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Defining a Contact Form Locally. Obstructions?
Posted:
Nov 19, 2012 12:01 AM


Hi,
Let M be a 3manifold with a contact structure C , i.e., a nowhereintegrable 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 1form 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.



