Existence of Engel structures [Elektronische Ressource] / vorgelegt von Thomas Vogel

Vogel Thomas

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• Mathematics

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• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2004
• Nombre de lectures: 25
• Langue: English

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Existence of Engel structures
Thomas VogelExistence of Engel structures
Dissertation zur Erlangung des Doktorgrades
an der Fakultat¨ fur¨ Mathematik, Informatik und Statistik
der Ludwig–Maximilians–Universitat¨ Munchen¨
Vorgelegt am 17. Marz¨ 2004 von
Thomas VogelErstgutachter Prof. Dieter Kotschick, D. Phil.
Zweitgutachter Prof. Dr. Kai Cieliebak
¨auswartige Gutachter Prof. Dr. Yakov Eliashberg (Stanford University, USA)
Prof. Dr. Michele` Audin (Universite´ Louis Pasteur, Strasbourg, France)
Tag der mundlichen¨ Prufung¨ 13. Juli 2004Contents
Chapter 1. Introduction 3
1.1. Contact topology 4
1.2. First results on Engel structures 6
1.3. Constructions of Engel manifolds 7
Chapter 2. Contact topology 13
2.1. Basic results on contact structures 14
2.2. Legendrian curves 18
2.3. Facts from the theory of convex surfaces 24
2.4. Bypasses in overtwisted contact structures 27
Chapter 3. First results on Engel structures 33
3.1. Even contact structures 34
3.2. Engel structures – Definition and first examples 37
3.3. Topology of Engel manifolds 49
3.4. Deformations of Engel structures 51
3.5. Engel vector fields 57
3.6. Analogues of Gray’s theorem 61
Chapter 4. Round handles 67
4.1. Generalities 68
4.2. Model Engel structures on round handles 76
4.3. Relations between the models onR andR 831 2
Chapter 5. Closed Engel manifolds from round handles 93
5.1. Gluing Engel structures 94
5.2. Vertical modifications of transversal boundaries 96
5.3. Doubles 99
5.4. Modifications of rotation numbers and framings 104
5.5. New Engel manifolds – Doubles 107
5.6. Connected sums 110
Chapter 6. The existence theorem 117
6.1. Model Engel structures on round handles of index2 118
6.2. Tori in overtwisted contact manifolds 124
6.3. Model Engel structures onR 1293
6.4. Proof of Theorem 6.1 136
Chapter 7. Geometric examples 143
7.1. contact manifolds 144
7.2. Geometric Engel – Prolongation 149
7.3. Engel manifolds – Remaining geometries 155
Bibliography 163
Lebenslauf 165
1CHAPTER 1
Introduction
Distributions are subbundles of the tangent bundle of a manifold. It is natural not to
consider general distributions but to make geometric assumptions, for example integrabil
ity. In this case the distribution is tangent to a foliation. Another possibility is to assume that
a distribution is nowhere integrable. Important examples of this type are contact structures
on manifolds of odd dimension. Contact structures are hyperplane fields on manifolds of
odd dimension which are maximally non–integrable everywhere. On3–dimensional mani
folds properties of contact structures reflect topological features of the underlying manifold
in a surprising way.
An Engel structure is a smooth distributionD of rank2 on a manifoldM of dimension
4 which satisfies the non–integrability conditions
rank[D,D] = 3 rank[D,[D,D]] = 4,
where[D,D] consists of those tangent vectors which can be obtained by taking commuta
tors of local sections ofD.
If one perturbs a given Engel structure to a distribution which is sufficiently close to
2D in theC –topology, then the new distribution is again an Engel structure. Moreover all
Engel structures are locally isomorphic, i.e. every point has a neighbourhood with local
coordinatesx,y,z,w such that the Engel structure is the intersection of the kernels of the
one–forms
(1) =dz xdy =dx wdy.
This normal form was obtained first by F. Engel in [Eng].
The stability property described above is called stability in the sense of singularity
theory. R. Montgomery has classified the distributions with this stability property.
THEOREM 1.1 (Montgomery, [Mo1]). If a distribution of rank r on a manifold of
dimensionn is stable in the sense of singularity theory, thenr(n r) n. It belongs to
one of the following types of distributions.
n arbitrary r = 1 foliations of rank one
n arbitrary r =n 1 contact structures ifn is odd,
even contact structures otherwise
n = 4 r = 2 Engel structures
So Engel structures are special among general distributions and even among the stable
distribution types in Theorem 1.1 they seem to be exceptional. On the other hand they
appear very naturally. For example a generic plane field on a four–manifold satisfies the
Engel conditions almost everywhere. Engel structures can also be constructed from con
tact structures in a natural way. Certain non–holonomic constraints studied in classical
mechanics also lead to Engel structures.
34 1. INTRODUCTION
