an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T12:34:17Z http://mathoverflow.net/feeds/question/116537 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116537/an-extended-question-of-gromov-every-generalized-open-almost-complex-manifold an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? Hassan Jolany 2012-12-16T16:57:43Z 2012-12-20T23:19:00Z <p><strong>Definition</strong> (<strong><em>Open Manifolds</em></strong>):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 ;</p> <p>M.Gromov proved Every <strong>open</strong> almost complex manifold admits a symplectic structure, </p> <p><strong>So My question is</strong> , how can we extend it for <strong>Generalized Almost Complex manifolds</strong>(in the sense of Hitchin and Gualtieri )?</p> <p>Every <strong>generalized open almost complex manifold</strong> admits a <strong>non trivial generalized symplectic structure</strong>? </p> http://mathoverflow.net/questions/116537/an-extended-question-of-gromov-every-generalized-open-almost-complex-manifold/116932#116932 Answer by Tim Perutz for an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? Tim Perutz 2012-12-20T22:51:00Z 2012-12-20T22:51:00Z <p>In his thesis</p> <p><a href="http://arxiv.org/abs/math/0401221" rel="nofollow">http://arxiv.org/abs/math/0401221</a></p> <p>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.</p> <p>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. </p>