Twistor Theory for co-CR quaternionic manifolds and related structures
aa r X i v : . [ m a t h . DG ] J un TWISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDSAND RELATED STRUCTURES
STEFANO MARCHIAFAVA AND RADU PANTILIE
Abstract
In a general and non metrical framework, we introduce the class of co-CR quaternionicmanifolds, which contains the class of quaternionic manifolds, whilst in dimensionthree it particularizes to give the Einstein–Weyl spaces. We show that these manifoldshave a rich natural Twistor Theory and, along the way, we obtain a heaven spaceconstruction for quaternionic-K¨ahler manifolds.
Introduction
Over any three-dimensional conformal manifold M , endowed with a conformal con-nection, there is a sphere bundle Z endowed with a natural CR structure [14] . Further-more, if M is real analytic then [13] the CR structure of Z is induced by a germ uniqueembedding of Z into a three-dimensional complex manifold e Z which is the twistor spaceof an anti-self-dual manifold f M ; accordingly, M is a hypersurface in f M , and the latteris called the heaven space (due to [18] ; cf. [14] ) of M (endowed with the given confor-mal connection).In [17] (see Section 2 ), we obtained the higher dimensional versions of these con-structions by introducing the notion of CR quaternionic manifold . Thus, the genericsubmanifolds of codimensions at most 2 k − k , are endowed with natural CR quaternionic structures. Moreover, assuming real-analyticity, any CR quaternionic manifold is obtained this way through a germ uniqueembedding into a quaternionic manifold [17] .Returning to the three-dimensional case, by [8] , if the inclusion of M into f M admitsa retraction which is twistorial (that is, its fibres correspond to a (one-dimensional)holomorphic foliation on e Z ) then the connection used to construct the CR structureon Z may be assumed to be a Weyl connection; moreover, there is a natural correspon-dence between such retractions and Einstein-Weyl connections on M . Furthermore,any Einstein–Weyl connection ∇ on M determines a complex surface Z ∇ and a holo-morphic submersion from e Z onto it; then Z ∇ is the twistor space of ( M, ∇ ) [8] .Furthermore, the correspondence between Einstein–Weyl spaces and their twistor Mathematics Subject Classification.
Primary 53C28, Secondary 53C26.R.P. acknowledges that this work was partially supported by the Visiting Professors Programme ofGNSAGA-INDAM of C.N.R. (Italy).S.M. acknowledges that this work was done under the program of GNSAGA-INDAM of C.N.R. andPRIN07 ”Geometria Riemanniana e strutture differenziabili” of MIUR (Italy). S. MARCHIAFAVA AND R. PANTILIE spaces is similar to the correspondence between anti-self-dual manifolds and theirtwistor spaces (see, also, [16] ). Also, from the point of view of Twistor Theory, theanti-self-dual manifolds are just four-dimensional quaternionic manifolds (see [9] ).This raises the obvious question: is there a natural class of manifolds, endowed withtwistorial structures, which contains both the quaternionic manifolds and the three-dimensional Einstein–Weyl spaces ?In this paper, where the adopted point of view is essentially non metrical, we answerin the affirmative to this question by introducing, in a general framework, the notion of co-CR quaternionic manifolds and we initiate the study of their twistorial properties.This notion is based on the (co-)CR quaternionic vector spaces which were introducedand classified in [17] (see Section 1 , and, also, Appendix A for an alternative definition)and, up to the integrability, it is dual to the notion of
CR quaternionic manifolds .An interesting situation to consider is when a manifold may be endowed with botha CR quaternionic and a co-CR quaternionic structure which are compatible . Thisgives the notion of f -quaternionic manifold, which has two twistor spaces. The sim-plest example is provided by the three-dimensional Einstein–Weyl spaces, endowedwith the twistorial structures of [14] and [8] , respectively; furthermore, the abovementioned twistorial retraction admits a natural generalization to the f -quaternionicmanifolds (Corollary 4.5 ). Also, the quaternionic manifolds may be characterised as f -quaternionic manifolds for which the two twistor spaces coincide.Other examples of f -quaternionic manifolds are the Grassmannnian Gr +3 ( l + 3 , R ) oforiented three-dimensional vector subspaces of R l +3 and the flag manifold Gr (2 n +2 , C )of two-dimensional complex vector subspaces of C n +2 (= H n +1 ) which are isotropicwith respect to the underlying complex symplectic structure of C n +2 , ( l, n ≥
1) . Thetwistor spaces of their underlying co-CR quaternionic structures are the hyperquadric Q l +1 of isotropic one-dimensional complex vector subspaces of C l +3 and Gr (2 n + 2 , C )itself, respectively. Also, their heaven spaces are the Wolf spaces Gr +4 ( l + 4 , R ) andGr (2 n + 2 , C ) , respectively (see Examples 4.6 and 4.7 , for details). Another naturalclass of f -quaternionic manifolds is described in Example 4.8 .The notion of almost f -quaternionic manifold appears, also, in a different form, in[10] . However, there it is not considered any adequate integrability condition. Also, in[5] , [1] and [4] are considered, under particular dimensional assumptions and/or in ametrical framework, particular classes of almost f -quaternionic manifolds.Let N be the heaven space of a real analytic f -quaternionic manifold M , withdim N = dim M + 1 . If the connection of the f -quaternionic structure on M is in-duced by a torsion free connection on M then the twistor space of N is endowed with anatural holomorphic distribution of codimension one which is transversal to the twistorlines corresponding to the points of N \ M . Furthermore, this construction also worksif, more generally, M is a real analytic CR quaternionic manifold which is a q-umbilical hypersurface of its heaven space N . Then, under a non-degeneracy condition, this dis-tribution defines a holomorphic contact structure on the twistor space of N . Therefore, WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 3 according to [15] , it determines a quaternionic-K¨ahler structure on N \ M (cf. [5] ,[7] ).It is well known (see, for example, [20] and the references therein) that the three-dimensional Einstein–Weyl spaces are one of the basic ingredients in constructions ofanti-self-dual (Einstein) manifolds. One of the aims of this paper is to give a first indica-tion that the study of co-CR quaternionic manifolds will lead to a better understandingof quaternionic(-K¨ahler) manifolds.1. Brief review of (co-)CR quaternionic vector spaces
