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A symplectic manifold is a pair $(M,\omega)$, where $\omega$ is a non-degenerate closed two-form. When $M$ is compact, Hodge decomposition implies that such manifolds have non-zero second betti-number, and one sometimes analogues of the Hard Lefschetz Theorem.

What happens if one drops the assumption that $\omega$ is closed. For sure $\omega$ will not give a class, but maybe something interesting will happen. What are examples of such spaces.

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  • $\begingroup$ If $\omega$ s not closed it will not give a class at all —zero or not. $\endgroup$ Oct 28, 2014 at 18:52
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    $\begingroup$ If $\omega$ is not closed, you get what is called an almost symplectic manifold. The structure is then simply a reduction of structure group (from $GL_{2n}(\mathbb{R})$ to the symplectic group) for the tangent bundle of the manifold. Essentially, this is more homotopy theory than geometry... $\endgroup$ Oct 28, 2014 at 19:12
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    $\begingroup$ While it's true that the condition of admitting such a reduction is topology, once you have one the situation is more geometrical: when $\omega$ is closed there is no local structure but now there is, namely $d\omega$ itself. These geometries can be classified into four types. $\endgroup$ Oct 28, 2014 at 21:23
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    $\begingroup$ Two more comments, concerning examples: the only spheres admitting an almost symplectic structure are $S^2$ and $S^6$, with $S^2$ being symplectic and $S^6$ not. Moreover, an application of the $h$-principle methods show that almost symplectic structures on open manifolds are homotopic (through almost symplectic structures) to symplectic structures, this is known as flexibility. $\endgroup$ Oct 29, 2014 at 8:22
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    $\begingroup$ @OldřichSpáčil, I believe it's in Gray and Hervella's The sixteen classes of almost Hermitian manifolds and their linear invariants (1980), but I can't seem to find the paper in my collection (and I don't belong to an institution). You get a finer classification when you have the full almost-Hermitian structure, which is the main part of the paper. The idea is to decompose $\Lambda^3V$ into irreducible reps of the non-compact symplectic group, and apply that to $d\omega$. Similar classifications exist for various other structures, and there are subsequent papers. $\endgroup$ Oct 29, 2014 at 22:27

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