Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy
aa r X i v : . [ m a t h . DG ] D ec HIGHLY SYMMETRIC -PLANE FIELDS ON -MANIFOLDSAND -DIMENSIONAL HEISENBERG GROUP HOLONOMY TRAVIS WILLSE
This article is dedicated to Mike Eastwood on the occasion of his 60th birthday.
Abstract.
Nurowski showed that any generic 2-plane field D on a 5-manifold M determines a natural conformal structure c D on M ; these conformal struc-tures are exactly those (on oriented M ) whose normal conformal holonomy iscontained in the (split, real) simple Lie group G . Graham and Willse showedthat for real-analytic D the same holds for the holonomy of the real-analyticFefferman-Graham ambient metric of c D , and that both holonomy groups areequal to G for almost all D . We investigate here independently interesting2-plane fields for which the associated holonomy groups are a proper subgroupof G .Cartan solved the local equivalence problem for 2-plane fields D and con-structed the fundamental curvature tensor A for these objects. He furthermoreclaimed to describe locally all D whose infinitesimal symmetry algebra has rankat least 6 and gave a local quasi-normal form, depending on a single functionof one variable, for those that furthermore satisfy a natural degeneracy condi-tion on A , but Doubrov and Govorov recently rediscovered a counterexample toCartan’s claim. We show that for all D given by Cartan’s alleged quasi-normalform, the conformal structures c D induced via Nurowski’s construction are al-most Einstein, that we can write their ambient metrics explicitly, and that theholonomy groups associated to c D are always the 5-dimensional Heisenberggroup, which here acts indecomposably but not irreducibly. (Not all of theseproperties hold, however, for Doubrov and Govorov’s counterexample.) Wealso show that the similar results hold for the related class of 2-plane fields de-fined on suitable jet spaces by ordinary differential equations z ′ ( x ) = F ( y ′′ ( x ))satisfying a simple genericity condition. Contents
1. Introduction 22. Generic 2-plane fields on 5-manifolds 42.1. The Cartan curvature tensor 52.2. The split octonions and the homogeneous model 52.3. G z ′ = F ( x, y, y ′ , y ′′ , z ) 103. Some metric and conformal geometry 113.1. Conformal tractor and ambient geometry 133.2. Holonomy 154. Nurowski conformal structures 18 Mathematics Subject Classification.
Key words and phrases. conformal geometry, Fefferman-Graham ambient metric, generic dis-tributions, special holonomy. z ′ = F ( y ′′ ) 285.3. Other examples 31References 321. Introduction
In a well-known but technically demanding 1910 paper, [Car10], Cartan solvedthe local equivalence problem for what in modern geometric language are called2-plane fields on 5-manifolds, and the most interesting such fields are those thatsatisfy a simple genericity condition. This class is the lowest-dimensional example of k -plane fields on n -manifolds that admit nontrivial local invariants, but already thegeometry of these fields is surprisingly rich and furthermore enjoys close connectionswith some exceptional geometric objects, including the algebra of the split octonionsand the exceptional Lie group G .One of the most striking realizations of these connections was described byNurowski [Nur05, Nur08] and Leistner and Nurowski [LN12], whose work exploitsthe geometry of generic 2-plane fields D on 5-manifolds M to produce metrics ofholonomy equal to G (here and henceforth, G denotes the split real form of theexceptional Lie group). They produce candidate metrics of this kind by concate-nating two constructions: First, Nurowski exploited Cartan’s solution of the localequivalence problem for these 2-plane fields to show that any such field D inducesa canonical conformal structure c D of signature (2 ,
3) on the underlying manifold[Nur05]. Second, the Fefferman-Graham ambient construction associates to anyconformal structure (
M, c ) of signature ( p, q ) an essentially unique metric e g of sig-nature ( p + 1 , q + 1) on a suitable open subset f M ⊆ R + × M × R , though for most c the metric e g cannot be identified explicitly [FG11]. Applying this latter construc-tion to a conformal structure c D produces a pseudo-Riemannian metric of signature(3 , D parametrized by R and found corresponding (polynomial) ambient metrics e g D of c D . By giving explicitly a certain object parallel with respect to e g D —namely, a 3-form of a certain algebraic type—they showed that the holonomy groups Hol( e g D ) ofthe metrics in this family all are contained in the stabilizer in SO(3 ,
4) of the 3-form,which turns out to be G , and moreover that for an explicit, dense open subset ofparameter values, Hol( e g D ) = G [LN12]. This is interesting in part because thereare relatively few examples of metrics with this holonomy group. Later, Grahamand Willse showed that for all real-analytic D on oriented 5-manifolds, there is anambient metric e g D such that Hol( e g D ) ≤ G and that, in a suitable sense, equalityholds generically [GW12].In this article we give an explicit infinite-dimensional family of 2-plane fields D and corresponding explicit ambient metrics e g D for which the containment ofholonomy in G is proper. The 2-plane fields in this family satisfy two stronginvariant criteria (but, pace Cartan’s claims, the family is not characterized by theseconditions). First, Cartan described the fundamental curvature quantity of 2-plane IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 3 fields D , which we may interpret as a tensor field A ∈ Γ( ⊙ D ∗ ) [Car10]. If wecomplexify A , then for each u ∈ M , we may regard the roots of A u ⊗ C ∈ ⊙ D ∗ u ⊗ C as elements of the complex projective line P ( D u ⊗ C ), and if A u = 0, we call thepartition of 4 given by the root multiplicities the root type of D at u ; for example,we say that D has root type [4] at u if A u ⊗ C is nonzero and has a quadrupleroot, or equivalently, if the line it spans is contained in the rational normal curvein ⊙ D ∗ u ⊗ C . Second, the (infinitesimal) symmetry algebra of D is the Lie algebra aut ( D ) of vector fields that preserve D . Cartan claimed that one can locally encodeany 2-plane field D such that (A) D has constant root type [4] (that is, root type[4] at every u ∈ M ), and (B) dim aut ( D ) ≥
6, in a principal bundle E → M with 2-dimensional structure group and a coframe (13) on E that depends only asingle function I , but Doubrov and Govorov have recently produced an interestingcounterexample to this claim. For any smooth function I the structure equations ofthis coframe determines a 2-plane field D I , and we produce explicit ambient metrics e g I (15) on spaces f M I for the conformal structures c I := c D I they induce. Theseconformal structures all enjoy additional special structures, including (exactly) a 2-dimensional vector space of almost Einstein scales (in fact, almost Ricci-flat scales).Each of these scales in turn corresponds to a parallel null vector field on f M I , andso Hol( e g I ) must be contained inside the common stabilizer of those vector fields; weshow that Hol( e g I ) is actually this full group, which roughly indicates that the onlyobjects parallel with respect to e g I are those arising from the parallel G -structureand the described null vector fields.We also compute for the conformal structures c I a closely related notion of ho-lonomy that has recently enjoyed heightened attention [Lei06, HS09, ˇCGGH]. Onecan encode any n -dimensional conformal manifold ( M, c ) in a rank-( n + 2) bundle T → M called the tractor bundle, together with some auxiliary data including acanonical (normal) conformal connection ∇ T on T ; we call the holonomy Hol( ∇ T )of this connection the (normal) conformal holonomy of c .Partly via explicit computation we prove the following: Theorem A.
Let I : U → R be a smooth function whose domain U ⊂ R is openand connected, and let e g I be the Ricci-flat ambient metric (15) of c I . Then, Hol( ∇ T I ) ∼ = Hol( e g I ) ∼ = H ,where ∇ T I is the tractor connection of c I and H is the -dimensional Heisenberggroup. The local holonomy Hol ∗ u ( ∇ E ) of a connection ∇ E on a vector bundle is definedin the paragraphs after Definition 13 below.Some algebra shows that these holonomy groups act indecomposably, and so theambient metrics e g I respectively provide new examples of metric holonomy groupsthat act indecomposably but not irreducibly; among metric connections this phe-nomenon can occur only in indefinite signature.The remainder of this article is organized as follows: Section 2 collects some gen-eral facts about generic 2-plane fields D on 5-manifolds, including about the Cartancurvature tensor A . We construct a homogeneous model of such 2-plane fields usingjust the algebraic structure of the split octonions, e O ; this connects the geometryof these fields to the exceptional Lie group G , which can be realized as the auto-morphism group of e O , and we recall some specific algebraic facts about G relevantto the geometry of the 2-plane fields D I . We also recall Goursat’s quasi-normal TRAVIS WILLSE form, which locally realizes any D as a differential equation z ′ = F ( x, y, y ′ , y ′′ , z )for some function F ( x, y, p, q, z ) defined on a neighborhood of the origin in R andfor which F qq is nowhere zero; we exploit this form in direct computations in Section5. Section 3 gives some basic facts about conformal geometry, including the con-struction of the conformal tractor bundle and ambient metric, and also recalls andrelates various notions of holonomy, including the normal holonomy Hol( ∇ T ) of aconformal structure. Section 4 discusses Nurowski’s construction of the conformalstructure c D from a generic 2-plane field D on a 5-manifold and describes brieflya manifestly invariant construction of that structure in the language of parabolicgeometry, for which we deliberately provide no other background; the standardreference for this topic is [ ˇCS09]. We give several key results about these confor-mal structures, including a characterization of them among all conformal structuresdue to Hammerl and Sagerschnig [HS09] and some facts about holonomy groupsassociated to them. Finally, in Section 5 we give explicit data for the 2-plane fields D I , including formulas for Ricci-flat ambient metrics e g and the parallel objects onthe tractor bundle and ambient manifold that guarantee the containment of theindicated holonomy groups in H . We use these data to prove Theorem A, andvarious other results related to these 2-plane fields, including that the holonomyof any Ricci-flat representative of c I is equal to R . We then consider a class of2-plane fields, namely the fields D F ( q ) for which the function F in the local normalform depends only on q , which thus correspond to differential equations z ′ = F ( y ′′ ).This class is closely related to that of the 2-plane fields D I , and we show that theholonomy results we prove for the 2-plane fields D I essentially hold for the 2-planefields D F ( q ) too. Finally, we discuss briefly Doubrov and Govorov’s striking coun-terexample, which behaves substantially differently from the 2-plane fields in theabove, but for which we postpone detailed discussion to a later paper.All objects are smooth by hypothesis except where stated otherwise.Many computations in this work were done with Maple, and in particular thepackage DifferentialGeometry . It is a pleasure to thank Ian Anderson, that pack-age’s primary author, for hosting the author at Utah State University in March 2012after the conference “Differential Geometry of Distributions,” when some of thiswork was done, as well as for his assistance with the package. The author thanksRobert Bryant for suggesting to the author Cartan’s 1910 paper as a possible sourceof 2-plane fields with interesting associated holonomy groups at the October 2010meeting of the Pacific Northwest Geometry Seminar at the University of Oregon.The author also thanks Mike Eastwood, Ravi Shroff, Dennis The, and the refereefor various helpful comments made during the paper’s preparation and revision,and Boris Doubrov for a helpful exchange regarding the counterexample to Car-tan’s classification he produced with Artem Govorov. Support from the AustralianResearch Council is gratefully acknowledged.2.
Generic -plane fields on -manifolds Definition 1.
A 2-plane field D on a 5-manifold M is generic if [ D, [ D, D ]] =
T M .If D is generic, then the derived plane field [ D, D ] has constant rank 3.Two generic 2-plane fields (
M, D ) and ( c M , b D ) are equivalent if there is a dif-feomorphism ϕ : M → c M such that T x ϕ · D = b D ϕ ( x ) for all x ∈ M , and they aremerely locally equivalent at x ∈ M and b x ∈ c M if there are neighborhoods V of x and b V of b x such that the restricted 2-plane fields ( V, D | V ) and ( b V , b D | b V ) are IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 5 equivalent. If (
M, D ) and ( c M , b D ) are locally equivalent at x and b x for all x ∈ M and b x ∈ c M , we simply say that the 2-plane fields are locally equivalent, withoutreference to a choice of points. When using any of the above notions of equivalence,we may suppress mention of the underlying manifolds when their identities are clearfrom context. Definition 2.
Let E be a k -plane field on a smooth manifold M . A vector field ξ ∈ Γ( T M ) is an infinitesimal symmetry of E if L ξ η ∈ Γ( E ) for all η ∈ Γ( E ).The (infinitesimal) symmetry algebra of E is the space aut ( E ) of infinitesimalsymmetries of E , and the compatibility of the Lie bracket of vector fields withthe Lie derivative ensures that aut ( E ) is a Lie subalgebra of Γ( T M ) under thatoperation.2.1.
The Cartan curvature tensor.
Cartan constructed the fundamental curva-ture invariant, which we call the
Cartan curvature (tensor) , of a generic 2-planefield D on a 5-manifold ( M, D ): a symmetric quartic form on D , that is, an ele-ment A ∈ Γ( ⊙ D ∗ ). This form is precisely the harmonic curvature of the type ofparabolic geometry that encodes this structure, and it vanishes identically iff D islocally equivalent to the homogeneous model of this geometry described in the nextsubsection.By complexifying and projectivizing, at each point x ∈ M we may regard theroots of A x ⊗ C ∈ ⊙ D ∗ x ⊗ C as elements of the complex projective line P ( D x ⊗ C ).If A x = 0, then A x ⊗ C has exactly four roots counting multiplicity, and we call thepartition Λ of 4 given by the multiplicities of the roots the root type of D at x .By convention, if A x = 0, we say that D has root type [ ∞ ] at x . If D has a givenroot type Λ at all x ∈ M , we just say that D has (constant) root type Λ.2.2. The split octonions and the homogeneous model.
