TITLE: 'Contactization' and Symplectization
QUESTION [10 upvotes]: Given any manifold $M$, we can get a symplectic manifold by taking the cotangent bundle $T^\ast M$ with symplectic form $\omega=\sum dp_i\wedge dq_i$. Given any manifold $M$, we can get a  contact manifold by taking the projectivization of the cotangent bundle $\mathbb{P}^\ast M=(T^\ast M-\lbrace0\text{-section}\rbrace)/{\sim}$ where the contact form arises from the tautological 1-form on $T^\ast M$.
Given any contact manifold $(N,\lambda)$, we can get a symplectic manifold by symplectization $\mathbb{R}\times N$ with symplectic form $d(e^s\lambda)$. Continuing in the same spirit:
Is there a "contactization" to pass from any given symplectic manifold to a contact one, making use of the symplectic data?
Aside: I came 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 exact, 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 integral cohomology class in $H^2(M)$, and this is some principal $S^1$-bundle called ''pre-quantization''.  Is ''pre-quantization'' the only way to contactize here?

REPLY [6 votes]: 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.