Compatibility of G2-structures and symplectic structures

Let $Q$ be a 7-dimensional 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?

edited May 20 2012 at 14:09

Spiro Karigiannis
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asked May 20 2012 at 7:01

Mohammad Shafiee
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It's not so easy to see what you mean. What is the symplectic structure on $Q \times \mathbb R$ exactly? – Spiro Karigiannis May 20 2012 at 14:11

As I said in my question, by Gromov's Theorem QcrossR admits a symplectic structure homotopic to omega, where omega is as above. All known examples of G2 structures are compatible with a symplectic structure in the above sense. But I could not find counterexample but did not see how prove it in general. – Mohammad Shafiee Jun 9 at 12:15