Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors
aa r X i v : . [ m a t h . DG ] A ug Towards a Classification of pseudo-RiemannianGeometries Admitting Twistor Spinors
Andree Lischewski
Department of Mathematics, Humboldt University, Rudower Chausse 25, 12489 Berlin,Germany
Abstract
We show that for pseudo-Riemannian conformal structures a totally lightlikesubspace fixed by the conformal holonomy representation is locally equivalentto having a Ricci-isotropic pseudo-Walker metric in the conformal class. Thisgeneralizes results obtained for lightlike lines and planes and naturally appliesto parallel spin tractors resp. twistor spinors on conformal spin manifolds. Infact, it clarifies which twistor spinors are locally equivalent to parallel spinors.Moreover, we study the zero set of a twistor spinor using the curved orbit decom-position for parabolic geometries. Generalizing results from the Lorentzian casewe can completely describe the local geometric structure of the zero set, con-struct a natural projective structure on it, and show that locally every twistorspinor with zero is equivalent to a parallel spinor off the zero set. An appli-cation of these results in low-dimensional split-signatures leads to a completegeometric description of local geometries admitting non-generic twistor spinorsin signatures (3 ,
2) and (3 ,
3) which complements the well-known description ofthe generic case. In contrast to the generic case where generic geometric dis-tributions play an important role, the underlying geometries in the non-genericcase without zeroes turn out to admit integrable distributions.
Keywords:
Conformal holonomy, Twistor spinors, Conformal Killing forms,Parallel spinors
1. Introduction
In this article we consider a space- and time-oriented, connected pseudo-Riemannian spin manifold of signature ( p, q ). One can canonically associateto this setting the real resp. complex spinor bundle S g with its Clifford mul-tiplication, denoted by µ : T M × S g → S g , and the Levi-Civita connectionlifts to a covariant derivative ∇ S g on this bundle. Besides the Dirac operator D g , there is another conformally covariant differential operator acting on spinorfields, obtained by performing the spinor covariant derivative ∇ S g followed byorthogonal projection onto the kernel of Clifford multiplication, P g : Γ( S g ) ∇ Sg → Γ( T ∗ M ⊗ S g ) g ∼ = Γ( T M ⊗ S g ) proj ker µ → Γ(ker µ ) , Email address: [email protected] (Andree Lischewski) alled the twistor operator. Elements of its kernel are called twistor spinors andthey are equivalently characterized as solutions of the conformally covarianttwistor equation ∇ S g X ϕ + 1 n X · D g ϕ = 0 for all X ∈ X ( M ) . In physics, twistor spinors appeared in the context of general relativity andwere first introduced by R. Penrose in [1]. They became of interest in differentialgeometry as T. Friedrich observed that special solutions of the twistor equation,the so called Killing spinors, are related to the first eigenvalue of the Diracoperator on a compact Riemannian spin manifold, see [2]. Since then the twistorequation on Riemannian manifolds has been widely studied, e.g. in [3]. Inparticular, it is well-known that a Riemannian spin manifold admitting a twistorspinor without zeroes is conformally equivalent to an Einstein manifold whichadmits a parallel or a Killing spinor. The zero set in the Riemannian case hasbeen widely studied (cf. [4, 5]). It consists of isolated points and if a zero exists,the spinor is conformally equivalent to a parallel spinor off the zero set. Incontrast to the Riemannian and Lorentzian case (cf. [6, 7]), the investigation ofthe twistor equation in other signatures is widely open. The following generalquestions are of interest:1. Which pseudo-Riemannian geometries admit nontrivial solutions of thetwistor equation ?2. How are further properties of twistor spinors related to the underlyinggeometries ? In particular, what are the possible shapes of the zero set Z ϕ ⊂ M ?3. How can one construct examples of manifolds admitting twistor spinors ?Recently, a Spin c -version of the twistor equation became of interest in the con-text of the AdS/CFT correspondence in physics, see [ ? ? ], and in this contextthe above questions help to distinguish pseudo-Riemannain manifolds on whichsupersymmetric conformal field theories can be placed. Obviously, the simplestsubcase of twistor spinors are parallel spinors. [8] gives a complete classificationof all non-locally symmetric, irreducible pseudo-Riemannian holonomy groupsadmitting parallel spinors. The other extremal case to irreducible acting holon-omy is the case of a maximal holonomy invariant totally lightlike subspace. Thisleads to parallel pure spinors on pseudo-Riemannian manifolds which are stud-ied in [8]. In split signatures, an explicit normal form of the metric is known.Furthermore, there are many examples and classification results for geometriesadmitting Killing spinors (cf. [9], [8]). Another well-understood case are twistorspinors on Einstein spaces. [3] shows that in case of nonzero scalar curvature thespinor decoposes into a sum of two Killing spinors whereas in case of a Ricci-flatmetric the spinor D g ϕ is parallel. Also much is known about twistor spinors onLorentzian manifolds. The most general result was obtained by F. Leitner in[10]. One can give up to conformal equivalence a complete list of local geometriesadmitting a twistor spinor off a certain singular set: Depending on the causaltype of the associated conformal vector field V ϕ one has a parallel spinor on aBrinkmann space, a local splitting into a Riemannian and Lorentzian factor, aLorentzian-Einstein Sasaki structure or a Fefferman space. One can use this todeduce that the zero set of a twistor spinor with zero on a Lorentzian manifoldconsists either of isolated images of null-geodesics and off the zero set one has a2arallel spinor on a Brinkmann space, or the zero set consists of isolated pointsand off the zero set one has a local splitting ( R , − dt ) × ( N, h ), where the lastfactor is Riemannian Ricci-flat Kaehler, in the conformal class.[11] indicates that in signatures higher than Lorentzian there are new interestingrelations between twistor spinors and constructions of conformal structures outof projective structres. However, as there is no complete classification of man-ifolds admitting twistor spinors, one often restricts oneself to small dimensionsin order to find out which geometries play a role there. [12] classifies metricsadmitting parallel spinor fields in small dimensions. It is moreover known that aRiemannian 3-manifold admitting a twistor spinor is conformally flat, and a Rie-mannian 4-manifold with twistor spinor is selfdual ([3]). In Lorentzian geometry,there is a classification of all local geometries admitting twistor spinors withoutzeroes and constant causal type of the associated conformal vector field V ϕ fordimensions n ≤
7, which can be found in [6] or [7]. In signature (2 , Hol ( M, c ) ⊂ SL (3 , R ) ⊂ SO + (3 , generic twistor spinors in signature (3 ,
2) and (3 , h ϕ, Dϕ i 6 = 0 (signa-ture (3 ,
3) is also discussed in [15]). They are shown to be in tight relationshipto so called generic 2-distributions on 5-manifolds resp. generic 3-distributionson 6-manifolds, that means every generic twistor spinor gives rise to a genericdistribution, and conversely, given a manifold with generic distribution, onecan canonically construct a conformal structure admitting a twistor spinor, andthese two constructions are inverse to each other.Twistor spinors are objects of conformal geometry and (except the Riemanniancase) all mentioned results in the pseudo-Riemannian context are established bymaking use of conformal tractor calculus and by equivalently describing twistorspinors as parallel sections in the spin tractor bundle associated to a conformalspin manifold as presented in [16] or [6]. In this setting, geometries admittingtwistor spinors are equivalently characterized as those conformal spaces (
M, c )where the lift of the conformal holonomy group
Hol ( M, c ) ⊂ SO ( p + 1 , q + 1) to Spin ( p + 1 , q + 1) stabilizes a nontrivial spinor. A problem closely related to thetwistor equation is therefore the classification of pseudo-Riemannian conformalholonomy groups which is completely solved only in the Riemannian case (cf.[16]). In arbitrary signatures, one knows a conformal analogue of the local de-Rham/Wu-splitting theorem (cf. [10]) and all holonomy groups acting transitiveand irreducible on the Moebius sphere were classified in [17]. The most involvedcase is the situation when the holonomy representation fixes a totally lightlikesubspace H ⊂ R p +1 ,q +1 . The associated local geometries are only known incases dim H ≤ ≥ M, c ) there exists a totally lightlike, k -dimensionalparallel distribution in the standard tractor bundle, then every point of someopen and dense subset admits a neighborhood U , a metric g ∈ c U and a k − L ⊂ T U such that
Ric g ( T U ) ⊂ L, (1) L is parallel wrt. ∇ g . (2)Conversely, if U ⊂ M is an open set equipped with a metric g ∈ c U and a k − L ⊂ T U such that (1) and (2)hold, then L gives rise to a k − dimensional totally lightlike, parallel distributionin the standard tractor bundle over U .In the rest of the article we apply this result to the classification problem fortwistor spinors, and we show that a large class of twistor spinors is locally equiv-alent to parallel spinors off a certain singular set. In fact, if we have a parallelspin tractor, we can associate via the holonomy principle a holonomy invari-ant spinor v ∈ ∆ p +1 ,q +1 (up to conjugation). As an application of Proposition3.2 we show that if the kernel H v of this spinor under Clifford multiplicationwith vectors from R p +1 ,q +1 is nontrivial, the associated twistor spinor is locallyconformally equivalent to a parallel spinor off a singular set (Proposition 3.3).Also the converse is true. In the remainder of the article we then present twomain applications of these results: First, we are able to clarify the local struc-ture of the zero set of a twistor spinor in arbitrary signatures: In Theorem 4.3we show that for ϕ ∈ Γ( S g )a twistor spinor with zero, the zero set Z ϕ is anembedded, totally lightlike submanifold. Moreover, for every x ∈ Z ϕ there areopen neighborhoods U of x in M and V of 0 in T x M such that Z ϕ ∩ U = exp x (ker D g ϕ ( x ) ∩ V ) . Besides, we show that the conformal structure canonically induces a torsion-free projective structure on the zero set of a twistor spinor (Proposition 4.4).In this regad we mention [19] where a similar statement is proved for zero setcomponents of certain conformal vector fields. As a second application we study(real) twistor spinors in small dimensions. We are able to classify geometriesadmitting non-generic twistor spinors in signature (3 ,
2) and (3 ,
3) which com-plements the analysis of the generic case from [11]. We prove in Proposition5.1 that real twistor half-spinors in signature (2 ,
2) without zeroes and realtwistor (half-)spinors without zeroes in signatures (3 ,
2) and (3 ,
3) satisfyingthat h ϕ, D g ϕ i ≡ ϕ ⊂ T M are integrable (off asingular set). Finally, we can also obtain some results in the less studied signa-tures (4 ,
2) and (4 , p,q
4n low dimensions as known from [12] directly relate the general previous resultsto concrete statementes in low dimensions.
