20ε. In the scale σ, the components of the parallel tractor S itself are σ σ = 1, µe σ = 0, ρ σ = −1 2ε, (5.2) and substituting in (3.9) gives that KA B := −SCΦC A B σ =  1 2εθb 1 2(εφ ab + φ̄ ab ) | 0 θb  . (5.3) We have introduced the 2-form φ̄ab := −2ψab ∈ Γ(Λ2T ∗M [1]) for notational convenience. Note that ξb σ = θb. Substituting (5.2) in (3.13), (3.14), (3.15) gives that I, J , K simplify to Iab = 1 2 ( −εφ ab + φ̄ ab ) , Jab = ξcχ cab Kab = 1 2 ( −εφ ab − φ̄ ab ) . (5.4) The endomorphism component of K in the scale σ coincides with −Ka b. 38 K. Sagerschnig and T. Willse 5.4.1 The canonical splitting The components of canonical tractor objects on M5 in the splitting determined by σ are them- selves canonical (and so are, just as well, their trivializations); in particular this includes the components χabc, θc, ψbc of ΦABC . Moreover, via Proposition 2.10, D and the scale σ together determine a canonical splitting of the canonical filtration D ⊂ [D,D] ⊂ TM5: TM5 = D⊕ L⊕E. (5.5) The fact that ξ = θ gives the following: Proposition 5.8. The line field L in the splitting (5.5) determined by σ coincides with (the restriction to M5 of) the line field of the same name spanned by ξ in Section 3.2. The splitting (5.5) entails isomorphisms L ∼= [D,D]/D and E ∼= TM5/[D,D], and so the components Λ2D ∼=→ [D,D] and D⊗ ([D,D]/D) ∼=→ TM5/[D,D] of the Levi bracket L give rise to isomorphisms Λ2D ∼=→ L and D⊗ L ∼=→ E. The preferred nonvanishing section ξ = θ ∈ Γ(L) thus yields isomorphisms Λ2D ∼=→ R (equivalently, a volume form on D) and D ∼=→ E. We may use the volume form to identify D ∼= Λ2D⊗D∗ ∼= D∗ and dualize the previous isomorphism to yield a bilinear pairing D×E→ R. This splitting is closely connected with the notions of antipodal and null-complementary distributions introduced in Section 4.2.2. Theorem 5.9. The canonical distribution E spanned by the tractor component ψ in the splitting D⊕ L⊕ E determined by σ (equivalently, the distribution determined by φ̄), is (the restriction to M5 of) the distribution antipodal or null-complementary to D. Proof. From Section 4.2.2 the antipodal or null-complementary distribution is spanned by I −K, and substituting using (5.4) gives that this is φ̄. � Remark 5.10. Together D and L comprise the underlying data of another parabolic geometry on M5, namely of type (SL(4,R), P12), where P12 is the stabilizer subgroup of a partial flag in R4 of signature (1, 2) under the action induced by the standard action. The underlying structure for a regular, normal geometry of this type is a generalized path geometry in dimension 5, which consists of a 5-manifold M5, a line field L ⊂ TM5, and a 2-plane distribution D ⊂ TM such that (1) L ∩D = {0}, (2) [D,D] ⊆ D⊕ L, and (3) if η ∈ Γ(D), ξ′ ∈ Γ(L), and x ∈M together satisfy [η, ξ′]x = 0, then ηx = 0 or ξ′x = 0 [18, Section 4.3.3] (in our case, these conditions follow from the properties of the Levi bracket L). In this dimension, this geometry is sometimes called XXO geometry, in reference to the marked Dynkin diagram that encodes it. The restriction of L to Mξ −M5 = M4 is contained in D|M4 , so M5 is the largest set on which this construction yields a generalized path geometry. 5.4.2 The canonical hyperplane distribution Denote by C ⊂ TMξ the hyperplane distribution orthogonal to L := 〈ξ〉|Mξ . 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 M5: 1. The pullback gC of the metric g := σ−2g 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 null- complementary to D. The Geometry of Almost Einstein (2, 3, 5) Distributions 39 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 gC. Proof. (1) The conformal class has signature (2, 3), and Corollary 5.6(3) gives that L = C⊥ is timelike. (2) In the scale σ, ξ σ = θ and dξ[ σ = K[ = −1 2(εφ + φ̄). The decomposability of φ and φ̄ (the latter follows from Proposition 2.10(5)) implies ξ[ ∧ dξ[ ∧ dξ[ σ = −1 2εφ ∧ θ ∧ φ̄. By Propo- sition 2.10(8), φ ∧ θ ∧ φ̄ is a nonzero multiple of the conformal volume form εg and so vanishes nowhere iff ε 6= 0. (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 ∈M5 = {x ∈M : σx 6= 0}, the family Dx := {Dx : D ∈ D} is precisely the set of totally isotropic 2-planes in Cx self-dual with respect to the (weighted) bilinear form gC and the orientation determined by εC. Computing in an adapted frame shows that the images of I, J,K ∈ Γ(End(TM)[1]) are contained in C[1], so they restrict sections of End(C)[1], which by mild abuse of notation we denote Iαβ, Jαβ, Kα 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, IαγI γ β = −εσ2δαβ ∈ Γ(End(C)[2]) and Proposition 5.13 below. In the scale σ, this and the remaining equations become: IαγI γ β σ = −εδaβ, JαγK γ β = −Kα γJ γ β σ = Iαβ, JαγJ γ β σ = δaβ, Kα γI γ β = −IαγKγ β σ = −εJαβ, Kα γK γ β σ = εδaβ, IαγJ γ β = −JαγIγβ σ = −Kα β. (5.6) Proposition 5.13. Let (M,D) be an oriented (2, 3, 5) distribution and σ an almost Einstein scale σ for cD, and let E be the distribution antipodal or null-complementary to D determined by σ. Then, on M5: 1. I ∈ End(C) is an almost (−ε)-complex structure, and −I|D is the isomorphism D ∼=→ E determined by σ introduced at the beignning of the subsection. If σ is Ricci-flat, then I|E = 0, and if σ is not Ricci-flat, then I|E is an isomorphism E ∼=→ D. 2. J ∈ End(C) is an almost paracomplex structure, and its eigenspaces are D and E: J |D = idD, J |E = − idE. 3. K ∈ End(C) is an almost ε-complex structure, K|D = −I|D, and K|E = I|E. If ε = +1, the (∓1)-eigenspaces of K are the limiting 2-plane distributions D∓ defined in Section 4.2.2. In the non-Ricci-flat case, we can identify the pointwise U(1, 1)-structure on M5 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 struc- ture on (the restriction to M5 of) C, that is, an injective ring homomorphism H̃ ↪→End(Cx) for each x ∈M5, where H̃ is the ring of split quaternions. 