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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!

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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 $t$.

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  • $\begingroup$ Thanks that helped a lot!! On a side note; is the 'converse' also true? I.e. is there for every volume preserving diffeomorphism $\psi: M \rightarrow M$ a (compactly supported) divergence-free vectorfield whose flow satisfies $\phi_t=\psi$ for some time $t$? $\endgroup$ – Dennis Apr 6 '18 at 14:09
  • $\begingroup$ That's a very interesting question. A necessary condition on $\psi$ is that it is the identity outside a compact subset (what is usually called having contact support). If you allow time-dependent vector fields it can be shown that if $\psi$ is "close enough" (in a certain topology) to the identity then it is isotopic to the identity. The argument uses the famous Moser trick and its proof can be found in the book The Structure of Classical Diffeomorphism Groups by Banyaga. In general I don't know. $\endgroup$ – João Caminada Apr 6 '18 at 15:04
  • $\begingroup$ Just to be clear, the refered isotopy occurs within the group of volume preserving diffeomorphisms with compact support. $\endgroup$ – João Caminada Apr 6 '18 at 16:07
  • $\begingroup$ Thanks a again, that was a very informative answer! $\endgroup$ – Dennis Apr 6 '18 at 21:26

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