The group of automorphisms of the (unital) associative algebra of quaternions H isSO(3) acting trivially on R ( ⊆ H ) and canonically on Im H .A linear hypercomplex structure on a (real) vector space E is a morphism of as-sociative algebras σ : H → End( E ) . A linear quaternionic structure on E is anequivalence class of linear hypercomplex structures, where two linear hypercomplexstructures σ , σ : H → End( E ) are equivalent if there exists a ∈ SO(3) such that σ = σ ◦ a . A hypercomplex/quaternionic vector space is a vector space endowed witha linear hypercomplex/quaternionic structure (see [2] , [9] ).If σ : H → End( E ) is a linear hypercomplex structure on a vector space E thenthe unit sphere Z in σ (Im H ) ⊆ End( E ) is the corresponding space of admissible linearcomplex structures . Obviously, Z depends only of the linear quaternionic structure de-termined by σ .Let E and E ′ be quaternionic vector spaces and let Z and Z ′ be the correspondingspaces of admissible linear complex structures. A linear map t : E → E ′ is quaternionic ,with respect to some function T : Z → Z ′ , if t ◦ J = T ( J ) ◦ t , for any J ∈ Z (see [2] ).If, further, t = 0 then T is unique and an orientation preserving isometry (see [9] ).The basic example of a quaternionic vector space is H k endowed with the linearquaternionic structure given by its canonical (left) H -module structure. Moreover,for any quaternionic vector space of dimension 4 k there exists a quaternionic linearisomorphism from it onto H k . The group of quaternionic linear automorphisms of H k is Sp(1) · GL( k, H ) acting on it by (cid:0) ± ( a, A ) , x (cid:1) axA − , for any ± ( a, A ) ∈ Sp(1) · GL( k, H ) and x ∈ H k . If we restrict this action to GL( k, H ) then we obtain thegroup of hypercomplex linear automorphisms of H k .If σ : H → End( E ) is a linear hypercomplex structure then σ ∗ : H → End( E ∗ ) ,where σ ∗ ( q ) is the transpose of σ ( q ) , ( q ∈ H ) , is the dual linear hypercomplex structure .Accordingly, we define the dual of a linear quaternionic structure. Definition 1.1 ( [17] ) . A linear co-CR quaternionic structure on a vector space U is apair ( E, ρ ) , where E is a quaternionic vector space and ρ : E → U is a surjective linearmap such that ker ρ ∩ J (ker ρ ) = { } , for any admissible linear complex structure J on E .A co-CR quaternionic vector space is a vector space endowed with a linear co-CRquaternionic structure. S. MARCHIAFAVA AND R. PANTILIE
Dually, a
CR quaternionic vector space is a triple (
U, E, ι ) , where E is a quaternionicvector space and ι : U → E is an injective linear map such that im ρ + J (im ρ ) = E ,for any admissible linear complex structure J on E .A map t : ( U, E, ρ ) → ( U ′ , E ′ , ρ ′ ) between co-CR quaternionic vector spaces is co-CRquaternionic linear (with respect to some map T : Z → Z ′ ) if there exists a map e t : E → E ′ which is quaternionic linear (with respect to T ) such that t ◦ ρ = ρ ′ ◦ e t .By duality, we also have the notion of CR quaternionic linear map .Note that, if (
U, E, ι ) is a CR quaternionic vector space then the inclusion ι : U → E is CR quaternionic linear. Dually, if ( U, E, ρ ) is a co-CR quaternionic vector space thenthe projection ρ : E → U is co-CR quaternionic linear.By working with pairs ( U, E ) , where E is a quaternionic vector space and U ⊆ E is areal vector subspace, we call (Ann U, E ∗ ) the dual pair of ( U, E ) , where the annihilatorAnn U is formed of those α ∈ E ∗ such that α | U = 0 .Any CR quaternionic vector space ( U, E, ι ) corresponds to the pair (im ι, E ) , whilstany co-CR quaternionic vector space (
U, E, ρ ) corresponds to the pair (ker ρ, E ) . Theseassociations define functors in the obvious way.To any pair (
U, E ) we associate a (coherent analytic) sheaf over Z as follows. Let E , be the holomorphic vector bundle over Z whose fibre over any J ∈ Z is the − ieigenspace of J . Let u : E , → Z × ( E/U ) C be the composition of the inclusion E , → Z × E C followed by the projection Z × E C → Z × ( E/U ) C . Definition 1.2 ( [19] ) . U = U − ⊕ U + is the sheaf of ( U, E ) , where U − = ker u and U + = coker u .If ( U, E ) corresponds to a (co-)CR quaternionic vector space then U is its holomorphicvector bundle, introduced in [17] . In fact, ( U, E ) corresponds to a co-CR quaternionicvector space if and only if U is a holomorphic vector bundle and U = U + . Dually,( U, E ) corresponds to a CR quaternionic vector space if and only if U = U − (note that, U − is a holomorphic vector bundle for any pair). See [19] for more information on thefunctor ( U, E )
7→ U .Here are the basic examples of (co-)CR quaternionic vector spaces.
Example 1.3 (cf. [17] ) .
1) Let V k , ( k ≥
1) , be the vector subspace of H k formed of allvectors of the form ( z , z + z j , z − z j , . . . ) , where z , . . . , z k are complex numbersand z k = ( − k z k . Then ( V k , H k ) corresponds to a co-CR quaternionic vector spaceand its holomorphic vector bundle is O (2 k ) . Hence, the dual pair is a CR quaternionicvector space and its holomorphic vector bundle is O ( − k ) .2) Let V ′ = { } and, for k ≥ V ′ k be the vector subspace of H k +1 formed of allvectors of the form ( z , z + z j , z − z j , . . . , z k − + z k j , − z k j) , where z , . . . , z k are complex numbers. Then ( V ′ k , H k +1 ) corresponds to a co-CR quaternionic vectorspace and its holomorphic vector bundle is 2 O (2 k + 1) . Hence, the dual pair is a CRquaternionic vector space and its holomorphic vector bundle is 2 O ( − k −
1) .
WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 5
Also, by [17] , any (co-)CR quaternionic vector space is isomorphic to a product,unique up to the order of factors, in which each factor is given by Example 1.3(1)or (2) .
Definition 1.4. A linear f -quaternionic structure on a vector space U is a pair ( E, V ) ,where E is a quaternionic vector space such that U, V ⊆ E , E = U ⊕ V and J ( V ) ⊆ U ,for any J ∈ Z .An f -quaternionic vector space is a vector space endowed with a linear f -quaternionicstructure.Let ( U, E, V ) be an f -quaternionic vector space; denote by ι : U → E the inclusionand by ρ : E → U the projection determined by the decomposition E = U ⊕ V .Then ( E, ι ) and (
E, ρ ) are linear CR-quaternionic and co-CR quaternionic struc-tures, respectively, which are compatible .The f -quaternionic linear maps are defined, accordingly, by using the compatiblelinear CR and co-CR quaternionic structures determining a linear f -quaternionic struc-ture.From any f -quaternionic vector space ( U, E, V ) , with dim E = 4 k , dim V = l , thereexists an f -quaternionic linear isomorphism onto (Im H ) l × H k − l (this follows, for ex-ample, from the classification of (co-)CR quaternionic vector spaces [17] ).We end this section with the description of the Lie group G of f -quaternionic linearisomorphisms of (Im H ) l × H m . For this, let ρ k : Sp(1) · GL( k, H ) → SO(3) be the Liegroup morphism defined by ρ k ( q · A ) = ± q , for any q · A ∈ Sp(1) · GL( k, H ) , ( k ≥
1) .Denote H = (cid:8) ( A, A ′ ) ∈ (cid:0) Sp(1) · GL( l, H ) (cid:1) × (cid:0) Sp(1) · GL( m, H ) (cid:1) | ρ l ( A ) = ρ m ( A ′ ) (cid:9) . Then H is a closed subgroup of Sp(1) · GL( l + m, H ) and G is the closed subgroup of H formed of those elements ( A, A ′ ) ∈ H such that A preserves R l ⊆ H l . This followsfrom the fact that there are no nontrivial f -quaternionic linear maps from Im H to H (and from H to Im H ). Now, the canonical basis of Im H induces a linear isomorphism(Im H ) l = (cid:0) R l (cid:1) and, therefore, an effective action σ of GL( l, R ) on (Im H ) l . We definean effective action of GL( l, R ) × (cid:0) Sp(1) · GL( m, H ) (cid:1) on (Im H ) l × H m by( A, q · B )( X, Y ) = (cid:0) q (cid:0) σ ( A )( X ) (cid:1) q − , q Y B − (cid:1) , for any A ∈ GL( l, R ) , q · B ∈ Sp(1) · GL( m, H ) , X ∈ (Im H ) l and Y ∈ H m . Proposition 1.5.