A natural model forgeneric 2-plane fields on 5-manifolds can be efficiently and beautifully realized usingan 8-dimensional real algebra called the split octonions, which in turn is intimatelyrelated to several exceptional objects.Up to isomorphism there are exactly seven composition algebras over R , thatis, algebras over R with a unit and a nondegenerate bilinear form h· , ·i that satisfies h xy, xy i = h x, x i h y, y i for all elements x and y in the algebra; the facts here aboutcomposition algebras are given, for example, in [Har90], where they are callednormed algebras. Four of these are the celebrated normed division algebras, R , C , H , and O . The remaining three are the so-called split analogues of the latterthree; the largest and richest of these, both algebraically and geometrically, is the split octonions , which we denote e O . This algebra has dimension 8 over R , and itsbilinear form has signature (4 , a of a composition algebra A imaginary if h , a i = 0, andwe denote the set of such elements by Im A . Restricting h· , ·i to Im A defines anondegenerate bilinear form (which we again denote by h· , ·i ); we can then definethe cross product · × · : Im A × Im A → Im A on that subspace by x × y := − Im( xy ).Regarding × as a (2 , A and dualizing (and changing signs, to agreewith the convention in [Sag06]) defines a 3-tensor Φ ∈ ⊗ Λ (Im A ) ∗ byΦ( x, y, z ) := − h x × y, z i = h xy, z i , TRAVIS WILLSE and one can show that it is totally antisymmetric.We henceforth restrict our attention to A = e O ; for further details of most ofthe constructions in the rest of the subsection and Subsection 2.3, see [Sag06] and[HS09]. Since bilinear form of this algebra has signature (4 , e O has signature (3 , e O and hence Im e O from Φ alone; in particular, the bilinear form on Im e O satisfies(1) h x, y i = − tr( z x × ( y × z )).The algebra and geometry of e O are in some ways richer than that of its analogue, O , because the former contains zero divisors; these are exactly the nonzero vectorsnull with respect to the bilinear form. To exploit this structure, we define the (punctured) null cone to be the set N := { x ∈ Im e O − { } : h x, x i = 0 } of nonzero null vectors in Im e O and define the (null) quadric to be its projec-tivization, Q := P ( N ) (by construction, Q is diffeomorphic to ( S × S ) / Z , wherethe nonidentity element of Z acts by the antipodal map simultaneously on the twospheres). For any x ∈ Im e O we define the (algebraic) annihilator of x to be thevector spaceAnn x := { y ∈ Im e O : x × y = 0 } = { y ∈ Im e O : Φ( x, y, · ) = 0 } ,and one can show that dim Ann x = 3 if x is nonzero and null and Ann x = [ x ] if x is non-null, where [ x ] denotes the span of x .For x ∈ N , some easy algebra yields the inclusions[ x ] ⊂ Ann x ⊂ (Ann x ) ⊥ ⊂ [ x ] ⊥ ,which we can together regard as a vector space filtration of T x N = [ x ] ⊥ . Varying x defines a filtration of the tangent bundle T N by plane fields of constant rank 1,3, 4, and 6. This descends to a filtration of the tangent bundle of the quadric: { } = D ⊂ D − ⊂ D − ⊂ D − = T Q .In particular, rank D − = 2 and rank D − = 3. We denote ∆ := D − ; computingshows that [∆ , ∆] = D − and [∆ , [∆ , ∆]] = [ D − , D − ] = D − = T Q . (Givenconstant-rank plane fields V and W , the set [ V, W ], which a priori need not haveconstant rank, is [
V, W ] p = { [ ξ, η ] p : ξ ∈ Γ( V ) , η ∈ Γ( W ) } .)In particular, ∆ is a generic 2-plane field on Q , and we call the pair the homoge-neous model of the geometry of such fields. We say that a generic 2-plane field ona 5-manifold is locally flat if it is locally equivalent to ( Q , ∆), and one can showthat local flatness is equivalent to the Cartan curvature tensor A being identicallyzero [Car10].2.3. G . The Lie algebra of the automorphism group Aut e O of the split octonionsis simple, has dimension 14, and has indefinite Killing form, so it is the split realform of the simple complex Lie algebra of type G . We thus henceforth denote thisautomorphism group by G .Since G preserves { } and h· , ·i , it also preserves [1] ⊥ = Im e O and Φ ∈ Λ (Im e O ) ∗ .One can show that G is connected; then, since (1) realizes the bilinear form h· , ·i on Im e O in terms of its algebraic structure, G also preserves that form and henceadmits a natural embedding G ֒ → SO(3 , thus IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 7 restricts to an action on the null cone N , and by linearity it descends to an actionon the quadric Q .Conversely, the full algebra e O can be recovered from Φ, and so G is precisely thestabilizer of Φ under the induced action of GL(Im e O ) on Λ (Im e O ) ∗ . The stabilizerof any other 3-form in the orbit GL(Im e O ) · Φ is a conjugate of G in GL(Im e O ), andwe call the 3-forms in this orbit split-generic . (The modifier generic indicatesthat this orbit turns out to be open. In fact, the action of GL(Im e O ) on Λ (Im e O ) ∗ has exactly two open orbits. The stabilizer of any element in the open orbit thatdoes not contain Φ is just the compact real form of the complexification G C of G ,and this compact form can be realized as the algebra automorphism group of theoctonions, O .)Given a 7-dimensional real vector space V and any vector space isomorphism τ : Im e O → V , we say that a 3-form Φ ∈ Λ V ∗ is split-generic if and only if τ ∗ Φ is;by construction this characterization does not depend on the choice of τ . So, up toalgebra isomorphism we may realize the imaginary split octonions, endowed withthe cross product, by giving such a pair ( V , Φ) with Φ split-generic.Now, given such a realization ( V , Φ) of the imaginary split octonions, one canrealize the induced inner product h· , ·i explicitly in terms of the 3-form Φ. Any3-form ϕ on a 7-dimensional real vector space V induces a symmetric Λ V ∗ -valuedbilinear form, namely, ( X, Y ) ( X y ϕ ) ∧ ( Y y ϕ ) ∧ ϕ . One can show that thisform is nondegenerate if and only if ϕ is generic, in which case it distinguishesa nonzero volume form vol ∈ Λ V ∗ [Bry87, Hit01] and hence yields an R -valuedbilinear form on V : If we regard the bilinear form as a linear map ⊗ V → Λ V ∗ ,dualizing gives a map V → V ∗ ⊗ Λ V ∗ . Its determinant is a map det : Λ V → Λ ( V ∗ ⊗ Λ V ∗ ) ∼ = ⊗ Λ V ∗ , and dualizing again gives a map det : R → ⊗ Λ V ∗ which is nonzero because the bilinear form is nondegenerate, and which we mayregard as a distinguished element of ⊗ Λ V ∗ . Since V is real, there is a uniqueelement vol ∈ Λ V ∗ such that vol ⊗ = det. So, if ϕ is generic, we can define asymmetric bilinear form H ( ϕ ) ∈ ⊙ V ∗ by(2) H ( ϕ )( X, Y ) vol := √ X y ϕ ) ∧ ( Y y ϕ ) ∧ ϕ .The factor √ V . The form H ( ϕ ) is split-generic if and only if the bilinear form has signature (3 , , V = R , let ( E a ) denote the standard basis on V and ( e a ) its dual basis, and defineΦ := √ ( −√ e ∧ e ∧ e − e ∧ e ∧ e − e ∧ e ∧ e + e ∧ e ∧ e − √ e ∧ e ∧ e ).The matrix representation in the basis ( E a ) of the bilinear form H (Φ) is I
00 0 − I , TRAVIS WILLSE where I is the 2 × H (Φ) nondegenerate and hassignature (3 , h X, Y i := H (Φ)( X, Y ).The Lie algebra g is the algebra of derivations of the split octonions, which bythe discussion earlier in the subsection is just the annihilator of Φ in the Lie algebra gl ( V ) of derivations of V . Computing gives that, in the basis ( E a ), g has matrixrepresentation(3) tr A Z s W X A √ J Z T s √ J − W T r −√ X T J −√ Z J sY − r √ J √ J X − A T − Z T − Y T r − X T − tr A : A ∈ gl (2 , R ) X, Y ∈ R Z, W ∈ ( R ) ∗ r, s ∈ R ,where J = (cid:18) −
11 0 (cid:19) .One can use (3) to show that G acts transitively on the null cone N (and hence thenull quadric Q ), so we may regard Q as the homogeneous space G /P , where P is the stabilizer in G of a point in Q , that is, of a null line in Im e O . Furthermore,the 2-plane field ∆ on Q was constructed algebraically from Φ, so it is invariantunder this action, and this motivates the moniker homogeneous model for ( Q , ∆).We will be interested in determining the holonomy of certain 7-dimensional met-rics (see Section 5), all of which admit a parallel split-generic 3-form and two linearlyindependent null vector fields. (We say that a 3-form on a smooth 7-manifold issplit-generic if and only if its value at each point is split-generic; if a parallel 3-form is split-generic at one point, it is split-generic everywhere.) These holonomygroups will be contained in the common stabilizer in SO( H (Φ)) ∼ = SO(3 ,
4) of thesplit-generic 3-form Φ and two linearly independent null vectors x and y ; however,this stabilizer depends on the relative configuration of those three tensors. Proposition 3.
Suppose x, y ∈ N , and let Φ be the split-generic -form thatdefines the algebraic structure on V . The isomorphism type of the common stabilizer Stab
SO(3 , (Φ) ∩ Stab
SO(3 , ( x ) ∩ Stab
SO(3 , ( y ) is K, if [ x ] = [ y ]H , if Φ( x, y, · ) = 0 , [ x ] = [ y ] R , if h x, y i = 0 , Φ( x, y, · ) = 0SL(2 , R ) , if h x, y i 6 = 0 ,where [ z ] denotes the line in Im e O spanned by the vector z , K is the stabilizer of an(arbitrary) vector in N , and H is the -dimensional Heisenberg group.Proof. By the discussion earlier in this subsection, the stabilizer of Φ in SO( V ) isG , and so the common stabilizer is just the common stabilizer in G of x and y .Since G acts transitively on N , we may identify x with the null vector e ∈ V ;then, where K is the stabilizer in G of this vector, the common stabilizer is justStab K ( y ). IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 9
Consulting (3) shows that the Lie algebra of K is(4) k = Z s W A √ J Z T s √ J − W T −√ Z J s − A T − Z T : A ∈ sl (2 , R ) Z, W ∈ ( R ) ∗ s ∈ R .Using this realization it is easy to check that the action of K partitions N intothe following orbits; we also give a representative of each orbit in terms of the basis( e a ): • singletons { λx } , λ ∈ R ∗ ; λe • the set Ann x − [ x ]; e • the set ([ x ] ⊥ − Ann x ) ∩ N ; e • hypersurfaces { z ∈ N : h x, z i = λ } , λ ∈ R ∗ ; λe .Consulting (4) gives that the common stabilizer in k of e and e is(5) Z s W W
00 0 a −√ Z − √ s − W √ s − W −√ Z s a − Z ∼ = h ,where a , Z , s, W , W ∈ R ; this algebra is isomorphic to the 5-dimensional Heisen-berg Lie algebra. This and analogous computations for the three other orbit typesgive that the isomorphism types of the stabilizers in K of the given orbit repre-sentatives (and hence that of arbitrary elements y in their respective orbits) are,respectively, K , H , R , and SL(2 , R ). (cid:3) Remark 4.
For each of the four cases in the proposition, one can read some ofthe information about the above stabilizers directly from the root diagram of g .By choosing an appropriate Cartan subalgebra, we may identify the stabilizer Liealgebra k = stab g ( x ) as the subalgebra spanned by the root spaces of the roots inthe indicated box in the diagram below together with the 1-dimensional subalgebraof the Cartan subalgebra (which corresponds to the center node) that fixes x . Infact, if we identify x with e as in the proof, the Cartan subalgebra of g (3)comprising the diagonal matrices will do.For any y ∈ Ann x − [ x ], one can choose a basis of V so that, for example, x = e (again) and y = e , and if we again take the Cartan subalgebra to be the set ofdiagonal matrices, the stabilizer b k := stab g ( y ) of y in g is the one indicated. Thecommon stabilizer algebra stab g ( x ) ∩ stab g ( y ) is just the span of the root spacesin both k and b k , namely those of the circled roots (in particular, no nonzero elementof the Cartan subalgebra stabilizes both e and e ); the diagram shows that thisalgebra is isomorphic to h . k b k Proceeding analogously for y in the K -orbit ([ x ] ⊥ − Ann x ) ∩ N and any K -orbit { z ∈ N : h x, z i = λ } , λ ∈ R ∗ , respectively yields diagrams k b k and k b k .In the diagram for { z ∈ N : h x, z i = λ } , λ ∈ R ∗ , the circle around the center nodeindicates that the common stabilizer of x and y contains a 1-dimensional subalgebraof the Cartan subalgebra of g . Remark 5.
