Developing Maps and Engel Automorphisms
Koji Yamazaki

TL;DR
This paper investigates the automorphism groups of Engel manifolds, showing they embed into the automorphism group of a related contact orbifold's Cartan prolongation, and constructs examples with trivial automorphism groups.
Contribution
It proves the embedding of Engel automorphism groups into contact orbifold automorphisms and constructs Engel manifolds with trivial automorphism groups.
Findings
Automorphism group embeds into the automorphism group of the Cartan prolongation.
Constructed Engel manifold with trivial automorphism group.
Developing map's properties influence automorphism group structure.
Abstract
A completely nonintegrable -dimensional distribution on a -manifold is called an Engel structure. A -manifold with an Engel structure is called an Engel manifold. The developing map for an Engel manifold is very important tool to determine the Engel structure. Montgomery used it to prove that an Engel automorphism is determined by the values on a global slice. Moreover, Montgomery constructed Engel manifolds whose automorphism group is small. In this paper, we prove that the automorphism group of an Engel manifold is embedded into the automorphism group of the Cartan prolongation of a contact -orbifold, if the developing map is not a covering map. As an application, we will construct an Engel manifold whose automorphism group is trivial.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometry and complex manifolds · Homotopy and Cohomology in Algebraic Topology
Developing Maps and Engel Diffeomorphisms
Koji Yamazaki
Abstract
A completely nonintegrable -dimensional distribution on a -manifold is called an Engel structure. A -manifold with an Engel structure is called an Engel manifold. The developing map for an Engel manifold is very important tool to determine the Engel structure. Montgomery [5] used it to prove that an Engel automorphism is determined by the values on a global slice. Moreover, Montgomery [5] constructed Engel manifolds whose automorphism group is small. In this paper, we prove that the automorphism group of an Engel manifold is embedded into the automorphism group of the Cartan prolongation of a contact -orbifold, if the developing map is not a covering map. As an application, we will construct an Engel manifold whose automorphism group is trivial.
0 Introduction
An Engel structure is a completely nonintegrable -dimensional distribution on a -manifold , and the pair is called an Engel manifold (cf. Definition 1.1).
In 1889, Engel [2] proved a Darboux-type theorem for Engel manifolds, which states that every Engel manifold is locally Engel diffeomorphic to with standard Engel structure. In 1993, Montgomery [5] proved that every stable germ of -dimensional distribution on a -manifold must satisfy the inequality . (See [5].) A pair of positive integers satisfies and if and only if it satisfies , , or . The equality is realized by -dimensional foliations, while the equality is realized by contact structures if is odd and even contact structures if is even. The equality is realized by Engel structures.
An Engel automorphism (resp. a contact automorphism) of an Engel manifold (resp. a contact manifold) is a self-diffeomorphism preserving the Engel structure (resp. the contact structure). The automorphism group (resp. ) is defined as the set of Engel automorphisms (resp. contact automorphisms) of an Engel manifold (resp. a contact manifold ). An Engel manifold has a -dimensional distribution called the characteristic foliation (cf. Definition 1.5). Let be an Engel manifold, and let be a -dimensional submanifold intersecting transversally with the characteristic foliation of the Engel manifold . (Such a submanifold is called a slice of a foliation.) Then, the manifold has a contact structure . (See [9].) It is known that an Engel structure is completely determined locally by the characteristic foliation and the contact structure of a slice of the characteristic foliation.
Montgomery proved the following theorem.
Theorem** (Montgomery [6, Theorem 3 (b)]).**
Let be an Engel manifold, and let be a global slice of the characteristic foliation . (i.e. is a -dimensional submanifold intersecting transversally with all leaves of the foliation .) Two Engel automorphisms and of satisfy an equality if and only if they satisfy an equality for any .
Montgomery’s theorem may be useful for computing Engel automorphism groups. In fact, Montgomery [6, Theorem 6] used this theorem to discover an Engel manifold with small automorphism group. Moreover, in practice, the theorem can be improved. If we could construct a global slice functorially, we might obtain a homomorphism from an Engel automorphism group to a contact automorphism group. Then, Montgomery’s theorem should be rephrased as a claim that the group homomomorphism is injective. However, there is no hope to construct of global slices functorially. To modify this, we use the leaf space instead of a global slice.
