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?**

Mathematical methods of classical mechanics(1989 translation, p.368). $\endgroup$ – Francois Ziegler Sep 16 '12 at 6:22