In the language of [BH12], the four conditions on x and y in thestatement of Proposition 3 are equivalent to the pair ([ x ] , [ y ]) ∈ Q × Q of null linesbeing 0, 1, 2, and 3 rolls apart, respectively, so the proposition shows in particularthat for each k ∈ { , , , } , G acts transitively on the space of pairs of null linesthat are k rolls apart, that is, that these are precisely the four orbits of the inducedG action on pairs of null lines.2.4. Ordinary differential equations z ′ = F ( x, y, y ′ , y ′′ , z ) . Consider a second-order ordinary differential equation in the Monge normal form(6) z ′ = F ( x, y, y ′ , y ′′ , z ),where y and z are functions of x . Introducing coordinates p and q for the y ′ and y ′′ identifies the partial jet space J , ( R , R ) with R (with coordinates ( x, y, p, q, z ))and realizes the differential equation (6) as the exterior differential system(7) ω := dy − p dxω := dz − F ( x, y, p, q, z ) dx − F q ( x, y, p, q, z )( dp − q dx ) ω := dp − q dx on R : Explicitly, a triple ( x, y ( x ) , z ( x )) is a solution of (6) if and only if itsprolongation ( x, y ( x ) , y ′ ( x ) , y ′′ ( x ) , z ( x )) is an integral curve of the common kernelof (7). IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 11
Since { ω , ω , ω } is linearly independent, the common kernel ker { ω , ω , ω } isa 2-plane field D F on dom F , and checking directly shows that it is generic if andonly if F qq is nowhere zero. These defining 1-forms were chosen so that the derived3-plane field satisfies [ D F , D F ] = ker { ω , ω } .Goursat showed that all generic 2-plane fields arise this way, at least locally. Lemma 6. [Gou22, § Let D be a generic -plane field on a -manifold M , andfix u ∈ M . Then, there is a function F defined on an open subset of R containing such that D and D F are locally equivalent near u and . For accessible proofs of this lemma, see [BH93, p. 2.6], which proves a generaliza-tion of the lemma to manifolds of dimension 5 and higher, or see [Kru11a, Theorem3] or [Str09, § F in this lemma—somewhat moreprecisely, different functions F can yield equivalent 2-plane fields D F —so we referto (7) only as the local Monge (quasi-)normal form for generic 2-plane fieldson 5-manifolds. The field D q is locally flat [Car10].For later use, we augment (7) with two auxiliary forms to produce a local coframe( ω a ) of T M :(8) ω := dy − p dxω := dz − F ( x, y, p, q, z ) dx − F q ( x, y, p, q, z )( dp − q dx ) ω := dp − q dxω := dqω := dx .The frame ( E a ) of T M dual to ( ω a ) is(9) E := ∂ y E := ∂ z E := ∂ p + F q ( x, y, p, q, z ) ∂ z E := ∂ q E := ∂ x + p∂ y + q∂ p + F ( x, y, p, q, z ) ∂ z .In particular, D F = ker { ω , ω , ω } = h E , E i = h ∂ q , ∂ x + p∂ y + q∂ p + F ( x, y, p, q, z ) ∂ z i .3. Some metric and conformal geometry
Conformal geometry is the geometry of smooth manifolds in which one has anotion of angle but (in particular) not of length.
Definition 7. A conformal structure on a smooth manifold M is an equivalenceclass c of (pseudo-Riemannian) metrics under the relation ∼ , where g ∼ b g if andonly if b g = Ω g for some positive function Ω ∈ C ∞ ( M ), and the pair ( M, c ) is calleda conformal manifold . Any metric g ∈ c is a (conformal) representative of c . The signature ( p, q ) of a conformal structure c is just the signature of any(equivalently, every) conformal representative.An infinitesimal symmetry of a conformal structure c on an n -manifold M ,(alternatively, a conformal Killing field on ( M, c )) is a vector field ξ ∈ Γ( T M )that preserves the conformal structure in the sense that for a representative metric g ∈ c , L ξ g = λg for some λ ∈ C ∞ ( M ). Taking traces gives that this condition isequivalent to tf( L ξ g ) = 0, where tf( S ) denotes the tracefree part S ab − n S cc g ab of S , and direct computation shows that this condition is independent of the choiceof representative g . We denote the space of infinitesimal symmetries of c by aut ( c );checking shows that it is closed under the Lie bracket of vector fields, so it is a Liesubalgebra under that operation.The metric bundle of the conformal structure c on a smooth manifold M isthe ray bundle π : G → M defined by G := a x ∈ M { g x : g ∈ c } .By construction, the sections of G are precisely the representative metrics g of c .The action R + × G → G defined by s · g x = δ s ( g x ) := s g x naturally realizes G as a principal R + -bundle; we denote the infinitesimal generator of this dilation by T := ∂ s δ s | s =1 . The metric bundle admits a tautological degenerate, symmetric2-tensor g ∈ Γ( ⊙ T ∗ G ) defined by ( g ) g x ( ξ, η ) := g x ( T g x π · ξ, T g x π · η ), which wemay identify with the conformal structure itself.Fixing a representative g ∈ c yields a trivialization G ∼ = M × R + by identifyingthe inner product t g x with ( x, t ). In this trivialization, the tautological 2-tensoris g = t π ∗ g , the dilations are given by δ s : ( t, x ) ( st, x ), and the infinitesimalgenerator is T = t∂ t .For any w ∈ R the conformal density bundle of weight w is the bundle D [ w ] := G × ρ − w R associated to G by the R + -representation ρ − w ( y ) := s − w y . Wemay identify this bundle with ` x ∈ M { f : G x → R : δ ∗ s f = s w f, s ∈ R + } and henceits sections with real-valued functions on G of homogeneity w (with respect to thedilations δ s ). A choice of representative g ∈ c induces a trivialization of each densitybundle D [ w ] by identifying f ∈ Γ( D [ w ]) with f ◦ g ∈ C ∞ ( M ), where we regard g as a section M → G .Given a vector bundle E → M and any w ∈ R , we may form a conformallyweighted vector bundle E [ w ] := E ⊗ D [ w ], and again a choice of representativetrivializes any such bundle by identifying v ⊗ f ∈ Γ( E [ w ]) with ( f ◦ g ) v ∈ Γ( E ).By construction, g satisfies δ ∗ s g = s g and depends only on the T π -fibers ofits arguments, and unwinding definitions shows that we may view the conformalstructure itself as a canonical section g of ⊙ T ∗ M [2].The class of conformal structures considered in this work all admit an additionalspecial structure called an almost Einstein scale.A metric g on an n -manifold M is Einstein if Ric = 2 λ ( n − g for some smoothfunction λ ∈ C ∞ ( M ). If n ≥ M is connected, then λ is necessarily constant;this is the Einstein constant of g , though this term is usually used for the fullcoefficient 2 λ ( n − Definition 8.
Suppose (
M, c ) is a conformal manifold. An
Einstein scale for c isa (nonvanishing) weighted smooth function σ ∈ Γ( D [1]) such that the (unweighted)metric σ − g is Einstein, and if c admits an Einstein scale, we say that c is (con-formally) Einstein . A weighted smooth function σ ∈ Γ( D [1]) with zero set Σ isan almost Einstein scale for c if σ − g | M − Σ is Einstein. We denote the space ofalmost Einstein scales of c by aEs ( c ). If dim M ≥ M is connected, and if thealmost Einstein scale σ is not identically zero, every restriction of σ − g | M − Σ to aconnected component of M − Σ has the same Einstein constant, which we hencecall the
Einstein constant of σ ; if λ = 0, we say that σ is an almost Ricci-flat IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 13 scale . (For expository convenience, we declare the identically zero almost Einsteinscale to be almost Ricci-flat too.)3.1.
Conformal tractor and ambient geometry.
The conformal tractor bundleis a construction that encodes a conformal manifold (
M, c ) of signature ( p, q ) in arank-( p + q + 2) bundle T → M endowed with some auxiliary structure. Theclosely related ambient metric construction assigns to ( M, c ) a pseudo-Riemannianmanifold ( f M , e g ) of signature ( p + 1 , q + 1); this latter construction involves somechoices, but the nonuniqueness in its construction is manageable. Invariant dataextracted from either construction (and which, in the case of the ambient metric,do not depend on any choices made) are invariants of the underlying conformalstructure and hence can be used to analyze that structure; indeed, this was theoriginal motivation for the construction of the ambient metric [FG85].We first describe the ambient metric associated to c (for n odd, which is all thatwe need here), following the standard reference [FG11], and then use it to constructthe standard conformal tractor bundle as described in [ ˇCG03]. The conformaltractor construction was first given [Tho26] and was rediscovered and extended in[BEG94] using an approach different from the one here, and it can also be describedin the language of parabolic geometries [ ˇCS09]. The discussion in [GW12, §
2] issimilar to the one below but includes the case of even n .Henceforth in this subsection, ( M, c ) is a conformal structure of dimension n ≥ p, q ). Consider the space G × R , and denote the standard coordinateon R by ρ . The dilations δ s extend to G × R by acting on the G factor, thatis, by δ s ( g x , ρ ) = s · ( g x , ρ ) := ( s g x , ρ ), and we again denote its infinitesimalgenerator T := ∂ s δ s | s =1 . The map G ֒ → G × R defined by z ( z,
0) embeds G as ahypersurface in G × R , and we identify G with its image, G × { } , under this map.Again a choice of representative g induces a trivialization G × R ↔ R + × M × R by identifying ( t g x , ρ ) ↔ ( t, x, ρ ), which defines an embedding M ֒ → G × R by x (1 , x,
0) and yields an identification T ( G × R ) ∼ = R ⊕ T M ⊕ R .A smooth metric e g of signature ( p + 1 , q + 1) on an open neighborhood f M of G in G × R invariant under the dilations δ s , s ∈ R + , is a pre-ambient metric for ( M, c )if (1) it extends g , that is, if ι ∗ e g = g , and (2) if it has the same homogeneityas g with respect to the dilations, that is, if δ ∗ s e g = s e g (again, for all s ). A pre-ambient metric is straight if for all p ∈ f M the parametrized curve s s · p isa geodesic. Any nonempty conformal structure admits many pre-ambient metrics;Cartan’s normalization condition for a conformal connection [Car23] suggests thatRicci-flatness is a natural distinguishing criterion. Definition 9.
Let (
M, c ) be a conformal manifold of odd dimension at least 3. An ambient metric for (
M, c ) is a straight pre-ambient metric e g such that Ric( e g ) is O ( ρ ∞ ); the pair ( f M , e g ) is an ambient manifold for ( M, c ).Here, we say that a tensor field on f M is O ( ρ ∞ ) if it vanishes to infinite orderat each point of the zero set of ρ . We formulate Fefferman-Graham’s existence anduniqueness results for ambient metrics of odd-dimensional conformal structures asfollows: Theorem 10. [FG11]
Let ( M, c ) be a conformal manifold of odd dimension atleast . There exists an ambient metric for ( M, c ) , and it is unique up to pullbackby diffeomorphisms that restrict to id G and up to infinite order: If e g and e g areambient metrics for ( M, c ) , then (after possibly restricting the domains of both to appropriate open neighborhoods of G in f M ) there is a diffeomorphism φ such that φ | G = id G and φ ∗ e g − e g is O ( ρ ∞ ) . We now recover the standard tractor bundle from the ambient construction. Let e g be an ambient metric for ( M, c ). Since e g is straight, the fiber G x of G → M isa geodesic of e g (with geodesic parametrization s s · g x for any g x ∈ G x ). Directcomputation shows that a vector field ξ along G x in f M is parallel if and only if T δ s · ξ = sξ for all s ∈ R + (see [GW12, §
2] for details). We define the standardtractor bundle to be the bundle whose smooth sections are vector fields along G in f M with this homogeneity with respect to δ s . Definition 11.
