Symplectic submanifold - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold and let $\iota:S\hookrightarrow M$ be a smooth embedding. The image, or equivalently $S$ with this embedding, is a symplectic submanifold if the pulled-back form

\omega_S=\iota^*\omega

is nondegenerate. Since $d\omega_S=\iota^*(d\omega)=0$ , this makes $(S,\omega_S)$ a symplectic manifold. Pointwise, the condition is that $d\iota_p(T_pS)$ be a symplectic subspace of $T_{\iota(p)}M$ for every $p\in S$ . In particular, $S$ has even dimension.

Normal splitting

At each point of a symplectic submanifold,

T_{\iota(p)}M=d\iota_p(T_pS)\oplus d\iota_p(T_pS)^\omega.

The symplectic orthogonal spaces form a smooth vector bundle over $S$ , called the symplectic normal bundle, and the restriction of $\omega$ to that bundle is nondegenerate. This is the linear splitting behind symplectic neighborhood results.

Examples and contrasts

Every open symplectic submanifold is symplectic with the restricted form. In the product

(M_1\times M_2,\operatorname{pr}_1^*\omega_1+\operatorname{pr}_2^*\omega_2),

the slice $M_1\times\{p\}$ is symplectic. A complex submanifold of a Kähler manifold is symplectic for the restricted Kähler form.

An isotropic submanifold instead has zero restricted form. A positive-dimensional Lagrangian submanifold is therefore never symplectic. A coisotropic submanifold may have a nonzero kernel; if a submanifold is both symplectic and coisotropic, it is open in the ambient manifold.

Conventions and scope

Some sources permit symplectic immersions and speak of “immersed symplectic submanifolds.” This knowl uses the embedded convention: the map $\iota$ is an embedding, and its image carries the subspace topology. Nondegeneracy alone implies closedness of the restricted form only because the ambient form $\omega$ is already closed.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Oxford DOI record. Relevant: §3.3, symplectic submanifolds and symplectic normal bundles.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Springer DOI record. Relevant: symplectic submanifolds and local models.