First results on Engel structures
3.1. Even contact structures
DEFINITION3.1. LetM be a2n–dimensional manifold andEa distribution onM of codimension one. E is an even contact structure if for every local defining1–formα, the
2–formdαhas maximal rank onE.
In other words,E is an even contact structure if for every local defining formα, the
(2n−1)–formα∧dαn−1has no zeroes. In dimension4an equivalent formulation of this condition is[E,E] =T M. Here[E,E]atpconsists of all vector which can be obtained as commutators of local sections atpofE.
SinceE has dimension2n−1, the rank ofdαE is2n−2. Hencedα
E has a kernel
W ⊂ E of dimension one. Because
d(f α)E =f dα E
,
the line fieldW does not depend on the choice of a local defining formαforE.
DEFINITION 3.2. The line fieldW is the characteristic line field ofE. The foliation induced by this line field is called the characteristic foliation.
COROLLARY 3.3. A manifold which admits an even contact structure has vanishing
Euler characteristic.
Very simple examples of even contact structures can be obtained from contact mani- folds(N,C) as follows: Letπ : M = M → N be a fibre bundle with one–dimensional
fibre. Let
E=
V ∈T M π∗(V)∈ C(π(p))forV ∈TpM .
This distribution is an even contact structure onM. The tangent space ker(π∗)of the fibers
is contained inEand spans the characteristic line field ofE.
Now suppose thatW is a vector field tangent toW and letαbe a local defining form ofE. By definition ofW we have
(LWα)E = (iWdα) E = 0.
HenceLWα is a multiple ofα. This implies thatW preserves the even contact structure. Since we have chosenW arbitrary (but tangent toW) we have
LEMMA3.4. The characteristic foliation of an even contact structureEpreservesE.
Another important property of the characteristic line field is the next lemma.
LEMMA 3.5. LetE be an even contact structure onM andW be the characteristic line field ofE. IfN is a hypersurface transversal toWthenT N∩ Eis a contact structure onH.
IfN0 is another transversal such that two interior pointsp∈N andq∈N0lie on the same leafWp of the characteristic foliation, then the map obtained by following nearby
leaves, and thereby identifying a neighbourhood ofpinN with a neighbourhood of q in
N0, preserves the induced contact structures.
PROOF. Letp∈N andαa defining form forEon a neighbourhood ofp. Thenα
N is a defining form for the distributionT N∩ E onN. By the transversality assumption onN, dαis non–degenerate onT N ∩ E. HenceT N ∩ E is a contact structure.
The statement about the identification of contact structures follows immediately from
Lemma 3.4.
Ifnis even, a contact structure on a manifold of dimension2n−1induces an orienta- tion of this manifold. This has consequences for the relation between the orientability the characteristic line field of an even contact structure and the underlying manifold.
PROPOSITION3.6. LetEbe an even contact structure on a4n–manifoldM. Then an orientation ofM induces an orientation of the characteristic line fieldW and vice versa.
PROOF. Forp∈ M choose a local transversalN toW containingp. By Lemma 3.5,
E induces a contact structure onN. SinceN has dimension4n−1, the contact structure induces an orientation ofN. HenceTpN has a distinguished orientation. Moreover, again since N is transversal toW, we have TpN ⊕ Wp = TpM. Thus an orientation of Wp induces an orientation ofTpM and vice versa.
Since we can identify germs of transversals throughpusing W, this relation between the orientation of Wp andTpM is independent of the choice of the transversal throughp
by Lemma 3.5.
Although the definition of even contact structures on even dimensional manifolds is very similar to the definition of contact structures on odd dimensional manifolds, these two structures are of very different nature. One indication for this is the existence of a distinguished line field contained in an even contact structure. More evidence is contained in the following theorem. For the definitions see [ElM].
THEOREM3.7 (McDuff, [McD]). The property of distributions of corank one to be an
even contact structure is ample. All forms of theh–principle apply. In particular every even dimensional manifold with vanishing Euler characteristic admits an even contact structure.
By Corollary 3.3, the condition on the Euler characteristic of the manifold is necessary. The analogous theorem for contact structures or Engel structures is wrong.
Finally we give an example of how even contact structures may arise on exact sym- plectic manifolds. We will use it in the construction of model Engel structures later.
EXAMPLE 3.8. Let(M, ω) be a symplectic manifold andW a Liouville vector field without zeroes. Henceα=iWωis a nowhere vanishing1–form and
LWω =diWω=ω
by the definition of Liouville vector fields. SinceE = ker(α)has corank one,E contains a symplectic subbundle of codimension one inE. Sodαhas maximal rank on ker(α)and αdefines an even contact structure onM. SinceW is a Liouville vector field,α=iWdα vanishes on ker(α). SoW spans the characteristic line field of ker(α).
3.1.1. Local normal form for even contact structures. Just like contact structures,
even contact structures are locally isomorphic. Still there is a slight difference between the proof of the Darboux theorem for even contact structures and the proof of Theorem 2.9 : Unlike in the case of contact structures, a given defining form does not have a standard expression in general. This is due to the fact that vector fields tangent toWpreserveEbut they do not necessarily preserveα.
A slightly different proof of the Darboux theorem for even contact structures can be found in [BCG].
THEOREM3.9. LetM be a2n-dimensional manifold carrying an even contact struc- tureE andp ∈ M. Then there is a coordinate systemz, x1, y1, . . . , xn−1, yn−1, won a neighbourhood ofpsuch that
dz−
n−1
X
i=1
xidyi
definesEon this neighbourhood.
PROOF. Consider a foliated chart of the characteristic foliationW ofE on a neigh- bourhoodU ofp
ψ:U −→R2n−1×R
such thatψ(p) = (0,0). Letwdenote the coordinate of the second factor in R2n−1 ×R. Thenψ∗(W) = span(∂w). LetN be the hypersurface corresponding toR2n−1× {0}. It is transversal to the distinguished line field ofE. As was shown in Lemma 3.5, the distribution T N ∩ EonN is a contact structure.
By Theorem 2.9, there are coordinates z, x1, y1, . . . , xn−1, yn−1 on a neighbourhood
V ⊂N ofpin the hypersurfaceN such that the contact structureT N∩ EonV is defined by the form (16) α=dz− n−1 X i=1 xidyi.
Consider the product coordinate systemz, x1, y1, . . . , xn−1, yn−1, w on a product neigh-
bourhood diffeomorphic toV ×Rofpand let pr:V ×R→V be the projection on the first factor. Eis invariant under the flow of∂tby Lemma 3.4. So
pr∗α=dz−
n−1
X
i=1
xidyi