Remark, however, that the leaf space is not a manifold in general. We say that an Engel manifold has the trivial characteristic foliation if the leaf space of the characteristic foliation is a manifold.
Any (global) slice of the characteristic foliation of an Engel manifold has a contact structure, and any holonomy of preserve the contact structure. This means the leaf space of has a contact structure. (See [9].) We write for the pair of the leaf space of the characteristic foliation of an Engel manifold and the contact structure. The leaf space is a contact -dimesional manifold if has the trivial characteristic foliation (cf. Proposition 1.9). The correspondence induces a functor from the category of Engel manifolds with trivial characteristic foliations to the category of contact -dimensional manifolds (cf. Section 1.3).
There is also an inverse construction. For a contact -manifold , the projectization of the -plane bundle admits an Engel structure (cf. Section 1.1.3). This Engel manifold is called the Cartan prolongation of , and is denoted by . The correspondence induces a functor from the category of contact -manifolds to the category of Engel manifolds with trivial characteristic foliations (cf. Section 1.3). The functor is fully-faithful (cf. Corollary 1.16 and Section 1.3).
Cartan prolongations can also be defined for a larger category, étale Lie groupoids. In this paper, we only need to discuss Cartan prolongations of contact orbifolds (cf. Remark 1.22). Although the leaf space of the characteristic foliation of an Engel manifold is not a manifold in general, the Cartan prolongation of can still be a manifold. We say that an Engel manifold has the good characteristic foliation if the Cartan prolongation of the leaf space is a manifold (cf. Definition 1.19).
The composition of two functors and induces a group homomorphism between the automorphism groups. We want to find out conditions under which the group homomorphism is injective.
Let be an Engel manifold. (One can, suppose that the Engel manifold has the good characteristic foliation for simplicity. See [9] for the general case.) There exists a local Engel diffeomorphism called the developing map associated with the Engel manifold (cf. Definition 1.13), where a local Engel diffeomorphism is a local diffeomorphism between Engel manifolds preserving the Engel structures (cf. Definition 1.3). The developing map is natural. i.e. For any Engel automorphism , the following diagram is commutative:
[TABLE]
(Where is an Engel automorphism . Recall that is the group homomorphism .)
We now review an outline of the proof of Montgomery’s theorem. Let be a global slice of the characteristic foliation of the Engel manifold . Let be two Engel automorphisms. We suppose that is trivial for simplicity. The condition for any implies an equality . Then, we obtain an equality . Let be the Engel diffeomorphism . Fix a point . Let be a leaf of including the point . Let be a leaf of the characteristic foliation of the Engel manifold including the point . We discuss the following diagram:
[TABLE]
If it satisfies an equality , then it satisfies an equality for any , because is connected and and are lifts of a continous map with respect to the local diffeomorphism . If a local diffeomorphism is not a covering map, then the condition is not necessary. This is because, roughly speaking, a “point” is included in in this case, where is an “infinity point” of (cf. Lemma 2.5). Then, it satisfies the equality .
The main theorem in this paper is the following result.
Theorem 2.1**.**
Let be a connected Engel manifold with good characteristic foliation (cf. Definition 1.19). If the developing map associated with is not a covering map, then the group homomorphism is injective.
In Montgomery’s theorem, Engel automorphisms are determined by the values on a global slice. In Theorem 2.1, Engel automorphisms are determined by the values on the infinity points of the leaves. Roughly speaking, Theorem 2.1 is regarded as an application of Montgomery’s Theorem to
[TABLE]
An infinity point being bounded means that the point is included in the Cartan prolongation . The condition that is not covering implies .
Finally, as an application, we give a positive answer (Theorem 2.13) to the following question.
Question** (AIM Problem lists111http://aimpl.org/engelstr/3/).**
Is there an Engel manifold with trivial automorphism group?