There exists an isomorphism of Lie groups G = GL( l, R ) × (cid:0) Sp(1) · GL( m, H ) (cid:1) , given by ( A, A ′ ) ( A | R l , A ′ ) , for any ( A, A ′ ) ∈ G .In particular, the group of f -quaternionic linear isomorphisms of (Im H ) l is isomor-phic to GL( l, R ) × SO(3) . Note that, the group of f -quaternionic linear isomorphisms of Im H is CO(3) . S. MARCHIAFAVA AND R. PANTILIE A few basic facts on CR quaternionic manifolds
In this section we recall, for the reader’s convenience, a few basic facts on CR quater-nionic manifolds (we refer to [17] for further details).A (smooth) bundle of associative algebras is a vector bundle whose typical fibre isa (finite-dimensional) associative algebra and whose structural group is the group ofautomorphisms of the typical fibre. Let A and B be bundles of associative algebras. Amorphism of vector bundles ρ : A → B is called a morphism of bundles of associativealgebras if ρ restricted to each fibre is a morphism of associative algebras.Recall that a quaternionic vector bundle over a manifold M is a real vector bundle E over M endowed with a pair ( A, ρ ) where A is a bundle of associative algebras, over M , with typical fibre H and ρ : A → End( E ) is a morphism of bundles of associativealgebras; we say that ( A, ρ ) is a linear quaternionic structure on E (see [6] ). Standardarguments (see [9] ) apply to show that a quaternionic vector bundle of (real) rank4 k is just a (real) vector bundle endowed with a reduction of its structural group toSp(1) · GL( k, H ) .If ( A, ρ ) defines a linear quaternionic structure on a vector bundle E then we denote Q = ρ (Im A ) , and by Z the sphere bundle of Q .Recall [22] (see [9] ) that, a manifold is almost quaternionic if and only if its tangentbundle is endowed with a linear quaternionic structure. Definition 2.1.
Let E be a quaternionic vector bundle on a manifold M and let ι : T M → E be an injective morphism of vector bundles. We say that ( E, ι ) is an almost CR quaternionic structure on M if ( E x , ι x ) is a linear CR quaternionic structureon T x M , for any x ∈ M .An almost CR quaternionic manifold is a manifold endowed with an almost CRquaternionic structure.On any almost CR quaternionic manifold ( M, E, ι ) for which E is endowed with aconnection ∇ , compatible with its linear quaternionic structure, there can be defineda natural almost twistorial structure, as follows. For any J ∈ Z , let B J ⊆ T C J Z be thehorizontal lift, with respect to ∇ , of ι − (cid:0) E J (cid:1) , where E J ⊆ E C π ( J ) is the eigenspace of J corresponding to − i . Define C J = B J ⊕ (ker d π ) , J , ( J ∈ Z ) . Then C is an almost CRstructure on Z and ( Z, M, π, C ) is the almost twistorial structure of ( M, E, ι, ∇ ). Definition 2.2. An (integrable almost) CR quaternionic structure on M is a triple( E, ι, ∇ ) , where ( E, ι ) is an almost CR quaternionic structure on M and ∇ is an al-most quaternionic connection of ( M, E, ι ) such that the almost twistorial structure of(
M, E, ι, ∇ ) is integrable (that is, C is integrable). Then ( M, E, ι, ∇ ) is a CR quater-nionic manifold and the CR manifold (
Z, C ) is its twistor space .A main source of CR quaternionic manifolds is provided by the submanifolds ofquaternionic manifolds.
WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 7
Definition 2.3.
Let (
M, E, ι, ∇ ) be a CR quaternionic manifold and let ( Z, C ) be itstwistor space. We say that ( M, E, ι, ∇ ) is realizable if M is an embedded submanifoldof a quaternionic manifold N such that E = T N | M , as quaternionic vector bundles,and C = T C Z ∩ ( T , Z N ) | M , where Z N is the twistor space of N .Then N is the heaven space of ( M, E, ι, ∇ ) .By [17, Corollary 5.4] , any real-analytic CR quaternionic manifold is realizable.3. Co-CR quaternionic manifolds An almost co-CR structure on a manifold M is a complex vector subbundle C of T C M such that C + C = T C M . An (integrable almost) co-CR structure is an almostco-CR structure whose space of sections is closed under the bracket.Note that, if ϕ : M → ( N, J ) is a submersion onto a complex manifold then(d ϕ ) − (cid:0) T , N (cid:1) is a co-CR structure on M ; moreover, any co-CR structure is, locally,of this form. Definition 3.1.
Let E be a quaternionic vector bundle on a manifold M and let ρ : E → T M be a surjective morphism of vector bundles. Then (
E, ρ ) is called an almost co-CR quaternionic structure , on M , if ( E x , ρ x ) is a linear co-CR quaternionicstructure on T x M , for any x ∈ M . If, further, E is a hypercomplex vector bundle then( E, ρ ) is called an almost hyper-co-CR structure on M . An almost co-CR quaternionicmanifold ( almost hyper-co-CR manifold ) is a manifold endowed with an almost co-CRquaternionic structure (almost hyper-co-CR structure).Any almost co-CR quaternionic (hyper-co-CR) structure ( E, ρ ) for which ρ is anisomorphism is an almost quaternionic (hypercomplex) structure. Example 3.2.
Let (
M, c ) be a three-dimensional conformal manifold and let L = (cid:0) Λ T M (cid:1) / be the line bundle of M . Then, E = L ⊕ T M is an oriented vector bun-dle of rank four endowed with a (linear) conformal structure such that L = ( T M ) ⊥ .Therefore E is a quaternionic vector bundle and ( M, E, ρ ) is an almost co-CR quater-nionic manifold, where ρ : E → T M is the projection. Moreover, any three-dimensionalalmost co-CR quaternionic manifold is obtained this way.Next, we are going to introduce a natural almost twistorial structure (see [16] forthe definition of almost twistorial structures) on any almost co-CR quaternionic mani-fold (
M, E, ρ ) for which E is endowed with a connection ∇ compatible with its linearquaternionic structure.For any J ∈ Z , let C J ⊆ T C J Z be the direct sum of (ker d π ) , J and the horizontallift, with respect to ∇ , of ρ ( E J ), where E J is the eigenspace of J corresponding to − i . Then C is an almost co-CR structure on Z and ( Z, M, π, C ) is the almost twistorialstructure of ( M, E, ρ, ∇ ).The following definition is motivated by [9, Remark 2.10(2) ] . S. MARCHIAFAVA AND R. PANTILIE
Definition 3.3. An (integrable almost) co-CR quaternionic manifold is an almostco-CR quaternionic manifold ( M, E, ρ ) endowed with a compatible connection ∇ on E such that the associated almost twistorial structure ( Z, M, π, C ) is integrable (that is, C is integrable). If, further, E is a hypercomplex vector bundle and the connectioninduced by ∇ on Z is trivial then ( M, E, ρ, ∇ ) is an (integrable almost) hyper-co-CRmanifold . Example 3.4.
