'Contactization' and Symplectization

2012-09-16T05:50:42Z

Given a contact manifold $(M,\lambda)$ we can pass to the symplectization $(\mathbb{R}\times M,\omega=d(e^s\lambda))$ and this is great to bring the machinery of symplectic geometry into the contact world. [Edit]: As pointed out in comments, there is another notion of symplectization (although less used). I am then naturally curious about the other direction.
In particular, for Seiberg-Witten theory and Embedded Contact Homology you can talk about symplectic cobordisms (4-manifolds) with contact boundary. But this assumes that I already have some contact geometry at the ends of my symplectic world.

The immediate question in my mind was: Is there a contactization to pass from a given symplectic manifold to a contact one? Or something in the spirit of it?

And then I come across a paper of Eliashberg-Hofer-Salamon (Lagrangian Intersections in Contact Geometry), and in certain scenarios we do indeed have one. If our symplectic manifold $M$ is $\ast$exact$\ast$, i.e. $\omega=d\alpha$, then $(M\times S^1,dz-\alpha)$ is a contact manifold. Now if we don't have exactness, there is at least a way to contactize $M$ when some positive multiple of $\omega$ represents an $\ast$integral$\ast$ cohomology class in $H^2(M)$, and this is some principal $S^1$-bundle called ''pre-quantization''.

So my ultimate two questions:
1) Is ''pre-quantization'' the only way to contactize here?
2) Is there some useful notion of contactization in other scenarios?
I should clarify that I am not interested in getting contact manifolds in general; I'm not looking for an analog of $T^*M$ (which is a symplectic manifold for any $M$), or more explicitly $\mathbb{P}T^*M$ (which is a contact manifold for any $M$). So while symplectization requires the contact form, contactization should make use of the symplectic form.

Answer by Eugene Lerman for 'Contactization' and Symplectization

2012-09-17T14:43:09Z

The "pre-quantization" construction of a contact manifold out of symplectic manifold predates prequantization by a couple of decades: see Boothby, W. M.; Wang, H. C. On contact manifolds. Ann. of Math. (2) 68 1958 721–734. The analogue of the theorem for symplectic orbifolds is due to Thomas: Thomas, C. B. Almost regular contact manifolds. J. Differential Geometry 11 (1976), no. 4, 521–533.

You may think of the Boothby-Wang construction as constructing a contact fiber bundle over a symplectic manifold with fiber $S^1$. If we look at the construction this way, it can be generalized. See my paper Contact fiber bundles. J. Geom. Phys. 49 (2004), no. 1, 52–66.

'Contactization' and Symplectization - MathOverflow

'Contactization' and Symplectization

Answer by Eugene Lerman for 'Contactization' and Symplectization