An example of Engel manifolds with trivial automorphism group is constructed in Example 2.12. Our construction is very pathological. However, our intuition predicts that the automorphism group of a generic Engel manifold is trivial, or very small. Mitsumatsu gives the following question, which remains open.
Question** (Mitsumatsu).**
Is there a closed Engel manifold with trivial automorphism group?
In Section 1.1.1, we will give a definition of Engel manifolds. In Section 1.1.2, we will define the characteristic foliation and discuss the functor from Engel to contact. In Section 1.1.3, we will define the Cartan prolongation and discuss the functor from contact to Engel. In Section 1.1.4, we will define the developing map. In Section 1.2, we generalize the discussion in the previous sections. We prove the main theorem in Section 2.2. We give a positive answer to the AIM Problem in Section 2.3.
Acknowledgements
I would like to thank Professor Mitsumatsu and the members of the Saturday Seminar for their discussions.
1 Engel manifolds
In Section 1.1, we review some basic concepts and their properties of Engel manifolds. In Section 1.2, we introduce contact orbifolds to extend the relationship between Engel structures and contact structures. In Section 1.3, we will reformulate the previous discussion in terms of category theory.
1.1 Engel manifolds
An Engel structure is a completely nonintegrable -dimensional distribution on a -manifold . The pair is called an Engel manifold. Any Engel manifold has a -dimensional foliation called the characteristic foliation. The leaf space of the characteristic foliation admits a canonical contact structure. Conversely, a contact -manifold has an Engel manifold called the Cartan prolongation. An Engel structure determines the developing map.
This section is based on [6], [1] and [8]. The detailed proofs are summarized in [9].
1.1.1 Definitions of Engel manifolds
Definition 1.1** (Engel manifolds).**
Let be a -dimensional manifold. An Engel structure on is a -dimensional distribution such that subsets indexed by points determines -dimensional distribution and that subsets indexed by points determines -dimensional distribution , where the sets and are defined as:
[TABLE]
The pair is called an Engel manifold.
Remark 1.2*.*
An Engel manifold can also be defined as follows. Let be a smooth distribution on a manifold. We can regard as a locally free sheaf of some vector fields. Define the sheaf as the sheafification of a presheaf where the presheaf is inductively defined as follows.
[TABLE]
Suppose that is a -dimensional distribution on a -manifold. Then, is an Engel structure if and only if the sheaf is a locally free sheaf whose rank is and the sheaf is a locally free sheaf whose rank is .
A morphism between Engel manifolds is defined as follows.
Definition 1.3**.**
A local diffeomorphism is a smooth map such that the differential is an isomorphism for any .
Let and be Engel manifolds. A local Engel diffeomorphism is a local diffeomorphism with .
A local Engel diffeomorphism is an Engel diffeomorphism if the map is a bijection. An Engel automorphism is an Engel diffeomorphism from an Engel manifold to itself.
For example, developing maps are local Engel diffeomorphisms (cf. Section 1.1.4). A morphism of the category of Engel manifolds is defined as a local Engel diffeomorphism (cf. Section 1.3).
1.1.2 Characteristic foliations
We will discuss a correspondence from Engel to contact. The main subject of this subsubsection is the characteristic foliation. Any Engel manifold has a -dimensional distribution called the characteristic foliation. (Remark that any -dimensional distribution is integrable. An integrable distribution determines a foliation. In this paper, we do not distinguish an integrable distribution with the corresponding foliation unless it is confusing.)
Proposition 1.4** ([6], cf. [9, section 1.2.1]).**
Let be an Engel manifold. Then, there exists a unique -dimensional distribution on the manifold satisfying . Moreover, the distribution included in .
Definition 1.5**.**
The above -dimensional distribution is called the characteristic foliation of the Engel manifold .
The leaf space of the characteristic foliation has a contact structure. (See [9].) However, the leaf space may not be a manifold. In order to describe the correspondence between Engel manifolds and contact manifolds, we give a definition of contact manifolds and trivial characteristic foliations.