Let (
M, c ) be a conformal manifold. The (standard conformal)tractor bundle is the bundle π : T → M defined by T = a x ∈ M { ξ ∈ Γ( T f M | G x ) : T δ s · ξ = sξ, s ∈ R + } .We call a section of T a (standard) (conformal) tractor (field) .We construct some additional natural objects on T . The sections of T are thevector fields on T G that satisfy δ ∗ s X = s − X ; since by definition δ ∗ s e g = s e g , if X, Y ∈ T x , then e g ( X, Y ) is constant. So, the restriction of e g to G defines a fibermetric g T (of signature ( p + 1 , q + 1)) on T . Now, T has homogeneity 0, so we mayregard it as a section of T [1], and the span of T ∈ Γ( T G ) is invariant under δ s , soit descends to a distinguished line subbundle of T , which by mild abuse of notationwe call [ T ].The Levi-Civita connection e ∇ of the ambient metric defines the tractor con-nection ∇ T on T as follows. First note that the map T π : T G →
T M induces arealization of the tangent bundle by T x M = { η ∈ Γ( T G| G x ) : T δ s · η = η, s ∈ R + } / [ T | G x ].Now, for X ∈ Γ( T ) and ξ ∈ Γ( T M ), define ∇ T ξ X := e ∇ ξ X ,where ξ is an arbitrary lift of ξ to { η ∈ Γ( T G| G x ) : T δ s · ξ = ξ, s ∈ R + } in the aboverealization of T M . Direct computation verifies that the right-hand side is indepen-dent of the choice of lift, that it is a section of T , and that ∇ T is a vector bundleconnection. This construction of the conformal tractor bundle depends on the choiceof ambient metric e g , but different choices yield equivalent constructions. A proofthat this construction is equivalent to the standard tractor bundle of [BEG94] isgiven in [ ˇCG03]. It is a direct consequence of the compatibility of the ambient met-ric with its Levi-Civita connection that the tractor metric and tractor connectionare likewise compatible in that they satisfy ∇ T g T = 0.Given a tractor χ ∈ Γ( T ), counting homogeneities shows that we may regard g ( χ, T ) as a section of D [1], so this defines a tensorial canonical projection Π : Γ( T ) → Γ( D [1]); By construction, the kernel of this map is the space ofsections of Γ([ T ] ⊥ ), so Π descends to a natural bundle isomorphism T / [ T ] ⊥ ∼ = D [1].Conversely, there is a natural map L : Γ( D [1]) → Γ( T ), called the BGG splittingoperator , which is not tensorial but depends on the 2-jet of a section of D [1],that satisfies L (Π ( χ )) = χ for any parallel tractor X ∈ Γ( T ) (see, for example,[HS09, §§ IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 15 under the canonical projection. Furthermore, this image admits a natural geometricinterpretation.
Proposition 12.
The restrictions of the maps Π and L are isomorphisms (10) { parallel tractors } Π ⇄ L { almost Einstein scales } of vector spaces. In fact, the first modern formulation [BEG94] of the tractor bundle constructs T and ∇ T so that almost Einstein scales are exactly the (weighted) smooth functionswhose suitable prolongations yield sections of T parallel with respect to ∇ T .If σ is a nonzero almost Einstein scale of Einstein constant λ , the parallel tractor L ( σ ) satisfies h L ( σ ) , L ( σ ) i = − λ (this is immediate using a trivialization of thetractor bundle that we do not describe here; see, for example, [Lei06]). In particular, L ( σ ) is null if and only if σ is an almost Ricci-flat scale.3.2. Holonomy.
We briefly review and relate several notions of holonomy; see[KN96, Chapter 2] for a more detailed discussion.Let E → N be a vector bundle of rank k over a field F , and let ∇ E : Γ( E ) → Γ( E ⊗ T ∗ N ) be a general connection on E . For any piecewise smooth curve γ :[0 , → N , a local section σ of E is parallel along γ if ∇ Eγ ′ σ = 0. This is a first-order linear ordinary differential equation with piecewise smooth coefficients, so wedefine the linear parallel transport map P γ : E γ (0) → E γ (1) that sends ξ ∈ E γ (0) to σ ( γ (1)), where σ is the unique solution to the differential equation that satisfies σ ( γ (0)) = ξ . If γ is a loop based at u ∈ N , that is, if γ (0) = γ (1) = u , then since P γ is invertible (its inverse is just P γ − , where γ − ( t ) := γ (1 − t )), P γ − ∈ GL( E u ).Let Ω u denote the space of loops based at u . Definition 13.
Let E → N be a vector bundle with general connection ∇ E , andlet u be a point in N . The holonomy of ∇ E based at x is the groupHol u ( ∇ E ) := { P γ : γ ∈ Ω u ( N ) } ≤ GL( E u ).Picking a basis of E u identifies it with F k and so realizes Hol u ( ∇ E ) as an ex-plicit subgroup of GL( n, F ); any other choice of basis yields a conjugate subgroup,so without reference to bases we may regard Hol u ( ∇ E ) as a conjugacy class of sub-groups of GL( n, F ). If N is connected and v is another point in N , let α be anypath from u to v ; then by construction(11) Hol v ( ∇ E ) = P α Hol u ( ∇ E ) P α − ,so that conjugacy class is also independent of the base point u . So, for connectedmanifolds, we may suppress reference to a base point and call that conjugacy classthe holonomy Hol( ∇ E ) of ∇ E . For each x , we may regard the fiber E u as arepresentation of Hol u ( ∇ E ), whence (11) shows that the isomorphism type of therepresentation does not depend on the choice of base point.If U ⊂ N is an open subset containing u , then all the loops in U based at u areloops in N based there, and thus Hol u ( ∇ E | U ) ≤ Hol u ( ∇ E ). So, we define the localholonomy of ∇ E based at u to be the groupHol ∗ u ( ∇ E ) := \ β Hol u ( ∇ E | U β ) where β indexes the set of open subsets U β ⊆ N that contain u . The local holonomyis a connected subgroup of GL + ( T u E ), and there is some open set V containing u such that Hol ∗ u ( ∇ E ) = Hol u ( ∇ E | V ); we denote its Lie algebra by hol ∗ u ( ∇ E ).If a section σ is a parallel, then σ u is fixed by Hol u ( ∇ E ); conversely, any s ∈ E u fixed by Hol u ( ∇ E ) defines by parallel transport a parallel section σ such that σ u = s .Thus, for a parallel section σ , Hol u ( ∇ E ) ≤ Stab
GL( E u ) ( σ u ). Analogous statementshold for sections of tensor bundles ⊗ k E ∗ parallel with respect to the connectionsinduced by ∇ E .The holonomy of a vector bundle connection ∇ E is closely related to its curvature R E . Theorem 14 (Ambrose-Singer) . Let ∇ E be a connection on a vector bundle E → M , denote by R E its curvature viewed as a section of ⊗ T ∗ M → End( E ) , and fix u ∈ M . The underlying vector space of the Lie algebra hol u ( ∇ E ) ≤ End( E u ) of the holonomy group Hol u ( ∇ E ) is spanned by the elements P − γ R Eγ (1) ( X, Y ) P γ ,such that γ : [0 , → M is a smooth curve such that γ (0) = u , P γ is the paralleltransport map E u → E γ (1) defined by the connection that ∇ E induces on End( E ) ,and X, Y ∈ T γ (1) M . See, for example, [KN96, Chapter 2, Theorem 8.1] for the principal bundle ver-sion of this theorem.One can recover partial information about the holonomy of ∇ E at x to infiniteorder by differentiating curvature the curvature R E , and it will turn out that thisweaker version of holonomy will be all we need to prove the main result below. Definition 15.
Let ∇ E be a vector bundle connection on a vector bundle E → M , and fix u ∈ M . The infinitesimal holonomy (algebra) of ∇ E is the Liesubalgebra hol ′ u ( ∇ E ) ≤ gl ( E u ) generated by the endomorphisms( R Eu ) Cab D,m ··· m k X a Y b Z m · · · Z m k k for all k ≥ X, Y, Z , . . . , Z k ∈ T u N , where C and D are indices on the fiber E u , and where the covariant derivatives are taken with appropriate connections onΛ T ∗ M ⊗ E ⊗ ( ⊗ k T ∗ M ) given by coupling the connection on End( E ) induced by ∇ E with an arbitrary connection on M .By construction, hol ′ u ( ∇ E ) ≤ hol ∗ u ( ∇ E ) ≤ hol u ( ∇ E ) (see [KN96, Chapter 2,Proposition 10.4] for a proof of the first containment for principal connections). Wethen call the connected subgroup Hol ′ u ( ∇ E ) ≤ GL( E u ) with Lie algebra hol ′ u ( ∇ E )the infinitesimal holonomy (group) of ∇ E , and by construction, Hol ′ u ( ∇ E ) ≤ Hol u ( ∇ E ). (In fact, this becomes an equality when N is simply connected and theunderlying data is real-analytic.)We will invoke the holonomy construction for two particular connections. Givena pseudo-Riemannian metric h of signature ( p, q ) on an n -manifold N , the holonomyof h , which we denote Hol u ( h ) (or just Hol( h ) if we only care about its conjugacyclass) is just the holonomy Hol u ( ∇ h ) of its Levi-Civita connection, ∇ h . Since h itself is parallel, the induced action of Hol u ( h ) must preserve h u , that is, Hol u ( h ) ≤ O( h u ); passing to conjugacy classes gives Hol( h ) ≤ O( p, q ). IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 17 A k -plane field S is said to be parallel with respect to a connection ∇ if for allsections ξ ∈ Γ( S ) and all vectors η ∈ T N we have ∇ η ξ ∈ S . By construction,if a k -plane field S is parallel, then Hol u ( ∇ ) fixes S u ; conversely, if Hol u ( ∇ ) pre-serves a k -plane s ⊂ T u N , by parallel transport it defines a parallel plane field S such that S u = s . In particular, if S is parallel, then S u ≤ T u N is a subrepre-sentation of Hol u ( ∇ ); as usual if S is proper we say that Hol u ( h ) acts reducibly.One can show that any parallel plane field is integrable, so S defines a foliationof N by k -manifolds. Since the holonomy preserves h u and S u , the ( n − k )-planefield S ⊥ is parallel too. Then, if S is nondegenerate—that is, if S ∩ S ⊥ = { } ,which in particular is always the case if h is definite—the Hol u ( h )-representation T u N decomposes as T u N = S u ⊕ S ⊥ u . In this case, one can show that there is anopen set U containing u and integral manifolds V and W of S and S ⊥ through u , respectively, such that ( U, h | U ) is locally isometric to ( V × W, h V ⊕ h W ), where h V and h W are the respective pullbacks of h to those leaves. Then, by construc-tion, Hol( h | U ) = Hol( h V ) × Hol( h W ), so to understand pseudo-Riemannian metricholonomy (locally, anyway) it is enough to understand indecomposably acting ho-lonomy groups, that is, those for which the representation T u N of Hol u ( ∇ ) doesnot decompose as a direct sum of proper subrepresentations, where ∇ here denotesthe Levi-Civita connection. There is no full classification of these groups, but thereis a complete list of irreducibly acting holonomy groups of metrics, at least on sim-ply connected manifolds, and therefore of irreducibly acting metric local holonomygroups. (There are significant partial results, however, about which groups canoccur in the remaining case, that is, the local metric holonomy groups that actindecomposably but not irreducibly.) Theorem 16 (Berger’s List) . [Ber55] Let ( N, h ) be a simply connected pseudo-Riemannian n -manifold of signature ( p, q ) that is not locally a symmetric space(a pseudo-Riemannian manifold with parallel curvature tensor). If Hol(
N, h ) actsirreducibly, then up to isomorphism it is one of the following: SO( p, q ) , U( p , q ) , SU( p , q ) , Sp( p , q ) , Sp( p , q ) · Sp(1) , SO( n , C ) , G c (only in signatures (7 , and (0 , ), G (signatures (3 , and (4 , ), G C (only in real dimension 14), Spin(7) (signatures (8 , and (0 , ), Spin(3 , (signature (4 , ), and Spin(7 , C ) (real di-mension ). (In fact, Berger’s List originally included several other groups, but these were alllater shown to occur only for symmetric spaces [Bry87].) By the above discussion,if the holonomy Hol( N, h ) of a pseudo-Riemannian manifold acts indecomposablybut not irreducibly, it must admit some proper degenerate k -plane field S , and thusa proper parallel totally null plane field, namely, S ∩ S ⊥ . In particular, locally allholonomy groups of metrics that are not locally symmetric and that act indecom-posably but do not appear on Berger’s list must admit a proper parallel totally nullplane field.Now, given a conformal structure c of signature, say, ( p, q ), the conformalholonomy or tractor holonomy of c is the holonomy Hol( ∇ T ) of its tractorconnection. Since ∇ T is compatible with the (signature-( p + 1 , q + 1)) tractormetric g T , Hol( ∇ T ) ≤ O( p + 1 , q + 1).Given furthermore a choice of ambient metric e g for c , we call Hol( e g ) the ambientholonomy of c , though this choice in general depends on the choice of e g . Itfollows from the construction of the tractor connection from the ambient metricthat Hol( ∇ T ) ≤ Hol( e g ) for all ambient metrics e g of c [ ˇCG03]. We will use the following recent result of ˇCap, Gover, Graham, and Hammerl.
Theorem 17. [ ˇCGGH]
Let ( M, c ) be an conformal manifold of odd dimension atleast , fix u ∈ M , let e g be an ambient metric for c , and let z ∈ G be an element inthe fiber of G → M over u . Then, hol ′ u ( ∇ T ) ∼ = hol ′ z ( e g ) . There is an analogue of this theorem for even-dimensional manifolds, but thestatement is more subtle and we do not need it here.4.
