Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition

Given a maximally non-integrable 2-distribution D on a 5-manifold M, it was discovered by P. Nurowski that one can naturally associate a conformal structure [g]D of signature (2,3) on M. We show that those conformal structures [g]D which come about by this construction are characterized by the exist...

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Дата:	2009
Автори:	Hammerl, M., Sagerschnig, K.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2009
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149138
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition / M. Hammerl, K. Sagerschnig // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 41 назв. — англ.

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spelling	irk-123456789-1491382019-02-20T01:26:30Z Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition Hammerl, M. Sagerschnig, K. Given a maximally non-integrable 2-distribution D on a 5-manifold M, it was discovered by P. Nurowski that one can naturally associate a conformal structure [g]D of signature (2,3) on M. We show that those conformal structures [g]D which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of [g]D can be decomposed into a symmetry of D and an almost Einstein scale of [g]D. 2009 Article Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition / M. Hammerl, K. Sagerschnig // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 41 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34A26; 35N10; 53A30; 53B15; 53B30 http://dspace.nbuv.gov.ua/handle/123456789/149138 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Given a maximally non-integrable 2-distribution D on a 5-manifold M, it was discovered by P. Nurowski that one can naturally associate a conformal structure [g]D of signature (2,3) on M. We show that those conformal structures [g]D which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of [g]D can be decomposed into a symmetry of D and an almost Einstein scale of [g]D.
format	Article
author	Hammerl, M. Sagerschnig, K.
spellingShingle	Hammerl, M. Sagerschnig, K. Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Hammerl, M. Sagerschnig, K.
author_sort	Hammerl, M.
title	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition
title_short	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition
title_full	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition
title_fullStr	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition
title_full_unstemmed	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition
title_sort	conformal structures associated to generic rank 2 distributions on 5-manifolds – characterization and killing-field decomposition
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149138
citation_txt	Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition / M. Hammerl, K. Sagerschnig // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 41 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT hammerlm conformalstructuresassociatedtogenericrank2distributionson5manifoldscharacterizationandkillingfielddecomposition AT sagerschnigk conformalstructuresassociatedtogenericrank2distributionson5manifoldscharacterizationandkillingfielddecomposition
first_indexed	2025-07-12T21:28:57Z
last_indexed	2025-07-12T21:28:57Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 081, 29 pages Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition? Matthias HAMMERL and Katja SAGERSCHNIG Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria E-mail: matthias.hammerl@univie.ac.at, katja.sagerschnig@univie.ac.at Received April 09, 2009, in final form July 28, 2009; Published online August 04, 2009 doi:10.3842/SIGMA.2009.081 Abstract. Given a maximally non-integrable 2-distribution D on a 5-manifold M , it was discovered by P. Nurowski that one can naturally associate a conformal structure [g]D of signature (2, 3) on M . We show that those conformal structures [g]D which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of [g]D can be decomposed into a symmetry of D and an almost Einstein scale of [g]D. Key words: generic distributions; conformal geometry; tractor calculus; Fefferman construc- tion; conformal Killing fields; almost Einstein scales 2000 Mathematics Subject Classification: 34A26; 35N10; 53A30; 53B15; 53B30 Dedicated to Peter Michor on the occasion of his 60th birthday celebrated at the Central European Seminar in Mikulov, Czech Republic, May 2009. 1 Introduction and statement of results In this section we briefly introduce the main objects of interest and state the results of this text. 1.1 Generic rank 2-distributions on 5-manifolds Let M be a smooth 5-dimensional manifold and consider a subbundle D of the tangent bund- le TM which shall be of constant rank 2. We say that D is generic if it is maximally non- integrable in the following sense: For two subbundles D1 ⊂ TM and D2 ⊂ TM we define [D1,D2]x := span({[ξ, η]x : ξ ∈ Γ(D1), η ∈ Γ(D2)}). (1) Then we demand that [D,D] ⊂ TM is a subbundle of constant rank 3 and [D, [D,D]] = TM . In other words, two steps of taking Lie brackets of sections of D yield all of TM . It is a classical result of Élie Cartan [19] that generic rank 2-distributions on M can equiva- lently be described as parabolic geometries of type (G2, P ). This will be explained in Section 4. 1.2 The associated conformal structure of signature (2,3) and its characterization In [36] P. Nurowski used Cartan’s description of generic distributions to associate to every such distribution a conformal class [g]D of signature (2, 3)-metrics on M . ?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection is available at http://www.emis.de/journals/SIGMA/Cartan.html mailto:matthias.hammerl@univie.ac.at mailto:katja.sagerschnig@univie.ac.at http://dx.doi.org/10.3842/SIGMA.2009.081 http://www.emis.de/journals/SIGMA/Cartan.html 2 M. Hammerl and K. Sagerschnig There is a well studied result similar to Nurowski’s construction: This is the classical Feffer- man construction [22, 4, 6, 12, 13, 34, 35] of a (pseudo) conformal structure on an S1-bundle over a CR manifold. It has been observed by A. Čap in [8] that both Nurowski’s and Fefferman’s results admit interpretations as special cases of a more general construction relating parabolic geometries of different types. In Section 4.4 we will discuss conformal structures associated to rank two distributions in this picture. Furthermore, we prove that given a holonomy reduction of a conformal structure [g] of signature (2, 3) to a subgroup of G2, the conformal class [g] is induced by a distribution D ⊂ TM . Using strong techniques from the BGG-machinery [16, 5, 29] and tractor bundles [11, 10], we then proceed to prove our first main result in Section 5. Before we can state it we introduce some simple notation for tensorial expressions; this is slightly redundant since we will later use a form of index notation for such formulas. Take some g ∈ [g] and let η ∈ ⊗kT ∗M for k ≥ 2. The trace over the i-th and the j-th slot of η via the (inverse of) the metric g will be denoted tri,j(η) ∈ ⊗k−2T ∗M . For an arbitrary tensor η, alt(η) is the full alternation of η. Theorem A. Let [g] be a conformal class of signature (2, 3) metrics on M . Then [g] is induced from a generic rank 2 distribution D ⊂ TM if and only if there exists a normal conformal Killing 2-form φ that is locally decomposable and satisfies the following genericity condition: Let g ∈ [g] be a metric in the conformal class, D its Levi-Civita connection and P its Schouten tensor (11). Define µ := tr1,2Dφ ∈ T ∗M, ρ := +24φ+ 4alt(tr1,3DDφ) + 3alt(tr2,3DDφ) + 24alt(tr1,3P ⊗ φ)− 6tr1,2P ⊗ φ ∈ Λ2T ∗M. Here we use the convention 4σ = −tr1,2DDσ. Then one must have φ ∧ µ ∧ ρ 6= 0. To be precise, Theorem A assumes orientability of TM , but this is a minor assumption only made for convenience of presentation – see Remark 5. 1.3 Killing field decomposition Let sym(D) denote the infinitesimal symmetries of the distribution D, i.e., sym(D) = {ξ ∈ X(M) : Lξη = [ξ, η] ∈ Γ(D) ∀ η ∈ Γ(D)}. The corresponding objects for conformal structures are the conformal Killing fields cKf([g]) = {ξ ∈ X(M) : Lξg = e2fg for some g ∈ [g] and f ∈ C∞(M)}. Since the construction that associates a conformal structure [g]D to a distribution D is natu- ral, every symmetry ξ ∈ X(M) of the distribution will also preserve the associated conformal structure [g]D, i.e., it is a conformal Killing field. This yields an embedding sym(D) ↪→ cKf([g]D). We will show that a complement to sym(D) in cKf([g]D) is given by the almost Einstein scales of [g]D: a function σ ∈ C∞(M) is an almost Einstein scale for g ∈ [g]D if it is non-vanishing on an open dense subset U of M and satisfies that σ−2g is Einstein on U [24]. The natural origin of almost Einstein scales via tractor calculus will be seen in Section 3. Conformal Structures Associated to Generic 2-Distributions 3 Theorem B. Let [g]D be the conformal structure associated to a generic rank 2 distribution D on a 5-manifold M , and let φ be a conformal Killing form characterizing the conformal struc- ture as in Theorem A. Then every conformal Killing field decomposes into a symmetry of the distribution D and an almost Einstein scale: cKf([g]D) = sym(D)⊕ aEs([g]D). (2) The mapping that associates to a conformal Killing field ξ ∈ X(M) its almost Einstein scale part with respect to the decomposition (2) is given by ξ 7→ tr1,3tr2,4 ( φ⊗Dξ − 1 2ξ ⊗Dφ ) , where D is the Levi-Civita connection of an arbitrary metric g in the conformal class. The mapping that associates to an almost Einstein scale σ ∈ C∞(M) (for a metric g ∈ [g]) a conformal Killing field is given by σ 7→ tr2,3φ⊗ (Dσ)− 1 4tr1,2(Dφ)σ. We remark here that the constructions in this paper, both for characterization (Section 5) and automorphism-decomposition (Section 6), are largely analogous to the ones of [12] and [13] for the (classical) Fefferman spaces. This article incorporates material of the authors’ respective theses [39, 27]. 2 Preliminaries on Cartan and parabolic geometries In this section we discuss general parabolic geometries. These are special kinds of Cartan geo- metries. 