Definition 1.6** (contact manifolds).**
Let be a -dimensional manifold. A contact structure on the manifold is a corank distribution such that a -form does not vanish anywhere for any local -form with . The pair is called a contact manifold, and the above -form is called a contact form.
Let and be contact manifolds. A local contactomorphism is a local diffeomorphism with . A local contactomorphism is a contactomorphism if the map is a bijection. A contact automorphism is a contactomorphism from a contact manifold to itself.
Remark 1.7*.*
Any example of local contactomorphism does not appear in this paper. However, it is a necessary definition to neatly describe categorical arguments (cf. Section 1.3), such as functor.
Definition 1.8**.**
Let be an Engel manifold. We say that has the trivial characteristic foliation if the leaf space of the characteristic foliation is a manifold.
Proposition 1.9** ([6], cf. [9, Proposition 1.15]).**
Let be an Engel manifold with trivial characteristic foliation . Let be the leaf space of the foiation . Then, the distribution is well-defined on , and it is a contact structure on .
The pair of the above leaf space of and the above contact structure is a contact manifold. The contact manifold is denoted by . In this paper, we do not distinguish the contact manifold with the manifold unless it is confusing.
1.1.3 Cartan prolongations
We will construct an Engel manifold, called the Cartan prolongation, from a contact -manifold. Let be a contact 3-manifold. Define a manifold as , where is a projectization of a vector space . (i.e. \mathbb{P}(V)=\{\mbox{dimension 1V}\}) Let be the projection.
We define a -dimensional distribution on in the following way. For each with , is a -dimensional linear subspace of . Define a distribution as .
Lemma 1.10** ([6], cf. [9, Lemma 1.16]).**
**
- •
The above distribution is an Engel structure on .
- •
The distribution coincides with the distribution .
- •
The distribution is the characteristic foliation of the Engel manifold .
Definition 1.11** (Cartan prolongation).**
The above Engel manifold is called the Cartan prolongation of the contact manifold . This Engel manifold is denoted by . In this paper, we do not distinguish the Engel manifold with the manifold unless it is confusing.
1.1.4 Developing maps
The leaf space of the Cartan prolongation is contactomorphic to a given contact manifold . The Cartan prolongation is “minimal” object among such Engel manifolds. The “minimality” means a certain universality.
Let be an Engel manifold with trivial characteristic foliation . Let , and let be the projection. Define as for .
Lemma 1.12** ([6], cf. [9, Lemma 1.18]).**
The above map is a local Engel diffeomorphism.
Definition 1.13**.**
The above local Engel diffeomorphism is called the developing map associated with the Engel manifold .
Remark 1.14*.*
Let be an Engel manifold with trivial characteristic foliation . Let be the characteristic foliation of the Engel manifold . The leaf space of the foliation is contactomorphic to the contact manifold . Then, the developing map induces the following bijection .
[TABLE]
This holds even if the leaf space of the foliation is not a manifold (cf. Remark 1.20).
The developing map has a universal property. (Remark that the correspondence induces C from B.)
Proposition 1.15** (cf. [9, Proposition 1.19]).**
Take any contact -manifold and any local Engel diffeomorphism . Then, there exists a unique local contactomorphism such that the following diagram is commutative:
[TABLE]
(Where an Engel local diffeomorphism is induced from .)
Let and be contact manifolds, and let be a local contactomorphism. Then, an Engel diffeomorphism between the Cartan prolongations is induced.
Let be a set of local contactomorphisms from to . Let be a set of local Engel diffeomorphisms from to . The construction induces a map
[TABLE]
Corollary 1.16** ([8], cf. [9, Corollary 1.20]).**
The above map is a bijection.
Proof.
It is obvious because the counit of the adjunction is an isomorphism. (See Section 1.3 and [3].) ∎
The construction induces a group homomorphism . This homomorphism is an isomorphism for any contact manifold because of Corollary 1.16. (Corollary 1.16 means the functor is fully-faithful. cf. Section 1.3)
1.2 Engel manifolds and contact orbifolds
In this subsection, we generalize the discussion in the previous subsection to a relationship between Engel manifolds and contact -orbifolds. First, we define a contact orbifold. (See [9] for a more detailed definition.)