2. The almost split-quaternionic structure in (1) depends only on D. Proof. (1) This follows immediately from the identities (5.6). (2) This follows from computing in an adapted frame. � 40 K. Sagerschnig and T. Willse 5.4.3 The ε-Sasaki structure The union M5 of the open orbits turns out also to inherit an ε-Sasaki structure, the odd- dimensional analogue of a Kähler structure. Definition 5.15. For ε ∈ {−1, 0,+1}, an ε-Sasaki structure on a (necessarily odd-dimensional) manifold M is a pair (h, ξ), where h ∈ Γ(S2T ∗M) is a pseudo-Riemannian metric on M and ξ ∈ Γ(TM) is a vector field on M such that 1) habξ aξb = 1, 2) ξ(a,b) = 0 (or equivalently, (Lξh)ab = 0, that is, ξ is a Killing field for h), and 3) ξa,bc = ε(ξahbc − δacξb). 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 ξa,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. Theorem 5.16. Let D be a 1-parameter family of conformally isometric oriented (2, 3, 5) distri- butions related by an almost Einstein scale σ. On M5 := {σ 6= 0}, the signature-(3, 2) Einstein metric −g = −σ−2cD and the Killing field ξ together comprise an ε-Sasaki–Einstein structure. Proof. Substituting in (3.2) the components of K in the scale σ (5.3), using that the endomor- phism component of K in that scale is −Ka b, and simplifying leaves −ξcξc σ = 1, Ka cξ c σ = 0, Ka cK c b σ = ε(δab + ξaξb), (5.7) together with ξcKb c = 0, but this last equation follows from the second equation and the g- skewness of K. Similarly, expanding the left-hand side of ∇K = 0 using (5.3) with respect to the scale σ, eliminating duplicated equations, and rearranging gives ξb,c σ = Kb c, Ka b,c σ = −ε(ξagbc − ξbδac). (5.8) 1. Rearranging the first equation in (5.7) gives (−gab)ξaξb = 1. 2. Since Ka c is g-skew, symmetrizing the first equation in (5.8) with gab gives ξ(b,c) = K(bc) = 0; equivalently, ξ is a Killing field for g and hence for −g. 3. This follows immediately from substituting the first equation in (5.8) into the second. (Here indices are raised and lowered with g and not with the candidate Sasaki metric −g.) � Corollary 5.17. Let D be an oriented (2, 3, 5) distribution and σ a nonzero Ricci-flat almost Einstein scale for cD. The null-complementary distribution E they determine is integrable. Proof. Since ε = 0, the second equation of (5.8) says that K is parallel. By Section 4.2.2, K is decomposable and (as a decomposable bivector field) spans E|M5 , so that distribution is ∇-parallel and hence integrable. But M5 is dense in M , and integrability is a closed condition, so E is integrable (on all of M). � Proposition 5.18. Let (g, ξ) be an oriented ε-Sasaki–Einstein structure (with ε = ±1) of signature (2, 3) on a 5-manifold M . Then, locally, there is a canonical 1-parameter family D of oriented (2, 3, 5) distributions related by an almost Einstein scale for which the associated conformal structure is c = [−g]. The Geometry of Almost Einstein (2, 3, 5) Distributions 41 Proof. Set c = [−g], let σ ∈ Γ(E [1]) be the unique section such that−g = σ−2g, and S := LV0 (σ) the corresponding parallel tractor. Define the adjoint tractor K := LA0 (ξ). It is known that a Sasaki–Einstein metric g satisfies Pab = 1 2εgab, 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 ∇VK = 0, K2 = ε id + S⊗ S[, and S yK = 0. 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 Ωab C D ∈ Γ(Λ2T ∗M ⊗ End(V)). The curvature of the induced connection on the bundle Λ3 CεW of ε-complex volume forms on W is given by Ωab C C + εiεΩab C DKD C . Now Ωab C C = 0 by skew-symmetry, and, since K is parallel, Ωab C DKD C = 0 by [16, Proposition 2.1]. Thus, the induced connection on Λ3 CεW is flat, so it admits local parallel sections. Let Ψ be such a (local) parallel section normalized so that Ψ ∧ Ψ̄ = −4 3 iεK ∧ K ∧ K. Denote Re Ψ the pullback to V of the real part of Ψ. Then, by Proposition 3.3, the parallel tractor 3-form Φ = ReΨ + εS[ ∧K ∈ Γ ( Λ3V ) defines a parallel G2-structure on V compatible with H. By the discussion before Proposi- tion 2.10, its projecting slot defines a (2, 3, 5) distribution with associated conformal structure c = [−g]. Finally, parallel sections of Λ3 CεW satisfying Ψ ∧ Ψ̄ = −4 3 iεK ∧K ∧K are parametrized by {z ∈ Cε : zz̄ = 1} (that is, S1 if ε = −1 and SO(1, 1) if ε = 1). � 5.4.4 Projective geometry On the complement M5 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 struc- ture [∇g], where g is the Einstein metric σ−2g, 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 torsion- free 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, Sec- tion 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. 5.4.5 The open leaf space L4 As in Section 3.5, we assume that we have replaced M by an open subset so that πL is a locally trivial fibration over a smooth 4-manifold. Define L4 := πL(M5): By Corollary 5.6(2) M5 is a union of πL-fibers, so L3 := πL(M4) = L− L4 is a hypersurface. 42 K. Sagerschnig and T. Willse Since ξ|M5 is a nonisotropic Killing field, −g := −σ−2cD|M5 descends to a metric ĝ on L4 (henceforth in this subsection we sometimes suppress the restriction notation |M5). By Propo- sition 3.8 Lξσ = 0 and LξK = 0, so the trivialization K ∈ Γ(End(TM5)) is invariant under the flow of ξ. Since it annihilates ξ, it descends to an endormorphism field we denote K̂ ∈ End(TL4). Then, Proposition 5.6 implies K̂2 = −ε idTL4 , that is, K is an almost ε-complex structure on L4. This yields a specialization to our setting of a well-known result in Sasaki geometry. Theorem 5.19. The triple (L4, ĝ, K̂) is an ε-Kähler–Einstein structure with R̂ = −6εĝ. Proof. Since Ka bξ b = 0, the g-skewness of K implies the ĝ-skewness of K̂. Thus, ĝ and K̂ together comprise an almost Kähler structure on L4; the integrability of K̂ is proved, for example, in [5], so they in fact consistute a Kähler structure. Since πL|L4 is a (pseudo-)Riemannian submersion, we can relate the curvatures of g and ĝ via the O’Neill formula, which gives that ĝ is Einstein and determines the Einstein constant. � 5.4.6 The ε-Kähler–Einstein Fefferman construction The well known construction of Sasaki–Einstein structures from Kähler–Einstein structures im- mediately generalizes to the ε-Kähler–Einstein setting; see, for example, [44] (in this subsub- section, 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 construc- tion [18, Section 4.5] between the respective Cartan geometries. Further details will be discussed in an article in preparation [56]. An ε-Kähler structure (ĝ, K̂) of signature (2, 2) on a manifold L4 can be equivalently encoded in a torsion-free Cartan geometry (S → L4, ω) of type (S,A), where (S,A) = (SU(1, 2),U(1, 1)) if ε = −1 and (S,A) = (SL(3,R),GL(2,R)) if ε = 1, see, for example, [17] for the Kähler case. 