Let (
M, c ) be a three-dimensional conformal manifold and let (
E, ρ ) bethe corresponding almost co-CR structure, where E = L ⊕ T M with L the line bundleof M . Let D be a Weyl connection on ( M, c ) and let ∇ = D L ⊕ D , where D L is theconnection induced by D on L . It follows that ( M, E, ρ, ∇ ) is co-CR quaternionic ifand only if ( M, c, D ) is Einstein–Weyl (that is, the trace-free symmetric part of theRicci tensor of D is zero).Furthermore, let µ be a section of L ∗ such that the connection defined by D µX Y = D X Y + µ X × c Y for any vector fields X and Y on M , induces a flat connection on L ∗ ⊗ T M . Then(
M, E, ι, ∇ µ ) is, locally, a hyper-co-CR manifold, where ∇ µ = ( D µ ) L ⊕ D µ , with ( D µ ) L the connection induced by D µ on L (this follows from well-known results; see [16] andthe references therein).Let τ = ( Z, M, π, C ) be the twistorial structure of a co-CR quaternionic manifold( M, E, ρ, ∇ ) . Recall [16] that τ is simple if and only if C ∩ C is a simple foliation (thatis, its leaves are the fibres of a submersion) whose leaves intersect each fibre of π at mostonce. Then (cid:0) T, d ϕ ( C ) (cid:1) is the twistor space of τ , where ϕ : Z → T is the submersionwhose fibres are the leaves of C ∩ C . Example 3.5.
Any co-CR quaternionic vector space is a co-CR quaternionic manifold,in an obvious way; moreover, the associated twistorial structure is simple and its twistorspace is just its holomorphic vector bundle.
Theorem 3.6.
Let ( M, E, ρ, ∇ ) be a co-CR quaternionic manifold, rank E = 4 k , rank(ker ρ ) = l . If the twistorial structure of ( M, E, ρ, ∇ ) is simple then it is realanalytic and its twistor space is a complex manifold of dimension k − l + 1 endowedwith a locally complete family of complex projective lines { Z x } x ∈ M C . Furthermore, forany x ∈ M , the normal bundle of the corresponding twistor line Z x is the holomorphicvector bundle of ( T x M, E x , ρ x ) .Proof. Let (
Z, M, π, C ) be the twistorial structure of ( M, E, ρ, ∇ ) . Let ϕ : Z → T be the submersion whose fibres are the leaves of C ∩ C . Obviously, d ϕ ( C ) defines acomplex structure on T of dimension 2 k − l + 1 . Furthermore, if for any x ∈ M wedenote Z x = ϕ ( π − ( x )) then Z x is a complex submanifold of T whose normal bundle isthe holomorphic vector bundle of ( T x M, E x , ρ x ) . The proof follows from [12] and [21,Proposition 2.5] . (cid:3) WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 9
Proposition 3.7.
Let ( M, E, ρ, ∇ ) be a co-CR quaternionic manifold whose twistorialstructure is simple; denote by ϕ : Z → T the corresponding holomorphic submersiononto its twistor space. Then ( M, E, ρ, ∇ ) is hyper-co-CR if and only if there exists asurjective holomorphic submersion ψ : T → C P such that the fibres of ψ ◦ ϕ are integralmanifolds of the connection induced by ∇ on Z .Proof. Denote by H the connection induced by ∇ on Z . Then H is integrable if andonly if d ϕ ( H ) is a holomorphic foliation on T ; furthermore, this foliation is simple ifand only if E is hypercomplex and H is the trivial connection on Z . (cid:3) f -Quaternionic manifolds Let F be an almost f -structure on a manifold M ; that is, F is a field of endomor-phisms of T M such that F + F = 0 . Denote by C the eigenspace of F with respect to − i and let D = C ⊕ ker F . Then C and D are compatible almost CR and almost co-CRstructures, respectively. An (integrable almost) f -structure is an almost f -structure forwhich the corresponding almost CR and almost co-CR structures are integrable. Definition 4.1. An almost f -quaternionic structure on a manifold M is a pair ( E, V ) ,where E is a quaternionic vector bundle on M and T M and V are vector subbundlesof E such that E = T M ⊕ V and J ( V ) ⊆ T M , for any J ∈ Z . An almost hyper- f -structure on a manifold M is an almost f -quaternionic structure ( E, V ) on M such that E is a hypercomplex vector bundle. An almost f -quaternionic manifold ( almost hyper- f -manifold ) is a manifold endowed with an almost f -quaternionic structure (almosthyper- f -structure).With the same notations as in Definition 4.1 , an almost f -quaternionic structure(almost hyper- f -structure) for which V is the zero bundle is an almost quaternionicstructure (almost hypercomplex structure).Let k and l be positive integers, k ≥ l , and denote by G k,l the group of f -quaternioniclinear isomorphisms of (Im H ) l × H k − l . The next result is an immediate consequenceof the description of G k,l given in Section 1 . Proposition 4.2.
Let M be a manifold of dimension k − l . Then any almost f -quaternionic structure ( E, V ) on M , with rank E = 4 k and rank V = l , corresponds toa reduction of the frame bundle of M to G k,l .Furthermore, if ( P, M, G k,l ) is the reduction of the frame bundle of M , correspondingto ( E, V ) , then V is the vector bundle associated to P through the canonical morphismof Lie groups G k,l → GL( l, R ) . Example 4.3.
1) A three-dimensional almost f -quaternionic manifold is just a (three-dimensional) conformal manifold.2) Let N be an almost quaternionic manifold endowed with a Hermitian metric andlet M be a hypersurface in N . Then (cid:0) T N | M , ( T M ) ⊥ (cid:1) is an almost f -quaternionicstructure on M . Obviously, any almost f -quaternionic structures ( E, V ) on a manifold M correspondsto a pair ( E, ι ) and (
E, ρ ) of almost CR quaternionic and co-CR quaternionic struc-tures on M , where ι : T M → E and ρ : E → T M are the inclusion and projection,respectively.
Definition 4.4.
