Let $M$ be a compact [smooth manifold](/page/Smooth%20Manifold) without boundary. For each integer $k\geq 0$, let $\Omega^k(M)$ denote the space of smooth differential $k$-forms on $M$. Let $(\omega_t)_{t\in[0,1]}$ be a smooth family with $\omega_t\in\Omega^2(M)$ such that each $\omega_t$ is closed and nondegenerate. Suppose that the de Rham cohomology class $[\omega_t]\in H^2_{\mathrm{dR}}(M)$ is independent of $t$. Then there exists a smooth isotopy

\begin{align*} \psi:[0,1]\times M\to M \end{align*}

such that, writing $\psi_t:M\to M$ for $\psi_t(x)=\psi(t,x)$, each $\psi_t$ is a diffeomorphism of $M$, $\psi_0=\operatorname{id}_M$, and

\begin{align*} \psi_t^*\omega_t=\omega_0 \end{align*}

for every $t\in[0,1]$.