2. Twistor spinors and associated objects
We consider R p,q , that is, R n , where n = p + q , equipped with a scalarproduct h· , ·i p,q of index p , given by h e i , e j i p,q = ǫ i δ ij , where ( e , ..., e n ) denotesthe standard basis of R n and ǫ i ∈ {± } are fixed. In general, we should thinkof h· , ·i p,q as being the pseudo-Euclidean standard scalar product of index p , i.e. ǫ i = − ≤ i ≤ p and ǫ i = +1 for p + 1 ≤ i ≤ n . However, in order tosimplify the following calculations, we shall work with this more general notionof R p,q . We denote by Cl p,q the Clifford algebra of ( R n , −h· , ·i p,q ) and by Cl C p,q its complexification. It is the associative real or complex algebra with unitmultiplicatively generated by ( e , ..., e n ) with the relations e i e j + e j e i = − h e i , e j i p,q . It is well-known (see [21, 22]) that if p − q Cl p,q . If p − q ≡ Cl p,q . Furthermore, Cl C p,q admits up to equivalence exactly one irreducible complex representationin case n is even and two such representations if n is odd. In case that thereare two equivalence classes of irreducible real or complex representations, theycan be distinguished by the unit volume element as presented in [21]: Let ω R := e · .... · e n ∈ Cl p,q and ω C := ( − i )[ n +12 ] − p ω R ∈ Cl C p,q . If p − q ≡ Cl p,q or Cl C p,q maps ω R to Id or − Id . Bothpossibilities can occur and the resulting representations are inequivalent. Theanalogous statements are true in the complex case for Cl C p,q and n odd (cf. [23]).This opens a way to distinguish a up to equivalence unique real resp. complexirreducible representation for all Clifford algebras Cl p,q and Cl C p,q by requiringthat ω is mapped to Id in case n even ( K = C ) or p − q ≡ K = R ). Remark 2.1.
For concrete calculations we shall make use of the following ir-reducible, complex representation of Cl C p,q : Let E, T, g and g denote the 2 × E = (cid:18) (cid:19) , T = (cid:18) − (cid:19) , g = (cid:18) ii (cid:19) , g = (cid:18) −
11 0 (cid:19) . Furthermore, let τ j = ( ǫ j = 1 ,i ǫ j = − . Let n = 2 m . In this case, Cl C ( p, q ) ∼ = M m ( C ) as complex algebras, and anexplicit realisation of this isomorphism is given byΦ p,q ( e j − ) = τ j − · E ⊗ ... ⊗ E ⊗ g ⊗ T ⊗ ... ⊗ T | {z } ( j − × , Φ p,q ( e j ) = τ j · E ⊗ ... ⊗ E ⊗ g ⊗ T ⊗ ... ⊗ T | {z } ( j − × . n = 2 m + 1. In this case, there is an isomorphism e Φ p,q : Cl C ( p, q ) → M m ( C ) ⊕ M m ( C ), given by e Φ p,q ( e j ) = (Φ p,q − ( e j ) , Φ p,q − ( e j )), j = 1 , ..., m, e Φ p,q ( e m +1 ) = τ m +1 ( iT ⊗ ... ⊗ T, − iT ⊗ ... ⊗ T ) , and Φ p,q := pr ◦ e Φ p,q is an irreducible representation mapping ω C to Id .Fixing an irreducible real or complex representation ρ : Cl ( C ) p,q → End (∆ p,q )and restricting it to the spin group Spin ( p, q ) ⊂ Cl p,q ⊂ Cl C p,q yields a rep-resentation of Spin (+) ( p, q ) on the space of real or complex spinors ∆ p,q ∈{ ∆ R p,q , ∆ C p,q } , called the real or complex spinor representation . One possible re-alisation in the complex case is ∆ C p,q = C m , where n = 2 m + 1 or n = 2 m (cf.Remark 2.1). In case n even ( K = C ) or p − q ≡ K = R ), ∆ p,q splits intothe sum of two inequivalent Spin ( p, q ) representations ∆ ± p,q according to the ± ω (cf. [22, 23]). In our realisation from Remark 2.1 one can findthese half spinor modules as follows: Let us denote by u (1) the vector (cid:18) (cid:19) ∈ C ,by u ( −
1) the vector (cid:18) (cid:19) ∈ C and set u ( ǫ , ..., ǫ m ) := u ( ǫ m ) ⊗ ... ⊗ u ( ǫ ) for ǫ ν = ±
1. Then we have∆ C , ± p,q = span { u ( ǫ , ..., ǫ m ) | m Y ν =1 ǫ ν = ± } . Note further that Cl ( C ) p,q acts on ∆ p,q via the representation ρ , and as R n ⊂ Cl p,q ⊂ Cl C p,q , this defines the Clifford multiplication X · ϕ := ρ ( X )( ϕ ) of a vectorby a spinor which naturally extends to a multiplication by k -forms: Letting ω = P ≤ i <...
1, is a Hermitian scalarproduct on ∆ C p,q . If p, q >
0, it has neutral signature and it holds that h X · u, v i ∆ C p,q + ( − p h u, X · v i ∆ C p,q = 0 . (4)for all u, v ∈ ∆ C p,q and X ∈ R n . In the real case, we can proceed analogous (cf.[24]) by choosing an irreducible real representation of Cl p,q such that as vectorspace ∆ R p,q can be realised to be R N for some N (cf. [22]). We then let ( · , · ) ∆ R p,q denote the standard scalar product on this space and define h· , ·i ∆ R p,q as in (3),where me may now set d = 1. (4) still holds in the real case. Moreover, h· , ·i ∆ R p,q is symmetric if p = 0 , p = 0 and q = 0) or it isdefinite ( p = 0 or q = 0). In case p = 2 , R p,q , h· , ·i ∆ R p,q ) is asymplectic vector space.There is an important decomposition of ∆ p +1 ,q +1 into Spin ( p, q ) − modules. Let( e , ..., e n +1 ) denote the standard basis of R p +1 ,q +1 . We introduce lightlikedirections e ± := √ ( e n +1 ± e ). One then has a decomposition R p +1 ,q +1 = R e − ⊕ R p,q ⊕ R e + of R p +1 ,q +1 into O ( p, q ) − modules. We define the annihila-tion spaces Ann ( e ± ) := { v ∈ ∆ p +1 ,q +1 | e ± · v = 0 } . It follows that for every v ∈ ∆ p +1 ,q +1 there is a unique w ∈ ∆ p +1 ,q +1 such that v = e − w + e + w , leadingto a decomposition ∆ p +1 ,q +1 = Ann ( e − ) ⊕ Ann ( e + ) . (5)As xe ± = − e ± x for all x ∈ R p,q ∼ = span( e , ..., e n ) ⊂ R p +1 ,q +1 we see that R p,q and Spin ( p, q ) act on Ann( e ± ). We can thus realise ∆ p,q as being Ann ( e ± ).Now fix an isomorphism α : Ann ( e − ) → ∆ p,q of Spin ( p, q )-representations.Then there is an induced isomorphism β : Ann ( e + ) → ∆ p,q , v α ( e − v ) andan isomorphism ∆ p +1 ,q +1 | Spin ( p,q ) ∼ = ∆ p,q ⊕ ∆ p,q , (6) v = e + w + e − w ( α ( e − w ) , α ( e − e + w )) (7)of Spin ( p, q ) modules. One calculates that wrt. this decomposition the scalarproduct h· , ·i ∆ p +1 ,q +1 is given by h (cid:18) v w (cid:19) , (cid:18) v w (cid:19) i ∆ p +1 ,q +1 = − δ p √ (cid:0) h v , w i ∆ p,q + ( − p h w , v i ∆ p,q (cid:1) (8)where v j , w j ∈ ∆ p,q for j = 1 , δ = i in case K = C and δ = 1 in the realcase. If q > p we instead work with a scalar product which involves e i j with ǫ i j = 1 in thedefinition. It has analogous properties, cf. [22].
7n general, the orbit structure of ∆ p,q under the
Spin + ( p, q ) action becomesvery complicated as n = p + q increases. However, in small dimensions theorbits are well understood (cf. [12]), and there is one distinguished orbit whichturns out to be of particular importance here, namely the so called pure spinors (cf. [12, 8, 25]). In order to define them, we follow [8]. Let m = [ n/ C n × ∆ C p,q → ∆ C p,q . To each spinor v ∈ ∆ C p,q we associatethe subspacesker C v := { X ∈ C n | X · v = 0 } and ker v := { X ∈ R n | X · v = 0 } . One checks that ker C v is isotropic with respect to the complex linear extension h· , ·i C p,q of h· , ·i p,q , and in particular, ker v is isotropic with respect to h· , ·i p,q .Clearly, dim ker v is an Spin ( p, q )-orbit invariant and it holds that λ ( g )(ker v ) ⊂ ker v for all g ∈ Spin ( p, q ), where λ : Spin ( p, q ) → SO ( p, q ) denotes the doublecovering map. A complex spinor v ∈ ∆ C p,q is said to be pure if dim C ker C v = (cid:6) n (cid:7) ,i.e., if its kernel under (extended) Clifford multiplication is a maximally isotropicsubspace. In this case, dim R ker v is called the real index of v . Next, we considerthe real case and pure spinors in ∆ R p,q . The notion of a real pure spinor can bedeveloped for all signatures ( p, q ) as explained in [12], but we are only interestedin pure spinors in the split signatures ( m, m ) and ( m + 1 , m ), and in thesecases pure spinors in ∆ R p,q can be defined using the complex definition (cf. [8]):Consider the split signatures ( m, m ) and ( m + 1 , m ) and the inclusion of the realspinor module ∆ R p,q ⊂ ∆ C p,q = ∆ R p,q ⊕ i ∆ R p,q (using a real structure as explainedbefore). Then a spinor v ∈ ∆ R p,q is called (real) pure if it is the real or imaginarypart of a pure spinor in the complexified module ∆ C p,q which has real index m . So in the following, when talking about pure spinors, we mean either thecomplex case or real pure spinors in split signature. If n = 2 m the set of purespinors in ∆ p,q forms precisely one orbit under the Spin + ( p, q ) action, whereasin case n = 2 m pure spinors form one orbit in each half spinor module ∆ ± p,q .Given a real pure spinor χ ∈ ∆ R p,q in split signature, [8] shows that up toconjugation in Spin + ( p, q ) the stabilizer of χ under the Spin + ( p, q ) action isgiven by Stab χ Spin + ( p, q ) = SL ( m ) ⋉ N, (9)where N is a certain nilpotent group. In the Lorentzian case one can associate to every nonzero spinor a nonva-nishing vector, the so called Dirac current. Generalizing this construction, weassociate to every spinor χ ∈ ∆ p,q a series of forms α kχ ∈ Λ kp,q , k ∈ N , so called algebraic Dirac forms , given by h α kχ , α i p,q := d k,p · h α · χ, χ i ∆ p,q ∀ α ∈ Λ kp,q . (10) d k,p is a nonzero constant depending on the chosen representation but not de-pending on χ , ensuring that the so defined form is indeed a real form. Thefollowing properties of these forms turn out to be important and are easilychecked: 8 roposition 2.1. Let χ ∈ ∆ p,q and k ∈ N . α pχ = 0 ⇔ χ = 02. α kχ = d k,p P ≤ i
Let χ ∈ ∆ p,q \{ } and let k := dim ker χ ( ≤ p ) . Then α pχ can bewritten as α pχ = l ♭ ∧ ... ∧ l ♭k ∧ e α, (11) where l j ∈ R p,q for ≤ j ≤ k such that span { l , ..., l k } = ker χ (in par-ticular, this implies that the l j are lightlike and mutually orthogonal), e α ∈ Λ p − k (cid:16) ( ker χ ) ⊥ (cid:17) and (11) is maximal in the sense that there exists no lightlikevector l k +1 being orthogonal to l i for ≤ i ≤ k such that α pχ = l ♭ ∧ ... ∧ l ♭k ∧ l ♭k +1 ∧ ee α .Moreover, whenever α pχ can be written as in (11) for mutually orthogonal light-like vectors l , ..., l k , it follows that l , ..., l k ∈ ker χ . The Proof of Lemma 2.2 is a straightforward generalization of the proof forthe case p = 1 as presented in [6]. Lemma 2.2 generalize well known facts aboutthe associated Dirac current V χ to a spinor χ ∈ ∆ ,n − in the Lorentzian casefrom [6]: It holds that || V χ || = 0 implies that V χ · χ = 0 being is a special caseof Lemma 2.2, and V χ is always causal. Remark 2.2.
