Curl operators parameterized by the set of Riemannien metrics on a 3-manifold

Question

Let $M$ be an orientable 3-manifold. On $M$, fix a vector field $X$. The curl of $X$ relative to the Riemannien metric $g$ and the volume form $\mu$, $\nabla_{g,\mu}\times X$, is defined by the formula $$ di_X g=i_{\nabla_{g,\mu}\times X}\mu.$$

When is it possible to choose a metric and volume form such that $$ \nabla_{g,\mu}\times X=\lambda X,$$ where $\lambda$ is a nowhere vanishing function?

There are many $X$ for which such a metric and volume form can be found. In particular, $X$ that arise as Reeb vector fields relative to some contact 1-form on $M$ are all examples (http://www.math.upenn.edu/~ghrist/preprints/beltrami.pdf).

This post imported from StackExchange MathOverflow at 2015-05-28 07:26 (UTC), posted by SE-user Josh Burby

Comments

Do you require that the volume form be that given by the metric (and orientation), or is it chosen separately? I'm guessing the former but your notation makes me unsure.

This post imported from StackExchange MathOverflow at 2015-05-28 07:26 (UTC), posted by SE-user Paul Reynolds

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@Paul Reynolds : Sorry for not being clear. I'm not requiring the volume form to be the one determined by the metric.

This post imported from StackExchange MathOverflow at 2015-05-28 07:26 (UTC), posted by SE-user Josh Burby

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In that case if you can find $g$ and $\mu$ then you can find them so that $\lambda = 1$. I was going to say something about the case $X$ is non-vanishing, but it's pretty much what is said in the paper you linked to.

This post imported from StackExchange MathOverflow at 2015-05-28 07:26 (UTC), posted by SE-user Paul Reynolds