# On Some Transverse Geometrical Structures of Lifted Foliation to Its Conormal Bundle

Authors: Cristian Ida; Alexandru Oană
Journal: The Scientific World Journal
 (2015)

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## 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 framedf(3,ε)-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.


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## Body

## 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 f(3,ε)-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 considerM an (n+m)-dimensional manifold which will be assumed to be connected and orientable.Definition 1.
A codimensionn foliation F on M is defined by a foliated cocycle {Ui,φi,fi,j} such that (i)
{Ui}, i∈I, is an open covering of M;(ii)
for everyi∈I, φi:Ui→N are submersions, where N is an n-dimensional manifold, called transversal manifold;(iii)
the mapsfi,j:φi(Ui∩Uj)→φj(Ui∩Uj) satisfy(1)φj=fi,j∘φifor every(i,j)∈I×I such that Ui∩Uj≠∅.Every fibre ofφi is called a plaque of the foliation. Condition (1) says that on the intersection Ui∩Uj the plaques defined, respectively, by φi and φj coincide. The manifold M is decomposed into a family of disjoint immersed connected submanifolds of dimension m; each of these submanifolds is called a leaf of F.ByTF we denote the tangent bundle to F and Γ(F) is the space of its global sections, that is, vector fields tangent to F, and by QF=TM/TF we denote the normal bundle of F.In this paper, a system of local coordinates adapted to the foliationF means coordinates (x1,…,xn,y1,…,ym) on an open subset U on which the foliation is trivial and defined by the equations dxa=0,a=1,…,n.We notice that the total spaces of the conormal bundleQ*F of F carry a natural foliation F~ of codimension 2n such that the leaves of F~ are covering spaces of the leaves of F, and it is called the natural lift of F to its conormal bundle Q*F.If we denote by{dxa}, a=1,…,n, the corresponding local coframe on Q*F, then we can induce a chart (xa,pa,yu) on Q*F where p=padxa∈ΓQ*F, and the system of equations xa=const., pa=const. defines the foliation F~.LetQF~=T(Q*F)/TF~ be the normal bundle of the foliated manifold (Q*F,F~). The vectors ∂/∂xa,∂/∂pa, a=1,…,n, form a natural frame of QF~ at the point (xa,pa,yu)∈Q*F. The canonical projection π:Q*F→M given by π(xa,pa,yu)=(xa,yu) induces another projection π*:T(Q*F)→TM which maps the vectors tangent to F~ in the vectors tangent to F. Thus, π* induces a mapping π~*:QF~→QF and is denoted by V(Q*F)=kerπ~* which is a vertical bundle spanned by the vectors ∂/∂pa, a=1,…,n.Lemma 2.
Leto:M→Q*F be the zero section of the conormal bundle Q*F. Then the set o(M) is saturated on Q*F with foliation F~.