2.1 Cartan geometries Let G be a Lie group and P < G a closed subgroup. The Lie algebras of P and G will be denoted by p and g. Let G π→M be a P -principal bundle over a manifold M . The right action of P on G will be denoted by rp(u) = u · p for u ∈ G and p ∈ P . The corresponding fundamental vector fields are ζY (u) := d dt \|t=0 rexp(tY )(u) = d dt \|t=0 u · exp(tY ) for Y ∈ p. Definition 1. A Cartan geometry of type (G,P ) on a manifold M is a P -principal bundle G π→ M endowed with a Cartan connection form ω ∈ Ω1(G, g), i.e., a g-valued 1-form on G satisfying (C.1) ωu·p(Tur pξ) = Ad(p−1)ωu(ξ) for all p ∈ P , u ∈ G, and ξ ∈ TuG. (C.2) ω(ζY ) = Y for all Y ∈ p. (C.3) ωu : TuG → g is an isomorphism for all u ∈ G. We say that ω is right-equivariant, reproduces fundamental vector fields and is an absolute parallelism ω : TG ∼= G × g. It is easily seen that for a Cartan geometry (G, ω) of type (G,P ) the map G × g 7→ TM given by (u,X) 7→ Tuπω −1 u (X) induces an isomorphism TM = G ×P g/p. In particular, dim M = dim g/p. 4 M. Hammerl and K. Sagerschnig Cartan geometries can be viewed as curved versions of homogeneous spaces: The homogeneous model of Cartan geometries of type (G,P ) is the principal bundle G→ G/P endowed with the Maurer–Cartan form ωMC ∈ Ω1(G, g), which satisfies the Maurer–Cartan equation dωMC(ξ, η) + [ ωMC(ξ), ωMC(η) ] = 0 for all ξ, η ∈ X(G). For a general Cartan geometry (G, ω) the failure of ω to satisfy the Maurer–Cartan equation is measured by the curvature form Ω ∈ Ω2(G, g), Ω(ξ, η) = dω(ξ, η) + [ω(ξ), ω(η)]. (3) One can show that Ω vanishes, i.e., ω is flat, if and only if (G, ω) is locally isomorphic (in the obvious sense) with (G,ωMC). Since the Cartan connection defines an absolute parallelism ω : TG ∼= G × g, its curvature can be equivalently encoded in the curvature function κ ∈ C∞(G,Λ2(g∗)⊗ g), κ(u)(X,Y ) := Ω ( ω−1 u (X), ω−1 u (Y ) ) . One verifies that Ω vanishes on vertical fields ζY for Y ∈ p, i.e., it is horizontal. This implies, that κ in fact defines a function G 7→ Λ2(g/p)∗ ⊗ g. And since Ω is P -equivariant, so is κ. We denote by AM := G ×P g the associated bundle corresponding to the restriction of the adjoint representation Ad : G → GL(g) to P . It is called the adjoint tractor bundle (general tractor bundles will be introduced below). Note that since the curvature of a Cartan connection is horizontal and P -equivariant, it factorizes to a AM -valued 2-form K ∈ Ω2(M,AM) on M . Thus, Ω ∈ Ω2(G, g),K ∈ Ω2(M,AM) and κ : G → Λ2(g/p)∗ ⊗ g all encode essentially the same object, namely the curvature of the Cartan connection form ω, and technical reasons will determine which representation should be used at a given point. 2.2 Tractor bundles For any G-representation V , the associated bundle V := G ×P V is called a tractor bundle. Tractor bundles carry canonical linear connections: Extend the structure group of G from P to G by forming G′ := G ×P G. Then ω extends uniquely to a G- equivariant g-valued 1-form ω′ on G reproducing fundamental vector fields – i.e., to a principal connection form. Since V = G×P V = G′×GV one has the induced tractor connection ∇V on V. For computations we will use the following explicit formula: Let ξ′ ∈ X(G) be a P -invariant lift of a vector field ξ ∈ X(M) and fs ∈ C∞(G, V )P be the P -equivariant V -valued function on G corresponding to s ∈ Γ(V). Then ∇V ξ s corresponds to the P -equivariant function ξ′ · fs + ω(ξ′)fs. (4) 2.3 Parabolic geometries We now specialize to parabolic geometries. This class of Cartan geometries has a natural algeb- raic normalization condition which is employed in this text to describe conformal structures and generic distributions as parabolic geometries. For a thorough discussion of parabolic geometries we refer to [15]. Conformal Structures Associated to Generic 2-Distributions 5 Let us start with the algebraic background and introduce the notion of a \|k\|-graded Lie algebra g: this is a a semisimple Lie algebra together with a vector space decomposition g = g−k ⊕ · · · ⊕ gk such that [gi, gj ] ⊂ gi+j . Then g− = g−k ⊕ · · · ⊕ g−1 and p+ = g1 ⊕ · · · ⊕ gk are nilpotent subalgebras of g. The Lie algebra p = g0 ⊕ · · · ⊕ gk is indeed a parabolic subalgebra of g, and g0 is its reductive Levi part. The grading induces a filtration on g via gi := gi⊕· · ·⊕gk. Let G be a Lie group with Lie algebra g. Let P be a closed subgroup of G whose Lie algebra is the parabolic p ⊂ g and such that it preserves the filtration, i.e., for all p ∈ P we have Ad(p)gi ⊂ gi ∀i ∈ Z. Definition 2. A Cartan geometry of type (G,P ) for groups as introduced above is called a parabolic geometry. In the following, we will also consider the subgroup G0 := {g ∈ P : Ad(g)gi ⊂ gi ∀ i ∈ Z}. of all elements in P preserving the grading on the Lie algebra, which has Lie algebra g0, and the subgroup P+ := {p ∈ P : (Ad(p)− id)gi ⊂ gi+1 ∀ i ∈ Z}, which has Lie algebra p+. Actually P decomposes as a semidirect product P = G0 n P+; thus P/P+ = G0. 2.4 Lie algebra differentials and normality Let V be a G-representation. We now introduce algebraic differentials ∂ : Λi(g/p)∗ ⊗ V → Λi+1(g/p)∗ ⊗ V and ∂∗ : Λi+1(g/p)∗ ⊗ V → Λi(g/p)∗ ⊗ V. For the first of these, we use the G0-equivariant identification of g∗− with (g/p)∗ and define ∂ as the differential computing the Lie algebra cohomology of g− with values in V . For the Kostant codifferential ∂∗ we use the P -equivariant identification of (g/p)∗ with p+ given by the Killing form; it is then defined as the differential computing the Lie algebra homology of p+ with values in V . We include the explicit formula for ∂∗, which will be needed later on: On a decomposable element ϕ = Z1 ∧ · · · ∧ Zi ⊗ v ∈ Λip+ ⊗ V, ∂∗ is given as ∂∗(ϕ) := i∑ j=1 (−1)jZ1 ∧ · · · ∧ Ẑj ∧ · · · ∧ Zi ⊗ (Xjv) + ∑ 1≤j<k≤i (−1)j+k[Zj , Zk] ∧ Z1 ∧ · · · ∧ Ẑj ∧ · · · ∧ Ẑk ∧ · · · ∧ Zi ⊗ v. The G-representation V carries a natural G0-invariant grading V0 ⊕ · · · ⊕ Vr for some r ∈ N. The induced filtration V i := Vi⊕· · ·⊕Vr is even P -invariant. The grading on V and the grading g−k⊕· · ·⊕g−1 = g− ∼= g/p naturally induce a grading on the spaces Ci(V ) = Λi(g/p)∗⊗V , which is preserved by both ∂ and ∂∗. While ∂∗ is seen to be P -equivariant, ∂ is only G0-equivariant. 6 M. Hammerl and K. Sagerschnig We consider the spaces of cocycles Zi(V ) := ker ∂∗ ⊂ Λi(g/p)∗ ⊗ V , coboundaries Bi(V ) := im ∂∗ ⊂ Λi(g/p)∗ ⊗ V and cohomologies Hi(V ) := Zi(V )/Bi(V ). Let Πi : Zi(V ) → Hi(V ) be the canonical surjection. By [31], the differentials ∂ and ∂∗ are adjoint with respect to a natural inner product on the space Ci(V ). Via the Kostant Laplacian � = ∂∗ ◦ ∂ + ∂ ◦ ∂∗ this yields a G0-invariant Hodge decomposition Ci(V ) = im ∂ ⊕ ker �⊕ im ∂∗. Thus, as a G0-module, Hi(V ) can be embedded into Zi(V ) ⊂ Ci(V ). Since G×P g/p = TM the associated bundle Ci := G×P Λi(g/p)∗⊗V of Ci(V ) is ΛiT ∗M ⊗V, whose sections are the V-valued i-forms Ωi(M,V). The P -equivariant differential ∂∗ carries over to the associated spaces, ∂∗ : Ωi+1(M,V) → Ωi(M,V). We set Zi(V) := G ×P Zi(V ), Bi(V) := G ×P Bi(V ) and Hi(V) := G ×P Hi(V ). The canonical surjection from Zi(V) onto Hi(V) is denoted by Πi. If the tractor bundle V in question is unambiguous we just write Ci, Zi, Bi, Hi. The Kostant codifferential provides a conceptual normalization condition for parabolic ge- ometries: Recall that the curvature of a Cartan connection form ω factorizes to a two-form K ∈ Ω2(M,AM) on M with values in the adjoint tractor bundle. Definition 3. A Cartan connection form ω is called normal if ∂∗(K) = 0. In this case one has the harmonic curvature KH = Π2(K) ∈ H2(AM). In the picture of P -equivariant functions on G, the harmonic curvature corresponds to the composition of the curvature function κ with the projection Π2 : Z2(g) → H2(g), i.e., to κH = Π2 ◦ κ . There is a simple algorithm to compute the cohomology spaces Hi(V ) provided by Kostant’s version of the Bott–Borel–Weil theorem, cf. [31, 41]. Mostly, we will just need to know H0(V ), which turns out to be the lowest homogeneity of V , i.e., H0(V ) = V/V 1 = V/(p+V ). 2.5 The BGG-(splitting-)operators The BGG-machinery developed in [16] and [5] will feature prominently at many crucial points in this paper. The presentation here is very brief and the most important operators will later be given explicitly (see the end of the next section on conformal geometry). The highly useful Lemma 1 below can be understood without its relation to the BGG-machinery. The main observation is that for every σ ∈ Γ(H0) there is a unique s ∈ Γ(V) with Π0(s) = σ such that ∇Vs ∈ Γ(Z1), i.e., such that ∂∗(∇Vs) = 0. This gives a natural splitting LV0 : Γ(H0) → Γ(V) of Π0 : Γ(V) → Γ(H0) called the 1-st BGG-splitting operator and it defines the 1-st BGG-operator ΘV 0 : Γ(H0) → Γ(H1), σ 7→ Π1(∇Vs(L0(σ))). We only remark that this construction of differential splitting operators of the projections Πi : Zi → Hi proceeds similarly, and one obtains the celebrated BGG-sequence Hi Θi→ Hi+1. We will often need the following consequence of the definition of LV0 : If s ∈ Γ(V) is parallel, one trivially has ∂∗(∇Vs) = 0, and thus s = LV0 (Π0(s)). This is important enough to merit a Lemma 1. On the space of parallel sections of a tractor bundle V, LV0 ◦Π0 is the identity, i.e., if s ∈ Γ(V) with ∇Vs = 0, then s = LV0 (Π0(s)). Conformal Structures Associated to Generic 2-Distributions 7 In particular, if the projection of a parallel section s ∈ Γ(V) to its part in Γ(H0) = Γ(V/V1) vanishes, s must already have been trivial. We now proceed to discuss conformal structures as parabolic geometries in Section 3 and do likewise for generic rank two distributions in Section 4. 