*Let $X$ be the Hopf vector field on the three-sphere. Is there a smooth nowhere zero function $f$ so that the modified vector field $fX$ is not the Reeb vector field of *any* contact form on the three-sphere?*


This question is a follow-up on [aglearner's](https://mathoverflow.net/users/13441/aglearner) interesting [question](https://mathoverflow.net/questions/259390/reeb-flows-on-s3-versus-volume-preserving-flows) on how to construct volume-preserving flows on the three-sphere that are not Reeb vector fields. 


I'm also looking for evidence to support the following **guess**: *If $X$ is a vector field on the three-sphere, there exists a nowhere zero smooth function $f$ so that the vector field $fX$ is not a Reeb vector field for any contact form on the three-sphere.*



Non-Reeb vector fields on the three-sphere
The answer is yes. 


**Proposition.** *Let $\alpha$ be the standard contact form on the three-sphere (for which the Reeb vector field is the Hopf vector field $X$). If $f$ is a strictly positive function on $S^3$ such there exist two Hopf circles $\gamma\_1$ and $\gamma\_2$ for which
$$
\int\_{\gamma\_1} f^{-1} \alpha \neq \int\_{\gamma\_2} f^{-1} \alpha \ ,
$$
then the vector field $fX$ is not the Reeb vector field of any contact form on $S^3$.*


The key idea is that all solutions of $\dot{x}(t) = f(x(t))X(x(t))$ are periodic and that if $fX$ were a Reeb vector field, all these solutions would have the same period. 


Indeed, assume $fX$ is the Reeb vector field of a contact form $\beta$. If $\gamma\_1$ and $\gamma\_2$ are integral curves of $fX$ with periods $T\_1$ and $T\_2$, then
$$
T\_2 - T\_1 = \int\_{\gamma\_2} \beta - \int\_{\gamma\_1} \beta = \int\_\Sigma d\beta \ , 
$$
where $\Sigma$ is a cylinder with boundary $\gamma\_2 - \gamma\_1$ and foliated by Hopf circles (I'm being a bit sloppy with the word *foliated*). It follows that this last integral is zero and so $T\_1 = T\_2$. 


Now we just have to notice that the period of $\gamma\_i$ ($i = 1, 2$) equals
the integral of $f^{-1}\alpha$ over $\gamma\_i$:
$$
\int\_{\gamma\_i} f^{-1} \alpha = \int\_0^{T\_i} f^{-1}(\gamma\_i(t))\alpha(\dot{\gamma\_i}(t)) \, dt = \int\_0^{T\_i} f^{-1}(\gamma\_i(t))\alpha(f(\gamma\_i(t)) X(\gamma\_i(t))) \, dt = T\_i
$$


