Almost Robinson geometries
aa r X i v : . [ m a t h . DG ] F e b ALMOST ROBINSON GEOMETRIES
ANNA FINO, THOMAS LEISTNER, AND ARMAN TAGHAVI-CHABERT
Abstract.
We investigate the geometry of almost Robinson manifolds, Lorentziananalogues of Hermitian manifolds, defined by Nurowski and Trautman as Lorentzianmanifolds of even dimension equipped with a totally null complex distribution ofmaximal rank. Associated to such a structure, there is a congruence of null curves,which, in dimension four, is geodesic and non-shearing if and only if the complexdistribution is involutive. Under suitable conditions, the distribution gives rise to analmost Cauchy–Riemann structure on the leaf space of the congruence.We give a comprehensive classification of such manifolds on the basis of their intrin-sic torsion. This includes an investigation of the relation between an almost Robinsonstructure and the geometric properties of the leaf space of its congruence. We alsoobtain conformally invariant properties of such a structure, and we finally study ananalogue of so-called generalised optical geometries as introduced by Robinson andTrautman.
Contents
1. Introduction 22. Algebraic description 52.1. Notation and conventions 52.2. Linear algebra 62.3. The stabiliser of a Robinson structure 142.4. One-dimensional representations of Q G of algebraic intrinsic torsions 162.6. Isotypic Q -submodules of G Q -submodules of G G -structures 28 Date : Thursday 11 th February, 2021 at 01:36.2010
Mathematics Subject Classification.
Primary 53C50, 53C10; Secondary 53B30, 53C18.
Key words and phrases.
Lorentzian manifolds, almost Robinson structures, G-structure, intrinsictorsion, congruences of null geodesics, conformal geometry, almost CR structures.AF was supported by GNSAGA of INdAM and by PRIN 2017 “Real and Complex Manifolds:Topology, Geometry and Holomorphic Dynamics”. TL was supported by the Australian ResearchCouncil (Discovery Program DP190102360). ATC & TL declare that this work was partially supportedby the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 PolishMNiSW fund. ATC was also supported by a long-term faculty development grant from the AmericanUniversity of Beirut for his visit to IMPAN, Warsaw, in the summer 2018, where parts of this researchwas conducted. He has also received funding from the GA ˇCR (Czech Science Foundation) grant 20-11473S. G -structures 756. Generalisation to other metric signatures and odd dimensions 78Appendix A. Projections 78Appendix B. Generalised almost Robinson geometry — connections 81References 821. Introduction
In a recent article [24], the authors give a comprehensive review of the notion of optical structure on a Lorentzian manifold p M , g q , simply understood as a null linedistribution K on M . Many of the geometric properties of this distribution and itsorthogonal complement are encoded in terms of its screen bundle H K “ K K { K , whichis naturally equipped with a bundle metric h inherited from g . One may naturallywish to endow H K with further bundle structures. In the present article, where weassume M to have dimension 2 m `
2, we equip H K with a bundle complex structure J compatible with h . Such a structure was introduced by Nurowski and Trautmanin [64, 105, 106], where it is equivalently described in terms of a totally null complex p m ` q -plane distribution N . The real span of the intersection N X N then determinesthe line distribution K . Following their terminology, we shall refer to the pair p N, K q asan (almost) Robinson structure . The structure group of the frame bundle is reduced to p R ą ˆ U p m qq˙p R m q ˚ , which is a subgroup of the group Sim p m q , which characterisesoptical structures, and as in [24], we shall describe the geometric properties of an almostRobinson structure in terms of its intrinsic torsion . LMOST ROBINSON GEOMETRIES 3
In dimension four, as is well-known [71, 64, 105, 106, 24], an almost Robinson struc-ture p N, K q is essentially equivalent to an optical structure. The key point, here, isthat the involutivity of the totally null complex 2-plane distribution N is equivalent tothe congruence K of null curves tangent to K being geodesic and non-shearing , that is,the conformal class of the bundle metric h is preserved along the curves of K . Whatis more, the rank-one complex distribution N { C K descends to the leaf space M of K ,thereby endowing it with a Cauchy–Riemann (CR) structure . This CR geometricalaspect of Robinson manifolds was particularly emphasised by Robinson, Trautman andthe ‘Warsaw’ school [83, 59, 50, 61]. This property is useful when seeking solutions tothe Einstein field equations [52, 92, 62], a problem that is in turn linked to analyticquestions regarding the embeddability of CR manifolds [90, 91, 51, 35, 88].In higher (even) dimensions, the congruence of null curves of an involutive almostRobinson structure p N, K q is always geodesic, but shearing in general [106]. The leafspace of K nevertheless still acquires a CR structure [64, 105]. In addition, (almost)Robinson structures are Lorentzian analogues of (almost) Hermitian structures, to bor-row the expression from Nurowski and Trautman [64]. In both cases, the underlyinggeometric object is that of an almost null structure , that is a totally null complex p m ` q -plane distribution. This perspective allows one to have a unified approachto pseudo-Riemannian geometry in any signature. In dimension four, the analogiesbetween Lorentzian and Hermitian geometries were already pointed out in [59, 60] es-pecially in connection with important theorems of mathematical relativity. Worthy ofmention are the following. ‚ The
Goldberg-Sachs theorem [26, 25] relates the existence of non-shearing con-gruences of null geodesics on algebraically special Einstein spacetimes has aRiemannian counterpart [79, 59, 5, 28]. ‚ The
Mariot–Robinson theorem [53, 80] gives a one-to-one correspondence be-tween analytic non-shearing congruences of null geodesics and null (i.e. degen-erate) electromagnetic fields in vacuum. ‚ The
Kerr theorem [69] tells us how such congruences arise in Minkowski spaceas complex submanifolds of three-dimensional complex projective space, canbe interpreted in Riemannian signature: any local Hermitian structure on four-dimensional Euclidean space corresponds to a holomorphic section of its twistorbundle [20, 86].In split signature, one obtains analogous results – see e.g. [31, 28]. These are also inti-mately connected with the notion of pure spinors , and thus hark back to ´Elie Cartan’sseminal work [12], which was subsequently developed in [8, 9, 46, 45]. It is then nosurprise that in dimension four, the spinorial approach to general relativity promotedby Penrose and his school [110, 68, 70, 71] shed much light on the complex aspect ofcongruences of null geodesics, and were influential in the development of twistor theory[69]. These ideas were later developed in higher even dimensions in [37, 38, 39, 40, 42],and most notably in the article [41] by Hughston and Mason, where the Kerr andRobinson theorems are generalised in the context of involutive almost null structures.These results were expanded by the third author of the present article in [95, 96, 97, 98],
ANNA FINO, THOMAS LEISTNER, AND ARMAN TAGHAVI-CHABERT where a comprehensive study of almost null structures according to their intrinsic tor-sion is given in both even and odd dimensions. The recent articles [67, 23, 72] alsotouch on related topics on pseudo-Riemannian manifolds.In dimension four, non-shearing congruences of null geodesics are ubiquitous in thestudy of null electromagnetic fields and exact solutions to the Einstein field equations[80, 26, 25]. One question that arises is, which of an optical structure and an (al-most) Robinson structure has most relevance in higher dimensions? On the one hand,Robinson–Trautman and Kundt spacetimes, which are by definition characterised bythe existence of a non-twisting non-shearing congruence of null geodesics, have beenwell studied in arbitrary dimensions, see e.g. [76, 74]. On the other hand, the Kerrmetric and its variants admit a pair of twisting congruences of null geodesics, which arenon-shearing in dimension four, but fail to be so in higher dimensions [77]. Nonetheless,as was first brought to light in [54], these metrics admit several Robinson structures inany dimensions . Almost Robinson structures can also be defined in terms of a maximaltotally null complex distribution on odd-dimensional Lorentzian manifolds: the blackring in dimension five is equipped with a pair of Robinson structures, but does notadmit any non-shearing congruences of null geodesics [93] — see also [94]. In dimensionthree, one can similarly obtain analogous results – see e.g. [63].Another reason why non-shearing congruences of null geodesics in higher dimensionsare not as common as in dimension four has to do with curvature. Let us reviewthe various geometric interpretations of the following algebraic condition on the Weyltensor: W p k, v, k, v q “ , for any sections k of K and v of K K ,(1.1)where K is an optical structure with congruence of null geodesics K . These are asfollows: ‚ In dimension four , if K is geodesic and non-shearing regardless of whether K isnon-twisting or not , then condition (1.1) holds — see e.g. [89, 71]. ‚ In dimensions greater than four, if K is non-shearing and non-twisting , thencondition (1.1) holds — in fact, the Weyl tensor satisfies an even strongercondition [66]. ‚ In odd dimensions greater than four, if K is twisting and condition (1.1) holds,then K must also be shearing [66]. ‚ In even dimensions greater than four, if K is twisting and non-shearing, andcondition (1.1) holds, then the twist induces an almost Robinson structure on p M , g q [99].In fact, in even dimensions greater than four, all the Einstein metrics admitting a non-shearing twisting congruence of null geodesics satisfying a mild algebraic condition onthe Weyl tensor were found in [99]. These depend on three real parameters, and includethe Taub–NUT–(A)dS metric and the Fefferman–Einstein metric — see also [4]. Wethus see that the non-shearing condition is rather restrictive in dimensions greater thanfour. These are not explicitly referred to as Robinson structures there, but may be interpreted as such. In [93], these Robinson structures are referred to as optical structures in a sense similar to [60].This terminology is now obsolete by virtue of [24].
LMOST ROBINSON GEOMETRIES 5
The structure of the paper is as follows. Section 2 focuses on the algebraic descrip-tion of the intrinsic torsion of an almost Robinson structure: we classify the possibleintrinsic torsion modules in Theorems 2.15 and 2.18. This is then translated into thelanguage of bundles in Section 3, where we introduce almost Robinson geometries andinvestigate the geometric properties of their associated congruences of null curves andleaf spaces. Their relation to pure spinors fields is explained in Theorem 3.44, and isused in Section 3.11 to make contact with almost Hermitian manifolds. In Section 4,our definition of almost Robinson manifold is extended to the conformal setting. Inparticular, Theorem 4.4 gives all the conformally invariant classes of the intrinsic tor-sion. Finally, we introduce the notion of generalised almost Robinson geometry firstconsidered in [82] in Section 5 by weakening the conformal structure to a certain equiv-alence class o of Lorentzian metrics on a smooth manifold. Of notable interest Theorem5.4, which lists all the possible o -invariant classes of the intrinsic torsion. Section 6discusses the possible generalisations to other metric signatures. We have relegated toAppendices A and B a number of technical formulae that are used in the main text.We shall provide examples to illustrate some of the algebraic conditions that theintrinsic torsion of an almost Robinson structure can satisfy, focussing essentially ondimensions greater than four. Considering the rich range of classes of almost Robinsonstructures, this article does not aim to cover every possible case, but it leaves theconstruction of almost Robinson structures with prescribed intrinsic torsion as openproblems — see for instance Remark 3.10. We do not touch on questions related tothe curvature of almost Robinson manifolds, these being dealt with in [95]. Finally,we point out the recent articles [3, 4] whose content overlaps with some aspects of thepresent article. 2. Algebraic description
Notation and conventions.
We set up the notation and conventions used through-out this article by recalling some basic notions of algebra – see e.g. [85, 16] for furtherdetails. The fields of real numbers and complex numbers will be denoted R and C respectively, the imaginary unit by i, i.e. i “ ´ V and W be two real or complex vector spaces with duals V ˚ and W ˚ . Theannihilator of a vector subspace U of V will be abbreviated to Ann p U q . The tensorproduct of V and W will be denoted V b W , the p -th exterior power of V by ^ p V , its p -th symmetric power by Ä p V .If g is a non-degenerate symmetric bilinear form on V , the orthogonal complementof a subspace U of V with respect to g will be denoted U K . The subspace of Ä p V consisting of elements tracefree with respect to g will be denoted by Ä p ˝ V .Let us assume that V is complex and of dimension 2 m . Under the Hodge dualityoperator ‹ : ^ p V ˚ Ñ ^ m ´ p V ˚ , for p “ , . . . , m , the space ^ m V ˚ splits into thespace of self-dual m -forms ^ m ` V ˚ and the space of anti-self-dual m -forms ^ m ´ V ˚ , i.e. ^ m V ˚ “ ^ m ` V ˚ ‘ ^ m ´ V ˚ , where ‹ α “ ˘p i q α for any α P ^ m ˘ V ˚ .Suppose now that V is real. The complexification C b V – V ‘ i V of a real vectorspace V will be denoted C V . There is an induced reality structure, ¯ : C V Ñ C V on C V , which preserves the elements of V , i.e. for v P C V , we have that v P V if and ANNA FINO, THOMAS LEISTNER, AND ARMAN TAGHAVI-CHABERT only if ¯ v “ v . If A is a vector subspace of C V , its complex conjugate is defined by A : “ t v P C V : v P A u . We say that A is (totally) real if A “ A .Suppose now that V has dimension 2 m and is equipped with a complex structure J ,that is an endomorphism of V that squares to the identity on V , i.e. J ˝ J “ ´ Id. Then C V “ V p , q ‘ V p , q , where V p , q and V p , q are the ` i- and ´ i eigenspaces of J respectively, each m -dimensional complex vector subspace, and complex conjugate to each other, i.e. V p , q – V p , q . Similarly, we have a splitting of the dual space C V ˚ “ p V p , q q ˚ ‘ p V p , q q ˚ , and p V p , q q ˚ “ Ann p V p , q q and p V p , q q ˚ “ Ann p V p , q q . For any non-negative integer p , q , the space of all p p, q q -forms on V is defined to be ^ p p,q q V ˚ : “ ^ p p V p , q q ˚ b ^ q p V p , q q ˚ . Similarly, we define the spaces ä p p,q q V ˚ : “ ä p p V p , q q ˚ b ä q p V p , q q ˚ , p V ˚ q : “ ! α P ^ p , q V ˚ b ^ p , q V ˚ : π p , q p α q “ ) . (2.1)where π p , q is the natural projection from ^ p , q V ˚ b ^ p , q V ˚ to ^ p , q V ˚ . Thisnotation reflects the Young tableau symmetries of this irreducible GL p m, C q -module,where GL p m, C q is the complex general linear group acting on V p , q – C m .Since we are interested in real vector spaces, we also define, following the notationin [85], rr ^ p p,q q V ˚ ss b R C : “ ^ p p,q q V ˚ ‘ ^ p q,p q V ˚ , p ‰ q , r ^ p p,p q V ˚ s b R C : “ ^ p p,p q V ˚ , (2.2)This notation will be extended in the obvious way to Ä p p,q q V ˚ and p V ˚ q .Finally, we shall consider a Hermitian vector space p V , J, h q where J is a complexstructure compatible with a positive-definite symmetric inner product h , i.e. J ˝ h “´ h ˝ J . Then V p , q – p V p , q q ˚ and V p , q – p V p , q q ˚ so that V p , q and V p , q aretotally null with respect to h . The Hermitian 2-form on V is defined by ω “ h ˝ J .For pq ‰
0, the subspace of ^ p p,q q V ˚ and Ä p p,q q V ˚ consisting of all p p, q q -forms thatare tracefree with respect to ω ´ will be denoted ^ p p,q q˝ V ˚ and Ä p p,q q˝ V ˚ respectively.Note that ^ p , q V ˚ – u p m q and ^ p , q˝ V ˚ – su p m q , where u p m q and su p m q are the Liealgebras of the unitary group U p m q and special unitary group SU p m q respectively.2.2. Linear algebra.
Null structures.
Let r V be a p m ` q -dimensional oriented complex vector spaceequipped with a non-degenerate symmetric bilinear form r g . We introduce abstractindices following the convention of [24]: minuscule Roman indices starting with thebeginnning of the alphabet a, b, c, . . . will refer to elements of r V and its dual, andtensor products thereof, e.g. v a P r V and α ab P r V ˚ b r V . Round brackets and squared LMOST ROBINSON GEOMETRIES 7 brackets over groups of indices will denote symmetrisation and skew-symmetrisationrespectively, e.g. T p ab q “ ´ T ab ` T ba ¯ , β r abc s “ p β abc ´ β acb ` β bca ´ β bac ` β cab ´ β cba q . In particular, the symmetric bilinear form satisfies r g ab “ r g p ab q , and together with itsinverse r g ab , will be used to raise and lower indices. The tracefree (symmetric) part ofa tensor with respect to g will be adorned with a small circle, e.g. either as T p ab q ˝ or p T ab q ˝ . Definition 2.1. [9, 46, 64, 96] A null structure on p r V , r g q is a maximal totally null(MTN) vector subspace of r V , i.e. N “ N K . In other words, for any v, w P N , r g p v, w q “ N has dimension m ` N on p r V , r g q singles out the one-dimensional vector subspace ^ m ` Ann p N q of ^ m ` r V ˚ . Any element ν of ^ m ` Ann p N q satisfies ν aa ...a m ν ab ...b m “ . (2.3)Conversely, any p m ` q -form ν on p r V , r g q that is totally null, i.e. satisfies (2.3), definesthe MTN vector subspace N “ ! v P r V : v ν “ ) . A null structure N is either self-dual if ^ m ` Ann p N q Ă ^ m ` ` r V ˚ , or anti-self-dual if ^ m ` Ann p N q Ă ^ m ` ´ r V ˚ .The space of all MTN vector subspaces of r V , i.e. null structures, is a complex homoge-neous space of complex dimension m p m ` q , referred to as the isotropic Grassmannian Gr m ` p r V , r g q of p r V , r g q . This space splits into two disconnected components Gr ` m ` p r V , r g q and Gr ´ m ` p r V , r g q according to whether the MTN is self-dual or anti-self-dual. The com-plex Lie group SO p m ` , C q acts transitively on each of these components. Remark 2.2.
Any complement of N in r V must be dual to N with respect to r g . Wewill thus write any splitting of N Ă r V as r V “ N ‘ N ˚ , bearing in mind that in general such a splitting is not canonical. For consistency withthe notation introduced subsequently, we shall assume with no loss of generality that N is self-dual. In abstract index notation, elements of Ann p N q will be adorned with lowerRoman majuscule letters, and elements of N ˚ with upper Roman majuscule letters,e.g. v A P N ˚ – Ann p N ˚ q and α A P Ann p N q – N . These indices will be immovable.We also introduce splitting operators p δ aA , δ aA q , that is, projections δ aA : r V ˚ Ñ N ˚ and δ aA : r V ˚ Ñ N that satisfy δ aA δ Ba “ δ BA . These will also be used to inject elements of N and N ˚ into r V . ANNA FINO, THOMAS LEISTNER, AND ARMAN TAGHAVI-CHABERT
Robinson structures.
Let V be a p m ` q -dimensional real vector space equippedwith a non-degenerate symmetric bilinear form g of (Lorentzian) signature p m ` , q ,i.e. p` , ` , . . . , ` , ´q . As is customary, we call p V , g q Minkowski space. The abstractindex notation introduced in the previous section will equally apply to p V , g q .Denote by C V the complexification of V , and extend g to a non-degenerate complex-valued symmetric bilinear form C g on C V . By abuse of notation, we shall often denote C g by g . The complexification p C V , C g q of p V , g q thus gives rise to the complex space p r V , r g q considered in the previous section, together with a reality condition. Thus, byextension, there is a well-defined notion of null structure on p V , g q via p C V , C g q . Tomake this idea more precise, we note that the complex conjugate N of a MTN vectorsubspace N on p C V , C g q is also MTN. Definition 2.3 ([46]) . The real index of a null structure N on p C V , C g q is the complexdimension of the intersection of N and N . Definition 2.4. A Robinson structure on Minkowski space p V , g q of dimension 2 m ` N of real index one on p C V , C g q . We shall denote it by the pair p N , K q where(1) N is an MTN vector subspace of C V of real index one,(2) K is the real null line N X V .With this second condition, we have that C K “ N X N and N ` N “ C K K .It turns out that the real Lie group SO p m ` , q also acts transitively on each ofthe connected spaces of MTN vector spaces of p C V , C g q . In other words: Lemma 2.5 ([46]) . Let p V , g q be Minkowski space of dimension m ` . A null structureon p C V , C g q always has real index one, and hence is a Robinson structure. Remark 2.6.
Note that p N , K q and p N , K q define the same Robinson structure. TheirHodge duality is the same when m is even, but opposite when m is odd. We shall saythat a Robinson structure p N , K q is (anti-)self-dual if N is (anti-)self-dual. This entailsof course a preference of N over N when m is odd, but there is no ambiguity when m is even.Any element ν of ^ m ` N will be referred to as a complex Robinson p m ` q -form .In particular, when m is odd, if ν is a self-dual, so are its complex conjugate ¯ ν and itsreal span ν ` ν .2.2.3. Robinson structures and optical structures.
It is clear that a Robinson structure p N , K q on p V , g q determines in particular an optical structure, namely K , in the senseof [24]. We therefore have a filtration of vector subspaces t u Ă K Ă K K Ă V , (2.4)and the screen space H K “ K K { K inherits a positive definite symmetric bilinear form h given by h p v ` K , w ` K q : “ g p v, w q , for any v, w P Γ p K K q .Any element of K will be referred to as an optical vector , and any element of Ann p K q as an optical -form . LMOST ROBINSON GEOMETRIES 9
Now, define an endomorphism J of H K and its complexification C H K by J p v ` C K q “ ´ i v ` C K , for any v P N , J p v ` C K q “ i v ` C K , for any v P N .Then J is a complex structure on H K , and C H K splits into the eigenspaces of J , i.e. C H K “ H p , q K ‘ H p , q K , (2.5)where H p , q K : “ N { C K and H p , q K : “ N { C K . These can be shown to be maximal totallynull with respect to the bilinear form h on C H K “ p N ` N q{p N X N q , and thus J iscompatible with h , i.e. J is Hermitian [64].Conversely, suppose that p V , g q is equipped with an optical structure K together witha complex structure J on the screen space H K compatible with h . Define N “ v P C K K : J p v ` C K q “ ´ i v ` C K ( . Then N has dimension m `
1, and is totally null. Indeed, for any v, w P N , we have g p v, w q “ h p v ` C K , w ` C K q“ h p i J p v ` C K q , i J p w ` C K qq“ ´ h p v ` C K , w ` C K q“ ´ g p v, w q , since J ˝ h “ ´ h ˝ J , and thus g p v, w q “
0. The complex conjugate N is definedanalogously.In abstract index notation, elements of H K and of its dual, and tensor productthereof, will be adorned with minuscule Roman indices starting from the middle of thealphabet i, j, k, . . . . In particular, the screen space inner product and its inverse will beexpressed as h ij and h ij respectively, and will be used to lower and raise this type ofindices. The complex structure and the Hermitian 2-form will take the form J ij and ω ij : “ J ik h kj respectively. As before, symmetrisation and skew-symmetrisation will bedenoted by round and squared brackets around indices respectively, and the tracefreepart of a tensor with respect to h ij will be adorned by a small circle.We shall use upper and lower Greek indices to denote elements of H p , q K and p H p , q K q ˚ respectively, and upper and lower overlined Greek indices to denote elements of H p , q K and p H p , q K q ˚ respectively, e.g. v α P H p , q K and α ¯ β P p H p , q K q ˚ . As usual, symmetrisationand skew-symmetrisation will be denoted by round brackets and squared brackets re-spectively. The Hermitian form on C H K will then be expressed as h α ¯ β and its inverseby h α ¯ β , which will be used to convert indices, i.e. v ¯ β “ v α h α ¯ β for any v α P H p , q K . Thetotally tracefree part of a mixed tensor T αβ ¯ γ , say, with respect to h α ¯ β will be denoted p T αβ ¯ γ q ˝ . We shall also introduce for convenience splitting operators p δ iα , δ i ¯ α q on C H K , that isprojections δ iα : C H ˚ K Ñ p H p , q K q ˚ and δ i ¯ α : C H ˚ K Ñ p H p , q K q ˚ that satisfy δ iα J ij “ i δ jα , δ i ¯ α J ij “ ´ i δ j ¯ α ,h ij δ iα δ i ¯ β “ h α ¯ β , h ij δ iα δ iβ “ . Their dual versions p δ αi , δ ¯ αi q can be obtained by raising and lowering the indices with h ij and h α ¯ β . These will be used to inject from H p , q K to C H K , and so on. Thus, inparticular, we can express h ij and ω ij “ J ik h kj as ω ij “ h α ¯ β δ α r i δ ¯ βj s , h ij “ h α ¯ β δ α p i δ ¯ βj q . Remark 2.7.
From the discussion above, it is also conceptually useful to start withan optical structure K on p V , g q , and declare a null or Robinson structure on p V , g q be compatible with K if K “ V X N . In dimension four, there is a single Robinson structure(up to complex conjugation) compatible with an optical structure, but this is not truein higher dimensions – see Remark 2.14.2.2.4. Splitting.
As before let p N , K q be a Robinson structure on Minkowski space p V , g q . Any choice of splitting C V “ N ‘ N ˚ , (2.6)for some choice of dual N ˚ induces a splitting of the filtration (2.4) as V “ K ‘ H K , L ‘ L , (2.7)where K “ N X V “ N X V , L : “ N ˚ X V “ N ˚ X V , H K , L : “ K K X L K . In particular, L is a null line dual to K . Note that H K , L is isomorphic to the screenspace H K , but this isomorphism depends on the choice of L . Further, in completeanalogy with (2.5), we obtain the splitting C H K , L “ H p , q K , L ‘ H p , q K , L , (2.8)where H p , q K , L : “ N X N ˚ “ N X C H K , L , H p , q K , L : “ N X N “ N X C H K , L . We note that H p , q K – H p , q K , L and H p , q K – H p , q K , L , but these isomorphisms depend onthe choice of N ˚ We introduce splitting operators p ℓ a , δ ai , k a q with dual p κ a , δ ia , λ a q adapted to (2.7),where k a and ℓ a are elements in K and in L respectively such that g ab k a ℓ b “
1, and δ ai projects from V ˚ to H ˚ K , L and satisfies g ab δ ai δ bj “ h ij . Any change of splitting whichpreserves k a induces the transformations k a ÞÑ k a , δ ai ÞÑ δ ai ` φ i k a , ℓ a ÞÑ ℓ a ´ φ i δ ai ´ φ i φ i k a , (2.9)for some φ i in p R m q ˚ , and similarly for their duals. LMOST ROBINSON GEOMETRIES 11
Finally, combining the splitting operators p ℓ a , δ ai , k a q and p δ iα , δ i ¯ α q yield new ones p ℓ a , δ aα , δ a ¯ α , k a q where δ aα : “ δ ai δ iα projects from C V ˚ to p H p , q K , L q ˚ . We naturally obtaindual spitting operators p κ a , δ αa , δ ¯ αa , λ a q .2.2.5. Robinson -forms. For any choice of splitting operators p κ a , δ ia , ℓ a q , and recallingthat ω ij is the Hermitian form on H K , we set ω ab “ ω ij δ ia δ jb and define ρ abc “ κ r a ω bc s . (2.10)By Lemma 3.1 of [24], the definition of ρ abc depends only on the choice of optical 1-form κ a , and not on the choice of δ ai and λ a . One can check that the 3-form ρ abc satisfies ρ ab e ρ cde “ ´ κ r a g b sr c κ d s . (2.11)We shall refer to such a 3-form as a Robinson -form (associated to the optical -form κ a ) .Conversely, let ρ abc be a 3-form that satisfies the algebraic property (2.11) for somenull 1-form κ a . Then, one can check that κ r a ρ bcd s “ k a ρ abc “
0. To prove κ r a ρ bcd s “
0, we skew (2.11) with κ a to get κ r a ρ bc s f ρ def “
0. Now contracting with ρ egh and using (2.11) again yields κ r a ρ bc sr d κ e s “
0, and the result follows by skewingover the first four indices. That k a ρ abc “ ω ij “ ℓ a δ bi δ cj ρ abc for any choice of splittingoperators p ℓ a , δ ai , k a q where κ a ℓ b “
1, we see that ω ij is the required Hermitian form on H K up to sign. This sign can be fixed so that if k a “ g ab κ b , v c ρ cab “ ´ v r a k b s , for any v a P N , w c ρ cab “ w r a k b s , w a P N . Remark 2.8. ‚ In dimension four, ρ bcd is the Hodge dual of κ a , i.e. κ “ ‹ ρ . Thisreflects the fact that an optical structure is equivalent to a Robinson structure. ‚ In dimension six, the 3-form ρ abc defined above can be either self-dual or anti-self-dual under Hodge duality, consistent with the fact that the complex conju-gate N of a MTN vector space N has the same Hodge duality as N in this case– see Remark 2.6.2.2.6. Robinson spinors.