3 Conformal structures Two pseudo-Riemannian metrics g and ĝ with signature (p, q) on a n = p + q-dimensional manifold M are said to be conformally equivalent if there is a function f ∈ C∞(M) such that ĝ = e2fg. The conformal equivalence class of g is denoted by [g] and (M, [g]) is said to be a manifold endowed with a conformal structure. An equivalent description of a conformal structure of signature (p, q) is a a reduction of structure group of TM to CO(p, q) = R+×O(p, q), and the corresponding CO(p, q)-bundle will be denoted G̃0. The associated bundle to G̃0 for the 1-dimensional representation R[w] of CO(p, q) given by (c, C) ∈ CO(p, q) = R+ ×O(p, q) 7→ cw for w ∈ R is called the bundle of conformal w-densities and denoted by E [w]. We will use abstract index notation and notation for weighted bundles similar to [25]: Ea := T ∗M , Ea := TM , Eab = Ea ⊗ Eb, Ea[w] := Ea ⊗ E [w]. Recall the Einstein convention, e.g., for ξa ∈ Γ(Ea) = X(M) and ϕa ∈ Γ(Ea) = Ω1(M), ξaϕa = ϕ(ξ) ∈ C∞(M). Round brackets will denote symmetrizations, e.g. E(ab) = S2T ∗M and square brackets anti-symmetrizations, e.g. E[ab] = Λ2T ∗M . In the following we will not distinguish between the space of sections Γ(Ea···b[w]) and Ea···b[w] itself. Given a metric g ∈ [g], a section σ ∈ E [w] trivializes to a function [σ]g ∈ C∞(M) and one has [σ]e2fg = ewf [σ]g. Tautologically, the conformal class of metrics [g] defines a canonical section g in E(ab)[2] = Γ(S2T ∗M ⊗ E [2]), called the conformal metric, such that the trivialization of g with respect to g ∈ [g] is just g. The conformal metric g allows one to raise or lower indices with simultaneous adjustment of the conformal weight: e.g., for a vector field ξp ∈ Ep = X(M) one can form ξp = gpqξ q ∈ Ep[2] = Γ(T ∗M ⊗ E [2]), which is a 1-form of weight 2. 3.1 Conformal structures as parabolic geometries Let Mp,q be a given symmetric bilinear form of signature (p, q) on Rn = Rp,q, and define the symmetric bilinear form h of signature (p+ 1, q + 1) on Rn+2 by h = 0 0 1 0 Mp,q 0 1 0 0  . (5) We define P̃ ⊂ SO(h) ∼= SO(p + 1, q + 1) as the stabilizer of the isotropic ray R+e1, and one finds P̃ = CO(p, q) n Rn∗. The Lie algebra so(p+ 1, q + 1) = so(h) is \|1\|-graded so(h) = so(h)−1 ⊕ so(h)0 ⊕ so(h)1 = Rn ⊕ co(p, q)⊕ Rn∗. Realized in gl(n+ 2) it is given by matrices of the form−α −ZtMp,q 0 X A Z 0 −XtMp,q α  , α ∈ R, X, Z ∈ Rn, A ∈ so(Mp,q). (6) 8 M. Hammerl and K. Sagerschnig Let (G̃ →M, ω̃) be a Cartan geometry of type (SO(h), P̃ ). Define G̃0 := G̃/P̃+ = G̃/Rn∗. Then G̃0 is a CO(p, q)-principal bundle over M and TM = G̃ ×P̃ so(h)/p = G̃0 ×CO(p,q) Rn, i.e., G̃0 → M gives a reduction of structure group of TM to CO(p, q) and thus a conformal structure of signature (p, q). Since there are many non-isomorphic Cartan geometries of type (SO(h), P̃ ) describing the same conformal structure on the underlying manifold, one imposes a normalization condition on the curvature K ∈ Ω2(M, ÃM) of ω̃. Using the notion of normality introduced in Definition 3, one has: Theorem 1 ([18]). Up to isomorphism there is a unique P̃ -principal bundle G̃ over M endowed with a normal Cartan connection form ω̃ ∈ Ω1(G̃, so(h)) such that G̃/Rn∗ = G̃0 is the conformal frame bundle of (M, [g]). This provides an equivalence of categories between oriented conformal structures of signature (p, q) and normal parabolic geometries of type (SO(h), P̃ ). 3.2 Tractor bundles for conformal structures The standard tractor bundle of conformal geometry is obtained by the associated bundle T := G̃ ×P̃ Rn+2 of the standard representation of P̃ = SO(h) on Rn+2. P̃ preserves the filtration {0} ⊂ R 0 0  ⊂  R Rn 0  ⊂  R Rn R  (7) of Rn+2, and therefore gives a well-defined filtration {0} ⊂ T 1 ⊂ T 0 ⊂ T −1 = T . The associated graded of T is gr(T ) = gr−1(T )⊕ gr0(T )⊕ gr1(T ), with gr1(T ) := T 1 = E [−1], gr0(T ) := T 0/T 1 = Ea[−1], (8) gr−1(T ) := T −1/T 0 = E [1]. It is a general and well known fact of conformal tractor calculus that a choice of metric g ∈ [g] yields a reduction of the P̃ -principal bundle G̃ to the CO(p, q)-principal bundle G̃0, and this is seen to provide an isomorphism of a natural bundle with its associated graded space. In the case of the standard tractor bundle T , this gives an isomorphism of T with gr(T ), and a section s ∈ Γ(T ) will then be written [s]g =  ρ ϕa σ  ∈ E [−1] Ea[1] E [1]  . (9) For ĝ = e2fg one has the transformation [s]ĝ =  ρ̂ ϕ̂a σ̂  = ρ−Υaϕ a − 1 2σΥbΥb ϕa + σΥa σ  Conformal Structures Associated to Generic 2-Distributions 9 where Υ = df . The insertion of E [−1] into T as the top slot is independent of the choice of g ∈ [g] and defines a section τ+ ∈ T [1]. The insertion of E [1] into T as the bottom slot is well defined only via a choice of g ∈ [g] and defines a section τ− ∈ T [−1]. Let e1, . . . , en+2 be the standard basis of Rn+2. Then τ+ and τ− can be understood as the sections corresponding to the constant functions on G̃0 mapping to e1 ⊗ 1 ∈ Rn+2 ⊗ R[1] resp. en+2 ⊗ 1 ∈ Rn+2 ⊗ R[−1]. Since h ∈ S2T ∗Rn+2 is SO(h)-invariant it defines a tractor metric h on T . With respect to g ∈ [g] and the decomposition (9) of an element s ∈ Γ(T ) [h]g = 0 0 1 0 g 0 1 0 0  . Let D be the Levi-Civita connection of g ∈ [g], then the tractor connection ∇T on T is given by [∇T c s]g = ∇T c  ρ ϕa σ  =  Dcρ− P b c ϕb Dcϕa + σPca + ρgca Dcσ − ϕc  . (10) Here P = P (g) = 1 n− 2 ( Ric(g)− Sc(g) 2(n− 1) g ) (11) is the Schouten tensor of g. The trace of the Schouten tensor is denoted J = gpqPpq. The adjoint tractor bundle is ÃM = G̃×P̃ so(h), which can be identified with so(T ,h) = Λ2T . With respect to g ∈ [g], ÃM = TM ⊕ co(TM, g) ⊕ T ∗M , and in matrix notation a section [s]g = ξ ⊕ (α,A)⊕ ϕ ∈ X(M)⊕ co(TM, g)⊕ Ω1(M) will be written as−α −ϕa 0 ξa A ϕa 0 −ξa α  . The curvature form K̃ ∈ Ω2(M, ÃM) has in fact values in ÃM0; this is called torsion-freeness. It furthermore decomposes into Weyl curvature C ∈ Ω2(M, so(TM)) = E c c1c2 d and Cotton–York tensor A ∈ Ea[c1c2]: K̃c1c2 = 0 −Aac1c2 0 0 C a c1c2 b Aa c1c2 0 0 0  . (12) The Weyl curvature C is the completely trace-free part of the Riemannian curvature R of g. The Cotton–York tensor is given by A = Aac1c2 = 2D[c1Pc2]a. We will later need the first BGG-splitting operators for the tractor bundles T , ÃM = Λ2T and Λ3T , and therefore give general formulas from [29] for the space V := Λk+1T for k ≥ 0. The P̃ -invariant filtration (7) of Rn+2 from above carries over to the invariant filtration of the exterior power V = Λk+1Rn+2, written {0} ⊂ V 1 ⊂ V 0 ⊂ V −1 = V . Again, this yields filtrations of the associated bundles: {0} ⊂ V1 ⊂ V0 ⊂ V−1 = V := Λk+1T . The notion of the associated graded space is the same: we define gr(ΛkT ) as the direct sum over all gri(ΛkT ) := (ΛkT )i/(ΛkT )i+1. With respect to g ∈ [g], for k ≥ 0, one again obtains an isomorphism of Λk+1T with gr(Λk+1T ), and we will write [Λk+1T ]g =  E[a1···ak][k − 1] E[a1···ak+1][k + 1] \| E[a1···ak−1][k − 1] E[a1···ak][k + 1]  . 10 M. Hammerl and K. Sagerschnig This identification employs the insertions of the top slot τ+ ∈ T [1] and bottom slot τ− ∈ T [−1]: ρa1···ak ϕa0···ak \| µa2···ak σa1···ak  7→ τ− ∧ σ + ϕ+ τ+ ∧ τ− ∧ µ+ τ+ ∧ ρ. (13) To understand the map σ 7→ τ− ∧ σ better, observe via (8) that one has a canonical embedding of E[a1···ak][k] = ΛkEa[1] into (ΛkT )0/(ΛkT )1 = gr0(ΛkT ). Since τ− ∈ T [−1], σ 7→ τ− ∧ σ is thus seen to yield an isomorphism of Ea1···ak [k + 1] with gr−1(Λk+1T ) and analogously for the other components. The tractor connection on Λk+1T is given by ∇Λk+1T c  ρa1···ak ϕa0···ak \| µa2···ak σa1···ak  =  Dcρa1···ak − P p c ϕpa1···ak − kPc[a1 µa2···ak]( Dcϕa0···ak + (k + 1)gc[a0 ρa1···ak] +(k + 1)Pc[a0 σa1···ak] ) \| ( Dcµa2···ak −P p c σpa2···ak + ρca2···ak ) Dcσa1···ak − ϕca1···ak + kgc[a1 µa2···ak].  . (14) The first BGG-splitting operator LΛk+1T 0 : E[a1···ak][k + 1] → Λk+1T is given by LΛk+1T 0 (σ) (15) =  ( − 1 n(k+1)D pDpσa1···ak + k n(k+1)D pD[a1 σ\|p\|a2···ak] + k n(n−k+1)D[a1 Dpσ\|p\|a2···ak] +2k n P p [a1 σ\|p\|a2···ak] − 1 nJσa1···ak ) D[a0 σa1···ak] \| − 1 n−k+1gpqDpσqa2···ak σa1···ak . 3.3 Almost Einstein scales The first splitting operator for the standard tractor bundle is LT0 : E [1] → Γ(T ), σ 7→  1 n(4− J)σ Dσ σ  . (16) By (10), ∇T ◦ LT0 (σ) =  1 nDc(4σ − Jσ)− P p c Dpσ (DaDbσ + Pabσ) + 1 n(4σ − Jσ)gab 0  . Since 1 n(4σ − Jσ)gab is minus the trace-part of (DaDbσ + Pabσ) and H1(T ) = E(ab)0 we have that the first BGG-operator of T is ΘT 0 : E [1] → E(ab)0 , σ 7→ (DaDbσ + Pabσ)0. By computing the change of the Schouten tensor P with respect to a conformal rescaling one obtains that with U = {x ∈M : σ(x) 6= 0}, (DaDbσ + Pabσ)0 = 0 ⇔ σ−2g is Einstein on U. (17) Conformal Structures Associated to Generic 2-Distributions 11 This says that P (σ−2g), or equivalently Ric(σ−2g), is a multiple of σ−2g on U . U always has to be an open dense subset of M , and we call the set of solutions of (17) the space of almost Einstein scales [24], i.e. aEs([g]) = ker ΘT 0 ⊂ E [1]. (18) It turns out to be a differential consequence of (17) that 1 nDc(4σ − Jσ) = P p c Dpσ, and thus one has the well known fact Proposition 1. ∇T -parallel sections of the standard tractor bundle are in 1:1-correspondence with aEs([g]). 3.4 Conformal Killing forms Via (14) and (15) one computes that for σ ∈ E[a1···ak][k+1] the projection of ∇Λk+1T ◦LΛk+1T 0 (σ) to the lowest slot Ec[a1···ak][k + 1] in Ω1(M,Λk+1T ) is given by σa1···ak 7→ Dcσa1···ak −D[a0 σa1···ak] − k n− k + 1 gpqDpσqa2···ak . This is the projection of σa1···ak to the highest weight part of Ec[a1···ak][k+ 1] which is formed by trace-free elements with trivial alternation, we write E{c[a1···ak]}0 [k + 1] := {σa1···ak : 0 = σ[ca1···ak] and 0 = gca1σca1···ak }. One computes that in fact HΛk+1T 1 = E{c[a1···ak]}0 [k + 1] and obtains the first BGG-operator ΘΛk+1T 0 : E[a1···ak][k + 1] → E{c[a1···ak]}0 [k + 1], σ 7→ D{cσa1···ak}0 . Forms in the kernel of ΘΛk+1T 0 are thus the conformal Killing k-forms. Unlike the case of k = 0, it is not true for k ≥ 1 that always ∇Λk+1T (LΛk+1T 0 σ) = 0 for σ ∈ ker ΘΛk+1T 0 ⊂ E[a1···ak][k + 1]. However, given a section s ∈ Γ(Λk+1T ) with lowest slot Π(s) = σ ∈ E[a1···ak], one has by construction of LΛk+1T 0 that s = LΛk+1T 0 σ and that ΘΛk+1T 0 σ = Π ◦ ∇Λk+1T ◦ LΛk+1T 0 = 0; i.e., parallel sections of Λk+1T do always project to special solutions of ΘΛk+1 0 σ = 0. These solutions were termed normal conformal Killing forms by F. Leitner [33]. Thus, by definition, normal conformal Killing k-forms are in 1:1-correspondence (via LΛk+1T 0 and Π) with ∇Λk+1T -parallel sections of Λk+1T . Denote the components of the splitting LΛk+1T 0 σ given in (15) by ρa1···ak ϕa0···ak \| µa2···ak σa1···ak  . (19) A normal conformal Killing form satisfies ∇Λk+1T (LΛk+1T 0 σ). By (14), the resulting equation in lowest slot just says that σ is a conformal Killing form. Additionally, we get the following equations for the components ϕ, µ, ρ: Dcρa1···ak − P p c ϕpa1···ak − kPc[a1 µa2···ak] = 0, Dcϕa0···ak + (k + 1)gc[a0 ρa1···ak] + (k + 1)Pc[a0 σa1···ak] = 0, (20) Dcµa2···ak − P p c σpa2···ak + ρca2···ak = 0. 