Symplectic map - Imported Knowlpedia Content

Definition

Let $(M,\omega_M)$ and $(N,\omega_N)$ be symplectic manifolds. A symplectic map is a smooth map $f:M\to N$ such that

f^*\omega_N=\omega_M,

where $f^*\omega_N$ is the pullback of the differential form $\omega_N$ . This definition does not require $f$ to be injective, surjective, or a diffeomorphism. Nondegeneracy of $\omega_M$ implies that $d f_x:T_xM\to T_{f(x)}N$ is injective at every $x$ , so every symplectic map is a smooth immersion and necessarily $\dim M\leq \dim N$ .

Basic consequences

The identity map is symplectic, and a composite of symplectic maps is symplectic by functoriality of pullback. If $M$ and $N$ have the same dimension, a symplectic map is a local diffeomorphism. A diffeomorphism satisfying the pullback equation is a symplectomorphism; its inverse also preserves the symplectic forms.

These facts follow directly from the definition and the linear nondegeneracy argument in Cannas da Silva, §1.1.

Examples and non-examples

The inclusion of a symplectic submanifold with its restricted form is a symplectic map. A constant map from a positive-dimensional symplectic manifold is not symplectic, because its pullback of every positive-degree form vanishes. A diffeomorphism need not be symplectic: it must satisfy the displayed pullback identity, not merely preserve orientation or volume.

Conventions and scope

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §§1.1–1.2, symplectic linear algebra and symplectic maps.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Publisher record. Relevant: Chapter 1, conventions for symplectic manifolds and maps.