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 7→ 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 connection ω̂ of type (S,A) determines an isomorphism TL4 ∼= S ×A m and (via this isomorphism) an ε-Hermitian structure on TL4. 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 → L4 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 connection ω̂ ∈ Ω1(S, s) is given by the sum ω̂ = γ+ θ of the pullback of the Levi-Civita connection form γ ∈ Ω1(S, a) and the soldering form θ ∈ Ω1(S,m). For the construction we first build the correspondence space CL4 := S/A0 ∼= S ×A (A/A0), where A0 = SU(1, 1) if ε = −1 and A0 = SL(2,R) if ε = 1. Then, CL4 → L4 is an S1-bundle if ε = −1 and an SO(1, 1)-bundle if ε = 1. We can view ω̂ ∈ Ω1(S, s) as a Cartan connection on the A0-principal bundle S → CL4. Next we fix inclusions S ↪→ G2 ↪→ SO(3, 4), such that S stabilizes a vector S satisfying H(S, S) = −ε in the standard representation V of G2 (here H is the bilinear form the representation determines on V), the S-orbit in G2 /Q ∼= SO(3, 4)/P̄ is open and A0 = S∩Q = S∩P̄ . Consider the extended Cartan bundles G = S×A0Q The Geometry of Almost Einstein (2, 3, 5) Distributions 43 and Ḡ = S ×A0 P̄ . There exist unique Cartan connections ω ∈ Ω1(G, g2) and ω̄ ∈ Ω1(Ḡ, so(3, 4)) extending ω̂ [18]. Thus one obtains Cartan geometries of type (G2, Q) and (SO(3, 4), P̄ ), respec- tively, on CL4 that are non-flat whenever one applies the construction to a torsion-free non-flat Cartan connection ω of type (S,A). Proposition 5.20. Let (ĝ, K̂) be an ε-Kähler–Einstein structure, ε ∈ {±1}, of signature (2, 2) on L4 such that R̂ab = −6εĝab. Then the induced conformal structure c := ĝ on the correspon- dence space CL4 is a (2, 3, 5) conformal structure equipped with a parallel standard tractor S, H(S,S) = −ε, which corresponds to a non-Ricci-flat Einstein metric in c. (Here H is the canonical tractor metric determined by c.) Conversely, locally, all (2, 3, 5) conformal structures containing non-Ricci-flat Einstein met- rics arise via this construction from ε-Kähler–Einstein structures. Proof. We first show that the conformal Cartan geometry (Ḡ → CL4, ω̄) on the correspondence space is normal if and only if the ε-Kähler structure on L4 is Einstein with R̂ab = −6εĝab. The curvature of the Cartan connection ω̂ = γ + θ is given by Ω̂ = dγ + 1 2 [γ, γ] + 1 2 [θ, θ] ∈ Ω2(S, a). Computing the Lie bracket [[X,Y ], Z] for X,Y, Z ∈ m, and interpreting the curvature as a tensor field on L4 shows that it can be expressed as Ω̂ij k l = R̂ij k l + εĝjlδ k i − εĝilδkj + K̂jlK̂ k i − K̂ilK̂ k j − 2 K̂ijK̂ k l, where ĝij denotes the metric, R̂ij k l its Riemannian curvature tensor and K̂ij = ĝikK̂ k j the Kähler form. Tracing over i and k shows that Ω̂kj k l = R̂kj k l + 6εĝjl. Thus Ω̂kj k l = 0 if and only if R̂ab = −6εĝab. Further, for an ε-Kähler–Einstein structure R̂ij k lK̂ l k = 2 K̂ l iR̂kl k j holds. If R̂kj k l = −6εĝjl, this implies that Ω̂ij k lK̂ l k = 0. This precisely means that the Cartan curvature takes values in the subalgebra a0 ⊂ a of matrices with vanishing complex trace. Since a0 ⊂ p̄, the resulting conformal Cartan connection ω̄ is torsion-free. Vanishing of the Ricci-type contraction of Ω̂, i.e., Ω̂kj k l = 0, then further implies that that the conformal Cartan connection is normal. Conversely, normality of the conformal Cartan connection implies Ω̂kj k l = 0 and thus R̂ab = −6εĝab. By construction and normality of the conformal Cartan geometry (Ḡ → CL4, ω̄), the induced conformal structure on CL4 is a (2, 3, 5) conformal structure that admits a parallel standard tractor S, H(S, S) = −ε, with underlying nowhere-vanishing Einstein scale σ. The vertical bundle V CL4 for CL4 → L4 corresponds to the subspace a/a0 ⊂ s/a0, i.e., V CL4 = S×A0 (a/a0). Since this is the unique A0-invariant 1-dimensional subspace in s/a0, the vertical bundle coincides with the subbundle L spanned by the Killing field ξ. We now prove the converse. Let (G5 → M5, ω5) the Cartan geometry of type (S,A0) on M5 determined (according to Theorem 5.1) by the holonomy reduction coresponding to a parallel standard tractor S, H(S, S) = −ε, of a (2, 3, 5) conformal structure. Restrict to an open subset so that πL : M5 → L4 is a fibration over the leaf space determined by the corresponding Killing field ξ. Since ξ is a normal conformal Killing field, it inserts trivially into the curvature of ω5. Since ξ spans VM5, by [18, Theorem 1.5.14] this guarantees that on a sufficiently small leaf space L4 one obtains a Cartan geometry of type (S,A) such that the restriction of (G5 →M5, ω5) is locally isomorphic to the canonical geometry on the correspondence space over L4. Normality of the conformal Cartan connection implies that the Cartan geometry of type (S,A) is torsion- free and the corresponding ε-Kähler metric is non-Ricci-flat Einstein. � 44 K. Sagerschnig and T. Willse Remark 5.21. It is interesting to note the following geometric interpretation of the corre- spondence spaces: If ε = −1, the bundle CL4 → L4 can be identified with the twistor bundle TL4 → L4 whose fiber over a point x ∈ L4 comprises all self-dual totally isotropic 2-planes in TxCL4. If ε = 1, CL4 → L4 can be identified with the subbundle of the twistor bundle whose fiber over a point x ∈ L4 comprises all self-dual 2-planes in TxCL4 except the eigenspaces of the endomorphism K̂x. The total space CL4 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 L4 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). 