12 M. Hammerl and K. Sagerschnig 4 Generic rank two distributions and associated conformal structures 4.1 The distributions Let M be a 5-manifold. We are interested in generic rank 2 distributions on M , i.e., rank 2 subbundles D ⊂ TM such that values of sections of D and Lie brackets of two such sections span a rank 3 subbundle [D,D] and values of Lie brackets of at most three sections span the entire tangent bundle TM (recall (1)). In other words, these are distributions of maximal growth vector (2, 3, 5) in each point. Defining T−1M := D, T−2M := [D,D] and T−3M = TM , the distribution gives rise to a filtration of the tangent bundle by subbundles compatible with the Lie bracket of vector fields in the sense that for ξ ∈ Γ(T iM) and η ∈ Γ(T jM) we have [ξ, η] ∈ Γ(T i+jM). Given such a filtration, the Lie bracket of vector fields induces a tensorial bracket L on the associated graded bundle gr(TM) = ⊕ gri(TM), where gri(TM) = T iM/T i+1M . This bundle map L : gr(TM) × gr(TM) → gr(TM) is called the Levi bracket. It makes the bundle gr(TM) into a bundle of nilpotent Lie algebras; the fiber (gr(TM)x,Lx) is the symbol algebra at the point x. Note that a rank 2 distribution D ⊂ TM is generic if and only if the symbol algebra at each point is isomorphic to the graded Lie algebra g = g−1⊕g−2⊕g−3, where dim(g−1) = 2, dim(g−2) = 1, dim(g−3) = 2, and the only non-trivial components of the Lie bracket, g−1 × g−2 → g−3 and Λ2g−1 → g−2, define isomorphisms. Generic rank 2 distributions in dimension 5 arise from ODEs of the form z′ = F (x, y, y′, y′′, z) (21) with ∂2F ∂(y′′)2 6= 0 where y and z are functions of x, see e.g. [36] for that viewpoint. In his famous five-variables paper [19] from 1910, Élie Cartan associated to these distributions a canonical Cartan connection, a result that shall be stated more precisely in Section 4.4. 4.2 Some algebra: G2 in SO(3, 4) Let us recall one of the possible definitions of an exceptional Lie group of type G2. It is well known (e.g. [3]) that the natural GL(7,R)-action on the space Λ3R7∗ of 3-forms on R7 has two open orbits and that the stabilizer of a 3-form in either of these open orbits is a 14-dimensional Lie group. For one of these orbits it is a compact real form of the complex exceptional Lie group G2, and for the other orbit it is a split real form. To distinguish between the two open orbits, consider the bilinear map R7 × R7 → Λ7R7∗, (X,Y ) 7→ iXΦ ∧ iY Φ ∧ Φ, associated to a 3-form Φ. This bilinear map is non-degenerate if and only if Φ is contained in an open orbit. In that case, it determines an invariant volume form vol on R7 given by the root 9 √ D ∈ Λ7(R7∗) of its determinant D ∈ (Λ7R7∗)9, see e.g. [30]. Hence H(Φ)(X,Y )vol := iXΦ ∧ iY Φ ∧ Φ (22) defines a R-valued bilinear form H(Φ) on R7 which is invariant under the action of the stabilizer of Φ. It turns out that H(Φ) is positive definite if the stabilizer is the compact real form, and it has signature (3, 4) if the stabilizer is the split real form of G2. In the sequel, let G = G2 be the stabilizer of a 3-form Φ ∈ Λ3R7∗ such that the associated bilinear form H(Φ) has signature (3, 4). The above discussion implies that this G2 naturally includes into the special orthogonal group G̃ = SO(3, 4), an observation which will be crucial for what follows. Conformal Structures Associated to Generic 2-Distributions 13 Let us be more explicit and realize SO(3, 4) as SO(h), with h = 0 0 1 0 M2,3 0 1 0 0  , M2,3 =  0 0 0 1 0 0 0 0 0 1 0 0 −1 0 0 1 0 0 0 0 0 1 0 0 0  . Via this bilinear form we identify R7∗ ∼= R7. Consider the standard basis e1, . . . , e7 on R7, and define G2 as the stabilizer of Φ ∈ Λ3R7, Φ := − 1√ 3 e7 ∧ e2 ∧ e3 + 1√ 6 e5 ∧ e4 ∧ e2 + 1√ 6 e6 ∧ e4 ∧ e3 − 1√ 6 e7 ∧ e4 ∧ e1 − 1√ 3 e1 ∧ e5 ∧ e6. (23) Then, via the identification Λ3R7∗ ∼= Λ3R7, H(Φ)(X,Y ) = 1√ 6 h(X,Y ), (24) and this equation characterizes the SO(h)-conjugacy class of G2. That is, Φ has SO(h)-stabilizer conjugated to G2 if and only if H(Φ) is some non-zero multiple of h. The Lie algebra so(h) has the matrix representation (6). It contains the Lie algebra g of G2, which is formed by elements M ∈ gl(7,R) such that Φ(Mv, v′, v′′) + Φ(v,Mv′, v′′) + Φ(v, v′,Mv′′) = 0, as the subalgebra consisting of matrices tr(A) Z s W 0 X A √ 2JZt s√ 2 J −W t r − √ 2XtJ 0 − √ 2ZJ s Y − r√ 2 J √ 2JX −At −Zt 0 −Y t r −Xt −tr(A)  (25) with A ∈ gl(2,R), X,Y ∈ R2, Z,W ∈ R2∗, r, s ∈ R and J = ( 0 −1 1 0 ) . For later use, let us note here that the complement of g in so(h) with respect to the Killing form is isomorphic to the seven dimensional standard representation of G2. That means we have a G2-module decomposition so(h) = g⊕ R7. The sequence 0 → g ↪→ so(h) iΦ→ R7 → 0 (26) is G2-equivariant and exact. Here iΦ : so(h) = Λ2R7 → R7 is the insertion of so(h) into Φ. The factor of Φ as given in (23) was chosen such that the insertion iΦ : R7 → Λ2R7 = so(h) (27) splits sequence (26). Next we consider parabolic subgroups in G2 and SO(h). Let e1 ∈ R7 be the first basis vector in the standard representation. Then the isotropy group of the ray R+e1 is a parabolic 14 M. Hammerl and K. Sagerschnig subgroup P̃ in SO(h), and the intersection P = P̃ ∩ G2 is a parabolic subgroup in G2. To describe explicitly the corresponding parabolic subalgebra p ⊂ g, we introduce vector space decompositions of the Lie algebra. We consider the block decomposition g0 g1 g2 g3 0 g−1 g0 g1 g2 g3 g−2 g−1 0 g1 g2 g−3 g−2 g−1 g0 g1 0 g−3 g−2 g−1 g0  , of matrices (25); this defines a grading g = g−3 ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ g3. Note that the subalgebra g− = g−3⊕g−2⊕g−1 coincides with the symbol algebra of a generic rank two distribution in dimension five as explained in Section 4.1. The grading induces a filtration g3 ⊂ g2 ⊂ g1 ⊂ g0 ⊂ g−1 ⊂ g−2 ⊂ g−3, which is preserved by the action of P on g. The subalgebra p = g0 is the Lie algebra of the parabolic P , and the subalgebra g0 ∼= gl(2,R) is the Lie algebra of the subgroup G0 ⊂ P that even preserves the grading. The subgroup G0 is isomorphic to GL+(2,R) = {M ∈ GL(2,R) : det(M) > 0}. 4.3 The homogeneous model and associated Cartan geometries Let us look at the Lie group quotient G2/P next. The action of G2 on the class eP̃ ∈ SO(h)/P̃ induces a smooth map G2/P → SO(h)/P̃ . Since both homogeneous spaces have the same dimension, the map is an open embedding. Since G2/P is a quotient of a semisimple Lie group by a parabolic subgroup, it is compact, and the map is in fact a diffeomorphism. The group SO(h) acts transitively on the space of null-rays in R7, which can be identified with the pseudo-sphere Q2,3 ∼= S2 × S3. It turns out that the metric h on R7 defined in (5) induces the conformal class of (g2,−g3) on Q2,3, with g2, g3 being the round metrics on S2 respectively S3. The pullback of that conformal structure yields a G2-invariant conformal structure on G2/P . Explicit descriptions of the canonical rank two distribution on Q2,3 ∼= S2 × S3 can be found in [38]. In an algebraic picture the distribution corresponds to the P -invariant subspace g−1/p ⊂ g/p. Via the identification of T (G/P ) with G×P g/p, this invariant subspace gives rise to a rank two distribution, which is generic in the sense of Section 4.1. More generally, suppose (G, ω) is any parabolic geometry of type (G2, P ). Recall from Sec- tion 2 that the Cartan connection ω defines an isomorphism TM ∼= G ×P g/p. Hence, for any such geometry, the subspace g−1/p gives rise to a rank two distribution. This distribution will be generic if a regularity condition on the Cartan connection is assumed; we shall introduce this condition next: Let T−1M ⊂ T−2M ⊂ TM be the sequence of subbundles of constant ranks 2, 3 and 5 coming from the P -invariant filtration g−1/p ⊂ g−2/p ⊂ g/p. Consider the associated graded bundle gr(TM). This bundle can be naturally identified with G ×P gr(g/p) ∼= (G/P+)×G0 g−. Since the Lie bracket on the nilpotent Lie algebra g− is invariant under the G0-representation, it induces a bundle map {, } : gr(TM)× gr(TM) → gr(TM), the algebraic bracket. A Cartan connection form ω is said to be regular if Conformal Structures Associated to Generic 2-Distributions 15 the filtration T−1M ⊂ T−2M ⊂ TM is compatible with the Lie bracket of vector fields and the algebraic bracket coincides with the Levi bracket of the filtration. But this precisely means that the rank two subbundle D := T−1M is generic and T−2M = [D,D] (compare with Section 4.1 and the structure of g−). Regularity can be expressed as a condition on the curvature of a Cartan connection. Since g has a P -invariant filtration, we have a notion of maps in Λk(g/p)∗ ⊗ g of homogeneous degree ≥ l, and the set of these maps is P -invariant. A Cartan connection form is regular if and only if the curvature function is homogeneous of degree ≥ 1; this means that κ(u)(gi, gj) ⊂ gi+j+1 for all i, j and u ∈ G. Note that if the curvature function takes values in Λ2(g/p)∗ ⊗ p, i.e. the geometry is torsion-free, then it is regular. Now we can state Cartan’s classical result in modern language. In this paper we restrict our considerations to orientable distributions. Equivalently, this means that the bundle TM be orientable. Then we have the following: Theorem 2 ([19]). One can naturally associate a regular, normal parabolic geometry (G, ω) of type (G2, P ) to an orientable generic rank two distribution in dimension five, and this establishes an equivalence of categories. The above discussion explains that a regular parabolic geometry of type (G2, P ) determines an underlying generic rank two distribution D, and (for our choice of P ) the distribution turns out to be orientable. The converse is shown in two steps: First one