Abstract

We consider the lift of a foliation to its conormal bundle and some transverse geometrical structures associated with this foliation are studied. We introduce a good vertical connection on the conormal bundle and, moreover, if the conormal bundle is endowed with a transversal Cartan metric, we obtain that the lifted foliation to its conormal bundle is a Riemannian one. Also, some transversally framed -structures of corank 2 on the normal bundle of lifted foliation to its conormal bundle are introduced and an almost (para)contact structure on a transverse Liouville distribution is obtained.

1. Introduction and Preliminaries

The study of the lift of transversal Finsler foliations to their normal bundle using the technique of good vertical connection was initiated by Miernowski and Mozgawa [1] where it is proved that the lifted foliation is a Riemannian one. Also, using different methods, some connections between foliations and Lagrangians (or Hamiltonians) in order to recover Riemannian foliations are investigated in the recent papers [2–5]. Our aim in this paper is to extend the study from [1] for the case of lifted foliation to its conormal bundle. In this sense we introduce a good vertical connection on the conormal bundle and we give an application of it in order to obtain that the lifted foliation is a Riemannian one in the case when the conormal bundle is endowed with a transversal Cartan metric. Moreover, in this case, some transversally framed -structures and an almost (para)contact structure associated with lifted foliation are investigated.

The methods used here are similarly and closely related to those used in [1, 6] for the case of transversal Finsler foliations.

Let us consider an -dimensional manifold which will be assumed to be connected and orientable.

Definition 1. A codimension foliation on is defined by a foliated cocycle such that (i) , , is an open covering of ;(ii) for every , are submersions, where is an -dimensional manifold, called transversal manifold;(iii) the maps satisfy for every such that .

Every fibre of is called a plaque of the foliation. Condition (1) says that on the intersection the plaques defined, respectively, by and coincide. The manifold is decomposed into a family of disjoint immersed connected submanifolds of dimension ; each of these submanifolds is called a leaf of .

By we denote the tangent bundle to and is the space of its global sections, that is, vector fields tangent to , and by we denote the normal bundle of .

In this paper, a system of local coordinates adapted to the foliation means coordinates on an open subset on which the foliation is trivial and defined by the equations .

We notice that the total spaces of the conormal bundle of carry a natural foliation of codimension such that the leaves of are covering spaces of the leaves of , and it is called the natural lift of to its conormal bundle .

If we denote by , , the corresponding local coframe on , then we can induce a chart on where , and the system of equations ., . defines the foliation .

Let be the normal bundle of the foliated manifold . The vectors , , form a natural frame of at the point . The canonical projection given by induces another projection which maps the vectors tangent to in the vectors tangent to . Thus, induces a mapping and is denoted by which is a vertical bundle spanned by the vectors , .

Lemma 2. Let be the zero section of the conormal bundle . Then the set is saturated on with foliation .

2. Good Vertical Connection on

The purpose of this section is to define a linear connection related to considered foliated structure, where . Since we have the foliated manifold , we are looking for a Bott connection such that for any vector field tangent to and any transversal vector field we havewhere is the canonical projection and .

Let us consider now the -transversal Hamilton-Liouville vector field defined by , . It can be checked that this definition is well posed. From the definition of the Bott connection, the following lemma holds.

Lemma 3. Let be a Bott connection. Then for every vector field tangent to .

Now, consider the local frame of and recall that the vectors form the basis of . With these settings we putFrom the above formulas it follows thatThe Bott connection allows us to define a mappingwhere . If we denote by the restriction of the linear mapping to the bundle , then we can state the following.

Definition 4. The Bott connection is said to be a good vertical connection if is a bundle isomorphism.

Observe that is a good vertical connection if and only if the matrix is nondegenerated. If we put , then we can split the bundle into direct sum:The coefficients of the mapping in the basis of areIt is easy to check that the vectors , where , form a basis of . In the sequel we will use the basis , called adapted, as well as its dual . Using this coframe we can define the local connection forms bywhereNotice that . The formula defines a linear mapping . This mapping allows us to extend the connection to the horizontal bundle bywhere , . In this way we construct a linear connection in :where , and is the vertical projection from decomposition (6). In particular we havewhere is given in (9).

If is a -form with values in , locally given bythen, following [1, 7], we can define an exterior differential by puttingA straightforward calculus shows that the above formula is well defined.

The bundle admits a natural section given byIt is clear that the form is well defined.

Definition 5. The form is called the torsion form of the connection .