Let (
M, E, V ) be an almost f -quaternionic manifold. Let ( E, ι ) and(
E, ρ ) be the almost CR quaternionic and co-CR quaternionic structures, respectively,corresponding to (
E, V ) . Let ∇ be a connection on E compatible with its linearquaternionic structure and let τ and τ c be the almost twistorial structures of ( M, E, ι, ∇ )and ( M, E, ρ, ∇ ) , respectively. We say that ( M, E, V, ∇ ) is an f -quaternionic manifold if the almost twistorial structures τ and τ c are integrable. If, further, E is hypercomplexand ∇ induces the trivial flat connection on Z then ( M, E, V, ∇ ) is an (integrable almost)hyper- f -manifold .Let ( M, E, V, ∇ ) be an f -quaternionic manifold and let Z and Z c be the twistorspaces of τ and τ c , respectively (we assume, for simplicity, that τ c is simple). Then Z is called the CR twistor space and Z c is called the twistor space of ( M, E, V, ∇ ) .Let ( M, E, V ) be an almost f -quaternionic manifold and let ∇ be a connectionon E compatible with its linear quaternionic structure. Let C and D be the almostCR and almost co-CR structures on Z determined by ∇ and the underlying almostCR quaternionic and almost co-CR quaternionic structures of ( M, E, V ) , respectively.Then C and D are compatible; therefore ( M, E, V, ∇ ) is f -quaternionic if and only ifthe corresponding almost f -structure on Z is integrable.Let ( M, E, V ) be an almost f -quaternionic manifold, rank E = 4 k , rank V = l , and D some compatible connection on M (equivalently, D is a linear connection on M whichcorresponds to a principal connection on the reduction to G k,l , of the frame bundleof M , corresponding to ( E, V ) ). Then D induces a connection D V on V . Moreover, ∇ = D V ⊕ D is compatible with the linear quaternionic structure on E . Corollary 4.5.
Let ( M, E, V, ∇ ) be an f -quaternionic manifold, rank E = 4 k , rank V = l , where ∇ = D V ⊕ D for some compatible connection D on M . Denote by τ and τ c theassociated twistorial structures. Then, locally, the twistor space of ( M, τ c ) is a complexmanifold, of complex dimension k − l + 1 , endowed with a locally complete family ofcomplex projective lines each of which has normal bundle k − l ) O (1) ⊕ l O (2) .Furthermore, if ( M, E, V, ∇ ) is real analytic then, locally, there exists a twistorialmap from the corresponding heaven space N , endowed with its twistorial structure, to ( M, τ c ) which is a retraction of the inclusion M ⊆ N .Proof. By passing to a convex open set of D , if necessary, we may suppose that τ c issimple. Thus, the first assertion is a consequence of Theorem 3.6 . The second statementfollows from the fact that there exists a holomorphic submersion from the twistor spaceof N , endowed with its twistorial structure, to the twistor space of ( M, τ c ) , which mapsdiffeomorphically twistor lines onto twistor lines. (cid:3) WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 11
Note that, if dim M = 3 then Corollary 4.5 gives results of [13] and [8] . Example 4.6.
Let M l = Gr +3 ( l + 3 , R ) be the Grassmann manifold of oriented vectorsubspaces of dimension 3 of R l +3 , ( l ≥
1) . Alternatively, M l can be defined as theRiemannian symmetric space SO( l + 3) / (cid:0) SO( l ) × SO(3) (cid:1) . As the structural group ofthe frame bundle of M l is SO( l ) × SO(3) , from Proposition 4.2 we obtain that M l iscanonically endowed with an almost f -quaternionic structure. Moreover, if we endow M l with its Levi-Civita connection then we obtain an f -quaternionic manifold. Itstwistor space is the hyperquadric Q l +1 of isotropic one-dimensional complex vectorsubspaces of C l +3 , considered as the complexification of the (real) Euclidean spaceof dimension l + 3 . Further, the CR twistor space Z of M l can be described asthe closed submanifold of Q l +1 × M l formed of those pairs ( ℓ, p ) such that ℓ ⊆ p C .Under the orthogonal decomposition R l +4 = R ⊕ R l +3 , we can embed M l as a totallygeodesic submanifold of the quaternionic manifold f M l = Gr +4 ( l + 4 , R ) as follows: p R ⊕ p , ( p ∈ M l ) . Recall (see [15] ) that the twistor space of f M l is the manifold e Z = Gr ( l + 4 , C ) of isotropic complex vector subspaces of dimension 2 of C l +4 , wherethe projection e Z → f M is given by q p , with q a self-dual subspace of p C (inparticular, p C = q ⊕ q ). Consequently, the CR twistor space Z of M l can be embeddedin e Z as follows: ( ℓ, p ) q , where q is the unique self-dual subspace of ( R ⊕ p ) C whichintersects p C along ℓ .In the particular case l = 1 we obtain the well-known fact (see [3] ) that the twistorspace of S is Q (cid:0) = C P × C P (cid:1) . Also, the CR twistor space of S can be identifiedwith the sphere bundle of O (1) ⊕ O (1) . Similarly, the dual of M l is, canonically, an f -quaternionic manifold whose twistor space is an open set of Q l +1 . Example 4.7.
Let Gr (2 n + 2 , C ) be the complex hypersurface of the GrassmannianGr (2 n + 2 , C ) of two-dimensional complex vector subspaces of C n +2 (cid:0) = H n +1 (cid:1) formedof those q ∈ Gr (2 n + 2 , C ) which are isotropic with respect to the underlying complexsymplectic structure ω of C n +2 ; note that,Gr (2 n + 2 , C ) = Sp( n + 1) / (cid:0) U(2) × Sp( n − (cid:1) . Then Gr (2 n + 2 , C ) is a real-analytic f -quaternionic manifold and its heaven spaceis Gr (2 n + 2 , C ). Its twistor space is Gr (2 n + 2 , C ) itself, considered as a complexmanifold.To describe the CR twistor space of Gr (2 n + 2 , C ), firstly, recall that the twistorspace of Gr (2 n + 2 , C ) is the flag manifold F , n +1 (2 n + 2 , C ) formed of the pairs( ℓ, p ) with ℓ and p complex vector subspaces of C n +2 of dimensions 1 and 2 n + 1 ,respectively, such that ℓ ⊆ p .Now, let Z ⊆ Gr (2 n + 2 , C ) × Gr (2 n + 2 , C ) be formed of the pairs ( p, q ) such that p ∩ q and p ∩ q ⊥ are nontrivial and the latter is contained by the kernel of ω | q ⊥ , wherethe orthogonal complement is taken with respect to the underlying Hermitian metric of C n +2 . Then the embedding Z → F , n +1 (2 n + 2 , C ) , ( p, q ) ( p ∩ q, q ⊥ + p ∩ q ) induces a CR structure with respect to which Z is the CR twistor space of Gr (2 n + 2 , C ).Note that, if n = 1 we obtain the f -quaternionic manifold of Example 4.6 with l = 2 .The next example is related to a construction of [23] (see, also, [9, Example 4.4] ). Example 4.8.