constructs a regular parabolic geometry inducing the given distribution. Next one employs an inductive normalization proce- dure based on the proposition below, which we state explicitly here, since it will be needed it in Proposition 4. Proposition 2 ([15]). Let (G, ω) be a regular parabolic geometry with curvature function κ, and suppose that ∂∗κ is of homogeneous degree ≥ l for some l ≥ 1. Then, there is a normal Cartan connection ωN ∈ Ω1(G, g) such that (ωN − ω) is of homogeneous degree ≥ l. In the proposition the difference ωN − ω, which is horizontal, is viewed as a function G → (g/p)∗ ⊗ g, and the homogeneity condition employs the canonical filtration of (g/p)∗ ⊗ g. 4.4 A Fefferman-type construction The relation in Section 4.3 between the homogeneous models G2/P and SO(h)/P̃ suggests a relation between Cartan geometries of type (G2, P ) and (SO(h), P̃ ), i.e., generic rank two distributions and conformal structures. Indeed, it was P. Nurowski who first observed in [36] that any generic rank two distribution on a five manifold M naturally determines a conformal class of metrics of signature (2, 3) on M . Starting from a system (21) of ODEs, he explicitly constructed a metric from the conformal class. A different construction of such a metric can be found in [14]. In the present paper, we shall discuss Nurowski’s result as a special case of an extension functor of Cartan geometries, see [17, 8, 20]. Let i′ : g ↪→ so(h) denote the derivative of the inclusion i : G2 ↪→ SO(h). Given a Cartan geometry (G →M,ω) of type (G2, P ), we can extend the structure group of the Cartan bundle, i.e., we can form the associated bundle G̃ = G ×P P̃ . Then this is a principal bundle over M with structure group P̃ . We have a natural inclusion j : G ↪→ G̃ mapping an element u ∈ G to the class [(u, e)]. Moreover, we can uniquely extend the Cartan connection ω on G to a Cartan connection ω̃ ∈ Ω1(G̃, so(h)) such that j∗ω̃ = i′ ◦ ω. The construction indeed defines a functor from Cartan geometries of type (G2, P ) to Cartan geometries of type (G̃, P̃ ). We will later need the relation between the curvatures of ω̃ and ω, which is discussed in the next lemma. We use the inclusion of adjoint tractor bundles AM ↪→ ÃM via AM = G ×P g ↪→ G ×P so(h) = G̃ ×P̃ so(h) = ÃM. 16 M. Hammerl and K. Sagerschnig Lemma 2. 1. The curvature form Ω̃ ∈ Ω2(G̃, so(h)) of ω̃ pulls back to the curvature form Ω ∈ Ω2(G, g) of ω: j∗(Ω̃) = Ω. 2. The factorizations K ∈ Ω2(M,AM) of Ω and K̃ ∈ Ω2(M, ÃM) of Ω̃ agree: K̃ = K ∈ Ω2(M,AM). In particular, K ∈ Ω2(M, ÃM) has values in AM ⊂ ÃM . Proof. Since the exterior derivative d is natural, it commutes with pullbacks: j∗dω̃ = d(j∗ω̃) = dω. Since also j∗([ω̃, ω̃]) = [ω, ω], we thus see that by definition of curvature (3) we have j∗Ω̃ = Ω. Now the inclusion j : G → G̃ is a reduction of structure group from P̃ to P . Therefore, factorizing Ω̃ ∈ Ω2 hor(G̃, so(h))P̃ to the curvature form K̃ ∈ Ω2(M, ÃM) is the same as pulling back Ω̃ via j and then factorizing. � By Theorem 2, we can associate a canonical Cartan geometry (G, ω) of type (G2, P ) to a generic rank two distribution on a five manifold M . As discussed in Section 3, any Cartan geometry (G̃, ω̃) of type (G̃, P̃ ) determines a conformal structure on the underlying manifold M . Thus the above Fefferman construction shows that a generic rank two distribution D naturally determines a conformal class [g] of metrics of signature (2, 3). However, a priori we do not know whether ω̃ is the normal Cartan connection associated to that conformal structure (which will be important for applications); to see this requires a proof. We will give a proof based on the following result which is derived via BGG-techniques: Proposition 3 ([7]). Suppose E ⊂ ker(∂∗) ⊂ Λ2(g/p)∗ ⊗ g is a P -submodule, and consider the G0-module E0 := E ∩ ker(�). Let (G →M,ω) be a regular, normal parabolic geometry such that the harmonic curvature takes values in E0. If either ω is torsion-free or for any φ, ψ ∈ E we have ∂∗(iφψ) ∈ E, where iφψ is the alternation of the map (X0, X1, X2) 7→ φ(ψ(X0, X1)+p, X2) for Xi ∈ g/p, the curvature function κ has takes in E. Kostant’s version of the Bott–Borel–Weil theorem [31] provides an algorithmic description of the G0-representation ker(�). Doing the necessary calculations for the \|3\|-graded Lie algebra g of G2 from Section 4.2 (or consulting [40], if you want the computer to calculate for you) leads to: Lemma 3. The harmonic component κH of a regular, normal parabolic geometry of type (G2, P ) takes values in a G0 = GL+(2,R)-submodule of g1 ∧ g3 ⊗ g0 isomorphic to S4(R2)∗. It follows, that the geometry is torsion-free. Proof. An algorithmic calculation shows that the G0 = GL+(2,R)-representation ker(�) is an irreducible summand of g1 ∧ g3 ⊗ g0 isomorphic to S4(R2)∗. Since κH takes values in ker(�), it is, in particular, contained in Λ2p+ ⊗ p. Now we can apply Proposition 3 to conclude that the entire curvature κ takes values in that P -module, i.e., the geometry is torsion-free. � We now show that normality of ω implies normality of ω̃: Proposition 4. Let (G → M,ω) be a regular normal parabolic geometry of type (G2, P ), and let (G̃ → M, ω̃) be the associated parabolic geometry of type (G̃, P̃ ). Then ω̃ ∈ Ω1(G̃, so(h)) is normal. Conformal Structures Associated to Generic 2-Distributions 17 Proof. The Killing forms of g resp. so(h) provide natural identifications p+ ∼= (g/p)∗ and p̃+ ∼= (so(h)/p̃)∗. This allows us to view the curvature function of the geometry (G, ω) as a function κ : G → Λ2p+ ⊗ g and the curvature function of (G̃, ω̃) as a function κ̃ : G̃ → Λ2p̃+ ⊗ so(h). The inclusion i′ : g → so(h) and the isomorphism g/p ∼= so(h)/p̃ induce a map I : Λ2p+⊗g → Λ2p̃+⊗so(h). Since all maps involved in the construction are homomorphisms of P -modules, this is indeed a homomorphism of P -modules. In terms of I, the curvature functions are related via κ̃(j(u)) = I ◦ κ(u) (28) for all u ∈ G. By equivariance, this uniquely determines κ̃ on G̃. Let ∂̃∗ : Λ2p̃+ ⊗ so(h) → p̃+ ⊗ so(h) be the Kostant codifferential describing the conformal normalization condition, i.e., on decomposable elements U ∧ V ⊗A ∈ Λ2p̃+ ⊗ so(h) we have ∂̃∗(U ∧ V ⊗A) = U ⊗ [V,A]− V ⊗ [U,A]. (29) To prove that the geometry (G̃, ω̃) is normal amounts to showing that κ̃ takes values in ker(∂̃∗). By (28), this is equivalent to the fact that κ takes values in ker(∂̃∗ ◦ I). Proposition 3 allows to reduce this remaining problem to a purely algebraic one: Since, by Lemma 3, regular, normal parabolic geometries of type (G2, P ) are torsion-free, Proposition 3 implies that κ takes values in ker(∂̃∗ ◦I) if and only if this is true for the harmonic curvature component κH . By equivariance, the map ∂̃∗ ◦ I either vanishes on G0-irreducible components, or it is an isomorphism. We have observed that ker(�) is an irreducible G0-representation isomorphic to S4(R2)∗. Hence if we can show that the image of the map ∂̃∗ ◦ I : Λ2p+⊗ g0 → p̃+⊗ so(h) does not contain an irreducible summand isomorphic to S4(R2)∗, then κH has to be contained in that kernel. But looking at formula (29), we see that ∂̃∗ ◦ I(p+ ⊗ g0) is actually contained in p̃+ ⊗ p̃+, which is easily seen to contain no summand isomorphic to S4(R2)∗. � 4.5 The parallel tractor three-form and the underlying conformal Killing 2-form Let T be the standard tractor bundle for a conformal structure [g] associated to a generic 2-distribution D on a 5-manifold M . Then T is easily seen to carry additional structure: Proposition 5. 1. The standard tractor bundle T for a conformal structure [g] associated to a generic 2- distribution carries a parallel tractor 3-form Φ ∈ Γ(Λ3T ∗) = Γ(Λ3T ). 2. The tractor 3-form Φ determines an underlying normal conformal Killing 2-form φ ∈ Γ(Λ2T ∗M ⊗ E [3]) = E[ab][3], which is locally decomposable. Proof. 1. By construction, the conformal Cartan bundle is the associated bundle G̃ = G ×P P̃ , where G is the Cartan bundle for the distribution. Hence the tractor bundle can be viewed as T = G×P R7. It follows that the P -equivariant function fΦ : G → Λ3R7 mapping constantly onto the three-form Φ stabilized by G2 induces a section Φ ∈ Γ(Λ3T ). Proposition 4 implies that the normal tractor connection ∇Λ3T is induced from the normal Cartan connection ω ∈ Ω1(G, g). Hence, according to (4), ∇Λ3T ξ Φ corresponds to the function u 7→ (ξ′u · fΦ) + ωu(ξ′(u))(fΦ(u)), where u ∈ G and ξ′ ∈ X(G) is a P -invariant lift of a the vector field ξ ∈ X(M). Since fΦ is constant and ω takes values in the isotropy algebra g of Φ, this means that ∇Λ3T Φ = 0, i.e., Φ is a parallel tractor three-form. 2. Recall from Section 3.4 that we have a natural bundle projection Π0 : Λ3T → Λ2T ∗M⊗E [3] and that parallel sections of V := Λ3T project to normal conformal Killing 2-forms. 