Nurowski conformal structures
In [Nur05], Nurowski showed that to any generic 2-plane field D on a 5-manifold M one can associate a canonical conformal structure c D of signature (2 ,
3) on M ;we call any conformal structure that arises in this way a Nurowski conformalstructure . One can read from Nurowski’s original formulation that with respectto c D , D is totally null and [ D, D ] = D ⊥ .Given a 2-plane field D F defined by a smooth function F via the quasi-normalform, (7), we denote the conformal structure it induces via Nurowski’s constructionby c F := c D F . Nurowski’s remarkable formula for a representative g F of c F in thecoframe ( ω a ) is a sextic polynomial in the components of the 4-jet of F ; it containsmore than 70 terms, however, so we do not reproduce it here.One can realize Nurowski’s conformal construction much more compactly atthe cost of (substantial) abstraction, using the language of parabolic geometries.Cartan showed that a conformal structure of signature ( p, q ) on a manifold M canbe realized as a principal ˙ P -bundle over M endowed with a o ( p + 1 , q + 1)-valuedCartan connection satisfying a natural normalization condition (this connection iscalled the normal conformal connection), where ˙ P is the stabilizer of a null line in R p + q +2 endowed with an inner product of signature ( p + 1 , q + 1). The motivatingfeature of parabolic geometry (and more generally, Cartan geometry) is that itrealizes many geometries in a common framework, allowing them to be treated ina unified way.A generic 2-plane field D on a 5-manifold M can be realized as a principal P -bundle E → M endowed with a g -valued Cartan connection ω satisfying somenormalization criteria (recall that P is the stabilizer in G of a null line in Im e O ).Recall too that there is a natural embedding G ֒ → SO(3 , P =˙ P ∩ G , so we may extend any such E by forming the principal ˙ P -bundle ˙ E := E × P ˙ P ; then, we can extend ω to a o (3 , ω on ˙ E by equivariance, and thatequivariance guarantees that ˙ ω is a Cartan connection. One can furthermore showthat ˙ ω satisfies Cartan’s normalization condition, so it corresponds to a conformalstructure c D of signature (2 , D the Cartan curvature tensor A ∈ Γ( ⊙ D ∗ ) described in Subsection2.1: The conformal structure c D can be encoded in a CO(2 , M , and its Weyl curvature W (which is the harmonic curvature ofthe parabolic geometry corresponding to conformal geometry in dimension n ≥
4) can be regarded as a section of the bundle associated to the representationof Weyl-type symmetries tensored with an appropriate conformal density bundle.The underlying 2-plane field D determines a reduction of the frame bundle toGL(2 , R ) and hence induces natural decompositions of the corresponding CO(2 , , R )-representations. In particular, the Weyl curvature IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 19 decomposes into 15 pieces; all but five pieces turn out always to be zero, and thehighest-weight summand is exactly Cartan’s A up to a nonzero constant factor.One can use this Cartan connection approach to show that the conformal struc-ture c ∆ that the flat model ( Q , ∆) induces on Q is just the conformally flat structureinduced by the conformally flat structure on S × S via the projection S × S → Q .4.1. Characterization of Nurowski conformal structures.
Nurowski confor-mal structures (
M, c ) are characterized in terms of objects on the base manifold[HS09]; we give a version (that is essentially given in that reference) of that char-acterization in terms of tractor data. A tractor 3-form is a section of the bundleΛ T ∗ , and the tractor connection induces a connection, which we also denote ∇ T ,on that bundle. Given any vector bundle S → M and any section ϕ ∈ Γ(Λ S ∗ )such that ϕ x is generic for all u ∈ M , let H ( ϕ ) ∈ Γ( ⊙ S ∗ ) denote the bilinear formdefined respectively on each fiber S u by H ( ϕ ) u := H ( ϕ u ), where H is the map (2). Theorem 18. [HS09, Theorem A]
A conformal structure c on an oriented -manifold M is Nurowski (that is, there is a generic -plane field D on M suchthat c = c D ) if and only if there is a parallel tractor -form Φ ∈ Γ(Λ T ∗ ) compati-ble with the tractor metric g T in the sense that g T = H (Φ) . For any conformal manifold (
M, c ), just as for the (standard) tractor bundle T , there is a canonical projection Π : Γ(Λ T ∗ ) → Γ(Λ T ∗ M [3]) and an operator L : Γ(Λ T ∗ M [3]) → Γ(Λ T ∗ ) so that L (Π ( χ )) = χ for every parallel tractor3-form χ ∈ Γ(Λ T ∗ ). In particular such a 3-form is determined by its image underΠ , and the weighted 2-forms produced this way are exactly the so-called normalconformal Killing 2-forms of c (see [Lei05, HS09]). Given a 2-plane field D on anoriented 5-manifold M and a corresponding parallel tractor 3-form as in Theorem18, the compatibility condition forces the 2-form Π (Φ) to be nonvanishing andlocally decomposable, so its kernel is a 3-plane field on M , and this turns out tobe [ D, D ]. Since [
D, D ] ⊥ = D (where ⊥ denotes the orthogonal with respect tothe induced conformal structure c D ), D is exactly the 2-plane field spanned bythe (locally decomposable, weighted) bivector field given by raising both indices ofΠ (Φ).By naturality, every infinitesimal symmetry of a generic 2-plane field D on a 5-manifold is also an infinitesimal symmetry of the induced conformal structure c D ,defining a natural inclusion aut ( D ) ֒ → aut ( c D ). Hammerl and Sagerschnig showedone can realize the space of almost Einstein scales of c D as a natural complementto aut ( D ) in aut ( c D ): Theorem 19. [HS09, Theorem B]
Let D be a generic -plane field on a -manifold M , let c D denote the Nurowski conformal structure it induces, let Φ be a parallel -form characterizing D as in Theorem 18, and denote φ := Π (Φ) . There is anatural map aEs ( c D ) ֒ → aut ( c D ) defined by (12) σ φ ab σ b + φ ba,b σ where covariant derivatives are taken with respect to the Levi-Civita connection ofan arbitrary representative g ∈ c D , and where σ b := ∇ b σ . If we identify aEs ( c D ) with its image under this map, it is complementary to aut ( D ) in aut ( c D ) : aut ( c D ) = aut ( D ) ⊕ aEs ( c D ) . The projection aut ( c D ) → aEs ( c D ) defined by this decomposition is given by ξ a φ ab ξ b,a − ξ a φ bab, . Equation (12) corrects a sign error in the statement of the theorem in [HS09].4.2.
Holonomy groups of Nurowski conformal structures.
Let D be a generic2-plane field on an oriented 5-manifold. Since the tractor metric g T of the Nurowskiconformal structure c D is indefinite but is equal to H (Φ), the discussion before (2)implies that Φ is split-generic and that g T has signature (3 , c has signature (2 , V ) of a split-generic 3-form is G , so the characterization of holonomycontainment in terms of tensor stabilizers in Subsection 3.2 lets us reformulateTheorem 18 in the language of holonomy. Theorem 20.
A conformal structure c on a connected, oriented -manifold M isNurowski if and only if for some (equivalently any) x ∈ M , Hol x ( ∇ T ) ≤ G forsome copy of G contained in SO( g T x ) . Here, the containment is just the translation of the compatibility condition be-tween the split-generic parallel tractor 3-form and the tractor metric.
Remark 21.
We can extend the statement of Theorem 18 to nonoriented con-formal structures on 5-manifolds by replacing M with its orientation cover. Wecan correspondingly extend the statement of Theorem 20 to such conformal struc-tures by replacing G and SO( g T x ) with G · Z and O( g T x ), respectively, where thenontrivial element of Z is just − id on the underlying vector space T x .If D is real-analytic, so is the Nurowski conformal structure c D it induces, andthus we may associate to D the canonical real-analytic metric e g D of c D . Example 22.
For the 2-plane fields D F [ a ,b ] defined by the functions F [ a , b ]( x, y, p, q, z ) := q + a p + a p + a p + a p + a p + a p + a + bz ,where a := [ a , . . . , a ] and b are constants, Leistner and Nurowski computed the in-duced conformal structures and remarkably produced explicit corresponding Ricci-flat ambient metrics e g F [ a ,b ] and compatible split-generic 3-forms, which in particularshows that Hol( e g F [ a ,b ] ) ≤ G for all [ a , b ]. Computing directly using the proceduregiven in [GW12, §
5] shows that if at least one of a , a , a , or a is nonzero, then D F [ a ,b ] has root type [3 ,
1] everywhere except on at most three hyperplanes of theform { p = p } , where it has root type [4] or [ ∞ ]. Leistner and Nurowski showedfor any such [ a , b ] that Hol( e g F [ a ,b ] ) = G by eliminating the possibility of propercontainment using ad hoc methods and explicit tensorial data they computed forthis class.(For completeness, if a = a = a = a = 0 but a = − b , b , then D F [ a ,b ] has constant root type [4] and dim aut ( D F [ a ,b ] ) = 7, whence it is locally equivalentto D I for some constant I ; see Subsection 5.1 below. If a = a = a = a = 0and either a = − b or a = 2 b , then D F [ a ,b ] has constant root type [ ∞ ], that is, D F [ a ,b ] is locally flat.)Graham and Willse used Theorem 18 to show for all oriented, real-analytic D that Hol( e g D ) ≤ G , and extended Leistner and Nurowski’s arguments to show that,in a suitable sense, equality holds for nearly all D [GW12]. IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 21
In the following we are interested in the exceptions thereto: In Section 5, wewill give a broad class of 2-plane fields D for which the conformal holonomy andthe holonomy of suitable ambient metrics of the conformal structure c D are propersubgroups of G .To identify these groups explicitly, we will use the following proposition, which isjust a translation of Proposition 3 into the setting of holonomy using the stabilizercharacterization of holonomy in Section 3.2. Proposition 23.
Let ( N, h ) be a connected pseudo-Riemannian -manifold thatadmits a parallel split-generic -form Ψ that satisfies H (Ψ) = h and two linearlyindependent, parallel, null vector fields ξ and η . Then, the holonomy of h satisfies Hol( ∇ h ) ≤ H , if Ψ( ξ, η, · ) = 0 ; Hol( ∇ h ) ≤ R , if h ( ξ, η ) = 0 and Ψ( ξ, η, · ) = 0 ; Hol( ∇ h ) ≤ SL(2 , R ) , if h ( ξ, η ) = 0 . (See [Kat98, Corollary 2.5] for a related result about common stabilizers of non-isotropic spinors and related results in 2.4 there.)We can give an analogous result for tractor holonomy Hol( ∇ T ) by replacing( N, h ) with the tractor bundle T of a conformal structure and the tractor metric g T , Ψ with a section of Λ T ∗ , and ξ and η with tractors. Though we do notneed this result later, we record here a tractor holonomy version of part of thisproposition in terms of base data as an application of the BGG splitting operators. Proposition 24.
Let D be a generic -plane field on a -manifold M , and let φ ∈ Γ(Λ T ∗ M [3]) be the corresponding normal conformal Killing -form. If c admitstwo almost Ricci-flat scales σ, τ ∈ Γ( D [1]) such that neither is a constant multiple ofthe other (or equivalently, nonzero scales that correspond to non-homothetic metrics σ − g and τ − g ) and such that φ ab σ a τ b − φ aab, ( στ a − τ σ a ) = 0 ,then Hol( ∇ T ) ≤ H . Some special -plane fields on -manifolds Cartan’s class of highly symmetric -plane fields with root type [4] . In [Car10], Cartan solved the local equivalence problem for generic 2-plane fieldson 5-manifolds. He restricted attention to 2-plane fields with constant root typeand first classified 2-plane fields according to those types. To avoid a proliferationof cases, he (largely) further restricted attention to 2-plane fields whose symmetryalgebra has dimension at least 6. We summarize some of his results; for a relativelyaccessible but detailed exposition of Cartan’s arguments, see [Sto00, § Theorem 25. [Car10]
Let D be a generic -plane field on a -manifold M withconstant root type, and suppose that dim aut ( D ) ≥ . Then, one of the followingholds: • D has constant root type [ ∞ ] and so is locally flat, and thus aut ( D ) ∼ = g • D has constant root type [4] and dim aut ( D ) is either or • D has constant root type [2 , and dim aut ( D ) = 6 . Cartan produced what he claimed was a local normal form for D with constantroot type [4] and for which dim aut ( D ) ≥ § point in M one can find a neighborhood U of that point and construct a princi-pal bundle E → U and an adapted coframe ( η , η , η , η , η , π , π ) on E withstructure equations dη = 2 η ∧ π + η ∧ π + η ∧ η dη = η ∧ π + η ∧ η dη = Iη ∧ η + η ∧ π + η ∧ η dη = Iη ∧ η + η ∧ π + η ∧ π (13) dη = 0 dπ = 0 dπ = − π ∧ π − Iη ∧ η + η ∧ η ,where I is a smooth function on U ; by construction, D is the common kernel of thepullbacks of η , η , and η to M by any local section of E → U . Pulling back theforms π and π shows that the Lie algebra of the structure group of E is the uniquenonabelian 2-dimensional Lie algebra. Furthermore, Cartan constructed the frameso that the Cartan curvature of D is just A = ( η ) . Differentiating both sidesof the equations for dη and dπ shows that dI = Jη for some function J . Theinvariant I is fundamental in the sense that all other invariants, in particular J , arefunctions of I . In fact, Doubrov and Govorov have recently showed that Cartan’sclassification neglects at least one example: There is a generic 2-plane field on a5-manifold of constant root type [4] and with 6-dimensional symmetry algebra butwhich cannot be realized locally in the above form for any function I ; see Example38 for further discussion of this counterexample.Since dη = 0, locally we may take η = dx for a coordinate x ; substituting gives dI = J dx , so I is a function of x alone. One can then satisfy the pullback of thesystem (13) to M by an arbitrary section by setting η = dz + pI dy + q dp − [ q + Ip − (1 + I − I ′′ )] dxη = dy − p dxη = − dp + q dxη = dq − I dxη = dxπ = 0 π = − I ′ dy − I dp + [(1 + I − I ′′ ) y − I ′ p − Iq ] dx ,where we have suppressed pullback notation and suggestively used the variablesthat occur in the Monge normal form equation (6).The general solution to the system defining the 2-plane field D (which again isthe common kernel of the pullbacks of η , η , and η to M ) is y = f ( x ) p = f ′ ( x ) q = f ′′ ( x ) z = − Z [ f ′′ ( x ) + I ( x ) f ′ ( x ) + (1 + I ( x ) − I ′′ ( x )) f ( x ) ] dx . IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 23 (The coefficient corrects an arithmetic error in equation (6) in [Car10, § F I ( x, y, p, q, z ) = − [ q + Ip + (1 + I − I ′′ ) y ]for some smooth function I ( x ). Conversely, given any smooth function I on anopen subset of R , we denote by D I := D F I the generic 2-plane field ker { ω , ω , ω } that F I defines via (7); in particular, ∂ q F I = −
1, which is nowhere zero, so F I isgeneric.If I is constant, then (13) defines the structure constants for the (local) symmetryalgebra aut ( D I ) of D I , which thus has dimension 7. (Kruglikov analyzes most ofthese in detail in [Kru11b] but instead realizes these 2-plane fields using the Mongenormal form equation defined by F ( x, y, p, q, z ) = q m for constants m . Togetherthese 2-plane fields [excepting those for m = 0 ,
1, which are nongeneric] account forall of the 2-plane fields D I with constant I except the single case I = ± , whichis equivalent, for example, to the 2-plane field defined by the Monge normal formequation F ( x, y, p, q, z ) = log q [DG13].)If I is not constant, then since it is a function of x we can regard x itself as aninvariant of the structure. This invariant turns out to be inessential (see [Sto00, § D I by pulling back the structure equations (13) to a leaf { x = k } ; suppressingpullback notation, we get dη = 2 η ∧ π + η ∧ π + η ∧ η dη = η ∧ π dη = η ∧ π + η ∧ η dη = η ∧ π dπ = 0 dπ = π ∧ π ,so in this case dim aut ( D I ) = 6. Directly checking verifies Cartan’s claim [Car10,Subsection 44] that for constant I the symmetry algebra is solvable, and that thesame is true for nonconstant I . Remark 26.