We proceed to describe a Robinson structure in terms ofspinors following the treatment of [12, 71, 8, 9, 46, 47, 96].We first consider a p m ` q -dimensional complex vector space p r V , r g q equipped with anon-degenerate symmetric bilinear form. The double cover Spin p m ` , C q of SO p m ` , C q allows us to define the spinor representation S of p r V , r g q , which splits into a directsum of two 2 m -dimensional irreducible chiral spin spaces S ` and S ´ , the spaces ofspinors of positive and negative chiralities respectively. Elements of S ` and S ´ will beadorned with primed and unprimed majuscule Roman indices respectively, e.g. α A P S ` and β A P S ´ , and similarly for the dual spin spaces S ˚˘ with lower indices. The spinspace S is also equipped with Spin p m ` , C q -invariant bilinear forms, which allowthe following identifications: m odd m even S ˘ – S ˚˘ S ˘ – S ˚¯ (2.12) The
Clifford action of r V on S is effected by means of the van der Waerden symbols γ aAB and γ aA B , injective maps from V to the space of homomorphisms Hom p S ˘ , S ¯ q .These satisfy the Clifford property γ p aA C γ b q C B “ g ab δ B A , γ p a A C γ b q C B “ g ab δ BA , (2.13)where δ B A and δ BA denote the identity elements on S ` and S ´ respectively. Let ν A be a spinor, and consider the linear map ν Aa : “ ν B γ aB A : r V Ñ S ´ . Denote by N the kernel of ν Aa . By (2.13), N must be totally null. We say that ν A is pure if N hasmaximal dimension m `
1, i.e. N is a null structure on p r V , r g q . Any spinor proportionalto ν A defines the same null structure. More generally, Cartan showed [12] that there isa one-to-one correspondence between null structures on p r V , r g q and pure spinors up toscale. Further, self-dual null structures correspond to pure spinors of positive chirality,and anti-self-dual null structures to pure spinors of negative chirality. A spinor ν A ispure if and only if it satisfies the purity condition [41, 94] ν Aa ν aB “ . (2.14)Note that the image of ν Aa is isomorphic to any choice of complement N ˚ of N in r V .Moreover, any element v a of N is of the form v a “ ν aA v A for some v A P S ˚´ up to theaddition of an element in the kernel of the dual map ν Aa : S ˚´ Ñ C V . When m “ , , N thus singles out a one-dimensional vector subspace S N ` of S ` , any element ν A of which satisfies (2.14).The van der Waerden symbols generate the Clifford algebra C ℓ p r V , r g q of p r V , r g q , which,by virtue of (2.13), is isomorphic to the exterior algebra ^ ‚ r V – ^ ‚ r V ˚ as a vectorspace. The Clifford algebra is also a matrix algebra isomorphic to the space of endo-morphisms of S . These two properties allow us to construct invariant bilinear formson S with values in ^ k r V ˚ for k “ , . . . , m `
1. The case k “ m and k , these formsrestrict to non-degenerate forms on either S ˘ ˆ S ¯ or S ˘ ˆ S ˘ . Of relevance to thepresent article are the cases k “ , , m `
1. For k “ ,
3, we have m odd m even γ aA B : S ` ˆ S ´ Ñ r V ˚ γ aAB : S ` ˆ S ` Ñ r V ˚ γ aAB : S ´ ˆ S ` Ñ r V ˚ γ aA B : S ´ ˆ S ´ Ñ r V ˚ γ abcA B : S ` ˆ S ´ Ñ ^ r V ˚ γ abcA B : S ` ˆ S ` Ñ ^ r V ˚ γ abcAB : S ` ˆ S ´ Ñ ^ r V ˚ γ abcAB : S ` ˆ S ` Ñ ^ r V ˚ (2.15)For k “ m `
1, regardless of whether m is odd or even, we have the following twobilinear forms γ a ...a m A B : S ` ˆ S ` Ñ ^ m ` ` r V ˚ ,γ a ...a m AB : S ´ ˆ S ´ Ñ ^ m ` ´ r V ˚ , (2.16)where we recall ^ m ` ˘ r V ˚ are the self-dual and anti-self-dual parts of ^ m ` r V ˚ . LMOST ROBINSON GEOMETRIES 13
Now, if N is a self-dual null structure, the restriction of S N ` to the first display of(2.16) yields the isomorphism S N ` b S N ` – ^ m ` Ann p N q . We can thus think of S N ` asa square root of ^ m ` Ann p N q .At this stage, we return to the real picture by consider p m ` q -dimensional Minkowskispace p V , g q . We can then apply all the facts outlined above to the complexification p C V , C g q of p V , g q . In addition, the real structure ¯ on C V preserving V induces anantilinear map on S , which interchanges the chiralities of spinors when m is odd, andpreserve them when m is even. The image of a spinor under this antilinear map is re-ferred to as the charge conjugate of that spinor. Thus, the charge conjugate of a spinor ν A P S ` will be denoted ν A when m is odd, and ν A P S ` when m is even. Moreover,if ν A is pure so is its charge conjugate. We define the real index of a pure spinor tobe the real index of its associated null structure. In particular, if g has Lorentziansignature, all pure spinors have real index one. A Robinson structure p N , K q can thusbe defined by a pure spinor up to scale. We shall refer to any such spinor as a Robinsonspinor .The pure spinor ν A and its charge conjugate can be paired using the spinor bilinearforms to obtain invariants of the Robinson structure as shown in [12, 46]. These arelisted below: m odd m even κ a “ γ aA B ν A ν B κ a “ γ aA B ν A ν B ρ abc “ i γ abcA B ν A ν B ρ abc “ i γ abcA B ν A ν B ν a ...a m “ γ a ...a m A B ν A ν B ν a ...a m “ γ a ...a m A B ν A ν B ν a ...a m “ γ a ...a m AB ν A ν B ν a ...a m “ γ a ...a m A B ν A ν B (2.17)With these definitions, ρ abc is the Robinson 3-form associated to the optical 1-form κ a ,i.e. these satisfy (2.11), and ν a ...a m is a complex Robinson p m ` q -form. Forms of odddegrees can be constructed in a similar way. Details can be found in [46]. Remark 2.9.
Clearly, any given optical 1-form or Robinson 3-form is defined by aRobinson spinor up to a phase, and a given complex Robinson p m ` q -form by aRobinson spinor up to a sign. To see this we note that, for any non-zero complexnumber z , under the transformation ν A ÞÑ zν A , the charge conjugate of ν A getsmultiplied by ¯ z , and the forms defined in (2.17) transform as κ a ÞÑ rκ a , ρ abc ÞÑ rρ abc , ν a ...a m ÞÑ z ν a ...a m , where | z | “ r P R ą . Remark 2.10.
Any choice of splitting (2.6) of C V is equivalent to choosing a one-dimensional subspace of S ˚` consisting of pure spinors dual to S N ` . Elements thereofannihilate N ˚ .2.2.7. Characterisations of Robinson structures.
We summarise the findings of the pre-vious section in the following proposition.
Proposition 2.11.
Let p V , g q be Minkowski space of dimension m ` . The followingstatement are equivalent. Using the isomorphisms (2.12), the charge conjugate of a spinor ν A P S ` can always be identifiedwith ν A P S ˚´ regardless of the parity of m . (1) p V , g q is equipped with a Robinson structure p N , K q .(2) p V , g q is equipped with a complex totally null simple p m ` q -form ν a ...a m .(3) p V , g q is equipped with an optical structure K whose screenspace H K “ K K { K isendowed with a complex structure J ij compatible with the induced metric h ij .(4) p V , g q admits a -form κ a and -form ρ abc satisfying ρ ab e ρ cde “ ´ κ r a g b sr c κ d s . (5) p V , g q admits a pure spinor ν A of real index . Remark 2.12.
In Proposition 2.11, the 1-form κ a and the 3-form ρ abc are defined upto an overall real factor, while the pure spinor ν A is defined up to an overall complexfactor as explained in Remark 2.9.2.3. The stabiliser of a Robinson structure.
There are two approaches to describethe stabiliser of a Robinson structure p N , K q :(1) From its definition, it suffices to consider the respective stabilisers R and R of N and N in SO p m ` , C q . The stabiliser of a Robinson structure is then theintersection R X R X SO p m ` , q .(2) We characterise a Robinson structure as an optical structure together with aHermitian structure on the screen space to derive its stabiliser Q as a closedLie subgroup of the stabiliser P of K in G “ SO p m ` , q .For the present purpose, it will be more useful to use the second approach. We shallassume that K is oriented so that the stabiliser of K together with its orientation is P “ Sim p m q “ CO p m q ˙ p R m q ˚ “ p R ą ˆ SO p m qq ˙ p R m q ˚ . Note that CO p m q acts on the screen space H K “ K K { K as SO p m q does, that is, R ą acts trivially on H K . The nilpotent part of P will be denoted P ` . Since Q Ă P isrequired to stabilise in addition a Hermitian structure on H K , we obtain that Q “ p R ą ˆ U p m qq ˙ p R m q ˚ , “ $&% p e ϕ , A, ν q : “ ¨˝ e ´ ϕ ν ´ e ϕ ξξ J ι p A q ´ e ϕ ι p A q ξ J ϕ ˛‚ | ϕ P R ,A P U p m q ,ξ P p R m q ˚ ,.- , where U p m q is the unitary group, and we have used the standard embedding ι : U p m q Ñ O p m q . We set Q “ R ą ˆ U p m q . Clearly, P ` is also the nilpotent part of Q .To describe the Lie algebra q of Q , we shall refer to the notation already introducedin [24]. Setting V “ V “ K , V “ K K , we have a filtration of P -modules t u “ : V Ă V Ă V Ă V ´ : “ V , (2.18)which we shall conveniently split into a direct sum of G -modules, V “ V ‘ V ‘ V ´ , (2.19)where G “ CO p m q is the reductive part of P . For each i “ ´ , ,
1, we have V i – V i { V i ` as vector spaces. In terms of our earlier notation V ´ “ L and V “ H K , L . LMOST ROBINSON GEOMETRIES 15
Recall from [24] that the Lie algebra g – ^ V ˚ of G “ SO p n ` , q can then beexpressed as a direct sum of G -modules g “ g ´ ‘ g ‘ g , (2.20)where g ˘ – V ˚˘ b V ˚ and g – ` V ˚´ b V ˚ ˘ ‘ ^ V ˚ . Note that V ˚´ b V ˚ is theone-dimensional centre z of g .Now, the complex structure splits C V and its dual as C V “ V p , q ‘ V p , q , C V ˚ “ p V p , q q ˚ ‘ p V p , q q ˚ , (2.21)where V p , q “ N X C V and V p , q : “ N X C V . Writing V ˚ “ V ˚´ ‘ rr ^ p , q V ˚ ss ‘ V ˚ , (2.22)we find that the summands in the G -invariant decomposition of g given in (2.20) splitfurther into irreducible Q -modules: g ˘ “ V ˚˘ b rr ^ p , q V ˚ ss , g “ z ‘ ´ R ω ‘ r ^ p , q˝ V ˚ s ‘ rr ^ p , q V ˚ ss ¯ , (2.23)where we recall that ω is the Hermitian 2-form on V . We identify the Lie algebra q of Q as the Lie subalgebra q “ z ‘ R ω ‘ r ^ p , q˝ V ˚ s , which can be seen to be isomorphic to R ‘ u p m q . In addition, the Lie algebra q of thestabiliser Q of the Robinson structure is given by q : “ z ‘ ´ R ω ‘ r ^ p , q˝ V ˚ s ¯ ‘ ´ V ˚ b rr ^ p , q V ˚ ss ¯ . (2.24)As expected, q “ g X q . Remark 2.13.
In low dimensions, we note the following points: ‚ In dimension four, we have that p – q , i.e. an optical structure is a Robinsonstructure. ‚ In dimension six, the semi-simple part ^ V ˚ of g splits into a self-dual partand an anti-self-dual part. More explicitly, su ` p q “ R ω ‘ rr ^ p , q V ˚ ss and su ´ p q “ r ^ p , q˝ V ˚ s , where su ˘ p q are isomorphic to two copies of su p q . Remark 2.14.
Clearly, the space of all (oriented) Robinson structure on p V , g q isisomorphic to SO p m ` , q{ Q . In addition, for a given optical structure K on p V , g q , the space of all oriented Robinson structures compatible with K is isomorphicto P { Q – SO p m q{ U p m q .2.4. One-dimensional representations of Q . For any w P R , we define the one-dimensional representations R p w q and C p w, q of Q on R of weight w and on C ofweight p w, q by p e ϕ , A, ξ q ¨ r “ e wϕ r , for any r P R , p e ϕ , A, ξ q ¨ z “ p e ϕ det A q w z , for any z P C . We also define C p , w q : “ C p w, q . One can check that R p´ q – K , C p´ , q – ^ m ` Ann p N q , C p , ´ q – ^ m ` Ann p N q . These leads to the one-dimensional representations C p w, w q : “ C p w, q b C p , w q forany integers w , w . We also note C b R p w q – C p w, w q for any integer w .2.5. Space G of algebraic intrinsic torsions. Let us now consider the Q -module G “ V ˚ b p g { q q . We treat only the case m ą m “ Q “ P , which is alreadydealt with [24]. Theorem 2.15.
Assume m ą . The Q -module G “ V ˚ b p g { q q admits a filtration G Ă G Ă G ´ Ă G ´ (2.25) of Q -modules G i : “ p V ˚ q i ` b p g { q q for i “ ´ , ´ , . Its associated graded Q -module gr p G q “ gr ´ p G q ‘ gr ´ p G q ‘ gr p G q ‘ gr p G q , where, for each i “ ´ , ´ , , gr i p G q “ G i { G i ` , and gr p G q “ G . These decomposeinto direct sums of irreducible Q -modules gr j,ki p G q as follows: gr ´ p G q “ gr , ´ p G q , gr ´ p G q “ gr , ´ p G q ‘ ´ gr , ´ p G q ‘ gr , ´ p G q ‘ gr , ´ p G q ¯ ‘ ´ gr , ´ p G q ‘ gr , ´ p G q ¯ ‘ gr , ´ , gr p G q “ gr , p G q ‘ ´ gr , p G q ‘ gr , p G q ‘ gr , p G q ‘ gr , p G q ¯ , gr p G q “ gr , p G q . For each i, j, k , the Q -module gr j,ki p G q is isomorphic to the Q -module G j,ki as given inTable 1. Note that gr , p G q and gr , p G q do not occur when m “ .Proof. We use the same strategy as in Proposition 3.3 of [24]: the filtration (2.18)induces the filtration (2.25) of Q -modules on G Ă V ˚ b ` g ´ { g ˘ . We proceed as beforeusing the decompositions (2.22), (2.23) and (2.24) to find G – ´ V ˚´ ‘ rr ^ p , q V ˚ ss ‘ V ˚ ¯ b ´´ V ˚´ b rr ^ p , q V ˚ ss ¯ ‘ rr ^ p , q V ˚ ss ¯ . The result follows by distributing this expression and splitting each summand intoirreducibles: V ˚´ b V ˚´ b rr ^ p , q V ˚ ss “ G , ´ , V ˚´ b rr ^ p , q V ˚ ss “ G , ´ , rr ^ p , q V ˚ ss b V ˚´ b rr ^ p , q V ˚ ss “ G , ´ ‘ ´ G , ´ ‘ G , ´ ‘ G , ´ ¯ ‘ ´ G , ´ ‘ G , ´ ¯ , rr ^ p , q V ˚ ss b rr ^ p , q V ˚ ss “ G , ‘ G , ‘ G , ‘ G , , V ˚ b V ˚´ b rr ^ p , q V ˚ ss “ G , , V ˚ b rr ^ p , q V ˚ ss “ G , , where, recalling that V ˚˘ – R p¯ q , the modules are described in Table 1. (cid:3) LMOST ROBINSON GEOMETRIES 17 Q -module Description Dimension G , ´ V ˚ m G , ´ R p q G , ´ R p q G , ´ rr ^ p , q V ˚ ss b R p q m p m ´ q G , ´ r ^ p , q˝ V ˚ s b R p q p m ` qp m ´ q G , ´ r ^ p , q˝ V ˚ s b R p q p m ` qp m ´ q G , ´ rr Ä p , q V ˚ ss b R p q m p m ` q G , ´ rr ^ p , q V ˚ ss b R p q m p m ´ q G , V ˚ m G , rr ^ p , q V ˚ ss m G , rr ^ p , q V ˚ ss m p m ´ qp m ´ q G , rr V ˚ ss m p m ` qp m ´ q G , rr ^ p , q˝ V ˚ ss m p m ` qp m ´ q G , rr ^ p , q V ˚ ss b R p´ q m p m ´ q Table 1.
Irreducible Q -submodules of G Remark 2.16.
Observe that the P -module V ˚ b p g { p q represents the space of intrinsictorsions of the underlying optical structure. This can be viewed as a Q -submodule of G . In the next proposition, which is a direct consequence of Theorem 2.15, we singleout the irreducible P -submodules of gr p G q , thereby making contact with the intrinsictorsion of an optical structure described in [24]. Proposition 2.17.
Assume m ą . Let G “ V ˚ b p g { q q and consider the gradedmodule gr p G q given in Theorem 2.15. Define gr ´ p G q : “ gr , ´ p G q , gr ´ p G q : “ gr , ´ p G q , gr ´ p G q : “ gr , ´ p G q ‘ gr , ´ p G q ‘ gr , ´ p G q , gr ´ p G q : “ gr , ´ p G q ‘ gr , ´ p G q , gr p G q : “ gr , p G q . and in dimension six, gr , ` p G q : “ gr , ´ p G q ‘ gr , ´ p G q , gr , ´ p G q : “ gr , ´ p G q . Then, for each i, j , gr ji p G q is an irreducible P -module except in dimension six, where gr p G q is not irreducible, but gr , ˘ p G q are. Moreover, for any choice of splitting, define G ´ : “ G , ´ , G ´ : “ G , ´ , G ´ : “ G , ´ ‘ G , ´ ‘ G , ´ , G ´ : “ G , ´ ‘ G , ´ , G : “ G , , and in dimension six, G , ` : “ G , ´ ‘ G , ´ , G , ` : “ G , ´ . Then, for each i, j , G ji is an irreducible P -module except in dimension six, where G isnot irreducible, but G , ˘ are. Isotypic Q -submodules of G . Let us fix a splitting of G into Q -modules.Observe that the modules in each of the pairs p G , ´ , G , ´ q , p G , ´ , G , ´ q , p G , ´ , G , ´ q and p G , ´ , G , ´ q are isotypic, i.e. they have the same dimensions. This means that one canconstruct further irreducible Q -modules by assigning some algebraic relations amongthese in terms of parameters. To this end, we need to describe them by means of theprojections p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , r x : y s P RP , p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , r x : y s P RP , p Π ˆ ´ q r z : w s : V ˚ b g Ñ G , ´ ‘ G , ´ , r z : w s P CP , p Π ˆ q r z : w s : V ˚ b g Ñ G , ‘ G , , r z : w s P CP , (2.26)whose precise definitions have been relegated to Appendix A for convenience. Here, RP and CP are projective lines, respectively real and complex. We can then definethe following additional Q -submodules of G : p G ˆ ´ q r x : y s : “ im p Π ˆ ´ q r x : y s Ă G , ´ ‘ G , ´ , for r x : y s P RP . p G ˆ ´ q r x : y s : “ im p Π ˆ ´ q r x : y s Ă G , ´ ‘ G , ´ , for r x : y s P RP . p G ˆ ´ q r z : w s : “ im p Π ˆ ´ q r z : w s Ă G , ´ ‘ G , ´ , for r z : w s P CP . p G ˆ q r z : w s : “ im p Π ˆ ´ q r z : w s Ă G , ´ ‘ G , ´ , for r z : w s P CP .(2.27)Their descriptions and dimensions are given in Table 2. Q -module Description Dimension p G ˆ ´ q r x : y s R p q p G ˆ ´ q r x : y s r ^ p , q˝ V ˚ s b R p q p m ´ qp m ` qp G ˆ ´ q r z : w s rr ^ p , q V ˚ ss b R p q m p m ´ qp G ˆ q r z : w s V ˚ m Table 2. Q -submodules of G — here r x : y s P RP and r z : w s P CP Note that by definition, p G ˆ ´ q r , s “ G , ´ , p G ˆ ´ q r , s “ G , ´ , p G ˆ ´ q r , s “ G , ´ , p G ˆ q r , s “ G , ´ , p G ˆ ´ q r , s “ G , ´ , p G ˆ ´ q r , s “ G , ´ , p G ˆ ´ q r , s “ G , ´ , p G ˆ q r , s “ G , ´ . The Q -submodules of G . We are now in the position of determining all the Q -submodules of G . For this purpose, we shall appeal to the Q -module epimorphismsΠ j,ki : V ˚ b g Ñ G j,ki and (2.26) described in Appendix A with respect to some chosensplitting of G . Any Q -submodule of G must be a sum of the irreducible Q -submodules G j,ki given in Table 1, together with the families of irreducibles Q -modules given inTable 2. Not every such sum is a Q -module. To determine which Q -submodules of G are also Q -submodules, we compute how a change of splitting (2.9) transforms themaps Π j,ki . This will tell us how the various modules G j,ki are related under the actionof P ` , the nilpotent part of Q . We will then be able to determine the Q -submodulesof G accordingly.To facilitate the readability, we contract the projections (A.4) with suitable combi-nations of δ iα , δ i ¯ α to define γ α : “ Π , ´ p Γ q α ,ǫ : “ Π , ´ p Γ q ,τ ω : “ Π , ´ p Γ q ,τ αβ : “ Π , ´ p Γ q αβ “ Π , ´ p Γ q αβ , τ ˝ α ¯ β : “ Π , ´ p Γ q α ¯ β ,σ α ¯ β : “ Π , ´ p Γ q α ¯ β , σ αβ : “ Π , ´ p Γ q αβ ,ζ αβ : “ Π , ´ p Γ q αβ ,E α : “ Π , p Γ q α “ Π , p Γ q α ,G α : “ Π , p Γ q α , G αβγ : “ Π , p Γ q αβγ ,G αβγ : “ Π , p Γ q αβγ , G ˝ ¯ αβγ : “ Π , p Γ q ¯ αβγ ,B αβ : “ Π , p Γ q αβ . (2.28)Their complex conjugates are defined similarly. Note that with these definitions, σ αβ “ σ p αβ q , τ αβ “ τ r αβ s ,σ α ¯ β h α ¯ β “ , τ ˝ α ¯ β h α ¯ β “ ,σ α ¯ β “ σ ¯ αβ “ σ β ¯ α , τ ˝ α ¯ β “ τ ˝ ¯ αβ “ ´ τ ˝ β ¯ α ,ζ αβ “ ´ ζ βα , B αβ “ ´ B βα ,G p αβ q γ “ , G r αβγ s “ , G ˝ ¯ βαγ h α ¯ β “ . Theorem 2.18.
Define { G , ´ : “ t Γ P V ˚ b g : Π , ´ p Γ q “ u{ p V ˚ b q q , { G , ´ : “ t Γ P V ˚ b g : Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , { G ,i ´ : “ t Γ P V ˚ b g : Π ,i ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , i “ , { G ,i ´ : “ t Γ P V ˚ b g : Π ,i ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , i “ , , { G , ´ : “ t Γ P V ˚ b g : Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , { G , : “ t Γ P V ˚ b g : Π , p Γ q “ Π ´ p Γ q “ Π ´ p Γ q “ Π ´ p Γ q “ Π ´ p Γ q “ u{ p V ˚ b q q , { G , : “ t Γ P V ˚ b g : Π , p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q“ Π , ´ p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , { G , : “ t Γ P V ˚ b g : Π , p Γ q “ p Π ˆ ´ q r´ , s p Γ q “ u{ p V ˚ b q q , { G , : “ t Γ P V ˚ b g : Π , p Γ q “ p Π ˆ ´ q r , s p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , { G , : “ t Γ P V ˚ b g : Π , p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , and, { G , : “ t Γ P V ˚ b g : Π , p Γ q “ p Π ˆ q r p m ´ q i , ´ s p Γ q “ Π , p Γ q “ Π , p Γ q “ Π , p Γ q“ Π , ´ p Γ q “ Π , ´ p Γ q “ Π ´ p Γ q“ Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , when m ą , while { G , : “ t Γ P V ˚ b g : Π , p Γ q “ p Π ˆ q r , ´ s p Γ q “ Π , p Γ q “ Π , p Γ q“ p Π ˆ ´ q r , s p Γ q “ Π , ´ p Γ q “ Π ´ p Γ q“ Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , when m “ . In dimension six, i.e. m “ , we always have { G , “ { G , “ G . Define further, for any r x : y s P RP , r z : w s P CP , p { G ˆ ´ q r x : y s : “ t Γ P V ˚ b g : p Π ˆ ´ q r x : y s p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , p { G ˆ ´ q r x : y s : “ t Γ P V ˚ b g : p Π ˆ ´ q r x : y s p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q , p { G ˆ ´ q r z : w s : “ t Γ P V ˚ b g : p Π ˆ ´ q r z : w s p Γ q “ p z ` w q Π , ´ p Γ q “ u{ p V ˚ b q q , p { G ˆ q r z : w s : “ t Γ P V ˚ b g : p Π ˆ q r z : w s p Γ q “ p p m ´ q i w ` z q Π , ´ p Γ q“ p p m ´ q i w ` z q Π , ´ p Γ q “ p Π ˆ ´ q r z : w s p Γ q “ p w i ´ z q Π , ´ p Γ q“ p w i ´ z q Π , ´ p Γ q “ z Π , ´ p Γ q “ Π , ´ p Γ q “ u{ p V ˚ b q q . Then, for each i, j, k , { G j,ki is the largest Q -submodule of G that does not contain G j,ki ,and similarly for p { G ˆ ´ q r x : y s , p { G ˆ ´ q r x : y s , p { G ˆ ´ q r z : w s and p { G ˆ q r z : w s . In particular,any Q -submodule of G arises as the intersection of any of the ones above.In addition, there are inclusions of Q -submodules, which are denoted by arrows inthe diagrams below. LMOST ROBINSON GEOMETRIES 21
For m ą : { G , { G , { G , { G , { G , p { G ˆ q r´ p m ´ q i:1 s { G , p { G ˆ ´ q r´ s p { G ˆ ´ q r s p { G ˆ ´ q r p m ´ q i:1 s { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ G ✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ (cid:19) (cid:19) ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ (cid:21) (cid:21) ❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ G G ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ D D ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ H H ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ I I ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ B B / / ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ) ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ②②②②②②②②②②②②②②②②②②②②②②②②②②②②②② < < ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ E E ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ G G ②②②②②②②②②②②②②②②②②②②②②②②②②②②②②② < < ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ / / ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ) ) ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ " " ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (cid:28) (cid:28) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? / / ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ (cid:26) (cid:26) ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ (cid:23) (cid:23) ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ (cid:21) (cid:21) ✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯ (cid:20) (cid:20) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ (cid:26) (cid:26) ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ / / ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? For m “ : { G , { G , { G , p { G ˆ q r´ s { G , p { G ˆ ´ q r´ s p { G ˆ ´ q r s { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ G ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ (cid:22) (cid:22) ✇✇✇✇✇✇✇ ; ; rrrrrrrrrrrrrrrrrrrrrrrrrrr / / ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ * * ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ rrrrrrrrrrrrrrrrrrrrrrrrrrr ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ B B ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ D D ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ / / ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ * * ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ ! ! ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (cid:28) (cid:28) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ A A ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ; ; ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ) ) ●●●●●●●●●●●●●●●●●●●●●● ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ (cid:29) (cid:29) ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ (cid:26) (cid:26) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (cid:28) (cid:28) ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ * * / / ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ A A / / ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? For any m ą , p { G ˆ q r s p { G ˆ ´ q r´ s { G , ´ { G , ´ { G , ´ { G , ´ ♦♦♦♦♦♦♦♦♦♦♦♦ ❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩ , , ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❩❩❩❩❩❩❩❩❩❩❩ , , ❞❞❞❞❞❞❞❞❞❞❞ ♦♦♦♦♦♦♦♦♦♦♦♦ ❖❖❖❖❖❖❖❖❖❖❖ ' ' LMOST ROBINSON GEOMETRIES 23
For any m ą , any r z : w s P CP ztr´ p m ´ q i , s , r , su : p { G ˆ q r z : w s p { G ˆ ´ q r´ z : w s { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ { G , ´ ③③③③③③③③③③③③③③ < < ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩ , , ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ " " ❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❩❩❩❩❩❩❩❩❩❩❩ , , ❞❞❞❞❞❞❞❞❞❞❞ ♦♦♦♦♦♦♦♦♦♦♦♦ ③③③③③③③③③③③③③③③ < < ❉❉❉❉❉❉❉❉❉❉❉❉❉ " " For m ą , any r´
4i : 1 s ‰ r z : w s P CP , and any r x : y s P RP : p { G ˆ ´ q r x : y s p { G ˆ ´ q r x : y s p { G ˆ ´ q r z : w s { G , ´ ❖❖❖❖❖❖❖❖❖❖❖❖ ' ' / / ♦♦♦♦♦♦♦♦♦♦♦♦ Remark 2.19.