5.5 The hypersurface curved orbit M4 On the complement M −M5, σ = 0 (and so µ is invariant); this simplifies many formulae there. First, ε = −HΦ(S, S) = −µaµa. Substituting in (3.10), (3.13), (3.14), (3.15) and using that expression for ε yields (on M4) ξa = µbφ ba, (5.9) Iab = 3µcµ[cφab] = −εφab − 2µ[aξb], (5.10) Jab = 3µcφ[caθb] = µc(∗φ)cab, (5.11) Kab = −2µcµ[aφb]c = 2µ[aξb]. (5.12) Denote by MS the set on which the line field S is defined (recall from Section 5.3 that S := 〈µ〉 on the space where σ = 0 and µ 6= 0). By the proof of Proposition 5.5, this is M4 in the Ricci-negative case, M4∪M+ 2 ∪M − 2 in the Ricci-positive case, and M4∪M2 in the Ricci-flat case. Proposition 5.22. On the submanifold MS, S⊥ = TMS ⊂ TM |MS . Proof. The set MS is precisely where σ = 0 and µa = σ,a 6= 0, so σ is a defining function for MS and hence TMS = kerµ there. � 5.5.1 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〉, whereW := 〈S〉⊥, and if S is isotropic (if σ is Ricci-flat), it determines a filtration associated to (3.3): V ⊃ W ⊃ imK ⊃ kerK ⊃ 〈S〉 ⊃ {0}. On M4, S ∈ 〈X〉⊥ − im(X × · ), 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〉 ∼= TM [−1], the sublattice of vector bundles N satisfying 〈X〉 � N � W descends to a natural lattice of subbundles of TM |M4 . 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 TM |M4 admits a natural lattice of vector subbundles under the operations of span and intersection: If σ is non-Ricci-flat, the lattice is The Geometry of Almost Einstein (2, 3, 5) Distributions 45 D [D,D] L (D + E)⊥ D + E C 0 TM |M4 S E [E,E] TM4 L⊕ S (L⊕ S)⊥ 0 1 2 3 4 5 . In particular, this contains a full flag field 0 ⊂ L ⊂ (D + E)⊥ ⊂ (L⊕ S)⊥ ⊂ TM4 on TM4. The subbundles in the lattice that depend only on D[D;σ] and not D are 0, L, S, L⊕ S, (L⊕ S)⊥, C, TM4, TM |M4. If σ is Ricci-flat, the lattice is D [D,D] L C 0 (D + E)⊥ D + E TM |M4 S TM4 E E⊥ 0 1 2 3 4 5 This determines a natural sublattice (D + E)⊥ L E⊥ TM4 0 E S 0 1 2 3 4 of vector subbundles of TM4. The subbundles in the first lattice that depend only on the 1- parameter family D[D;σ] and not D are 0, L, S, E, E⊥, C, TM4, TM |M4. In both cases, the restriction L|M4 (which depends only on D[D;σ]) is the intersection of the distribution D and the tangent space TM4 of the hypersurface. In the lattices, all bundles are implicitly restricted to M4, 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. 5.5.2 The hypersurface leaf space L3 Recall that L3 := πL(M4). Since S is spanned by the invariant component µ of S and LξS = 0, S descends to a line field Ŝ ⊂ TL|L3 . This line field is contained in TL3 iff S is contained in TM4 = S⊥, that is (by Proposition 5.23) iff σ is Ricci-flat. 46 K. Sagerschnig and T. Willse Similarly, since the flow of ξ preserves g, it also preserves C ∩ TM4 = (L ⊕ S)⊥. Then, because L ⊂ C ∩ TM4 ⊂ S⊥ = TM4, C ∩ TM4 descends to a 2-plane distribution H ⊂ TL3. Proposition 5.24. The 2-plane distribution H ⊂ TL3 defined as above is contact. Proof. Since C ∩ TM4 = (ker ξ[) ∩ TM4, we can write this bundle as ker ι∗M4 ξ[, where ιM4 : M4 ↪→M denotes inclusion. By construction, ι∗M4 ξ[ (where we have trivialized ξ[ with respect to an arbitrary scale τ) is also the pullback π∗Lβ of a defining 1-form β ∈ Γ(TL3) for H. Thus, π∗L(β ∧ dβ) = π∗Lβ ∧ d(π∗Lβ) = ι∗M4 ξ[ ∧ d(ι∗M4 ξ[) = ι∗M4 (ξ[ ∧ dξ[), but computing in an adapted frame shows that ξ[ ∧ dξ[ vanishes nowhere, and hence the same holds for β ∧ dβ; equivalently, H is contact. � Now, consider the component ζab := −σψab − µcχcab − ρφab ∈ Γ(End◦(TM)) of KA B in the splitting (3.9) with respect to a scale τ . Let J : H → TL|L3 be the map that lifts a vector η̂ ∈ Hx̂ to any η ∈ TxM4 for arbitrary x ∈ π−1 L (x̂), applies ζ, and then pushes forward back to Tx̂L by πL. 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, LξK = −[LA0 (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′ ∈ TxM4 differ by an element of kerTxπL = 〈ξx〉; 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 ζbaξ a = αξb for some smooth function α, so TxπL · ζ(ξ) = 0, and hence J is well-defined. Finally, under a change of scale, ζ is transformed to ζab 7→ ζab + Υaξb − ξaΥb for some form Υa ∈ Γ(T ∗M) [4]. A lift ηb of η̂ ∈ H is an element of C ∩ TM4 ⊂ C = ker ξ[, Υaξbη b = 0. The term ξaΥbη b is again in kerTxπL, and we conclude that J is independent of the scale τ . Now, in the notation of the previous paragraph, we have µbζ b cη c = µb(−µdχdbc − ρφbc)ηc = −ρνbφbcηc. This is −ρξcηc, and we saw above that ξcη c = 0, so µbζ b cη c = 0, that is, ζbcη c ∈ kerµ = S⊥. Using the g-skewness of ζ gives ξbζ b cη c = −ηbζbcξc = −ηb(αξb) = −αηbξb, but again ηbξ b = 0, so we also have ζbcη c ∈ ker ξ[ = L⊥. Thus, ζ(η) ∈ L⊥ ∩ S⊥ = C ∩ TM4, and pushing forward by πL gives J(η̂) ∈ H, so we may view J as an endomorphism of H. Proposition 5.25. The endomorphism J ∈ Γ(End(H)) defined as above is an ε-complex struc- ture. Proof. In the above notation, unwinding (twice) the definition of J gives that J2(η̂) = TxπL · ζ2(η). Now, another component equation of (3.2) is ζacζ c b− ξaνb− νaξb = εδab +µaµb for some ν ∈ Γ(T ∗M). The above observations about the terms Υaξb and ξaΥb apply just as well to ξaνb and νaξb, and since η ∈ C ∩ TM4 ⊂ S⊥ = kerµ, we have µaµbη b = 0, so the above component equation implies J2 = ε idH. 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 (L3,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. 