Engel vector fields

In this section we want to investigate the set of vector fields preserving a given Engel structure on some manifoldM. We have already treated the case of contact vector fields in Section 2.1.2. The results we obtain for Engel structures are similar.

DEFINITION3.44. A vector field preserving the Engel structure is called Engel vector

field. We denote the Lie algebra of Engel vector fields by χ(D). A vector field which preserves an even contact structure is an even contact vector field.

Of course a vector field which preserves D also has to preserve the associated even contact structureE = [D,D]. Conversely, starting from a vector field preservingEwe can always find an Engel vector field.

LEMMA3.45. LetXbe a vector field preservingE. Then there is a unique sectionW

of the characteristic line fieldWsuch thatXe =X−W preservesD.

PROOF. LetU be an open subset ofMsuch thatWadmits a sectionW without zeroes onU and such that there is a1–formβwith the property

D

U =ker(α)∩ker(β).

We choose a1–formγsuch thatγvanishes onWsuch thatα, β, γare linearly independent at each point ofU. The characteristic foliationW ofE is defined by the3–formα∧dα. Since Xpreserves the even contact structure it also preserves the characteristic foliation. The conditions onXeto preserveDare

(i) Xe preservesE, i.e. there is a functiongsuch thatL

e

Xα=gα, and (ii) L

e

Xβ =g1α+g2β for smooth functionsg1, g2.

LXβis a linear combination ofα, βandγbecause it vanishes onW by

(LXβ)(W) =LX(β(W))−β(LXW) = 0.

On the other handLWβ =iWdβalso vanishes onW. Hence this form can also be written asaα+bβ +cγ with differentiable functionsa, b, c on U. We fix a local sectionY of

Dwhich is linearly independent ofW. Then the Engel conditions imply[W, Y]6∈ D but

[W, Y]∈ E. Therefore

(LWβ)(Y) =−β([W, Y])

has no zeros. This implies means that h has no zeroes on U. Hence there is a unique functionf with the property that

LXβ−f LWβ =LX−f Wβ

is a linear combination ofα, β. By definition ofW,Xe =X−f W also preservesE. Hence e

Xsatisfies condition (ii), soXe is an Engel vector field

Now we can coverM by open sets with the properties ofU. By the uniqueness of the local construction we obtain a smooth global Engel vector fieldXe =X−W for a unique

sectionW of the characteristic line field.

We assume thatE = [D,D]is a coorientable even contact structure with an orientable characteristic foliation. Letαbe a defining form ofE. As in the case of contact structures treated in Section 2.7 we can associate the function α(X) to each vector field X which preservesE. Unlike in the case of contact structures this function is not arbitrary but it has to satisfy a condition concerning its behaviour along the leaves ofW. LethW be the function with the property

(27) LWα=hWα .

IfXpreservesE, thenα(X)satisfies the identity

LW(α(X)) =iWdiXα=iWLXα−iWiXdα

=iXiWdα=hWα(X).

DEFINITION3.46. We define the subspaceC∞(α)ofC∞(M)by C∞(α) =f ∈C∞(M)LWf =hWf .

Note that if we useW0=gW with a nowhere vanishing functiongthen LW0α=gh_Wα .

If f satisfiesLWf = hWf then this function also satisfies LW0f = h_W0f. So C∞(α) depends only on the choice ofα. The functions inC∞(α)play the same role forχ(D)as C∞(H)for the space of contact vector fields.

THEOREM 3.47. The map which assigns to each Engel vector field X the function

α(X)is a bijection ontoC∞(α).

PROOF. Suppose thatα(X) ≡ 0. Then X is tangent to E and it has the properties which we used to defineW. Therefore it is tangent to W. On the other hand the proof of Lemma 3.45 shows that if a vector field is tangent toW and non–zero, then it does not preserveD. SoX ≡0. This shows injectivity.

In order to prove surjectivity, choose a setTi of hypersurfaces transversal to W such that every leaf ofW intersects at least one of these hypersurfaces. Now letf ∈ C∞(α). We apply Proposition 2.7 to f

Ti and the contact form α

Ti in order to obtain a contact

vector fieldXionTi. Using the flowϕtofW we can extendXito an even contact vector fieldX_i0 on the orbit ofTi.

We now show thatα(X_i0) =f. As a consequence ofLWα=hWαandLWf =hWf we obtain (α(X_i0))(ϕt(p)) = (α(ϕt∗Xi)) (ϕt(p)) = ((ϕ∗tα)(Xi)) (p) =exp Z t 0 hW ◦ϕsds (p)·(α(Xi))(pi) =exp Z t 0 hW ◦ϕsds (p)·f(p) =f(ϕt(p)).

forp∈Ti. HenceX_i0satisfiesα(X_i0) =f. By Lemma 3.45 we can find Engel vector fields

e

Xiby subtracting appropriate local sectionsWiofW fromXi0.

It remains to show that the vector fieldsXie are restrictions of one global Engel vector

field. This follows from injectivity which is already proved. Hence there is a global Engel

vector fieldXe withα(Xe) =f.

The setC∞(α)depends on the choice ofα. A very simple situation occurs when we can chooseαsuch thatLW(α∧dα) = 0. SinceLW(α∧dα) =iW dα2

this assumption impliesW ∈ ker(dα). SoLWα = 0andC∞(α)consists of smooth functions which are constant along the leaves ofW. Whether or not such a choice ofα is possible depends only on the characteristic foliation. If W admits a closed defining form it is said to be

volume–preserving. Under these assumptions the Engel structure admits an Engel vector

field whose properties are similar to those of Reeb vector field, cf.Lemma 2.6.