Let M be a quaternionic manifold, ∇ a quaternionic connection on itand Z its twistor space.Then Z is the sphere bundle of an oriented Riemannian vector bundle of rank three Q . By extending the structural group of the frame bundle (cid:0) SO( Q ) , M, SO(3 , R ) (cid:1) of Q we obtain a principal bundle (cid:0) H, M, H ∗ / Z (cid:1) .Let q ∈ S ( ⊆ Im H ) . The morphism of Lie groups C ∗ → H ∗ , a + b i a − bq inducesan action of C ∗ on H whose quotient space is Z (considered with its underlying smoothstructure); denote by ψ q : H → Z the projection. Moreover, ( H, Z, C ∗ ) is a principalbundle on which ∇ induces a principal connection for which the (0 ,
2) component of itscurvature form is zero. Therefore the complex structures of Z and of the fibres of H induce, through this connection, a complex structure J q on H .We, thus, obtain a hypercomplex manifold ( H, J i , J j , J k ) which is the heaven spaceof an f -quaternionic structure on SO( Q ) (in fact, a hyper- f structure). Note that, thetwistor space of SO( Q ) is C P × Z and the corresponding projection from S × SO( Q )onto C P × Z is given by ( q, u ) (cid:0) q, ψ q ( u ) (cid:1) , for any ( q, u ) ∈ S × SO( Q ) .If M = H P k then the factorisation through Z is unnecessary and we obtain an f -quaternionic structure on S k +3 with heaven space H k +1 \ { } and twistor space C P × C P k +1 .Let ( M, E, V ) be an almost f -quaternionic manifold, with rank V = l , and ( P, M, G k,l )the corresponding reduction of the frame bundle of M , where rank E = 4 k . Then T M = ( V ⊗ Q ) ⊕ W , where W is the quaternionic vector bundle associated to P through the canonical morphisms of Lie groups G k,l −→ Sp(1) · GL( k − l, H ) . Notethat, W is the largest quaternionic vector subbundle of E contained by T M . Theorem 4.9.
Let ( M, E, V ) be an almost f -quaternionic manifold and let D be acompatible torsion free connection, rank E = 4 k , rank V = l ; suppose that ( k, l ) =(2 , , (1 , . Then ( M, E, V, ∇ ) is f -quaternionic, where ∇ = D V ⊕ D . Moreover, W is integrable and geodesic, with respect to D (equivalently, D X Y is a section of W , forany sections X and Y of W ).Proof. Let ι : T M → E be the inclusion and ρ : E → T M the projection. It quicklyfollows that we may apply [17, Theorem 4.6] to obtain that (
M, E, ι, ∇ ) is CR quater-nionic. To prove that ( M, E, ρ, ∇ ) is co-CR quaternionic we apply [17, Theorem A.3]to D . Thus, we obtain that it is sufficient to show that for any J ∈ Z and any X, Y, Z ∈ E J we have R D ( ρ ( X ) , ρ ( Y ))( ρ ( Z )) ∈ ρ ( E J ) , where E J is the eigenspace of J , with respect to − i , and R D is the curvature form of D ; equivalently, for any J ∈ Z and any X, Y, Z ∈ E J we have R ∇ ( ρ ( X ) , ρ ( Y )) Z ∈ E J , where R ∇ is the curvatureform of ∇ . The proof of the fact that ( M, E, V, ∇ ) is f -quaternionic follows, similarly WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 13 to the proof of [17, Theorem 4.6] . The last statement, follows quickly from the factthat ( ∇ X J )( Y ) is a section of W , for any section J of Z and X, Y of W . (cid:3) From the proof of Theorem 4.9 we immediately obtain the following.
Corollary 4.10.
Let ( M, E, V ) be an almost f -quaternionic manifold and let D be acompatible torsion free connection, rank E ≥ . Then ( M, E, ρ, ∇ ) is co-CR quater-nionic, where ρ : E → T M is the projection and ∇ = D V ⊕ D . Next, we prove two realizability results for f -quaternionic manifolds. Proposition 4.11.
Let ( M, E, V, ∇ ) be an f -quaternionic manifold, rank V = 1 , where ∇ = D V ⊕ D for some compatible connection D on M . Then ( M, E, ι, ∇ ) is realizable,where ι : T M → E is the inclusion.Proof. By passing to a convex open set of D , if necessary, we may suppose that thetwistorial structure ( Z, M, π, D ) of the co-CR quaternionic manifold ( M, E, ρ ) is simple,where ρ : E → T M is the projection. Thus, by Theorem 3.6 , we have that (
Z, M, π, D )is real analytic. It follows that Q C is real analytic which, together with the relation T M = ( V ⊗ Q ) ⊕ W , quickly gives that the twistorial structure ( Z, M, π, C ) of ( M, E, ι )is real analytic. By [17, Corollary 5.4] the proof is complete. (cid:3)
The next result is an immediate consequence of Theorem 4.9 and Proposition 4.11 .
Corollary 4.12.
Let ( M, E, V ) be an almost f -quaternionic manifold, with rank V =1 , rank E ≥ , and let ∇ be a torsion free connection on E compatible with its linearquaternionic structure. Then ( M, E, ι, ∇ ) is realizable, where ι : T M → E is theinclusion. We end this section with the following result.
Proposition 4.13.
Let ( M, E, V, ∇ ) be a real analytic f -quaternionic manifold, with rank V = 1 , where ∇ = D V ⊕ D for some torsion free compatible connection D on M .Let N be the heaven space of ( M, E, ι, ∇ ) , where ι : T M → E is the inclusion, anddenote by Z N its twistor space. Then Z N is endowed with a nonintegrable holomorphicdistribution H of codimension one, transversal to the twistor lines corresponding tothe points of N \ M .Proof. By passing to a complexification, we may assume all the objects complex ana-lytic. Furthermore, excepting Z , we shall denote by the same symbols the correspondingcomplexifications. As for Z , this will denote the bundle of isotropic directions of Q .Then any p ∈ Z corresponds to a vector subspace E p of E . Let F be the distributionon Z such that F p is the horizontal lift, with respect to ∇ , of ι − ( E p ) , ( p ∈ Z ) . As( M, E, V, ∇ ) is (complex) f -quaternionic F is integrable. Moreover, locally, we maysuppose that its leaf space is Z N . Let G be the distribution on Z such that, at each p ∈ Z , we have that G p is the horizontal lift of ( V x ⊗ p ⊥ ) ⊕ W x , where x = π ( p ) . Define K = G ⊕ ker d π . Then the complex analytic versions of Cartan’s structural equationsand [11, Proposition III.2.3] , straightforwardly show that K is projectable with respectto F . Thus, K projects to a distribution H on Z N of codimension one. Furthermore,by using again [11, Proposition III.2.3] , we obtain that H is nonintegrable. (cid:3) Quaternionic-K¨ahler manifolds as heaven spaces A quaternionic-K¨ahler manifold is a quaternionic manifold endowed with a (semi-Riemannian) Hermitian metric whose Levi-Civita connection is quaternionic and whosescalar curvature is assumed nonzero.Let ( M, E, ι, ∇ ) be a CR quaternionic manifold with rank E = dim M + 1 . Let W bethe largest quaternionic vector subbundle of E contained by T M and denote by I the(Frobenius) integrability tensor of W . From the integrability of the almost twistorialstructure of ( M, E, ι, ∇ ) it follows that, for any J ∈ Z , the two-form I| E J takes valuesin E J / ( E J ∩ W C ) ; as this is one-dimensional the condition I| E J nondegenerate has anobvious meaning. Definition 5.1.