## 2. Good Vertical Connection on(Q*F,F~)

The purpose of this section is to define a linear connection∇:X(V(Q*F))→X(T*(Q~*F)⊗V(Q*F)) related to considered foliated structure, where Q~*F=Q*F-o(M). Since we have the foliated manifold (Q~*F,F~), we are looking for a Bott connection such that for any vector field X tangent to F~ and any transversal vector field Y we have(2)∇XY=pQF~X,Y~,where pQF~:T(Q*F)→QF~ is the canonical projection and pQF~(Y~)=Y.Let us consider now theF~-transversal Hamilton-Liouville vector field defined by C*:Q*F→V(Q*F), C*(xa,pa,yu)=pa∂/∂pa. It can be checked that this definition is well posed. From the definition of the Bott connection, the following lemma holds.Lemma 3.
Let∇:X(V(Q*F))→X(T*(Q~*F)⊗V(Q*F)) be a Bott connection. Then ∇XC*=0 for every vector field tangent to F~.Now, consider the local frame∂/∂xa,∂/∂pa,∂/∂yu of T(Q*F) and recall that the vectors ∂/∂pa form the basis of V(Q*F). With these settings we put(3)∇∂/∂xa∂∂pb=Γacb∂∂pc,∇∂/∂pa∂∂pb=Γcab∂∂pc,∇∂/∂yu∂∂pb=Γucb∂∂pc.From the above formulas it follows that(4)Γucb=0,∇∂/∂paC*=δca+pbΓcab∂∂pc.The Bott connection ∇ allows us to define a mapping(5)L:XQF~⟶XVQ*F,LX=∇X~C*,where pQF~(X~)=X. If we denote by Λ the restriction of the linear mapping L to the bundle V(Q*F), then we can state the following.Definition 4.
The Bott connection∇ is said to be a good vertical connection if Λ:V(Q*F)→V(Q*F) is a bundle isomorphism.Observe that∇ is a good vertical connection if and only if the matrix δca+pbΓcab is nondegenerated. If we put H(Q*F)=kerL, then we can split the bundle QF~ into direct sum:(6)QF~=HQ*F⊕VQ*F.The coefficients of the mapping L in the basis ∂/∂xa,∂/∂pa of QF~ are(7)L∂∂xa=pbΓacb∂∂pc,L∂∂pa=δca+pbΓcab∂∂pc=Lca∂∂pc.It is easy to check that the vectors δ/δxa=∂/∂xa+Nab∂/∂pb, where Nab=-L-1bcpdΓacd, form a basis of kerL. In the sequel we will use the basis δ/δxa,∂/∂pa, called adapted, as well as its dual {dxa,δpa=dpa-Nabdxb}. Using this coframe we can define the local connection forms by(8)∇∂∂pb=ωab⊗∂∂pa,where(9)ωab=Γcabdxc+Γabcdpc=Γcab+ΓabdNdcdxc+Γabcδpcωab=Hcabdxc+Γabcδpc.Notice that Hcabpb=Nca. The formula θ∂/∂pa=δ/δxa defines a linear mapping θ:V(Q*F)→H(Q*F). This mapping allows us to extend the connection ∇ to the horizontal bundle H(Q*F) by(10)∇XY=θ∇Xθ-1Y,where Y∈Γ(H(Q*F)), X∈Γ(T(Q*F)). In this way we construct a linear connection in QF~:(11)∇XY=∇XvY+∇XY-vY,where Y∈Γ(QF~), X∈Γ(T(Q*F)) and v:QF~→V(Q*F) is the vertical projection from decomposition (6). In particular we have(12)∇δδxa=ωab⊗δδxb,where ωab is given in (9).Ifφ∈ΓQ*F~⊗QF~ is a 1-form with values in QF~, locally given by(13)φ=φa⊗δδxa+φb⊗∂∂pb,then, following [1, 7], we can define an exterior differential Dφ by putting(14)Dφ=dφa-φa∧ωac⊗δδxa+dφb-φb∧ωcb⊗∂∂pc.A straightforward calculus shows that the above formula is well defined.The bundleQ*F~⊗QF~ admits a natural section η given by(15)η=dxa⊗∂∂xa+dpb⊗∂∂pb=dxa⊗δδxa+δpb⊗∂∂pb.It is clear that the form η is well defined.Definition 5.
The formζ=Dη is called the torsion form of the connection ∇.Locally the formζ can be expressed as follows:(16)Dη=-dxa∧ωac⊗δδxc+dδpb-δpc∧ωbc⊗∂∂pbDη=ζc⊗δδxc+ζb⊗∂∂pb,where(17)ζc=12Heac-Haecdxa∧dxe-Γacedxa∧δpe,ζb=-dNcb∧dxc-Hebaδpa∧dxe-Γbaeδpa∧δpe.


## 3. Transversal Cartan Metrics onQ*F 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 Q*F is a basic function (with respect to the lifted foliation F~) K:Q*F→[0,∞) which has the following properties:(i)
K is C∞ on Q~*F;(ii)
K(x,λp)=λK(x,p) for all λ>0;(iii)
then×n matrix (gab), where gab=1/2∂2K2/∂pa∂pa, is positive definite at all points of Q~*F. Also K(x,p)>0, whenever p≠0. As usual, [8], the properties of K imply that(18)pa=gabpb,pa=gabpb,K2=gabpapb=papa,Cabcpc=Cacbpc=Ccabpc=0,where (gab) is the inverse matrix of (gba) and we have put pa=1/2∂K2/∂pa, Cabc=-1/4∂3K2/∂pa∂pb∂pc.Also,gab determines a metric structure on V(Q*F) by setting(19)GvX,Y=gabx,pXax,y,pYbx,y,p,for every X=Xa(x,y,p)∂/∂pa and Y=Yb(x,y,p)∂/∂pb∈Γ(V(Q*F)).Similar reasons as for transversal Finsler foliations (see Theorem 3.1 from [1]) lead to the following result.Theorem 6.
LetK:Q*F→[0,∞) be a transversal Cartan metric and let Gv be the Riemannian metric on V(Q*F) induced by K as in (19). Then there exists exactly one Bott vertical connection ∇:X(V(Q*F))→X(T*(Q~*F)⊗V(Q*F)) such that (i)
∇ is a good vertical connection;(ii)
ifX,Y∈Γ(V(Q*F)) and Z∈Γ(T(Q~*F)), then(20)ZGvX,Y=Gv∇ZX,Y+GvX,∇ZY;(iii)
ζ(X,Y)=0 for every X,Y∈Γ(V(Q*F));(iv)
ζ(X,Y)∈Γ(V(Q*F)) for every X,Y∈Γ(H(Q*F)).Also, the isomorphismθ does not depend on the coordinates along the leaves of F~; so the Riemannian metric in QF~ defined by G=Gh+Gv, where Gh(X,Y)=Gv(θ-1(X),θ-1(Y)) for every X,Y∈Γ(H(Q*F)) and G(X,Y)=0 for every X∈Γ(H(Q*F)) and Y∈Γ(V(Q*F)), is a transversal Riemannian metric for the lifted foliation F~ to the conormal bundle Q*F of F. Hence, we can consider the following.Theorem 7.
If the conormal bundle of foliationF is endowed with a transversal Cartan metric, then the lifted foliation F~ to the conormal bundle Q*F is Riemannian.


