I am trying to learn about Contact Geometry and Open Book Decompositions. I went through the example of the Hopf Fibration for $S^3$ and how you can see a contact structure.
I am now trying to do the same with $S^1 \times S^2$ using the open book decomposition with binding B=$S^1$ x {N,S} (North and South Poles) and pages consisting of annuli the sweep out the inside of the picture:
I have been reading through Etnyre's "Lectures on Open Book Decompositions", and am trying to show that this specific open book supports a contact structure.
By support I mean that given the contact form $\alpha$ for my contact strucutre:
1) d$\alpha$ is a positive area form on each page and
2) $\alpha$ > 0 on B
Now I thought I could do this locally using Darbouxs Theorem and choose the contact strucutre on $\mathbb{R}^3$ given by the form $\alpha$ = dz + $r^2$d$\theta$.
It makes sense on the binding that $\alpha$ would be positive since we could view the binding as just pointing in the z direction in a neighborhood of a point on it (green line).
But given any page of the book I can't see how d$\alpha$ = 2r dr$\wedge$d$\theta$ is an area form on the page. Locally it seems like at a point on the page this would not be true as for each page I am fixing my angle going around the $S^1 \times S^2$.
Is this the correct way to go about things?
I tried to also attack this problem by trying to put $S^1 \times S^2$ into $\mathbb{R}^5$ and take a contact form there and restrict it down to my space.
I also thought there might be some standard contact form for this 3- manifold in coordinates ($\psi,\theta,\phi$), but I have no idea what that would look like.
Thanks for any help, and sorry the pictures are so big.