We note that(1) There is some redundancy in the inclusions: e.g. { G , X { G , ´ X { G , ´ “ { G , X { G , ´ “ { G , X { G , ´ , and, for m ą { G , Ă p { G ˆ q r´ p m ´ q i:1 s X { G , Ă p { G ˆ ´ q r s X p { G ˆ ´ q r p m ´ q i:1 s “ { G , ´ X { G , ´ , (2) The only Q -submodules of G that are not contained in { G , ´ are { G , Ă p { G ˆ ´ q r ´ s . Proof.
The theorem is a direct consequence of the following transformation rules of thequantities defined by (2.28) under the change (2.9): γ α ÞÑ γ α ,ǫ ÞÑ ǫ ` γ α φ α ` γ α φ α ,τ ω ÞÑ τ ω ´ i p γ α φ α ´ γ α φ α q ,τ αβ ÞÑ τ αβ ´ γ r α φ β s ,τ ˝ α ¯ β ÞÑ τ ˝ α ¯ β ` ˆ ´ γ α φ ¯ β ` γ ¯ β φ α ˙ ˝ ,σ α ¯ β ÞÑ σ α ¯ β ` ˆ γ α φ ¯ β ` γ ¯ β φ α ˙ ˝ ,σ αβ ÞÑ σ αβ ` γ p α φ β q ,ζ αβ ÞÑ ζ αβ ´ γ r α φ β s ,E α ÞÑ E α ` τ αβ φ β ´ σ αβ φ β ` τ ˝ α ¯ β φ ¯ β ´ σ α ¯ β φ ¯ β ´ i2 m τ ω φ α ´ m ǫφ α ´ γ α φ β φ β ,G α ÞÑ G α ´ τ ˝ α ¯ β φ ¯ β ` σ α ¯ β φ ¯ β ` m ´ m τ ω φ α ´ m ´ m i ǫ φ α ` φ β ζ βα ´ φ β γ β φ α ` φ β φ β γ α ,G αβγ ÞÑ G αβγ ` ` ´ τ r αβ ` ζ r αβ ˘ φ γ s ,G p αβ q γ ÞÑ G p αβ q γ ´ ` τ γ p α ` ζ γ p α ˘ φ β q ` σ γ p α φ β q ´ σ αβ φ γ ´ φ p α γ β q φ γ ` φ α φ β γ γ G ˝ ¯ αβγ ÞÑ G ˝ ¯ αβγ ` ´ τ ˝r β | ¯ α φ | γ s ´ σ r β | ¯ α φ | γ s ` φ ¯ α ζ βγ ´ φ ¯ α γ r β φ γ s ¯ ˝ ,B αβ ÞÑ B αβ ` ˆ m ´ G r α ´ E r α ˙ φ β s ´ φ γ G γαβ ´ φ γ G γαβ ´ φ ¯ γ G ˝ ¯ γαβ ` φ γ τ γ r α φ β s ´ φ γ φ γ ζ αβ ` φ γ σ γ r α φ β s ´ φ ¯ γ τ ˝r α | ¯ γ φ | β s ` φ ¯ γ σ r α | ¯ γ φ | β s ` φ γ φ γ γ r α φ β s From these transformations, we immediately deduce, for any r x : y s P RP , r z : w s P CP , xǫ ` yτ ω ÞÑ xǫ ` yτ ω ` p x ´ i y q γ α φ α ` p x ` i y q γ α φ α ,xσ α ¯ β ´ y i τ ˝ α ¯ β ÞÑ xσ α ¯ β ´ y i τ ˝ α ¯ β ` ˆ ´p x ` y i q γ α φ ¯ β ` p x ´ y i q γ ¯ β φ α ˙ ˝ ,zτ αβ ` wζ αβ ÞÑ zτ αβ ` wζ αβ ´ p z ` w q γ r α φ β s ,z E α ` w G α ÞÑ z E α ` w G α ´ z φ β σ βα ` p´ z τ βα ` w ζ βα q φ β ´ p w i ´ z q τ ˝ α ¯ β φ ¯ β ` p w i ´ z q σ α ¯ β φ ¯ β ´ i2 m p p m ´ q i w ` z q τ ω φ α ´ m p p m ´ q i w ` z q ǫ φ α ´ w i φ β γ β φ α ` p w i ´ z q φ β φ β γ α . The result follows as in the proof of Proposition 3.4 of [24].
LMOST ROBINSON GEOMETRIES 25
In dimension six, one has φ γ φ γ τ αβ “ ´ φ γ τ γ r α φ β s , G γαβ “ , G ˝ ¯ γαβ “ , so that B αβ ÞÑ B αβ ` ` G r α ´ E r α ˘ φ β s ´ φ γ G γαβ ´ p τ αβ ` ζ αβ q φ γ φ γ ` φ γ σ γ r α φ β s ´ φ ¯ γ τ ˝r α | ¯ γ φ | β s ` φ ¯ γ σ r α | ¯ γ φ | β s ` φ γ φ γ γ r α φ β s This completes the proof. (cid:3)
Finally, for future use, and to make contact with the intrinsic torsion of an opticalstructure given in [24], we also define { G ´ : “ { G , ´ , { G ´ : “ { G , ´ , { G ´ : “ { G , ´ X { G , ´ X { G , ´ , { G ´ : “ { G , ´ X { G , ´ , { G : “ { G . (2.29) 3. Almost Robinson manifolds
Almost Robinson structures.
Throughout we shall follow the notation andconventions of Section 2 now translated into the bundle setting.
Definition 3.1. [64, 96] Let p M , g q be an oriented pseudo-Riemannian manifold ofdimension 2 m `
2. An almost null structure on p M , g q is a complex distribution N ofrank p m ` q and totally null with respect to C g . When N is involutive, i.e. r N, N s Ă N ,we call N a null structure .In other words, an almost null structure is a smooth assignement of a null structureto the tangent space at each point, and according to Definition 2.3, one may talk of thereal index of an almost null structure at a point. When g is of Lorentzian signature,we make the following definition. Definition 3.2.
Let p M , g q be a time-oriented and oriented Lorentzian manifold ofdimension 2 m `
2. An almost Robinson structure on p M , g q consists of a pair p N, K q where N is a complex distribution of rank m ` C g , and K a real line distribution such that C K “ N X N . We shall refer to the quadruple p M , g, N, K q as an almost Robinson manifold or almost Robinson geometry .On addition, we call p N, K q‚ a nearly Robinson structure when r K, N s Ă N , and ‚ a Robinson structure when r N, N s Ă N , i.e. N is involutive.We shall accordingly refer to p M , g, N, K q as a nearly Robinson manifold or as a Robin-son manifold .Clearly, a Robinson manifold is a nearly Robinson manifold. Definitions involvingweaker assumptions on orientability are possible.
Remark 3.3.
Equivalently put, an almost Robinson structure is an almost null struc-ture of real index one. By Lemma 2.5, any almost null structure on a Lorentzian mani-fold defines an almost Robinson manifold. The terminology ‘almost null structure’ willnevertheless be favoured in the case when we wish to emphasize the geometric aspects of the almost Robinson structure not particularly tied to the geometry of the real nullline distribution K , as will be done in Section 3.10.An almost Robinson structure p N, K q induces an optical structure on p M , g q in thesense of [24], namely the filtration of vector bundles K Ă K K Ă T M . (3.1)The orientation and time-orientation on M induce an orientation on K , and the screenbundle H K : “ K K { K of K inherits a positive-definite bundle metric h from g . Anysection of K will be referred to as an optical vector field , while any section of Ann p K K q will be referred to as an optical 1-form.The bundle complex structure J on the screen bundle H K induces a splitting of itscomplexification C H K “ H p , q K ‘ H p , q K , C H ˚ K “ p H p , q K q ˚ ‘ p H p , q K q ˚ , where H p , q K and H p , q K denote the ` i- and ´ i-eigenbundles of J respectively. In ab-stract index notation, we shall denote the bundle complex structure and the bundleHermitian structure on H K by J ij and ω ij respectively, so that ω ij “ J ik h kj . Followingthe notation of Section 2.1, we also define the bundles, for any non-negative integer p , q , ^ p p,q q H ˚ K : “ ^ p p H p , q K q ˚ b ^ q p H p , q K q ˚ , ä p p,q q H ˚ K : “ ä p p H p , q K q ˚ b ä q p H p , q K q ˚ . For pq ‰
0, the subbundles of elements of ^ p p,q q H ˚ K and Ä p p,q q H ˚ K that are tracefreewith respect to the bundle Hermitian structure will be denoted by ^ p p,q q˝ H ˚ K and Ä p p,q q˝ H ˚ K respectively. Similarly, we introduce the subbundle H ˚ K .As in Section 2, we split the complexified tangent bundle as C T M “ N ‘ N ˚ for some chosen complement N ˚ of N in C T M , dual to N via C g . This splitting is notcanonical in general. This induces a splitting of the filtration (3.1) T M “ K ‘ H K,L ‘ L , (3.2)where L : “ N ˚ X T M , H K,L : “ K K X L K . Note that N ˚ defines an almost Robinson structure p N ˚ , L q on p M , g q , where L is thereal span of N ˚ X N ˚ and is dual to K . In addition, C H K,L “ H p , q K,L ‘ H p , q K,L , C H ˚ K,L “ p H p , q K,L q ˚ ‘ p H p , q K,L q ˚ . where H p , q K,L “ N X N ˚ , H p , q K,L “ N X N ˚ . We also have isomorphisms of vector bundles H K,L – H K , H p , q K,L “ H p , q K , and so on,which depend on the choice of N ˚ . LMOST ROBINSON GEOMETRIES 27
The splitting operators and their duals, introduced in Section 2, that is, p δ aA , δ aA q , p ℓ a , δ ai , k a q , p δ iα , δ i ¯ α q , p ℓ a , δ aα , δ a ¯ α , k a q , p δ Aa , δ aA q , p κ a , δ ia , λ a q , p δ αi , δ ¯ αi q , p κ a , δ αa , δ ¯ αa , λ a q , will be used throughout the article to convert index types, with the convention that k a α a “ α and ℓ a α a “ α for any 1-form α a – in effect, we may write ℓ a “ δ a , k a “ δ a , κ a “ δ a and λ a “ δ a .In order to avoid ambiguity when taking components of covariant derivatives of sometensor α b...d , we shall often write p ∇ α q ab...d : “ ∇ a α b...d . For instance, p ∇ α q α i “ k a p ∇ a α bcd q δ bα ℓ c δ di . The splitting operators will also be used as injectors. Thus, if ω ij is the Hermitian2-form on H K , we can set ω ab “ ω ij δ ia δ jb for some chosen splitting operators, andconstruct the Robinson -form ρ abc : “ κ r a ω bc s associated to the optical 1-form κ a .Finally, we shall take covariant derivatives of the splitting operators viewing themas a frame, noting the useful relation p ∇ ρ q aαβ “ p ∇ a δ b r α q δ bβ s , for future reference.3.2. Pure spinors.
Whenever p M , g q is assumed to be spin, we introduce a spin bun-dle S and translate the theory of spinors summarised in Section 2.2.6 to the languageof bundles. We shall denote the spinor bundle by S and its irreducible parts by S ` and S ´ . We shall not distinguish notationally between the Levi-Civita connection ∇ andthe induced spin connection. We now view the van der Waerden symbols γ aA B and γ aAB , the bilinear forms (2.16) as fields on M , parallel with respect to ∇ . There is alsoan antilinear map on each fiber of S , induced from the reality structure on p C T M , C g q ,and thus, a notion of charge conjugate of a spinor field.An almost Robinson structure p N, K q , where we assume for specificity that N isself-dual, can therefore be expressed by a non-vanishing section ν A of S ` that is pureat every point, i.e. the kernel of the map ν Aa : “ γ aB A ν B : Γ p T M q Ñ Γ p S ´ q is precisely N . Since the kernel N of the map ν Aa is invariant under rescaling of thespinor ν A , the almost Robinson structure is in fact equivalent to the existence of acomplex line subbundle S N ` of S ` , which is spanned by ν A . The bundle S N ` canbe viewed as a square root of the line bundle ^ m ` Ann p N q . Indeed, we have anisomorphism of bundles ^ m ` Ann p N q – S N ` b S N ` . (3.3)Any spinor ν A annihilating the almost null structure N will be referred to as a Robinsonspinor , and any section of ^ m ` Ann p N q a complex Robinson p m ` q -form . A completely parallel analysis can be carried out starting with the charge conjugateof ν A , which spans the complex conjugate bundle S N ` . The invariants 1-form κ a , 3-form ρ abc , p m ` q -forms ν a ...a m ` and ν a ...a m ` of the almost Robinson structure canthen be recovered from ν A and its charge conjugate using (2.17).We are now in the position of stating the direct translation of Proposition 2.11 intothe language of manifolds: Proposition 3.4.
Let p M , g q be a time-oriented and oriented smooth Lorentzian ma-nifold of dimension m ` . The following statements are equivalent.(1) p M , g q is endowed with an almost Robinson structure p N, K q .(2) p M , g q admits a complex simple totally null p m ` q -form.(3) p M , g q is endowed with an optical structure K whose screen bundle p H K , h q isequipped with a bundle complex structure compatible with the bundle metric h .(4) p M , g q admits a null -form κ a and a -form ρ abc such that ρ ab e ρ cde “ ´ κ r a g b sr c κ d s . (3.4) (5) when p M , g q is spin, it admits a pure spinor ν A (of real index one). Almost Robinson structures as G -structures. From the discussion of [24],we shall also view an almost Robinson structure as a reduction of the frame bundle tothe Q -bundle F Q where Q “ p R ą ˆ U p m qq ˙ p R m q ˚ , and given any Q -module A ,we will construct associated vector bundles F Q p A q : “ F Q ˆ Q A . Similarly, a choiceof splitting gives rise to the Q -invariant vector bundles where Q “ R ą ˆ U p m q . Itwill often be more convenient to deal with the reduced coframe bundle F ˚ Q . A sectionof F ˚ Q will then consist of a null complex coframe t θ , θ α , θ ¯ α , θ u “ t κ, θ α , θ ¯ α , λ u suchthat(1) κ annihilates K K , or equivalently, κ “ g p k, ¨q for some section k of K ;(2) t κ, θ α u annililate N , or equivalently, t κ, θ ¯ α u annililates N ;(3) t θ α u are unitary with respect to the screen bundle metric;(4) the metric takes the form g “ κλ ` h α ¯ β θ α θ ¯ β . We shall refer to t κ, θ α , θ ¯ α , λ u as a Robinson coframe .Any two Robinson coframes t κ, θ α , θ ¯ α , λ u and t r κ, r θ α , r θ ¯ α , r λ u are related by a trans-formation of the form r κ “ e ϕ κ , r θ α “ φ βα θ β ` φ α κ , r λ “ e ´ ϕ ` λ ´ φ α θ α ´ φ ¯ α θ ¯ α ´ φ α φ α κ ˘ , (3.5)where ϕ is a smooth real-valued function, and φ α , φ βα are smooth complex-valued func-tions on M with φ αβ being a U p m q -transformation at any point, i.e. h α ¯ β “ h γ ¯ δ φ αγ φ ¯ β ¯ δ .Associated to the representations R p w q and C p w, w q defined in Section 2.4, where w and w are integers, we define the bundle E p w q of boost densities of weight w , alreadyintroduced in [24], and the bundle E p w, w q of boost-spin densities of weight p w, w q . Inparticular, we have the identifications K – E p´ q , L – E p q , E p´ , q : “ ^ m ` Ann p N q , E p , ´ q : “ E p´ , q “ ^ m ` Ann p N q . (3.6) LMOST ROBINSON GEOMETRIES 29 If p M , g q is assumed to be spin, we define the smooth complex line bundles E p , q : “ ` S N ` ˘ ´ , E p , q : “ ´ S N ` ¯ ´ , E p´ , q : “ ` E p , q ˘ ˚ , E p , ´ q : “ ` E p , q ˘ ˚ . (3.7)In particular, we recover (3.6) by virtue of (3.3). Our definitions are consistent withthose of the real line bundles E p w q in the sense that E p w q “ E p w , w q{ S , for any integer w .This can be readily be checked using (2.17).3.4. Intrinsic torsion.
As explained in Section 2 of [24], the intrinsic torsion of analmost Robinson structure is given by a section T ˚ M b F Q p g { q q , which we identify withthe Q -invariant subbundle G : “ F Q ˆ Q G where G : “ V ˚ b g { q . We shall accordinglycall G the bundle of intrinsic torsions of p N, K q . Its Q -invariant subbundles and Q -invariant subbundles will presently be defined with reference to Section 2, and inparticular Theorem 2.15.The filtration (2.25) on G induces a filtration of Q -invariant subbundles G “ : G ´ Ą G ´ Ą G Ą G ´ , where G i : “ F Q ˆ Q G i . Correspondingly, the associated graded vector bundlegr p G q “ gr ´ p G q ‘ gr ´ p G q ‘ gr p G q ‘ gr p G q , where gr i p G q : “ F Q ˆ Q gr i p G q , splits into irreducible Q -invariant subbundles gr j,ki p G q : “ F Q ˆ Q gr j,ki p G q . For each choice of splitting, these are isomorphic to the Q -invariantsubbundles G j,ki : “ F Q ˆ Q G j,ki , and we introduce further p G ˆ ´ q r x : y s : “ F Q ˆ Q p G ˆ ´ q r x : y s , for each r x : y s P RP , p G ˆ ´ q r x : y s : “ F Q ˆ Q p G ˆ ´ q r x : y s , for each r x : y s P RP , p G ˆ ´ q r z : w s : “ F Q ˆ Q p G ˆ ´ q r z : w s , for each r z : w s P CP , p G ˆ q r z : w s : “ F Q ˆ Q p G ˆ q r z : w s , for each r z : w s P CP ,with reference to Table 1 and (2.27).We correspondingly define the Q -invariant subbundles { G j,ki : “ F Q ˆ Q { G j,ki , p { G ˆ ´ q r x : y s : “ F Q ˆ Q p { G ˆ ´ q r x : y s , for each r x : y s P RP , p { G ˆ ´ q r x : y s : “ F Q ˆ Q p { G ˆ ´ q r x : y s , for each r x : y s P RP , p { G ˆ ´ q r z : w s : “ F Q ˆ Q p { G ˆ ´ q r z : w s , for each r z : w s P CP , p { G ˆ q r z : w s : “ F Q ˆ Q p { G ˆ q r z : w s , for each r z : w s P CP ,where the Q -modules are all defined in Theorem 2.18. Finally, in order to make contact with the intrinsic torsion of an optical geometrydescribed in [24], we define, for each i, j , G ji : “ F Q ˆ G G ji , { G ji : “ F P ˆ P { G ji , where G ji are defined in Proposition 2.17, and { G ji are defined by (2.29).To determine the algebraic properties of the intrinsic torsion of an almost Robinsonstructure, we choose an optical 1-form κ a and its associated Robinson 3-form ρ abc ,compute their respective covariant derivatives ∇ a κ b and ∇ a ρ bcd , and project thesetensors onto the various Q -invariant subbundles of G by means of splitting operators t ℓ a , δ aα , δ a ¯ α , k a u . This is achieved by extending the projections Π j,ki defined in AppendixA to bundle projections, where we identify the tensor Γ abc as a connection 1-form of ∇ a adapted to the splitting operators. We shall make use of the notation alreadyintroduced in section 2, mirroring the definition (2.28). Let us set γ i : “ p ∇ κ q i ,ǫ : “ p ∇ κ q ij h ij , τ ij : “ p ∇ κ q r ij s , σ ij : “ p ∇ κ q p ij q ˝ ,E i : “ p ∇ κ q i , (3.8)These components split further into irreducible Q -components. In particular, thecomponent τ α ¯ β splits as τ α ¯ β “ m τ ω h α ¯ β ` τ ˝ α ¯ β , where τ ω : “ ω ij τ ij “ τ α ¯ β h α ¯ β , τ ˝ α ¯ β : “ ` τ α ¯ β ˘ ˝ , In addition, the components of the covariant derivative of the Robinson 3-form ρ abc ofinterest are ζ αβ : “ p ∇ ρ q αβ ,G iβγ : “ p ∇ ρ q iβγ ,B αβ : “ p ∇ ρ q αβ . (3.9)the second of which splits further into the irreducible components G αβγ : “ G r αβγ s , G αβγ : “ ` G p αβ q γ ´ G p αγ q β ˘ ,G ˝ ¯ γαβ : “ p G ¯ γαβ q ˝ , G α : “ h β ¯ γ G ¯ γαβ , so that G αβγ “ G αβγ ` G αβγ , G ¯ αβγ “ G ˝ ¯ αβγ ´ m ´ G r β h γ s ¯ α ,G iβγ “ δ αi G αβγ ` δ ¯ αi G ¯ αβγ . The other relevant components of ∇ a ρ bcd can be found below: p ∇ ρ q a jk “ , p ∇ ρ q aαβγ “ , LMOST ROBINSON GEOMETRIES 31 p ∇ ρ q α “ ´ i γ α , p ∇ ρ q αβ “ ´ i τ αβ ´ i σ αβ , p ∇ ρ q ¯ αβ “ ´ i 12 m ǫh β ¯ α ` i τ β ¯ α ´ i σ β ¯ α , p ∇ ρ q α “ ´ i E α , p ∇ ρ q αβ ¯ γ “ γ r α h β s ¯ γ , p ∇ ρ q αβγ ¯ δ “ ` τ α r β ` σ α r β ˘ h γ s ¯ δ , p ∇ ρ q ¯ αβγ ¯ δ “ ˆ m ǫh r β | ¯ α ´ τ r β | ¯ α ` σ r β | ¯ α ˙ h | γ s ¯ δ , p ∇ ρ q αβ ¯ γ “ E r α h β s ¯ γ , p ∇ ρ q β ¯ γ “ i p ∇ κ q h β ¯ γ , p ∇ ρ q i β ¯ γ “ i p ∇ κ q i h β ¯ γ , p ∇ ρ q β ¯ γ “ i p ∇ κ q h β ¯ γ . The elements defined above should be regarded as trivialisations of sections of G j,ki .Such sections generally carry a boost weight. These will be adorned with a breve accent.To be clear, we have collected them below. G -bundle Description Tensor G ´ H ˚ K,L b E p q ˘ γ i G ´ E p q ˘ ǫ G ´ ^ H ˚ K,L b E p q ˘ τ ij G ´ Ä ˝ H ˚ K,L b E p q ˘ σ ij G H ˚ K,L ˘ E i Table 3.
Irreducible G -submodules of G Q -bundle Description Tensor G , ´ H ˚ K,L b E p q rr ˘ γ α ss G , ´ E p q ˘ ǫ G , ´ E p q ˘ τ ω G , ´ rr ^ p , q H ˚ K,L ss b E p q rr ˘ τ αβ ss G , ´ r ^ p , q˝ H ˚ K,L s b E p q r ˘ τ ˝ α ¯ β s G , ´ r ^ p , q˝ H ˚ K,L s b E p q r ˘ σ α ¯ β s G , ´ rr Ä p , q H ˚ K,L ss b E p q rr ˘ σ αβ ss G , ´ rr ^ p , q H ˚ K,L ss b E p q rr ˘ ζ αβ ss G , H ˚ K,L rr ˘ E α ss G , rr ^ p , q H ˚ K,L ss rr ˘ G α ss G , rr ^ p , q H ˚ K,L ss rr ˘ G αβγ ss G , rr H ˚ K,L ss rr ˘ G αβγ ss G , rr ^ p , q˝ H ˚ K,L ss rr ˘ G ˝ ¯ γαβ ss G , rr ^ p , q H ˚ K,L ss b E p´ q rr ˘ B αβ ss Table 4.
Irreducible Q -submodules of G We shall also introduce the quantity˘ τ α ¯ β “ m ˘ τ ω h α ¯ β ` ˘ τ ˝ α ¯ β , which we identify as a section of G , ´ ‘ G , ´ .Finally, to characterise sections of the bundles { G j,ki , we simply apply the results ofTheorem 2.18. For instance, the intrinsic torsion is a section of { G , if and only if itsweighted components in any splitting satisfy˘ γ i “ ˘ σ αβ “ τ ˝ αβ ` ˘ ζ αβ “ ˘ G αβγ “ . Remark 3.5.
We can also express some of the properties of the almost Robinsonstructure in terms of a Robinson spinor — this will be done in Section 3.10.3.5.
Congruences of null geodesics.
Since an almost Robinson manifold p M , g, N, K q defines in particular an optical geometry, there is a congruence of null curves K asso-ciated to it. The algebraic properties of the intrinsic torsion of the optical geometry p M , g, K q can be related to the geometric properties of K as reviewed in [24]. Wesummarise the correspondence in Table 5. LMOST ROBINSON GEOMETRIES 33
Intrinsic torsion Congruence { G ´ “ G ´ geodesic { G ´ non-expanding { G ´ non-twisting { G ´ non-shearing { G “ t u parallel Table 5.
Geometric properties of K and intrinsic torsion ˚ T We also record the following lemma, whose proof is straightforward.
Lemma 3.6.