One–dimensional foliations are extensively studied in the theory of dynamical systems.
Contact structures have attracted much interest during recent years. On manifolds of di
mension3 the distinction between overtwisted and tight contact structures due to Y. Eliash
berg has lead to many interesting results. Using convex integration, one can find even con
tact structures on all manifolds with vanishing Euler characteristic. Therefore even contact
structures seem to be less interesting. In contrast to this, and just like for contact struc
tures, the standard conditions which ensure the validity of anh–principle are not satisfied
by Engel structures.
An Engel structure induces a flag of distributions
(2) W DE = [D,D]TM
such that each distribution has corank one in the next one. Here E is an even contact
structure. We say that the foliationW is associated to the even contact structure. Usually
it is called the characteristic of the even contact structureE. The flow of vector
fields tangent to the characteristic foliation preservesE.
The existence of the flag (2) implies strong restrictions for the topology of Engel ma
nifolds. The following theorem can be found in [KMS]. It was known already to V. Gersh
kovich. Unfortunately his preprint [Ger] was not available to the author.
THEOREM 1.2. An orientable4–manifold which admits an orientable Engel structure
has trivial tangent bundle. Every Engel manifold admits a finite cover which is paralleliz
able.
According to [KMS] the preprint [Ger] suggests an incomplete proof of the converse
of Theorem 1.2. The Euler characteristic of an Engel manifold vanishes since there is a
non–singular line field onM, or by parallelizability.
In the literature one can find two constructions of Engel structures. The first one is
1called prolongation. With this method one finds Engel structures on certain S –bundles
over three–dimensional contact manifolds. The Engel structures obtained in this way
are relatively simple, for example their characteristic foliations are given by the fibers
1of the S –bundle. This method is described in [Mo2]. The second construction is due
to H. J. Geiges, cf. [Gei]. It yields Engel structures on parallelizable mapping tori. Its
major disadvantage is that one can say nothing about the characteristic foliation or other
properties of the Engel structure.
In this thesis we develop three new constructions of Engel manifolds. Our main result
is the converse of Theorem 1.2
THEOREM 1.3. Every parallelizable4–manifold admits an orientable Engel structure.
Note that Theorem 1.3 can be proved on open manifolds using theh–principle for open,
Diff–invariant relations, cf. [ElM]. Thus our proof of Theorem 1.3 treats the case of closed
manifolds.
1.1. Contact topology
In Chapter 2 we discuss contact structures. Contact structures are maximally non–
integrable hyperplane fields on manifolds of odd dimension. In Engel manifolds contact
structures appear naturally on hypersurfaces transverse to the characteristic foliation and
the theory of contact structures on three–dimensional manifolds will play an important
role in our constructions of Engel structures. Therefore we are mostly concerned with the
case of manifolds of dimension 3. Much of the material presented here can be found in
[Aeb, EH, Gir1, Ho].
One of the most important properties of contact structures on closed manifolds is
Gray’s stability theorem which is valid in all odd dimensions.1.1. CONTACT TOPOLOGY 5
THEOREM 1.4 (Gray, [Gr]). Let C be a smooth family of contact structures on at
compact manifold. Then all contact structuresC are isotopic.t
We will use this theorem frequently. In particular in our first construction of Engel
structures we need the construction of the isotopy. We also show that there is a one–to–
one correspondence between contact vector fields and differentiable functions on a contact
manifold. In Section 2.1.3 we derive the local normal form of contact structures from Dar-
boux’s theorem about local normal forms for symplectic manifolds. Like Gray’s theorems
these results are valid for contact structures on odd dimensional manifolds.
For the remaining part of Chapter 2 we discuss contact structures on3–manifolds.
In Section 2.2 we discuss Legendrian curves. Legendrian curves are curves which are
tangent to the contact structure. We show that every curve is isotopic to a Legendrian one
relative to the endpoints. The classical invariants of null–homologous Legendrian curves
in a contact manifold are the Thurston–Bennequin number and the rotation number from
[Ben]. These invariants allow us to distinguish between Legendrian curves up to isotopy
through Legendrian curves. Stabilization of Legendrian curves is an efficient method to
modify the Le isotopy type of a Legendrian curve. It is explained in S