Defining a Contact Structure Globally -- Obstructions?

In summary, there is a global 1 form whose kernel is a distribution of 2 planes but there is an obstruction to this being a contact structure.
  • #1
WWGD
Science Advisor
Gold Member
6,910
10,293
Hi,
Let ##M^3## be a 3-manifold embedded in ##\mathbb R^3## and consider a 2-plane field ( i.e. a Contact Structure) assigned at each tangent space ##T_p##. I am trying to understand obstructions to defining the plane field as a 1-form ( Whose kernel is the plane field/ Contact Structure) Given a specific point we can define a local form w as a linear map ##\mathbb R^3 \rightarrow \mathbb R ## whose kernel is the contact plane. I am curious about the obstructions to defining the contact structure through a global 1-form. I suspect it may have to see with the triviality of either the global (tangent) bundle or the 2-subbundle of the tangent bundle . Is this correct? Can anyone add anything and/or give examples?
Thanks. This seems like @lavinia could know the answer.
 
Last edited:
Physics news on Phys.org
  • #2
Hi WWGD

I don't know anything about contact geometry but it seems if there is a smooth distribution of 2 planes ##V## on a 3 manifold that is the kernel of a global 1 form ,then the tangent bundle splits as ##TM=V⊕ξ## where ##ξ## is a trivial line bundle.

Conversely if with respect to some Riemannian metric ##<,>## , one can choose a globally non-zero vector field ##s## that is orthogonal to ##V##, then the 1 form ##ω= <s,>## is a global 1 form whose kernel is ##V##.

Notes:

- Every oriented 3 manifold has trivial tangent bundle.
- If the 3 manifold is orientable and the 2 plane distribution is the kernel of a global 1 form. then the 2 plane distribution is also orientable.
- I think a contact distribution is automatically a sub-bundle since locally it is the kernel of a 1 form. Yes?
- It is possible in a oriented 3 manifold to have an unorientable 2 dimensional sub-bundle with non-trivial normal line bundle. It would be interesting to find one that is a contact structure.

Sorry not to be of more help.
 
Last edited:
  • Like
Likes WWGD
  • #3
lavinia said:
Hi WWGD

I don't know anything about contact geometry but it seems if there is a smooth distribution of 2 planes ##V## on a 3 manifold that is the kernel of a global 1 form ,then the tangent bundle splits as ##TM=V⊕ξ## where ##ξ## is a trivial line bundle.

Conversely if with respect to some Riemannian metric ##<,>## , one can choose a globally non-zero vector field ##s## that is orthogonal to ##V##, then the 1 form ##ω= <s,>## is a global 1 form whose kernel is ##V##.

Notes:

- Every oriented 3 manifold has trivial tangent bundle.
- If the 3 manifold is orientable. then the 2 plane distribution is also orientable.
- I think a contact distribution is automatically a sub-bundle since locally it is the kernel of a 1 form. Yes?
- It is possible in a oriented 3 manifold to have an unorientable 2 dimensional sub-bundle with non-trivial normal line bundle. It would be interesting to find one that is a contact structure.

Sorry not to be of more help.
Actually pretty helpful, Lavinia, thanks, and sorry to put you on the spot :).
 
  • #4
It was fun to learn a little about it.

Here is a different idea of obstruction in a special case. This time there will be a global 1 form whose kernel is a distribution of 2 planes but there will be an nice obstruction to this being a contact structure.

For a closed orientable Riemannian 2 manifold such as the sphere, the tangent unit circle bundle is a closed 3 manifold and is also a principal ##SO(2)## bundle. ##SO(2)## acts on a fiber circle by rotation. A connection 1 form is dual to the line bundle along the fiber circles - this is a trivial line bundle - and its kernel is a distribution of 2 planes that is called the horizontal space of the connection. In general this horizontal space not a contact structure. The obstruction is the curvature. The curvature form must be everywhere non-zero in order for the horizontal distribution to be a contact structure. So for instance, the horizontal distribution on the tangent circle bundle to the unit sphere in ##R^3## is a contact structure.
 
Last edited:
  • Like
Likes WWGD
  • #5
Just to add a few things in the unlikely case you don't know them yet: Contact structures are nowhere-integrable , meaning there is no open set the restriction of whch is the tangent bundle of a manifold. The condition ## \theta \wedge d \theta \neq 0## follows from Frobenius theorem, as the needed condition for a distribution to be somewhere-involutive.
 
  • #6
WWGD said:
Just to add a few things in the unlikely case you don't know them yet: Contact structures are nowhere-integrable , meaning there is no open set the restriction of whch is the tangent bundle of a manifold. The condition ## \theta \wedge d \theta \neq 0## follows from Frobenius theorem, as the needed condition for a distribution to be somewhere-involutive.

Right. Thank you. I read the proof. It is a nice way to characterize it.

In the curvature case, if ##ω## is the connection 1 form then ##dω## is the curvature 2 form and is equal to ##π^{*}-KdV## where ##K## is the Gauss curvature so ##ω∧dω## is zero at a point only if ##K \circ π## is zero at that point. So if ##K## is never zero, ##ω∧dω## is never zero.
 
  • Like
Likes WWGD

1. What is a contact structure?

A contact structure is a geometric structure that describes the local behavior of smooth surfaces in three-dimensional space. It is defined by a smooth vector field that is everywhere tangent to the surface and satisfies a non-degeneracy condition.

2. How is a contact structure defined globally?

A contact structure is defined globally by extending the local vector field to cover the entire surface in a consistent way. This can be done using a partition of unity, which allows the local vector field to be smoothly glued together.

3. What is the purpose of defining a contact structure globally?

Defining a contact structure globally allows us to study the behavior of the surface as a whole, rather than just locally. It also allows us to make comparisons and draw conclusions about different surfaces with similar contact structures.

4. What are some obstructions to defining a contact structure globally?

There are several obstructions that may prevent a contact structure from being defined globally. These include topological obstructions, such as the existence of non-trivial loops on the surface, and analytic obstructions, such as the existence of singularities in the vector field.

5. Can all surfaces have a globally defined contact structure?

No, not all surfaces can have a globally defined contact structure. Some surfaces may have obstructions that prevent a contact structure from being defined globally, while others may not have a contact structure at all. It is an ongoing area of research to determine the conditions under which a surface can have a globally defined contact structure.

Similar threads

  • Differential Geometry
Replies
10
Views
639
  • Differential Geometry
Replies
11
Views
672
Replies
5
Views
2K
Replies
4
Views
2K
  • Differential Geometry
Replies
10
Views
2K
  • Differential Geometry
Replies
8
Views
2K
  • Differential Geometry
Replies
8
Views
2K
  • Differential Geometry
Replies
2
Views
511
  • Differential Geometry
Replies
9
Views
2K
  • Differential Geometry
Replies
4
Views
3K
Back
Top