For all smooth functions I , the symmetry algebra aut ( D I ) containsthe 2-dimensional subalgebra h ∂ z , y∂ y + p∂ p + q∂ q + 2 z∂ z i , but for general I it isdifficult to identify the other symmetries explicitly. With computer assistance onecan identify for constant I an explicit basis for the full 7-dimensional symmetryalgebra, but any such basis is surprisingly complicated, so we do not give one here.Nurowski’s formula for a representative g F of the conformal structure c F inducedby the 2-plane field D F specializes dramatically for the functions F I : Evaluating it(and rescaling by a constant for convenience) gives the representative metric(14) g I := g F I = − I ( ω ) + 3 ω ω − Ip ω ω − ω ω − ω ) ;the coframe ( ω i ) is that defined in (8). Here and henceforth, we identify a function I ( x ) with its pullback to the space M I := { ( x, y, p, q, z ) : x ∈ dom I } by the projection ( x, y, p, q, z ) x .We will show that the conformal structures c I := c F I = [ g I ] determined by thefunctions F I all have tractor and ambient holonomy isomorphic to a particularproper subgroup of G .For use in our below proof of the main theorem, we give explicit formulae forsome of the objects constructed above for the 2-plane fields D I . Proposition 27.
Let I ( x ) be a smooth function whose domain is open and con-nected, and set f M I := R + × M I × R .The generic -plane field D I on the -manifold M I is given by D I = (cid:10) ∂ q , ∂ x + p∂ y + q∂ p − [ q + Ip + (1 + I − I ′′ ) y ] ∂ z (cid:11) ,and the Nurowski conformal structure c I it defines contains the representative g I defined by (14) .The metric e g I on f M I defined by (15) e g I = 2 ρ dt + 2 t dt dρ + t (cid:0) g I − ρI dx (cid:1) is a Ricci-flat ambient metric for c I (we suppress the notation for pulling back g I by the projection Π : f M I → M I defined by Π : ( t, u, ρ ) u ).The -form e Φ I ∈ Γ(Λ T ∗ f M I ) defined by e Φ I := C [ − t dt ∧ ω ∧ ω − t dt ∧ ω ∧ dρ − t ω ∧ ω ∧ ω + 10 t Ipω ∧ ω ∧ ω − t Iω ∧ ω ∧ dρ + 3 t ω ∧ ω ∧ ω + t ω ∧ ω ∧ dρ + ( − t I dt ∧ ω ∧ ω + t dt ∧ ω ∧ ω ) ρ ] is parallel and satisfies H ( e Φ I ) = e g I , where C = 2 − / − / , and where we suppressthe notation for the pullback of I and the coframe forms ω a by Π . In particular, e Φ I is split-generic.The parallel tractor -form associated to D I is Φ I := e Φ I | G , and the trivializationof the normal conformal Killing form φ ∈ Γ(Λ T ∗ M [3]) with respect to g I is (16) φ I := − C ω ∧ ω .Proof. The formula for D I is given in Subsection 2.4. The metric e g I has homogene-ity 2 by construction, and inspection shows that pull it back to G yields g = t g I ,so it is a pre-ambient metric for c I ; computing directly shows that it is Ricci-flat, soit is in fact an ambient metric for c I . Computing directly shows that e Φ I is paralleland satisfies H ( e Φ I ) = e g I , and φ I = Π (Φ I ). (cid:3) Proposition 28.
Let I ( x ) be a smooth function on an open interval. The functions σ ( x ) in the -dimensional solution space S of the homogeneous, linear, second orderordinary differential equation (17) σ ′′ − Iσ = 0 ,are almost Ricci-flat scales of c I (trivialized with respect to the representative g I ). Inparticular, every tractor L ( σ ) in the corresponding -dimensional vector subspace L ( S ) ⊂ Γ( T ) is null.For each solution σ ∈ S , the vector field (18) ξ σ := d ( σt ) ♯ = t − ( − σ ′ ∂ z + σ∂ ρ ) ∈ Γ( T f M I ) IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 25 is parallel and null, the tractor ξ σ | G produced by restricting it is just L ( σ ) , and theinfinitesimal symmetry of c I corresponding to σ via (12) is given in the frame (9) by − ( σE + 4 σ ′ E ) .In the definition of ξ σ , we have suppressed the notation for the pullback of σ by theprojection R + × M I × R → M I .Proof. The trivialized Ricci-flat scales are exactly the functions σ of ( x, y, p, q, z )such that Ric( σ − g I | M − Σ ), where Σ is the zero set of σ . Consider functions σ thatdepend only on x ; computing directly gives that on M − Σ,Ric( σ − g I ) = 3 σ − ( σ ′′ − Iσ ) dx ,where a prime ′ denotes the derivative ∂ x . For such σ , the Ricci curvature vanishesif and only if the quantity σ ′′ − Iσ does. The remaining claims follow from directcomputation. (cid:3) Remark 29.
In fact, direct (but tedious) analysis of the equation Ric( σ − g I | M − Σ ) =0 for general σ (that is, not just those that depend only on x ) shows that these ac-count for all of the almost Einstein scales of g I , but we will see this follows indirectlyfrom below results.With the above data in hand, we are prepared to prove the main results. Proof of Theorem A.
By Proposition 27 e g I admits a parallel split-generic 3-form e Φ I that satisfies H ( e Φ I ) = e g I , and by Proposition 28 it admits two linearly independentparallel null vector fields, say, ξ and η . Computing gives that e Φ( ξ, η, · ) = 0, so byProposition 23, Hol( e g I ) ≤ H .We now show that the infinitesimal holonomy hol ′ ( e g I ) of the ambient metric e g I has dimension at least 5. Then, since Hol ′ ( e g I ) ≤ Hol( e g I ), both holonomy groupsare equal to H .We compute the infinitesimal holonomy of e g I using the definition; its curvatureis(19) e R = 60 t ( ω ∧ ω ) .Fix u ∈ f M I ; then, hol ′ u ( e g I ) admits a filtration ( V ru ) by the vector spaces V ru ⊂ End( T u f M ) spanned by endomorphisms generated by at most r derivatives of cur-vature: More precisely, V ru = V r | u , where V r := { e R CAB D,J ··· J k X A Y B Z J · · · Z J k k : k ≤ r ; X, Y, Z , . . . , Z k ∈ Γ( T f M I ) }⊆ Γ(End( T f M I )).We compute the filtered pieces one at a time. Respectively the definition and aneasy induction using the Leibniz rule give (cid:26) V = { e R CAB D X A Y B : X, Y ∈ Γ( T f M I ) } V r = V r − ∪ { S CD,J Z J : S ∈ V r − ; Z ∈ Γ( T f M I ) } , r > V = h ψ i , where ψ := E ⊗ ω + E ⊗ ω ∈ Γ(End( T f M I )),and h · i denotes the span over C ∞ ( T f M I ). Next, computing ( ψ ) CD,J and contracting with an arbitrary vector field Z J givesthat V = h ψ , ψ , ψ i , where ψ := t − E ⊗ dt + 15 ∂ ρ ⊗ ω ψ := 4 E ⊗ ω − E ⊗ ω − t − E ⊗ dt + 90 ∂ ρ ⊗ ω .Continuing gives that V = h ψ , ψ , ψ , ψ i , where ψ := 3 E ⊗ ω +3 E ⊗ ω − IpE ⊗ ω +9 t − E ⊗ dt +6 IE ⊗ ω +180 ∂ ρ ⊗ ω .and that V = h ψ , ψ , ψ , ψ , ψ i , where ψ := t − E ⊗ dt + t − IE ⊗ dt + 15 I∂ ρ ⊗ ω − ∂ ρ ⊗ ω + 50 Ip∂ ρ ⊗ ω .In particular, the given generating set of V is linearly independent at every pointin f M , so dim hol ′ u ( e g I ) ≥ dim V u = 5. (In fact, since the Lie algebra of Hol u ( e g I ) hasdimension at most 5, this gives hol u ( e g I ) = hol ′ u ( e g I ) = V p .)If we take u to be a point in G , then Theorem 17 shows that hol ′ ( ∇ T I ) = hol ′ ( e g I ) = h . Since M I is simply connected, Hol( ∇ T I ) = H .We could have avoided using Theorem 17 and instead showed directly the equal-ity of the infinitesimal holonomy algebras by expanding the derivatives e R Cab D,J ··· J k to third order in Christoffel symbols; using the relationship between the tractorand ambient curvature tensors, one can then show that the restrictions of ψ a to G ,1 ≤ a ≤
5, which by construction are sections of End( T ), can all be produced bytaking derivatives of tractor curvature. (cid:3) Corollary 30.
Let I be a smooth function on an open interval. Then, the space aEs ( c I ) of almost Einstein scales of D I is exactly the -dimensional space S ofalmost Ricci-flat scales identified in Proposition 28. Then, the dimension of theconformal symmetry algebra of c I is dim aut ( c I ) = (cid:26) , I not constant , I constant .Proof. Analyzing the representation V of G in Subsection 2.3 shows that thevectors preserved by the restriction of that representation to H = Stab G ( e ) ∩ Stab G ( e ) are exactly those in the 2-dimensional subspace h e , e i . Since Hol( ∇ T I ) =H , the space of parallel sections of T is 2-dimensional, and by the correspondence(10), aEs ( c I ) has dimension 2 and hence must coincide with S .Now, Theorem 19 gives thatdim aut ( c I ) = dim aut ( D I ) + dim aEs ( c I ) = dim aut ( D I ) + 2.By the discussion at the beginning of the section, the symmetry algebra aut ( D I )has dimension 7 if I is constant and dimension 6 if not. (cid:3) Remark 31.
Metrics admitting the types of parallel objects the metrics e g I do ad-mit many additional parallel objects. Let ( N, h ) be a connected pseudo-Riemannian7-manifold that admits a parallel split-generic 3-form Ψ that satisfies H (Ψ) = h and a special null plane field S comprised of parallel vector fields, then S itselfis parallel, as is the conull 5-plane field S ⊥ ⊃ S . Given any nonzero parallel nullvector field ξ ∈ Γ( S ), all plane fields in the complete flag field0 ⊂ [ ξ ] ⊂ S ⊂ Ann ξ ⊂ (Ann ξ ) ⊥ ⊂ S ⊥ ⊂ [ ξ ] ⊥ ⊂ T N
IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 27 on N are parallel (and null or conull) and hence are subrepresentations of Hol( h );here, Ann ξ is the 3-plane field with fiber (Ann ξ ) x := Ann( ξ x ). Given a secondparallel null vector field that is not a multiple of ξ , one can produce further parallelplane fields by forming the intersections and spans (and the orthogonal plane fieldsthereof) of the pieces of the corresponding flag fields.If we fix u ∈ N and identify ξ u with e ∈ V and S with h e , e i , thenAnn ξ u = h e , e , e i ,Ann ξ ⊥ u = h e , e , e , e i , S ⊥ u = h e , e , e , e , e i , and[ ξ ] ⊥ u = h e , e , e , e , e , e i .Then, consulting (5) shows that even though the holonomies of the metrics e g I andthe corresponding connections ∇ T I do not act irreducibly, they do act indecompos-ably. Remark 32.