All possible algebraic Dirac forms α ϕ for 0 = ϕ ∈ ∆ C ,n − havebeen classified in [10]. Precisely one of the following cases occurs:1. α ϕ = l ♭ ∧ l ♭ , where l , l span a totally lightlike plane in R ,n − .2. α ϕ = l ♭ ∧ t ♭ where l is lightlike, t is a orthogonal timelike vector.3. α ϕ = ω (up to conjugation in SO (2 , n − ω is the standardKaehler form on R ,n − . In this case n = 2 m and Stab α ϕ O (2 , n − ⊂ U (1 , m − E ⊂ R ,n − such that α ϕ | E = 0and α ϕ is the standard Kaehler form on the orthogonal complement E ⊥ of signature (2 , m ) (again, this is up to conjugation in SO (2 , n − Stab α ϕ O (2 , n − ⊂ U (1 , m ) × O ( n − m + 1)).Lemma 2.2 implies that the first case occurs iff ker ϕ is maximal, i.e. 2-dimensional. The second case occurs iff this kernel is one-dimensional whereasthe last two cases can only occur if the kernel under Clifford multiplication istrivial. We fix some notations about basic objects from spin geometry (follow-ing [23]) and recall the definition and properties of twistor spinors as intro-duced in [3]. Let (
M, g ) be a space-and time-oriented, connected pseudo-Riemannian spin manifold of index p and dimension n = p + q ≥
3. By P g + we9enote the SO + ( p, q )-principal bundle of all space-and time-oriented pseudo-orthonormal frames s = ( s , ..., s n ). A spin structure of ( M, g ) is then givenby a λ − reduction ( Q g + , f g ) of P g + to the group Spin + ( p, q ). In particular, f g : Q g + → P g + is a 2-fold covering. The associated bundle S g := Q g + × Spin + ( p,q ) ∆ p,q is called the real or complex spinor bundle . In case that ∆ p,q = ∆ + p,q ⊕ ∆ − p,q , itholds that S g = S g, + ⊕ S g, − , and one then has the notion of half-spinor fields.The algebraic objects introduced in the last section define fibrewise Cliffordmultiplication µ : T ∗ M ⊗ S g → S g and an inner product h· , ·i S g . Clearly, theproperties of h· , ·i ∆ p,q translate into corresponding properties of h· , ·i S g . Finally,the Levi Civita connection ∇ g on ( M, g ), considered as a bundle connection ω g ∈ Ω ( P g + , o ( p, q )), lifts to a connection e ω g ∈ Ω ( Q g + , spin ( p, q )) which in turninduces a covariant derivative ∇ S g on S g . Locally, ∇ S g is given by the formula ∇ S g X ϕ = X ( ϕ ) + 12 X ≤ k 3. We call a frame ( s , ..., s n ) over x ∈ M a conformalframe if there is g ∈ c such that the vectors s , ..., s n are pseudo-orthonormalwith respect to this metric. Collecting all these frames, we obtain the con-formal frame bundle ( P , π , M ; CO + ( p, q )) with structure group the identitycomponent of the conformal group CO ( p, q ) ∼ = R + × O ( p, q ). Using the generaltheory of parabolic geometries (cf. [27]), one shows that the oriented confor-mal structure ( M, c ) is equivalently encoded in a normal parabolic geometry ( P , π, M, ω nc ) of type ( G, P ) over M , where we have the following objects: G = SO ( p + 1 , q + 1), and the parabolic subgroup P ⊂ G is defined as follows:Let ( e , ..., e n +1 ) denote the standard basis of R p +1 ,q +1 , introduce two light-like directions by setting e ± := √ ( e n +1 ± e ) and let P := Stab R + e − G , where G acts on R p +1 ,q +1 via the standard matrix action. These algebraic objectsdescribe the flat model ( G → G/P, ω MC ) (with ω MC being the Maurer Car-tan form) for conformal structures, being a double cover b Q p,q := G/P of the(pseudo-)Moebius sphere Q p,q equipped with a flat conformal structure b c . Weset b C p,q := ( b Q p,q , b c ). For a general conformal structure ( P , π, M, ω nc ) the Car-tan connection ω nc ∈ Ω ( P , g ) on P is uniquely determined by certain nor-malisation conditions (cf. [16]) and called the normal conformal Cartan connec-tion . It describes the deviation from the flat model. It´s extension to a principalbundle connection on the principal G -bundle P := P × P G induces a covari-ant derivative ∇ nc on the standard tractor bundle T ( M ) := P × P R p +1 ,q +1 .Furthermore, h· , ·i p +1 ,q +1 induces a bundle metric on T ( M ), and ∇ nc turns outto be metric. Therefore, it seems natural to define the conformal holonomy ofthe conformal structure to be the holonomy of this connection : Hol x ( M, c ) := Hol x ( T ( M ) , ∇ nc ) ⊂ SO + ( T ( M ) x , h· , ·i ) ∼ = SO + ( p + 1 , q + 1) . The null line I = R e − ⊂ R p +1 ,q +1 which defines the parabolic subgroup P induces a filtration I ⊂ I ⊥ ⊂ R p +1 ,q +1 which leads to a filtration I ⊂ I ⊥ ⊂T ( M ) of T ( M ).Fixing a metric g ∈ c leads to a natural reduction σ g : P g → P to the structuregroup SO + ( p, q ). The standard representation of SO + ( p + 1 , q + 1) on R p +1 ,q +1 splits into the direct sum of 3 SO + ( p, q ) representations: R p +1 ,q +1 ∼ = R ⊕ R p,q ⊕ R with ae − + Y + be + ( a, Y, b ) . This in turn leads to an isomorphism T ( M ) g ∼ = M ⊕ T M ⊕ M . (13)With respect to this identification we have that I = M ⊕ ⊕ I ⊥ = M ⊕ T M ⊕ 0. In particular, we can use g to identify sections s ∈ Γ( T ( M )) with In fact, for our purposes one does not need to introduce the general concept of parabolicgeometries for this equivalence. For an explicit construction via first prolongation we refer to[16] or [26] For a slightly different, but in our case equivalent approach to define the holonomy of aCartan connection we refer to [16] α, Y, β ), where α, β ∈ C ∞ ( M ) and Y ∈ X ( M ). Under this identification,the bundle metric is given by h ( α , Y , β ) , ( α , Y , β ) i T ( M ) = α β + α β + g ( Y , Y ) , (14)and one has the following formulas for the metric description of the tractorconnection ∇ nc and its curvature R ∇ nc : ∇ ncX αYβ = X ( α ) + K g ( X, Y ) ∇ gX Y + αX − βK g ( X ) ♯ X ( β ) − g ( X, Y ) , R ∇ nc X ,X αYβ = C g ( X , X ) YW g ( X , X ) Y − βC g ( X , X ) ♯ , (15)where C g ( X, Y ) := ∇ gX ( K g )( Y ) − ∇ gY ( K g )( X ) defines the Cotton-tensor and W g is the Weyl-tensor of the conformal structure. Under a conformal change e g = e σ g , the metric representation of a standard tractor transforms accordingto (cf. [16]) αYβ e α e Y e β = e − σ ( α − Y ( σ ) − β || grad g σ || g e − σ ( Y + β grad g σ ) e σ β . (16)An analogous first prolongation procedure can be carried out in the confor-mal spin setting (cf. [16]). Let CSpin (+) ( p, q ) = R + × Spin (+) ( p, q ) denote the(identity component of) the conformal spin group, coming together with a dou-ble covering λ : CSpin + ( p, q ) → CO + ( p, q ). ( e G = λ − ( G ) , e P = λ − ( P )) is thepair on which conformal spin structures are modelled as parabolic geometries.The Cartan geometry ( e G → e G/ e P ∼ = b Q p,q , ω MC ) is the flat model and can beviewed as the space b C p,q equipped with a conformal spin structure (cf. [10]).For concrete calculations we use a realisation of the flat model preseted in [6]: b Q p,q is isomorphic to the set of time-oriented null directions in R p +1 ,q +1 . It isnaturally embedded in R p +1 ,q +1 via i : b Q p,q ֒ → R p +1 ,q +1 , where R + · x s h x, x i n +2 · x (17)and where h· , ·i n +2 denotes the standard Euclidean inner product. One checksthat i ( b Q p,q ) = S p × S q ⊂ R ,p +1 × R ,q +1 . It holds that b c = [ i ∗ h· , ·i p +1 ,q +1 ],yielding the conformally flat conformal spin manifold b C p,q = ( b Q p,q , b c ), whichrealises the flat model for conformal spin structures of index p . Suppose nowthat ( M, c ) carries a conformal spin structure, being a λ -reduction ( Q , f )of the bundle P to CSpin + ( p, q ). In analogy with the previous case, thisgeometric structure is via first prolongation equivalently encoded in a parabolicgeometry ( Q , e π, M, e ω nc ) of type ( e G, e P ) such that one has the following doublecoverings:( Q g + ; Spin + ( p, q )) ֒ → ( Q ; CSpin + ( p, q )) ↔ ( Q , g ω nc ) ( e G, e P ) f g ↓ f ↓ f ↓ λ ↓ ( P g + ; SO + ( p, q )) ֒ → ( P ; CO + ( p, q )) ↔ ( P , ω nc ) ( G, P )The normal conformal spin connection e ω nc ∈ Ω ( Q , spin ( p +1 , q +1)) induces acovariant derivative - also denoted by ∇ nc - on the (real or complex conformal)12 pin tractor bundle S T ( M ) := Q × e P ∆ p +1 ,q +1 . Furthermore, one has in analogyto the metric setting an inner product h· , ·i S on this bundle, and a pointwiseClifford multiplication µ ( X, ψ ) := X · ψ of sections X ∈ Γ( T ( M )) and spinorfields ψ ∈ Γ( S T ( M )).Fixing a metric g ∈ c leads to a reduction σ g : Q g + → Q of ( Q , e P ) to( Q g + , Spin + ( p, q )). We let Q denote the enlarged Spin + ( p + 1 , q + 1)-principalbundle, and as S T ( M ) ∼ = Q × Spin + ( p +1 ,q +1) ∆ p +1 ,q +1 , we may use g to identify Q g + × ρ ◦ i cs ∆ p +1 ,q +1 ∼ = S T ( M ) , [ l, v ] [[ σ g ( l ) , e ] , v ] , where i cs : Spin ( p, q ) ֒ → Spin ( p + 1 , q + 1) denotes the natural inclusion, and e ∈ Spin ( p + 1 , q + 1) is the neutral element. The decomposition (7) of ∆ p +1 ,q +1 induces projections proj g ± : S T ( M ) → Q × Spin ( p,q ) Ann ( e ± ) and an vectorbundle isomorphism Φ g : S T ( M ) → S g ( M ) ⊕ S g ( M ) , (18)[[ σ g ( l ) , e ] , e − w + e + w ] [ l, β ( e + w )] + [ l, α ( e − w )] . (19)One calculates that under this identification, ∇ nc is given by the expression (cf.