# 'Contactization' and Symplectization

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?

• A remark: What you say in the cases where $\omega$ is exact or integral is precisely the content of Appendix 4L "Contactification" in Arnol'd's Mathematical methods of classical mechanics (1989 translation, p.368). Sep 16, 2012 at 6:22
• IMO (and I am perhaps a crank about this), there are two things we call "symplectization". The first should be thought of canonically as consisting of all elements of $T^*M$ whose kernel is the contact structure. If the contact structure is co-orientable, there are two connected components, each of which is naturally an $R^+$ bundle. Choosing one and taking the log gives you the symplectization you wrote with the symplectic form you wrote. Sep 16, 2012 at 10:50
• The second symplectization, also $\mathbb{R}\times M$ is where a translation invariant $J$ should live -- this makes sense for a stable Hamiltonian structure, and should really be thought of as a kind of blown up or stretched out version of $(0,1) \times M$. If you consider e.g. the Hofer energy we use for pseudoholomorphic curves (or even better, the definition in the case of a stable Hamiltonian structure), you see that we take a sup over a family of forms that tame $J$, each of which have (uniformly bounded) finite volume. These comments may or may not be relevant to you. Sep 16, 2012 at 10:53

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.

• But this contact fiber bundle need not have a symplectic manifold as its base, so I don't see an explicit connection between symplectic $M$ and contact fiber bundle $N\to M$. For instance, I can build a symplectic space out of any manifold by taking the cotangent bundle, but it defeats the spirit of what I'm after. Sep 17, 2012 at 21:26
• You can build a contact manifold out of any manifold $M$ by taking the projectivization $P(T^*M)$ of its cotangent bundle . Is this what you are after? Sep 17, 2012 at 21:41
• Sorry but what I'm looking for is the exact opposite, i.e. something that depends on $\omega$ (in the same way that the symplectization $M\times \mathbb{R}$ required $\lambda$). Sep 18, 2012 at 3:33
• The contact structure on the contact fiber bundle does depend on the symplectic form on the base. Contact fiber bundles have natural connections and symplectic forms on the base are part of the curvature, if I remember correctly. Sep 18, 2012 at 11:07
• I agree with Chris that a contact fibre bundle doesn't have to have a symplectic base, even when the total space is contact. For example, take the Hopf fibration $S^{3} \to S^{7} \to S^{4}$ with the standard contact forms on $S^{3}$ and $S^{7}$. Sep 19, 2012 at 14:37