Let p M , g, N, K q be an almost Robinson manifold with congruence of nullcurves K . Denote by J ij the complex structure on the screen bundle H K . Suppose thatthe intrinsic torsion ˚ T of p N, K q is a section of { G ´ so that K is geodesic with twist ˘ τ ij and shear ˘ σ ij . Then ˚ T is a section of ‚ { G , ´ , i.e. ˘ τ αβ “ , if and only if ˘ τ ij and J ij commute, i.e. J r ik ˘ τ j s k “ ; ‚ { G , ´ , i.e. ˘ σ αβ “ , if and only if ˘ σ ij and J ij commute, i.e. J p ik ˘ σ j q k “ ; ‚ { G , ´ X { G , ´ , i.e. ˘ τ α ¯ β “ , if and only if ˘ τ ij and J ij anti-commute, i.e. J p ik ˘ τ j q k “ ; ‚ { G , ´ , i.e. ˘ σ α ¯ β “ , if and only if ˘ σ ij and J ij anti-commute, i.e. J r ik ˘ σ j s k “ . It is well-known, see e.g. [81, 24], that the bracket condition r K, K K s Ă K K isequivalent to K being geodesic, and the condition r K K , K K s Ă K K to K being geodesicand non-twisting. The latter can split into the two following obvious subcases. Proposition 3.7.
Let p M , g, N, K q be a p m ` q -dimensional almost Robinson ma-nifold with congruence of null curves K with leaf space M . The following statementsare equivalent.(1) r N, N s Ă C K K ;(2) the intrinsic torsion is a section of { G , ´ X { G , ´ , i.e. ˘ γ i “ ˘ τ ω “ ˘ τ ˝ α ¯ β “ (3) K is geodesic and its twist anti-commutes with the screen bundle complex struc-ture. Proposition 3.8.
Let p M , g, N, K q be a p m ` q -dimensional almost Robinson ma-nifold with congruence of null curves K with leaf space M . The following statementsare equivalent.(1) r N, N s Ă C K K ;(2) the intrinsic torsion is a section of { G , ´ , i.e. ˘ γ i “ ˘ τ αβ “ (3) K is geodesic and its twist commutes with the screen bundle complex structure. We shall deal with the remaining bracket conditions, namely r N, N s Ă C K and r N, N s Ă N in Propositions 3.14 and 3.45 respectively.Locally, we shall identify an almost Robinson manifold p M , g, N, K q of dimension2 m ` p m ` q -dimensional smooth manifold M ,namely, the local leaf space of K . This leaf space will be endowed with various geometricstructures depending on the geometric properties of K . In the next sections, we shallexamine the relation between the intrinsic torsion of p N, K q with the induced geometricstructures on the leaf space. As a notational rule followed in this article, tensor fieldson M will be underlined to distinguish them from tensor fields on M . Remark 3.9.
We briefly recall the results of [24] that pertain to the optical structure K associated to an almost Robinson manifold p M , g, N, K q . In the following, we shalldenote the conformal class of g by r g s . If the congruence of null curves K tangent to K is geodesic, there is a subclass r g s n.e. of metrics in r g s for which K is also non-expanding.By extension, r g s and r g s n.e determine conformal subclasses of bundle metrics r h s and r h s n.e respectively. ‚ If K is geodesic, the screen bundle H K descends to a rank-2 m distribution H on M . In this case, there is a one-to-one correspondence between optical vectorfields k such that £ k κ “
0, where κ “ g p k, ¨q , and sections of Ann p H q . ‚ If K is geodesic and non-twisting, the induced distribution H on M is involutive. ‚ If K is geodesic and non-shearing, the screen bundle metric H K induces a bun-dle conformal structure c H on p M , H q . There is a one-to-one correspondencebetween metrics in r g s n.e. (or equivalently, bundle metrics in r h s n.e. ) and metricsin c H . It follows that if K is already non-expanding for g , then h descends to adistinguished bundle metric h on H . Remark 3.10.
It is clear from Theorem 2.18 that the only proper subbundles of G that are not contained in G ´ are { G , Ă p { G ˆ ´ q r ´ s . (3.10)This means that the congruence of null curves associated to an almost Robinson struc-ture whose intrinsic torsion lies in any proper subbundles of G except for (3.10) mustbe geodesic. We leave it as a conjecture whether one can construct almost Robinsonmanifolds whose intrinsic torsion lies in (3.10) but whose congruence is not geodesic.3.6. Almost CR geometry.
The geometric structure that an almost Robinson struc-ture may induce on the leaf space of its associated congruence of null curves is analmost CR structure, of which we now recall some notions.3.6.1.
General definitions. An almost CR structure on a p m ` q -dimensional smoothmanifold M consists of a pair p H, J q , where H is a rank-2 m distribution equipped witha bundle complex structure J . A CR structure is an almost CR structure for whichthe ´ i-eigenbundle H p , q of J , or equivalently its i-eigenbundle H p , q , is in involution.We call an almost CR structure p H, J q together with a choice of 1-form θ annihilating H an almost pseudo-Hermitian structure . LMOST ROBINSON GEOMETRIES 35
Following [17], we define the
Levi form of the almost CR structure p H, J q to be thebundle homomorphism L : Ann p H q Ñ ^ p T ˚ M { Ann p H qq given by the compositionAnn p H q ÝÑ T ˚ M d ÝÑ ^ T ˚ M ÝÑ ^ p T ˚ M { Ann p H qq . Concretely, for any section θ of Ann p H q , θ ˝ L p v, w q “ d θ p v, w q for any sections v and w of H . We shall refer to L : “ θ ˝ L as the Levi form of the pseudo-Hermitianstructure p H, J , θ q .A coframe θ , θ α , θ ¯ α ( adapted to the almost CR structure p H, J q on M consists ofa real 1-form θ and m complex 1-forms θ α with θ ¯ α “ θ α , where H “ Ann p θ q , and H p , q “ Ann p θ , θ α q . Any other coframe t p θ , p θ α , p θ ¯ α u adapted to p H, J q is related to t θ , θ α , θ ¯ α u by p θ “ e ϕ θ , p θ α “ φ βα θ β ` φ α θ , (3.11)where ϕ , φ βα , and φ α are smooth functions on M , with the requirement that the deter-minant of φ βα be non-vanishing. Even with a choice of an almost pseudo-Hermitianstructure θ on p M , H, J q , there is no canonical choice for t θ α u in general. The choiceof a vector field e dual to θ splits T M as T M “ H ‘ span t e u . (3.12)The structure equations for a given CR coframe θ , θ α , θ ¯ α ( can be expressed asd θ “ L α ¯ β θ α ^ θ ¯ β ` L αβ θ α ^ θ β ` L ¯ α ¯ β θ ¯ α ^ θ ¯ β ` α ^ θ , d θ α “ θ β ^ Γ βα ` A α ¯ β θ ^ θ ¯ β ´ N ¯ β ¯ γ α θ ¯ β ^ θ ¯ γ , d θ ¯ α “ θ ¯ β ^ Γ ¯ β ¯ α ` A ¯ αβ θ ^ θ β ´ N βγ ¯ α θ β ^ θ γ , (3.13)for some complex functions L α ¯ β , L αβ , A ¯ αβ , N βγ ¯ α and 1-forms Γ βα and α on M , where L α ¯ β is skew-hermitian, α is real, and the remaining quantities are defined by complexconjugation.Here, we identify L α ¯ β and L αβ as the p , q -part and p , q -part of the Levi form L : “ θ ˝ L of the pseudo-Hermitian structure p H, J , θ q respectively. The Levi formallows us to derive two invariants of p H , J q at a point of M , namely ‚ its rank , that is the largest integer r for which θ ^ ` d θ ˘ r does not vanish atany point; ‚ its signature , defined to be the signature of the Hermitian form ´ i L α ¯ β .Clearly, these do not depend on the choice of CR coframe. We shall assume regularityof the rank and signature throughout the article, i.e. these will be constant everywhere.We say that the almost CR structure p H , J q is ‚ partially integrable if the Levi form is of type p , q , i.e. L αβ “ ‚ integrable or involutive if H p , q is involutive, i.e. L ¯ α ¯ β “ N ¯ β ¯ γα “ ‚ contact or non-degenerate if H is a contact distribution, i.e. θ ^ ` d θ ˘ m doesnot vanish at any point, i.e. the Levi form has maximal rank; One could also include negative rescaling of θ in more generality. ‚ totally degenerate if H is involutive, i.e. θ ^ d θ “ L αβ “ L α ¯ β “ Example 3.11.
The model for a contact CR manifold is the CR sphere S m ` viewedas a hypersurface in C m ` . More generally, any real hypersurface in C m ` is a CRmanifold. Example 3.12.
As a special case of almost CR manifold, consider an almost complexmanifold p B , J q . Then, one can take M to be a bundle over B with one-dimensionalfibers such as R ˆ B , R ą ˆ B and S ˆ B , and choose t θ α u to be a unitary frame for B , extend it to a coframe on M Ñ B by adjoining a horizontal 1-form θ . Then clearly p M , H, J q is an almost CR manifold, where H “ Ann p θ q , and the ´ i-eigenbundle of J is H p , q “ Ann p θ , θ α q .3.6.2. Partially integrable contact almost CR manifolds.
Suppose that p H, J q is bothcontact and partially integrable. Then p H , J q is equipped with a subconformal structure c H,J compatible with the bundle complex structure J . Indeed, the Levi form defines a subconformal metric h : “ i L ˝ J on p M , H, J q , whose signature is inherited from thatof ´ i L . Note that H p , q and H p , q are totally null with respect to c H,J . In particular,we have a one-to-one correspondence between contact forms in Ann p H q and metrics in c H,J , each metric being given by h : “ i L ˝ J where L “ θ ˝ L for some contact form θ .An equivalent description of the Levi form in terms of density bundles can be obtained– see e.g. [27, 10].Furthermore, each choice of contact 1-form θ determines a unique vector field e ,the Reeb vector field, satisfying θ p e q “ θ p e , ¨q “
0, which induces a canonicalsplitting (3.12). In addition, one can choose an adapted coframe t θ α , θ ¯ α u for H , forwhich d θ “ L α ¯ β θ α ^ θ ¯ β . Any two such coframes must be related by a change (3.11) where φ α “ L α ¯ β ∇ ¯ β ϕ . Withno loss, one can always choose t θ α , θ ¯ α u to be unitary with respect to ´ i L α ¯ β .Finally, to each choice of contact 1-form θ , there exists a unique connection ∇ ,namely the Webster–Tanaka connection , that preserves θ and d θ , with prescribedtorsion [100, 108]: with reference to the structure equations (3.13), we identity Γ αβ with the connection 1-form of ∇ , A αβ “ h α ¯ γ A ¯ γβ with the so-called pseudo-Hermitiantorsion tensor , and N βγα “ N βγ ¯ δ h α ¯ δ with the so-called Nijenhuis torsion tensor . Byvirtue of the Bianchi identities, these satisfy A r αβ s “ N r βγα s “
0. While N βγα is aCR invariant – it is the obstruction to the involutivity of H p , q – the torsion tensor A αβ depends on the choice of contact form, and is an invariant of the almost pseudo-Hermitian structure only. It may be interpreted as the obstruction to the Reeb vectorfield being a transverse symmetry of the CR structure. If A αβ “ p H , J , θ q a Sasaki almost pseudo-Hermitian contact manifold.Further invariants of the partially integrable contact structure p H, J q can be obtainedfrom the curvature of ∇ . Of interest are the Chern–Moser tensor when m ą
1, and the Q -curvature when m “
1. If p H, J q is integrable, then the vanishing of these tensors isequivalent to the CR manifold being locally diffeomorphic to the CR sphere – see e.g. LMOST ROBINSON GEOMETRIES 37 [13]. When p H, J q is integrable, one can also formulate a pseudo-Hermitian analogue ofthe Einstein condition in terms of the θ -dependent Webster-Ricci tensor , which leadsto the notion of
CR–Einstein manifold – see e.g. [48, 10].
Example 3.13.
Following [48], take B to be a K¨ahler–Einstein manifold, and M itsanti-canonical bundle. Then M inherits a pseudo-Hermitian contact structure p H , J q that is CR–Einstein. The distribution H is annihilated by the connection 1-form on M . Conversely, any CR–Einstein manifold locally arises in this way.We shall leave aside analytical questions related to CR manifolds, especially in con-nection with embeddability, and we refer the reader to [105] and references therein forfurther details.3.7. Flow conditions on almost Robinson structures.
In this section, we obtainconditions on the intrinsic torsion that characterises the almost Robinson structure p N, K q when Lie dragged along the flow of any section of K . Proposition 3.14.
Let p M , g, N, K q be a p m ` q -dimensional almost Robinson ma-nifold with congruence of null curves K with leaf space M . The following statementsare equivalent.(1) p N, K q is a nearly Robinson structure, i.e r K, N s Ă N , i.e. for any opticalvector field k , and any v P Γ p N q , £ k v P Γ p N q ;(2) any complex Robinson p m ` q -form ν is preserved along K , i.e. for any opticalvector field k , £ k ν “ f ν for some smooth function f ;(3) the intrinsic torsion is a section of p { G ˆ ´ q r´ s X { G , ´ , i.e. ˘ γ i “ ˘ σ αβ “ , ˘ ζ αβ “ τ αβ ; (4) p N, K q induces an almost CR structure p H, J q on M .If any of these conditions holds, K is geodesic and its shear commutes with the screenbundle complex structure. Remark 3.15.
Condition (1) tells us that the splitting of C H K into H p , q K and H p , q K is preserved along the integral curves of K . In particular, the distribution H on M inherits this splitting, which is equivalent to an almost CR structure, as claimed by (4). Proof.
The equivalence between (1) and (2) is clear. For the almost Robinson structureto be preserved along K , i.e. £ k v P Γ p N q for any k P Γ p K q , v P Γ p N q , we must have0 “ p £ k δ aα q δ aβ “ p k b ∇ b δ aα q δ aβ ´ p δ bα ∇ b k a q δ aβ “ ζ αβ ´ τ αβ ´ σ αβ , “ p £ k δ aα q κ a “ ´ ´ k b ∇ b κ a ¯ δ aα “ ´ γ α , for any splitting operators p ℓ a , δ aα , δ a ¯ α , k a q . Now taking the symmetric and skew-symmetricparts yield ´ τ αβ ` ζ αβ “ σ αβ “
0, and γ α “
0. This computation shows that condi-tions (1) and (3) are equivalent.The equivalence between (2) and (4) can be proved following [64]. One can alwaysfind a complex Robinson p m ` q -form ν such that £ k ν “ k .So ν is the pullback of a complex p m ` q -form ν on M . This ν clearly shares the same algebraic properties of ν . In particular, it is simple, and θ ^ ν “
0, where θ is a real 1-form that pulls back to an optical 1-form on M , and annihilates a rank-2 m distribution H on M . This means that span t ν u “ ^ m ` Ann p H p , q q for some complex rank- m vector subbundle H p , q of C H . The story for the complex conjugate of ν is entirelyanalogous, and yields a complex rank- m vector subbundle H p , q of C H . It is thenstraightforward to check H p , q X H p , q “ t u at any point, i.e. C H “ H p , q ‘ H p , q ,which defines the almost CR structure on M as required.The last claim of the proposition follows from Lemma 3.6. (cid:3) That the almost Robinson structure induces an almost CR structure on the leaf spaceis not equivalent to a Robinson 3-form being preserved along the congruence. Instead,we obtain the following conditions:
Proposition 3.16.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K with leaf space M . The following statements are equivalent.(1) any Robinson -form ρ abc is preserved along K , i.e. £ k ρ abc “ f ρ abc for somesmooth function f , and any optical vector field k .(2) the intrinsic torsion is a section of p { G ˆ ´ q r s X { G , ´ , i.e. ˘ γ i “ ˘ σ α ¯ β “ , ˘ ζ αβ “ ´ τ αβ . (3) any Robinson -form induces a -form on the leaf space p M , H q .If any of these conditions holds, K is geodesic and its shear anticommutes with thescreen bundle complex structure.Proof. Choose splitting operators p ℓ a , δ aα , δ a ¯ α , k a q and let ρ abc be the Robinson 3-formassociated to the optical 1-form κ a “ g ab k b . The condition that ρ abc be preserved along K gives k d ∇ d ρ abc ´ k d ∇ r a ρ bc s d “ f ρ abc . Now, contracting with δ aα δ bβ δ c ¯ γ , δ aα δ bβ ℓ c and δ aα δ b ¯ β ℓ c yields γ α “ ζ αβ “ ´ τ αβ and σ α ¯ β “ (cid:3) We now take a slight detour to consider the parallelism of N along K . Proposition 3.17.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) the intrinsic torsion is a section of { G , ´ , i.e. ˘ γ i “ ˘ ζ αβ “ . (2) any Robinson -form is recurrent along K .(3) N is parallel along K .If any of these conditions holds, K is geodesic.Proof. The equivalence between (1) and (2) is tautological, while the equivalence be-tween (2) and (3) follows from the definition of ˘ ζ αβ . That K is geodesic in this casefollows from ˘ γ i “ (cid:3) LMOST ROBINSON GEOMETRIES 39
Combining Propositions 3.14 and 3.16, we obtain:
Proposition 3.18.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K with leaf space M . The following statements are equivalent.(1) N and any Robinson -form are preserved along K .(2) The intrinsic torsion is a section of { G , ´ X { G ´ X { G , ´ , i.e. ˘ γ i “ ˘ σ ij “ ˘ τ αβ “ ˘ ζ αβ “ . (3) p N, K q induces an almost CR structure p H, J q on M and the screen bundlemetric induces a bundle conformal structure c H on p M , H q .If any of these conditions holds, K is geodesic and non-shearing, its twist commuteswith the screen bundle complex structure, N is parallel along K , and p M , H, c H , J q isa conformal Hermitian vector bundle. Remark 3.19.
It is important to note, in anticipation of the next section, and withreference to Section 3.6, that the conformal structure c H in Proposition 3.18 does notnecessarily arise from the Levi-Civita of p H , J q – see Remarks 3.23 and 3.32 below.3.8. Nearly Robinson structures and almost CR structures.
In [64], the authorsshow how to construct Robinson manifolds as lines bundle over CR manifolds. Here, wegeneralise the construction to nearly Robinson manifolds, which, we recall, are almostRobinson manifold for which the almost Robinson structure is preserved along the flowof any optical vector field. Their characterisation in terms of their intrinsic torsion isgiven in Proposition 3.14. The proof of the following result is self-evident.
Proposition 3.20.
Let p M , H, J q be a p m ` q -dimensional oriented almost CR ma-nifold, and M : “ R ˆ M ̟ ÝÑ M be a trivial line bundle over M . Fix a triplet ` t θ , θ α , θ ¯ α u , h α ¯ β , θ ˘ where ‚ t θ , θ α , θ ¯ α u is a CR-coframe on p M , H, J q , ‚ h α ¯ β is a smooth field of positive-definite Hermitian matrices on M , and ‚ θ is a -form on M such that θ ^ ε does not vanish at any point for any p m ` q -form ε on M .Then p M , g q is a time-oriented and oriented Lorentzian manifold with metric g “ ̟ ˚ θ θ ` h α ¯ β ̟ ˚ θ α ̟ ˚ θ ¯ β , (3.14) and p N, K q , where N “ Ann t ̟ ˚ θ , ̟ ˚ θ α u and K “ Ann t ̟ ˚ θ u K , defines a nearlyRobinson structure on p M , g q . In particular, M is the leaf space of the congruence ofnull geodesics tangent to K .Any two such triplets ` t θ , θ α , θ ¯ α u , h α ¯ β , θ ˘ and ´ t p θ , p θ α , p θ ¯ α u , p h α ¯ β , p θ ¯ where t θ , θ α , θ ¯ α u and t p θ , p θ α , p θ ¯ α u are related by (3.11) define the same metric (3.14) if and only if p θ “ e ´ ϕ θ ´ φ αβ φ β θ α ´ φ ¯ α ¯ β φ ¯ β θ ¯ α ´ φ β φ β θ , p h α ¯ β “ h γ ¯ δ p φ ´ q αγ p φ ´ q ¯ β ¯ δ . In particular, φ αγ is an element of U p m q at every point. Remark 3.21.
Variations of the above construction are possible by replacing the R -factor of M by R ą or S for instance. Definition 3.22.
We shall refer to any almost Robinson structure constructed on M : “ R ˆ M ÝÑ M as in Proposition (3.20) as a lift of the almost CR manifold p M , H, J q . The 1-forms t θ , θ α , θ ¯ α u will be referred to as horizontal , and the 1-form θ as vertical (with respect to the fibration M Ñ M ).We emphasise that the metric constructed in Proposition 3.20 is not canonical ingeneral. To do away with the choice of CR coframe and 1-form θ , while fixing theconformal class r h α ¯ β s of the field of Hermitian matrices, one needs to introduce thenotion of generalised almost Robinson geometry , which is dealt with in Section 5 – seeProposition 5.3. Remark 3.23.
One application of Proposition 3.20 is to construct an almost Robinsonmanifold p M , g, N, K q with prescribed intrinsic torsion from a chosen almost CR ma-nifold p M , H, J q . As already pointed out, p H, J q entirely determines the twist of theassociated congruence K and the involutivity property of N . However, the p , q -partof the shear and the expansion of K depend on the choice of Hermitian form h α ¯ β inthe metric (3.14), and one has two possible options:(1) either we choose h α ¯ β to depend on the fibers of M Ñ M in a non-trivial way,in which case K is shearing;(2) or we set h α ¯ β “ e ϕ h α ¯ β for some Hermitian form h α ¯ β on p M , H , J q and smoothfunction ϕ on M , in which case K is non-shearing, and, if ϕ is a function on M ,non-expanding too. If the p , q -part of the Levi form L α ¯ β is non-degenerate,we may take h α ¯ β “ ´ i L αβ . This is the case if p H, J q is partially integrable andcontact: the resulting p N, K q is then said to be twist-induced , of which we shallsay more in Section (3.9).Various considerations may dictate the choice of screen bundle Hermitian form h α ¯ β .Example 3.49 is a case in point.Clearly, if the almost CR structure is totally degenerate, choosing h α ¯ β “ e ϕ h α ¯ β gives rise to a non-twisting non-shearing congruence of null geodesics, i.e. the resultingmanifold p M , g q is either a Kundt spacetime or a Robinson–Trautman spacetime —see Sections 3.12 and 3.13.There are further, intermediate, situations where the Levi form of the CR structureis degenerate but not identically zero. This allows for screen bundle metrics to beconstructed partly from the Levi form.The remaining properties of the intrinsic torsion of p N, K q will also depend on thechoice 1-form θ as shall be seen later – see Lemma 3.53.Before we proceed, we give the converse of Proposition 3.20 – see [64] for the invo-lutive case. Proposition 3.24.
Let p M , g, N, K q be a nearly Robinson manifold with congruence ofnull curves K . Then M is locally diffeomorphic to the trivial line bundle R ˆ M ̟ ÝÑ M ,where M is the local leaf space of K and is equipped with an almost CR structure p H , J q . Further, locally, g takes the form (3.14) for some CR coframe t θ , θ α , θ ¯ α u , fieldof Hermitian matrices h α ¯ β on M , and -form θ on M that never vanishes on K . LMOST ROBINSON GEOMETRIES 41
Proof.
Note that M is locally diffeomorphic to the line bundle R ˆ M ̟ ÝÑ M , whereeach fiber is a null curve of K . By Proposition 3.14, the assumption on the intrinsictorsion tells us that the almost Robinson structure descends to an almost CR structure p H , J q on the local leaf space M of K . Following our convention, N { C K and N { C K descend to the eigenbundles H p , q and H p , q of J respectively. Let t θ , θ α , θ ¯ α u bea coframe on M where span t θ u “ Ann p H q and span t θ , θ α u “ Ann p H p , q q . Thenspan t ̟ ˚ θ u “ Ann p H q and span t ̟ ˚ θ , ̟ ˚ θ α u “ Ann p N q , and it follows immediatethat g must be of the form (3.14) where h α ¯ β is a field of Hermitian matrices on M ,and θ a 1-form on M that never vanishes on K . (cid:3) Propositions 3.20 and 3.24 allow us to relate the geometric properties of an almostRobinson manifold with those of the almost CR structure on the leaf space of itsassociated congruence of null geodesics. To understand this relation better, a fewremarks are in order.
Remark 3.25.
It is crucial to bear in mind that the coframe t θ , θ α , θ ¯ α u on M does not in general pull back to a Robinson coframe on M by simply adjoining the 1-form θ , in the sense that t θ α u are not unitary with respect to h α ¯ β . This essentially dependson the choice of h α ¯ β . To be precise, a Robinson coframe t θ , θ α , θ ¯ α , θ u for the metric(3.14), so that t θ α u is unitary for h α ¯ β , is related to t θ , θ α , θ ¯ α u via θ “ ̟ ˚ θ , θ α “ ψ β α ´ ̟ ˚ θ β ¯ ` ψ α θ , (3.15)for some smooth functions ψ α and ψ β α on M , where ψ β α takes values in GL p m, C q .Note that one can always choose our Robinson coframe such that ψ α “ m “
1, we have the decomposition GL p , C q – C ˚ – R ą ¨ U p q , in which casewe always have ψ αβ “ r e i φ δ βα where r, φ are real with r ą
0. Now, the space of allHermitian forms on C m is isomorphic to the homogeneous space GL p m, C q{ U p m q ofreal dimension m . Thus, the failure of t θ α u to be unitary with respect to h α ¯ β at anypoint is measured by an element of GL p m, C q mod U p m q .From (3.15), it is straightforward to compute the twist of K as2 L α ¯ β “ τ γ ¯ δ ψ αγ ψ ¯ β ¯ δ , L αβ “ τ γδ ψ αγ ψ β δ . From the first of these equations, the G , ´ ‘ G , ´ -component of the intrinsic torsionencodes the signature of the Levi form of p H, J q , while the G , ´ -component of theintrinsic torsion, the partial integrability of p H, J q . From these remarks, we are ableto draw the following conclusions. Proposition 3.26.
Let p M , g, N, K q be a nearly Robinson manifold with congruenceof null curves K and almost CR leaf space p M , H, J q . The following statements areequivalent:(1) The intrinsic torsion has non-degenerate { G ´ -component ˘ τ ij ;(2) K is maximally twisting;(3) p H, J q is contact. Proposition 3.27.
Let p M , g, N, K q be a nearly Robinson manifold with congruenceof null curves K and almost CR leaf space p M , H, J q . The following statements areequivalent: (1) ˚ T is also a section of { G , ´ , i.e. ˘ γ i “ ˘ τ αβ “ ˘ σ αβ “ ˘ ζ αβ “ , (2) the twist of K commutes with the screen bundle complex structure;(3) p H, J q is partially integrable.If any of these conditions holds, K is a non-twisting congruence of null geodesics, and N is parallel along K . Proposition 3.28.
