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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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    $\begingroup$ As long as you are giving definitions of things, perhaps you could include definitions of the generalized objects. $\endgroup$
    – S. Carnahan ♦
    Dec 17, 2012 at 0:44
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    $\begingroup$ Haskell, by a "generalized symplectic structure" do you mean a generalized complex structure in the sense of Hitchin and Gualtieri? $\endgroup$
    – Tim Perutz
    Dec 20, 2012 at 14:57
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    $\begingroup$ 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? $\endgroup$
    – Dmitri Panov
    Dec 20, 2012 at 17:15
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    $\begingroup$ 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 mathoverflow.net/questions/94114/symplectic-leaves 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. $\endgroup$
    – Tim Perutz
    Dec 20, 2012 at 23:59
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    $\begingroup$ 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. $\endgroup$
    – Tim Perutz
    Dec 21, 2012 at 0:12

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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.

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  • $\begingroup$ I have studied his thesis arxiv.org/abs/math/0401221, in fact with your previous explanation you again repeated my question $\endgroup$
    – user21574
    Dec 20, 2012 at 23:00
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    $\begingroup$ Isn't this an answer to the question you ask in the last line of your post? $\endgroup$
    – Tim Perutz
    Dec 20, 2012 at 23:07