Locally the form can be expressed as follows:where

3. Transversal Cartan Metrics on and Riemannian Foliations

As in the case of transversal Finsler metrics on the normal bundle of a foliation, [1, 3], a transversal Cartan metric on is a basic function (with respect to the lifted foliation ) which has the following properties:(i) is on ;(ii) for all ;(iii)the matrix , where , is positive definite at all points of . Also , whenever . As usual, [8], the properties of imply thatwhere is the inverse matrix of and we have put , .

Also, determines a metric structure on by settingfor every and .

Similar reasons as for transversal Finsler foliations (see Theorem 3.1 from [1]) lead to the following result.

Theorem 6. Let be a transversal Cartan metric and let be the Riemannian metric on induced by as in (19). Then there exists exactly one Bott vertical connection such that (i) is a good vertical connection;(ii)if and , then(iii) for every ;(iv) for every .

Also, the isomorphism does not depend on the coordinates along the leaves of ; so the Riemannian metric in defined by , where for every and for every and , is a transversal Riemannian metric for the lifted foliation to the conormal bundle of . Hence, we can consider the following.

Theorem 7. If the conormal bundle of foliation is endowed with a transversal Cartan metric, then the lifted foliation to the conormal bundle is Riemannian.

4. Transversally Framed -Structures on

The study of structures on manifolds defined by a tensor field satisfying has the origin in a paper by Yano [9]. Later on, these structures have been generically called -structures. On the tangent manifold of a Finsler space, the notion of framed -structure was defined and studied by Anastasiei in [10] and on the cotangent bundle of a Cartan space the study is continued in [11, 12]. Taking into account that the conormal bundle has a local model of a cotangent manifold, in this section we extend the study concerning -structures in our context.

Let . A framed -structure of corank on a -dimensional manifold is a natural generalization of an almost contact structure on (for ) and of an almost paracontact structure on (for ), respectively, and it is a triplet , , where is a tensor field of type , are vector fields, and are -forms on such thatwhere denotes the Kronecker tensor field on . The name of -structure was suggested by the identity . For an account of such kind of structures, we refer to [13].

Let us consider now that is endowed with a transversal Cartan metric . The linear operator given in the local adapted basis of bydefines an almost complex structure on for and an almost paracomplex structure on for , respectively. We also haveLet us put and . Thus, we have two global transverse vector fields on which are linearly independent. The first is transversally horizontal and the second one is transversally vertical.

From the definition of it followsNow, if we consider the dual transverse -forms of and , respectively, locally given by and , then we easily check thatNext, using , , and , , we construct the transverse tensor field of type on by puttingUsing a similar argument as in [6, 10–12] by direct calculus, we obtain the following.

Theorem 8. The triple , , provides some transversally framed -structures of corank on ; that is, the following hold: (i), , ;(ii);(iii) is of rank and .

Theorem 9. The transversal Riemannian metric on satisfies

For we putBy using Theorems 8 and 9, we obtainThus, is a transverse -form on . It is degenerate with null space .

Also, using the calculus in local coordinates, we easily obtainOn the other hand, we havewhere in the second relation we have used which follows from the homogeneity conditions (18) of . Comparing now with we obtainwhere . Thus, is transversally closed if and only if is transversally closed. Concluding, is in general an almost transversally presymplectic structure on .

Similarly, for , we can put

We have the following.

Theorem 10. The mapping is a symmetric bilinear form on and the annihilator of is .

Proof. The symmetry and bilinearity are obvious. Also, the null space of iswhich end the proof.

Locally, we obtainwith , since , and similarly , since .

Remark 11. The map is a transversally singular pseudo-Riemannian metric on .

5. An Almost (Para)Contact Structure on Transverse Liouville Distribution of

Denote by the line vector bundle over spanned by and we define the transverse vertical Liouville distribution as the complementary orthogonal distribution to in with respect to ; namely, . Hence, is defined by ; that is,Thus, any transverse vertical vector field can be expressed aswhere is the projection morphism of on . By direct calculations, one gets the following.

Proposition 12. For any transverse vertical vector fields , one has

Theorem 13. The transverse vertical Liouville distribution is integrable.

Proof. The proof follows using an argument similar to Theorem 3.1 [14] (see also Theorem 2.1 [6] or Theorem 4 [15]).

In the following, we will consider the transverse Liouville distribution of as the complementary orthogonal distribution to in with respect to ; that is, .

Let us restrict to all the geometrical structures introduced in Section 4 for all . We indicate this by overlines. Hence, we have(i) since lies in ;(ii) since for every transverse vector field ;(iii);(iv) is an endomorphism of since We denote now and .