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.

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

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:

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:

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 ${\textstyle g(\cdot ,\cdot )=\omega (\cdot ,J\cdot )}$.