Robert Wolak

Foliated and associated geometric structures on foliated manifolds

This paper presents a unified approach to geometric structures on foliated manifolds. Three types of structures are distinguished : : foliated, transverse and associated. This unified approach allows to simplify proofs and to obtain new results concerning foliations with additional geometric structure. In this paper we present a definition of a transverse geometric structure as well as those of foliated and associated structures, and study relations between them. Then we propose a unified approach to these geometric structures. We obtain generalizations of results and simplifications of proofs of R.A. Blumenthal as well as new results on foliations admitting foliated and associated geometric structure. For simplicity’s sake all the objects considered are smooth and manifolds are connected, unless otherwise stated. For another approach to geometric structures on foliated manifolds see P.Molino [26,28]. In the presentation of transverse structures the author is indebted to Prof.A.Haefliger and his vision of transverse properties. The author had (1) Departamento de Xeometria e Topoloxia, Facultade de Matematicas, Universidade de ’ Santiago de Compostela, 15705 Santiago de Compostela, Spain (2) Instytut Matematyki, Uniwersytet Jagiellonski, Wl. Reymonta 4, 30-059 Krakow, Poland the privilege of discussing these problems with him. In fact, without his comments the definition would have been more restrictive than the one presented in this paper. The author would like to express his gratitude to Prof. P. Molino who helped him understand better foliated structures and whose treatment of foliated G-structures was an inspiration. During the preparation of this paper the author was a guest of Institut d’Estudis Catalans, Centre de Recerca Matematica in Bellaterra (Barcelona). , 1. Transverse structures Let (M, F) be a foliated manifold of dimension n whose foliation ~’ is of codimension q. The foliation ,~ is said to be modelled on a q-manifold No if it is defined by a cocycle U = f=, modelled on No, i.e. 1. is an open covering of M, 2. Ii: Ui -~ No are submersions with connected fibres, and = f on The q-manifold N = U Ni, Ni = /,(!7,), is called the transverse manifold associated to the cocycle U and the pseudogroup 1£ of local diffeomorphisms of N generated by g=~ the holonomy pseudogroup representative on N (associated to the cocycle U). N is a complete transverse manifold. The equivalence class of ~-C we call the holonomy pseudogroup of F (or (M, ,~) ) . In what follows, we assume that ~’ is defined by a cocycle U and we denote by N and ?-C the transverse manifold and holonomy pseudogroup associated to U, respectively. It is not difficult to check that different cocycles defining the same foliation provide us with equivalent holonomy pseudogroups, cf. [16]. In general the converse is not true. Example 1. Let F be a transversely affine foliation whose developing mapping is surjective, cf. [14]. Its holonomy pseudogroup has a representative the pseudogroup defined by the action of the affine holonomy group r on Rq. . If the fibres of the developing mapping are not connected and some connected components correspond to different leaves of F, then Rg cannot be a complete transverse manifold of the foliation F, and ?r is not a representative of the holonomy pseudogroup associated to any cocycle defining the foliation 7. As an example of such a foliation we can take the one-dimensional Hopf foliation of S2 x S1. The precise statement of the converse result is best expressed using the notion of a K-foliation, cf. [15]. Let K be any pseudogroup of local diffeomorphisms on a q-manifold No. A ~-foliation is a foliation defined by a cocycle U modelled on No with gij being elements of the pseudogroup ~C. Then the holonomy pseudogroup associated to U is equivalent to a subpseudogroup of J~. With this in mind we have the following. LEMMA 1.2014 Let F be a foliation defined by a cocycle U and let (?-~’, N’) be a pseudogroup equivalent to (?nC, N). . Then F is an H’-foliation. To introduce the definition of a transverse geometric structure we use the notion of a natural bundle, cf. [32]. DEFINITION 1.2014 Let N be a transverse manifold of the foliation ,F. An H-invariant subbundle E of a natural fibre bundle F(N) is called a transverse geometric structure of the foliation ~. It is not difficult to see that the definition does not depend on the choice of a cocycle U defining the foliation (i.e. on the choice of a transverse manifold and holonomy pseudogroup). Let (x’ N’) be a pseudogroup equivalent to (~-C, N) and (~«) be the equivalence. Then the subbundle E~ = E F(N’): v’ = F(~«)(v)~ v E F} (resp. E~ = ~u’ E F(N’): F(~«)(v’) E F~ for a contravariant functor) is an H’-invariant subbundle of F(N’ ) which is locally isomorphic to the subbundle E. Therefore, it is possible to talk about holonomy invariant subbundles on the transverse manifold. Moreover, when solving a particular problem, we can choose a cocycle making our foliation a K-foliation for a suitable pseudogroup K,. Example 2. 1. Let L be the functor associating to a manifold its bundle of linear frames. Then any ?i-invariant G-reduction of L(N) is a transverse geometric structure. Such a foliation is called a G-foliation, cf. [19,24,33]. 2. Any H-invariant section of a fibre bundle F(N) is a transverse geometric structure. Thus, if we take the functor of the tangent bundle we get holonomy invariant vector fields on the transverse manifold. For the contravariant functor of the cotangent bundle we get ?-~-invariant 1-forms and as x-invariant sections of the tensor products of these bundles(functors) we get ,}{-invariant tensor fields. Using a suitable associated fibre bundle to the bundle of linear frames (such functors are natural fibre bundles), we obtain that any H-invariant linear connection in an M-invariant G-structure is a transverse geometric structure. Such foliations we have called B7 G-foliations, cf. [39]. Riemannian and transversely affine foliations belong to this class. We obtain holonomy invariant connections of higher order or Cartan connections as holonomy invariant sections of associated fibre bundles to the bundle of frames of higher order. Such connections exist for transversely conformal or transversely projective foliations, cf. [8]. . 3. Let F be a K-foliation. If JC is a Lie pseudogroup, then the subbundle of r-jets from the definition of a Lie pseudogroup, cf. [34], is a transverse geometric structure. Foliations admitting foliated systems of differential equations are of this type, cf. [42]. Many geometric constructions provide useful ways of constructing new transverse geometric structures. In fact, the holonomy pseudogroup is a pseudogroup of local automorphisms of a given transverse geometric structure. Any geometric object related to this structure and invariant under the action of the pseudogroup of local automorphisms of the initial structure is itself a transverse geometric structure. For example, take a holonomy invariant G-structure B(N, G). Then its prolongations and their structure tensors are also holonomy invariant, cf. [20,35]. If V is a holonomy invariant connection in B(N, G), then its torsion and curvature tensor fields T and R, respectively, are holonomy invariant as well as and 2. Foliated structures A foliated geometric structure, intuitively, is a geometric stucture on a foliated manifold which is locally constant along the leaves ; for example, in local adapted coordinates it can be expressed in transverse coordinates only. It is the case of a foliated vector field or the Riemannian metric induced on the normal bundle by a bundle-like Riemannian metric. To present the definition of a foliated structure we need the following. Let Fol be the category of foliated manifolds with codimension q foliations. Global mappings which preserve foliations and which are transverse to them are the morphisms in this category, i.e. f E iff C and f *.~’Z = . DEFINITION 2.A covariant contravariant) functor F on the category Folq into the category of locally trivial fibre bundles and their fibre mappings is called a foliated natural bundle if the following conditions are satisfied : i~ for any foliated manifold (M, .~’), is a locally trivial fibre bundle over M ; ii~ let f E ~’I ), (M2, ~’2 )). Then the fibre mapping F( f ) has the following properties: :

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on a closed manifold M, it is known that