Fibrations and contact structures

HAMIDOU DATHE AND PHILIPPE RUKIMBIRA

Received 18 May 2004 and in revised form 8 October 2004

We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms.

1. Introduction

A contact form on a (2n+ 1)-dimensional manifoldMis a 1-formαsuch thatα∧(dα)n is a volume form onM. The system of equationsα(Z)=1 anddα(Z,X)=0 for arbitrary Xuniquely determines a vector fieldZcalled the Reeb vector field or the characteristic vector field ofα. The tangent subbundleξ=kernαof rank 2nis called the contact struc-ture associated withα. In general, a contact structure on a (2n+ 1)-dimensional manifold is a rank 2ntangent subbundle which is locally determined by contact forms (see Blair’s book [1] for more details about contact structures).

The manifolds in this paper will be assumed to be oriented and all the plane fields con-sidered herein are supposed to be transversely orientable. Letξbe a hyperplane field on a manifoldM. Whenξis a foliation, we say thatξis deformable into contact structures if there exists a one parameter familyξt of hyperplane fields satisfyingξ0=ξand for all t >0,ξtis contact. It is well known from Eliashberg and Thurston’s work [3, page 31, The-orem 2.4.1] that any oriented codimension 1,C2_{-foliation on an oriented 3-manifold can} be perturbed into contact structures, except the product foliation ofS2_×S1_{by spheresS}2_. It was then unknown if this approximation can always be done through a deformation. In this paper, we deal with particular deformations called “linear.” For a foliationξdefined by a 1-formα0, a deformationξtdefined by 1-formsαtis said to be linear ifαt=α0+tα whereαis a 1-form onM(independent oft). We point out that our definition of linearity is weaker than that of Eliashberg and Thurston [3, page 23]. In [2], the following theorem was proved.

Theorem1.1. LetMbe a closed,(2n+ 1)-dimensional manifold,α0a closed1-form onM, andαany1-form onM. Then, the following two conditions are equivalent.

(i)The1-formsαt=α0+tαin a linear deformation ofα0are contact for allt >0. (ii)The1-formαis contact andα0(Z)=0whereZis the Reeb vector field ofα.

Copyright©2005 Hindawi Publishing Corporation

Using this theorem, we will obtain a characterization ofT2_{-bundles overS}1_.

2. Torus bundles over S1

In dimension 3, one has the following characterization ofT2_{-bundles overS}1_.

Theorem2.1. LetMbe a closed,3-dimensional manifold. Then the following two condi-tions are equivalent.

(i)There is onMa closed nonsingular1-formα0and a1-formαsuch that the1-forms αt=α0+tαin a linear deformation ofα0are contact for allt >0.

(ii)Mis aT2_{-bundle overS}1_.

Proof. First we prove that (i) implies (ii). Ifα0andαare as in (i), then, byTheorem 1.1,α is a contact form whose characteristic vector fieldZsatisfiesα0(Z)=0. Arbitrarily close toα0, there is a closed, rational 1-formβwhich is obtained as follows. By the Hodge-de Rham theorem,α0=h+df wherehis a harmonic 1-form and f is a smooth function. Let h1,. . .,hlbe a basis of the vector space of harmonic forms onM. Thenh=λ1h1+···+ λlhlwhere the numbersλi’s represent the periods ofh. One can arbitrarily approximate each ofλi by a rational numberqiand consider the 1-formβ=q1h1+···+qlhl+df. If the approximationsqiare close enough, thenβis a nonsingular, closed 1-form with rational periods. The leaves of the foliation kernβform a fibration ofMby closed surfaces ΣoverS1_{. Letg be any metric onM in whichβis of unit length and letN denote the} metric dual unit vector field ofβ. Let Pr(Z)=Z−β(Z)Nbe the orthogonal projection ofZonto kernβ. Suppose that at some pointx∈M, Pr(Z)=0. Then, at that point,Z=

β(Z)Nand therefore 0=α0(Z)=β(Z)α0(N). Ifβandα0are sufficiently close, thenα0(N) is close toβ(N)=1 and henceα0(N)=0, which implies thatβ(Z)=0. So, Pr(Z)=0 at some point implies thatZ=0 at that particular point, contradicting the fact that Z is a nonsingular vector field. Therefore, Pr(Z)=Z−β(Z)N is a nonsingular vector field. Moreover,β(Pr(Z))=0, hence Pr(Z) is everywhere tangent to the fiberΣ, which means thatΣis a 2-torusT2_.

