# flexibility of almost contact Reeb'' vector fields

New version of the question:

Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form whose restriction to $\ker \alpha$ is non-degenerate. In particular, I don't ask that $\omega$ be closed. From this pair, I define a vector field $R$ with the properties that $\alpha(R) = 1$, and $\omega(R, \cdot) = 0$.

A class of examples come from hypersurfaces in a symplectic manifold. In those examples, the 2-form is the restriction of the ambient symplectic form (and is thus closed), and the 1-form is obtained by contracting the symplectic form with a normal vector field.

Suppose $V$ is a closed manifold. Can I deform the pair $(\alpha, \omega)$ so that the only minimal invariant sets of the resulting vector field are non-degenerate periodic orbits?

More generally, does this class of vector fields have any rigidity to it?

Explanation of the change: The first version of this question restricted attention to the class of examples of hypersurfaces in a symplectic manifold. From what Alvarez Paiva commented, it seems unlikely that this is doable. I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.)

Why I am asking this: I started thinking about this question in trying to understand some examples related to the question of putting a contact structure on an almost contact manifold. I don't think that my line of thinking is related to the original question anymore, but I am still curious to understand how soft almost contact is (compared to contact).

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Could you please give a reference to Giroux's paper? –  alvarezpaiva Mar 22 '12 at 12:42
Could you please explain how you use $\nu$ to define $\Sigma$? Is $\nu$ the kernel of the restriction of $\omega$ to $M$? –  Dmitri Mar 22 '12 at 13:55
@Dmitri: I want $\Sigma$ to be a co-oriented hypersurface in $M$, so I have chosen a vector field $\nu$ in a neighbourhood of $\Sigma$ that is transverse to $\Sigma$. I use it to define $R$ by the relationship that $\omega(\nu, R) = 1$ and $R$ is in the kernel of $\omega|_{\Sigma}$. I'm sorry I was a bit terse in my question. –  Sam Lisi Mar 22 '12 at 15:05
What you're asking sounds hard: if you take your hypersurface to be the unit co-sphere bundle of some Riemannian or Finsler metric, then just perturbing it to a metric where all closed geodesics are non-degenerate is the bumpy metric theorem. Of course, the theorem says that such metrics are residual, which is much stronger that what you need. But in exchange, you would need the bumpy metric to have ergodic geodesic flow in a very strong way. –  alvarezpaiva Mar 22 '12 at 17:08
@alvarezpaiva: thanks for this observation. I think it means that I am trying to approach this lemma the wrong way. –  Sam Lisi Mar 22 '12 at 17:33