Suppose $\kappa$ is a no-where vanishing 1-form, then its kernel is integrable is equivalent to condition $d\kappa \wedge \kappa = 0$.


My question is, can such foliation smoothly deformed such that actually $d\kappa = 0$?


**EDIT: Smooth means real $C^\infty$, and I want the $\kappa$ to be nowhere-vanishing along the deformation.** Thanks Loïc Teyssier pointing out the lack of context.


Thank you.



Deformation of foliation
If $\mathcal F$ is a smooth codimension one foliation on $S^3$ then $\mathcal F$ cannot be defined by a closed $1$-form. This follows from [Novikov's compact leaf theorem](http://en.wikipedia.org/wiki/Novikov%27s_compact_leaf_theorem) which says that one of the leaves of $\mathcal F$ is a torus $T$ bounding a Reeb component. If $\mathcal F$ were defined by a closed $1$-form then every leaf near $T$ would also be compact. Indeed, at a neighborhood of $T$,
 any closed $1$-form defining $\mathcal F$ would admit a primitive as we can normalize local primitives by setting them to be zero at $T$. Level sets of this primitive would be other compact leaves for $\mathcal F$. [Reeb global stability Theorem](http://en.wikipedia.org/wiki/Reeb_stability_theorem) implies that every leaf of $\mathcal F$ is compact, and that $Sˆ3$ admits a map to $S^1$ with connected fibers. But this is impossible since from it we get a surjective map from $\pi\_1(S^3)=0$ to $\pi\_1(S^1)=\mathbb Z$.


The same argument applies to codimension one foliations on compact $3$-folds with finite fundamental group.


