Symplectic embedding - Imported Knowlpedia Content

Definition

Let $(M,\omega_M)$ and $(N,\omega_N)$ be symplectic manifolds. A symplectic embedding is a smooth embedding $\iota:M\hookrightarrow N$ satisfying

\iota^*\omega_N=\omega_M.

Equivalently, it is a symplectic map that is an embedding. Thus the differential $d\iota_x$ identifies each tangent space $T_xM$ with a symplectic linear subspace of $T_{\iota(x)}N$ , while the embedding condition also controls the topology of the image. The definition permits positive codimension but forces $\dim M\leq\dim N$ .

Basic properties

The composite of symplectic embeddings is a symplectic embedding. If the source and target have equal dimension, a symplectic embedding is a symplectomorphism onto an open subset of the target. Restricting the target form to the image recovers the source form exactly; preservation merely up to a nonzero scalar is not enough.

Examples and non-examples

The standard inclusion $\mathbb R^{2k}\hookrightarrow\mathbb R^{2n}$ , with coordinates paired in the same order and the standard forms, is symplectic. The inclusion of a symplectic submanifold equipped with the restricted form is another example. An embedding whose tangent images are isotropic subspaces cannot be symplectic in positive dimension, because the pulled-back form then vanishes.

Role in symplectic geometry

Existence of a symplectic embedding is more rigid than existence of a smooth embedding. Symplectic capacities and Gromov's non-squeezing theorem supply obstructions invisible to dimension and topology; see McDuff and Salamon, Chapter 12.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Publisher record. Relevant: Chapters 1 and 12, symplectic embeddings and embedding obstructions.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, symplectic maps and submanifolds.