5.5.3 Ricci-negative case: The classical Fefferman conformal structure If σ is Ricci-negative, then by Example 5.2, M −M5 = M4 inherts a conformal structure cS of signature (1, 3). We can identify the standard tractor bundle VS of cS with the restrictionW|Σ of The Geometry of Almost Einstein (2, 3, 5) Distributions 47 the ∇V parallel subbundle W := 〈S〉⊥, and under this identification the normal tractor connec- tion on VS coincides with the restriction of ∇V to W|Σ [33]. In particular, Hol(cS) ≤ SU(1, 2), but this containment characterizes (locally) the 4-dimensional conformal structures that arise from the classical Fefferman conformal construction [16, 51], which canonically associates to any nondegenerate partially integrable almost CR structure of hypersurface type on a mani- fold a conformal structure on a natural S1-bundle over that manifold [30], [18, Example 3.1.7, Section 4.2.4]. Proposition 5.26. Let D denote a 1-parameter family of conformally isometric oriented (2, 3, 5) distributions related by a Ricci-negative almost Einstein scale σ. 1. The conformal structure cS of signature (1, 3) determined on the hypersurface curved or- bit M4 is a Fefferman conformal structure. 2. The infinitesimal generator of the (local) S1-action is ξ|M4 = ι7(σ)|M4, so the line field it spans is L|M4 = D ∩ TM4 for every D ∈ D. 3. The 3-dimensional CR-structure underlying the Fefferman conformal structure (M4, cS) is (L3,H,J). Proof. The first claim is deduced in the paragraph before the proposition. The latter claims follow from (1), unwinding definitions, and the proof of [16, Corollary 2.3]. � 5.5.4 Ricci-positive case: A paracomplex analogue of the Fefferman conformal structure This case is similar to the Ricci-negative case, but differs qualitatively in two ways. First, the endomorphism J ∈ End(H) is a paracomplex structure rather than a complex one; let H± denote its (±1)-eigendistributions, which are both line fields. A contact distribution on a 3-manifold equipped with a direct sum decomposition into line fields is the 3-dimensional specialization of a Legendrean contact structure, the paracomplex analogue of a partially inte- grable almost CR structure of hypersurface type [18, Section 4.2.3]. These correspond to regular, normal parabolic geometries of type (SL(3,R), P12) where P12 < SL(3,R) is a Borel subgroup. The analog of the classical Fefferman conformal structure associates to any Legendrean con- tact structure on a manifold N a neutral conformal structure on a natural SO(1, 1)-bundle over the manifold [43, 53]. By analogy with the construction discussed in Section 5.5.3, we call a conformal structure that locally arises this way a para-Fefferman conformal structure. Second, the conformal structure cS is defined on the union M4 ∪M+ 2 ∪M − 2 , but only its restriction to M4 is induced by the analogue of the classical Fefferman construction (indeed, recall from Section 5.5 that the vector field ξ whose integral curves comprise the leaf space L vanishes on M±2 ). The paracomplex analogue of Proposition 5.26 is the following: Proposition 5.27. Let D denote a 1-parameter family of conformally isometric (2, 3, 5) distri- butions related by a Ricci-positive almost Einstein scale σ. 1. The conformal structure cS|M4 of signature (2, 2) determined on the hypersurface curved orbit M4 is a para-Fefferman conformal structure. 2. The infinitesimal generator of the (local) SO(1, 1)-action is ξ|M4 = ι7(σ)|M4, so the line field it spans is L|M4 = D ∩ TM4 for every D ∈ D. 3. The 3-dimensional Legendrean contact structure underlying (M4, cS|M4) is (L3,H+⊕H−), where H± are the (±1)-eigenspaces of the paracomplex structure J on H. 48 K. Sagerschnig and T. Willse The geometry of 3-dimensional Legendrean contact structures admits another concrete, and indeed classical (local) interpretation, namely as that of second-order ordinary differential equa- tions (ODEs) modulo point transformations: We can regard a second-order ODE ÿ = F (x, y, ẏ) as a function F (x, y, p) on the jet space J1 := J1(R,R), and the vector fields Dx := ∂x + p∂y + F (x, y, p)∂p and ∂p span a contact distribution (namely the kernel of dy − p dx ∈ Γ(T ∗J1)), so 〈Dx〉⊕〈∂p〉 is a Legendrean contact structure on J1. Point transformations of the ODE, namely those given by prolonging to J1 (local) coordinate transformations of R2 xy, are precisely those that preserve the Legendrean contact structure (up to diffeomorphism) [25]. 5.5.5 Ricci-f lat case: A fibration over a special conformal structure In this case, Example 5.2 gives that the hypersurface curved orbit M4 locally fibers over the space L̃ of integral curves of S (nota bene the fibrations πL|M4 : M4 → L3 in the non-Ricci- flat cases above are instead along the integral curves of L), and that L̃ inherits a conformal structure c L̃ 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 L̃ that we write as 0 ⊂ E/S ⊂ E⊥/S ⊂ T L̃ it depends only on D. Since E is totally c-isotropic, the line field E/S is c L̃ -isotropic, and by construction it is orthogonal to E⊥/S with respect to c L̃ . Thus, we may regard the induced structure on L̃ 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 ⊂ TL3. 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. 