Recently I read that divergence-free vectorfields give rise to volume-preserving flows, but I fail to prove this statement. Let $M$ be an oriented,finite dimensional, smooth manifold equipped with a volume form $\omega$. Furthermore let $X$ be a divergence-free vectorfield on $M$ with respect to $\omega$ with compact support. We know that the vectorfield $X$ gives rise to a global flow $\phi_t(x)$.
Claim: This flow preserves the volumeform, i.e. for any fixed time $t\in \mathbb{R}$ the smooth function $\phi_t:M\rightarrow M$ fulfils $\phi^{*}_t \omega=\omega$, where $f^{*}$ denotes the pullback of a form for a smooth function $f:M\rightarrow N$ between two smooth manifolds $M$ and $N$.
Attempt: So far I tried to work in local coordiantes, hoping that a straightforward calculation yields the desired result.
In local coordinates we can write $\omega_x=f(x) dx^1 \wedge \dots \wedge dx^n$ for a smooth function $f:U\subset M\rightarrow \mathbb{R}$. Then:
$\phi^{*}_t \omega_x=f(\phi_t(x)) d\phi^1_t \wedge \dots \wedge d\phi^n_t$,
where $\phi^i_t=x^i\circ \phi_t$ and $x^i$ are the coordinate functions. Further $d\phi^1_t=\partial_i \phi^1_t dx^i$ (using the summation convention) and so on:
$\Rightarrow \phi^{*}_t \omega_x=f(\phi_t(x)) \partial_{i_1}\phi^1_t \dots \partial_{i_n}\phi^n_t dx^{i_1} \wedge \dots \wedge dx^{i_n}=f(\phi_t(x)) \epsilon^{i_1 i_2\dots i_n}\partial_{i_1}\phi^1_t \dots \partial_{i_n}\phi^n_t dx^1\wedge \dots \wedge dx^n$
We want this to be equal to $\omega_x$ and so we need to show that
$f(x)=f(\phi_t(x)) \epsilon^{i_1 i_2\dots i_n}\partial_{i_1}\phi^1_t \dots \partial_{i_n}\phi^n_t$.
Since $X$ is divergence free, we have:
$0=L_X \omega=d \iota_X \omega$, by Cartans magic formula and since $\omega$ is an $n$-form and therefore closed. Now to establish a connection between $X$ and $\phi_t$ we need to exploit the fact that $\phi_t$ is its flow: $\frac{d}{dt}\phi_t= X(\phi_t)$. In particular $\frac{d}{dt}\phi^i_t(x)=X(\phi_t(x))(x^i)$ for all $x\in M$.
My problem now is that in order to establish a connection between the flow and the vectorfield, we need the time derivative of the flow, which does not occur in the calculations above so far. So how exactly can I exploit this connection?
If it is of any help, we may assume that the manifold $M$ is compact and Riemannian and $\omega$ the Riemannian volume form.
Thanks a lot in advance!
We have $$ \frac{d}{dt}\bigg|_{t = t_0}(\phi_t^*\omega) = \phi_{t_0}^*(\mathrm{L}_X\omega) = \phi_{t_0}^*((\mathrm{div}\;X)\omega) = 0. $$ Hence, $$ \phi_t^*\omega = \phi_0^*\omega = \omega $$ for all $t$.