Examples of Engel structures Apart from the constructions we present in

later chapters, there are two other known construction methods for Engel structures. The first one – called prolongation – is based on contact structures on3–manifolds. The second construction yields Engel structures on certain mapping tori induced by diffeomorphisms ψ:N→N of3–manifolds.

Starting from a contact structure C on a3–manifold N one can construct an Engel structure. We consider the equivalence relation

v∼wforv, w∈ C \ {0} ⇔v =λwfor someλ∈_R

onC \ {0}. Then the spacePC=C \ {0}/∼of Legendrian lines is a closed4–dimensional manifold. By construction, there is a fibration pr:PC →N sending each Legendrian line to the corresponding base point inN. The fiber isRP1.

Letε:C \N →_PC. One can define a distribution of rank two onPCby

DC=v∈Tε(l)PC

pr_∗(v)∈ε(l) .

DEFINITION3.14. This construction of a distribution onPCis called prolongation. Prolongation really yields Engel structures.

PROPOSITION3.15. DCis an Engel structure onPC.

PROOF. Letp∈N. The fibers ofPCare clearly tangent toDC. ThusDCis a subbundle

of rank two ofTPC. Forε(v) ∈PCchoose a local trivializationW, XofDCsuch thatW

is tangent to the fibers. Letϕtbe the local flow ofW. Then by definition pr∗ X ϕt(ε(v))

∈ C(pr(ε(v)))

is a curve transversal to the lineε(v)inC. Hence d dt t=0 pr_∗(X(ϕt(ε(v)))) =pr∗([W, X])6∈ε(v),

so[W, X](p)is not contained inDC. Thus[DC,DC] =pr∗C. This shows that the leaves of

the characteristic foliation ofDCare the fibers of pr:PC −→N.

We have shown that pr_∗(X)and pr_∗([W, X])spanC . Now we restrict pr to a hyper- surface throughpwhich is tangent toX. This suffices for the calculation of[X,[W, X]]. When we restrict pr to this hypersurface we obtain a local diffeomorphism. Then

pr_∗([X,[W, X]]) = [pr_∗(X),pr_∗([W, X])]6∈ C

by the definition of contact structures. This shows that[D,[D,D]]has full rank.

The Engel structures obtained this way are not orientable since the restriction ofDC to

a fiber ofPCis the the Whitney sum ofTRP1and the tautological bundle overRP1. While the first bundle is trivial, the tautological bundle is not orientable. One obtains orientable Engel structures when one does the same construction using oriented Legendrian lines.

Engel structures constructed by prolongation provide local models for the Engel struc- ture on tubular neighbourhoods of transversal hypersurfaces (cf. Theorem 3.19) and one can obtain automorphisms of these Engel structures from diffeomorphisms a contact struc- ture.

LetN1 andN2be3–manifolds with contact structuresC1,C2 and letϕ:N1 →N2 be

a contact diffeomorphism. From ϕone can construct a diffeomorphismϕe : PC1 → PC2

which preserves the induced Engel structures D1,D2. Fori = 1,2 we denote the maps C_i\Ni →PCi byκi. The following proposition can be found in [Mo2], according to this paper it was known before.

PROPOSITION3.16. The diffeomorphism

e

ϕ:PC1−→PC2

κ1(v)7−→κ2(ϕ∗(v))

mapsD₁ toD₂. Every diffeomorphismPC1 → PC2 preserving Engel structures is of this form.

PROOF. Consider the map

e

ψ:PC2 −→PC1

κ2(w)7−→κ1(ϕ−∗1w).

The compositionψe◦ϕ_eis the identity ofPC₁since e

ψ◦ϕ_e(κ1(v)) =κ1(ϕ∗−1(ϕ∗(v))) =κ1(v)

and similarly forϕ_e◦ψ. Thuse ϕ_eis a diffeomorphism. Now letY with base pointκ₁(v)be

tangent to the Engel structureD1 onPC1. The base point ofϕe∗(Y) isκ2(ϕ∗(v)). On the

other hand

(17) pr₂∗(ϕe∗(Y)) =ϕ∗(pr1∗(Y))

is contained inϕ∗(κ1(v)) =κ2(ϕ∗(v))and this is the basepoint ofϕe(Y). Thusϕepreserves

Engel structures.