To prove that (ii) implies (i), we will exhibit on everyT2_{-bundle overS}1_{a contact form} αwhose characteristic vector fieldZis tangent to the fibersT2_.α

0can then be taken to be the pullback ofdθ, the volume form onS1_{, under the fiber projection map. Suppose that} Mis an orientableT2_{-bundle overS}1_{with monodromyA∈SL2Z. Following Geiges and} Gonzalo [5], we consider 3 different cases.

Case 1(traceA≥3). The manifoldM is a left quotient of Sol3, the solvable Lie group defined as a split extension ofR2_{byR. The groupRacts onR}2_{byz·(x,y)=(e}z_x,e−z_y). Sol3can be identified withR3_{with the following multiplication:}

x0,y0,z0

·(x,y,z)=x0+ez0x,y0+e−z0y,z0+z

. (2.1)

The bundle projection is given by (x,y,z)→z(see [7]). On Sol3, set

α=cos

2πn γ z

e−g(z)_dx−sin

2πn γ z

n∈N,gis a smooth, monotone increasing function that satisfies

(i)g(z)=znearz=0, (ii)g(z)=γ/2 nearz=γ/2, (iii)g(z+γ)=g(z) +γfor allz.

The 2-formdαis given by

dα= −

_2πn

γ sin _2πn

γ z

e−g(z)+g(z)e−g(z)cos _2πn

γ z

dz∧dx

− 2πn γ cos 2πn γ z

eg(z)_+g(z)eg(z)_sin

2πn γ z

dz∧d y,

α∧dα=

_2πn

γ +g(z) cos _2πn γ z sin _2πn γ z

dx∧d y∧dz.

(2.3)

Since the coefficient inα∧dαis bigger than 2πn/γ−g(z), one sees that fornlarge so that 2nπ/γ−g(z)>0,αdefines a contact form on Sol3that is invariant underΓ, the discrete subgroup generated by elements of the form (α1,β1, 0), (α2,β2, 0), (0, 0,γ). Moreover, the characteristic vector fieldZof the induced contact form is given by

F(x,y,z)∂x+G(x,y,z)∂y (2.4)

for some functionsFandGonM. Therefore,Zis tangent to the fibers of the projection map Sol3→Γ\Sol3.

Case 2(traceA<−3). The manifoldMis diffeomorphic to a quotient of the formΓ\Sol3 whereΓis generated by elements (α1,β1, 0), (α2,β2, 0) in Sol3and a generator (see [4])

(x,y,z)−→−eγ_x,−e−γ_{y,z+γ. (2.5)}

Set

α=cos _nπz

γ

e−g(z)_dx−sinnπz γ

eg(z)_{d y (2.6)}

for oddn. The same calculations as in the previous case show thatαinduces a contact form onΓ\Sol3whose characteristic vector field is tangent to the fibers of the projection map.

Case 3(the trace of the monodromy matrixAsatisfies|traceA| ≤2). In this case (see [6]), eitherAis a periodic matrix or it is conjugate to±Ak,

Ak= 1 k 0 1 , (2.7)

Subcase 1(Ais conjugate toAk). The manifoldMis a left quotient of Nil3. Nil3can be thought of asR3_{with the multiplication}

x0,y0,z0

·(z,y,z)=x0+x,y0+y,z0+z+x0y

. (2.8)

Every compact left quotient of Nil3is diffeomorphic to one of the formΓk\Nil3,k∈

Z/{0}, whereΓkis generated by elements (k, 0, 0), (0, 1, 0), (0, 0, 1). The left quotients are precisely theT2_{-bundles overS}1_{with monodromyA}

k, the bundle projection being given by (x,y,z)→y.

Set

α=cos(2π y)dx−sin(2π y)dz−f(x)d y, (2.9)

where f is a smooth, monotone increasing function that satisfies (i) f(x)=xnearx=0,

(ii) f(x)=k/2 nearx=k/2, (iii) f(x+k)= f(x) +kfor allx.

One sees thatdαis given by

dα= −2πsin (2π y)d y∧dx−2πcos (2π y)d y∧dz+f(x) sin (2π y)dx∧d y,

α∧dα= − 2π+f(x) sin2(2π y)dx∧d y∧dz. (2.10)

Since 2π+f(x) sin2(2π y) is strictly positive, the 1-formαinduces a contact form on Γk\Nil3whose characteristic vector fieldZis given by

F(x,y,z)∂x+G(x,y,z)∂z (2.11)

for some functionsFandGonM.Zis therefore tangent to the fibers of theT2_fibration. Subcase 2 (T2_{-bundles over S}1 _{with periodic monodromy). These are compact left} quotients of E2, the universal cover of the group of Euclidean motions in the plane. The Lie algebra ofE2 admits a basis{e1,e2,e3}with brackets [e1,e2]=0, [e3,e1]=e2, [e2,e3]=e1.

