The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Foliations with Transversal Quaternionic Structures Paolo Piccinni Izu Vaisman Vienna Preprint ESI January Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat Foliatio ns with Transversal Quaternionic Structures by Paolo Piccinni and Izu Vaisman ABSTRACT We consider manifolds equipp ed with a foliation F of co dimen sion q and an almost quaternionic structure Q on the transversal bundle of F After discussing conditions of pro jectability and integrability of Q we study the transversal twistor space Z F which by denition consists of the Qcompatible almost complex structures We show that Z F can b e endowed with a lifted b foliation F and two natural almost complex structures J J on the transversal 1 2 b bundle of F We establish the conditions which ensure the pro jectability of J 1 and J and the integrability of J J is never integrable 2 1 2 Preliminaries We recall the basic denitions of quaternionic geometry eg The general framework is the C category An almost hypercomplex structure on a q dimensional dierentiable man 4q ifold N is an ordered triple H I I I of almost complex structures 1 2 3 satisfying the quaternionic identities I I I for and cyclic p ermutations If the structures I I I are integrable H is said to b e 1 2 3 a hypercomplex structure Mathematics Subject Classication C C Key words and phrases FoliationQuaternionic Structure Twistor Space Acknow ledgements This pap er was started while the second author visited the Istituto di Matematica Universita di Roma under the auspices of the CNR Italy He is ex pressing here his thanks to the hosts The rst author acknowledges supp ort by MURST of Italy and supp ort and hospitality by the Erwin Schrodinger Institute for Mathematical Physics in Vienna If H I I I is an almost hyp ercomplex structure any triple J J 1 2 3 1 2 J obtained from I I I by multiplying by a matrix of SO is again 3 1 2 3 an almost hyp ercomplex structure Moreover there exists a set of compati ble almost complex structures asso ciated with a given almost hyp ercomplex structure namely the set of all J a I a I a I where a a a are 1 1 2 2 3 3 1 2 3 2 2 2 functions satisfying a a a 1 2 3 4q An almost quaternionic structure on the manifold N is a rank vec tor subbundle Q of the endomorphism bundle E ndTN lo cally spanned by almost hyp ercomplex structures H I I I which are related by SO 1 2 3 matrices on the intersections of trivializing op en sets A quaternionic struc 4q ture on the manifold N is an almost quaternionic structure such that there exists a torsionless connection r of TN which when extended to the vector bundle E ndTN preserves the subbundle Q ie rQ Q The existence of the Qpreserving torsionless connection r is not equivalent with the inte grability of Q as a Gstructure The existence of a at torsionless connection which preserves Q implies that Q can b e obtained from lo cal quaternionic 4q co ordinates on N If an almost quaternionic structure Q is xed on N the local bases I I I which span the vector bundle Q are also called local com 1 2 3 patible almost hypercomplex structures and any lo cal J a I a I a I 1 1 2 2 3 3 2 2 2 is called a local Qcompatible almost complex structure a a with a 3 2 1 A Riemannian metric g on a almost hyp ercomplex manifold N H is almost hyperhermitian resp ectively almost hyperkahler if it is almost Hermitian resp ectively almost Kahlerian with resp ect to all the structures I of H Then g also is compatible with any H compatible structure J Similarly on an almost quaternionic manifold N Q the met ric g is quaternion Hermitian if it is Hermitian with resp ect to the lo cal bases I of Q and it is quaternion Kahler if it is quaternion Hermitian and Q is parallel ie rQ Q with resp ect to the LeviCivita connection r of g In b oth cases the prop erty says nothing ab out the integrability of the struc tures I The terms are also used for manifolds endowed with the resp ective structures Of course a hyp erkahler manifold necessarily is hyp ercomplex and a quaternion Kahler manifold necessarily is quaternionic The twistor space ZN of an almost quaternionic manifold N Q is de ned as the manifold of the Qcompatible almost complex structures of the 2 tangent spaces of N Thus ZN is an S bundle asso ciated with the vector bundle Q where Q has the metric which makes the lo cal bases H I I I 1 2 3 orthonormal bases p+4q Now let us consider a C manifold M equipp ed with a pdimensional foliation F Denote by L T F the tangent bundle of F and by F T M L its transversal vector bundle of rank q We will