[16]) ∇ ncX (cid:18) ϕφ (cid:19) = (cid:18) ∇ S g X − X · K g ( X ) · ∇ S g X (cid:19) (cid:18) ϕφ (cid:19) . Together with (12) this yields a reinterpretation of twistor spinors in terms ofconformal Cartan geometry: Theorem 2.3. Let ( M, c ) be a connected, space- and time-oriented conformalspin manifold of dimension n ≥ . For any metric g ∈ c , the vector spacesof twistor spinors in Γ( S g ) and parallel sections in Γ( S T ( M )) are naturallyisomorphic viaker P g → Γ( S g ( M ) ⊕ S g ( M )) (Φ g ) − ∼ = Γ( S T ( M )) , ϕ (cid:18) ϕ − n D g ϕ (cid:19) (Φ g ) − ψ ∈ P ar ( S T ( M ) , ∇ nc ) , i.e. a spin tractor ψ ∈ Γ( S T ( M )) is parallel iff for one (and hence for all g ∈ c ),it holds that ϕ := Φ g ◦ proj g + ψ ∈ ker P g and D g ϕ = − n · Φ g ◦ proj g − ψ .2.4. The twistor equation on forms In the Lorentzian case, it has paid off to associate to every spinor field ϕ ∈ Γ( S g ) a vector field V ϕ ∈ X ( M ), the so called Dirac current, as done in[7, 6]. The zero sets of these objects coincide, i.e. Z ϕ = Z V ϕ . If ϕ is a twistorspinor, then V ϕ is a conformal vector field and Lorentzian geometries admittingtwistor spinors can partially be classified by studying the behaviour of V ϕ , cf.[7] for details. This procedure can be generalized to arbitrary signatures bymaking use of the nc-Killing form theory as presented in [28] or [10]. We listsome later needed facts:A global version of (10) associates to a spinor field ϕ ∈ Γ( S g ) a k − form α kϕ ∈ Ω k ( M ) for each k ∈ N with Z ϕ = Z α pϕ . In the special case k = 1, it holds that13 ϕ = V ♭ϕ . If ϕ is a twistor spinor, the forms α kϕ turn out to be normal conformal(nc-)Killing k-forms , meaning that they are conformal Killing forms, ∇ gX α kϕ − k + 1 ι X dα kϕ + 1 n − k + 1 X ♭ ∧ d ∗ α kϕ = 0 , for all X ∈ X ( M ) , which satisfy additional normalisation conditions as to be found in [6]. Onechecks that if α = α kϕ is a nc-Killing k -form wrt. g , then the rescaled form e α := e ( k +1) σ α kϕ = α ke σ e ϕ (20)is a nc-Killing k -form wrt. e g = e σ g .On the other hand, if we view the twistor spinor as parallel spin tractor ψ ∈ P ar ( S , ∇ nc ), we can also associated to this object a series of forms. In order todefine them, we introduce the tractor k-form bundle Λ k T ( M ) := P × P Λ kp +1 ,q +1 .Sections, i.e. elements of Ω k P ( M ) := Γ(Λ k T ( M )) are called tractor k -forms on M . Clearly, the standard scalar product on Λ kp +1 ,q +1 induces a bundle metric onthis space and the normal conformal Cartan connection ω nc leads to a covariantderivative ∇ nc . Again, (10) can be applied pointwise, and in this way, a seriesof tractor forms α kψ is associated to every spin tractor ψ ∈ Γ( S ). In the specialcase of ψ being parallel, α kψ turns out to be parallel as well. Parallel tractor k − forms are called (normal) twistor k-forms .Fixing a metric in the conformal class leads to the following description oftractor k -forms: First, note that every form α ∈ Λ k +1 p +1 ,q +1 decomposes into α = e ♭ + ∧ α − + α + e ♭ − ∧ e ♭ + ∧ α ∓ + e ♭ − ∧ α + for uniquely determined forms α − , α + ∈ Λ kp,q , α ∈ Λ k +1 p,q and α ∓ ∈ Λ k − p,q . Whence, we can reduce P to P g + with structure group SO + ( p, q ) and see that there is an isomorphismΛ k +1 T ( M ) g ∼ = Λ k ( M ) ⊕ Λ k +1 ( M ) ⊕ Λ k − ( M ) ⊕ Λ k ( M ), and consequently, eachtractor ( k + 1)-form α ∈ Ω k +1 P ( M ) uniquely corresponds via a fixed metric g ∈ c to a set of differential forms α g ↔ ( α − , α , α ∓ , α + ), where α − , α + ∈ Ω k ( M ) , α ∈ Ω k +1 ( M ) , α ∓ ∈ Ω k − ( M ). We can also write this as α g = s ♭ − ∧ α − + α + s ♭ − ∧ s ♭ + ∧ α ∓ + s ♭ + ∧ α + , (21)i.e. the s ± are global lightlike sections in the line bundles in T ( M ) g ∼ = M ⊕ T M ⊕ M induced by e ± . With respect to the splitting (21), the covarianttractor derivative ∇ nc on Γ(Λ p +1 ,q +1 T ( M )) is given by ∇ ncX g = ∇ gX − ι X X ♭ ∧ − K g ( X ) ♭ ∧ + ∇ gX X ♭ ∧ ι K g ( X ) ∇ gX ι X ι K g ( X ) K g ( X ) ♭ ∧ ∇ gX . Using this expression, it is straightforward to calculate that if α is a normaltwistor ( k + 1)-form, then α + is a nc-Killing k -form and α , α ∓ , α − are uniquelydetermined by α + . On the other hand, given a nc-Killing k − form α ∈ Ω k ( M ) In comparison to [28] the roles of α + and α − are interchanged since the reference realisesthe parabolic subgroup P as stabilizer of the line R e + . g , there is a unique twistor ( k + 1) − form α k +1 ∈ Ω k +1 P ( M ) suchthat (cid:0) α k +1 (cid:1) + = α holds. Now let ϕ ∈ Γ( S g ) be a twistor spinor with associatedparallel spin tractor ψ ∈ Γ( S ). Then it is straightforward to calculate that wrt. g ∈ c (cid:16) α k +1 ψ (cid:17) + = d · α kϕ and (cid:16) α k +1 ψ (cid:17) − = d · α kD g ϕ (22)where d , are nonzero constants not depending on ψ (cf.[29]). Moreover, itholds that ϕ ( x ) = 0 ⇒ α k +1 ψ ( x ) = d s ♭ − ( x ) ∧ α kD g ϕ ( x ) , (23) D g ϕ ( x ) = 0 ⇒ α k +1 ψ ( x ) = d s ♭ + ( x ) ∧ α kϕ ( x ) . Note that these formulas determine the SO ( p +1 , q +1) orbit type of the parallelform α k +1 ψ on all of M . The existence of certain normal twistor forms has manyinteresting implications on the (local) geometry of M as studied in [10]. Tosummarize, one has the following implications between the objects introducedso far: ϕ ∈ Γ( S g )TS o o g ∈ c / / nc-Killing (cid:15) (cid:15) ψ ∈ Γ( S T ( M )) Hol-Pr. / / normal twistor (cid:15) (cid:15) v ψ ∈ ∆ p +1 ,q +1alg. Dirac (cid:15) (cid:15) cond. for ( ( PPPPPPPPPPPP Hol ( M, c ) α pϕ ∈ Ω p ( M ) o o g ∈ c / / α p +1 ψ ∈ Ω p +1 P ( M ) Hol-Pr. / / α p +1 v ψ ∈ Λ p +1 p +1 ,q +1cond.for ♥♥♥♥♥♥♥♥♥♥♥♥ 3. Geometries admitting totally lightlike, holonomy-invariant sub-spaces Let ψ ∈ Γ( S ( M )) be a parallel spin tractor. We set H ψ ( x ) := { v ∈ T x ( M ) | v · ψ ( x ) = 0 } . This leads to a totally lightlike and parallel distribution H ψ ⊂T ( M ). We want to prove that the twistor spinor induced by ψ via fixing ametric in the conformal class is locally equivalent to a parallel spinor iff H ψ is nontrivial. Main ingredient is the following statement about totally lightlikeparallel distributions in the standard tractor bundle: Proposition 3.1. Let ( M, c ) be a conformal manifold of dimension n ≥ andlet H ⊂ T ( M ) be a totally lightlike distribution of dimension k ≥ which isparallel wrt. the Cartan connection ∇ nc . Then there is an open, dense subset f M ⊂ M such that for every point x ∈ f M there is an open neighborhood U x ⊂ f M and a metric g ∈ c | U x such that wrt. the splitting (13) H is locally given by H | U x g = span K , ..., K k − , for lightlike vectorfields K i ∈ X ( U x ) . roof. If k = 1, this is a well known fact (cf. [16]). We can adopt parts ofthe (first steps of the) proof and the notation from [18] where the statementis proved for k = 2 and we may then also assume that k > 2. However, welater use a different method. To start with, we set L := I ⊥ ∩ H , where I is theisotropic line defining the parabolic subgroup P . With respect to g ∈ c one hasthat L = X ∈ H | X = αY . Note that L := pr T M L ⊂ T M is conformallyinvariant. Step 1: We claim that there is an open, dense subset f M ⊂ M such that rk L | f M = k − L 6 = { } as otherwise H would have rank 1. Moreover, there is noopen set in M on which rk L = k as follows from Lemma 2 in [18]. Consequently,there is an open, dense subset (which we again call M ) over which 0 < rk L < k .Now let x ∈ M and fix a basis L , ..., L s of L x , where s ≤ k − 1. We may addtractors Z l = a l Y l for 1 ≤ l ≤ k − s such that L , ..., L s , Z , ..., Z k − s is a basisof of H x . We know that k − s ≥ 1. If k − s > Z + Z and Z − Z . However, Z − Z ∈ L x . Thus, k − s = 1, which showsthat rk L x = k − Step 2: We claim that also L = pr T M L has rank k − x ∈ M .To this end, let g ∈ c be arbitrary. Then we choose generators of L around x such that locally L g = span σ e K , ..., σ k − e K k − . We may assume that σ ( x ) = 0. Otherwise, we find ϕ ∈ C ∞ ( M ) with e K ( ϕ )( x ) = 0 and considerthe metric e g = e ϕ g instead (cf. (13)). Moreover, we may by rescaling thegenerators assume that there is a neighborhood U of x on which σ ≡ | σ i | < i = 2 , ..., k − 1. By linear algebra we then see that there are lightlikevectorfields K i for i = 1 , ..., k − g on U L g = span K , K , ..., K k − . (24)Suppose now that there is an open set on which ∈ L . Differentiatingin direction X ∈ T M yields that ∇ ncX = X ∈ H for all X as H isparallel. This is not possible for dimensional reasons. Consequently, on anopen and dense subset the vectors K , ..., K k − are linearly independent and as In this proof, in order to keep notation short, whenever we restrict to an open, densesubset we again call it M . = span( K , ..., K k − ) this shows that there is an open and dense subset of M on which the rank of L is maximal. Step 3: It follows precisely as in the k = 2-case from [18], Lemma 3 and 4, that L ⊥ = pr T M (cid:0) H ⊥ ∩ I ⊥ (cid:1) Step 4: In the setting of Step 2 we consider the local representation (24) of L wrt. g and set L ′ := span ( K , ..., K k − ). Both L and L ′ are integrable distributions:Let i, j ∈ { , ..., k − } . As H is parallel and totally lightlike we have that ∇ K i K j = − P g ( K i , K j ) ∇ gK i K j − g ( K i , K j ) ∈ Γ( L ). Switching the roles of i and j andtaking the difference yields K i , K j ]0 ∈ Γ( L ). Thus [ K i , K j ] ∈ L ′ . Similarlyone shows that even [ K , L ′ ] ⊂ L ′ (25) Step 5: We now apply Frobenius Theorem: For every (fixed) point y of (an open anddense subset of ) M we find a local chart ( U, ϕ = ( x , ..., x n )) centered at y with ϕ ( U ) = { ( x , ..., x n ) ∈ R n | | x i | < ǫ } such that the leaves A c k ,...,c n = { a ∈ U | x k ( a ) = c , ..., x n ( a ) = c n } ⊂ U are integral manifolds for L for every choice of c j with | c j | < ǫ . It