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Topic: Defining a Contact Form Locally. Obstructions?
Replies: 2   Last Post: Nov 22, 2012 7:36 AM

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musatov@att.net

Posts: 5
Registered: 11/19/12
Defining a Contact Form Locally. Obstructions?
Posted: Nov 19, 2012 12:01 AM
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Hi,

Let M be a 3-manifold with a contact structure C , i.e., a nowhere-integrable hyperplane distribution.

I am trying to show that we can define locally (in a neighborhood Wx of each point x in M) a

form w , whose kernel is/defines the contact structure.

So, the idea is to define a 1-form whose kernel is precisely the hyperplane distribution.

My idea: for each x , we select first a basis B= {v1,v2} for the plane/hyperplane defined at x. We then

extend the basis B into a basis B' ={v1,v2,v3} for the tangent space at x, and we declare the form

w to satisfy w(v1)=w(v2)=0, and w(v3)=1 (every subspace is the kernel of linear map ). Now,

I don't see what the obstruction is to defining a global contact structure. Any ideas?

Thanks.



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