Let I be a smooth function on an open interval, and let ξ be a parallelvector field on ( f M I , e g I ) (see (18)). Again if we fix u ∈ f M I and identify ξ u with e ∈ V , then [ ξ u ] ⊥ = h e , e , e , e , e , e i , and consulting (5) again shows that thissubrepresentation of H is faithful but that no proper subrepresentation thereofis. (Restricting to G ⊂ f M I yields the analogous statement for parallel tractors andtractor holonomy.) Since [ ξ u ] is null, however, the pullback of e g I to any leaf L of thefoliation defined by the plane field [ ξ ] ⊥ (which is integrable because it is parallel)via the inclusion ι : L ֒ → f M I is degenerate: By construction, at each point u ∈ L , ξ u is in T u L and is orthogonal to every vector in that space.The 2-form ι ∗ e g I degenerates only along this direction, however, so it descendsto a pseudo-Riemannian metric g on the space of integral curves of ξ | L ∈ Γ( T L ).The representation Hol u ( e g I ) | [ ξ u ] ⊥ fixes ξ u , so it descends to a representation on thequotient space [ ξ u ] ⊥ / [ ξ u ], which by construction we may identify with the holonomyrepresentation Hol [ u ] ( g ) of g at the integral curve [ u ] through u . If we yet againidentify ξ u with e ∈ V , then [ ξ u ] ⊥ / [ ξ u ] ∼ = h e , e , e , e , e i , and consulting (5)shows that Hol [ u ] ( g ) ∼ = R .Now, let S be the parallel 2-plane field comprising the null parallel vector fieldsof e g I . By construction, each plane field in the complete flag field0 ⊂ S/ [ ξ ] ⊂ Ann ξ/ [ ξ ] ⊂ (Ann ξ ) ⊥ / [ ξ ] ⊂ S ⊥ / [ ξ ] ⊂ T L/ [ ξ ]on the space of integral curves is parallel (and null or conull). (In fact, (5) showsthat every 2-, 3-, or 4-plane P u such that S u / [ ξ u ] ⊆ P u ⊆ S ⊥ u / [ ξ u ] extends to aparallel k -plane field.)Again, S ⊥ u / [ ξ u ] is a faithful representation of H but no proper subrepresentationis. Furthermore the pullback of g to any leaf of the foliation determined by theplane field S ⊥ / [ ξ ] is again degenerate, but at each point in the leaf, the pullbackdegenerates only along the direction S/ [ ξ ]. So, it descends to a metric on the (3-dimensional) space of integral curves of the line field S/ [ ξ ], which we may alsointerpret as the space of leaves of the foliation determined by the plane field S | R ina leaf R of the foliation determined by S ⊥ ⊂ T M , and again consulting (5) showsthat the holonomy of this metric is trivial.One can interpret the ideas in the previous remark to determine the metricholonomy of distinguished representatives of the conformal classes c I . Proposition 33.
Let I be a smooth function on an open interval, and let g ∈ c I bea Ricci-flat representative. Then, g admits a parallel null vector field and Hol( g ) ∼ = R .Proof. Let σ ∈ Γ( D [1]) be the Ricci-flat scale so that g = σ − g (in particular, σ vanishes nowhere, and so by changing sign if necessary, we may assume that σ is everywhere positive). By Proposition 28, σ must be in the 2-dimensionalvector space identified therein, and the vector field ξ σ ∈ Γ( T f M I ) is parallel withrespect to the Levi-Civita connection of e g I . Then, the orthogonal plane field [ ξ σ ] ⊥ is ker( ξ σ ) ♭ = ker d ( tσ ) (where, as in that proposition, σ has been trivialized bythe representative g I ∈ c I ), and the leaves of the foliation defined by this planefield are the hypersurfaces L C := { t = C/σ ( x ) } , C >
0. On L C , the integralcurve [ u ]( τ ) of ξ σ | L C satisfying the arbitrary initial condition [ u ](0) = u :=( C/σ ( x ) , x , y , p , q , z , ρ ) is γ ( τ ) = ( C/σ ( x ) , x , y , p , q , z − C σ ( x ) σ ′ ( x ) τ, ρ + C σ ( x ) τ ),which is defined for all time τ . In particular, every integral curve [ u ] intersectsthe hypersurface L C ∩ G = L ∩ { ρ = 0 } exactly once, so we may identify it with thespace of integral curves [ u ], but by construction L C ∩ G is the image of the metric g regarded as a section M → G . Unwinding definitions shows that (1) the inducedmetric on the space of integral curves of ξ σ | L is just the pullback of e g I to L C ∩ G ,and (2) if we identify M with L C ∩ G , this pullback is just g itself. (cid:3) The referee observed that parts of the proof of this proposition can be simplifiedsome using some easy facts about the ambient metrics of Ricci-flat metrics, includ-ing that for such a metric g , there are coordinates r and s on the ambient space forwhich the metric e g = 2 dr ds + r g is an ambient metric for [ g ], where as usual wesuppress pullback notation.5.2. Plane fields defined by ODEs z ′ = F ( y ′′ ) . Many of the above results holdjust as well for the class of 2-plane fields D F ( q ) defined via (7) for the smoothfunctions F ( q ) that depend only on q and for which F ′′ ( q ) is nowhere zero, so that D F ( q ) is generic; by Subsection 2.4, these 2-plane fields encode ordinary differentialequations z ′ = F ( y ′′ ), where y and z are functions of x . (Nurowski consideredthis class of 2-plane fields as an example in [Nur05], essentially gave equations (23)and (26), and observed that for generic functions F ( q ) the root type of D F ( q ) isgenerically equal to [4]; see below.) We identify F ( q ) with its pullback to the set M F ( q ) := { ( x, y, p, q, z ) : q ∈ dom F } by the projection ( x, y, p, q, z ) q . These 2-plane fields again all have symme-try algebra with dimension at least 6, and we can now identify these symmetriesexplicitly:(20) aut ( D F ( q ) ) ≥ (cid:28) ∂ x , ∂ y , ∂ z , x∂ x + 2 y∂ y + p∂ p + z∂ z , x∂ y + ∂ p ,F ′ ∂ x + ( pF ′ − z ) ∂ y + ( qF ′ − F ) ∂ p + Z F ′′ F dq · ∂ z (cid:29) .As mentioned earlier, if D is a generic 2-plane field for which dim aut ( D ) = 7, then D is locally equivalent either to D q m for some constant m or to D log q . Conversely, if m
6∈ {− , , , , , } , then dim aut ( D q m ) = 7 and the symmetry algebra is spanned IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 29 by the right-hand side of (20) and y∂ y + p∂ p + q∂ q + mz∂ z [Kru11b]; the symmetryalgebra of D log q is spanned by the right-hand side of (20) and y∂ y + p∂ p + q∂ q + x∂ z .If m ∈ {− , , , } , then D q m is locally flat, and if m = 0 or m = 1, then ∂ q ( q m ) = 0 and so D q m is not generic.Computing directly using the procedure given in [GW12, § D F ( q ) is(21) A F ( q ) = ( F ′′ ) − Ψ[ F ′′ ] dq ,where Ψ : C ∞ (dom F ) → C ∞ (dom F ) is the nonlinear differential operator(22) Ψ[ U ] := 10 U (4) U − U ′′′ U ′ U − U ′′ ) U + 336 U ′′ ( U ′ ) U − U ′ ) .So, these 2-plane fields are closely related to the class of 2-plane fields D I describedabove, but the root type of a 2-plane field D F ( q ) need not be constant: D F ( q ) hasroot type [ ∞ ] at ( x, y, p, q, z ) ∈ M F ( q ) if Ψ[ F ′′ ]( q ) = 0 and root type [4] at thatpoint otherwise. To the knowledge of the author, it is unknown whether every D I can be locally realized (at each point) as a 2-plane field D F ( q ) for some F ( q ).For completeness, we collect some explicit data for the 2-plane fields D F ( q ) intwo propositions; they are produced in the same way as are their analogues inPropositions 27 and 28, so we suppress the proofs. Proposition 34.
Let F ( q ) be a function on an open interval such that F ′′ vanishesnowhere, and set f M F ( q ) := R + × M F ( q ) × R .The generic -plane field D F ( q ) on the -manifold M F ( q ) is given by D F ( q ) = h ∂ q , ∂ x + p∂ y + q∂ p + F ( q ) ∂ z i ,and the Nurowski conformal structure c F ( q ) it defines contains the (again, relativelysimple) representative (23) g F ( q ) = 30( F ′′ ) ω ω + [ − F (4) F ′′ + 4( F ′′′ ) ] ω − F ′′′ ( F ′′ ) ω ω + 30( F ′′ ) ω ω − F ′′ ) ω .The metric e g F ( q ) on f M F ( q ) defined by (24) e g F ( q ) = 2 ρ dt + 2 t dt dρ + t (cid:18) g F ( q ) − ρ F (4) F ′′ − F ′′′ ) F ′′ ) ( ω ) (cid:19) is a Ricci-flat ambient metric for c F ( q ) (we suppress the notation for pulling back g F ( q ) and ω by the projection Π : f M F ( q ) → M defined by Π : ( t, x, ρ ) x ).The -form e Φ F ( q ) ∈ Γ(Λ T ∗ f M F ( q ) ) defined by e Φ F ( q ) := C ′ [( F ′′ ) t dt ∧ ω ∧ ω + F ′′′ t dt ∧ ω ∧ dρ − ( F ′′ ) t dt ∧ ω ∧ dρ + F ′′′ ( F ′′ ) t ω ∧ ω ∧ ω − ( F ′′ ) t ω ∧ ω ∧ ω − ( F ′′ ) t ω ∧ ω ∧ ω + ( F (4) − F ′′′ ) ( F ′′ ) − ) t ω ∧ ω ∧ dρ + F ′′′ F ′′ t ω ∧ ω ∧ dρ + ( F ′′ ) t ω ∧ ω ∧ dρ + [ (103 F (4) − F ′′′ ) ( F ′′ ) − ) t dt ∧ ω ∧ ω + F ′′′ F ′′ t dt ∧ ω ∧ ω + ( F ′′ ) t dt ∧ ω ∧ ω ] ρ ] , is parallel and satisfies H ( e Φ F ( q ) ) = e g F ( q ) , where C ′ = 2 / / / , and where wesuppress the notation for the pullback of F and the coframe forms ω a by Π . Inparticular, e Φ F ( q ) is split-generic.The parallel tractor -form associated to D F ( q ) is Φ F ( q ) := e Φ F ( q ) | G , and thetrivialization of the normal conformal Killing form φ ∈ Γ(Λ T ∗ M [3]) with respectto g F ( q ) is (25) φ F ( q ) := C ′ ( F ′′ ) ω ∧ ω . The form of the ambient metric (24) is different in [Nur08]; there, Nurowskistarts with a different representative metric and uses supplemental variables t and u := − ρt on the ambient space instead of t and ρ . Proposition 35.
Let F ( q ) be a function on an open interval such that F ′′ vanishesnowhere. The functions σ in the -dimensional solution space S of the homoge-neous, linear, second order ordinary differential equation (26) 10( F ′′ ) σ ′′ − F ′′′ F ′′ σ ′ + ( − F (4) F ′′ + 56( F ′′′ ) ) σ = 0 ,are almost Ricci-flat scales of c F ( q ) (that have been trivialized with respect to therepresentative g F ( q ) ). In particular, every tractor L ( σ ) in the corresponding ( -dimensional) vector subspace L ( S ) ⊂ Γ( T ) is null.For each solution σ , the vector field ξ σ := d ( σt ) ♯ = ( F ′′ ) − t − σ ′ ∂ y + t − σ∂ ρ ∈ Γ( T f M F ( q ) ) is parallel, and the tractor ξ σ | G produced by restricting it is just L ( σ ) , where wehave suppressed the notation for the pullback of σ by the projection R + × M F ( q ) × R → M F ( q ) . (Equation (26) could also be recovered by the corresponding Ricci-flatness equa-tion in [Nur08, §
3] by rescaling the variable Υ there by the appropriate conformalfactor and then changing variables via Υ = − log σ .)We also give an analog of Theorem A for the 2-plane fields determined by func-tions F ( q ). Theorem 36.