Let p M , g, N, K q be a nearly Robinson manifold with congruenceof null curves K and almost CR leaf space p M , H, J q . The following statements areequivalent:(1) The intrinsic torsion is also a section of { G ´ , i.e. ˘ γ i “ ˘ τ ij “ ˘ σ αβ “ ˘ ζ αβ “ , (2) K is non-twisting;(3) p H, J q is totally degenerate.If any of these conditions holds, N is parallel along K . For the expansion and shear, we compute ǫm h α ¯ β ` σ α ¯ β “ h γ ¯ β ψ δγ p ψ ´ q αδ ` h α ¯ δ p ψ ´ q ¯ γ ¯ δ ψ ¯ β ¯ γ , where ψ δγ : “ £ k ψ δγ for the optical vector field k “ g ´ p θ , ¨q . In particular, weinterpret the G , ´ -component ˘ σ α ¯ β of the intrinsic torsion as the infinitesimal obstructionto t θ α u being unitary with respect to h α ¯ β . This is consistent with the fact that anyinfinitesimal element of GL p m, C q{ U p m q consists of a Hermitian matrix. Moduling outby any elements proportional to h α ¯ β yields tracefree Hermitian matrices. The followingpropositions are immediate consequences of these considerations.In the next result, the assumptions of Proposition 3.18 are weakened. Proposition 3.29.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K and leaf space M . The following statements are equivalent.(1) The intrinsic torsion is a section of p { G ˆ ´ q r´ s X { G ´ , i.e. ˘ γ i “ ˘ σ ij “ , ˘ ζ αβ “ τ αβ , (2) p N, K q induces an almost CR structure p H, J q on M , and r g s n.e. induces abundle conformal structure c H on H .If any of these conditions holds, K is a non-shearing congruence of null geodesics, and p H , J , c q is a conformal Hermitian vector bundle. Proposition 3.30.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K and leaf space M . The following statements are equivalent.(1) The intrinsic torsion is a section of { G , ´ X { G , ´ X { G , ´ with non-degenerate G , ´ ‘ G , ´ -component, i.e. ˘ γ i “ ˘ τ αβ “ ˘ σ αβ “ ˘ ζ αβ “ , with non-degenerate ˘ τ α ¯ β . (2) p N, K q induces a partially integrable contact almost CR structure p H, J q on M . LMOST ROBINSON GEOMETRIES 43
If any of these conditions holds, K is a maximally twisting congruence of null geodesics. Proposition 3.31.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K and leaf space M . The following statements are equivalent.(1) The intrinsic torsion is a section of { G , ´ X { G ´ X { G , ´ with non-degenerate G , ´ ‘ G , ´ -component, i.e. ˘ γ i “ ˘ τ αβ “ ˘ σ ij “ ˘ ζ αβ “ , with non-degenerate ˘ τ α ¯ β . (2) p N, K q induces a partially integrable contact almost CR structure p H, J q on M ,and r g s n.e. induces a conformal structure on p M , H, J q .If any of these conditions holds, K is a maximally twisting congruence of null geodesics. Remark 3.32.
If any of the conditions of Proposition 3.31 holds, the distribution H on M is equipped with two conformal structures: ‚ one, c H , inherited from r g s n.e. ; ‚ the other, c H,J , induced from the Levi form of p H , J q and ultimately the twistof K .These two conformal structures are distinct in general, and the measure of the failureof them coinciding is given by the { G , ´ -component ˘ τ ˝ α ¯ β of the intrinsic torsion.Further, from Remark 3.9, metrics in c H are in one-to-one correspondence withmetrics in c H , and metrics in c H,J are in one-to-one correspondence with optical vectorfields k such that £ k κ “ κ “ g p k, ¨q . These two conformal structures arerelated in the following way. Since K is maximally twisting, we know from [24] thatthere exists a unique optical vector field k a such that the twist of k a is normalised to τ ij τ ij “ m for any metric g in r g s n.e. . With this normalisation, the Levi form of theCR structure is related to the twist of k by ´ i L α ¯ β “ h α ¯ β ` τ ˝ α ¯ β . (3.16)The tensor τ ˝ α ¯ β could equivalently be viewed as a deformation of the complex structure J on H .In the next section, we shall focus on a special case of the aforementioned resultswhere the only non-vanishing of the twist, i.e. of the G ´ -component of the intrinsictorsion, lies in G , ´ .3.9. Twist-induced almost Robinson structures.
We shall now present a specialcase of an almost Robinson structure arises in the following context.
Proposition 3.33.
Let p M , g, K q be an optical geometry of dimension m ` withcongruence of oriented null curves K . Let κ a be an optical -form and set τ abc : “ κ r a ∇ b κ c s . The following two conditions are equivalent.(1) The -form τ abc satisfies τ abe τ ecd “ ´ m τ r aef g b sr c τ d s ef ‰ , (3.17) (2) K is geodesic and twisting, and there exists a unique optical vector field k whosetwist endormorphism h ´ ˝ τ is a bundle complex structure J compatible with h on the screen bundle p H K , h q , i.e. J “ h ´ ˝ τ . In particular, the twist of K induces an almost Robinson structure p N, K q on p M , g, K q , and κ “ g p k, ¨q determines a unique Robinson -form given by ρ abc “ κ r a ∇ b κ c s . (3.18) Proof.
Choose splitting operators t ℓ a , δ ai , k a u . Then we can write ∇ r a κ b s “ λ r a γ b s ` τ ab ` α r a κ b s for some γ a , τ ab and α a , where ℓ a γ a “ ℓ a τ ab “ k a γ a “ k a τ ab “
0. Contracting(3.17) with k a ℓ b k c ℓ d and δ ai ℓ b δ cj ℓ d yields γ i γ i “ , (3.19) τ ik τ kj “ ´ m τ kℓ τ kℓ δ ji ‰ , (3.20)respectively, where γ i “ γ a δ ai and τ ij “ τ ab δ ai δ bj . Equation (3.19) tells us that γ i “ h ij is positive-definite, i.e. K is geodesic. Equation (3.20) tells us that we canrescale k by ? m } τ } so that the twist of the rescaled optical vector field satisfies τ ik τ kj “ ´ δ ji , i.e. h ´ ˝ τ is a bundle complex structure on H K . The uniqueness of k follows from theassumption that K is oriented. (cid:3) Remark 3.34.
That equation (3.18) singles out an optical 1-form also follows fromthe fact that LHS has boost weight 2 and the RHS boost weight 1.
Definition 3.35.
We shall refer to the almost Robinson structure given in Proposition3.33 as a twist-induced almost Robinson structure . Remark 3.36.
Let us re-emphasise that by Proposition 3.33, the congruence associ-ated to a twist-induced almost Robinson structure is always geodesic and (maximally)twisting.It is clear that an almost Robinson structure p N, K q is twist-induced if and only ifits intrinsic torsion is a section of { G , ´ X { G , ´ with non-vanishing G , ´ -component, i.e.˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ , ˘ τ ω ‰ . However, the following proposition tells us that the intrinsic torsion must in fact be asection of a subbundle thereof.
Proposition 3.37.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) p N, K q is a twist-induced almost Robinson structure;(2) the intrinsic torsion of p N, K q is a section of { G , ´ X { G , ´ X { G , ´ X { G , withnon-vanishing G , ´ -component, i.e. ˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ ˘ ζ αβ “ ˘ G αβγ “ , ˘ τ ω ‰ . If any of these conditions holds, N is parallel along K .Proof. That (2) implies (1) is immediate since { G , ´ X { G , ´ X { G , ´ X { G , is a subbundle of { G , ´ X { G , ´ . For the converse, we note that a twist-induced almost Robinson structure LMOST ROBINSON GEOMETRIES 45 singles out a preferred optical 1-form κ such that p d κ q ab “ ω ab ` κ r a α b s for some 1-form α a , where ω ab is a representative of the screen bundle Hermitian structure. Notethat the associated Robinson 3-form is given by ρ abc “ κ r a ω bc s . Taking the exteriorderivative of d κ yields 0 “ p d ω q abc ` ω r ab α c s ´ κ r a p d α q bc s . (3.21)Choose a splitting t ℓ a , δ aα , δ a ¯ α , k a u . Contracting (3.21) with k a δ bα δ cβ leads to0 “ p d ω q αβ “ p ∇ ω q αβ ´ ω ab δ aα δ bβ “ p ∇ ρ q αβ “ ζ αβ , i.e. ˚ T is a section of { G , ´ . Now, contract (3.21) with δ aα δ bβ δ cγ yields0 “ p ∇ ω q αβγ “ p ∇ ρ q αβγ “ G αβγ , i.e. ˚ T is a section of { G , , which completes the proof. (cid:3) Remark 3.38.
Proposition 3.37 tells us that if the intrinsic torsion of a given almostRobinson manifold is a section of { G , ´ X { G , ´ but not of { G , ´ , then it must be a sectionof { G , ´ X { G , ´ X { G , ´ X { G , . This should be contrasted with the situation regarding theGray–Hervella classification of almost Hermitian manifolds [30]: the sixteen classes ofalmost Hermitian manifolds can be naturally arranged in terms of inclusions, whichare shown to be strict in the sense that each class contains an almost Hermitian metricthat does not belong to any of the other fifteen classes.The next result is a direct consequence of Propositions 3.31 and 3.37. Proposition 3.39.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) p N, K q is a twist-induced almost Robinson structure and the shear of K com-mutes with the screen bundle complex structure;(2) the intrinsic torsion is a section of { G , ´ X { G , ´ X { G , ´ X { G , ´ X { G , with non-vanishing G , ´ -component, i.e. ˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ ˘ σ αβ “ ˘ ζ αβ “ ˘ G αβγ “ , ˘ τ ω ‰ . If any of these conditions holds, p N, K q is a nearly Robinson structure and the inducedalmost CR leaf space p M , H , J q of K is partially integrable and contact. Reference [99] investigates twist-induced almost Robinson structures with non-shearingcongruences of null geodesics. The next proposition collects some of the results foundtherein.
Proposition 3.40 ([99]) . Let p M , g, N, K q be an almost Robinson manifold with con-gruence of null curves K . The following statements are equivalent.(1) p N, K q is a twist-induced almost Robinson structure and K is a non-shearingcongruence of null geodesics; (2) the intrinsic torsion ˚ T is a section of { G , ´ X { G , ´ X { G ´ X { G , ´ X { G , X { G , withnon-zero G , ´ -component, i.e. ˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ ˘ σ ij “ ˘ ζ αβ “ ˘ G αβγ “ ˘ G ˝ ¯ αβγ “ , ˘ τ ω ‰ . If any of these conditions holds, p N, K q is a nearly Robinson structure and the almostCR leaf space p M , H , J q of K is partially integrable and contact. In addition, p H , J q is equipped with a conformal structure which is induced by both the Levi form and theconformal class r g s . Remark 3.41.
In both Propositions 3.39 and 3.40, the G , -component ˘ G αβγ of theintrinsic torsion of p N, K q can be identified with the Nijenhuis tensor of p H, J q . Remark 3.42.
Continuing from Remark 3.32, in the case of a twist-induced almostRobinson structure, using the existence of a distinguished optical vector field, equation(3.16) reduces to ´ i L α ¯ β “ h α ¯ β , and there is essentially a single conformal structure induced from either p H , J q or r g s n.e. ,i.e. c H “ c H,J . Proposition 3.43.
Let p M , g, N, K q be a twist-induced almost Robinson manifold withnon-expanding non-shearing congruence of null geodesics K . Then there exists a uniquegenerator k of K and a null vector field ℓ such that g p k, ℓ q “ and κ “ g p k, ¨q satisfies d κ p k, ¨q “ , d κ p ℓ, ¨q “ . Further, the almost Robinson structure induces a partially integrable almost CR struc-ture on the leaf space p M , H q with a distinguished almost pseudo-Hermitian structureinherited from κ .Proof. Since the twist of K induces the almost Robinson structure, K must be maxi-mally twisting. Applying Proposition 4.35 of [24] yields the first part of the proposition.For the second part, we note that by Corollary 4.38 of [24], p M , H q inherits a contactsub-Riemannian structure h with distinguished contact form θ inherited from κ . Thecorresponding Levi form is a bundle Hermitian bilinear form ω on H . The compositionof h ´ and ω is precisely the bundle complex structure of a partially integrable almostCR structure as required. (cid:3) One can also drop the non-expanding assumption on the congruence K in the propo-sition above. We will return to this point in Section 4.3.10. Almost Robinson structure as almost null structures.
To obtain furthergeometric interpretations of the subbundles of G , we shall presently regard the almostRobinson structure p N, K q as an almost null structure N on p M , C g q in its own right,i.e. without any reference to the complex conjugate N . The structure group of N is thestabiliser R of an MTN vector subspace of C m ` in SO p m ` , C q . As we shall beusing a spinorial approach, we shall assume with no loss of generality, at least locally,that R is a subgroup of Spin p m ` , C q . We shall then identify N as the kernel of themap ν Aa : “ γ aB A ν B : Γ p T M q Ñ Γ p S ´ q for some Robinson spinor ν A . In particular,with a choice of dual N ˚ , the image of ν Aa is isomorphic to N ˚ . The intrinsic torsion LMOST ROBINSON GEOMETRIES 47 of such a structure is already investigated in [94], and we shall appeal to the resultscontained therein for the subsequent analysis.
Theorem 3.44.
Let p M , g, K, N q be a p m ` q -dimensional almost Robinson manifoldwith intrinsic torsion ˚ T . Let ν A be a Robinson spinor and set ν Aa : “ γ aB A ν B . Then,for m ą , ˚ T P Γ p { G , q ðñ ´ ν a r A ∇ a ν bB ¯ ν C s b “ , (3.22) ˚ T P Γ p { G , q ðñ ´ ν a p A ∇ a ν bB q ¯ ν Cb “ , (3.23) ˚ T P Γ p { G , q ðñ ´ ∇ a ν bB ¯ ν Cb ` m ´ ν r Ba ∇ b ν bC s ` ν b r B ∇ b ν C s a ¯ “ , (3.24) ˚ T P Γ ´ p { G ˆ q r s ¯ ðñ ν A ∇ a ν aB ´ ν aB ∇ b ν A “ . (3.25) In dimension six, i.e. m “ , equivalences (3.23) and (3.25) hold, but equivalences (3.22) and (3.24) are replaced by ˚ T P Γ pp { G ˆ ´ q r ´ s q ðñ ´ ν a r A ∇ a ν bB ¯ ν C s b “ , (3.26) and (3.27) ˚ T P Γ p { G , q ðñ ´ ∇ a ν bB ¯ ν Cb ` ´ ν r Ba ∇ b ν bC s ` ν b r B ∇ b ν C s a ¯ ´ ν bA ∇ b ν A γ aBC “ , respectively. For the last equation, we have used the bundle isomorphisms S ˘ – S ˚¯ and C T M – ^ S ` – ^ S ´ .Proof. Set Γ
ABC “ p δ aA ∇ a δ bB q δ Cb and Γ ABC “ p δ aA ∇ a δ bB q δ Cb . Then, one can show [96]RHS of (3.22) and (3.26) ðñ Γ r ABC s “ , RHS of (3.23) ðñ Γ p AB q C “ , RHS of (3.24) ðñ ` Γ ABC ˘ ˝ “ Γ ABC “ , RHS of (3.25) ðñ Γ BBA “ Γ ABC “ , and in dimension sixRHS of (3.27) ðñ ` Γ ABC ˘ ˝ “ Γ p AB q C “ . Here ` Γ ABC ˘ ˝ “ Γ ABC ´ m δ r B | A Γ DD | C s .We can immediately deduce thatΓ r ABC s “ ðñ G ¯ α ¯ β ¯ γ “ ζ ¯ α ¯ β ` τ ¯ α ¯ β “ , Γ p AB q C “ ðñ G ¯ α ¯ β ¯ γ “ σ ¯ α ¯ β “ τ ¯ α ¯ β ´ ζ ¯ α ¯ β “ γ ¯ α “ , ` Γ ABC ˘ ˝ “ ðñ B ¯ α ¯ β “ G ˝ α ¯ β ¯ γ “ p m ´ q E ¯ α ` G ¯ α “ τ ˝ α ¯ β “ σ α ¯ β “ , Γ BBA “ ðñ ´ E ¯ α ` G ¯ α “ τ ω “ ǫ “ , where B ¯ α ¯ β , G ¯ α ¯ β ¯ γ , G ¯ α ¯ β ¯ γ , G ˝ α ¯ β ¯ γ , ζ ¯ α ¯ β , τ ¯ α ¯ β , E ¯ α and γ ¯ α are the complex conjugates of thosedefined by equations (3.9). The result now follows immediately from the definition ofthe Q -invariant bundles and Theorem 2.18. (cid:3) As a consequence, we obtain
Proposition 3.45.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) Any Robinson spinor ν A satisfies ´ ν aA ∇ a ν bB ¯ ν Cb “ . (3.28) (2) Any Robinson spinor ν A is recurrent along N , i.e. ´ ν aA ∇ a ν r B ¯ ν C s “ . (3.29) (3) The intrinsic torsion of p N, K q is a section of { G , X { G , , i.e. ˘ γ i “ ˘ τ αβ “ ˘ σ αβ “ ˘ ζ αβ “ ˘ G αβγ “ ˘ G αβγ “ . (4) N is in involution, i.e. for any v, w P Γ p N q , r v, w s P Γ p N q .(5) N is parallel along itself, i.e. for any v, w P Γ p N q , ∇ v w P Γ p N q .(6) p N, K q induces a CR structure on the leaf space of K .If any of these conditions holds, K is geodesic, its twist and shear commute with thescreen bundle complex structure, and N is parallel along K .Proof. One can show [96] that equations (3.28) and (3.29) are equivalent, from whichfollows the equivalence (1) and (2). The equivalence between (1) and (3) is a directconsequence of Theorem 3.44. The equivalence between (1) and (6) is already givenin [41, 96], while the equivalence between (4) and (5) is established in [94], and thatbetween (4) and (6) in [64]. (cid:3)
Remark 3.46.
It is important to note that the involutivity of the almost Robinsonstructure p N, K q does not imply that the Robinson 3-form is preserved along K , andthus, may not descend to the leaf space M — see Proposition 3.16. If it did, it wouldimply that the congruence of null geodesics is non-shearing, which is not true in generalexcept in dimension four. Remark 3.47.
In the analytic category, one may complexify p M , g q to a complexRiemanian manifold p Ă M , r g q , and extend N analytically to p Ă M , r g q [109, 111, 18]. By theFrobenius theorem, condition (4) of Proposition 3.45 is equivalent to the local existenceof a complex foliation of p Ă M , r g q by p m ` q -dimensional complex submanifolds on which r g is totally degenerate. Condition (5) then tells us that these leaves are totally geodesicwith respect to the Levi-Civita connection of r g . See also [71, 41, 98] for further details.Another interesting class of almost Robinson manifolds is given by the next propo-sition, which weakens the assumptions of Proposition 3.40. Proposition 3.48.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent. LMOST ROBINSON GEOMETRIES 49 (1) Any Robinson -form ρ abc satisfies d ρ “ α ^ ρ for some -form α . (3.30) (2) The intrinsic torsion is a section of { G , X { G , , i.e. ˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ ˘ σ α ¯ β “ ˘ ζ αβ “ ˘ G αβγ “ ˘ G ˝ ¯ αβγ “ . If any of these conditions holds, K is geodesic, its shear anticommutes with the screenbundle complex structure, and p N, K q induces an almost CR structure p H , J q on theleaf space M of K .Further, if K is twisting, then p N, K q is twist-induced, and p H , J q is contact andpartially integrable.Proof. Choose an optical 1-form κ a with Robinson 3-form ρ abc . Then, using p d ρ q abcd “ ∇ r a ρ bcd s , we compute the various components of (3.30) to find p d ρ ´ α ^ ρ q αβ ¯ γ “ ðñ γ α “ , p d ρ ´ α ^ ρ q αβ “ ðñ τ αβ ` ζ αβ “ , p d ρ ´ α ^ ρ q α ¯ β “ ðñ σ α ¯ β “ , p d ρ ´ α ^ ρ q αβγ “ ðñ G αβγ “ , p d ρ ´ α ^ ρ q ¯ αβγ “ ðñ G ˝ ¯ αβγ “ , p d ρ ´ α ^ ρ q αβγ ¯ δ “ ðñ τ αβ “ , p d ρ ´ α ^ ρ q α ¯ β “ ðñ τ ˝ α ¯ β “ . The remaining contractions are vacuous. The equivalence between (1) and (2) nowfollows. The properties of K and its leaf space M follow from ˘ γ i “ ˘ τ αβ “ ˘ τ ˝ α ¯ β “ ˘ σ α ¯ β “ ˘ ζ αβ “
0. If K is twisting, the only non-vanishing component of the twist is ˘ τ ω , whichtells us that p N, K q is induced from the twist (by definition), and the associated almostCR structure on M is partially integrable as in Proposition 3.39. (cid:3) In the twisting case, Proposition 3.48 is thus slightly stronger than Proposition 3.39.
Example 3.49 (The Kerr–NUT–(A)dS metric) . In [15], the authors present a higher-dimensional generalisation of the
Kerr–NUT–(A)dS metric . In dimension 2 m `
2, incoordinates p r, y α , t, φ i q , where α, i “ , . . . m , this Einstein metric takes the form g “ UX d r ´ XU ˜ W d t ´ m ÿ i “ γ i d φ i ¸ ` m ÿ α “ U α X α d y α ` m ÿ α “ X α U α ˜ p ` Λ r q W ´ Λ y α d t ´ m ÿ i “ p r ` a i q γ i a i ´ r d φ i ¸ , where U “ m ÿ α “ p r ` y α q , W “ m ź α p ´ Λ y α q ,X “ p ` Λ r q m ź i “ p r ` a i q ´ M r ,U α “ ´p r ` y α q m ź α ‰ β “ p y β ´ y α q , p α “ , . . . , m q ,γ i “ m ź α “ p a i ´ y α q , p i “ , . . . , m q ,X α “ ´p ´ Λ y α q m ź i “ p a i ´ y α q ´ L α y α , p α “ , . . . , m q . Here, Λ is the cosmological constant, M the mass, L α are the m NUT parameters, and a i the m rotation parameters.Define κ “ d r ` XU ˜ W d t ´ m ÿ i “ γ i d φ i ¸ ,λ “ d r ´ XU ˜ W d t ´ m ÿ i “ γ i d φ i ¸ ,θ α “ d y α ` i X α U α ˜ p ` Λ r q W ´ Λ y α d t ´ m ÿ i “ p r ` a i q γ i a i ´ r d φ i ¸ . The set of null 1-forms t κ, θ α u , where κ is real and θ α are complex, defines a Robinsonstructure p N K , K q . Similarly, the set of null 1-forms t λ, θ α u defines a Robinson struc-ture p N L , L q , where L is dual to K . More generally, it is shown in [54] that this metricadmits 2 m ` almost null structures, which yield 2 m Robinson structures associated toeach of the optical structures K and L , and any of their Robinson spinors satisfies(3.28).The respective congruences K and L of null geodesics associated to K and L arenon-shearing for m “
1, but shearing for all m ą p N K , K q descends to a CRstructure on the leaf space of K , which is contact since K is maximally twisting. Thesame goes for p N L , L q .These findings also apply to other related metrics such as the Myers–Perry metric[57] that may be viewed as a special case of the Kerr–NUT–(A)dS metric in the limitwhere the NUT parameters and cosmological constant tend to zero. Example 3.50 (The Taub–NUT–(A)dS- metric) . It is shown in [99] that the Taub–NUT–(A)dS metric [101, 58] generalised to even dimensions as in [6] admits a pair oftwist-induced Robinson structures. In fact, the intrinsic torsion of any these Robinsonstructure is a section of { G , . In particular, any of their Robinson spinors satisfies (3.24). LMOST ROBINSON GEOMETRIES 51
The associated congruences of null geodesics are non-shearing as is already pointed outin [99, 4].3.11.
Analogies between almost Robinson geometry and almost Hermitiangeometry.
Recall that an almost Hermitian manifold consists of a triple p M , g, J q ,where p M , g q is a p m ` q -dimensional smooth Riemannian manifold, and J is analmost complex structure compatible with g , i.e. J ˝ g “ ´ g ˝ J . An equivalent definitionof an almost Hermitian structure on p M , g q is as an almost null structure N of realindex zero, i.e. the complexified tangent bundle splits as C T M “ N ‘ N [64] – sincein Riemannian signature, an almost null structure always has real index zero, we maydispense of this attribute. The equivalence between the two definitions is establishedby identifying N and N as the eigenbundles of J . Dually, one can express the almostHermitian structure in terms of a non-vanishing section of ^ m ` Ann p N q , which can benormalised up to a phase against its complex conjugate. Locally, or globally if p M , g q isspin, this section is the ‘square’ of a pure spinor field (of real index 0). It annihilates N ,while its charged conjugate annihilates N . Their pairing yields the almost Hermitian2-form of p M , g, J q and its powers [46].Thus, as emphazised in [64], the point of contact between almost Robinson geometryand almost Hermitian geometry is their underlying almost null structure, and the onlydistinguishing feature between them is the real index of the underlying null structure,which is itself determined by the metric signature.The relation between almost Robinson structures and almost null structures was al-ready investigated in the previous section, especially in Theorem 3.44, using the resultsof [96]. One can play the same game by studying the geometry of an almost Hermitianmanifold p M , g, J q in the light of its underlying almost null structure p M , C g, N q . Tothis end, we recall the Gray–Hervella classification of almost Hermitian manifolds givenin [30]. Following the notation of that reference, the bundle W of intrinsic torsions of p M , g, J q splits into irreducible U p m ` q -invariant subbundles as W “ W ‘ W ‘ W ‘ W , (3.31)where at any point W – rr ^ p , q p R m ` q ˚ ss , W – rr p R m ` q ˚ ss , W – rr ^ p , q˝ p R m ` q ˚ ss , W – rr ^ p , q p R m ` q ˚ ss . (3.32)There are various ways to characterise the intrinsic torsion ˚ T of p M , g, J q . For instance, p M , g, J q is almost K¨ahler, i.e. ˚ T is a section of W , if and only if the almost Hermitian2-form ω “ g ˝ J is closed. It is Hermitian, i.e. ˚ T is a section of W ‘ W , if and only ifthe Nijenhuis tensor of the complex structure vanishes. But only a subset of the Gray–Hervella classes will be relevant to the present discussion, namely those that reflectthe geometric properties of the underlying almost null structure. For instance, one cancharacterise a Hermitian manifold in terms of a pure spinor field that is recurrent alongthe totally null distribution it defines [41, 96, 47]: this is equivalent to the eigenbundlesof J being in involution — see Proposition 3.45 for the Robinson analogue. Proceedingas in the proof of Theorem 3.44, one can easily prove the equivalence between the firstand last columns of Table 6, which summarizes the correspondences between the variousclasses of almost null structures, almost Robinson structures and almost Hermitian structures. The equivalence between the first and second columns follows directly fromTheorem 3.44. We leave the details for the reader.Almost null structures Almost Robinson structures Almost Hermitian structuresEq. (3.22) { G , W ‘ W ‘ W Eq. (3.23) { G , W ‘ W ‘ W Eqs. (3.22) and (3.23) { G , X { G , W ‘ W Eq. (3.24) in dim ą { G , W Eq. (3.27) in dim 6 { G , W ‘ W Eq. (3.25) p { G ˆ q r s W Eq. (3.24) and (3.25) t u t u Table 6.
Comparison of the intrinsic torsion of almost Robinson struc-tures and the Gray–Hervella classification of almost Hermitian manifoldson the basis of the properties of their underlying null structure.3.12.
Almost Robinson manifolds of Kundt type.Definition 3.51.
An almost Robinson manifold p M , g, N, K q is said to be of Kundttype if p M , g, K q is a Kundt spacetime, i.e. the congruence of curves tangent to K isgeodesic, non-expanding, non-shearing and non-twisting.Equivalently, the intrinsic torsion of such manifolds is a section of { G ´ X { G ´ X { G ´ ,i.e. ˘ γ i “ ˘ ǫ “ ˘ τ ij “ ˘ σ ij “ . For most of this section, however, we shall restrict ourselves to nearly Robinsonmanifolds of Kundt type, in which case the intrinsic torsion is a section of G “ { G ´ X{ G ´ X { G ´ X { G , ´ , i.e. ˘ γ i “ ˘ ǫ “ ˘ τ ij “ ˘ σ ij “ ˘ ζ αβ “ . Note that this implies that N is parallel along K . By Proposition 3.28, p N, K q inducesan almost CR structure with totally degenerate Levi form on the leaf space p M , H q ,and the screen bundle metric h on H K descends to a bundle metric h on H . This tellsus that p H , h, J q is a Hermitian vector bundle. In addition, since K K is in involution,so is H . Thus, p M , H , h q admits a local Riemannian foliation. Putting these twofacts together allows us to characterise nearly Robinson manifolds of Kundt type in thefollowing terms. Proposition 3.52.
