[proofplan] Moser's strategy is to join $\omega_0$ and $\omega_1$ by the straight-line path $\omega_t = t\omega_0 + (1-t)\omega_1$ and produce an isotopy $\psi_t$ with $\psi_t^*\omega_t = \omega_0$ by integrating a cleverly chosen time-dependent vector field $X_t$. Differentiating the target identity reduces the problem to finding $X_t$ solving $\mathcal L_{X_t}\omega_t + \dot\omega_t = 0$. Cartan's formula converts this into the exact-form condition $d(\iota_{X_t}\omega_t) = -\dot\omega_t$, which is solvable precisely because the cohomology class $[\omega_0 - \omega_1]$ vanishes in $H^n_{dR}(M)$ — itself a consequence of the equal-integral hypothesis via the top-cohomology theorem for closed orientable manifolds. Volume-form non-degeneracy then produces a unique $X_t$, and compactness of $M$ guarantees completeness of the flow up to time $1$. [/proofplan]

[step:Interpolate via the convex combination $\omega_t = t\omega_0 + (1-t)\omega_1$] Define \begin{align*} \omega: M \times [0,1] &\to \bigwedge\nolimits^n T^*M, \\ (p, t) &\mapsto t\, \omega_0|_p + (1-t)\, \omega_1|_p, \end{align*} and write $\omega_t := \omega(\cdot, t) \in \Omega^n(M)$. Fix $p \in M$. Since $\omega_0$ and $\omega_1$ are volume forms compatible with a fixed orientation, there exist positive constants (depending on $p$ and a local frame) such that $\omega_0|_p$ and $\omega_1|_p$ are positive multiples of the same top form at $p$. Any convex combination $t\omega_0|_p + (1-t)\omega_1|_p$ with $t \in [0,1]$ is therefore also a positive multiple of that top form, in particular nowhere zero. Hence $\omega_t$ is a volume form for every $t \in [0,1]$. The path derivative $\dot\omega_t := \frac{d}{dt}\omega_t = \omega_0 - \omega_1 \in \Omega^n(M)$ is independent of $t$. [/step]

[step:Reduce the equation $\psi_t^*\omega_t = \omega_0$ to a pointwise ODE for $\psi_t$]We seek a smooth family of diffeomorphisms $\psi_t: M \to M$ for $t \in [0,1]$ with $\psi_0 = \operatorname{id}_M$ and \begin{align*} \psi_t^*\omega_t = \omega_0 \quad \text{for all } t \in [0,1]. \end{align*} Differentiating in $t$ and applying the chain rule for pullbacks (see [Pullback Derivative Identity](/theorems/???)), \begin{align*} \frac{d}{dt}(\psi_t^*\omega_t) = \psi_t^*\!\left(\mathcal L_{X_t}\omega_t + \dot\omega_t\right), \end{align*} where $X_t$ is the time-dependent vector field generating $\psi_t$ by $\frac{d\psi_t}{dt}(p) = X_t(\psi_t(p))$ for all $p \in M$. Since $\psi_t$ is a diffeomorphism for each $t$, $\psi_t^*$ is injective on forms. Hence the condition $\frac{d}{dt}(\psi_t^*\omega_t) = 0$ is equivalent to \begin{align*} \mathcal L_{X_t}\omega_t + \dot\omega_t = 0 \quad \text{for all } t \in [0,1]. \end{align*}[/step]

[guided]We want a diffeomorphism $\psi: M \to M$ pulling $\omega_1$ to $\omega_0$ — but writing down such a diffeomorphism directly is hard. Moser's insight is to search for a **family** $\psi_t$ that gradually deforms the identity into the desired $\psi$, always keeping $\psi_t^*\omega_t = \omega_0$ along the way. By the end ($t = 1$): $\psi_1^*\omega_1 = \omega_0$. A smooth family of diffeomorphisms $\psi_t$ starting at the identity is the same as the flow of a time-dependent vector field $X_t$, via the ODE $\frac{d\psi_t}{dt} = X_t \circ \psi_t$, $\psi_0 = \operatorname{id}$. Differentiating the target equation $\psi_t^*\omega_t = \omega_0$ in $t$ should produce an equation for $X_t$. The general formula for the time derivative of a pullback of a time-dependent form under a time-dependent flow is \begin{align*} \frac{d}{dt}(\psi_t^*\omega_t) = \psi_t^*\!\left(\mathcal L_{X_t}\omega_t + \tfrac{\partial \omega_t}{\partial t}\right), \end{align*} which follows from the chain rule: the first term captures how $\psi_t$ changes (producing the Lie derivative along its generator), the second captures how $\omega_t$ itself changes. Setting the right-hand side equal to $0$ and noting that $\psi_t^*$ is injective (as $\psi_t$ is a diffeomorphism), we reduce to: \begin{align*} \mathcal L_{X_t}\omega_t + \dot\omega_t = 0. \end{align*} This is the **Moser equation**. If we can solve it for $X_t$ at each $t$, we can integrate to produce $\psi_t$.[/guided]