Let F ( q ) be a function on an open interval such that F ′′ vanishesnowhere, and let e g F ( q ) be the Ricci-flat ambient metric (24) of c F ( q ) . Let ∇ T F ( q ) denote the tractor connection of c F ( q ) . • If A F ( q ) = 0 , that is, if D F ( q ) is locally flat, then Hol( ∇ T F ( q ) ) ∼ = Hol( e g F ( q ) ) ∼ = { e } . • If A F ( q ) = 0 , then Hol( ∇ T F ( q ) ) ∼ = Hol( e g F ( q ) ) ∼ = H .Proof. By Proposition 34 e g F ( q ) admits a parallel split-generic 3-form e Φ F ( q ) thatsatisfies H ( e Φ F ( q ) ) = e g F ( q ) , and by Proposition 35 it admits two linearly independentparallel null vector fields, say, ξ and η . Computing gives that e Φ( ξ, η, · ) = 0, so byProposition 23, Hol( e g F ( q ) ) ≤ H .Computing gives that the curvature of f M F ( q ) is e R = t ( F ′′ ) − Ψ[ F ′′ ]( ω ∧ ω ) ,where Ψ is the differential operator given by (22). By (21), if A F ( q ) = 0 thenΨ[ F ′′ ] = 0 and so e R = 0. Since f M F ( q ) is simply connected, Hol( e g F ( q ) ) = { e } , andsince Hol( ∇ T F ( q ) ) ≤ Hol( e g F ( q ) ), Hol( ∇ T F ( q ) ) = { e } . IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 31
If Ψ[ F ′′ ] = 0, pick u ∈ M F ( q ) such that Ψ[ F ′′ ]( q ) = 0. Proceeding as in the proofof Theorem A, one can show that dim V u = 5, so dim hol ′ u ( e g F ( q ) ) ≥
5, and thusHol( e g F ( q ) ) = H . Again, Theorem 17 gives that Hol( ∇ T F ( q ) ) = H too. (cid:3) Other examples.
Some further examples of 2-plane fields reveal constraintson the possible relationships among root type [4], symmetry algebra dimension,and holonomy.The first example shows that having symmetry algebra of dimension at least 6is not a necessary condition for the holonomy of the tractor connection to be equalto H . Example 37.
Example 6 of [Nur05] states that, according to Cartan, every 2-plane field with root type [4] can be (presumably locally) realized as D F ( q ) forsome function F ( q ). The is untrue: Consider the 2-plane fields D F [ r ] defined forconstant r by F [ r ]( x, y, p, q, z ) = e y (cid:2) e − y q − e − y p ) r (cid:3) on { q > p } ; except when r = 0 , aut ( D F [ r ] ) ≥ h ∂ x , x∂ x − ∂ y − p∂ p − q∂ q ,x ∂ x − x∂ y − xp + 1) ∂ p − p + 2 xq ) ∂ q , ∂ z i ∼ = gl (2 , R ),Strazzullo computed that if r ∈ {− , } [Str09, Example 6.7.1] then D F [ r ] has roottype [4], and tedious analysis shows that for these values equality holds in (27)and hence in particular that dim aut ( D F [ r ] ) = 4. Since dim aut ( D F ( q ) ) ≥ F ( q ), the 2-plane fields D F [ − and D F [2] are not locally equivalent to D F ( q ) for any F , nor to D I for any function I . One can still find for both of theseexamples, however, an explicit Ricci-flat ambient metric for the induced conformalclass and, proceeding as in the proofs of Theorems A and 36, show that the tractorconnection and the ambient metric both have holonomy H . In particular, thisexample suggests that there might be a much broader class of 2-plane fields withassociated holonomy groups equal to H than the classes D I and D F ( q ) consideredin this paper. Example 38. [Doubrov & Govorov’s Counterexample] Recently Doubrov and Gov-orov constructed the 2-plane field D ∗ := D F defined by the function [DG13] F ( x, y, p, q, z ) = y + q / .They computed that D ∗ has constant root type [4] and that aut ( D ∗ ) ∼ = sl (2 , R ) ⋊ h ,where h denotes the 3-dimensional Heisenberg algebra; in particular, aut ( D ∗ ) isnonsolvable and dim aut ( D ∗ ) = 6. All of the 2-plane fields D I have solvable symme-try algebra, however, so D ∗ is not locally equivalent to D I for any function I , whichdisproves Cartan’s longstanding claim that the 2-plane fields D I (locally) exhaustthe 2-plane fields of constant root type [4] and symmetry algebra of dimension atleast 6.Strazzullo claims that D F [2 / also has root type [4] and 6-dimensional symmetryalgebra [Str09, Example 6.7.2]; this, together with the fact aut ( D F [2 / ) containsa subalgebra isomorphic to sl (2 , R ) and so is nonsolvable, would mean that it,too, would be a counterexample to Cartan’s claim, though apparently this was notnoticed until later. Computing, however, gives that D F [2 / has constant root type [2 , ,
1] and that the claimed symmetry algebra is not correct (in fact, this wouldviolate Theorem 25). Presumably this is a typo: Checking shows that the 2-planefield D F [1 / does have constant root type [4] and the (6-dimensional) symmetryalgebra indicated for D F [2 / —the algebra is spanned by the right-hand of (27) and h e − y/ [4 ∂ x + 2 p ∂ p + (6 pq − p ) ∂ q − e y ∂ z ] ,e − y/ [4 x∂ x − ∂ y + 2 xp ∂ p + (6 xpq − q − p x + 2 p ) ∂ q − e y x∂ z ] i —so D F [1 / is also a counterexample to Cartan’s claim. In fact, Doubrov andGovorov state [DG13, Remark 3] that they will prove in a forthcoming paper [DG]that up to local equivalence, D ∗ is the unique 2-plane field with transitive symmetryalgebra of dimension at least 6 that Cartan did not identify. Checking shows thatthe symmetry algebra of D F [1 / is transitive, and so by that result D F [1 / is locallyequivalent to D ∗ .This is the only example of which the author is aware of a 2-plane field of constantroot type [4] for which the associated holonomy groups are G ; in particular, itshows that constant root type [4] is not a sufficient condition for having holonomyequal to H , even if one also assumes that the symmetry algebra has dimension atleast 6. It remains possible, however, that having root type [4] or [ ∞ ] at each pointis a necessary condition for an (oriented) 2-plane field to have associated holonomygroups equal to H , or even for those holonomy groups to be a proper subgroup ofG . Remark 39.
Analyzing the Nurowski conformal structure c ∗ induced by D ∗ revealsthat its behavior differs substantively from that of the structures c I induced by the2-plane fields D I in Cartan’s class and also enjoys additional unusual properties.First, the induced conformal structure c ∗ := c D ∗ is not almost Einstein, which,by Theorem 19 implies that aut ( c ∗ ) = aut ( D ∗ ), and by the discussion in 3.2 thatthe tractor and ambient holonomy groups associated to c ∗ are not contained in thestabilizer in G of any nonzero vector in the standard representation. Remarkably,with substantial effort one can solve explicitly for the Ricci-flat ambient metric e g ∗ of c ∗ —there are relatively few known classes of examples of conformal structuresthat are not almost Einstein for which this is true. (Moreover the exact expressionone most easily obtains for e g ∗ is not polynomial in ρ .) Using this expression onecan show using the techniques in this paper that the tractor and ambient holonomygroups associated to c ∗ are in fact the full group G , and hence this yields anotherexplicit example of a metric with this exceptional holonomy group. Because thesefeatures are of independent interest, we postpone further discussion of (and explicitdata for) this unusual example to a dedicated article currently in preparation [Wil]. References [BEG94] T. N. Bailey, Michael G. Eastwood, and A. Rod Gover. Thomas’s structure bundle forconformal, projective and related structures.
Rocky Mountain Journal of Mathematics ,24(4):1191–1217, 1994.[Ber55] Marcel Berger. Sur les groupes d’holonomie homog`enes de vari´et´es `a connexion affine etdes vari´et´es riemanniennes.
Bulletin de la Soci´et´e Math´ematique de France , 83:279–330,1955.[BH93] Robert L. Bryant and Lucas Hsu. Rigidity of integral curves of rank 2 distributions.
Inventiones mathematicae , 114(1):435–461, 1993.[BH12] John C. Baez and John Huerta. G and the Rolling Ball. Arxiv preprintarXiv:1205.2447 , pages 1–28, 2012.
IGHLY SYMMETRIC 2-PLANE FIELDS ON 5-MANIFOLDS 33 [Bry87] Robert L. Bryant. Metrics with exceptional holonomy.
Annals of mathematics ,126(3):525–576, 1987.[Car10] ´Elie Cartan. Les syst´emes de Pfaff ´a cinq variables et les ´equations aux d´eriv´ees partiellesdu second ordre.
Annales scientifiques de l’ ´E.N.S. 3e s´erie , 27:109–192, 1910.[Car23] ´Elie Cartan. Les espaces `a connexion conforme.
Ann. Soc. Polon. Math. , 2:171–221,1923.[ ˇCG03] Andreas ˇCap and A. Rod Gover. Standard tractors and the conformal ambient metricconstruction.
Annals of Global Analysis and Geometry , 24(3):231–259, 2003.[ ˇCGGH] Andreas ˇCap, A. Rod Gover, C. Robin Graham, and Matthias Hammerl.
In preparation .[ ˇCS09] Andreas ˇCap and Jan Slov´ak.
Parabolic Geometries I . Mathematical Surveys and Mono-graphs. American Mathematical Society, Providence, 2009.[DG] Boris Doubrov and Artem Govorov. Classification of generic 2-distributions on 5-manifolds with simply transitive symmetry algebras.
In preparation .[DG13] Boris Doubrov and Artem Govorov. A new example of a generic 2-distribution on a5-manifold with large symmetry algebra.
Arxiv preprint arXiv:1305.7297 , pages 1–6,2013.[FG85] Charles L. Fefferman and C. Robin Graham. Conformal invariants.
The MathematicalHeritage of ´Elie Cartan (Lyon, 1984), Ast´erisque, Numero Hors Serie , pages 95–116,1985.[FG11] Charles L. Fefferman and C. Robin Graham.
The Ambient Metric . Annals of Mathe-matics Studies. Princeton University Press, Princeton, 2011.[Gou22] ´Edouard Goursat.
Le¸cons sur le probl`eme de Pfaff . Librairie Scientifique J. Hermann,Paris, 1922.[GW12] C. Robin Graham and Travis Willse. Parallel tractor extension and ambient metrics ofholonomy split G . J. Diff. Geom. , 92:463–505, 2012.[Har90] F. Reese Harvey.
Spinors and Calibrations . Academic Press, Boston, 1990.[Hit01] Nigel Hitchin. Stable forms and special metrics. In
Global Differential Geometry: TheMathematical Legacy of Alfred Gray , Contemporary Mathematics. American Mathe-matical Society, Providence, 2001.[HS09] Matthias Hammerl and Katja Sagerschnig. Conformal Structures Associated to GenericRank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decompo-sition.
Symmetry, Integrability and Geometry: Methods and Applications , 5, August2009.[Kat98] Ines Kath. G ∗ -structures on pseudo-Riemannian manifolds. Journal of Geometry andPhysics , 27:155–177, 1998.[KN96] Shoshichi Kobayashi and Katsumi Nomizu.
Foundations of Differenetial Geometry Vol-ume I . Wiley-Interscience, 1996.[Kru11a] Boris Kruglikov. Lie theorem via rank 2 distributions (integration of PDE of class ω =1). Arxiv preprint arXiv:1108.5854 , pages 1–26, 2011.[Kru11b] Boris Kruglikov. The gap phenomenon in the dimension study of finite type systems.
Arxiv preprint arXiv:1111.6315v2 , pages 1–20, 2011.[Lei05] Felipe Leitner. Conformal Killing forms with normalisation condition.
Rend. Circ. Mat.Palermo (2) Suppl , 75:279–292, 2005.[Lei06] Thomas Leistner. Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds.
Differential Geometry and its Applications , 24:458–478, 2006.[LN12] Thomas Leistner and Pawe l Nurowski. Conformal structures with G -ambient met-rics.
Ann. Sc. Norm. Super. Pisa Cl. Sci. , 11:407–436, 2012.[Nur05] Pawe l Nurowski. Differential equations and conformal structures.
Journal of Geometryand Physics , 55(1):19–49, 2005.[Nur08] Pawe l Nurowski. Conformal structures with explicit ambient metrics and conformal G holonomy. In Symmetries and overdetermined systems of partial differential equations ,pages 515–526. Springer, 2008.[Sag06] Katja Sagerschnig. Split octonions and generic rank two distributions in dimension five.
Arch. Math. (Brno) , 42:329–339, 2006.[Sto00] Olle Stormark.
Lie’s Structural Approach to PDE Systems . Encyclopedia of Mathemat-ics and Its Application. Cambridge University Press, Cambridge, 2000.[Str09] Francesco Strazzullo.
Symmetry Analysis of General Rank-3 Pfaffian Systems in FiveVariables . PhD thesis, Utah State University, 2009. [Tho26] Tracy Yerkes Thomas. On Conformal Geometry.
Proceedings of the National Academyof Sciences of the United States of America , 12(5):352–359, 1926.[Wil] Travis Willse.
In preparation . Mathematical Sciences Institute, Building 27, Australian National University, ACT2601, Australia
E-mail address ::