18 M. Hammerl and K. Sagerschnig Let V = Λ3R7 be endowed with the canonical P̃ -invariant filtration discussed in Section 3.2. We see from our explicit formula (23) for Φ that, up to a factor, a representative in V/V 0 = gr−1(V ) is given by e7 ∧ e2 ∧ e3. Around every point x ∈ M we can choose a local section σ : U → G. On σ(U) ⊂ G ⊂ G̃ we define 3 constant functions, mapping to e7, e2 and e3; these correspond to sections s7, s2 and s3 of the standard tractor bundle T . s7 is simply τ−; to be precise, we use that σ : U → G gives in particular a trivialization of the conformal weight bundles, and we can regard τ− as an (unweighted) section of T . The tractors s2 and s3 lie in T 0 and therefore project to elements ϕ2 and ϕ3 in T 0/T 1 = gr0(T ) = Ea, where we again use the trivialization of the conformal weight bundles. Thus, τ− ∧ ϕ2 ∧ ϕ3 is a representative of ϕ = Π0(Φ) ∈ V/V0 = gr−1(V), and the identification (13) of V/V0 = gr−1(V) with E[ab] tells us that ϕ = ϕ2 ∧ ϕ3 ∈ E[ab]. � Remark 1. The parabolic subgroup P of G2 preserves a finer filtration of the standard repre- sentation, which yields the following refinement of the ’conformal’ filtration of standard tractor bundle: T 2′ ⊂ T 1′ ⊂ T 0′ ⊂ T −1′ ⊂ T −2′ . The isotropic line bundle T 2′ corresponds to the subspace generated by e1 ∈ R7. The bundle T −1′ is the orthogonal complement to T 2′ with respect to the tractor metric. The explicit form of the three-form Φ (see (23)) shows how to characterize the additional filtration components in terms of Φ. Let τ+ ∈ Γ(T ⊗ E [1]) be the canonical insertion of E [−1] into T 2′ ⊂ T which was already defined in Section 3.1. Then the subbundle T 1′ can be described as the set of all tractors s ∈ Γ(T ) such that isiτ+Φ = 0. The subbundle T 0′ is the bundle orthogonal to T −1′ . Remark 2. The distribution D can be recovered from the conformal class [g] associated to the distribution and the conformal Killing 2-form φ. The kernel of the 2-form is the rank three distribution [D,D]. The rank two distribution can be recovered as the kernel of the restriction of a metric g to the rank three distribution, in other words, the set of isotropic elements in the kernel of φ. 5 Holonomy reduction and characterization The aim of the following section is to characterize conformal structures arising from generic rank 2 distributions in dimension five in terms of normal conformal Killing 2-forms satisfying certain additional equations. See Theorem A for a precise statement of the result. We proceed as follows. First, we prove that a conformal manifold of signature (2, 3) whose conformal holonomy is contained in G2 is obtained from a generic rank two distribution via a Fefferman construction. Then we aim for a characterization of the conformal structures in terms of underlying conformal data; we derive conditions to distinguish those normal conformal Killing 2-forms coming from parallel tractor 3-forms defining holonomy reductions to G2. This is done analogously to [12], where the authors arrive at a version of Sparling’s characterization [26] of Fefferman spaces in terms of a conformal Killing field. Remark 3. Recently T. Leistner and P. Nurowski showed on examples that in some cases the conformal structures constructed in this way have explicit ambient metrics with holonomy group G2, see [37] and [32]. 5.1 Conformal holonomy Let (M, [g]) be a conformal structure of signature (2, 3) encoded in a Cartan geometry (G̃, ω̃) as described in Section 3. The standard tractor bundle T of [g] is endowed with the tractor Conformal Structures Associated to Generic 2-Distributions 19 connection ∇T , and we define the conformal holonomy of [g] as Hol([g]) := Hol(∇T ). See also [2]. Now T comes about as associated bundle to G̃′ := G̃ ×P̃ SO(h) and ∇T is induced from the principal connection form ω̃′ ∈ Ω1(G̃′, so(h)). Thus, we have that Hol(∇T ) = Hol(ω̃′). By construction, the pullback of ω̃′ to G̃ ⊂ G̃′ is the Cartan connection form ω̃. In the Fefferman-type construction of Section 4.4 we started with a parabolic geometry (G, ω) of type (G2, P ) encoding a generic rank 2 distribution D, and we associated to this the parabolic geometry (G̃, ω̃) of type (SO(h), P̃ ) by equivariantly extending ω to ω̃. If we add the extended bundles G′ = G ×P G2 and G̃′ = G̃ ×P̃ SO(h) = G ×P SO(h) to the picture, we obtain the commuting diagram of inclusions (G′, ω′) � � // (G̃′, ω̃′) (G, ω) ?� OO � � // (G̃, ω̃) ?� OO In particular, this yields a holonomy reduction of (G̃′, ω̃′) to (G′, ω′), and thus Hol(ω̃′) = Hol(ω′) ⊂ G2. By Proposition 4, ω̃ is normal. Hence, Hol(ω̃′) is indeed the conformal holonomy Hol([g]D), which is thus seen to be contained in G2. We are now going to show the converse: if for a conformal structure (M, [g]) of signature (2, 3) the conformal holonomy Hol([g]) is contained in G2, then there is a canonical generic rank 2- distribution D on M such that [g] = [g]D. Let π : G̃′ → M be the surjective submersion of the SO(h)-principal bundle G̃′. The next proposition covers the holonomy reduction of a conformal Cartan geometry to G2. A similar result has been obtained in [1]. Proposition 6. Let (G̃, ω̃) be such that (G̃′, ω̃′) has holonomy in G2 and let H ⊂ G̃′ be a reduction of (G̃′, ω̃′) to G2. Then: 1. H ⊂ G̃′ and G̃ ⊂ G̃′ intersect transversally. We denote the resulting submanifold by G := H ∩ G̃. 2. For every u ∈ G, Tuπ(TuG) = Tπ(u)M . 3. G is a P -principal bundle over M . 4. Let ω be the pullback of ω̃ ∈ Ω1(G, so(h)) to G. Then ω ∈ Ω1(G, g) is a Cartan connection form. Proof. 1. We have that TuH + TuG̃ ⊃ u · g + u · p̃ = u · so(h) = ker (Tuπ). Since Tuπ : TuG̃ → Tπ(u)M is surjective, we have that dim(TuH+ TuG̃) = dim(so(h)) + dim(g/p) = dim(TuG̃′). 2. Take u ∈ G = H∩ G̃ and ξ ∈ Tπ(u)M . Since the restrictions of π to H and G̃ are surjective submersions, there exist ξ1 ∈ TuH and ξ2 ∈ TuG̃ such that ξ = Tuπξ1 = Tuπξ2. Then ξ1 − ξ2 ∈ ker Tuπ = u · so(h) = u · g + u · p̃. Thus, there exist η1 ∈ u · g and η2 ∈ u · p̃ such that ξ1 − ξ2 = η1 + η2. Let ξ′ = ξ1 − η1 = ξ2 + η2 ∈ TuG. Then indeed Tuπξ ′ = ξ. 3. Assume first that for an x ∈ M there is a u ∈ Hx ∩ G̃x = Gx. Then evidently Gx = u · (G2 ∩ P̃ ) = u ·P . It therefore remains to show that Hx ∩ G̃x is always non-empty: let u ∈ Hx. 20 M. Hammerl and K. Sagerschnig Then there is a g ∈ SO(h) such that u · g ∈ G̃. Since G2/P = SO(h)/P̃ (see Section 4.2), there is a p ∈ P̃ such that gp = g′ ∈ G2, and then u · g′ ∈ H since H is a G2-subbundle and u · g′ = (u · g) · p ∈ SO(h); i.e., u · g′ ∈ G. 4. We now consider G as a reduction of the P̃ -principal bundle G̃ to P and denote by ω the pullback of ω̃ ∈ Ω1(G̃, so(h)). By construction, H ⊂ G was obtained by holonomy reduction of (G̃′, ω̃′) to G2. In particular, ω̃′\|TH has values in g, and thus ω ∈ Ω1(G, g). P -equivariance and reproduction of p-fundamental vector fields is clear since G is just a P -principal subbundle of G̃ and ω̃ is a Cartan connection form satisfying (C.1)–(C.2) by assumption. We thus need to check that also (C.3) holds for ω. I.e., for every u ∈ G we need that ωu : TuG → g is an isomorphism. We have seen that Tuπ(TuG) = Tπ(u)M . Since u · p̃ = ker (Tuπ) ⊂ TuG̃ we see that TuG ⊂ TuG̃ must span at least dim(g/p)-complementary dimensions and thus already TuG + u · p = TuG̃. But then ωu(TuG/u · p) = ω̃u(Tug̃/u · p̃) = g̃/p̃ = g/p. This, together with ωu(u · p) = p by reproduction of fundamental vector fields, gives that indeed ωu(TuG) = g. � Now suppose (G̃, ω̃) is a normal parabolic geometry of type (G̃, P̃ ) associated to [g] with Hol([g]) ⊂ G2, and let (G, ω) be the parabolic geometry of type (G2, P ) obtained via reduction as explained in Proposition 6. Since every normal conformal Cartan connection is torsion-free and p ⊂ p̃, ω is torsion-free. This evidently implies that ω is regular, and therefore it determines an underlying generic 2-distribution. We will show in Theorem 3 that the canonical conformal structure associated to this distribution is [g]; this will employ the following Lemma 4. Let (G̃, ω̃) be a normal parabolic geometry of type (G̃, P̃ ) with Hol(ω̃′) ⊂ G2, and let (G, ω) be the parabolic geometry of type (G2, P ) obtained via reduction. Then there is a normal Cartan connection ωN ∈ Ω1(G, g) such that the difference (ωN−ω) is of homogeneous degree ≥ 3. Proof. Let κ : G → Λ2p+ ⊗ g be the curvature function of ω. We will verify that κ is of homogeneous degree ≥ 3. Since the Kostant codifferential preserves homogeneities, then also ∂∗κ is of homogeneous degree ≥ 3, and by Proposition 2 this shows that there is a normal Cartan connection ωN that differs from ω at most in homogeneous degree ≥ 3. Torsion-freeness of ω means that for any u ∈ G, κ(u) is contained in Λ2p+ ⊗ p. Hence the only component of κ(u) of homogeneous degree < 3 that remains to be investigated is κ2(u) ∈ g1 ⊗ g1 ⊗ g0. We show that this component vanishes as well. Let ∂̃∗ : Λ2p̃+⊗ so(h) → p̃+⊗ so(h) be the conformal Kostant codifferential. Choose linearly independent elements X1, X2 ∈ g−1, X3 ∈ g−2 and X4, X5 ∈ g−3; then these elements give a basis of so(h)/p̃. Consider the dual basis Z̃1, . . . , Z̃5 ∈ p̃+ with respect to the Killing form on so(h). By construction, κ(u)(Xi, Xj) = κ̃(u)(Xi, Xj) for all u ∈ G. Thus normality of the conformal Cartan connection implies∑ i<j ([Z̃i, κ(u)(Xi, Xj)]⊗ Z̃j − [Z̃j , κ(u)(Xi, Xj)]⊗ Z̃i) = 0. (30) Recall that we have a g-module decomposition so(h) = g⊕V. The projection πg : so(h) → g maps Z̃j to an element Zj ∈ p+ dual to Xj ∈ g− with respect to a multiple of the Killing form on g. Equivariance of the projection implies πg([Z̃i, κ(u)(Xi, Xj)]) = [Zi, κ(u)(Xi, Xj)]. It follows that (30) also holds with Z̃i’s replaced by Zi’s. The only part of that sum contained in lowest homogeneity, i.e. in g1 ⊗ g1, is [Z1, κ2(u)(X1, X2)]⊗ Z2 − [Z2, κ2(u)(X1, X2)]⊗ Z1, and so this expression vanishes as well. Since the representation of g0 on g1 given by the Lie bracket is faithful, this indeed implies that κ2(u)(X1, X2) = 0. � Conformal Structures Associated to Generic 2-Distributions 21 Having Proposition 6 and Lemma 4, we can now show: Theorem 3. Let (M, [g]) be a conformal structure of signature (2, 3) with conformal holonomy Hol([g]) ⊂ G2. Then [g] is canonically associated to a generic rank two distribution D via a Fefferman-type construction. Proof. Let (G̃, ω̃) be the normal parabolic geometry of type (SO(h), P̃ ) associated to the con- formal structure [g]. Let (G, ω) be the parabolic geometry of type (G2, P ) constructed in Propo- sition 6. Then we know that ω is regular, and by Lemma 4 there is a normal Cartan connection ωN ∈ Ω1(G, g) that differs from ω at most in homogeneous degree ≥ 3. Recall that the Cartan connection ω determines an isomorphism G×P g/p ∼= TM . Regularity of ω implies that the image of G×P g−1/p under this isomorphism is a generic rank two distribu- tion D. Furthermore, we have a P -invariant conformal class of bilinear forms of signature (2, 3) on g/p, and the conformal structure induced via the above isomorphism on M is just [g]. On the other hand, the Fefferman construction associates a conformal structure [g]D to the distri- bution D. This is the conformal structure induced via the isomorphism G×P g/p ∼= TM defined by the normal Cartan connection ωN ∈ Ω1(G, g) associated to the distribution D. Since ωN −ω is of homogeneous degree ≥ 3, the difference (ω−ωN ) takes values in p. But this implies that ω and ωN induce the same isomorphism TM ∼= G ×P g/p and hence the same conformal structure on M ; i.e., the