holds that L U = span (cid:16) ∂∂x , ..., ∂∂x k − (cid:17) and moreover thecoordinates may be chosen such that K = ∂∂x over U . After applying somelinear algebra to the generators of L ′ and restricting U if necessary, we mayassume that generators of L ′ are given on U by K i ≥ = α i ∂∂x + ∂∂x i (26)for certain smooth functions α i ∈ C ∞ ( U ). The integrability condition (25)implies that ∂∂x α i = 0 for i = 2 , ..., k − . (27)The integrability of L ′ and (27) then yield that ∂∂x i α j − ∂∂x j α i = 0 for i, j = 2 , ..., k − . (28)For fixed c k , ..., c n as above we consider the submanifold A c k ,...,c n and the differ-ential form α := − P k − i =1 α i dx i ∈ Ω ( A c k ,...,c n ) , where the α i ≥ are restrictionsof the functions appearing in (26) and we set α ≡ − 1. (27) and (28) yield that dα = 0. Whence there exists by the Poincare Lemma (after restricting U if nec-essary) a unique σ c k ,...,c n ∈ C ∞ ( A c k ,...,c n ) with σ c k ,...,c n ( ϕ − (0 , ..., , c k , ..., c n )) =17 and ∂∂x σ c k ,...,c n = 1 , (29) ∂∂x i σ c k ,...,c n = − α i for i = 2 , ..., k − . (30)We then define σ ∈ C ∞ ( U ) via σ ( ϕ − ( x , ...., x n )) := σ x k ,...,x n ( ϕ − ( x , ..., x n ))and observe that on U∂∂x σ = 1, ∂∂x i σ = − α i for i = 2 , ..., k − . (31) Step 6: (26) and (24) imply that on U K ( σ ) = 1 and K i ( σ ) = 0 for i = 2 , ..., k − 1. Wenow consider the rescaled metric e g = e σ g on U . The transformation formula(16) and (31) then show that wrt. this metric L is given by L U = span K , ..., K k − . We may add one generator βX ∈ Γ( U, H ) such that pointwise (wrt. e g ) H = L ⊕ span βX . It follows that X ∈ L ⊥ . By step 3 there exists a smoothfunction b on U with X = pr T M bX and bX ∈ H ⊥ . As H is lightlike, itfollows that b = β . Therefore we have that ∈ H ⊥ over U . However, thisimplies that β = 0. dim H + dim H ⊥ = n + 2 and dimension count show that X has to be a linear combination of the K i , i = 1 , ..., k − 1, and passing to newgenerators then proves the Proposition.We study some consequences. In the setting of Proposition 3.1 we have that H is parallel iff H ⊥ is parallel. Locally, we have wrt. the metric appearingin Propostion 3.1 that H ⊥ = span Xσ | X ∈ L ⊥ . It follows that H ⊥ isparallel iff ∇ ncY Xσ = − P g ( X, Y ) ∇ gY X + σP g ( Y ) − g ( X, Y ) ∈ Γ( U, H ⊥ )for all X ∈ Γ( U, L ⊥ ) and Y ∈ X ( U ). This is equivalent to parallelity of L , P g ( X ) = 0 for all X ∈ L ⊥ and P g ( T M ) ⊂ L . As in [30] we conclude that thescalar curvature is zero. Thus we have proved the following Proposition.18 roposition 3.2. If on a conformal manifold ( M, c ) there exists a totally light-like, k -dimensional parallel distribution in T ( M ) , then every point of some openand dense subset admits a neighborhood U , a metric g ∈ c U and a k − -dimensional totally lightlike distribution L ⊂ T U such that Ric g ( T U ) ⊂ L, and L is parallel wrt. ∇ g . (32) Conversely, if U ⊂ M is an open set equipped with a metric g ∈ c U and a k − -dimensional totally lightlike distribution L ⊂ T U such that (32) holds, then L gives rise to a k − dimensional totally lightlike, parallel distribution L ⊕ span in T ( U ) . In case k = 1, this means that there is locally a Ricci-flat metric in theconformal class. In case k = 2 this describes conformal pure radiation metricwith parallel rays as discussed in [18]. Proposition 3.2 also generalizes resultsfrom [28] where the statement is proved under the additional condition thatthe totally lightlike distribution arises from a decomposable, totally lightliketwistor k − form. One proves precisely as in [18], Remark 2, that in the settingof Proposition 3.1 one gets the conformally invariant curvature condition W g ( L, L ⊥ , · , · ) = 0for the Weyl tensor for the existence of a totally lightlike, parallel null-plane inthe tractor bundle.We apply these results to the case of twistor spinors on conformal spin man-ifolds. Let ψ ∈ Γ( S ) be a parallel spin tractor on ( M p,q , c ) and for g ∈ c let ϕ ∈ Γ( S g ) be the associated twistor spinor. As ψ is parallel, the pointwisekernel of Clifford multiplication ker ψ ( x ) = { v ∈ T x ( M ) | v · ψ ( x ) = 0 } yieldsa totally lightlike and parallel distribution H ψ ⊂ T ( M ). Similarly, if even ϕ isparallel, we get a totally lightlike, parallel distribution L ϕ ⊂ T M . One thenhas the following immediate consequence from Proposition 3.1: Proposition 3.3. If ψ ∈ Γ( S ) is a parallel spin tractor with H ψ = 0 , thenthere is an open and dense subset f M ⊂ M such that on f M the associatedtwistor spinor is locally conformally equivalent to a parallel spinor. Proof. We notice that Proposition 3.1 yields the desired f M and for x ∈ f M a neighborhood U and a local metric g U ∈ c U such that s + ∈ H ψ U . Ifwe decompose ψ on U wrt. the metric g as in Theorem 2.3, i.e. ψ | U =[[ σ g ( l ) , e ] , e − w + e + w ] for some function w : U → ∆ p +1 ,q +1 , it follows that e + e − w = 0 on U which implies that e − w = 0. However, by Theorem 2.3 itfollows that on U we have D gϕ = proj g − ( ψ ) = 0. Thus, ϕ is on U both harmonicand a twistor spinor and therefore parallel wrt. g .Note that by the same argumentation as in the last proof every parallelspinor Γ( S g ) ∋ ϕ g ↔ ψ ∈ Γ( S ) satisfies H ψ = 0. Whence, Proposition 3.3 yieldslocally an equivalent characterisation of those parallel spin tractors which lead19o parallel spinors in terms of conformally invariant objects. In terms of theoriginal data the Proposition can be rephrased as follows: Note that wrt. thedecompositon (18) the requirement H ψ = 0 is equivalent to say that there is x ∈ M , g ∈ c and a nontrivial tripel ( α, X, β ) ∈ R ⊕ T x M ⊕ R such that X · ϕ ( x ) + αD g ϕ ( x ) = 0 ,X · D g ϕ ( x ) + βϕ ( x ) = 0 . Thus, if, for example, D g ϕ vanishes at some point for some metric in the con-formal class, then the twistor spinor is already locally equivalent to a parallelspinor locally around every point (up to a singular set). Moreover, Proposi-tion 3.3 admits several further consequences and applications which contributeto the classification problem for local geometries admitting twistor spinors onpseudo-Riemannian manifolds.We first describe how it is related to and generalizes other results obtained forthe Riemannian and Lorentzian case. For a Riemannian spin manifold ( M n , g )with twistor spinor ϕ one has that (cid:16) M \ Z ϕ , e g = || ϕ || (cid:17) is an Einstein space ofnonnegative scalar curvature e R . If e R > 0, then the rescaled spinor decomposesinto a sum of two Killing spinors whereas in case e R = 0 one has a Ricci-flatmetric with parallel spinor. Proposition 3.3 precisely describes the last casein which dim H ψ = 1. For the Lorentzian case, Lemma 2.2 yields a relationbetween Proposition 3.3 and the classificaion of twistor spinors on Lorentzianmanifolds using the nc-Killingform theory in [10]. Theorem 3.4 ([10]; Thm.10) . Let ϕ ∈ Γ( S g C ) be a spinor on a Lorentzian spinmanifold of dimension n ≥ . Then one of the following holds on an open anddense subset f M ⊂ M : α ψ = l ♭ ∧ l ♭ for standard tractors l , l which span a totally lightlike plane.In this case, ϕ is locally conformally equivalent to a parallel spinor withlightlike Dirac current V ϕ on a Brinkmann space. α ψ = l ♭ ∧ t ♭ where l is a lightlike, t is an orthogonal, timelike standardtractor. ( M, g ) is locally conformally equivalent to ( R , − dt ) × ( N , h ) × · · · × ( N r , h r ) , where the ( N i , h i ) are Ricci-flat K¨ahler, hyper-K¨ahler, G -or Spin (7) -manifolds. α ψ is of Kaehler-type at every point (cf. Remark 2.2).The following cases can occur: (a) The dimension n is odd and the space is locally equivalent to a LorentzianEinstein-Sasaki manifold. (b) n is even and ( M, g ) is locally conformally equivalent to a Feffermanspace. (c) α ψ is a volume form on a nondgenerate subbundle V ⊂ T ( M ) . Thenthere exists locally a product metric g × g ∈ [ g ] on M , where g is aLorentzian Einstein-Sasaki metric on a space M of dimension n =2 · rk ( α ( ϕ )) + 1 admitting a Killing spinor and g is a RiemannianEinstein metric with Killing spinor on a space M of positive scalarcurvature scal g = ( n − n )( n − n − n ( n − scal g . Lemma 2.2 shows us that H ψ = 0 occurs exactly in the first two cases inwhich we get a parallel spinor as also follows from Proposition 3.3. In the third20ase, it holds that dim H ψ = 0 and the spinor cannot be rescaled to a parallelspinor. In particular, we have shown that the Killing spinors defining LorentzianEinstein Sasaki structures (cf. [9]) can never be rescaled to parallel spinors.We describe further geometric consequences implied by Propositon 3.3. If ϕ can locally be rescaled to a parallel spinor, the vanishing of the torsion of ∇ g implies as a global consequence that L ϕ is an integrable , distribution on f M .Now fix x ∈ f M and let U ⊂ f M be an open neighborhood with metric g ∈ c | U such that ϕ is parallel wrt. g on U . One has that Ric g ( T U ) ⊂ L ϕ | U as impliedby Proposition 3.2. However, note that this also follows from the well-knownfact that Ric g ( T M ) · ϕ = 0 for any parallel spinor ϕ . Moreover, it followsfrom Lemma 2.2 or Proposition 3.1 that dim H ψ = dim L ϕ | U + 1. In casethat k := dim L ϕ | U > Hol ( U, g ) acts reducible with a fixed totally lightlike k − dimensional subspace. If k = p , i.e. ϕ is a pure spinor on U , it follows fromLemma 2.2 that even a totally isotropic p − form is fixed. If k = p − Hol ( U, g )fixes a p − form of type α pϕ = l ♭ ∧ ... ∧ l ♭p − ∧ t ♭ , where t is not lightlike and itfollows that even then totally lightlike form l ♭ ∧ ... ∧ l ♭p − is fxed by the holonomyrepresentation. If k = 0 it follows from Proposition 3.3 or Ric g ( X ) · ϕ = 0 that g is a Ricci-flat metric on U . There is a complete list of possible irreducible,non locally symmetric holonomy groups for this case as to be found in [8]. Wesummarize these results as follows: Proposition 3.5. Let ψ be a parallel spin tractor with H ψ = 0 . Then thereis an open, dense subset f M ⊂ M such that L ϕ is an integrable distribution on f M . Moreover, any x ∈ f M admits an open neighborhood U ⊂ f M and a metric g ∈ c | U such that ϕ is a parallel spinor on ( U, g ) and one of the following casesoccurs: k := dim L ϕ = 0 . In this case, Hol ( U, g ) acts reducible with fixed k − dimensionaltotally lightlike subspace L and Ric g ( T U ) ⊂ L . Moreover, if k = p, p − there is a totally isotropic parallel k − form. k := dim L ϕ = 0 . The space ( U, g ) is Ricci-flat. If it is not locallysymmetric and Hol ( U, g ) acts irreducible, then it is one of the list in [8]. We further remark that similar integrability conditions for pure twistorspinors have been derived in [31] and [32]. In split signature ( m + 1 , m ) where∆ C m +1 ,m admits a real structure and one can speak about real spinor fields onecan say even more about parallel pure spinor fields by using results from [8]which give an explicit normal form for the metric for this case. More concretely,let ( M, h ) be a pseudo-Riemannian spin manifold of split signature ( m + 1 , m )admitting a real pure parallel spinor field. Then one can find for every pointin M local coordinates ( x, y, z ) , x = ( x , ..., x m ), y = ( y , ..., y m ) around thispoint such that h = − dz − m X i =1 dx i dy i − m X i,j =1 g ij dy i dy j , (33)where g ij are functions depending on x, y and z and satisfying g ij = g ji for i, j = 1 , ..., m , m X i =1 ∂g ik ∂x i = 0 for k = 1 , ..., m. (34)21onversely, if one uses (33) and (34) to define a metric h on a connected openset U ⊂ R m +1 , then ( U, h ) is spin and admits a real pure parallel spinor. Sim-ilar statements hold in case ( p, q ) = ( m, m ), where one has to omit the lastcoordinate etc.As a special application we consider twistor spinors equivalent to parallelspinors in case p = 2. If H ψ = { } , then in the above notation one has aparallel 2-form α ϕ on ( U, g ). The SO + (2 , n − α ϕ = l ♭ ∧ l ♭ corresponds toa parallel pure spinor. In the second case, α ϕ = l ♭ ∧ t ♭ , we can conlcude thatthere is a nontrivial lightlike, parallel vectorfield and thus ( U, g ) is a Brinkmannspace. In the third case, ( U, g ) is Ricci-flat (as L ϕ = 0) and Hol ( U, g ) leavesinvariant a (possibly trivial) n − m dimensional nondgenerate subspace E ⊥ and α ϕ is Kaehler on E . It follows with Remark (2.2) that there is a localsplitting ( U, g ) ∼ = ( U , g ) × ( U , g ), where the first factor is Ricci-flat pseudoKaehler of signature (2 , m − 2) and the second factor (which might be trivial)is Riemannian Ricci flat. Moreover, by Leitners argument from [10] both factorsadmit parallel spinors. 4. The zero set of a twistor spinor In this section we want to describe the possible local shapes of the zeroset Z ϕ of a twistor spinor ϕ ∈ Γ( S g ) and study the properties and relatedlocal geometries of the spinor off its zero set. It is shown in [3] that in theRiemannian case the zero set consists of a countable union of isolated points.For the Lorentzian case, [29] shows that Z ϕ - if nonempty - consists either ofisolated points or of isolated images of lightlike geodesiscs. Moreover, one hasthat for a given x ∈ Z ϕ , there is an open neighborhood U of x in M and V of0 in T x M such that Z ϕ ∩ U = exp x (ker D g ϕ ( x ) ∩ V ) . (35)The proof of (35) relies on the investigation of the zero set of V ϕ , being aconformal vector field which additionally satisfies ι V ϕ W g = 0. We show that(35) holds in all signatures by making use of the holonomy reduction procedurefor general Cartan geometries as introduced in [20]. Applied to our setting,this reads as follows: Let ψ ∈ Γ( S ( M )) be a ∇ nc -parallel spin tractor. Weview S ( M ) = Q × e G ∆ p +1 ,q +1 . By standard principle bundle theory, ψ thencorresponds to a e G − equivariant smooth map b ψ : Q → ∆ p +1 ,q +1 . As ψ isparallel, the image O := b ψ (cid:16) Q (cid:17) ⊂ ∆ p +1 ,q +1 is a orbit wrt. the e G -action, calledthe e G -type of ψ . We now bring into play that ∇ nc is induced by (cid:0) Q , e ω nc (cid:1) :Let x ∈ M . We define the e P − type of x wrt. ψ to be the e P − orbit b ψ (cid:0) Q (cid:1) ⊂O ⊂ ∆ p +1 ,q +1 which may change over x ∈ M . M then decmposes into a disjointunion according to e P − types, each of which is an initial submanifold of M . ThenProposition 2.7 from [20] applied to our setting immediatly yields the following: Proposition 4.1. Let ( M p,q , c ) be a conformal spin manifold and let ψ be aparallel spin tractor on (cid:0) Q → M, e ω nc (cid:1) . For given g ∈ c denote by ϕ ∈ Γ( S g ) the corresponding twistor spinor. Let x ∈ M . Then there is a parallel spin ractor φ on the homogeneous model (cid:16) e G → e G/ e P = b Q p,q , ω MC (cid:17) for which x ′ := e e P ∈ e G/ e P has the same e P − type wrt. φ that x has wrt. ψ . Further, let ϕ ′ correspond to φ via a conformally flat metric g St on b Q p,q . Then there areopen neighborhoods N of x in M and N ′ of x ′ in b Q p,q and a diffeomorphism Φ : N → N ′ such that Φ( x ) = x ′ and Φ ( Z ϕ ∩ N ) = Z ϕ ′ ∩ N ′ . As locally all possible shapes of the zero set already show up in the homo-geneous model, we are led to study the zeroes of twistor spinors on b C p,q . Using(17) we identify b Q p,q with the product S p × S q ⊂ R p +1 ,q +1 equipped with theconformally flat metric g St := − g S p + g S q . We follow [6] in order to constructall twistor spinors on b C p,q . We decompose every x ∈ R n +2 ∼ = R p +1 × R q +1 into x = ( x , x ). There is a natural, globally defined orthonormal frame field onthe normal bundle N b Q p,q , given by ζ ( x ) = ( x , 0) and ζ n +1 ( x ) = (0 , x ) for x ∈ b Q p,q . The spin structure on b C p,q is then naturally induced by a standardspin structure on R p +1 ,q +1 , and the spinor bundles are related by S R p +1 ,q +1 | b Q p,q ∼ = Ann ( ζ + ζ n +1 ) | {z } ∼ = S b Qp,q,g ⊕ Ann ( ζ − ζ n +1 ) | {z } ∼ = S b Qp,q,g . Wrt. this splitting, every spinor ϕ on R p +1 ,q +1 decomposes into ϕ = ϕ + ϕ . Forgiven v ∈ ∆ p +1 ,q +1 \{ } we let ϕ v ( x ) := x · v for x ∈ R p +1 ,q +1 , yielding a twistorspinor on R p +1 ,q +1 . Using the relation between the spinor derivatives ∇ R p +1 ,q +1 and ∇ b Q p,q it is straightforward to calculate that the induced spinor ϕ v, is atwistor spinor on (cid:16) b Q p,q , g (cid:17) (with ϕ v, ≡ Proposition 4.2. Let ϕ := ϕ v, be a twistor spinor on ( b Q p,q , g = g St ) , inducedby a twistor spinor ϕ v on R p +1 ,q +1 as explained above. Suppose that there is x ∈ Z ϕ . Then it holds that Z ϕ = exp x ( ker D g ϕ ( x )) or Z ϕ = { x, − x } . Proof. First, one shows using the formulas in [6] that D g ϕ v, ( y ) = n ( − v + ζ · y · v ) for all y ∈ b Q p,q . In particular, x ∈ Z ϕ v, implies that ker D g ϕ ( x ) = { t ∈ T x b Q p,q | t · v = 0 } . Now let b ∈ T x b Q p,q \{ } with g x ( b, b ) = h b, b i p +1 ,q +1 =0. One checks that the geodesic through x in direction b is given by δ b ( t ) =cos( t || b || ) · x + sin( t || b || ) · b || b || with || · || being the standard Euclidean normon R p +1 . If now additionally b · v = 0, we have that δ b (1) · v = 0 as x ∈ Z ϕ ,i.e. x · v = 0. This shows the ⊃ direction. On the other hand, suppose that y ∈ Z ϕ . As y · v = x · v = 0, it follows that 0 = ( yx + xy ) · v = − h x, y i p +1 ,q +1 v .Since h y i , y i i = 1 for i = 1 , 2, we find α i ∈ [0; π ] and d ∈ R p +1 , d ∈ R q +1 with h x i , d i i = 0 and || d || = || d || = 1 such that y i = cos( α i ) · x i + sin( α i ) · d i for i = 1 , 2. The condition h x , y i = h x , y i then leads to α = α = α . Thus, y = cos( α ) · x + sin( α ) · d d = d + d ∈ T x b Q p,q . If sin( α ) = 0, we conclude that d · v = 0, andthus d ∈ ker D g ϕ ( x ). As moreover || d || = 1, we see that there is t ∈ R with y = x · cos( t || d || ) + d || d || · sin( t || d || ) = δ d ( t ) = δ td (1). If sin( α ) = 0, we haveeither that y = x where the statement is trivial or y = − x . If ker D g ϕ ( x ) isnontrivial in this situation, we may choose arbitrary d ∈ ker D g ϕ ( x ) \{ } fora geodesic δ d joining x and − x . Otherwise ker D g ϕ ( x ) = 0 and the situation Z ϕ = { x, − x } occurs.The proof further yields the following for the flat model: Let x ′ ∈ Z ϕ ′ and suppose that for some w ∈ T x ′ b Q p,q ∩ W , where W := { w ∈ T x ′ b Q p,q | p h w, w i n +2 < π } it holds that y = exp x ′ ( w ) = δ w (1) ∈ Z ϕ ′ . As h y, y i = 0it follows that w = 0 and w = 0, and the geodesic δ w is thus given by δ w ( t ) = cos( t || w || ) · x ′ + sin( t || w || ) · w || w || + cos( t || w || ) · x ′ + sin( t || w || ) · w || w || . Now x ′ , y ∈ Z ϕ ′ implies that h x ′ , δ w (1) i p +1 ,q +1 = 0 which yields thatcos ( || w || ) = cos ( || w || ). However, w ∈ W implies that || w || = || w || . Con-sequently, h w, w i p +1 ,q +1 = 0. Now y · v = x · v = 0 leads to w · v = 0 as inthe proof of the previous Proposition. This shows that w ∈ ker D g ϕ ′ ( x ′ ) . Inthe notation of Proposition 