## 4. Transversally Framedf(3,ε)-Structures on (Q*F,F~)

The study of structures on manifolds defined by a tensor field satisfyingf3±f=0 has the origin in a paper by Yano [9]. Later on, these structures have been generically called f-structures. On the tangent manifold of a Finsler space, the notion of framed f(3,1)-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 Q*F has a local model of a cotangent manifold, in this section we extend the study concerning f-structures in our context.Letε=±1. A framed f(3,ε)-structure of corank s on a (2n+s)-dimensional manifold N is a natural generalization of an almost contact structure on N (for ε=1) and of an almost paracontact structure on N (for ε=-1), respectively, and it is a triplet (f,(ξi),(ωi)), i=1,…,s, where f is a tensor field of type (1,1), (ξi) are vector fields, and (ωi) are 1-forms on N such that(21)ωiξj=δji,fξi=0,ωi∘f=0,f2=-εI-∑i‍ωi⊗ξi,where I denotes the Kronecker tensor field on N. The name of f(3,ε)-structure was suggested by the identity f3+εf=0. For an account of such kind of structures, we refer to [13].Let us consider now thatQ*F is endowed with a transversal Cartan metric K. The linear operator ϕ given in the local adapted basis of QF~ by(22)ϕδδxa=-εgab∂∂pb,ϕ∂∂pa=gabδδxbdefines an almost complex structure on QF~ for ε=1 and an almost paracomplex structure on QF~ for ε=-1, respectively. We also have(23)GϕX,ϕY=GX,Y,∀X,Y∈ΓQF~.Let us put ξ1=pa/Kδ/δxa and ξ2=pa/K∂/∂pa=1/KC*. Thus, we have two global transverse vector fields on (Q*F,F~) which are linearly independent. The first is transversally horizontal and the second one is transversally vertical.From the definition ofϕ it follows(24)ϕξ1=-εξ2,ϕξ2=ξ1.Now, if we consider the dual transverse 1-forms of ξ1 and ξ2, respectively, locally given by ω1=pa/Kdxa and ω2=pa/Kδpa, then we easily check that(25)ω1∘ϕ=ω2,ω2∘ϕ=-εω1,ω1X=GX,ξ1,ω2X=GX,ξ2,∀X∈ΓQF~.Next, using ϕ, ξi, and ωi, i∈{1,2}, we construct the transverse tensor field f of type (1,1) on (Q*F,F~) by putting(26)fX=ϕX-ω2Xξ1+εω1Xξ2,X∈ΓQF~.Using a similar argument as in [6, 10–12] by direct calculus, we obtain the following.Theorem 8.
The triple(f,(ξi),(ωi)), i∈{1,2}, provides some transversally framed f(3,ε)-structures of corank 2 on (Q*F,F~); that is, the following hold: (i)
ωi(ξj)=δji, f(ξi)=0, ωi∘f=0;(ii)
f2=-ε(I-ω1⊗ξ1-ω2⊗ξ2);(iii)
f is of rank 2n-2 and f3+εf=0.Theorem 9.
The transversal Riemannian metricG on (Q*F,F~) satisfies(27)GfX,fY=GX,Y-ω1Xω1YGfX,fY=-ω2Xω2Y,X,Y∈ΓQF~.Forε=1 we put(28)ΦX,Y=GfX,Y,X,Y∈ΓQF~.By using Theorems 8 and 9, we obtain(29)ΦX,Y=-ΦY,X,X,Y∈ΓQF~.Thus, Φ is a transverse 2-form on (Q*F,F~). It is degenerate with null space span⁡{ξ1,ξ2}.Also, using the calculus in local coordinates, we easily obtain(30)Φδδxa,δδxb=0,Φδδxa,∂∂pb=-δab+papbK2,Φ∂∂pa,∂∂pb=0.On the other hand, we have(31)dω1δδxa,δδxb=δδxapbK-δδxbpaK,dω1δδxa,∂∂pb=1K-δab+papbK2,dω1∂∂pa,∂∂pb=0,where in the second relation we have used pa=K∂K/∂pa which follows from the homogeneity conditions (18) of K. Comparing now Φ with dω1 we obtain(32)1KΦ=dω1+Ψ,where Ψ=δ/δxbpa/K-δ/δxapb/Kdxa∧dxb. Thus, 1/KΦ is transversally closed if and only if Ψ is transversally closed. Concluding, 1/KΦ is in general an almost transversally presymplectic structure on (Q*F,F~).Similarly, forε=-1, we can put(33)HX,Y=GfX,Y,X,Y∈ΓQF~.We have the following.Theorem 10.
The mappingH is a symmetric bilinear form on (Q*F,F~) and the annihilator of H is kerf.Proof.
The symmetry and bilinearity are obvious. Also, the null space ofH is(34)X∈ΓQF~∣HX,Y=0,∀Y∈ΓQF~=X∈ΓQF~∣GfX,Y=0=kerfwhich end the proof.Locally, we obtain(35)H=gab-papbK2dxa⊗dxb-gab-papbK2δpa⊗δpbwith det⁡gab-papb/K2=0, since gab-papb/K2pb=pa-pa=0, and similarly det⁡gab-papb/K2=0, since gab-papb/K2pb=pa-pa=0.Remark 11.
The mapH is a transversally singular pseudo-Riemannian metric on (Q*F,F~).


