Coisotropic Embeddings

Definition

Coisotropic embeddings are a special type of submanifold in a symplectic manifold where the symplectic form vanishes when restricted to the tangent space of the submanifold. These embeddings are significant as they help in understanding the structure of symplectic manifolds and play a role in symplectic reduction and the study of Lagrangian submanifolds.

5 Must Know Facts For Your Next Test

1. Coisotropic embeddings must satisfy the condition that the symplectic form $\\omega$ vanishes when restricted to the tangent space at each point in the submanifold.
2. The dimension of a coisotropic submanifold is constrained such that it is less than or equal to half the dimension of the ambient symplectic manifold.
3. Coisotropic submanifolds can be seen as generalizations of Lagrangian submanifolds but with less stringent conditions on their dimensions.
4. The concept of coisotropic embeddings is closely linked to the idea of moment maps in symplectic geometry, particularly in relation to Hamiltonian actions.
5. Studying coisotropic embeddings leads to interesting results in areas such as deformation theory and algebraic geometry, revealing deeper insights into the structure of symplectic manifolds.

Review Questions

• How do coisotropic embeddings relate to Lagrangian submanifolds in symplectic geometry?
  • Coisotropic embeddings are similar to Lagrangian submanifolds in that they both involve conditions on how the symplectic form behaves when restricted to certain submanifolds. However, while Lagrangian submanifolds must have half the dimension of the ambient manifold and their symplectic form restricts to zero, coisotropic embeddings can have dimensions up to half but do not require this strict relationship. This flexibility allows coisotropic embeddings to serve as an important bridge between various types of submanifolds in symplectic geometry.
• Discuss how coisotropic embeddings facilitate the process of symplectic reduction and why this is significant.
  • Coisotropic embeddings play a crucial role in symplectic reduction by providing a framework for understanding how certain structures within a symplectic manifold can be simplified or quotiented by group actions. When one performs symplectic reduction with respect to a coisotropic submanifold, it helps in obtaining new symplectic manifolds that retain some properties of the original one while reducing complexity. This reduction process is significant because it allows mathematicians to analyze complex systems in more manageable settings, leading to insights in areas like classical mechanics and dynamical systems.
• Evaluate the implications of studying coisotropic embeddings on broader fields like algebraic geometry or deformation theory.
  • Studying coisotropic embeddings reveals important connections between symplectic geometry and other mathematical areas such as algebraic geometry and deformation theory. For instance, insights gained from examining how these embeddings interact with algebraic varieties can lead to advancements in understanding moduli spaces and stability conditions. Furthermore, in deformation theory, analyzing how coisotropic structures behave under perturbations can yield information about stability and rigidity properties of families of geometrical objects, ultimately enriching our comprehension of geometric transformations across various contexts.

Related terms

Lagrangian Submanifolds: These are special types of submanifolds where the dimension is half that of the ambient symplectic manifold, and the symplectic form restricts to zero on them.

Symplectic Reduction: A process used to construct a new symplectic manifold by taking a quotient of a symplectic manifold by a group action, often simplifying the analysis of the original manifold.

Contact Structures: These structures are related to symplectic geometry and can be thought of as odd-dimensional analogs of symplectic forms, playing an important role in the study of coisotropic submanifolds.