What foliations are symplectic foliations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:37:09Z http://mathoverflow.net/feeds/question/119586 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119586/what-foliations-are-symplectic-foliations What foliations are symplectic foliations? Giuseppe 2013-01-22T17:41:18Z 2013-01-23T16:46:42Z <p>On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poisson structure on $M$?</p> http://mathoverflow.net/questions/119586/what-foliations-are-symplectic-foliations/119677#119677 Answer by Nicola Ciccoli for What foliations are symplectic foliations? Nicola Ciccoli 2013-01-23T16:46:42Z 2013-01-23T16:46:42Z <p>Even though each leaf can carry a symplectic structure still there may be no global Poisson structure. Some steps in understanding the problem (and as far as I know the only) were contained in a work by Bertelson <a href="http://arxiv.org/abs/math/0010191" rel="nofollow">http://arxiv.org/abs/math/0010191</a>, also Commun. Contemp. Math. 3 441-456 (2001). This paper gives some indications on what may go wrong.</p> <p>On the positive side some results are obtained by the same author at the end of her paper "A h-principle for open relations invariant under foliated isotopies",J. Symplectic Geom. Volume 1, Number 2 (2002), 369-425. </p> <p>The underlying idea is that if the leaves are open manifolds then the most easy obstructions vanish, but nevertheless the "openess" of the leaves should be somehow uniform,</p> <p>All this, of course, taking for granted that "foliation" means "regular foliation". </p> <p>For the non regular case I doubt anything interesting, apart from examples, is known.</p>