conformal structure [g] is the one induced by the distribution D: [g] = [g]D. � 5.2 Characterization via the tractor 3-form We have seen that conformal structures associated to generic rank two distributions in dimension five precisely correspond to reductions in conformal holonomy from SO(h) to G2. As a next step towards the desired characterization result, we explain that such a holonomy reduction can be encoded in terms of a parallel tractor 3-form satisfying a certain compatibility condition with the tractor metric. Let G̃′ be the extended SO(h)-principal bundle over M , and let ω̃′ ∈ Ω1(G̃′, so(h)) be the extension of the Cartan connection to a principal connection form. We consider the holonomy group Holu = Holu(ω̃′) for an arbitrary point u ∈ G̃′. Then, for g ∈ SO(h), one has Holu·g = gHolug−1, and Hol([g]) is well defined up to conjugation in SO(h). Let Hu ↪→ G̃′, u ∈ G̃′, be the reduction of the SO(h)-bundle G̃′ to Holu. If Ψ ∈ Λ3T is parallel, it corresponds to a SO(h)-equivariant function f : G̃′ → Λ3R7 which is a constant Ψu ∈ Λ3R7 on Hu. Hence Holu ·Ψu = Ψu, or Holu ⊂ SO(h)Ψu . If u′ is another point in G̃′ one has Holu′ = gHolug−1 for some g ∈ SO(h) and Ψu′ = g ·Ψu. Thus f(G̃′) = SO(h) ·Ψu. We say that SO(h)Ψu is the orbit type of the parallel tractor Ψ. To be precise, the orbit type is defined up to conjugation in SO(h). Recall that the group G2 ⊂ SO(h) has been realized as the stabilizer SO(h)Φ for Φ ∈ Λ3R7 given by (23). Compatibility condition (24) singles out the SO(h)-orbit of Φ ∈ Λ3R7. Hence Hol([g]) reduces to G2 if and only if there is a ∇Λ3T -parallel Φ ∈ Λ3T satisfying the global version of (24), i.e., H(Φ) = λh for a λ ∈ R\{0}, (31) where, for s1, s2 ∈ Γ(T ), H(Φ)(s1, s2)vol = is1Φ ∧ is2Φ ∧Φ, (32) and vol ∈ Λ7T ∗ is the tractor volume form. 22 M. Hammerl and K. Sagerschnig 5.3 Characterization in terms of the underlying conformal Killing 2-form We want to express compatibility condition (31) of the ∇Λ3T -parallel tractor Φ ∈ Λ3T in terms of the underlying normal conformal Killing 2-form φ = Π0(Φ) ∈ Γ(Λ2T ∗M ⊗ E [3]). According to (19) we have Φ =  ρa1a2 ϕa0a1a2 \| µa2 φa1a2  =  ( − 1 15D pDpφa1a2 + 2 15D pD[a1 φ\|p\|a2] + 1 10D[a1 Dpφ\|p\|a2] +4 5P p [a1 φ\|p\|a2] − 1 5Jφa1a2 ) D[a0 φa1a2] \| − 1 4gpqDpφqa2 φa1a2  . We will also write this splitting as Φ = τ− ∧ φ+ ϕ+ τ+ ∧ τ− ∧ µ+ τ+ ∧ ρ =  ρ ϕ \| µ φ  . (33) By (20) the conditions for a conformal Killing 2-form φ ∈ Ω2(M)⊗ E [3] to be normal are Dcρa1a2 − P p c ϕpa1a2 − 2Pc[a1 µa2] = 0, Dcϕa0a1a2 + 3gc[a0 ρa1a2] + 3Pc[a0 φa1a2] = 0, Dcµa2 − P p c φpa2 + ρca2 = 0. We now consider the map H : Λ3T → S2T ∗ defined in (32). As a SO(h)-representation S2R7∗ decomposes into the irreducible components S2 0R7∗ of trace-free symmetric 2-forms and the space Rh of multiples of h. The corresponding decomposition on the tractor level is S2T ∗ = S2 0T ∗ ⊕ Rh. Accordingly, H(Φ) decomposes into H(Φ)0 and H(Φ)tr. Compatibility condition (31) then means that H(Φ)0 = 0 and H(Φ)tr 6= 0. Lemma 5. A parallel tractor 3-form Φ satisfies H(Φ)0 = 0 if and only if φ ∧ φ ∧ µ = 0. In particular, H(Φ)0 = 0 whenever φ is locally decomposable. Proof. S2 0T ∗ is the tractor-bundle associated to the irreducible representation of SO(h) on S2 0Rn+2∗. By assumption, Φ is ∇Λ3T -parallel. The mapping Φ 7→ H(Φ) 7→ H(Φ)0 is algebraic, and thus naturality of the tractor connection implies that H(Φ)0 is ∇S2 0T ∗-parallel. The section H(Φ)0 ∈ Γ(S2 0T ∗) can thus by Lemma 1 be recovered via the BGG-splitting operator LS2 0T ∗ 0 from its projection to H0(S2 0T ∗) = E [2]. This projection is achieved by inserting twice the top slot τ+ into H(Φ)0, but since h(τ+, τ+) = 0 this is the same as evaluating H(Φ)(τ+, τ+). Now according to (32) H(Φ)(τ+, τ+)vol = (iτ+(Φ)) ∧ (iτ+(Φ)) ∧Φ. Using the representation (33) with respect to a metric g ∈ [g], we have iτ+Φ = φ− τ+ ∧ µ and thus H(Φ)(τ+, τ+)vol = (φ− τ+ ∧ µ) ∧ (φ− τ+ ∧ µ) ∧ (τ− ∧ φ+ ϕ+ τ+ ∧ τ− ∧ µ+ τ+ ∧ ρ) = φ ∧ φ ∧ τ+ ∧ τ− ∧ µ− τ+ ∧ µ ∧ φ ∧ τ− ∧ φ− φ ∧ τ+ ∧ µ ∧ τ− ∧ φ = 3τ+ ∧ τ− ∧ φ ∧ φ ∧ µ. This vanishes if and only if φ ∧ φ ∧ µ = 0. � Conformal Structures Associated to Generic 2-Distributions 23 Assume now that H(Φ)0 vanishes, i.e. H(Φ) = H(Φ)tr = λh, and since 0 = ∇S2T ∗0 (λh) = (dλ)h we have that λ ∈ R is a constant. Lemma 6. Suppose that H(Φ)0 = 0. Then, H(Φ) = λh for a constant λ ∈ R, λ 6= 0, if and only if φ ∧ µ ∧ ρ 6= 0. (34) Proof. We check that λ 6= 0 by inserting τ+, τ− since H(Φ)(τ+, τ−) = λh(τ+, τ−) = λ: H(Φ)(τ+, τ−)vol = (iτ+Φ) ∧ (iτ−Φ) ∧Φ = (φ− τ+ ∧ µ) ∧ (τ− ∧ µ+ ρ) ∧ (τ− ∧ φ+ ϕ+ τ+ ∧ τ− ∧ µ+ τ+ ∧ ρ) = φ ∧ τ− ∧ µ ∧ τ+ ∧ ρ+ φ ∧ ρ ∧ τ+ ∧ τ− ∧ µ− τ+ ∧ µ ∧ ρ ∧ τ− ∧ φ = 3τ+ ∧ τ− ∧ φ ∧ µ ∧ ρ. Thus, λ 6= 0 if and only if φ∧µ∧ ρ 6= 0. Note that this fixes the constant λ and φ∧µ∧ ρ either vanishes globally or nowhere. � We are now ready to prove Theorem A: Theorem A. Let [g] be a conformal class of signature (2, 3) metrics on M . Then [g] is induced from a generic rank 2 distribution D ⊂ TM if and only if there exists a normal conformal Killing 2-form φ which is locally decomposable and satisfies the genericity condition φ ∧ µ ∧ ρ 6= 0. Proof. Suppose [g] is induced from a generic rank 2 distribution D ⊂ TM . Then Proposition 5 shows that there is a parallel tractor 3-form Φ ∈ Γ(Λ3T ) that projects to a locally decomposable normal conformal Killing 2-form φ = Π0(Φ). Furthermore, the discussion in Section 5.2 shows that H(Φ) is a nonzero multiple of the tractor metric h. This implies φ∧µ∧ρ 6= 0 by Lemma 6. Conversely, suppose we have a locally decomposable normal conformal Killing 2-form φ ∈ Γ(Λ2T ∗M ⊗ E [3]) satisfying the genericity condition. According to Section 3.4, φ corresponds to a parallel tractor 3-form Φ given by (33). Lemmata 5 and 6 show that the assumptions on φ imply H(Φ) = λh, for λ 6= 0, and thus Hol([g]) ⊂ G2, as explained in Section 5.2. By Theorem 3, this means that the conformal structure is canonically associated to a 2-distribution D. � Remark 4. It is a well known consequence of the classical Plücker relations (cf. [21]) that a two form ϕ is locally decomposable if and only if ϕ ∧ ϕ vanishes globally. Remark 5. Throughout this paper we have assumed orientability of TM . This however, is only a minor point: If we leave this assumption and denote by O the 2-fold covering of M which is the orientation-bundle, we would obtain a twisted normal conformal Killing 2-form φ ∈ Γ(Λ2T ∗M ⊗ E [3]⊗O). 6 Decomposition of conformal Killing fields of [g]D The goal of this section is to prove Theorem B. We will show that every conformal Killing field of [g]D decomposes into a symmetry of the distribution D and an almost Einstein scale. The space of almost Einstein scales aEs([g]) was defined in (18) of Section 3.3. Now we discuss symmetries. Since D and [g] are equivalently described by Cartan geometries (G, ω) resp. (G̃, ω̃) we can determine their symmetry algebras by determining the symmetries of their corresponding Cartan 24 M. Hammerl and K. Sagerschnig geometries – in fact, one can define them in this way. For this purpose, we give the general description [9] of the Lie algebra of infinitesimal automorphisms of a parabolic geometry below in Section 6.2. Before this Cartan-geometric description, let us discuss the classical notions. An infinitesimal automorphism or symmetry of the distribution D ⊂ TM is a vector field on M whose Lie derivative preserves D, i.e., sym(D) = {ξ ∈ X(M) : Lξη = [ξ, η] ⊂ Γ(D) ∀ η ∈ Γ(D)}. (35) Remark 6. In this text we won’t show directly that the symmetries of the distribution sym(D) defined via (35) agree with the infinitesimal automorphisms inf(ω) of the corresponding Cartan geometry discussed below. We just use the fact that associating a regular normal parabolic geometry of type (G2, P ) to a generic rank 2-distribution D is an equivalence of symmetries. The explicit form of the splitting from vector fields on M into the adjoint tractor bundle relating the classical and the Cartan-viewpoint is only given in the conformal case, since there we will later need the explicit formula. 6.1 Conformal Killing fields The symmetries of the associated conformal structure [g] = [g]D are the conformal Killing fields cKf([g]) = {ξ ∈ X(M) : Lξg = e2fg for some g ∈ [g] and f ∈ C∞(M)}. Since Lξg decomposes into a multiple of g and a trace-free part one can equivalently demand that Lξg is pure trace. Now, with D the Levi-Civita connection of g ∈ [g], Lξg being pure trace is equivalent to D(cξa)0 = D(cga)0pξ p = 0; i.e., the symmetric, trace-free part of Dcξa vanishes. As an equation on 1-forms of conformal weight 2 this is in fact described by the first BGG- operator of Λ2T : By (15), the splitting operator LΛ2T 0 : X(M) = Ea[2] → Λ2T = ÃM is given by σa ∈ Ea[2] 7→  ( − 1 10D pDpσa + 1 10D pDaσp − 1 5Jσa +2 5P p aσp + 1 25DaD pσp ) D[cσa] \| − 1 5D pσp σa  . (36) Now the first BGG-operator ΘΛ2T 0 of Λ2T defined by the composition ΘΛ2T 0 : X(M) = Ea[2] → E(ab)0 [2], ΘΛ2T 0 = Π1 ◦ ∇Λ2T ◦ LΛ2T 0 is seen by direct calculation employing (14) to be ξa 7→ D(cξa)0 for ξ ∈ X(M); i.e., ΘΛ2T 0 is the conformally invariant operator governing conformal Killing fields. We now proceed to prove a technical lemma to be used in the proof of Theorem 4 below. It is a general fact for parabolic geometries satisfying a certain homological condition that parallel sections of the adjoint tractor bundle insert trivially into curvature [9], but it is easy to see this directly for conformal Killing fields, where this has first been observed in [23]. We only sketch a simple proof for conformal structures of dimension ≥ 4, which is all we need: Conformal Structures Associated to Generic 2-Distributions 25 Lemma 7. Let s ∈ Γ(ÃM) such that ∇Ãs = 0. Then K̃(Π(s), ·) = 0. Proof. Since ∇Ãs = 0 one has Rc1c2s = ∇Ã [c1 ∇Ã c2]s = 0, with R ∈ Ω2(M, gl(ÃM)) the curvature of ∇Ã. But since ∇Ã is the induced tractor connection of the Cartan connection form ω̃, one has that Rs is the algebraic action of K̃ ∈ Ω2(M, ÃM) = Ω2(M, so(T )) on s; thus, K̃ annihilates s. Via the projection Π : Γ(Λ2T) → Ea[2] = X(M), s projects to a conformal Killing field σa ∈ X(M), and from the explicit formula (12) one obtains