Definition 1.17**.**
Let be an orbifold. A contact structure (resp. an Engel structure) on is a family of contact structures (resp. Engel structures) on each chart such that all local group actions and all transformation maps are contactomorphisms (resp. Engel diffeomorphisms). The pair of the orbifold and a contact structure (resp. a Engel structure) on is called a contact orbifold (resp. an Engel orbifold).
Let , be contact orbifolds (resp. Engel orbifolds). A local contactomorphism (resp. a local Engel diffeomorphism) is a local diffeomorphism , which is a local contactomorphism (resp. a local Engel diffeomorphism) on each chart.
As with Proposition 1.9, the following proposition holds.
Proposition 1.18** (cf. [9, Proposition 2.4]).**
Let be an Engel manifold. Suppose that the leaf space of the characteristic foiation of is an orbifold. Then, the orbifold admits a contact structure.
The pair of the leaf space of the characteristic foiation of the above Engel manifold and the above contact structure is a contact -orbifold, which is denoted by . The Cartan prolongation of a contact -orbifold is a patching of the Cartan prolongations of charts. In general, the Cartan prolongation of a contact -orbifold is an Engel orbifold.
Definition 1.19**.**
Let be an Engel manifold. Suppose that the leaf space of the characteristic foiation of is an orbifold. (This assumption is not essential. See Remark 1.22.) We say that the Engel manifold has the good characteristic foliation if the Cartan prolongation of the contact orbifold is a manifold.
Let be an Engel manifold with good characteristic foliation. The developing map associated with is a patching of the developing maps associated with charts.
Remark 1.20*.*
Let be the characteristic foliation of the Engel manifold . The leaf space of the Cartan prolongation is contactomorphic to the contact orbifold . Then, the developing map induces the following bijection (cf. Remark 1.14).
[TABLE]
An orbifold can be considered as a proper étale Lie groupoid. (See [4].) We will not discuss Lie groupoids in this paper, but we will give you some remarks. In fact, the leaf space of a foliation is generally regarded as an (étale) Lie groupoid by being identified with the holonomy groupoid. The holonomy groupoid of the characteristic foliation of an Engel manifold has a contact structure. (See [9].) We write the holonomy groupoid with contact structure as . The Cartan prolongation of a -dimendional contact étale Lie groupoid is generally an Engel étale Lie groupoid. The necessary and sufficient condition that the Cartan prolongation of a contact étale Lie groupoid is an Engel manifold is given by the following proposition.
Proposition 1.21** ([9, Corollary 4.5]).**
The Cartan prolongation of a -dimendional contact étale Lie groupoid is a manifold if and only if all of the following conditions are satisfied.
- •
* is an orbifold.*
- •
* is positive (cf. [9]).*
- •
* is odd for all where is the isotoropy group at .*
Remark 1.22*.*
If the Cartan prolongation of a -dimendional contact étale Lie groupoid is a manifold, then is an orbifold. That is why there is no need to discuss étale Lie groupoids in this paper.
Remark 1.23*.*
The orbifold is the leaf space of the characteristic foliation of the Engel manifold . In particular, the all holonomy groups of the characteristic foliation of are finite. The all leaves of the characteristic foliation of are compact. Then, the Reeb local stability theorem can be applied (cf. Theorem 2.2).
1.3 Categorical viewpoints
In this subsection, I will reconsider the previous discussion in terms of category theory. Some concrete categories are defined as follows.
Contact the category of contact -manifolds.
the category of Engel manifolds with trivial characteristic foliation. (See Definition 1.8.)
the (higher) category of contact -orbifolds whose Cartan prolongation is a manifold.
the category of Engel manifolds with good characteristic foliation. (See Definition 1.19.)
The construction defines a functor . Let be a functor which sends each Engel manifold to the leaf space of the characteristic foliation. Then, there exists an adjunction . The unit map of the adjunction is the developing maps (cf. Proposition 1.15). Corollary 1.16 means that the functor is fully faithful.