5.6 The high-codimension curved orbits M± 2 , M2, M ± 0 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 M2 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), P1) and (SL(3,R), P2) 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 un- parametrized geodesics). Again, the formulae for various objects simplify on this orbit: By the proof of Proposition 5.5 we have σ = 0, ξ = 0, and µcθc = ±1 here, and substituting in (3.13), (3.14), (3.15) gives I = −φ, J = ±φ, K = 0. These specializations immediately give the Ricci-positive analog of Corollary 5.7: Proposition 5.28. Let (M,D) be an oriented (2, 3, 5) distribution and σ a Ricci-positive scale for cD. Then, the limiting normal conformal Killing forms φ∓∞ := ±I + J respectively vanish precisely on M±2 , so the distributions D∓∞ they respectively determine are respectively defined precisely on M5 ∪M4 ∪M∓2 . Proposition 5.29. For all x ∈M±2 , TxM ± 2 = Dx (for every D ∈ D). The Geometry of Almost Einstein (2, 3, 5) Distributions 49 Proof. By Proposition 5.5, M±2 = {x ∈ M : ξx = 0} and so TM±2 ⊆ ker∇ξ. On the other hand, as in the proof of that proposition we have ξa,b = −ζab − θcµcδab, and computing in an adapated frame shows that on M±2 , ξa,b has rank 3. Equivalently, the kernel has dimension 2 = dimTM±2 = 2, so ker∇ξ = TxM ± 2 . Writing ∇Vc KA B = 0 in components gives ξb,c = −ζb,c − µdθdδ b c, and as in the proof of Proposition 5.5, −ζbc = σψbc + µdχ db c + ρφbc. Since x ∈ M±2 , µdφd = ±1 and σ = 0. For η ∈ Dx, Proposition 2.10(2) gives that φbcη c = 0, and computing in an adapted frame gives that µdχ db c restricts to idD on M±2 . Substituting then gives ξb,cη c = 0, so by dimension count TxM ± 2 = Dx. � 5.6.2 The curved orbit M2 As for the hypersurface curved orbits, forming the intersections and spans of the components of the filtrations determined by S and X × S in this case yields a lattice of (14) vector subbundles of V, and determining the lattice of (10) vector subbundles of TM |M2 this induces shows in particular that one has a distinguished line field S = D∩TM2 on M2. Specializing the formulae for I, J , K as in the previous cases gives that on M2, I = J = K = 0. 6 Examples In this section, we give three conformally nonflat examples, one for each sign of the Einstein constant; each is produced using a different method. To the knowledge of the authors, before the present work there were no examples in the literature of nonflat (2, 3, 5) conformal structures known to admit a non-Ricci-flat almost Einstein scale.7 In particular, these examples show that none of the holonomy reductions considered in this article force local flatness of the underlying conformal structure. Example 6.1 (a distinguished rolling distribution). We construct a homogeneous Sasaki– Einstein metric of signature (3, 2) whose negative determines a Ricci-negative conformal struc- ture. Each (2, 3, 5) distribution in the corresponding family is diffeomorphic to a particular special so-called rolling distribution. Let (S2, h+, J+) and (H2, h−, J−) respectively denote the round sphere and hyperbolic plane with their usual Kähler structures, rescaled so that their respective scalar curvatures are ±12. In the usual respective polar coordinates (r, ϕ) and (s, ψ), h+ := 2 3 · 1( r2 + 1 )2 (dr2 + r2dϕ2 ) , J+ := r∂r ⊗ dϕ− 1 r∂ϕ ⊗ dr, h− := 2 3 · 1( s2 − 1 )2 (ds2 + s2dψ2 ) , J− := s∂s ⊗ dψ − 1 s∂ψ ⊗ ds. Then, the triple (S2 ×H2, ĝ, K̂), where ĝ := h+ ⊕−h− and K̂ := J+ ⊕ J−, is a Kähler structure satisfying R̂ab = 6ĝab. The Kähler form ĝacK̂ c b is equal to (dα)ab, where α := 2 3 ( − ϕr dr (r2 + 1)2 + ψs ds (s2 − 1)2 ) . The infinitesimal symmetries of the Kähler structure are spanned by the lifts of the infintesi- mal symmetries of (S2, h+, J+) and (H2, h−, J−), and so the infinitesimal symmetry algebra is aut(ĝ, K̂) ∼= so(3,R)⊕ sl(2,R). 7We recently learned from Bor and Nurowski that they have, in work in progress, also constructed examples [8]. 50 K. Sagerschnig and T. Willse On the canonical S1-bundle π : M → S2 × H2 defined in Section 5.4.6, with standard fiber coordinate λ, define β := dλ−2π∗α. Then, the associated Sasaki–Einstein structure is (M, g, ∂λ), where g := π∗ĝ + β2. The normalizations of the scalar curvatures of the sphere and hyperbolic plane were chosen so that Rab = 4gab. The 1-parameter family {Dυ} of corresponding oriented (2, 3, 5) distributions, which in particular induce the conformal class [−g], is Dυ = 〈 3 ( r2+ 1 ) s∂r + 3 ( s2− 1 ) s cos γ∂s + 3 ( s2− 1 ) sin γ∂ψ + 4s ( sψ cos γ s2 − 1 − rϕ r2 + 1 ) ∂λ, 3 ( r2 + 1 ) s∂ϕ + 3 ( s2 − 1 ) s sin γ∂s − 3r ( s2 − 1 ) cos γ∂ψ + 4s s2 − 1 rψ sin γ∂λ 〉 , where γ := r2 − 1 r2 + 1 ϕ+ s2 + 1 s2 − 1 ψ − 3λ+ υ. 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(Dυ) ∼= aut(ĝ, K̂) ∼= so(3,R)⊕ sl(2,R) and aut([−g]) ∼= aut(g) ∼= aut(ĝ, K̂)⊕ 〈ξ〉 ∼= so(3,R)⊕ sl(2,R)⊕ R. One can show that every distribution Dυ is equivalent to the so-called rolling distribution for the Riemannian surfaces (S2, g+) and (H2, g−). The underlying space of this distribution, which we can informally regard as the space of relative configurations of S2 and H2 in which the surfaces are tangent at a single point, is the twistor bundle [2] over S2 × H2 whose fiber over (x+, x−) is the circle Iso(Tx+S2, Tx−H2) ∼= S1 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 auto- matically entail a reduction to a smaller group. Let (L2, h, J) denote the para-Kähler Lorenztian surface with h := 2 3 ( r2 + 1 )2 (−dr2 + r2dϕ2 ) , J := r∂r ⊗ dϕ+ 1 r∂ϕ ⊗ dr. Then, the triple (L2 × L2, 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 (L2, h) and (L2,−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 specify- ing a para-Fefferman conformal structure cN 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 cN . 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. 