A CR quaternionic manifold (
M, E, ι, ∇ ) , with rank E = dim M + 1 ,is nondegenerate if I| E J is nondegenerate, for any J ∈ Z .Let M be a submanifold of a quaternionic manifold N and Z the twistor space of N .Denote by B the second fundamental form of M with respect to some quaternionicconnection ∇ on N ; that is, B is the (symmetric) bilinear form on M , with values in( T N | M ) /T M , characterised by B ( X, Y ) = σ ( ∇ X Y ) , for any vector fields X , Y on M ,where σ : T N | M → ( T N | M ) /T M is the projection. Definition 5.2.
We say that M is q-umbilical in N if for any J ∈ Z | M the secondfundamental form of M vanishes along the eigenvectors of J which are tangent to M .From [9, Propositions 1.8(ii) and 2.8] it quickly follows that the notion of q-umbilicalsubmanifold, of a quaternionic manifold, does not depend of the quaternionic connec-tion used to define the second fundamental form.Note that, if dim N = 4 then we retrieve the usual notion of umbilical submanifold.Also, if a quaternionic manifold is endowed with a Hermitian metric then any umbilicalsubmanifold of it is q-umbilical.The notion of q-umbilical submanifold of a quaternionic manifold can be easily ex-tended to CR quaternionic manifolds. Indeed, just define the second fundamental form B of ( M, E, ι, ∇ ) by B ( X, Y ) = σ ( ∇ X Y + ∇ Y X ) , for any vector fields X and Y on M , where σ : E → E/T M is the projection.
Theorem 5.3.
Let N be the heaven space of a real analytic CR quaternionic manifold ( M, E, ι, ∇ ) , with rank E = dim M + 1 . If M is q-umbilical in N then the twistor space Z N of N is endowed with a nonintegrable holomorphic distribution H of codimensionone, transversal to the twistor lines corresponding to the points of N \ M . Furthermore, WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 15 the following assertions are equivalent: (i) H is a holomorphic contact structure on Z N . (ii) ( M, E, ι, ∇ ) is nondegenerate.Proof. By passing to a complexification, we may assume all the objects complex ana-lytic. Also, we may assume ∇ torsion free. Furthermore, excepting Z , which will besoon described, below, we shall denote by the same symbols the corresponding com-plexifications.Let dim N = 4 k . As the complexification of Sp(1) · GL( k, H ) is SL(2 , C ) · GL(2 k, C ),we may assume that, locally, T N = H ⊗ F where H and F are (complex analytic) vec-tor bundles of rank 2 and 2 k , respectively. Also, H is endowed with a nowhere zerosection ε of Λ H ∗ and ∇ = ∇ H ⊗ ∇ F , for some connections ∇ H and ∇ F on H and F ,respectively, with ∇ H ε = 0.Then, by restricting to a convex neighbourhood of ∇ , if necessary, Z N is the leafspace of the foliation F N on P H which, at each [ u ] ∈ P H , is given by the horizontallift, with respect to ∇ H of [ u ] ⊗ F π H ( u ) , where π H : H → N is the projection. Let Z = P H | M and let F be the foliation induced by F N on Z . Note that, the leaf spaceof F is Z N .Let P H + P F ∗ be the restriction to N of P H × P F ∗ . Then ([ u ] , [ α ]) [ u ] ⊗ ker α de-fines an embedding of P H + P F ∗ into the Grassmann bundle P of (2 k − N . As ∇ = ∇ H ⊗ ∇ F , this embedding preserves the connec-tions induced by ∇ H , ∇ F and ∇ on P H + P F ∗ and P . Let F P be the distributionon P which, at each p ∈ P , is the horizontal lift, with respect to ∇ , of p ⊆ T π P ( p ) N ,where π P : P → N is the projection. Then the restriction of F P to P H + P F ∗ is adistribution F ′ on P H + P F ∗ .The map Z → P , [ u ] T M ∩ (cid:0) [ u ] ⊗ F π H ( u ) (cid:1) , is an embedding whose image iscontained by P H + P F ∗ . Moreover, the fact that M is q-umbilical in N is equivalentto the fact that F is the restriction of F P to Z .If for any ([ u ] , [ α ]) ∈ P H + P F ∗ we take the preimage of ker( ε ( u ) ⊗ α ) through theprojection of P H + P F ∗ we obtain a distribution of codimension one G ′ on P H + P F ∗ which contains F ′ . Furthermore, G = T Z ∩ G ′ is a codimension one distribution on Z which contains F .To prove that G is projectable with respect to F , firstly, observe that this is equiv-alent to the fact that the integrability tensor of G is zero when evaluated on the pairsin which one of the vectors is from F . Thus, as F is integrable, F = F ′ | Z and G = T Z ∩ G ′ , it is sufficient to prove that, at each p ∈ P H + P F ∗ , the integrabilitytensor of G ′ is zero when evaluated on the pairs formed of a vector from a basis of F ′ p and a vector from a basis of a space complementary to F ′ p .Let SL( H ) and GL( F ) be the frame bundles of H and F , respectively, and letSL( H ) + GL( F ) be the restriction to N of SL( H ) × GL( F ) . Then the kernel ofthe differential of the projection of SL( H ) + GL( F ) is the trivial vector bundle over SL( H ) + GL( F ) with fibre sl (2 , C ) ⊕ gl (2 k, C ) . Also, note that, for any ( u, v ) ∈ SL( H ) + GL( F ) , we have that u ⊗ v is a (complex-quaternionic) frame on N .Let G be the closed subgroup of SL(2 , C ) × GL(2 k, C ) which preserves some fixedpair (cid:0) [ x ] , [ α ] (cid:1) ∈ C P × P (cid:0)(cid:0) C k (cid:1) ∗ (cid:1) . Then P H + P F ∗ = (cid:0) SL( H ) + GL( F ) (cid:1) /G andwe denote F ′′ = (d µ ) − ( F ′ ) and G ′′ = (d µ ) − ( G ′ ) , where µ is the projection fromSL( H ) + GL( F ) onto P H + P F ∗ .For any ξ ∈ C ⊗ C k we define a horizontal vector field B ( ξ ) which at any ( u, v ) ∈ SL( H )+GL( F ) is the horizontal lift of ( u ⊗ v )( ξ ) . Then F ′′ is generated by the Lie alge-bra of G and all B ( x ⊗ y ) with α ( y ) = 0 . Also, G ′′ is generated by sl (2 , C ) ⊕ gl (2 k, C )and all B ( ξ ) with (cid:0) ε ( x ) ⊗ α (cid:1) ( ξ ) = 0 , where ε is the volume form on C .Further, similarly to [11, Proposition III.2.3] , we have (cid:2) A ⊕ A , B ( x ⊗ x ) (cid:3) = B ( A x ⊗ x + x ⊗ A x ) , for any A ∈ sl (2 , C ) , A ∈ gl (2 k, C ) , x ∈ C and x ∈ C k . Also, because ∇ is torsion free we have that, for any ξ, η ∈ C ⊗ C k , thehorizontal component of (cid:2) B ( ξ ) , B ( η ) (cid:3) is zero. These facts quickly show that, at each( u, v ) ∈ SL( H ) + GL( F ) , the integrability tensor of G ′′ is zero when evaluated on thepairs formed of a vector from a basis of F ′′ ( u,v ) and a vector from a basis of a spacecomplementary to F ′′ ( u,v ) . Consequently, G is projectable with respect to F .Next, we shall prove that G is