A developing map may not be an Engel diffeomorphism, but is a local Engel diffeomorphism. Morphisms of the category must be local Engel diffeomorphisms. Then, morphisms of the category Contact must be local contactomorphisms for the sake of the above argument. (cf. Remark 1.7)
Similarly, there exists an adjunction . The unit map of the adjunction is the developing map.
Remark that the class may not be a category, but is a higher category. (If an orbifold is defined as a Lie groupoid, then is not a category, but is a -category.) Then, and may not be functors, but are lax functors. However, is a functor, because it is a lax functor between (-)categories. Therefore, we only need to discuss (-)functors in this paper.
2 The main result
We will show the following theorem.
Theorem 2.1**.**
Let be a connected Engel manifold with good characteristic foliation (cf. Definition 1.19). If the developing map associated with is not a covering map, then a group homomorphism is injective.
2.1 Foliations and lemmas
We will use the following theorem without proof (cf. [7]).
Theorem 2.2** (Reeb local stability theorem).**
Let be a manifold, and let be a foliation on . Suppose that a leaf of is compact, and that the holonomy group at a point is finite. Then, for any open neighborhood of , there exists a tubular neighborhood of with a projection such that
- •
,
- •
the open set is a union of some compact leaves of ,
- •
a fiber intersects transversally with for any point .
Remark 2.3*.*
Fix any Riemannian metric on the manifold . A tubular neighborhood of the leaf is constructed by starting geodesics from to normal direction.
Suppose that there exists a slice of . (i.e. The submanifold intersects transversally with the foliation , and the dimension of the manifold is .) Take a point . There is a locally defined metric around such that the slice is totally geodesic around in local. The domain of the metric can be extended to the whole domain by a partition of unity. Then, the tubular neighborhood of the leaf with a projection satisfies . This discussion will be used later.
Moreover, we will prepare some technical lemmas for the proof of Theorem 2.1.
Lemma 2.4**.**
Suppose that , , and and the holonomy is the same as in the Reeb local stability theorem (Theorem 2.2). Moreover, suppose that the dimension of the foliation is . Let be any tubular neighborhood of with a projection as in the Reeb local stability theorem. Then, there exists a complete non-singular vector field on tangent to such that if and only if for any point , where is the flow of .
Proof.
Because , there exists a complete non-singular vector field on the leaf such that if and only if . We regard the foliation as an integrable distribution. Then, is a vector bundle on . The following diagram is a fiber product diagram because a fiber and the foliation intersect transversally for any :
[TABLE]
Let be the pullback of along the map . The vector field is tangent to . The open set is a union of some compact leaves of . The vector field is complete because all leaves in the open set are compact. The vector field is non-singular because is non-singular. Take any point . For any point ,
[TABLE]
Therefore, if and only if . ∎
Lemma 2.5**.**
Let be a local diffeomorphism. There exists an open embedding such that the following diagram is commutative, where :
[TABLE]
Proof.
The affine space is simply connected. There exists a lift of the local diffeomorphism for the universal covering . Then, the local diffeomorphism is monotonic. The map is an open embedding. ∎
2.2 Proof of the main result
First, we prepare several symbols.
Definition 2.6**.**
Let be an Engel manifold, and let be the characteristic foliation of . For any point , denote a leaf of the foliation including as . For any subset , define a subset as .
Let be a connected Engel manifold with good characteristic foliation , and let be the developing map of . Theorem 2.1 follows immediately from the following two lemmas.
Lemma 2.7**.**
If there exists a leaf of the characteristic foliation such that the map (cf. Definition 2.6) is not a covering map, then a group morphism is injective.
Lemma 2.8**.**
If the map (cf. Definition 2.6) is a covering map for any leaf of the characteristic foliation , then the developing map is a covering map.
Proof of Theorem 2.1.
Suppose that the developing map is not a covering map. By the counterpart of Lemma 2.8, there exists a leaf of the characteristic foliation such that the map is not a covering map. Then, the group homomorphism is injective by Lemma 2.7. ∎
We prove these lemmas in turn.
Proof of Lemma 2.7.
