I'm interested in 1-forms $\alpha$ on a 3-dimensional manifold $M$ with the following properties:
1) $\alpha \wedge d \alpha \leq 0$
2) The set $Z(\alpha)$ for which $\alpha \wedge d \alpha = 0$ is controlled in some sense. Initially I would like to consider $Z(\alpha)$ to consist of finitely many points, perhaps to be extended to a situation where $Z(\alpha)$ is compact.
When $Z(\alpha)$ is empty such a 1-form determines (at least locally) a contact structure on $M$. When $Z(\alpha)$ it (locally) determines a contact structure on the manifold minus these 'bad' points. My knowledge of contact geometry is limited and a cursory reading of the literature suggests that what I'm interested in is not covered. I have come across 'almost contact structures', but I do not think these cover my situation either.
My question: is there anything in the literature about contact forms with isolated zeros (by which I mean, there is an $x$ such that $\alpha(x) = 0$ and an open neighbourhood around $x$ on which $\alpha$ is not zero)? Has any work been done on the local breakdown of the non-integrability condition $\alpha \wedge d \alpha \neq 0$ for a contact form $\alpha$?
Motivation: I'm studying structures that arise in chiral nematic liquid crystals. The alignment of the liquid crystal molecules is given by a vector field $v$ (or equivalently 1-form) that describes the orientation at each point. The structures that arise are chiral in the sense that they have a well-defined handedness at each point. The chirality is defined by $v \cdot \nabla \times v$, which is equivalent to the condition I gave for the corresponding 1-form.
One problem with using contact structures in this situation is that the liquid crystal will have defects, line or points where the orientational order is not defined. The vector field $v$ will be zero at these points, and thus $v \cdot \nabla \times v$ will also be zero there. Hence my interest in the kinds of structures I have described.
I have a few ideas and have made some progress on local properties of point defects by looking at the singularity theory of Arnold, I was just wondering if there was any work already out there that might be applicable. I can't find anything, but maybe I don't know the right terms to search for.