an extended question of Gromov: Every generalized open almost complex manifold admits a generalized symplectic structure?

2012-12-16T16:57:43Z

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that every symplectic manifold admits an almost complex structure but for open manifolds , the inverse is also correct and infact ;

M.Gromov proved Every open almost complex manifold admits a symplectic structure,

So My question is , how can we extend it for Generalized Almost Complex manifolds(in the sense of Hitchin and Gualtieri )?

Every generalized open almost complex manifold admits a non trivial generalized symplectic structure?

Answer by Tim Perutz for an extended question of Gromov: Every generalized open almost complex manifold admits a generalized symplectic structure?

2012-12-20T22:51:00Z

In his thesis

http://arxiv.org/abs/math/0401221

Marco Gualtieri explains that a generalized almost complex structure on an $n$-manifold $M$ is a reduction of the structure group of $TM \oplus T^\ast M$, which has its canonical hyperbolic quadratic form, from $O(n,n)$ to $U(n,n)$. He points out (p. 48) that since $U(n,n)$ retracts to its maximal compact subgroup $U(n)\times U(n)$, such a reduction implies a reduction of structure for $TM$ to $U(n)$, hence an almost complex structure. By Gromov's symplectic h-principle, an open manifold with a generalized almost complex structure therefore admits a symplectic form, which is an example of a generalized complex structure.

I have nothing to say, however, about the more substantial question of whether the inclusion of the generalized complex structures into the generalized almost complex structures is a highly connected map.

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? - MathOverflow [closed]

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**?

Answer by Tim Perutz for an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**?

an extended question of Gromov: Every generalized open almost complex manifold admits a generalized symplectic structure? - MathOverflow [closed]

an extended question of Gromov: Every generalized open almost complex manifold admits a generalized symplectic structure?

Answer by Tim Perutz for an extended question of Gromov: Every generalized open almost complex manifold admits a generalized symplectic structure?