Symplectic embeddings

Definition

Symplectic embeddings are smooth, injective mappings between symplectic manifolds that preserve the symplectic structure, meaning the differential of the embedding takes the symplectic form on the source manifold to that on the target manifold. This concept is essential for understanding how different symplectic manifolds relate to each other and has significant implications in the study of symplectic capacities and Gromov's theorem, which link the geometry of these structures with topological and analytical properties.

5 Must Know Facts For Your Next Test

1. Symplectic embeddings provide a framework for comparing different symplectic manifolds and determining whether one can be 'fit' into another while preserving their symplectic structure.
2. The existence of a symplectic embedding from one manifold to another can often be determined using symplectic capacities, which serve as obstructions to embedding.
3. Gromov's theorem highlights that if a symplectic manifold is non-compact, there might be no embeddings into certain compact manifolds, emphasizing constraints on dimensions.
4. The study of symplectic embeddings is closely tied to questions of flexibility and rigidity within symplectic geometry, impacting how we understand geometric structures.
5. Constructing explicit examples of symplectic embeddings often involves techniques from Hamiltonian dynamics and requires careful analysis of the underlying symplectic structures.

Review Questions

• How do symplectic embeddings relate to the concept of symplectic capacities and what role do they play in determining whether one manifold can embed into another?
  • Symplectic embeddings are intimately connected to symplectic capacities, as these capacities serve as invariants that measure the 'size' of a symplectic manifold. If the capacity of one manifold is less than that of another, then no symplectic embedding can exist between them. This relationship helps mathematicians understand the limitations and possibilities of how different manifolds interact within the realm of symplectic geometry.
• Discuss how Gromov's theorem impacts our understanding of symplectic embeddings in both compact and non-compact manifolds.
  • Gromov's theorem asserts that certain non-compact symplectic manifolds may not allow embeddings into compact ones. This means that when dealing with non-compact manifolds, we must consider additional constraints when exploring potential embeddings. The implications of this theorem guide researchers in identifying boundaries and limitations within the study of symplectic embeddings, ultimately influencing how we visualize and work with these structures.
• Evaluate the significance of constructing explicit examples of symplectic embeddings in advancing our understanding of the underlying geometric properties of symplectic manifolds.
  • Constructing explicit examples of symplectic embeddings serves as a powerful tool for advancing our knowledge of the geometric properties inherent in symplectic manifolds. These constructions illuminate how various manifolds can interact under embedding conditions, revealing insights into flexibility and rigidity within the field. By analyzing these examples, mathematicians can develop deeper theories about morphisms in symplectic geometry and enhance their understanding of foundational concepts such as Hamiltonian dynamics and their applications.

Related terms

Symplectic Manifold: A smooth manifold equipped with a closed, non-degenerate 2-form known as the symplectic form, which is crucial for defining symplectic geometry.

Gromov's Theorem: A fundamental result in symplectic geometry stating that every symplectic manifold contains a Lagrangian submanifold, which has profound implications for the study of embeddings.

Symplectic Capacity: An invariant associated with a symplectic manifold that measures its size or volume in a way that is preserved under symplectic embeddings.