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 ${\displaystyle M}$. If, in addition, ${\displaystyle \omega }$ is closed, then it is a symplectic structure.^[1][2]

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

Relation to other geometric structures

An almost symplectic manifold is a pair ${\displaystyle (M,\omega )}$ of a smooth manifold and an almost symplectic structure. The manifold ${\displaystyle M}$ can be equipped with extra structures, such as a positive-definite bilinear form ${\displaystyle g}$ (i.e. a Riemannian metric) or an almost complex structure ${\displaystyle J}$. Furthermore, these extra structures can be required to be compatible with each other, making the quadruple ${\displaystyle (M,\omega ,g,J)}$ 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: ${\textstyle \omega }$ closed. Can always be extended to an almost Kähler structure.
3. complex manifolds: ${\textstyle J}$ integrable. Can always be extended to an Hermitian structure.
4. Kähler manifolds: ${\textstyle \omega }$ closed and ${\textstyle J}$ 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 ${\displaystyle 4\subset 3\cap 2}$ and ${\displaystyle 3\cup 2\subset 1}$. 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 ${\displaystyle b_{1}=3}$, but any compact Kähler manifold must have even odd-Betti numbers.^[3]
• Fernández, Gotay and Gray described a compact 4-manifold which is symplectic but not complex, hence not Kähler.^[4]
• The Hopf surface, and more generally Hopf manifolds, are compact complex manifolds but not symplectic, hence not Kähler.
• For a small ${\displaystyle \epsilon >0}$, the 2-form ${\displaystyle \omega _{\varepsilon }=\sum _{i=1}^{n}dp_{i}\wedge dq_{i}+\varepsilon q_{1}dp_{2}\wedge dq_{2}}$ makes ${\displaystyle \mathbb {R} ^{2n}}$ into an almost symplectic manifold that is not symplectic nor complex.

From almost symplectic to almost Hermitian manifolds

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

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