Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question
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 '12 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 '12 at 12:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.