[step:Apply Cartan's formula to convert the Moser equation into an exact-form condition] Since $\omega_t \in \Omega^n(M)$ and $\dim M = n$, the exterior derivative vanishes by degree: $d\omega_t \in \Omega^{n+1}(M) = 0$, hence $d\omega_t = 0$. By [Cartan's Magic Formula](/theorems/1535), \begin{align*} \mathcal L_{X_t}\omega_t = \iota_{X_t}(d\omega_t) + d(\iota_{X_t}\omega_t) = d(\iota_{X_t}\omega_t). \end{align*} The Moser equation $\mathcal L_{X_t}\omega_t + \dot\omega_t = 0$ therefore becomes \begin{align*} d(\iota_{X_t}\omega_t) = -\dot\omega_t = \omega_1 - \omega_0. \end{align*} So it suffices to find an $(n-1)$-form $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \omega_1 - \omega_0$, and then solve $\iota_{X_t}\omega_t = \alpha$ for $X_t$. [/step]

[step:Solve the cohomological obstruction using $\int_M(\omega_0 - \omega_1) = 0$]We claim $[\omega_0 - \omega_1] = 0$ in $H^n_{dR}(M)$, equivalently, $\omega_0 - \omega_1$ is exact. The form $\omega_0 - \omega_1 \in \Omega^n(M)$ is closed (automatically, as $\dim M = n$). By the [Top Cohomology of Closed Orientable Manifolds](/theorems/???), $M$ closed, connected, orientable, and $n$-dimensional implies \begin{align*} H^n_{dR}(M) \xrightarrow{\;\;\sim\;\;} \mathbb R, \qquad [\eta] \mapsto \int_M \eta. \end{align*} (If $M$ is not connected, apply this component-by-component; the equal-integral hypothesis should be interpreted on each connected component after a straightforward reduction, or verify that the theorem extends additively. We proceed assuming connectedness; the general case follows by summing.) The hypothesis $\int_M \omega_0 = \int_M \omega_1$ gives \begin{align*} \int_M (\omega_0 - \omega_1) = \int_M \omega_0 - \int_M \omega_1 = 0. \end{align*} Hence $[\omega_0 - \omega_1] = 0$ under the isomorphism, so there exists $\alpha \in \Omega^{n-1}(M)$ with \begin{align*} d\alpha = \omega_0 - \omega_1 = -\dot\omega_t. \end{align*} Setting $\beta := -\alpha$, we get $d\beta = \omega_1 - \omega_0$, the desired right-hand side.[/step]

[guided]This is the **cohomological step**. The Moser equation $d(\iota_{X_t}\omega_t) = \omega_1 - \omega_0$ asks us to exhibit an $(n-1)$-form whose exterior derivative is the prescribed $n$-form $\omega_1 - \omega_0$. An arbitrary closed $n$-form on $M$ need not be exact — the obstruction is its cohomology class $[\omega_1 - \omega_0] \in H^n_{dR}(M)$. For a *closed* orientable connected $n$-manifold, Stokes-type arguments (together with Poincaré duality) yield \begin{align*} H^n_{dR}(M) \xrightarrow{\;\cong\;} \mathbb R, \quad [\eta] \mapsto \int_M \eta, \end{align*} so the integral completely detects the cohomology class. The hypothesis $\int_M\omega_0 = \int_M\omega_1$ is precisely the statement that $[\omega_0 - \omega_1] = 0$, which is equivalent to the existence of an $(n-1)$-form $\alpha$ with $d\alpha = \omega_0 - \omega_1$. Without the equal-integral hypothesis, Moser's theorem would fail: the integral $\int_M \omega$ is a diffeomorphism invariant, since diffeomorphisms preserve volumes up to sign.[/guided]