Suppose for two Engel diffeomorphisms . Let be the local Engel diffeomorphism , and let . We discuss the following diagram:
[TABLE]
is a leaf of the foliation because of Remark 1.20. Subsets and are leaves of the foliation because the maps and are Engel diffeomorphisms. We obtain an equality . The maps and are not covering maps. Both maps and can be regarded as local diffeomorphisms . By Lemma 2.5, the maps and may be regarded as the map respectively. Reversing the orientation, if necessary, we can assume . We discuss the following diagram:
[TABLE]
A congruence equality mod holds for any point . A function is a constant function because of a differential equality . Then, there exists a integer such that a equality holds. Take a sequence converging to the point . An equality holds. Both sequences and converge to the point because a sequence converge to the point and the maps and are diffeomorphisms. An equality holds. We obtain a solution . Then, we obtain an equality . The two lifts and of a map are identical on the connected domain . Therefore, the group morphism is injective. ∎
Proof of Lemma 2.8.
Denote the characteristic foliation of the Engel manifold as .
Suppose that the map is a covering map for any leaf of the characteristic foliation . We will show that the developing map is a covering map.
Let and be the sets as the following.
[TABLE]
Remark . The subset is open because of the Reeb local stability theorem (cf. Theorem 2.2). (See Remark 1.23 for reasons why the Reeb local stability theorem can be applied.)
Let be the set of connected components of . Then, the map is proper. Therefore, the map is a finite covering map.
We will show that the subset is closed and open in the whole space . It is obvious that the subset is open in the whole space . We have only to show that the subset is closed in the whole space . Take any point in the closure of the subset in the whole space , and let . We will show that the point is included in the subspace . We have only to show that the point is included in the subspace because the subset is closed in the subspace . Let , and let . (Remark, the leaf of the foliation may be not compact, but the leaf of the foliation is compact.) Take a slice of including the point such that the map is an embedding,. (There exists such a slice because the map is a local diffeomorphism.) By Lemma 2.4, there exist
- •
a tubular neighborhood of the leaf with a projection and
- •
a complete non-singular vector field on the open set tangent to the foliation
such that these satisfy the conditions in Lemma 2.4. Specifically these satisfy the following conditions.
- •
,
- •
a fiber intersects transversally with the foliation for any point ,
- •
if and only if for any point , where is the flow of the vector field .
Furthermore, by Remark 2.3, we may assume the following.
- •
.
Let . (Then, the map induces a diffeomorphism .) Let , and let be a pullback of the vector field by the developing map . (The vector field is a non-singular vector field on the open set .) The vector field is complete because the map is a covering map for any leaf of the foliation . Let be the flow of the vector field . Then, a map is a surjective local diffeomorphism. There exists a sequence converging to the point , by definition of the point . Moreover, we can assume that there exists a sequence such that equalitys , because the map is a local diffeomorphism. The sequence converges to the point because of an equality . Then, the sequence converges to the point . The point is included in the set because of a inclusion relation . Let be the order of the holonomy group of the leaf of the foliation . Then, we obtain an equality , where the integer is the degree of the finite covering map . If we take the limit with respect to the index on both sides, we get an equality . This means that a trajectory of the vector field through the point is periodic. In other words, the subset is compact. Therefore, the point is included in the subspace . The component of the subspace is closed in the subspace . The point is included in the component . Therefore, the subset is closed in the whole space .
We will show that the developing map is a covering map.
Case.1
.
Then, we obtain an equality () because the manifold is connected and the subset is closed and open. The developing map is proper. The developing map is a finite covering map.
Case.2
.
Then, we obtain an equality . In particular, all leaves of the foliation are simply connected. All of holonomy groups are trivial. (Then, the leaf space is a manifold.) The developing map is surjective because the map is surjective for any leaf of the foliation (cf. Remark 1.14). Take any point in the Cartan prolongation , and fix a point . Let , and let . Same to the above, take , , , , , , , . The following diagram is commutative:
[TABLE]
We will explain that the map is a diffeomorphism. The map is obviously a local diffeomorphism by its construction. We have to show that the map is bijective. We will construct an inverse map of the map . Take any point . There exists a unique point such that , because the holonomy group at the point is trivial. Then, an equality holds. We obtain an equality because of the correspondence in Remark 1.14. There exists a point such that . Moreover, the point is unique because the vecor field is not periodic. (If it is periodic, then the foliation has a compact leaf. This is in contradiction to .) We define a map by . The map is the inverse map of the map . The map is bijective. Therefore, the map is a diffeomorphism.