The following proposition does not require thatEis induced by an Engel structure. PROPOSITION3.48. Let E be a coorientable even contact structure on a4–manifold

M and letWbe the characteristic foliation. Then the following conditions are equivalent.

(i) There is a defining form α forE and a vector fieldR such thatα(R) = 1and

iRdα= 0. The vector fieldRis well defined only up to addition of a vector field

tangent toW.

IfE = [D,D]is induced by an Engel structureDthen there is a unique Engel vector field with the same properties asR.

(ii) W can be defined by a closed form.

PROOF. (i) ⇒ (ii) Let α be a defining form for E and let R be a vector field as in (i). The characteristic foliation is tangent to the kernel of the3-formα∧(dα). Then d(α∧dα) = (dα)2is a form of top degree onM. It is zero becauseiR((dα)2)≡0. Thus

W can be defined by a closed form.

(ii) ⇒ (i) There is a closed defining form η for W. Let α_e be a defining form for

E. Thenα_e∧dα_eis another defining form forW. Hence there exists a functionf without zeroes such thatη =f αe∧(dαe)

n−1

. Since bothηand−ηare closed and defineW, we may assumef =eg >0. Thenα=ef /2α_eis a defining form forEsuch that

α∧dα=fα_e∧dα_e=η

is closed. Hence (dα)2 = 0 and the kernel of dα is 2–dimensional. Using the non– integrability ofE and the properties of the characteristic foliation one can show thatE ∩

ker(dα) =W.

Choose a complement ofW in kerdα. This is also a complement ofEinT M. In par- ticular it is orientable. Thus we can find a nowhere vanishing sectionRof this complement such thatα(R) = 1. By construction we haveiRdα = 0soRpreservesαand the even contact structure.

IfE = [D,D]is induced by an Engel structure we use Lemma 3.45 to obtain a Reeb vector field for the Engel structure which depends on the choice of the defining form α within the class of one–forms whose exterior derivative has rank2.

Letα be a contact form on a 3–manifold N. When we apply the prolongation con- struction discussed in Section 3.2.2 to the contact structureC=ker(α)we obtain an Engel structureDon the total space of the circle bundle pr:PC →N. Then pr∗αis a form onPC which definesE = [D,D]. Obviouslydpr∗αhas rank two everywhere. The characteristic foliation of Dis volume preserving since it corresponds to the fibers of a fibers bundle. Among the different lifts of the Reeb vector fieldRofαtoPCthere is one unique liftRe

which preservesD.

The following more interesting example is due to R. Montgomery. In [Mo2] it is used to show that the space of infinitesimal automorphisms of an Engel structure can have finite dimension. We use Theorem 3.47 to prove this fact.

EXAMPLE 3.49 ([Mo2]). LetΣ be an orientable surface of genusg(Σ) ≥ 2with a hyperbolic Riemannian metric and letN =S1Σ ⊂ T∗Σbe the circle bundle of1–forms

of unit length. OnN there is a1–formλdefined by

λ(V) =α(pr_∗(V))forV ∈TαN .

The contact structure kerλis trivial because it is coorientable and it is tangent to the ori- entable circle bundleS1T∗Σ.

We fix a trivializationC1, C2ofC. LetRbe the Reeb vector field ofλ. The horizontal

lifts of these vector fields toN×S1are denoted by the same symbols. We writeϕfor the coordinate on the second factor ofN ×S1. The vector fields

Wε= ∂ ∂ϕ +εR

X= cos(ϕ)C1+ sin(ϕ)C2

span an Engel structure D_ε if |ε| is small enough. The characteristic foliation ofD_ε is spanned byWε. A defining form ofEε= [Dε,Dε]is

λε =pr∗λ−εdϕ .

The characteristic foliation W_ε is volume preserving becausedαεhas rank two for all ε. Since

αε(R) =α(R) = 1 iRdαε =pr∗(iRdα) = 0,

RpreservesEε. HoweverRdoes not preserveD_εin general. By Lemma 3.45 we can find a vector field preservingDεif we subtract an appropriate multiple ofWε. SinceRis a Reeb vector field it preservesC. With

Y = [∂ϕ, X] =−sin(ϕ)C1+ cos(ϕ)C2

we can decompose[R, X] =f X+gY as linear combination ofX, Y. Then

R− g 1 +gεWε, X = f− f g 1 +εg X− LX g 1 +εg Wε

is tangent toDε. So the Engel vector field corresponding toReis e

R− g

1 +gεWε.

We can view the characteristic foliation ofDεas the foliation on the mapping torus of the diffeomorphismψ2πεwhereψtis the flow ofRonN. The flow ofRonNis conjugate to the geodesic flow ofΣon the circle bundleS1T M. Since geodesic flow of a hyperbolic

surface is ergodic, cf. [Pat], the onlyψ2πε–invariant functions onN are constant. Hence C∞(αε)contains exactly the constant functions ifε6= 0.

By Theorem 3.47 this implies that the space of diffeomorphisms preserving the Engel structureDεis one–dimensional forε 6= 0. It has infinite dimension ifε= 0by Proposi- tion 3.16.