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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?

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closed as not a real question by Alexandre Eremenko, Chris Gerig, Hassan Jolany, Lee Mosher, Tim Perutz Dec 20 '12 at 23:55

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

As long as you are giving definitions of things, perhaps you could include definitions of the generalized objects. – S. Carnahan Dec 17 '12 at 0:44
Haskell, by a "generalized symplectic structure" do you mean a generalized complex structure in the sense of Hitchin and Gualtieri? – Tim Perutz Dec 20 '12 at 14:57
This is a good question, I don't understand why someone would like to close it. Anyway, do you know an example a generalised almost complex structure that is not isotopic to an ordinary almost complex structure? – Dmitri Dec 20 '12 at 17:15
I just cast the last vote to close, even though I agree with Dmitri that asking about h-principles for generalized almost complex structures is a perfectly good topic, for the following reasons. Haskell has a track record of asking unclear questions such as Here, he ignored Scott's reasonable request for clarification, and did not correct the non-standard term "generalized symplectic" to "generalized complex" when I pointed it out. When I answered the question, rather trivially, he changed it to a pretty meaningless question. – Tim Perutz Dec 20 '12 at 23:59
Basically, I think Haskell ought to put a bit more effort into making his questions clear and precise, and ought not to change them when someone else has tried to give a clear and precise answer. – Tim Perutz Dec 21 '12 at 0:12
up vote 1 down vote accepted

In his thesis

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.

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I have studied his thesis, in fact with your previous explanation you again repeated my question – Hassan Jolany Dec 20 '12 at 23:00
Isn't this an answer to the question you ask in the last line of your post? – Tim Perutz Dec 20 '12 at 23:07

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