Almost symplectic manifold

In differential geometry, an almost symplectic structure on a differentiable manifold $M$ is a non-degenerate two-form $\omega$ on $M$. If, in addition, $\omega$ is closed, then it is a symplectic structure.

An almost symplectic manifold is equivalent to an Sp-structure; requiring $\omega$ to be closed is an integrability condition.

Relation to other geometric structures

An almost symplectic manifold is a pair $(M,\omega )$ of a smooth manifold and an almost symplectic structure. The manifold $M$ can be equipped with extra structures, such as a positive-definite bilinear form $g$ (i.e. a Riemannian metric) or an almost complex structure $J$. Furthermore, these extra structures can be required to be compatible with each other, making the quadruple $(M,\omega ,g,J)$ into an almost Hermitian manifold.

However, this definition does not assume any further integrability condition. With increasing assumptions on integrability, one gets increasingly rigid (i.e. less generic) geometric structures:

1. Almost symplectic manifolds. Can always be extended to an almost Hermitian structure.
2. symplectic manifolds: $\omega$ closed. Can always be extended to an almost Kähler structure.
3. complex manifolds: $J$ integrable. Can always be extended to an Hermitian structure.
4. Kähler manifolds: $\omega$ closed and $J$ integrable.

Note that, for instance, an almost symplectic manifold might be extensible to inequivalent almost Hermitian manifolds, which is why they are different concepts.

The inclusion relations are $4\subset 3\cap 2$ and $3\cup 2\subset 1$. All these inclusions are strict, due to the following counterexamples:

• The Kodaira–Thurston 4-manifold is symplectic and complex but not Kähler. Indeed, it is a compact nilmanifold and its first Betti number is $b_1=3$, but any compact Kähler manifold must have even odd-Betti numbers.
• Fernández, Gotay and Gray described a compact 4-manifold which is symplectic but not complex, hence not Kähler.
• The Hopf surface, and more generally Hopf manifolds, are compact complex manifolds but not symplectic, hence not Kähler.
• For a small $ϵ>0$, the 2-form ${\omega }_{\epsilon }=\sum _{i=1}^{n}d{p}_i\wedge d{q}_i+\epsilon q_{1}d{p}_2\wedge d{q}_2$ makes ${\mathbb{ℝ}}^{2n}$ into an almost symplectic manifold that is not symplectic nor complex.

From almost symplectic to almost Hermitian manifolds

Given an almost symplectic manifold $(M,\omega )$, an almost Hermitian structure $(\omega ,g,J)$ can be constructed by means of a structural group reduction from $\mathrm{S}\mathrm{p}(2n,\mathbb{ℝ})$ to $\mathrm{U}(n)$ and the associated bundle construction.

Indeed, the $\omega$-symplectic frame bundle $P_{\mathrm{S}\mathrm{p}}\to M$ has structure group $\mathrm{Sp}(2n,\mathbb{ℝ})$. The subgroup $\mathrm{U}(n)=\mathrm{Sp}(2n,\mathbb{ℝ})\cap \mathrm{O}(2n)$ is the stabilizer of a compatible pair $(J,g)$, with $J^2=-I$ and $g(\cdot ,\cdot )=\omega (\cdot ,J\cdot )$.

The associated bundle$${{𝒥}}_{\omega }:=P_{\mathrm{S}\mathrm{p}}{\times }_{\mathrm{S}\mathrm{p}(2n,\mathbb{ℝ})}(\mathrm{S}\mathrm{p}(2n,\mathbb{ℝ})/\mathrm{U}(n))\to M,$$whose fiber at $x$ is the set of ${\omega }_x$-compatible almost complex structures. The homogeneous space $\mathrm{S}\mathrm{p}(2n,\mathbb{ℝ})/\mathrm{U}(n)$ is nonempty and contractible.

A reduction $P_{\mathrm{U}}\subset P_{\mathrm{S}\mathrm{p}}$ exists if and only if ${{𝒥}}_{\omega }$ has a global section. Contractible fiber implies no obstruction classes; a section exists. Its value is a smooth bundle morphism $J:TM\to TM$ with $J^2=-I$ and $\omega (\cdot ,J\cdot )$ positive-definite.

Further reading

Hazewinkel, Michiel, ed. (2001), "Almost-symplectic structure", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page