Now let Φ : PC1 → PC2 be a diffeomorphism preserving Engel structures. ThenΦ

preserves the characteristic foliations or – equivalently –Φtakes fibers ofPC1 to fibers of

PC2, thus the map

ϕ:N1 −→N2

p7−→pr₂ Φ pr−₁1(p)

is well defined. The inverse ofϕcan be constructed in the same manner soϕis a diffeo- morphism. The diagram

PC1 _Φ // pr1 PC2 pr2 N1 ϕ _// N2

commutes. As Φ preserves Engel structures, Φ also preserves the induced even contact structures. The even contact structureE_ionPCi satisfies pri∗Ei =Cifori= 1,2. Hence

ϕ∗(C1) =ϕ∗(pr1∗E1) =pr2∗(Φ∗(E1)) =C2

so ϕ is a contact diffeomorphism. Let ϕ_e : PC1 → PC2 be the induced Engel diffeo-

morphism. We want to show that ϕe

−1 _{◦Φ is the identity map of}

PC2. It is clear that

e

ϕ−1◦Φpreserves each fiber. We want to show that each fiber is preserved pointwise. Let v∈ D1(κ1(l))be such that pr1∗(v)6= 0. Recall

D₁(ε1(l)) = w∈T_ε1(l)PC1 pr_1∗w∈ε₁(l) . Nowϕe −1_{◦ΦpreservesD} 1. Suppose thatϕe −1_◦Φ(ε 1(l)) =κ1(l0). By (17) pr1∗(ϕe −1 ∗ (Φ∗(v))) =ϕ∗−1(pr2∗(Φ∗(v))) =ϕ−∗1(ϕ∗(pr1∗(v))).

While on the left we have an element ofκ1(l0), the expression on the right is an element of

κ1(l). Thusϕe

−1_{◦Φpreserves the fibers of}

Another construction is due to H.–J. Geiges, [Gei]. It shows that parallelizable map- ping tori of compact3–manifolds admit Engel structures without using contact structures. Suppose thatψ:N −→N is a diffeomorphism of a compact3–manifold. Let

M = (N×[0,1])/(x,1)∼(ψ(x),0).

be the mapping torus ofψ. The projection ofN ×[0,1]onto the second factor induces a fibration M −→ S1 = [0,1]/0 ∼ 1. We writetfor the coordinate on[0,1]. The vector field∂tonN ×[0,1]induces a vector fieldX0onM.

Now we assume thatM is parallelizable. In order to construct a framing ofT M such thatX0is a component, we fix an arbitrary almost quaternionic structureT M 'M ×H. Then we obtain a framing

X0, X1=iX0, X2 =jX0, X3 =kX0 .

PROPOSITION 3.17 (Geiges, [Gei]). If n ∈ _N is large enough, the distributionD_n spanned byX0and Yn= 1 n cos n 2_t X1+ sin n2t X2 +X3 is an Engel structure.

PROOF. In order to verify thatDnis an Engel structure for large n, we calculate the commutators [X0, Yn] =n −sin(n2t)X1+ cos(n2t)X2 + 1 n cos(n 2_t)[X 0, X1] + sin(n2t)[X0, X2] + [X0, X3] [X0,[X0, Yn]] =n3 −cos(n2t)X1−sin(n2t)X2 + [X0,[X0, X3]] + 2n −sin(n2t)[X0, X1] + cos(n2t)[X0, X2] + 1 n cos(n 2_t)[X 0,[X0, X1]] + sin(n2t)[X0,[X0, X1]]

Notice that asngrows to infinity

Yn−→X3 1 n[X0, Yn]∼ −sin(n 2_t)X 1+ cos(n2t)X2 1 n3[X0,[X0, Yn]]∼ −cos(n 2_t)X 1−sin(n2t)X2 .

SinceMis compact, we can choosenso big that

X0, Yn,[X0, Yn],[X0,[X0, Yn]]

is a framing ofT M.

Unlike in the case of prolongation it is not possible to determine explicitly the charac- teristic foliation of Engel structures obtained this way. This is a major disadvantage of this construction.

REMARK3.18. A mapping torus has vanishing Euler characteristic since there is a vec- tor field without zeroes. One can show that the signature of a four dimensional orientable mapping torus is always zero. However the following example shows that orientable map- ping tori do not necessarily admit spin structures.

LetE →T2be a complex line bundle overT2 with odd first Chern class and letCbe the trivial complex line bundle. Consider theCP1–bundleM =P(E⊕C)obtained from E by fiberwise one–point compactification. Then the normal bundle of the image of the

zero sectionσofEinM is the pull back ofE underσ. Alongσthe tangent bundle ofM decomposes as a direct sumT M

σ =T σ⊕σ

∗_{E. HenceT M}

σ has odd first Chern class and thereforeT M does not admit a spin structure.

This shows that the condition on orientable mapping tori to be parallelizable is not redundant in dimension4and higher.

3.2.3. Tubular neighbourhoods of transversal hypersurfaces. LetMbe a manifold