Almost symplectic manifold

In differential geometry, an almost symplectic structure on a differentiable manifold ${\displaystyle M}$ is a non-degenerate two-form ${\displaystyle \omega }$ on ${\displaystyle M}$.^[1] If in addition ${\displaystyle \omega }$ is closed, then it is a symplectic form.

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

Relation to other types of manifolds

An almost symplectic structure is simply a tuple of ${\displaystyle (M,\omega )}$. The manifold can be equipped with extra structure: a positive-definite bilinear form ${\displaystyle g}$ (i.e. Riemannian metric) and an almost complex structure ${\displaystyle J}$. Furthermore, the extra structure can be all compatible with each other, making it into an almost Hermitian manifold.

However, so far we do not assume integrability. With increasing assumptions on integrability, we get increasingly rigid (i.e. less generic) geometric structures:

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

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

The inclusion relations are ${\displaystyle 4=3\cap 2,\;3\cup 2\subset 1}$. The inclusions are strict, in that:

• The Kodaira–Thurston 4-manifold is symplectic but not Kähler. It is a compact nilmanifold. It cannot have a compatible Kähler structure, since its first Betti number is ${\displaystyle b_{1}=3}$, but a compact Kähler manifold must have even odd-Betti numbers.^[2] More here.^[3]
• The Hopf surface, and more generally Hopf manifolds, are compact complex but not Kähler.
• For a small ${\displaystyle \epsilon >0}$, ${\displaystyle \omega _{\varepsilon }=\sum _{i=1}^{n}dp_{i}\wedge dq_{i}+\varepsilon q_{1}dp_{2}\wedge dq_{2}}$ makes ${\displaystyle \mathbb {R} ^{2n}}$ into an almost symplectic manifold that is not symplectic or Hermitian.

Almost Hermitian manifold

The almost Hermitian manifold is constructed by structural group reduction from ${\textstyle \mathrm {Sp} (2n,\mathbb {R} )}$ to ${\textstyle \mathrm {U} (n)}$, and the associated bundle.

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

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

A reduction ${\textstyle P_{\mathrm {U} }\subset P_{\mathrm {Sp} }}$ exists iff ${\textstyle {\mathcal {J}}_{\omega }}$ has a global section. Contractible fiber implies no obstruction classes; a section exists. Its value is a smooth ${\textstyle J}$ with ${\textstyle J^{2}=-I}$ and ${\textstyle \omega (\cdot ,J\cdot )}$ positive-definite.