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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.

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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