Symplectic Folding

Definition

Symplectic folding is a technique in symplectic geometry where a symplectic manifold is 'folded' along certain submanifolds, creating a new manifold that retains the symplectic structure. This process is significant because it allows for the construction of new symplectic manifolds from existing ones and plays a crucial role in understanding the behavior of symplectic forms under various geometric transformations.

5 Must Know Facts For Your Next Test

1. Symplectic folding can create new manifolds that exhibit different topological and geometrical properties compared to their original forms.
2. The process often involves identifying certain submanifolds within a larger symplectic manifold and defining how they interact under folding.
3. Symplectic folding is particularly useful in proving the existence of certain types of symplectic structures that would otherwise be difficult to construct directly.
4. This technique is applied in various contexts, including the study of Lagrangian submanifolds and in the context of Gromov's theorem on symplectic fillings.
5. Symplectic folding can lead to insights about the relationships between different symplectic manifolds and their respective topological features.

Review Questions

• How does symplectic folding contribute to the construction of new symplectic manifolds, and what implications does this have for symplectic geometry?
  • Symplectic folding allows mathematicians to create new symplectic manifolds by folding existing ones along specific submanifolds. This process retains the symplectic structure, enabling researchers to explore new geometric properties and relationships between manifolds. The ability to generate new examples helps deepen our understanding of symplectic geometry and its applications in areas such as Hamiltonian dynamics.
• In what ways does Gromov's theorem relate to the technique of symplectic folding, and what are some outcomes of this relationship?
  • Gromov's theorem highlights the existence of symplectic fillings for certain manifolds, which can be achieved through techniques like symplectic folding. By folding a manifold, one can often construct a filling that meets the conditions set by Gromov's theorem, demonstrating how folding can yield new insights into the topology and geometry of these spaces. This interplay showcases the power of folding as both a tool and a conceptual framework in symplectic geometry.
• Evaluate how symplectic folding influences our understanding of Hamiltonian dynamics and its geometric implications.
  • Symplectic folding has a profound impact on our understanding of Hamiltonian dynamics by revealing how changes in the underlying geometric structure can affect dynamical behavior. By manipulating symplectic manifolds through folding, researchers can study how Hamiltonian systems evolve under different configurations. This exploration sheds light on the intrinsic relationships between geometry and dynamics, leading to richer insights into both fields and their interconnections.

Related terms

Symplectic Manifold: A smooth manifold equipped with a closed non-degenerate 2-form known as a symplectic form, which allows for the study of geometric properties related to Hamiltonian dynamics.

Gromov's Theorem: A fundamental result in symplectic geometry stating that every symplectic manifold can be filled with a symplectic form, leading to important implications for the topology and geometry of these spaces.

Hamiltonian Dynamics: A mathematical framework used to describe the evolution of dynamical systems governed by Hamilton's equations, which arise naturally in the context of symplectic geometry.