Let p M , g, N, K q be a p m ` q -dimensional nearly Robinson ma-nifold of Kundt type with congruence of null curves K . Then the local leaf space LMOST ROBINSON GEOMETRIES 53 p M , H, h, J q of K is foliated by a smooth one-parameter family of m -dimensionalalmost Hermitian manifolds, each tangent to H . We can express the almost Hermitian foliation as a triple p H p u q , h p u q , J p u qq depend-ing on a smooth parameter u , such that for each u “ ˚ u constant, p H p ˚ u q , h p ˚ u q , J p ˚ u qq isan almost Hermitian manifold tangent to H at any point. For each choice of param-eter u , the 1-form κ “ d u is an optical 1-form for K . The Riemannian foliation on p M , H, h, J q thus singles out a family of optical 1-forms.We can apply Proposition 3.24 to express the metric g in terms of a Robinson coframeadapted to the Kundt geometry. We first note that we can choose a coframe t θ , θ α , θ ¯ α u on p M , H, h, J q where(1) θ is a (locally) exact 1-form annihilating H , and(2) t θ α u is a one-parameter family of unitary coframes on p H , h, J q .The 1-form θ determines a smooth parameter u (up to an additive constant) such that θ “ d u . Any two coframes t θ , θ α , θ ¯ α u and t p θ , p θ α , p θ ¯ α u adapted to p M , H, h, J q mustbe related via p θ “ e ϕ θ , p θ α “ φ βα θ β ` φ α θ , (3.33)where ϕ is a smooth function on M constant along H , and φ βα and φ α are smoothcomplex-valued functions on M , where φ βα is an element of U p m q at each point.Note that setting e “ BB u splits T M as (3.12). This fixes the freedom in choos-ing t θ α , θ ¯ α u , up to unitary transformations, by requiring that θ α p e q “
0. This nowdetermines a local Robinson coframe t κ, θ α , θ ¯ α , λ u “ t θ , θ α , θ ¯ α , θ u where θ “ ̟ ˚ θ “ d u , θ α “ ̟ ˚ θ α . We shall refer to such a coframe as a (complex) Kundt coframe .Clearly, any two Kundt coframes t θ , θ α , θ ¯ α , θ u and t p θ , p θ α , p θ ¯ α , p θ u are related bythe transformations (3.5) where ϕ “ ϕ , φ α “ φ α , φ βα “ φ βα , ϕ being constant along K K .Since κ is closed, £ k κ “ k “ g ´ p κ, ¨q , we can conveniently choose an affineparameter v along the geodesics of K such that k “ BB v . This allows us to write λ inthe form λ “ d v ` λ α θ α ` λ ¯ α θ ¯ α ` λ θ , (3.34)where λ α , λ ¯ α and λ are smooth functions on M .We now streamline the notation by setting t θ , θ i , θ u “ t θ , θ α , θ ¯ α , θ u . We introducea connection ∇ on M that preserves h ij , θ and e with torsion tensor A ij “ A p ij q , i.e. p ∇ ∇ i ´ ∇ i ∇ q f “ ´ A ij ∇ j f , for any smooth function f on M .Note this connection depends on the choice of θ i up to orthogonal transformations,and thus on the choice of e . It is related to the Levi-Civita connection ∇ of g in the following way: ∇ θ “ θ d E , ∇ θ i “ ∇ θ i ´ B ij θ j d θ ´ C i θ b θ ´ A ij θ j b θ ´ E i θ d θ , ∇ θ “ ´ E ^ θ ´ E θ d θ ` θ b C ` A ´ B , (3.35)where A “ A ij θ i b θ j , B “ B ij θ i b θ j , E “ E i θ i ` E θ , C “ C i θ i , with B ij “ B r ij s , E i , E and C i being functions on M satisfying B ij “ ´ ∇ r i λ j s ` λ r i λ j s ,C i “ ´ ´ ∇ i λ ´ ∇ λ i ´ λ i λ ` λ λ i ´ A ij λ j ¯ ,E i “ λ i ,E “ λ . This choice of notation is of course not fortuitous since we can identify the componentsof the intrinsic torsion E α and B αβ . In fact, reading off the components of the Levi-Civita connections from equations (3.35) leads to a proof of the next result. Lemma 3.53.
Let p M , g, N, K q be a nearly Robinson manifolds of Kundt type withcongruence of null geodesics K and leaf space p M , H, h, J q . Then, for any splitting p ℓ a , δ ai , k a q , the -form λ a “ g ab ℓ b satisfies E i “ ´p ∇ λ q i “ ´p d λ q i . If the intrinsic torsion is in addition a section of p { G ˆ q r´ p m ´ q i:1 s X { G , X { G , X { G , ,then B αβ “ p ∇ λ q αβ “ ´p d λ q αβ . is an invariant of p N, K q . Since both κ a and ρ abc are the pull-backs of some 1-form and 3-form from M to M ,we can relate the present classification of almost Robinson structures with the Gray–Hervella classification [30] of almost Hermitian manifolds. To this end, we simply notethat for any splitting p k a , δ ai , ℓ a q , using (3.35), we have p ∇ ρ q ijk “ ∇ i ω jk , (3.36)where ω ij “ J ik h kj is the smooth family of Hermitian 2-form associated to the almostHermitian foliation on p M , H , h, J q . One can readily check that this does not dependon the choice of coframe — this essentially follows from equations (A.1) and the factthat the optical invariants ˘ γ i , ˘ ǫ , ˘ σ ij and ˘ τ ij all vanish. The Gray–Hervella classes can beeasily obtained by comparing the LHS and the RHS of equation (3.36) with referencesto (3.31) and (3.32). We have collected the findings in Table 7. LMOST ROBINSON GEOMETRIES 55
Type of almost Hermitian structure Intrinsic torsion ˚ T Intrinsic torsion ˚ T on M W W W W { G , { G , { G , { G , almost Hermitian X X X X G X X X X G X X X XX X X X semi-K¨ahler
X X X X
Hermitian
X X X X incl. locally conformally almost K¨ahler
X X X XX X X XX X X XX X X X quasi-K¨ahler
X X X X nearly K¨ahler
X X X X almost K¨ahler
X X X X special Hermitian
X X X X incl. locally conformally K¨ahler
X X X X
K¨ahler
X X X X
Table 7.
Relation between the intrinsic torsions ˚ T and ˚ T for nearlyRobinson manifolds of Kundt type.The following proposition gives a characterisation of an almost Robinson manifold ofKundt type in the case where the leaves of the Riemannian foliation on the leaf spaceare K¨ahler manifolds. We leave the proof to the reader. Proposition 3.54.
Let p M , g, N, K q be a nearly Robinson manifold of Kundt type withcongruence of null geodesics K and leaf space p M , H , h, J q . The following statementsare equivalent.(1) the intrinsic torsion is a section of { G , X { G , X { G , X { G , , i.e. ˘ γ i “ ˘ ǫ “ ˘ τ ij “ ˘ σ ij “ ˘ ζ αβ “ ˘ G α “ ˘ G αβγ “ ˘ G αβγ “ ˘ G ˝ ¯ αβγ “ . (2) any Robinson -form ρ abc is recurrent along K K , i.e. κ r a ∇ b s ρ cde “ κ r a α b s ρ cde , for some -form α a ;(3) N is parallel along K K . (4) Each leaf of the almost Hermitian foliation on p M , H, h, J q is a K¨ahler mani-fold. Besides the classes of nearly Robinson manifolds of Kundt type enumerated in Table7, other interesting degeneracy conditions on the intrinsic torsion are also possible. Infact, these can be partly characterised by the covariant derivative of the 1-form λ usingLemma 3.53. Proposition 3.55.
Let p M , g, N, K q be a nearly Robinson manifolds of Kundt typewith congruence of null geodesics K and leaf space p M , H, h, J q . Let r z : w s P CP such that z ‰ . The following two statements are equivalent.(1) the intrinsic torsion ˚ T is a section of p { G ˆ q r z : w s , i.e. ˘ γ i “ ˘ ǫ “ ˘ τ ij “ ˘ σ ij “ ˘ ζ αβ “ , z ˘ E α ` w ˘ G α “ . (2) for any choice of Kundt coframe and affine parameter v along K , the vertical -form λ is given by (3.34) where λ α “ λ p q α ´ wz vλ p q α , (3.37) for some complex-valued functions λ p q α and λ p q α on M such that λ p q α “ G α “ δ kα h ij ∇ i ω jk . (3.38) In addition, the -form λ p q i does not depend on the choice of Kundt coframe.Finally, for optical vector field k , with κ “ g p k, ¨q , the Weyl tensor satisfies k a W abc r d κ e s k c “ . (3.39) Proof.
This is a straightforward computation. By definition, the condition zE α ` w G α “ z λ α ` w G α “
0, which has solution given preciselyby (3.37) and (3.38).For the last part, it is shown in [75] that condition (3.39) on Weyl tensor is equivalentto λ i being linear in v . (cid:3) Remark 3.56.
Weaker conditions where one takes z and w to be non-constant complex-valued smooth functions in Proposition 3.55 are possible. In this case, it is no longertrue that λ i is linear in v .The next proposition follows from the interpretation of the vanishing of the intrinsictorsion as the reduction of the holonomy of the Levi-Civita connection to Q , or equiv-alently, to the parallelism of the distribution N . The last item follows from Lemma3.53. Proposition 3.57.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) the intrinsic torsion vanishes identically, i.e. ˘ γ i “ ˘ ǫ “ ˘ τ ij “ ˘ σ ij “ ˘ ζ αβ “ ˘ E i “ ˘ G α “ ˘ G αβγ “ ˘ G αβγ “ ˘ G ˝ ¯ αβγ “ ˘ B αβ “ . (2) the holonomy of the Levi-Civita connection is reduced to Q “ p R ą ˆ U p m qq ˙p R m q ˚ , the structure group of p N, K q . LMOST ROBINSON GEOMETRIES 57 (3) any Robinson -form ρ bcd is recurrent, i.e. ∇ a ρ bcd “ α a ρ bcd , for some -form α a .(4) N is parallel, i.e. for any v P Γ p N q , ∇ v P Γ p N q .(5) any Robinson spinor ν is recurrent, i.e. ´ ∇ a ν bA ¯ ν bB “ , i.e. ´ ∇ a ν r A ¯ ν B s “ . (6) p M , g, N, K q is of Kundt type with leaf space p M , H, h, J q where h is a smoothone-parameter family of K¨ahler metrics, and for any choice of Kundt coframeand affine parameter v along K , the vertical -form λ is given by (3.34) where λ i “ λ p q i is a -form on M , and, at any point, ∇ r i λ p q j s is an element of u p m q . Remark 3.58.
We note that if a Robinson 3-form is recurrent, so is any optical 1-form.
Remark 3.59.
One can also weaken the assumptions of Proposition 3.57, by supposingthat on each leaf tangent to H , the metric h ij is almost K¨ahler, i.e. the Hermitian 2-form is closed, rather than K¨ahler. Then locally one may take λ p q i to be any potential1-form for the Hermitian 2-form. Then ∇ r i λ p q j s is an element of u p m q . This implies that B αβ “
0, i.e. the G , -component of the intrinsic torsion vanishes.Finally, the next two results, stated without proofs, are concerned with further ho-lonomy reduction. Proposition 3.60.
Let p M , g, N, K q be an almost Robinson manifold with congruenceof null curves K . The following statements are equivalent.(1) the holonomy of the Levi-Civita connection is reduced to a subgroup of U p m q ˙p R m q ˚ .(2) p M , g, N, K q admits a parallel Robinson -form ρ bcd , i.e. ∇ a ρ bcd “ .(3) p M , g, N, K q is of Kundt type with leaf space p M , H , h q where h is a smoothone-parameter family of K¨ahler metrics, and for any choice of Kundt coframeand affine parameter v along K , the vertical -form λ is given by (3.34) where λ i “ λ p q i and λ “ λ p q are -forms on M , and, at any point, ∇ r i λ p q j s is anelement of u p m q .If any of these conditions, p M , g, N, K q admits a parallel optical vector field. Proposition 3.61.
Let p M , g, N, K q be an almost Robinson spin manifold with con-gruence of null curves K . The following statements are equivalent.(1) the holonomy of the Levi-Civita connection is reduced to a subgroup of SU p m q˙p R m q ˚ .(2) p M , g, N, K q admits a parallel Robinson spinor ν , i.e. ∇ a ν “ .(3) p M , g, N, K q is of Kundt type with leaf space p M , H , h q where h is a smoothone-parameter family of Ricci-flat K¨ahler metrics, and for any choice of Kundtcoframe and affine parameter v along K , the vertical -form λ is given by (3.34) where λ i “ λ p q i and λ “ λ p q are -forms on M , and, at any point, ∇ r i λ p q j s isan element of su p m q .If any of these conditions, p M , g, N, K q admits a parallel optical vector field. Example 3.62.
Parallel Robinson structures are relevant to the study of solutionsto the supergravity equations. These equations are rather restrictive. For instance,it is shown in [14] (see also [32]) that solutions known as p , q -vacua (up to localisometry) are six-dimensional Lie groups admitting a bi-invariant Lorentzian metricand an anti-self-dual 3-form induced from the Lie bracket. In the same reference, itis proved that these must be either Minkowski space, a one-parameter family of so-called Freud–Rubin vacua on AdS ˆ S with equal radii, or a six-dimensional Nappi–Witten vacuum. The latter is locally isometric to a certain Cahan–Wallach space. Incoordinates t u, v, x , x , x , x u , the metric is given by g “ d u ˆ d v ´ h ij x i x j d u ˙ ` h ij d x i d x j , (3.40)where h ij is the standard Euclidean metric on R . The anti-self-dual 3-form inducedfrom the Lie bracket takes the form ρ “ u ^ ` d x ^ d x ` d x ^ d x ˘ . (3.41)It can be checked that ρ defines a Robinson 3-form with optical 1-form κ “ d u . Both κ and ρ are parallel with respect to the Levi-Civita connection.One may start with a Kundt geometry p M , g, K q with congruence of null geodesics K such that the leaf space M of K is a fiber bundle over a smooth 2 m -dimensionalRiemannian manifold p B , h q . Then any almost complex structure on B compatible with h lifts to an almost Robinson structure on p M , g q compatible with K . Depending on itstopology, p B , h q may admit many almost Hermitian structures with non-generic intrinsictorsion, or even families thereof. For instance, let us take p B , H q to be Euclidean space.Then locally there is infinitely many Hermitian structures (see e.g. [20]) that can belifted to an almost Robinson structure p N, K q on p M , g q .A less trivial example follows. Example 3.63.
Let B be the Iwasawa manifold, that is, the quotient of the three-dimensional complex Heisenberg group by a discrete subgroup. In [2], the authorsconstruct almost Hermitian structures in the following Gray–Hervella classes W ‘ W ‘ W , W ‘ W ‘ W , W ‘ W ‘ W , W ‘ W ‘ W . Of these, the set of all invariant Hermitian structures on this six-dimensional real ma-nifold is known to consist of the union of a point (its bi-invariant Hermitian structure)and a 2-sphere [1, 44]. These are in fact special Hermitian, i.e. their intrinsic torsionis of class W . For topological reasons, B canno admit any K¨ahler structure. It is alsoconjectured that B cannot admit almost Hermitian structures in the classes W and W .The Kundt geometry p M , g, K q associated to p B , h q admits almost Robinson struc-tures corresponding to the almost Hermitian structures on p B , h q , and their classes ofintrinsic torsion can be read off from Table 7.Not every almost Robinson structure on a Kundt spacetime is a nearly Robinsonstructure, i.e. descends to the leaf space of the associated congruence of null geodesics.In the following example, we explains how one may arrive at such an almost Robinsonmanifold. LMOST ROBINSON GEOMETRIES 59
Example 3.64.
Take M “ R ˆ R ˆ R m – R ˆ R ˆ C m “ t v, u, z α u , and let g bethe metric on M given by g “ u d v ` λ α d z α d u ` λ ¯ α d z ¯ α d u ` λ p d u q ` h α ¯ β d z α d z ¯ β , where λ α , λ ¯ α and λ are arbitrary smooth functions on M , and h α ¯ β is the standardhermitian form on C m . Let k “ BB v and set K : “ span t k u . Then p M , g, K q is a Kundtspacetime, and any almost Robinson structure p N, K q on M compatible with K isannihilated by the set of 1-forms κ “ g p k, ¨q “ d u , θ α “ d z α ` φ α ¯ β d z ¯ β . for some complex-valued functions φ αβ on M with φ αβ “ φ r αβ s – the functions φ αβ areessentially the components of a section of Gr ` m ` p M , g, K q . Note that t θ α u does not constitute a unitary coframe for h α ¯ β in general. Choosing the φ αβ such that £ k φ αβ ‰ M of K . Take forinstance, φ αβ “ f p v q φ αβ for some smooth function f of v and smooth functions φ αβ on M . For definiteness, let us assume m “
2. The bundle of all almost Hermitianstructures on R has fibers isomorphic to CP . Any unitary frame t θ , θ u for h takesthe form θ “ ` a ¯ a ` b ¯ b ˘ ´ ` a d z ` b d z ˘ , θ “ ` a ¯ a ` b ¯ b ˘ ´ ` a d z ´ b d z ˘ . for some smooth complex-valued functions a and b on M with a, b not both vanishing.Note that this expression is invariant under non-zero rescaling of p a, b q , so at any point, r a, b s defines an element of CP as expected. Take a “ b any smooth complex-valued function depending on v , i.e. £ k b ‰
0. Then p N, K q is an almost Robinsonstructure on p M , g, K q that does not descend to p M , H, h q .3.13. Almost Robinson manifolds of Robinson–Trautman type.
In completeanalogy with Definition 3.51, we make the following definition.
Definition 3.65.
An almost Robinson manifold p M , g, N, K q is said to be of Robinson–Trautman type if p M , g, K q is a Robinson–Trautman spacetime, i.e. the congruence ofcurves tangent to K is geodesic, expanding, non-shearing and non-twisting.The intrinsic torsion of such a manifold is a section of { G ´ X { G ´ , with non-degenerate G ´ -component i.e. ˘ γ i “ ˘ σ ij “ ˘ τ ij “ , ˘ ǫ ‰ . Again, it is natural to consider nearly Robinson manifolds of Robinson–Trautman type.These manifolds enjoy properties similar to those of their Kundt counterparts. Inparticular, the leaf space M of the congruence K tangent to K is foliated by 2 m -dimensional almost Hermitian manifolds. The intrinsic torsion of p M , g, N, K q canalso be related to the Gray–Hervella class of the almost Hermitian foliation as in Table7 except in cases where { G , is involved — this would contradict the fact that K isexpanding. This means that the only Gray–Hervella classes allowed would be thosecontaining W .However, since a Robinson–Trautmann spacetime is conformal to a Kundt spacetime[78, 24], one may still consider the full Gray–Hervella classification applied to the almost Hermitian foliation on M . For instance, if the intrinsic torsion ˚ T is a section of { G , X{ G , , then the intrinsic torsion of the almost Hermitian foliation on M is either a sectionof W ‘ W or a section of W . The following example due to [65] is a case in point.It is also particularly interesting due to the fact that the Einstein-Maxwell equationsmay single out an almost Robinson structure on a Robinson–Trautman spacetime. Example 3.66.
Let p M , g, K q be a Robinson–Trautman optical geometry of dimension2 m `
2. Suppose g satisfies the Einstein–Maxwell equations with an electromagneticfield F ab “ F r ab s , that is, F ab is closed and co-closed, and the Einstein field equationstake the form Ric ab “ m Λ g ab ` πT ab ` m ´ m g ab T cd T cd , where Λ is the cosmological constant, and the energy-momentum tensor is given by T ab “ π ˆ F ac F bc ´ g ab F cd F cd ˙ . We assume further that F ab satisfying k a F a r b k c s “ k of K . Fordefiniteness, assume m ą
2. Then [65], there exist coordinates t r, u, x i u such that themetric takes the form g “ ´ u d r ´ H p r qp d u q ` r h ij p x q d x i d x j , where2 H p r q “ K ´ m p m ` q r ´ µr m ´ ` Q m p m ´ q r p m ´ q ´ } F } m p m ´ q r , and the electromagnetic field is given by F “ Qr m d r ^ d u ` F ij p x q d x i ^ d x j . Here, K P t´ , , u , µ , Q and } F } “ F ij F ij are constants, and k “ BB r is a null vectorfield tangent to K . The vector field ℓ “ BB u ´ H p r q BB r defines a optical structure L dualto K . Set κ “ g p k, ¨q “ d u and λ “ g p ℓ, ¨q “ ´ d r ´ H p r q d u . Assuming } F } ‰
0, theelectromagnetic 2-form F ab determines two almost Robinson structures p N K , K q and p N L , L q : their associated 3-forms are proportional to κ ^ F and λ ^ F respectively. Byvirtue of the Maxwell equations, it can be shown [65] that the metric h ij is (almost)K¨ahler–Einstein. It follows from Table 7 that the intrinsic torsion of each of thesealmost Robinson structures must be a section of { G , X { G , .The six-dimensional case is similar.As for Kundt spacetimes, one can associate to any smooth 2 m -dimensional Riemann-ian manifold p B , h q a Robinson–Trautman geometry p M , g, K q with congruence of nullgeodesics K such that the leaf space M of K is the trivial line bundle R ˆ B . Thealmost complex structure on B compatible with h lifts to an almost Robinson structureon p M , g q compatible with K . Example 3.67 (The Tangherlini–Schwarzschild metric) . The smooth manifold M “ R ˆ R ą ˆ S m admits the Tangherlini–Schwarzschild metric – see e.g. [73]. Then p M , g, K q is a Ricci-flat Robinson–Trautman spacetime which does not admit any LMOST ROBINSON GEOMETRIES 61 global Robinson structure except in the case m “
1. This follows from the fact thatthe 2 m -sphere admits a Hermitian structure if and only if m “
1. However, since S m is conformally flat, it locally admits infinitely many Hermitian structures – these corre-spond precisely to holomorphic sections of the twistor bundles over S m , a Riemannianarticulation of the Kerr theorem – see e.g. [20].3.14. Compatible linear connections.
We end Section 3 with a brief considerationof linear connections compatible with a given almost Robinson structure.
Proposition 3.68.
Let p M , g, N, K q be an almost Robinson manifold. Fix a splitting p ℓ a , δ ai , k a q . Define a linear connection ∇ with ∇ a ξ b “ ∇ a ξ b ´ Q abc ξ c , (3.42) where Q abc is a tensor such that Q “ Q i “ Q “ Q “ ,Q j “ ´ Q j “ γ j ,Q ij “ ´ Q i j “ m ǫ h ij ` τ ij ` σ ij , Q α ¯ β “ ´ m ǫ h α ¯ β ´ τ α ¯ β ,Q i p jk q “ ´ p m ´ q G i h jk ,Q j “ Q j “ ´ Q j “ ´ Q k “ E j ,Q p jk q “ ´ m f h jk , and Q βγ “ ´ τ βγ ` ζ βγ ,Q αβγ “ G αβγ , Q ¯ αβγ “ G ¯ αβγ ,Q αβ ¯ γ “ ´ p m ´ q G p α h β q ¯ γ , Q α ¯ γβ “ ´ p m ´ q G α h β ¯ γ ,Q βγ “ B αβ . Then ∇ is a connection compatible with K and r h s with torsion tensor satisfying T j “ ´ γ j ,T ij “ ´ τ ij , T j k “ ´ σ jk ,T “ T k “ T j “ , and T r βγ s “ ´ ζ βγ ´ τ βγ , T p βγ q “ ´ σ βγ ,T r αβγ s “ ´ G αβγ , T α p βγ q “ G p βγ q α ,T ¯ α r βγ s “ ´ G ˝ ¯ αβγ ,T αβ ¯ γ “ T ¯ α p βγ q “ T r βγ s “ . Proof.
This is a straightforward computation using (3.42) and the fact that the torsiontensor is given by T abc “ ´ Q r ab s c . (cid:3) Remark 3.69.
The linear connection defined in the proposition above depends onlyon the choice of k and λ . Note however that even with fixed k and λ , ∇ as definedin the proof of the proposition is not unique, the undefined components of Q abc in theproposition above being entirely arbitrary and not affecting the property of the torsion.We may set all the remaining components of Q r ab s c to zero, and obtain the followingcorollary. Corollary 3.70.
Let p M , g, N, K q be an almost Robinson manifold. Suppose that theintrinsic torsion of p N, K q is a section of { G ´ X { G X { G X { G . Then p M , g, N, K q admits a torsionfree connection that preserves p N, K q and (the conformal class of ) thescreen bundle metric. Conformal almost Robinson structures
As for optical geometry and almost Hermitian geometry, the notion of almost Robin-son structure, or indeed almost null structure, finds a very natural setting in conformalgeometry. We shall follow the conventions set up in [7, 24], and denote a conformalstructure on a smooth manifold M by c . For each w P R , the bundle of conformaldensities of weight w is denoted by E r w s . In particular, E r s is referred to as the bundleof conformal scales. The conformal structure can be encoded by means of the conformalmetric g ab , that is a non-degenerate global section of Ä T ˚ M b E r s . For each g in c , we extend the Levi-Civita connection ∇ of g to a linear connection on E r w s for each w P R . The exterior covariant derivative will be denoted by d ∇ . Further details canbe found in the aforementioned references. Definition 4.1.
Let p M , c q be an oriented and time-oriented Lorentzian conformalmanifold of dimension 2 m `
2. An almost Robinson structure on p M , c q consists of apair p N, K q where N is a complex distribution of rank p m ` q totally null with respectto g , and K a real line distribution such that C K “ N X N . We shall refer to thequadruple p M , c , N, K q as an almost conformal Robinson manifold . Remark 4.2.
As in the metric case, one can describe an almost conformal Robinsonmanifold as an almost null structure (of real index one).The conformal metric g ab induces a conformal bundle metric h ij on H K . Proposition3.4 can immediately be translated into the conformal setting as follows: LMOST ROBINSON GEOMETRIES 63
Proposition 4.3.
Let p M , c q be an oriented and time-oriented Lorentzian manifold ofdimension m ` . The following are equivalent.(1) p M , c q is endowed with an almost Robinson structure p N, K q .(2) p M , c q admits a complex simple totally null p m ` q -form ν a a ...a m of conformalweight m ` .(3) p M , c q is endowed with an optical structure K whose screen bundle H K “ K K { K is equipped with a bundle complex structure compatible with the inducedconformal structure.(4) p M , c q admits a -form κ a of conformal weight , and a -form ρ abc of confor-mal weight such that ρ ab e ρ cde “ ´ κ r a g b sr c κ d s . (5) p M , c q admits a pure spinor ν A of real index one. As in the metric case, we shall also refer to k a , κ a , ρ abc , ν a a ...a m and ν A as an opticalvector field , an optical -form , a Robinson -form , a complex Robinson p m ` q -form and a Robinson spinor respectively.With reference to the proposition above, the bundle complex structure J ij yields abundle Hermitian structure ω ij “ J ik h kj of conformal weight 2. For a given optical1-form κ a we obtain a Robinson 3-form ρ abc “ κ r a ω bc s of conformal weight 4, where ω ab is such that k c ω c r a κ b s “ ω ij “ ω ab δ ai δ bj . The complex p m ` q -form ν a a ...a m isrequired to have conformal m ` ν aa ...a m ν ba ...a m κ a κ b . The relation with pure spinor fields is analogous to the Lorentzian case. Here, neither ν A nor its charge conjugate are conformally weighted, but the van der Waerden symbolscarry some conformal weight. This means that for each k “ , . . . , m `
1, the spinorbilinear form with values in (complex) k -forms has conformal weight k `
1. We omitthe details, which will play no rˆole in the subsequent discussion.4.1.
Conformal invariants of an almost Robinson structure.