Letω1,ω2,ω3be the dual coframe of left invariant 1-forms onE2. ω1andω2are contact forms,ω3is a closed 1-form. The foliation defined byω3passes down to any left quotient ofE2 into the fibration byT2_{. Each of the contact formsω1,ω}

2 passes down to theT2 -bundle, with characteristic vector field tangent to the fibersT2_.

On aT2_{-bundle overS}1_{, it is not just the pullback of the volume form onS}1 _{that is} linearly deformable into contact forms. One has the following proposition.

Proposition2.2. LetM be aT2_{-bundle overS}1_{. Then any rational, nonsingular, closed} 1-form onMis linearly deformable into contact forms.

Proof. Letα0be a rational, nonsingular, closed 1-form onM.α0defines aΣ-fibration of MoverS1_{, whereΣis a closed surface. Since the homotopy groupsπ}

exact sequences:

0−→π1(Σ)−→π1(M)−→Z−→0, 0−→π1

T2−→π1(M)−→Z−→0. (2.12)

It follows thatΣhas to be a torus. The rest of the proof is as the proof of (ii) implies

(i) fromTheorem 2.1.

Corollary2.3. Any fibration over the circleS1_{on the three-dimensional torusT}3_{is linearly} deformable into contact structures.

In [2], we give an example of transversely affine foliation which is linearly deformable into contact structures. Here, we will prove the following generalization of this example. Proposition2.4. On a three dimensional closed manifold, every transversely affine folia-tion of codimension 1 with holonomy is linearly deformable into contact structures.

Proof. The proof is based on the principe of deforming a foliation near a curve of non-trivial holonomy (see [3]). It is known that (see [8]) a transversely affine foliation with nontrivial holonomy on a compact manifold has only finitely many compact leaves and each of them has nontrivial holonomy, hence a curve with nontrivial holonomy. A natural question is the following: can one generalizeTheorem 1.1to all transversely affine foliations? The answer is no. Here is a counterexample.

LetM be aT2_{-bundle over the circle with monodromyAsatisfying TrA >2.M can} be identified with some left quotient of Sol3(see notations in the proof ofTheorem 2.1,

Case 1). Letλbe the eigenvalue ofAsuch thatλ >1,uandvthe eigenvectors correspond-ing toλand 1/λ.

On Sol3, set the vector fields X= −1/(Logλ)∂/(∂z), Y =λ−zu, and Z=λz_{v. These} vector fields induce three vector fields onM also denoted byX,Y,Zand which satisfy the identities [X,Y]=Y, [X,Z]= −Z, [Y,Z]=0. The duals of these vector fields are nonsingular 1-formsω1,ω2, andω3satisfyingdω1=0,dω2=ω1∧ω2, anddω3= −ω1∧ ω3. LetᏲdenote the codimension 1 foliation defined by kernω2.Ᏺis a transversely affine foliation onM. For allt≥0, the 1-formsαt=ω2+tω3define a linear deformation ofᏲ into contact structures; but the 1-formω3is not contact.

References

[1] D. E. Blair,Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathemat-ics, vol. 203, Birkh¨auser Boston, Massachusetts, 2002.

[2] H. Dathe and P. Rukimbira,Foliations and contact structures, Advances in Geometry4(2004), no. 1, 75–81.

[3] Y. M. Eliashberg and W. P. Thurston,Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Rhode Island, 1998.

[4] H. Geiges,Symplectic structures onT2_{-bundles overT}2_{, Duke Math. J.67(1992), no. 3, 539–} 555.

[6] K. Sakamoto and S. Fukuhara,Classification ofT2_{-bundles overT}2_{, Tokyo J. Math.6(1983),} no. 2, 311–327.

[7] P. Scott,The geometries of3-manifolds, Bull. London Math. Soc.15(1983), no. 5, 401–487. [8] B. Seke,Sur les structures transversalement affines des feuilletages de codimension un, Ann. Inst.

Fourier (Grenoble)30(1980), no. 1, 1–29 (French).

Hamidou Dathe: D´epartement de Math´ematique, Physique et Informatique, Facult´e des Sciences et Techniques, Universit´e Cheikh Anta Diop (UCAD), Dakar, Senegal

E-mail address:[email protected]

Philippe Rukimbira: Department of Mathematics, Florida International University, Miami, FL 33199, USA