Construction of an Engel manifold with trivial automorphism group
aa r X i v : . [ m a t h . S G ] J a n Construction of an Engel manifold with trivialautomorphism group
Koji YamazakiFebruary 10, 2020
Abstract
An Engel manifold is a 4-manifold with a completely non-integrable2-distribution called Engel structure. I research the functorial relationbetween Engel manifolds and Contact 3-orbifolds. And I construct anEngel manifold that the automorphism group is trivial.
A 2-distribution D on a 4-manifold E is an Engel structure if D : def = D + [ D , D ] has rank 3, and D : def = D + [ D , D ] has rank 4. The pair ( E, D )is called an Engel manifold. A Engel manifold has an unique 1-distribution L ⊂ D , called the characteristic foliation, such that [ L , D ] ⊂ D .In 1999, Montgomery[2] gives an example with “small” automorphismgroup. Question (AIM Problem lists ) . Is there an Engel structure with trivialautomorphism group?We construct such an Engel manifold. However, the following questionis open.
Question (Mitsumatsu) . Is there a closed Engel manifold with trivial au-tomorphism group?
Acknowledgements
The results is obtained thanks to discussions with members of SaturdaySeminar (Dosemi). I grateful and would like to thank them. http://aimpl.org/engelstr/3/ Engel Manifolds
Definition 1.1.
Let E be a 4-manifold. An Engel structure on E is asmooth rank 2 distribution D ⊂
T E with following condition: D : def = D + [ D , D ] has rank 3, and D : def = D + [ D , D ] has rank 4.The pair ( E, D ) is called an Engel manifold .Let ( E , D ) , ( E , D ) be Engel manifolds. A Engel morphism f : ( E , D ) → ( E , D ) is a local diffeomorphism f : E → E with df ( D ) ⊂ D . Example . Let (
M, ξ ) be a contact 3-manifold, let E = P ( ξ ) def = a x ∈ M P ( ξ x ),and let π : E → M is the projective map. Now, we define a rank 2 distribu-tion D on E in the following way: For each l ∈ E with π ( l ) = x , l ⊂ P ( ξ x )is a line that cross the origin. By the way, we define D l def = dπ − l ( l ) ⊂ T l E .Then, D is an Engel structure on E . This Engel manifold ( E, D ) is calledthe Cartan prolongation , and we denote this P ( M, ξ ).The functor (
M, ξ ) P ( M, ξ ) is fully faithful. In particular,
Aut ( P ( M, ξ )) ∼ = Aut ( M, ξ ) is a “very big” (infinite dimmensional) group.
Propositoin 1.3 ([2] , [1] , [3]) . Let ( E, D ) be an Engel manifold. Thereexists a unique rank distribution ∃ ! L ⊂ D such that [ L , D ] ⊂ D . The above L is called the characteristic foliation of ( E, D ).Any 3-dimensional submanifold M ⊂ E intersecting transversally L hasa contact structure D ∩ T M . And any vector field tangent to L preservesthe “even contact structure” D . In particular, any holonomy of L is a germof contact morphism. Immediately, E/ L has a contact structure D / L .For simplicity, suppose that E/ L is a manifold, Let π : E → E/ L bethe quotient map. We define an Engel morphism φ : E → P ( E/ L , D / L )as φ ( e ) def = dπ ( D e ). This φ is called the development map . And, the functor( E, D ) ( E/ L , D / L ) P ( E/ L , D / L ) induces a group morphism Φ : Aut ( E, D ) → Aut ( E/ L , D / L ) ∼ = Aut ( P ( E/ L , D / L )). Theorem 1.4 (Y.[4]) . Let ( E, D ) be a connected Engel manifold. Supposethat E/ L is a manifold. If the development map φ is not covering, then theabove Φ is injective.Example . E def = R ∋ ( x, y, z, θ ). D def = h ∂ θ , cos( θ ) X + sin( θ ) Y i ( X def = ∂ x − y∂ z , Y def = ∂ y ). Then, ( E, D ) is an Engel manifold. L = h ∂ θ i is the2haracteristic foliation. And, E def = D = h ∂ θ , X, Y i .The leaf space is M def = E/ L ∼ = R ∋ ( x, y, z ). And the contact struc-ture is ξ def = E / L ∼ = h X, Y i . So, the Cartan prolongation is M × S ∼ = P ( M, ξ ); ( x , [ θ ])
7→ h cos( θ ) X x + sin( θ ) Y x i .Then, the development map φ : E → M × S ; ( x , θ ) ( x , [ θ ]) is covering. Example . Fix a point x ∈ R and a number n ∈ Z ≥ . E def = R − { x } × (( −∞ , − nπ ] ∪ [ nπ + ǫ, ∞ )) ( ǫ ∈ (0 , π ]). D is above. Then,( E, D ) is an Engel manifold.Then, the development map φ : E → M × S ; ( x , θ ) ( x , [ θ ]) is not cover-ing. Let ( E, D ) be a connected Engel manifold. Suppose that E/ L is a man-ifold.Define σ : E/ L → Z ≥ ∪ {∞} by σ ( L ) def = min { φ − ( y ) | y ∈ P ( D / L ) L } . Definition 1.7.
The above σ is the twisting number function of ( E, D ). Propositoin 1.8.
For any f ∈ Aut ( E, D ) , the induced automorphism E/ L → E/ L preserves the twisting number function σ .Proof. Obvious.
Take a countable dense subset Q = { x n } ∞ n =1 ⊂ R . E def = R − S n { x n } × (( −∞ , − nπ ] ∪ [ nπ + ǫ, ∞ )) ( ǫ ∈ (0 , π ]). D is as with Example 1.5. Theorem 2.1.
Aut ( E, D ) is trivial.Proof. The twisting number function σ is the following: σ ( x ) = ( n ( x = x n ) ∞ ( otherwise )For any f ∈ Aut ( E, D ), the induced automorphism f : R → R is theidentity on Q . Because Q ⊂ R is dense, f is the identity on R . So f is theidentity. 3 eferences [1] Jiro Adachi. Engel structures with trivial characteristic foliations. Alge-braic & Geometric Topology , 2(1):239–255, 2002.[2] Richard Montgomery. Engel deformations and contact structures.
Trans-lations of the American Mathematical Society-Series 2 , 196:103–118,1999.[3] Thomas Vogel. Existence of engel structures.
Annals of mathematics ,pages 79–137, 2009.[4] Koji Yamazaki. Non-covering development maps and engel automor-phisms. arXiv preprint 1903.02362arXiv preprint 1903.02362