4.1 we can therefore choose N ′ = exp x ′ ( V ) to be asufficiently small normal neighborhood of x ′ for some open neighborhood V of0 in T x ′ b Q p,q with V ⊂ W , and get thatΦ( Z ϕ ∩ N ) = Z ϕ ′ ∩ N ′ = exp x ′ (ker D g St ϕ ′ ( x ′ ) ∩ V ) . (36)We now return to general twistor spinors on ( M p,q , c ). We claim that in thenotation of Proposition 4.1 for the zero x ∈ Z ϕ it holds thatdim ker D g ϕ ( x ) = dim ker D g St ( x ′ ) . (37)Indeed, in the notation of Proposition 4.1 and subsection 2.3 we have that ψ ( x ) = [ σ g ( l ) , e − w ] ⇒ D g ϕ ( x ) = [ l, α ( e − w )] φ ( x ′ ) = [ σ g St ( l ′ ) , e − w ′ ] ⇒ D g St ϕ ′ ( x ′ ) = [ l ′ , α ( e − w ′ )]for spinors w, w ′ ∈ ∆ p +1 ,q +1 . As the e P − types coincide, there is e p ∈ e P such that e p · ( e − w ) = e − · w ′ . (38)We therefore invetigate the e P -action on Ann ( e − ) ⊂ ∆ p +1 ,q +1 more closely.Consider the 2-fold covering λ : Spin ( p + 1 , q + 1) → SO ( p + 1 , q + 1) which isexplicitly given by λ ( u )( x ) = u · x · u − (cf. [23]), i.e. e p · x = λ ( e p )( x ) · e p. (39)Now we can find a ∈ R + , A ∈ O ( p, q ) and v ∈ ( R ) ∗ such that wrt. the splitting R p +1 ,q +1 ∼ = R e − ⊕ R p,q ⊕ R e + we have that λ ( e p ) = a − v − a h v, v i p,q A − aAv ♯ a . Itis then a straightforward calculation using (39) and the formulas for the actionof x ∈ R n +1 on Ann ( e − ) ⊕ Ann ( e + ) that (wrt. appropriate bases) e p acts as (cid:18) X Y aX (cid:19) on Ann ( e − ) ⊕ Ann ( e + ) for some X ∈ GL (∆ p,q ) with µ ( Ax ) · X = X · µ ( x ) (40)24or x ∈ R p,q , where we identify Ann ( e ± ) ∼ = ∆ p,q as explained in (7). (38) and(40) then imply that A (ker e − w ) = ker e − w ′ which proves (37).Another interesting observation is that the quantity ker D g ϕ ( x ) does not dependon the zero x ∈ Z ϕ . One way to see this is the structure of the parallel tractorform α p +1 ψ . We have already observed that α p +1 ψ ( x ) = d · s ♭ + ( x ) ∧ α pD g ϕ ( x ) for x ∈ Z ϕ . Lemma 2.2 then yields that dim ker D g ϕ ( x ) = dim ker ψ ( x ) − x ∈ Z ϕ as ψ is parallel. The zero set Z ϕ now turns out to be an embedded submanifold of M : Let x ∈ Z ϕ be arbitrary. Inthe setting of Proposition 4.1 and (36) we choose neighborhoods N and N ′ wherewe may assume that N ′ = exp x ′ ( V ) is a normal neighborhood of x ′ as in (36).We then consider e Φ := (exp x ′ ) − | V ◦ Φ : N → V . Propositions 4.1 and (36) yieldthat e Φ( Z ϕ ∩ N ) = ker D g ϕ ′ ( x ′ ) ∩ V . We may compose this map with a linearisomorphism A x ′ : T x ′ b Q p,q → R n satisfying A x ′ (ker D g ϕ ′ ( x ′ )) = R k × { } , andin this way we obtain a submanifold chart for Z ϕ around x . This submanifoldis totally lightlike, since for every curve γ with Im γ ⊂ Z ϕ the twistor equationyields that γ ′ ( t ) · D g ϕ ( γ ( t )) = 0. As D g ϕ ( γ ( t )) = 0, we have that γ is isotropic.In addition, Lemma 3.4.1 from [6] says in our notation that for every x ∈ Z ϕ one has that exp x (ker D g ϕ ( x ) ∩ D x ) ⊂ Z ϕ , where D x is the maximal domain ofdefinition for the exponential map at x . For dimensional reasons, one then hasthat exp x (ker D g ϕ ( x ) ∩ V ) is an open submanifold of the embedded submanifold Z ϕ ∩ U ′ for appropriate neighborhoods V of 0 ∈ T x M and U ′ of x ∈ M . Thisyields (35) for arbitrary dimensions. We summarize our observations: Theorem 4.3. Let ϕ ∈ Γ( S g ) be a twistor spinor with zero. Then the zero set Z ϕ is an embedded totally lightlike, totally geodesic submanifold of dimensionker D g ϕ ( x ) , where the last quantity does not depend on the choice of x ∈ Z ϕ .Moreover, for every x ∈ Z ϕ there are open neighborhoods U of x in M and V of in T x M such that Z ϕ ∩ U = exp x ( ker D g ϕ ( x ) ∩ V ) . (41)More loosely speaking, the connected components of the zero set consisteither of an isolated point or of the image of a null-geodesic or of a totallynull-plane etc. A mixture of two of these geometric objects cannot occur as thezero set of one single twistor spinor. The whole local geometry of the zero setis encoded in the quantity ker D g ϕ ( x ). In case of a Ricci-parallel metric in theconformal class one has stronger results about the shape of the set V appearingin (41) as explained in [6].We next show that the conformal class naturally induces a projective struc-ture on the zero set of a twistor spinor. Recall that two connections ∇ and b ∇ on a manifold N are called projectively equivalent iff there exists a 1-formΥ ∈ Ω ( N ) such that b ∇ X Y = ∇ X Y + Υ( Y ) X + Υ( X ) Y ∀ X, Y ∈ X ( N ) . Moreover, it does not depend on the chosen metric in the conformal class as can be seendirectly from the transformation formulas. 25 more geometric interpretation is that two linear connections with the sametorsion are projectively equivalent if and only if they admit the same geodesicsas unparametrized curves. A projective structure on N is an equivalence classof connections. Proposition 4.4. Let ϕ ∈ Γ( S g ) be a twistor spinor with Z ϕ = ∅ on ( M, c ) .Then for every g ∈ c the Levi Civita connection ∇ g descends to a torsion-free linear connection ∇ on Z ϕ . If g and e g are conformally equivalent, theinduced connections ∇ and e ∇ are projectively equivalent, i.e., there is a naturalconstruction ϕ on ( M, c ) → ( Z ϕ , [ ∇ ]) from conformal structures and a twistor spinor with zero to torsion-free projec-tive structures on the zero set. Proof. It follows directly from (41) that for x ∈ Z ϕ the tangent space to thezero set is given by T x Z ϕ = ker D g ϕ ( x ) ⊂ T x M . In particular, ker D g ϕ ( x )does not depend on the choice of g ∈ c . For given X, Y ∈ X ( Z ϕ ) we thenset ∇ X Y ( x ) := (cid:0) ∇ g dt ( Y ◦ γ ) (cid:1) (0), where γ : ( − ǫ, ǫ ) → Z ϕ ⊂ M is the maximalgeodesic in M with γ (0) = x and γ ′ (0) = X ( x ). ∇ g dt is the induced derivativealong γ . We have to check that ∇ X Y ∈ X ( Z ϕ ). As ( Y · D g ϕ ) ◦ γ = 0, it followsthat0 = ∇ S g dt (( Y · D g ϕ ) ◦ γ ) = (( ∇ X Y ) · D g ϕ ) ◦ γ + ( Y ◦ γ ) · ∇ S g dt ( D g ϕ ◦ γ ) | {z } = n K g ( X ◦ γ ) · ( ϕ ◦ γ )=0 Consequently, ∇ X Y ( x ) ∈ ker D g ϕ ( x ) = T x Z ϕ . Clearly, this holds for everymetric in the conformal class. The fact that ∇ is torsion-free follows directlyfrom the corresponding property of ∇ g . Now let e g = e σ g be a conformallyequivalent metric. There is the well-known transformation formula ∇ e gX Y = ∇ gX Y + X ( σ ) Y + Y ( σ ) X − g ( X, Y )grad g σ. As for x ∈ Z ϕ the space ker D g ϕ ( x ) is totally lightlike, it is a direct consequenceof the definition of ∇ that for all X, Y ∈ X ( Z ϕ ) we have e ∇ X Y = ∇ X Y + d b σ ( X ) · Y + d b σ ( Y ) · X, where b σ := σ | Z ϕ . It follows that ∇ and b ∇ are projectively equivalent.Note that as a direct consequence of the definitions it holds that i ∗ R g = R ∇ ,where i : Z ϕ ֒ → M and R ∇ is the curvature tensor of the connection ∇ . In par-ticular, if c admits a flat representative then so does [ ∇ ].It is now natural to ask what can be said about the spinor and associatedlocal geometries off the zero set if one knows the (local) structure of Z ϕ . Inthe Riemannian case, a twistor spinor is always parallel on a Ricci-flat spaceoff the zero set. For Lorentzian signature F. Leitner showed that in case ofan isolated zero the Lorentzian metric is locally off the zero set isometric to a26tatic monopole − dt + h where h is a Riemannian Ricci-flat metric with parallelspinor. If the zero is not isolated, then off the zero set the space is locally con-formally equivalent to a Brinkmann space with parallel spinor. Our results fromsection 3 show that in every signature the spinor is locally equivalent to a par-allel spinor off the zero set. In fact, let ψ ∈ Γ( S ( M )) be a parallel spin tractorwith associated twistor spinor ϕ ∈ Γ( S g ) for g ∈ c . Let x ∈ Z ϕ . It then holds at x that ψ ( x ) = [[ σ g ( l ) , e ] , e − w ] for some w ∈ ∆ p +1 ,q +1 . However, thismeans that s − ( x ) ∈ ker ψ ( x ). In particular, since the dimension of this kernelis constant over M , Proposition 3.3 applies and yields the next statement. Theorem 4.5. Let ϕ ∈ Γ( M, S g ) be a twistor spinor admitting a zero. Thenthere is an open dense subset f M ⊂ M with Z ϕ ⊂ M \ f M such that for every x ∈ f M there is an open neighborhood U x ⊂ f M such that ϕ can be rescaled to aparallel spinor on U x . Our discussion from section 3 implies further consequences relating the shapeof the zero set to local geometric structures off the zero set: Proposition 4.6. Let ϕ ∈ Γ( S g ) be a twistor spinor with nonempty zero set Z ϕ . Then there is a set of singular points sing ( ϕ ) ⊂ M with Z ϕ ⊂ sing ( ϕ ) such that the following holds: There is ≤ k ≤ p such that Z ϕ is an embed-ded k − dimensional totally lightlike submanifold. On M \ sing ( ϕ ) , the spinor islocally conformally equivalent to a parallel spinor and the corresponding met-ric holonomy representation fixes a lotally lightlike subspace of dimension k . If k = p or k = p − there is even a fixed totally lightlike k − form. If k = 0 , i.e.the zero is isolated, there is locally a Ricci-flat metric in the conformal class. For the proof we observe first that for the number k appearing in the Proposi-tion it holds that k = dim ker D g ϕ ( x ), where x ∈ Z ϕ . It follows in the notationof Lemma 2.2 and Proposition 3.5 that k = dim H ψ − L ϕ | U − L ϕ is parallel and totally lightlike (wrt. to a suitable metric in the conformal class).In case p = 2, the discussion from the end of section 3 together with the