I have a smooth compact oriented manifold without boundary, $M$, with the volume form $\Omega$ and a Riemannian metric $g$.


Given a function $\phi$ is there any canonical way to obtain a divergence-free smooth vector field $V$ which is orthogonal to $\nabla \_{g} \, \phi $?


Optimally, I would like $V$ to be globally defined and orthogonal to $\nabla \_g \, \phi $, but eventually even locally could do it,


thanks in advance,


n.



construct divergence free vector field manifold
Your problem is perhaps better stated as asking, given a manifold $M$ with a volume form $\Omega$, and a smooth function $\phi$, if there is a vector field $X$ whose flow preserves $\Omega$ and for which $\phi$ is constant along the flow lines of $X$. This problem has no Riemannian metric. Locally, near a point where $d\phi \ne 0$, we can find coordinates in which $\Omega$ is the Euclidean volume form, and in which $\phi$ is the first coordinate function. Why? Use the Moser homotopy lemma to arrange coordinates $x^1, \dots, x^n$ in which $\Omega$ is the Euclidean volume form. Then find a function $f$ so that $\partial f/\partial x^2 \partial \phi/\partial x^1-\partial f/\partial x^1 \partial \phi/\partial x^2=1$, locally, and then replace $x^1$ with $\phi$ and $x^2$ with $f$. Now your vector field can be any divergence free vector field in $x^2,\dots,x^n$, and it gives a vector field tangent to level sets of $\phi$, since it doesn't involve $x^1$.


