Defining a Contact Structure Globally -- Obstructions?

  • #1
WWGD
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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.
 
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  • #2
lavinia
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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.
 
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  • #3
WWGD
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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
lavinia
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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.
 
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WWGD
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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
lavinia
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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.
 
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