often identify the transversal bundle F with a complementary distribution E of L ie a split ting of the exact sequence F L T F TM Almost hyp ercomplex and almost quaternionic structures can b e dened similarly on vector bundles of rank q Accordingly they will b e reductions of the structure group of the bundle to G where the group G GLq H for the almost hyp ercomplex structures and GLq H S p G GLq H S p I d for the almost quaternionic structures H is the algebra of the quaternions Furthermore almost hyp erhermitian and quaternion Hermitian structures corresp ond to the structure groups S pq and S pq S p resp ectively We will consider structures of these typ es on the transversal bundle F of a foliation F and refer to them as transversal almost hypercomplex transver sal almost quaternionic etc structures of the foliation F Such structures sp oradically app eared in the literature eg In what follows we use Bott connections D TM F F A Bott connection is a connection on the transversal bundle F which extends the partial connection D L F F given by D s Y X s Y where Y is a tangent vector eld of the leaves of the foliation F and X s is any vector eld on M such that X s s F always denotes s spaces of global cross sections of vector bundles Notice that an identication F E where TM E L implies the replacement of by D X Y X X E Y b eing the pro jection TM E A Riemannian metric g splits TM T F T F and we will take E T F F Then in particular D can b e extended to a Bott connection D by dening D r X E where r is the LeviCivita connection of X X g For Riemannian foliations this Bott connection D is the unique torsionless metric connection of the normal bundle T F F Denition a A transversal almost hyp ercomplex structure H I I I of F is pro jectable if the partial connection D preserves the struc 1 2 3 tures I i e DI b A transversal almost quaternionic structure Q E nd F is pro jectable if D preserves Q DQ Q The pro jectability condition of Q can b e formulated in terms of lo cal bases H I I I Namely Q is pro jectable i for any choice of a lo cal 1 2 3 basis H I I I there exist lo cal forms such that 1 2 3 DI I I DI I I DI I I 1 2 3 2 1 3 3 1 2 As a matter of fact if equations of the typ e hold for some choice of H similar equations hold for any choice of H If J E nd F we may also see it as a cross section of E nd E and for Y T F s F we have D J s D J s J D s Y J X J Y X Y Y Y s s for any X TM such that s X A cross section s is projectable if s s s pro jects to a tangent vector eld of any lo cal space of slices of the foliation F From it follows that J is pro jectable i J s is pro jectable whenever s is pro jectable Therefore pro jectability in the sense of Denition a means that we have a structure which is the lift of almost hyp ercomplex structures of the lo cal slice spaces The same is true in the case of Denition b see Prop osition later on Accordingly we will give Denition A pro jectable transversal almost hyp ercomplex or almost quaternionic structure of a foliation F is integrable if the pro jected structures of the lo cal slice spaces are hyp ercomplex or quaternionic resp ectively If integrability holds the word almost will b e omitted Examples We b egin by an example of a foliation with a transversal almost hyp er complex structure Let F b e a transversally holomorphic foliation of real p+2q co dimension q on a manifold N This means that on N there are lo cal u a a u a co ordinates y z z where y are real co ordinates and z are complex co ordinates with holomorphic transition functions such that F is dened by a a z const z const Furthermore assume that g is a bundlelike Rie mannian metric on N which is F transversally Kahler ie a b a b g g z zdz dz g z zdz dz ab ab where the rst two terms dene a Kahler metric in the co ordinates z and u the remaining unexplicited terms contain dy In this pap er we use the Einstein summation convention Now consider the manifold M dened by the total space of the conormal bundle of the foliation F ie the annihilator of the tangent bundle T F If E is the g orthogonal bundle of F then M E On M there exists a natural lift F of F such that the leaves of F are covering spaces of the leaves of F Moreover the lo cal slice spaces of F are cotangent bundles of Kahler manifolds It is not dicult to construct an almost hyp ercomplex structure on such a cotangent bundle If we do this on the lo cal slice spaces the obtained structures glue up to a pro jectable almost hyp ercomplex structure transversal to F The construction of the almost hyp ercomplex structure of the cotangent bundle of