Embedding problems

Definition

Embedding problems refer to questions regarding the possibility of smoothly and symplectically embedding one geometric or topological space into another. These issues are particularly significant in symplectic geometry as they explore the limitations and capabilities of certain spaces to contain or represent others while preserving their essential geometric structures.

5 Must Know Facts For Your Next Test

1. Embedding problems highlight the differences between topological and symplectic embeddings, as some embeddings may be topologically possible but fail to meet symplectic criteria.
2. The complexity of embedding problems increases with the dimensionality of the spaces involved, often leading to intricate classifications of what can be embedded into what.
3. Gromov's non-squeezing theorem serves as a cornerstone example within embedding problems, showcasing how size constraints can limit the possibilities of symplectic embeddings.
4. These problems also have implications for various fields, including physics, where understanding the relationships between different phase spaces is critical.
5. The study of embedding problems involves both constructive approaches (finding explicit embeddings) and obstructions (demonstrating when embeddings cannot exist).

Review Questions

• How do embedding problems differentiate between topological and symplectic embeddings, and what implications does this have in symplectic geometry?
  • Embedding problems illustrate that while certain embeddings may be permissible in topology, they may not satisfy the more stringent requirements of symplectic geometry. For example, a space might be able to fit within another geometrically but fail to preserve essential structures such as the symplectic form. This differentiation is crucial for understanding how geometric shapes interact within the framework of phase spaces and influences applications across various domains.
• Discuss the significance of Gromov's non-squeezing theorem in relation to embedding problems and its implications for symplectic geometry.
  • Gromov's non-squeezing theorem is pivotal in addressing embedding problems because it establishes that a larger symplectic ball cannot be embedded into a smaller symplectic cylinder. This result not only emphasizes limitations in symplectic embeddings but also highlights the importance of size and dimensionality in such configurations. The theorem impacts further research and applications by providing clear boundaries on what can and cannot be achieved in symplectic geometry.
• Evaluate the role that Lagrangian submanifolds play in embedding problems, particularly how they contribute to our understanding of symplectic structures.
  • Lagrangian submanifolds are integral to embedding problems as they serve as environments where certain geometric properties are preserved. Their unique characteristic of having the symplectic form vanish allows them to act as critical tools for examining potential embeddings within symplectic manifolds. Analyzing how these submanifolds interact with various spaces leads to deeper insights into both obstructions and constructive methods for embeddings, thus enriching our understanding of overall symplectic structure.

Related terms

Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional manifold equipped with a closed non-degenerate differential 2-form, which provides a geometric structure that allows for the study of Hamiltonian dynamics.

Gromov's Non-Squeezing Theorem: This theorem states that a symplectic ball cannot be symplectically embedded into a smaller symplectic cylinder, emphasizing the intrinsic limitations of embedding spaces in symplectic geometry.

Lagrangian Submanifold: A Lagrangian submanifold is a special type of submanifold in a symplectic manifold where the symplectic form restricts to zero, often playing a crucial role in embedding problems.