The map is obviously a covering map. Then, the developing map is a covering map in local around the point . Therefore, the developing map is a covering map in global, because the point was taken arbitarily. ∎
2.3 Application: An answer of AIM problem
Let be a connected Engel manifold with trivial characteristic foliation (cf. Definition 1.8). Let be the developing map.
Define the function as . (For a set , a number is a cardinal of the set . If the set is a infinite set, then we define .) We call the function the twisting number function.
Proposition 2.9**.**
For any Engel diffeomorphism , the induced diffeomorphism preserves the twisting number function .
Proof.
Let be a point of the leaf space . is a leaf of the foliation . We consider the following diagram:
[TABLE]
The maps and are isomorphic. Then, we obtain an equality . This means that an equality holds. ∎
The following example is the universal covering of the Cartan prolongation of the affine space with standard contact structure. Then, the twisting number function is trivial.
Example 2.10*.*
Let . Denote its coordinates as . Let , (for , ). The pair is an Engel manifold.
The distribution is the characteristic foliation. The distribution is represented as . The leaf space of the foliation is represented as , and the induced contact structure on is represented as . We have an identification . A map is identified with the developing map. Then, the twisting number function is a constant function .
We slightly modify Example 2.10.
Example 2.11*.*
Fix a point and an integer .
Let , for some constant . The subset is open. Then, the subspace is an open submanifold. Let be as in Example 2.10. The pair is an Engel manifold.
A map is identified with the developing map. Then, the twisting number function is the following.
[TABLE]
We further modify these examples to make an Engel manifold whose automorphism group is trivial. This implies that we give a positive answer to the AIM Problem.
Example 2.12*.*
Take a countable dense subset . Let , for some constant . The subset is open. Then, the subspace is an open submanifold. Let be as in Example 2.10. The pair is an Engel manifold.
A map is identified with the developing map. Then, the twisting number function is the following.
[TABLE]
The author had come up with this construction by himself, but he mistakenly thought that the above would not be a manifold. When the author introduced this AIM problem and our main result to Mitsumatsu, he proposed the same construction independently of the author, which led to the final solution.
Theorem 2.13** (Mitsumatsu, Y.).**
Let be the Engel manifold as in Example 2.12. The autorphism group is trivial.
Proof.
For any Engel diffeomorphism , the induced diffeomorphism is the identity on by Proposition 2.9. The diffeomorphism is the identity on the leaf space because the subset is dense. The Engel diffeomorphism is the identity by Theorem 2.1 (or Lemma 2.7). ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Jiro Adachi. Engel structures with trivial characteristic foliations. Algebraic & Geometric Topology , 2(1):239–255, 2002.
- 2[2] Friedrich Engel. Zur invariantentheorie der systeme pfaff’scher gleichungen. Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse , 41:157–176, 1889.
- 3[3] Saunders Mac Lane. Categories for the working mathematician , volume 5. Springer Science & Business Media, 2013.
- 4[4] Ieke Moerdijk and Janez Mrcun. Introduction to foliations and Lie groupoids , volume 91. Cambridge University Press, 2003.
- 5[5] Richard Montgomery. Generic distributions and lie algebras of vector fields. Journal of differential equations , 103(2):387–393, 1993.
- 6[6] Richard Montgomery. Engel deformations and contact structures. Translations of the American Mathematical Society-Series 2 , 196:103–118, 1999.
- 7[7] Itiro Tamura. Topology of Foliations: An Introduction , volume 97. American Mathematical Soc., 2006.
- 8[8] Thomas Vogel. Existence of engel structures. Annals of mathematics , pages 79–137, 2009.
