Designer metric for a vector field

Question

A vector field $V$ on a manifold $M$ admits an invariant metric if there exists a Riemannian metric $g$ with $L_Vg = 0$. How can one characterize the vector fields on $M$ that admit an invariant metric? Note that if $V$ and $W$ both admit invariant metrics then the two metrics need not be the same.

If $V$ is the infinitesimal generator of a circle action then it admits an invariant metric. For if $g$ is any metric on $M$ then the average of $g$ along the $V$-flow is an invariant metric for $V$. However, this averaging trick seems like it might run into trouble when $V$ doesn't generate a circle action due to the $V$-flow stretching $M$.

If $V$ is any vector field and $m\in M$ is a non-singular point for $V$ then there is a neighborhood $U\ni m$ in which there is an invariant metric. This is a simple corollary of the straightening-out theorem for vector fields. Therefore it would seem that this question is really global in nature. Note that I can't just patch together these local $g$ using a partition of unity in the naive way because the partition is not necessarily constant along $V$-lines.

I am particularly interested in understanding whether $V$ on a $3$-manifold $M$ admits an invariant metric when $V$ has a non-trivial first-integral $p:M\rightarrow\mathbb{R}$ and an invariant volume form $\Omega\in\Omega^3(M)$.

Answer 1

Let's focus on the case when $M$ is a compact $3$-dimensional manifold. In such case one can propose a kind of criterion. In some sense this criterion is tautological, but it gives a kind of classification of such vector fields.

Criterion. Consider the subgroup $A$ of $Diff(M)$ generated by the field. Denote by $\overline A$ the closure of $A$ in $Diff(M)$. Then $v$ preserves a metric if and only if $\overline A$ is a compact subgroup of $Diff(M)$. Moreover there exist only three cases. 1) $\overline A\cong S^1$, 2) $\overline A\cong T^2$, 3) $\overline A\cong T^3$.

To understand that $\overline A$ should be a compact group $S^1$, $T^2$ or $T^3$ (in case invariant $g$ exists), note that in case $v$ preserves a metric $g$ on $M$, $\overline A$ is contained in the compact subgroup of isometries $Iso(M,g)$. The latter group is a compact Lie group and the closure of $A$ in it is a compact Abelian subgroup.

In the opposite direction, in case $\overline A$ is compact, we can take any metric $g$ on $M$ and average it over $\overline A$ to get an invariant subroup.

Now consider three possible cases.

1. $\overline A\cong S^1$. This is clearly the case when the vector filed generates a periodic flow on $M$. Here $M$ can be any Seifert fiber spaces in case $A$ is acting without fixed points. If you have fixed points, then they form a collection of circles on $M$ and the complement to such a collection is a Seifert fiber space that fibers over an open surface.

2. $\overline A\cong T^2$. In this case $M$ is either a lens space (a quotient of $S^3$ by an Abelian group) or $T^3$. And the action of $T^2$ is standard, while $A=\mathbb R^1$ is dense in $T^2$.

3. $\overline A\cong T^3$. In this case $M$ is $T^3$ and the action of $v$ is linear.

So, we see that in practice, unless $v$ is periodic we have very few examples.