laststatement directly leads to the following relation between the shape of the zeroset and local geometries: Proposition 4.7. Let ϕ ∈ Γ( S g ) be a twistor spinor with zero on ( M ,n − , g ) .Then exactly one of the following cases occurs: Z ϕ consists locally of totally lightlike planes. In this case, the spinor islocally equivalent to a parallel spinor off the zero set and gives rise to aparallel totally lightlike 2-form. Z ϕ consists of isolated images of lightlike geodesics. In this case, the spinoris off the zero set locally conformally equivalent to a parallel spinor on aBrinkmann space. Z ϕ consists of isolated points. In this case there is for each point off thezero set an open neighborhood and a local metric in the conformal classsuch that the resulting space is isometric to a product ( U , g ) × ( U , g ) where the first factor is Ricci-flat pseudo-Kaehler and the second factor(which might be trivial) is Riemannian Ricci-flat. Both factors admit aparallel spinor. . Low dimensions One important application of the main statement, Proposition 3.3, is thecase that ψ is a pure spinor, i.e. dim H ψ = p + 1. It immediatly follows withLemma 2.2 that α p +1 ψ is totally lightlike in this case, and thus Proposition 3.3and its Corollaries apply (if p = 0) yielding important consequences in smallsplit-signatures due to the following mainly algebraic observations concerningthe orbit structure of ∆ p,q under the Spin + ( p, q )-action as discussed in [12]:1. In signatures (2 , 2) and (3 , 3) every real half-spinor ϕ ∈ ∆ R , ± m,m \{ } is pure.Consequently, every real twistor half-spinor on ( M , , g ) without zeroesis double pure , by which we mean that both ϕ and the associated spintractor ψ are pointwise pure.2. In signature (3 , 2) every nonzero real spinor is pure. In signature (4 , ψ ∈ ∆ R , is pure iff it is nonzero and h ψ, ψ i ∆ R , = 0. Withthe scalar product formula (8) one concludes that a twistor spinor ϕ ∈ Γ( M , , S g ) is double pure iff ϕ admits no zeroes and h ϕ, D g ϕ i ≡ ϕ ∈ Γ( M , , S g ) without zeroesis double pure iff h ϕ, D g ϕ i ≡ Proposition 5.1. Real twistor half-spinors in signature (2 , without zeroesand real twistor (half-)spinors without zeroes in signatures (3 , and (3 , sat-isfying that h ϕ, D g ϕ i ≡ are locally conformally equivalent to parallel spinors(off a singular set). Their associated distributions ker ϕ ⊂ T M are integrable(off a singular set). Moreover, in the mentioned cases, the locally parallel spinor is real andpure at every point and the considered signatures are split signatures. In viewof this, (33) gives a local normal form for the metric. Consequently, one hasa complete local description of the geometries admitting non-generic twistorspinors in signatures (3 , 2) and (3 , , 2) and (3 , 3) from[14], i.e. h ϕ, D g ϕ i 6 = 0, where the associated distribution ker ϕ turns out to begeneric. Moreover, since the mentioned non-generic twistor spinors are pure,Proposition 4.6 applies, yielding the following about the zero set structure: Proposition 5.2. Let ϕ ∈ Γ( S g ) be a real twistor (half-)spinor in signature (2 , or (3 , . Then the zero set Z ϕ -if nonempty- consists locally of totallylightlike planes. For a real twistor half-spinor with zero in signature (3 , thezero set is locally an embedded 3-dimensional totally lightlike submanifold.5.2. Twistor spinors in signature (4,3) We start with algebraic observations: As known from [8], Spin + (4 , 3) actstransitive on each of the level sets M c := { v ∈ ∆ R , | h v, v i = c } for c = 0 and M \{ } is precisely the space of pure spinors. Moreover, it holds for τ ∈ M c =0 that ker τ = { } and h X · τ, τ i = 0 for all X ∈ R , . We use this in order todescribe the orbit structure of ∆ R , . According to [12] each N c := { v ∈ ∆ R , |h v, v i = c } constitutes for c = 0 a single orbit. However, N c \{ } decomposesinto at least 2 orbits wrt. the Spin + (5 , − action.28 emma 5.3. It holds that ker v = { } for all v ∈ N ⊂ ∆ R , . Proof. We realise ∆ , = Ann ( e − ) ⊕ Ann ( e + ) ∼ = ∆ , ⊕ ∆ , as described in(5). With respect to this identification, we write v = (cid:18) τχ (cid:19) for τ, χ ∈ ∆ R , . Itfollows that a vector x = αe − + y + βe + acts as x · v = (cid:18) y · τ + αχ − y · χ − βτ (cid:19) . (42)The scalar product formula (8) implies that v ∈ N ⇔ h τ, χ i ∆ R , = 0. If oneof τ, χ is trivial, the claim is obvious. Otherwise, we distinguish two cases:Suppose that h χ, χ i = h τ, τ i = 0. In this case τ and χ are pure spinors withtrivial pairing. It is a classical fact (cf. [25]) that in this case ker τ ∩ ker χ = { } .(42) implies that each nonzero element of this intersection lies in ker v . Thusit remains to consider the case where (wlog.) h τ, τ i 6 = 0. If χ / ∈ R , · τ , wewould for dimensional reasons have that ∆ R , = R , · τ ⊕ R χ which impliesthat h ∆ R , , τ i = 0 in contradiction to τ = 0. Therefore, we find y ∈ R , with y · τ = χ . It follows that y · χ + || y || τ = 0 yielding that e − − y − || y || e + ∈ ker v .Moreover, one calculates using orbit representatives that ker v = 0 if v ∈ N c =0 . Proposition 5.4. Let ϕ be a real twistor spinor on a conformal space ( M , , c ) .Let g ∈ c . Then exactly one of the following cases occurs: It is h ϕ, D g ϕ i ≡ . In this case the spinor is locally equivalent to a parallelspinor off a singular set. One either has locally a parallel pure spinor fieldwith a normal form of the metric given by 33 or the spinor is locally aparallel spinor on a space whose holonomy representation is contained in G . It is h ϕ, D g ϕ i 6 = 0 . Up to singular points there is locally around each pointan Einstein metric with nonzero scalar curvature in the conformal class.The twistor spinor cannot be rescaled to a parallel spinor but decomposesinto the sum of two Killing spinors. Proof. Let ψ ∈ Γ( S ) be the parallel spin tractor associated to ϕ . The scalarproduct formula (8) shows that h ψ, ψ i = const. · h ϕ, D g ϕ i . Thus, the previousalgebraic discussion yields that h ϕ, D g ϕ i ≡ ψ admits a nontrivial kernel H ψ .It follows that ϕ is locally parallel by section 3. Thus, ϕ is locally of constant Spin + (4 , 3) orbit type. This means it is either pure or has trivial kernel. Thestabilizer of a spinor v ∈ M c =0 ⊂ ∆ R , is isomorphic to a copy of the exceptionalgroup G (cf. [8]). This proves the first part. h ϕ, D g ϕ i 6 = 0 is equivalent to h ψ, ψ i 6 = 0. It is known from [12] that for every v ∈ N c =0 ⊂ ∆ R , one has λ ( Stab v Spin + (5 , ∼ = Spin + (4 , ⊂ SO + (4 , ⊂ SO + (5 , . (43)The conformal holonomy representation thus stabilizes a non-null vector yield-ing an Einstein metric on an open, dense subset. It is a classical fact (cf. [9])that on an Einstein space every twistor spinor decomposes into the sum of twoKilling spinors. (cid:3) , 3) arewell understood (cf. [8]). Moreover, since dim ker v ∈ { , } for all v ∈ ∆ , \{ } one has that the zero set of a real twistor spinor in signature (4 , 3) with zero con-sist locally either of isolated points or of 3-dimensional totally lightlike planes.In the first case one has locally G -holonomy off the zero set, in the secondcase one locally has a parallel pure spinor off the zero set as follows from thedimension of L ϕ and the proof of the last Proposition. We again start with some algebraic observations. The complex spinor mod-ule ∆ C , admits a real structure commuting with Clifford multiplication givingthe real module ∆ R , . The same applies to ∆ R , . We consider the map i : ∆ R , | Spin + (5 , → ∆ R , , v v being an isomorphism of Spin + (5 , R , as Spin + (5 , − module and it holds that h i ( v ) , i ( v ) i ∆ R , = h v, v i ∆ R , . Let v ∈ ∆ R , with h v, v i 6 = 0. (43) then yields that λ ( Stab v Spin + (5 , ⊂ λ ( Stab v Spin + (5 , ∼ = Spin + (4 , ⊂ SO + (4 , ⊂ SO + (5 , . However, as also λ ( Stab v Spin + (5 , ⊂ SO + (5 , 3) we see that in fact up toconjugation λ ( Stab v Spin + (5 , ⊂ SO + (4 , ⊂ SO + (5 , R , .If h v, v i = 0 we cannot make a general statement about ker v . There is thesubcase of pure spinors but it is also possible for v to have trivial kernel. Thereis no complete orbit classification available.In complete analogy to the second case of Proposition 5.4 one now shows thefollowing: Proposition 5.5. Let ϕ ∈ Γ( M , , S g R ) be a twistor spinor with h ϕ, D g ϕ i 6 = 0 .Then there is on an open, dense subset an Einstein metric with nonzero scalarcurvature in the conformal class. Moreover, the spinor cannot be resclaed to aparallel spinor. We cannot completely describe the case h ϕ, D g ϕ i = 0. There is a subcasein which h ϕ, ϕ i ≡ ≡ h D g ϕ, D g ϕ i in which ϕ is locally conformally equivalentto a parallel spinor. This follows since the assumptions gurantee the existenceof 0 = X ∈ T M with X · ϕ = X · D g ϕ = 0 which implies that H ψ = 0. Thereis another subcase when ϕ is parallel and L ϕ = { } where one has a Ricci-flatpseudo-Kaehler metric in the conformal class (cf. the discussion of the p = 2case at the end of section 3).For a nonzero spinor v ∈ ∆ R , it holds that dim ker v ∈ { , } . Thus, thezero set of a real twistor spinor with zeroes in signature (4 , 2) consists either of Note that in contrast to the complex case, ∆ R , is irreducible as Spin + (4 , − module.Hence there are no real half-spinors in signature (4 , Z ϕ consists of isolated points and the geometry off the zero set isRicci-flat pseudo-Kaehler. 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