# Behavior of Reeb vector field (or almost contact 1-form), and “Contact instanton”

I am having trouble "visualizing" the behavior of the two objects mentioned in title. And the question I'm raising might be vague in some sense (or too specific). Also I am hope for some reference related and accessible for a physics student. Another question that I asked a few days ago related is HERE

I see that it is rather easy for a manifold to have an almost contact structure (ACS), obstruction being 3rd integral S-W class. However, it is not always possible to find one for any given nowhere-vanishing vector fields (or a 1-form), trivial example being $S^1 \times S^4$, with vector field (or 1-form) pointing along $S^1$.

So, ACS only select some of 1-forms.

(1) So I am wondering if there is some characteristic property of such 1-forms?

To me, the 1-form along $S^1$ of $S^1 \times S^1$ and $S^1 \times \mathbb{CP}^2$ doesn't have much difference (they are both very nice), but they have different fate. This is one reason that puzzles me.

Also, (2)can some seemingly "problematic" vector fields, say, the ones generating irrational flows, correspond to some ACS?

Finally, on some recent papers on 5d gauge theory and Killing spinors <Twisted supersymmetric 5D Yang-Mills theory and contact geometry>, <Supersymmetric Gauge Theories on the Five-Sphere>, among others, a concept "contact instanton" arises: a (anti-) contact instanton is defined through equation

$$*(\kappa\wedge F) = \pm F$$ where $F$ is field strength of some gauge field. $\kappa$ is contact 1-form of certain K-contact structure.

It seems that the equation only uses a unit-length-nowhere-vanishing 1-form $\kappa$, without using further information from contact structure. (3)So I am wondering if such concept makes sense (or interesting) for any given nowhere-vanishing $\kappa$? (worrying about those "problematic" 1-forms) Is this kind of object studied in math community?

Thank you!

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An almost contact structure is not just a nowhere vanishing one-form. This is true in dimension 3, but in higher dimension you also need a 2-form which is non-degenerate in the kernel of the 1-form. –  Paolo Ghiggini Jun 10 '13 at 4:23