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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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The existence of invariant volume form for a vector field $X$ is cohomological problem, i.e. one has to solve a cohomological equation to guarantee the existence of such a form.

In fact, if $\Omega_0$ denotes an arbitrary volume form, one defines the $\Omega_0$-divergence of $X$ as the function $\mathrm{div} X\colon M\to\mathbb{R}$ such that $\mathcal{L}_X\Omega_0 = (\mathrm{div} X)\Omega_0$, where $\mathcal{L}_X$ denotes the Lie derivative. Then, if one considers a function $u\colon M\to\mathrm{R}$ and the volume form $\Omega := e^{-u}\Omega_0$, one can show that $\Omega$ is $X$-invariant if and only if $$Xu = \mathrm{div}X.$$ So, in order the existence of an invariant volume form is equivalent to prove the existence of a solution $u$ for this PDE, and there is no general theory to study such equations.

As you mention above, there is some clear obstruction to solve such an equation, but in general we don't know how to characterize all of them. In some particular cases, for instance when the vector field induces an Anosov flow, we know that periodic orbits are all the obstructions to solve any (sufficiently smooth) cohomological equation, and so, such a vector field admits an invariant volume form if an only if the integral of $\mathrm{div} X$ on every periodic orbit is equal to zero. This is the so called Livsic theorem.

A very good survey about cohomological equations is the following one by A. Katok and E. Robinson: http://www.personal.psu.edu/axk29/pub/KR.pdf

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  • $\begingroup$ Thanks a lot Alejandro. One more question: I'm actually interested in the flow (induced by the vector field) up to orbit equivalence. I think this is equivalent to saying that the vector field can be multiplied by a positive function. Would it then be easier to give a topological charecterization (hopefully something like there are no transverse surfaces)? $\endgroup$
    – Tali
    Commented Apr 30, 2019 at 8:08
  • $\begingroup$ Tali — Multiplying a (smooth) vector field by a positive (smooth) function is only one of several ways to possibly obtain another vector field with equivalent orbits. For instance, for any "structurally stable" vector field V in the sense of Smale after undergoing a sufficiently small C<sup>1</sup> perturbation will result in a field V' that is orbit equivalent to V. But be aware that the equivalence may be only by a homeomorphism, and not a diffeomorphism, of the underlying manifold that is close to the identity. Another way is to carry V to a new one V' by any diffeomorphism of the manifold. $\endgroup$
    – Daniel Asimov
    Commented Oct 29, 2019 at 0:51

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