Let $Q$ be a 7dimensional smooth manifold endowed with $G_2$structure $\varphi$. It is easy to see that $Q \times \mathbb R$ admits an almost symplectic structure $\omega$ such that reduction of $\omega$ on $Q$ is equal to contraction of $\varphi$ by $Y$, for some vector field $Y$ on $Q$. By Gromov's Theorem , $Q \times \mathbb R$ admits a symplectic structure. Can we replace "almost symplectic structure" by " symplectic structure" in above statement?
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