8This para-Kähler–Einstein structure is isometric to [22, equation (4.21)], which is attributed there to Nurowski. The Geometry of Almost Einstein (2, 3, 5) Distributions 51 Consider the projective structure [∇] on R2 xy containing the torsion-free connection ∇ char- acterized by ∇∂x∂x = 3xy2∂x + x3∂y, ∇∂x∂y = ∇∂y∂x = 0, ∇∂y∂y = x3∂x − 3x2y∂y. Eliminating the parameter in the geodesic equations for ∇ yields the ODE ÿ = (xẏ− y)3, which corresponds (recall Section 5.5.4) to the function F (x, y, p) = (xp − y)3. The point symmetry algebra of the ODE (that is, the symmetry algebra of the Legendrean contact structure on J := {xp − y > 0} ⊂ J1(R,R)) is sl(2,R) and acts infinitesimally transitively. Hence, we may identify J with an open subset of (some cover of) SL(2,R). With respect to the left-invariant local frame EX := −(xp− y)2∂p, EH := x∂x + y∂y, EY := 1 xp− y (∂x + p∂y). of J , the line fields spanning the contact distribution are 〈∂p〉 = 〈EX〉, and 〈Dx〉 = 〈EY −3EX〉. The Fefferman conformal structure (N, cN ) is again homogeneous: Its (5-dimensional) symmetry algebra aut(cN ) contains an infinitesimally transitive subalgebra isomorphic to gl(2,R). A (local) left-invariant frame of N realizing this subalgebra is given by ÊX = EX + x(xp− y)∂a, ÊH = EH − ∂a, ÊY = EY , ∂a, where a is the standard coordinate on the fiber of N → J and our notation uses the natural (local) decomposition N ∼= J ×Ra. In the dual left-invariant coframe {χ, η, υ, α}, the conformal structure cN has left-invariant representative gN := −χυ − η2 + ηα− υ2. The scale σN := ea/2 √ xp− y (given here with respect to the scale corresponding to gN ) is an almost Einstein scale, and hence gE := σ−2 N gN is Einstein (in fact, Ricci-flat). The conformal class c on M := N × Rr containing g′ := gE − dr2 admits the almost Einstein scale r (here given with respect to g′): g := r−2g′|{±r>0} ∈ c|{±r>0} is a Poincaré–Einstein metric for cN , and in particular is Ricci-positive, and cN is a conformal infinity for g; see [29, Section 4]. (We suppress the notation for the pullback by the canonical projection M = N × R→ N .) So, the curved orbits are M±5 = {(p, r) ∈ N × R : ± r > 0}, M4 = N × {0} ↔ N , and M±2 = ∅. On M5, g := r−2g′ is Ricci-positive, and (N, cN ) is a conformal infinity for either of (M±5 , g|M±5 ). The infinitesimal symmetry algebra aut(c) of c has dimension 6, and is spanned by X := y∂x−p2∂p+p∂a, H := −x∂x+y∂y + 2p∂p−∂a, Y := x∂y +∂p, Z := e−a[(xp−y)∂p−x∂a], A := −2∂a + r∂r, ∂r. Now, X ∧ H ∧ Y ∧ A ∧ ∂r = −2(xp − y)2∂x ∧ ∂y ∧ ∂p ∧ ∂a ∧ ∂r, which vanishes nowhere on M , so (M, c) is homogeneous. Computing the compatible parallel tractor 3-forms, and in particular using (4.6), gives that one 1-parameter family of conformally isometric oriented (2, 3, 5) distributions D∓t that induce c and are related by the Einstein scale r is given on M5 as D∓t := 〈 ± re−a∓t xp− y ÊX + 2∂a,∓ [ 2e2a±t(xp− y)2 + 1 2 e∓tr2 ] ÊX + ear(xp− y)ÊH ± 2(xp− y)2e2a±tÊY + 1 r ∂a + ∂r 〉 . Computing the wedge product of the two spanning fields shows that this span extends smoothly across M4 to a (2, 3, 5) distribution on all of M . By definition this family is D(D−0 ; r), and the corresponding conformal Killing field is ι7(r) = A. The infinitesimal symmetry algebra of D±t is aut(D±t ) = 〈X ,H,Y,±e±tZ − 2∂r〉 ∼= gl(2,R). In particular, this furnishes an example of an inhomogeneous (2, 3, 5) distribution that induces a homogeneous conformal structures. The metric g′ is itself Ricci-flat, so the conformal structure c admits two linearly independent almost Einstein scales. In the scale of c determined by g′, aEs(c) = 〈1, r〉, and the corresponding 52 K. Sagerschnig and T. Willse conformal Killing fields are spanned by ι7(1) = Z + ∂r and ι7(r) = A. These scales correspond to two linearly independent parallel tractors, which reduces the conformal holonomy Hol(c) to a proper subgroup of SL(3,R); computing gives Hol(c) ∼= SL(2,R) nR2. Example 6.3 (submaximally symmetric (2, 3, 5) distributions). In Cartan’s analysis [21] of the equivalence problem for (2, 3, 5) distributions, he showed that if the dimension of the infinitesimal symmetry algebra of a (2, 3, 5) distribution D has infinitesimal symmetry algebra of dimension < 14 (equivalently, if it is not locally flat) and satisfies a natural uniformity condition, then dim aut(D) ≤ 7. (It was shown much more recently, in [48], that the uniformity condition is unnecessary.) Moreover, equality holds iff the distribution is locally equivalent, up to a suitable notion of complexification, to the distribution DI := 〈 ∂q, ∂x + p∂y + q∂p − 1 2 [ q2 + 10 3 Ip 2 + ( 1 + I2 ) y2 ] ∂z 〉 (6.1) on R5 xypqz for some constant I.9 The almost Einstein geometry of the distributions DI is discussed in detail in [64]: The induced conformal structure cI := cDI contains the representative metric gI := [ −3 2 ( I2 + 1 ) y2 + 2Ip2 − 1 2q 2 ] dx2 − 4Ip dx dy + q dx dp − 3p dx dq − 3 dx dz − 3I dy2 + 3 dy dq − 2 dp2. The trivializations by gI of the almost Einstein scales of cI are the pullbacks by the projection R5 xypqz → Rx of the solutions of the homogeneous ODE σ′′− 1 3Iσ = 0 in x, and all of these turn out to be Ricci-flat. In particular the vector space of almost Einstein scales of cI is 2-dimensional, so by Theorem 3.5 dim aut(cI) = dim aut(DI) + dim aEs(cI) = 9. For all I, Hol(cI) is isomorphic to the 5-dimensional Heisenberg group. Unlike for the non-Ricci-flat cases, the authors are aware of no example of a (2, 3, 5) distribution D for which cD is equal to the full (8-dimensional) stabilizer SL(2,R) nQ+ in G2 of an isotropic vector in the standard representation. These distributions are contained in the first class of examples of (2, 3, 5) distributions whose induced conformal structures locally admit Einstein representatives [52, Example 6].10 9The coefficient 10 3 corrects an arithmetic error in [21, Section 9, equation (6)]. Also, note that we have specialized the formula given there to constant I. 10In that reference, these distributions were given in a form not immediately recognizable as diffeomorphic to those in (6.1). For I 6= ± 3 4 , the distribution DI is diffeomorphic to the distribution defined via [52, equation (55)] by the function F (q) = qm, where k = 2m− 1 is any value that satisfies I2 = (k2 + 1)2 (k2 − 9)( 1 9 − k2) ; when I = ± 3 4 , one may take F (q) = log q [26]. T h e G eom etry o f A lm ost E in stein (2,3 ,5) D istrib u tion s 53 A Tabular summary of the curved orbit decomposition. Ma ε Section S S ∩ P structure A leaf space structure X L S −1 0 +1 M±5 • Section 