Almost symplectic manifold - Imported Knowlpedia Content

Definition

An almost symplectic manifold is a pair $(M,\omega)$ , where $M$ is a finite-dimensional smooth manifold and $\omega$ is a smooth differential $2$ -form such that $\omega_p$ is nondegenerate at every $p\in M$ . Thus

\omega_p(v,w)=0\text{ for every }w\in T_pM \quad\Longrightarrow\quad v=0.

Equivalently, each tangent space $(T_pM,\omega_p)$ is a symplectic vector space. Closedness is not required: an almost symplectic form may have $d\omega\ne0$ . Requiring $d\omega=0$ gives a symplectic manifold.

Structure and consequences

Nondegeneracy forces $M$ to have even dimension, say $2n$ . The top-degree form $\omega^n$ is nowhere zero, so $\omega$ canonically orients $M$ . These are pointwise consequences of symplectic linear algebra; they do not use $d\omega=0$ .

The bundle map

\omega^\flat:TM\longrightarrow T^*M,\qquad v\longmapsto\iota_v\omega

is a vector-bundle isomorphism. Consequently every one-form determines a unique vector field through contraction with $\omega$ , even when the form is not closed.

Examples and non-examples

Every symplectic manifold is almost symplectic after forgetting closedness. On $\mathbb R^4$ , let

\omega=dx_1\wedge dx_2+e^{x_1}dx_3\wedge dx_4.

Its square is $2e^{x_1}dx_1\wedge dx_2\wedge dx_3\wedge dx_4$ , so it is nondegenerate, while

d\omega=e^{x_1}dx_1\wedge dx_3\wedge dx_4\ne0.

Thus it is almost symplectic but not symplectic. An odd-dimensional manifold is a decisive non-example because an alternating form on an odd-dimensional tangent space is necessarily degenerate.

Conventions and scope

Some authors use “nondegenerate $2$ -form” without naming the resulting structure, while others reserve “almost symplectic” for this precise pointwise condition. The word “almost” records only the missing closedness axiom; it does not specify an almost complex structure, although compatible almost complex structures can be constructed from such a form.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Oxford DOI record. Relevant: Chapter 2, nondegenerate skew forms, and Chapter 3, symplectic forms.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Springer DOI record. Relevant: pp. 3–8, nondegenerate and closed $2$ -forms.