[step:Uniquely determine $X_t$ from the volume-form isomorphism $\iota_\bullet \omega_t: \Gamma(TM) \to \Omega^{n-1}(M)$] With $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \omega_0 - \omega_1$, we set $\beta := -\alpha$ so $d\beta = \omega_1 - \omega_0$. The Moser equation reduces to the *pointwise linear* equation \begin{align*} \iota_{X_t}\omega_t = \beta \quad \text{for all } t \in [0,1]. \end{align*} *Claim.* For each $t \in [0,1]$ and each $p \in M$, the map $T_pM \to \bigwedge^{n-1}T^*_pM$, $v \mapsto \iota_v(\omega_t|_p)$, is a linear isomorphism. *Proof of claim.* Both vector spaces are $n$-dimensional (since $\dim T_pM = n$ and $\dim \bigwedge^{n-1}T^*_pM = \binom{n}{n-1} = n$), so it suffices to check injectivity. Suppose $\iota_v(\omega_t|_p) = 0$ for some $v \in T_pM$. For any $w \in T_pM$, pair with $w$: \begin{align*} \iota_v(\omega_t|_p)(w, \ast) = \omega_t|_p(v, w, \ast), \end{align*} in particular, for any basis $w_2, \dots, w_n$ of a complement of $\operatorname{span}(v)$, \begin{align*} \omega_t|_p(v, w_2, \dots, w_n) = \iota_v(\omega_t|_p)(w_2, \dots, w_n) = 0. \end{align*} If $v \ne 0$, we could extend $v, w_2, \dots, w_n$ to a basis and $\omega_t|_p$ would vanish on this basis, contradicting the non-vanishing of the volume form $\omega_t|_p$. Hence $v = 0$, so the map is injective, hence bijective. *Consequence.* For each $t$ and $p$, there exists a unique $X_t(p) \in T_pM$ with $\iota_{X_t(p)}(\omega_t|_p) = \beta|_p$. This defines a vector field $X_t$ on $M$, which is smooth jointly in $(p, t)$ because the assignment is inverse to a smooth linear isomorphism depending smoothly on parameters. [/step]

[step:Integrate the time-dependent vector field using compactness of $M$ to obtain the isotopy]We have a smooth time-dependent vector field $X: M \times [0,1] \to TM$, $(p, t) \mapsto X_t(p)$. Since $M$ is closed (compact without boundary), by the [Completeness of Smooth Vector Fields on Compact Manifolds](/theorems/???), every smooth time-dependent vector field on $M$ has a flow defined for all time on which $X$ is defined, in particular on $[0,1]$. Hence there exists a smooth family of diffeomorphisms \begin{align*} \psi: M \times [0,1] &\to M, \\ (p, t) &\mapsto \psi_t(p), \end{align*} with $\psi_0 = \operatorname{id}_M$ and $\frac{d\psi_t}{dt}(p) = X_t(\psi_t(p))$. By construction of $X_t$, \begin{align*} \mathcal L_{X_t}\omega_t + \dot\omega_t = d(\iota_{X_t}\omega_t) + \dot\omega_t = d\beta + \dot\omega_t = (\omega_1 - \omega_0) + (\omega_0 - \omega_1) = 0. \end{align*} Therefore, by the reduction of the Moser equation in step 2, \begin{align*} \frac{d}{dt}(\psi_t^*\omega_t) = \psi_t^*(\mathcal L_{X_t}\omega_t + \dot\omega_t) = 0. \end{align*} Hence $t \mapsto \psi_t^*\omega_t$ is constant on $[0,1]$. Evaluating at $t = 0$: \begin{align*} \psi_0^*\omega_0 = \operatorname{id}_M^*\omega_0 = \omega_0. \end{align*} So $\psi_t^*\omega_t = \omega_0$ for every $t \in [0,1]$. Setting $\psi := \psi_1$, \begin{align*} \psi^*\omega_1 = \psi_1^*\omega_1 = \omega_0. \end{align*} The family $(\psi_t)_{t \in [0,1]}$ is a smooth isotopy from $\psi_0 = \operatorname{id}_M$ to $\psi_1 = \psi$, completing the proof.[/step]

[guided]Having constructed $X_t$, the final task is to flow. For a time-dependent vector field $X_t$ on $M$, the flow $\psi_t$ is defined by the ODE \begin{align*} \tfrac{d\psi_t}{dt}(p) = X_t(\psi_t(p)), \qquad \psi_0 = \operatorname{id}_M. \end{align*} A priori the flow is only defined for short time — it might blow up if integral curves run off to infinity. On a **compact** manifold this cannot happen: integral curves stay inside the compact set $M$, and the standard existence-uniqueness ODE theorem on manifolds guarantees that the flow is defined for all $t$ in the interval on which $X_t$ is defined. Since $X_t$ is defined on $[0,1]$, so is $\psi_t$. This is precisely where the hypothesis "$M$ is closed" is consumed: without compactness we could not integrate up to $t = 1$. By construction, $X_t$ satisfies the Moser equation $\mathcal L_{X_t}\omega_t + \dot\omega_t = 0$, hence $\frac{d}{dt}(\psi_t^*\omega_t) = 0$. The function $t \mapsto \psi_t^*\omega_t$ is constant; at $t = 0$ it equals $\omega_0$, so at $t = 1$ it still equals $\omega_0$: \begin{align*} \psi_1^*\omega_1 = \omega_0. \end{align*} Defining $\psi := \psi_1$, the family $(\psi_t)_{t \in [0,1]}$ provides a smooth isotopy from $\operatorname{id}_M$ to $\psi$. This completes the proof.[/guided]