5.4 SU(1, 2) SU(1, 1) Sasaki–Einstein U(1, 1) Kähler–Einstein ±HΦ(X,S) > 0 L ⊂ [D,D] L t D –• SL(2,R) nQ+ SL(2,R) null-Sasaki–Einstein GL(2,R) null-Kähler–Einstein • SL(3,R) SL(2,R) para-Sasaki–Einstein GL(2,R) para-Kähler–Einstein M4 • Section 5.5.3 SU(1, 2) P− Fefferman conformal (sig. (1, 3)) PSU(1,2) 3-dim. CR structure H(X,S) = 0 (X × S) ∧X 6= 0 L ⊂ D S t [D,D]• Section 5.5.5 SL(2,R) nQ+ Rn R3 fibration over conformal (sig. (1, 2)) + isotropic line field ∗† complete flag field‡ • Section 5.5.4 SL(3,R) P+ para-Fefferman conformal (sig. (2, 2)) P12 second-order ODE M±2 • Section 5.6.1 SL(3,R) P1 or P2 2-dim. projective – – X × S = ±X – S ⊂ [D,D] S t D M2 • Section 5.6.2 SL(2,R) nQ+ Rn (R4 nR) line field – – X × S = 0 X ∧ S 6= 0 – S ⊂ D M±0 • – SL(2,R) nQ+ SL(2,R) nQ+ trivial – – X ∧ S = 0 – – † This is the solvable 5-dimensional group Rn ((R2 nR)⊕ 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. Ma ξa Iab Jab Kab M±5 θa 1 2(−εφab + φ̄ab) −θcχcab 1 2(−εφab − φ̄ab) M4 µbφ ba −εφab − 2µ[aξb] 3µcφ[caθb] 2µ[aξb] M±2 0 −φab φab 0 M2 0 0 0 0 M±0 0 0 0 0 The formulae for the open orbits M± 5 are given in the scale σ. 54 K. Sagerschnig and T. Willse Acknowledgements It is a pleasure to thank Andreas Čap for discussions about curved orbit decompositions and natural operators on 3-dimensional CR and Legendrean contact structures, Boris Doubrov and Boris Kruglikov for discussions about the geometry of second-order ODEs modulo point trans- formations, Rod Gover for comments about conformal tractor geometry, John Huerta for com- ments about the algebra of G2, Pawe l Nurowski for a suggestion that gave rise to Example 6.1, and Michael Eastwood and Dennis The for comments about various aspects of the project. Ian Anderson’s Maple package DifferentialGeometry was used extensively, including for the derivation of Proposition 4.1 and Algorithm 4.12 and the preparation of Example 6.2, and it is again a pleasure to thank him for helpful comments about the package’s usage. Finally, the authors thank the referees for several helpful comments and suggestions. 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Tokyo 19 (1943), 444–453. https://doi.org/10.1016/S0393-0440(97)00073-9 https://doi.org/10.1007/978-3-662-02950-3 https://doi.org/10.1007/BF00759678 http://arxiv.org/abs/1303.1307 http://arxiv.org/abs/0904.0186 http://arxiv.org/abs/math.DG/0406316 https://doi.org/10.1007/978-0-387-73831-4_23 http://arxiv.org/abs/math.DG/0604393 https://doi.org/10.1016/j.geomphys.2004.11.006 http://arxiv.org/abs/math.DG/0406400 https://doi.org/10.1088/0264-9381/20/23/004 http://arxiv.org/abs/math.DG/0306331 https://doi.org/10.1007/978-3-662-12927-2 https://doi.org/10.1007/978-3-662-12927-2 https://doi.org/10.1007/s00209-003-0549-4 http://arxiv.org/abs/math.DG/0206117 https://doi.org/10.1063/1.531249 http://arxiv.org/abs/hep-th/9409089 https://doi.org/10.1098/rspa.1963.0124 https://doi.org/10.1016/j.difgeo.2013.10.010 http://arxiv.org/abs/1302.7163 https://doi.org/10.1007/BF02566943 https://doi.org/10.2307/1990341 https://doi.org/10.2307/1990341 https://doi.org/10.3792/pia/1195573364 1 Introduction 2 Preliminaries 2.1 epsilon-complex structures 2.2 The group G2 2.2.1 Split cross products in dimension 7 2.2.2 The group G2 2.2.3 Some G2 representation theory 2.3 Cartan and parabolic geometry 2.3.1 Cartan geometry 2.3.2 Holonomy 2.3.3 Tractor geometry 2.3.4 Parabolic geometry 2.3.5 Oriented conformal structures 2.3.6 Oriented (2,3,5) distributions 2.4 Conformal geometry 2.4.1 Conformal density bundles 2.4.2 Conformal tractor calculus 2.4.3 Canonical quotients of conformal tractor bundles 2.4.4 Conformal BGG splitting operators 2.5 Almost Einstein scales 2.6 Conformal Killing fields and (k-1)-forms 2.7 (2,3,5) conformal structures 2.7.1 Holonomy characterization of oriented (2,3,5) conformal structures 2.7.2 The conformal tractor decomposition of the tractor G2-structure 3 The global geometry of almost Einstein (2,3,5) distributions 3.1 Distinguishing a vector in the standard representation V of G2 3.1.1 Stabilizer subgroups 3.1.2 An varepsilon-Hermitian structure 3.1.3 Induced splittings and filtrations 3.1.4 The family of stabilized 3-forms 3.2 The canonical conformal Killing field xi 3.3 Characterization of conformal Killing fields induced by almost Einstein scales 3.4 The weighted endomorphisms I, J, K 3.5 The (local) leaf space 3.5.1 Descent of the canonical objects 4 The conformal isometry problem 4.1 The space of G2-structures compatible with an SO(3,4)-structure 4.2 Conformally isometric (2,3,5) distributions 4.2.1 Parameterizations of conformally isometric distributions 4.2.2 Additional induced distributions 4.2.3 Recovering the Einstein scale relating conformally isometric distributions 5 The curved orbit decomposition 5.1 The general theory of curved orbit decompositions 5.2 The orbit decomposition of the flat model M 5.2.1 Ricci-negative case 5.2.2 Ricci-positive case 5.2.3 Ricci-flat case 5.3 Characterizations of the curved orbits 5.4 The open curved orbits M5+/- 5.4.1 The canonical splitting 5.4.2 The canonical hyperplane distribution 5.4.3 The varepsilon-Sasaki structure 5.4.4 Projective geometry 5.4.5 The open leaf space L4 5.4.6 The varepsilon-Kähler–Einstein Fefferman construction 5.5 The hypersurface curved orbit M4 5.5.1 The canonical lattices of hypersurface distributions 5.5.2 The hypersurface leaf space L3 5.5.3 Ricci-negative case: The classical Fefferman conformal structure 5.5.4 Ricci-positive case: A paracomplex analogue of the Fefferman conformal structure 5.5.5 Ricci-flat case: A fibration over a special conformal structure 5.6 The high-codimension curved orbits M2+/-, M2, M0+/- 5.6.1 The curved orbits M2+/-: Projective surfaces 5.6.2 The curved orbit M2 6 Examples A Tabular summary of the curved orbit decomposition. 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