An optical geom-etry with a congruence of null geodesics has two conformal invariants, the shear andthe twist, which we may view as fields of conformal weight two [82, 24]. To determinethe conformal invariants of an almost Robinson manifold p M , c , N, K q , we examinehow the covariant derivative of the Robinson 3-form changes under a change of metrics p g “ e ϕ g for some smooth function ϕ as given below: p ∇ a κ b “ ∇ a κ b ` r a κ b s ` g ab Υ c κ c , p ∇ a ρ bcd “ ∇ a ρ bcd ` a ρ bcd ´ r b ρ cd s a ` g a r b ρ cd s e Υ e , where Υ “ d ϕ . Projecting these tensors into their components with splitting operators,we find that p γ i “ e ϕ γ i , p τ ij “ e ϕ τ ij , p σ ij “ e ϕ σ ij , p ǫ “ ǫ ` n Υ c k c , p ζ βγ “ e ϕ ζ βγ , p E i “ e ϕ p E i ´ Υ i q , p G γ “ e ϕ p G γ ´ p m ´ q iΥ γ q , p G αβγ “ e ϕ G αβγ , p G αβγ “ e ϕ G αβγ , p G ˝ ¯ αβγ “ e ϕ G ˝ ¯ αβγ , p B βγ “ e ϕ B βγ . Note that 2 p m ´ q i p E γ ´ p G γ “ e ϕ p p m ´ q i E γ ´ G γ q . We also find that for any r z : w s P CP , z p τ αβ ` w p ζ βγ “ e ϕ p zτ αβ ` wζ βγ q . From these computations, we immediately conclude:
Theorem 4.4.
Let p M , c , N, K q be an almost conformal Robinson manifold. Let g bea metric in c so that p M , g, K q is an optical geometry with bundle of intrinsic torsions G . Any conformally invariant subbundle of G must be an intersection of the following: { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , , { G , , { G , , p { G ˆ q r p m ´ q i: ´ s , { G , , p { G ˆ ´ q r x : y s , r x : y s P RP , p { G ˆ ´ q r z : w s , r z : w s P CP . As for conformal optical geometries, there is a subclass n.e. c of metrics in c with theproperty that whenever g is in n.e. c , the congruence K is non-expanding, i.e. for any k P Γ p K q with κ “ g p k, ¨q , κ div k ´ ∇ k κ “
0. An application of this fact proves thefollowing result.
Proposition 4.5.
Let p M , g, N, K q be an almost Robinson manifold with intrinsictorsion ˚ T . Suppose that ˚ T is a section of p { G ˆ ´ q r x : y s for some real x and y with x ‰ ,i.e. x ˘ ρ ` y ˘ τ ω “ .Then y “ , i.e. ˚ T is a section of { G , ´ .In other words, ˚ T can never be a section of p { G ˆ ´ q r x : y s with xy ‰ .Proof. This is a simple consequence of Proposition 5.6 of [24]. We can always find ametric p g in n.e. c so that p ǫ “
0. But this means that p τ ω , and since this is conformallyinvariant, we have that τ ω “ (cid:3) LMOST ROBINSON GEOMETRIES 65
We also know from [24] that there exists a family of optical vector fields k such that £ k κ “ κ “ g p k, ¨q . If the corresponding Robinson 3-form ρ abc is preservedalong K , then £ k ρ abc “
0, where the Lie derivative is given by £ k ρ abc “ k d ∇ d ρ abc ` ρ d r ab ∇ c s k d ´ n ` ρ abc ∇ d k d . Here ∇ is the Levi-Civita connection of any metric in c . The details are left to thereader.4.2. Conformally parallel Robinson structures.
In [24], we saw that under certainconditions, one can find metrics in c for which the optical structure is parallel. Weextend this result to the Robinson setting. Proposition 4.6.
Let p M , c , N, K q be a conformal Robinson manifold with congruenceof null curves K . Suppose that the intrinsic torsion of p N, K q for some (and thus any)metric g in c is a section of { G ´ X { G , , i.e. ˘ γ i “ ˘ τ ij “ ˘ σ ij “ ˘ ζ αβ “ ˘ G αβγ “ ˘ G αβγ “ ˘ G αβγ “ ˘ B αβ “ , p m ´ q i ˘ E γ ´ ˘ G γ “ . (4.1) Equivalently, any Robinson spinor satisfies (3.24) and any optical -form κ satisfies κ ^ d ∇ κ “ . Suppose further that the Weyl tensor W abcd satisfies k a W ab r cd κ e s “ . Then locally, there is a subclass par. c of metrics in c with the property that whenever g isin par. c , the almost Robinson structure is parallel, i.e. any Robinson spinor, and thus anyoptical -form and Robinson -form, are recurrent. In particular, the holonomy of theLevi-Civita connection of any metric in par. c is contained in Q “ p R ą ˆ U p m qq ˙ p R m q ˚ .Any two metrics in par. c differ by a factor constant along K K .Proof. The hypothesis can be expressed by saying that the only three non-vanishingcomponents of the intrinsic torsion are ˘ ǫ , ˘ E i and ˘ G α , the latter two being related by(4.1). The subclass par. c was already already found in [24], and gave ˘ E i “ ǫ “ G α “
0. Hence, the intrinsic torsion vanishes and the resultfollows. (cid:3)
Example 4.7.
As a special, conformally invariant, case of Proposition 3.55, supposethe intrinsic torsion of the almost Robinson structure is a section of { G , X { G , X { G , Xp { G ˆ q r p m ´ q i: ´ s so that 2 p m ´ q i E γ “ G γ . It then follows that the horizontal 1-form λ of the Kundt metric satisfies λ ℓ “ λ p q ℓ ´ v m ´ h ij ` ∇ i ω jk ˘ J ℓk . As expected, this does not depend on the choice of Kundt coframe.4.3.
Conformal lift of almost CR structures.
In this short section, we revisitthe lift of almost CR structures considered in Section 3.8. A conformal version ofProposition 3.20 can be formulated in the following terms:
Proposition 4.8.
Let p M , H, J q be a p m ` q -dimensional oriented almost CR ma-nifold, and M : “ R ˆ M ̟ ÝÑ M be a trivial line bundle over M .Let ` t θ , θ α , θ ¯ α u , h α ¯ β , θ ˘ and ´ t p θ , p θ α , p θ ¯ α u , p h α ¯ β , p θ ¯ be two triplets that give rise totwo almost Robinson geometries p M , g, N, K q and p M , p g, N, K q as in Proposition 3.20.Suppose t θ , θ α , θ ¯ α u and t p θ , p θ α , p θ ¯ α u are related by (3.11) . Then p g “ e ϕ g , if and only if p θ “ θ ´ φ αβ φ β θ α ´ φ ¯ α ¯ β φ ¯ β θ ¯ α ´ φ β φ β θ , p h α ¯ β “ e ϕ h γ ¯ δ p φ ´ q αγ p φ ´ q ¯ β ¯ δ . In particular, φ αγ is an element of U p m q at every point. Again, the proof is pretty much tautological.4.4.
Non-shearing twist-induced almost Robinson structures.
As a direct con-sequence of Proposition 5.16 of [24], we obtain:
Proposition 4.9.
Let p M , c , K q be a p m ` q -dimensional conformal twist-inducedalmost Robinson manifold with non-shearing congruence of null geodesics K . Then foreach g P n.e. c , there exists a unique pair p k a , ℓ a q where k is a generator of K and ℓ a nullvector field such that g p k, ℓ q “ and κ “ g p k, ¨q satisfies d κ p k, ¨q “ , d κ p ℓ, ¨q “ , and the twist of k satisfies τ ij τ ij “ m .Let p k a , ℓ a q and p p k a , p ℓ a q be any two such pairs corresponding to metrics g and p g in n.e. c , with p g “ e ϕ g for some smooth function φ constant along K . Then, p k a “ k a , and κ “ g p k, ¨q , p κ “ p g p p k, ¨q , λ “ g p ℓ, ¨q , p λ “ p g p p ℓ, ¨q , p δ ia and δ ia are related via p κ a “ e ϕ κ a , p λ a “ λ a ` J j k Υ k δ ja ´ h ij Υ i Υ j κ a , p δ ia “ δ ia ´ ω ij Υ j κ a , where Υ i : “ δ ai ∇ a φ . Almost all the results of Sections 3.7, 3.8, 3.9 and 3.10 can be safely formulated inthe conformal setting. In particular, we give the conformal version of Proposition 3.40below.
Proposition 4.10.
Let p M , c , N, K q be a p m ` q -dimensional conformal twist-inducedalmost Robinson manifold with non-shearing congruence of null geodesics K . Then p N, K q induces a partially integrable contact almost CR structure on the leaf space p M , H, J q of K . In particular, there is a one-to-one correspondence between metrics in n.e. c and contact -forms for p H, J q . LMOST ROBINSON GEOMETRIES 67
Remark 4.11.
It is shown in [99] how the Levi-Civita connection of a metric in n.e. c relates to the Webster–Tanaka connection of its corresponding almost pseudo-Hermitianstructure. Example 4.12 (Fefferman construction) . There is a well-known canonical construc-tion, originally due to Fefferman [21, 22] and later characterised by Sparling, Graham[29] (see also [10, 11]), which associates to any contact CR structure p M , H , c q a con-formal structure p M , c q of Lorentzian signature on the total space of a circle bundleover M . One of the characterising features of such conformal structures is the exis-tence of a null conformal Killing field. It generates a congruence of null geodesics, andits twist induces a Robinson structure. The construction was later generalised to thepartially integrable case in [49]. An appropriate modification of this construction yieldsTaub-NUT metrics as shown in [4] and [99].We end this section with the following proposition regarding the most degenerateconformally invariant condition on the intrinsic torsion. Its proof can easily be obtainedfrom Theorem 2.18. Proposition 4.13.
Let p M , c , N, K q be an almost Robinson manifold with congruenceof null curves K . Let g be any metric in c , and suppose its intrinsic torsion ˚ T is asection of { G , . Then K is a non-shearing congruence of null geodesics, and ‚ if K is non-twisting, the leaf space is foliated by Hermitian manifolds of Gray–Hervella class W . ‚ if K is twisting, the almost Robinson structure is induced by the twist of K , anddescends to a contact CR structure on the leaf space of K . The Mariot–Robinson theorem.
Recall that a solution to the vacuum Maxwellequations on a four-dimensional Lorentzian manifold p M , g q is a 2 form F that is bothclosed and co-closed, i.e. d F “ d ‹ F “ ‹ is the Hodge duality operator.Note that these equations are conformally invariant, which justifies the inclusion ofthis section at this point. Such a solution is said to be null if F satisfies the algebraicconstraint κ ^ F “ κ .The Mariot theorem [53] states that any solution to the vacuum Maxwell equationsgives rise to a non-shearing congruence of null geodesics. The congruence is generatedby k “ g ´ p κ, ¨q . The converse, known as the Robinson theorem [80], is also true,provided that we work in the analytic category: one can construct an analytic nullsolution to the vacuum Maxwell equations from any analytic non-shearing congruenceof null geodesics. We can clearly substitute non-shearing congruence of null geodesicsby Robinson structure here, and this move allows for generalisation of the theoremto irreducible spinor fields [71]. In fact, it makes its understand more transparent aswe shall now explain. We note that the assumption of analyticity is crucial for theimplication part of the theorem as references [90, 91] make clear.Following [19, 55], we first note that any null 2-form F must be the sum of a self-dual totally null simple complex 2-form ν and its (anti-self-dual) complex conjugate ν . In the language of the present paper, ν is called a complex Robinson 2-form, andannihilates an almost null structure N . The condition that F be both closed and co-closed is equivalent to ν being closed. But if ν is closed, N must be integrable, i.e. N is a null structure (or Robinson structure). Conversely, if N is an analytic self-dual null structure, it gives rise to a foliation N by two-dimensional totally null complexleaves on the complexification p Ă M , r g q of p M , g q – see Remark 3.47. Take any 2-form ν on the two-dimensional local leaf space Ă M of N . Then ν is necessarily closed, and sois its pullback from Ă M to Ă M . A completely parallel argument applies to the complexconjugate of ν , and their sum gives rise to an analytic null solution to the vacuumMaxwell equation on restriction to p M , g q .The Mariot–Robinson theorem was later generalised to even dimensions in [41] andto odd dimensions in [97]. Its proof hinges on the same reasoning. We work in theanalytic category with a complex Riemannian manifold p Ă M , r g q of dimension 2 m `
2: atotally null simple p m ` q -form ν defines an almost null structure N , and if ν is closed, N is integrable. Conversely, any (self-dual) null structure N gives rise to a foliation N by p m ` q -dimensional totally null complex leaves on Ă M . The pullback of any formof top degree on the p m ` q -dimensional local leaf space of N is a totally null simple p m ` q -form that is necessarily closed.If we now start with an analytic Lorentzian manifold p M , g q of dimension 2 m ` p M , g q . The result is a Lorentzian articulation of the Mariot–Robinson theorem.4.6. The Kerr theorem.
We now describe all local analytic Robinson structures oneven-dimensional Minkowski space M . This problem is conformally invariant, and assuch, is most elegantly formulated in the language of twistor geometry. The result,now known as the Kerr theorem , was initially motivated by the search for Kerr–Schildsolutions to the Einstein field equations [43], but came to play a seminal rˆole in Penrose’sthen-nascent twistor geometry of [69].We first review the story in dimension four, where analytic Robinson structures areidentified with analytic non-shearing congruences of null geodesics. The appropriateframework is the so-called
Twistor correspondence (also referred to as the
Klein corre-spondence ), which we shall presently describe – see [69, 107, 36] for details. We considera four-dimensional complex vector space T . The Grassmannian of two-planes in T isa smooth four-dimensional complex projective quadric Q in P ` ^ T ˘ – CP , and assuch, is naturally equipped with a complex holomorphic conformal structure. Thereare two disjoint families of two-dimensional linear subspaces of Q , the α -planes and the β -planes , according to whether these planes are self-dual or anti-self-dual. The α -planesof Q are parametrised by the points of the projective space PT – CP , known asthe twistor space of Q , and the β -planes are parametrised by the points of dual twistorspace PT ˚ . Twistor space contains an analytic family F of complex lines parametrisedby the points of Q . We thus have a geometric correspondence between Q and PT . Inthis complex setting, a local null structure on Q is simply a foliation by α -planes. Theleaf space of such a foliation can then be interpreted as a hypersurface in PT . A firstversion of the Kerr theorem can then be formulated in the following terms:Locally, any null structure on Q gives rise to a complex hypersurface in PT thatintersect the lines of F transversely. Conversely, any null structure on Q arises in thisway.To consider real Minkowski space, we introduce a Hermitian inner product x¨ , ¨y on T of signature p , q . Under the action of the stabiliser SU p , q of x¨ , ¨y , Q decomposes LMOST ROBINSON GEOMETRIES 69 into six orbits, one of which we identify as compactified Minkowski space M c : “ S ˆ S .In other words, Q is the complexification of M c . This comes as no surprise consideringthat SU p , q is the double cover of the conformal group SO p , q . Similarly, PT admits the orbit decomposition PT “ PT ` \ PN \ PT ´ , (4.2)where PT ` : “ tr Z s P PT : x Z, Z y ą u , PN : “ tr Z s P PT : x Z, Z y “ u , PT ´ : “ tr Z s P PT : x Z, Z y ă u . (4.3)Here, Z can be viewed as complex coordinates on T – C . As a real hypersurface in CP , PN is the five-dimensional CR hypersphere, i.e. PN has topology S ˆ S andis equipped with a contact CR structure of signature p , q . In fact, PN is the space ofnull lines in M c . In the present context, the Kerr theorem tells us that if p N, K q is ananalytic Robinson structure on some subset of M c with congruence of null geodesics K , the leaf space of K can be identified as a CR submanifold of PN that arises asthe intersection of a complex hypersurface in PT with PN . Any analytic Robinsonstructure arise in this way.Non-analytic Robinson structures on Minkowski space can also be dealt with as alimiting case [90, 91, 71].The generalisation of the Kerr theorem in the complex case was carried out in [41,33, 34, 98] and is analogous. We consider a smooth projective complex quadric Q in CP m ` , which we may identify as the isotropic Grassmanian Gr p C m ` q . An α -planein Q is now a self-dual linear subspace of Q of dimension m ` β -plane its anti-self-dual counterpart. As before, we define the twistor space PT of Q to be the space ofall α -planes of Q , and the primed twistor space PT of Q to be the space of all β -planesof Q . This is the isotropic Grassmanian Gr ` m ` p C m ` q . When m is odd, PT – PT ˚ ,while when m is even PT – PT ˚ and PT – p PT q ˚ . From an algebraic viewpoint, itis convenient to realise PT and PT as the spaces of pure spinors, up to scale, for thedouble cover Spin p m ` , C q of the complex conformal group SO p m ` , C q .Twistor space is a complex manifold of dimension p m ` qp m ` q and contains ananalytic family F of m p m ` q complex submanifolds parametrised by the points of Q . In this complex setting, the Kerr theorem states [41, 98] that any local analyticnull structure N on Q locally gives rise to a complex submanifold N of dimension m ` F transversely, and every null structure arises in this way. In effect, thesubmanifold N is none other than the leaf space of the foliation tangent to N .From this complex description, it is only a small step to obtain the Lorentzian versionof the Kerr theorem. Just as in dimension four, under the action of the real form SO p m ` , q of SO p m ` , C q or its spin analogue, Q decomposes into six orbits,which includes compactified Minkowski space M c : “ S m ` ˆ S . To deal with PT , wenote that the spin representation for the conformal group SO p m ` , q is equippedwith a Hermitian inner product x¨ , ¨y of split signature [8], which restricts to a Hermitianform on PT . Using the results of [46], we find that PT admits the decomposition (4.2)where its orbits are defined just as in (4.3). Their interpretation is as follows: ‚ PN is of real dimension p m ` qp m ` q ´
1, and consists of self-dual p m ` q -dimensional linear subspaces of Q of real index 2. These are the pure spinorsthat are null with respect to the hermitian inner product on PT ; ‚ PT ` and PT ´ are of real dimension p m ` qp m ` q , and consist of self-dual m ` Q of real index 0. These are the purespinors that are positive, respectively negative, with respect to the hermitianinner product on PT .In particular, by virtue of being a real hypersurface defined by the vanishing of thehermitian form on PT , the orbit PN is a CR manifold . Remark 4.14.
It is crucial to note that for m ą PN cannot be identified with the p m ` q -dimensional space of null lines NL in M c . In general NL is a homogeneousspace equipped with a Lie contact structure [87], and does not admit any distinguishedCR structure unless m “
1. One can still identify the leaf space of the congruence K as a submanifold of NL , but NL alone cannot encode its CR structure.We are now in the position of stating the real version of the Kerr theorem: anyanalytic Robinson structure on p m ` q -dimensional Minkowski space gives rise to acomplex m ` PT whose 2 m ` PN is a CR submanifold thereof. Conversely, any analytic Robinson structurearises in this way. Remark 4.15.
As stated, the Kerr theorem is only concerned with the involutivity ofthe almost null structure be it in the complex case or in Lorentzian signature. Furtherdegeneracy conditions on the intrinsic torsion on the null structure have not yet beeninvestigated, but these would impact on the way the leaf space of the foliation tangentto the null structure sits in PT or PN . Generalised almost Robinson geometries
Generalised Robinson structures.
We now present a variant of the notionof almost Robinson structure, which in dimension four corresponds to the notion ofoptical geometry presented in [102, 103, 82, 83, 84, 56, 104], and which was referred toas generalised optical geometry in [24].
Definition 5.1.
Let M be a smooth manifold of dimension 2 m `
2. A generalisedalmost Robinson structure consists of a triple p N, K, o q , where N is a complex p m ` q -plane distribution, K : “ N X T M is a real line distribution on M , and o an equivalenceclass of Lorentzian metrics such that(1) for each g in o , N is null with respect to the complex linear extension of g .(2) any two metrics g and p g in o are related by p g “ e ϕ p g ` κ α q , (5.1) for some smooth function ϕ and 1-form α on M , and κ “ g p k, ¨q for somenon-vanishing section k of K . In fact, this was already demonstrated in the odd-dimensional analogue of the Kerr theorem in[98]. In even dimensions, complex case, this was established in unpublished work by Jan Gutt (Privatecommunication with the third author.)
LMOST ROBINSON GEOMETRIES 71
We shall say that the generalised almost Robinson structure is ‚ restricted if any of the 1-forms α in (5.1) satisfies α p k q “ ‚ of Kerr–Schild type if any of the 1-forms α in (5.1) satisfies α ^ κ “ p M , N, K, o q as a generalised almost Robinson geometry .We see in particular that a generalised almost Robinson structure determines a gen-eralised optical geometry p K, o q in the sense of [24]. It is straightforward to check thatthe two conditions above are well-defined. In particular, the property of K and N being totally null does not depend on the choice of metric in o , and neither does thenotion of orthogonal complement K K of K . A generalised almost Robinson geometry p M , N, K, o q also has an associated congruence of null curves K tangent to K .The following lemma is immediate. Lemma 5.2.
Let p M , N, K, o q be a generalised almost Robinson geometry. For eachmetric g in o , p N, K q is an almost Robinson structure on p M , g q . We shall therefore re-employ the terminology used in the previous section. In par-ticular, any non-vanishing section of K will be called an optical vector field and any1-form annihilating K K an optical 1-form. The definition of a Robinson p m ` q -formas a section of ^ m ` Ann p N q does not depend on the choice of metric in o , and neitherdoes the notion of Robinson 3-form. To see this, we take, for specificity, two metrics g and p g in o related via p g ab “ g ab ` κ p a α b q , for some 1-form α a . We can choose splitting operators for p N, K q and N, K q foreach of the metrics: p κ a , δ αa , δ ¯ αa , λ a q and p ℓ a , δ aα , δ a ¯ α , k a q for g ab , and p p κ a , p δ αa , p δ ¯ αa , p λ a q and p p ℓ a , p δ aα , p δ a ¯ α , p k a q for p g ab such that p κ a “ κ a , p δ αa “ δ αa , p λ a “ λ a ` α a , p k a “ βk a , p δ aα “ δ aα ´ βα α k a , p ℓ a “ ℓ a ´ βα k a . (5.2)Here, β “ p ` α q ´ . Let ω ij be the bundle Hermitian structure for p N, K q . Then itsassociated Robinson 3-form ρ ab “ κ r a ω bc s where ω ab “ ω ij δ ia δ jb remains invariant underthe change (5.2).Generalised almost Robinson geometry arises naturally in the context of lifts ofalmost CR manifolds as described in Section 3.8. Proposition 5.3.
Let p M , H , J q be a p m ` q -dimensional almost CR manifold, and M : “ R ˆ M be a trivial line bundle over M . Then M is naturally equipped witha generalised almost Robinson structure p N, K, o q such that for any metric g in o , p M , g, N, K q is a nearly Robinson structure, i.e. the intrinsic torsion of the correspon-ding almost Robinson structure p N, K q on p M , g q is a section of p { G ˆ ´ q r´ s X { G , ´ .Proof. This is a direct consequence of Proposition 3.20: the equivalent class o of metricson M related via (5.1) is simply the set of all lifts of p M , H , J q to M for a fixed choice ofa conformal class of Hermitian forms r h α ¯ β s . The distributions N and K do not dependon the lift, and by Lemma 5.2, p N, K q is an almost Robinson structure on p M , g q . (cid:3) The key idea of the next theorem is that it allows us to construct families ofLorentzian metrics equipped with almost Robinson structures sharing the same geomet-ric properties. To be precise, choosing a metric g in o determines an almost Robinsonstructure on p M , g q , and one may ask which subbundles of the bundle of intrinsic tor-sions G do not depend on the choice of metric g in o . We shall call those that remaininvariant under such a change o -invariant subbundles. Theorem 5.4.
Let p M , N, K, o q be a generalised almost Robinson geometry. Let g be a metric in o so that p N, K q defines an almost Robinson structure for p M , g q withbundle of intrinsic torsions G .(1) The following subbundles of G are o -invariant: { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , p { G ˆ ´ q r x : y s , r x : y s P RP , { G , ´ X { G , ´ , p { G ˆ ´ q r´ s , { G , ´ X { G , , { G , ´ X { G , , { G , ´ X { G , .Any o -invariant subbundle of a generalised almost Robinson geometry that isnot restricted must be an intersection of these.(2) Assuming that the generalised almost Robinson geometry is restricted, in addi-tion the following subbundles of G are o -invariant: { G , ´ , p { G ˆ ´ q r z : w s , r z : w s P CP , { G , .Any o -invariant subbundle of a restricted generalised almost Robinson geometrythat is not of Kerr–Schild type must be an intersection of these and the ones in(1).(3) Assuming that the generalised almost Robinson geometry is of Kerr–Schild type,in addition the following subbundles of G are o -invariant: { G , , { G , , p { G ˆ q r´ p m ´ q i:1 s ,and { G , , when m ą , { G , ´ X { G , , when m “ .Any o -invariant subbundles of a generalised almost Robinson geometry of Kerr–Schild type must be an intersection of these and the ones in (1) and (2).Proof. Let g be a metric in o . Any subbundle of G that is invariant under changes ofmetrics in o must also be conformally invariant. Thus, it is enough to consider thesubbundles given in Theorem 4.4, and a metric p g in o related to g by p g ab “ g ab ` κ p a α b q , for some 1-form α a . Denote by p ∇ and ∇ their corresponding Levi-Civita connections.Then for any 1-form ξ a , we have p ∇ a ξ b “ ∇ a ξ b ´ Q abc ξ c , where Q abc “ Q abd g dc is given explicitly in Appendix B. From (B.8), (B.1), (B.2) and(B.3), we immediately find that the subbundles { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ , { G , ´ do LMOST ROBINSON GEOMETRIES 73 not depend on the choice of metric in o . The same clearly applies to the subbundles p { G ˆ ´ q r x : y s for any r x : y s P RP .We now proceed with the remaining subbundles given in Theorem 4.4. ‚ Suppose ˚ T is a section of { G , ´ . Then by (B.9) and (B.4), we have p p ∇ ρ q βγ “ α τ βγ . The LHS is zero if and only if either τ αβ “ α p k q “
0. The former isequivalent to ˚ T being a section of { G , ´ X { G , ´ . ‚ Suppose ˚ T is a section of p { G ˆ ´ q r z : w s where r z : w s P CP . Then by (B.9),(B.10) and (B.4), we havei z p p ∇ ρ q r αβ s ` w p p ∇ ρ q αβ “ p z ` w i q α τ αβ . We immediately conclude that p { G ˆ ´ q r´ s is o -invariant. Suppose now r z : w s ‰ r´
2i : 1 s . Then, we must have either τ αβ “ α p k q “ ‚ Suppose ˚ T is a section of { G , . This subbundle is contained in p { G ˆ ´ q r ´ s ,which we know is o -invariant provided either α p k q “ τ αβ “ ζ αβ “
0. Inaddition, by skew-symmetry we find that p p ∇ ρ q r αβγ s “ , which does not yield any further conditions. Hence, { G , is o -invariant. ‚ Suppose ˚ T is a section of { G , . This is a subbundle of { G , ´ , i.e. σ αβ “
0. It isalso contained in p { G ˆ ´ q r ´ s . So, for o -invariance, we must have ˛ either τ αβ “ ζ αβ “
0. In this case, by (B.11), (B.5) and (B.2), we find that p p ∇ ρ q p αβ q γ “ . ˛ or α p k q “
0, from which we find p p ∇ ρ q p αβ q γ “ α p α τ β q γ , which tells us that one must impose in addition α p v q “ v P Γ p K K q for invariance. ‚ Suppose ˚ T is a section of { G , . This is also a subbundle of { G , ´ and { G , ´ , i.e. τ ˝ α ¯ β “ σ α ¯ β “
0. It is also contained in { G , ´ , so for o -invariance, we must have ˛ either τ αβ “ ´ p p ∇ ρ q ¯ αβγ ¯ ˝ “ . If we assume τ αβ “
0, then no further conditions are necessary. ˛ or α p k q “
0. Then, again, by (B.12), (B.6) and (B.3), we have ´ p p ∇ ρ q ¯ αβγ ¯ ˝ “ p α ¯ α τ βγ q ˝ , from which we immediately conclude that α p v q “ v P Γ p K K q for o -invariance. ‚ Suppose ˚ T is a section of p { G ˆ q r´ p m ´ q i:1 s . This is contained in p { G ˆ ´ q r p m ´ q i:1 s ,which is o -invariant if α p k q “ τ αβ “ ζ αβ “
0. In particular, by (B.15),(B.16), (B.4) and (B.6), we have2 h γ ¯ α p p ∇ ρ q βγ ¯ α ` h γ ¯ α p p ∇ ρ q ¯ αβγ “ ´ p m ´ q i Q β ´ i h γ ¯ α p Q ¯ αβγ ´ Q ¯ αγβ q . Comparing the two terms on the RHS shows that for invariance to hold, oneneeds α ^ κ “ ‚ Suppose ˚ T is a section of { G , . In dimension greater than six, this is a subbundleof { G , ´ , i.e. τ αβ “
0. Then, by (B.14), (B.7), (B.4), we have p p ∇ ρ q βγ “ Q r βγ s . where Q r βγ s “ ´ Q r β α γ s ´ α r β p ∇ κ q γ s ` p ∇ α q r βγ s ,Q r β α γ s “ ´ β p d α q r β α γ s ` β ` p ∇ κ q r β ` p ∇ κ q r β ˘ α γ s α For o -invariance, we need α p v q “ v P K K , i.e. α ^ κ “
0. Then we areleft with p p ∇ ρ q βγ “ p d α q βγ “ α τ βγ , since α a “ α κ a , which we know it zero. Invariance then follows.In dimension six, τ αβ is not necessarily zero. Suppose it is not. Since { G , Ăp { G ˆ q r´ s , we must have α p k q “
0, and by (B.14), (B.7), (B.4), we find Q r βγ s “ ´ Q r β α γ s ´ α r β p ∇ κ q γ s ` p d α q r βγ s ` α τ βγ , and Q r β α γ s “ ´p d α q r β α γ s For o -invariance we must also assume α p v q “ v P K K . Then we areleft with p p ∇ ρ q βγ “ p d α q βγ ` α τ βγ “ α τ βγ , where we have made use of the fact that α a “ α κ a . So for o -invariance, wemust have either α a “
0, which we rule out, or assume in addition τ αβ “ ζ αβ “ (cid:3) Remark 5.5.
