Let $X$ be a non-singular $C^\infty$ vector field on a three manifold $M$. There are some obvious obstructions for finding a volume form that is preserved under the flow given by $X$: If $X$ is singular, or If there's a sphere to which $X$ is transverse (which implies a singularity), or if $X$ is transverse to a torus. Could it be that these are all the obstructions? Or is there a theorem characterizing all topological obstructions?

I have a Riemannian metric on the manifold to begin with, and I want to change the metric so that the vector field (that avoids the above obstructions) becomes divergence free, but is still smooth or at least $C^1$.

Thanks for any hints/answers/references!


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