## 5. An Almost (Para)Contact Structure on Transverse Liouville Distribution of(Q*F,F~)

Denote by{ξ2} the line vector bundle over Q*F spanned by ξ2 and we define the transverse vertical Liouville distribution as the complementary orthogonal distribution S(Q*F) to {ξ2} in V(Q*F) with respect to Gv; namely, V(Q*F)=S(Q*F)⊕{ξ2}. Hence, S(Q*F) is defined by α:=ω2|V(Q*F); that is,(36)ΓSQ*F=X∈ΓVQ*F;αX=0.Thus, any transverse vertical vector field X∈Γ(V(Q*F)) can be expressed as(37)X=PX+αXξ2,where P is the projection morphism of V(Q*F) on S(Q*F). By direct calculations, one gets the following.Proposition 12.
For any transverse vertical vector fieldsX,Y∈Γ(V(Q*F)), one has(38)GvX,PY=GvPX,PY=GvX,Y-αXαY.Theorem 13.
The transverse vertical Liouville distributionS(Q*F) 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(Q*F,F~) as the complementary orthogonal distribution S¯(Q*F) to {ξ2} in QF~ with respect to G; that is, S¯(Q*F)=H(Q*F)⊕S(Q*F).Let us restrict toS¯(Q*F) all the geometrical structures introduced in Section 4 for all QF~. We indicate this by overlines. Hence, we have(i)
ξ1¯=ξ1 since ξ1 lies in S¯(Q*F);(ii)
ω2¯=0 since ω2(X)=G(X,ξ2)=0 for every transverse vector field X∈S¯(Q*F);(iii)
G¯=G|S¯(Q*F);(iv)
f¯(X)=ϕ¯(X)+εω1¯(X)⊗ξ2 is an endomorphism of S¯(Q*F) since (39)Gf¯X,ξ2=Gϕ¯X,ξ2+εω1¯XGξ2,ξ2Gf¯X,ξ2=ω2(ϕ¯(X))+εω1¯(X)=0.We denote now ξ¯=ξ1¯ and η¯=ω1¯.By Theorem8, we obtain the following.Theorem 14.
The triple(f¯,ξ¯,η¯) provides an almost (para)contact structure on S¯(Q*F); that is, (i)
f¯3+εf¯=0, rankf¯=2n-2=(2n-1)-1;(ii)
η¯(ξ¯)=1, f¯(ξ¯)=0, η¯∘f¯=0;(iii)
f¯2(X)=-ε(X-η¯(X)ξ¯), for X∈S¯(Q*F).Also, by Theorem9 we obtain the following.Theorem 15.
The transversal Riemannian metricG¯ verifies(40)G¯f¯X,f¯Y=G¯X,Y-η¯Xη¯Y,for every transverse vector fields X,Y∈S¯(Q*F).Concluding, the ensemble(f¯,ξ¯,η¯,G¯) is an almost (para)contact Riemannian structure on S¯(Q*F).



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