nonintegrable. For this, firstly, observe that those( u, v ) in (cid:0) SL( H ) + GL( F ) (cid:1) | M for which u ⊗ v preserves the corresponding tangentspace to M form a principal bundle, which we shall call ‘the bundle of adapted frames’,whose structural group K can be described, as follows. We may write C ⊗ C k = gl (2 , C ) ⊕ (cid:0) C ⊗ C k − (cid:1) so that K is the closed subgroup of SL(2 , C ) × GL(2 k, C )which preserve Id C . Thus, K contains SL(2 , C ) acting on gl (2 , C ) ⊕ (cid:0) C ⊗ C k − (cid:1) by( a, ( ξ, η )) ( aξa − , η ) , for any a ∈ SL(2 , C ) , ξ ∈ gl (2 , C ) and η ∈ C ⊗ C k − .Note that, T M is the bundle associated to the bundle of adapted frames through theaction of K on sl (2 , C ) ⊕ (cid:0) C ⊗ C k − (cid:1) . Also, Z ( ⊆ P ) is the quotient of the bundle ofadapted frames through the closed subgroup of K preserving C ξ ⊕ (cid:0) ker ξ ⊗ C k − (cid:1) ,for some fixed ξ ∈ sl (2 , C ) \ { } with det ξ = 0 .If we, locally, consider a principal connection on the bundle of adapted frames thenwe can define, similarly to above, the corresponding ‘standard horizontal vector fields’ B ( ξ ) , for any ξ ∈ sl (2 , C ) ⊕ (cid:0) C ⊗ C k − (cid:1) , so that G corresponds to the distributiongenerated by the Lie algebra of K and all B ( ξ ) with ξ ∈ C ⊗ C k − or ξ ∈ sl (2 , C )such that ξ (ker ξ ) ⊆ ker ξ . Thus, if we take ξ ∈ sl (2 , C ) with ξ (ker ξ ) ⊆ ker ξ and A ∈ sl (2 , C ) such that [ A, ξ ](ker ξ ) * ker ξ then A and B ( ξ ) determine sections of G whose bracket is not a section of G .Finally, the equivalence of the assertions (i) and (ii) is a straightforward consequenceof the fact that if we denote by W the largest complex-quaternionic subbundle of T N | M contained by T M then F + (d π ) − ( W ) = G , where π : Z → M is the projection. (cid:3) The next result follows immediately from [15] and Theorem 5.3.
WISTOR THEORY FOR CO-CR QUATERNIONIC MANIFOLDS 17
Corollary 5.4.
The following assertions are equivalent, for a real analytic hypersurface M embedded in a quaternionic manifold N : (i) M is nondegenerate and q-umbilical. (ii) By passing, if necessary, to an open neighbourhood of M , there exists a metric g on N \ M such that ( N \ M, g ) is quaternionic-K¨ahler and the twistor lines determinedby the points of M are tangent to the contact distribution, on the twistor space of N ,corresponding to g . If dim M = 3 then Corollary 5.4 and [17, Corollary 5.5] give the main result of [13] .Also, the ‘quaternionic contact’ manifolds of [5] (see [7] ) are nondegenerate q-umbilicalCR quaternionic manifolds. Appendix A. The intrinsic description of linear(co-)CR quaternionic structures A conjugation , on a quaternionic vector space, is an involutive quaternionic au-tomorphism (not equal to the identity); in particular, the corresponding orientationpreserving isometry on the space of admissible complex structures is a symmetry in aline. Example A.1 ( [6] ) . Let U H = H ⊗ U be the quaternionification of a vector space U (the tensor product is taken over R ), endowed with the linear quaternionic structureinduced by the multiplication to the left.If q ∈ S then the association q ′ ⊗ u
7→ − qq ′ q ⊗ u , for any q ′ ∈ H and u ∈ U , definesa conjugation on U H .In fact, more can be proved. Proposition A.2.
Any pair of distinct commuting conjugations τ and τ on a quater-nionic vector space E determine a quaternionic linear isomorphism E = U H , for somevector space U , so that τ and τ are defined, as in Example A.1 , by two orthogonalimaginary unit quaternions.Proof. Let T , T : Z → Z be the orientation preserving isometries corresponding to τ , τ , respectively, where Z is the space of admissible linear complex structures on E .As T and T are commuting symmetries in lines ℓ and ℓ , respectively, it followsthat either ℓ = ℓ or ℓ ⊥ ℓ . In the former case, we would have T T = Id Z which,together with the fact that τ and τ are commuting involutions, implies τ = τ , acontradiction. Thus, if ℓ and ℓ are generated by I and J , respectively, then IJ = − IJ ;denote K = IJ .Now, E = U + ⊕ U − , where U ± = ker (cid:0) τ ∓ Id E (cid:1) . Furthermore, as τ τ = τ τ ,we have U + = V + ⊕ V − and U − = W + ⊕ W − , where V ± = ker (cid:0) τ | U + ∓ Id U + (cid:1) and W ± = ker (cid:0) τ | U − ∓ Id U − (cid:1) .A straightforward argument shows that IV + = V − , J V + = W + and KV + = W − . Thus, if we denote U = V + then E = U ⊕ IU ⊕ J U ⊕ KU and the association q ⊗ u q u + q Iu + q J u + q Ku , for any q = q + q i + q j + q k ∈ H and u ∈ U ,defines a quaternionic linear isomorphism from U H onto E which is as required. (cid:3) The quaternionification of a linear map is defined in the obvious way. Then aquaternionic linear map between the quaternionifications of two vector spaces is thequaternionification of a linear map if and only if it intertwines two distinct commutingconjugation.Let U be a vector space and let Λ be the space of conjugations on U H .The next proposition reformulates a result of [6] . Proposition A.3.
There exist natural correspondences between the following: (i)
Linear quaternionic structures on U ; (ii) Quaternionic vector subspaces B ⊆ U H such that U H = B ⊕ P τ ∈ Λ τ ( B ) ;(iii) Quaternionic vector subspaces C ⊆ U H such that U H = C ⊕ T τ ∈ Λ τ ( C ) . Furthermore, the correspondences are such that C = P τ ∈ Λ τ ( B ) and B = T τ ∈ Λ τ ( C ) . We can now give the intrinsic description of linear CR quaternionic structures.
Proposition A.4.
There exists a natural correspondence between the following: (i)
Linear CR quaternionic structures on U ; (ii) Quaternionic vector subspaces C ⊆ U H such that (ii1) C ∩ T τ ∈ Λ τ ( C ) = 0 , (ii2) C + σ ( C ) = U H , for any σ ∈ Λ .Proof. If (
E, ι ) is a linear CR quaternionic structure on U then C = (cid:0) ι H (cid:1) − ( C E ) satisfiesassertion (ii) , where C E is the quaternionic vector subspace of E H given by assertion(iii) of Proposition A.3 .Conversely, if C is as in (ii) then on defining E = U H /C and ι to be the compositionof the inclusion of U into U H followed by the projection from the latter onto E weobtain the corresponding linear CR quaternionic structure. (cid:3) Finally, by duality, we also have.
Proposition A.5.
There exists a natural correspondence between the following: (i)
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