that the Weyl curvature C annihilates σ. Employing the symmetries of C, one then immediately has that this is equivalent to trivial insertion of σ. Then K̃(σ, ·) = 0 follows from (n− 2)Aabc = DpCpabc. � 6.2 Infinitesimal automorphisms of parabolic geometries In this subsection we will use the description [9] of the symmetry Lie algebra of a parabolic geometry (G, ω) of type (G,P ). This will be applied for (G, ω) being, as above, the geometry of type (G2, P ) describing the generic rank 2 distribution D and for the conformal geometry encoded in the Cartan geometry (G̃, ω̃) of type (SO(h), P̃ ). For more details see [9] or [28]. Since G → M is a P -principal bundle over M and the geometric structure is encoded in the Cartan connection form ω ∈ Ω1(G, g), one defines an automorphism of (G, ω) as a P - equivariant diffeomorphism Ψ of G preserving ω, i.e., Ψ∗ω = ω. It is well known that the automorphism group of a Cartan geometry is always a Lie group (see for instance [15]). The space of infinitesimal automorphisms is given by the P -invariant vector fields on G preserving ω, i.e., inf(G, ω) = {ξ ∈ X(M)P : Lξω = 0}. The Lie algebra of the automorphism group of (G, ω) then consists of those vector fields in inf(G, ω) that are complete. Now ω : TG → g is a P -equivariant trivialization of TG, and thus, for a ξ ∈ inf(G, ω) the function f = ω ◦ ξ : G → g is P -equivariant. The function f therefore defines a section of the adjoint tractor bundle AM = G ×P g. We have an explicit formula for the tractor connection ∇A on AM , see (4): Let s ∈ Γ(AM) be the section corresponding to the P -equivariant function f ∈ C∞(G, g)P . To compute ∇A η s for η ∈ X(M) we take a P -invariant lift η′ ∈ X(G) of η. Then ∇A η s corresponds to the P -equivariant map u 7→ η′u · ω(ξ) + [ωu(η′), ωu(ξ)]. For the next lemma, first note that the natural projection Π : AM = G ×P g → G ×P g/p = TM projects s ∈ Γ(AM) to a vector field Π(s) ∈ X(M), which is in fact just the projection of the P -invariant vector field ξ ∈ X(G)P . Thus one can insert Π(s) ∈ X(M) into the curvature form K ∈ Ω2(M,AM). Proposition 7 ([9]). Let s ∈ Γ(AM) be the adjoint tractor corresponding to the P -invariant vector field ξ ∈ X(G)P . Then Lξω = 0 iff ∇A η s+K(Π(s), η) = 0 ∀η ∈ X(M), i.e., ξ ∈ inf(G, ω) if and only if the corresponding adjoint tractor s ∈ Γ(AM) is parallel with respect to the connection ∇̂As = ∇As+K(Π(s), ·). 26 M. Hammerl and K. Sagerschnig 6.3 Decomposition of the conformal adjoint tractor bundle. Let AM := G ×P g be the adjoint tractor bundle of a generic 2-distribution D and let ÃM := G̃ ×P̃ so(h) be the adjoint tractor bundle of the associated conformal structure. The tractor connection on AM will be denoted by ∇A and the one on ÃM by ∇Ã. Recall from Section 4.2 that as aG2-representation, so(h) = R7⊕g, i.e., so(h) decomposes into the direct sum of the standard representation ofG2 ⊂ SO(h) on R7 and the adjoint representation Ad : G2 → GL(g). This decomposition was realized by the exact sequence (26) of Section 4.2 and its splitting (27). On the level of associated bundles it yields a decomposition of the conformal adjoint tractor bundle: ÃM = G̃ ×P̃ so(h) = G ×P so(h) = (G ×P R7)⊕ (G ×P g) = T ⊕ AM. (37) I.e., the conformal adjoint tractor bundle ÃM is the direct sum of the standard tractor bundle T and the adjoint tractor bundle AM of the generic distribution. Let us check that we can also decompose the tractor connection into ∇Ã = ∇T ⊕∇A. (38) Take a vector field ξ ∈ X(M) and its horizontal lift ξ′ to a vector field on the extended bundle G′ = G ×P G2. Then, for s ∈ Γ(ÃM), the tractor derivative ∇ξs is defined by differentiating the G2-equivariant function f : G → so(h) corresponding to s in direction ξ′. But taking this derivative evidently commutes with the algebraic projections of f to its components fT : G → R7 and fAM : G → g; thus (38) holds. To prove the decomposition of conformal Killing fields we need the following theorem about the deformed connections ∇̂As = ∇As+K(Π(s), ·) and ∇̂Ãs = ∇Ãs+ K̃(Π(s), ·) whose parallel sections describe infinitesimal automorphisms. Recall that according to Lemma 2 we have K̃ = K ∈ Ω2(M,AM). Theorem 4. Let s ∈ Γ(ÃM) be a section of the conformal adjoint tractor bundle which splits according to decomposition (37) into s1 ∈ Γ(T ) and s2 ∈ Γ(AM). Then s is parallel with respect to ∇̂Ã if and only if s1 is ∇T -parallel and s2 is ∇̂A-parallel. Proof. Let s1 ∈ Γ(T ) and s2 ∈ Γ(AM) be ∇T - resp. ∇̂A-parallel sections. Since the restriction of∇Ã to Γ(T ) ⊂ Γ(ÃM) is just∇T , the section s1 includes as a∇Ã-parallel section into Γ(ÃM). Lemma 7 shows that that K(Π(s1), ·) = 0, and thus also ∇̂Ãs1 = 0. For s2 we have ∇̂Ãs2 = 0 by Lemma 2, and thus s := s1 + s2 satisfies ∇̂Ãs = 0. Conversely, we take s ∈ Γ(ÃM) with ∇̂Ãs = 0 and decompose s = s1 ⊕ s2 ∈ Γ(T )⊕ Γ(AM) according to (37). Since K has values in AM we have that s1 ∈ Γ(T ) is parallel with respect to the standard tractor connection ∇T by (38). We still need to show that s2 is parallel with respect to ∇̂A, while so far we know that ∇As2 +K(Π(s1), ·) +K(Π(s2), ·) vanishes. But since s1 is parallel as a section of ÃM with respect to the usual adjoint tractor connection ∇Ã according to (38), we can again apply Lemma 7, which tells us that s1 inserts trivially into the curvature K̃ = K. Thus also ∇̂As2 = 0. � Conformal Structures Associated to Generic 2-Distributions 27 Proposition 7 provides an identification of inf(G̃, ω̃) with parallel sections of ÃM with respect to ∇̂Ã and an identification of inf(G, ω) with parallel sections of AM with respect to the connection ∇̂A. We can thus translate Theorem 4 into a decomposition of conformal Killing fields: Theorem B. Let [g] be conformal class of signature (2, 3)-metrics on M and φ ∈ E[ab][3] a locally decomposable normal conformal Killing two form that satisfies the genericity condition (34). By Theorem A, there is a generic distribution D with [g] = [g]D. Then every conformal Killing field decomposes into a symmetry of the distribution D and an almost Einstein scale: cKf([g]) = sym(D)⊕ aEs([g]). (39) The mapping that associates to a conformal Killing field ξ ∈ X(M) its almost Einstein-scale part with respect to the decomposition (39) is given by ξa 7→ φpq(Dξ)pq − 1 2ξ pDqφpq, (40) where D is the Levi Civita connection of an arbitrary metric g in the conformal class. The mapping that associates to an almost Einstein scale σ ∈ E [1] a conformal Killing field is given by σ 7→ φapD pσ − 1 4σD pφpa. (41) Proof. By Proposition 7 conformal Killing fields of [g] are in 1:1-correspondence with ∇̂Ã- parallel sections of ÃM . By the theorem above every such section decomposes into a parallel standard tractor in Γ(T ) and a ∇̂A-parallel section of AM . By Proposition 1 and again Propo- sition 7, now for AM , this yields the decomposition (39). It is now straightforward to make this decomposition explicit in terms of the normal conformal Killing 2-form of Theorem A encoding the generic distribution D. To map an almost Einstein scale σ ∈ E [1] to a conformal Killing field we use the splitting operator LT0 : E [1] → Γ(T ) given in (16), contract this section into the characterizing ∇Λ3T - parallel 3-form Φ ∈ Γ(Λ3T ) given by (33) via the tractor metric h, and project the resulting section of Λ2T = ÃM down to X(M). This yields (41). To project a conformal Killing field ξ ∈ X(M) to its almost Einstein scale-part we proceed similarly: we map it to Λ2T via (36), contract it into Φ ∈ Γ(Λ3T ) and project the resulting standard tractor to E [2]. This gives (40). � To be precise, one has to use a constant scalar multiple of φ ∈ E[ab][3] such that the compo- sition of (40) with (41) is the identity. Remark 7. Mapping (41) actually works more generally: in the presence of an almost Einstein scale it was shown in [25, Corollary 5.2] that one can associate to every conformal Killing 2-form, not only to normal ones, a conformal Killing field. Acknowledgements We cannot underestimate the value of discussions with Andreas Čap on various technical proce- dures used in this paper. The concept of orbit types of parallel tractors was introduced to the first author by Felipe Leitner, who moreover suggested to check for simplicity of the underlying 2-form. We thank the referees for their careful reading and various valuable suggestions for improvements. The first author was supported by the IK I008-N funded by the University of Vienna. 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Lie Theory 14 (2004), 481–499. http://arxiv.org/abs/0811.4122 http://arxiv.org/abs/math.DG/0107101 http://arxiv.org/abs/0904.0186 http://arxiv.org/abs/math.DG/0604393 http://arxiv.org/abs/math.DG/0406400 http://arxiv.org/abs/math.DG/0701891 http://bart.math.muni.cz/~silhan/lie/lac/formR.php 1 Introduction and statement of results 1.1 Generic rank 2-distributions on 5-manifolds 1.2 The associated conformal structure of signature (2,3) and its characterization 1.3 Killing field decomposition 2 Preliminaries on Cartan and parabolic geometries 2.1 Cartan geometries 2.2 Tractor bundles 2.3 Parabolic geometries 2.4 Lie algebra differentials and normality 2.5 The BGG-(splitting-)operators 3 Conformal structures 3.1 Conformal structures as parabolic geometries 3.2 Tractor bundles for conformal structures 3.3 Almost Einstein scales 3.4 Conformal Killing forms 4 Generic rank two distributions and associated conformal structures 4.1 The distributions 4.2 Some algebra: G2 in SO(3,4) 4.3 The homogeneous model and associated Cartan geometries 4.4 A Fefferman-type construction 4.5 The parallel tractor three-form and the underlying conformal Killing 2-form 5 Holonomy reduction and characterization 5.1 Conformal holonomy 5.2 Characterization via the tractor 3-form 5.3 Characterization in terms of the underlying conformal Killing 2-form 6 Decomposition of conformal Killing fields of [g]D 6.1 Conformal Killing fields 6.2 Infinitesimal automorphisms of parabolic geometries 6.3 Decomposition of the conformal adjoint tractor bundle. References

Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition

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