The o -invariance of the subbundles p { G ˆ ´ q r´ s and { G , X { G , in The-orem 5.4 comes as no surprise. These correspond to N being preserved by the flow ofany section of K , and the involutivity of N respectively, and these geometric propertiesdo not depend on the conformal structure. Remark 5.6.
Theorem 5.4 should be contrasted with the situation regarding gener-alised optical geometries, where the o -invariants are precisely the conformal invariants,namely the shear and twist, of the optical geometry of some metric g in o as pointedout in [82, 24]. LMOST ROBINSON GEOMETRIES 75
Generalised Robinson geometries as G -structures. Following the originaldefinition of a generalised optical structure of [102, 103, 82, 83, 84, 104], we express anequivalent definition of a generalised almost Robinson geometry in the following terms:
Proposition 5.7.
Let M be a smooth oriented p m ` q -dimensional manifold. Thenthe following statements are equivalent.(1) M is endowed with a generalised Robinson structure p K, o q .(2) M is endowed with a pair of distributions K p q and K p m ` q of rank and m ` respectively such that K p q Ă K p m ` q (5.3) and its associated screen bundle K p m ` q { K p q is equipped with a conformalstructure of Riemannian signature together with a compatible bundle complexstructure.Proof. Recall from [24] that a generalised optical structure p K, o q is equivalent to theexistence of a filtration (5.3), where K p q is identified with K , together with a confor-mal structure on the screen bundle K p m ` q { K p q . Since a generalised almost Robinsonstructure is in particular a generalised optical structure, it suffices to exhibit a com-patible bundle complex structure on K p m ` q { K p q . But this follows directly fromProposition 4.3 and the fact that the screen bundle does not depend on the choice ofmetric in o . (cid:3) A generalised almost Robinson structure on a smooth manifold M can thereforebe regarded as a G -structure where the structure group of the frame bundle of M isreduced from SL p m ` , R q (or GL p m ` , R q if we drop the assumption that M isoriented) to the closed Lie subgroup H that stabilises the filtration (5.3), together witha conformal structure and compatible bundle complex structure on the screen bundle.One can easily check that H has dimension p m ` q .Under the assumption of real-analyticity, the generalised almost Robinson geometry p M , N, K, o q is integrable as a G -structure if and only if there exist local coordinates t u, v, z α , ¯ z ¯ α u on M , where u and v are real, z α complex, and ¯ z ¯ α “ z α such that(1) BB v spans K , or equivalently d u annihilates K K ,(2) t d u, d z α u annihilate N , and(3) o contains the Minkowski metric g “ u d v ` h α ¯ β d z α d z ¯ β , where h α ¯ β is thestandard Hermitian metric on C m . Example 5.8.
The metric (3.40) together with its Robinson 3-form (3.41) in Example3.62 belongs to the equivalence class of metrics of an integrable generalised almostRobinson geometry of Kerr–Schild type.The characterisation of integrable generalised optical structure was dealt with in[82, 24]. In the case of generalised almost Robinson geometries, we have the followingresult.
Theorem 5.9.
Let p M , N, K, o q be a generalised almost Robinson geometry with con-gruence of null curves K . The following statements are equivalent. (1) There exists a torsionfree linear connection ∇ compatible with p N, K q and o , ∇ u v P Γ p N q , for any v P Γ p N q , u P Γ p T M q , (5.4) ∇ u g p v, w q 9 g p v, w q , for any g P o , v, w P Γ p K K q , u P Γ p T M q , (5.5) (2) For any metric g in o , p M , g, N, K q is a nearly Robinson manifold whose in-trinsic torsion is a section of { G ´ X { G , X { G , X { G , . In particular, it is ofRobinson–Trautman or Kundt type.Further, in the neighbourhood of any point in M , there exists smooth functions u and v such that ‚ BB v spans K , or equivalently, d u annihilates K K , ‚ t d u, d z α u annihilate N , and ‚ o contains the metric g “ u d v ` h , where h is a family of conformally flatHermitian metrics of Gray–Hervella class W smoothly parametrised by u ,if and only if any of the conditions (1) and (2) holds together with the condition ` κ r a W bc sr de κ f s ˘ ˝ “ , (5.6) for any -form κ annihilating K K , where W abcd is the Weyl tensor of any metric in o .Proof. We show that (1) implies (2). Let ∇ be as given in (1). Note that we canreexpress (5.5) equivalently as ∇ a g bc “ β a g bc ` γ a p b κ c q , (5.7)for some tensor fields β a and γ ab , and where κ a anniliates K K . Now, the differencebetween ∇ and the Levi-Civita connection ∇ for some g in o can be expressed uniquelyby ∇ a α b “ ∇ a α b ´ Q abc α c , for any 1-form α a ,where Q abc “ β c g ab ` γ c p a κ b q ´ β p a g b q c ´ γ p ab q κ c ´ κ p a γ b q c . This follows from the requirement that ∇ be also torsion-free, and (5.5) holds. Nowchoosing splitting operators t ℓ a , δ ai , k a u “ t ℓ a , δ aα , δ a ¯ α , k a u for g , we find p ∇ κ q j “ , p ∇ κ q ij “ β h ij , p ∇ κ q j “ ` β j ` γ j ´ γ j ˘ , p k a ∇ a δ bβ q δ bγ “ , p δ aα ∇ a δ bβ q δ bγ “ , p δ a ¯ α ∇ a δ bβ q δ bγ “ β r β h γ s ¯ α , p ℓ a ∇ a δ bβ q δ bγ “ γ r βγ s , and it follows immediately that the intrinsic torsion of the almost Robinson structureof p M , g, N, K q is a section of { G ´ X { G , X { G , X { G , . We note that this is independentof the choice of metric in o by Theorem 5.4.To prove that (2) implies (1), we note that since { G , X { G , X { G , is a Q -invariantsubbundle of { G , ´ X { G ´ X { G ´ , we can simply take the linear connection given inCorollary 3.70.The final part of the proof follows from Theorem 7.6 of [24] and Table 7. (cid:3) LMOST ROBINSON GEOMETRIES 77
Remark 5.10.
Consider now a generalised almost Robinson geometry p M , N, K, o q for which the Minkowski metric η belongs to o , i.e. any metric in o is conformal to ametric of the form g “ η ` κα , for some optical 1-form κ and 1-form α . This means that p N, K q is an almost Robin-son structure for Minkowski space p M , η q . In the particular case where the 1-form isproportional to κ itself, we recover the original Kerr–Schild metric, considered by Kerrand Schild in [43] in dimension four, and which is an exact first-order perturbation ofthe Minkowski metric. Using the Kerr theorem of Section 4.6, one can then generatemany Robinson manifolds with non-zero curvature. Example 5.11 (The Myers–Perry metric) . Let us write the Minkowski metric in stan-dard coordinates t t, x α , y α , z u α “ ,...,m in dimension 2 m ` η “ ´p d t q ` m ÿ α “ ` p d x α q ` p d y α q ˘ ` p d z q , and let κ “ d t ` m ÿ α “ r p x α d x α ` y α d y α q ` a α p x α d y α ´ y α d x α q r ` a α ` zr d r , and f “ M r ´ ř mα “ a α pp x α q `p y α q qp r ` a α q ś mα “ p r ` a α q . Here, the radial coordinate is defined by m ÿ α “ p x α q ` p y α q r ` a α ` z r “ . Then, the Kerr-Myers-Perry metric in Kerr-Schild form is given by [57] g “ η ` f κ . Being a relative of the Kerr-NUT-(A)dS metric – see Example 3.49 – the Myers–Perrymetric admits two sets of 2 m Robinson structures corresponding to two optical struc-tures.As shown in [54], these Robinson structures are defined by the eigenspinors of aso-called conformal Killiang-Yano -form φ ab that is also closed, i.e. φ ab satisfies theoverdetermined system of linear first order partial differential equations: ∇ a φ bc “ m ` g a r b ∇ d φ c s d . Since these Robinson structures exist for η , by the Kerr theorem, they must arise froma complex submanifold of dimension m ` Generalisation to other metric signatures and odd dimensions
The setting of the present article can be easily adapted to any pseudo-Riemannianmanifold p M , g q of signature p p ` , q ` q for any even integer p, q with p ` q “ m ,where there is also a notion of almost null structure N , defined to be a totally nullcomplex distribution of rank m . In this general case, the real index r of N can takeany of the values [46] r ” min p p ` , q ` q p mod 2 q . When pq ‰
0, it is therefore necessary to define an almost Robinson structure as analmost null structure of real index one . Equivalently, these can be characterised asan optical geometry whose screen bundle is endowed with a bundle complex structurecompatible with the screen bundle metric. One has to be cautious in the definition ofa twist-induced almost Robinson structure since the screen bundle metric is no longerpositive definite. This difference is also reflected in the pure spinor approach, whichnow may be of different real indices: beside the purity condition (2.14), pure spinorsof real index one now satisfy further algebraic conditions [12, 46]. Other than theseconsiderations, the properties of the intrinsic torsion given in the present article willapply to different metric signatures.Finally, one can also define an almost Robinson structure p N, K q on a p m ` q -dimensional smooth Lorentzian manifold p M , g q (or its conformal analogue). Here, N is a totally null complex p m ` q -plane distribution, i.e. an almost null structure,of real index one so that K is the rank-one null distribution arising from the realspan of N X N . In dimension three, they are equivalent to optical geometries. Unlikein even dimensions, the real index has to be specified here, since generically, in odddimensions, the real index is zero [45]. Another crucial difference is that N is nowstrictly contained in its rank- p m ` q orthogonal complement N K , which makes thealgebraic classification of its intrinsic torsion significantly more involved. For instance,one may require the integrability of either N or N K , or both. Nevertheless, these arealso relevant to the study of solutions to Einstein field equations in higher dimensionsas was shown in [54, 93]. The geometry of almost null structures in odd dimensionsis investigated in [94, 98, 97, 63]. Almost Robinson structures can also be defined insignatures p p ` , q ` q for any p, q with pq ‰ Appendix A. Projections
In this appendix, we define the projections from the modules of intrinsic torsion G toits irreducible Q -modules as given in Theorem 2.15. We choose a Robinson 3-form ρ abc associated to an optical 1-form κ a , and splitting operators p ℓ a , δ ai , k a q “ p ℓ a , δ aα , δ a ¯ α , k a q , p δ iα , δ i ¯ α q , with κ a “ g ab k b . We shall be using these to convert index types, with theadditional convention that ℓ a “ δ a , k a “ δ a . Thus, if α a is a 1-form, we shall write k a α a “ α , ℓ a α a “ α . As usual, the screen inner product, Hermitian form, and complex structures will bedenoted h ij , ω ij and J ij respectively. Now let Γ abc of V ˚ b g and ω ab “ ω ij δ ia δ jb . so that LMOST ROBINSON GEOMETRIES 79 ρ abc “ κ r a ω bc s . We streamline notation by setting p Γ ¨ κ q ab : “ ´ Γ abc κ c , p Γ ¨ ω q abc : “ a r b d ω c s d , p Γ ¨ ρ q abcd : “ ´ a r be ρ cd s e . In particular, p Γ ¨ ρ q abcd “ p Γ ¨ ω q a r bc κ d s ` p Γ ¨ κ q a r b ω cd s . Note also that p Γ ¨ κ q ab k b “ p Γ ¨ ω q abc k c “ ´p Γ ¨ κ q ac J b c . Then it is easy to check p Γ ¨ ρ q a jk “ , p Γ ¨ ρ q a jk “ p Γ ¨ ω q ajk ` p Γ ¨ κ q a ω jk , p Γ ¨ ρ q aijk “ p Γ ¨ κ q a r i ω jk s , p Γ ¨ ρ q a k “ ´p Γ ¨ κ q aj J kj . (A.1)Let us first recall the projections from the module of intrinsic torsions to its irre-ducible G -modules from [24]:Π ´ : V ˚ b g Ñ G ´ , Γ abc ÞÑ Π ´ p Γ q i : “ p Γ ¨ κ q i , Π ´ : V ˚ b g Ñ G ´ , Γ abc ÞÑ Π ´ p Γ q ij : “ p Γ ¨ κ q ij , Π ´ : V ˚ b g Ñ G ´ , Γ abc ÞÑ Π ´ p Γ q : “ Π ´ p Γ q ij h ij , Π ´ : V ˚ b g Ñ G ´ , Γ abc ÞÑ Π ´ p Γ q ij : “ Π ´ p Γ q r ij s , Π ´ : V ˚ b g Ñ G ´ , Γ abc ÞÑ Π ´ p Γ q ij : “ Π ´ p Γ q p ij q ˝ , Π : V ˚ b g Ñ G , Γ abc ÞÑ Π p Γ q i : “ p Γ ¨ κ q i . (A.2)With reference to equations (A.1), we also define alternatives to Π ´ and Π :Π : V ˚ b g Ñ G ´ : Γ ÞÑ Π p Γ q ij : “ p Γ ¨ ρ q ij , Π : V ˚ b g Ñ G : Γ ÞÑ Π p Γ q i : “ p Γ ¨ ρ q ijk h jk . We note the relation Π p Γ q ij “ ´ Π ´ p Γ q ik J j k .In dimension six, there are two further projections from V ˚ b g to the self-dualand anti-self-dual parts of G ´ — see [24]. These will not be needed as they will besubsumed in the projections to G , ´ ‘ G , ´ and G , ´ given below.We shall presently introduce, for each i, j, k , a Q -module epimorphism Π j,ki : V ˚ b g Ñ G j,ki with the properties that V ˚ b q lies in the kernel of Π j,ki , and Π j,ki descends toa projection from G to G j,ki . By construction, the kernel of Π j,ki mod V b q is preciselyisomorphic to the complement p G j,ki q c of G j,ki in G as Q -modules, i.e.ker Π j,ki { p V ˚ b q q – p G j,ki q c , i.e. ´ ker Π j,ki { p V ˚ b q q ¯ c – G j,ki . (A.3)Let us define the projectionsΠ , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q jk : “ p Γ ¨ ρ q ℓ ℓ ´ δ ℓ j δ ℓ k ´ J j ℓ J k ℓ ¯ , Π : V ˚ b g Ñ G , Γ ÞÑ Π p Γ q ijk : “ p Γ ¨ ρ q iℓ ℓ ´ δ ℓ j δ ℓ k ´ J j ℓ J k ℓ ¯ , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q jk : “ p Γ ¨ ρ q ℓ ℓ ´ δ ℓ j δ ℓ k ´ J j ℓ J k ℓ ¯ . To guide the reader, we shall note thatΠ , ´ p Γ q αβ “ p Γ ¨ ω q αβ , Π p Γ q ijk “ p Γ ¨ ω q ijk Π , p Γ q jk “ p Γ ¨ ω q jk . We can define the remaining projections Π j,ki : V ˚ b g Ñ G j,ki by the propertiesΠ , ´ p Γ q : “ r Π ´ p Γ q α ¯ β h α ¯ β s “ Π ´ p Γ q ij ω ij , Π , ´ p Γ q ij : “ rr Π ´ p Γ q αβ ss , Π , ´ p Γ q ij : “ r ` Π ´ p Γ q α ¯ β ˘ ˝ s , Π , ´ p Γ q ij : “ r Π ´ p Γ q α ¯ β s , Π , ´ p Γ q ij : “ rr Π ´ p Γ q αβ ss , Π , p Γ q i : “ rr Π p Γ q α ss , Π , p Γ q i : “ rr Π p Γ q ¯ αβγ h β ¯ α ss “ Π p Γ q jki h jk , Π , p Γ q ijk : “ rr Π p Γ q r αβγ s ss , Π , p Γ q ijk : “ rr Π p Γ q p αβ q γ ss , Π , p Γ q ijk : “ rr ` Π p Γ q ¯ αβγ ˘ ˝ ss . (A.4)That these are indeed projections is not too difficult to check. We also define a variantof the maps Π , ´ and Π , byΠ , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q ij : “ rr iΠ p Γ q r αβ s ss , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q i : “ m ´ p Γ q ijk ω jk “ rr´ i m ´ p Γ ¨ ρ q αβ ¯ γ h β ¯ γ ss . One can indeed verify that these satisfy Π , ´ p Γ q ij “ Π , ´ p Γ q ij and Π , p Γ q i “ Π , p Γ q i .We are now in the position to introduce the following families of maps: for any r x : y s P RP , r z : w s P CP , p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r x : y s p Γ q : “ x Π , ´ p Γ q ` y Π , ´ p Γ q , p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r x : y s p Γ q : “ r x Π , ´ p Γ q α ¯ β ´ y i Π , ´ p Γ q α ¯ β s , p Π ˆ ´ q r z : w s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r z : w s p Γ q : “ rr z Π , ´ p Γ q αβ ` w Π , ´ p Γ q αβ ss , p Π ˆ q r z : w s : V ˚ b g Ñ G , ‘ G , , Γ ÞÑ p Π ˆ q r z : w s p Γ q : “ rr z Π , p Γ q α ` w Π , p Γ q α ss . Finally, we give alternative forms of the maps defined above in terms of algebraicrelations with J ij , ω ij and h ij :Π , ´ : V ˚ b g Ñ G . ´ , Γ ÞÑ Π , ´ p Γ q i : “ Π ´ p Γ q i , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q : “ Π ´ p Γ q , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q : “ Π ´ p Γ q ij ω ij , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q ij : “
12 Π ´ p Γ q kℓ ´ δ ki δ ℓj ´ J i k J j ℓ ¯ , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q ij : “
12 Π ´ p Γ q kℓ ˆ δ ki δ ℓj ` J i k J j ℓ ´ n ω ij ω kℓ ˙ , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q ij : “
12 Π ´ p Γ q kℓ ´ δ ki δ ℓj ` J i k J j ℓ ¯ , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q ij : “
12 Π ´ p Γ q kℓ ´ δ ki δ ℓj ´ J i k J j ℓ ¯ , Π , ´ : V ˚ b g Ñ G , ´ , Γ ÞÑ Π , ´ p Γ q jk : “ p Γ ¨ ρ q ℓ ℓ ´ δ ℓ j δ ℓ k ´ J j ℓ J k ℓ ¯ , LMOST ROBINSON GEOMETRIES 81 Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q i : “ Π p Γ q i , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q k : “ Π p Γ q ijk h ij , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q ijk : “
12 Π p Γ q ℓm r i ´ δ ℓj δ mk s ´ J j ℓ J k s m ¯ , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q ijk : “
12 Π p Γ q ℓmk ´ δ ℓ p i δ mj q ´ J p i ℓ J j q m ¯ , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q ijk : “
12 Π p Γ q ℓm r k ˆ δ mj s δ ℓi ` J j s m J i ℓ ´ m ´ ´ h j s i h ℓm ´ ω j s i ω ℓm ¯˙ , Π , : V ˚ b g Ñ G , , Γ ÞÑ Π , p Γ q jk : “ p Γ ¨ ρ q ℓ ℓ ´ δ ℓ j δ ℓ k ´ J j ℓ J k ℓ ¯ . Note that for the map Π , , we use the identity ω ij p Γ ¨ ω q ijk “ h ij p Γ ¨ ω q ijℓ J k ℓ , orequivalently h ij p Γ ¨ ω q ijk “ ´ ω ij p Γ ¨ ω q ijℓ J k ℓ . Other useful identities includeΠ , ´ p Γ q ki J j k “ Π ´ p Γ q k r i J j s k , Π , ´ p Γ q ki J j k “ Π ´ p Γ q kℓ ˆ J p i k δ ℓj q ` n ´ h ij ω kℓ ˙ , Π , ´ p Γ q ki J j k “ Π ´ p Γ q k r i J j s k , Π , ´ p Γ q ki J j k “ Π ´ p Γ q k p i J j q k , Π , p Γ q j J k j “ ω ij p Γ ¨ ω q ijk . For the remaining modules, we record p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r x : y s p Γ q : “ x Π , ´ p Γ q ` y Π , ´ p Γ q , p Π ˆ ´ q r x : y s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r x : y s p Γ q : “ x Π , ´ p Γ q ik J j k ` y Π , ´ p Γ q ij , p Π ˆ ´ q r z : w s : V ˚ b g Ñ G , ´ ‘ G , ´ , Γ ÞÑ p Π ˆ ´ q r z : w s p Γ q : “ ℜ ´ z Π , ´ p Γ q ij ` w Π , ´ p Γ q ij ¯ ` ℑ ´ z Π , ´ p Γ q ik ` w Π , ´ p Γ q ik ¯ J j k , p Π ˆ q r z : w s : V ˚ b g Ñ G , ‘ G , , Γ ÞÑ p Π ˆ q r z : w s p Γ q : “ ℜ ´ z Π , p Γ q i ` w Π , p Γ q i ¯ ` ℑ ´ z Π , p Γ q j ` w Π , p Γ q j ¯ J ij , where ℜ p¨q and ℑ p¨q denote the real and imaginary parts respectively. Appendix B. Generalised almost Robinson geometry — connections
Let p M , N, K, o q be a generalised almost Robinson geometry, and let g and p g be twometrics in o related via p g ab “ g ab ` κ p a α b q , for some 1-form α a . Denote by p ∇ and ∇ their corresponding Levi-Civita connections.Then for any 1-form ν a , we have p ∇ a ν b “ ∇ a ν b ´ Q abc ν c , where Q abc “ Q abd g dc is given by Q abc ` Q abd α c κ d ` Q abd κ c α d “ ´p ∇ c κ p a q α b q ´ p ∇ c α p a q κ b q ` p ∇ p a κ b q q α c ` p ∇ p a α b q q κ c ` α p a p ∇ b q κ c q ` κ p a p ∇ b q α c q Set β “ p ` α q ´ . Contracting this expression with instances of k a , δ aα , δ a ¯ α and ℓ a ,and using the definitions (3.8) and (3.9) yields Q a “ , (B.1) Q αβ “ ´ βγ p α α β q ` βα σ αβ , (B.2) Q ¯ αβ “ ´ β p ∇ κ q α β ´ βγ β α ` α β p d κ q β ` β p d α q β . (B.3) Q βγ “ α p γ γ β q ` α τ βγ , (B.4) Q αβγ “ ´ Q αβ α γ ` α p α τ β q γ ` α γ σ αβ , (B.5) Q ¯ αβγ “ ´ Q ¯ αβ α γ ´ α r β p ∇ κ q γ s ¯ α ` p ∇ κ q ¯ α p β α γ q ` α ¯ α τ βγ , (B.6) Q βγ “ ´ Q β α γ ´ α r β p ∇ κ q γ s ` p d α q r βγ s ` p ∇ κ q p β α γ q ` α τ βγ (B.7)Now, p ∇ a κ b “ ∇ a κ b ´ Q abc κ c , p ∇ a ρ bcd “ ∇ a ρ bcd ´ Q a r be ρ cd s e , so that p p ∇ κ q ab k a δ bi “ γ i , p p ∇ ab q δ a p i δ bj q ˝ “ σ ij ´ Q p ij q ˝ , p p ∇ ab q δ a r i δ bj s “ τ ij , (B.8) p p ∇ ρ q abcd q k a δ bβ δ cγ ℓ d “ ζ βγ ` Q r βγ s , (B.9) p p ∇ ρ q abcd q δ a r α δ bβ s k c ℓ d “ ´ i τ αβ ´ i Q r αβ s , (B.10) p p ∇ ρ q abcd q δ aα δ bβ δ cγ ℓ d “ G αβγ ` Q α r βγ s , (B.11) p p ∇ ρ q abcd q δ a ¯ α δ bβ δ cγ ℓ d “ G ¯ αβγ ` Q ¯ α r βγ s , (B.12) p p ∇ ρ q abcd q ℓ a δ bβ δ cγ ℓ d “ B βγ ` Q r βγ s , (B.13) p p ∇ ρ q abcd q ℓ a δ bβ δ cγ δ d ¯ α “ E r β h γ s ¯ α ´ Q r β h γ s ¯ α , (B.14) h γ ¯ α p p ∇ a ρ bcd q ℓ a δ bβ δ cγ δ d ¯ α “ p m ´ q ` i E β ´ Q β ˘ , (B.15) h γ ¯ α p p ∇ a ρ bcd q δ a ¯ α δ bβ δ cγ ℓ d “ ´ G β ` i h γ ¯ α p Q ¯ αβγ ´ Q ¯ αγβ q . (B.16)In particular, ´ h γ ¯ α p p ∇ a ρ bcd q ℓ a δ bβ δ cγ δ d ¯ α ´ h γ ¯ α p p ∇ a ρ bcd q δ a ¯ α δ bβ δ cγ ℓ d “ G β ´ p m ´ q i E β ` p m ´ q i Q β ´ i h γ ¯ α p Q ¯ αβγ ´ Q ¯ αγβ q . References [1] E. Abbena, S. Garbiero, and S. Salamon. Hermitian geometry on the Iwasawa manifold.
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Universit`a di Torino, Dipartimento di Matematica “G. Peano”, Via Carlo Alberto, 10- 10123, Torino, Italy
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