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 μ, ∇g,μ×X, is defined by the formula diXg=i∇g,μ×Xμ.
When is it possible to choose a metric and volume form such that ∇g,μ×X=λX,
where
λ 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