# Showing existence of a diffeomorphism preserving volume forms

I want to show that given $$\alpha,\beta$$ two volume forms on a closed manifold such that $$\int_M\alpha = \int_M\beta$$ how can we show that there exists a diffeomorphism $$\phi:M\rightarrow M$$ such that $$\phi^*\beta = \alpha$$.

Attempt: I suspect we have to use Moser's lemma which states that

If $$(\omega_t),t\in[0,1]$$ is a smooth family of symplectic forms which are cohomologue (i.e. $$[\omega_t] = [\omega_0]$$ for $$t\in[0,1]$$). Then there exists a smooth family of diffeomorphisms $$(\phi_t),t\in[0,1]$$ such that $$\phi^*_t\omega_t = \omega_0$$ for $$t\in[0,1]$$.

The idea is to build an isotopy $$\phi_t : M \to M$$, for every $$0\leq t \leq 1$$, such that $$\phi_0^* = Id$$ and $$\phi = \phi_1$$. To do so, we notice that

$$\int_M \alpha = \int_M \beta \implies \exists \ \eta \in \Omega^{(n-1)} (M), \ \beta = \alpha + d \eta$$

Define an $$n$$-form by $$\lambda_t = \alpha + t d\eta$$. This is a volume form (why?). Now, since $$M$$ is compact (you forgot to mention this on your question) an isotopy can be generated by the flow of a time-dependent vector field given by

$$X_t = \left(\frac{d}{dt} \phi_t\right) \circ \phi_t^{-1}$$

It is sufficent to find $$\phi_t^* \lambda_t = \alpha$$, so

\begin{aligned} 0 = \frac{d}{dt} \phi_t^* \lambda_t &= \phi^*_t \left(\mathcal L_{X_t} \lambda_t + \frac{d}{dt}\lambda_t\right)\\&=\phi_t^* (d\iota_{X_t}\lambda_t + \iota_{X_t}\underbrace{d\lambda_t}_{0} + d\eta)\\&=\phi_t^* d\left(\iota_{X_t}\lambda_t + d\eta\right) \end{aligned} where you should try to understand each equality.

So the problem boils down to solving

$$\iota_{X_t}\lambda_t + d\eta = 0 \tag{1}$$

But since $$\lambda_t$$ is a volume form the map $$\mathfrak X(M) \to \Omega^{(n-1)}(M)$$, given by $$X \mapsto \iota_{X}\lambda_t$$ is an isomorphism for every $$t$$, so there exists a unique $$X_t$$ satisfying (1). The flow of $$X_t$$ defines an isotopy satisfying $$\phi^*_t \lambda_t = \alpha$$, letting $$\phi = \phi_1$$ yields the desired isotopy.